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SemigroupTheoryandEvolutionEquations
PUREANDAPPLIEDMATHEMATICS AProgrGF714MonogrGphsyTextbooks,mdLectureNotes EXECUTIVEEDITORS EarlJ.Taft ZuhairNashed R2uit榕gerS【U/Fn1i1v7eUyr.飞3AsdSNewBrunswickNewJersey 【U/FM1di1v7eUyr.飞3AsdSJN＼Vfeωl4yαrkk-Dedlαωl4yαre CHAIRh4ENOFTHEEDITORIALBOARD S.kobayashiUnivenjoyQ/CGlifor-niGyBerkeleyBerkeley,CGl{fbrniG EdwinHcwitt UnivenjoyQ/WGSJIiF1gtonSGGttleyWGshington EDITORIALBOARD MS-BGOttendiDonGldPGssmGnUnivenjoyQ/CGl{fbrniGPSGnDiegoUnivenjoyQ/Wisconsin-MMison JGCKK-HGleFreds-RobertsGeorgiGInstituteQ/TechnologRutgey-sUnivenjoy MGrvinMGrcusGiGF1-CGrloRotGU协ni仰vWe仍rnωSdiO砂yQ4/Cωαl{价f沟brniωαyS归αntωαMαωAs"SBαrbαrα W.S.MGSSGYDGlyjdLRussellYαdle【U/Fn1i1v7eUyr.飞3AsdSαFn1dStωαt印e【U/FM1di1v7eUyr.飞3AsdS LeωOpOlMdOJN＼VfαCyh1bi切Fn1JαFn1eCοyr.ηwOFn1i切Fn1SCαFn1l归OFn1CeFn1t扩FrAηOBrαSdilhedi衍yr.ηOdePeωAsSS币q2uiihsS币αSFjhJSfi比CαSR2uit榕gerS【U/Fn1i1v7eUyr.飞3AsdSαFM1ddU协ni仰vW阳e旷仍rnωSdiO砂yQ/ROChester- AnilJVerodewdterSchemppCOrFn1edlHl【U/FM1di1v7eUyr.飞3AsdS MGrkTepbyUnivenjoyQ/Wisconsin-MilMωkee
LECTURENOTES INPUREANDAPPLIEDMATHEMATICS 1.N.JGCobsonyExceptionalLieAlgebras2.L.-ALindGhlGndF.PoufsenyThinSetsinHarmonicAnalysis3.ZSGIGKGyClassincationTheoryofSemi-SimpleAlgebraicGroups 4.F.HirzebruchyW.D.JVGwynGnnyGndS.S.koJIyDifferentiableManifoldsandQuadraticForms(outofprint) 5.I.CihGvel.RiemannianSymmetricSpacesofRankOne(outofprint) 6.RB.Burckef-CharacterizationofC(X)AmongItsSubalgebras7.B.R.McDonGldyA.R.MGgidyGndk.C.Smith,RingTheory:ProceedingsofthcOklahomaConferenCC 8.Y;-ZSitiyTechniquesofExtensiononAnalyticObjects 9.S.RCGrGdztSyW.E-Rf句。专nbergeryGndB.YoodyCalkinAlgebrasandAlgebrasofOperatorsonBanachSpaces 10.E.0.RoxinyP.-T.LitiyGndR.LSternbergyDifferentialGamesandControlTheory11.M;orzechGndCSmGlliTheBraucrGroupofCommutativeRings12.S.Thomeiu:TopologyandItsApplications13.lM;Lopezmdk.A.Ross,SidonSets 14.WW.ComjbrtmdS.JVegrepontisyContinuousPseudometrics 15.K.MckmnonGndlM;RobertsonyLocallyConvexSpaces 16.M;CGY7716liGndS.MMinyRepresentationsofthcRotationandLorentzGroups:AnIntroduction 17.6.B.SeligmGnyRationalMethodsinLieAlgebras 18.D-6.deFigMeir-edoyFunctionalAnalysis:ProceedingsofthcBrazilianMathematicalSocietySymposium 19.LCesmtR.KGnnGF17GndlD.ScJ11414ryNonlinearFunctionalAnalysisandDifferentialEquations:ProceedingsofthcMichiganStateUniversityConference 20.llSchQ矿tryGeometryofSpheresinNormcdSpaces21.K.YGnOGndM;konAnti-InvariantSubmanifolds 22.W.V;VGSConcelosyTheRingsofDimensionTwo 23.R.E.CJIGndleryHausdOrffCompactincations vd4EEWQJUriρlvvnuρlv4EEWa4EEWO飞UQJU.,,ALHPmcMρlvLH4ELf-OQJUσbnJUρlvρlvcoripivdσboopOTC内αmOLMTCUV…BFAMnGn7'tL儿nGrFPCJA斗斗牛
ConferenCC 25.S.K.JGinyRingTheory:ProceedingsofthcOhioUniversityConference 26.B.R.A4cDonGldGndR.A.Morris,RingTheoryII:ProceedingsofthcSecondOklahomaConferenCC 27.R.B.A4urGGndA.RJIGF71tuffGyOrderableGroups28.lRGm4StabilityofDynamicalSystems:TheoryandApplications 29.Hi-CWGngyHomogeneousBranchAlgebras 30.E.0.RoxinyP.-T.LitiyGndR.LSternbergyDifferentialGamesandControlTheoryII 31.R.D.Porter,IntroductiontoFibreBundles 32.MAltmGnyContractorsandContractorDirectionsTheoryandApplications 33.lS.GolGF17DecompositionandDimensioninModuleCategories 34.6.FGirltYGGtheryFiniteElementGalerkinMethodsforDifferentialEquations 35.JD.SdbjyNumbersofGeneratorsofIdealsinLocalRings36.SS.Miller,ComplexAnalysis:ProceedingsofthcS.U.N.Y.BrockportConference 37.R.Gordon,RepresentationTheoryofAlgebras:ProceedingsofthcPhiladelphiaConference 38.M;GotOGndF.D.GrossJIGFZSPSemisimpleLieAlgebras39.A.ZArrudGPN.CA.dGCostGyGndR.CJIZJGqttiyMathematicalLogic:ProceedingsofthcFirstBrazilianConference
40.FVGnOystGGyenyRingTheory:Proceedingsofthc1977AntwerpConference 41.F.VGnOystGGyenGndA.VeUyr.飞3AsκS 42.M;SGOYGnGrGYGnGPPositivelyOrderedsemigroups 43.D.LRussell;MathematicsofFinite-DimensionalControlSystems 44.P.-ZLitJGndE.RoxinyDifferentialGamesandControlTheoryIII:ProceedingsofthcThirdKingstonConference,PartA 45.A.Gey-GFnitGGndlseberryyOrthogonalDesigns:QuadraticFormsandHadamardMatrices 46.lCigleryv;LoserliGndP.A4ichoryBanachModulesandFunctorsonCategoriesofBanachSpaces 47.P.-ZLitJGndlG.StitinenyControlTheoryinMathematicalEconomics:ProceedingsofthcThirdKingstonConference,PartB 48.CByrnesyPartialDifferentialEquationsandGeometry 49.6.kbmbGtieryProblemsandPropositionsinAnalysis 50.lknoMnGCFIeryAnalyticArithmeticofAlgebraicFunctionFields 51.F.VGnOystGGyenyRingTheory:Proceedingsofthc1978AntwerpConference 52.B.kedey717BinaryTimeSeries 53.lBGrros-JVetoGndR.A.ArtinoyHypoellipticBoundary-ValueProblems54.R.LSternbergyA.lKGlinowskiyGndlS.PGPGdGKisyNonlinearPartialDifferentialEquationsinEngineeringandAppliedScience 55.B.R.McDonGldyRingTheoryandAlgebraIII:ProceedingsofthcThirdOklahomaConference 56.lS.GolGF17StructureSheavesoveraNoncommutativeRing 57.T-v;JVGrGJYGnGylG.WilliGFnsyGndR.M;MGthsenyCombinatorics,RepresentationTheoryandStatisticalMethodsinGroups:YOUNGDAYProceedings58.TA.Burton,ModelingandDifferentialEquationsinBiology 59.K.H.kimGndF.W.RotidyIntroductiontoMathematicalConsensusTheory 60.lBGnGSGndk.GoebdyMeasuresofNoncompactnessinBanachSpaces 61.0.A.J＼fiefsonyDirectIntegralTheory62.lESmith,G.0.Kenny,GndR.N.BGllyOrderedGroups:ProceedingsofthcBoiseStateConferenCC 63.lCroninyMathematicsofCellElectrophysiology 64.lW.BreweryPowerSeriesOverCommutativeRings 65.P.K.KGFnthGnGndM;GuptGySequenceSpacesandSeries
66.T-6.A4cLGttghlinyRegressiveSetsandtheTheoryofIsols 67.T.LHerdmmyS.M;RGnkinyIHPGndH.W.StechyIntegralandFunctionalDifferentialEquadons 68.R.DrGperyCommutativeAlgebra:AnalyticMethods 69.W-6.McKGYGndlPGtemyTablesofDimensions-Indices,andBranchingRulesforRepresentationsofSimpleLieAlgebras 70.R.LDGVGneyGndZEJViteckLClassicalMechanicsandDynamicalSystems 71.lVGF1666lyPlacesandValuationsinNoncommutativeRingTheory 72.CFGi的,IqjcctivcModulesandIqjcdiveQuotientRings 73.A.FiGCCOyMathematicalProgrammingwithDataPerturbationsI 74.P.SchtiltzyCPrGGgeryGndR.StilliVGF17AlgebraicStructuresandApplicationsProceedingsoftheFirstWesternAustralianConferenceonAlgebra 75.LBiCGnyT.kepkGyGndP.JVGF716CyRings,Modules,andPrcradicals 76.D.C.KG)/αndM;By-Gen,ConvexityandRelatedCombinatorialGeometry:ProceedingsofthcSecondUniversityofOklahomaConference 77.P.FletcherGndW.F.Lindgrey17Quasi-UniformSpaces 78.C-C.YGngyFactorizationTheoryofMcromorphicFunctions 79.0.TGUSS妙,TernaryQuadraticFormsandNorms80.S.P.SinghGndlH.Burry.NonlinearAnalysisandApplications81.K.B.HGFmsgenyT.LHerdmmyEW.Stech.mdR.LWheeler,VolterraandFunctionalDifferentialEquations
82.N.LJohnson,M-lkdlGher--GndCZLong,FiniteGeometricgProceedingsofaConferenceinHonorotT.G.Ostrom 83.6.ZZGPGIGyFunctionalAnalysis,Holomorphy,andApproximationTheory 84.S.GrecoGndG.VGIMPCommutativeAlgebra:ProceedingsofthcTrentoConference 85.AV;FiGCCOPMathematicalProgrammingwithDataPerturbationsII 86.l-B.HiriGrt-UrrttOYyW.OettliyGndlStoeryOptimization:TheoryandAlgorithms 87.A.FigGTGlGmGnCGGndM;A.PiCGrdelloyHarmonicAnalysisonFreeGroups 88.M;HGrGdGyFactorCategorieswithApplicationstoDirectDecompositionofModules 89.V;zfstrGIGSCZJ子StrictConvexityandComplexStrictConvexity:TheoryandApplications 90.V;LGkshmiKGnthGF717TrendsinTheoryandPracticeofNonlinearDifferentialEquations 91.ELMGnochGGndlB.SrilYGSIGWyAlgebraandItsApplications 92.D.V;(7114dF10178句YGndG.v;(7114dnolyskyyClassicalandQuantumModelsandArithmeticProblcms 93.lW.LongleyyLeastSquaresComputationsUsingOrthogonalizationMethods 94.LP.deAlCGFUGF刀,MathematicalLogicandFormalSystems 95.C.E.AtillyRingsofContinuousFunctions 96.R.CihztGquLAnalysis,Geometry,andProbability 97.LFuchsGndLSdceyModulesOverValuationDomains 98.P.FischerGndW.R.Smith,Chaos,Fractals,andDynamics 99.W.BPowellGndC.TsinGKisyOrderedAlgebraicStructures loo-6.M;Rω;stGSGndZM;Rω;stGASPDifferentialGeometry,CalculusofVariations-andTheirApplications 101.R-f.HdnnGnnGndk.H.HQ斤nGFmyContinuousLatticesandTheirApplications 102.lH.LiglubottY7167HLGndS.M;RGnkinyHLPhysicalMathematicsandNonlinearPartialDifferentialEquations 103.CA.BGkerGndLM;BGttenyFiniteGeometrics 104.lW.BrewerylW.BzinceyGndES.VGnVIeckLinearSystemsOverCommutativeRings 105.CMc0·oryGndzsh沪的,GeometryandTopology:Manifolds,Varieties,andKnots106.D.W.ktideryE.G.K.Lopez-EscobGryGndCH.Smith,MathematicalLogicandTheoreticalComputerScience 107.B.-LLinGndS.SimonsyNonlinearandConvexAnalysis:ProceedingsinHonorofkyFan 108.S.lLee,OperatorMethodsforOptimalControlProblems
109.V;LGkshmikGnthGF刀,NonlinearAnalysisandApplications 110.S.F.McCormickMultigridMethods:Theory,Applications,andSupercomputing111.M;CTGngomyComputersinAlgebra112.D.V;(7114dn0178句ymdG.v;(7114dF10178'妙,SearchTheory:SomeRecentDevelopments 113.D.Chudnovs均YGndR.D.JenksyComputerAlgebra 114.M;C.TGngomyComputersinGeometryandTopology 115.P.Nelson,v;FGberyT.A.Mmteq矿剖,D.LSethyGndA.B.White,k.TransportTheory,InvariantImbedding,andIntegralEquations:ProceedingsinHonorofG.M.Wingfs65thBirthday 116.PCldyneMPS.InlyernizziyE.AdtidierLGndZZVTGbieySemigroupTheoryandAp-plications 117.lViF11468现OrthogonalPolynomialsandTheirApplications:ProceedingsofthcInternationalCongress 118.CMDqftrynos-G.LGdGSyGndG.PGPGnicolG0147DifferentialEquations:ProceedingsofthcEQUADIFFConference 119.EO.RoxinyModernOptimalControl:AConferenceinHonorofSolomonLcfschctzandJosephP.LaSalle120.lCDiGZ.MathematicsforLargeScaleComputing 121.PS.A4ilQ/617扣子NonlinearFunctionalAnalysis
122.C.SGdos妙,AnalysisandPartialDifferentialEquations:ACollectionofPapersDedicatedto此4ischaCodar 123.R.M;shortliGeneralTopologyandApplications:Proceedingsofthc1988NortheastConferenCC 124.R.WongyAsymptoticandComputationalAnalysis:ConferenceinHonorofFrankW.J.Olvcrfs65thBirthday 125.D.V;(7114dnovs句YGndR.D.JenksyComputersinMathematics126.W.D.WdlisyEShenyW.WeiyGndLZJ1147CombinatorialDesignsandApplications 127.S.ElGydLDifferentialEquations:StabilityandControl 128.6.ChenyE.B.Lee,W.LittmGF17GndLMGrk245',DistributedParameterControlSystems:NewTrendsandApplications 129.W.N.E17GritltjInequalities:FiftyYearsOntkomHardy,LittlewoodandPOlya130.EG.KGper-GndM;GGrbeyyAsymptoticAnalysisandtheNumericalSolutionofPartialDifferentialEquations 131.0.ArinoyD.E.AXGIrodyGndM;kiynyndyMathematicalPopulationDynamics:ProceedingsoftheSecondInternationalConference 132.S.ComyGeometryandComplexVariables133.lA.GoldsteinyF.KGppdyGndW.ScJIGPPGCFIeryDifferentialEquationswithApplicationsinBiology,Physics,andEngineering 134.S.lAndimGyR.koppey771αF17P.R.A4isrGylZReichynGF17GndA.R.Todd,GeneralTopologyandApplications135.P.CldynentjE.Adtidier-iyB.dePGgterySemigroupTheoryandEvolutionEquations:TheSecondInternationalConference OtherVolumesinPrepGrGtion
Pagei SemigroupTheoryandEvolutionEquations TheSecondInternationalConference 4ELnρivriley泪gbkadcpiu比2cu山阴dEvmimumBpi De伊LUniversityqfTechnologyDe伊l,TheNetherlands Enzoh4itidieri UniversityqfUdineUdine,ItaOY MaHE到口eME霄,Inc. HHYU『k·自由"HHongkozzg
Pageii LibraηrofCongressCatalogingEEin--PublicationData Semigrouptheoryandevolutionequations:thesecondinternationalconference/editedbyPhilippeCHmem-BendePagtCI--Enzo此4itidieri.p.cm.Includesbibliographicalreferencesandindex.ISBNO-8247-8545-2(alk.paper) 1.Semigroups-EE-Congresses.2.Evolutionequations--E-Congresses.1.CMmem-Philippe.ILPagtcr,Bende.III-h4itidied-Enzo- QA171.S5241991512『2一一dc20 91-3662CIP Thisbookisprintedonacid-tkccpaper- Copyright@1991byMARCELDEKKER,INC.AllRightsReserved Neitherthisbooknoranypartmaybereproducedortransmittedinanyformorbyanymeans,electronicormechanical,includingphoto-copying,microfilming,andrecording,orbyanyinformationstorageandretrievalsystem,withoutpermissioninwritingtkomthepublisher. MARCELDEKKER,INC.270MadisonAvenue,NewYork,NewYork10016 Currentprinting(lastdigit):10987654321 PRINTEDINTHEUNITEDSTATESOFAh4ERICA
Pageiii Preface TheSecondInternationalConferenceonTrendsinSemigroupTheoryandEvolutionEquationswasheldSeptember25t029919899attheDepartmentofTechnicalMathematicsandInformaticsofthcDelftUniversityofTechnology,Delft,TheNetherlands.Thetopicstreatedinthisconferenceincludedrecentdevelopmentsinsemigrouptheory(c.g·-positive,dual,integrated),andnonlinearevolutionequations(c.g·-maximalregularity,NaviCI--Stokesequations,Thomas-Fermiequations),controltheory,andboundaryvalueproblems.IncomparisonwiththepreviousconferenceinTrieste(1987),moreemphasiswasgiventononlinearaspectsofthcsubjects. OnbehalfofthcOrganizingCommittee(C.J.vanDuijILC.A.Timmermans,andtheeditors),weexpressourthankstotheScientificCommittee(H.AmanILM.G.Crandall-G.DaPrato,0.DickmanILandW.vonWahl)fortheiradvice. Theorganizationofthisconferencewasmadepossiblebythennancialsupportof: -FacultyofTechnicalMathematicsandInformatics,TUDelft -VertrouwcnSCOInmissic -koninklijhNcdcrlandscAkademevan认Fetenschappen -TheNetherlandsOrganizationforScientificResearch 一-DelftGcotechnics -InstitutdcCalculMathcmatiquc -IBMNcdcrlandN.V. -OdNcdcrlandscVcrkoopmaatschappijB.V. -RabObank -RankXerox Inaddition,theOrganizingCommitteegratefullyacknowledgesthesupportofthcdean,ProfessorD.Wohers-andtheManager,Ir.H.vanIpcrcILofthcFacultyofTechnicalMathematicsandInformatics,TUDelft.
Pageiv SpecialthanksarcduetoTiniNicnhuh-HerassistancewasessentialintheorganizationofthcconferenceandthepreparationofthcProceedings. Finally,wethankthecontributors,thereferees,MarcelDekker,Inc.,especiallyMs.MariaAllcgI孔fortheircooperationduringthepreparationofthisvolume. PHILIPPECLEMENT ENZOh4ITIDIERI BENDEPAGTER
PageV Contents Preface 111 ContributorS 1X OnaFamilyofGeneratorsofAnalyticSemigroupsPGoloAcqMistGPGce SomeRealizationsofInteractionProblemsFdixAliMdmetiGndSergeJViCGise 15 SobolevImbeddingsandIntegratedsemigroupsW.Arendt 29 CompletelyAccrctivcOperatorsPh.BenilGnGndM;6.CrGndGll 41 UnboundedOne-ParameterSemigroups-FI-6chctSpaces,andApplicationsReinhGrdBurger 77 DimensionsofContinuousandDiscreteSemigroupsontheV-SpacesThjerryCOMfyIon 93 ANewShortProofofanOldFolkTheoreminFunctionalDifferentialEquadons0.DiekmGnnGndS.MVerduynLunel 咽,IAυ咽,I PerturbationTheoryforDualsemigroupsV.VariationofConstantsForII111las0.DidmGnnyM;GyUmbergyGndHiR.Thiey716 107 MaximalRegularityforAbstractDifferentialEquationsinHigherOrderInterpolationSpacesGioVGnniDorGGndAlbertoVenni 125
Pagevi ApplicationofSpectralDistributiontoSomeCauchyProblemsinLP(RqH.EmGmir-GdGndMJGZGr 143 SomeGlobalExistenceandBlow-upResultsforSemilinearParabolicSystemswithNonlinearBoundaryConditionsJOGchimESCher- 153 Time-DependentSchrδdingerOperatorsandSimulatedAnnealingAlbertoFrigeriOGndGGbriefeGrillo 165 ParabolicProblemswithStrongDegeneracyattheSpatialBoundaryJeromeA.GoldsteinGndchin-YZJGnLin 181 LinearandsemilinearBoundaryConditions:TheAnalyticCaseGuntherGminerGndkbusG.kzthn 193 ExactControllabilityofthcWaveEquationinPresenceofCornersandCracksPierreGrislyGrd 213 AbstractLinearParabolicProblemswithNonhomogeneousBoundaryConditionsDGVideGuidetti 227 NonlocalNonlinearSchrδdingerEquationsB.HeiynsoethmdH.Lmge 243 AnOperator-TheoreticalApproachtoDiracfsEquationonLP-SpacesMGtthiGSHiebey- 259 ExactControllabilityandUniformStabilizationofkirchoffPlateswithBoundaryControlOnlyonAw|ZandHomogeneousBoundaryDisplacementZLGd 267 C-ExistenceFamiliesRGb7hDelGttbeF?ftls 295 OnSomeSpectralPropertiesofthcStreamingOperatorwithMollifiedBoundaryConditionsG.LGtiroGndAldoBelleni-A4orGnte 311 OnaNonlinearHyperbolicIntcgrodifferentialEquationwithSingularKernelBig-OlQ/Loden 325 GeneralizedEvolutionOperatorsand(Generalized)C-Semigroups 337
61InterLtimer- ExamplesandResultsConcerningtheBehaviorofGeneralizedSolutions,Integratedsemigroups-andDissipativcEvolutionProblemsG1InterLtimer- 347 SomeRegularityResultsforLinearVariationalSecond-OrderParabolicEquadonsAlessGndrGLztnGrdi 357 GeneralizationofthcHillc-YosidaTheoremfSGOMiyGdrGGndJVGokihmkG 371
CertainSemigroupsonBanachFunctionSpacesandTheirAqjointslM;A.M.VGnNumenmdBenDePGgter- 383 PhaseSpaceforannthOrderDifferentialEquationinBanachSpaceEnricoObrecht 391 QuasilinearParabolicVolterraEquationsinSpacesofIntcgrableFunctionsJGnPrdt 401 AlmostPeriodicityPropertiesofSolutionstotheNonlinearCauchyProbleminBanachSpacesW.M;Ruess 421 AsymptoticBehaviorofSomePerturbedC。-SemigroupsW.ScJIGPPGCher- 441 SemigroupsDennedbyAdditiveProcessesLSmitSGndlA.VGnCGM 463 UniformlyBoundedSolutionsofQuasilinearParabolicSystemsBrunelloTerreni 483 SemigroupsGeneratedbyFirst-OrderDifferentialOperatorsonJVEDimensionalDomainsA4αdeneGGbriefeUIynet 499 ANonhomogeneousDirichletProblemforaDelayDifferentialEquationPGolGVery10le 511 LocalExistenceforaParabolicProblemwithFullyNonlinearBoundaryconditionArisinginNonlinearHeatConduction:AnLPEApproachP-WeidemGier 519 Index 523 Pagevii
Contributors PAOLOACQUISTAPACEDepartmentofMethodsofMathematicalModelinginAppliedScience,UniversityofRome"LaSapienza,"Rome,ItalyFELIXALIMEHMETITechnischeHochschuleDarmstadt,Darmstadt,Germanyw.ARENDTDepartmentofMathematics,UniversityofFranchbComte,Besancon,FranceALDOBELLENLMORANTEDepartmentofCivilEngineering,SchoolofEngineer-ing,Florence,ItalyPH.BENILANFacultyofSciences,UmversityofFranchbCorn时,Besancon,FranceREINHARDBURGERInstituteforMathematics,UniversityofVienna、Vienna,AustriaJANA.VANCASTERENDepartmentofMathematicsandComputerScience,Uni-versityofAntwerp(UIALWilrijk/Antwerp,BelgiumTHIERRYCOULHONDepartmentofAnalysis,UniversityofParisVLParis,FranceM.G.CRANDALLDepartmentofMathematics,UniversityofCalifornia,SantaBar-bara,California0.DIEKMANNCentreforMathematlcsandComputerScience,Amsterdam,TheNetherlands,andInstituteforTheoreticalBiology,Leiden,TheNetherlandsGIOVANNIDOREDepartmentofMathematics,UniversityofBologna,Bologna,ItalyH.EMAMIRADDepartmentofMathematics,UniversityofPoitiers,Poitiers,France LX
X Confributors JOACHIMESCEERMathematicalInstitute,UniversityofZurich,ZUriel1,SwitzerlandALBERTOFRIGERIODepartmentofMathematicsandInformationSciences,Uni-versityofUdine,Udine,ItalyJEROMEA.GOLDSTEINDepartmentofMathematics,TulaneUniversity,NewOr-leans,LouisianaGUNTHERGREINERMathematicalInstitute,UniversityofTUbingen,TUbingeILGermanyGABRIELEGRILLODepartmentofMathematicsandInformationScience,Univer-sityofUdine,Udine,ItalyPIERREGRISVARDI.M.S.P.,MathematicsLaboratory,UniversityofNice,Nice,FranceDAVIDEGUIDETTIDepartmentofMathematlcs,1JIllvemityofBoiogna,Bologna,ItalyM.GYLLENBERGInstituteforAppliedMathematics,LuleaUniversityofTechnol-ogy,Lulea,SwedenB.HEIMSOETHMathematicsDepartment,UniversityofCologne,Cologne,GermanyMATTHIASHIEBERMathematicsInstitute,UniversityofTUbingerhTUbingeIUGer-manyM.JAZARDepartmentofMathematics,UniversityofPoitiers,Poitiers,FranceKLAUSG.KUHNMathematICSInstitute,UniversityofTUbingen,TUbingen,Ger-HIanyH.LANGEMathematicsDepartment,UniversityofCologne,Cologne,GermanylLASIECKADepartmentofAppliedMathematics,UniversityofVirginia,Charlottes-ville,VirginiaRALPHdeLAUBENFELSDepartmentofMathematics,OhioUniversity,Athens,OhioG.LAURODepartmentofAppliedMathematics,SchoolofEngineering,Florence,ItalyCHIN-YUANLINDepartmentofMathematics,TexasA&MUniversity,CollegeSta-tion,TexasSTIG·OLOFLONDENInstituteofMathematics,HelsinkiUniversityofTechnology,Espoo,FinlandGUNTERLUMERMathematicsInstitute,UniversityofMons,Mons,BelgiumALESSANDRALUNARDIDepartmentofMathematlcs,UniversityofCagliari,Cag-liari,ItalyISAOMIYADERADepartmentofMathematics,SchoolofEducation,WasedaUni-versity,Tokyo,Japan
Contributors xt J.M.A.M.VANNEERVENCentreforMathematicsandComputerScience,Am-sterdam,TheNetherlandsSERGENICAISEDepartmentofPureandAppliedMathematics,UniversityofSci-encesandTechnologyofLille,VilleneuvedFAscq,FranceENRICOOBRECHTDepartmentofMathematics,UniversityofBologna,Bologna,ItalyBENDEPAGTERDepartmentofMathematics,DelftUniversityofTechnology,Delft,TheNetherlandsJANPRussDepartmentofMathematics,PaderbornUniversityofTechnology,Pad-erborn,GermanyW.M.RUESSDepartmentofMathematics,UniversityofEssen,Essen,GermanyW.SCHAPPACEfERInstituteforMathematics,UniversityofGraz,Graz,AustriaLSMITSDepartmentofMathematicsandComputerScience,UniversityofAntwerp(UIALAntwerp/Wilrijk,BelgiumNAOKITANAKADepartmentofMathematics,KochiUniversity,Kochi,JapanBRUNELLOTERRENIDepartmentofMathematics,"F.Enriqlies,"UniversityofMilaILMilan,ItalyH.R.THEMEDepartmentofMathematics,ArizonaStateUniversity,Tempe,ArizonaR.TRIGGIANIDepartmentofAppliedMathematics,UniversityofVirginia,Char-huesvine,VirginiaMARLENEGABRIELEULMETMathematicsInstitute,UniversityofTUbingen,TUbingeILGermanyALBERTOVENNIDepartmentofMathematics,UniversityofBologna,Bologna,ItalyS.M.VERDUYNLUNELDepartmentofInformationSciences,FreeUniversity,Am-sterdam,TheNetherlandsPAOLAVERNOLEDepartmentofMathematics,UniversityofRome,"LaSapi-enza,"Rome,ItalyP.WEIDEMAIERFacultyofMathematicsandPhysics,UniversityofBayreuth,Bay-reuth,Germany
OnaFamilyofGeneratorsofAnalyticSemigroups PAOLOACQUISTAPACEDepartmentofMethodsofMathematicalModelinginAppliedScience,UniversityofRome"LaSapierlza,"Rome,Italy 0.INTRODUCTION Let{A(t),te[0,T]}beafamilyofgeneratorsofanalyticsemigroupsinacomplexHiIbertspaceH,andsupposethatboth{A(t)}and.{A(t)}fulfiItheassumptionsof(AcquistapaceandTerreni,1987)inasomewhatstrengthenedform,i.e.assumethat: foreachte[0.T].A(t):DEH一今HiSaClosedlinear"A{t)operator;inadditionthereexistoe]π/2,π[andM>Osuchthatρ(A(t))25T日,whereS(。):={zd:|argz|φ},and}(0.1) ||[λ-A(t)]-1||E(H)三M[1+|λ|]-1VM百页,vte[0,T]; thereexistN>Oandα,ρε]0,1]withα+ρ〉1,suchthat 、‘,,,9ιnu,,..、,、ro'『IE」飞八+噜EA「ELαS&LM川〈-}UH,,、ωι『...」-A、..,,s,,..、AA-A、...,&LrE飞AAFZEE』4,..『IE」、‘.,,&L,,..、AA飞八FEEE』、‘.,,&L,,..、AA VλeS(0),Vt,sε[0,T]; -theoperators{A(t),te[0,T]}satisfy(0.1)and(0.2) 叫Jnu飞BEt-·》EIE』,, withthesameconstantso,M,N,α,ρ. REMARK0.1By(0.1),thedomainsDarenecessarilydenseinH,A(t)-SOthatA(t)iSWelldefined(anddenselydefinedtoo).口 DenotebyZ(H)thesetofself-adjointboundedlinearoperators
2 Acquisfapace onH:Z(H)isaBanachspacewiththeZ(H)norm.Considerforeachte[0,T]the。perator 、..,H,,..、守,-epa、..,,&L,,.、AADa+PA.、..,,&L,,..、AA--P‘、..,,&L,,..、A (0.4) whoseprecisedefiniti。nwillbegiveninSection1.ItiSknown(seeSections6.1,6.2in(DaPrato,1973))thatf。reachte[0,T],A(t)generatesananalyticsemigroupinz(H),andinadditionA(t)preservespositivity,i.e.ifpeDandPEE0,thenA(t)PEE0.AU}OurgoaliStoshowthatundertheaboveassumptionsthefamily{A(t),tε[0,T]}fulfilStheassumptionsof(Acq1listapaceandTerreni,1987),or,moreprecisely,satisfies(0.1)and(0.2),withρreplacedbyanysmallernumber,intheBanachspaceZ(H).AsanapplicationofthiSresult,weareabletoshowexistenceofclassicalSOILItionsforanabstractnon-autonomousRiccati equationarisinginthestudyoftheLinearQuadraticRegulatorProblemforparabolicsystemswithboundarycontrol.Duetolackofspace,thiSapplicationwillappearinaforthcomingpaper(AcquistapaceandTerreni,inpreparation). REMARK0.2Memayreplace(0.2)bytheSlightlyweakercondition ··Aρ『BEEd飞八+唱,..FEEE』....αS&L....k?L=M川SMut-,」-A、‘..,S''-z、An-A、..,,&ELV,,..、AAFEEE』噜A『..,」、..,,+L,,,、AA飞八FEEE』、...,&,.",,..、An VλeS(0),Vt,se[0,T], whereα,ρε]0,1]andα+ρ〉1fori=1,...,k;whatiScrucialhereiSiiiithatρi>0,andthiSrequireIIlentmakessuchassumptionstrongerthanthatof(AcquistapaceandTerreni,1987),whereonthecontrarytheρi'Sareallowedtobepossibly0. 1.THEOPERATORA(t)FORFIXEDt. Aprecisedefinitionoftheoperator(0.4),forfixedte[0,T],canbegiveninthefollowingway(comparewith(DaPrato,1973)).FixpeZ(H)andconsiderthesesquilinearformdefinedonD×Dby:A(t}A{t) ··.AnυevdxMH、-EEJVJ、‘.,,&L,,.、AnxPA,,,,‘、+""飞··JVdPAX、...,&L,,..、A,,..‘、=、..,yx&L,,..、PAY (1.1) Weset
GeneratorsofAnalyticSemigroups 3 D:={PeZ(H):3C(t;P)〉OsuchthatAU} (1.2) 、、,,、‘,,&-uaanuεVJXMV曰"vdUHX、..,,P·&L,,..、C〈-、‘..,VJX+L,··E、PAV IfpeD,thenφ(t…)hasauniqueextensionφ(t;·,·)t。AU}P"H×Hsuchthat φ(t;x,y)=φ(t;x,y)Vx,yeDPA{t} (1.3) 口uεvdxvuuyHX、...,PA&L,,..、C〈-飞,,vdx&L,...、P-AV hencebyRiesz'RepresentationTheoremthereexistsanoperator QP(t)eJE(H)suchthat 口uεVJXVH、‘..,,vdx、..,,&L,,a、pn吨'''E飞=、..,,vdx&L,,..、PAY (1.4) Nowwedefine A(t)P:=Q(t)VPεD,A{t}' (1.5) i.e. 口uevdxwv、...,VJX&Lra、P-AV--UH、,EEJVdxPA、lJ+凰",,,‘、A,,..‘、 (1.6) ra'E·-uc-A&Lrapan.,4neh&L+··Anuexdna--UAHnuεPAFA.、4&Lah+Lkramerew 〈-HH、‘..,,VJDAX、..,&LrE飞AA,,..‘、、...,VJX&L,,.、P-AV--H、...JVd、..,&L,,.‘、AAXPA,,..飞 uuvd『···」HX、..,,&L,...、AA+HX、‘..,P·+L,,..、crEEL〈- thiSmeansPxeD-andA{t) ·tw,,、AnuePAMV、.,,-E.·Hnuexvvx、..,&L,,..、AADA+XP品'、..,,&L,,,.、A--XD且、...,&L,,.、A (1.7) 1.e.(0.4)holdswhenevaluatedatanyxeD.Inparticular,byA(t}(1.4),(1.3),(1.1)and(1.7)itfollowseasilythat MH、IEJVd、..,,&L,..、P-nu哥x,,..‘、=H、,EEJVdx、..,,+L,,..、pnu,,..飞 Vx,yeDA(t) andthereforeA(t)PEQ(t)eZ(H)foreachpeDPAU)-eA{t)TheoperatorA(t)dgeneratesthesemigroup{e,￡兰O}EJE(Z(H)),definedby
4 Acquistapace 、‘..,uu,...、zePAφlwanp、epa--LACKde--PA--、APζe (1.8) indeed,wehave: PROPOSITION1.1Denoteby1theidentityoperatoronZ(H).Mehave: (i)D={PeZ(H):A{t) rfA{t}唱飞3;Jiee-ipx,t(A(t)凡y)HVx,问;3;2ll(eeA{tL1)叽(H}=0}; H?--ep&,,、‘=-···-ARυ.、4.,-a --、‘,,-··"-A-nueDA、..,,&··",...、AU---AnuGD且,,、‘、‘.,,.,ι.,.&.、·品,,..、 、、,,nu--UH?中PA&LAp-川一旦与mo--MA--&红、34-LAAnuepa,,、、= Proof.(i)By(1.8)and(0.1)-〔0.3)itfoIlowsthat 、...,TAnu,....‘ε&··WMvnu〉C飞vv、..,,MAV,,..、C〈-、..,,H,,、艺,,..、ψι+·-AHCKde hencetheargumentofChapter9,Remark1.5of(kato,1966)showsthatifP,QeZ(H)and r-EA(t}唱、lim|eF-Apx,y|=(Qx,y)"vx,yeH,PO飞、JHHthenPεDandA(t)P=Q.SupposeconverselythatpeD:thenbyA{t)A{t)(1.7)itiSeasytogetforeachXe[)andyeH:A{t} 曰"、‘..,,,vdXDa+11&LXrt1JA,,..飞、‘EE,,H1&=-uuu1川川JA",卢、xe--J,,..‘、、‘EE,,,'AMH-4E··pζ-「EIL、,PU飞IJAR-Hε414e-rl飞.·A飞-(卢飞AHrEELCKdp晶efl、+,,EBEE--‘、mox--uspa-AC飞llUH--咱iH-飞EEEEEJ.y)&LXAPA红、1-e--rst飞、,,-····.."--{-eF、AA一红、一+e-,,EEEE---‘、mo--ual户ζ henceby(1.6)wegettheresultsinceDisdenseinH.A{t}(ii)-(iii)SeeProposition1.2(i)-(iii)。f(Sinestrari,1985).口 EXAMPLE1.2DiSnotdenseinZ(H)ingeneral(unIess,ofAU}course,theA(t)'Sarebounded.Indeed,setH:=L2(O,π),and22.2,21.2A(t)三A:=d/dx,wlthD:=M(O,π)nM(0,π);thenwehaveAo HeFAMvnu飞fpkdwvnenri、..JFζq6nra飞nrxe-E品∞FFω=n--FA-AHPζe--F&AHCKJe
GeneratorsofAna1yticSemigroups 5 1/2r飞wheree(x):=(2/π)sin(nx).f:={f.el.NowifDweredenseinn'n飞'rzJHAZ(H),thenweshouldhave,choosingP:=1H: lim||(eEA-1艺(H))1H||Z(H}=0,伫￥oi.e.f。reachε〉Otheresh。111dexistδ〉Osuchthatε ,...‘εXUnue户巳evvε〈飞lrJ4E·.--UHFAUHri、‘..,,UM·E·矗AH户电句,"e,,..‘、rl吨E飞P-us +hencebytakingf:=e,ne肘,wewouldgetn ||(e2￡A-h)en||H=1-exp(-2n亏)〈εvnem＼v￡ε]0,δε[, whichiSimpossible.口 REMARK1.3DespiteofExample1.2,weobviouslyhave linl||(eEA(t}-1Z{H})PX||H=ovpeZ(H),vxeH,vte[0,T]·口(13)伫=O 2.MAINRESULT By(0.1)-(0.3)andtheresultsof(AcquistapaceandTerreni,1986),(AcqLlistapace,1988),(Acquistapace,FlandoliandTerreni,1990,inpress),(AcquistapaceandTerreni,1990)wecanconstructtheevolutionoperatorU(t,S)associatedto{A(t)},andthefollowingpropertieshoIdtrue: PROPOSITION2.1ForOSS〈tzETwehave: (i)U(t,S)=U(t,r)U(r,s)Vre[S,t],U(t,t)=1H;(ii)U(t,S)eZ{H,D}and3du(t,s)/dt=A(t)U(t,S);飞A(t}J.(iii)U(tJ)d(H,DA(S)·)and3dU(tJ)/ds=-AU)U(tJ);(iv)3dU(tJ)/ds=-[A(S)川tJ)γ;.(V)||U(t,s)||￡{H}+||U(t,s)||￡(日)+(t-s)||dU{t,s)/dt||E(H)++(t-s)||dU(t,s)/ds||￡{H}sc(0,M,N,α,ρ,T)-Proof.(i)-(ii)SeeTheorem2.3of(Acquistapace,1988).(iii)See(6.11)of(AcquistapaceandTerreni,1990).(iv)SeeTheorem6.4of(AcqllistapaceandTerreni,1990).
6 Acquistapace (v)SeeTheorem2.30f(Acquistapace,1988)andThe。rem6.40f(AcquistapaceandTerreni,1990).口 Considern。wthe。Perat。rE(·,·):z(H)一今艺(H)definedby -E(t,s)P:=U(T-s,T-t)Pu(T-s,T-t),OssstsT,PGZ(H).(2.1) AstraightforwardcomputationshowsthatE(t,s)isstronglycontinuousinz(H),andinadditionifOSS〈tsT E(t,S)=E(t,r)E(r,s)Vre[s,t],E(t,t)=1z{H}'生E(t,s)P=A(T-t)EU,s)PVPGZ(H),dtiE(t,s)P=-E(t,s)A(T-s)PVPeDA{T-s}' (2.2) henceE(t,S)iSthe(necessarilyunique)evolutionoperatorassociatedto{A(T-t),te[0,T]}.MewillshowinourmainTheorem2.3belowthatthefamily{A(T-t)}satisfies(0.1)and(0.2)(withpreplacedbyanysmallernumber)inthespaceZ(H).AsaconsequenceofTheorem2.3,thereSIlltsof(AcquistapaceandTerreni,1987),(AcquistapaceandTerreni,1986)and(AcqLlistapace,1988)immediatelyimplyseveralregularitypropertiesfortheevolutionoperatorE(t,s).REMARK2.2Ofcourse,manysm。。thnesspropertiesforE(t,S)and-E(t,S)mayalSObedirectlyderivedby(2.1),usingtheregularity-resultsforU(t,s)andU(t,s)pr。vedin(Acquistapace,1988),(Acquistapace,FlandoliandTerreni,1990,inpress),(AcquistapaceandTerreni,1990).HoweverwebelievethatTheorem2.3hassomeinterestinitself,sinceitprovidesanewclassofgeneratorsofanalytiCSemigroupshavingago。ddependenceont(i.e.satisfying(0.1)and(O.2));thisclassisnotthe,eusualHabstractversionofsomeellipticoperatorwithtime-dependentcoefficierltsandhomogeneousboundaryconditions,actingonsoneconeretefurlctionspace,althoughitsconstructioninfactstartSfromanoperatorofthatkind.口 THEOREM2.3Underassumptions(0.1)-(0.3)theoperatorsA(t),definedby(1.2),(1.6),enjoythefollowingproperties:
GeneratorsofAnalyticSemigroups 7 (i)A(t):DG(H)→Z(H)isaclosedlinearoperator;inAU}additi。nthereexistoe]π/2,。[andM〉0,dependingono,M,oosuchthat ||[λ-A(t)]-1||zmH})SM。[1+|λ|]-1Vλe百百丁,vtε[0,T]; (ii)foreachεe]0,1[thereexistsN〉0,dependingonO,M,N,α,p,oε,suchthat ||A(t)[λ-A(t)]-1[A{t)-LA(s)-1]||E(Z{H})三SNolt-s|α[1+|λ|]-ρ(1-uvλε百气丁,叭,se[0,T] Proof.SeeSection3. 3.PROOFOFTHEOREM2.3 Assume(0.1)-(0.3)andletA(t)bethe。peratordefinedinZ(H)by(1.2),{1.6).Firstofallweneedarepresentationoftheresolventoperat。r[λ-A(t)]-1 PROPOSITION3.1Part{i)。fTheorem2.3holdstrueand,inaddition,wehave [川(t)]-1p=jyA(tf]-1p[叶A(t)]-1dμVPGZ(叽VM罚歹, (3.1) 。obeingdefinedinThe。rem2.1(i);hereγisanycurvelyinginρ(A{t))咿(A(t).)andj。ining+ωe-1ηt。+ωe171fors。阴阳]π/23[,andthesymbolfγmeans(27Ei)-1I宫.EA{t}Proof.Clearly,ift:〉OwehaveePεDf。reachpeZ(H),andA{t} nu、fpF丐vv.-wAH户毛e、...,+L,,-E·、AAPA.、..,+··AA户ζe+-EWAH户ζePA--tw,..、AH户毛e.、...,&L,,..、AA--PA--、AFL3e、‘..,&L,,-E、A sothat nu、/C飞vv.--户ζ、...,M00,...、cs、...,uuz,...、EP·-LACK3e、..,+·",,..、A part(i)ofTheorem2.3thenf。1lowsbystandardarguments.Fixn。wPεZ(H)andλeET百丁.ThenbytheLaplacetransformo
8 AcqtdISfapace formulaweget [川(t)]'1p=re-飞机产pJA{~e'O Onthe。therhand,wehave -eEA(t}=fe￡ν[ν-A(t)]飞ν,eEA{t}=feeμ[μ-A(t)丁1dμ,'苦':r2where古1,:r2obeytherequiremerltslistedabove;hencebyFubini'STheoremandtheresolventidentityweget --vdu'AU-A『EEE」、..,,&L,...、AAVFEEE』P··A『EEE」.、..,,&L,..‘、AAμFEE'』νpr、eμ产r、e户骂、八e『6TS俨tT1J...y俨ifld∞optttJ=P··A『...」、...,+L,,..、AH飞八FEZL VAUU',d....『2··J、..,,争L,,..、AnvFEELP‘....『lE」.、‘..,,+L,,..、AAμFEEL---、..,,飞八μ+ν,,..、叮4万ra+IJ....γu「TJ-- Mecanseleetthecurves宫,百insuchawaythat:(a)foreach12λeS(0)andμε:rthepointλ-μliesontherigTIt-handSideof苦,and2similady(b)f。reachλεgT百丁andv町thepointλ-vliesontheo1right-handSideofir.ThiScanbeachievedbychoosing,for2instance, 飞Itrlj'A'dVILj&L叮ιllosog--2Hl主OllrS〈-SS.、.品'+Snu-mAr+1-2.O-r=-z=rl4l飞ZIts-1UU飞ttrtJ1、1〉ll--A-A-1·'。2SAV。scolleo--ro--2r〉-〉-rr1200nrnyxxeerr--=ZZf斗tf升飞'&叮,』γMWγu (orientedfrom+∞exp(-101)to+∞exp(io1),j=1,2),whereoo<01〈02<Oandroe]0,bf|tgo|-1[,so,thatby(0.1)'both飞andz2arecontained,,.inρ(A(t))np(A(t))foreachte[0,T].NowifλeS(0)wemay··ClosethecurveirontherightH,and1evalLlatetheintegraloverz1bymeansofresidues'theorem,obtairling(3.1).Theproofiscomplete.口 REMARK3.2OfcoursewemightalsoHClosethecurveirontheright"2(insteadof万1),obtainingsirnilarly 而oe飞八VVHF'AU1J1H-rE飞1」艺)Ft+lup&rtVVAAVFEEE』P·...『...」.、..,,&L,,E·、Anv飞八FEEL守orιIJ--p·A-A1,」、..,,&L,,..、A飞AriL (3.2)
GeneratorsofAnalyticSemigroups 9 where:rsatisfiestherequirementslistedinProposition3.1.口 Fixn。wPεZ(H),s,te[0,T]andλεgT百丁;considertheoperatoro Z:=[λ-A(t)]-1P(t)-14(s)叮P. (3.3) +Lah&LWohs。&Levahuew、.,,&凰MAnue7'-VJ唱E·-suo-、AVbo 、‘..,A哇叫J,,.‘、,......,..nu嘈···aeεvvε-AnF『,,,4、八+-E··FEEE』αs&L、...,εnFαM川MAU,,..、cguuz7I】、...,&L,,..、A andthiswillprovepart(1i)。fTheorem2.3.Weremarkthatif(3.4)holdswithλ=1,thenforeachλεS(0)O withlλ|〈1wehave ||A(t)ZIE{H}==||[14(t)][λ-A(t)]九(t)[14(t)]-1[A(t)-14(S)叮P||￡{H)三主C(0,M,N,α,ρ,ε)|t-s|α主C(0,M,N,α,ρ,ε)|t-s|α[1+|λ|]训1-ε), i.e.(3.4)holdsforeachλe百百丁with|λ|〈1aswell.Henceitissufficienttoprove(34)foreachλε百瓦丁with|λ|主1Tothispurposeusing(3.2)wesplitp(t)-14(S)叮Pinthefollowingmanner: P(t)'14(s)叮P= =j扣-A(t)丁1巾(t)]飞[-rA(叶1P[rA(S)]-Vv=)=jj[[-rMJ]-L[-rA(S)丁-hrA(t)]飞+[-rω丁14[ν-A(t)]可 Clearly,thecurve百heremustbec。ntainedinp(A(t))叩(A(t).).、andinρ(A(s))呻(A(s)),andinaddition吁:={zeC:-z可}musthavethesameproperty;forinstancewemaytake宫:=θS(0),。rientedfromo+ωexp(-ioo)to+∞exp(ioo).NowletusfixMgT百7with!λ|注1.Using(3.5),(3.3)andtakingintoaccount(3.1)and(3.2)wesplitZasfollows:
lo Acquistapace Z=j」Jr川丁1[[fA(t)丁1-[叫(S)丁-1lp.·[ν-A(t)]飞λ-μ-A(t)]飞μdv+ (3.6) +jγi[忖A(t)·]吁叫S)·]-1p.·[[川)]叮叫(S)]-忻州)]-1d附=:Z1+Z2 wherewemaychoose古:=δS(0)andz:=θS(0),withO<0<02〈0;for122instancewemaychooseO:=(20+0)/3,。:=(0+20)/3,sothatoando2OO2depend。nly。no,M.Next,werewriteZ1usingtheresolventidentity: -vdpi1川1114dl二1『...J.吨,E」、..,,CM+L,,..、AArr.F'EE』「L斗-、J4ν~JY、tJ-+L飞八,,..、,,..、AA--VFEZLFEllzL··A-1-」.、..,,&L,..‘、AAU!FEE-』?』γVFiElv--··γP1+1J---A7L (3.7) -jJ飞机(t)叫[个川丁1-[-rω丁于··(λ-μ-v)飞λ-μ-A(t)]-1dμdv=:Z11+Z12; ofcoursebothZandZareabsolutelyconvergentintegralS.1112InZwemayevaluatetheintegraloverZbys,Closing:ron112therightHandusingresidues'theorem:wefind(sincethepointλ-vliesontheright-handSideof古2) 、..,,QuvnJd(-A『...」、...,&L,,-z、AAVFEELPA『,,,,,,」-z&『aBEd-、..,,S,,,.、AAVFEEL-A吨,,」.、..,,&L,fkAnVFEEKFElt-L-A『EEJ.、..,,&L,...、Anv飞八FEEL--a-TO「4lJ=-sa·-EAZ Similarly,inZweevaluatetheintegralover古byHClosing12:r1ontheleft",finding(Sinceλ-μliesontheright-handSideof飞): Z=0.12 (3.9) ConsidernowZ.BythechangeofVariableν=-2,wehave2
GeneratorsofAnalyticSemigroups Z2=jγλ[忖州)丁1[H(S)·]-1p.[k[-吃?叫刊zr叫叫-4圳AM州川叫("川tυ叶)吁]丁-1-f[吃?叫叫-4圳AM(hs)吁r]丁γ-→1忏AM州川("ωtυ)]丁-1臼 where-γ1:={zee:-ze百1},。rientedfrom+ωexp(i(π-01))to+∞exp(-i(π-01)).Butthefurlcti。n 『BEE--J-A『,,」、...,S,..、AAZFEEE』-A『...」、...,&L,,.、AnZFEEE』FEllELPA-A『...J..、..,,S,,..、AZFEEL→z isabsolutelyintegrableandh。1。morphicintheregion 、‘,,、....a..,-AV-E-AUπ,···kezgora俨LVεZ,,田、 ll (3.10) sothatin(3.10)wecanreplace-古1byz1;thus,writingagainvinplaceofz, Z2=j古人[λ-rA(t)·]-1[rA(S)丁1p.[[-rA(t)]叮-rA(S)]机 (3.11) Next,usingtheresolventidentitywerewriteZ2asthesumofthreeabsolutelyconvergentintegrals: Z2=j飞J丛jLγ1[λ-1训叫μy川川-4圳A川("川川tυJ)户.丁叫][I[-叫叫ν俨川川-4圳A川(叫tυ叮)吁]吁俨川叫A川h川叫(怡ωs剖叶)吁]r-117]μ扒-4A("叫tυ叮)吁]丁-→飞马1、讪d伽叫ν-j宫飞J丛2jLjLy1J(忖ν川川川)γ户叫-→飞1飞[忖州川).丁丁]丁-1P.[[-rA(t)]飞[-rA(S)rlh(t)]-1叫++jJV1(忖V)叩-A(t)·]-1p.[[-v-A(t)]飞[-rA(S)]-忻州)]-1叫40+Z21+Z2 (3.12) andasbeforewecanevaluateinZtheintegraloveryandinZ21122theintegraloverir2'obtaining
l2 Acqtdistapace Z=0.21' (3.13) 、‘..,A哇4E·A叫JVft,d-A『,E·」、...,&L,...、AAV、八rEE』吨,EEE·-EEd··A『···」、...,SFE飞AnvFEEL‘A『aEEd、...,4LFZ飞AAVF'''』FEE--EELDa-A『lJ.、...,&LrE飞AnVFEEL--守。r4lJ--9tE2Z By(3.6)-(3.9)and(3.12)-(3.14)wefinallyhave Z=Z+Z+Z.O1122' (3.15) whereZiSdefinedin(3.12)andZ,Zaregivenby(3.8),(3.14).1122Letuscomputenow,accordingto(1.1〕,thequantityφz(t;x,y) 喻,..,Tnu,.....e&LAUna·'MAAnuevdxrOFA --UH、‘IJVd、..,,&L,..飞AAX7L,,..飞+H、...,,vdzx、..,,+LFE飞AA,,..‘、=yx+L,,..、ZAV (3.16) HH、‘EE,,vdx?ι.、...,&L,,.、An,,..飞+UH、‘..,,vd?L血'、...,&L,,,.、AAx,,,.‘、= ...sothatby(3.15)wehavetoestimateA(t)Z,A(t)Z1andA(t)Z22Ointhet(H)norm. Tothispurposeweneedtwolemmas. LEMMA3.3Ifλe百瓦丁with|λ|Uandμ句1UV2,then 、1'nuAV唱AAU,,..、n.、4S1,」HFV飞AFEEL〉-HF飞八 Proof.Quiteeasy,口 LE附fA3.4Ifλ6(0)with|λ|剖,μ町UZandν町,thenforeachO··121 εε]0,1[wehave 叫JU/ερ叮ESEJV+4EA「EL?-sγ}EHρ,、-E1」τl-zttJHP··且..,-E·E·-+-11」1‘rtL、..JSUU阳卜'(νρFEEa』--『EEJ噜At--E-飞八『Eld--E·+1114+ιFEELMα-sν-FEEE』+LFEEEEEEE』'A、‘.,,-n俨『EBEJ.IJα&L,rtM川A,-MHH俨,-AU‘-E.,,..、「l』C.飞J〈-&L,,..、AA Proof.Wewrite l|川·[川(t)·]-1[[俨川叫A川川(=||Mt).[川
GeneratorsofAnalyticSemigroups I3 句6J'ε+、..,同JhJ'ε)-MM·E·-,,、rtωι-A吨EEJ.、‘.,,s,,.‘、AVF''L.、..JS,,E飞AA. 'andusinghypothesis(0.2)for{A(t)}weget |l|川.[川("川旷叶tυ川叫)户巧.丁]巾俨川川AM川(三c(M,N,α,ρ)lt-s|α[1+|λ-μ|]-ρ(1-ε/2)[1+|ν|]-ρε2; byLemma3.3wegettheresult.口 -LetusnowestimateA(t)Z.By(3.12)wehaveo 川·zo=j宫2ly)·[忖川丁-1[[rMUT-[PA(S)·]-1lp. sothatbyLemma3.4and(0.1)wegetforeachεε10,1[ .ε-Aρ1,」飞八+咽,..「''LαS+L、.,,""?,-PA、..,,ραM川MHAV,,..、C〈-、,,,uuznu7L.、‘..,&L,,E·、A -j41[1+|μ|]-ρ气1+|V|]-M|伽|ldμ|三三C(0,M,N,α,p)||P||Z{H}|t-siα[1+|λ|]-ρ(14). (3.17) 'ConcerningA(t)Z11'by(3.8)wehave 州)211=jJ(t)·[λ-rA(t)丁1A(t)·[-川(t〕·]-1.[h(t)丁-1-h(S).]-1lA(仆rA(sfr1p[川(t)]-1dν andbyLemma3.4weeasilygetforeachεε]0,1[ .、..,,pαM川MAV,,..、C〈-曰",,‘、E·E·-....z·、..,,&,、,,E‘、AA ·||Plzmlh|α[1+|λ|丁ρ{川j宫[1+|ν|]-PMW|三三C(0,M,N,α,PJ)||P|iz(H}|t-s|α[1+|λ|]利1f} (3.18) -TheestimateforA(t)ZisquiteSimilar:by(3.14)wehave22anal。g。usly
l4 Acquistopace .αs&LMHZDA、..,,ραM"MAvfE飞cs、,MHE?-9』Z.、...,&L,,..、A -j歹[1+|ν|]-ρ[1+|叫]-1|dv|三主C(0,M,N,α,ρ,ε)||Pl|Z(H)ltES|α〔1+|λ|]训1f} Estimates(3.17)-(3.19}showthat(3.4)h。ldstrue:this concludesthepr。ofofTheorem2.3.口 REFERENCES 1.P.Acquistapace,Diff.Irzt.Eq.1:433(1988).2.P.Acquistapace,F.FlandoliandB.Terreni,SIANJ.ControlOptimiZ.(1990;inpress).3.P.AcquistapaceandB.Terreni,Rend.Sem.Mat.Univ.Padova78:47(1987). 4.P.Acq1listapaceandB.Terreni,"DifferentialEquatiorISinBanachSpaces,,(A.FaviniandE.Obrechteds.),LectureNotes,Springer-Verlag,Berlin,1223:1(1986).5.P.Acq1listapaceandB.Terreni,preprintDip.Hatem-Univ.Nilano12:(1990).6.P.AcquistapaceandB.Terreni,inpreparation.7.G.DaPrato,J.Nath.PuresAppl.52:353(1973).8.T.kato,"perturbationTheoryforLinearOperators",Springer-Ver-lag,Berlin,1966.9.E.Sinestrari,J.Hath.Anal.Appl.107:16(1985).
SomeRealizationsofInteractionProblems FELIXALIMEHMETITectmischeHochschukDarmstadt,Darmstadt,GermanySERGENICAISEDepartmentofPureandAppliedMathematics,UniversityofSci-encesandTechnologyofLille,Villeneuved'Ascq,France 0.INTRODUCTION. InmathematicalphysicsoftenarisesthenecessityofdescribingsystemswhereevolutionphenomenaofpossiblydiferentrlaturehaveaIIinnuenceononeaIIother-ImportantrealizationsaretransmisdoIIproblemsonraminedspaces(notionintroducedbyG.Lumerin[10],cf.alsoS.Nicaise[121,[13],FAIiMehmetilll,肉,B.Gramsch闷,J.vonBelow[4landothers). Letusconsideraaexample:amodelforavibratiIlgcross: I5
l6 AliMdunettandNicatse FO盯COIP抖〉让ie臼Soft山lh1r町ee!i川川川t价e盯盯Ir打.S以〈O)让lh1u川1川ttliih〈O〉xm1口盯1mSItE:R+×Qi→Rof (IV)拷问(t,z)-cθ:1勺(t92)=0,V(t、z)ιR+×Qz,Vi(I)uz(0、r)=山(叫,θtuz(0、2)三0、V27ιQz,Vi(To)IlI(t吨。)二tb口,0)=u3(t,0)=tt4(t、0),VtιR+ (丑)汇θz咐,0+)=0 ThismodeldescribesrrHrctioIIandtransmissionofwavesattlleramiEmtifmmdrofacross.(ThemactsohltlionofthissystrIIlisgiveIIin[吁,formmyrpmltsOI川、liipticuldparabolk·problrmsonnetworksff.tlrPIll}licationsofS.NK-ziiFEPmdG.L1mld). IngeneralLranliaedsparcsaresystemsofdoIIlai口叭飞vherepartsofthel川mdaripsarribMi丑ml(cf.IlOLl12],[13]). 011robjectiveistolmwthisframcworkaIIdtostudyproblemsofthefollowiIlgkiIKi:ITplareSOIllfoftlrequatiOIlsunder(W)byparabolicorellipticmluMions(cf.secti0113);COIMiderdomaiIISQZofdifcrtIItdimeIlsioII(d.sectiOII2);COIMiderIIOIIliIIEYlrequationsa且(lIIOILliIlear(oralsoROIl10(1l)COIlditionsfor(To)(cfsrrtio口2、foItlrli时arcaseseel2]) OUI-stratcgyistodes叮ihetileflift?rentltiIKIF-ofinteractionsbyaclospdrollvrxsubsetofaproductofSobolevspacesoIltilt‘doI11aiIIS.飞17egettilespatialpartloftheprobkTIIassulxlifereIltialof〈-ertainconvexf11IlctlionahORtllis冈山川(byatirorcmofhiiMy[1叫11lL,IP〉OSSi川山山lbM〉才ly、w飞V毗rpο川州I川t旧Ib〉m川tal川tUiOIlm1山Si1口1[3剖]吨℃dfSe凹川CCd?吨tUiOI口l1)E飞VFOlhu川1tUiO口e町叫呻(q甲lμ1mtUiO口S叭呼wi江tIhltIhrmf芒?Sfe3SIP)χm川a川川1刊ti归ta1lIP川川;a1ItSartm1tedm吨[3laml[14l(世geneIateli肘ar叫se,cf.sectioI13) Ac11nottyledgemeηts· FAlihfrllIIletilikestothankthcuDUltscheForsell11IlgsgeIIleiIlschaftnfortilt、irSIll〉port(rrspareilscholarship). 飞17ptllfull〈proff、ssol·G.L11IIlerandprof俨SSorB.GritIIIsellforvaluablesl1pport.
RealizationsofInteractionProblems l7 1.GENERALFRAMEWORK. 1.1.Assumptions- (i)LetnεN*={1,2,3,.}beExed- (0ii斗)FOralltε{1'γ...川'η}L,C∞O丑mmSdiderrealHilbertS盯lP】aCe臼S(1;『(←.＼、.))1川:)an(H瓦th『(七.＼'.)MfHf‘)μS1u1CdhtIhh1matl只7L〕→Hzi.e-kisdemelyadcoatimmlslyimlm1&diMOHz. (iii)飞快denote v=IIVtand主=HHz, z=1 z=1 whichareHilbertspacesfortheiIlducedinnerproductsdcI1otedrespectivelyby(·,·)FaM(-,12·(iv)Letl/beaclosHlsubspaceofVmdHtheclosureofVinH.SoIY」→H (v)Foralli=1,...,凡letαi:vz×只→Rbeam口mmIn1Ee?gpa剖tUiVeC∞On川t竹i日川1u1Oulb〉i口山l口ih11remmEaa盯11口一IIr、f扣Or口mInm1丑1.飞飞7esupposethatα,isVt-coerciveintileweaksells飞thatis、TIllfI?existα,3>Osuchthat F/ZY,,ζ」,,&咽,.、Vv,zq,"FrNUQμFJ,句,",..uα>一、、lJ'uUJ'z·‘、α Deaneα:V×V→Rby (11) α((叫,(叫)):=艺创(uzfz Clcarly,αisaIIOIInegative,continuousbilinearform,whirllislrcorrrhvpi11the飞veal王seIISe. (vi)Letψzbeaproper(#+∞)comyexlowersc111icontimomO.s.cjfmdionfromHzintoRU{+∞} (vii)LetKbea口onemptyclosedconvexsubsetofV(KiscalledtlriMeradio口的).
l8 A/iMehmefiandNraise 1.2.Deanitions- LNusddIlethefunctionsO,OKandψonHasfollows: ''产-K」、飞飞叫Hι」rAU、tlfAU"w∞1吨2+/BEEf、l飞=机川川AVJ川=,IAVAV UJUn汇M一一川札γ 1.3.Lenln1a. φ?φI飞'α74dψαTEPTopeTContyez13.c.functions. 1.4.Theorem- 30.θ白、Fandθ非αTe771miTrzd7710710tone- Proof:DirectcomequenceofMintykTheorem([3!,Theorem3).. Now,wewanttoKIlowwhetherθ￠+θ47orθ￠κ+θψismaximalIIIOIIotoIlf-ForKIsta时飞1山吨Theorem9of[3l,wecanstatethefollowi吨result. 1.5.Theorem- AmumethdtheTeczistsaconstαntCsuchthd (12) ￠((I+λθψ)-Iu)三￠(u)+Cλ foTαll入>OαπddluεV.Thcπθ。+θ￠isTTbαzimdTnOTtoto71e.AssumemoTCoverthdfoTαlJUεIL(I+λθ。)一11tεK,foTαllλ>OJtb,enθ￠κ+θOismαzimd 7rl0710tone, 1.6.Remark. IIItllepImThustheorem、wecanreplacethecondition(12)byalocalonei.e.tlltyeexistsaCOIlstantCZsuchthat (13) ￠t((I+入θ如)-IUi)三仇(ut)+Cz入, forallλ>0,uzE叭,iε{1,...,n}.Ifweass旧时thatH=HaBdthat(I+λθ￠)一1IlεV、forallA>OandallUεV,thentheconchIsiOIlsofTIleor川Il1.5remaintrue.
RealizationsofInteractionProblems I9 Now,weareabletoconsidertheevolutionequatioaassociatedwithθ￠+δ4?orθ帆'+θψ.OwingtoTheorem2lofi3],wehavethe 1.7.Theorem. UMeTtheαsswrzpti0718of1.50T1.69letf=(fJL1beαbsolutelucontimousfmm[0,TiintoHαnduoεD(θ￠K+θψ).ThentheTeezistsαmiqmftmctionUεC([0,Tl;H)sdishing u(t)εD(θ￠K+θ圳,Vtε[0,Tl, UisLipschitzcontinuouson[0,Tl,生+(θ仰+御)(uHfαe.m(0、T)、dtu(0)=u0· 1.8.Deanitions- 飞几7ecallIfinteractionset,θ功Kinteractionoperatorandtheproblemin1.7interactionproblem. Letusremarkthatourmaingoalsarethefollowing:giveII句,仇,VaIIdIf, then i)HowtocharacterizeD(δ￠ff+δ￠)?ii)Whatisthemeamlgoffε(θ￠K+θψ)(u),whenUξD(θ￠K+θψ)? Thiswillbetheobjectiveofparagraph2toanswertothesetwoquestionsiIIsomeparticularsituations.Thiswillshowwhatkindofproblemswecansolvewiththisverysimpletheory. 2.DESCRIPTIONOFTHEOPERATORSINSOMEAPPLICA-TIONS. 2.1.Commondata. Throughoutthisparagraph,weusethenotationsofthepreviousoneandwemakethefollowingassumptiOIls: i)Weconsiderboundedopeasubsets511ofRkt,wherekgεN"＼t=1,..1·
20 AliMehmetiandNicatse ii)WetakeHz=L2(Qt),Vt=H1(Qt)andtheformαiistheformoftilegradienti.e.forallth1yξVz: 向(tm)=儿vu川Z 2.2.Anonlinearexampleonatopologicalnetwork. LetusCOIlsidcrthreecopiesofthesameinterval(0,1).Thatmeansthatn=3f山iO1=Q2=03=(0,1).Asinteractionset,wecomi巾r V={(uz)?=lEIIKsaMying (21)(22) ti1(0)=U2(0)=U3(0);uz(1)=0,Viε{1,2,3}}. LetkbeaconvexlS-CfuactionfromRiMoRU{+∞}aMwedenoteby?=θlt、thesubdiferentialofk.Forsimplicity,wesupposethatD(γ)=R.飞机〈ldIletlrfunction仇onHzasfollows: Z,dukt∞flQ+/EEEJ、EEE飞一-u仇V ifk(u)εL1(Qz), else. ItisclassicalthatψsisconvexandLs.c.,therefore飞iydeanedby12isautomaticallyconvex1.s.c.(owingto1.3).Moreover,itisprovedbyBr位isin[3]tl川inequality(1.3)isfulmledforall问εH1(Qt).ThisallowsustoS}hl(O〉wt吐imttllh1efa1SS11uImI丑1PtiOInlSof1.6aalrref且1u1dl白mlHled. Usingvariationalinequalities,wecanshowthat D(彻)={(ttJLεIImq)satisfying(21),(22)and (23) nuh-zno-33了μ问 30issinglevalmdandforallU仨D(θ的: (θ￠)(u)=(-AUJL1·
RealizafionsofInferactionProblems 2I OwiIIgtoTheorem1.7,wehadprovemthatthefollowingproblemisuniquelysolvable: (2.4) 旦旦+A+γ()3fQdtuz(0,·)=ti02,Vi=1,2,3, ThisevolutionequationisactuallyaIIevolutioaequationonthetopologicalaetworkformedbythreestraightlinesegmentsQt,whichhaveiIlformmOIIOIIeoftheirextremity(seeEgue1) 什1 咱EAeruσbF Evolutionequationsontopologicalnetworksofthistypewerestudiedimpol.[1317[4],[2](seealsootherpapersoftheseauthorsL 2.3.SeEniPenneableinterface. Letn二3,QI=02=Q3=(0,1)andletusset V={(udLIεIIlfsatisfying(22)and (25) U1(0)=U2(0)}.K={uεVMallingu3(0)三u1(0)}. SinceKisaclosedcO盯excomewithvertex0,byTheorem112of[叫,wededmthattileinteractiomoperatorθ￠IIPhasthefollowingform: D(θ阳')={uεKfulElli鸣(2.3)and 32(0)剑,37川(0)-u1(0))=。}
22 AliMehmetiand入fraise Forallti巳D(θ￠K), θ￠K(u)=(-Aut)?=I. Aspreviously,wecmalsoperturbθ似'byδ￠.Inthatcase,weobtainanoperatorwithIIOIIiiIlearitiesoneachbranchQZbutalsoELtthecommonvertex. ToEIlistlthissectioIUletusgivealinearexample,whichisacoupledproblembetweenaone-dimensionaldomainandatwo-dimensioIlaldomain.WeIIlentioII thisexamplesinceithassomecOBIlectionswithaproblemofelasticity,obtainedbyCiarlet,LeDretandNzengwain[71,couplingaplateequationina2-Ddomainandthelinearelasticitysystemina3-Ddomam- 2.4.Acoupledproblem. Letusset r={(z,0):0<Z<1},。1={(ZJ)εR2:z2+ν2<1}＼F,。2={(2,0):0<Z<2}, thislastonebeingidmtiEedwiththerealinterval(0,2). V={(uhh)εHF(Qz)fumingγ+UI=γ-U1=u2onr U1=Oon301＼r, U2(0)=u2(2)=0}, whereδQIdemtestheboundaryofQ11ndγ叭,n￡t(respγ-u,γ一步ι)(lmotethetraceofttandthetraceoftheoutwardnormalderivativeofumr fromabove(resp.frombelow)inQ1(seeague2) 。10% Figure2
RealizationsofInteractionProblems 23 UsingTheorem112of[6landTheorem1.52.3of[9l,wecmprovethatUED(δ￠)andf=(θ￠)(u)ifU=(uhU2)εVfulElsu2(1)=Oand -A==fiQ2U2θu1θu1一一τ+γ+一一+γ-T=hinr,U品θν+-Uν132aa一二字=hinQ2＼r. Thismeansthatthisoperatorsplitsintotwoputs:thearstoneisaaoperatorcouplingO1withitscutr(withDirichletboundaryCOIlditionsforu10日θQ1＼raMforγ+UIatOand1),whilethesecondoneistheLaplaceoperatoron(1,2)withDirichletboundarycomditions- 3.MIXEDHYPERBOLIC-PARABOLICPROBLEMS. Inthissection,wewanttoapplyparagraphVI4in[14ltointeractionproblemswithhyperbolicequationsonsomedomainsandparabolicequationsonothers.Letusrecalltheabstractresult: 3.1.Assumptions. (i)LetVbeaHilbertspaceandA:V→V,theRjeszmap- (ii)LetCε￡(V,VObesuchthatthesesqdimarformCdennedby c(2,u):=Cz(u),Vz,νεV, issymmetricandnonnegativeonV.Then||z|lw:=c(2,z)1/2isaseminormonVWwilldenotetheseminormspace(V,||·||w). (iii)LetD(B)beasubspaceofVandBεL(D(B),VF)bemonotone. (iv)AssumethatB+CisstrictlymonotoneandA+B+C:D(B)→VFis aSUI-JectIOIl- 3.2.Theorem([141,CorollaryVI4.B,p.140). UMeTtheassumptions3.1,foTαllfεCI([0,∞),IV),uoεV,uIεD(B)
24 AliMehmetiandNraise stdLthdAtto+Btt1εI$719theTeezistsαwziqMsolutionUEC1([0,∞),V)of (31) CUFεcl([0,∞),WF),u(0)二Uo,cd(0)=CU1,UJ(t)εD(BLVt主0,Ati(t)+Btf(t)εIV',Vt主0,(CUF)F(t)+Btf(t)+Au(t)=f(吟,Vt主0. 飞VealpplythisresultinthefollowiIIgcontext. 3.3.Data. 飞飞hmaketheassumptiom1.1(i)to(v),withthefollowingdata (i)LetQsCRKUt=1,..,n,beboundeddomainswithLipschitzbomduim (ii)飞;vetakeHz=L2(QB),vz=H1(Qa),thebiliIlearformαtisddIledby α(ut)=咱川 1vlrreG:J=tzjl,CaεL∞(flt)arerealvalued.Wesupposethatthebiii时arformGisV-coercivei.e.thereexistsγ>Osuchthat α(u,u)主7||u||号,VuεV (iii)LetI={1尸..,叫andJbeasubsetofI (iv)Considerinitialconditionsuozεva,tiIzεKS11chthatULtiIξV (v)C0日siderfzEC∞([0,∞),Qt). 3.4.Deanition- fijDeFneA:b,→VFbu A叫。):=α仙,υ). ItistheRicszmapofVP1tyhenVisequippedttyiththeinneTPToductG·
RealizationsofInteractionProblems 25 。"DeFMC:V→Hα叫B:V→Hbu 7JC」-q&ρbVJJ'MMatρbuor--、l自飞一-ctu'儿'''u-a&ω、、..,,,CZ们UJ,,‘‘、一一、‘,,,,,UJ''E、、PL B(ut)=(叫一υf). smceVisαclosedsdspαceofVmdVHHFtρeseethdCyBε￡(l气VF). ftttjDeFMf(t):=(A(t,·))t=1,.,". 3.5.Lemma. ftjC,Bεζ(V,VF)αTesumwzetTicα叫monotone. fttjB+CisstTidlymonotone.ftttjA+B+C:V→VFtsasu甘mUon- 3.6.Theorem- WithDαtα3.3andDcj17Bitt0713.4pωehαtyetheconclusionsofTheOTETn31. 3.7.Proposition. ifH二1D(Qz)CVaMthesolution(ui)offiIjischssicdfi.e.utεC2([0,∞)×Qt)),thenftIjimplies (32) θ?ut(tJ)-2二θJ(α:1θIUt)(tJ) +ctUz(t,z)=A(t,吟,V(t,z)ε[0,∞)×仇,Viελ (3.3) θtut(M)-2二θj(α:jaut)(们) (3.4)(3.5) +ciut(t,z)=fi(t,叫,V(t,z)ε[0,∞)×仇,VtεI＼J,uz(0,z)=uω(z)VzεQhvtεI,δtUz(0,z)=UIz(z)VzεQhviεJ.
26 AliMehmetiandMeatse 3.8.Remark. Forexamplesofconcreteinterpretationoftheinteractionconditions(i.e.u(t)εV,Vt三0)seesection2. ItispossibletostudymixedParabolic-EllipticequaIionsusingtheresultsofparagraphV4of[14].Thiswillbemadeelsewhere- REFERENCES [1lFALIMEETMETI,Probi-mesdetrammissioIlpoudesAquationsdesomlesliII兰airesetquasilindaires,SAminureJVaillant198243,p.75-96,Herm:mvParis,1984. [2lFALIMEETMETLRegularsolutionsoftrammissionandiIIteractionprob-lemsforwaveequations,Math.Meth-Appl.Sci.,11,1989,p.665-685. [3iHBREZIS,MomtomcitymethodsinHilbertspacesandsomeapplicatiomtoIIonliIlearpartialdifferentialequations,inE.Zarantonello,ed.9COIltril311tioIIStOIIOIIlineazfunctionalanalysis,1971,p.101·156. [4lJVOIIBELOW,Classical叫vabilityoflinearequatiomonmtworks,JDif-Eq.,72,1988,p.316,337. [5lHBREZIS,OpdrateursmaximauxmOIIotonesetsemi-groupesdecontrac-tionsdanslesespacesdeHilbert,Math.Studies5,North-HollaIId,AIRster-dam,1973. [GlPG.CIARLET,TheanitedemeIIImethodforellipticproblems,Studiminhfath.andAPPI-,North-Ifoliand,Amsterdam,1978. [71P.G.CIARLET,HLEDRETandR.NZENGWA,JunctionsbetweentILIfeamenSionalandtwo-dimensionallineariyelasticstructures,J.Math.PuresetAppl.,68,1989,p.261-295. [slBGRAMSCH,ZUIIIEi由ettmgssatzvonIKllichbeiSobolevriiuIIlen,MathZeitSChr.,106,1968,p.81-87. [9lPGRISVARD,Ellipticproblemsinmmmoothdom出nRMonographs:IndstudiesinMath.,24,Pitman,1985. [10lG.LUMEI飞,Espacesrami且是setdifusionSSurlesrbeauxtopologiq时S,CRAcad-Sc.Paris,291,sdrieA,1980,p.627-630.
RealizationsofInteractionProblems 27 [111GJ.MINTY,OnthemonotOIlicityofthegradientofaconvexfmction,PaciEcJ.Math.14,1964,p.243-247. [12lS.NICAISE,Probl主mesdeCauchyposdsennormeuniformes盯lesespacesramindsdldmeIItaims,C.R.Acad-Sc.Paris,303,SArieI,1986,P-443446. [131S.NICAISE,LelapladenSurlesdseauxdemdmemiomelpolygonauxto-pologiq11凹,J.Math.PuresetAPPI-,67,1988,p.93-113.[14iRESHOWALTER,HilbertSpaceMethodsforPartialDifere川alEq1mtio川、MonographsandStudiesinMath.,1,Pitma口,1977.
SobolevImbeddingsandIntegratedSemigroups W.ARENDTDepartmentofMathematics,UniversityofFranck-Comte,Besancon,France 1.INTRODUCTION TheconceptofintegratedsemigroupshasbeendeveIopedinordertoprovideaframeworkforCauchyprob1emswhicharenotgovernedbyaC。-semigroup(seeArendt(1987)andNeubrander(1988)).Itmaybebrieflydescribedasfollows.LetAbeaIinearoperatoronaBanachspace.ConsidertheCauchyprob1em U'(t)=Au(t)u(0)=x. (t主0) 飞,/tEA.唱4·品'''飞 IfAgeneratesaC-semigroupT,thenforXED(A)thefunctionu(t)=T(t)xisaOc1assicalsolutionof(1.1).LetkEmandconsidertheoperators ,sd飞-JS/·飞中A、,,,-ALa/·飞,,,咱E4-K、‘EJS+L,,.‘、tortt'=、‘,Jf」,.‘、QU thek-timesintegratedsemigroup.NowitmavhappenthattheseoperatorsexistwithoutthatAgeneratesasemigroup.Makingthismoreprecise1eadstothenotionofk-timesintegratedsemigroupsandtheirgenerators. 29
30 Arendf Ofcourseao-tiITIesintegratedsemigroupisthesameasaco-semigroup;andifAgeneratesaK--tiITIesintegratedsemigroup,thenAgeneratesant-tiITIesintegratedsemigroupforallt主k.Sothenumberkisanexpressionforthere8111arityoftheCauchyproblemdefinedbyA.ItiSofgreatinteresttodeterminethebestpossibIeconstantkforcon-cretedifferentia1operators.Atypicalexamp1eiSgivenbytheSchrtjdingerequatlOI1.DenotebyAPtheLap1acianonLP(EN)Withmaximaldomaininthesenseofdistributions(wherel三p三∞).IthadbeenshownbyHUE-mander(1960)thatiApgeneratesaC。-semigrouponlyifp=2.HoweverBalabaneandEmaIIlirad(1983)provedthatiAPisthegeneratorofasmoothdistributiongroupoforderkif111』k>N|一,-|,1<p<∞,andthisimpliesthatiAgeneratesak-timesintegratedlp21psemigroup(seeArendtandke11erITlann(1989)).RecentlyHieber(1989)extendedthiSresu1ttothecasewhenp=1,∞,andheshowedthatitisoptiIIlal:iAP11doesnotgenerateak-timesintegratedseMgroupifk<N|E-z|·SotheexactNorderOfre8111arityisdeterminedonR.However,theresultsarebasedonmultip1iertheory,andSOanextensiontoboundeddomainsorVariabIecoeffi-CientsiSnotpossib1ebythismethod.Ontheotherhand,inArendt,NeubranderandSch1otterbeck(1989)integratedsemigroupsaredescribedbyinterpoIationofsemigroups.InthepresentnoteweusethiSmethodtogetherwithellipticregularityandSobolevimbeddingstoNinvestigatetheSchr6dingeroperatoronLP(口),whereQCEisaboundedopenset.ThereSUItissurprisir18·OnboundedsetstheSchrtjdingerequationiSmoreNreg111arthanonR.Infact,iA(withDirichletorNeumannboundaryconditions)generatesak-timesintegratedseMgrouponLP(Q)(1<p<∞)ifk三旦|i-i|and-22p-∞1.Nonco(口),C(口),L(Q),L(Q)1fk>Z·OurmethodstiIlworksiftheLaplacianisreplacedbyastrict1ye11ipticoperatorwithsmoothcoefficient. 2.INTEGRATEDSEMIGROUPSANDTHECAUCHYPROBLEM LetAbeanoperatoronaBanachspaceEandkEmU{O}.WesaythatAgeneratesak-timesintegratedsemigroup,ifthefollowingiSsatisfied:ThereexistWERandastrong1ycontinuousflinetionS:[0,∞)→4(E)satis-fying sup|e-wts(t)|<∞t主O (2.1) suchthat(w,∞)cp(A)(theres01ventset)and
SobolevImbeddings 3I -1kr∞-λtR(λ,A):={λ-A)=λ儿eS(t)dt (2.2) forallλ>w.Notethattheintegral(2.2)existsasanimproperstrongRiemann-integral.ThefunctionSisunique1ydeterminedbythepropertiesabove.Itisca11edthek-timesintegratedsemigroupgeneratedbyA.AO-tirnesintegratedsemigroupisthesameasaC-semigroup,seeArendt(1987).OIfAgeneratesak-timesintegratedsemigroupSandmEm,thenAgeneratesak+m-tiIIlesintegratedsemigroupVgivenby (t-s)m-lv(t)=Jo(m-1)!S(t)dt (t呈O). (2.3) Thef011owingexistenceanduniquenesstheoremh01ds. THEOREM2.1.LetAbethegeneratorofak-timesintegratedsemigrouponE,wherekemU{O},1etfEC([0,τ],E)withT>OandUEE.ThentheprobIemO 、‘.,,,、‘,,,,A/,‘、nu.,寸ltJτ,nUFEl--』'''飞、户UnH、‘,,,Eb,「EEEd宁、,nu「lL,,.飞、'iFUr-、u U'(t)=Au(t)+f(t) (tE[0,τ]) (2.4) u(O)=UO hasatmostonesolution. ThereexistsaSOILItionwheneveroneofthefollowingconditionsissatisfied. (a) ,、、.,,AA,,‘、nurtou,、、,,,EL,1,」T.,nurstL,,.飞、电BA+kpbrtzi u噜:=Au+f(O)ED(A),iO 、‘.,JAA/,.、、nυrp、、‘E/nu/,、、、、,,,,唱,ikfa·、FA+咱-4·KUA--L民u、‘,,,A且,,.、nur巳、‘aynurs飞、rTA+唱Ba晶UAA--?』u (b) 、h/唱IA牛,KA/·飞hurtou and 飞,/'EA+LHAAAJ,‘、、nu「lJτ,nu「『Li·、户Urtzi FortheproofwerefertoArendt(1987)exceptforpart(b)whichwi11beprovedinSection3.Itispossibletocharacterizegeneratorsofk,timesinte-gratedsemigroupsbywell-posednessofthehomogeneousCauchyproblem,seeNeubrander(1988). Final1y,wesaythatanoperatorAgeneratesak-timesintegratedgroup(wherekEmU{0}),ifAand-Agenerateak-timesintegratedsemigroup. 3.INTEGRATEDSEMIGROUPSANDINTERPOLATION LetEandFbeBanachspaces.WewriteEc,FifECFandiftheinjediorlofEintoFiscontinuous(whichmeansthat|x|F三comt.|x|Efora11XEE).IfBisanoperatoronFsuchthatp(B)工队thenforkEmU{O}thespace
32 Arendt D(BK)isaBamchspaceforthemrmjxlbt=|x|宁lfixl+...+iBkxlandweBMkalwaysmeanthis(oranequivalent)normifwelookatD(B)asaBanachspace.Obviously,D(BK)马F.Moreover,foreveryhp(B)theoperator(λ-B)kisanisomorphismofD(BK)ontoF.NowassumethatE乌F.ByBEwedenotethepartofBinE;i.e.D(BE):={XTD(B)nE:=BXEE}·IfλEP(B)suchthat(λ-B)4ECE,thenλEP(BE)l-l-1and(k-BE)4=(λ-B)lE(therestrictionof(λ-B)itoE).ThepurposeofthepresentnoteistocomparetheCauchyproblemwithrespecttoBonthespaceFwiththeonedefinedbyBEontheinterpoIationspaceE.kThecasewhereE=D(B)playsaparticularrole.Assmethatp(B)1￠.ThenB1,:=BLissimi1artoB-Moreprecisely,lettingU=(λ-B)kwhereAEP(B)ιD(Bb)k+1onehasBk=UBUwithdomainD(B)=D(B).HenceBgenerateskaC。-semigroupifandor11yifBkgeneratesaCO-semigroup(Cf.Nage1(1984)).KHowever,ifBgeneratesaC1emigroupandifD(B)CE乌F,ingeneral,BEdoesOnotgenerateaCO-semigroup,butitalwaysgeneratesak-timesintegratedsemi明group;moreover,a11k-tiIIlesintegratedsemigroupstzar1beobtainedinthatway.ThisiSshownbythef011owingtheoremwhichistheIIlainresultofArendt,NeubranderandSchlotterbeck(l989).Moreprecisely,theequivalenceof(i)and(ii)isprovedinthearticlejustmentioned,theequivalenceto(iii)beinganeasyconsequenceprovedbelow. THEOREM3.1.LetAbeanoperatoronaBanachspaceEandkER.Thefollowingareequiva1ent. (i)Ageneratesak-timesintegratedsemigroupS. (ii)ThereexistaBanachspaceFandageneratorBofaC-semigroupVonFOsuchthat (a)D(BK)ζE电F,(b)(A-B)lECEforsomeAερ(B), (C)A=BE· (iii)ThereexistsaBanachspaceGsuchthat (a)D(AK)CG-E,(b)(λ-A)1GCGforsomeλερ(A), (c)Ar、generatesaC-semigroupUonG.bO
SobolevImbeddings 33 REMARKS3.2.1)TheoperatorAinTheorem3.1isnotnecessarilydense1ydefined. 2)IntheSituationofTheorem3.lonehas U(t)x=V(t)xfora11XEGand rtk-lstt)x=儿(t-s)/(k-1)!VU)xds fora11xtE. 3)Conditionb)in(11)and(iii)isautomatica11ysatisfiedifk=1.4)If(ii)h01ds,then(λ-B)-lECEfora11λEP(B)(by[5,(2.1)],forexample)andsop(B)cp(A).If(iii)holds,thenp(A)cp(AG). 5)Aswillbeseenintheproof,undertheassumption(ii),in(iii)onecantakeG=D(BK).ThenA=Bandsoby4),p(B)Cp(A)cp(A)=p(B)=p(B);GkGi.e.p(A)=p(B). 的IfH1,H2'H3arethreeBanachspacessuchthatHICH2CH3andH2乌鸟,thenitf011owsfromtheclosedgraphtheoremthatHl乌H2·Thusin(ii)kkweactual1yhaveD(B)乌E,andD(A)乌Gin(iii). PROOF.Theequivalenceof(i)and(ii)isprovedinArendt,Neubranderandsch1otterbeck(1989).k(ii)=>(iii).If(11)holdsktG=D(BLThenA=BisthegeneratorofGkaC。-semigroupand(iii)(a)and(b)areobvious.(iii)=>(i〉.Assumethat(iii)holds.LetC=AnandH=DUK).Then-lvD(C)cH.ThereexistsλEp(A)suchthat(λ-A)GCG.HenceλEp(C)and-1-1kk(λ-C)H=(λ-A)D(A)cD(A)=H.Bytheequiva1enceof(1)and(ii)itf011owsthatC=Ageneratesak-tirnesintegratedsemigroup.SinceAKandAHkareSiIIliIar(1)f011ows.口 Theorem3.lassertsthatintheinterp01ationsituation(ii)onemayloseatmostkdegreesofregularitybyconsideringAinsteadofB.Thisismoregenera11ytrue. THEOREM3.3.LetE,FbeBanachspacessatidyingEQF.LetBbeanoperatoron-lkFsuchthatO-B)ECEforsomekEP(B)andD(B)CEwherekEIN.LetA=BE-Thenfor且EmU{O}thef011owingh01ds. (a)IfBgeneratesan且-timesintegratedsemigrouponF,thenAgeneratesak+且-timesintegratedsemigrouponE. (b)IfAgeneratesanE-timesintegratedsemigrouponE,thenBgeneratesak+t-tiITIesintegratedsemigrouponF.
34 Arendt PROOF.Firstweremarkthatthehypothesisimpliesthatp(B)cp(A)and-1(A-A)=(λ-B)lforall入Ep(B).lE(a)ByTheorem3.lthereexistaBanachspaceHandthegeneratorCofa且-lc。-semigrouponHsuchthatFιH,D(C)cF,(μ-c)HCHfora11μEP((;)(Cf.-1Remark3.2.4)andB=CItf011owsthatA=C,(μ-C)ECEforallμ4p(C)andH.Et+kD(A)cH.SothecIaiInfo11owsfromTheorem3.1.(b)LetG=D(Bt).ThenD(AK)ζGandA=B=BItf011owsfrom(a)thatGGK-Bkgeneratesak+1-tirnesintegratedsemigroup.SinceBkisSimilartoB,thesameistrueforB.口 REMARK3.4.OnecangiveanalternativeproofofTheorem3.3inthecasewhenAandBaredenseIydefined.Infact,Bgeneratesani-timesintegratedsemigrouponFifandonlyifforsomew呈0,(w,∞)cp(B)and W＼/飞Ant、‘,,,,RU,HH/E飞pun、‘.,,RU,、λ,,.、、D且n、‘,,,,wλ,,,‘、f‘LPUS--M n=0,1,...}<∞ whereμEp(B)isfixed(Cf.[19]).SoundertheassumptionsofTheorem3.3a),for a11XEE,入>w,n=0,1,...,onehas n且+knn且+k|(入-w)rIR(λ,A)R(μ,A)x|E=|(λ-w)R(入,B)R(μ,B)x|E,nn且ka三const.|(λ-w)R(λ,B)R(μ,B)hR(μ,B)bx|LD(Bb)三const·|(λ-w)nR(λ,B)nR(μ,B)且x|F 豆comt-M|x|F豆const.|xlEwheretheconstantsdonotdependonλandn.ThisshowsTheorem3.3a)tohold.口 ThefollowingconsequenceofTheorem3.lisknown(Cf.ke11ennarmandHieber(1989),Neubrander(1988)andHieber(1989)). COROLLARY3.5.LetAbethegeneratorofak-tiIIlesintegratedsemigrouponEandBEd(E,D(AK)).ThenA+Bgeneratesak-timesintegratedsemigroup. PROOF.LetGandCbeasinTheorem3.1(1ii).ThenBnEd(G)SOthatvkk(A+B)G=AG+BGgeneratesaco-semigrouponG.ButD((A+B))cD(A).SotheC1aiITlf011owsfromTheorem3.1.口 FinallywegivetheproofofTheorem2.lb)asaconsequenceofTheorem3.1.
SobolevImbeddings 35 PROOFOFTHEOREM2.1bj.LetGandCbeas8iveninTheorem3.1(iii).Hencek+111OED(A)CD(C)andfEC([0,τ];D(C)).Sobypazy(1983),4.2.6,thereexistsIUEC([0,τ],D(C))nC([0,τ],G)suchthatUE(t)=Cu(t)+f(t)fortE[0,τ]andu(0)=110.HenceUisasolutionof(2.4)sinceG乌EandD(C)句D(A).口 4.THESCHRODINGEREQUATION Inthefo11owingweapp1ytheinterPolationtheoremstodifferentia1operators-Wewi11usetheSob01evimbeddingtheoreminthef011owingform. NNTHEOREM4.1.LetQ=Ror1etQCRbeaboundedopensetofclassC.Letkfm. U-(a)Ifk〉N,thenH气口)CC(Q).(b)Let2三p<∞andk〉N(i-i).ThenHK(Q)CLP(Q).=2p(c)Letl〈q三2andk〉N(i-i).Thenwk,q(Q)CL2(口).=q2Fortermin01ogyandproofofthisresultwerefertoBrezis(1983)(seealsoAdams(1975)). A.THESCHRODINGEREQUATIONONBOUNDEDDOMAINSLetQCRNbeaboundedset·WedefinetheoperatorA2onL2(51)byD(A2)={UEH;ω):AUELh)}, A211=Au.ThenA2isaself-adjointoperator(whichgeneratesaC-semigrouponL2(Q),OseeBrGzis(1983),chapterX).ItfollowsfromtheBeurIing-Denycriterion(seeReed-simon(1978),XIII·12)thatthereareuniqueoperatorsAPonLP(Q)(l豆p豆∞)suchthat (a) (0,∞)CP(AP);PAisthepartofAinLif∞运p>q运1,qllR(λ,A)=R(λ,A)'ifλ〉0,-+-=1,qpq (b) (c) 1豆p,q豆∞,q其∞. TheseoperatorsgeneratecontractionsemigroupsonLP(Q).Hereweareinterestedintheoperators A:=iAPP (1豆p豆∞), fAOY-iev.吨4+Lcepser,、、,,,-口,,.、、FUJUna、、.,,nu,,.、、.0、lpunMJ,‘、nqι-lvL?』nAozinrouocrAAUDdynraa+LO--Anus+Larasp品e+」eahrtenne--gs7』aA1i'1&eeswrusoac
36 Arendt 2InthesequelweassumethatQisofcIassC(seee.g.Br臼is(1983),IX.6).2ThenD(A2)=H。(Q)nH(口)(seeBr岳zis(1983),IX.6,forexamp1e).ldefirstconsiderthecaseN三3.ThenbytheSobolevimbeddingtheorem2一-1Hιcc(口)andSOD(A)cC(口)nH(口)=C(口).Itf011owsfromTheorem3.1thatA2OOOandAgenerateanintegratedgrouponC(Q)andC(fJ),respectively.SinceOD(A2)cc。(Q)cLP(Q)cL2(Q)(2豆p豆∞),Ageneratesanintegratedgroupifp2三p三∞forthesamereason.2SinceR(λ,A叮)L(Q)cC(口)andR(λ,A叮)isselfFadjoint,itf011owsthat4O4R(λ,A)LI(口)cl气。).HenceD(A)cL气。)乌Ll(Q).SinceD(A)cD(A)cL2(Q)乌Lq(Q)lq1月forl豆q<2,itf01lowsthatAgeneratesanintegratedgrouponL可(Q)forql三q三2(byTheorem3.3(b)).Sofarwehaveprovedthefollowing. 2THEOREM4.2.AssumethatQisofClassC.IfN三3,thentheoperatorsA=iAnpp(1三p三∞)aswe11asAandAgenerateaonceintegratedgrouponLY(Q),C(Q)OCa71dC(Q),respectively. OONow1etNbearbitrary.IfQisofClassCwecanusee11iptiCregularity.k2KInfact,inthatcaseD(A)CH(口)(KEIN)(see,e-8·,Br邑zis(1983),以.6)andSObythesobolevimbeddingtheoremD(AK)cLP(口)ifk主旦(上,i),2三p<2and一22phND(A)cC(Q)ifk>-HenceAgeneratesak-timesintegratedsemigrouponLY(Q)4.PNllfork三-(---),2〈p<∞andAAandAgeneratek-tiIIIesintegratedsemigroups-22p=∞'OConLU),c。(口)andC(百),respectively,ifk>:1lFor1三q<∞wearguebyduality.Letl<q<2,一+-=1,andletpqk〉旦(i-l)=旦(i-l).LetA〉0.SinceR(λA)kL2CLPitf011owsthat-=2q222p'2'R(λ,Aq)kLqcL2andsoD(AK)CL气。)QLq(Q)·Itfol]owsfromtheequivalenceofq(i)and(iii)inTheorem3.lthatAgeneratesak-timesintegratedsemigrouponqLq(Q)-k2k1Ifk>Z,thenR(λ,A2)L(日)cco(Q).consequentlyR(λ,Al)L(Q)CL(il)andsoD(Al)CL(口)乌L(口)·ThusAlgeneratesak-timesintegratedgrouponL1ω)fork2lk>lWehaveprovedthefollowingtheoreminthecasewhenQisC.ForthegeneralcasewehavetoapplyTheorem3.3successively(seetheproofbe1ow). NTHEOREM4.3.LetQCRbeaboundedopensetofClassC.Nll(a)Let1〈p<∞.Ifk〉一|---|thenAgeneratesak-timesintegratedgroupon=22p'pLP(Q).
SobolevImbeddings 37 (b)Ifk>LthenA,A,AandAtgene川ek-timesintegratedgroupsonC。(Q),·OC∞i-∞1C(口),L(口)andL(Q),respective1y. Intheproofwewi11usethatforl<p〈∞ D(Ap)=w;,pnw2,p2sinceQisofc1assC(seeAgmon,D011811sandNirenberg(1959)). PROOF.Let见ENU{O}suchthat且-1<N/4豆1.Fork三t-lletpk=2N/(N-4k)andletpE=∞.1.LetkE{0,1,...,且}.ThenAgeneratesak-timesintegratedgroupifp<∞andP2豆p豆pk·Infact,letmbethemaximumofthenumbersksuchthatthisassertionholds.Supposethatm<t.BytheSobo1evimbeddingtheorem(BrGzis(1983)Cor011ary2,p-EPEIX.15orAdams(1975),Theorem5.4)onehasD(A)cWMCLYcLMforpmPm三p豆Pm+1'p<∞·HenceAPgeneratesanIII+1-timesintegratedgrouponLP(Q)byTheorem3.3.Thiscontradictsthemaximalityofmand1.isproved. 2.Let2豆p<∞andk主N/2(1/2·1/p).ThenAgeneratesak-timesintegratedgroup.PThisisclearfork兰克(sinceby1.Ageneratesant-timesintegratedgroupforpa11p<∞).If见,1主k,thenN/2(1/2-1/p)豆k=N/2(1/2-1/pk),sothatp豆PK'andtheCIairnfo11owsfrom1. 3.IfE>N/4,thenbytheSobolevimbeddingtheorem2,PE-11-1-∞PE-lD(A)cwnHcC(Q)门H(Q)=C(Q)CC(Q)cLCL(Q).SinceAOOOrE-lrE-1generatesant-1-timesintegratedgroup(by2.),itfollowsfromTheorem3.3thatA0'ACandA∞generate且-tiInesintegratedgroups. 4If且=N/4letPE(P川,∞)Then叫)42,pnH;(Q)cCJEI)SinceAPgeneratesanE-tirnesintegratedgroupby2.,itfollowsthatA,AandAgeneratet+1-timesOC∞integratedgroups. 5.Fir1allylet1二q<2.LetpE(2,∞]suchthat1/p+1/q=1.Ifq>1andk呈N/2(l/q-1/2)=N/2(1/2-1/p)orq=landk>N/4,thenAE=Ageneratesaqpk-tiTIlesintegratedgroupby2.-4.SinceAhasdensedomain,itf011owsfromtheqgenerationtheoremArendt(1987),Theorem4.3,thatAgeneratesak-timesintegra-qtedgroupaswe11.口 Thestep-by-stepmethodpresentedintheprecedingproofhasanotheradvantagebesidesneedingIessregularityofQ:Onemayperturbbyboundedoperatorsinterpo-
38 Arendt 1atingonLP(口).Moreprecise1y,letBEzt(LP(Q))(1豆p三∞)suchthatBf=Bffora11ppqfELP(QMLq(口),(1句,q三∞).Thenweobtainbythesameproof:A+Bgeneratesppak-timesintegratedgroupifk主N/2|1/2-l/plandl〈p〈∞ork>N/4andp=l,∞.Inordertogiveaconcreteexamplewenotethef011owingexistenceandunique-nessresultwhichisaconsequenceofTheorem3.1andTheorem2.1. 32COROLLARY4.4.LetQCRorRbeopen,boundedofClassC.Let24fEC([0,τ],C(Q))suchthatf(0)EC(Q).ThenforUEC(Q)thereexistsauniqueOOOSO111tionof 、‘,,J、‘,,,36iiE飞OFU.,叮EEEJT.,nU「E『'』,,.、.、吨俨Un川、、.,,r,、.,,J,nM,,E‘O户U.,、E14JT.,nUF'EE』,,.‘、噜EAFU,,、u u(t)ECO(Q),Au(t)ECO(Q) (tE[0,τ]) UB(t)=iAu(t)+f(t) (tE[0,T]) u(0)=U.O kαttHereC(Q)={UEC(Q):DUEC(Q)for|α|三k},wherek=1,2,...O··TheresultsobtainedinthissectionarenotrestrictedtoDirichletboundaryconditions.Completeana108011sresultsarevalidifNeLunarmconditionsareconsi-deredinstead.Moreover,theresultsremaintrueiftheLaplacianisreplacedbyanysymetricstrictlye11ipticoperatorofordertwowithsmoothcoefficients. B.THESCHRODINGEREQUATIONINRNInthefollowingwe1etLP=LP(RN)(l豆p三∞).ThemethodusedforboundeddomainsNp2doesnotworkonR.HoweveroneobtainsresultsonLnLfor2豆p豆∞andonq2L+Lforl三q豆2.22NLetA2bedefinedonLbyD(A2)=H(R),A2f=Af.ThenA2=iA2generates22paunitarygrouponL.For2豆p〈∞denotebyAthepartofAinL11L.=2p THEOREM4.4.N11(a)If2豆p<∞andk〉-(---)thenAgeneratesak-timesintegratedgroup=22p'2p2pL门L(b)Ifp=∞andk>Z,thenA2∞generatesak-tiInesintegratedgrouponL2门L∞.N p2PROOF.Forλερ(A2)theoperatorR(λ,A2)leavesLnL1IIvariant;hencesodoesR(λ,A)forλεp(A).ItfoIlowsfromtheSobolevimbeddingtheoremthat2叫)cL2门队L2incase(a)and(b)SotheclaimfollowsfromTheorem33口
SobolevImbeddings 39 Theorem4.4hasconsequencesformultipliers.Let 勿lp={mEL∞:maE3(LPnL2)and|3'1(mG)|τ豆const|u|TforallUELPnL2},yPTP'"={mEL∞:mGE3(LPnL2)forallUELPnL2},2p22Awhereg:L→L,U+U,denotestheFourier-transform.ForkEm,tER,XERNlet(x)=ft(t-s)kal/(k-l)!e-is|x|2ds-KtJoNllNCOROLLARY4.5.Let2〈p<∞andk〉-(---)orp=∞andk>-ThenmEmfora11==22p4·Kt2ptER. 11However,itisknownthatIILEm(tER)iIIlp1iesk〉N卜--|(seeHieber"王tpz2p(1989)andSjOstrand(1970)).Finally,letl〈q<2i+i=1.Then(Lq+L2)'=LPnL2.DefiningA='pqq+2Lq+L2byAf=iAfwithmaximaldomain(inthesenseofdistributions),onehasq+2(A).=A.Sooneobtainsq+22,p N11COROLLARY4.6.Letl豆q<2.If1<qandk〉-(---)orq=1andk>一,thenA=2q24q+2q2generatesak-tiTIlesintegratedgrouponL+L. REFERENCES 1.R.A.Adams(1975):SobolevSpaces.AcademicPress.New-York.2.S.AgrnoruA.Dou811s,L.Nirenberg:Com.PureApp1.Math.马:623(1959)-3.W.Arendt.IsraeIJ.Math.59:327(1987).4.w.Arendt,H.ke11ermarln(1989):In:VolterraintegrodifferentiaIequationsinBanachspacesandapplications.DaPrato,Ianne111,eds.Pitman.5.W.Arendt,F.Neubrander,U.Schlotterbeck:SemesterberichtFunktional-analysisTUbingeni2:1(1988/89).ToappearinSemigroupForm.6.M.Balabane,H.Emamirad:J.Math.AnalysisandAPMicationsZ旦:61(1979).7.M.Ba1abane,H.Bnamirad(1983):Contributionstonor111nearpartialdiffe-rentialequations,C.Bardos,A.DamIamiar1,J.Diaz,J.HerrIarldezeds.Pitman,Bostonp.16.8.M.Balabane,H.A.EIIIamirad:Trans.Mer.Math.Soc.291:357(1985)-9.H.Brezis(1983):AnalyseForletiormelle.Massor1.Paris-10.G.DaPrato,E.Giusti:Ann.Mat.PuraApp1.76:377(1967).11.M.Hieber:IntegratedsemigroupsanddifferentiaIoperatorsonLP.Thesis.TUbinger1.1989.
40 Arendf 12.L.HUrrnander:ActaMath.104:93(1960).13.H.ke11ermann,M.Hieber:J.Funct.Anal.84:160(1989)14.F.Neubrander:PacificJ.Math.135:111(1988).pN15.M.Pang:ResolventestimatesforSchr&dingeroperatorsinL(R)andthetheoryofexponentia11yboundedC-semigroups.preprint(1989).16.A.Pazy(1983).semiRroupsofLinearOperatorsandAPPIicationstoPartialDifferentialEquations.Springer.Berlin.17.Reed,B.Simon(1978):MethodsofMathematica1PhysicsIV.AnalvsisofOperators.AcademicPress.18.S.Sj己strand:Ann.ScuoUNorm.Sup.Pisa丝:331(1970).19.N.Tanaka,I.Miyadera:Proc.JapanAcad.尘,S住.A:139(1987).
CompletelyAceretiveOperators PH.BENILANFacultyofSciences,UniversityofFrancheComte,Besancon,FranceM.G.CRANDALL*DepartmentofMathematics,UniversityofCalifornia,SantaBar-bara,California OIntroduction Itiselementarythatthetranslationsemigroup 、‘,,,-EAnu,,..‘飞 (S1(t)υ)(z)=叫z+t),t,z主0, dehesasemigroupoflinearorder-preservingcontractionsonP(0,∞)forl三p三∞.Thesamepropertiesareenjoyedbytheheatsemigroup (02)(乌(制 inL13(IRN)andotherlinearsemigroups-Manynordinearsemigroupsarealsoorder-preservingandcontractineveryLP;asimpleexplicitexampleis (0.3) 叫z+t)(S3(t)u)(z)=‘ t,z220, *Supportedinpartbythefollowinggrantsandcontracts:ArmyResearchO而ceDAAL03-81K-0043and03-9O.G·-0102.NationalScienceFoundationDM$8505531and90-02331、omceofNavalResearchNO001488.K-0134. 4I
42 BenilanandCrandall whichddnesasemigro叩oforder-preservingcontractionsonLP(0,∞).Ofcourse.slarisesasthesolutionoperatorfortheCauchyproblemforut-uz=0,t,z>0、whileS2correspondsinthesamewaytout-Au=0,zεIRlV,t>0.S3correspondstothenonlinearequationut-tiz+叫u|=0;thesimplicityoftheformulaisduetothecornmutativityofFlandthesemigroup (0.4) (z)(T(tM)(z)=??一十E|ulz whichcorrespondstout+叫u|=0.IMeed、S3(t)=SI(t)T(t)=T(t)Sl(t).Thiscommuta-tionpropertyfailsfor52(川、T(t),butonemaystillddneasemigroupwhichshouldbethesolutionoperatorforttt-Au+uh|==ObyusingtheLiE」Trotter-katoformula (0.5) 马(thA(S2(j)T(DYu; itcanbeshownthatthislimitexistslocallyuniformlyif(forexample)f|z|u(r)εL1(阪入F)aIIdthenfort>O吗ε-h|(34(t)tt)(z)iscontinuous‘130UIldedandintegral3le.That341)definesanorder-preservingsemigroupofcontractionsoneachLP(IRN)canbeseeIIfromtilefactthatF(t)212(t)T(t)isorderpreservingandacontractiononeachLPaIXitiwexistenceofthelimit.TheproofoftheexistenceofthelimitmaybebasedontherelatioTI |52(t)T(t)u-52(t)T(t)d|三32(t)|u一也| whichfollowsfromthefactthatS2(t)islimarandorder-preservingandpointwispcoMrarmtioIlpropertiesofT(t).Thisieadsto 保uu、、‘,,,,,,,,也,,I‘飞7"QU<-Aun＼lt/＼1111/t-n/lt＼T＼lI/t-n/III-＼吨,-CU/i11＼un＼lf/、111I/t一η/It--＼T＼111J/t-n/it--＼巾,dMQU/III-＼ Moregenerallyif(口,B,μ)isameasurespace(thatis,μisanonnegativemeasureonth俨σ-algebraBofsubsetsofO),(F(t))t>oisafamilyoforder--preservingcontractionsonLP(fl)forl三pf二∞,wemayoftenddneasemigroupoforder-preservingcontractionso口LP(Q)bytheformula (0.6) S(t)tt=nl鸣。F(t/n)nu byprovingthatthelimitexists.Thediscussionaboveprovidessimpleinstanceswiththefeaturesdescribednext.Intheexamples(01),(03)thesemigroupsSI(t),33(t)arenaturallyddnedforanymeasurablefunctionUonl0,∞[;intheexamples(02),(0.5)themostgeneralclassoffunctionsonwhichSz(t)ora(t)canbenaturaliydennedisnotclearfromthisdiscussion,butitincludesfunctionsintegrablewithtilew-eighte一|z|.Inanycase,tilesemigroupsSi(tLz=1,2、3可4aredeEnableonalinearspacecontainingtheLPspacesforl三p55∞,theyareorder-preserving,mapLPintoitselfandtheycontractintheLPnorm-
Complete〈yAceretiveOperators 43 Inthispaperweconsiderageneralmeasurespace(口,8,μ)andseektost叫ysemi-groupsS(t)onsomesubsetCofthespaceM(Q)ofthemeasurablefunctionson(Q‘βμ)飞飞'ithpropertieslikethoseabove.Forexample, (0.7) (V俨广尸尸贝印俐川(μ川t仆州)川(Cm川川n川dLpm川川p盯mm(仰川mQ创m)川)CC门m川e们忖………r陀m时叫e臼创仙s剖仙叫tbrisanorder-preservingcontractioninLμFp'(川fQ])fort;三三0、1三p三∞. Itturnsoutthatthecorrectwaytoformuiatetheclumsyideaofan"order-preservingcontractiononLPforeveryp"isasfollows:westMysemigro叩sS(t)withthepropertythateachmapS=S(t)satiSEes (08)j(h-sa-k)+4(川-k)+forUaECK>O (whichisinfactequivalentto(0.7)IIIidermildassumptions).飞W飞V7eCaiHlamapS丘;CC几f川(Q)→几MI(Q)Sa川tih5f乌々υ3y川.才imrn1g(仰0.8剖)aCωOImIn1pμlet伦eC∞OI川l川trmaCdtibOIn川1盯;飞wvest1u1d句〉y,s町eImni厄grωO1u1psofCt(川:(讯w'hiChdοI口1Ottlal巩w'a〉y,slea飞Y.PLPiIn1飞VFariarn1t)L.飞、V?ewillalsobeinterestedinthegenera10rs-4ofsuchsemigroupsor‘moreprecispiy,inoperatorsAsuchthaltthesemigroupsε-tAth叮ge11eratehtllesenseofrlomlinearsemigrouptheoryaresemigroupsofcompletecontrartio口5·飞飞-ealsoReeka"largespace"inwhichoursemigroups叭'iHact.人I(Q)istoolarge(invit3%-ofF川anrisubtletiesarisecorrespondingtotilefactthatfori=L2、:14、lirIIt↓oSi(t)ti=UfailsinL艾JforsomηUELX)(QLTilerightlargespaceforvariouspurposesturnsouttobetilespa℃ecalledLo(fl)bρlow-飞Verecallsori1ebasicfactsofnonlinearseraigrouptheory-IfXisaBanachspaceandAisalloperatorinx飞thenAissaidaccrdiveiftheresolveTIts(I+λ4)一1arecontractioI15inλ'(definedonR(I+人,4))for入>0.For‘example-iffJ(t)isasemigroupofcontractionsOIlCC矗IaIId-Aisitsinanitesill1algenerator-then,4isanaceretiveoperator-TileoperatorAissaidr714cereth-eifAi$accreth-eandtheresolvents(I+λ4)一laree飞它rywheredefinedonay:ifAi5771-acereti飞飞thenwemaydehethesemigroupr-1.4ofcontractionsonr=D(A)bytheexponentialformula )口可υnu'''E‘、 川u=A(I+io→ti 川lichconvergesunifomlyfort三Obounded([13)).飞飞-emimictheabm-eterminology:anoperator44inM(Q)iscompletelyaccretiveiftileresolvents(I十人A)一Iarecompletecontractionsfor入>0;ifXisalinearsubspaceofAf(QLanoperatorAin,Yissaidm-completelyaccrdiveinXifAiscompletelyaccretiveandtheresoh-ents(I+人A)-1areeverywheredennedonX.InthearstsectionwestudycompletecontractionsS:CCM(0)→M(fl).ThemaintoolusedhereistherelationU《tydeanedbv (010)kuif/(u-k)+三/(υ-k)+and/(u+k)-4(川k)一川>O fInthiscontext‘anoperatorine飞'isamappingfromXtoitssubsets-
44 BenilanandCrandall ThisrelationiscloseiyrelatedtotheHardy-Little飞Voodinterpolationtheoryasdevt?lopedforinstancein[9i;rearrangementsappearpmnim1tlyinthistheoryIncontmt、附叭'iHIMitieMprimarilyinitiatedin[12iaMdeveloptheresultswewiilneedwithoutanympofrearrangements.Thesecondsectionconcernscompletelyaccretiveoperators-AprimarypointbeCOInpstilestudyofthepropertyU《U+入tyfor入>0、andtbmaintoolisacharacterizationofthisproperty.IfUεLo(QLUεL(QLwhere 剧。)=Ll(Q)+俨(Q)={UEm);f(|til-k)+<∞formmk>0} and M)={叫川);f(lul-KY<∞hra川k>0}lhthpclosureofL1(fl)nLχ(0)inL(flLIndeed,weprovethatlifUεLo(QLt'εL(QLthplIU《:tt十人t'holdsforA>Oifandonlvif )'EEEA-BEAnu{ fpp(u)三obHPl) wilerePi)={pεCχ(R);0三Pr三Lspt(pf)iscompact、0￠5pt(p)).spt(p)bpi鸣the3upporiofp.OnecanGndprecursorsofthischaracterizationinthetheoryOi-Beurling-DeIIYCOIl-四r川吨SPIIIigro叩soflimaτoperatorsonLJ(fl)h叭'hichinterpolateH(see[IOU-AnotheriMC川50ImaybefouMiI1thePape仔rb〉Br位ωihs-St阳i山In1L1(s川e俨&ιl5O5ωOnm1rP5句}y归川.3吼5引tema川tlY叮de肘们、V.edlkh问O叫〉iP〉me创mIn川1ts1mn[口2lL.[F7]川).HO附Ver.thisC}h1amr忖t忖e肝叫r川i泣Za川川tUihEO)a盯寸hiht℃叭肝革气吓刊飞V刊.刁e凹5i山t忖sfullsCωQ叮ip严〉咒eO∞nlh〉y.i江f』μ川i叫(Qm)工乞忧∞汇Ulandifone飞叭飞Y.孔础a剖川l口川川1让刊ttl8tωO叽w.Or此ik〈inO川tihlr町e盯r5P归aC白e们s川fharlLν2(川口)。IL1(f}}(5e何e‘fori川taIn1Ce.[2)for山。ofotherspacesinaconcrete5it1latioI1)飞飞-ero町i1ldηSrction21viththeddiIIitiOIlaIIdsomepropertiesofthegeneralspace5iIIUvilich认-eu-il11ωrk、namel3'tilenormalBanachspaces-TilesearethrBaIlachspaces(凡|l||)embeddfyiiIl1I(Q)whichsatisfy (0.12)Ltε1I(Q)-rεX,U《1'均UεXand|lu||三||叫|. iIISertions3and4westudy771-completelyacτrdliveoperatorsinnounaiBanaril-pacesalldii10挝、IIIigroupsthe〉-generate-Twotypesofresultsaredeveloped:日15t,weIIIF114eprecisethefactthatamcompletelyaceretivpoperatorandtilesemigroupitgmeratesareuniquelydefinedbytheirtraceonL1(fl)nLCC(Q);5ecoIIdly、inanormalBaIIJtchspawxsatisfyingthecondition (0.l3) un《tiεX‘un→1ta.e.=争Tin→UinX weestablishpropertiesenjoyedby771-completelyaceretiveoperatorswhicharefamiliarinthetheoryofm-accrdiveoperatorsinuniformlyconvexanduniformlysmoothBanachspacesweremarkthatinthecaseX=LI(Q)(whichsatisHes(0.13))theseresuitshavebeenobtaimdin[15i).
CompletelyAcerefiveOperators 45 Section5isdevotedtotheextensionoftheunonlinearffille-Yosidatheoremntosem-groupsofcompletecontractions.Section6isdevotedtotheperturbationof771-completelyaceretiveoperatorsbymaximalmomtonegraphs(thisextendssomeofoftheresultsof[l2];anabstractversionoftheresultsof[3lisinpreparation([51)).Finally,inSection7、盯studycompletelyaceretivesubdiferentialswhichleadsustosomeconcreteexamplesfortheabstractthmryfrompartialdibrentialequations(andseeI5lformoreexamples). 1SPACES,CONTRACTIONSANDEYTERPOLATION Let(Q、B,μ)beaσ-hitemeasurespaceandletAf(Q)denotethespaceof(a.e-equivalenceclasses)ofmeasurablemappingsfromQintoIR:eqmiit〉飞inequalit3飞etc.‘飞飞Fillalwaysmeanμa.e.onQandM(Q)isalinearspaceinthenaturalway.IfEisasubsetofM(Q)附willputE+={uεE:U主0}while(incontrast)r+二TVO=rmx{r,0}andr-=一(r〈O)=-min{r＼O}.Theintegral (114) fU=:=儿u is附ll-〈iefinedoIIAI(Q)+andtakesitsvaluesin[0,∞].Forl三p<∞thespaceLP(Q)isjustthesubspaceoftf(口)on飞vtlichthefunctional (1.15) ||叫|p=(fh|P)1/P isanite.飞飞tilethefunctionalassociatedwithLχ(口)is (1.16) ||叫|∞=inf{kε[0.叫:|u|三k}. 飞VedenotebyL(fl)thespace (1.17) L(Q)=LI(Q)+L∞(Q); L(Q)isexactlythesubsetofM(Q)onwhichthefunctional (118)||叫|1+∞=inf{||f||1+||g||∞:fJελI(QLf+g=u ishiteandL(Q)equippedwith||||1+∞isaBanachspace-ItisstraightforwardtostIOU-that (1.19) ||u||1+∞=以(k+/(|叫一k)+) andthatif{tin}isasequenceinL(Q)andUεL(0),then||un一叫|1+∞→Oifando吨vif (120) /(|un-u|-k)+→0岛rk>O
46 BenilαnandCrundall IIIPEtIticular,wernaydiagonalizetoconcludethatifun→UinL(口)‘thent}hhl阳町e盯r陀eisaS贝刊1ullb〉旧(q甲l俨1阳ttlIhIhM1让i5entailsIui竹切7n叫l勺J→Ua.e.Ifμ(Q)<cc,thenL(fl)=Ll(Q)(thenormsaredibrent,butequivalent).Ifμ(口)=∞、thenL(fl)containsUI三p三∞LP(fl)andtheinclusionisstrictifμisnonatomic-Theconvexfu盯tiomj(r)=|r户,j(T)=(|r|-k)+‘etc.,whichoccurabovearetypicalelementsoftheclassJogivenby(121)Jo={convexlower-semicontimommapsJ:m→[0,∞lsatisfyingj(0)=0}.飞飞-eareconcernedabouttheinterpiaybetweeRtik·functionalsN(e.g.,NmightbpOIIeofthoseintroducedintroducedabove)andmappingsS(e.g.、SmightbethevalueofSPITIigroupS(t)oraresolventJAassociatedwithanacereti刊OPPratorA).DEFINITION1.1.LetXbealinearspaceand入「:X→[-cι∞iandS:D(5)CX→XFisan入?-contraction(equivalentlyLacontractionibr入r)if N(Su-sa)三八lT(u-d)fort4、dεD(3)Thenrstlresultwewillformulate,Propositionl2below、isastraightforwardextmSionoftirBrezis←Strauss([l21)variantoftheRieszinterpolationth∞reIII-Part(iii)ofthispropositionshowsthatanorderpmerviIlgcomractionfor||||land||||义℃ωmO创I川ra巾nm1a口Ot}h1芒盯rf1uImIn1CtiOInlalSa5巩WvelHlL. PROPOSITION12.LetS:D(S)CAI(Q)→M(Q)-fi)ldetSbeacontractionfor||llIand(l.22)UεD(5)andk>OimplyU八kED(SLThenSsatjsβes (l.23) fjuh)+)三f川)forjεJ〕叫Dm) ifandoIIlvif (124) Sti三||tt+|loeforIt巳D(S). {ii)LrtSbeacontractionfor||!i∞and(l.23)UξD(F)andk>Oimply(ti←k)+εD(FLTJIPIISsatiSEes(1.23)ifandonlyif(l26)/(h)+三fu+hruξD(S)(iii)LetS:D(S)CM(Q)→M(Q)andassume(127)队QξD(S)andk主OimpliesU八(d+L)or(U-k)〉aED(S)-ThenSsatiSEes(128)/仙一Sd)三fj(u-G)forjdoand川εD(S)ifaMonlyifSisorderpreservingandacontractionfor||||Iand||||∞·OneessentiaipointoftheproofofPropositionl2iscontainedinthefoilowiRgwdLKnownlemma:
CompletelyAcerefiveOperators 47 LEMMA1.3.LetU,UεM(Q).Then (129) f川)三fM)forjdo if(andonlyif) (130) f(17-k)+4(u-k)+扣rk>O Toillustratethearguments,weprove(iii).Choosingj(r)successivelytober+、|rLand(r一||u-d||∞)+in(128),weseethatitimpliesSisorder-preservingandacontractioninLl(Q)andL∞(0).Weprovethesumciency.InviewofLemma1.3、itisenoughtoshowthat(131)f((Su一")-k)+4((u→)-k)+岛rk>0,川εD(町, since,bysymmetry,thesameinequalityholdswithUandareversed-飞飞-eGxu,aεD(S)andk>OandassunlethattIY=U〈(也+k)isinD(5).飞hclaimthat (1.32) Sω三Sfi+k Assumingthisforthemoment‘wehaveSu-Sfi-k三Su-Stuandthen(Sti-58-k)+三|Stt-SIL-l:integratingthisinequa!ityandusingtheL1(Q)-contractionpropertyofFtogetherwith|u-r|=(u-a-k)+、(1.31)follows-Toprove(132),using(127)飞vithk=0、川haww〈也εD(S)orwvaED(SLIfω〈aED(SLusingtheorderpremvi鸣and||||∞contractionpropertieswehave Sw-Sd三Sw-3(w〈台)三||w-w〈白||x=||(tu-G)+||坦 andifu-vaεD(SL Stu-Sfi三S(ωVG)-Sd三||ωVG-Q||∞=||(町-d)+||dc; sincew三d+k,(1.32)holds.InthecaseinwhichwV(d-k)εD(SLonehasStu三sa-kandtheproofisentirelysimilar.口Itwillbeconvenienttohaveanotationforthereiationexpressedby(129). DEFINITION1.4.Letti.uεM(0).ThenU《Uif (1.33) /j(U)三fj(u)岛rjEJo REMARKS1.5.ItisobviousthatU《Uifandonlyifu+《U+andu-《u-.Therelation《isclearlytransitiveandonecanthinkofitωasortofordering‘butitisrlotantisymmetric-U《Uandt?《UdoesnotimplyU=ty-Forinstance,ift,εM(Q)+＼L(Q),thenU《UforanyUεM(Q)+andthereforeU《Uandt?《Uholdwhenevertt、t?εM(Q)+＼L(0).Ontheotherhand,ifu,tyεL(fl户,thenU《vandu《UbothholdexactlywhenUandt'havet}wsamedistributionofvalms,thatisμ({Ti>人})=μ({t7>人})
48 BenilanandCrandall Ior入>0、町、equivalent13＼theIIOIIdecreasingrearrangementsofUandrareideIitical-RecalldlλttheIIOIldecreasingrearrangementw*oftrεM(口)+isgivenby (l:14) J(3)=inf{σε[0.叫;μ({w〉σ})三s} TIleseassertionsareaconsequenceofthecharacterizations:(a)iflt、IYEAI(Q)+.ttrnr《ttifandonlyif儿t俨(8)ds寸ttfh)dδfort三0 aIKi(b)iftiξM(0)+‘tImltiξL(口)ifandonly,ifM1fh)ds<CClfor。<1.19i盯e认edonot旧e(a)inthispaper‘wewillnotproveit(we[p9])k;(b)i臼冒Ob飞V.i巾O饥川1ull.￥S盯i吗thecharacterizationofconvergenceinL(fl)andLemmal3itisnothardtoprovotileIlextresultaIld飞velpavethistothereader- PROPOSITION1.6.TheFet{阳、r)εL(Q)×L(口):1·《u}isclosdinL(Q)×L(Q). 飞VeemphasizethatwehavenotaIldwillIIotusereal-raIIgemeIitsinthi5pappr;byCOIltras仁itisaprincipaltooliIItileclassicalHardy-LittlewoodapproachtoiIlterpolationωPIPE-rm叫Lfore以丁X〈anm叫1吁IP〉le巳‘iUIn1[p阴9川lA∞tth阳e町Idiπe阳t什ih1r℃t山ihle佣Or叮i山IHl[归问9叫]isthat阴山etherelationν《UasdeHnedabove-whiletheHardy-Li11ip飞Wfxlrelation1·〈Umpamht|《|r|.Ofcous巳thereisaparalleldNPlopmeMfor--气'HinplaceofH《mofII11lchofwhatlwediscuss叭'hichalsoavoidsuseofrmrrangerTlentstllhhstraidIdol-叭'ardand飞vewiHIIotsaymoreaboutithere-Observethatr《UisaitroIigerconditionthanu〈t·:飞vewillusethefullforceofitoccasionallv.Justasoneonlyneedstocheck(1.33)forasubclassoffstoverifytilerelationγ《ti‘1·《tiiIIIplipsthatX(r)三N(ti)formorefu盯1iomlsthanthoseofthpformfj(u)forjEJo.飞飞'piIItroduff、anamefortilffrfunctionaJRbeiouvaswellasmapping-飞VIIIchHC0日tractmthem. DEFINITION1.7.{ijAifu盯tJimalNJI(Q)→]一∞‘∞lisnormalifN(fJ)三N(tt)whenewrt'《tt.(ii)rlmapS:D(5)CM(口)→M(fl)isacompletecontractionifitisaIINcontractionibr叭'ervnormalfunctional入'. Forexample‘||||lχisanormalfmctiomlwhichismotoftheformfj(u)forsomejεJo.B〉'thedeHIIitions‘amapSasiII(ii)isacompletecontractionifaIld0日i〉-if 5u-Sfi《U-dfortt‘dεD(S); observothat‘inthislanguage‘Proposition1.2(iii)assertsthatS(undertilecondition(1.27))isacompletecontractionifandoniyifitisorder-preservingandacontractionfor||||land|||ipREMARKl.8.TheinterpolationresultPropositionl.2maybeextended、withexactiythesanrproof、tothesettingiIluhichF:D(F)ζ几川口1)→JV(Q2)where(Qi、B1·jiz)吗i=L2.
Complefe〈yAcereIIveOperators 49 aretwomeasurespaces-Ofcourse,U《UforU巳Jt盯fb)andtiεM(QI)means儿州dMAj(川1hrjdo-42da61andthisisequivalenttothecorrespondingvariantof(1.30).ThenotionofacompletecontractiontD(S)CM(Ql)→AI(fh)isnowmostco盯mientlydennedbySu-Sd《U-dforU,QεD(S)-ThespiritoftheproofofProposition1.2(augmentedasinRemark1.8)providesaneasierroutetomanyresultsthantheirclassicalpresentations.Forexampie,observingthatthemapS:LImy→LI(IR+)givenbySu=♂satidestheassumptionsofthefollowingProposition1.99onehasanextremelysimpleproofoftherelationti*-r《ti-ttheclassicaltheoremofLorentz-S}limogaiti([91heo陀m37.4l)assertsthe啊!altermultu*-G*-〈U-G.Notealsothatf(J(s)一内))2ds三儿(u一巾μyieldstheclassicalHardy-Littlewoodinequality儿l二∞】Thef扣OllhO川吨resultih5avariantofaikee町m!I叽InmIPROPOSITION1L.9.LetS:D(5)CLl(QI)→Ll(02)satis〈v(135)/Sudμ2=lUdμlforUεD(S)Jfl2JQland (l36)u咱也ξD(S)‘k主0=字U〈(白+k)εD(S)andS(ti〈(d+k))三31i〈(S白+k). ThenSisacompletecontraction.Proof:TILerelation(l.36)caTIbewritten(137)(Su-sa-k)+=Su-Su八(Sfi+k)三31i-3(u八(白+k)) andintegrationfollowedbyuseof(1.36)}Fields儿(SUE-Sb-k)+d向三AStt-S(u〈(a+k))dji2=儿(u-a-k)+ditlda‘2JEtl 2COMPLETELYACCRETIVEOPERATORSANDMOREFUNCTIONALANALYSIS IfAisamappingfromM(0)intothesubsetsofM(QLwewillcallitsimply"anoperatorh.AnoperatorAwillbeidentinedwithitsgraphandthen(ti,1')εAandrEAttmeanthesamething.TheHefeetivedomain,,D(A)ofAis{UεAf(Q):Au#0)-nfeconsideroperators44w-hoseresol飞回ltsJA=(I+入A)-larecompletecontractions;thatis啕theclassesofoperatorsde自Ilednext.
50 BenilanandCrandull DEFINITION21.LetAbeanoperatorinM(0)-Aiscompletelyaceretiveifu-6《tt-a+入(u-L)forλ>Oand(ti,u),(白,台)εA.Inotherwords、AiscompieteiyaceretiveifN(u-G)三JW(u-G+入(u-P))for(队叫?(亿b)εAandeverynormalfunctionalNonM(Q).ObservethattheddMUonofcompietelyaceretiveoperatorsdoesnotreferexplicitlytotopologiesornorm.However,ifAiscompletelyaccrdiveinMm)andACX×X,whereXisasubspaceofAf(Q)whichisaBanachspacewhosenormisgivenbyanormalfuIICtional,thenAisaccrdiveinX.ChoicesforXmightbeLP(Q),155p三∞orL(Q).Itwillbeimportanttocharacterizecompleteaccretivityandforthistheessentialpointistocharacterizethosepairsoffunctionsu、vsuchthatU《U+bholdsfor人>Oinamefuway-ThiswillrequireustorestrictUtoiieinL(Q)andUtolieinthe511135pacpLo(Q)ofL(Q)givenby (2.l) Lo(口)={uεL(口):μ({|u|>k})<∞fork>O}. ToformulatletheIlextresultwewil!employthefollowingclassofmappings:(22)Po=(PEC∞(囚):0三pf三Lspt(j)isml阳t、andO￠sptp)wherespt(p)isthesupportofp.PROPOSITION22.LettiεLo(QLuξL(Q).ThenthefollowingCOMitiomareeq山v-aleIIf; 、hBEE-,,「α( U《U+入1·for入>0; (b) nu>'KFAor-'hA>urI-叫<-nu<一'L'K<uftI门叫 (c) f(P(tOU)主Oforan川Po; andfd)ifp:R→]一∞、∞[ismdecreasingandp(0)=0,thenf(p(tt)ν)+三f(p(u)11一observethattheconditionUεLo(Q)istoinsurethattheexpressionsin(b)and(c)aredennedindeed、becauseUELo(IILin(b)theintegralsareoversetsof丑llitemeasureandin(c)theconditionsonpguaranteethatp(u)isboundedandsupportedonasetofhitemeasure.Theintegralsin(d)takevaluesin[0‘∞lforarbitraryU,UεM(口).ProofofProposition2.2.Let flifT>Osigno(T)={Oifr=Ol-lifr=0.Bytakingp(r)=sig问(T-k)+andp(r)=-sig问(-r一-k)+in(d),oneimmediatelyseesthat(d)斗(b).飞hprove(b)=〉(a).ForA,k>O(u+入ν-k)+主(tt-k)++人叫{u>k}
CompletelyAccretiveOperators 5I whereXEisthecharacteristicfunctionofE.SinceUεLo(QLAUFX{u>k}εL1(Q)and、by,(b), fhw}=人才。k}吃。 SO f(u-k)+三f(u+λu-k)+Inthesameway,oneshowsthat f(川)-4(u+川k)- andSOU《U+加.Wenowprove(a)斗(c).LetpεPo,6>Oandsptpc[6,∞(.Set忡)=Ir川sLetk250besuchthatu1=(u-k)+εLI(Q).Wehave 。三j(u+入υ)三j(u++人U1+入k)三j(入k)+(u++入U1)p(u++入171+人k) andsincej(入k)=Oif人k三6、ji(u+人u)εL1(Q)ifλk三6.Invoking(a)andjεJowehavefj(u)三fj(u+加)f…yλ>O andthen 0<fj(u+b)-j(u)lforA>OwithAK<6.Bytheconvexity,(j(ti+入u)-j(叫)/入decreasestop(叫νεL1(Q)as人↓OandsoO三/p(u)U InordertoobtainthisinequalityforageneralpεPLwerepeattheaboveanalysisforp‘ssupportedin]一∞,0[andthen,byadditionofthesetwocases,theresultfollows.Itremainstoprovethat(c)斗(d).Letp:E→[一∞,∞]benoMecreasingandp(0)=0.Werequirethefollowingelementarylemma.LEMMA2.3.Hp:]0,∞[→[0,∞lisnodecread昭·thenthereisasequence{f川inD(]0,∞[)+suchthatzfn(s)dsincreasestop(T)asn→∞forT>OByLemma2.3,thereisasequencepnsuchthat|pn(r)|increasesto|p(r)|asn→∞andpra/||pLll∞εPo·By(c) fωn(喇 andbythepropertiesofthepn,(pn(u)υ)+and(pn(u)u)-increaseto(p(u)vyand(p(ti)飞')-asn→∞,so(d)isprovedbyiettingn→∞above.口Asanimmediatecorollaryofthecharacterization(c)ofProposition22、wehave:
52 BenilanandCrandall COROLLARY2.4.LetABbecompletelyaccrdiveoperatorsinL(Q)withD(ALD(B)CLo(Q).ThenA+BisacompletelyaceretiveoperatorinL(Q). Completecontractionsgiverisetocompletelyaceretiveoperatorsintheusualwaythatcontractionsgiverisetoacereiiveoperators- PROPOSITION2.5.LetS:D(S)cLo(Q)→几I(Q)beacompletecontractionandμ;三0.ThenA=μ(I}S)iscompletelyaceretive.Proof:飞lVeneedtoshow,thatfor入>O (23)U-u《U-1'+入μ(u-SIi~(u-Su))=(l+忖)(u-u)+人μ(Sr-Su); h01vever、ifjεJ0、 fj(14一P)=fj(」一(((l+均)(u一U)+入μ(Su-h))+-L(仙一b))l+入μ1rAuf<一二-lj((l+人μ)(u-17)+人μ(Su-Su))+一」二lj(SIA一仇').人μjl+入μj三了←fj((1+均)(u-17)+Mt(仇'一h))+忐fJ(u-t)目, Ifk>Oandj(r)=(T-k)+orJ(T)=(r+k)一,thenfj(u-u)<∞andrea fj(li一U)三fj((1+均)(u-u)+人μ(Sv-h)) whencetheresult.口飞飞、continuethissectionbyrecallingvariouspropertiesofthespaceLo(Q)飞VP叭'illneed.First‘ (2.4) M)={uUJ(Q):f(|u|-k)+<∞fork>O}、 asisimmediatefromthedearIition-ItfollowsthatLo(Q)isaclosedsubspaceofL(Q);infact‘itistheclogureinL(Q)ofthelinearspanofthesetofcharacteristicfunctionsofsetsoffiIlitemcasu陀He凹after‘Lo(fl)carriesthenormi|||1+∞;itisthenaBamchspace-Withthenaturalpairing(u,17)=fut,ofUεLln∞(QLt,εL(QLthedualspaceofIdO(fl)isisometricailyisomorphicto (2.5) Lln∞(Q)=L1(Q)nL∞(Q) w-henLln∞(fl)isgiventhenorm (2.6) ||tt||ln∞=max{||u||1,||叫|∞}. Hereafterweusew-Lo(fl)todenoteLo(Q)withitsweaktopology.TheutilityofthecharacterizationProposition22(c)furtherenhancedbythenext result.
CompletelyAceretiveOperators 53 PROPOSITION2.6.IfpεPo,then((UJ)εLo(Q)×川):fvp(u)三。) issequentiallyclosedinLo(Q)×w-Lo(0).Proof:Webstprovethatif{un}CLo(Q)andun→UinLo(Q),thenthereexistsasetkwithEI山emeasureandasubsequence(stiHdenotedby{ttn})suchthat (2.7)p(un)=OonQ＼Kforn=1,2,...andp(un)→p(u)a.e. Indeed、fixksothatp(r)=Oif|T|<k.Sincej(|un-u|-k/2)+→0,wemaypasstoasubseqm盯e(denotedagainbyhn})forwhich艺nf(|h-叫-k/2)+<∞andthen 、tt飞rttjk-4>url』JEIKU、、EEISIf飞tIBflJM仙一4>unurl』fEEt∞UK/FEEt-飞、C、..,,,'K>nu,,‘1∞Uh--pt‘aFI while +＼1111f/k-2unu/,ttttt飞rt'fd<-k-41t飞fJ扯一4>unurl』,、BE飞 soji(K)<∞.Passingtoafurthersubsequence,wemayassumetin→tAa.e.andthen(2.7)holds.UsingthisandreplacingQbyk,thepropositionfollowsfromtheclassicalresultCO盯errliIIg附akconvergenceinL1whichassertsthatifQisofhitemeasure、{1、}CLl(Q)convergesweaklyt01·inL1(口).{wn}isaboundedsequencehL∞(Q)withwn→1ra-e.,thenf1·711L?yl-bft?ω.口ThenextτeSUItfollowsatoncefromPropositions2.2and2.6. COROLLARY2.7.Let44beancompletelyaceretiveoperatorinLo(Q).Thenthem-qllentialclosureofAinLMQ)×tiy-Lo(Q)iscompletelyaceretive.ObservethattheBanachspaceX=Lo(Q)hastheproperty (2.8)UεX‘UεM(町、t'《U=争UεXand||ν||x三||u||x· Itwillbeconvenienttointroduceanameforsetsandspaceswiththeseinvarianceproperties- DEFINITION2.8.AsubsetCofM(0)isnormalifuεC,νεM(0)andU《UimplyUεC.Moreover,aBamchspaceX,||||xwithXCM(0)isanormalBamchspaceifx#{0}and(2.的holds-eq山valentlFiXisanormalBamchspaceiftheunitballBofxisanontrivialnormalset. SimpleexamplesofnormalBamchspacesare:LP(Q),1三p三∞、L(Q),Lo(0),Lln∞(0).REMARKS2.9.AnormalBamchspaceXsatidesLln∞(Q)CXCL(0)andtheinjectionsarecontinuous.Inthisregard,notethatifX,||||isanormalBanachspaceandυεX,thenty+《叽U-《-uimplyu+,f,|u|εX.Also,ifUεM(Q),u,ωεXand1U三U三u,then|叫:三|1'|+|tu|soti+,u-,UεX.Now、wehavealreadyremarkedthat
54 BenilanandCrandall ifuEM(Q)+＼L(Q)‘thenU《UholdsforallUξM(flyanditfollowsthatifanormalBanachspaceXcontainsanelementofM(Q)＼L(QLthentheunitbal!ofXcontainsM(QLwhichisimpossible.?归xt‘ifUεLln∞(fl)+,UεX+＼{0},thenwemaychooseA>Osuchthatf(人u一||u||∞)+>||u||l·ThenforO<k三||叫|∞,f(IY-k)+三||叫|1三f(入u-k)+andfork三||叫|∞、f(ν-k)+=0三f(tt-k)+,sowehaveU《λti·ItfollowsthatLln∞(Q)CX.ToseethattheinjectionXCL(fl)iscontinuous,arstobservethatifu,tinεX,U三tinandun→νinX,thenU三ti.Thisfollowsfromh-u)+=lim(tJ-un)+=Osincew→w+iscontinuousinX(observethat(1,+一ω+)+《(tY-w)).Iftheinjectionisnotcontinuous,therewillbeasequencetinεX+suchthat||tin||→OaM||un||1+cc=1.However、川thenhave k+f(un-k)+三1orf(ttn-k)+三l-kforbOIfEisasetofRIliteandpositivemeasureandA>0,wehave f(hE-k)+=(入-k)+μ(E); sincel-k主(入-k)+μ(E)forO<入三min{l、l/jt(IZ)}andO三k三lwealsohave入飞E《unandthen人||XE||三||un||→0,whichisacontradiction-ToseethattileinjectionLI内;文(口)CXiscontinuous,U-emaynowinvoketheclosedgraphtheoremandthrCOIltir111ityoftheinjectionXCL(fl)justproved.RE1IARtt210.ThenotionofanormalBanachspaceiscloselyrelatedtothenotionofararrangementi川ariantBaMchfunctionspaceωddmdin[9];however,thenotionsdomtcoincide-First-wedonotassumethatanormalBanachspaceisaBanachfunctionspace(forexample‘Lo(Q)isanormalBanachspacebutnotaBanachfunctionspace、asitisnothducedbyaBanachfunctionnorm).Secondl〉＼theinvariaIIcepropertyofarearrangementiIlumntlinearspaceXrequiresonlythatifUεXand|u|*=|u|气thenUξX.Itiseasytoseethatanormallinearspaceisrearrangementinvariant‘buttheconversedoesnotholdwithoutfurtherassumptions.飞Vewillneedthefactscollectedinthefollowingproposition. PROPOSITION211.LetUεLo(0).Then的七、εM(口):t?《u}isaweaklysequentiallycompactsubsetofLo(口).{ii)LetX,|||!beanormalBanachspacesatis〈vingXCLo(0)andhavetheproperty (2.9)un《UEX,n=1,...andun→Uα-e.斗||un-u||→0. Munisasequencesatis秒ingun《UεXforn=1,...,andun→Uinu-Lo(0),then||ttn-ul|→0. REMARK212.Theassumption(2.9)onXiscleariysatidedforX=LP(Q),1<p<∞、owingtotheuniformconvexityofthesespaces-ItaisoholdsforX=Ll(0),sinceun→tia-e.andlims叩n→∞fhn|三f|U|implies||thz-u||1→0.AnotherspacewiththispropertyisLo(Q)‘sinceiftheassumptionsof(2.9)holdandk>0,wehave(|un|-k)+《(|u|-k)+‘80(|un|-k)+→(ltt|-k)+inL1(Q).Since(|tin-It|一批)+三(|h|-k)++(|lt|-k汁,1W
Complete{yAceretiveOperators 55 concludethatfhn-ul一批)+→Obydominatedconvergence.Incontrast,(2.9)doesnotholdforX=Lm∞(fl)(exceptintrivialcases)orfortheubigOrlicz"spacesingeneral.ProofofProposition211.Weestablish(i).ObserveRrstthatifUεL(Q),thenCu={uεM(0):U《U}isaclosedconvexsetinL(Q)andthereforeitiswealdyclosedinL(Q)(andSOinLo(0)ifUεLo(Q)).Let问εCuforn=1,....Form>O‘putwn,m=((|un|-击)+八m)sign(tVI (2.10) |川-UYI|=|un|〈去+(|un|-m一士)+ andso,usingtyylεCu wm-un||1+∞三士+j(|问|-m)+三士+j(|u|-m)+ whichimplies (2.11) YYJIBLlMn,m-un||1+∞=Ouniformlyinn. Now,formaxed,tiynmisboundedinLm∞(Q)(duetothedominationtk《u)andsoinL气。)andhenceweaklyprecompactinL2(fl)andtheninLo(flkThusastarIdarddiagonalargumentallowsustopasstoasubsequenceofthet771(whichwestilldenotebytVl)andassumethatlimn→∞tun-m→tumexistsweaklyinLo(fl).Then(211)impliesthatumisCauchyinLo(OLSOWm→tuholdsforsomewεLo(Q).Using(2.11)again、1‘n→wweakl3·inLo(Q).Weestablish(iikItisenoughtoprovethatthereisasubsequenceofthetinwhichCOI1vergea.e.toU(fortheneverysubsequencehasasubsequence、etc.).Tothisend.wearstsho叭'thatfork>0,(un-k)+→(u-k)+weaklyinLo(fl).Since(un-k)+《(u-k沪、weTRayassumethat(tin-k)+→zweaklyinLo(fl)andz《(u-k)+;inparticular‘zεLI(Q)andfz三f(U-k)+.Ontheotherhand,since(tin-k)三(un-k)+呼wehaveU-k三zandthen(U-k)+EEz-itfollowsthat(U-k)+=z.Observethattheaboveimpliesthatf(un-k)+→f(u-k)+.Since-un《-u,wealsohave(tln+k)一→(u+k)-weaklyinLo(Q)andf(tAn+k)-→f(U+k)-.Next,letpεC∞(R)berlondecreasingandsatisfyO￠spt(p)andspt(pf)iscompact- Then 'KJU'hnnr,hh+T+、‘』F,h此,,E·、nr+'KT''1、∞∞flb一-T,,,‘、p ",byFubinhtheorem, j仰71)=IJ二lj(un-WFF(k)+-j(un+川(问-idk andso,bydominatedconvergence (212) f阳)→fMU) Inthesameway,oneshowsthatp(ttn)→p(u)weaklyinLo(Q).
56 BenilanandCrandall Observingthatp(h)isboundedinLln∞(Q)(andhenceinL气。)),weconcludethatphtn)→p(u)weaklyinP(Q).Now,applying(212)withp(r)2inplaceofp、weseethat||p(un)||2→||p(u)|bandsop(un)→p(u)stronglyinL2(fl).PassingtoasubsequeIICtp(un)→p(u)a.e.GivenR〉0、wemaychoosep(r)sothatitisstrictlyincreasingon击三|T|三ftandthen、ifp(un)→p(u)a-twetmveun→Uae.on{击<|叫<If}Diagonalizing、weobtainthedesiredresult. 3m-COMPLETELYACCRETIVEOPERATORS Inthissection飞vewiHstudycompletelyacereti飞吧OperatorsAwhichsatisfyrangeCOIlElitions. DEFINITION31.LetXbealinearsubspaceofM(QiAIioperatorJIinXism-completelyaceretiveinXifAiscompletelyaceretiveandfZ(I+入A)=Xfor入>0.RE丁、fARK3.2.Theddmuondoesnotrequirea飞'tobeaIiaIlachspaeeaIIdmdo们I101IequireAtobem-accretiveinanyBaIlachspace-However-ifAiscompletei3accreti飞飞thenitisaccrdiveinL(Q)aIIdifAis771-completelyaccrdiveinasubspaceXofL(QLtheIItileciOSUI-eAof44inL(fl)iscompletelyaccretive(thisiseasytosee)aIKl7714ceretiveiIItileclosureaxofXiIIL(Q)(astandardfactaboutaccretiveoperatorsforwhichR(l+A)isdmse).飞飞恒alsonotethatlifAiscompletelyaccrdiveina$11bspacpXofAf(Q)aIKIR(I+λ4)=Xforsome入>0、thentheonlycompletel:vaccr肘liveoperatorBinXwhichextendsAisA;i.e.、ifBiscompletelyaceretiveandACBCX×X、thenB=A-Thisfollows-asusual、fromthefactthatcompletelyaccrdiveoperatorshavesiIlgle-valuedresolvents,aIldif(I+λ4)一1C(I+人B)一landbothha飞℃domainX、thpIItir;vaI℃equalaRdsoA=B.Hence771-completelyaccrdiveoperatorsinXaremaximalinti川、sense.Sincewewillbeconsideringextensionsandrestrictionsofoperators、tilefoliowingIIotatioIIforthepartofanoperatorinasetY×Zwillbeuseful- DEFINITION3.3.Let44beanoperatorinAJ(fl)aIIfjl;ZbesubsetsofM(Q)-Then AyxZ=An(γ×Z)and443'=4γ×3' Sincethereareseveraltopυlogiestoconsider‘weadoptthefollowiηgconventions(whichwerealreadyusedabove):Unlessother飞visestated、tllenotationsA‘X‘dc,willdenotetheclosureofoperatorsAandsetsaxintiletopologyofL(Q).Toindit、atetileclosureofaIIoperatorAinaBanachspaceY(respectivel3飞theclosureofasubsetSIofY)inthetopologyofE,wewiHw-rite"theclosureAofAinY"(respedivel}＼"theclosureXofXiRYH). PROPOSITION3.4.LetXbealinear511bspaceofLo(Q)whichl-SnormalspaceandaIId4beacompletelyaceretiveoperatorinX.(i)S叩posethereisaAo>OforIvhichR(I+入04)jsdenseiIILo(口)TflmtileopPrator
CompletelyAceretiveOperators 57 AX=(互)xistheuniquem--completelyaceretiveextensionofAinX.Moreover,AXisdeEIIedby合εAXfiifandonlyif飞UεXand (31)U-d《U一台+入(u-6)fora1l(台,b)εAandλ>0. (iOIfAism-completelyaceretiveinX,thenAissequentiallyclosedinX×XequippedwiththerelativetopologyfromLo(Q)×ω-Lo(Q);moreoverAuisconvexforUεD(A)-piOItmoreover,XisanormalBanachspacesatisfyi昭(2.例,thenD(A)nxisexactlytheclosureD(A)xofD(A)inX. Proof:For(i)weusethestandardfactthatAism-aceretiveinLMQ)inviewoftheassumptionsandthereforeAism-completelyaceretiveinLo(fl)inview-ofCorollary2.7.Choosingd,fεXSOthatQ+AQ3f,iffεXandU+Au3f,wehave U-d《U-Q+(f-u)一(f-G)=f-f (sincef-UεAthetc.).SinceXisnormal,UεX.Thus(A)xisTFZ-completelyaccrdiveinxandextendsA.ThesameconstructioncouldhavebeencarriedoutwithanycompletelyaccrdiveextensionBofAinXandwewouldconcludethatEwasanm-completelyacCTeth'eextensionof立inLo(QLS0万二万and(万)JYDB.IfBis771-completelyaceretiveinX、thenequalitymustholdbyRemarks32.Toestablishthecharacterization(3.1),letAbetheoperatorde白nedby A={(u,ν)εX×X:(31)holds}. ItisclearthatAXcA.Ontheotherhad、if(u、ν)ελthenB=AU{(u,u)}isacompletelyaccrdiveextensionofAinXandthusBCAXbytheabove;weconcludethatA=AX.For(ii),wenotethatthesequentialclosureofAinX×XinthetopologyofLo(Q)×ω-Lo(Q)iscompleteiyaccrdivebyCorollary2.7andSOcannotproperlyextendA.TheconvexityofAufollowsfromA=AandtheevidentconvexityofAu‘whereAisgivenabove-For(iii),letJA=(I+入A)-1and7A=(I+入主)一1.Notethat百五丁nx=页苟nx={uεX:7λu→UinLom)as人↓0)、 while 页ZTX=(uεX:7λu=JAti→uinX挝人↓0)sinceJλistherestrictionofJλtoX.LetuoεD(A)andUεX;wehaveJAti-JλUO《u-uoandJAUo→uoinXandthereforeJλtt→UinXifandonlyifJ入ti→UinL(fl)byProposition2.11(ii).口 DEFINITION3.5.(i)LetCbeasubsetofLo(Q).Then (3.2) c。={t,εC:U《UforallUεC}.
BenilanandCrandall 58 (ii)LetAbeanoperatorjnLo(51).ThenAOistherestrjdiorIofAdeEnedby UεD(A). for Aou=(Au)。 (3.3) LEMMA3.6.LetCζLo(Q)beconvex-Thencoconsistsofatmostoneelement. Proof:Ifu,νεCo,then f们)=fj(U)三fj(于)三jf兴叫户(U) foreveryjεJ0·Theequalityandthearstinequalityareduetotheassumptionsonti,t'aIIdtheconvexityofC,whilethesecondinequalityisduetotheconvexityofj.ItiseasytoseethatwemaychooseastrictlyconvexjεJosuchthatfj(u)<∞andtllmthesecondinequalityisstrict(acontradiction)unlessU=ν.口 PROPOSITION3.7.LetXbeanormalBariachspacewithXCLo(fl)andAbeanmcompletejyaceretjveoperatorjIIX.(jjForA>OtheEUsidaapproximationAA=(I一(I+入A)-1)/入ofAisanm-completelyaceretiveoperatorinJY.(iijForμ>入>OaIIdUεX,AμU《Aλtt;moreover-ifvεAu,thenAλti《t'(iii)AOissingle-valuedandD(A。)=D(A). D(A)={uξX:thereexistst?εXsuchthatAAU《uforai!人>0} UξD(A). forall limAEU=AOUλ↓on 正iIId (3.4) Proof:WerecallthattheYosidaapproximationAλandtheresolventJAmaybeddnpdbv AAUεAJAtt. U=JAIl+人Aλu, (3.5) Gd、、‘-a,,FJ4'EE‘、ρa>-nyn3、λ+r,,J,,..‘飞paecncd、‘..,FCJ,..‘飞qLqLnOA'LGOonro俨APVMLU&EUa'h&IueA'Lon&EUQUrA 人η>0. for 人>O=争f+人g《f+(人+η)g for f《f+λ9 (3.6) Therefore‘ifu,aεD(Aλ),then(sinceAλt4巳AJAthetc.), U-d=JAU一Jλ也+入(Aλu-AAd)《Jλu-JAIl+人(Aλu-Aλd)+η(Aλu-AAQ)=U-Q+句(AλU-Aλ&) for可>0.IffollowsthatAAiscompletelyaccrdive;sinceAλisTY卜aceretiv吧,AAis771complpte!yaceretive.proving(i).
CompletefpPAceretiveOperators 59 Next,ifUεAu,wehave U一JAtA=JA(tt+加)-Jλtt《U+b-U=加 sinceJAisacompletecontraction.HenceAλu=(u一JAU)/λ《U.Ifμ>入>0、Aμ=(Aλ)μ-λandthen,applyingtheresultjustprovedwithAλinplaceofA、oneobtainsAμt4《AλuforUεX,proving(ii).LetUεX.SinceAuisconvexbyProposition34,AOU=(Ati)oisatmostasingletonbyLemma3.6.LetUξXandAAtt《UforA>0;wemaychooseasequence入n→OsothatAAnu→zinω-Lo(fl)andz《UbyProposition2.ll.SinceXisIIormaLzεX;OBtheotherhand、Jλnu-U=人71Aλnti→OinLo(Q)andweconcludethatzεAubyProposition34.Inparticular,UεD(A)and,using(ii),z《UforanyuεAu一i.e.,zεA。14.SinceAOissingle-valued,weconcludethatAAtt→AOuinω-Lo(Q)andtheninLo(Q)byinvokingProposition2.11(ii)inthespaceLo(Q)(Remark2.12).口 4SEhfIGROUPSGENERATEDBYm-COMPLETELYACCRETIVEOPER-ATORS LetXbean(arbitrary)Banachspace,Abeanm-accrdiveoperatorinXandε-tA一-一-xbethesemigroup-AgeneratesontheclosureD(A)ofD(A)inX;itisgivenbytheexponentialformula (4.1) ftA14二nkgc(I+jA)-7114for叫万TZ7 contractsinX,andforuε写写xwehaveftAu→UinXast↓0;wesaythatftAuiscontinuousinX"whenthislastconditionholds.IntilecasethatXisanormalBanachspaceandAism-completelyaceretiveinX,wemayalsoconsider万,whichisTFIe-completelyaccrdiveinYendow-edwiththenormofL(0)ThuswemayconsiderftAonboth万两xand苟言;=页75Todistinguishthesetwosemigroups、wereservethenotatione-tAfortheRrstandε-tAforthesecondf.飞飞?earstciaEIfytherelationbetweenthesetwosemigro叩s.NotethatftAiscontimominxandftAiscontinuousinL(0)(equivalently,inYwiththetopologyofL(Q)).PROPOSITION41.LetXbeanormalBamchspace,Abem-completely,aceretjveinxandftA,rtAbethemIigroupsgem-atedbyAandZon万古?and页互了inXandXrespectively-TIJefollowingassertionsthenholdfort230.。)ftAisacompletecontractionfort三0. ↑ThisisabitinCOIlnictwiththegeneraltheoryofmildsolutions,whichdoesnotrequirethatAbem-aceretivetodiscusse-tAandassociatestilesamesemigroupwithanoperatoranditsclosure.
60 BenilanandCrandall (山iμωiυjεff-一4-(扩ljωiυ)ff一t江:立I(♂万T曰万7nX川)KC万T币万7nXProof:TilefirstassertionfollowsfromtheisaCωOImTη1IP〉lMedtecontractionforη=l尸川..andthepassagetothelimitn→∞intileI-elation 、、‘,,,,4吨,,E,‘、、DC」UUFAor--'LU《vn＼111/At-n+FI/It--＼,4倍tn＼1lI/4吨t-n十rl/FI--＼ whichiseasyusingLemma13.TIlenrstpartofassertion(ii)followsfromtheexponentialformulasincetileinjectioneycL(fl)iscontinuous-ToseethevalidityoftheseCOIldpartoftheassertion-认e$implymmIKthatftAisacontractioniIIthemmofL(fl)JLIId写写isthtT巾meof写写λinL(fl).Assertion(iii)isprovedbychoosinguo巳D(A)andUεD(A)nxandnotingthat(])impliesc-11tt-ft才uo=ε-t立u一-ftAuo《U-uo,soε-t11iEaYbythenormalityof1.口Agsertion(iii)ofproposition4.lmayberestatedasfollows:therpstrictionofftAt0万口才广1XdpfineRamligroupon万737nX飞VuviilusethenotatioI114(t)forthisWIIIiιroup-ltisgiveIIbytileexponeIItialformula (42) 344(f)lt=L(Q)-limlI+二A)-lttforUξ万厅hxn→∞飞yI/ WINY-fAthenotationemphasizesthatthe℃onvergenceisMthvtopologyofL(fl). THEOREIU4.2.LetaybeanormalBariachspacewjlhXCLo(IILa4bθ711-cmnpletdyMfrtMIPiIIaymdf4(t)brtliesemigmuponBtzlnxfiefiMdby(JY/Thm.(l)11干Bllavθ {一-SA(t)1i←li)D(A)=〈UεD(4)nXJuεxsuchthat一寸一《1'formallt>Oj (ij)FJ1(t〉D(4)CD(J1)fort;三0.(iii)lftiED(A).tlien (4.3) i-544(t)u厂一-《tyfort>Oand1·εAti AIid (4.4) SA(t)ti-UL(Q)-lim=-Aou.tlot Proof:Forsimplicit3＼wewriteF(t)inplaceof544(t).FirstIIJρtethat、accordiIlgtoPropo-sition4.l(iL AQ(h)F(t)t』-F(t)tt=F(f)1(h)ti-F(t)lt《F忡)ti-Ufortfl兰。
CompletelyAceretiveOperators 6I andthus(i)implies(ii).Weprove(43).FixUξAu,t‘k>0.飞怕have f(u-J川k)+=噜(Jtlu-J川。+兰f(Jtltt-I忻州+三三f(u-JrAK)+=nj(uJrw Ontheotherhand,j(u-JAU-W=f(JA(叫人υ)一Jλti一汕)+三人f(u-k)+ SOwehave /(u-J;u-n人k)+三忖(t?-k)+Dividingthisrelationby71λ,putting入=t/nandpassingtothelimityields f(tt-f伸一。+三f(ν-k)+ Oneestimatesj((u-S(t)u)/t+k)一isasimilarwayandtheproofof(43)iscomplete.fo11timing‘itisnowclearfrom(4.3)thatD(A)iscontainedinthesetontheright-haIKlsideoftlmequalityassertedin(i).LetUεD(A)nxand1·εXbesuchthat(S(t)u-u)/t《Uforsmallt>0.Then,byProposition2.11(i),thereexistsh↓Osuchthat S(tn)u-Ut→zinw-Lo(Q)nforsomez《uandthuszεX.SinceS(t)u=ftAuaM万ismcompietelyaceretdiveinLo(Q),standardfactsaboutsemigro叩sgeneratedbymaceretiveoperatorsyield-zε44u.ButA=AxbyProposition3.4(i)、SOzε-Att.Inparticular-ttεD(A)andthettha山stoTheorem42,-z=Aou(recallProposition3.7(iii)).L75i吨Proposition2.1landRemark2.12,weseethat(4.4)holds.口REMARK4.3.LetusobservethattheaboveargumentsshowthatifUεD(A)andthereisasequencef177t↓Osuchthat -hnAU-eHU llyz -→z 1nw一Lo(QL t山}h1eI川=阶uand(守u一εf-叫川zhmU川1ι巾Lh0叫川(刊川IQ川lz=A。u.Thefollowingisavariantofaresultof[4linthem-completelyaccrdivecase-THEOREM4.4.InadditiontotheassumptionsofTheorem42,assumethatAisPOSEithrelyhomogeneousofdegreeO<m#1,i.e.,A(入u)=入mAtiforUεD(A).ThenforUε万日丁nxadt>0,附haveS440)uεD(A)and (4.5) A|叫|AOSA(t)uk2一-|m-l|t
62 BenilanandCrandaii Itmoreover,U220,then45AV)u(4.6)(m-l)AOSA(t)三一-7一-Proof:WeputS(t)=SA(t).ThehomogeneityconditionimpliesS(t+fl)ti=入一15(t)(入ti)where入=(1+h/t)1/(m-1)andthus |S(t+h)u-S(tMl三|l一入||S(t+hM|+|S(t)u-S(t)(入u)|.ObservethatS(t+fz)u《U(sinceOEAO)andS(t)(Au)-S(t)u《(人-l)uimply|1人||S(t+hM|《|l一人|!uland|S(t)(川)-S(tM|《|l一入||14|.Since{f:f《|l一人||[I|}isconvex、itfollowsthat|5(t+h)-S(t)叫《2|l一人||u|.Since(l一人)/h→l/(川-l)ash↓0‘usingTheorem42(i)and(iii)weobtainS什)uεD(A)and(4.5)holds.TheiIlmlmlity(4.6)ifU主Ofollowsfrom[4l.口 5GENERATORSOFSE肌fIGROUPSOFCO岛fPLETECONTRACTIONS Weturntotheproblemofattemptingtodaractednthegeneratorsofstmigroupsofcompletecontractionsandwewillsucceedinaspecialcase.AnaturalstructureconditioIIOIIthesetsonwhichthesesemigroupsaretobeddIIHiisdiscussedhst.IIarisesasaconditionsatisaedbyD(A)wheneverAism,completelyaccrdiveinLo(fl).LetLip(IR)betheLipschitzcontinuousmapsfromRtoRandset (3l) Po={pεLip(IR):p(0)=OandO三pff三la.e.}; thisistileclosureofPoinC(IRLWeconsidersubsetsCofLMQ)withthefollowmgproperty: (5.2) thdεC,pεPo斗U+p(Q-u)εC. Forexample、ifu?台εCandp(r)=αrforsomeαε[0,lL(5.2)requiresthatU+α(白-tt)=(1一α)u+αdεC;(52)impliesCisconvex‘butitisastrongerconflitiorlthanconvexity-Otherpossiblechoicesofparep(r)=TV(-k)andp(T)=r〈kfork三oaIldthesechoicesshowthatifrsatides(52),thenitalsosatides (5.3) u咱自εtandk三0=字dV(It一川、b〈(ti+k)εC. Infact‘ifCisaclosedconvexsetsatisfying(5.3)thenitsatisam(5.2)(seeRemark77)THEOREM51.LetAbem-completelyaccrdiveinLMIlkThenD(A)satjs币臼(5.2)-Proof:LetJλ=(I+入A)-lbetheresolventofA.JAisacompletecontractionaIKllimλloJλu=UholdsinLo(fl)ifandonlyifUεD(A).Letu、GεD(A)andpιPWemRttoshowthatU+p(Q-u)εD(A).WehaveJA(u+p(fi-u))-Jλu《U+p(fi-u)-U=p(。-u)‘JA(u+p(白-tt))-JAIl《u+p(白一叫-L=(u-a)+p(自一叫. (54)
CompletelyAceretiveOperators 63 AR(byrlow)usual,thisimpliestileexistenceofAn↓OsuchthatJ儿(ti+p(。-ti))→zinw-LMO)and (5.5) z-U《p(fi-u),z-a《(u-a)+p(ft-u). Weseektoshowthat(5.5)impliesz=U+p(ft-u).ItthenfollowsbyargumentsgivenseveraltimesabovethatJA(74+p(d-u))→U+p(fi-u)inLo(0).飞Aferecordtheessentialpointinalemma. LEMMA52.HAUεLo(0),pεPandz《p(u),U-z《U-p(u),thenz=p(叫. Proof:Beforetheproof,letusnotethattousethelemmaon(55)、wereplacez-UbyzandQ-UbyUin(5.5).Ifk>Owehave f(u-k)+=f(u一z一(k一州))+(z一p(k)))+/56)三f(u-z一(k-P(k)))++f(z一州)))+4(卜帅)一(k-p(k)))++f(P(u)-P(k)))+=f(u-k)+ wherethesecondinequalityisdmtotheassumptionsaIIdthehalequalityisduetothefactthat f(f+9)+=ff++fg+ifandonlyif的。togetImwith(1t-p(u)一(k-p(k)))(p(u)-p(k))三OsincepεPimpliesthatbothT→p(r)aINl1·→T←p(r)areIlOIidecreasing-Thusw-emusthaveequalityeverywherein(5.6)andtlwsimplefactthenimplies((u-k)一(z-p(k)))(z-p(k))三0.Itfollowsfromthisthat{u〉k}n{z〈p(k)}and{U<k)n{z>p(k)}are口11llandthisforeveryk>OAparallelargUTIleMestablishesthesamefork<Oandalimittakescareofk=0,SO lt({1i>k}门{z<p(k)})=μ({u<kln{z>p(k)})=0 ft飞rallkIfμ({z>p(u)})>0,山IIμ({z>P(u)+去))>Oforsomei附ger71>O川Thenμ({z>p(ttNjt三U<lti)>OforsomeintegerlHO附ver, (z>P(ti)寸,卡以干)cL>叶)+jJ<干)c(z>P(Lfi),u<干) yieldsacontradiction.Thuszf二p(u)andz;三p(u)followsinthesameway.口ClosedsubsetsCofLo(Q)satisfying(52)admitnaturallydeEI1edprojectionsP:Lo(0)→Casdetailedinthenextresult. PROPOSITION5.3.LetCCLo(Q)beanonempty,closedandsatis秒(52).ThenforUξLMQ)thereexistsauniqueuεCsuchthat (57)tiY-U《w-U+入(u-u)forA主OandtcεC.
64 BentlanandCrandal/ Moreover,ifPc:Lo(fl)→CisddIIedbyPcu=U讥FIrn(57jholds,thenPcisacompletecontractionandPcu=UforUEC.Proof:飞Veshowarsttheuniqueness;theproofofthisalsoestablishesthatPcdeRRedasaboveonthesetofUεLo(口)forwhichthereexistsUεCsuchthat(57)hoidsisacompletecontraction.SincePcu=UisobviousforUεC,itwillremaintoestablishtheexistence(orD(Pc)=Lo(Q)),whichwillbedonelast.SupposeU,也,飞合巴Cand(w一…一叫人(…)扣r入主{}and时CYψ-f《论一合+入(合-d)for入主()andψεC. (5.8) IfpεPo‘weobservethat-p(-r)hasthesameproperty.Choosingw=LiIlthe白rstrelatioTland丘'=UintheseCOIldaIIdinvokingProposition22、weconcludethat f(l'一州 AddiRgyields j((t-L)一(ti-W(f-v)三0;appPaliIlgtoProposition22again,wehave L-1'《b-17+入((17-L)一(14一的)for入三0 andsetting入=lweconcludethat台-1'《也-u.Ifd=Uwelearnri511日iqueandso飞YV111坷'dehePcu=νaIIdthenwehavejustshownPcft-PCIi《Q-14:thatis‘Pisacompiptecontractiononitsdomain-TO行川cluEie、we口lustsilowthatPcisddmdOIlLo(Q).Itwillsdi口?toasRIlmpoεC‘日iIicelfzξCandA'=f'-z、thenAFalsosatidi何(5.2)aIKiitiseasy10挝、号tiItttIYII=PIJU-z)+z飞Veanalyze(5.2)further叭'henOεC.LEMMA5.4.LetCCLo(0)satisOW2jandoεCFIlmforuεCandl吗k>O (39) (|叫-k)+〈l)sig叫u)εc Inparticular.Cl寸LInx(fl){andthereforef门L2(0)jisdenseinC. Proof:Since0.UECand p(T)=r-((|T|-k)+八l)sign(T) liesin乒0吨(52)implies叫l=U+p(-14=(h'|-k)+〈l)sig口(叶ε仁Since叫,lELl门提(Q)aIIdtkl→UiIILo(Q)ask↓Oandl→∞,thedensityclaimfollows-口EndofProofofProposition5.3.WereturntothePI-oofthatD(Pc)=Lo(fl)ifoεC.LetusnrstnotethatD(Pc)isclosedinLo(QLsincetinCD(Pc)andtin→UinLo(Q)impliesPCIAnisconvergentinLo(Q)sincePcisacompletecontraction.Moreover‘itis叭-klentthatthenPctt=limn→∞PC1473;inparticular,l)(Pc)isclosediIILMQ)Nowlet
CompletelyAceretiveOperators 65 C2=CnL2(fl);thisisadense(byLemma5.4)andconvexsubsetofCwhichisclosedinL2(Q)andevidentlysatisaes(52).LetP2betheL2(0)projectiononC2·ThenforU巳L2(QLwξC2andpεPowehaveP21A+p(ω-P2u)εC2andso,bythevariationaicharacterizationofP2ti, (5.10) /(乃u-u)P(ω一乃们0扣rMC2 Itfollowsthat(5.7)holdsforwεC2withU=P214.Bydensity,thisrelationholdsforallwεkandweconcludethatPctt=P2uforUεL2(fl).SinceD(Pc)isbothdenseandclosedinLMO),wearedone.口Inview,ofTheorem42,thefollowingresultestablishesabijectionbetweenthem-completelyaceretiveoperatorsAinLo(Q)forwhichApm)#eandstronglycontinuoussemigroupsofcompletecontractionsS(t)onclosedsetsCsatisfyi鸣(5.2)forwhichthereexistsUsuchthatS(t)u-U→OinL2(Q)ast↓0.Thebijectionisgivenbyeitherofthecorrespondences:A仲S(t)ifandonlyifS(t)=ftAifandonlyif-A。istheidnitesimalgeneratorofS(f).Inthestatementweusetheoperatorliminft↓o巧constructedfromafamilyofoperators刀,t>0,inL(0).Itisde自nedb371·εliminft↓o巧uifthereexistsut,Utε巧utfort>Osuchthat(叫‘ty)→(tu)inL(口)×L(Q)ast↓0.THEOREM5.5.LetCCLMQ)beclosedandsati54·(52)andS(t)beastrong1:vcon-tiI1110115semigroupofcompletecontractionsonCforwhich (511)thereexistszεCsuchthatS(t)z-z→0εL2(口). LetPcbetheprojectiononCgivenbyProposition53.Then (5.12) I-S(t)PCA=liminf一一一-t↓ot is771-completelyaceretiveandS(t)=ε-tA.Moreover,AistheuniqueT71·completelyac-crdiveextensionofAosatis秒ingD(A)CCwhere-AoistheiII币IIitestiIIIalgeneratorofS(t). Proof:飞VemayassumethatOεCasabove;indeed,ifzεCisthepointforwhich(5.11)holdsandk=C-Z吨thenS(t)(tt-z)=S(t)u-zforUεCddmsasemigro叩ofcompletecontractionsonI(andtheargumentsbelowallowustoworkwithIfandS(t).LetC2=CnL2(QLS2(t)=S(t)|C2andP2betheprojectionofL2(Q)onC2·Clearly,S2(t)isasemigroupofcompletecontractionsonC2;infact,S2(t)isstronglyωIltiI111011s(inL2(0))byProposition2.llbecauseS(t)(u-z)-S(t)O《U-zandS(t)0→OinL2(Q)by(511).AccordingtotheHilbertspacetheoryofcontractionsemigro叩s, I-S2(t)P2liminft=AL wherethetopologyofL2(Q)isusedtotakethelimit,is7TIl-aceretiveanditistheuniquem-accrdiveextensionoftheinRIlitesimalgeneratorof52(t)satisfyingD(A2)CC2·Since
66 BenilanαndCrandall 3(t)P2isacompletecontraction,A2isTYZ-completelyaceretiveinL2(Q)byLemma2.5‘SincetheoperatorAddnedby(5.12)iscompletelyac〈fretiveforthesamereMOIl-,dosed(inLo(fl))andextendsAuAism-completelyaccrdiveinLo(0).Theotherclaimsfolloweasily(recallProposition3.4(i),etc.)口 6ATHEOREMONA+β IIIthissectionwepresentageneralperturbationtheoremwhichproducesm-completelyaeel-diveoperators.LetdbeamaximaimonotonegraphinRwithoε3(0).IhelldiII『d1l臼sanoperatorBinLo(fl)accordingtowεBuifti,wELo(Q)and叫z)ιd(u(z))a-P-I巳Q.SincezuεBthtbεBaentails(U-d)(w一曲);三0,itisclearthatBiscompletely盯crdiveinLo(Q).Infact,Bism-completelyaceretiveinLo(OLsinceU+w=f,u7ξBuislIIIiqtlelysol刊dby叫z)=(I+d)-1(f(z)),ω(z)=f(z)-u(z).Moreover,ifXisanormalSIlbspa何ofLo(QLBλ'ism-completelyaceretiveiIIX,sinceU+w=fandtiIUJ20impliesfl<<fandw<<f.THEOREM6.1.LetAYbealinearsubspaceofLo(fl)whichisnormal.lrt44beaI1171刊IIIpletelyaccrethveoperatorinXandOεA0.LetdbeaInaxjmalmonotοIIegraphinIR.OEd(0)andBbetheoperatorinLo(fl)inducedby3.ThenA+Bxisn卜completelyat-fretiveiIIX.AIoreover,thereisaneverywheredeβnedorder-preservingmapTinXRtJClithatforeveryfεX,U=(I+A+BX)一lfandw=TfsatisOY 、)l.、){r ti+w+Au3f、ω(z)ε3(u(z))a.e.OIIQ aIld (6.2) ti+ω《f. Let115prefacetheproofwiththeobservationthatu,EUEXand(6.1)uniquelydrIVrI旧时SItbutmfneces5arilywandthat(62)Ileodnothoidforeverychoiceofω.llldeed‘ifQ=lR,β(O)=mandA=3,thenU三OforanyfandwcXcaIlbechosenarhi11111·ily.Noticethat(61)and(62)togetherstilldonotuniquelydctermimu-inthiscxtimpie-However‘itisnotKIlowIIifthemappingTwUhalltheabovepropertiesisuniqueiIlgeneral.Proof:FirstletusnotethatA+BiscompletelyaccretivebyCorollarv2.4.XextobservethatwemayaswellassumethatX=Lo(fl).Indeed,byRemarks2.9R(I斗-A)=XDLliAIde(QLwhichisdenseinLo(QLandthenZism.CωOmIp3iket忧时edlb〉y,aCCredti忖飞V?ei阳口L0叫(Q)aUIn14=AXbyProposition3.4.IffεXandwesolveU+四十z=f‘tuεBuandzEAusubjecttoU+w《fasin(6.2)、thenU《faIIdtty《fsincettw三OandSOti‘1UGXantithenzεX.HencewεBJ气'14andzεAu.ThuswrassumeX=Lo(口)belowLet入>OanddλbetheYosidaapproximationOi-戌Since!LfλisLipschitzcontinuousantiIIOIIdecreasing,theoperatoritinducesinLo(Q)isLipschitzcontinuousarlfim-completplJva(vrtiveaIlciitclearlycoincideswiththeYosidaapproximationofB.InviewoftbfJVts
CompletetyAceretiveOperators 67 thatthes旧nofma-completelyaceretiveoperatorsism-completelyaceretiveandthesumofanTTI--acceretiveoperatorandaLipschitzcontinuousaceretlveoperatorIS771-accretlve、A+Bλism-completelyaceretiveinLo(Q).ThusiffεLo(Q)wemayuniquelysolve(6.3)uλ+BλUλ+Atiλ3fforuλ=ZλfwhereZλ=(I+BA+A)-1.Wewishtoshowthat(6.4)叫+BλUλ《f.Tothisend,letk>0,P1(u)=signo(tt-k)+andp2(u)=P1((I+9λ)一1(u)).Clearlyp2ismdecreasi吨,boundedandp2(u(z))=Oon{u<k+应λ(k)},SOP2(u)issupportedontheset{U>k+3入(k)},whichhasEI山emeasure.Fromtheequation(6.3)wehaveuλ=f-(uλ+BλUλ)εAUλ,byassumptionwehaveOεAOandSOUλ《Uλ+入叫咱whichimplies(usingProposition22(d))thatfm(问)主oor.withz=uλ+FA(uλ)=uλ+BAUλ,fmigno(z一k)+三ffsigno(z-k)+ Itfollowsthat Similarly? /(z-k)+纣(f-k)+扣rk>Of(z刊)-4(川)-fork>O a叫SO(Proposition1.3) z=uλ+BλUλ《fand(6.4)holdsasclaimed-From(6.4)andweconcludethatuA《fandBλtu=点(ttλ)《λsouλanddA(ziλ)a11dlieinweaklysequentialiyprecompactsubsetsofLo(fl)(Proposition2.11)·Let入n↓Oanduλn→thFAn(tun)→ωintu--Lo(fl).FromtheaboveandProposition211weconcludethatU+TU《fandu,ωεLo(0).IOCOInI山tetllep100fitisenoughtoshowthattUn→uinLo(Q).Indeed,themt-n=f-Wλn+Fλn(ILAn))εAUAn'un→f一(u+ω)inw-Lo(flhandthenUεAubyProposition3.4(ii).Moreover,郎"(tAAn)εP(9n),gn=(I+灿的-11tAnand||gn-uλn|lLo(0)=人n||应λn(uλn)||Lo(Q)三人n||fllLom),(becauseLo(Q)isnormalandFλ(uλ)《f)SOgn→UinLo(口)andωεB(u)forthesamereasonsasabove.SinceZA=(I+BA+A)-lisIIOnexpansive,toshowthatlimλ↓OZAfexistsinLo(ft),weneedonlytreatadensesetoffs.Forthiswemayrelyontheknown(Hilbertspace)factthatthelimitexistsinL2(0)(andthereforeinLo(0))iffεL2(Q).WehavecompletedtheproofuptotheexistenceofthemappingT.SinceZAisacompletecontractionandBλisorder-preserving,叫=T入f=Bλ(Z入f)exp陀sses叫asanorder-preservingfunctionoff.Usingtheaboveproof,thefactthatweal〈convergencepreservesorderandamaximalitya1811ment,weconclude.口飞飞fenotethefollowingcorollaryofTheorem61.
68 BenilanandCrundall COROLLARY62.LettheassumptionsofTheorem6.lholdwithX=LMQ)andfιLo(Q).Iffε(A+B)(u),thentherejsawεBusuchthatω《fandw+Au主f Proof:itfollowsfromTheorem6.lthatforeachE>0,thereisasolutioII屿,1ikof Etif+uy+Auf主f+EU,μYEBtihEtiE+uk《f+EU. Si盯eEu+(A+B)143f+Euaswell‘tif=U.HencetL-EεB队叫+Au3f,叫isseql川1tiallyprecompactinu-Lo(0)aIIdifWEn→winw-Lo(Q),thenwεBIi吨w+Au5fandtu《f.口 7COMPLETELYACCRETIVESUBDIFFERENTIALS LetXbealimarsubspaceofMm)andφ:X→]一∞冲∞i.Weddnetkoperatorθ4飞'中inXhy1'ε句,φ(u)丰斗UεD(中)、1yεXand(7.1)〈f)@(w)三φ(u)+/(tu-u)t?for四ξXwith(tI'-u)1,εL1(Q).HereD(φ)={tiεX;中(u)<∞}istheefectivedomajIIofφ.FOI-example、ifXCL气。;landD(φ)#ιtheIIUξθx@(ti)exactlywhenu、1?εXaIKitb(叫主φ(u)+J(117-II)t'fortuξX;inotherwordsOx@=(θφ)x,whereδφisthesubdiiTerentialinL2(fl)oftheexteIMOIl夺ofφtoL2(0)叭'tlichis+∞onf(0)＼X.Moregeneraliy、ifyisalinearRubspaceofM(Q)withXC}Pandφistheextensionofφtoywhichis+∞o仔ofY.tilH1θλ'φ=(句,φ)x.Thefollowinglemmaissimplebutbasic.飞Veusebelow,PoandPofrom(2.川a川i (5.lL LEMMA71.LetXbealinears山spaceofLom)whichiE口MInalandφ:X→l-l工、+叫-J155UHlethat (7.2)φ(u+p(fi-u))+φ(6-p(fi-u))三φ(u)+φ(Q)forU,dEXholdsforpEP0·ThenO,飞'φisCOInpletelyaceretive.Proof:NoticethatsinceXisnormai,p(w)εxfortuexsincep(w)《饥'·Nowletrε04飞'φ(u),奋εδxφ(fl)andpεPo-Sincep(fi-u)εLl门∞(flLbydefinitionofθλ'φ‘wehave φ(u+P(&-u))沙(们fp(&-u)u,争(卜州一川φ(&)一/p(&-u)fyAdding,using(72)andφ(ti)+驯的<∞,weobtainfp(i-u)(u-b)三OandconcludebyProposition22.口AsoneexpectsfromtheclassicalHilbertspacethtpr3飞toobtainmaximalmono-tOIlicitvoftilesubdibrentialofaconvexfunctional,onerlf-edslo飞ver--semiCOIltir111itvoftirfunctional-Inthisdirection、weobserve:
CompletelyAceretiveOperators 69 PROPOSITION72.LetXbeanormalBanachspacewithLIn∞(fl)asadensestlbspaceandφ:X→]一∞,+∞i.Assumethatφisls.c,inXand(7.2)holdsforpεP0·Then(a)(72)holdsforanypεP0·(b)φisconvex.(c)ForUεD(φ),UεX,thefollowingassertionsareequivalent:(i)Uξθxφ(u),。i)φ(u+p(ω-u))三φ(u)+fp(ω-tt)tyforwεX,pεPo,。ii)φ(w)三φ(ω-p(ω-u))+fp(ω-ti)tyforωεX,pεP0· NoticethatsinceXisanormalBanachspacewithLln∞(Q)denseinX,wehaveXCLo(0)(indeedXCL(0)withcontinuousinjectionbyRemark2.9andLMO)istheclosureofLI门∞(fl)inL(0)).WerequirealemmaLEMMA7.3.LetXbeanormalBmachspacewithLm∞(Q)denseinX,{ωη}beasequenceinXconverginginXtouyand{pn}beasequenceinPo的thpn(T)→p(r)forrεRThenpεPoandpn(tiJn)→p(ω)inX Proof:Thebstassertionoftheconclusionisclear.Now,bytlleAscolitheorem‘pn(r)→p(T)uniformlyforTinaboundedsubsetofIR;hence,ifz巳L∞(fl)wehave|lpn(z)-p(z)||如→0.Moreover,since|pn(r)|三|叫,ifzεLI(0),then|lpn(z)-p(z)||1→0;itfollowsthatifzεLln∞(fl),thenpn(z)→p(z)inX(sinceL1「l∞(0)CXwithcontinuousinjection).Nowthemapsz→pn(z)arecompletecontractionsandthencontractionsinX;sinceLln∞(fl)denseinX,theresultfollows.口ProofofProPOSItIOI17.2:Toprove(a),letU,aεX,pεPo.UsingLemma3.2thereexistspnξPowithlpn|三|plsuchthatpn(r)→p(r)forrεRUsingLemma73andlower-semicontinuityofιonecanpasstothelimitin(72)forpntoobtain(72)forp.Toshow(h),apply(7.2)withp(r)=T/2toobtain φ((也+u)/2)三(1/2)(φ(u)+@(。))foru、也ξX. Sinceφisl-51.,itisclassicalthatthisimpliesconvexityofφ.Finally.weprove(c).Itisclearthat(i)implies(ii).Using(72)(withd=ω)andφ(u)<∞,oneseesimmediatelythM(ii)implies(iii)-Now,,usingthesameproofasforpart(a),if(iii)holdsthen 州 Choosingp(r)=T,oneobtainsυεθxφ(u).口UndertheassumptionsofProposition7.2,θxφmaybeempty;forinstance‘takeX=Lln叫QLφ(u)=futywithUεL(Q);alltheassumptionsofProposition7.2aresatided(φislinearcontinuousonX)andθxφisemptyift7￠Lln∞(0).However,wehavethefollowingresult: THEOREM7.4.LetXbeanormalBaIIachspacewithLm∞(fl)denseinXand争:X→l一∞『+∞lAssumethatf72)holdsforpEPo,0εθxφ(0)andφisl.s.c.forthe
70 BentlanandCrandall topologyofX+L2(Q)(inthesenseφ(u)三iiminf@(un)forany(un),UinXwithun→UinX+L气。)).ThentheclosureinXofδxφism-completelyaceretiveinX Toprovethistheoremrequiressomepreliminaries;inparticular,wewillneedtotaketheiS-Ce盯elopeofafunctional.Ifφ:X→l-∞,+culandXisaBmachspace,wewilldenotebyφXthefunctionaldennedby (7.3) φX(u)=lninf忡忡):tuεxand||tiJ一u||三r}; φXistheLs.c.enveiopeofφinX.ItisclearthatφXisa1.s.c.functionalonXwithvaluesin[一∞,+∞],φX三φ,D(φX)C万市丁xandforUεx,φaY(u)=φ(u)ifandonlyifφisl-R.C.inXatthepointti.Wehavethefollowingsimplefacts: LEMMA7.5.LetXbeanormalBamchspacewithXCLo(町,φ:X→]一忧,+∞laMφXbrTthel.s.ceIIvelopeofφinX.AssmethatφXtakesitsvaluesinl一∞,+∞l(lttjLetpkiIh;if(7.2)holdsforφ,thenitholdswithφXinplaceofφ(bjlfLI门∞(il)isdenseiIIXand(72)holdsforpεP0.thenθxφXisanextensionof/ixφ ProofofLemma7.5:Forthepart(a)letU,GεX.Byde白muonofφXthereexistslU川、11‘71ξXsuchthattun→u‘曲n→QinXandφ(wn)→φX(叫,φ(d7n)→φX(G);弓iIIce飞hemaptt→p(u)isacompletecontraction,itisacontractiononaysuchthat川川一叫l)→p(G-u)inayandF0、using(72)‘4vI(u+p(。-u))+φX(&-p(&-u))三liminfφ(wn+p(4'n-wn))+liminfφ(曲n一p(ltiyn-UK))三liI11inf(φ(wn+p(dyn-wn))+φ(曲n一p(d?η-wn)))三iiminf(φ(h)+φ(ψn))=φX(叫+φX(白).IopIO飞·{part(b),letuεδ,飞'φ(叫andpιP0·UsingProposition72(c)we}laveφ(u)汁(Ii-p(江'一u))+fp(w-Wf…εX BIltasIIltileproofaboveof(aLthisisstilltruewithφXinplaceofφ,andtheIlbyproposition7.2(c),wehaveUεδxφX(uk巴ProofofTheorem7.4:InviewoftheassumptionOεOxφ(0),wemayreplaceφ(u)byφ(It)一φ(0)andassumeφ三0‘φ(0)=0.AssumenrstthatX=Lm∞(Q)andletφzbeth川.sz.envelopeinP(Q)oftheextensionofφtoL气。)by+∞onL2(Q)＼Lln∞(口).℃learly-φ2主0,φ2(0)=0,φ2SFttides(72)forpEPoandisconvex(useLemma7.5(a)andProposition72(b)).ThesubdiEerentialδφ;Jisthencompletelyaccretiveandmaximalmonotone,thatism-completelyaccrdiveinL气。).Wealsohaveoεθφ2(0)andthenA=(θφz)LIn∞(Q)isr71-completelyaccrdiveinLInz(fliNow,sinceφisl-S-c.forL2(fl),φ2=φonLln∞(fl)aMthenclearlyACθLln∞(Q)φ.Bymaximality(orusingLemma7.5(b)),A=δLln∞(Q)φaINitilenθLm∞(Q)φis711-completelyaceretiveinLIndU(0)·
CompletelyAceretiveOperators 7l Considernowthegeneralcase.LVestillmayassumeφ22Oandφ(0)=0;wemayrestrictφtoL1门∞(fl)andcomputeθLm∞(51)φ.Bytheproofabove、θLIneu(Q)φisTTt-completelyaccrdiveinLm∞(0)anditsclosureinXis771·completeiyaccrdiveinX(Lln∞(Q)isdenseinX).Wewillhavethesamefortheclosureofδx也becauseitextendsθLM∞(Q)φ,aswenowshow.HUεθLIn∞(fl)φ(14)andpεPo‘then,byassumption φ(u+P(ω-u))主争(u)+fu(P(ω-u)) foruεXandwemayuseProposition72(c)tocOMi叫ethatuεθxφ(u).口REMARKS7.6.(a)Withthesameproofasabove(inthecaseX=Lln∞(fl)),oneseesthatifXisalinearsubspaceofF(0)whichisnormalandφ:X→]-∞,+∞]iU.s.cforL2(Q),satisnes(72)forpεPoandOεOxφ(0),thenOxφitselfism-completelyaccrdiveinX.However,withassumptionsofTheorem7.4,ifXisnotcontainediIIL2(QLingeneralδxφisnotclosed(seeExample711).(b)TheassumptionOεδxφ(0)mayberelaxedtotheassumptionthatthereexistsGεX、FEL勺。)nxsuchthatbεθxφ(创.OnecanthenapplytheaboveargumentstothefuMionai中OIIL1n∞(fl)givenby审(叫:中(u+6)-φ(的一jPti-REhfARK7.7.Alitheresultsaboveapplytothecaseφ=ICWithCCX.HereICistheindicatorfunctionofC;itisgivenbyIc=OonCandIc=+∞ofofC.Theproperty(7.2)forpεPoisthenexactiy(5.2).Inthiscase,θλ'φisthegraph{(tiJ?)εC×X;f(u-w)t'主OforwεCwith(tt一ω)uεLI(fl)}soforUξX、(I+θxφ)-17i={νεC;f(t?-uy)(u-u)三OforωεCwiih(u一ω)(u-υ)εL1(Q)}.IfXisaBamchspace‘thefuIIctioIIalφisls.c.inXifandonlyifCisclosedinX.IfXisaIIormalBanalchspacewithLm∞(11)deIlseinXandCisaclosedsetinXsatisfying(52)、PropmitiEm7.2(c)statesthat δ飞,φ={(tIJ)εcd;/内斗')t20f…εC‘pεPo}whichisclosed,andthenTheorem7.4(augmentedbyRemark7.6)(b)-notethattyεδφ(u)forU仨C)andthecomputation (I+机γlu={t,εC;f(u→)(…)三Of…εCW灿(u-w)(u一们内)} implythatforUεX,thereexistsoneandoniyoneuεCsuchthat f帅一ω)(u-悦。如rωεc,pεPo whichisequivalentto(5.7).Inotherwords,Proposition5.3isacorollaryoftheresultsabove.REMARK7.8.LetXbealirmrspacewhichisnormal,φ:X→]一∞,+∞)andsetPφ={pεPo;(72)holdsforp}.Thefollowingpropertiesareclear:(a)pεPφ#(I-p),pεPφwherep(T)=-p(-r)(b)pJεPφ功p+户。(I-2p)εPφ(apply(72)for户withU+p(G-u),d-p(fi-u)inplaceofti,的.
72 Beni/αFIandCrandG// (c)lfφisconvex,thenPφisconvex〈d)IiLln∞(Q)isdenseinayandφisLB·C.inX‘thenPφisclosedinC(R)(useLemma73)-kowPoisaconvexcompactsetinC(R)whoseextremepointsarethefunctionsPB(7·)=MXBh)山、whereBisaBOrelsetinR.OntheotherhaM、I-PB=PB',wilerBr=R＼ff‘户B=P-BandpBI+PB2O(I-2PB1)=PBwithB=(B;门(I-2PBl)-1(B2))U(BI门(I-2PBI)一1(B;)).UsingtheseremarksonecanseethatifLln∞(Q)isdenseinxaIldφis1.5·(二i口XandsatisEes (7.4)φ(Q〈(u-k))+φ((也-k)VU)三φ(u)+φ(。)for队QEX、k>0(飞飞'hichis(72)forp=P]k戊luk>0)、then(72)holdsforanypεP0·Actually‘intherasFX=L2(Q),thereisanotherproofofthisresultbasedOIlPropositionl.2、thefactthat(7.J)impliesT-acCIetivityinLlandLCClofthesubdifferentialθφaIIdtilefactthat(72)holdsifmdonlyifjp(fi-u)(b-u)主OforUξθφ(u),合εθφ(也)(sm[7]).飞VeconchIdethissectionwithconcreteexamples‘beginningwithageneralframpwork-EXAMPLE7.9.LetQbeanopensetinRNwiththeLebesgueI11ωsure吁J:11×R入'→(0.+ccjbemeasurableinh,已)εQ×RNandl-5.caIIEltO盯曰:in55RNwithj(LO)=oa.e川μbeamnnegativeBowlnleasueonQ,jo:0×R→[0,+父llbeli-IMamabkBiIIIξflforaIIyrεRandi.5.c.andconvexinrεRwithJo(AO)=Ojttp..andDbηaliLIeumbspmofutJm)lineuiyembeddedin川口,jt)DefinethefunctionalφonDby(7.5)φ(u)=lj(l＼gradu(z))dr+/jo(r,叫r))djt(r)fortiεDJQJfl飞vhereintilesecondintegral、UisactuallytheimageofubythePIllbpddingofDiI11041f(Q、ji);thissecondintegraliswell-definedsincetileassumptionsOIlJoimplythaltjo(·『tt)ξ1l(flJt)forttEAI(口,μ)Recall山tiftiξlGJ(Q)aIlflpE沪otilfIip(u)ξliLKfl)mfi(7.6)gradp(u)=pr(u)gradUa-e. 飞Vehavethefollowinggeneralresult:LEMMA7.10.IViththeassumptionsabove,letXbealinearsubspacpofM(Q)11'hiclijsnormalandcontainsDandpεPo.IfDsatis币es (7.7) UεD=〉p(u)εD thentheextensionofφtoXby+∞onX＼DsatiSEes(72).Proof:Letu趴、&ε,X;(口7.22幻)istrivialifoneofUO创Ir.aisnotinD(φ)L'sωOweImIn1扎叮ya剖sS盯川1u1mI口mTUleu白εD(φ)L.飞叭W飞V/ehavethenU,aεDandthefunctionsj(z,gradu),j(z,grada)andjo(11i)守Jo(2、a)areEIlitea.e.onQandμ-a.eonQrespectively-By(77)、(7.6)and(7.5)‘if1、=U+p(tl-u),毛=d-p(台-u)εD,thenφ(17)=jj(z,人gradQ+(l一人)gradu)dz+ljo(ια也+(l-o)叫dμJQJQφ(b)=lj(z,人gradtt+(l一人)gradd)d.r+ljo(29αu+(l-Q)的dμJQJQ
Complete{yAccrefiveOperafors 73 叭'}1ereλ=py(也-u),α=X{u#}P(&-u)/(也-u).wehaveAεM(口),0三入三la.e.andαζAf(口号μ),0三α三lμ-a.e.Byconvexityofj(z,已)withrespectto己a.e.2、weiIa飞-ej(z,λgradd+(l一人)gradu)+j(z,λgradU+(l-A)gradd)三入j(2,grada)+(1-A)j(2,gradu)+人j(-,gradU)+(l一人)j(z,grada)=j(z,gradU)+j(Agrad企)a.e. Inthesameway jo(z,α也+(1一α)u)+jo(z,αu+(l一α)d)三jo(2、u)+jo(2,也)μ-a.e. Then,integrating,weobtainφ(ν)+φ(的三φ(u)+φ(a).口AssumenowthatDiscontainedinLo(0)and(7.7)holdsforpεPoandletXbeamrmalBamchspacewithDCXCLo(fl)andLIn∞(0)denseinX.By1T、}h1eOr陀em7.4‘WknowthatifφXi扫slL.s.C.forL2(Q),thentheclosureinXofδxφXis肝completelyacc削iveinX(observethatOε句,φ(0)andrecallthatθxφC句,φX,byLemma7.5).ToiMerp削thisoperator,considertheparticularca优D=D(Q)whichclea肉'satisaes(7.7)forpεPo:inthiscase、θxφcontainsthecouples(u、u)εD(0)×XsuchthattimeexistshεLjoc(fl)NandhoεL习jLOC」(Q,μ)sa剖汕tUihsf乌々削〉y川?才inh(2)εδj(Z'grad1u』(Z))a.e巳.Zι、h0(z叫)εθj0(Z,u(Z))μ.a.e巳.ZαηtU7=-di忖Vh+h0μi阳nDf气(Q)L、 (7.8) whereθJ(2,已)(resp.。jo(2,叶)isthevalueat5(resp.r)ofthesubdibrentia!ofthels.c-convexfunctionjh,.):5εRJN→j(zi)ε[0,∞](resp.TER→jo(-T.r)ε[0、。c])-Indeedif(队1h、ho)εP(0)×X×LLc(Q)入'×Ljoc(0、μ)satisnes(7.8)andtu巳?(QL附have(byddnitionofthesubdiferential) j(2,gradw)主j(Agradu)+hgrad(ω-u)a.e.,jo(2,ω)主jo(2,u)+ho(w-u)μ-a.e. sobyintegration(anddeMUonofderivationinDI(Q)) φ(ω)主φ(u)+f巾-u) InthegeneralsituationtheoperatorOxφXis"someinterpretation,,ofthediferentialoperator-divθj(2,gradU)+θjo(z,14)μwithsomeboundary(ortransmission)conditions.Thisassertionisquiteheuristic;asfarasweknow,thegeneralsituationisfarfromresolved-w-eonlygivehereaparticularcasetoillustratedimcultiesinmakingthisgeneralassertionpreelse.EXAMPLE7.11.LetQbeasmoothboundedopensetinRNwiththeLebesguemmsuI飞j:RN→{0,+∞[beconvexcontinuouswithj(0)=Oandsatisfying (7.9) ,.j(己).··…一一一时二乌|己|一,…
74 BenilanondCrtIndall fft-xanol-malBarlachspaceX(necessariiyincludediIlLo(Q)=L(Q)=Ll(fl)since。isbounded)deHneφxby ?飞'(← Thisω川spo毗toφdd时inExample79withjo=OandD=xnwjl(Q)whichsatis白es(7.7)forpεPo(recall(7.6)).Adaptingtheproofin[llforthecaseX=L2(OLonecanshowthatφxislsiforLl(fl)(dueto(7.9);1hi-canbeextendedtoamuchmorpgfneralsituation、butstillrequiressomecoercivityassumptionas(7.9)).Itfollowsthenfromti1号resultsabovethatR(l+δxφx)〉XnL2(Q).IfXCL气。),θxφλ'ism芒OITIPIetely盯CIPtivpinXaMtirm山sofillsilowthat θ,1φλ'={(uju)巳l111(Q)xX;thereexistshEL1(0)Nwithhιθj(gMu)ae,1工-diviIIDI(markifhgM…f叫 IfYiHDotCOIltaiI1miinL气QLasfftrasweknow‘tiledescriptionofOλ'φλ'andafortiori、ofit-closureinX、isfarfromresoivpd.Toexhibittheproblems、considertheparticularcasel(f)二(l/p)|己|飞X=Lq(口)withl<p<∞‘1三q<2.[↑singclassicalvariatioI1aianftipisiiIKlSoI)olevcII11rddings、ifeittmrp>八γorp=N,q>lorp<八二l/q+1/p三1+l/八二iiwIIδ入'φxisTTI-completelyaccretiveinX、single-valuHlaIIEide自nedby θ入'φx(u)=-Apli=-div(|gradtip--2gradu) ()11 DMx?y)={uElvJ♂(mnAY;二kpti巳X}.It-p=入＼q=iorp<凡l/q+l/p>l+1/JV,itfollowsfromtheresultsof[lllandFjtlIMtlrclosureAofθ飞'φλ'inX(whichism-completelyaccrdiveiBX)isstillsmgie-valuedaMdennedbyAu=-ApthbutD(A)isnolongercontai时inWJP(fl);iIiparticular4手tf}xφ,Y·Indeed,ifD(4)c叫,p(fl),thenAtt=Apuεw一15(0)withs=p/(p-l)fortiξD(A)andsoLq(Q)=R(I+A)CLVJ,p(fl)+lV45(Q)whichisi订m叫In叩1可IP》O归blkeb》y,Ouia1-川川附ES引-门1门川l门川I口mIn1ψiFp:〉刘川》叫川tUiOIn1sOIn1P‘q.Actuall}＼thesameargumentshowsthatifl/q主p-1+1/N‘thenD(1)(annotbemIMinedin111J(fl);onemustthenmakeprecisethemaMEofApitiIltiliR叫se(see[8l). REFERENCES 1.H.AttouchandA.DamlamiaIhApplicationsdesmdthodesdeconvexi时etmonotoIIieArAudedecertainesequationsquasi!indaires,Proc.RoyalSoc-Edinburgh、79:107129(1977).2.P.BAnilaI1、EquationdvvolutiondansunespacedeBanachquelCOIlqueetapplica-1ions,Til-sed'Etat、Orsay、1972.
ComplefetyAccretiveOperafors 75 3.P.BGniiaIhHBr位isandhi.G.Crandan,seTRilinearellipticequationinLI(RN).Ann.Sc.Norm.S叩.Pisa,33:523-555(1975).4.P.Benilanandk1.G.Crandan,RegularizingefTechofhomogrIIetXISevolutionequaFtiOIhContributionstoAnalysisandGeometry,D.N.ClarketaL、eds.,JohnsI1opUnsUniversityPress‘Baltimore(1981)、23-39.5.P.BAnilanandkf.G.Crandan,OntheoperatorALPinL1:abstractresultsandconcreteexamples,inpreparation.6.P.BdnilaI1,KIf.G.CrandanandA.Pazy,Evolutiongovernedbyaceretiveoper·ators,InpreparatIOn-7.P.BAnilanandC.Picard,QuelquesaspectsnoniindftireduPrincipedumaximum-SAminuredeTheoriedupotentielParism4:Lect.NotesMath.713、Springer-NPWYork(1979),1·37.8.P.BGIIiianetal.,QuasilinearellipticequationsinL1,inpreparation-9.C.BennettandR.Shafpley‘Interpolatlonofoperators,AcademicPress‘Ne叭'York(1988).10.A.BeurlingandJ.fJPIly?Dirichletspaces吧Prοc.Nat,Acari-Sc.、45:208-2l5(1959)-11.LBoccardoandT.Gallouet、OnsomenonlinearellipticandparabolicequatiOIlsinvolvingmeasuredata,J.Func-An.,87:149-169(1989).12.H.Brdzisand叭7.Strauss,SemilineuellipticequationiIILI、J.JLfath.Soc-Japan、25:565-590(1973).13.M.G.CrandanandTLiggett,GenerationofsemigroupsofnonlineartransformatiOIlsingeneralBanact15paces、AI11.J-Afath.、93:265-298(1971).14.JVIG.CrandanandLTartar,SomerelationsbetweennonexpansiveaIIdorderpre-servingmappings,Proc.Amer.Math.Soc.,74:385-390(1980).15.C.H.L是,Ddrival3ilitdd'11IIsemigroupsengendrdparunop三rateurmaccrdtifdaIML1etaccrdtifdansLdc、C.R.Ac.Sc.Paris,283:469472(1976).
UnboundedOne-ParameterSemigroups,FE-6chetSpaces,andApplications REINHARDBURGERInstituteforMathemetics,UniversityofVienna,Vienna,Austria 1.INTRODUCTION TheCauchyproblem θv(z,t)=ψ(zj)(m(z)一仇(t))+f川ψ(归2叽'0创)=ψ0以(Z叫)?Z巳R (1.la) subjecttotherestrictions 川)主0,j川dz=17Md×[叫(11b) arisesinpopulationgenetics.Itdescribestheevolutionoftypedensitiesv(zj)inalargepopulationsubjecttomutationandselection.Therealvaluedfunctionm(z)isameasureofatmssforindividualsoftypeZand仇。)=fm(z)ψ(zj)dzdenotesthemeanatmss.ThefractionofindividmlsoftypeZoriginatingthroughmutationfromindividualsoftypeUduringthetimeintervaldtisdenotedbyu仰,u)dt.Itisassumedthatμ1(z)=ju(z,z)dzisinL∞(R). Variousspecialcasesaadaspectsofthisintegro-diferentialequationoritsdiscretetimeanaloguehavebeenstudiedintheliterature(cf.Barger,1988, IthankW.ArendtfordrawingmyattentiontoC-semigroupsandRdeLaubenfelsforsendingmeunpublishedmaterial.FinancialsupportbytheAustrian"FondszurF&rde-rungderwissensehaftlichenFOI-schurlg",ProjectP6866,andbythe"OsterreichischeFor-schungsgemeinschaftmisgratefullyacknowledged. 77
78 Burger 1989;I〈arliIL1988;Kimura,1965;TUI-elli71984,aILdreferencestherein).AmathematicallyrigoroustreatmentoftwoquitediferentspecialsituatiOIlshasbeenprovidedbythepresentauthor(loc.cit.)involvingthetheoryofone-parametersemigroups-ThedynamicsoftheaboveequatioIIdependscriticailyonthechoiceofm.Ifmisboundedaboveandifmdecreasesto一∞ash|→∞thmithasbeenshowninBCEger(1988)thatuderadditionaltechnicalassumptionsuniquesolutionsexistforailinitialvaluesf。εL1.Moreover,thereexistsauniqueiydeterminedstationarysolutioIlwhichisgloballystable.Themainstepintheproofwasaperturbationtheeoremforgeneratorsofpositivesemigroups-AtypicalrepresentativeofthistypeofatIlessfuIIctionsism(z)=-822,whereSdenotestheselectioIIco-emdent.Biologicall弘suchfuIIdiomdescribestabilizingselectionwithORetypehthepopulationbeingsuperiortoallothers.AnotherimportaIItformofselectioIIisdirectiomlselection,wheretypeswithincreasing(orequiva-lentlydecreasing)z-valuesareselectivelyfavoured.TheprototypeofatIlessfmctionforthiskindofselectionism(z)=sz.ThisspecialcasewastreatedinBarger(1989)usingsemigro叩sinFrdchetspaces-There,existeMeanduniquenessforintialvaluesinacertainFrdchetspacewhichiscontinuouslyembeddedintoL1wasproved.InthiscasenostationarysolutioncaI1ex-ist,iIIsteadsolutionsmovetotilerightandthedynamicsmayberelativelycomplicatedasshowIlbysomepreliminaryI111日1时icalsimulatiOIls(unpubl.).InthepresentpaperpartoftheresultsofBbrger(1989)aregeneralizedaMimproved.Inordertoinvestigatethemnlimarequation(1.1)itisIlecessaryandsumcKIlttoconsiderthelinearCauchyproblem (W)fθt=m(z)内,t)+f巾,uM(W)句(zj)εR×[03)v(z,0)=ψo(叫,ZεR (12a) subjecttotherestriction ψ(zj)三0,(zj)εm×[0,∞), (12b) (seeSection4).InSection2atheoryofmultiplicationsemigroupsofopera-torsonFrdelletspacesforunboundedfunctionsmisdeveloped.InSection3
One-ParameterSemigroupsandFreehetSpaces 79 theperturbationtheoremofBC1rger(1989)isgeneralized.SectiOI14explorestherelationbetweenthepresentapproachandtheC-semigroupapproachtotreatuaboundedsemigroups-ItturnsoutthatthemethodspresentedhereleadtomoregeneralresultsinconnectionwiththeCauchyproblemandhavetheadditionaladvantagethatasatisfactoryperturbationtheoryisavailable.Finally,theresultsareappliedtothenonlinearequation(1.1). 2.MULTIPLICATIONSEMIGROUPS Inthissectiontwodiferentsettingsaredevelopedtotreatsemigroupsthataregeneratedbymultiplicationoperatorswhosespectrumisnotboundedabove.FirstsomeIlotationsandpreliminariesarestated.Throughout,MdeIIotesalocallycompactspacecarryingaσ-ftnitemea-sureUand侣,||||B)isaBmachspaceofmeasurablefunctionsonMwiththefollowingproperties:(i)βiscontinuouslyimbeddedintoLjAM,ν)(ii)BisaBmachmoduleoverL∞(M,ν)withrespecttopointwisemultiplication(i.e.fεB,hεL∞(M)=〉llfεBand||fzf||B三lM||∞|lf||B). SuchspacesarecalledBamchfunctionspaces(Zaanen,1967)orsolidBFEspaces(Feichtinger,1979).Banachfunctionspaceshave,amongothers,thefollowingniceproperties:(i)|||f|||B=Oifandonlyiff=0ν-a.e.,(ii)iffεBthmfisaI山eν-a.e.,(iii)iffnεBandfn主OonBforalln主1andfn↑fa-e.then,eitherfεBand||fJB↑||f||B<∞,ori|fn||8↑||fh=∞ThemostimportantexamplesareLP-spaces.Itisthepurposeofthisnotetoinvestigatemultiplicationoperators(andtheassociatedsemigroups)oftheform Af(z)=m(z)f(z)a.e.OIlM (2.1) forappropriatefε8.ThefunctionmissupposedtosatisfythefollowingCOIlditiOIls (ml)misreal-valuedandmεLj认M,ν)
80 Burger (m2)esssupzufm(z)=+∞.Condition(m2)isrequiredbecauseotherwisethesemigro叩gmeratedbytheoperatorAisstronglycontinuousandthepresenttheoryisnotnec-essary.From(m2)itisclearthatingeneralAdoesBotgenerateastronglyCOIltiII11011ssemigroupofboundedoperatorssince,forexample,inanLP.spacethespectrumofAisnotbomdedabovebecauseitequalsessra吨e(m)(cf.Nagel,1986).MotivatedbytheexampleiIItheIntroductionIamin-terestedtoando川iIIwhichsenseAgeneratesasemigro叩{T(t)}which,obviously,shouldbeoftheformT(t)f(z)=etm(z)f(z).Thisddmsasemi-groupofunboundedoperatorsoneveryLP-space?p;21.ItturnsoutthatcertainFrdelletspacesaretheappropriateframeworktodealwiththisprob-leI11,siIIceonalocallyconvexspaceanoperatOImaybethegeneratorofamullgyCOIltiI111011ssemigroupaadyethavenoresolvents- 2.1.TIIEDIFFERENTIABLECASE.InthissectionaFrdchetspaceisiIItroe巾cedwhichadmitsthatthemultiplicationoperatorAof(2.1)ge时ratesadiferentiabiesemigro叩inthesenseofHegmr(1981)(seebelow).Totl山E1imdeE口e PAεα、、,,,,VZIRAJ'a,、、、plvorLUεFJBPV,JmαFU--、h,,JF,,,,dj''1飞aPA (2.2) Ihttlf企扣创O叫)ihlhOW盯St山ha川tβ瓦α:芦={υfεL叮jMJ(M)伫:e俨俨aMvn叮FisaBaI且laC}hlf1u1umI日1CtUiOIn1SPaCe巳.NowdetaiI日1e E={fεLjoc(M):fεAforallαεEZ} (2.3) withtopologydd肘dbythesystemofS优阳e创ImIisCOI口ltUiI口11u1O1u1Slw〉yrembeddedi沁ntωOBanditisaFF、r眨dC}h1edtSPaCebecauseitistileIIP!〉〉1r.OjeCttliiVelUiInmInlditoftheBaI口lage臼mIn1芒盯rate叫db问Jy?acountableIn11u1ImIn1be盯rofS优e臼臼I日mIn1inorms,since αε[b,cl斗pdf)三||fαm-fcM-18十||eam+fcM+||8三pb(f)+pc(fL(2.4) HereaMthro鸣IloutM+={zεM:m(z)主0},M一={ZEM:川(z)<0)andm+andnl一denotethepositiveandmgativepartof771,respectively-
One-ParameterSemigroupsandFreehetSpaces 8I ThecharacteristicfunctionofthesetSisdenotedbyCS-ItfollowsthatAisacontinuousoperatoronEwithD(A)=ε,because pa(Af)=||eammf||B三max{pa+1(fLpa-1(f)}(2.5) (usingm(z)三max{emh),fm(z)}).TheaboveconstructionisadirectgeneralizationofthatinBf吨er(1989)-However,thetopologyonEisalsogeneratedbythefollowingfamilyofseminormswhichmakesmostcalculationseasier: Q={qα}α>0, BFTFVmapu--、、,,/rJJ''E飞αQA (2.6) Thisfollowsfrom max{Pα(fLP-a(f)}三q|α|(f)三max{PH|(f),P一|a|(f)}. Thenweobtaintheestimates 。三α三b=学qα(f)三qb(f),fεε (2.7) and 。。今"J,,‘、、、‘,,/rJJ'a‘飞+any〈一BFφ,dm+ae<一BrrJmmae〈-FJA/''飞飞aoz Inthesequelwewillthereforeonlyconsiderthepair(5,Q).BeforestatiIlgtheErstresultwerecallthatamapT:R+→L(E)(thecontinuousoperatorsOIlE)iscalledadtFETeMMblesemigTOTLponE(intlrsenseofHegner,1981)ifithasthefollowingproperties(D1)T(0)=1E,theidentitymaponE,(D2)T(s+t)=T(s)T(t),Vs,tεRh(D3)Tispointwisediferentiable,i.e.,limt→0+[(T(t)f-f)/tlexistsforeachfεE,(D4){T(t)}islocallyequicontinω1风i.e-7ε>Oexistssuchthatforallα主Othereissomeb三Osatisfyingqa(T(t)f)三qb(f)foralltε[OJl.
82 Burger ThentheoperatorgTdeanedbygT(f)=limt→0+(T(t)f-f)/tiscalledtilein缸litesimalgeneratorofthesemigro叩{T(t)}.ItsatisaesgTεL(ε)-BasicpropertiesofsuchsemigroupsandcorrespondingCauchyproblemshavebeeninvestigatedbyHegmr(1981).ThereadermayalsonoticethatintheBELILachspacecasethetermdiferentiablesemigroupisusedinadiferentway(cf.Goldstein,1985orNapl,1986). PROPOSITION2.l.ThemultiplicationoperatorAgeneratesadiferentiablesemigro叩onEgivenbyT(t)f(z)=etm(z)f(z)ν-a.e. PROOF:VLTeomittheproofof(D1)aM(D2).Toshow(D3)observethat B＼1111/rId-FJ-tm-e-FJm/III-＼mapu--＼飞lII/川十rT1νA/II--＼QA nuω-FJ/'st飞+aQA,?ι一一BIIdm+aε,?ι<一 SIIICe ()-dmh)-1|1|(呻)t)2(巾)t)3||mh)||mztt|2!+丁一+|<te sureAisω川n11omAfεεandthereforeliHIt-→0+巳产iEEThisimpliesthatAisthegeneratorof{T(t)}andTispointwisedifereMiable LocaleqUcontimity(D4)isaconsequenceof Qdq,,】,,..、飞C」<-s?ι〈-nu、、.,/rJ/at、、,LV+α04<一、、,,/FJJ''飞飞+aQA〈一BIIumplvnae--、、‘,,,r'J、‘.,,/4ι/,..飞T,,,‘飞、aGA using(2.7). 2.2.TIIESTRONGLYCONTINUOUSCASE.IIIcertainapplications,forexam-PIewIlenapplyingtheperturbationtheoremdiscussedinthenextsectioILtilediferentiablecasecanonlyberealizedunderrestrictiveassumptionsOIl771.Moreover,whendealingwithinitialvalueproblemsitisofteIIdesirabletohaveexisteIIceanduniquenessofsolutionsinaspacewhichisaslargeaspossible.Theseconsiderationsleadtothesubsequentdevelopments.
One-ParameterSemigroupsandFrkhetSpaces 83 Thefamilyofseminormsdeanedby(22)or(2.6)isnowreplacedby r={γα}α主0, γα(f)=||eam+f||B· (210) Itfollowsthat 。三α三b=争γa(f)三γb(f). (211) Then F={fεLj认M):γa(f)<∞forallα主0}(2.12) isaFrdchetspacewithtopologydeanedbyr.However,theoperatorAdennedby(2.1)isnolongercontinuousonF.ConcerningthefuIMUonmwerequireadditionallythat(II13)m-isboundedoncompactsets- LEMMA.ThemultiplicationoperatorAisaclosedoperatoronFwithdensedomainD(A)={fεF:||m-f||B〈∞}. PROOF:ItisobviousthatD(A)isdenseinF.NextobservethatforfεD(A)wehave γa(Af)=||eam+mf||B三||eam+m+f||8+||εam+m-f||B三γα+1(f)+||m-flls<∞. Finally,Aisclosed.SiIIceFisaFrkhetspace,andthereforequashormed,itissumcieMtoshowthat{fn}gD(A),limfn=fandlimAfn=gimpliesfεD(A)andAf=g(Yosida,1978,Ch.IL6).Chooseasubsequencefnkthatconvergesa.e.tof.Itfollowsthat771fnk→mfa.e.andhencemf=ga.e.Therefore,Af=ga.e.,mfεFandhencefεD(A).HereithasbeenusedthatFiscontinuouslyimbeddedintotheBamchfunctionspaceB. PROPOSITION2.2.TheoperatorAgeneratesastronglycontinuousmulti-plicationsemigro叩onF,givenbyT(t)f=etmf. PROOF:Firstweshowthat ADεrJrJ-f-tm一pu-m叫一-FJA
One-ParameterSemigroupsandFreehetSpaces 85 andthearstexpressionontherighthandsideof(216)is三tγα+1(fhwhichisshownasin(213).ToestimatethesecondtermweproceedasaboveandchooseacompactsetKEMsothat||m-fCM-w||B<ε/3aMtsothat||(ftm--1)CM-nk||∞〈ε/(3||f||B)andt||m-f||B<ε/3.Thenthesecondtermin(216)canbeestimatedas =||(ftm-一1)|f|CM-||B三||(ftm-一1)|f|CM-wlB+ε/3三t||771-fCM-w||B+ε/3三2ε/3fora11fεD(A).Togetherwiththeestimateforthearsttermoftherighthandsideof(216)thisyieldsstrongcontinuityof{etm}t20· 3.PERTURBATIONSOFMULTIPLICATIONSEMIGROUPS IBBCIrger(1989)ageneralperturbationtheoremfordiferentiablesemigroupsoaFrdchetspaceswasproved.Infactthesameproof,uptoobviouschaIIges,showsthatitalsoholdsforstronglycontinuoussemigroupsandthatitcanbeslightlygeneralized.Forthesakeofcompletenesswestateitbelow.LetDdenoteaFrdchetspace(ormoregenerallyaseqmItiallycompletelocallyconvexspace).AdenotesaclosedoperatoronDthatgeneratesastronglycontinuoussemigro叩{TO)}andBisacontinuouslinearoperatoronD.Inparticular,itisassumedthatacalibrationHindexedbyrealnumbersexists,i.e.H={u:αεI},witheitherI=RorI={zεR:Z主0}suchthatforeveryinterval{bAEIandαε[b,clandforallfεEthereisaconstantKbCWith sup凡(f)三kb,c(同(f)+πc(f))aε[b,c] (31a) andsuchthat πa(T(s)f)三πα+s(f), fεE, S〉0, (3.lb) holds.Furthermore,itisassumedthatalocallyboundedfunctionF:I→[0,∞)existssuchthatforeachαεmandb三0 F(α?b):=supF(α+s)<∞sε{0,bl (3.2a)
86 BUrger and 作a(Bf)三周(α)πa(f), fεD,αεR (3.2b) holds. IfAisasinSection2andHisoneofthefamiliesofseminormsdeanedinSection2thenconditions(3.1a)and(3.lb)aresatiSEed-Condition(32b)justmeansthattheoperatorBis7TaBContinuousforeach7raξH.ThenthefollowingtheoremcanbeprovedbyimitatingtheproofofTheorem2inBarger(1989). THEOREM3.1.SupposethatD,日,AandBMeasabove.ThenA+Bistheidnitesimalgenerator。fastronglycontimoussemigro叩{V(t)}onDpwhichisgivenby CXD V(t)=汇凡(t), t;三0, (3.3a) where %(t)=T(t)ωdh仲fT(t-s)叫 fεD,t主0.(3.3b) IfAisdigerentiablethenA+Bisdig-rentiable- Theseriesdeanedbyeqs.(3.3)issometimescalledthePhillipsperturba-tioaseries(cf.Goldstein,1985).InviewoftheapplicationsmentionedintheIntroduction,Iamiaterestedinperturbationsofmultiplicationsemigroupsbyintegraloperators.Therefore,deane Bf(z)=f川)f(ν)句, fεF, (3.4) whereFisasintheprevioussectionandfMz)dzalwaysmeansfMMz)d叭z).Toapplytheabovetheoremitisnecessarytohavetheesti-mate(32b).Weconsidertwocases.(i)Misalocailycompactgroupandu(ZJ)=u(z-u)(byabuseofaotatiOB),i.e.,Bisaconvolutionoperator.(ii)u(z,ν)=u(z)(againbyabuseofmtation).
One-ParameterSemigroupsandFreehetSpaces 87 Bothcaseshavebeenconsideredinpopulationsgenetics,althoughinaslightlydifereMcontext(cf.BfIrger,1988,1989;kadin,1988;Kimura1965;Tuelli,1984)-Cωe(i)Assumethatmsatisaes(m4)em+isS1山n11ltiplicative,i.e.thereexistsaconstantC>Osuchthatem+(z+ν)三Cem+(z)em+(U)locallya.e.Thisimpliesthateam+(z+ν)三CaeamAzham+(ν)holdsforallα>Olocallya.e.Infact,itissumdenttorequirethateam+isweaklysubmultiplicative,i.e.thereexistconstantsC17CLC>OandafunctionwonM,0<ω(z)<∞forallzεM,suchthatC1ω(z)三eam+(z)三C2ω(z)locallya.e.andω(z+ν)三Cω(z)ω(ν)forallz,νεM.ThefunctionwcmbechoseIltobecoatimom(cf.Feichti吨曰,1979).Thereadermayalsonotethatsubadditivityofm+impliessubmultiplicativityofem+butnotconversely. SupposethatAisaBanachfunctionspaceandadditioaallyaBanachalgebrawithrespecttoconvolutionandthatBisaBanachfuIlctioaspaceandaleftBaaachmoduleoverAwithrespecttoconvolution.ThereforegεA,fεBimpliesg*fεBand||g*f||8三||glA||f||8·DenotebyrA={γf}α主0andrB={γfh>ofamiliesofseminormsdeE时dasiB(210)andbyFAandFBthecorrespondingFrdelletspacesasdeanedia(212).Ifωisweaklysubmultiplicativethenaneasycalculationshowsthat |f*g|ω三const(|f|ω*|g|ω) holds(providedtheconvolutionmakesseme).ItfollowsthatforfεFBandUεFA γf(Bf)=Ileam+u*f||8三ca||eam+u||A||eam+fllB=caγf(u)γf(f) isvalid.Therefore(32b)issatisaedwithF(α)=Caγf(u)and(32a)holdsdueto(211)sinceUεFA. Remα7·k.Itshouldbenoticedthatitismuchlessrestrictivetoassumethatem+isweaklysubmultiplicativethantoassumethatemis.Therefore,
88 BUrger thestronglycontinuouscaseispreferableoverthedifferentiablecaseformanyapplications. case(ii)LetBbeaBamchfunctionspacecontinuouslyembeddediMOL1(M)andletU巳F.Thenoneobtains,using||f||1三||fl!B=γo(f)三γa(f), γα(Bf)=||eam+Bf||8三||eαm+u||8||f||1三γa(u)||f||B三F(α)γα(f) withp(α)=γa(u).Itfollowsthatconditions(32)hold. 4.RELATIONTOCPSEMIGROUPSANDTHECAUCHYPROBLEM C-semigroupswereintroducedindependentlybyDaPrato(1966)andDaviesandPang(1987)totreatCauchyproblemswheresemigroupsofunboundedoperatorsoccur.ThedeanitionisasfollowsDeF7zition-LetCbeaninjectiveboundedoperator(onaBamchspaceBLAfamilyofbouIIdedoperators{S(t)}t>oiscalledaC-semigmupifithasthefollowingproperties: (Cl)S(0)=C.(C2)SO)S(s)=CS(t+吟,fort,s主0.(C3)S(t)isstronglycontinuous,i.e.,forallfEBthemapt→SO)ffrom[0,∞)intoBiscontinuous. If,furthermore,thereexistω三OandM〉Osuchthat||S(t)||三krtthen{S(t)}iscalledexpomatiallybounded.C-semigro叩shavebeenstudiedinsomedetailinaseriesofpapersbydeLaubmfeis(1990ahc).TypicalexamplesofC,semigroups,whicharenotexponentialiybounded,arisethroughmultiplicationsemigro叩Sasfoilows(comparealsoDaviesandPang,1987;deLaubmfds,1990a,b).LetBbeaBanachfunctionspaceasinSection2andlet{εtm}beamultiplicationsemigro叩withmsatisfying(ml),(m2)and(m3).ChooseCf(z)=e-amh)2f(z)forsomeα>0.TheII{S(t)}deanedbySO)f(z)=Cetm
One-ParameterSemigroupsandFMCIVetSpaces 89 α>Owhichisnotexpomntiallybounded(duetocondition(m2)).Infact{S(t)}isanentiree-ad-gro叩(cf.deLa山eIhls,1990bLbutweshallnotpursuethelatterconcepthere.GivenanoperatorAwithdomainD(A)inaBaaachspaceB,considerthefollowingabstractCauchyproblem: f'。)=Af(吟,t三0,f(0)=hεD(A).(4.1) deLaubedels(1990a)provedthefollowing THEOREM41.SupposeAisthegeneratorofaC-semigroup-Then(41)hasauniquesolutionforallfεC(D(A))givenbyf(t)=S(t)C-1fo·ThesolutionsdependcontinuouslyontheinitialvaluefointhefbMowingsense.If||c-1(ft-h)||convergestozeromntendstoidnit只thenthesolutionsfn(t)of(4.Owithfn(0)=ftεC(D(A))convergetofO)uniformlyon compactsets- Ifthisresultisappliedtomultiplicationsemigroups,thatis,withAdeanedasin(2.1),OIleobtainsexistenceanduniquenessforinitialvaluessatisfyiag meam2foεB (4.2) forarbitrarybutftxedα〉0.Itisobviousthatthisresultisnotoptimalbecausethesolutionsinthisspecialcasearegivenbyf(t)=emtfoandtherefore(42)isasufEdentconditionensuringexistenceanduniquenessbutnotanecessaryone. UsingsemigroupsinFdchetspacesandaresultofkomura(1968)thefollowingcanbeproved. THEOREM42.LetDbeaFrdclmtspaceandletAbethemarlitesimalgen-erator。fastronglycoMirimmsemigro叩{T(t)}ThentheCaMlzyproblem(4.OhasuniquesolutionsfbrallhεD(A).Moreover,thesolutionsdependcontinuouslyontheinitialdatarelativetothetopologyonD. PROOF:ExistencefollowsimmediatelyfromProposition12ofkOII111ra(1968),whichsaysthatfεD(A)impliesT(t)fεD(A)foranyt主0,
90 Burger thatT(t)fiscontinuouslydifereMiableintrelativetothetopologyofDandthat￡T(t)f=AT(t)f=TO)Afforallt主0.Toproveuniquenesssupposethatf(t)εD(A)isanarbitrarysolutionoff,(0例t吟)=Af贝(0例t吟)Sa创tihSd吨f々3yrihI丑1f只(0创)=hεD(A)L.Then二T(t一吟f(s)=一川-M(s)+T(t-s)f(s)=OThisimpliesT(t)f(0)=T(0)f(t)=f(t).Finally,supposeft,hεD(A)andft→foinD.SinceDisaFrdchetspacestrongcontinuityofthesemigro叩{T(t)}impliesthatitislocallyequicontinωus(seeProposition1.lofkomm,1968).Therefore,givenarbitrarybutaxedε>0,foranyseminormπεH(whereHdehesthetopologyofD)thereexistsaseminorIII扩εHsuchthat叫T(t)ft-TO)fo)三扩(ft-fo)holdsforalltε[0,ε]-Thisimpliescontinuousdependenceofsolutionsontheinitialdata. Theorem(42)showsthatformultiplicationsemigroupsexisteIIceaaduniquenessofsolutionsof(41)isassuredforallinitialvalueshεεinthedifferentiablecaseofSection21andforallhεD(A)EFinthestronglycontinuouscaseofSection22.Clearly,thesetoffunctionssatisfying(42)isapropersubsetofbothEandD(A)gF.AfurtheradvantageofthepresentapproachusingFrdchetspacescom-paredtoC-semigroupsisthataperturbationtheoryofC-semigroupshasonlybeendevelopedunderrestrictiveassumptions(seedeLa由enfels,1990吟,sothatananalogueofTheorem3.lisnotobtainableatthemoment.AsacorollarytoTheorem3.laIIdTheorem42weobtain COROLLARY4.3.AssumethatFisdeanedby(212)withBgL1(EZ),thatmsatiSEes(mOto(m4)andthatUisgivenbyomofthecasesofbctioI13withUEF.ThentheCauchyproblems(1.Oand(1.2)admituniquesolutionsforallinitialvalueshεD(A)EF.Moreover,thesolutionsdependcontinuouslyontheinitialdata. PROOF:Theassertionfor(12)isanimmediateconsequenceofTheorems3.1and42.TheassertionforEq.(1.1)canbevcrinedasfollows(comparealsoBarger,1989).
One-ParameterSemigroupsandFreehetSpaces 9l Ifv(-J)isasolutionof(1.1)satisfying|仇。)|<∞(whichisthecaseforvεF)then W):=e叫tM+μ(叫价川 isasolutionof(1.2).Conversely,ifhisasolutionof(12),theIIVO)=h(t)/jh(zj)dzisasolutionof(11)aslongas仇。)isbite- REFERENCES1.R.Biirger,Mdh.Z.197:25$272(1988).2.R.Biirger,Preprint,UIliversitatWien(1989).3.G.DaPrato,RiceTcheMαt.15:225248(1966).4.EB.DaviesandM.M-H.Pang,Proc.LondonMdh.Soc-55:181-208(1987)-5.H.G.Feichunger,SitztLngsbeT.OsteTTetch-AKαd.阴气83.188:451-471(1979)-6.J.A.Goldstei日,SemigroupsofLinearOperatorsandApplications.Ox-ford:Univ.Press.1985.7.S.JHegmr.SIAMJ.Mαth.Aml12:243-273(1981).8.S.kadin,Non-gaussianphenotypicmodelsofquantitativetraits-In:22dCongressonQuantitativeGenetics.(edt.byBWeiretal.).1988.9.M.kimurAPToe-NMl.Acαd.Sci.USA54:731-736(1965).10.TKomura,J.FtLMt.Amlym2:25$296(1968).11.R.deLaubedels,J.Fmet.Amlysis-Toappear(1990a)-12.R.deLaubenfels,SemigTOTLPFOTum-Toappear(1990b).13.R.deLaubenfels,Preprint,UniversityofAthens,Ohio(19900.14.R.Nagel(ed.).One-ParameterSemigro叩sofPositiveOperators.Lect.NotesMath.1184.Berlin-Heidelberg-NewYork:Springer.1986.l臼5.M.Tu盯Ir时.16.K.YOSdida趴?FunctionalAnalysis.5tt1.ed.Berlin-Heidelberg-NewYork:Springer.1978.17.A.C.ZaaneIUIntegration.Amsterdam:NorthHolland.1967.
DimensionsofContinuousandDiscretesemigroupsontheLP-Spaces THIERRYCOULHONDepartmentofAnalysis,UniversityofParisVI,Paris,France hthisrepo此,Iahdlconsideracontinuous(resp.disczete)semigroup(1:)130(resp.(TK)KEm-Lac也IgontheLFspaces,mdIshdldiscusstherelationshipbetweenestimatesofthetype||Ti||l『∞三Cf鸭j气Vt>0(resp.|iTKl|l『∞三CK→/气vkEN-LandLP-Lqinequalitiesinvolvingtheidmite-imalgenerator-A(reap.theoperatorI-T),whichmaybeseenudiEezentiatedvezsiOIlsofthefoEXIlezestimates. Indeed,Vampouloaprovedizt[91仙",证(Tth〉oiaasymmetricmbmarkovianmnigmzpactingonU(X,t),where(XJ)isameasurespace-and证一AistheixdRitesixnalgeneratorof(TtLphthen; !|R||i4∞三〈71-现/气vt〉0φllf||L/(川)全C(Af,f),VfεD(A) (事) (串串). Onesaysthat{Ttb〉oisofdimensionn江oneofthesetwoconditionsissatided.V町opoulosusedthistheorem,andthefactthat(叫iamudmiertoh扭曲thanh)3toobtain,amongotherthing,,sharpMdgenerale,timate,ontheheatkernelMdontheωavolutionpowmofmeaaumona1血modulazLieFoupG(间,Hon.IShantrytopresenthereaverysimpleandmoregeneralapproachtothefunctionalanalytictool,whichleadtotheseestimates.Thismaterialispartlyajoint,workwithLaurentsalog-Coste.ForsomerelatedresultsseeDavies(6iadCa巾t品moka-Stroock(1;. 93
94 Coulhon OUEWoof-w诅relyon·LANEXTRAPOLATIONRESULT.hthe岛Uowing,(XJ)毗beaσ-ERitemeωuzespace- THEOREM.Let(TibcbeasemigroupwhichiseqtdcontinuousonLL(XI)andL叫X,。-Ifthereexistα〉Oand1芒P〈q5+∞suchthat: ;:z:iF『q三cra,Vt〉0, then ;|TJl→:〈l三Ct-8,Vt〉OJJUhJ=α/(1/P-1/qi TH.re-Uthasitsorigininatheoremofvazopoulos(问,p.242),whichhadasomewhat明nplicatedproof,因ktgideasgoingbacktoHa均-Litti-wood例,viaFderma川tdn(川、p.172)-Ishallgivehereasimpleproof,duetoYvesR叮naud,tftdmyself,whichappeared坦问. LetfEU(XI)门Le(Xl)/{0}andebemchthatj=于+9ForT〉O‘let Cf-T=sup{俨fe;|贝flL/!|fh}.叫{0,Tiawehmvt〉川引f!|,三cre||R/2filq三et--i|引/2flluluf||;-8,byHaderSince(11)仆。is叫uicommIOU-onLl(XI)、onehas !lmlq三C尸||贝ρfll76llfll!, andtorH[0,2Tjp iiTJ||q主Ct叫叫j|1f|li HenceC-I-T三cc;1'"ldsince840,1[,Cj-TisboundedindependentlyoffandT.ThereforelTilll叶q三et--16,Vt〉0.Tbdualsen-groupT;-thusBaud"!!T;·!1、→J;三Cfhd,andonemayapplytheSMIleu伊nBentUbeforetogetiruH伺二Ct-M881.vt〉0.forasuableOl,hellue:lth→∞二Ct-ot6h,Vt〉0.Onectledseasilythatthisistheannounced凹tiznat吧
zh-mensionsof&migroupsoniFESmces 95 AnumlogomresultcanbeprovedforthepowersofmoperatorTwhichizpower-boundedonL1(XI)andL∞(XJ),andsendsLi(XI)削。L∞(XJ).see[4). II.TWOWAYSTOESTIMATE||TJl→∞Isttalinowgvetwompiewaystoobtain(*)告。m(叫OEin叩autiesofthesumtype,undezweakaasumption-on(11)t〉0·Theycomzpondtotwopossibler-formulations。f(叫:ontheomhand,(Af,f)=ilAljvU,md(叫叩山atflniabomdedhmL2(XA)toLh/(川)(XJ)On山othertlmd,{AfJ)=-拮hdTif||!.ThusanaturalLPanalog帆forpd1,+∞{,oftbL2inequaliq(")hgiveneitherby:A-af2isboundedhomLF(Xi)toU(xi),wherej=j-t,orby:|lf||:三CR叫Af,儿),vfEP(A),whenfp='"(f)ifp-1,mc噎Re(AfA)=一站|叫|Zf||; ThehtreformulationwasexploitedM[2j(malso[3j):TE王BOREM1.Let(Ti)仆obeeqdcoMhuOMonLI(xi)mdL∞(XJ),andboundedanalyticωLP(XJ),品raaxedpe]1,+∞[.Th制,ifA-anjsboundedboznLF(X,t)toU(X,们,町thp〈q〈十CUAaIIdα〉0,onehas: i|贝ii14ω三Cf现/2,vt>0. wherenhgivenby1/q=1/p-α/n. 旦旦1:Since(TiboisboundedanalyticonU(XI),onehastheestimate: ||A叫|!"叫Vbolp『p ,,"'伊HAveSAAau唱俨n4。A"〈.-a··叶,,Rarwp、na、LHE-A4也A民4也AA--伊μ「且,、&冒·.nwwoinhMgmgdnmbo i!T;ilF叫三Cfaf2. TheextrapolationargumentofSInowendstheproof. Notic-thatwecouldaswellhavesupposed(1:)t20tobeboundedanalyticonLJ(X{)·
96 CouIhon IzhallnowexploittheEefondT-formulation· TEEOREM2.Letm}仆『]beeqUcontauouonLl(Xi)mdL∞(Xi).Suppose他at:|ifitsch(Af,儿),vfGD(A),的thl<p<qThen ilzll川豆川,bω…=击 旦旦1:LetfεD(A).Onehas:ftdft||f|!ρ||flNZf!::=儿万llzfltds=PIRe(AZfdtfh)dsTherefore,byhypothesis,l!f!!;三去点!!只fltds.Now,since(TiboU叩ticomauouaonLqxd),tllZflt三UE11TJll:"‘andonegets: |!引f||,三crh||f||p TheextrapolationargumentofSIagainendstheproof. 旦旦皿础皇;1)Aconsequenceoftheorem1uditsconveneisthat,证-AisthegeneratorofamnigroupwhichiaeqmCORMuousonLI(XFOmdL∞(X{)?md证A-曲'isboundedhomu·{X,t)toLh(Xi),withαo〈Oaztdl〈po〈qo(+∞,thenA-αUboundedhozriLF(XJ)toLf(XJ),aasoonuα〉OFP〉Lmdα/(1/p-1/q)=α。/(1/po-1/qo).Ifthe-emigroupgeneratedby-Aisnotanalytic,itmfRcestocomidHthes凹nigroupgeneratedb,-Alf2.2)Thehypothesisoftheorem2implies,byttalder,thatllfuq三CllAfllJfir＼thereforethatA-iisboundedhornL'(XFt)toLq(Xpt),wherel/r=1/q+2,/r.ThusithmtuxaltoukwhetherasemigEoupwhichiaequicoMimIOU-onL1(XI)aadLG二(X.0‘and"tid凹A-a:Lq(X,t)→Lq(X,。,hasadimension(i1whetheronecandroptheasmxzptionofanalytidyintheorem1)TheuazzpledTJ=fuf.with(X,t)disq哥怡,showsthatitisnottheuase.
DimensionsofsemigroupsonU-Spaces 97 3)ThemethodofVampoulostodeduce{")告om(叫msilygivestheconverseoftheoremLsinceitonlyrelieson让leformulaA'曲=lf仲俨-Ind.where描IdozUtknow证theconverseoftheorem2holdsornot(atanyrate,oneshouldsupposethat(ZL〉oiscontractiveonLP(XJ)).4)OI四cmlocalizetheorem1atzeromdatMnity,ii.e.Cddth1amrma缸川4Cd俯丁1tO(rtγ-叫叽/叫2竹)'t→0.mdi|Z||1『∞=0(γ饲/勺,t→+∞.mteZEIUoffe/2.Itisobviousfozt→0,aMitttasbeendoz川叫21fort→+∞.Anapplicationwasuventheretoheatkernelsonquasi-isometricriemanxIianmmifolds-Asfortheorem2,itiseasytoseethatE(71)t〉OactsonLl(Xi)mdL∞(X咱们,thenl|ftgqh(AfJF)+||f||:1.vf正刷刷,由ψHesllR||l『∞=0(rn勺,t→0,udthatif(Ti)t兰。iseqUcontinuousonLl(XA)mdL∞(XJ),aMTlisbomdedbmLi(XI)toL∞(XI),thm||f!|:三CIRe(AfJp)+R叫AfJJlpVIED(A),implies||Ti||1→∞=0(f四/2),t→+∞,withn=三? 皿.DISCRETESEMIGROUPS-Thisp缸agraphhtaken告om[41ω[51,utddbe削ctly.Togetthe扭dopeoftheorem1foradiscretemnigrouppmintroducedM问thenotionofanalyticityofmopmtor:TiamdtobeaMyticonLF(XJ)证iiTK-THliip→F三CA,VKε副·.ThoughasymmetricsubmarkoviMIoperatormaynotbeanalyticonL2(XJ),onecanperturbitsoutomakeitmalyticFmdonecmprove: TEEOREMV.LetTbeasJFREmdrirsuMarkovianoperatoronU(XJ)pwhirhisregularizing,i.e.boundedbomL1(XA)toL∞(Xi).Then||TKill4∞三ct叭VKE别＼Handonjyif(I-Tyaj2-IisboundedbomL2(XJ)toLf(Xi).where1/q=1/2-α/肌forsome(evem)α叫OA/2[. Theanalogueofthemm2ismnstraightfomazd; TEBORZMT.LetTapowuboundedoperatMonLI(xi)ωdL∞(XJ),w扣-dhregulazizing-ThenllTflt三司llflt-||Tf||:),如1〈P〈q〈+∞,irnPEesllTKli1→PF三CK-川,wherenisgivenbyl/n=1/p-1/q Acozr-2inationofthmtwoth-or-mismduiinthefoUowmgsituation:Let(PJPlJbetwobimarkoviankernelsonX,i.e.twoEEleasuablemappingsfromXλXtoR+·such
98 Coulhon th剖JPAZEZ)dtb)=JPAZJ)dtb)=1,VZGX,andmppAZJ)〈+∞,,=1,2Wezhaiidehethjbytlj=p.mdpikjhJ)=fp?-ij(z,z沽,(ZJ)々(zLandweshallsaythatpl《P2证thereexi",csuchthatpl忡,u)三CbhJ),V(矶,)EXYX Oneth臼Ehu; COROLLARY.Let(psL剖,2betwotpimarkovimkernelsonXsuchthatpiissyznZFEetrirandp1久〈P2·Then ,upp俨(SJ)二O(k斗/巧,for"〉0, izzzPISH 叫P;k)(SFV)=ο(k-"/2)· 坠四1」Letmzupp0·en〉2(see[5ifoztheothezcase).LetTlandT2betheopezatomrespectivelyassociatedtoplmdm,i.e. M(z)=fμp酌'川Byhypothesisl|Tfi|14回=问pikJ(ZJ)=0(t"/2)·ApplyingtheoremItoTf,oneseesthat(I-TfyiI2一Iisboundedfromf(XJ)toLhl(←2l(XJ).Sincempp2(ZJ)〈+∞,RisreFIlmingmdonehas: ‘‘,.,,,P、X,,f飞向2·zurt,,JWV。.,,JAd,,,,噜h·、‘,,,,q4··AT,,A,..‘飞C/二,,,ampaE飞,,,,饨,,a,,JR ...,F‘....,,..‘、 N侧,mceEissymetric-l|(I-Tf)lfV|!;二;!fIZ-:NIt-Letusc叫mtMslastquantityto!!fu:-linf!l;;thehypothesisPI《P2impii何that,forαsmallenough,P2=叩1+(1-α)阳,when"isagainabimukoviankernel.Itfollow-that: |!几fl!;三(αllTIfli2+(1-α)!|鸟flh)=三αllTIf||仨(1-α)!!只f!!;三α!!T:f!!;+(1-α)!!f!!;, since13.theoper剖orassociatedtop3·isacontractionofL2(Xi).Finaliy!!f!!;-!mf!!;三α-l(!!fll;-JZfJCLand!!T2fJL/t←2j三C(!!fy;-lJTJflj),即EefonliTJill『∞=qk→j2).aftertheoremz Notic哥thatTheorem1d。"notapplytl盯吧,sihIn孙lU.刊T=isnots盹1ulPPOs肘哥dtω0b哥s叼yImIn1zmrn1咂叫trdiC.
o-men-Sion-SofhrnigroupsonU-Spaces 99 REFERENCES [1lCARLENE,KUSUOKAS,STROOCKD,UpperboundsforsymmetricMazkovtEMEntionhmctions,Am.Inst.H.poizBEardFproba-etstat.,mPPI-auno2‘1987,pp245·287.[2]COULHONT.,Dimension&rmhid'unsemi-groupeaItalytique,APMaimauBUl.Sc.Math.,1990.[slCOIlLHONT,SALOFF-COSTEL,ThAomdeHardy-Littlewood-Sobolevpou1e··臼ni-Foupesd'。pArateursetapplicationauSToupesdeLieunixnodu-IMres,S6minaired'AnaivsedeISUEdvemit6deClermont-FerrandII,87/88,expos- nu21,1989.[4]COULHONT,SALON-COSTELJuUsance-dhopA川eudFIlm-smt,&pMaitreauAm.Inst.H.Poincar6,proba-etstat-FV01.26,n。371990.[slcOULHONT,SALOFF-COSTEL.,Marchesal臼toimnongym-trique-surlesgroupesu山nodulures-c.RA.5.Paris.t310、民rieLpp.627·630、1990.[6]DAVIESEB,Heatkernelsandspectraltheory,Caz由ridgeUE,1989.[7lFEFFERMANC-tSTEINBBHPspa忡,in胃m叫vamble,FActamath.,129.1972‘PP·137.193.[81HARDYJ-H.,ILu,」IrvmIJi.ReihneAznEgew.Math.,18771932,pp405·423[91VAROPOULOSN-ladyωlewoodtheoryfoE阳nigmps、J.F、ZEd-Anal..vol-63,n。2,1985,pp.240·260.[10lVAROPOULOSN,AnalysisonLiepups,1.阳此And--叫76.no2,1966,pp346410. 7
ANewShortProofofanOldFolkTheoreminFunctionalDifferentialEquations 0.DIEKMANNCentreforMathematicsandComputerScience,Amsterdam,TheNetherlands,andInstituteforTheoreticalBiology,Leiden,TheNetherlandsS.M.VERDUYNLUNELDepartmentofInformationSciences,FreeUniversity,Am-sterdam,TheNetherlands ABSTRACT WcusethetheoryofWeiIistein-ArOIlszajndeterminantstoprovethatthemultiplicityofzMaroutuftbdaracteriMicequationsequalsthealgebraicmultiplicityofzasanfigfnvalufoftheinhlitesimalgenerator- 1.INTRODUCTION Let〈beaIln×TL-matrixvaluednormalizedboundedvariationfunction.WithtileretardedfunctionaldimrentialHillation1Ph二(t)=jd〈(7)z(t-7)(1.1)dtjoonecanassociatethecharacteristicequation detA(z)=0, (12) IOI
l02 DiekmannandVerduynLunel where PhA(z)=zI-ldC(7)fn.(1.3)Zerosofthecharacteristicequationyieldexponentialsolutionsof(11)andhigherorderzerosyieldpolynomial-EEXponentialsolutions.Thedeanition (T(t)ψ)(7)=z(t+7;ψ)for-h三7主0, (1.4) wheretHZ(t;ψ)denotestheuniquesolutionof(1.1)correspondingtotheinitidconditionz(7)=ψ(7)for-h三7三0(1.5)yieldsaCo-seIMgmpofbo1MedliI阳roperatorsonCzC([-fhOl;cn).ItsintiI山es-imalPIle刚orisgivenby(cf.Hale[6l) 、,、,J川vf'、一一、IJOU,,,.飞·ψ月LE川Y,,4,.、一一、、..,,,AD Aψ=ψ, (1.6) where(〈J)denotesthef肌川i江fandOInllwyi江f(υ1L.22刽)IhhlK灿(O}THEOREM.ThealgebraicmultiplicityοfzasaneigenvalueofAr?qualsthemultiplicityοfzasarootdthecharackEristiccquation(1.2)ThefirstproofofthisthcomIlisdmtoLevinger[7l.AsecondproofwaspublishedbykappelandWimmer[91.TheaimoftImpresentnoteistoshowthatthetheoremactu-allyisastraightforwardcoIisequenceofthegeneraltheoryofthe矶reinstein-ArOIlszajIideterIIlimnt(kato[10l)OIICCaliIMkethisobservationassoonasonerealizestt削(1.l)canbecomderedasafinitemlkpert旧bationofthehtrivial"functionaldiffer-entialeEIllation 生(t)=0·(1.7)dtThccorrectsettingforsuchaperturbationpoiIitofviewinvolvesdualsemigroups(CidHmtetal[123l,DickIIlann[5i)andtheembeddingofCintothespacecn×L∞[-h‘oj.He盯r陀ewCt8呻§扑出ihl凶叫adlHlhtfm、X}pμ}Alh扣Mυ山叫}βitt山}hi山idualSeImIn1igrO1u1ptIhite飞OIr.y『butnottiletheoryitself. 2.PROOFOFTHETHEOREM LetD:L∞→L况betileu川x川川)讥1ulIequivalenceclassesCOIu川i川tuωa山iI川iu山川ih川In吨l咆gaILUJJiP阴8Cdthli忱tZCOIn川l刊川tUiInm1u1u1O1u1sfh1ulIn眈l比CtUiOωInl、JaIn叫lK叫【dia缸《C!允灿tUiOmInl Dψ=伊. (21) ForeveryzECwcdefineapseudo-inverse σJUσ川YZσ7T'''''''nu--7ψDZP (2.2)
TheoreminFunctionalDWbrentialEquafions I03 Notethatindeed(z-D)Ps(z-D)-1=I,butthat(Ps(z-D)一1(z-D)ψ)(7)=ψ(7)-ez(7)ψ(0)forψεP(D),(2.3) where ez(T):=ε?(2.4)Nextweddmthe"11npert川叫'operatorA2*:Cn×L∞→Cn×L∞by 、EBEJ、‘ZE,,,nu,,,‘‘、ω'一一α、、...,,D,,,.‘‘、PCLω'＼11EFI/αψ/FE--＼,EeEt--、‘..,,,曲'。OAD 、11』,/ODOUAU/FE--1、一一事。oA (2.5) Astraightforwardcalculationyields(zI-Ar)-l=(二川!D)J (2.6) WeconcludethatzEρ(Ar)ifandonlyifz#Oandthatz=Oisaneigenvalueofalgebraicmultiplicityn,witheigenvectorseoα.TheperturbationhasdomainP(B)=Cn×C(whereCdenotesthesetofequivalenceclasseswhichcontainacontinuousfunction)andaction 飞飞EEF/户'、nununυ/,ttt飞、一一B (2.7) where〈denotestilef11时 D(AG*)=D(Ar), AG*=Ar+B. (2.8) NotethatBisrelativelyboundedandhasHIlitedimensionalrange-Theidentity 、‘EE,,n39",,,‘‘、、‘..,,,电i、...,,*。OAF-z,,...‘、Brt,,...、、唱,..、‘...,,*。OAF-?巾,,··z、、一一、‘.E,,,*GAF-馆b,,,..‘、 showsthatwecananalysethespectrumofAG*bycombininginformationaboutthespectrumofArandinformationconcerning det(I-BOI-Ar)-1) ThisisexactlythekeyideaoftheWEINSTEIN-ARONSZAJNTHEOREM(kato[10lTheoremIV62). 、‘..,,"U唱EAqL,,,‘‘、、、..,,,、1EF、‘..,,,*。。AFt,IM,,,-E‘、BFt,,,..‘、合aueAU少剧,,..‘、ν+*。。AZ~ν一一*。AZ,,,‘‘、~ν whereforanyclosedoperatorT foifzEρ(T)P(z;T)={dim冗(spectralp叫ection)ifzisanisolatedpointofσ(T)1∞otherwise (211)
l04 DiekmannandVerduynLunei andforanymcromorphicfunctionffkifzisazerooforderkoffν(z;f)={-kifzisapoleoforderkoff(2.12)tootherwise- So,inparticularAfoforz手。D(z;Ar)={T(2.13)Lnforz=0.Combining(2.6)and(2.7)wehdI-BUI-Arγ14I一?,ez)-M(γD)-1))ρ and 。*-1-tA(z)一-ndet(I-B(zI-A))-de一一-zdetA(z).(215)z COROLLARY. D(z;AG*)=ν(z;detA(z))ToconcludetheproofitoIilyremainstoshowtilerelationshipbetweenA(Hand A. Letj:C→Cn×LocbctileembeddiIig ＼、-ts/'仰川YU'/ttI飞、一一ψ-qJ (2.16) SinceD(Ar)CjCthemolventmpsintojCandeig阴阳ctorsbelo吨toJC.Sowithoutlossofgenerality,weIIlayrestrictourspectralanalysistojC.NowAOVψεjCifandonlyifvεcland白(0)=忆,叫.Moreover,inthatcasej-lAG--jψ=φ.ItfollowsthatAis,IIIOdulotileembeddingi,thepartofAG*inC 3.REMARKS (i)InkaAShoekandVerduynLuneli81,thea川orsdevelopageneralproceduretoconstructcharacteristicmatrixfunctionsandusetileideaofequivalencetoprovetheabovemultiplicitytheoremt-orvariousclassesofequatioII8.Fortheopera-torsappearinginthepresentpaperthcequivale配ein[8lleadstotheformulasmentionednext-(ii)Onecanwrite ＼l'/OUYIZ,,,.、、AO'A今/M/r『11＼ZF一一、‘...,,*①OAF-z,,...、、BFt (31) and ＼11』F/nUFt噜EA-nu,~/lt＼、‘..,,,z,,az飞E一一唱EA、、..,,,*GOAF47巾,,,..、、 (3.2)
TheoreminFunctionalDWbrenttalEquations I05 where F(Z)=(;-m (3.3) and E(z)=(I。}飞Ps(z-D)-1/areregularoperatorvaluedfunctions.Theformula (34) ,,..ZF＼、-EF/ouviz10UA/tlt＼、‘..,,少命,,..‘飞E一一、‘...,,*。AFtz,,...‘、 (3.5) thenciea句showstheequivalenceof(zI-Ah)一1andA一1(z)andoneCanamongotherthings?derivethepreciserelationshipbetweengeneralizedeigen-vectorsofAG*andJordanchainsofAfromthatformula.(Seekaashoekandver出ynLunell8lj(iii)AspectraltheoryofunboundedoperatormatricesiscurrentlybeingdevelopedinTflbingenbyNapl[11landothers.(iv)ArelatedbutsomewhatdifcrentperturbationpointofviewispresentedintheworkofDeschandSchappa( ACKNOWLEDGEMENT WethankA.vanHartenforbringingtheWeinstein-ArOIlszajndeterminanttoollrattentlOI1. REFERENCES [1lCLtMENT,PH.,DIEKMANN,0.,GYLLENBERG,M.?HEIJMANS守H-J-A.M.ANDH.R.THIEMEPerturbationtheoryfordudsemigroups;Thes1III-reaexivecase?Math.Ann.277(1987),709·725.[2]-1Time-dependentperturbationsinthesun-reHexivecase,Proc.Roy.Soc-Edinb.109(1989),145-172.[3j一,Nonlinearupsetlitzcontinuousperturbationsinthesun-rebxivecase‘VolterraIntegro-DigerentialEquationsinBanacltSpacesMdApplications{G.DaPratoandM.IanIICllj,θdsj,PitmanResearchNotesinMathematics190(1989),67·89.[41DESCH,W.ANDWSCHAPIMHER,SpectMpropertiesofahIn旧l让i旧t悦e+叫.吐ddiImIn阳1perturbedlinear8肘阳e衍ImI[5叫lDIEEK〈MANN,O.,PerturbeddualS肥eImni堪gmIhhndIHhnM川it阳θDimensionalSySt衍阳CCω?牙1mISerdiesF37(l987)L,67.73.
I06 DiekmannandVerduynLunel [6lHALE,J-K.,TheoryοfFUI川ionalDiHUrcmaJEquations,Springer-Vedag,NewYork,1977.[7lLEVINGER,B.W.,Afolktheomminfunctionaldibmtidequations,J.Difer-entialEqm.4(1968)呼612·619.[8lKAASHOEK,M.A.ANDS.M.VERDUYNLUNEL,Characteristicmatricesandspectralpropertiesofevolutionarysystems,toappearintheIMAprpprintseries‘September1990.[9lKAPPEL,F.ANDHKWIMMER,Anelementarydivisortheoryforau川tOnmO仰mmlOuii肘arfmCti臼O叫dωifπfe阳i此川tUia叫lequations,J.DiEereωInMItuiadlEqn8.21(1976创)'1u34个川.」147.[H10叫lKATO,TT.'IP3、讪切e臼臼Ir阳.York,1976.[11lNAGELFR.,TheS叩Pe川肘削(C川.吃trnlu旧l口mInmIlof1uIIdomain,J.Funet-And.89(1990),291·302.
PerturbationTheoryforDualSemigroupsV-VariationofConstantsFormulas 0.DIEKMANNCentreforMathematicsandComputerScience,Amsterdam,TheNetherlands,andInstituteforTheoreticalBiology,Leiden,TheNetherlandsM.GYLLENBERGInstituteforAppliedMathematics,LuleaUniversityofTechnol-ogy,Lulea,SwedenH.R.THIEME来DepartmentofMathematics,ArizonaStateUniversity,Tempe,Arizona 1.INTRODUCTION IfAoistheinhlitesimalgeneratorofastrongiycontinuoussemigroupTb(t),t主0,onaBamchspaceX,thedualA;ofAoistheweal♂genera-torofthedualsemigro叩T♂(t)=(Zb(t))飞t20,onX*inthefollowin sense: 串SupportedbyaHeisenbergscholarshipfromDeutselleForschungsgemeinschaft I07
l08 Diekmanneta/. fεD(AUandA;f=扩 、、..,,,'EA.咽'i/zt飞 if 兰(z,写(t)f)=(z,Tt(t)扩),Zεx,t三0·dtFOI-anyz飞X气t三OwehavefmfdTED(AU(12) and AdtTJ(T)f叫=勾(旷一♂ HereMTt(T)fdThastobeinterpretedastheweak*integ时 (才tmT)fdT)=f(巧(川*)dr (1.3) InCldmeMetal.(1989b)itisshownthattheperiurbedoperatorA×=AZ+C?whereC:D(A;)→X*isbou时edandlinear,gemaweaikMtdiyf*continuoussemigro叩T×onX*intilesenseof(1.l)and(12)-NotethatingeneMX①:=万曰:7乒X＼InCldmmtetal.(1989b)therestrictionT①ofT×onX①isconstructedErstviaavariationofconstantsf扣Ornm盯mI丑1川aandthenextendedtothespaceX＼InCldmeTLtetal.(1989c)ageneralHille-Yosidatypecharacterizationisderivedfortheweak-generatorsofweakly*continuoussemigm叩S(inthesenseof(1.1)and(1.2)). Inthispaperwederivevariationofconstantsformulasforthesemi-groupT×(ratherthanforitsrestrictionT①toXθ).Thecomtr1川ioIIpre-sentedhere-whichisindependentoftheapproachinCldmeMetal.(1987,1989均占)-reliesontheobservationthatA×generatesantintegratedsemigro叩'S×onX*suchthatSX(t)islocallyLipsctlitzintileoperatornorm.SXcanalsobedescribedbyavariationofconstantsformula.SeeATendt(1987),kcllemzαm(thesis)?Keller77lameHiebeT(1989)7NC1吵TmαndCT(19S8创)?ThieTM(toappear)forsomebackgroundmaterialconcerIling‘integratedsemigroups-' Analternativeapproach,whichdoesnottaketheoperatorCasastartingpointbutconsiders4multipliedintegrals'ofthedualsemigroupTtinstead,ispresentedbyDiekmαmpGyllenbeTggTilteme(preprint).
VariationofconstantsFormulas I09 Theformulasderivedinthispaperwillalloweasyderivationofcer-tainpropertiesofT×.Weexpectthattheywillplayacrucialroleinex-tendingtheperturbationtheoryfromdualsemigroupstodualevolutionarysystemsandinhandlingquasilinearCauchyproblemsonnon-rekxivedualBanachspaces-Suchproblemsarisefromphysiologicallystructuredpopula-tionmodels-seeMetzeDiekmαm(1986)forreference-inwhichpopula-tiongrowthcouplesbacktoindividualdevelopment. 2.BASICIDEASANDRESULTS InCldmeMetal.(1989b)astronglycontinuoussemigroupT①isconstructedonP=页互57viathevariationofconstantsformulaT①(t)hmt)内户。一切T①(T)JdT,JεX①(21)withTFdenotingtherestrictionofTttoX①.ThenT①isextendedtoX食bytheso-calledintertwiningformula T×(t)=(入I-A×)T①(t)(入I-A×)-1. Avariationofconstantsformulaoftype(21)isnotpossibleforT×becauseCisassumedtobedennedonX①only.Inordertoovercomethisdimcultyweshalljustifythefoliowingformulainsection6: T×(t)♂=汀T勾mmJ川川川(0例t均巾旷)MZ♂*+川ω旷*L1一寸J立坦马2二OJJ才fμt￥ηm(0t一仰(入一A均;D)叮×(忡T什巾)讪Z♂ω*=TJ(t)♂+旷-AfT×(t-T)CM一4)叮(T)♂dT TX(t)canberepresentedbya4generation'expansion T×(t)=汇TJ(t)(2.3) withTJ=TtandTL(旷=旷-J出oho-仰(λ-4)叮J(均川(24)
IIo Diekmannetal. Theseries(2.3)convergesintheoperatornorm.飞hshallseeinsection6thatthe旷-limλ→∞in(22)and(2.4)holdsuniformlyfortinboundedintervals,||f||三landthatT川(t)zisacontinuousx··valuedfunctionoftforanyZεX.Strangelyenoughwehavenotbeenabletoprovetheseresuitsdi·rectly.Sowetakeadetourwhichisofitsowninterest.ItiswellknowIIthatA×=A;+CsatisEestheresolventestimatesandthereforegeneratesan4integratedsemigroup'S×(t),t;三0,onX*whichislocallyLipschitzintwithrespecttotheoperatornorm.SeeATendt(1987),kelleTmαm,(thesis),kellemzamgHUbeT(1989).ActuallyitispossibletowritedownavaHathnofconstantsformulaforS气namely with 俨(t)=导(t)+fmt一讨dT(C俨(T))=SJO)+儿tS×(t一讨dT(CSJ(T))SJ(t)=fmr)dTbeingthe4integratedsemigroup'generatedbyA;.TheStieluesintegralsin(25)holdintheoperatornorm.Fromthenrstform1山in(2.5)werealizethatS×(t)fcanbediferentiatedintheweak*semeyieldingT×(t)♂:=主s×(t)fdt=勾mm(0例t仿)忡♂ι*+儿fμt》、hη职m(0t一什训训dιT(￠CωωS俨忻×=Tt(t)f+儿tT×(t一讨dT(CSJ(TY)Thearstintegralin(2.6)isaweak*Stieltjesintegral.Thesecondeq1lai--ityin(2.6)willrevealthatX忖3T×*(t)zisacontinuomfunctionoftforzεX.Thiswillimplythattilesecondintegrali11(2.6)makessenseasaweak*Stieluesintegral.Aswewillseeinsection6thesecondequalityin(2.6)alsoshowsthat(入-A5)-IT×(t)islocallyLipschitzintwithrespecttotheoperatornormbecause(λ-AU-1Tt(t)hasthisproperty-(2.6)wiHthenimply(22). (2.5) Thegenerationexpansion(2.3),(24)isderivedsimilarlyusingagenerationexpansionforS×.Thefoliowingformulaisparticularlyhelp-fulinstudyingthedependenceofTXonCandTtSetV∞(t)=CS×(门,
VariationofConstantsFormulas III 1、(t)=CSJ(t)andCOMMerthesecondequalityin(2.6)and(2.5):T×(t)♂=TJ(t)♂+归。一什dAWTY)V∞(t)40)+fmt-TMTV∞(T)(27)=%(t)+fM一训问(T) InthenextsectionaconvolutioncalculusforlocallyLipschitzoperatorker-nelswillbedevelopedinwhichV∞playstheroleofaresolventkernelfor问. 3.ACONVOLUTIONCALCULUSFORLOCALLYLIPSCHrrzCONTINUOUSOPERATORKERNELS 3.1.LIPSCHITZKERNELSANDTHEIRCONlyOLUTION ByakeTMl(ofoperators)wemeanafamilyU(t),t主0,oflinearboundedoperatorsonaBanachspaceYwhichsatisEes U(0)=0 (3.1) andislocallyLipschitzint(withr臼pecttotheoperatornorm),i.e.fora口t〉0thereexistsaAt〉0suchthat |iU(T)-U(s)||三At|T-s|,0三7刊三t. (3.2) Thekernelsformavectorspaceinanobviousway.WedeEneseminorms||-||tby ||U(r)-U(s)||||U||t:=。三呼ψ|r-s|By(3.l),U(0)=0?wehave (3.3) sup||UO)||三t||U||t-O<r〈t (3.4) WiththeseseminormsthekernelsformaF176chetspacewhichbecomesanalgebrainthefollowingway:FortwokernelsU,Vwedennetheconvolution女by TV,αTUflA一-a?bvi宵U (3.5)
II2 Diekmannefal. TIleintegralin(3.5)isaStieitj臼integralintheoperatornorm,i.e.itisthelimitofsums ZU(t-sj)(V(TJ+1)-V(Tj)),sjε[勺,rj+ll withO=TO<...〈Tn+1=t,whenthepartitionro,...,Tn+1getsnneI·.Byreorderingthesumsoneeasilychecksthat TVTUFα,llA一一,,bv女U (3.6) withtheintegralbeingthelimitofsums 汇(U(TJ+1-U(Tj))V(t-sj),δjε[rjrj+ll, 0=TO〈...<Tn+1=t.ItisconvenienttoextendthekernelstoRbysettlng U(t)=0,t三0. (3.7) ThentheyareiocallyLipschitzonRand 〈-STV,dTSUflA一-quvi宵U (3.8) U女Visakernelagain;actuallywehavethefollowinginequalitiesintermsoftilesemimms||-||t· Idemn1a3.1. vua'ι<一俨',GTVUflA<一/YE'食U Proof.LetO三r,S三t.Then(U女V)(s)一(U*V)(T)isapproximatedby SUHIS 汇(U(s一句+1)-U(T一σj+1))(V(σ1+1)-V(σ川 witil 0=σo<…〈ση+1=t. Thenormofthesum(3.9)canbeestimatedby ··Jσ+,Jσ+σF/YE'TS+σun汇时'
VariationofConstantsFormulas ll3 Takingthelimitbyrenningtilepartitionsweobtain 飞111E』,/σ,dσVσUfl·-A/''ZEE-飞TGU〈-TV女Uquv食U,,,‘飞 ThisimpliestheErstestimate.Thesecondistrivial. Wecanintegrate(U女V)(t)andobtainamorefamiliarconvolution. Lemma3.2.fJ(U女V)(r)dr=fJU(t-T)VO)dr=:(U*V)(t).Inother words, (U女V)(t)=兰(U*V)(t)dtwithtlmdiferentiationholdingintheoperatornorm. Proof. f(UJ)(T)dr=fItU(T-sMsV(材=f(fu(r一叫ιV(s)=才ft(才fft←~-Thesecond,fourthandafthequalityfollowbyapproximatingtheintegralsbysumsandrearrangingthese,thef岛i让rs"tequalityholdsb均yd白e白hIn旧1让it"iO∞In叽1U,thethirdbystandardintegralCadiC1uU1d山lh1u肌1Notingthat 兰(U*V)=Ut*V=U*V1,dtprovidedtherespectivederivativesexist,weEndthat J2U女(V女W)=二τ(U*(V*W))GVJ2(U女V)女W=二τ((U*V)*W).GEM Astheassociativityof*iswell-knownandeasilycheckedbystandardinte-grationtheory,wehave Lemma3.3.女isassociative,i.e.theFTdchetspaceofJEernelsismalge-bra. InviewofLemma3.1,theFrdchetspaceofkernelsdeservesthenameFreehetalgebra.
ll4 Diekmannetal. 32RESOLVENTKERNELS TheresoiventkernelV∞ofakernell/bisdeterminedbytherelation V∞=比+1/b食V∞=l/b+V∞食问. (3.10) Ifitexiststheresolventkernelisuniquebyitsalgebraicproperties.SeeGTipeπbeTgetal.(1990),Section9.3,Lemma3.3.Remark.OftentheresolventkernelofakernelWoisdennedby WK=Wb-Wo女IV∞=Wo-W∞食Wo.(311) SeeGTipe讪eTgetal.(1990),Section9.3.Notethat(310)transiatesinto(311)bysettingW∞=-V∞,Wo=一问.Theconceptof(311)seemstobemorenaturalwhen‘frequencydomainmethods,areusedwhereastileconceptof(3.10)ismoreconvenientwhenexpioiti吨orderrelationsincasethatYisanorderedBanachspace-Thestandardconstructionoftheresolventkernelistheseriesofmul-tipleconvolutions: V∞=汇VHZ (312) with vd=町,V食(n+1)=V食"食问. (313) Themainpointisshowingtheconvergenceoftileseries.(3.10)thenfollowsfrom Lemma3.4.V州女问=比女V州 whichisimmediatebyinduction.FromLemma3.lweobtainbyinduction Lemma35.!lv巾+1)||t三二||问iuz+1,n主1. S02二;二1||V叫|t三||Vb||texp(t||问||t)andtheseries(312)CO盯ergesintilesemimmsll-|it.By(3.4),ε;二IV刊(t)COI附rgesintheope川ornormuniformlyfortinboundedintervals.Asacorollarywehavetheestimate Lemma3.6.||l匀。||t三||叫|texp(t|问||t).
VariationofConstantsFormulas II5 Theimportanceofresolventkernelsconsistsinsolvingconvolutionequat1OIls. Lemma3.7.(GTipenbeTgetal.(1990),Section93,Lemma3.5) Theconvolutionequation U=Uo+比女U isuniquelysolvedf气Y U=Uo+V∞*Uo, whereas W=Wo+W食% isuniquelysojvedby MY=阴/o+日/o*V∞· BeforeweestimatethesolutionsofconvolutionequationswemakethefollowingsimpleobservationwhichfollowsfromLemma3.6. Lemma3.8.l+fJ||V∞||rdr三expo||Vb||t). Notethat||%||tisamonotonenon-decreasingfunctionoft.Thefol-lowingisnoweasilyderivedfromLemma3.7andLemma31. Lemma3.9.LetWsolveW=日句+W*VborW=IVO+比女W.Then ||W||t三||IVb||texpo||%||t). Weusethislemmaforstudyingthedependenceoftheresolventker-nelV∞on讥.Let U∞=Uo+Uo女U∞V∞=问+Vb食1年。. Then U∞-V∞=(Uo一问)+(U。一问)女U∞+问食(U∞-V∞). ByLemma3.9 ||U∞-1名。||t三||Uo一问+(U。一问)女U∞lltexpo||问||t).
ll6 Diekmanneto/. ByLemma3.l ||儿一V∞||d||Uo一问||t(1寸t||U∞||J卡p(t||问||t) ByLemma3.8, ||U∞-V∞||t三||Uo-Vb||texp(t||Uo||t)expo||%||t). Sowehave Lemma3.10.LetU∞?V∞betheresolventkernelsofUo,比respectively-Then ||U∞-V∞lit三||Uo一问||texp(t[||Uo||t+||%||tl). 4.PERTURBATIONOFLOCALLYLIPSCHITZCONTINUOUSINTEGRATEDSEMIGROUPS Itiswell-knownthatanope川orA;onaBamchspaceYgeneratesan‘iII-tegratedsemigro叩'SJ(t),t主0,onYwhichislocallyLipschitz(withmpecttotheopeMornorm)ifλ-Atcanbecontinuouslyinvertedforλ〉ωandtheresolventestimates …λf||(λ-AJ)-n||三八-w,λ〉ω,nεN(4.1) aresatisned.Actually ||SJ(t)-SJ(T)|!三Meωt(t-T),t主T主0.(42) Moreoverwerecallthatbydenmuon SJ(t)阶)=f(导(rh)一吓(T))d飞导(0)=0(43) SeeATendt(1987),kellemzαm(thesis),kelleTmamgHiebeT(1989).ThefollowingrelationsholdbetweenA×andS×: Lemma4.1a)Letz,νεYThenZεD(AJ)andAJz=uif￡SJ(t)z=Z+SJ(t)uforallt主0.b)(λ-AJ)-1=入fre-MSJ(t)dt=fFfMdtSJ(t).c)ForanyUεY,t主0,Msif(T)udrεD(AJ)andAtMSJ(TMdT=SJ(t)ν-tu.
VariationofconstantsFormulas II7 See,e.g.,Thime(toappear). IfC:D(At)→Yisaboundedlinearoperator,theope川orA×=At+Calsosatidestheestimates(4.1)(withdiferentω).SeetheproofofTheorem1.linPazy(1983),Section3.1.SoA×generatesalocallyLipschitzcontinuousintegratedsemigroupS×.CompareProposition3.3inkelleTTnameHUKT(1989).ActuallyitispossibletonMSXasthesolu-tionofthevariationofconstantsformula 川)=S附mF川川川(0例t付)+儿fρt、》S俨川×=们)+fmt一讨dT(C川 (4.4) TakingLaplacetransformsonerealizesthat λfJts×(t)dt=(λ-f)-1ApplyingCto(4.4)werealizethatCS×(t)coincideswiththeresolventker-melV∞ofl旬,凡(t)=CSJ(t).Hencem=SJ(t)+fm Inotherwords, S×=SJ+SJ女V∞· (4.6) Inturn,wecan自rstconstructV∞astheresolventkernelof问andddne5·×by(4.6).Ifwemultiply(4.6)byCandcomparewith(3.10)weandthatl尘。=CS×.Usingtheexpansion(3.12),(3.13)weobtainthegenerationexpanslon S×(t)=ZSJ(t)导+1(t)=fsJ(t一讨dT(C吓(T))=fsJ(t一付dT(cmT)) (4.7) Infactthedennition SJ=SJ食V飞n主1,yieldsCSLI=V巾+1)-see(4.7)andLemma34-andso SJ+1=SJ女(CSJ),SJ+1=SJ女(V叫女问)=(s;女Vm)女问=SJ女(CSJ).
II8 Diekmannetal. Asabyproduct,weobtaintheestimate z||SJ+1||t〈∞,foranyt〉0. (48) 5.PERTURBATIONOFDUALSEMIGROUPS IfT♂(t)isthedualsemigrouponX*associatedwithastronglycontinuoussemigroupTbonX-theinhitdmalgeneratorofwhichisAo一,then SJ(t)♂=fmr)♂dT (5.l) denmsthelocaliyLipschitzcontinuous4integmtedsemigroup'SJ(t)onX*whichisgeneratedbyA;.LetC:D(A;)→X*beaboundedlinearop-erator-ThentheperturbedoperatorAX=A;+CwithD(A×)=D(AUgeneratestheintegratedsemigroupgivenby(4.4).Fromthesecondequationin(4.4)werealizethatS×(t)fcanbediferentiatedintheweak-sensea口that 俨(旷:=2俨(t)♂=勾(t)♂+归(t-什dT(C阶 (52) Theintegralontherighthandsidehastobeinterpretedintheweak*sense.WenotethatXH3T阳(t)zisacontinuousfunctionoftforZξX.Takingthisintoaccountweobtainfromthearstequationin(4.4)that T×(旷=η(旷+fTX(t一什dT(CSJ(T)♂),(53) wheretheintegralontherighthandsidehastobeinterpretedinaweak* sense: (寸tTX(t一什dr(cmTY))isthelimitofthesums 汇((CSJ(71+1)-CSJ(刊z＼TX*(t一σj)抖, ζ」、ι.,,+nTnUTno&冒U&EUVAape'H&Ehvne'nw+目,J''ι.,Ja'ιC」.·Jσ''ι一一··A+nT<T··en<户口qu、JATIUTmr='ON
VariationofconstantsFormulas II9 From(12)andtilesecondequalityin(4.4)werealizethatSX(t)z·εD(AUand A;s×(t)♂=-♂+T×(t)♂一CSX(t)2·. Inotherwords TX(t)♂=♂+A×sx(02·. (5.4) Thisisproperty(12)forTX,AX.Using(4.3),(5.4)andLemma4.lc)wecanverifythatT×(t)isasemigroup-FromLemma4.la)weobtainthat ♂εD(A×),AXf=fifT×(t)♂-f=SX(t)扩,t主0. Thisiseqm咀lentto(1.1)forT〉〈,A×.Hencewehaveshownthattheweakly*continuousSemmi旭gmtuai阳nme"dbytheformulas(2.6). ItisnoweasytoobtainagenerationexpansionforT×.ProceediIlgasbeforewecandiferentiate(4.7)intheweak*senseobtaining TAI(旷:=ZSLI(旷J(5.5)=fmt一讨dT(cmT))=fmt一什dT(cmT))Itfollowsfrom(48)that 三二TJ(t)n=0convergesintheoperatornormuniformlyonboundedintervals,hencetheseriesin(4.7)canbediferentiatedintheweak*sensesuchthat 户。vhdz×nT∞汇叫一-Z心pu-a∞?'但叫一-zxsr-a一-Z×T 6.THEVARIATIONOFCONSTANTSFORMULA(22) Inordertogiveameaningtotheintegralsin(2.2)weprove Lemma6.1.a)(入一AD-1Tt(t)islocallyLipschitzintwithrespecttotheoperatornorm. b)Thesameholdsfor(入-4)-1T×(t).
I20 Diekmannetal. Proof:a) (λ-4)-113(t)=勾(t)(λ-4)-1=h(s)A;(λ一A;)-1ds+(λ-4)-1=-h(s)ds+fms)入(入一A;)-lds+(λ-4)-lb)By(52)(λ-4内×(t)=(λ-4)叮(t)+f(J-4)叮(t-付dT(CS×(T)) Parta)andLemma3.lnowimplytheassertion. Inordertoshowthenrstequalityinformula(22)weuseformula(52)andprovethat旷-J立zfTJ(t-T)CM-4)叮×(T)zys=归(t-T)ι(C州)♂) Notethattheintegralsonthelefthandsidecanbeapproximatedintheweak*sensebysums 汇TJ(t-71)C入(入-4)-1(SX(TJ+1)-SX(Tj))♂ uniformlyforlarge入anduniformlyfor|lfi|三1,tinboundedintervals- Theintegralontherighthandsidecanbeapproximatedintheweak*sensebysums 2二TJ(t-Tj)C(S×(Tj+1)-S×(73))♂uniformlyforlarge入anduniformlyfor||f||三1,tinboundediMervais-Soweonlyneedtosilowthat λ(λ-A;)-ls×(T)→S×(T),人→∞ uniformlyforTinboundedintervals.But 人(λ-A;)-IS×(T)-S×(r)=(入-A;)-1A;s×(T)=(入-A;)-1(T×(T)-I-CS×(r))-see(5.4)-and ||(入-A;)-1||→Oforλ→∞.
VariationofconstantsFormulas I2I Thesecondequalityin(22)isshownsimilarlyusing(53).NotethatTX*(t)2,ZεX,isacontinuousx··valuedfunctionoft主0.(2.4)isderivedfrom(5.5)inthesameway.Notefrom(4.7)and(1.2)thatSJ+10)fεD(A;)and A;SJ+10)♂=TJ+l(t)♂-2·-CSJ(t)♂.
I22 Diekmartnetal. REFERENCES ATendtpW.(1987):VectorvaluedLaplacetransformsandCauchyproblems.IsraelJ.Afath.59,327·352 Cldme7Ltpph.jDiekmannp0.;GyllenbeTgpA4.;HeijmanspH.J.A.A4.;Thie7FleyH.R.(1987):Perturbationtheoryfordualsemigroups-I.Thesun-Idexivecase.Math.Ann.277,709·725 Cldme714Ph.;DiekmannpO.p-GyllenbeTgyAr-jHeijmanspH.J.A.AZ-jThiemeyH.R.(1988):Perturbationtheoryfordualsemigroups-IITime-dependentperturbationsinthesun-rebxivecase-Proc.RoyalSoc-Edinburgh109A,145-172 Cld77lentpphJDiekmannp0.JGulle7zbeTgyA4.JHeijmansyH.J.A.A4.;ThiemepH.R.(1989a):PerturbationtheoryfordualsemigmIps.III-NonlinearLip-schitzcontinuousperturbationsinthesun-rdexivecase-VblterraIntqroeDiELrentidEquationsinBmachSpacesandApplications(G.daPrato,M.Iarlmliieda),67·89.PitmanResearchNotesinMathematicsSeries190,Longman Cldmentyph-JDiek7nanny。-JGyllenbeTgpAtjHeijmansyH.J.A.Af.p-ThiemepH.R.(1989b):Perturbationtheoryfordualsemigro叩s.IV.TheintertwiI卜ingformulaandthecanonicalpairing.SemigroupTheOIγaIIdApplications(CMInenLPh,Invernizzi,S.,Mitidieri,E.,Vrabie,I1.,eda),95116.Lee-tureNotesinPureandAppliedMathematics116,MarcelDekker Cld77bentyph-JDiek7nannyO.p-GullenbeTgpM-JHeijmansyH.J.A.M-JThieme,H.R.(1989c):AHille-Y。"atheoremforaclassofweakly*continuous阳11i-groups.SemigroupForum38,151177 Diek77lampO.p-GullenbeTgyM.p-Thieme,H.R.(preprint):SemigroupsandrenewalequationsondualBanachspaceswithapplicationstopopulationdvnaIRics- GTipenbeTgyG-jLonde71yS.·0.JStagαnsy0.(1990):VolterraIntegralandFunctionalEquations.CambridgeUniversityPress kelleTTMmyH.(thesis):IntegratedSemigroups-Tabingen1986
VariationofConstantsFormulas l23 kelleman7LyH.;HiebmM.(1989):Integratedsemigroups-J.FUnd-And.84,160·180 MetzyJ.A.J.P-Di伽αmy0.jds.(1986):TheDynamicsofphymlogicallyStructuredPopulations.LectureNotesinBiomathematics68.Springer NedTanhTyF.(1988):Integratedsemigro叩sandtheirapplicationstotheabstractCauchyproblem.Pac-J.Math.135,111-155 PazyyA.(1983):SemigroupsofLinearOperatorsandApplicationstoPar-tialDifFez-entialEquations.Springer TMemeyH.R.(toappear):4IntegratedSemigro叩S'andintegratedsolutionstoabstractCauchyproblems.JMAA
MaximalRegularityforAbstractDifrerentiaIEquationsinHigherOrderInterpolationSpaces GIOVANNIDOREDepartmentofMathematics,UniversityofBologna,Bologna,ItalyALBERTOVENNIDepartmentofMathematics,UniversityofBologna,Bologna,Italy 1Introduction hthispaperwebuildanabs位actmodeltostudymaximalregularityresultsforparabolicinitial-boundaryevolutionproblemsunderassumptionsontheoperatorinthespacevariableswhichdonotallowtostudythemintheclassicalsemigroup-theoreticalway-LetusconsideranevolutionproblemoftheUnd (d(t)=川)+川)u(O)=O 、‘.r-EA---AJ,‘‘、 whereAistherealizationinasuitablespace,withrespecttosuitableboundaryconditions,ofane111pucdifTerentialoperatoronaboundedopensubsetofRn-WhentheresolventoftheoperatorAdecreasesatinanityinthemaximalway(i.e.||(入-A)一ll|=O(|λ「1))theproblem(1.1)wasstudiedbyseveraldifferenttech-niques,includingtheapproachthroughtheinversionofthesumoftwooperator3forwhichwereferto[1].Howeverthefactthat||(入-A)-1||==O(|入|-1)部队|→∞holdsonlyinspaceswithalowdegreeofregularity.ForinstancetheLaplaceoperatoronasuf-adentlyregularboundedopensetQ,withhomogeneousDMIdletboundarycon- l25
I26 DoreandVenni ditions,hasthispropertyinLP(Q),butnotinwk3(Q)withk21.When1<p<∞thiscmbeseenbythedementaryremarkthat,inarehxiveBa-nachspace,themaximaldecreaseoftheresolventholdsonlywhentheoperatorisdenselydeaned,whichisfalseinthementionedcase.What,indeed,cmbeobtainedisanestimateofthewk,P-normof(入-A)-IUbymeansofthenormsofUinWJ,P(j兰的multipliedbyapowerof1+|入|whichdecreaseswhenjincreases.ThereforetheideaweexploitistobuildadiscretescaleofBmachspacesX03·..,X饨whichareinvariantforthedifferentialoperatorAandsuchthatthebehaviourof(入-A)-linXKbecomes"worse"askincreases.Nev-erthelessourintentionistoandmaximalregularityspacescontainedinXn(whichistheworstspace):thiswillbeperfonnedbyassumingthat(λ-A)-1canbede-composedintothesumofsuitableoperator-valuedfunctions.Afterasectionofpreliminaryresultsandtem1inology,in§3,makinguseofthedecompositionof(入-A)-1,weprovearegularitytheoremforthesolutionofanabstractoperatorequationoftheUnd(A+B)U=f.In§4weshowhowsuchade-compositioncmbeperfo口nedonmabstractlinearoperatorwitha"good"resolventoperator-Finally,in§5weapplyourabstractresulttoprovehigherorderregular-ityresultsforthesolutionofaparabolicequation,withhomogeneousboundaryandinitialdata. 2Preliminaries Inthesequel,wheneverαεRwedenoteby[α]thegreatestinteger三αandput(α)=α一[α].Wearegoingtomakeuseofrealmterpolationspacesofthekind(Y,D(Tm))92PwhereYisaBmachspace,TisaclosedlinearoperatoractinginY,misapositiveinteger,。ε]0,1[,pε[1,∞].IftheresolventsetofTcontainsahalf-line{te叭t20}andsup||(1+t)·t>O(te叩-T)-1||<∞then(YFD(Tm))6,pcanbecharacterizedasthespaceofallνεYsuchthatifweaxtheintegernumberskandTwith05二k〈Om<γ+k,thenyεD(TK)削阳fmtiontHtem-k(T(俨-T)一1)TTkyklongstodtthespaceLPwithrespecttothemeasure-onR+.Onthisspaceanequivalentt
MaximalRegularity I27 norn11S ny,,,飞飞-tIJ/a-tpY们udL晶TT、、,』,J、‘,FTMF。LVι'"',,E飞T,fl飞ι-mmaHW4,",∞+f·····A/It--1＼--PAV们ue (withtheusualrnodiacationwhenp=∞),allthisbeingindependentofkandT(sec[4]section1.14).Asaconsequenceofthischaracterizationweobtaintheequdity(YFD(Tm))ozP=(D(TK),D(Th))em-kwithequivalentnormsprovidedthathandkminte-飞/τ士TFPgemwith05;k<Om<h:hencethespaceinfactonlydependsonOm.Thustosimplifynotationsweadoptthefo11owingone(see[1]): D(T;SFP)=(Y,D(Tm))立nWBFF whenevermisapositiveintegerandsε]0,m[.AnobviousremarkisthatifO〈81〈S2thenD(T;句,p)iscontinuouslyembeddedintoD(T;s1,p). Wenowrecallthefollowingresultconcerningthe"maximalregularity,,forthesolutionoftheoperatorequation(A+B)U=Z(see[1]th.3.11). Theorem2.1LetXbeacomplexBamchspaceandlerA,BbeclosedlinearopemωnactinginX,sati功ingthejbMowingconditions-- 1.theresol1pentsetqfAcontainsthesetZA={ρem;p;2079A三PE二2何-OA}andonZA||(λ-A)-1||ι(X)gC(1+|入|)一12.theresolventsetqfBcontainsthesetZB={pew;p2070B三ψ三2何-OB}aFZdOFZZB川(入一B)-1||ι(X)三C(1+|入|)-13.OA+OB〈7r4.jbr入εp(A)andμεp(B),(入一A)-lcommute-Swith(μ-B)-1. Supposetharoε]071[,pε[1,+∞].ThenVzεD(A;87P)(resp.VzεD(B;87P)jtheequation(A+B)U=ZhasauniquesolutionU=Sz.MoreoνerASzandBSzbelongtoD(AJ,p)(resp.D(B;6,p)). Actuallymthepaper[1]theauthorsEndtheinverseaoftheoperatorA+B+入forsuitablevaluesof入;howeverwecantake入=OsinceinourcasethespectraofAand-Barcdisjoint.
I28DoreandVenni 3Theabstractmodel Supposethatfor05;kgηXKisacomplexBanachspacewithxblcontinuouslyembeddedintoXK·LetA:D(A)→Xo,B:D(B)→Xobeline缸OperatomwhosedomainsaresubspacesofX0·Weare1ookingfor"maximalregularityspacesofthehighestorder,,fortheoper-atorequation(A+B)U=Z,thatisspacesYembeddedinXnsuchthatforZξYtheequationhasauniquesolutionUεY,withAuεYandBuεY.HerewelisttheassumptionsonAandB. (H1)TheresolventsetofAinthespaceXocontainsthesetZA={pe叫p207。A至ψ三2何-OA}andonZA||(入-A)-l||ι(Xo)三C(l+|λ|)一1.(H2)TheresolventsetofBinthespaceXocontainsthesetZB={pe叩;p2076B三ψ三2何-8B}andonZB||(入一B)一1||￡(Xo)三C(1+|入|)一1. (H3)OA+OB〈何.(H4)Forλεp(A)andμεp(B),(入-A)一lcommuteswith(μ-B)-1. (H5)VK=07…,nXKisinvariantunder(μ-B)-1andonZB ||(μ-B)-1||ι(XK)gC(1+|μ|)一1. (H6)For05;jgk三ηthereexistsanoperator-valuedfunctionRKJ:三→￡(XjFXK),dennedandholomorphiconanopensetzwhichcontainsZA,such削(入一A)-lz=忌。RU(入)z,V入εZAandvzεXK· (H7)Thereexistα07…,αnεRwithO=αo<…〈αnsuchthatonZAitis||RKJ(入)||ι(Xjh)gC(l+|入|)一1+何一句.(H8)For入ξEandμεp(BLRkgj(入)(μ-Bj)-1=(μ-Bk)-lRKJ(λ),whereBJistheoperatordeanedbyD(Bj)={zεXjnD(B);BzεXj},Bjz=Bz. Weremarkthatfromassumption(H8)itfollowsthatVzεD(Bj)VAεzRKJ(入)zεD(Bk)andBKRKJ(入)z=RKJ(入)BJZ(05;j三k兰州.WedsoremarkthatthedomainofBjequals{zεD(BKBzεXj}.IndeedfromBzεXjandassumption(H5)itfollowsthatzεXJ·.
MaximalRegularity l29 InthesequelspaceslikeD(Bj;SFP)willappe缸.WeagreethatD(Bj汀,p)isaninterpdationspacebetweenXjandthedomainofsomepowerofBj(seeabove). Nowwestatethemaintheoremoftheabstractmodel. Theorem3.1Supposethattheasswnptiom(Hljto(Hmhold,letpε[1,∞]andletObeapositiverealnumbersuchthatO+问一αj￠zjbrO三j三η.Letus set "Y=nD(Bj;0+α"一αjpp)j=OThenVzεYtheequation(A+B)U=ZhasauniquesolutionSzεY,andmoreoverASzεY,BSzεY.Finally,ASandBSareboundedoperatorsjFomYtoY. Theuniquenessofthcsolutionof(A+B)U=Zfollowsfromlemma3.6of[11SinceYED(Bo;p,p)forsomepε]071[,fromtheorem3.11ofthesamepaperitfollowsthatifasolutionexistsitmustbeoftheforrnsz=-Etd(入-A)一1(hB)-12d入whereristhecurveobtainedbyjoini鸣thehalf-lines{ρe土叩;p主0},。A〈ψ〈7r-6B,orientedaccordingtomereasmgimaginaryparts. Fortheproofofthetheoremweneedsomepre1iminaryresub- Lemma3.2LetX3VbecomplexBanachspaces,andletEbetheopensetqfas-sumption(H6).Supposethatφ:z→乙(X,V)isaholomomhichFZCHonandthat||φ(入)||=O(|入尸)as|入|→∞·LetBbealinearoperatOMCtinginXsari吵ingassumption(H匀,andlet入obeacomplexnumber-。7ing"atright,,qfFLetqFTbenon-negativeintegerssuchthat丁<q〈T·ThenvzεD(BT一1)J队o+B)q1f队o+B)T1φ(λ)|一一一一|(入+B)-AZd入=lφ(入)!!(λ+B)-IZdλ.飞λo一入jJr飞入。一入/Pr04.Itisenoughtoconsiderthecaseγ=q+landthentoapplyinduction.Remarkthattheassumptionsγ<qandzεD(BT一1)ensurethattheintegralsconvergeabsolutely.NowJ~扩1φ(入)|-二一-i(入+B)-lzd入=飞入。一入/
I30 Dor-eandVertrti J/:+B)qf队0+B)qφ(入)|-一-i(λ+B)一1zd入+lφ(入)|)(λ。一λ)-lzd入,飞ojJr飞入。一λ/andthelastintegralvanishes,sincetheintegrandisholomorphicontheleftofranddecreasesappropdately. λ[αrα]Lemma3.3扩theassumptions(HIjω(H8)holdandzε[1D(Bj汀,thenj=O Sh-2元儿川)(汇到[αAj]飞+乌)讪 jbrartychoiceofε1·ε{0,1}(j=0,...,时αFZdtheintegral-SCOFZνergeinthenormqfxn· pmqf.Bylemma32thej-thintegraldoesnotdependonεj·Moreoveritis伽ious削itconmgesi川ermnofXnktuεD(B俨])men,byawl问lemma32,weget,,/飞..D＼[α.-αj]在;儿RnJ(入)山二?)(λ+B川yd入=LfR··(入)尸。+Bj}[问](入+Bj)~d入=mJrw飞入。一入/Jf一1/入。+Bd飞[问]一-lAj(入)(入+B--)|一一一一乙|ν队mJr'JJ飞入。一入/WhenUεD(BiαJ)wecansmwithmpccttojandget去21儿们0+乌)[…]-lbJ(λ)l)(入+Bj)-lud入=j=omk飞入。一λ/f一1f入o+Bn＼[问]-l(入一A)-10+Bn)|一一-iν队mJr＼入。一入/NowweapplythesamelemmaandobtainSU·Inthegeneralcasewehave Bdiα,]sz=SBJ[αJz==,./飞ED＼[α哪一αjl=-2亏LTLR时(入)(?于气j)(hBJ)-1BJIαJzd入=j=obH&JI飞λ0一λ/
MaximalRegularity I3I ,a/L·D＼[问-αj]=-BJ[αJT三气/RTZJ(入)(OT?)(入+Bj)-lzd入";二百μrz汀Muo-λ/whencef/λo+BJα"一αjl1sz=-ZE儿R饨,州i入。一二)(λ+Bj)td入 Thenextstepintheproofoftheorem3.1isthefollowinglemma: Lemma3.4ForZεYwehawSzεD(Bt[9]).[8+α,一αjlpr04.SinceZεD(Bj;9+α"一αj)豆D(Bjf汀,weC饵'-sz=-2二元7L(入。一入)-σjRW(λ)(hBj)-1(λo+Bj)~d入j=OMH·"Awhereσj=[0+αn一αj]一[0]=[α"一αj]+εjwithεjε{0,1}.NoweachintegrandtakesitsvaluesinD(Br[叫aManerapplyingBtwltoeachoneofthem,thej-thmtegImdbecomes(入。一入)一σjRTZJ(Mj(λ+马)-146l(λo+BjyjzSinceBJO](入。+Bj)%ξD(Bμ(0+αn一α儿p)wh灿MIIIbeddedintoD(BJ;(6+问一αj),∞),theXrnormofthe1astexpressioncanbeestimatedbyapowerofl+|入|whoseexponentis一1+α"一αj一[8+αn一αj]+[8]一(6+αn一αj)=-1.-(6)<一1as。￠Z.ThereforeszεD(BtW]). Proqfoftheorem3.IFirstofa11,lemma310of[1]saysth剖wheneverZξD(B;ε,p)wehaveSzεD(A)nD(B)andASZ+BSz=Z.FromthisfactandfromtheobviousinvarianceofYundertheoperatorB-litfollowsthatwhenZandBSzbelongtoY,alsoSzandASzbelongtoY.ThusweonlyprovethatZεY=字BSzεY,andtothisendweproceedbyinductiononn.Supposethatn=0.T11erlthespaceYisD(B;6,p).ThereforeifZεY,thenZεD(B[el)andBWlzεD(B;(8)FP).Bytheorem311of[1],wehavethatBSBmzεD(B;(的,p)and||BSBW]z||D(B;(OM)三C||BWlz||D(B;(的,p)
I32 DoreandVenni AstheCOInmutativityassumptionyieldsBSB[elz=B[6]BSz,weobtainthatBSzbelongstotheimageofD(B;(的,p)undery[町,whichisD(BJ,p),togetherwi由themqukedcontinuity.Nowwesupposethatthemstiltholdsforn-1,andEWoceedtoproveitforn.Remarkthattheresultforη-1issupposedtoholdVO>O(withaarlitenumberofexceptions),mdso阳factthatBSzbelongston到D(BjJ+α"一句,p)followsatoncefromtheinductivehypothesisbyreplacingOwithO+αn一αη一1·Thereforewhatistobeprovedis出at: nMnD(Bj;0+问一αjFP)斗BnSzεD(Bn;63)j=OInordertoprovethis,wehavetoshowthatBtW]SZεD(BU(的,p),i.e.thatthefunction tMt(句"(t+BJlBt[咔zdt.belongstothespaceLPwithrespecttothemeasure-onR+.Indeedwehave,twithσj=[6+αn-αj]一[6}:t(忖n(t+B饨)一lBt[何z=t(6)(1-t(t+BJ1)Bt[咔z= f饵,at(0)(-Ezi;儿(入o一λ)-叽j(λ)Bj(入+Bj)一1BY](入o+Bjyjzd入+丢东儿(入。一入)飞·(入)击Bj(t+旷lBje](入0+马)M入+TZta飞买主;儿(入。一λ)飞J(均占B仙可146](入0+Bj)句"λ)Noweachintegralofthesecondsumvanishessincetheintegrandsareholomorphicontheleftofranddecreaseappropriately.Thereforewhatremainscanbewrittenas nf入-t(6)艺-j(入。一入)一σjRW(入)一-BJ(入+BJ)叫。](λo+Bj)町zd入=j=07rzJr入-t1r1/t＼(6)2儿7寸(x)入(6)-(e叫-αj)(入。一入)一σjRqj(入)J=07TZJIA-x＼二/入(hAj)乌(λ+乌)-lBjel(入。+Bj)~d入=
MaximalRegularity I33 一去LL∞1/)(6)fil)ρ(9)-(hrαj)+1(入。一ρ川一σjRTZJ(ρ的j=OM扣1一步＼P/p(0+α,一αj)Bj(petψ+BJ)-1Bj叫入0+Bj)%etψ坐+p去土1∞lt/)(0)fil}p(的一(hAj)+1(入o-pe→ψ)-叩时(pe叩)jz02们J01一声士:FU/p(6叫一αj)Bj(ρe-叩+Bj)一1BY](入。+Bj)σjge-叩坐PThus阳Xrno口Iloft(0)Bn(t+Bn)-1BrwlszisboundedabovebyHM,/λ(6)C失1-J7i二iρ(8)-(hAj)+1(1+ρ)-MA广σjJZBAI+;＼P/(印|川|伊μ川pd卢(伊6h+惜叫α叫)恒吗吗B乌乌Jjρ.|川μ|怡ωpd(…叩f♂严川4叩hψh+马旷)γ-」叫甘1叩4坪6町叫~h]气U(队ω入沁o+乌)%|lxj)乡Nowremarkthat (的一(0+的一αj)+1一1+问一句一句=O andthat Bje](入。+Bj)%εD(Bμ(6+α吼一αj),p)since[6]+σj=[6+α"一αj]andZεD(BjJ+α饨一αjFP).Thereforethefunctions P时||p(hrαj)Bj(ρe土叩+马)一lBje](入o+Bj)σjz||XJL卢(6)belongtoLP(R+旦)and巳叶立一-issummabkwithrespecttothesamemea-'t1+Esure,sothatanapplicationofYoung'sinequalityforthemultiplicativegroupR+showsthatBt[elszεD(Bn;(8),p)andthat ||B;+[0]sz||D(BU(8)dgC艺||z||D(BI;9叫-α13)j=O rioOFApehu--··FBeAUU-EE--pknopuvp3···AKHT 4Adecompositionoftheresolventoperator Wenowexhibitadecompositionoftheresolventoperator(入-A)-lwhichsatidestheassumptions侣。to(H8).
I34 DoreandVenni SupposethatX,ZarecomplexBmachspaces,withZcontinuouslyembeddedinX.ImAbeaclosedlinearoperatoractinginX,sais勾ringassumption(H1)andsuchthatZisinvariantunderA-1.I￡tnbeapositiveintegerandsupposethatD(P)withitsgraPhnom(i.e.||叫|P(A')=||Anz||x)iscontinuouslyembeddedinZ.Now,forj=07…,"wecallXjthespaceofallZεxsuchthatAJ-饨zεZandgiveXjthenorm|忡忡=||AJ-~||z+||z||x.Itiseasytocheckthat||lljisaBmachnomandthatXKL→XK-1for15二kgn(where"ι→"meanscontinuousembedding).MoreoverXo=XandXn=Z,inbothcaseswithequivalentnorms.Supposenowthat入εp(A).ThenforO<k三η:k-1(入-A)-1=-2:ν-lA-j+入k-1A-bl(入一A)-1ascmbecheckedbyapplying(入-A)tobothsidesoftheequality.For入εp(A)wesetRKK(入)=0,RKO(入)=λk-1A-bl(入-A)-landfor15;j<kgnRAj(入)=一入k-j-1Aj-k.NowweshowthatRKj(入)ε乙(XLXj)andestimateitsnominthisspace-Supposethat15二j〈k三nandZεXJ·.ThereforeA142ξZ.ThusAK-nRKJ(入)Z=一入k-j-1AK-nAI--kz=一入k-j一1Aj-nzξZ,whichprovesthatRKJ(λ)ZεXK·Moreover||RKJ(入)z||XK=||入k-j-1Aj-kz|ix+||入k-j-1Aj-nz|lz三(1+|λ|)k-j-1(||Aj-k||￡(X)||z||x+HAJ--nz||z)ζC(1+|λ|)k一j-1||z||Xj-Incasej=kthereisnothingtoprove-FinallyVzεXwehavethatAK-nRK,o(入)Z=入k-lA-附1(入-A)-1ZεD(An)ζZ and||RKO(入)z||XK=||入k-1A-bl(入一A)一lz||x+||入k-lA-n+1(λ-A)一lz||z三cl(1+|λ|)叫|z||x+(1+|λi)叫|A叩l(入一A)-l巾(A")lgC(1+|λ|)k-1||z||x·Thereforetheassumptions(H6)and(H7)缸esatunedtakingαk=k.NowletBbeanotherclosedlinearoperatoractinginX.Letussupposethatwερ(B)Zisinvariantfor(μ-B)-1with||(μ-B)一1||ι(Z)gC(1+|μ|)一landthat(μ-B)一lcommuteswith(入-A)-1.ThenitiseasytoprovethateachXKisinvariantunder(μ-B)-1andthatassumption(H8)issatisaed.MoreoverasimplecomputationshowsthatVK=07…,η||(μ-B)-1||￡(XK)三C(1+|μ|)-1.
MaximalRegularity I35 Tostatetheomrr13.1inthiscase,wewritedowntheintmpolationspacesappear-ingMitbymakinguseonlyofthespacesX,ZandofthedomainsoftheoperatomBandBz,whmBzistheoperatordeanedbyD(Bz)={zεZnD(B);BzεZ},Bzz=Bz.WeagreethatBjhasthesamemeaningasinsection3,sothatBo=BandBn=Bz. Lemma4.1LetX,ZbecomplexBamchspaceswithZL→X.LdtAFBbeclosedlinearoperatomactinginX.SupposethatAandBsati吵theassumptionsfHOtofH句,tharD(An)t→ZandtharZisinvariantunderA-land(μ-B)-1with||(μ-B)一1||ι(Z)gC(1+|ILO-1onZB·Letpξ[1,∞]andoεR+＼ZThen,扩BjandBzhaIFethemeaningprecisedaboνe,wehave-- nD(BJJ+η-jpp)=(zεD(B;6+W);A-kzεD(Bz;6+kyp)Jrk=077斗 Proof.OwingtothecornmlitaivityoftheresolventoperatomofAandB,itiseasytoprovethatD(Bj)={zεD(B);ArnBzεZ}andbyinductiononmoneseesatoncethatD(BT)={zεD(BTFB);A1一ηB~εZ};moreoverthegraphnomofD(B广)isequivdentto||Bmz||x+||AJ-nBmz||z·Nowif巳ε]0,1[,wehavethat||tEBj(t+Bj)-1z||Xj=||tEBj(t+Bj)-12||x+||tEBz(t+Bz)-lAI--nz||z;thereforeZεD(Bj;《,p)ifandonlyifZξD(B;己,p)andAJ-nzξD(Bz;己,p)-MoreoveronD(Bj;己,p)mequiv剖entnomish||D(B;53)+HAY-~||D(Bz;53)·Hence D(马;6+η-jFP)=(zεD(48]+叫);Bjω-jzεD(B川的,p))=(zεD(B[6]叫);BWl+←jzO(B;(OM),Aj-v[川-jzσ(Bz;(8),p))Thereforezεn?=OD(Bj;6+η-jJ)ifaMonly江Zε门?=OD(B;6+n-jJ)=D(B;6+饨,p)andforogj三nAJ-饨zεD(Bz;0+η-j,p). RemarkthattheconditionA-nzεD(Bz;6+η,p)cmbeobtainedbyin-teIpolationfromzεD(B;6+",p).IndeedA-nεζ(X,Z)andA-nει(D(Bm)yD(BJP))owmgtotheCOInmutativityoftheresolventoperatorsofAandB.Thustheorem3.1inthepresentsituationbecomes:
I36 DoreandVenni Theorem4.2LetX,ZbecomplexBanachspaceswithZL→X.LetA,BbeclosedlinearoperatorsactinginX.SupposetharAandBsari功ytheassummons(Hljto(H句,tharD(An)ι→ZandthatZisinmriantunderA一land(μ-B)-lwith||(μ-B)一l||ι(Z)三C(1+|μ|)一1onZB-Lerpε[1,∞]andeεR+＼Z.Set Y=(zεD(BJ+mp);A-kzεD(Bz;6+KJ)卢rk=0,,η-1) ThenVzεYtheequation(A+B)U=ZhasauniquesolurionSzεY.MoreoνerASzεY,BSzεY. 5Applications Wearegoingtoapplytheabstractresultoftheprevioussectiontostudytheregularityofthesolutionsofaninitial-boundaryproblemforaparabollcequationinSobolev-SlobodeckiTspaces-LetQbeaboundeddomaininRNwithC∞boundaryand1<p<∞-Let N532N2=〉丁αjk(z)一」一一+飞飞b(z)」一+c(z)j:tzlhbjfOZK知JbjbeaproperlyellipticopemtorinQwithcoefacientsinC∞(Q).LetGbetheoperatorintheBmachspaceLP(Q)dennedby: D(G)=(uεw23(Q);U|rO) Gu=Lu-WesupposemoreoverthatGhlalsassumption(H1)withOG〈贸/2.Wereferto[4]§4.9.1and§52.lforcomments.Takingintoaccount[4ltheorem5.4.1,itiseasilyprovedthatD(G♂饨勺)=(uεW2μ饨叫'whichisaclosedS印1uIbhspaCeof冈FA2饨风叨,♂p(Q)上.WesetX=LP(0,T;LP(Q)),Z=LP(OFRW2叩(Q)),andddneAastheoperatorinXinducedbyG(i.e.(Au)(t)=G(u(t)))with D(A)=LP(0,T;D(G))
MaximalRegularity I37 SinceD(Gn)isaclosedsubspaceofWAnp(Q),D(An)isaclosedsubspaceofZ-MoreoversinceGfu161stheassumption(H1)thesameholdsforAwithOA=6G·NowletBbetheoperatorinXdeanedbyD(B)=(uεw13(0,T;LW));u(O)=0)BtL=u1.Thisoperatorallalstheassumption(H2)foreveqOB〉贺/2;hencetheassumption(H3)issatidedandalso(H4)holdsobviously.Finally,fromthefactthatBcom-muteswiththederivativesinthespacevariables,itfollowseasilythatZisinvariantfor(μ-B)-1andthatBzisthederivativeoperatorwithdomainD(Bz)=(uεw13(0,T;whP(Q));u(O)=0)thusitsresolventoperatorbehavesintherequiredway-归i旭saBan灿s叩mp归aCe削SisanmO∞rnm1卜H.4咀In1咿W附edenotebyW;♂气(0'T伫;Em)t阳h忱es叩pa倪CeOdft阳ih忱1回efunctionsUεW咿叫(0,T;Em)sucht由ha创tu(0创)=...=ud(k均)(0创)=0wherekisthenon-negativeintegersucht出ha创tk〈S-t〈k+1(whenogS〈twr(0,TJ)=wm(0,TJ))·Itmoreover,S￠ZthenWJ3(0,T;E)=D(BLSFP)whereBEisthededvativeinLP(01;E)withD(BE)=W俨(OFEm.Thisstatementispmedin[3]§6.1forO〈S<1andcmbeobtainedinthegeneralcasefromtheequa1ity(LmFEmp(BPLp=(D(BPP(BF))吁叼withη〈σm<η+1Thuswecanprovethefollowing: Theorem5.1LetnεNandOε]0,1[,。手1/p.PurY=wr饵,p(0,T;LP(Q))nwJP(0,T;W2叩(Q)) ThenVfεYtheinitial-boundaryvalueproblem生+LhfM[OT]×Qθtu(OFZ)=OzεQu(t,z)=00,z)ε[0,T]×θQha-SGmquedutiOFZUεYGFZdmoreover鱼,LuεYθt'Proof.Becauseoftheremarkswemadeabove,theorem42appliestothepresentsituationwithX,Z,A,BalreadydearIed.Thereforetocomphtetheproofweneedonlytoprovethatinoucase
I38 DoreandVennt wrw(0,T;LP(Q))nwf'mr;W2叨(Q)) Now,aswehavejustseen, D(B;9+叫P)=wrqp(or;LP(Q))D(Bz;6+KJ)=wrksP(0,T;W2饵,P(Q))andGKismisomorphismfromaclosedsubspaceofwh3(Q)ontow2(仙一的,P(Q)(see[4]theomi71)·ThereforeA-Kuεwr"(0,T;whP(Q))ifandonlyifuεwfkP(or;WXH)♂(Q)),sothat(uεD(B;6+η,p);A-KuεD(Bzj+kpp)fork==:O,,η-1)=/η-1＼wJ+叩(01;LP(Q))ninwrk,p(0,T;W2{叫)3(Q))i=飞k=O/TBnwFK,p(01;W2(叫)3(Q))k=0NowputtingXj=LP(0,T;WAj,P(Q)),j=O,1,...?nanddenotingbyBJthePMtofBinXj(sothatBo=B),theabovesetcanbewrittenas n(XjFD(BT)LELnj=OmF foranmεN,m〉6+η.Ontheotherhandeachomofthespaces(XjmBm))9+叫Withj、,,一百一,p12…,η-lcontains (XP(Bm))旦旦Mn(XJ(BrumWBFFWBFFhfactbytheorem4.1of[匀,denotingby[,]thecomplexinterpolationfurlCtor,wehave: [M叫+饨,川(Br))el=亏汇,PE言,PL([XlnltP(BKljJ)旦旦,pBut,bytheorems1.18.4,43.1/land2.42/1of[4],[XFXAL=ILP(or;L叩门,LP(or;W2饵,P(Q))lL= 吨,T;lLmwM(叫
MaximalRegularity I39 SOthat l卡仅川'ρρ川))也'川(臼叫叩B哼mrb川)m'♂pm'wF」"(LmFT;W2j,叩门FD(Bnh叶"mFPandeverycomplexinterpolationspacebetweentwoBmachspacescontainstheir1ntersect1or1.Therefore/伽-1飞wf+饵,p(0,T;LP(Q))ninwFK,p(0,T;wh-k)3(Q))l=飞k=O/wfmp(0,T;LP(Q))nwf,p(0,T;WKP(Q))andthetheoremisproved. Nowweapplyagaintheorem42exchangingther6leoftheopemtorsA,Bwithmspecttothepreviousapplication.WesupposethatQ,GandXarethesameasabove,butweput Z=WF饨,p(0,T;LP(Q))(Bu)(t)=G(u(t))withD(B)=LP(OFT;D(G))(Au)(t)=山)withD(A)=WJP(0,T;LP(Q)) Inthe叫uelweShanuse阳no剧OIIWFP(Q)(whereS主OFt(s-i)￠z)withthefollowingmeaning:ifS〉landkisthenon-negativeintegersuchthatp2k<S-t〈2k+2,thenw;3(Q)={uεWW(Q);Lju|BQ=0,forogj三k};ifO三S〈;thenWJP(Q)=WW(Q)-Weremarkthat D(An)=WFP(0,T;LP(Q)) and D(Bn)=LP(0,T;W俨P(Q))ThetopologicalinchmonD(An)ι→Zisobvious,sinceD(A")isaclosedsubspaceofZ.FUItherwenoticethattheconditions(H1)to(H4)stHlhold,sincetheyweretrueintheprecedingframeworkandamsymmetricmAandB.More-over,from((μ-B)一lu)(t)=(μ-G)-1(u(t))andthealreadyknownpropc口iesof(μ-G)一linLP,itfollowseasilythatZisinvariantunder(μ-B)-1and
I40 DoreandVennt ||(μ-B)-1||ι(Z)gC(1+|ILO-lonasuitablesector(infacteverysectorofthekind{petO;p>0,ε<O<2何一ε}).Inordertoapplytheorem42wehavetocharactedzethespacesD(B;6+矶纱,13(BLO+KFP)andY.ForsimplicityweonlydealwiththecaseO<O<1,evenifthisassumptionisirlessentidMall(providedthato￠Z).Wehave: D(BJ+矶P)=(X,D(BTL+l))9+n=TZ+lEY (L气02LP(Q))yLP(OyT;W俨23(Q)))ω=百了7pLρp叫巾(仰07rT巳;(μLpp(Qω),W2附ω2,3p气(Qω))h〕=Lpp(仰07rTh;W2μη伽ω+d2M6叫,飞飞w/!百百T7pvJwpmidedthato￠{j去}(sec[问4句]由阳ωωOmS1184吼M,j433an时dd由eda估h削rn旧削1让iNowUε1D〉(BZ)i江fandonlyifUεZnD(B)andBtuLεZ,t由ha剖ti比sUεW风叩叫p盯m(仰07rT伫;Lp汽(仙仙Qω))nLLp,yp叫m(0?1rTT伫'飞V;jL号俨(Qω))andBuεW叩叫(仰07J1TT巳T飞刊;la阳C∞On叫di刨HωO∞ni臼s"叫e叫叩quival创tOUεW叩叫(仰0FrT伫;W;;俨,♂p气(仰仙Qω))sωO由D凯(B马归Zρ)=WKP(or;w;叭2))Fmm阳lastequalityitfollowsD(B2)=wm(01;wr,p(Q));henceD(Bz;6+KJ)=(仰W叽叨p(0'rT伫;Lp阳))'yW叩叫(0?rT伫;W时;俨俨kb以+EE=T,pNow,ifEisanyBmachspacethefunctionUH(u(时,u(O),...,u(η一l)(O))isanisomorphismofWFW(0,T;E)ontoLP(OFT;E)×En.Thus(bytheorem433of[4]),ifEandFMeBanachspaces,wehave (WF饨,p(07T;E),即叫P(0,T;F))6,p=即川,p(0,T;(E,F)93)26+2kDThereforeD(Bz;8+k,p)=即叫P(0,T;WOJ(Q)).ThentheconditionsthatdeanethespaceYinthepresentsituationarethefol-1owing: "27钟26DUεLP(0,T;W0,y(Q))A-Kuεwm(or;wr+293(Q))forO三k三η-1; therefore nf、nkn2k+29DY=门wnKJ(OFT;WoJ(Q))k=OAsabove,weshowthatthearstandlastspacesinthisintersectionaresufacie时,astheotherscanbeobtainedfromthemthroughcomplexinterpolation.
MaximalRegularity I4l IfXj=冈/j3P(0,T;LP(Q))with05;jg饨,thenXjisnaturallyisomorphictoLP(Q;WJ3(OFT)),sothat,byalreadyquotedresults,[XopXJL=Xj·Inthe饨 presentsduauon (XjP(BfoiFP=WM(OFtwf叫 pmvided阳t2s￠{17;};阳3fo削hemea耶rn创ωintl叫mdingappli-cationshowsthat lLP(Optwjn+263(Q)),W吼叫Optwr,P(Q))lL=" (XjP(坪。但in=WB'tr,(伊Wvv仰呐jι川叫咐.＼♂叫1♂叫咐p气m(ω0'r川TR川;汕JLpmp竹B,rwjsP(or;WF叫)叫P(Q))Thuswehaveprovedthefollowing: Thmm5.2LetηεNandOξ]071[＼{去今}PutY=LP(Ophw俨2句(Q))nwm(OFtwr,p(Q)) ThenVfεYtheinitial-boundaryvalueproblem些+Lu=fδtu(07z)=O in[OFT]×QzεQ(tFZ)ε[0,T]×OQ hGMuniquesoluωnUεYanMdmmOreωer鱼,LUξYOt References [1]G.DaPrato,RGrisvard:Sommesd'op6ratellrslintaimsettquationsdiff6rer1·tielksop6ratioINICHes-J.Math.PuresAPPI-(9)54(1975)305-387.[2]RGrisvard:CornrTIlltativit6dedeuxfonctellrsd'interpolationetapplications;1Math.PIKesAPPI-(9)45(1966)143-290.[3]RGrisvard:Equationsdifftrentiellesabs位aites;Ann.Sci.EcoleNonn.Sup.(4)2(1969)311-395.[4]H.Triebel:Interpolationtheory,如1CHonspaces,dtiobentialoperaωmNorthHolland,Amsterdam,NewYork,Oxford,1978.
ApplicationsofSpectralDistributionstoSomeCauchyProblemsinLP(Rn) H.EMAMIRADDepartmentofMathematics,UniversityofPoitiers,Poitiers,FranceM.JAZARDepartmentofMathematics,UniversityofPoitiers,Poitiers,France 1.INTRODUCTION In[FLFoiasintroducedtheclassofboundedoperatorsinBanachspaceswhichinlieuofspectralmeasureadmitspectraldistributions.Someextensionofspectraldistributiontotheunboundedoperatorswasmadebykritt[Kl],[K2]andBalabane[B].Inourpreviouspaper[EJ]werevivedtheBalabane'sformalismwhichhasacloserelationwiththetheoryofsmoothdistributionsemigroups(see[B-El],[B-E2],[B-E3]).InfactitwasobservedbyArendtandKellermarl[A-K]thattheinterestingfeatureofthistheoryisitslinkwiththetheoryofintegratedsemigroups-TheconceptofintegratedsemigroupswasintroducedbyArendt[A汀,anddevelopedextens1velyin[A2],[A-K],[De汀,[De2],[De巧,[Hi],[K-H]and[N].In[EJlwehaveprovedthefollowingtheorem: Theorem1.LetAbeGlineardenselyde/inedoparatoronGBonachspaceX.Thefollowingareequivalent.fijAgeneratesGsmoothdistributiongroupoforderk.fiijAgeneratesak-timesintegratedtemperategroupcftjsuchthatl|Cftjll三Cfl+|t|jkforanyteRfiiij/oranypurelyimaginarynumberα,αAadmitsGspectraldistributtW1ofdegreek. I43
I44 Emamir-adandJazar InthisnoteweshowhowonecanretrievethestandardversionofStone'stheoremfromtheaboveresultandthenweapplythattosomeclassicalCauchyproblemsinLpfRnj 2.SPECTRAIdDISTRIBUTIONANDITSBASICPROPERTIES LetXbeaBanachspaceandtfxjthealgebraofboundedlinearoperatorswithuniformoperatortopology.ByaspectraldutF协tdIIonweα3meanalinearcontinuousmappingEfromD=:E(R)intoJE(X)whichosatisfies: i)E(ψ.ψ)=ε(ψ)E(ψ)forallψ,ψεpii)ForanyfunctionψεDsuchthatψ(o)=1,E(飞1)convergesstronglytoidentity,whereψFJtj=ψft/nj. ∞Definition1.ForanyfeE(R)letusdefineE(/)asanunbounded linear operatorby:DfEifjjzfxeX;limEff.ψjxexistsforanyψεDwithwojzlnn}.ThenforxeDfEffjj, Ef/jxzlimEff.p.jx.nnOneshowsthatE(f)isadenselydefinedclosableoperator,andin∞thesequelforanyfεE(R)wedenotebyE(/)thesmallestclosedextensionofE(f).WesaythatanunboundedHnearoperatorBadmitsthespectraldistributionE,orBisthemomerttumofE,ifthereisaspectraldistributionEsuchthatB=ε(t).HeretdenotetheidentityfunctioninR Nowweintroduceanintegerwhichmeasuresinsomesensetheregularityofaspectraldistribution.Forlε剧,letPlbethefollowingnormonD: PJWJHIkf生产ψlly1.oZKSlH·』 LetgldenotethecompletionofEforpl-WedesignatebyfgflftjzIe-MII5ffsjdsdR
ApplicationsofspectralDistributions l45 theFouriertransformationandby于=717=fL·罗fegj.WesaythataspectraldistributionEisofdegreelifEcanbeextendedasalinearcontinuousmappingon苦lequippedwiththenormnf/jz了HskfιNfll1L,dsL。2EKZEt Theorem2.InaHilbertspaceH,ifEisGspectraldistribtdionsuch that HEfψjHSH罗ψ||L1, qpeD (l) GFtdforanyψpositivefunctionin苦,EfWisself-adjoint-ThenEftjisOGself-Gdjointscalartypespectraloperator. EEg♀LForanyτ主OletusdefineT(τ)=E(ψ)whereψ(I)司'πτtdOτandIIO(ψτ)=1.Theproperties(i)and(ii)ofspectraldistributionεimplythatT(τ)isaC。-semigroup-Sinceψτftj主OforanyteRandfrom(l) --、』,/τmv/'-on〈-kz'''τewJ''飞εzt』,,,τ/'-T thissemigroupisofcontractionself-adjointoperators-ThereforeBthegeneratorofTisalsoself-adjoint(see[G,corollary45.]).AccordingtotheLumer-Phillipstheorem-Bism-aceretive.LetusdesignatebyAtheuniquenonnegativesquarerootof-B.Aisalsoself-adjoint.ThefinalpointistoprovethatAzEfd百tjJthisoperatorism-aceretiveandbyuniquenessofthesquarerootitsufficestoshowthatA2zEfπt2j.ForxeHandψasin(ii)wehaven-πτt22·1-πτt|iEffπt-τfe-ljjψ/x||三Hπt-τfe-ljH∞HfWl|X|l,(2)nLO 2·1-πτISIncet→πt-τfe-ljisaPOSItivefunctionandH(ψ)=H仰).LetOnτ→0; weget Efπt2ψjxzA2Efψj-XZEfψjA2Xnnn2....22foranyxεDfAj.TaKIngn→∞,thISImpllesAcE(πt).ConverselyforxeDfEfπt2jjandn→∞themequalIty(2)yieldsthat-πτt22-1-πτI2llEfπtjx-Efτfe-ljjxllsHπt-τfe-ljHL∞IIofψjilW1,
I46 EmamiradandJmar- sincet→τ-b-m-ljbelongstof0·HenceEfπt2jxzlimEfτ-1fe-πτt-ljjxτ→oandthereforexεDfA2j.. Remark1.IntheproofoftheTheoreml(preciselyforfiij斗fiiijj,wedefineforany/eP,「k(k)Ef/jz|(-l)(?/)(t)C(t)dt(3)ιRandweprovethatEisaspectraldistributionoforderk.Itisworthwhiletonotethatfork=0,thisdefinitioncoincidewiththoseof[D;Chapter81 Corollary1.InTheoreml,iftheambiantspaceXisaHilbertspacewithCzlandkzOinfiij,thentheequivalencebetweenfiijandfiiijnthestandardversionoftheStone'stheorem.E旦旦LAO-timesintegratedgroupisaCFgroupandC=limpliesthatthisC。-groupisofisometries-Fork=oandC=1,(3)yieldsHE(/)HS||罗/|lL1.*FurthermoreC(t)=C(-t)andforanypositivefunction/in步。wehave 〈ε(/)XJ〉=〈jwω =叽jW蚓附(0SwhichgiveusthepossibilityofapplyingTheorem2andsayingt由ha创tAi臼saself-adjointoperator.. 3.APPLICATIONS A1.TransportOperators. LetTobetheadvectionoperatorontheBanachspaceX=Ll(Rn×Rn),TO:巾,v)εX→TOKXJ)=-v.VJ(XJ)Itiswellknownt仙ha剖t仙operatorgeneratesagroupofibsOImIn1e创trdie臼s[U(οt)/凡](μx,川v吵)=/(μx-t川v,1v吵)onX.FurthermorethespectrumofTisσ叫(T)抖=σ(T)扣=im!RR,whereσ(A)istheOOrO
ApplicatimsofSpectralDistributions I47 residualspectrumofA(see[H]).ButtheresidualspectrumofascalartypeoperatormalwaysVOId(see[DUD-SotheremnospectralmeasureforwhichTOisascalartypeoperatorinx.AsimilarargumentisgivenbyRicker[R]fornon-spectralityoftheLaplaceoperatorinLP(RL IInthespiteofthefactthatthisoperatorisnotspectralofthe16scalartypeinL(R),Theorem1isapplicablewithC=landk=0,soTO= -v.v-Xadmitsaspectraldistributionofdegreezero. nILLetS={veR;0〈V-s|vlsv〈∞}beasphericalsellinnunznaxRn.ThepenerationtransportoperatorisT=T+A,whereAf(x,v)=1Ol-σ(x,v)/(x,v)andD(T)=D(T).LetKbeacompactsubsetofRn.Wea1Oassumethattheabsorptionrateσ(x,v)isameasurablepositiveboundedafunctionwhichvanishesoutsideofk×S.TheboundedperturbationtheoremimpliesthatT1generatesac。-groupUJt)inLI(Rn×SLwhichcanbeexplicitlyrepresentedby[U.(t)/](XJ)=expli-σfx-sv,vjdslffx-tv,vj.ittaI、"OJItfollowsfromestimateat.+∞||σfx-sv,vjds|三|v|'1|σfx-sv/|v|,vjdsIaEawow-∞三||σ川-fdiamkj/V.zC,au司EIEEn that Ifort兰OHU(t)H54lxpfcjfort〈OOncemoreTheorem1showsthatTIadmitsaspectraldistributionof1nthedegreezeroinL(R×S).Aparticularcaseofthisisthetransportoperatorwhichintervenesinthereconstructionofimagesfromprojectionsincomputerizedtomography(see[K-L-H;p.316]). III-LetusdefinethemultiplescatteringtransportoperatorasaperturbationofthepenetrationoperatorT.bytheproductionoperatorA2fM=jvkdvm巾'nwhereViseitherRorS.WeassumethatσpMzjfv州
I48 EmamiradandJazar hasthesamepropertiesofσ,andwedefineaMfσj.·zesssupfσfx,vj/|v|;仰,vjεRn×Vj and Ffσjfzesssupflσfx+tv,vjdtJfx,vjeRn×VjιRwhereσiseitherσorσ.Umeda[U]showedthatifexpfffσjjfdiamapakj儿ffσpj〈1,thenT=T+AgeneratesaequIboundedCo-group.Thus,Theorem121isalsoapplicable.AnothercaseforwhichTgeneratesalsoaequiboundedgroupmwhenTgeneratesthedynammsofafinitecollisionsystemintransportprocesses(forthedetailssee[E;Theorem1.4]). A2.TheSchr6dingerOperator. ThestudyofSchrbdingeroperatoriAinLP(Rn),p#2isalwaysmotivatedtheconceptsofsmoothdistributiongroups(see[B-El]and[B-E3]),integratedsemIgroups(see[A-K],[Hi],[K-HIand[N])andexponentiallyboundedC-semigroups(see[P]).Itisshownthat(seepn..2,pn[B-Ell)theoperatoriAinL(R)WMhdomamW(R)generatesasmoothdistributiongroupoforderk主nll/p-l/匀,henceitadmitsaspectraldistributionofthesamedegree. A3.TheWaveEquationOperator. Weconsiderthewaveequationinthefollowingform duZTziAU AzlL71 InordertoshowthatAadmitsaspectraldistribution,letusrecallthefollowingtheoremwhichisadirectapplicationoftheTheorem6,of[B-E21 Theorem3.LetAf￡jbeanN×Nmatrix-叹Glued/unctiononRn,ninfinitelydifferentiableonRIfOj,positivelyhomogenousofdegreeα〉0,Gnd/orDzf-iδ/δx,...,-iδ/δxj,AfDjistheoperatorfromSchwartznNspace/YfIRj/irtto/YYRj/de/inedby/:FfAfDjψj/fUzAfxj/iFψ/f￡j.TheniAfDjgeneratesGsmoothdistributiongroupoforderkinfLPflRFIj/N,/or
ApplicationsofSpectralDistributions l49 k兰n|l/p-1/2|fhenceAfDjisthemomentumofaspectraldistributionoforderkj,ifAfEjsatis-fiesthefollowingassumptions--(H1)AfEjisdiagonalizable--i.e.denotingbySIheunitspheren-1‘'‘'。fRn,的eremsttwoFouriermultipliermatricespfejandp-kjin;p×川andNE∞-mappingsλλhfromStoCsuchthat1'2,...,NvI l,/户ζ/''"'AP叮EBB-,,,,EEEdh,,,,产气ofN飞八、,/pζ/'110飞八FEEEEE』'''''』飞,,,,C与/,『飞pts/C飞/'-A Ees-nei (Hnforanyizl,...NGManyeesi叫 ForthewaveequationoperatorA, 『E·E·-··EJ￡OOE「EBB-L=户ζA and PfejzIl/d|l/d -ij习 A4.TheLinearElasticityOperator. Thelinearelasticityequationmaybewrittenas 。=u、..FAU句&饨,"cn,.4''-nOJJ'句,.no,,..、、..,,AU04'AC。66''-RU,JI句za到U,,.‘、 (forthederivationandmorephysicaldetailssee[C-H;p706]).Inordertoapplytheprevioustheoremwerepresentitina4×4matrixevolutionarysystemδU/θtziAfDjU,wherethesymbol IOb.fUObjrul|bf￡jO晶bfejOA|Afejzl/2|12||ObJ￡jObJU||b2fej0·b1fEjo-lwithb1f￡jzfCJC2j|笔|andb2fejzfcfc2j|引.Thismatrixisdiagonali灿lebyaconstantmatrixp.Wefindλ严jz-λJEjzC1|言|andλ3fEjz-λJ￡jzC2|￡|.HencetheTheorem3,viaTheorem1,showsthatAfDjisthemomentumofaspectraldistributionofdegreek主n|l/p-l/2|in[LP(Rn)]4.
l50 EmamiradandJazar REFERENCES {Al]W.Arendt,Resolventpositiveoperators,Proc.LondonMath.Soc-54(1987),32l-349.[A2]W.Arendt,VectorvaluedLaplacetransformsandCauchyproblems,IsraelJ.Math.59(l987),327-352.[A-K]W.ArendtandH.keHermann,IntegratedsolutionofVolterraintegro-differentialequationsandapplications,semesterberichtFunctionalanalysis,TtlbingeF1.Sommersemester1987.[B]M.Balabane,QuelquespropositionspouruncaKIllsymbolique,Thesed'EtaLUniversitddeParis7,1982.7,1982.{B-El]M.BalabaneandH.Emamirad,SmoothdistributiongroupandpnschrodingerequationinL(RLJ.Math.Anal.APPI-70(l979),6l-7l [B,E2lM.BalabaneandH.Emamirad,Pseudo-differentialparabolicsysteminLP(IRn),"Contributionstonon-linearP.D.E.",Re5earchNotesinMath.N。89,Pitman,NewYork,(l983),[B-E3]M.BalabaneandH.Emamirad,LPestimatesforSchrodingerevolutionequations,Trans.Amer.Math.SOC-292(l985)357-373.[C-H]R.CourantandD.Hilbert,"MethodsofMathematicalPhysics,,VolIIWiley,Interscience,NewYork,1962.rk,1962.[D]E.B.Davies,"One-ParameterSemigroups",Acad-Press,LondoF1,l980 [Del]R.deIdaubenfels,Integratedsemigroups,C-semigroupsandtheabstractCauchyproblem,semigroupForum(toappear).[De21R.deIdaubenfels,Integratedsemigroupsandintegrodifferential [De3] equations,Math.Z.(toappear).R.deIdaubenfels,Polynomiaisofgeneratorsofintegratedsemigroups,Proc.AMS.(toappear).N.Dunford,Spectraltheorey1ntopologicalvectorspaces,"Function,serles,operators"Colloq.Afath.Soc-JGF105Bolyat, [Du] 35,North-Holland,Amersterdam,i983,39l-422.39l-422.[E]H.Emamirad,ScatteringtheoryforlinearizedBoltzmannequation.Surveyandnewresults,TransportTheoryStatist.
ApplicationsofspectralDistributions l5l [F] Phys.16(l987),503-528.H.EmamiradandM.Jazar,SpectraldistributionsandgeneralizationofStone'stheoremto自LVhu--··Banachspaces.submittedtoJ.Math.Anal.APPI-C.Foias,Uneapplicationdesdistributionsvectoriellesala [E-J] theoriespectrale,Bull.Sci.Math.84(l960),147-158.[G]J.A.Goldstein,"Semigroupsoflinearoperatorsandapplications",Ox/ordUniversityPress.NewYork,1985.rk,1985.[H]J.Hejtmanek,Dynamicsandspectrumofthelinearmultiple133kscatteringoperatorintheBanachlatticeL(R×R丁,TransportTheoryStatist.Phys.16(1979),29-44.[Hi]M.Hieber,IntegratedsemigroupsanddifferentialoperatorsonLP,Dissertation,Tbbingen,1989.en,l989.[K-L-H]H.G.kapehC.G.LekkerkerkerandJ.Hejtmanek,"Spectralmethodsinlineartransporttheory",BirkhduserVerlag,Basel,1982.[K-H]H.keHermannandM.Hieber,IntegratedsemIgroups,J.Fund-Anal.84(1989),l60-180.l60-180.[Kl]B.kritt,Atheoryofunboundedgeneralizedscalaroperator,Proc.AAfS,32(1972),484-490.484-490.[K2]B.kritt,Atheoryofunboundedgeneralizedscalaroperator,Proc.AAfS,35(1972),74-80.),74-80.[N]F.Neubrander,Integratedsem1groupsandtheirapplicationstotheabstractCauchyproblem,PacificJ.Math.135(1988),l11-155. [P]M.M.H.Pang,ResolventestimatesforSchrbdingeroperatorinLP(Rn)andthetheoryofexponentiallyboundedC-groups,SemigroupForum,(toappear).[R]W.J.Ricker,FunctionalcalculifortheLaplaceoperatorinLP(町,MiniconferenceonHarmonicAnalysis,Conberra(Australia),Proc.CentreMath.Anal.15(l987),242-254.[U]T.UmedASmoothperturbauoninorderedBanachspacesandsimilarityforthelineartransportoperators,J.Math.Soc- 38(l986),617-625.
SomeGlobalExistenceandBlow-upResultsforsemilinearParabolicSystemswithNonlinearBoundaryConditions JOACHIMESCHERMathematicalInstitute,UniversityofZurich,Zurich,Switzerland Inthisnotewereportsomerecentresultsconcerningtheasymptoticbehaviourof solutionofthefollowingsystem(cf.[7])θtti-OJ(αJKθku)+αou=f(u)1nQ×(0,∞),αJKUJδKU+bu=g(u)onL×(0,∞),(A)U=0onr2×(0,∞),u(·,0)=uoonQ. HereQisaboundeddomaininRn,n;21,withsmoothboundaryθQ,F1andr2arebothopenandclosedsubsetsofθQwithr1UF2=θQandrInr2=的aIIdν=(川,...,νn)istheouternormalonOO.Forthecoemcentsweassumeαjk二αKJ=(αkj)T,αo=(αo)Tεc2(否,ζ(RN)),1三j,k三n,b=bTEC1(r1,ι(RN)), 、‘,,/''AJ''E飞、 l53
I54 ESCher whereι(RN),N主1,denotesthealgebraofallreal(N×N)-matricesandαTstandsforthetmVu叫adlh1时functionsU=(u1,...,UN)and(A)iswrittenasastronglycoupledsystem.Furthermoreitfollowsfrom(1)thattheellipticpartof(A)isformallyself-adjoint.ForsimplicityweassumethattheuniformLegendreconditionissatided,i.e. α;;(z)〈f〈?>0,ZE否,〈=(〈j)εRnN＼{0}. (2) Furtherlet (αo(z)5|5)主0αnd(b(u)5|5)主0,ZEQ,UEr1,5εRN Forthenonlineartermsfandgweassumethatf,gEC1(RN,RN)and,ifn主2,thatthereexistconstantsαε[1,ri),9ε[1,去言)with NRζ」户户、3··AAμ、‘,,,,,rp、,+''A,,..‘飞FL<一、‘.,Fprp、J,..‘、nyno--Aα、‘,,,,F,井、+'EA,,,,‘飞FL<一、、..,,,rp、,,,..‘、、FJ兔U Toinvestigateproblem(A)itisnaturaltoassociatewith(A)aseIdimarCauchyproblemoftheform 也+Au=F(u),u(0)=uo、 (B) whereAisanappropriaterealizationofthediEemltialoperatorappearingin(A)andFisasumcentlysmoothnonlinearoperatorwhichcontainsnotonlytheinhomogenityfinthedomaiIIQ,butaIsothe"boundaryterm,,g.ItfollowsformthebasicworkduetoH.Amann(cf.[3,4])thatsucharealizMionispossible(eveninmoregeneralsituations)andthat-AgeneratesananalyticSem川igr阳OuO∞ntheHilbertS叩PaCωeE一→1:==(σlη,βB以(Q)门)FwhereWJ,B(Q):={uεWJ(0,RN);u|F2二。}.Furt}阳mOIeitisshowninHithatthereisaone-to-onecorrespondencebetweensolutionsof(A)and时11tiomof(B)andthatforeachuoε1418(Q)theredI山auniquesolution岭,uo)of(B).Moreoveru(·,·)deHIlesalocalseIIlinowonIVJ,8(Q)suchthatboundedorbitsarerelativelycompact.HenceitfollowsfromthegeneraltheoryofseIIlinowsthatasolutionUof(A)existsglobally,providedanaprioriboundforUinWJ(Q)isknown.IIIthefollowi鸣wespecifyconditionsonf,gandtileinitialdatauowhichimplysuchanaprioriboundinWJ(Q).Tothisendweassumethatthereexistfu盯tiomO,ψεC2(RN,R)suchthat 功(0)=ψ(0)=Oandf=Vφ,g=Vψ. (3)
GlobalExistenceandBlowupResults l55 ThenwedeEmthepotentialenergyof(A)by 只hjα川 (4) HereαdenotestheDirichletformassociatedtotheellipticpartof(A)J.e. 们,hf(川 andPisapotentialoperatorgivenby nhfMZ+户(仇 (6) UsingthisenergyfunctionalEitispossibletoconstructaboundedandpositivelyinvariantneighbourhoodMoftheorigin-aSOcalledpotentialwell.Thusasolutionexistsglobally,providedwechoosetheinitialdatauoinAf.OntheothersideitiswellknowIIthatsolutionsofsemilinearparabolicproblemsdonotexistforalltimeingeReral,SOthatblowupphenomenaoccurs.Alsousingsomeenergyargumentsandthesocalled"concavity"method(cf.[14)weshowtl时tilesolutionof(A)mayblowupinhitetimeevenifonetermontherighthandsidehuadampingefeet,andsecondlythatthereisanotherinvariaMsetNsuchthatu(-Jo)blowsupinanitetimeifuobelongstoN.Denotingbyt+(uo)thepositiveexittimeoftheinitialdatauoandbyu(·,uo)themaximalsolutionof(A)onJ(tAo):=[0,t+(uo))wehavethefollowing THEOREMl:向jSupposethereisacomtαntk主Osuchthd NRC」户户、。"户户、-K<一、‘,,,一户户、、、..,,,rr、,,,.‘‘、03,,..‘‘、9·rp、'比<一、‘,,,,rr、3、‘..,,,rp、''''飞、rsJ,,a·‘、、 Thent+(t川=∞foreαcflUOEWJs(Q)pi.e.u(-Jo)iSGglobαlsolution. fbjAssumethM 。(5)ι-(f(5)|5),ψ(5)<-L(g(5)|Chrsom7,6>1(7)γ一1+6 αndthαt dtf
l56 ESCher Thenthesoluti071blowsupinFFlitetime,thMis, t+(uo)<∞αnd||u(t,uo)||wJ(Q)→∞αst↑t+(uo), proMdedthereisαtoEJ(tto)suchthαtu(to,uo)并0αndE(u(to,uo))<0.MO陀ouerifeither 。U>一、、.,,,rr、、‘..,,,,r、,,,,、飞。34'E‘、、JUnαnu>、llr、、、..,Frp、,,..、、FFJ,,..‘、 OT (9) 、EFJnurs、飞、飞飞、NRr℃r户、nυ>、、..,,,rc'、、、..,,,卢、,,..、、03,,.‘飞,anαnu>一、1IFr户、、‘.,,yrp、4,..、飞rtdJ,,,‘飞 thmygiuenαnyuoεWJn(Q)＼{0},thεTCezistsso>1suchthME(sco)<OfOTS主So. ForaproofofTheoremlwereferto[7,Theorem1.1]. Nextweassumethatoneofthefollowingconditionsholds: (i)r2并的, (ii)thereisasubsetQoofQwithpositivemeasuresudlthat (αo(z)引5)>OforZE50,5εRN＼to}, )AU喃自AP,,..‘、 (iii)tlmeisasubsetFoofuwithpositivemeasuresuchthat (b(u)5|5)>OforyEFo,5εRN＼{0}. ''A>4μαemonbv且。俨TA&EUa-n&ELV'dnayG·B·-··OLH、‘..,,VA可υ,,..、、、1,F同i4''t、、、、..,,qdJ'EE‘、、&ELUa'且6EUecoOPAPAUPDOLweviomVAρL、...····A&EUTAUF NRCLrr、-sμE'、C<一、IE/C'、,,..飞nyα卢、户L<一、‘lff'、/,a,、、、rId )''l唱'EA,,..‘飞 Notethat(11)impliesthattheoriginisacriticalpointof(A).NowwewanttoconstructthisboundedandpositivelyinvariantneighbourhoodMofthecriticalpointOmentionedabove.IfsuchaIIOIItrivialMexists,weknowthattilesolutionwithinitialdatauoξJLfexistsglobally.OntheotherhandweKIlowfrompart(a)ofTheorem1thatthesolutionblowsupinfinitetimeifE(uo)<0.ThereforethebestdloiceforMwouldbetllesetofailuoε117,6(Q)forwl山lIE(uo)主0.
GlobalExistenceandBlowupResults l57 ljnfortuIladythissetisneitherboundednorinvariantingeneral.Butthefollowingresultholds(cf.[7,Theorem1.2]): THEOREM2: fWd:=inf{E(u);uεWJn(Q)＼{0}ωthF(u):=VE(u)u=0}>0αndMd:={uEIVJB(Q);0三5(u)<d,F(u)三0}isαnopenboundedMigftbottrhoodoftheorigin.Mdispositiuelyin川、iGntunderthesen3177owu(·,·)αssocmtcdwith(A)αndoisαηαsgryzptoticdlustαbleequilibriumofu(·,·).Furtf旧、mo阿ω(uo)={0}fOTtioεMd,ωKTeω(uo)dmoksthcω-limitsdofu0· fbjNd:={uEw;6(Q);E(u)<d,F(u)<0}isdsoinmriαnttiMeru(·,·)-lfyinαdditi071, θf(0)=θg(0)=0,(5|f(5))三(5|θf(5)5),(5|g(5))三(5|θg(5)5),EERN,(12) thenu(·,1川bloωupinFMtetime,PTOMddtioENd· TogiveanexampleofTheorem1andTheorem2assumethaltN=2;λ,b主0;ε。,ε1εR;α,应,γεcv2(否,R)withα(z)>Oand(α7-92)(z)>0,zε5;pε(1,rj);qE(1,王E)(ifn主2)andco,问ξIf?(QR)＼{0}.Thenweconsiderthefollowingsystem 。tty-diu(αVu+dvw)+λu=εo(u2+ω2)P-1U}θttu-diu(γVw+SVu)+λω=εo(J+ω2)P-1wj(αVu+9Vw|ν)+bty=ε1(u2+ω2)q一1U1(γVw+PVu|ν)+btu=ε1(u2+w2)q一1ω)(u,四)(·,0)=(Uhwo) lnQ×(0,∞)‘ onOQ×(0,∞), onQ. InthissituationTheorem1showsthatthesolution(飞ω)existsgloballyif句,ε1三0.IfE(tYo,Wo)<Oandeither(p-qko主Oor(q-pk1主0,then(u,w)blowsupinaI山etime.Ifweassumethat入+b>Oandε0,ε1主Owithεo+ε1>OthenitfollowsfromTheorem2thatfor(句,Wo)εMdthesolution仙,ω)existsforallt主Oand(飞ω)(t)→OinIVJ(Q)ast→∞.For(tyo,叫)εNdthe叫utionblowsupinhite time.
I58 ESCher REMARKS:(a)Detailedworkontheasymptoticbehaviourofsemilinearparabolicproblemshasbeendonebymanyauthorsusingdiπerentmethods,cf.[2,5,6,9,10, 11,12,131. Ourresultsgivesomenewcontributionsinceweallownotonlynonlineartermsonthedomainbutalsononlinearboundaryconditions.Furthermoreoneshouldkeepinmindthat(A)isastrong1ycoupiedsystem.Thereforeallresultswhicharealreadyknownforsingleequationsorweaklycoupledsystemsandforwhichthemaximumprincipleisessentialdonotcarryovertooursituation,sinceitisknownthatforstronglycoupledsystemsthereisnogeneralmaximumprinciple. (b)AsalreadymentionedaboveweobtainTheorem1andTheorem2bystMyi吨theabstractCa时1yproblem(B).Hencesecondorderparabolicsystemslikesystem(A)areoniyonecasetowhichtheseabstractresultsapply.Forexamplethesamemethodsworkforparabolicinitialboundaryvalueproblemsofevenorderinwhichtheellipticpartisformallyself-adjoint.(Observeagainthatevenforsimplequαtionsofhigherorderlikeθtu-A214=finQ×(0,∞)withhomogeneousDirichletboundaryconditionsthereisnogeneralmaximunprinciple.)FurthermoreTheorem1canbegeneralizedtotimedependentproblems. (c)SupposethatN=1.Thencondition(3)isalwayssatinedsimlpybytaking￠(hff(S)巾,咐:=f仰)ds,EER If,inaddition,fandgarepositivelyhomogeneousofdegrt男7and6,respectively、thenassumption(7)isalsosatisfied. (d)TIleideatouse"potentialwells,,toproveglobalexistencewasnrstintroducedbySatti鸣er[15lforamIIlimarwaveequationwithhomopmomDirichletbouMaryCOMitions-Likewisein[15ltheconjectureismentionedthatthismethodSdIhlO1u川1dl(dia叫lSωOworkfOrIp}油阳a剖IabOdlikCP时〉山归ikhe凹ImIresultsinthisdirection.Timeintileauthorsconsiderproblem(A)intilesituationwhereN=1,f=0,αjk=6JK,αo=OonQ,b=OonF1andF27tO.Sincethegrowthandstructuralconditionsongaresimilartotheonesabm飞tileresultsin[14lareaspecialcaseofTheorem2.Globalexistencein[14]isprovedbyapproximatingtl四solutionbymeansofeigedInc--tionexpansions.Duetoourgeneralapproachweareinapositiontogiveacompletely
GlobalExistenceandBlowupResults I59 diferentproofwhichfurthermoregivesaratherprecisedescriptionofthedynamicalpropertiesoftheseminowu(·,·)nearthecriticalpoint0. (e)NotethatTheorem2iscloselyrelatedtotheprincipleoflinearizedstability.Indeed,itcanbeshownthateachoftheconditionsin(10)impliesthatthereisaω>Osuchthatσ(-A)c{zεC|Rez<一ω}(cf.[7,section3]).ThereforetheprincipleoflinearizedstabilityshowsthatOisauniformlyasymptoticallystableequilibriumandthereexistc,T>Osuchthatllu(t,140)||wJ(Q)三cf叫|tio||wJmfort主Oand||uo||wJ(Q)<T.HoweverTheorem2givessomemoreinformation,e.g.thatMdliesinthedomainofattractionofOandthatr<dist(0,Nd). (f)TheproofofTheorem1andTheorem2usesessentiallyanewregularityresultofthesolutionof(BLwhichcanbemotivatedbythefollowingconsideration.LetEbearealBamchspaceanddenoteby(·,·)E:EF×E→RtiledualitypairingbetweenEr-thedualofE-andE.NowwesetE11w;B(则,Eo:=L2(0)andweidentifyEjwithE0·TUihlmeHahn一BaInlaCdIh1theoremimplies E1〕EoL→E-1and (u,u)EI=(UJ)Eo,UεEo,UεE1· (13) FurtherwementionthattheDirichletformαconnectsproblem(A)withproblem(B)bytheformula(cf.[4,section10]) α(u,u)=(Au,u)E1,U,UEE1· (14) NowdenotebyUthesolutionof(B).Then,using(13),(14)andthefactthatPisanpotentialoperatorforF,wecalculateformdluthetimederivativeoftheenergyfunctionalE(u(·)): 在(u(t))=们(川))一川t均州州)川队川)L,=(Au(归例tO)一F(u(0例tO)川),A也(0t)川)E1=-(也(t),也(t))Eo=一||也(t)||毛。三0. (15) ThiscalculationwouldshowthatEisaLjapunovfunctionalfortheseIIlinowu(·,·)-SiIIcethedomainoftheenergyfunctionalEisE1aIIdsincethegeneraltheoryof
l60 ESCher semigro叩SonlygivesthatUEC(J(uo),EOnc1(J(uo),E一1)weshowinthefol-lowingthatUisdifrerentiablewithrespecttothetopologyofE1tojustifytheformalmanipulationabove. Tothisendweput几:=(E-1,E1)z,εε(0,1),whe刊(·,-hdenotesthecomplexinterpolationfuIICtor-Furtherwedenotebyζ(E,F)theBamchspaceofallboundedlinearoperatorsfromtheBanachspaceEtotheBanachspaceF.Nowassumethat一Ageneratesananalyticsemigro叩{ftA;t主0}onE一1 with||ftA||三Me川,t主OforsomeM主1,ω<0.FECI(EulQforsomeεε(0,1/4). (16) (17) From(l6)andawenkmWIIcharacterizationforgeneratorsofanalyticsemigro叩S(cf.[8,TIleomII4.1.5])weconclude 。U>a7ι'EAa7bFL<-EEPLAρLA (18) InthefollowingCdenotesapositiveconstantwhichmayvaryindiferentestimates-SinceZH||A叫|E-lisanmp山'alentnormonE1itfollowsfrom(18)that nu>a,,.咽,.'a''bc<一、,FEE,LAρL 、,,FQd'EAv,,,,‘飞、 Ontheotherhandwealsohave llftA||ζ(EhE1)三C,t主0. (20) Thusbyi川erPolationweobtainfrom(19)and(20) nu>44...-EAEbc<-E几f··~AFlv (21) Itshouldbementionedthatestimates(19)一(21)areSPNialcasesof!4,TlmmII81]-Nowwenxto,6>Oandput ω(t,h):=u(t十h)-u(t)-u(t)h,t主to,|h|<6, whereudenotestheuniquesolutionofthelinearimdCauchyproblem 台+Au=FF(u(t))υ,t>旬,。(to)=F(u(to))-Au(to).(C)
GlobalExistenceandBlowupResults l6l Thenaccordingtoassumption(17)weusethemeanvaluetheoremforFandinequality(21)inthevariationofconstantformulafor(B)州)=fMUO+ffh)A川队t主0,toobtainthefollowingestimate(cf.theproofTheorem2.3in[71)||川)||山川cf川←1||川)||川t三川 where,againthankstoestimate(21),α(-,h)isintegrableon(to,T)foreachhE[-6,叫,Tε(旬,t+(uo))aMα(t,h)=o(|h|)foreachtE(to,t+(uo)).NowweapplyageneralizedGro川allinequality[1,Lemma2.2ltoestimate(22)andconcludethatfortε(to,t+(uo)) nu→'hMPDa、‘.,,,'hJ,,.‘飞。一-E、、‘,,F'n4Etw,,,,‘飞ω Summerizingweobtainforthesolutionuof(B)(cf.[7,Tlmmn2.3]):UEC1(J(uo)＼{OLE1)and也istheuniquesolutionoftheiimarkedCauchyproblem(C).Itshouldbementionedthattheassumptionthatthesemigroupgeneratedby-Aisexponentiallystablein(16)wasoniyiMrohmdforsimplicityaMcanbedropped. (g)Finallyweshow,asanapplicationoftheconsiderationabove,thatunderthehypothesesofTheorem2theω-limitsetforuoεMdisgivenbyω(uo)={0}.DenotingbyDE(tto):=limiI山→o(E(u(t,uo))-E(问))/tthederivativealongorbitsofzatuo,itfollowsfrom(15)aM(13)that DE(u(s,uo))=一||也||L。三-c||Au(s)-F(u(s))||L-1'Sε(0,t+(uo)). SinceEisaLjapmovfunctionalwehave(cf.[16,TheoremIV,1.1l):们(tJO))-ho)斗机(SJO))ds,tε[OJ(uo))andthereforebythecontinuityof||Au(·)-F(u(·))||E-1 DE(uo)三-c||Auo-F(uo)||乞1'ttoεE1· ConsequentlytheinvarianceprincipleofLaSallegivesthat ω(tio)C{UεEUPE(u)=0}C{UεE1;Au-F(u)=0}.
I62 Est-her BythedeRMUonofFandfrom(14)itfollowsthat ω(uo)c{uεEUF(u)=0}. (23) Nowtakeuo巳MdandsupposethatthereisauEω(uo)＼{0}.Thenthereissequenceh→∞withu(儿,uo)→u,k→∞.SinceEisdecreamgalongorbitswehaveE(u(tk,uo))三E(uo)<dandthereforeε(u)<d.Ontheothersided=inf{E(u);UεE1＼{0}withF(u)=0}whichgives,accordingto(23),thecontradictionE(u)主d.Thisshowsthatω(tto)={0}foruoEAfd·
GlobalExistenceandBlowupResults l63 Referezlees 1.H.AMANN:Periodicsolutionsofsemilinear-parabolicequations.In"NonlinearAnaIysis:AcollectionofPapersinHonourofErichH.Rotlle".AcademicPress,NewYork,1978,1·29. 2.H.AMANN:Globalexistenceforsemilinearparabolicsystems.J.reineu.a鸣ew.Math.366(1985),4184. 3.H.AMANN:Semigroupsandnonlinearevolutionsequations.LinearAlgebra&APPI-84(1986),$32. 4.H.AMANN:Puabolicevolutionsequationsandnonlinearboundaryconditions.J.DiEEq皿72(1988),201369. 5.J.M.BALL:Remarksonblow-upandnonexistencetheoremsfornonlinearevo-lutionsequations.Quart.J.Math.,OxfordSeries,28(1977),473486. 6.M.CHIPOT,F-B.WEISSLER:Someblowupresultsforanonlinearparabolicequationwithagradientterm,toappear- 7.J.ESCHER:Globalexistenceandnonexistenceforsemilinearparabolicsystemswithnonlinearboundaryconditions.Math.Ann.284(1989),285305. 8.H.0.FATTORINI:TheCauchyProblem.Addison-Wesley,Reading,MA,1983. 9.M.FILA:BoundedIlessofglobalsolutionsfortheheatequationwithnonlinearbmMaryconditions.Comm.Math.Univ.Carolime30(1989),479-485. 10.A.FRIEDMAN,JB.McLOED:Blow-upofpositivesolutionsofsemilinearheatequations.IndianaUniv.Math.J.34(1985),425447. 11.H.KIELHOFER:Globalsolutionsofsemilinearevolutionsequationssatisfyingaenergyinequality.J.DitEqm.36(1980),188322. 12.H.A.LEVINEL.E.PAYNE:Somenonexistencetheoremsforinitial-boundaryvalueproblemswithnonlinearboundaryconstraints.Proc.Am.Math.Soc-46 (1978),277384. 13.H.A.LEVINE:StabilityandinstabilityforsolutionsofBurger'sequationwithanonlinearboundarycondition.SIAM.J.Math.Anal.19(1988),312,336.
I64 EJECher- 14.H.A.LEVINE,R.SMITH:Apotentialwelltheoryfortheheatequationwithanonlinearboundarycondition.Toappear- 15.D.H.SATTINGER:Onglobalsolutionofnonlinearhyperbolicequations.Arch.Rat.Mech.And.30(1968),148172. 16.J.A.WALKER:DynamicalSystemsandEvolutionEquations,TheoryandAp-plications.PlenumPress,NewYork,1980.
Time-DependentScdIhEuESimulatedAEnmEHInEeaIHiInEg ALBERTOFRIGERIODepartmentofMathematicsandInformationSciences,Uni-versityofUdine,Udine,ItalyGABRIELEGRILLODepartmentofMathematicsandInformationScience,Univer-sityofUdine,Udine,Italy 1.INTRODUCTION LetU:Rn→[0?∞)beaC2function-ItwillhelpintuitionifVisinterp时edas(poteIltiaLl)PIlergyofsomeEctitiousphysicalsystem-Apointmovingac-cordi吨totilepquationdzt/dt=-VIfm-iilclimbdomthe"emrgylaMscape刊{(叽Z)εRTT+1:z二〔T(z)}untilitmclmalocalminimumofU.Ifwewishtoallowhtoescapefromlocalminimaandvisittheglobalmilli111aofff,飞vemayakisomrsmallmMomco川ributioIItodzt/dt(tlm111aihctmtiom)an l65
I66 FrigerioandGrillo COIlsidertlwstochasticdiferentialequati011 dzr=-VU(Zt)dt+[2T(t)lid叫?Zo=ZF )唱Ei.唱'A,,a··、、 wlm-e{tut:tε[0,∞)}isTI-dimemionaistaMEM-dBrownianmotionandT(t)>Oistobei川erpretedastemperatureattimet.IfT(t)iscomunt(=1/F)andZ(9)=je一βUdz<∞,thenfora町initialconditionzεIRVIZLIIEiforanyBOrelsubsetBofIRηonehas且已ProbhtεBl=PβIBl=Z(9)-1jfβUdz;(1.2)JBTilemeasurePβdescribesthermalequilibriumattemperatureT=1/βfortileemrgyfuI川ioIIIf.If9islarge(temperatureTnearabsolutezero),thentilemeasurePβisconcentratedneartheglobaiminimaofI7.Thenone111ayllopethat?bylettiIlgT(t)increaseto∞ast→∞,onecanobtain 2:LProbiff(Zt)三mIIU+6]=Oforail6>0.(1.3) However,itlhlaSlb}e何e1n1SlhlOwIn1tih1a剖ttileratea剖twihM1diCih1tulhlernInm1口1a剖leq1u山1札i1U川i川lb〉r口i1u1Inm1口1isaPPrmOaCie一βm'where,atleastincertaincircumstances,TYlmaybeinformallydecribedastllemaximumdepthofalocalminimumwhichisnotaglobalminim11111(seebelow).TileIIitisappare川thatT(t)mustmtbereducedtooquickly:tileactitiousphysicalsystemwhoseenergyfunctionisUmustbeannealed,11otquenelled.Thisprocedureiscalledsimuhtedα7171ealingorLα71geumαlgw·f-fltn1.Acomputersimulationofequation(1.1),withT(t)gentlydeereasi鸣toOast→∞,isoftenL15edinpractice,inordertolookforaglobalminimumofacomplicatedfunctionU.Resultsofumkindof(1.3)havebeenobtainedbyseveralauthors1III-derSOIlletechnicalassumptionsOIlffaINiundertileconditiontilatT(t)二三c/lII(1+t)?wiler1lleCOIlstaMCislarger(oratleasttIIotsmaller)than711.WeshallmentionAium-Pentinidd.(1985)、GemallandHwang(1986),Gicias(1985),Clm吨dι(1987),HoneyaMStmok(1988LTfolkydd(1989)-TIlepresentworkisatechnicalvariationoftileaboveresults;basicali3气wewantedtounderstandinourOWI1waywhatilasbeendone,inviewofpossiblegeneralizatiOIlstootllersituations-飞叫Teshallreducetileproblemtotilestudyofatime-depende111familyofSellr&di11gel-operators,whichweshallhandlebytlMUseoftl四FUllIMI1-kacform川a(seee.g.Simon(1979))aMofultracon-tractivebounds,1altellfromtllebookofDavies(1989).
SchrbdingerOperatursandSimulatedAnnealing I67 2.REFORMULATIONOFTHEPROBLEM Lettherebegivenamnine肥asingfunctiontHT(t)of[07∞)tol0,∞).Wecanre-expresstasafunctiont=t(s)ofanewparameterSS时lIthatt(0)=oand dt一=9(s)=一一一-dsT(t(s))?thenSHF(s)isanoMecreasingfunctionof[0,∞)intol0,∞),with1d181(s)=JJ9(s)=页百孟页77|问归)· (2.1) (2.2) 9(s)willhavetoincreaseto∞asS→∞,butslowlyem吨l1.Forthetimebei吨,weassumethatF(s)isbomdedbyacomtaMbforalls,sotIM9(s)三9(0)+bsandt(s)三9(0)s+扫82.Weddmafamily{Ts:。三S<∞}ofpositivity-preservinglinearmapsofL∞=ζ∞(Rn,dz)intoitselfby [Tsgi(z)=E;W)[g(zt(s))]:gει∞, (2.3) (W)whereEzstandsforexpectationwithrespecttoWienermeasurestartingatzεRn?Ztisthesolutiontothestochasticdibrentialeqmtion(1.1)andt(s)isddmdin(2.1).ByHasYImskii'sstochasticLiapmovtheorem(Has'minskii(1980)),uMerverygeneralcOMitiomonU,ztdoesmtblowupto∞inEIlitetimewithprobabilityone,sothatTs(1)=1foralls.Itsumcesthat(z-VU)一三c|z|2forM|largeenough.ForginasuitabledO创InmI1a出iIn1(C∞O川a缸iIn川1让山inatleastthespaceCrofC∞functionsofcompactsupport),wehavebyIt07sformula 二Tsg=一Ts阳 (2.4) where Aβg=-Ag+9(?IT)·(飞79). (2.5) Aβextendstoapositiveself-adjointoperatorinC(Rnι;dz)?WIle1·eφβisthefu旧tionzHφβ(z)dennedby @β(z)=Z(3)-je-EU(2); (2.6) indeed?Aβistheself-adjointoperatorassociatedwiththeclosureofthepositivequadraticform Aβ(俨f|巧|2 (2.7)
l68 FrigerioandGrillo TileHilbertspaceU(Rn,吗dz)ismappedunitanu-o川0ιU2=ιU2(Rn飞,dzb均〉y?g-9〈φbβ川;1ulIn1de创盯1r.tt山'lhlhlihS1nm1口1a叩pp严iIn1咆gAβgoesi川1n1tωOHβ厚pih飞.mb均y HJ3f=φβAβ(φjlf)=(-A-l〉)f, (2.8) 飞vilerel;3is1lleoperatorofmultiplicationbytoilefunction 叼(Z)=zlVIV)|2-2AI(Z) (2.9) SellIGdingeroperatorsoftileform(2.8)arewellunderstood.ItisKIlowIIthatoisaIIOINiqmerateeigenvalueofHlhwithtileuniquepositiveeigenvectorφβ,aMthattilespectrumofHβliesiII[0,∞)-1Ioreo刊r,ifu;3(z)→∞as|z|一」实J?asistrueiI1011rassumptions?tileIIHβhascompactresoivent,witheigeIlvalues 。=Eo(β)<E1(β)三E2(β)三… TIleasymptoticbehaviouroftilespεctmlgαpE1(9)hasbeenstudiedinseveralcontexts(Gidas(1985),Simon(1983,1984)?Helbr(1988),HelbraMSjOFtrand(1984,1985ab叫HoneyandStrook(1988).Holi叮etd.(1989)).ItisrelatedMthtileproblemoftllesemiclassicallimit(asPlanck'sconstanth→0)inquantummechanics?where?however,OIleusuallyconsidersoperatorsoftlleformH=-A+h217aIIdIlotoftheformH=-L-R21主+hl气.IIIseveraldrcuIllstaIlces‘1hereexistsaI111111ber771suchthat j江-jlIliEM)l=mT (2.10) aIIdtilellumbermcallbedescribedasfoilon·s.LetI-haveEIlitelymany,isolated?11011degeIleratecriticalpointSZl…·?ZK1theseass11111ptionscallbeconsiderablyweakened).飞叫Titi1110lossill思eneralit〉-u-e111ayassumetlh1a1tlh1ea汕1b〉S叫Odlhu川1te1m川1口山1让山山ihIn山1 、..ZF)电airt))AU,,az‘、马气l,,..‘、It))CU,J,t.tEt.飞~,,,EE1、vj,'''LXJaomε-d一一)气t,,a·飞、、,‘、,,, Foranytwopointstyε配＼let!m(ιykm11{m(A!可(0)=z,γ(1)=y}-Finall〉＼letm=nlax{叫27y)|r?yεRVI}Itmaybeseenthatifmisattainedby771(r,y)?tolleIleiUlm-zory(orboth)isaIlabsolute111iniI1111111forU?inwhichF=0;sothat-ifIflIasauniqueglobalminimummisthemaximumheidIta
suhrodingerOperatorsandSimulatedAnnealing l69 pointmustgaininordertoreachtheglobalminimumonapathwhichtakestilelowestpasses-Thenaturalsettingfortheseresultsisthatofdiferentiablemanifolds.SupposethatwearegivenaninitialprobabilitydensityfunctionpoinC=C(Rnjz)suchthat?foranyBOrelsetBinR飞 叫MBl=Lh(z)dz=户。(zbM,(2川 whereIBdenotestileindicatorfunctionofthesetB.Wemaketheansatzthatthereexistsafamily{P87S主0}ofprobabilitydensityfunctionsinζ1,oftlrform 吨,"ιζ」sfJ、‘..,FZ,,..‘飞-nwaμAY、‘.,/z/,...飞-uIJ一一、‘,,/z/,,‘飞SDA (2.12) suchthat 叫叫(S)εB]=户。(z)(孔与)(z)dz==fh叫 (2.13) ForgεCr,wehaveontheonehand 二υ87gh)=二f山dz=-fP01(AM)dz==一(儿,[Aβ(d]φβ(s))=一(fsJβ(s)[g@β(s)l),(2.14) andontheotherhandwehave,atleastformally, 二(人79%))=(二M叫+(儿,小刚S))==(二川剧8))一(叶以(S)lF一({7)β(s斗争β(S)),ρwlh1ere9Ff(υS)=di93(υS)ν/dSandwlh1 (Eh Itfollows?atleastformally,that 二fs=一的(S)fs+俨(8)(U一(UMS))fs=一(-A+叩门儿,(2川1
I70 FrigerioandGrillo whereV(s)istileoperatorofmultiplicationbyV(s,z)givenby V(SJ)=hs)-jr(S)(印)一叫(S))(218) NotethatEq.(217)involvesafamilyoftime-ciepeMentSch而di吨eroperators.IIISection3weshallprovethat-7undersuitableconditionsonU,ithasauniquesolutionwhichisi11ζ2fol·allS>0,andthatthisremaintruealsowilentheinitialconditionhisa11111ltipleoftileDirac6?atleastwhend(s)isconstantinaneighbourhoodofzero.Insection4weshallgiveaproof(seealsoGemanandHuang(1986),Holleyetαl.(1989LCim鸣etαl(1987))that,undersuitableassumptiom011tilecoolingscheduleS←→β(s)(equivalenttotileusualformT(t)主c/ln(1+t)withC>771),tilequantity iVlt(s)!=||fs一φβ(s)||: (2.19) staysbounded?oreventendstozerointlrlimitS→∞.Thisresultshowsthatsimulatedannealingworks:indeedTwehave Problhs)εBl-Pβ(s)IBl=(fs一φ凹,),IBφβ(s)).(220) BytheSchwartzinequality,1lleabsolutevalueof(220)ismajorizedby Nit(s)li(IBφβ(S)?IBφβ(S))i=lV(t(s))§Pβ(s)IBli.(2.21) Si盯ePβ[U三mIIIf+6)→Oasd→∞forall6>0,itfollowsthat(1.3)holdswilemver(2.17)ilasasolutioninζ2andN(t)staysbounded. 3.ULTRACONTRACTIVEBOUNDSANDTHEFEYNhfAN-KACFORMULA LetSHβ(s)beacoMim川lydiEKm11.iableimctioIIEleamdon(0.∞),whosederivativeβ1(s)satisaesO主jr(s)三bforalls.ByciIa吨i鸣tileemrgymitsaIldtilescaleofAifnecessary?we111ayassumethatd(O)主2aIldbf;2,sothatalso dh)主2,lLT(s)2〉19(SLβ(s)兰州s)Vs.-2飞IVeshallfeelfreetousetilefollowingnotation: (31) 几=l》=2;4'2=φβ=2;A2=Aβ=2;H2=Hβ=2·(3.2)
SChr-bdingerOperatorsandSimulatedAnnealing I7l ConcerningU,weassumethatitisaC2function,andthattherearepositiveconstantsk,h,l,7,71,c,d?67withγ<k,suchthat U(z)主k|z|h-l,|VU|2+|AU|三meγzh-n(3.3) forallzinRn,andsuchthat,forallE>O(cf.Davies(1989))。三U三E(|VUP-AU)+c+; (3.4) EdenEIIEa3.1.TIlepotential 、、EE'''-uβUuti-E飞euaμti--qh-uwqμV一-euv (3.5) satisaestilebounds V(s)三;w(s)一内)l几-jr(巾+d),V(s)主一;护扫凤仰(υ忡8Proof.ForallS〉0,wehave (3.6) (3.7) ,"vsaμ-i一。"〉一,"veuaμ1i一。白+吨,"UV『EE'BEa--EEEd、‘.,/sjtE飞aμti一。"qasaμ1i-A哇rEEEE'』lEEEE』一-s司μV Inparticular?itfollowsfrom(3.4)that-U主一几一(c+d).Si川e(ITMispositiveforallAtheestimate(3.6)follows.From(3.4)itfollowsalsothat几兰一(c+d);recallingthatF(s)三β'(shweobtain(3.7).口 Usingthetheoryofultracontractivesendgroups,asinthebookofDavies(1989),wecanprovethefollowingLemma3.2.Tlmsemigroupexpl-tH2llωanintegralltermlIG(tJ,ν),a缸In1tih回阳eI阿existpositiveconstantsα,b,αsuchthat,forallz,yinRnandfin[0,叫,onellas 。三Ib(t,ZJ)三αexpH(t一αV1)]φ2(z)φ2(y).(3.8) Proof.LetA2bethepositiveself-adjointoperatorinC(Rn,钊dz)whichisunitarilyequivalenttoH2accordingto(2.6).Assumption(3.4)ispreciselytlrassmIptionoftheorem4.7.1ofDavies(1989)L?wlmlh1aSal 。三kt2(t,ZJ)三αexp[b(t一αV1)] (3.9)
I72 FrigertoandGrillo forsomepositiveα?b,α.Tile11(38)followsasiIIlle111Im(422)ofDavies(1989),si盯eIb(hz,ν)=λ'舍,2(t,z,ν)φ2(z)@2(y).口 Lemma3.3.ForailfinCaMtiII(0,∞),exp[-tHAfisinDomvh)forails.proof-Itsumcestoprovetilele111111aforfJEOiIIζ2.Tilen(3.8)impliesthat 。三(exp[-tH2lf)(z)三αexp[b(t一αVl)]φ2(z)(φ2,fh(3.10) aIldtheclaimfollowsupontakingintoaccount(33).口 NextwerecallSOIlmbasicfactsabouttheFeyIHIlan,kacformula,followingSimon(1979)witllud1101·lnodiacatiOIls-LetQbetilesetofBrownianpathsw:[07。。)→Rn-1飞·ith叫shr+duVJMietdμbetileproductmmsue011QofLebesgueIIleasuredzOIltilestartingpointzεE飞7n1withtulh1meprobabilitynm1e臼aSu盯redP叽W,ihM1让iCdihlma础lkt臼削eS{tu川tB岛1r.O阶飞叭W飞V门,1咀1n川1Ummi归山ma创mInlIm1川I〈〈O川叫;】刘川}川川tUiO∞1口l.Letly-beareal-valuedfuIICtiondeanedonRn(COIltinousaIldbounded,forsimplicityLandlet,Hbetileself-adjointrealizationinζ2oftileoperatorH=→L;二+IY.Then?forallf7giIICandforaliti11(0,∞),wehave (exp[-tHif,g)=(f,exp(-tHjg)==Lf(ω(0))up[-JMU)呻w(t)川,(311) whichmaybealso川itteniIltermsofaheatkernelk(t,z,y)主Oas 叫-tHlfJ)=(川-tHig)二f制而K(t川)9(y) NotethatthefactorofdiIIfro川of川iII1kddI山ioIlof叫s)servestoobtainH=→A+17insteadofH=一;A十17.IIIparticular,ifV三0,sothatH=Ho=-A?wehave (叶 ko(t?Z?Y)=(4πt)一号exp[-iz-y|2!-4t (3.12)
-SChrodingerOperatorsandSimulatedAnnealing l73 Le111111a3.4.Thereexistsa(11otmcessarilysimmetric)ileatltermik(f92.y)suchthat宁forallf,giI1乙2? 一-uμJuunyu,αv'ffIIlOPAxe-nu-u-IJpt''''nu-- =f7[EK(tAY)g(y)川? (3.13) andsuchthat,forailz?yiIIRnaMSiII(0,∞)?omlm 。三I((S?Z?y)三expITl(s)lIfo(s,z?U), (3.14) 。三K(S?Z,y)三expIT2(s)iIG(T3(s),z,ν), (3.15) whereIfoisgiveI11337(312)‘IY2isasi11le111111a32,aIIdwhere eu''ι,α+C1-2一一命,飞,α3μftlIO,α+C-i一。"一一ST (3.16) 72(S)=j(c+叫内)d→(c+仰),T3(s)=;fld(u)-F(u)ldu=;lt(s)-d(s)] (3.17) (318) proof-ForpositiverealT?let{ 民(s,z)=V(s,z)〈T. (3.19) Tilenz-1年(s,z)isac011tlinousboundedfunction,aIIdthesameistrueforSHV二(Au-s);SOthereisnoproblemingeneralizing(3.11)todeane nu。ivμ'.,α门可uydfIIIOxe-nu--taufl'。 Si肌ealsol二(s)satisEesthebomd(31)?itfollowsfromcomiderationofpos-itivef,ginC2COIlverginginthesenseOi-distub111ionstotlleDiraccieitaa1randatyrespectivelythatthereexistsa(110tmcessarilysimmetric)heatltermlIfF(t,Z?Y)suchthat (320)=f例而儿(tJJM(川 (321)
I74 FrigerioandGrillo 。三Iι(s,z,y)三expiTl(s)lIU(队Z?y). (3.22) Now-letT→∞T叽hlh1eIn…1fO1r.e佣aCih1turtU'.Forpositivef,gi11ζ2?(320)isamollotonicnomIlcrmsi吨functionofT?aMtilesameistrueforKF(t-AY),pointwiseintJ,y.Then(313)aM(3.14)follo飞withk(t,叽y)=limF→∞K俨(tJ?y).IIIordertoobMill(3.l川、weusetilebomd(3.6)onV(s).口 Inthefollowing?forthesakeofsimplicity,weshallassumethattileinitialCOIlditioIIfattimet=OisobtainedbystartiIlgfrolllanarbitraryfoεζ2at{1imet=-E13(OLandlettingtheprocessJUteVOlh飞VFefrOnm1t=一εβ(0)tot=0飞WVitlh1CωOmtan川t怆阳e创In1叩e川ureT(0创)=lL///β(0创).Tlh阳阳e创Ilf=eXp[-εdHβ刷(0)Jlfh0S臼i盯eJ圳(0创)三2,welhlaVel/汪切》趴(0)三l几包.Itfoilowsthatforpositivefowehave ody)=fh(r)Kβ(0)(EJJ)dr三 三fh(仙(飞ZJKMXP[b(〔 叭'ilichisinDOINV(s)forails.Sinceφ2isaCOIltinousfunction,asimilarboundholdsalso飞vilenfoisreplacedbyamultipleoftheDiracdelta.Theorem3.5Forf=exp[-EHβ(O)]元,foinζ2aMtin[0,叫,let (附)(ν)=fM(t,zJ)dz (3.24) ThellP(t)fisinDomV(s)5乙2foralls、ti11[0‘叹).andtimfollowi吨duHamelformulaholds: 叩)f=叫[-tHolf-f叫[十叫Holl(叫)fds(325) Tileconclusionremainstrueiffoεζ2isreplacedl巧'a11111l1ipleoftileDiracdeltaatSOIllepointz.PI-oof.认feadapttileproofiIlSimon(1979).Firstu-eworkwithlL(s).认7eilave GUFF/1,包,αuuvPIttlonrxe--U,α也flIOPAxed一'ω
&hrbdingerOperatorsandSimulatedAnnealing I75 andintegratingbetweenOandtweobtain SJUGUωeufvu'duufvFt''''''nuDAxer--'。一-1tAU,αNUWvhftt''ortBEEttttt』DAxe Takef,gεζ气n11山iplybothsidesoftheaboveequalitybyf(叫0))g(w(t)),aMintegrateonQwithrespecttodμ(uy).ByiMepeMe盯eoftllei旧时111entsofBrownian1110ti011indisjointtiIIIeintervals,weobtain,usingthedeanitpion(3.21)ofk(cf.Simon(1979)),f户μd伽zd而kιT(t冉川?=寸才ft〉》刊dω叫咐3斗[/叫7而{百向Z巧讯7让kιT(t式川' Let 川刀(U)=ff(z)KT(川d (3.27) (326)tellsusthat R(小 (tilepassagefromweaktostrongformisallowedsi旧ealloperatorsi盯olvedarebounded).ThereremainstωOPrmOv刊et山ha创tonema叮3yrletT→∞imn(仰3.2羽8)a圳In1obtain(325).Notethatforfoftheformexp[-EHβ(川fo,wehave 但(川 wherek(t,z,y)isobtainedbyreplacingvh,叫s))i11(3.13)byV(s-E?ω(s)),withV(t,z)=V(0,z)fort<0.AnapplimtionofLemmas3.2and3.4simwstllat [P(f)fj(y)三MXPH(fαVl)](fo?也)φ2(yLyεIR717tι[0,∞),(3.30) implyi吨thatP(t)faMV(s)P(t)farei11ζ2withζ2-mr11113omdeduni,formlyinsj.Bydominatedconvergence,wecantaketilelimitaST→∞in(3.26)?thusobtaini吨(3.25)inweakform-Uponmti吨thatV(s)P(s)isboundedbytheclosedgraphtheorem,thepassagefromweaktostrongformisallowed.口
l76 FrigerioandGrillo Finaliy,inordertosilowthatthepropagator(324)isasolutiontotheoriginaldiferentia-lequation(2.17)itsumcestorecallthefollowingclassical时suit(d.LionEaMMagems(1968));supposethatAisthegemmtorofananalyticSe创mI口山ItileCOIn1VOlh1u1tUiOIn1 TAVζ」GUFαGUJJACLfII,。一一ny (3.31) withfεζ2((07TL行)?where对isaHilbertspace,ddmsandemeMgofζ2((0,TLD(A)),whereD(A)isthedOma出iIn1ofAendowedwi让tlh1t址lh1回egraipμ〉才lh1mrnm1n瓜1a剖1h1nmI丑1OStallt7aIIdthisisexactlywhatweneedtobeallowedtotakederivativesiIIduHamei'sformulacorrespondingtotilepotentialV(s)atleastforalmostallt.SiIIcethisderivativewillbeusedusedtoderiveadiferentiaiinequalitywhichisafterwardsintegrated(seenextsection),thisisallthatweneedtocompletetimproveoftheexiste盯eofasolutiontotileevolutionequation(2.16)foraclassoftimedependentpotentialsarisingfromsimulatedannealing. 4.CONVERGENCEOFTHEANNEALINGALGORITHM. Inthissectionweshallprove?undersuitablehypothesesFthatsimulateda11-Ilealingworksasatoolforglobal111iainlization-Asexplainedintheendofsection2,theonlythingtodoistokeepundercontroltheasyI11ptoticbe-haviourofthequantityN(t)deSmdby(2.19),undersuitableconditionsonUandonthecoolingscheduleβ(t),Inordertoodothis,itissumdenttoproceedasfollows;letuscompute N(s)=(fs一φj3(S),fs-争β(S))=|lfs|li一(fg、争13(8))一(争j3(S)?fs)+|143(8)l!!= =||fUM-l‘ (4.1) si盯e||@β(8)||2=1and(fs,φβ(s))=jpAz)dr=1.Next}wecomputethetimederivativeof(4.1Ltryi11gtoderiveadiEerentialimequalitywhichwillgive118tllerequireduniformboundonlV(s). 11|fs一岛(S)||!=手||fsM=αsv、rαs=-2(儿,Hβ(s)fs)+ir(fsF([f一({f)β(S))fs)· (42)
SChr-odingerOperatorsandSimulatedAnnealing I77 WerecallthatHβispositiveandself-adjoiM?a11dthat岛isaneigenvectorofHβwitheigenvaluezero.Thereforeonecanrewrite(4.2)as 手||fs-φ川|12=αs、f=-2(fs一φ/3(S)?Hβ(s)(fs-争β(s)))+F(f87([f一(U)β(s))fs).(4.3) ThelmI Uf二αHβ+b (4.4) inthesenseofquadraticforms,whereα,barestrictlypositiveconstantsIIotdepeMi吨OUF;wemaychooseα<1sothat2一βf(s)α>Oforalls.17时ertilesehypotheses?omcanworkoutequation(4.3)intilefollowi吨way; 手l|fs一岛(S)|12三αs可F三一(2-db)(fs一φβ(S),Hβ(s)(fs一φβ(s)))+pfb||fs一φβ(s)||2+trb,(4.5) wlmewehaveused(4.1).Atthispoint,thespectralgapcOMitiomimpliesthat 丰|川旧|Mfs一φ刷阳川川s川川J)川|αSE飞F三一((2-db)E1(9(s))-bF)||fs一争β(s)ll2+bp1.(48) COIningbacktotheoriginaltiII1evariabletandintegratingthediferentia-linequality(4.6),weobtainanordinaryinequalityforlV(t(s))=||fs一φβ(s)||2.Set叫t(s))=-L((2-Fh)α)E1(F(s))-bms)),(41)d(s) aINlobservethat(4.6)canberewrittenas iN(t)=-c(t)N(t)+b主土dtT(t) (4.8) Itfollowsthat A叫UA哇,d＼飞tFF/1一川/t』E11飞d一'刷,αFLrIJ,fflu川'hv+ov''ιNHU-ducρl们CL<-N Asisusualintheliterature,wetakeacoolingscheduleofthefoE111 T(t)=C (4.10)
l78 FrigerioandGrillo Itfollowsfrom(2.10)that?foranypositivenumberm*〉m?onehas El(,)三re一βm'forsomer>0. (4.11) TileIlwehave c(t)=l2T(t)一α兰J一!E1(」一)-b主J一>＼飞dtTt/飞T(t)/dtT(t)一 〉l一一一一一l「it牛d}一?一＼ln(t+d)c(t+d)/-飞E), (4.12) ForC〉TYZwemaytakeC>m*〉msothatc(t)主h(t十d)-Pforsomepositivecomtantk,forsomepε(m/c,1),andfortlargeemugh-EEe盯e,thearstterminthelh.s.of(4.9)vanishesinthelimitast→∞.Withsuitablechangesofvariables,tilesecondtermisreducedtoanexpressionoftheform e才二d27 (4.13) whereb→∞;inthislimit,(4.13)isoftheorderoft(cf.Batemaa(1953))-Thisshowsthatalsothesecondintegralvanishesforlarget,thuscompletingourproof.NotethatourestimatesshowtheconvergenceoftheannealingalgorithmforC>771?whiletheydonotgiveanyinformationconcerningthecaseCf二m;however,inanalogywiththeresultsobtainedbyvariousauthorsindiferentcontestssuchascompactmanifolds(seeHoneyetd.(1989))orMarkovchainsonEIlitestatespace(seeHajek(1988)),onewouldsuspectthattheprocessdoesnotconvergeinprobabilitywhenC<mandconvergesfor C==η1.
aSChr-adingerOperatorsandSimulatedAnnealing l79 RefereIlees [1lF.Alum-PmtiRi,G.Parisi,FZirilli(1985).J.Opt.Them、UAppl.47:1.I2lH.Bateman(1953)·HigherTmnscendenMlFundi07叫Uol.HMcGraw-Hill-[3lT-S-Chiang,C.R-Hwang,S.J.Sheu(1987).MAllfJ.C071tmlα叫Opti-TIzizαti07125:537.[4lEB.Davies(1989).HmtkeTMlsα叫SpectmlThe07VCambridgelhliver-sityPress[5lS.Gema叽C.R-Hwang(1986).SIAMJ.Controlα叫Optimizdi07124:1031.[6iB.Gidas(1985).In:Pmceedingsofthe24thIEEEConfεTεMε071Decisionα71dControl(1985).[7lBffajek(1988).Mdh.OpeT.Res-13:311.[8叫lR.HaωS'、1nm11mSdkiHi(1980创).StochαsticStαbilitgofDigemztidEqMtions(Si-MilofandNoordhoE.[9lB.Etelfer(1988).semi-ChssicdAmlust{sfo俨the5Chddi7叼e俨OPe俨mαtO俨αnApplμiCωαt"iOT帆(LecturesNotesinMatlmmties1336,SpringerVedag).110lB.TTelfer,J.SjostraM(1984).ComT凡inP.D.E.9:337.[11lBfklfer,J.SjostraM(1985a).Am.LH.PP.(PhU归si叮qtu4"eThdωωO例7俨叫、245.[12lBHelferj.Sjoshad(1985b).Mdh.lVαchtTichte124:263.[13lB.Etelfer,J.Sjostrand(19850.Commun-inP.D.E.10:245.[14lR.A.Honey,D.W.Strood(1988).Commm.Mdhphus.115:553.[15lR.A.ETolley,S.kusuoka,D.W.StroodE(1989).JFtmet.And-83:333.[16lJ.LLio肘,E.Magenes(1968).probld771εα川Lim-te871011HomogdMSα叫Applicαti07以uolJJI(Dunod).[17jB.Simon(1979).FunctiomlIMegmuonαMQuα71tumphgsics(AcademicPress).[18lB.SiHml(1983).Ann.I.H.P.38:295.[19lB.Simon(1984).Am-dMdh.120:89.
ParabolicProblemswithStrongDegeneracyattheSpatialBoundary JEROMEA.GOLDSTEINDepartmentofMathematics,TulaneUniversity,NewOr-leans,Louisiana CHIN-YUANLINDepartmentofMathematics,TexasA&MUniversity,CollegeSta-tion,Texas 1.INTRODUCTION 、、,,,,咱E·-r'E‘、、 Ofconcernareparabolicpartialdifez-entiaiequationoftheform仇/δt=杂1(叭叭仇/θz)]+ψ2(ZJJU/θz) or (2)θu/θt=ψ1(z,u,δu/θz)δ2u/θz2+ψ2(z,u,θu/θz) I8I
l82 GoldsteinandLtn fort三OadZξQ=(071).EverythingwedowillhaveanaloguesirlthemultidimensionalcasewhenQCCRnTbutforsimplicityofexpositionweshallrestrictthespacedimensionton=1.Withequation(1)or(2)areassociatedaaiIIitialconditionandavarietyofboundaryconditions.Ofespecialinteresttousisthecasewheathedifusioncoemdentvan-ishesonthespatialboundary.Thatis?X=θ34Y1(in(1))aMx=ψ1(i口(2))bothsatisfy x(-71·)εC([0711×R×RL χ>Oin(071)×R×R,butx(z,矶。→OasZ→OorZ→1isallowed-OurapproachisbasedontheCrandan-Liggett-BeIlilaIItheorem,tilecomerstoneoftileIIonlinearsemigroupapproach.InordertousethistheoremFwemaketherestrictivehypothesisthatvlandψ1depeIIdonlyonZandtiz,110tonU.TheabseIIceofUdepeIIdeIIceleadstoadissipativeoperator.Undercertaincircumstances,Udependenceleadstoalocallyquasi-Elissipativeoperator,andrecentlyDOE-rohandRieder?inadeepandveryinteresti吨paper[7!?establishedlocal(intime)well-posedmssforthemixedinitial-boundaryvalueproblemwithUdepeadeIlceinthediffusioncoefEdent. 2.HISTORYOFTHEPROBLEM Ourresearchiathisproblemareahasevolvedinfourparts.WeIlextdescribethembridy. PartI[9].Wenrststudied(2)withv2三Oandψ1=v(z,tiz).Weassumedv(275)→03l们dyasZ→{070.TheboundarycOMitiomcouldbeofvarioustypes?including (-l)Ju(j)εpj(θu/θz) forjε{071}wherePjisamonotonei肌remingfunctionin5satisfyingpj(5)=OwhenE=0.(Moregenerally,Fjcanbeamaximalmonotonegraphinsomecases.)Itisassumedthatψ(z,。三vo(z)holdsforailz75ε[071l×R?whereψ。εC[0,llTVO〉Oon(0,1),andvo→Oslowly
ParabolicProblemswithDegeneracy I83 iIlthesensethat兀vo(z)一阶<∞forasuitablep主1(Seelloiforabriefexposition.)PartII[11.Thiswasthesubjectofourpresentationattheprede-cessorofthismeetinginTriesteHol.Againconsider(2)withv2三OaMψ1(z,u,ttz)=ψ(z,uz)主Vo(zhvoεCIO?lLvo〉Oon(0?1).Noassump-tioaismadeonhowfastvo(z)→OasZ→{0,1};theconvergemecanbearbitrarilyrapid.TherelevantboundarycOMitioaistheWe口tzelbomdaryCOMitionl13l-ThespaceisX=C[0,llandtheassociatedoperatorAisgiveIIby Au(z)=ψ(z,ut(z))u气zL Zε(071). FunctionsinthedomainofAsatisfyAuεXaMAu(z)→OaSZ→{0?1)-TheWentzelbomdarycondition,Au|θ。=0?involvestheLaplacianofU-ThisboundaryCOIlditioIlwillplayanimportantroleinthesequel. PartIII[12l.PartsIandIIdealwithanL∞theory.TheretheCraadail-LiggetttheoremwasappliediathesupremumBorn1spaceX=C(Q).TogetmLPtheory,onemustworkwithproblemshdiverge旧efom.Thusconsider(1)withψ2(2,η75)nOIlinereasi吨inηaMθ3ψ1(27η,已)三ψo(z)whereψoisasinPartI.Thusvo(z)→OslowlyaSZ→θQ.IBthiscasethemixedproblemisgovermdbyacontractionsemigro叩OBLP(0)forallp71三p<∞.InthendimensioIlalcase,thedetailsiIIVolvedwiththeelliptictheorygetabitmessy(see[12]). PartIV.Thisisourcurrentresearchprogram.Wewanttosolve(1)or(2)butwithvo(z)→OrapidlyasZ→θ0.Wehave五rstbegmtomakeprogressonthisproblem.Wehavenodeanitiveresultyet,butwepresenthereaErstresultwhichisamMrivialextensionofourrecentpaper[11l(iatheORedimensionalcase). 3.THESEh4IGROUPAPPROACH Thesemigroupapproachtotheinitialva111eproblem (3) du(t)/dt=Au(t) (t三0),u(0)=uo
I84 GoldsteinandLin inaBaaachspaceXistousethebackwarddifferencescheme UA一-F」-u一εu whenceuAt)=(I一εA)-1ue(t一ε).Thusforε=t/nandtk(0)=ti09 nuun、、,,,,At-nriJ,,‘飞一一、、,,/''ι/,,‘、EU InordertohavetAEconvergetothedesiredsolutionU?itisconveIIieIIttoassumethatAisesse7ztiallum-disstpαtiM.Dissipathitymeansforall入〉0, (4) ||(I一λA)-1|lLip三1. Essentialmdssipativitymeansinadditonto(4)?冗(I一人A)isdenseiIIXforsome(henceail)λ〉0.(Here7之denotes"range".) CRANDALL-LIGGETT-BENILANTHEORE岛1.LetAbee33￠7zttallym-dissipGtiMonX.LctAbetheclosueoffthegmp1bofjAmX×X.ThmfoTGllfεD:=D(AL T(t)f=lim(I-F)-nf ezistg.MomMTyT={T(t):t兰O}isα3t的TmO叼l切yCωOηt刮tπ川包Ouω3COηM时tt,肌r3Me肝W7WmrnLt叨g7Tm、刀Ot叩LψpO7ηZDPi.e, T(t):D→DT(t+s)=T(t)t(s)T(0)f=ft→T(t)fεC([0,∞);D)||T(t)||Lip三1 foTαllt250,foTαllt,S主OforαllfεD,for-eαchfεD,for-eαcfzt220. FindlyyfoTalluoεD,u(t)=T(t)uoisthetmqMmildsolutionof(3). TheexistenceandpropertiesofTwereestablishedbyCrmdallmdLiggettl6].ToBedan[2lisduethefactthatthereisasuitablenotionofwell-posedmssfortheCauchyproblem(3)sothattheuniquesolutionisgivenbythesemigroupappliedtotheinitialdata.WeshallRotdwell02
ParabolicProblemswithDegeneracy I85 theexactmseinwhichthesemigrO叩givesa"mildsolution,,of(3),ortheexteasiOIlstomultivaluedA,ortootheraspectsofthetheory-See,forexample,[1],[3],问,[8l. 4.THEFRAMEWORK Westudyavariantof(匀,namely (5) Ut=ψ(z,tLz)tizz+ψ(z,u,tAz) iatherealBamchspaceX=C[0,1].Webeginbystatingtwommi让mI口m1hypotheses. (HOvεC([071]×R);v(Z75)三vo(z)forαll(z,已)ε[0711×R;ψoεC[0?ll;vo>Oon(0,1).(H2)ψεCI([0,1!×R×R);η→ψ(z,η,已)ismMM阿αsi叼onRforαll(z75)ε[0711×R;ψfzαsdmostlineαrgmt川fzin已i.e.foreαcflα〉OthereisαconstαTltKasuchthd |￠(2,η,5)|:三ka(1+|5|) forαllZε[0,日,ηε[一α,叶,αndEεR;￠(z,070)=OforαllZε[071]. DeaneamaximaloperatorAby Au(z)=v(2,uF(z))u气z)+ψ(z,u(zLu'(z))forZE[071landUε17(A)={uεC2(0,1)门X:AuεX}.NextdeEmthreerestrictionsofAusingtheWentzelandhomogeneousDirichletboundaryconditions,viz., B=A|D(B) forB=A,Ao,Ahwhere ?(A)={uεD(A):Au(z)→OaSZ→{0,1}},17(Ao)={uεD(A):u(z)→OaSZ→{0,1}},D(A1)={uεD(A):u(吟,Au(z)→Oasz→{0,1}}=D(A)nD(Ao).
I86 GoldsteinandLin Considertheuniformellipticityhypothesis: (H3)vo(z)主ε〉OholdsfoTsomeε>OmdallZε[071]. 5.THECASEWITHNOBOUNDARYDEGENERACYPROPOSITION.17(HI)-(H3)holdytheηA[Tesp.AojTesp,A1ι]i归3臼3εωηt刮i叼αdllωUm.d副i3‘呻αdttMOπXlT陀eS叩p.onXJTesp.。πCo(0?1)={uεC[0?1!:叫0)=u(1)=0}]. Wesketchtheproof.Thedissipativityiseasyanddoesnotrelyon(H3);infactitonlyrequiresv主0.Lettihhε17(A)andletha=ut一λAtihi=1,2.Letω二叫一句(orω=U2-U1ifmcessuy)andchoosezof:[071lsuchthat叫zo)=||ω||.Ifzoε(0,1),then (6) ω1(zo)=07ω气zo)三0, whence ||(I一λ4)-Ih1一(I一人A)-Ih||=||ω||=町(zo)(7)三ω(zo)一人v(Z?u;(zo))ω气zo)一[ψ(zo,u1(zoLu;(zo))一ψ(zhU2(zoLu;(zo))]=(h1-h2)(zo)三||h1-b||. HereiII(7)weused(6)andthemomtonicityofψ(z,-J).IfzoE{0,1)theIlAu(zo)=OaMinequality(7)continuestoldd.ForAo,zoε{0,1}impliesω三0;thustlledissipativityofthethreeoperatorsA?Ao,AIfoliows.WenextshowthatAoism-dissipative,(Thatis,冗(I一λAo)=Xforλ>0.)Let入>0,hεC[071.WeseekasolutionUof U-入Aott=h, thatis, (8) 、、,,/咱EAnUJ,,‘飞n0、,",,,,,一一、1,/uuzJ,.‘飞AWT飞AHnUM尸=U飞-4tAμ叫均=、、BE/-0,,a飞、、UHUrl〈EL LetLbethenegativeDirichletLaplacimon[071.ThusLu=gmeans-u"=gFU(0)=u(1)=O?whence u(z)=才川)g(U)
ParabolicProblemswithDegeneracy l87 whereGistheGreen'sfunctionofL.Consequently(8),isequivalenttotheintegralequation (9创)u叫巾(μZ吟←)=才fρh1}马Gq(M川'Jj邱刷川u旷凶内州唰F气切ω旧(ωωωνω叫川)刀旷rl「尸川-→叮1飞[hκ崎(ωU川(ωωνω川)忡+刊￠川)L川μ7JjIu旷叽1气(ωνD阮ed负缸hn肘1e(SM)(μZ)tobetherighthandsideof(9).Using(H1)"(H3),omcaIlshowthatShisacontinuouscompactself-mapofsomeballinCI[071iFwhenceshhasaaxedpointbytheSctlauder丑xedpointtheorem.ThusAismdssipative. 6.NEMFRESULTS NowwewaMtoelimmate(H3).FormεNletJryz=max{v,1/m)ThenletAmbeAo,butdennedwithqvniaplaceofv.Amisdissipative.LethεC1[071!?人〉0.Takeλ=lfordeanitemss.Tosolve um一λAmUm=h? letf(z)=门+Sbethelinearfunctionon[0?1lsatisfyingf(z)=l巾)?Zε{0,1}.TheIlforg=h-2,um=14m-Esatisaes (10) (υm一h叫川(μ川Zυm(0创)=υm(1)=0' wherevm(z,5)二Gm(z,5+吟,f(z,η,已)=ψ(z,η+门+s,5+T).Sime(Hl)-(H3)hold,thereisa(unique)solutionumof(10)bytheProposition.Thedissipativeestimateimplies|川|tυJ与怡7Wm7nJEacompactmssanddiagonalizationargumentofthetypedusedin[11],wecaIIEndasubsequence{tywzJof{tywJconvergingtosomeυiaCAc(071).Thislimitusatisaes u-v(z,UF+T)υ"一f(z,υ,UF)=g in(0?l).MoreoverJHl|三||g||.Wewanttoshowt山ha创ttJ与与7W叫ron[p0'1订l.Inthatcaseu(0)=υ(1)=0,andU=U+tsatiSEesU一λAtL=h.Thuswemustcontrolum(z)Beartheboundary{0?1}.Eveninthelinearcase,suchcontrolcaanotbecontainedingerleraLmTerecallanoldresultofFeller,inapreciseformgivenbyCl臼lentandTimIIlemans[4].
l88 GoldsteinandLtn OnX=C[0,llconsiderthelimaroperator 、、BE/-i-i/''飞、、 Lu(z)=α(z)u气z)+F(z)uF(z) forZξ(0707whereα,3εC(0,1)的thα〉0,Fredvalued.Let17(L)={uεC2(0,1)nX:LuεX,Lu(z)→OasZε{0,1}};thusLhastheIVeMzeiboundarycondition.Then([4l)Lisclosed7dissipath?eanddemelydeamd.Lism-disspativeifandonlyif冗(I-L)=XifaMonlyifboth(Co)aM(C1)hold:(Co)WεLI(OJ/2)orf2W(z)fJα(s)-1W(s)一1dsdz=∞7(C1)WεL1(1/2,1)orji;2W(z)j;Jα(s)-IW(s)-1ds=∞?where ,-zIV(z)=exp{/-P(s)α(s)一1ds}.J1/2TheideaofadaptingthisconditiontothemnlimarsitmtioIIof[11]wasasfollows.Au=v(2,uf)uFFisan∞OIn1lUim、wvi江t}h1ψOPlaJy/i鸣theroleofαaMFbeingreplacedbyzero.[Ofcourse7Ais∞Illimaroperator,butthisisimIristicdiscmSion-Soweco川nmJThusthecorrespoMi吨WisW三1,wheIIceFIYεLI(0,1)aMthera鸣econditionisexpectedtohold.WithAu=v(2,uf)u11+ψ(凯u,1if)withψ#0,againwecmthinkthatψocorrespondstoα,b川日owthereisanaIIalogueofFcomingfromψ.TIlesimplestanaloguewhichwillinsurethatwεL2(0,1)nomatterhowfastα(z)→OaSZε{0,1}isF(z)三OforZε[1-AlLF(z)三Oforzε[074forsomeε〉0.Tosimplifythesubsequentproofsweshalltakeε=1/2.Weassumethatψ(ZJ,tf)isapproximatelylimarinufiathefollowing sense. (H4)ψ仰,η75)=Ml(z?η75)+Mo(ZFη)57四heTefOTαllα>OtheTEisαconsUMKasuchthd|Mo(z?η)L|M1怡,η75)|三Ifam(πz)foTαllZξ[071,ηε[一α,αLEεRy-Tno陀OMTMo(27η)三Oon[071/21×R,Mo(2,η)三Ooni1/2,1]×R.Finallyψ仙,070)=Oon[0?1l. THEOREM.Let(Hl),(H2)?(H4)hold.ThmAisesse7zttallγm-dtmpatiMGndde7川ludeβMdoηX=C[071].
ParabolicProblemswithDegeneracyI89 Letλ>0,hεC1[0717g=h-fasbefore.Recallthatummspreviouslydennedtobetheuniquesolutionof (12) η3一-fnυ、、E,/muz/,,‘＼nuμ'、1·'/FmυmuZJ''1、、唱iμHmOM尸=Im、lγυ才」/'1、巾hm/1、uh=、‘,/'-mvmmuurl〈ll、 where μ。(z7η)=M。作,η十门+s)?μ1(z,η,5)=M1(z,η+TZ+sj+T)+TMo(Z7η+门十s). Letω(z)=ksi叫门)?Zε[0?1].Herekisalargepositivecon-staIlttobedeterminedmomentarily.Notethattu;20,ω11=-7T21175二。?ω(0)=ω(1)=0.Also?mcegεC1[091iaMg(0)=g(1)=OTwehavehsin(M)三|g(z)|on[0?llforsome171〉0.Let7m(17)=Vm(凯υL(z))-17Zξ(0,1).From(12)wededuce |υ二一γm[υm一μo(27υm)uLl|=γm|g+μ1(z,υm,吟)!三γm[|gl+|μ1(z,υm7υL)|] 三γmω三7mω一ωFF-7mμo(2,um)ωF foras1山ablylargek〉hby(H4)(since!|tY771||三||g||)aMsi盯eμ。(27Um)ω1三0.Consequentlyforzm=ω土υm7 (13) z二一γmlzm一μo(z,um)zLlf三0? Zm(0)=zm(l)=0. Wecuimthatzm主Oon[071.IfnotthenzmhasanegativemiI山IIU111atsomepointzmε(071).ThenzL(zm)=07z二(zm)三0.Thm(13)becomes (14)z二(zm)三γm[zm(zm)一μo(zm,um(zm))zL(zm)]. ThelefthaMddeof(14)isnonnegative-Therighthandsideisstrictlynegative.Thiscontradictionestablishestheclaim.Fromzm主Oitfollowsthat |υm(z)|三ω(z)
I90 GoldsteinandLin forallzε[071landforallmεNThisisthedesiredcontrolontymm町theedpoints-Thusu=limumksatisaesuεC[07日,u(0)=υ(1)=0.Thetheoremfoilows-WeassumedMo(z,η)三OforZε[0,εlandM。但,η)主OforZε[1-Aliwithε=1/2.ItisactuallysumcieMtoassumethisforsome丑xedε>0.ImprovedonedimeIISionairesultsandhigherdimensiORalresultswilleventually?wehope,becompletedandappearinafuturepublicatiOIl-引fegratefuliyacknowledgethepartialsupportofanNSFgraataIIdthegracioushospitalityofourhostsinDelft. REFERENCES 1.V.Barb11?NOTLliTLeaTSemigTOTLpsαηdDtiTeTeMidEquαtionsmBαnGchSpαce,Noordhof,Leyden?1976.2.PhBdnilaIUEquationsd'evolutiondmsunespacedeBamchqueLconqueetapplications,Thesis,URiversitddeParisXLOssay?1972.3.PhB臼ilan,M.G.Crandall?andA.Pazy,Nod7zeαTEvoltLtioTlGov-eTnedbyACCTetityeOpeTαtoTS,toappear-4.PhClAmentandC.A.TimmermanROnCo-semigroupsgeneratedbydiferentialoperatorssatisfyingVentcel'sboundaryCOIlditiOIls、Iηdag-MMh.89(1986),379-386.5.M.G.Crandall,AIlintroductiontoevolutiongovernedbyaccrdiveoperators,inDummiedsustems,Vol-l(ed.byLCesari,J.K.HaleandJ.P.LaSalle),Academic,NewYork(1976)?131-165.6.LI.G.CrandaiiandTM.Liggett?GenerationofsemigroupsofnORlim町tramformationsonGeneralBmachspaces,Am肌J.Mdh.93(1971),265-298. 7.iR.DorrohandG.R.Riedez飞AsingIlarquasilinearparabolicprobleminonespacedimension,J.Dig.Eqm,,toappear-8.J.A.Goldstein,SemigTOtLpsofN07zlineα7·OpemtomandAppliCGti07"'toappear-9.J.A.GoldsteiaandC.-Y.Lin,Singularnonlinearparabolicb011日daryvalueprobleminonespacedimensions,J.Dig.Eq7队68(1987)7429"443.10.J.A.GoldsteinandC.-Y.Lin,Degenerateparabolicproblemsandthe飞AfeIItzelboundarycondition,inSemigroupTheoryandApplications(ed.byPhh.Cαl眨豆白臼I口mI1en风S.IhIn盯lV回e口m1i垃ZZ纣i,EE.Mi汹tUid&ie臼盯Ir叽.Dekker,NewYork(1989)7189-199.
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LinearandSemilinearBoundaryConditions:TheAnalyticCa-e GUNTHERGREINER*MathematicsInstitute,UniversityofTUbingeIUTUbingeruGermanyKLAUSG.KUHNMathematicsInstitute,UniversityofTUbingerhTUbingeILGer-many 1.INTRODUCTION IIIthispaperwestudyabstractinitialboundaryvalueproblemsinBaaacllspacesoftilffollowi口gform d(t)=Au(t)+F(t,"(t))Lu(t)=φ(t,u(t))u(0)=u0· PA飞、BYEA/'''飞~ ThearstequationrepresentsthedtgeTentidequation.HereAisaunboundedlinearoperatorandFisnonlinearandrelativelyboundedwithrespecttoA.Thesecondeq1latioIIdescribestheboundαTycondition.AgainLislinealrunboundedand@isIIOIlullfaralxl *TIllspaperwasfinishedduringastayattileCentrumfor飞VisKIlndEEIIInformatica(C飞VI)inArustekiaIIIIamverygrateful,bothtotheDutchNatIonalScienceFoundation(NO飞V)andtothec、VIforfinancialsupport l93
l94 Greinerandkuhn (inacertainsense)relativelyboundedwithrespecttoLFinally,thelastequationistheinitialcondition,uoistheprescribedintialvdue-Equationsoftheform(IBVP)arediscussedbymanyauthors,IAmll,[Am2l,[DLZLIG02l,IGrll,[Gr2],[Lal.In[LallinearproblemsinHilbertpacesareconsidered.III[Am2lsemilimarproblemsarediscussedandiniAm2lthe(moregemral)timedependentsituation.Bothresultsarebasedonextrapolationtechniques,i.e.oneisinterestedinsolutionsincertaindistributionspacesofnegativeorder.Withthispaperwesomehowcomplementtheworkof[GrliandlGr2l.Whileinthosepapersmainlyhyperbolicproblemsareconsidered(i.e.analyticityofthecorrespondi吨semigroupisnotrequired)inthispaperweheavilyusetheanalyticityandcanobtainmoregeneralresults.Ontheotherhandthetechniquesundmethodsaresimilartothosedevelopedintheearlierpapers-InparticuiaIwearelookingforsolutionsintheinitialspace-Beforegiventhepreciseassumptions,welookataconcreteexample.Considerthefol-lowingnonlineardifusionequationonaboundeddomainQinRnhavingsmoothh011IxhtryθQ 生(t,z)=AU(t,z)+f(t,z,队gradu),(t>0,刊。)θt学归,←ψ(tJ,u),(t>0,叫例。νu(0,z)=uo(z),(zεQ) Inordertohandlethisproblem(underappropriateassumptionsonfandv)iIl0111frameworkweconsideritasanabstractIBVPinthefollowingway.ConsiderthethrmBBamchspacesX:=Lp(Q)(1<p<∞),theSobolevspaceY:=I叮(Q)Sobolev-Slobodeckii(orBcsov)spaceθX:=WJ-1/P(θQ)(see[TI-lforaddMion)·Fh1u盯1r忖tlhk1rern口I∞r【e?weconsidertwolinearoperators,theLaplacianandthetraceoperator A:Y→X,Att:=Au ,L:Y→缸,Lu:=些。ν BothareboundedasoperatorsOIlY(equippedwiththeIVJ-norm),butunbom{icd川rllCOIlsMeredasoperatorsonXwithdomainY.Furthermore,theNemitskyi(orsuper-position)operatorF:[0,T)×Y→Xassociatedwithf,isdeanedbylF(t,叫i(z):二f(tJ,u(z),gradu(z))similarly@:[0,T)×Y→θXbyl@(t,u)l(z):=ψ(t,2,u(1)).WiththesedeaIlitionsthenonlineardifusionproblemcanbeconsideredasanabstractIBVPasdescribedabove.Throughoutthepaperwemakethefollowingbasicassumptions-X,YandθJEareBaBachspaces,YiscontinuouslyanddenselyembeddediIIX;(Hl)A:Y→XmdL:Y一→θXareboundedlinear,Lisasurjection;而2)theoperatorAo,whichistilerestrictionofAtokerLgeneratesan(H3)analyticsemigro叩(Tb(t))onX.
Lineαrandsemilinear-BoundaryConditionsl95 F:[01)×Y→Xmdj[01)×Y→θXmlocallyLipschitz;但4) Notethatthehypothesis(H3)impliesthatthehomogenouslinearproblemd=Au,Lu=0,u(0)=tAOiswell-posed. 2.THELINEARCASE Inthiscapterweconsiderthecasewhere@aadFarelinear.Theproblemthenisto乱时conditionsonlinearmappingsF:Y→Xandφ:Y→δXsuchthattheoperatorBdeEmdbyD(B):={νεY:LU=φu},BU:=AU+Fugeneratesananalyticsemigro叩-Forthecase@=0(i.e.theboundaryconditionsarenotchanged)therearewellknownperturbationresults.DuetothefactthatAogeneratesananalyticsemigroup,onemayevenallowunboundedperturbations.InfactoneonlyhastorequirethatFisrelativelyboundedwithrespecttoAandhassumdentlysmallA-bound(seee.gpoll66)Thismayleadtotheconjecture,thatfortheperturbationoftheboundaryconditionasimilt江附1山holds.Thishoweverisnottrue(seeExample21)andthequestionarises灿atkimlofperturbationsareallowed.WewillmaidycOImiderthecaseF=Oandde自nefor@:Y→θXtheoperatlorAφasthefollowingrestrictionofA: D(Aφ):={UEY:LU=句}andAφU:=AU (2.l) 2.1Example.WeconsiderontheBamchspaceX=LPI0,1],(1三p<∞)theoperatorAdennedbyAf=尸,D(A)=Y=WMI0,1]=={fεClio,1]:ftisabsolutelycontim01队ffIεLP[0,ll}.Iaordertogetawell-posedproblemonehastorestricttiledomain(toas111)spaceofcodimension2)byrequiringappropriateboundaryCOIlditiOIls-TilespaceofboundaryvalueswillbeZdimensional,thusθX=R2.WeconsiderDirichlet-aIKINe1Imamboundaryconditionsrespectively.I.e.,LD:Y一→R2'LDf=(f(0)、f(l))andLN:Y一→R2,LNf=(f'(0),f'(1)).ItiswellknownthattheC∞Or盯rrem叫S呵Ip〉On川d巾1ErCStUIr.ikCtUiOIn1S(wedenoteitbyADandANrespectively)generateanalyticSeImI口li地grmO1u1lp〉SonX.TheCOr口re臼SPOInld也iI口1gresolventoperatorscanberepresentedusingtheGIr‘eeIn1'、Sfunctions. [R(沪,AD州=JIKD川)f(U)du,(Rd>O)冈山N州才1k川,u)fM,(Rd>0)tilekernelskDandkNaregivenby 一1jS剖inh入(1一Z)S剖ih口lh1入νi江f2一khD(川吵=(入S灿λ)ijcoshλ(l一Z)coshλUi江fZ兰νkN(入,Z,U):=(λS目inh入)-1{lcosh入zcosh入(1-u)ifZ三ν
I96 Greinerandkuhn Nowwewillperturbtheboundaryconditionbyalinearfunctional φ:Y→R2givmby@(f):=(f仲f(jLO) (2.2) 讯repointoutthatφisrelativelyboundedwithbound0.WearstCOIlsiderthecaseofDirichletboundaryconditions,i.e.weassumethatLDf=φf.ThusweconsidertheoperatorBf=fFFwithdomainD(B)={fεW2,p[0,ll:f(j)=f(1)=0}.BisnotthegeneratorofaCo-semigroup-InfactitsresolventR(λ,B)isunboundedforλ→∞(itevengrowsexponentially).WeonlysketchtheproofaMomitthelengthycomputations.R(沪,B)istherankoneperturbationofR(沪,AD)givenby 阴阳)刀(Z)=lR川D州-fh(U)f(MMZ) wherethefunctionsfzλaadkλaregivenby M):=s叫(l一!22-1|) kλ(z):=(λsinh入)-Isinh入(1-z).SinceADisagenerator,||R(入,AD)||tendstoOasλ→∞.TheoperatornormoftherankoneP仙(ωLpη)-nmOωr口Imnoffhh2uλ(伙kλ川)oneobtains ∞→-Aro户,..入一2nrxeiq-A~Aqa、A'h llhl|p~λ-(1+t)forλ→∞Thusthenormoftherankoneperturbationgrowsofexponentialorder.IfweperturbtheNeumaanboundaryconditionbythesamefunctional@deanmiiB(22),i.e.weconsidertheoperatorCf=f"withdomainD(C)={fεW2,pl0,li:f'(0)=f(0)-f(jLfF(1)=0}thenweobtainagainageneratorofananalyticsemigro叩·ThiscanbeprovedbyshowingthatλR(入,C)isboundedforRe入>入。.AgaintileresolventisasuitablerankoneperturbationofR(λ,AN).Andoneonlyhastoestimatethemrmoftherankoneperturbation.Weomitthelengthycomputations,becauseitalsofollowsfromThIII-2.6(c)below.Foradditiveperturbationofageneratoritisenoughthattheperturbationisrelativelyboundedwithboundzero.Theaboveexampleshowsthatthisisnotenoughforperturba-tionoftheboundarymap-Itseemsthatitratherdependsonthe"degreeofdiscoIIHautynoftheboundarymapping.InsomesensethemappingLNrepresentingNeumanncondi-tionshasahigher"degreeofdiscontinuity,,thanLD-Inthefollowingwewanttomakemoreprecisehowtomeasurethe"degreeofdiscontinuity,,.ItdependsonthedecayrateoftheDirichletmappingLλfor入→∞.Alternatively,OIlecandeaneitusinginterpolationspaces,
LinearandSemilinearBoundaryConditions l97 FirstwerecallthedeanitionsoftheDirichletmap(smIGrll,)andinterpolationspace(附lB町,[CHADPlor[Trl).WeassumethatthehypothesesH1-H4aresatiSEed-Forλεe(Ao),therestrictionL|ker(λ-A)hasaninverseLλ:δX→ker(λ-A).(2.3)ThismappingiscalledDirichletmapping,becauseforhεθXtheelementZ:=LλfzisthesolutionoftheabstractDirichletProblem"(A一λ)z=0,Lz=h".Theinterpolationspaceoforderα(0三α<1),correspondingtotheSeImI丑1i地grOu(T巧0(0例t均))isdeanedasfollows(see[CHADPl,Sec.6)Xα:={zεX:limfα(Zb(t)z-z)=0}={zεX:limλα(入R(λ,Ao)z-z)=0}.λ→∞(2.4) Xαequippedwithoneofthefollowing(equivalent)normsisaBamchspace: ||z||α:=supt-α||T(t)oz-z||。<t〈1 ||z||α:=supλα||z一λR(λ,Ao)z||λ>λ。ObviouslywehaveXo=X(withanequivalentnorm)andwedeEmX1:=D(Ao)equippedwiththegraphnorm.ItcanbeshownthatXα,(O<α<l)coincideswiththeinterI灿lationspace(X,D(Ao)):∞C(X,D(Ao))α,∞obtainedfromXandD(Ao)by1山吨tlleco川nuomrealinterpolationmethod.(seelCHADPloriBBLSec.3.4) 2.2Proposition-Forαε(0,1)thefollowingassertionsareequivalent:(i)limλ→∞入α||Lλz||==OforeveryZεθX;。i)YCXαandtheinclusionisacontinuousmap;"i)ker(μ-A)CXOforsomeμεg(Ao);EZ旦旦f.ForμεQ(Ao)wehaveY=D(Ao)@ker(μ-A).MoreoverD(队A0)Cisobviousthat(ii)implies(iii)and(iii)impliesYCXα.SincetheinclusionsY→XaMXo→XareCOIltiI111011s,theclosedgraphtheoremimpliesthatY→XαisCOIltiI111011saswell.(i)件(iii):Wehaveforλ,μεe(Ao)(seelGrILl.3)LλZ=[id一(入一μ)R(入,Ao)lLμz=(id一λR(入,Ao))LμZ+μR(入,Ao)LμZ-Since||μR(入,Ao)||==。(入一1)wehaveforeveryZξθX||Lρl|=o(λ-α)φ||(id一人R(λ,Ao))LμZ||=o(λ-α)件L川εXα. Fromthisequivalencetheclaimfollows. 。 Webdenycheckassertion(i)oftheLemmaincaseoftheExample2.1.InthiscasetheDirichletmapsaregiveaasfollows:ForDirichletboundaryconditionsU:=(LD)λ(α,b)isgiveaby u(z)=(dnM)-1(b-mtIJAZ+α-dnh(1-z))(入>0,μ=JX)
I98 Greinerandkuhn andforNemannbound町conditionsU:=(LN)巾,b)isgimby u(z)=(μsinhμ)-1(bcoshμz一αcoshμ(1-z))(λ>0,μ=JX). Usingtheseexplicitrepresentationsonecmeasilyshow ||(LD)λ||=o(λ-α)字斗α<上2p11||(LN)λ||=o(λ-α)仁=〉α<-(1+-).2pThegeneralidea(sowedonotgiveaformaldeaMUon)isthatthelargerαcanbechosensothatoneoftheassertionsinProp.2.2remainstrue,thehigheristhedegreeofdiscontinuityofL.Inordertoprovethethemainresultweneedestimatesonthedecayoftheresolventfor人→∞.SinceR(入,Ao)isboundedfromXintoD(Ao)itcanbeconsideredasbOUR出dlinearoperatorfromXαintoXβforanytwoα,3ε[0,ll.WedenotetheaorIIlofthisoperatorby||R(λ,Ao)|lZ- 2.3Lemma.(a)HO三α三3三lthen||R(入,Ao)||2=0(川一α一1)for人→∞·(b)HO三F三α三lthen||id一λR(λ,Ao)||2=0(入β-o).EZ旦旦f.Theproofusesthreeingredients:Somebasicestimatesoftheresolvent,thefactthatthe(contimousreal)interpolationmethodisexact(seelCHADPis-2)andtherriterationtheorem(seeIBBl,3220or[Trl,1102).(a)Itiswellkmwnthat ||R(λ,Ao)!|;=||R(λ,Ao)||i=0(λ-1) and ||R(λ,Ao)||?=||AoR(λ,Ao)||;=||入R(入,Ao)-id||;=0(l). ExactIlessoftheinterpolationmethodthenimplies: ||R(λ,Ao)||;-α三(||R(λ,Ao)||;)1+α一β.(||R(λ,Ao)||i)β-α=0(入-(1+αJ))||R(λ,Ao)||;一β+α三(|iR(λ,Ao)||?)β-α(||R(入,Ao)||i)1+α一β=0(λ一(1+α一β))ForO:=百仨百wehaveα=(1-8)-0+8(1+α-3)andF=(1-8)(3一α)+6·llr旧CthereiterationtheoremimpliesJYα=(Xo,XI+α一β)8aMXβ=(X9一α,X1)8·17si吨theestimatesaboveandtheexactIlessoftheinterpolationmethodweobtain 4μ。+、λJ,,‘飞。一-AO、、,,,/α+aμ、‘,,,,nuA、λ,,..飞R/,,EE飞、AV、‘,,/αoβ飞,,/。UA、λit飞R'''1飞<一αaμ、‘.,/nuA-Ad,,,‘飞R
LinearandSemilinearBoundaryConditions I99 (b)Thispartworksdongthesamelinesm(a)startingwiththeestimates ||id一入R(入,Ao)||:=|itd一λR(λ,Ao)||i=0(1) and ||id一λR(λ,Ao)||;=||AoR(λ,Ao)||;=||R(λ,Ao)||i=0(入一1).Thenoneestimates||td一λR(λ,Ao)||;-Pand|lid-λR(λ,Ao)||i+卜。FinallyreiteratioIIMthexponent8:=d=石incombin山nw让hexactnessgivesthe的iMestim InthenextlemmaweinvestigatethedecayoftheDirichletmapping.Firstwerecallthefollowingwell-knowncharacterizationofgeneratorsofanalyticsemigroups-AdenselydehedoperatoristhegeneratorofmanalyticsemigroupifmdonlyifthereexistconstantsT,MsuchthatA:={λε￠:Re入>0,|入|>T}C以B)md||R(入,B)||三芮for入εAThesecondconditionwillbesubsequentlyabbreviatedby ||R(入,B)||=0(|λ|-1)for|λ|→∞,Reλ>O (2.5) withobviousInodiEcationsincaseofdiferentdecayrates(seee.g.thefollowinglemma). 2.4Lemma.AssumeO三3三α<landYCXα·ThenthenormoftheDirichletIlia尸LλconsideredMmapping企OIIIθXinXβsatides||Lλ||β=0(|入|β-α)for|入|→∞,Re入>OEZ旦旦f.FirstwerestricttorealλWeExA。εe(Ao)nIR.By[Grll,1.3wehaveLλ+λ。二(id一λR(λ+机,Ao))Lh·ByProp.22(b)LλoisacontinuousmapfromθXintoX。,heMe||Lλ+λ。||β三||(id一入R(入+入。,Ao))||3||Lh||α·ThenLemma2.3(b)implies|lLλ+λ。||3=0(λβ-α)andtherefore||Lλ||β=0(λ9-α)aswell.IfRe入>0,|入|bigenough,wehavebyIGr1l,1.3Lλ=(id一(入一|入|)R(入,Ao))Llλ|hence||山Lμ削川λ礼才|川|Mβ三|川M|HMiωd一(υλ一斗|λ衬均|O)R琐(λ,A勾0)M|川M|d白刷3引纠|川仙问|问问L纠句h|队川λ抖|巾|川h|MP三引(1忏+Jd5|川队|队λR蚓(λ,JA0)|阶|d阶们阳3如纠仙)川川|川川|问叫L句hh|队川叫川λ川川础|川川h|川h|MβS旦i口||入川R(队入,JA岛州创4幻创州0ω训)川||3isboundedbyLemma23(a)theC∞On叫Clhhu1mSm1f扣OlHlOws。 NowweformulateaarstcriterionforφwhichensuresthatAφ(see(2.1))isage时r-ator-FirstwerecallfromIGr1l,14howAφandAoarerelatedtoeachother.Fixλεg(Ao).ThenforUεYonehas uεD(Aφ)仲(td-Lλφ)νεD(Ao)and(λ-Aφ)u=(λ-Ao)(id-Lλφ)ν(26) 2.5Proposition.AssumethatYCXαforsomeα,055α<1.If+isboundedmthrespectto||·||X由andsatiSEeslimsuP|λ|→∞,Reλ>o||φLλ||<lthenAφisageneratorofanandyticsemigroup-
2ω Greinerandkuhn E王22f.SinceYisdenseinXα,(D(Ao)isdenseinXa!)φhasauniquecontinuousextensioIItoxhwhichwillalsobedenotedbyφ.Moreover,fromProp.22(b)itfollowsthatLλisaboundedlinearmapfromθXintoXα·Byassumptionthereexistconstantsq<1,T>Osuchthatfor人εA:={λε￠:Reλ>0,|λ|>T}wehave||φLλ||三q.Moreover,byenlargingrifnecessar3飞wecaIlassumethatACQ(Ao).ThenidθX一@LλisiI附rublefor人εAand q401<一、ALφ∞艺叫<一、ALφ∞汇叫一一、ALφxno,a Itfollowsthatidxa-Lλφisinvertibieaswell-Infact,oneeasilyVedaesthatitsinverseisgivenbyidxa+LiidM一φLλ)一1φFTom(2.6)weconcludethatλ-AφisinvedibleforλεAanditsinverseisgivenby R(λ,Aφ)=(tdx。一Lλ争)一IR(入,Ao)=R(λ,Ao)+Lλ(idθx-Lλ争)一1φR(入,Ai}).(28) Wehavetoshowthat||R(入,AU||=0(|入|-1)for|入|→∞,Reλ>0.FortllefirsthIIIin(2.8)thisistruesinceAogeneratesananalyticsemigro叩·ForthesecondtmIIwelmvp oonut--ARXαθφXXθθφλruXRUEG,,,‘飞X且unu、AL<一、、lynuA-ARφφ、ArLUxno,d/'E‘飞、ArLU UsingtheresolventequationweobtainR(λ,Ao)二R(|人|,Ao)(td一(入一|入|)R(入,A川)ir旧eusingLemma2.3weget ||R(入,Ao)||:三||R(|抖,Ao)||:(l+JE||入R(入,Ao)||2)=O(|λ|α一1). ByLemma24wehave||Lλ||fx=0(|λ「α)and(27)statesthat||(idθx-Lλ争)-IMZ=0(1).Combiningtheseestimatesweobtainthattherighttlandsideof(28)dcc叮SasO(|λ|-1)whichimpliesthat||If(入,AU||=0(|入|一1).ThusAφgeneratesaILtlmlyticsemigroup-。 InProp.2.5@hadtobecontinuouswithrespectto||·||xa·Ontheotherhand,theassumptionYCXαimpliesthatLisnotcontinuouswithrespectto||-||x臼·Infact,si旧CLvanishesonD(Ao)whichisdenseinXα,continuitywouldimplythatL=OTlmo只CmaysaythatφhasahigherdegreeofcontinuityasLInthemainresultofthissectionwegivesomeconditionswhichareeasiertoverifythanthehypothesisofProp.2.5. 2.6Theorem.AssumethatYCXαforsomeα,0三α<landφiscontinuous11·itlirespectto||·||xiEachofthefollowingconditionimpliest由ha剖tAφge臼I肘ratθ白Sa川川I川it山Se臼ImIn11grO1u1p.(a川|φ||ZxissumcieMlysmall;(b)φ:Xα→θXiscompact;(c)ibrsomeF<α@iscontinuouswithrespectto||·||Xβ·
LinearandSemilinearBoundaryConditions 20I Emf.UsingLemma2.4thecasesof(a)and(c)canbeeasilyreducedtoProp.2.5.(b)requiressomeextraarguments.(a)ByLemma2.4weknowthatlims叩|λ|→∞,Reλ>o||Lλ||3X<∞·Henceif||@||gx<(limsup||Lλ||2x)-1thehypothesisofProp.2.5isfulSHed-(c)Wehave||缸,λ||三||φ||3x||LAjx=0(|λ|-(α-m)byLemma2.4.ThusProp.2.5canbeapplied.(b)ForsimplicityweMsumeOεe(Ao).Wehave||λR(入,Ao)-id||2三Mfor|λ|sum-cieMlylargeandRe入>0.AndforzεD(Ao):||(入R(入,Ao)-id)z||α=!|(λR(λ,Ao)←td)A。-1Aoz||α三||(λR(λ,Ao)-id)Ao-1||:||Aoz||=||R(λ,Ao)||:||Aoz||=0(|λ|α-1).SinceD(Ao)isdenseweconcludethat||(λR(λ,Ao)-id)z||α→OforeveryZεXα·UsinglGrll1.3weobtainl|Lμ||α=||(入R(入,Ao)-id)Loz||αforallzεθX.Moreover,duetoequiCOIltiEmity,theconvergenceisuniformonrelativelycompactsubsetsofXα·Since@isassumedtobecompact,itfollowsthat||Lλφ||2→Ofor|入|→∞,Reλ>0.Itf扣OlH比lkO队】w飞w飞V,thatidX白一Lλφisi盯ertibleandby(2.6) R(λ,Aφ)=(idxo-Lλφ)-1R(λ,Ao)=R(λ,Ao)+Lλφ(idxa-Lλφ)-1R(λ,Ao).(2.9) ItremainstoshowthatthesecondtermontherighthadsidedecaysasO(|λ|一1).1UjS白i川Lemma2.ι4weobtainthedesiredestimate ||Lλ争(tdxo一Lλφ)-IR(λ,Ao)||三||Lλ||fx||@||Zx||(tdxa一Lλφ)-1||2||R(λ,Ao)||:=。(|λ|-α)0(1)-0(|λ|α-1)=O(|λ|一1). 。 Obviouslyonecancombineboundaryperturbationswithadditiveperturbations-飞17eonlywanttogiveonetypicalexample. Corollary.Letα,βbeCOIlstmtssatis今ing05二β<α<1MdassumethatYCXOIfF:Y→XisA-boundedwithA-boundOMidφ:Xp→θXiscontinuous,thentheoperatorBdehedbyD(B):={νεY:Lu=φν},BU:=Au+Fuisthegenerator。faIIanalyticsemigrouponX,EE旦旦f.Itiswellknown,thattheoperatorBo,Boz:=Aoz+Fz,withdomainD(Bo)=D(Ao)isthegeneratorofananalyticsemigro叩(seee.g.[Gol,66).ThespacesXocorrespondingtoAoandBorespectivelycanbecharacterizedasinterpolationspacescorrespondingto(X,D(Ao)).Thustheyareidentical.ThereforewecanapplyTIIIII-2.6theoperatorA+F,andtheboundaryoperatorsLandφ.。 AsimpleapplicationofthiscorollaryistheproblemconsideredinEx2.1.OIIeevenmayallowmoregeneraldiferentialoperators-E.g.theproblemcorrespondingtoAf=f"+bf'+Cf,f'(0)=αf(0),f'(1)=Ff(1)consideredasperturbationofAf=f",fF(0)=0,f'(1)=OsatisaesthehypothesesoftheCorollary.Inthefollowingwediscusstwofurtherexamples.Firstaone-dimensionalproblemwhichisrelatedto
202 Greinerandkuhn Schr凸di吨eroperatorswithaDiracfunctionalaspotentiai(seeITilformoregeneralresultsintheHilbertspacesetting).ThenwediscusstheLaplacimonboundeddomms.2.7Example.(a)LetX=Lp(E),1<p<∞.WeidentifyXwithLAIR-)@Lp(IR+),ie-weidentifyfEXwiththepair(f-,f+)wheref芋aretherestrictionsofftoR+:=[0,∞)andIR-:=(一∞,olrespectively.WeconsiderthesubspaceY:={(久,f+)εWJ(E一)@W;(R+):f一(0)=f+(0)}anddeanetheoperatorA:Y→XbyAf=(ff,fl)-Theboundaryspaceisone-dimensionalθX=RandLisgivenbyL(久,f+):=fL(0)一九(0).Thenker(λ-A)=span{fzμ},whereμ=JXandlU(z)=exp(一μ|z|).TheDKiellletmapisgivenbyLM=tkh抖,(μ=JX)andonehas|lLλ||=||hμ||p=2t一1p-tμ一(Ht).ThusbyProp22(a)YCXoforα<i(1+j)Weconsideraperturbationφ:Y→Rdennedbyφf=γf(α)where7,αεRareaxedCOIlstants-FromSobolevembeddingtheoremitfollowsthatXβCC(IR)forP>去ThuswecanapplyThm26((a)01(c))andconcludethatthefollowingoperatorgeneratesananalyticsemigrouponLp(R) A1f=(ff,f:),D(A1)={f:hεw;(E剖,f一(0)==f+(0),fL(0)一凡(0)=yf(α)}. NotethatforfεD(Al)thederivativeoffhasatOajumpofheightγf(ο).Infact,inthesenseofdistributionswehaveA1f=Af+γf(α)6owhere60denotestheDiracfunctionalat0.Incaseα=0,AIcanbeinterpretedasSellr凸dingeroperatorwithtilt、(highlysingular)potentialγ60(seeiTiu-(b)NowweassumethatQCRn,(n三2)isaboundeddomainwithsmoothb011IIElaryθQ.WeconsiderX=Lp(Q),theSobolevspaceY=wj(Q)andforandfortbb01111dary1-iva11mtheSobolevSlobodeckiispace147PP(θQ)(seee.g.[Trlfordetails).飞bcomidertheLaplacianamiasboundaryoperatorthetraceofNeumanntype,i.e. /δf＼A:Y→X,AfpAf,LN:Y→缸,LNf:=|7i＼vn/|θQThenLNisacontinuoussurjmtion(see[Trl,471)andAogeneratesaaanalyticS优emmi厄grm℃O〉凡川叫1u叫l可I(忖e.g.[Fal,4.5.1and4.83).UsingTriebelsnotation,itsdomainisB;,p,{LN}andfrom([TrL433and461)itfollowsthatforα<αF<j(1+j): 唁亭'-20α川WP一-Q,α♂吨,-nrBQNFLUdnrq,-nrB一一nrατIX〉∞0αYX一一αX ThusfromThIII-2.6wecanconcludethatforalinearperturbationφ:Y→θXwhichiscontinuouswithrespect||||xaforsomeα<j(1+j),theLapacianwithdOImiIL{fεw;(Q):LNf=町}generatesananalyticsemigro叩onLp(Q)DuetothemutualembeddingpropertiesbetweenSobolevandBesovspaces,thisistrueifφismM11011swith时specttosomelvfmmforsomed,2<j(1+j)Let'sconsiderforexampletheDirichlettraceLD:fHf|拍·FromlTrj,5.52itfollowsthat
LinearandSemilinearBoundaryConditions 203 LD:wf(Q)→YiscontinuousprovidedthatF>(1-j)+jorF>1ThusLDisaperturbationwhichsatidestheassumptionsofThIII-2.6(c).Moregeneral,onecaII1-icomposeLDwithaboundedmultiplicationoperatoronY=W〉F(θQ)e.g.byaCLfunctionm.Thentheperturbedproblemhasboundaryconditionsofmixedtype,i.e.oftheformgf(z)=m(z)f(吟,forZεθQ. TheargumentsofEx.24(b)givenaboveextendtomoregeneralsituations.InfacttheLaplacimAmaybereplacedbyanydibrentialoperatoroforder2whichisregularellipticinthesenseof[Trl.ThenormalderivativemaybereplacedbymyderivativealongasmoothnontangentialvectorEeldνonθQ. Weconcludethissectionwitharemarkconcerningtheinterpolationspacesassociatedtotheperturbedproblem.2.8旦旦旦旦ElE·InthesituationofThIII-2.6onecanmakesomestatementsontheinterpolationspacescorrespondingtotheperturbedoperatorAφ·Letsdenotethembyx?,(1三F三1).Infactfromtherepresentationoftheresolvent(2.8)and(2.9)respectivei〉气onecanconcludethatfor3<αonehasXβCX3Moreover,itisaotdime1山toverifythattheDirichletmappingscorrespondingtotheperturbedproblemaregivenbyLT=Lλ(idθX一φLλ)一1.FromthisonecanconcludethattheArproblemcanalsobeconsideredasaperturbationoftheAφ-problem-HenceonehasthereverseinchlsioIlxfcXβforF<αmwell
204 Greinerandkuhn 3.THESEMILINEARCASE InthissectionwediscussthegeneralproblemIBVPasstatedatthebeginniIIgofthepaper-WewillallowthatFandφarediscontinuousonXbutarecontinuousonsomespaceXα·AgaintheorderαisdeterminedbythedecayrateoftheDirichletmapping-Animportantstepisthedetaileddiscussionoftheinhomogenous,linearproblem.I.e.wewillarstconsider d(t)=Au(t)+f(t)Lu(t)=ψ(t)u(0)=u0· (31) wheref:[0,T)→Xandv:[0,T)→θXaregivencontinuousfunctions.Themuonofsolutionwhichwemainlyuseisthefollowing:31Deanition-AcontinuousfunctionU:[0,T)→XiscalledmildsolutioIiof(31)providedthatforeverytthefollowingthreeconditionsaresatisaed:(a)ku(s)dsεY;(b)u(t)一uo=A(兀u(s)ds)+j;f(s)ds;(c)L(du(s)ds)=fJv(s)ds Webdenymentionsomeofthebasicpropertiesofmildsolution-Tileproofsarestraightforward--AmildsolutionsatisEestheinitialcondition.-Everyclassicalsolution(i.e-JtεC1([0,TLX)门C([0,T),Y)aM(3.1)hoidspoiM飞me)isamildsolution-Ifamildsolutionisdiferentiableatsomepointto,thenu(to)εYaIIEi(31)holdsfort=t0·Inparticul町,amildsolutionUεC1isaclast-icaisolution.Ifuisamild叫utio凡thenthemappingtHjjtt(s)dδisC∞O川l|川|.||y. Whenmildsolutionsare"gluedtogether,,oneobtainsamildsolutionagain. 飞Vedonotknowifmildsolutionsalwaysexistwithoutanyfurtherassumptions-ThefollowingresultgivesaveryusefulcharacterizationofmildsolutionswhichalsocaIIbeIlse(ltoprovethatundercetrainassumptionsmildsolutionexistforanychoiceoftto‘f『V-IIIfactitgivesaIlexplicitformula(incaseamildsolutioIlexists).
Linearandsemilinear-BoundaryConditions 205 3.2Theorem.FixλξQ(Ao).ForafunctionUεC([0,TLX)thefollowingassertionsareequivalent:。)Uisamildsolution;(iod瓦(t-s)LM(s)dsεD(队A岛州0ω)川fOrev陀eU叼r叫附←t川0什+才f扑ht》￥巧阳(川只灼削仰Ms吟M忡)M灿d出创叫s叶+刊(队λ一斗4训A岛蚓0ω)才f扑μt〉」叽阳(川川)川dsσ Beforewegiveaproofwediscusssomeconsequences-Uniquenessisanimmediatecon-sequeacesinceassertion(32)givesanexplicitrepresentation.Moreover,amildsolutionexistsifandonlyifd瓦(t-s)LMs)dsεD(440)foreveryt Corollary1.Mildsolutionsof(3.1)areunique. Corollary2.Assumethattbrsomeα,0<α三lwehaveYCXothen(3.ljhasa(unique)mildsolution.坠旦fInmwdThm32附呐hwetoensurethat归(t一仙ψ(s)dsεD(Ao)fOIeverytBy[Pal,4.3.6thisistruewhenevertHLλψ(t)isacontinuousfunctioninafractionalpowerspace-ForO<旷<αwehaveD((-AoY)〉Xα〉Yandailinclusionsarecontinuous.SinceLλisacontinuousmappinginYtheassertionfollows.。 BeforeprovingThIII-32werecallaresultoninhomogenuousinitialvalueproblems(seee.g.[Pal,Sec.42). 川 4(才几(t-s)MS)忡JI(t-s州川(t)+丸。川)ρ Proofo门M320)斗(ii)Wedeane咐):=jh(s)仇,F(t):=fJf(s)dsj(t):=j;仰)dsThenu(0)=F(0)=0,@(0)=0,u,F,φεC1,d=uo+Au+F,Lu=φ.(3.4)FUrtherwedeEne U2:=LλLty=LλφanduI=U-U2· (3.5) Thenu1hasvaluesinD(Ao)andhinkex-(入一ALMoreover,sinceφεC1andLλiscontinuous,U2εC1andthereforeUIaswell-Wehave 61=uo+AU1+A问+F-62=uo+AOU1+λLλφ+F-Lλψ,U1(0)=0.(3.6)
206 Greinerandkuhn Thewell-knownvariation-of-constantsformulaforinitidvaiueproblemsthenimplies 叫(们才巧(t一吨。+入M(s)+mods-J民(t一仙以s)ds口 U10)εD(Ao)andby(33)theErsttermontherighthaMsideisinD(Ao)aswell-Consequentlyjd;T勾b(μt-S吟)L乌λM川ψ叭(υωS吟)μdSεD(忱A0)Fh飞、1川川r川t}mn川Oω屹e,a叩PmPl切〉y抖F叶in吨gA岛0O∞nbothSdi(d&岱的ik扣hEe臼T泪Saηusing(3.33)weobtain 川一叫t马￥、瓦阳川(0t一向(8)ds orequivalently (λ-AUJl川川)ds+扑(t-叫fh)ds+巧(tM= =Ao叫(t)+uo+λLλφ+F(t)==Au+uo+F=b(t)=u(t)Inthelastlineweusedtherehtions(3.3-5).Thisshowsthat(32)holdsandtllf自rstpartoftheproofiscomplete.(ii)=争(t).Atarstwepointout,that(32)impliesthattH(λ-AUd凡(t-s)LMs)dsiscontinuous,hezmtHfJ月(t-s)LMs)&iscoIMim-011swithrespecttothegraphnorm.Thisallowsustointerchangetheorderofintegrationinthesubsequentderivation-Weintegrate(32)andobtain 才叫s)ds= =才f扑ht飞￥瓦阳(川=才ft巧阳川川υ例S)川u圳0=Jt阳SM+f(lt几(s-付出)阶)+入ω(T))dT-jt川一川d)LM(T)dIWeusedthefactthatjf瓦(s)dsmapsi川时omainof山gematOrandAoffTo(δ)dδ二TUM)-Tb(α).ThisargumentalsoshowsthattheiirstandthesecondsumlandareeleIMItsofD(440)MomoverdLN
LinearandSemilinear-BoundaryConditions 207 byassumption,Thusweconcludethatfh(s)dsεD(A)andsinceD(Ao)=kerLandLLλ=idθxweobtain 叫tti(s叫=0+0+0+比ILO-)dr=JL(7·)dTand A(fu(s叫=Ao(f协)忡。+fAo(f到(s一付出)(附)+λω(T))dr一叫马(t-仙以r)dr+λJft〉μL乌ω川λ川ψ=附-idM+f附一T)-id州+山(T))dT一叫马(t一仙仲)dr+入jtM)dr==u(t)一uo-ff(T)dTThuswehaveallthreeconditionsofDef.3.lveHEedandtheproofthatUisamildsolution。 NowweCOIlsiderthesemilinearproblemIBVPstatedattheverybegiIIIling.Tilegeneralhypotheses(H4)willbemodinedinthefollowingway.ForO三3三lwecomiciel F:[0,T)×Xβ→Xandj[0,T)×Xp→θXarecontinuous(H49) 叽feusethefollowingnotionofsolutionforIBVP. 33DdMUon-Assume(H1)-(H3)and(H勾)uesMissed-ThenU巳C([0,T'),Xβ)where0<TF三Tiscalledalocalmijdsolution(oforderF)providedthefollowingholds(a)ku(s)dsεY;(bht(t)-uo=A(ku(s)ds)+f;F(M(s))ds;(c)L(兀u(s)ds)=Kφ(M(s))ds IncaseofTF=T,Uiscalledaglobalmildsolution. AsanimmediateconsequenceofThm.32andthedeanitionsofmildsolutionwehalvethefollowingcharacterization. 3.4Proposition.If(HIj-(H3)ad(H勾)tuesatidedthenforUεC([0,TF),Xβ)tllefollowingassertionsareequivalent.(i)Uisamildsolution;
208 Greinerandkuhn 。i)耳巧(t-s)LJ(M(s))dsεD(Ao)foreverytMIdu(hmuo+扑川川(S忡 FromDef.3.3onecanconcludeeasily,thatboundedlineartermscanbeiIIterchangedbetweenthelinearandthenonlinearpartwithoutchangingthesolutions.Moreprecisely:ForBει(Xβ)and审ε￡(Xβ,θX)the(formallydiferent)IBVP-problemcorrespondingtoA1:=A-B,F1:=F+B,L1:=L+齿,φ1:=φ+审hasthesamemildsolutionsastheinitialproblem.ChoosinginparticulartE=OandB:=μ-idweobserve,thatwithoutlossofgemralityonemyassumethat{λε￠:Re入>-1}cg(Ao)Toproveexistenceofsolutionsoneusesaxedpointarguments.InordertoapplythccontractionmappingprincipleonehastoassumethattheIlOIIliaearitiesareLipschitzCOIltiI111011s.OneobtainslocalorglobalmildsolutionaccordingtorequiringglobalorlocalLipsctlitzcontinuity-飞飞Tearstconsidertheglobalcase-Fromthisthelocalo110ULII1}{?deducedeasily. 3.5Theorem.Assumethat(HO-(H3)aresatideclaIIdYcxoforsomeQ>0.IfforsomeS,3<αthemappingsF:[0,T)×Xβ一→Xandφ:[0,T)×Xi3一→θxaregloballyLipscllitzcontinuousthenthereexistforeveryinitialvalueuo巳XβallIiiqllcglobalmildsolutionoforderi1.Aforeowr,ifuoεXy,3三γ三α,thenUεC([0,T),XJ-EE旦旦f.Aspointedoutabovewem叮W.l.0.gassumetlm{λε￠:Reλ>-1}cp(Ao).InparticularOεp(Ao).WehavetoshowthatforeveryTF<Tthereisafmcti0日UεC([0,TFLXs)satisfying(34)(ii)with入=0.InviewoflP斗,4.3.6adtheassumptionimLoCYCXGtheco毗iond几(t-s)LJ(川(s))dsεD(Ao)isam川1U川1SatUiSaed.飞民considerZ:=(C([0,T丁,Xβ),1||l)where|lfl||:=sup{ε-ktl|u(t)lh:fε[0、TIl}aIKitheCOIlstantκ>Owillbedeterminedlater.飞;飞fewillapplythecontractionIIlappiIIgprincipleintheBanachspaceZtothemapping由deEnedbytherighthandFddeof(3.8)i.e.,wedeane飞ZFtufollows (忡审如川叫叫u叫叫咐)X附削(μ例t忖)=凡川川(υ例川t忖)ObhSe盯rV刊e,thatforUεZtheIIlappi吨SHLo@(s,tl(j))isCωOI口l此川tUiInm1N1ω1u1Sw叭it出h飞Vm?唁币Ea山Llh1u肘1e臼SinX0TH1e臼m阳In阳1K(叫肥…aω以川〉X对m〈diagaininZMomver,tHK瓦(t一δ)F(M(s))dδisamIaled由巾l』he臼ImITIhl1u1S飞q审z壶1iswedlHi-de白In1ed.飞Rfehavetoshowthat审isacontraction.GivenU,tYEZtileII ||(审(叫一审(υ))(t)||β=sllp入β||(入R(λ,Ao)-td)(由(叫一审(t7))(t)||三λ>0
LinearandSemilinearBoundaryConditions 209 三supλβ||(λR(λ,Ao)-td)AojTb(t-s)Lo(@(s,u(s))一@(81(s))ds||+λ>OJO+:13Y||仰,Ao)一叫tm-s)盹 =:s叩Jλ+supJi.λ>0λ>0 (3.10) Wechooseapathr:={T-e士"+号):T主0}withd>OsuchthatthespectrumofAoiscontainedintheleftangulardomaindennedbyr.Onehas||R(μ,Ao)z||1三C1|μ|-α||zLforμεr,ZEXαadasuitableconstantc1·MoreoverTb(t)andR(λ,Ao)cmberepresentedasDunfordintegralalongr.TakingthisintoaccountwecanestimateJλasfollows: Jλ=||YjtzLj7Lεμ(t-s)R(川o)。(Lo(φ(川(s))一φ(们(s))))仙||1三J04π1Jrλ一μ λβftf|μ|it}-jl一一一·|ε川一叫|||R(μ,Ao)oLo(φ(s,u(s))一φ(81(s)))||1|dμ|ds三-2πhh|λ一μ|λβftf|μ|{}-jj一一一-|♂(t-s)|C1|μ|一α·||LoMX||φ(SJ(s))一@(SJ(s))||θx|dμ|ds三-2πoh|入一μ|1βrtr三||Lo||俨-二-qrjj|μ|1一α|λ一μ|-1|♂(t→)|产|lu-u川|dμ|ds.中"JOJFHereristheLipsdlitzconstantforφThemalintegralI1:=卫|eit川)|♂sdswithμ=T-eidcanbeestimatedasfollows I1=(κ+r-cos。)-1exp(-rtcosd)(expo(κ+r-cosd))一1)三♂t(κ+r-COS17)-1. Thusweobtain 九三句卜 where6=α-F>Oandc3isasuitableconstant.Thesecondtermisboundedbyc4·κ-6andtheterm入βfr|入一μ|-IM|一β|dμ|=儿|l一μ|-1|μ|叫dit|isindepeMmtofλTIhlu巩wve白InlalHl3yrarriveat supJλ三C5·κ-6ekt|1u-ty|lλ>0 (311) wherec5isaconstantindependentofκandU,u.ForthetermJLasimilarprocedureyields ←内lRM
2IO Greinerandkuhn 三去叫tJ|λ一μ|Vt-s)|||RMo)||it||帅)-u唰 三C6·κβ-leHttu-u川-Combining(3.10)'(311)weaIlallyhave (312) l|审(u)一曲(u)|l=sup{fM||(由(u)一曲(u))(t)||β}三(C5κ-6+C6κβ一1)|||u-ty|l. BychoosingκsumcKIltlylarge,oneachievesthat审isacontractiononZThenthereisauniqueaxedpoint.Incaseγ>FandtheinitialvalueuoisinXγCXβthen,asshownabove,thereexistasolutionUεC([0,T),Xβ)satisfying 叫t)=%(t问+jl(t-s)川(s))ds-叫马(t-sMM(S))ds TheErstsummandisinXysinceXyisinvariantunderthesemigroup-MOreO飞Vfe创Ir.atthebe-gihnm川lumIn旧l让iInl蚓af丘t忧阳e盯rd命ed自mg审)wepoiIIMoutthatbothd巧(t-s)F(M(s))dsaIIdAod瓦(t-s)Loφ(s,叫s))dsarecontinuousfunctionsfrom[0,T)→Xα(duetωO[CHADP町]‘569an6.110川).IHfγ三αthenXαC;X汇7.。 Theglobalversioncanbeusedtoprovealocalversion.Theprocedureisstandard-OIlereplacesFandφbyfunctionF1andφ1respectively,whichcoincideina||·||β-ILeighborhoodof(0,tto)withFandφaadaregloballyLipschitzcontinuous.ThiscanbeachievedforexamplebymultiplicationwithascalarfunctionXEdeanedasfollow-s: εF』内/】<一<-38paμHHHHHHHZZZHHtt-tl|/二<一<-r-nUFLvn,,"俨'APφAF'A...............、‘,,,,raμzε9",,..‘飞11-EOil〈1飞一一、‘,,FZ/t飞EVA叫 矶、omitthedetailsOIIlvmentiontheresult. 3.6Proposition.Assumethat(HI)-(H3)aresatiSEedandYCXαforsomeα>0.Ifforsome3,3<αthemappingsF:[01)×Xp→Xmdφ:[0,T)×均→θXarelocallyLipsditzcontinuousthenthereexistforeveryinitialvalueuoεXβauniqut?localmildsolutionoforderi7.Moreover,ifuoεxmF三75二α,thenUiscontinuousinXy· Bytheusualmaximalityargument,onecanshowthatthesolutionscanbeextendedtoamaximaltimeintervall0,Tmaz(tto)).Moreover,ifFandφareuniformlyLipschitzoIlboundedsubsetsthenitispossibletoproveablow-uptheorem.Thatis,eitherTmαz(tto)=Torlimt→Tmaz|lu(t)lh=∞.Insteadofprovingtheseresultswebridylookattheproblemformulatedattheverybeginning.RecallalsoEx2.7(b)
LinearandSemilinearBoundaryConditions 2II 3.7Example.LetX,Y,θX,AandLNbedeanedasinEx.2.7.TherewehaveshownthatYCXαprovidedthatα<i(1+j).AsImlineaωesweconsiderNemitskyiope时orsinducedbybyscalarfunctionsf:[0,T)×Q×R×Rn→Randψ:[0,T)×θQ×R→R.Theeondi阳lsonfandψhavetobechominsuchawaythatformmp,1<9<1+jtheNemHskyiop町MorsFandφareLipschitzfmmwf(Q)inLp(Q)aMfm川v;-t(θQ)intoitselfrespectively-[AplgivesasurveyofcriteriaforLipschitzcontinuityofNemitskyi(orsuperposition)operators,seealso[KZPSl,Sec.17.E.g.ifψεC2andthederivatives1-i1-ihandtpuzareuniformlybounded,thenjWFP(θX)→WpF(Q)inLp(51)isgloballyLipschitz. REFERENCES[AmllT1.Amam,LinearAlgebraAPPI-,84:3(1986)[Am2lH.Amann,J.DiEEquat,72:201(1988)IAplJ.Appell,Expo-Math.,6:209(1988)[BBlP.LButzerH.Berens(1967).Semi-GroupsofOperatorsandApproximation,Sprin-ger-Vedag,BerlinlCHADPlPhClAmeat,H.J.A.M.Heijmans,S.Aagem时,C.JvaaDuijIlaMBdePager(1987)One-ParameterSemizroms[DSZlW.Desell,W.SchappaCherandkangPeiZhang,HoustonJMath.15(1989),527-552[FblH.0.bttori叽TheCamhvProblem,Addison-Wesley,London[GolliA.Goldstein(1985)SemizromsofLinearOperatorsandApplications,OxfordUni-versityPress,NewYorkIG02liA.Goldstein,"NodimarSemigro叩s,PartialDiferentialEquationsandAttrac-tors"Proceedings,Springer-LNM1248:71(1987)IGrllG.Greiner,HoustonJ.Math.,13:213(198η[Gr2lG.Gminer,"TrendsinSemigroupTheoryandApplications",Proceedings,THeste1987IGr3lG.Greimr,Semigro叩Forum,38:203(1989)II〈ZPSlM.A.kramoselskii,P.P.ZabreikqELPUSWInkandP.ESobolevskii(1976)IntezralOperatorsinSpacesofSumEnableFUnctions,Nordhof,LeydeniLallLasiecKAAppj.Math.OptiIII-J:287(1980)[PalA.Pazy(1983),SemiEromsofLinearODerMorsandApplicationstoPartialDife·rentialEquations,Springer-Vedag,Berlin[TilA.Tip,J.Math.Phys.,31:308(1990)[tklH.TEiebel(1978).InterpolationTheorv.FUnctionSpaces.DifereMdOperators,North-Holland,Amsterdam
ExactControllabilityoftheWaveEquationinPresenceofCornersandCracks PIERREGRISVARDI.M.S.P.,MathematicsLaboratory,UniversityofNice,Nice,France Thisisthetranscriptoftheta1klqaveintheConferenceon"Trendsinsemi-qouptheorlJandev01utionequations"atDelft(H01land)inseptember1989.Unfortunateluvhenenterinqthe1ectureroomIbumpedintheportab1eblackboardvNchbeqanvibratinqandsqeakinqinaveruunpleasantvau.HureactionvasnatLJra11utotrlJtoholdnrm1lJtheedqesoftheblackboardinordertosilenceitassoonaspossib1e.OfcourselvashapplJnottohavetoholda11theedqesatthesametimeandlendeavoredtotakeittorest1ntheshortestpossibleume.Thisisal1vhate×actcontrollabilltuiSaboutineverlJdalJ-slanquaqe.AlsotheVho1estorlJVOLjldhavebeenevenmoreembarass1nq1ftheb18ckboardhadbeenfLlllofcracks.Thisisvhatthistalkisabout. 1.Theprob1em1nmathernaUca1vords: GivenLjoandu1tvorealvalLjedfunctionsdeflnedinaboundedopenSUbsetQofRnwe10okforatimeTanda··control··Vrealvaltieddefinedin艺T:=]0,T[×广vhere「=aQ,suchthatthe S01LJtionUofthemi×edproblem u"-Au=OinQT:=]0,T[×QU(O)=uo,u·(O)=u1, (1) U=vonZT fullfi1s u(T)=u·(T)=0. 2l3
2l4 Grisvard Herethesuperscript-denotesdifferentiationintimeandUOmeansthefunction×→U(t,×),×εQ.inothervordsvetakeavibratinqs1Jstem(qovernedbuthevaveequation)torestattimeT(andSUbsequenttimes)buactinqonitthrouqhitsboundarlJ(〉1athech01ceofthenon-homoqeneousDirich1etconditionU=von艺T).HereVVilldependonLJoandLj1vhileT hJillnot. OfcoursetheproNemiSmorerealisticvhenthecontrolisontheNeumanndatarnear11nq au/av=vonZT vherevdenotestheoutvardnorrna1unitvectoron「.ButNeLImanncontrolisalSOmore complicated.ThisisvhlJe×posit1onvi11bemadeonDirichletcontrol.Correspond1nqreSUMsforNeumanncontrolvi11bemere1Vstatedattheend. 2.S01v1nqthewaveequat1on: Webeqinvithabriefrevievofe×iStenceresultsforthevaveequation.itiscon〉ementtointroducetheunboundedse1fad]OintoperatorAu=-Au vithdomaln DA=(UεH10(QKAuεL2(Q)}intheHilbertspaceH=LZ(Q).!t1sknovnfromLions(1961)thatDA1/2=刊。(Q) andthatAcanbee×tendedintoalinearcontintJOLjsoperator,stllldenotedbuA,fromH10(Q)ontoH-1(Q).AccordinqlUVecanre、川tethevaveequation(vithhomoGeneOLJSDlflCMetcond1tions)asfollovs: ψεC([O,TLW。(Q))nc1([0,T],L2(Q))ψ··(t)+Aψ(t)=f(t),tε[0,T](2)
ExactControllabilityofwaveEquation 2I5 ψ(O)=ψ0,ψ·(O)=ψ1·LjsinqancornPIeteorthonormalslJsternofeigenfunctionsofAoneeas1lUfierivesthatfor everu ψ。εH10(Q),hεL2(Q),fεLt(0,T;L2(Q))theaboveproblem(2)hasaUniquesolution.Wesha11denotebuxthespaceL1(0,TiL2(Q))invhatfo11ovs.itisthenpossibletoprovethat aψ/avεL2(Z)andthatatp/avdependscontinLJOLJSlUinL2(艺)onthedatatpoεH10(Q),hεL2(Q)andfε×.Thisischeckedviththehelpofac1assicaltechniqueofRellichbasedonthefollovinqidentitlJVherethevectorfieldmbelonqstowt∞(Q)n:ForeveruψεH10(Q)SUChthatAψεL2(Q),onehasJQAψm.Vψd×=-ZKJjQDKrnlDKLPD1ψd×+(1/2)IQdivmlVψ|2d×+(1/2)J「m.v|部/avFdσ(3) Choosinqformavectorfieldsatisfuinqm.V〉Oontheboundaru,a11ovstoshovthataψ/avεL2(「).lnasim11arvalJforeveruψfulfil1inq(2)onehasJQfm.Vψd〉〈dt=jQDttpm.Vψd×|oT+(1/2)jQd1vrn((DW)2-lVψ|2}d×dt+艺人lIQDKfnlDktpDWd×dt-(1/2)jεm.v|aψ/av|2dσdt. (4) Thenchoosinqthesamemasaboveonechecksthataψ/avεL2(艺). Thismakesitpossibletotransposetheabovee×iStenceresultandtoderiveaneve×iStenceresu1tinaveal〈sense飞laLions-问aqenes"(1968).Wesha1lsa-thatUξX·=L∞(0,T;L2(Q))1saveaksolutionofthehomoqeneousequationvithinitia1datau。εL2(Q),问εH-1(Q)andDirichletdatavεL2(「×[0,T]),inotherwordsaveaksolutionof(1),1ftheree〉〈ists
2I6 Grisvard ψ。εL2(Q),ψTεH-1(Q)SUChthatJoLJfd×dt-〈ψ。,ψT〉+〈ψf,向〉=〈Ljf,ψ(O)〉-〈Uoj(O)》-j艺〉(aψ/av)dσdtforeverufεX,ψ。εHTo(Q),ψ1εL2(Q)vhereψisthesolLjtionofψεC([0,T];H10(Q))nd([0,T];LZ(Q))ψ..(t)+Aψ(t)=f(t)ψ(T)=ψ0,ψ·(T)=ψ1Novgivenu。εL2(Q),u1巳H-1(Q)424vεL2(z),thereexistsaLjnifqtje(u,中1,ψo)εx·×H-1(Q)×L2(Q) SUChthatUisaνeaksolutionActua111JuεC([0,T];L2(Q)),u·εC([0,T];H-1(Q))andbothdependcontinLlOLjS1uonthedatauoεL2(Q),UTεH-1(Q)andvεLZ(Z)ReferencesforthisrestJ1t,inthecaseofsmoothboundarlJ,areLasiecka-TrlGQlan1(1981),Las1ecka-Tr100iani(1983)andL且且豆(1983).SeealsoLasiecka-Tr1001ani(1986)fordatavinotherspacesthanL2(艺).ThiSiSObta1nedblJtransposinqtheoperatorψ←→Lψ=(ψ"+仰冲(T)j(T)} from DL={ψεC([0,Tl,H10(Q))nc1([0,T],L2(Q))j'+Aψε×) 1ntoX×H10(Q)×L2(Q)andSOIL11f1q(u,ψ1,ψ。)εx·×H-1(Q)×L2(Q)L恬{u,ψT,ψ。)=α hJhereαisthe1inearform &VKAUσAU、、,,,,UV飞σ//的甲飞。,,E‘、V?今,,,J、J、、-EJnu,,E、、的YnuHUJ、、,,、、‘,,,nu,,E‘、的YHU〈]口的Y which1scontinuOLjsonDL 3.TheH门bertUniquenessrtethod LetusassumeforthetimebeinqthattheboundarlJ「ofQisreGularenouqf1.Thenlt1sknovn
ExactControllabilityofWaveEquation 2I7 fromaseriesofvorksbuLas1ecka(1987),Laqnese(1983),Lions(1988),Triqqian1(1986)thatthetimeTa11ovse×actcontro11abilituforeveruuoεL2(Q)andU1εH-1(Q),(1)beinqmeant1ntheaboveweaksense,ifthetvofol1ovinqnormsareequivalent (ψ0月1}H||ψo||H10(Q)+||ψ1||L2(Q)(ψ0,ψJH(jεlM/avFdσdt)1/2vhereψεC([0,TLH10(Q))nc1([0,T];L2(Q))so1vestheaLJ×1liaruproblemψ··-Aψ=OinQTψ(O)=ψ0,ψ·(O)=ψ1ψ=OonZT·ThecontrolVisinL2(艺)andmoreoveritssupportmauberestrictedtoasubset艺oofZTifthereisalsoequivalenceofthenorms (ψoN1)同||ψo||H10(Q)+||h||L2(Q)(ψ0月1)同(jz。|aψ/avFdσdt}1/2 Theinequalitu(jε|aψ/avpdσdt)1/24C(|忡。||H10(Q)+||h||L2(Q)}follovsfrom(4).Wesha11thereforefocusontheconverseinequalitunamellJ||h||H10(Q)+||州|L2(Q)4C(jz。|aψ/aυ|2dσdt)1/2. 4.Us1nqmu1t1p11ers: Goinqbackto(3)vecannovconsidertheveruspecialvectorfie1d,alsocalledmuIUp11er,m(×)=×-×0、川thasuitabluchosen×0·Onehasdivm=nandDKfTI1=SK,landforeνefuψεH10(Q)SUChthatAψεL2(Q),onehasJQAψm.Vψd×=(-1+n/2)JQ|Vψ|2d×+(1/2)j「m.v|aψ/avpdσ-Ands1m11arlufor(4):foreveruψfLJ1fi11inq(2)onehasjofm.Vψd×dt=
218 Grisvard JQDttpm.Vψd×|oT+(n/2)jQ((DW)2-|Vψ|2)d〉〈dt+IQ|Vtp|2d〉〈dt-(1/2)Iεm.v|aψ/avpdσdt. intheparticularcasevhenf=O(thehomogeneouswaveequation)veqet(1/2)jεm.v|aψ/av|2dσdt=JQDttpm.Vψd×|oT+(n/2)jo((DW)2-iVψi2}d×dt+jQ|Vψ|2d×dt. Theriqhthandsideisalsoequa1tojQDttpm.Vψd×|oT+([n-1]/2)Jo((DW)2-|Vψ|2)d〉〈dt+(1/2)JQ((DW)2+|Vψ|2)d×dt ThisisTtimestheenerquE=(1/2)(||Vψ||2LZ(Q)n+||DW||2L2(Q))=(1/2)(||Vψo||2L2(Q)n+||ψ1||2L2(Q)) p1ussome"junk··sincetheterm IQDttpm.Vψd×|oT canbeestimatedusinqEvhiletheterm Jo((DW)2-|Vψ|2)d×dt reduces,afterinteqrationblJparts,toboundarlJtermsthatareeasiluestimatedsincetpsolvesthehomoqeneousvaveequation.Putt1nqeverlJthinqtoqetherveobtainaToandaconstantCSLJchthat c(T-To)E4(1/2)jzm.v|aψ/av|2dσdt (5) lnordertoGetanupperboundforEon1uthepartofzwherem.visstrictlupos1t1velsuseful.Denot1nq Z+(×。)=(×ε艺;m.V〉O)weqetthee×iStenceofaconstantCsuchthat(T-To)E4Cjz+(×o)laψ/av|2(1σdt inconelUSionvehaveshovntheequivalenceofthenorms (ψ0月1)同||ψo||H10(Q)+||h||L2(Q)(ψ0月1)H(jz+(×o)|aψ/avpdσdt)1/2, andtheHi1bertUniquenessNethodimpliese×actcontrollabilituforanuT〉ToV1thacontrolsupportedbuz+(×。)(vhlch,buthevau,isneveremptlJ)×obeinqSOfararbitraru(andthereforepossib1lJchosentomake艺+(×。)assma1laspossib1e).
ExactControllabilityofWaveEquation 2I9 LetUSobservefina111Jthata11theaboveholdsprovidedthe1nteqrationblJpartsin(3)andin(4)arealloved.ThisiscertainlutruevhentheboundaruofQisreqularenouqhimpluinqH2(Q)reqularituofthefunctionsψthatfulfil1ψεH10(Q)andAtpεL2(Q)(6) or(2). nuBRH-mnuRUed-nHnu-A,,、HUlnueuwFgau-HUnvnH-FaEJ AssumethatQisaplanepoluqonora3-dpollJhedron,QbeinqOnones1deofitsbounda「lJon1u.Thise×cludesCLjtsorcracks.ThenitisknovnfromGrisvard(1975)(1985),Dauqe(1986)that(6)imp11es ψεHS(Q)vhere3/2〈S〈S(Q),S42vherethenLlmbers(Q)dependsonQandisalvaus〉2vhenQisconve×.LjnfO「tunate1Umthenonconve〉〈cases(Q)maubearbltrarilucloset03/2.Onehasthereforeaψ/avεHS-3/2(「)cL2(「) andtheinteqrationsbupartsthatprove(3)areactuallUa11oved.何aine×actcontrollabilituh01dsforatimeTlarqeenouqhwithacontrolsupportedblJ艺+(×o),×oarb1trarlJ·WethLJSObtainthestatementthatqeneralizesinastra19htforva「dmanne「thereSUMsprO〉enbvuons(1988)onreqLl1ardomains: Theorem:LetQbeap1aneopendoma1nv1thp01ugonalboUndaruOrathreed1mens1onr181opendoma1nv1thp01uhedra1boUndaru.ASSLJmeQ1sononesideofitsboundaru「.Foranuxotheree×istsatimeToSuchthatforeveruT〉ToandeveruuoεL2(Q),UTεH-1(Q),thereexistsvεL2(z)vithsupportinz+(×o),suchthatUtheweaksolutionof(1)fulf11lsu(T)=u·(T)=0. 6.Cracks:LetUSconsiderforsimp1icituaplanedomainvithon1uonecrack
220 Grisvard supportedbuthepositivehalf×-a×is.WeassumethatthecrackendspreciselUatOandthattherestoftheboundaruissmoothaccordinqtothefiqurebelov: u 。 x Theninsuchadoma1n(巳)doesnotimpluthatψεHS(52)vithS〉3/2.Weon1uknowthattheree×istCεR纽且ψRεH2(Q)凶♀且iaalψ=ψR+CJFS1n(0/2)vhere×+iu=re10.ltisclearthattps:=J?sin(0/2)doesnotbelonqtoH3/2(Q)af1dthatatps/avisnotsquare1nteqrableon「s1nceltbehavesasr-1/2neartheO「1qin.Th1smakesthe1nteqral jε|aψ/aν|2dσdt mean1nqlessasvellastheinteqra1 Izm.v|部/aν|2dσdt 且凶监iim.V→Oasr→0.ThisforcesUStochoosean×oonthelinethatSUpportsthecracknamelutheO×a×1Slnthecaseofthefigure.Underthisassumptionitisnatura1totrutoprovetheidentitlJ(3)blJperform1nqtheinteqrationsblJpartsonthesubdomain
ExactControllabilityofwaveEquation 22I Qε=Qn(r〉ε), wherea11theinteqra1sinvolvedmakesense,andthenhttinqc→O(asitlSLjSLja1inthetheorlJofsinqLl1arinteqra1sforinstance).Une×pectedluinsteadoftheaboveidentituIQAtpm.Vψd×=(1/2)I「m.v|aψ/av|2dσ veobtainthef011ovinqthatinvolvesanaddit1onaltermdependinqonC.jQAψm.Vψd×=(1/2)j「m.v|aψ/avpdσ-(何/8)c2m.τ,(7) vhereτistheunitvectortanGenttothecracktipandp01ntinqinthed1rectionofthecrack(e×tendinq)lnconclLJSionveshallobtaintheinequalitu(5)requiredforbelnqabletoperformtheH11bertUniquenessnethodon1lJVhen m.τ〉O asinthecaseofthePictUre.HoveveritisusefultopointoutthattheapplicationoftheHilbertUniquenessHethodisnotSOstraiqhtforvard.lndeedinstudu1nqtheequivalenceofthenorms (ψ0,ψJH||ψo||H10(Q)+lWJ|L2(Q)(ψ0月1}H(Jε。|aψ/av|2dσdt)T/2vehavelosttheRellichtupeesumate.F011ovinqideasbuμ皿豆(1988)veintrodLJcethespaceEoflmua1data(归,th)suchthatthecorrespond1nqs01utionofthehomoqeneousvaveequationfLJlfi11saψ/avεL2(艺) equippedviththenorm (ψoNJH(jz。|aψ/avpdσdt)1/2luovtheinequalitlJ(5)shovsthatEisasubspaceofH1O(Q)×L2(Q).AsaconsequencetheH门bertUniquenessNethodShovse×actcontrollabilitufO「initialdatainE·.lmt1aldata1nL2(Q)×H-1(Q)areaparticularcase.Thee×tenslonofthepreviouscasetoamoreqeneralcasev1thseveralCrackslspossibleifonecanfindapoint×osuchthatm.v=0ω且m-t〉O
222 Grisvar-d ateachcracktip.Ane×arnpleofsuchasituationisqiven1nthef01lovinqfiqure. Summlnqupvehaveproventhe Theorem:LetQbeaplaneopendomainvithSmoothboundaru「e×ceptforafinitenumberofClJts.AssUmetheree×ists×oStjchthatm.v=Oa且dm-t〉Oateachcrackt1p.Thentheree×1stsaumeToslJChthatforeveruT〉ToandeverutJoeL2(Q),UtεH-1(Q),thereexistsvεL2(Z)v1thsUpport1nz+(×。),SUChthatUtheveaksolutionof(1)fulf11lsu(T)=u·(T)=0. LetusobserveflnallUthatthereSUMsofsections5and6canbecornMnedtodealv1thdoma1nsthathavecornersandcracks. 7.NeLImanncontro1:lfvereplacetheboundarucondition U=vonZT blJ
ExactControllabilityofWaveEquation 223 au/ijv=von艺Tthesiqnofm.τin(7)1schar19ed.WehavejQAψm.Vψd×=(1/2)J广my|aψ/aν|2dσ+(节/8)c2m.τ instead-Henceveareleadtoassumethat m-hOω且m-UOateachcracktipasshovnonthefollovinqfiqure.Thestatementofthetheoremrema1nsthesameasaboveprovidedthesiqnofm.τischanqed. S1milarresuMsareava1lab1eform1×edDirichlet-Neumannboundaruconditions1nGU豆半缸il(1989)andfortheequationofvibratinqplatesinU且且(1989) 8.F1r181suggest1onofProfessorTr1991an1:Nuchmorefe×1b111tlJ1Sa11oved1ntheGeometricd1stributionofthecracksifonevorksouttheaboveHilbertUniquenessHethodvithmoreqenera1mLIltip1iersm.ltisenouqhtoassumethatthematr1×Hvhoseentriesare hkl=DKml iSSUchthat H+H"
224 Grisvar-d ispositivedefiniteeverlJvhereinQ.OneneedsinadditionaH01mqref1·stlJpeumqueness theoremfordoma1nsvithCracksvhiche×tendstheclassica1One.Technica1detailsvillbeprovidede1sevhere- 9.B1b1109raphu: Qa且且阜(1988):ElllpticBoundaruva1ueProblemsonCornerDoma1ns,LectureNotesmr7athematics,n01341,Sprinqer-Verlaq.Grisvard(1975):BehaviorofthesolLJUonofanellipticboundaruva1ueprob1em1nap0118onalorpouf1edraldomain,SlJnspade门!,Hubbarded.AcademicPress,p.2O7-274♀且豆xaL且(1985):E!1ipticProblemsinnonsmoothdofTiains,忖onoqraphsandstud1es1n忖athematics24,Pitman.Grisvard(1989):Contr61abilitee×actedess01utionsde1·6quationdesO「Idesenpr6sencedesinGularit6s,J.问ath.PuresetAppl.,68,p.215-259.La♀且豆豆豆(1983):Contr01ofvaveprocessesvithdmt广ibLItedcontrolssupportedonaSUbreq1on,SlA问J.oncont「OlandOptimization,p.68-85las1ecka-Tr1001an1(1981):AcosineoperatorapproachtomodellinqL2(O,T;L2(「))-boundarlJinputhlJperbPONeequations,App1iedNathematicsandOptim1zation,7,p35-93.Las1ecka-Trioo1ani(1983):Reql」laritvofhuperb011cequationsunderL2(O,丁jL2(「))-Dirich1etboundaruterrns,AppliedHathernatiesandOpt1mization,1O,p.225一286.Lasiecka-Trlooian1(1986):Non-homoqeneousboundaruvalueproblemsforsecondorderellipticoperators,Journa1de阿athematiqLIesPuresetApp1iqLlees,65,p.149-152.Lasiecka-Triooiani(1989):E×actBoundaruControllab111tuofthevaveequationwithNeLJmannboundarlJcontrol,App11edHathematicsandopt1mizat1on,19,p243-290.μ♀且ii(1961):EqLJat1onsoperat1onne11esetprobl色mesau×limltes,Sprinqerver1aq.μω豆(1983):ControledesslJsternessinquliers,GautNer-V1llars,Pans-L且且iL(1988):Contr61aMUtee×acte,Perturbatio「lsetstabilisat1ondeslJstemesdistrlbu6s,Tome1,C011ectionRHA,「1asson,Paris.L1ons-问aGenes(1968):Prob18fTIesaLJ×lim1tesnonhomoqenesetapp11cations,DLJnod, Par1S.
ExactControllabilityofWaveEquation 225 且且且豆:(1988)Contmabilitee×actede1·6qtmuondesplaquesvibrantesdansunPOIωhe,CRASParis,307,i,p.517-521.Tr100iani(1988):E×actBoundarlJControllabilitlJonL2×H-1ofthevaveeqLlat10nv1thDirichletboundarucontr01actinqonaportionoftheboundaruaQandrelatedproblems,AppliedHathematicsandOptimization,18,p.241-277.
AbstractLinearParabolicProblemswithNonhomogeneousBoundaryConditions DAVIDEGUIDETTIDepartmentofMathematics,UniversityofBologna,Bologna,Italy I-STATEMENTOFTHEPROBLEMANDBASICASSUhfprIONS Inmanyconcreteparabolicproblems(seesectionV)oneisconcernedwiththefollowinggeneralsituation:(h1)E。,ElareBanachspaces,withEIgEo(continuousinclusion)andnorms||.||oand||.||1·(h2)Forj=lv·-J(reN)阳,...,μrarerealnumbers,withogLilg.--gLir〈1,E1·μ1,...,E1·μr,Fo。,FOo+μ1,...,FOo+μrareBanachspacessuchthat,ifA,B,Cε{E,F},己,η,pe{0,1,1·μ1,...,1·μr,80,Oo+μ1,...,80+μr},Oe]0,1[and(1·0)巳+问=p,CPisoftypeObetweenA巳andBTl-Further,if巳〈η,BηEA巳-WeShanindicatewithtkthenorminF巳·(h3)0。+μr<1.(h4)O<TO吨。:[0,T]→8(E1,Eo),forj=1,...,r$3j:[0,T]→8(E1,E1·的)nd(Fo。+町'Fo。)andthereexistpe]0,汀,C〉0,suchthat,forogtEESgT, 227
228 Guidetfi 、‘,FMdmE·e、,、a,..,...··Aat--aEd--抽,t飞、..JRHS卜ZP川ti飞,ewCH阿由M协〈-4-MEO由.ω2·α······4扒dJrem目μ-τ1』川Jdπrzduq-mn问·.J气eπ二4·dGFLV币,』UW尸LmseELSE0'也J俨〉hAVmmw阳wm、dA-1-国·····H‘)+MX、.F1··v·'ELashpka叫aelm+-MUM}-ipZT仆uvryι=飞J、J、Jhu.,JζJ正U句InE、-LULULUHHH,寸''飞''飞''飞 (λ-q(t))u=f 、在··/-azAJ,‘‘、 τ(记j(t)u·白)=0,j=lv·-J hasauniquesolutionMEIVfeEoAg1,...,gr)eHFI叫,KCwithiArgλlgOoorλ=Oandthefollowingestimateisavailable: 川)l|U||o+ll川C仲杂+|λ|川0|岛|Oo+护同) (2) (C〉0,independentoff,(g1,...,gr),λ)-Wewanttostudythefollowingproblem: 杂如t)=4α川)+f(0ωtτ(记jρ(0t)讪u(0ωtO)-g白j(0ωtO)》)=0,j=1L,v.川….川.J,Jr,tε[hS,η,u(s)=lu1O. (3) HereSe[0,TLfeC([s,TKEo),forje{1,...,r}句eC(ls,T];E1·μjLuobelongstosomesubspaceofEo-η1etipicalsituationiswhenq(t)isanellipticoperator,forj=1,...,rτ把j(t)areboundaryoperatomHcompatible"withq(t)(seetheapplicationsinthenhhseCHon).Ourgoalistodevelopalineartheorywhichcanbeusefultotreatquasilinearproblemswithnonhomogeneousboundaryconditionsonthelinesof(forex缸nple)[3].AsfarasIknowtheapproachIsuggestisnew.However,manyauthorshavestudiedtheconstructionofanevolutionoperatorintheparaboliccasethatallowtotreatourprobleminthecase岛=Oforanyj.ThemostgeneralconditionsseemthosecontainedinapaperbyAcqllistapaceandTerTeni(see[1]).Ofcourse,ourfinalresultsareapplicable(inourpudculusituation)to由econsmICHorlofanevo1116onoperatorunder
ParabolicProblemswithNonhomogeneoωBoundaryConditions 229 conditionswhich,inconcretecases,areanalogoustotheconditionsof[1](seeremark4.15andtheobsewationsfollowingtheorem5.1).Weshallconsiderseveraltypesofsolutions: D吃fJnitiOF11.I.AStrictsolutionof(3)isafunctionUeC1巾,T];Eo)(C([s,T];El)satisfying(3)pointwiseVtε[s,T](ofcourse由isimplies由atuoεE1).Aclassicalsolutionof(3)isafunctionUεC1(]s,T];EdnC(]s,η;El)(C([s,T];Eo)satisfyingthetwoErstequationsof(3)Vte]s,T]andsuchthatu(s)=110·Astrongsolutionof(3)isanelementUofC([s,T];Eo)suchthatu(s)=uoandforwhich阳reexisM叫uem(u(k))L1inC1([s,T];Eo)(C([s,T];El)convergingtodu(k)UinC([s,η;Eo)andsuchthat丁「-Gt(t)u(k)convergestofinC([s,T];Eo)and,forj=lv·-ιτ(把j(t)u(k)·句)convergestoOinC([s,T];Z3. II-SOMEBASICESTIMATES Inthissection,weshallcollectsomebasicfactsandestimateswhichwillbefundamentalforourpu叩oscsandcanbegenerallyobtainedbystandardmethodsbee,forexampleHOD-Wepose:D(A(t))={ueEl|τ$3j(t)u=Oforj=lv·-J},A(t)u=。(t)u.Inanycase,wecandefine,forte[0,TLS运0,feEo, expoA(t))f=(2πi)-1jexp(λs)(λ,A(t))-1dλifS〉0,exp(OA(t))f=f(4)YHereyistheusualregularpathwithsupponinp(A(t))from∞exp(-iOo)to∞expoo。),withπ/2〈Oo〈00·ThepropeItiesofexpoA(t))arethoseofanalyticsemigroupswiththepossibleexceptionofthestrongcontinuityin0,whichissatisfiedifandonlyiffbelongstokclosureofD(A(t))inEo(foraproofsee[9]).WestartwithsomesimpleestimatesinspacesoftypeO;westipulatetheconventionthat,ifO=0,G=Eo,ifO=1,G=E1· Lemma21.LetGbeaspaceqfopeO(ε[0,1])betweenEoandE1.Then--(a)thereexistsC〉0,such的arVIIeELVλεC,|ArgMgo。,Vtε[0,T],|川|邮ωλ川均州机!0俨俨)沪归0归1阳)卜川山.扎句州λ均灿圳M)讪川川附uω凶|H|F1.(b)thereexistsC〉OsuchthatVλεC,|Argλ|剑。,OgSgtgT,fε鸟,(1+|λ|)1·O||(λ-A(t))-lf-(λ-A(s))-1日|GgC(t·s)P||nlo;(c)thereexistsC〉Osuchthat,fbrs,te[0,η,σ〉0,
230 Guidetti |lexp(σA(s))·exp(σA(t))lld伍。,G)gCσ0|t-slR,||AO)exp(σAO))·AO)exp(σA(t))l|J伍。)gCσ-1|t-slR. Aconsequenceofedmate(b)inlemma2.lis Lemma2.2.ThemappingU:V={(tA,。ε[0,T]x[0,+∞[xEols=OimpliesfeD(A(t))(closureinEoj}→鸟,U(tAO=exp(sA(t))fiscontinuous-Thesamemapping,restrictedω[0,T]x]0,+∞[xEoiscontinuotuwithvaluesinE1· Lemma23.LetReC(]0,η;Eo),suchthatthereexistsye]0,1[jbrwhich|lR(t)||ogCt-YandjbranySe]0,T[thereexistsC(8)〉Osuchthat,hr-sgsgtgr,||R(忱。)|loge(训,s)Y.putV(t)=j叫((t-s)A(S川训SThen,(a)vεC([0,T];Eo)(C100,Tl;Eo)(C(]0,T];E1);(b)jbrte]0,T],v'(t)=j[A(s)cxp((t-s)A(s))-A(t)exp((t-s)A(t))]R(s)ds+。 +jA(t)exp((t-s)A(t))[R(s)-R(t)]ds+expoA(t))R(t);。 (C)。(t)v(t)=j[q(t)-G(s)]exp((t-s)A(s))RO)ds+j[Ab)exp((t-s)AO))·OOA(t)exp((t-s)A(t))]RO)ds+[expoA(t))·1]R(t);(d)jbrj=1,...,rτ把j(t)VO)=τ(j[Bj(t)·'3j(s)]exp((t·s)A(s))R(s)ds);。(e)tfurther-,RεCY([0,T];民)(Y〉O)andR(O)eD(A(O)kvεC1([0,T];Eo)nC([0,T];El). (a)and(b)followfromlemma22,thedifferentiabilityofvε(t)==Jf叫叩叭(仰(0t卜.斗S圳恻训f1mv叫吨(0thJ叼仙帆S吵))M[RM州s+JJ[h问以闪x邓p附.斗州S吵训)队A阶饵附.寸S圳川(0ωtO)ds+J闪(SA(t))R(t)ds")isaco叫uenceOf刊诩叫3与j(ωs)汩e侃叫xfrom(仙b)and(ωCο)1阳e创td归ng引t→0.InthefollowingweshallcallNj(λ,t)gthesolutionof(1)withlArgλlgo。,tε[0,T],je{1,...,rLgeE1·内,f=0,gk=Oifk#j,句=g.Wepose,forS〉0,te[0,T],
ParabolicProblemswithNonhomogeneousBoundaryConditions 23l Kj(tJ)g=(2πi)-1jexp(λs)Nj(λ,t)gdλY (5) Lemma2.4.LetGbeaspaceqfηpeOe[0,1η]betweenE乌oandE问1.Assumej扣ε{扎1LL,v….川…叫.川叮.吁J,J儿r(ωa)汀Th加er陀eeαxi归St归sC〉0,independentqfτ,t,λ,g,suchthat||[Nj(λ,t)-Nj(λ,t')]g||GgC|t·t'|P(1+|λ|)0·1[(1+|λ|)1·问-Oolglo。+||gl|1·问)(b)Vte[0,T],S〉0,Kj(tJ)ed(E1·的,G)and|lkj(ts)gllGgCOμj+00·61lg100+s-Ol|gll1·的),vgeE1·μj,withC〉0,independentqf g,s,t;(c)themapping[0,T]x]0,+∞[xE1·问→E1,(ts,g)→Kj(tA)giscontinuous;(d)ll[Kj(t,s)-kj(τ,s)]g|lGgC(t-OP[sμj+Oo·e-1|gloo+s-Ollg||1·问];(e)lqq(t)Kj(tA)-q(t)KjOA)lg|iGgC(t-τ)Pbμj+Oo-62lgloo+S·o-1||gill-町 Lemma23.Assumeje{lv·-,r},SeC(]0,T];E1·μj);assumethatthereexistpε]0,1[,suchthat||S(t)||1·μjgCt-pand,jbranyse]0,TLthereexistsC(8)〉0,suchthat,jbrsgt'三tgT,HS(t)-S(t')川1·μjgC(8)(t,t')P,Ihereexistsy〉1·00·μjsuchthatjbranyse]0,TLthereexistsC(8)〉0,suchthat,jbrsgt'gtgT,||S(t)-S(t')||005C(8)(t-mp川(t)=俨队t-s)S(MTh矶(a)weC1(]0,T];EdnC(]0,T];E1);(b沪rte]0,η,w'(t)=Kj(tA)S(t)+j[q(S)Kj(s,t-s)-q(t)Kj(t,t-s)]S(s)ds+。 jq(t)Kj(t,t-s)[S(s)·S(t)]ds;。 (ωο训巳阳(川[S(bωS吵).S(t)]ds;(州rkε{υ1,巾m叭kωw(ωOh=τ咐呐(份倚8句帧帧jk川k凶趴S凯(们J[倪叭阳k以仙((怡ωe功)扩扒|S创(0t)沟|h0o三Cαt.币p,JjhbFr.somep〈0o旷+μ均j,wεC(m[阴0,Tη];丑Eo趴)k;的扩SεCY([0,η;E1·问)nCP([0,T];EoJhrsomep〉1·00·问,wec1([0,T];Eo)(C([0,T];EI). Theprooffollowsthelinesoflemma23,usingthefact由at
232 Guidetft (2πi)-1jt(jexp(λ(t-s))Nj(λ,t)S(t)dλ)ds=(2πi)·ljklexp(λt)Nj(λ,t)S(t)dλOyy III-MILDSOLUTIONS Tosolve(3),weshalllookforsolutionsintheform u(t)=αp((t·州S))uo+J口附·σ)A(σ))R(σ)dσ+jEJWσ川)Wdσ(6), withRecos,T];Eo),||R(σ)||ogC(σ-s)-Y(Y〈1),SjeC(]s,TKE1·的),||Sj(σ)||1·μjgC(σ,s)-Y.ItiseasilyseenthatanyUoftheform(6)belongstoC(ls,T];Eo).Owingtolemmata23and25,formally,tf(t)=寄仲 -σ)A(σ))R(创dσ+j三J[巳(σ)-q(Olkj(σ川)制低 and,forj=1,...J τgj(t)=τ$3j(t)u(t)=τ(Sj(t)+[把j(t)-mj(s)lexp((t-s)A(s))uo+j[$3j(t)-mj(σ)lexp((t-σ)A(σ))R(σ)dσ+ZI[$3i(t)-mi(σ)]KK(σ,t-σ)SK(σ)dσ).k=ljvThiscarriestothefollowingsystemofintegralequations: R(t)=f(t)+[q(t)-q(s)]exp((t-s)A(s))uo+j[。(t)-q(σ)lexp((t--σ)A(σ))R(σ)dσ+zf[q(σ)-q(t)]KK(σ,t,σ)SK(σ)dσ,k=li Sj(t)=句(t)+[$3j(s)-mj(t)]exp((t-s)A(s))uo+j[33j(σ)-mj(t)]exp((t-(7)-σ)A(σ))R(σ)dσ+立[[mi(σ),把i(t)]KK(σ,t-σ)SK(σ)dσ)k=1jG=1,...,r). 币1epreviousdiscussionmotivatesthefollowingdefinition
ParabolicProblemswithNonhomogeneousBoundaryConditions 233 D41Ftition31.LetUGC(]s,η;Eo).Uisamildsolutionqf(3){fitisqftheform(6)withR,SjG=lv·-J)solving(7)in]s,η. Byconvenience,weput,forogs〈tgT,N∞(tA)=[CX(t)·。(s)]exp((t·s)AO)),fork21,Nok(tA)=[。(s),。(t)]KK(s,t-s),forj注1,Njo(tJ)=[mj(s)·'3j(t)]exp((t·s)AO)),forj主1,k21,Njk(tA)=[$3j(s)-mj(t)]KK(s,t-s).WeshallindicatewithN(tJ)theopemtorma位ix(Njk(tA))但jglO盐生·Tostudytheexistenceandtheregularityofmildsolutions,weneedsomeestimatescontainedinthefollowinglemma: Lemma32.PutA={(tA)eR2logS〈tgT}AssumeogS〈σgtgT.Then.-(a)N∞εC(A;在?伍。)),|lN∞(tA)||af(Eo)gC(t-QP-landVεε{0,P[thereexistC(ε)〉0,8(ε)〈lsuchthat||N∞(tJ)·N∞(σ,s)||8伍。)gC(ε)(t·σ)ε(σ-s)-8(ε);(b)jbrk21,NokεC(A;8(E1·阳,Eo),||Nok||d(E1·μ♂o)三C(t-OR+μk+Oo气Vεε[0,P+μk+00·1[thereexistC(ε)〉0,8(ε)〈1,suchthar||Nok(tA)·Nok(σ,s)||J但1·阳,Eo)gC(ε)(t-σ)ε(σ-s)-8(ε);(C)jbrj21,NjoεC(A;2(E。,E1·问)),|lNj003)HJ(E0,El训j)gC(t·S)R-1,||Njo(tA)||d(EoFOo)三C(t-s)R·h-μj,Vεε[0,P[thereexistC(E)〉0,8(ε)〈1,suchthat||Njo(t,s)'Njo(σ,s)||J(E0,E1·μj)gC(ε)(t·σ)ε(σ-s)-8(ε);||Njo(t,s)-Njo(σ,s)||8(EoFOo)gCO-s)R-00·问;(d)jbrj21,k21,NjkeqAS(E1·阳,E1·问)),||Njk(tA)l|d但卫)gC(t-QR+μ肘。。气1·μk1·μj||Njk(t,s)||d(E1·μKFOo)三C(t·S)R+μk·问-1,Vεε[0,P+μk+θ0·1[thereexistC(ε)〉0,8(E)〈1,suchthat||Njk(t,s)-Njk(σ,s)||8(EE)gC(ε)(t·σ)ε(σ-s)-8(吟,1·μk'1·μj||Njk(t,s)-Njk(σ,s)||8(E1·阳,FOo)gC(t·σ)自(σ-s)μk·问-1扩μj〈μk,VEe[0,P-问+μk[thereexistC(ε)〉0,8(E)〈1,such仇ar||Njk(tJ)·Njk(σ,olla?(E1·μkpoo)gC(ε)(t·σ)ε(σ-s)-80)ifμkgμj·
234 Guidetft Nowwewanttosolvethesystemofintegralequations(7).Themaintoolwillbethefollowing Proposition32.AssumeXo,...,XrareBanachspaces,jbr(tA)eAN(tJ)=(Njk(tJ))j=OV-J-k=0,....rwithNjkeqA;6(XK,Xj)),||Njk(t,s)||d(XK,Xj)gC(t-syYJkwithyjk〈1.Letσ=(σ。,...,σr)withσj〈lVj,cdX)=HC句(Xj),withCσj(Xj)={ueC(]0,T];Xj)||ltσjU||∞〈+∞}.Asswnetharj=0σk·句+飞jkglVG,k).Thentheequation u(t)=中(们JN(ts)u(S)dshasauniquesolutionUeCdX)jbranyoeCdX). (8) 阶∞fP川*飞=m叫句B助附y川[阴剖mIHI4心2(ο刚0阶仰)川}川urn叩lεC乌σ(仪〉X〈)and,forj=0,,r,可(hZuy)(队withdf)(t)=Oj(川f叫t)=t=0=￡J、λN问怅阳阳j抹μ川k以山仙ω(O仇ωtLωω,A列S吵叫)k=均0AUt=pm叫kmso,itissuf创ent川metMVZHJf)(t)川Xjget-σjT}川obtainedbyiIn1dtuICHOrn1. Corollary33.Undertheassumptions(h1)-(h7),(3)hasauniquemildsolutionjbranyfeC([s,T];Eo),他,4r)eHC([s,Tl;E1·),uoeEoj=OJ IV.STRICT,CLASSICALANDSTRONGSOLUTIONS InthisscCHonweshallstudytheexistenceandtheunicityofsMct,classicalandstrongSOIlIdons,startingfromthemildSOIlidonsconstructedinsectionIH.Webeginwiththefollowing Proposition41.Themildsolutionuqf(3)iscontinttotuin[s,T]ifaFtdonly扩u。ε D(A(s))(closureinEoj.
ParabolicProblemswithNonhomogeneousBoundaFYConditions 235 Proof.From(6)onehasthatUiscondnuOIlsinsifandonlyifexp((t·s)AO))uo→110(t→s),asthetwointegalsin(6)convergetoOast→s. Proposition42.扩uoeD(A(s)),feCY([s,T];Eo),jbrsomeye]0,1],jbrj=1,...,r句eCY([s,T];E1·的)nCPj([s,T];Fo。),forsomepj〉1·00·μj,themildsolutionUqf(3)isaclassicalsolution. Proof.Byproposition41,UeC([s,T];Eo)andu(s)=110.Further,owingtotheestimatesoflemma32,onecanvedfythatRsatisfiestheassumptionsoflemma25and,forj=lv·-几Sjsatisfiestheassumptionsoflemma2.8. Now,welookforconditionsguaranteeingthatthemildsolutionisamictsoludon.Wehave: Lemma43.ThefbMowingcordinmsguaranteethatthemildsolt4ZioniSStrict-- (a)uoeD(A(s)),A(s)uoeD(A(s)),f=0,jbrj=lv·-J句=0;(b)uo=0,feσ([s,η;Eo)jbrsomey〉0,f(sMD(A(s)),白=0;(c)uo=0,f=0,jbrj=lv·-J句eCY([s,T];E1叫)(CPj([s,T];FOo),jbrsomey>0,Pj〉LOo-μjVj,句(s)=OVj. Proof.Ineachcaseonecanuselemma32toshow阳tRsatisfiestheassumptions(e)oflemma22,while,forj=lv·-J,Sjsatisfiestheassumptions(Ooflemma25. Lemma4.4.Assumeq(t)=αindependentqfLjbrj=lv·-Jmj(t)=记jindependentqft,110GE1,f=0,jbrj=lv·-,rgj(t)=mjllofconstantfunction).Then,themildsolutionqfO)isaclassicalsolution.扩quoeD(A)irisasfriersolution. pr∞f.Itfollowsfromthefactthatu(t)=uo·A-muo+exp((t·s)A)A-lquo- Corollary45.Foranyse[0,T]theclosureqfD(A(s))inEocoincideswiththeclosureqfE1inL. Pr∞f.Itfollowsfromproposition4.1andlemma4.4.
236 Guidefft Lemma4.6.Assume(hl)-(h7)aresari4fIedand,further,巳eC1([0,T];矿(E1,Eo))zhrj=lv·-,rmjeC1([0,TKS(E1,E1·问))ncl([0,η;8(FOo+屿,Fo。))·Then,thestrictsolutionqf(3)fifexistingjisunique. Proof.Onecanverifythat,undertheseconditions,t→(λ-A(t))-leC1([0,T];dd(Eo))Vλec,Reλ20,andHE(λ-A(t))-1HJ(Eo)gC(1+lλl)-1.Thisallowstoapply73in[41 Lemma4.7.Undertheassumptionsqflemma4.6,theclassicalsolutionof(3)(扩existing)isunique- Pr∞f.Itfollowsfromlemmas43and4.6thatanyclassicalsolutionwithvanishingdataisamildsolutionon[s+h,TLforh〉0.So,fort注s+h,||u(t)||。三Cl|uO+h)||oconvergingtoOash→0+. Soonehas Lemma4.8.Undertheassumptionsqflemma4.6,扩uoε町,feCY([s,Tl;Eo)forsomey〉0,forj=lv·-J句εCY([s,Tl;E1·叫)(CPj([s,T];Fo。),fbrsomepj〉1-80·屿,themildsolutionistheuniqueclassicalsolutionqf(3). Lemma49.Undertheassumptionsoflemma4.6,ellerystrictsolutionof(3)coincideswiththecorrespondingmildsolution. Proof.UtUbethe(bylemma4.6)strictsolutionof(3).IfωεCZ飞R)andjω(t)dt=R1,weput,forE〉0,向(t)=ε-1ω(ε-10,udt)=EIK*川,withu*εC1(R;EdnC(R;E1)proloungingUtoR(suchaU*canbeconstructedbytheusualmethodofrefIexions(see[11}293)).uε(resMctcdto[s,T])belongstoC∞([s,T];E1).Weposeω)=等妙的川,g州4j(川)udsME1,fHCl(川],阳,g川cl([s,TKE1·μjLSo,bylemma4.8,uεisthemildsolutionwithdatauε(O),fε,gj,ε-As||udt)·u(t)||o→0,||生(t),f(t)Ho→0,屿,E(t)·gj(t)HWj(ε→0+)uniformlyin[s,T],theresultfollows.
ParabolicProblemswithNonhomogeneousBoundaFYConditions 237 Nowwewanttoextendthepreviousresultstothecasewhenonly(h1)·(h7)aresadded(andnotnecessarilytheassumptionsoflemma4.6). ZAemma4.I0.Undertheassumptions(h1)-(h7),thestrict"fexistingjsolutionqf(3)isuntque. pr∞fIfωεc:RR)andjω(t)dt=1,weput,forE〉0,ωε(t)=E-1ω(ε-10.Next,weRputq(t)=巳(O)ift〈0,q(t)=et(T)ift〉T,$3j(t)=臼j(O)ift〈0,$3j(t)=$3j(T)ift〉T,qε=ωε*q,$3js=ωε*把j.Itiseasilyseen,byperturbationmethods,thatforεsmallenough,。εandmj,εGe{lv·-J})satisfy(hl)·(h7)in[0,TLwithconstantsindependentof巳Further,。εeC∞(R,ag(E1,Eo)),mjzeC∞(R,d(E1,El训j))nC∞(R,af(FOo+u,Fo))-KtUbeas时ctsolutionof(3)withuo=0,f=0,白=OforμlOj=lv·-J.Then,forε>0,Uistheunique(bylemma4.9)strictsolutionof寄争争(0thαωεd沁(0t)讪V(ωtO们川)μ+叫[α阳ω(0t).ιι创α比ωεd(州tO)τ(tm3jμ￡(0t)忡V(ωtO)-〈(g倪3jμ,￡ε(ωtO).'把3句jK(tω)》)u叫(tω)》)=0(θ9)V(0ωs吟)=0.Bylemma4.9,Uisthemildsolutionof(14).Thisimpliesr||u||C([01];Eo)gC(||[巳-qdu||C([0,T];Eo)+ZH(把jz-把j)u||C([01];EtJVj=li叫whichtendstoOasεtendsto0. Now,asinthecaseofclcoefficients,onecanshow Lemma411.Undertheassumptions(hl)-(h7),theclassicalsolutionqf(3)(扩existing)isunique-扩11oeELfeCY([s,T];Eo)forsomey〉0,forj=1,...,rgjeCY([s,T];E1·μj)(CPj([s,η;Fo。),hr-somepj〉1·80·μj,themildsolutionof(3)istheuniqueclassicalsolutionqftheproblem-Everystrictsolutionqf(3)coincideswiththecorrespondingmildsolution. Lemma412.Assume(hl)-(h7)aresatMed,feCY([s,T];Eo)jbrsomep0,jbrj=1,...,r句εCY([s,T];E1·μj)nCPj([s,T];Fo。),jbrsomepj〉1·00·μj,110εEl,。(s)uo+f(s)eE1,jbrj=lv·-rτ(Bj(s)uo·句(S))=0.Then,themildsolutionqf(3)istheuniquestrict-Solutionqfrheproblem-
238 GuideIIt Proof.Bylemma4.11,itissufficienttoshowthatas町ictsolutionexists.Considertheproblem 害杂华〈0t)=4巳川川(0ωtO们)+巳阳ω阳ω(0ω伽tO归)讪肌uτ叫('把3jK(tO)V叫(tO)a(岛(tO).'m3j(0t)讪uOω))=0,j=1L,v.H….川.J,Jr,tε[hs,T]L,v(ωωs吟)=0. (10) Bylemma43(b)-(c),themildSOILIHonof(10)isasEictsoMUon-So,11(t)=v(t)+uoisas町ictso1116onoftheproblem- Lemma413.Problem(3)hasastrongsolution{fandonly扩uoeElJthestrongsolutionisuniqueandcoincideswiththemildsolutionqftheproblem. Pr∞f.ItisclearthattheCOMMonuoeElisnecessaqtohaveastrongsolution.On theotherhand,assumeuoeE1.Thereexistsasequence俨inC∞([s,T];Eo)conv吨ingtofine([01];Eo)叫伽j=lv·-J,a叫uenceg?inC∞([s,Tl;E1·μj)convergingtogjinC([SJ];EWj)Bylemma411tt叫roblemwithdatauo,归,可G=lv·-J)hasaclassicalsolutionuncoincidingwiththemildsolution.Thesequence(un)nεNconvergestothemildsolutionof(3)inC([0,TKEo).So,ωprovetheexistenceofastrongsolution,itissufficienttoshowthat,undertheassumpHonsoflemma4.11,theclassicalsolutionisstrong-Firstofa11,weremarkthatD(A(s)2)=E1.Infact,by corollaIy45,D(AO))=El.TakevoeD(A(s)).ThenexpoA(s))voeD(AO)2)Vt>Oand||expoA(s))vo-vo||。→0.S臼O,thereexistsaSequenCe(仅xkU)kkεE剧4inD职(A剧(怡ωS吟)2勺)C∞O∞In盯1W附V町叼gUirn鸣1毡ψgtωO川lu咐1+2立:N叫j(ω0,A砂砌s吵观湖)泡均g岛j(ωS吵).IhtiSωi均l忖yseen由阳创川tu阳kεE叫lV收比川kkU,J川|H|α(ωS)讪uk+fK(S吵)=A(0ωS吟)kxkεD(A(0s)》)Vk,τ叫(gm3jK(0ωS吵)uk.句(ωωs吵)η)=0,Vj,kkk.Takingtheinitialvalueukinsteadofuo,onehasastrictsolutionoftheproblem,whichisalsothemildsolutionofthesameproblemandSOConvergestothemildsolutionof(3)inC([s,T];Eo)ask→∞.So,theexistenceofastrongsolutionisestablished.'I-heuniquenessfollowsfromlemma4.11. SowecansummarizeallthepreviousmSIlltsinthefollowing
ParabolicProblemswithNonhomogeneousBoundaryConditions 239 Theorem4.I4.Undertheassumptions(hl),(h7),problem(3)hasastrongsolution扩 andonbytfuoeE1.Suchastrongsolutionisuniqueandcoincideswiththemildsolutionqftheproblem.扩feCY([s,T];Eo)jbrsomey〉Oardphr-j=lv·-J旬巳CY([s,T];E1·μj)nCPj([s,η;Fo。),jbrsomepj〉l-00·μj,thestrongsolutionisclassical,-{f,further,uoeE1,q(s)uo+f(s)εEland,hr-j=lv·-ιτ($3j(s)uo-句(s))=0,thestrongsolutionisstrict. Remark4.15.Ifoneisinterestedonlyinproblem 告协Aωu(川,teh 、‘,,,···A··且,,‘飞 u(s)=uo onecancommetafundamentalsolutionU(ts)relaxingthecondition(的)somewhat:itissufficienttoaskthatP+Oo+μj〉lonlyforthej-ssuchthatkeromj(t))isnotconstantint-Thiscanbeeasilyseen,remarkingthatinsystem(7)itisinthiscaseunnecessuytoconsiderthej-ssuch由atker(τ$3j(t))isindependentoft. V-APPLICATIONS Inthissectionweshallindicatesomeapplicationsofourabstractresultstomixedparabolicproblemswithnonhomogeneousboundarycondidons-WerecallthatonecanEndinliteratureacertainamountofmaximalregularityresults(seeLetQbeaboundedopensubsetofRn,withboundaryaQofclassC2r,lyingononesideofaQ.IfTe[0,T](O〈T〈+∞),weposeA(t爪,a)=zaα(t,x)a?witht→lαlg2r aα(tJeCR([0,T];C(画)),forsomep〉0,forj=lv·-JBj(tAJ)=zbjα(tl)ay,withmj〈2rVjandt→bjα(tJeCR([0,T];CB-mj(。))·lα|豆mjAssume由at(k1)Vte[0,T]A(t,xb)isstronglyelliptic;(k2)Vte[0,T]theoperatorA(tAJ)withtheboundaryconditionsBj(t,xb)formsaregularellipticproblemandvoeLπ/2,π/2]Agmon'sconditionsoee[10]3.8)缸esatisfied.
240 GUIdeIIi Indicatewithτthetraceoperator-Lctl<p<+∞.Then,VλeC,withReλ三Oandlλ|SLImeientlyl盯醉,Vte[0,ηtheproblem(λ-A(t,x,a))u(x)=f(x)inQ, (12) τ(Bj(t,x,a)u(x)·gj(x))=Oh川U叫Ieω111tionuew2明Q)VfεLm),(gj)j=1,,JrεIhiW2衍川rFNm.4刊rm叩njF=1thefollowingestimateisavailable(see[GUI1]prop.15): |ωλ汕川川灿川川|川川山州|川阳M川|hMMu川凶|H比|bO叼咐川川p俨川+川叶|H|川gα蚓ωC引川(0川|V鸣巳ε[阴0,p.1叫[(ω|H川|l川川.j川|川k|ks轧叩,3pf=nominWws叩,♂p叫(Qω)》).WeputEo=Fo=W2坤,P(Q)(Oe[0,1]),forj=1,...,rμj=mj(20,。(t)=A(tAA),记j(t)=Bj(tA,a).Then,wehavethefollowingresult,applyingtheorem4.14andremark4.15: Theorem51.Under-theassumptions(kl),(k2),扩P〉1·(2pr)-1.minjμj,(3)hasauniquestrongsolutionVuoeLP(QLfeC([s,Tl;LP(Q)),句eC([0,Tl;W2r-mj,P(Q))G=1,...,r).扩feCY([s,T];LP(Q))forsomey>0,句εCY([s,Tl;W2r-mj,P(Q))(CPj([s,tl;w己,P(Q))forasomepj>1-U(20·μj(20,O〈巳<p-lpsuchthatP〉1,巳/(2r),minjmj/(20,thestrongsolutionisclassical.扩,further,lloeW2r,P(Q)and,jbrj=lv·-J,τ($3j(s)uo-gj(s))=0,thestrongsolutionisstrict.InthecaseqfhomogeneousboundaryconditionsanelJoltuionoperatorcanbeconstructedjustassumingtharp>l-(2pr)-l-μjforanyjsuchthatker(tBJ(t))varieswitht Weremark由at,undertheassumpdonsoftheorem5.1,theredinterPolaHonspaces(LP(Q),D(A(t)))0,∞donotdependontif2mO〈nu+p-1(see[5]),foranyjsuchthatkeroaj(t))varieswitht.Inthiscasethetheoryof[llcanbeappliedtoconstructanevolutionoperator,underassumptionssimilartoours(seei2]73). NowassumethatO〈S〈1,QisopenandboundedinRn,lyingononesideofaQ,whichisasubmanifoldofRnofclassC1+S.Considerthefollowingsituation:iftenn[0,Tl(O〈T〈+∞),A(t人a)=zzaxi(aij(tA)ax;·),withFli=1]1
ParabolicProblemswithNonhomogeneousBoundaryConditions 24l nnt→附,)εCP([0,T];C1叫Q)),Re(j三PijOA)巳j巳i)主V|巳|2,forsomevpOSIUve.nNext,letB(txb)=zbj(tl)气,witht→bj(t,.)εCP([0,η;C1+S(Q)).j=1JAssume(k2)issatisfiedbyA(tλa)withtheboundaryconditionB(tl,a)Vte[0,TlThen,forλεC,Reλ220,|λllargeenoughtheproblem (λ,A(tl,a))u(x)=f(x),xeQ (14) τ(B(t,x,a)u-g(x))=0 hasauniquesolutionUeC2+S(Q)VfεCS(QLVgeCI+S(Q)andthefollowingestimateisavailable(sm[7]prop.216): (19)|AJHuHa+|lu||2+sgC(Hn|S+|λ|(1+s)β||g||∞+||g|ll+SL (HJlt=nominCt(QLHJ|∞=nominL∞(Q)).WeputEo=CS(QLE1=C2+S(QLOo=-s/2,Fo。=L∞(Q).weindicatewithct(Q)thespaceoflittleholdercontinuousfunctionsofexponentt.. S0,wehavethefollowingresult: Theorem5.2.Considertheproblem 字。,x)=A(LXAMOANf(tA)dtτ(B(t人a)u(tl)-g(tl))=OU(0,x)=uo(x) (15) Underthedeclaredassumptions,扩P〉lβ+s/2,(15)hasauniqueStrongsolutionvuoecS(Q),feC([0,T];CS(Q)),geC([0,T];Cl+S(Q)).扩feCY([0,T];CS(Q))forsomey〉0,geCY([0,T];Chs(Q))(CP([0,T];L∞(Q)),forsomep〉1/2+s/2,thestrongsolutionisclassical.Further,扩11oeC2+S(Q),
242 Guidetti A(0,a)uo+f(0,.)ecS(Q)andτ(B(0,-b)110(x)-g(0,.))=0,thestrongsolutionisstrter. REFERENCES [llP.Acquistapace,B.Terreni,LecturenotesinMath.1223,SpringerVerlag,1-11(1986).[2]P.Acqllistapace,B.Tenmi,Rend.Sem.Mat.Univ.Padova,vol-78,47-107(1987).[3]H.Amarln,Arch.Rat.Mech.Anal.92,153·192(1986)[4]G.DaPrato,P.Grisvard,J.Math.puresetappl,54,305387(1975).[5]P.Grisvard,Ann.Sci.Ec-Nom.Sup-,4cs61,2,311·395(1969).[6]D.Guidetti,"Amaximalregularttyresultwithapplicationstoparabolicproblemswithnonhomogeneousboundaryconditions",toappeuinRend.Sem.Mat.Univ.Padova.[7]D.Guidetti,"OnellipticproblemsinBesovspaces",toappe缸inMath.Nach--[8]S.G.krein,"LineardifferentialequationsinBanachspaces",TranslationsofMathemaHealMonogaphs,vo1.29(1971).[9]ESinestrari,iMath.Anal.APPI-107(1985),16·66.[10]H.Tanabe,"Equationsofevolution",MonographsandStudiesinMathematics,Pitman(1979).[11lV.ASolonnikov,"BoundaryvalueproblemsofmathematicalphysicsIII",Proceed-SteklovInst.,A.M-S.Providence(1967).[12]B.TeITeni,StudiaMath.92,141-175(1989).
NonlocalNonlinearSchrodingerEquations B.HEIMSOETHMathematicsDepartment,UniversityofCologne,Cologne,GermanyH.LANGEMathematicsDepartment,UniversityofCologne,Cologne,Germany LINTRODt兀TION Intheusualev。lutionaryn。nlinear、Schrδdingerequati。n。fsemi-linear。rquasi-linear、type iut=-Au+W{u,vu,...)U (1.1) thenonlinearpotentialWis。fl。caltypewhereWiss。memappingintocdepending。nthewave-functionU=u(×,t)anditsderivativesuptoacertain。rder1。cally,i.e.t。c。mputeW(u(x,t),...)atagivenpoint(x,t)。fspace-timecharacteritwouldonlybenecessaryt。kn。wu(x,t)(anditsderivatives)ina 243
244 HeimsoethandLange smallneighbourhood。f(x,t).TherecentinterestininvestigatingdissipativequantumIYiechar11calsystemshasleadtos。men。nlinear-schrδd1rEger‘equati。ns(asm。delsfordescribingdissipativepr。cessesinfissi。n,heavyi。nphysics,nuclearvisc。sityandfricti。nalquantization〕whichareinc。ntrasttothoseJustdescribednonlocalinthen。nlinear、ityie.t。evaluateW(u,...)f。racerta1nwave-furlctionU(x,t)onehastoknowU(x,t)global1yovers。Ineunb。urldeddomain.MOdelsofthistypecanbef。LindinAlbrecht(1975),Hasse(1975),kostin(1972),kan/Griffin(1974),Gisin(1981),Messer(1979).Ingeneralthenonlocalnonlinear、potentialis。ftype nr、W(u)=77fJ(BUU)(BUU)(Bu,u)LC(1.2)44Jl1"2,,...'ilJ=1、JJhereB,Caresomelinearoperat。rsinacomplexHiIbertspaceKJN-Hwithinnerproduct(.,.),andfa:Cd→Cs。Inen。nlinear、U2Nc。ntinuousfunctions;usuallyonehasH=L(R).s。rriemathe-rnaticalattemptst。investigatenonlinear、Schrδdingerequationsoftype(1.1)withapotentialoftheform(1.2)havebeenmadeinLange(1985)andBazley/Lange(1986).InthiSnoteweconsiderthesolvabilityoftheCauchyproblemf。ranabstractn。nlinear、Schr6dinger-equati。noftype iut=Au+kg[(Bu,u)]·Cuu(O)=U。 (1.3) wherethewavefuncti。rlUisamappingofs。metimeinterval[0,T)int。acomplexHilbertspaceHwithscalarpr。duct(.,.)A,B,CaresomelinearseIf--adJoint。Perat。rsinH,geC(C,c)kec,uoeH.Ourmainp。intistogetexplicitfOTITIUlasf。rthesolutions。f(1.3)interms。fthe11near‘Schr6dingergr。upsgeneratedbyAandC,andinterms。fsolution。fan。rdinaryfunctionaldifferentialequationfort→(Bu(t),U(t))
NonlocalNonlinearSChr-adingerEquations 245 II.ABSTRACTNONLINEARSCHRODINGEREQUATIONSOFNONLOCALTYPE:COMMUTATIVECASE LetHbeacomplexHilbertspacewithinnerpr。duct(.,.);letA,B,Cden。teself-adJointlinear、。Perat。rs。nHwithdenselydefinedd。mainsD(A),D(B),D(C)cHresp.;wesetD=D(A}nD(B)nD(C),andassumeDt。bedenseinH。furthermoreletgeC(R,R),kecandUεH.Wew。uldliket。。 solvetheCauchyproblem iut=Au+kg[(B11,u)]·Cuu(O)=U。 (2.1) f。rtwocases,namelyinthecasewhereA,B,Cmutuallyc。mmute{commutativecase),andotherwisewhenC=I(=identity。nH)andA,Bmaynotcommute(rIon-COBIRRItativecase).Intheapplicati。nsthesecondcaseseemst。bemoreinteresting. Theorem1fcommutativeCase)LetA,B,Cbeself-adJoirItlinearoperatorswhichmutuallycommute,letCbeeitherboundedorboundedfromaboveandIIII(k)g(s)EEOorboundedfrombelowandIm(k)g(s)三O(VsεR).ForUeDletψ(t)betheuniqm。。SOILitiononS=[0,T)oftheordinaryftinctiorlaldifferentialequation FMSd、..,,、...,s,,..、ωv,,..、σ。trJ。、..,k,,..、mT,..句乙ergs-EEE』、..,,、..,,&L,,.、ω',,..、g)1J。'KU,,..、..moIUBqLrt==、..,、IJ&Lnu,,.、,,.、·@vωv (2.2) Thentheuniquesolutionof(2.1)withintheclassC(S,D。)nc1(S,H)isgivenby t-ikIg(ψ(S))dsC U(t)=e 。 -itAeU。 (2.3) ProofLetU{t)begivenbyf。rElula(2.3).ThenU(t)isasolutionof
246 HeimsoethandLange u俨M、.E.,、.E,伽』,rE.、mv,,.‘、σ。k+u。AU----&-M、,,uo,,..、iu (2.4) intheclass。ffuncti。nsconsidered(strongsolutiorzJ.Topr。vethatU(t)isastrongs。luti。nof(2.1)itsufficest。sh。wthat ψ(t)三(Bu(t),U(t)).Letψ(t)Inenti。nedpropertieswehave (Bu(t),U(t)).Thenbytheab。ve ψ(t)=2Re(u,Bu)=2Re(-iAu-ikg(ψ{t))Cu,Bu) =2Irn(Au+kg(ψ(t))Cu,Bu)= =2Im(Au,BLi)+2g(qp(t))Im(k)·(Cu,Bu) =2g(ψ(t))IIII(k)·(Cu,Bu) since(Au,Bu),{Cu,Bu)arerealasaconsequence。f。urassumptions.Furthermore,Sinceexp[-1tA]andtexp[-iRe(k)Ig(qp(S))dx·C]areunitary。Perat。rsweget。 收(t)=2Im(k)g(ψ(t)) 2Im(k)Ig(ψ(s))ds·C(。e wherewehaveusedthecornmutativity。fA,B,andCagain.Thuswehavebydefinitionofqp(t) ψ(t)=ψ(t) forallte[O,T).SinceW(O)=ψ(O)wegetψsqp.TheuniquenessofthesolutionU(t)。f(2.1)f。11。wsfr。mthefactthatanystrongsoluti。rlof(2.4)isgivenbyf。rmula(2.3);butifv(t)isastr。ngsolution。f(2.1)thenitisals。astr。ngsolutionof{2.4)with争(t)=(Bv{t),v(t))andthusv(t)isgivenby(2.3)withψ(t)replacedby争(t).ButbythederivationwehaveJustdoneitfollowsthatals。争(t)satisfiesthesystem(2.2).Bytheassumption。ftheuniques。lvabilityof{2.2)thiSimplies争(t)三ψ(t)onSandfinallyv(t)gU(t)。nS.
NonlocalNonlinearSChr-bdingerEquations 247 Remarksardexamples (i) Theorem1maybeappliedegf。rA=-83(free-HhaEm剧』dilμt阳O∞rn』iMarn1札B=iδ(即配ntumopera阳·),C=-气,XH=L2(R) (ii)IfUisaneigenelement。fC(i.e.Cu=λU)then。。equati。n(2.2)。fTheorem1maybesimplifiedt。 ψ=2λ。Im(k)g(qp)-qp. (iii)Equati。n(2.2)。fTheorem1maybereducedt。an。rdinar、ydifferentialequati。nins。me。thercases,t。。-Namely,ifCsatisfiesarelati。n。ftype P(C)=O(2.5) withs。Inep。lyn。mialPthen(2.2)canbewrittenasafinitesystem。f。rdinar、ydifferent1alequati。nsf。rkthefLInetionsqp(t)=(CU(t),Bu(t)),k=0,...,N(whereKNdependsonthe。rder。fP).Wegivetwoexampleswhere{2.5)issatisfied,namelyif2Cisunitary,(i.e.C=I)。rifCisaproJection(i.e.C2=C).Inb。thcasesletuswiteψ(t)=(U(t),Bu(t)),ψ(t)=(Cu(t),Bu(t)).Inthefirstcase。faunitaryCwegettheequati。ns (r伫ψ户…=寸2h川…(比ω川kU川…)Mgψ=2IIπm『ni(k)g(ψ).ψ,ψ(O)=ψ。 (2.6) 22whichimply非非-ψqp言。。r收-qp=c。nst.=α。22(whereαh=收-ψ,收=(Cu,Bu).ψ=(U,B11。)).。。。。。。21/2Thus,ifα注Owegetψ=(ψ+α),and(2.6)。reducest。 (俨2h(UW)(ψ2+α)ν。qψP(0)=qψp。
248 ffeimsoefhandLange whichisequivalentt。(2.2). IfαsOwemayintroduceβ=-α.Thenif收(t)。。isasoluti。n。f (2ψ=2Im(k)g((收+β。)1/2)(ψ+自。)1/2ψ(O)=ψ。21/2thenbysettingψ(t)=(收(t)+β)wegetas。lution。。f(2.2). Analogously,ifCisapr。Jection,wearTiveatthesystem qp=2IIn(k)-g(ψ)收,收=2IIn(k)-g(ψ)收 whichmeansthathb-qp=c。rlst.=γ(γ="。-ψ)。 and(2.2)iSequivalentt。 (;(;)2乓?(ψ)(仰。) (iv)Itiseasilyseenunder。urassumpti。ns。nCinTheorem1thesystem(2.2)canbes。lveduniquely。nalocaltimeintervalS=[0,T)ifgisinadditi。nlocallyLipschitzian. (v)Toprovetheexistenceofas。111tionof(2.1)inTheorem1thecondition。nCandg(s}statedtherearenotnecessary.Itsurficest。assumethatthereisau。ε1DoandafunctionqpgC[O,T)thatsatisfies(2.2)suchthattheexpresssiontr2Im(k)Ig(ψ(s))ds·C、t。lle·Cu.Bul、。。J makessense,esp.that CuandBuareinthedomain。。 tofexp[Im(k)Jg(ψ(S))ds·C]f。ranyteS.Butthent。。
NonlocalNonlinearSChr-bdingerEquations 249 defineasuitableuniquenessclassseemst。bedifficult;nne.g.。nesh。uldassumethatUeD(C),BuheD(CU)f。r。。 anyneN. (vi)Theresults。fTheorem1maybegeneralizedt。abstractn。nlinear、Schr6dingerequati。ns。ftype iutzAu+F(u) whereFisofthef。rrn F(u)=kg{(BU,U),...,BU.U)}Cu1norF(u)=zkg((Bu,u))CJJJJ=1whereB,CCden。tevar-i。usseIf--adj。intlinear、JJ'。Perat。rs。nacomplexHiIbertspace,whichc。IIImutemutually-Als。,wecangeneralizeTheorem1tosystems。ftype VUFUF」、..,、EE,、...,、‘..,vuvu咱iqLRU口U,..、,,.、,,.、,,E飞唱AqLσDCO4A呵,"kk++UVAA--=+L+LUV.、-a·、4 III.NONCOMMUTATIVECASE Inthissectionweconsidertheinitialvalueproblemf。ranonlinear、Schr6dingerequati。nofthef。r111 (iUt=Au+kg((阳,11))UU(O)=UO (3.1) whereA,Bareself-adJ。intlinear。Perat。rs。nac。ITIPIexHiIbertspaceHwithinnerpr。duct(.,.),geC(R,R),kec,uoeH.Letusintr。ducetheCOInITILltat。rs[A,B]n(n主O)by 电E·E-nnuAAAA---E··+nnuAB--。BA (3.2) andletD:=D([A,B]).Oneeasilyseesthat[A,B]·=(-1)n[A,B]-nnn(n主O).IfU(t)issomefuncti。n。ft主OwithvaluesinHweset(forn主O):
250 HeimsoethandLange 伊rl(t)=(-i)n{U(t),[A,B]nu(t)) (3.3) (ifU(t)eD).Analog。usly-letf。ru。eDrlnqp:=(-i)n(U,[A,B]U)(3.4),。。n。(andqp=qp=(UBu)qp=qp=-i(UJA,B]U)。。'。'1.。1=2I111(Au。,Bu。)).Meremarkthatinanycasethefuncti。rlsqpn(t)arerealvalued,namelybytheproperties。f[A,B]nwehave ψrI(t)=(-1)kRe(u,[A,B]nu) (3.5) ifn=2kandqpn(t)={-1)KIIII(u,[A,Blnu) (3.6) ifn=2k+1(since(u,[A,B]u)=iIm(u,[A,B]u)ifnisodd).nn F。raneleInentUeHsuchthatUeD(Vrl主O)we。nintr。ducefunctionsqpn{t)by∞.kh(t)=-ztT飞+k.。.{37)K=。'E andset 争。(t)=争(t) (wheneverthosefuncti。nsexist).T。formulatethenexttheoremletusc。nsiderthef。11。wingassumptiorIsontheinitialvalueU,。ntherlorIlinear-ityg(s),on。theiinearoperatorsA,B,and。nthec。nstantkeC: (A1JThereexistsaT〉Osuchthat -itAeUεDon forallnzOandtεS:=[0,T);furthermoreletthefunction争(t)existonSsuchthat 革(t)求O forO〈t〈T
NonlocalNonlinearSchrodingerEquations 25l (A2)Let(A1)besatisfied.Thereexistsauniquesolutionof(户[γ(t:+2川仰)]ψ(3.8)qp(O)=qp-.-qp{t)onS=[0,T)(whereγ(t):--一一)争{t)Byastrictsolution。f(3.1)wemeanafLIneti。nU:S→H (S=[O,T))suchthatU(O)=U,thedifferentialequati。n。f。(3.1)issatisfiedon(0,T)and Uec1(S,Dn) (Vn主O) Theorem2fnon-commutativecaseJLettheassumption(A2)bevalidonS=[0,T)andqp(t)betheuniquesolutionof(3.8)onS.Thentheuniquestrictsolutionof(3.1)isgivenbyt-ikfg(qp(S))ds。-itA=eeU。 U(t) (3.9) Furthermorewehaveψ(t)三(U(t),Bu{t))onS. ProofLetqp(t)betheuniques。111ti。n。f(3.8)onS=[O,T)anddefineU(t)by(3.9).ThenU(t)isastricts。lutionof (iUt=Au+炮制t))uu(O)=U。 (3.10) (byusing(A1))-Wehavet。showthatψ(t)三(U{t),BU(t))。nS Thusletψ。(t):=(U(t),Bu(t))andψ(t)bedefinedasinn(3.3)-(3.6).Furthermorelet争(t)bedefinedasin(3.7);since 、...,&L,,..、-E··+nω'=、..,,+L,,.‘、nNmv (3.11) itfollowsthatallfuncti。ns争(t)exist。nSbyassmptionn(A),(A).1'2Furthermorebyadirectcalc111ati。nusing(3.10)onegetse.g.
252 Heir71soethandLange f。revenn 伊n=(-i)n(ut,[A,Blu)+{-i)n(u,[A,B]nut)n(-1)n+1{Au,[A,B]u)+11([A,B]nu,Au)n+1n+(-i)1g(ψ)k(u,[A,B]u)+ig(ψ}k([A,B]u,u)n+1-n=(-i)+1(u,[A,B]u)+(-i)g(ψ)(u,[A,Blu)(-1)(k-k)nn+1 =qpn+1+2IIri(k)g(qp)qp andanal。g。uslyforoddn.ThuswehaveforallnEEO nω'、1'ω'g、..JkmT··AnJL。+'n唱Aω'+n=ω'=nu,,.、nn·ω'ωvr··'飞L (3.12) Asolutionsequence。f(3.12)isgivenby飞{t)=h(t)叫FIm(k)忖(S))ds] (3.13) whichmeansthatforn=Owehave 『EEll」S-G、...,、‘..,s,,.‘、ω',,‘‘、σ。trJO、1''K,..、myt--。ιFallELpxe、..,&-LW,,,.、ON@'=、‘..,&L,,.‘、。ωv (3.14) asaSOlutionof(3.13)forn=0.Butsinceby(3.12),(3.14),(3.15) 。-。@'-N@'句,‘Hmv--1llE」SAU、‘.,,ω',,..、σ。trJO、..,,k,,..、mTEA叮FMFEEEBEES』Daxe4EAHω'---EAω' 。ω'γu--。ωvNmv-Mmv=。ω'。一ONωv-Nmv= weseethatqpisalsoasolutionof。 (ψ。=[百(t)+2Im(k)川ψ。qpo(O)=qpo
NonlocalNonlinearSchrbdingerEquations 253 Bysettinghh=qp-qpwearriveat。 (ψ=[钊t)+2h(k)仰)]收ψ(O)=0 whichimpliesthatqpogqp。nS.ThisprovesthatU(t)givenby(3.9)isastrictsolutionof(3.1).Ifv(t)isanyotherstricts。luti。n。f(3.1)thenletqpv=(v,Bv)andinc。rrespondancet。。urpreviousnotationψ=(-i)n(v,[A,B]v),ψ=qp(theselastfunctionsdependNNnv,nnonlyonU!).Thenwehaveqp=qpandasaboveV,。v nvω'、..,vω',,,、σD、..,,'K,..、m?EA「4+-E··+nvω'=nvω' andesp.forn=O OVω'、..,,v@',...、σ0、..,,'K,,..、mT,.-nJh+-EAVω'=。vψ· whichiITIPIies (ψv=[宫(t)+2h(k)Wv)]ψv伊v(O)=ψ。 andthusby(A2)that qpVEqponS.ThiSmeansUZV· RemarksandExamples2(ijTheorem2appliese.g.forH=L(R),A=-θx,B=XP(Pε剧;xpositionoperator),foritiseasytoseethat [A,B]n=Oforn主p+1 (ii)If[A,B]=Oforn注2(ase.g.in(ijforn A=-δx,B=x)andifgeC(R,R),thenonemaycomputeqp(t)fromtheintegr。-differentialequation
254 ffeiFYLsoethandLange s-dωvu。trd。、..,bAmy---呵,"P-xe4,..MV+mv、..,,mvσ口、..,bEAf飞。mmVT,..--呵,"、ZE,,--nu,,.‘..·ω'ω'rIJ-L (qp,=2IIIl(Au,Bu))。rfr。mtheequivalentn。nlineari。。second。rderequation 22.(ψ户……+叫川4川(…IhM川川…mM毗ω…(仪ω…kU…)j!g仆川川(忡ω忡ψ钊仰)沟…ψr川…-斗叩刊2川川Ih…m-4IImTnI(k)g(9ψpqψp=0ψ(O)=ψ,ψ(O)=ψ1· (3.15) (3.15)1ssolvableexplicitlyinsomecases,e.g.if g(qp)=-qp/2,k=iittakesthef。rm 4·qp+qpv+3qpqp=0 '--withgenerals。lutionψ(s)=(2αs+自)(αs-+RS+古)‘11111(withsomeconstantsα自古GR).Thisleadstothe1'F1'1explicitsolutionof(3.1)r、-1/2I12l-itA11(t)={一ψt+ψt+1}eUl21。l (iii)IfHissomeHamiltonianinaquantummechanical。2nsystem,thecasewhereH=L(R),A=(1-ik。)-H。B=H-,C=I,keR(3.1)isGisin,smodelof。。dissipativequantummechanics.AslightvariationofTheorem2leadstothesolution 。u。口"+L、‘E,,。-K+.,..,,..、e呵'』/'-E·.、-BEBEE--d。uouOH+LOKnJ』eFEEEaESE、=、IJ&L,,,‘、u ofacorrespondingCauchy-problemfornormalizedinitialfunctionsuo(whereg(s)=s,k=ik。). (iv) 222FortheharmorliCOScillator-HamiltorlianA=-δ+ωxxandthepositionoperatorB=x。necancomputethe
NonlocalNonlinearSchrbdingerEquations 255 s&Legoen。y、...&L,、ACixlnop-x-exnnωnωn]n2B2-ArJlE、--snr1』。B&L'aA&L,,‘umm。c neven n。dd. Further、m。re,。nehas qpn,。=(-1)k(2ω)nqp。(neven,n=2k)kn-1ψ=(-1)(2ω)qp(n。dd,n=2k+1)1whereqp=(u,xu),qp=-2Im(u,δU},md。。1号(t)=cos(2ωt)ψ+LsiMht)仇,。tEωiqp噜-2ωqPA.tg{2ωt)γ(t)=2ω&M2ωqp。+9P1.tg(2ωt)(f。rω=oonehas争(t)=ψ+札t;see(i))-oi (v) Let dXA=-δ】.B=e.Onecanseethatx- 、..,,nu〉-n,,.‘、n、...,....+X2υ吨'』,,..、xen、EJ咽,..,,..、=nDUA,... Ifwechooseasinitialfuncti。n -x/2-x2U=e。 、..JX,,.‘、n口μ呵'』x叮LJfxen叮L--RJ』xteahnxtnon2。/ixtemn‘dndnin、..,y1b-,,.、ne--eSOUVMIn.、A唱E』SBa'eA,...‘S''A&L,、.‘ withtheHer-mitepolyrzomialsH(x).Fr。mthisusingnformulasfr。Inthethe。ry。fEfermitepolyn。mials(seeAbramowitz/Stegun(1965),chap.22)wegetrOn。ddψn,。=l/π'(2k)!nk、、r二一一-r-2n=2keven;'2民.- thisimplies/π-2t2ψ(t)=45e,宫(t)=4t,andthedifferentialequati。nf。rqp(t)readsas
256 HeirnsoethandLange ψ=[4t+2IIII(k)g(qp)]ψ. (vi〕Thegeneralcasewherewelookf。rs。luti。ns。fanevolutionaryequationoftype(2.1) (ut=Au均((uh))Cuu(O)=U。 withseIf--adJointlinear、operat。rsA,B,C。nHwhichd。notc。『IImuteismuchmoredifficultt。treatasthecase C=I,SinceallVariantsofc。IIUnlitat。rsbetweenA,B,Cappearintheanalysis.ThereiSOrleexampleofthattypewhichcanbesolvedexplicity;thiSisonsagerJsequati。n(inR1)(iUt=Au+kg((BUU))BUU(O)=U。2whereA=-3,B=xonH=L(R),andkreal.Forx90(t)。nederivestheequation ψ+2kg(ψ)=O(where|1UH=1).Thecaseofac。Triplexkismuch2Lmorecomplicated- (vii)Physicallyanequati。noftype(3.1)maybeinterpretedasadampedquantummechanicalsystem(f。rkpurelyimaginary)withdampingcoefficientdepending。ntheexpectationvalueofsomequantummechanicaloperatorBIfkiSrealthismaybeseenasanoscillatorysystem.
NonlocalNonlinearSchrodingerEquations 257 REFERENCES 1.Abramowitz,M.,Stegun,I.A.(1965).Handb。。k。fMathematicalFunctions.Dover,Newy。rk.2.Albrecht,K.(1975).Anewclass。fSchrδdinger。Peratorsf。rquantizedfriction.Phys.Letters56B:127-129.3.Bazley,N.,Lange,H.(1986).TheoriginalSchrδdingerwaveequati。nrevisited.ApplicableAnal.212225-233.4.Gisin,N.(1981).Asimplenonlineardissipativequantumev。111t10nequati。r1.J.Phys.A:Hath.Gen.14:2259-2267.5.Hasse,R.M.(1975).Onthequantummechanicaltreatmentofdissipativesystems.J.Hath.Phys.16:2205-2011.6.kan,K.-K.,Griffin,J.J.(1975).QuantizedfrictionandthecorTespondanceprinciple:singleparticlewithfriction.Phys.Letter-s50B:214-243.7.kan,K.-K.,Griff、in,J.J.(1976).Collidingheavyions:Nucleiasdynamicalfluids.Rev.Hod.Phys.48:467-477.8.kostin,M.D.(1972).OntheSchrδdingerLangevinequation.J.Chem.Phys.5723589-3591.9.Lange,H.(1985).OnnonlinearSchrδdingerequati。nsinthetheoryofquantummechanicaldiSSipativesystems.NonlinearAnalysisT.M.A.9:25-39.10.Messer,J.(1979).Fricti。ninquantummechanics.ActaPhys.AEAstriaca50:75-91.
AnOperator-TheoreticalApproachtoDirac'sEquationonLP-Spaces MATTHIASHIEBERMathematicsInstitute,UniversityofTubingen,TUbingeIUGer-n1any 1.Introduction LineardifereIltialequationsinBmachspacesarecloselyconnectedwiththeconceptofone-parametersemigroups-IIIparticular,thistheoryallowsadetailedaaalysisoftheinitialvalueproblemu'(t)=Au(吟,u(0)=uoforadifereIItialoperatorAOIlafunctionspaceE.Ontheotherhand,thereexistmanyexamplesofdibrentialoperatorsgeneratingCo-semigroupsonU(IRn)onlyifp=2.AnexampleofsuchanoperatorisgivenbytheSchr凸dingeroperatoriA,whereAdenotestheLadman(see[H凸l).InordertotreatsuchoperatorsinmLP-settingintegratedsemigro叩swereintroducedinrecentyears(seelALIA,Nesl,[AXl,[BZl,[KZMNel)Tobemoreprecise,itwasshownin[HlthattheSchr凸dingeroperatoriAgeneratesaaα-timesintegratedsemigro叩onLP(Rn)ifα>n|1/2-1/p|.Similarresultsholdforsymmtrie,hyperbolicsystemsofarstorder.Inthisnotewewillconsiderindetailaaequationfromrelativisticquantummechanics,namelytheDiracequation.Therelativisticdescriptionofthemotionofaparticleofmassmwithspinl/2isprovidedbytheDiracequation(see[GMG,坷,[Fi) 、、,,,,咱EA.咱B··,,l飞 ny「Tnc2扣(叫)=c兰AJDJAU(M)-A4τW)+V巾,tLZEE3,t三0 HereUisafunctiondeanedonm3×R+whichtakesvaluesinr,Cisthespeedoflight,hisPlanck'sconstantandA1,岛,心,Aare4×4matricesgivenby 、、EEEE,,/1。υounu/,tEBB-飞飞、、‘EEa『,,,nυnUOUJ''BEEt‘飞0010000A-1一0100'Aq=40-too1000tooo1。υl/'EEE-飞。。。。01。A2一'-dlO。0'AA=哩。。咽'A。一1。。。UOU。一1259
260 ffieber IfV=Oandunitsarechoosenmthatallconstantsareequalt01,thentheequation(1.l)canbewrittenasasymmetric,hyperbolicsystemoftheform 二(;)=00亿)+t(;3)(;),(;也)=(;:) where =(D3D1-i川landD1·=一一(j=1,2,3)-D1+iD2一D3/JJAsunderlyingBmachspacewechooseE=LP(E3,￠)4(1<p<∞)anddeanethedomainofAtobeD(A):={fELP(E3,￠)2;AfELP(R3,￠)2}.ItfollowsfromaclassicalresultofBrenner([BLThm2)thattheaboveDiracoperator (1.3) D=(:1)+t03) isthegeneratorofaCo-semigro叩onLP(R3)4ifandonlyifp=2.InourmainresultwewillprovethattheDiracoperatorDgeneratesmα-timesintegratedsemigrouponLP(IR3)4wheneverα三2|1/2-1/p|+1.Asaconsequenceofthisthereexistsaunique,classicalsolutionof(12)whenevertioεD(A3)×D(A3). 2.Integratedsemigroupsandsymmetric,hyperbolicsystems Westartwiththedeanitionofintegratedsemigroupsandtheirgenerators. Deanition2.1.Letα三0.AlinearoperatorAonaBanachspaceEiscalledthegeneratorofmα-timesintegratedsemigroup,ifthereexistsaconstantω22Osuchthat(λ-A)isinvertibleforallλ>ωandifthereexistsastronglycontinuousmappingS:[0,∞)→L(E)satisfying||S(t)||三Mrt(t主0)forsomeM主OsuchthatPCXDR(λ,A)z=λαjfMS(t)zdt(zεE,人>ω).JO ThefunctionSiscalledtheα-timesintegratedsemigroupgeneratedbyA.NextwedeamthesymbolP(·)ofaarstorderdifferentialoperator T「θ二、T「P(D)=飞AOJ|ν卜l)byRZ):=i〉JAdv.在1v(仇)U1(θzn)vn匀J-k1 HerethecoefEcieIItsAUareelementsofAfN(￠),theringofallN×Nmatricesover￠.
Dirac'sEquationonLP-Spaces 26l Conversely,givenasymbolPoftheaboveform,weassociatewithPalinearoperatorAPonU(Rn)Nasfollows. SetD(Ap):={fεLP,F-→1(PFfη)εLp}anddedEnApf:1=F一→1(PFfη). HereFdenotestheFouriertransforminthespaceSFofalltempereddistributions.ThefollowingtheoremshowsthatintegratedsemigroupsoccurnaturallyifonestudiessystemsofdiferentialoperatorsonLP-spaces. Theorem2.2.(lHl)Letl<p<∞andP:En→MN(￠)begivenbyP(己):=iE二二145J,whereA1,,AnMehermiteanN×NmatricesEthereexistconstantsC1,..JNεRsuchthatσ(P(己))={C145|,..,cMK|}forallEERn,thentheoperatorAPgeneratesanα-timesintegratedsemigro叩onLP(IRn)Nwheneverα三(n-1)|l/2-1/p|. 3.TheDiracequationonLPspaces Considertheequation(1.2).Then,bysimilaritytransformations,weconcludearstthattheoperatorA:=(:1)withdomainD(A):=D(A)×D(A)generatesaaα-timesintegratedsemigro叩oaLP(R3)4ifandonlyifAad-Agenerateα-timesintegratedsemigroups0日LP(m3)2.NexttheeigenvaluesofthesymbolofAcanbecomputedtobeλ13(巳)=土i|5|.There-fore,byourTheorem2.2itfollowsthattheoperatorAgeneratesanα-timesintegratedsemigro叩onLP(EV)4ifα三2|1/2-1/PLTheusualwaynowtoprovethewellposedmssofequation(12)istoconsidertheopemorB:=i(J斗)asaboundedperturbationofA.IncontrasttothesituationofCrsemigroupswedonothavea"boundedperturbationtheorem,,forintegratedsemigro叩S(see[K,Hlforacounterexample).Neverthelesswewillshowinthefollowingthat,duetothespecialstructureofAandB,theDiracoperatorD=A+BwithdomainD(A)isthegeneratorofanα-timesintegratedsemigro叩onLP(IR3)4wheneverα三2|1/2-1/p|十1.Westaztwithalemma.ForthetimebeingletAbealinearoperatoronaBanachspaceF,AonF×Fbegivenby AD×A/''飞、D一-AD＼lt1/AOOA/rtl＼一-A andletBbetheboundedopemoronF×FdennedbyB=i(JJ)
262 Hieber Lemma3.1.AssumethattheoperatorAgeneratesmα-timesintegratedsemigroupforsomeαwith05二α三1.ThentheoperatorA+BwithdomainD(A)isthegenerator-ofaF-timesintegratedsemigroupfbrall822α+1. Prootweshowarstthattheoperator(λ-A-B)isinvertibleforlargeenough入aIIdconstructthenexplicitlyanexponentiallyboundedfamily(S(t))QoofboundedoperatorsonF×Fsuchthat acu、AρU∞rl1oaμ'EA--nu+A-AR holdsforalllargeλTothisendnoticethattheoperatorAgeneratesanα-timesintegratedsemigrouponF×FifandonlyifAand-Agenerateα-timesintegratedsemigroupsonF.Then,byaslightmodiacationofTheorem5loflA,KLweconcludethatA2generatesanα-timesintegratedcosinefunction(C(t))t主oonFsatisfying||C(t)||三MrtforsomeM,ω主0,i.e.theoperator(入2-A2)isiImrubleforallA>ωandthereexistsastronglycontinuousfunctionC:[0,∞)→L(E)suchthat尸R(沪,A2)=ffMm)dtholdsforallλ>ω.Hencetheoperator(λ-A-B)isalsoinvertibleforall入>ωanditsinverseisgivenby/入+tA飞2(λ-A-B)-1=(Aλ-tjR(λ+1,对)(λ>ω)Letnow(C10))t主obetheonceintegratedcosinefuIIctiongeneratedbyA2andletBbeaboundedoperatoronF.ConsideringtheintegralequationCB(t)=C(们dCB(t一s)BC1(s)dsweshowexaztlyasintheproofofTheorem5.3of[AXlthattheoperatorA2+Bgeneratesanα-timesintegratedcosinefunction(CBO))t主oonF.Inparticular,A2-Igemratesanα-timesintegratedcosinefunction(CI(t))t>oonF,Therefore入1可(λ2+LAZ)=λ1可(沪,A2一Iη)=才f∞fJ产Aλ川句川t吃飞CG叫I川(忖holdsforallλ>ω.UsingthisnotationwedeaneforaIlt220thefollowingoperators:乌川川1川山州(μ例t叫):1=才f俨俨t马》CC川'乌ω川2纣山州(μ例t忖):==才f俨t与与CGI川一f扑μ川川一→叫8吟州川)阳闷阳CG叫ωI川巾(υS吟)乌ω川2纣州州(μ例t阶):=马1(t):=40一川)ds
Dirac'sEquationonU-Spaces 263 NotethatS120)andSn(t)arewellddmd,sinced(t一叫cd收dsεD(A2)forailZεEandallt220.Nowitiseasytoverifythatthefamily(S(t))t主oofboundedoperatorsonF×Fgivenby 贝h(213230 isexponentiallyboundedandsatisSes JUS、Aρlv∞fIJO+α飞A一-B+A飞AR forallλ>ω.ThereforetheoperatorA+Bgeneratesanα+l-timesintegratedsemigrouponF×F.口 ApplyingthislemmatothesituationoftheDiracequationweobtainthefollowingresult. Theorem3.2.TheDiracop盯atorD=A+BonLP(R3)4(1<p<∞)ddnedasin(1.3)withdomainD(A)×D(A)generatesmα-timesintegratedsemigroupwheneverα主2|1/2-1/p|+1. Thegeneraltheoryofintegratedsemigroupsimpliesmowthefollowingcorollary. Corollary3.3.TheDiracequationtf(t)=Du(t),u(0)=tioonLP(IR3)4hasaunique,classicalsolutionforallinitialvaluesuoεD(A3)×D(A3). 4.Anapplicationtocosinefunctions Let(C(t))t>obeacosinefunctiononaBamachspaceEwithgeneratorA.Formanyproblems(seeforexample[Fi)itisintemtingtoknowwhetherthegivencosinefunctionallowsagrouprepresentation,i.e.whetherA(aftersuitablerescaling)possessesasquuerootandthecosinefunctioncanbewrittenashalfofthesumofthetwosemigroupsgeneratedbyA1/2and-A1/2.ItwasshownbyBNagy[Nalthattheanswerisnoingeneral.In1969H0.Fattoriniintroducedacondition,todaytheso-calledcondition(F)(see[Fior[T,Wl),underwhichacosinefunctionguaranteesagroupdecomposition.InparticularheshowedthatthisconditionisfulalledwhenevertheunderlyingBanachspaceisoftheformU(Q)(1<p<∞).ForgeElemiBmachspacesweobtainthefo1lowing- Proposition4.1.AssumethatAisalinearoperatoronaBmachspaceEsuchthat(0,∞)cρ(A)andlimsupλ→∞||λR(入,A)||<∞.HAgeneratesanα-timesintegrated
264 Hifeber cosinefunction(C(t))t主oforsomeαwithO三α三1,thentheoperatorAl/2generatesan(α+1)-timesintegratedsemigroup- ProotByhypothesisweknowthatlims叩λ→∞||λfZ(λ,A)||<∞.ThereforethroperatorJ1/2givenforallZεD(A)by唱PCXDJ1/22:=二jλ一1/2R(λ,AX--A)zdλπJo iswelldenned.Moreover,followingsectionIIIofl町,weconcludethatJ1/2isclosable-DenotingbyA1/2theclosureofiJ1/2itfollowsthat(Al/2)2=Aandthat入2ερ(A)impliesthatλερ(A1/2).NotethatinthiscaseR(λ,A1/2)=(λI+A1/2)R(入29ALNowsettmg 印):=fc仙+μJt(t一川)dsweverifythat(S(t))t>oisanexponentidlybounded,stronglycontinuousf川lb3y?Odf们仙lb〉刀mm川Oα∞川1u旧1口阳Ioperatorssatisfying fJtS(t)dt=击川川)R(λ2,A)=击川,A1/2) foralllargeenoughλ 口 Example4.2.ThesymbolPoftheDiracoperatorDissimilar(inthesenseofmatrices)tothefunctionH:R3→M4(￠)givenby ＼11tEEst-''、、..,,F、nununU/t飞4哇'h川、,,/卢、nunU/l飞nuqdL川、飞,/C'、nu/t飞nunuq,eL川、l/,户、/1飞nUAUnu'EA'h/,tt』tt飞飞一一、‘.,,,卢、/,..飞、H ThevalueshJ(己)(j=1,2,3,4)aretheeigenvaluesofP(巳)andaredeterminedby hI(5)=h2亿)=-hdE)=-h45)=i(lt|2+1)1/2. SincetheLaplacimAonLP(IR3)(l<p<∞)satishstheassumptionsoftheabovepropositionwheneverα三2|1/2-1/p|(seeIH])itfollowsthattheoperator(A-I)1/2、havingthefunctionhassymbol,generatesaF-timesintegratedsemigroupORLP(IR3)wheneverSZEα+1.
Dirac￥EquaIiononLP-Spaces 265 References [AlW.Arendt,VectorvduedLaplacetrmsformsandCauchyproblems.IsraeliMath.59(1987),327·352.[A,邸,SlW.Arendt,F.Neubrander,U.Schlotterbeck,Intezpolationofsemigroupsandintegratedsemigro叩s.preprint(1989)[AXlW.Arendt,H.Kellernmn,Integratedsolutionsofvblterraintegrodferentialequationsandapplications.In:G.DaPrato,hf.Iamlli(edsj,IMegrodifereatialEquations.Proc-Conf-Trento(1987)(toappear).[BZlM.Balabane,HA.Emamirad,LPestimatesforScl时di鸣erevolutionequation.Trans.Amer.Math.Soc-291(1985),357·373[BlP.Brenner,TheCauchyproblemtbrsymmetrichyperbolicsystemsinLP.Math.ScaM.19(1966),27·37.[FlH0.Fattori巾,SecondOrderLinearDiferentialEquationsinBM肌IISpaces.North-HollaIId1985.[GlJA.Goldstein,Semigro叩sofLi肘arOperatorsandApplications.OxfordUni-versityPress1985.[G,SlJ.A.GoldsteinandJ.T.SandefuzvEquipadiorIofEnergytbrSymmetricHyper-bolicSystems.In:C.V.Cofman,G.J.Fix(eds.),ConstructiveApproachestoMathematicalModels,AcademicPress1979.IHlM.Hider,IntegratedsemigroupsMidtheCauchyproblemtbrsystemsonLP-spaces-(toappearinJMath.Anal.APPI-)lH凸lLHKIIlander,EstimatesfortranslationinvariantoperatorsinLPspaces-ActaMath.104(1960),93-139.[KZlHKeHermann,M.Hieber,Integratedsemigro叩s.J.Fmet.Anal.84(1989),160-180.[NalB.Nagy,CosineoperatorfunctionsandtheabstractCauchyproblem.Period.Math.Hungar-7(3)(1976),15,18.[NelFNe1巾aIIder,IntegratedsemigroupsandtheirapplicationstotheabstractCauchyproblem.PaciacJMath.135(1988),Ill-155.[T,WlC.C.Travis,G.FWebb,CosineFmiliesMIdabstractnonlinearsecondorderdifemitialequations.ActaMath.Acad-Sci.Hung.32(1978),7596.
ExactControllabilityandUniformStabilizationofkirchoffPlateswithBoundaryControlOnlyonAw|ZandHomogeneousBoundaryDisplacement lLASIECKADepartmentofAppliedMathematics,UniversityofVirginia,Charlottes-vine,Virginia R.TRIGGIANIDepartmentofAppliedMathematics,UniversityofVirginia,Char-lottesvine,Virginia 1.INTRODUCTION,PRELIMINARIES,ANDSTATEMENTOFMAINRESULTS I,l.INT.政ymfCTIONANDPRELIMINARIES nLetQbeanopenboundeddomaininR,ntypjcaIIy22,withsuffiCientIysmoothboundary「.InQweconsiderthefoIIowingkirchoffpIateinthesoIutionw(t,x): 267
268 e-'YAw+A2w=ottttw(0,x)=w;w(0,x)=wOtw|立三OAw|τ·=O-OAwly=U-1 LaswekaandTriggiani in(0,T]×Q=Q; (1.1a) inQ; (1.1b) in(0,T]×「=立; (1.1C) in(0,T]×「=200.in(0,T]×「=211' (1.1d) (1.1e) withonlyoneCOIltrolactionueL(2)exercisedintheB.C.21(1.1e).ThiSiSadistinetivefeatureoftilt?probIemhereconsideredoverexistingliteratLIrepseebelow.III(1.1d-e)wehave「「opensetsof「;「non-empty「U「=「.AISOin(1.1a)p70'11'01 iSaPoSitiveconstant,$4hichradiCallychangesthedyElamiCaibehaviorOfthesystemoverthecase7=0,inthat-…unlikethelatter---theformercase守〉OgivesrisetoahyperboliCdynaIIiiCswithfinitespeedofpropagation.InthiSpaperwesha11beconcernedwiththeissueOfexactcontrollabilityinthespaceOfoptiIIlaIreg111arity,whiIefortheprobIemofuniformstabiIizationweshaIICOMimtothemainstateme川(seeproofin[IrT12]).ItwillbeexpedienttoSingleoutthecorrespondingproblemwithhomogeneousboundarydata 中tt斗A中tt+A2中=f中(0,x)=中。;中t(0,x)=中1中|22OA中|z=O inQ; (1.2a) inQ; (1.2b) in2; (1.2c) in2. (1.2d) TodesCribeourresUlt,weshallfirstIetAbethepositiveseIf-adjointoperatoronL(Q)definedby2 4A中=A中;2(A)={中GH(Q):中|「=的|「=0},A气=-衅,如(A兑)=H2(Q)n巾。) (1.3)
ControllabilityandStabilizationofkirchqfyplates 269 ThefoIIowingspaceidentificatIOIlsareknown(withequivaIentnorms)[G.1]: 5-8〈er、1-8P、‘、,.dnv--FlψloeAHB,aaeψtr〈LeA吼UF 4em(A)={中eH(Q):中|「zA中|「=0},云<eg1 (1.4) Tllefo11owingspecia]izationsthereofwjIIbeneededbeIow: 。14nuuuevbVAori。1·AnuUH--UMmAHEU屑一-e f2%llgllv=!|内||L(Q)={!|Vg|叫,叫11ivaleMto如(Ai)2J。 ,m....0、..、...,。,,...、嘈inuHnσoeh+L (1.5a) f22%2妈iIItmlequvaMIltto{jg+叫Vg|Em)=(llg||L2(Q)冲||A用g||L2(Q)〉Q =||gl|υ如(A:) (1.5b) 妈theIatternormbeingdenotedby2(A)-norm,orH(Q)-nγ0,γ e=%:2(俨)=(中eH(Q):中|「=0);forgeS(A%)提2 %2「2ilgll诏三llAMgl|=l(凸g)"412(A)Ln(Q)j-Q妈2%2equivalentto|lAg||+γ||Ag||三||g||性L(Q)L(Q)ib(AM)2γ (1.6a) (1.6b) tilelatternormbeingdenotedby2(牛-…;
270 LasieckaandTriggianil e=施:2(A)={中eH(Q):中|r=A中|r=0};forgeS(A元)也3 黑始f|ig||?=||rg||L(Q)=HAAgHL(Q)=(jW(Ag)|41}(17)2(AI)22J。 by(1.3)and(1.5a).ThebasiCSpacefortheSOlutiOIlsOfproblem(1.1)withueL(立)willbethespace21 。咱4nUUH×〔M1onQqrHrE·E·‘UMmAunE×UHMAHU27L (1.8) l,2.REGF/LARITYANDEXACTCONnoLIABILITY WebeginbyStatingtilemain(optimal)regularityrestIltsforthesoIutjOIlsofprob]ems(1.1)and(1.2). Theoreml,l,(FundamentaltraceregularityforthehomogeneousprobIem(1.2))Withreferencetoproblem(1.2)withf=OpwehaveforanyO<T<∞, AUAW'ELMTAPUG叮BEES--''SEatEdn4+L-AYEA-l阳、吨,-AU-uv倍」l、「li--LrlltJTL (1.9) E中(0)三jlmA中o)|2+lW1!2+叫钟1|2而。=||中J|2驭+H中J|2以+γ||中J|2Kum(A用)A2(A)AS(AFZ) {1.10) (1.11) =||中nH2敬+|l中J|2UV如(Am)42(AJequivaleMtoll{中0'中1)l|27%ib(At)×30(A) (1.12) ..、..,,吁,咱,A,,..、.,、..,,民V4·』,,..飞,FO咱-A,,..飞vdhu
ControllabilityandStabilizationofkirchq矿Plates 27I ByduaIity,ortransposit1on,onTheorem1.1,oneobtajns Theoreml,2.(Irlteriorregularityforproblem(1.1))WIthreferencetoprobIem(1.1),wehavethatforallyfinjteTandpsay, 「1=「: %%{w0,w1,u)→(w(t),wt(t)):continuous2(A)×2(A)×L2(0,T;L2(「))%妈4C([0,T];2(A)×如(A))..(1.13) Themapin(1.13)iS,infact,surjective.Indeed,bytimereversalofprobIefII(1.1),evenmoreiSCOIltainedinthefolIowingrestIlt,whichdoesnotreqtliregeometricalconditionsif「1=「. 兑VTheoreml.3.(ExactCOIltroIIab111tyonZ=ib(A)×E(At))(a)Letr=「.ThenthereexistsatimeT>O(whichcanbeoexpIiCit1yestimatedIntheproof,see(4.26)beIow)suchthatif%34T>TO,then:givenany{w0,w1}GZ=2(A)×3(A),themexistsastlitableCOIltrolueL2(0,T;L2(「))suchthatthecorrespondingsolutionof(1.1)satisfies MMmAA身×、..,,unAME、...」中A,nur--··‘,....、pue、t〉J+LW,wrdLnυ、..,,,中A+LW‘..,,.,中且,,..飞w (b)Moregenera11y,1et「#φ.Then,thesameexactOTcontrollabiIityrestIltasin(a)hoidstruewithcontroIUeL(0,T-L(「)),providedthat:2'21(b)thereexiStsavectorfieIdh(x)=[h(x),...,h(x)]G1[C2(η)]nsuchthat(i)h(X)·ν(X)SOon「。;ν=outwardunitnormaI;(1.14)[[2(ii)|H(x)v(x)-v(x)412ρ||v(x)|n41forS01附constantJJRQQnρ〉0,vve[L(0)],(1.15)2
ControllabilityandStabilizationofkirchqfyplates 273 -γAW+A2w=Ottttw(0,x)=w;ω(0,x)=W嘈ot1w|22OAw|zh=ovawAW|21=-ZL on(0,T]×Q=Q; (1.17a) OI10; (1.17b) on(0,T]×「EZ; (1.17c) AUFL=-nUFt×、...J中且,nu,,..、no (1.17d) OIl(OPT]×「三三11 (1.17e) Let(see(1.3)and(1.5b)), ‘...FU荆州,A,za--、2×、-EFUnMAME--时,7h (1.18) Theorem1.4.(UniformstabilizationonZ)Let「=中in(1.17d).70TLetQsat1sfythefoIIowingcondition:ThereexistsavectorfieIdh(X)e[C2(白)]nsuchthat:(i)h(x)iSparaIIe]tothe(unit)normaIvectorν(x)on「;(1.19)(ii)hsatiSfiescondition(1.15).(1.20)Then,thereexiStCOIlstants￡>OandM=Mr〉1suchthattheh-solutionsOfthefeedbackprobIem(1.17)satiSfy ,nu、2-+L,时,7Lnu咱iMHW+LrhueMJ飞-Ht7bnυtiww+-LPIAe--州,qiu、..,,、,,&L+L,t、dt·、+LWW (1.21) whereZγiSdefifledin(1.18)·. Remarkl,2.TheCIasSofdomajnsQtowhichTheorem1.4iSappIicabIe,i.e.,satisfyingcorkiitions(i)=(1.19)and(11)=(1.20)inc11IdesthecIassofstrictIyconvexdomains,orsettileoreticdiffereIleesthereof[L-T.7]..
274 LasieckaandTriggiant 1.4.LITERATURE Reforerlees[L-L.11and[Lag.1]considerkirchoffpIates.MorepreciseIy,[L-L.1,CI1.vlgivesexactcontrollabilityresultseitherunderdifftfrontboundaryCOIlditiOIls,ore1sewiththesameboundary1ditionswl=vAwl-wbutwiththeuseofbothCOIltroISlZ0'ulz-1' v,V,whichmoreoveraretakenindjffort号ntspaces.ReferenceO1 [Lag.1,Sect.4.41givesalIniformstabiIizationres111t,againusingdifferent(higher)boundaryCOIldjtiOIlsthantheonesusedhere,aIIEimoreover,withuseoftwofeedbackCOIltro]S.ItisweIIknownthattheprobIemsofexactCOIltro]IabiIIty/uniformStabiIizationaremuchdependentonthetypeofB.C.andthat,moreover,thepresenceofonlyonecontrolactionOfthetwoPotentiallyavailableintrodtleesadditionaIdiffiCUIties.ItisinterestingtocomparetheresuItsofthepresentpaperwiththosein[L-T.3]forthecorrespondingEuIer--Bern01111iprobIem(whichisnothyperbo11C)obtainedbySimpIysettingγ=Oin(1.1a).In[Id--T.3]exactCOIltroIIabjIityisobtainedinJP(FML(Q)tlsingCOME、olSofClassL(0,T;r(「))OIIly2… inAW|立,aMwithnogeO『制川calCOMitionsOIlQCOmpa陀withOupIr、eSeIn1tTih1eOIr、eImn1.3.ontileOtherhand,uniformStabiIizationiSobtaifledfortheEuler一BernoulliproblemundergeometriCalCOIlditionsfarmoreseverethantheonesgivenhereInourTheorem1.4forthecorrespondingresUIt.Moreover,thetwoproblemsentaiIdifferenttechniCaltooIs,e.g.,differentmuItipliers. 2.OPERATORMODELSFORPROBLEMS(1.1)AND(1.2) Boundaryhomogeneousproblem(1.2j.Theabstractequationwhjchdescribesthehomogeneousproblem(1.2}iSreadiIyseentobe(AppendixCof[L-T.12]) 弛-1中tt=A中+(I叫A)f,中(0)=中。,中t(O)=中1; (2.1}
ControllabilityandStabilizationofkirchqfYPlates 275 AAbAAbAH4··品以"Auq于·+TEA,,E、、A凡 (2.2) TheoperatorA,whichisreadiIyseentobeaboundedperturbation性-Aoftilenegativeoperator-一(AppendixCof[L-T.12]),istileγtg…川Ofa叫0 Non-homogeneousproblemfl.lj.Byproceedingasin[IfT.1},[L一T.5],{T.1],wehavethat-t-heabstractrljfferentiaIequationinfactorPrespectiveIy,additiveformwhich『nodeIsprobIem(1.1),iSgivenby =A(w-132u)onL2(Q);tt =Aw-ABUon,say,[￡(A)]仁tt (2.3) In(2.3),G21StheGreenoperatordefinedby ~r2y=Ggo{Ay=OinQ;yzOon「;22LAy=Oon「;Ay=gon「}21J (2.4) WereadiIyhaveforfutureuse[L-T.1,Remark3.2],[L…T.2], --叮阮咽,、,,、,G2=-ARO;y=Dvo{的=oinQ;yrOon「。;y=von「1},(2.5) andby(2.2)and(2.9)and[L-T.1,Lemma3.1],[L-T.2], 之A中峭♀ on「。; (2.6) on「1
276 LasieckaandTriggiani InourStudyofregularity(Section3)andOfexactcontroIlability(Section4),weshallneedtowritetheSOIutionattimetzTofthenon-homogeneousprobIem(1.1}wjthPsay,=w=OpwhichisgivenexpliCitlyby[L-T.11-[L-T.9],[T.1],O1 w(T;t=Q;W=w=O)01(T;t=0;w=w=0)01 =JZTU= ,+L+LAUAU、..,,、...,φLφt,,..飞,,,.、uu内,-qζ~户U~户U、-EF、..,φt+LTA中A((aduTAnu中AAufESE-tt,JFEEEEE-,JAH叫AA (2.7) *andcharacterizethe(Hiibertspace)adjOint工TOftheaboveopepatorJtTP qL、-I·e-J14呵,"7"7LU.,UTAZFEBEEE,‘、 〔叫|:;12(匀'Z=川)浦(A (2.8) interIIISOftilecorrespondinghomogeneouspartialdifferentiaIequation. 比于12mma2J,For{飞,z2)eZ=2(A)×2(AI),wehave 守,"+LAψE八】仰所h飞+L飞ltEJ'i呵,』ZZ*TA唱'晶rEEEEEEK onZ1onZ0' (2.9) where中(t)=中(t;中0'中1)isthesoI川iorlofthefoIIowi吨backwar problem f中tt-U中tt+A2中=0;忡|t=T=中。;中tlt=T=中1;忡|z=A中|22。 (2.1Oa) (2.1Ob) (2.10c)
ControllabilityandStabilizationofkirchqIYPlates 277 中。=(I+川%)-1z2e2(A元); ,、..,,U及A茹e噜iqu-URA噜SAU及AA时,+YEa---噜iAY (2.11) whoseSOlutioniSexpliCitly -E··AV'TA6LOAM+hUAψ'ml+LψL争LAψt (2.12) Proof.(i)From(2.7)and(1.8),wecomputeasusual[L-T.2],[IfT.4],{L-T.9],etc., qb飞BElld咆i内,』ZZ,u中Aψ比FEEZEEE、 TTZ一(AiA(T-t)BU(t)dt,AZ)-(Ar吃(T-t)EU(t)dtpA%z)j21Ln(0)j22L2(0)。-oT=-j~;坦。(11(t),GA[品(T--t)AZ+吃(T-t)AY])·(2.11)12Lh(「)' (z;i:;|)(t)zψ吧创(tt七ν川川、}J斗f-Jff,TT训Ir. (%1%1%-3ZA[吃(t-T)(I+γA)Z2+38(t一T)(I+7A)(-AZ1)]onZ1oonZ。 by(2.6),and(2.9)followsvia(2.10)-(2.12).TheproofofLemma2.1mayaIsobegivenbymuItipIyingthew-problem(1.1)by中andthe中-probIem(210)bywandintegratingbyparts..
278LasieckaandTriggiani 3.REGULARITY.PROOFOF(TRACE)THEOREM1.1ANDOFTHEOREM1.2 3,leAFUNDAMENTAl,IDENTITYFORPROBLEMfl,2j Twoofthethreefundamentalidentitiesforproblem(1.2)neededinthiSpaperareusedinthiSsection.ThethirdwiIIbeusedinSection4.Wefirstformalizethepropertythatproblem(1.2)withf三O(freesystem)isconservativeorenergy--preserving.Forproblem(1.2)withf三0,wedefine,inlinewith(1.10), 与(t)三jw(叫(t))|2+|叫")|2叫|吨(t)|2而。=ii中(t)ii2被+ll中+(t)ll2K如(俨)、如(AP (3.1) proFX〉Sition3.1.ForprobIem(1.2)withf三0,wehave Durt+LV,nu,EEZ、Aψ『n巳---+LS、..OC---+飞Aψ'EU (3.2) ProofOne川1tip1ieSEq(12a)byA中t,in川teg川teSbypartSanusestheB.C.(1.2C-d)卜.. promsttion3.2.WithreferencetoprobIem(1.2),Ieth(x)beasmoothvectorfieIdon白suchthath=ν(outwardunitnormal)on「.Then,thefollowingidentityholdstruewithHdefinedin(1.16); nyAUφLAV'v-LAYWrtlIJnv+nvAUAWIAVAψ-AwrtEEEJnyd叮EEEEEEI''EJ吨,"飞BE叶EEJ+L-AY-UAIM+内,』飞EE斗EEJA-uvmr、rat--IIl』rtllJTL以" +哈iWtl2叫中t)2-lV(喇2)仙h叫中tV(divh)WtdQQQ-[(飞,hV(呻))Q呼叫产V叫时
ControllabilityandStabilizationofkirchq矿Plates 279 Proof,Asketchoftheproofbasedonthem川tipIierh-V{A中)isprovidedinAppendixA:Eq.(A.7)therewithh·ν=1yieldsthelefthandsideof(3.3},whiIetherighthandsideof(A.5)iSpreciseIytherighthandsideOf(3.3).· 3.2.COMPLETIONOFTHEpm刀FOFTHEOREMI.l From(3.1),therighthandside(R.H.S.)ofidentity(3.3)canbewritten,bypoinear§inequalityon飞,as RHSOf(33)tφ中(t)川中(川 (3.4) aftertlsingtheenergypreservingproperty(3.2).Thus,(1.9)followsfrom(3.3)-(3.4).. 3.3.PRCOFOFTHEOREM1.21INTERIORREGULARITY WesketchaproofwhichadaptspastreasoningstopresentCIrc111nstarlees. *Stepl,WithreferencetothemapZcharacterizedinLemma2.1,TEq.(2.18),Theorem1.1saysafortiorithat *+4441T:continuousZ=iD(A)×2(A)→L2(2). (3.5) Then,withreferencetothemapZdefinedby(2.16),itfollowsTfrom(3.5)that 始VZ:continuousL(立)→Z=2(A)×ib(AI).2 (3.6)
280 LasieckaandTriggiani IOIlstep2SincelAoiisansc.gro叩generatoronz,wecannowappIySteps3··-4asintheabstractproofoftheTheoremin[L-T.11,p.747](originallygivenforsecond-orderhyperboliCequationsin[L-T.6]byabstractmethods),wheretheoperatorZherecorrespondsT*totheoperatorJiII(1.15)of口,一T.11].WethenCOIlc111dethatif=w=Oin(1.1b),thenthemap01 tl→{w(t),wt(t)):continuous→C({0,T];Z) asdesired.FinaIIy,[wweZandU=oin(1.1e)implythatthe0'1-semigroupSOIlltion(orcosine/sjnesoIutionasin(2.12))satisfiesllOIllN(t)叫(t)]=|exp(|Ao|t)|[w0月1]eC([0,T];Z).Theorem1.2isproved.· 4.EXACTCONTROLLABILITY:PROOFOFTHEOREM1.3 Stepl.BythereglllarityTheorem1.2theinput-s0111tionoperator-%讯工definedby(2.7)iSContiI111OUSL(2)→Z=2(A)×JD(A).Bythe21 timereversibilityofproblem(1.1),exactCOIltroiiabilityattimeT%%InthespaceZ=2(A)×2(A)wIthintheL(2)-cIassofCOIltroISiS21 。quivaIenttosllrjtactivityof工,inturnequivaIenttothepropertyT*thatJtThaveaCOIltinuOIlsinverse;i.e,,thereiSC>OsuchthatT .,7L飞、『-d9』Z,噜EAZrJ、、中A户U、/.、』,4『AFL,,..、qLEU噜i呵,"ZZ*Tψ比 (4.1) *whereZTiSdefinedin(2.8)andcharacterizedinLemma2.1,Eq. (2.9).AccordingIy,anequivalentpartiaIdjfferentiaIequationcharacterizationofinequality(4.1)e---andhenceofexactcontro11abiIityatT<∞overthespaceZwithintheC1assof
ControllabilityandStabilizationofkirchqfyplates28I L(2)-controlSUin(1.1e)--isasfoIIows:Thereexistsaconstant21c二〉Osuchthat 扳MAMAM却"阿AH22飞J噜..-AVanUAψ·,,,、.、''巾且户U、/-xu呵,"、E'E吁,EJAU「3-fEE卜ttk噜ift1lJT4 (4.2) where中soIvesthehomogeneouspmbImn(2.10)WIthinitiaICOIlditionsasin(2.11).Notethatby(2.11), L%4%%-1V||中HK=||AA中H=llA(I+γA)Alz||Y07自γOL(Q)2L(Q)JD(Am)equivalenttol|A弘z||=Hz||2L(0)2(A啕) 比%-134ll中11|%=||A中1||L(Q)=||A(I叫A)AZ1||L{0)JD(A)2equivalenttoilA飞||=||z||L(Q)'(A) Step2,Itremainstoshowiforwhen(4.2)hoIdstrue.ThiSiSdoneinthefoIIowingpropositionwhichiSthekeytechnjCEilissueofthepresentexactcontroIIabilityproblemfor(1.1). Proposition4.1.UndertheassumptionsofTheorem1.3,thereexistsatimeTO>O(estimatedbe1owin{4.26))suchthatifT〉TO,then forasuitabIeconstantC〉0,wehaveTF j(弩但j2。12CTVO);E中(0)equivalentto||(中0,中1)ll如(A施)×如(A兑)' (4.3) WIthE中(0)asin(110)一(112)or(31)wherebytimereversaIPwemaytake中tobethesoIutionofprobImn(1.2)withinjtiaIdata(中。,中1〉asin(211)att=O
282 LG2EieckaandTriggiGFTi Proofofpromsition4,i,stepfij,RostoftheproofIsreportedinAppendicesAandBforCOIlvenierlce.WetlseEq.(A.5),withlefthandsideasin{A.7},and{B.5}insertedintherighthandsideof(A.5).WeobtainwithHdefinedin(1.16), 才(阳i)气SEfikdzj即丛中}叫叫即中tm+吟伊{仙的专呻 bOT=阳A中t中t,A中divhV叫t-中γh-V(呻)叫(45) stepfiijaUsingassumption{1,15}ontkeFIlatrix在{x},weo垣tainfortherighthandside(R.H.S.)of(4.4),巳〉0: 2(怡叫ρ川川叫叫巳叫忖吵巾巾)4小仆归j户扣h川川|W阳阳川叫叫v叭叫w叫叫(ω叫呻叫A呻呻喇中如刺)川俨|f2+寸叫叫|W阶叫叫vw吮川中飞忖tJ|f2d州Q一5j(A呻中)卢2呻〈巾:ω仲川叫叫+巾%~b~h0ωTQQ (4.fi) GhzCTWMivh)i {4.7) WealSOrecall(B.6)(withfgO)inAppendixBandwrite nyAund争··、AVzvndAYAVFJ叫JQ且咆znyAunJU争LAW,RV+叫JHAWsavfU叫JQ、,,一-nyAU44&LAV'v+内,阳AVa-AVrliiJny (by(B.6)) r对|V(忡)|2+lWt|2Mjm中t)2dQ%{(叫中tA中时(48) Thus,using(4.8}in{4.6}andrecalli吨E中{t}in{3.1)resultsin
ControllabilityandStabilizationofkirchqfYPlates 283 RHSOf(44)主乓丘jvt)叫OT-Ej(A中)2忡Q(49) ρOT=bOT-号[(叫AA中时 (4.10) UsingtheCOIlservationofE(t),(3.2),wereadilyget中 nu,,.、AV'UU句'AWHJ飞-、...』、..EFnu,E·E·、AW'节巳+、...F中AAW『n巳FEE--‘PUJ飞-mInuny (4.11) WenowuseassumptjOIl(1.14)onhonthe1efthandsideof(4.4),whiIeontherightof(4.4)wecombine(4.9)and(4.11)andUeE(t)EE(0)toobtain中中 j〔学lj2h心jGE)2hud 2[(p一ε)T一2K1]E中(O), (4.12) whereifh(x)=x-xthenG=Oin(4.7).InthecaseofaOPhgeneralh(x),wecompletetheproofbyabsorbingallIowerordertermsin(4.12)byacompactnessargumentofthetypeusedinotherwaves-orpIatesproMems[L10.1],[Li0.2],[LIt.1],[L-T.2]-[L-T.4],[L-T.9]asadaptedtopresentcircumstances.InthecaseofaradiaIfieIdwhereGh=0,weon1yneedtoabsorbtheboundaryterm a中t/au. stepfiiij,Iβmma4.2.Undertheuniquenessproperty(b2)of Theorem1.3,inequaIity(4.12)impIiesthat:thereexistsapositive
284 LasieckaandTrtggiani timeTu=Tu(Q}dependingOIIQ(subscrIpt'U'standsfor 'uniqueness,'seeIater)suchthatifT>T>0,thereexistsaU constantCTsuchthat 。内,』rlu甲inu吨,"户lv+LAVEQqr-TUTanuqrHFUAYA+G呵,"飞tE斗lfJ+L-AwrttliJT4 径。ιA-um「L-rlllJTι中AFU . (4.13) Proof,TIleproofiSbycontradiction,asusua1.Letthereexistasequence(中n(t)}ofsoIutiomtopmbIem(2.10)withiMtiaIdata弟坦(中川,中n1)e2(A)×2(A)asin(211)suchthat(d/dt=') AUZ噜EAAUZ噜EAQ9-EU巾Anu呵,』FlurnAVB4,oqGrbTanu9』户UnAVEA+。呵,』飞lt寸,fJrn-A甲-UAl阳、fi--JT,h jf342C40 (4.15) Then(中Jsatisfyinequality(4.12)aMby(414),(415)wehave元%that(中n0,中n1)isluIIMfOrn川Isubsequence,中→中weaklyin2(A兑)and中呻some中weakIyiIITnOTOTn1TE(AWenext川Sider山?SOlution平(t)ofthesa1附P川}IeIIl(210)generatedbytheinitialdata{平。,平1),explicitly哥们)=吃(t)平04(t)飞Then(中n(t),中卢(t))→(哥们),平(t))in14%L(0,T;如(A)×2(A))weakStar.Hence, ∞7%(中n(t),中i(t))11IIifonnIyboundedinL(OJ;如(AI)必(A))·(4.16)
COF1trollabilitymdStabilizationq/KirchQIYPlates 285 L34-1iUsiIlgtheequation(2.1),中'=(I+刊)A中,weobtainvia(4.16),'n'n 。。也.{屹(t),忆"(t)}uniformlyboundedinL(0,T;2(A)×L2(Q)),(4.17) andthusafortiorivia(1.6)and(1.7), 3,3中nuniformlybO1u1MI2,2%uMfomIyboundedinH(Q)· (4.18) Then,bytracetheory[L-M,飞701.II,p.9], a中i钱说37uniformIyboundedinH'(2). (4.19) Then,bycompactnessfrom(4.16)[S.1]and,respectivelypfrom(4.19),wehaveforasubsequencethat 中n→平stronglyinC({川'];2(庐)=的。)n功。)); (4.20) 、,,、..,,。,...飞巧,缸'L、....,中A.,nu,.....,,..、户Un.,AVd'Blob、..or争LS,,?@t→''nAWf (4.21) a中JL否工→远去stronglyinL2(刀, (4.22) andby(4.20)-(4.22),asweIIas(4.14),weobta111 4·A(und-A唱'EJTAnura--‘,,,‘、内4户lu,,AVE+(M呵,"Tiu、...』minurEE--‘qr』PUT①『A+。呵L、ttttfJ侈rlllJ?ι (4.23) Ontheotherhand,平givenbeIOW(415)soIvesthehomogeneousproblem,
286 LasieckaandTriggiani 平tt1A平ttd平=oinQ; (4.24a) 学L=0 inz; (4.24b) inZ1' (4.24c) theIatteridentity(4.2c)from(4.15).Bycha昭ngvariabIe中=啡=A飞,see(13),weobtain 中一γA中+A2中=Otttt (4.25a) 中|z=A中|z=Oinz;&vt|-oin乏,3Zlz1一 (4.25b) (4.25c) yhu?Lneoh+Lsephr-S.,A=、..a-V「hscu--hhu+LWAU'nta+L守mYAYφL=-L冯「。忖,hu=+L·唱-AT①lwnJ』AU、JFhd=叫r"中41(nmoe.、,品、EA+LKUaourn-nyereohunr+LVd··0、1Lυe4C?"n··l4cudE飞 uniquessquestionhasapositiveanswer[L-L,p.127]:IfT>someTvthen中=A飞三OinQ,hence平三OinQPWhichprovidesa solutiontothetIMquemssquestionforprobIefIl(424)Ifro#中, theuniquenesscone111sion中三Owasassumedin(b2),Theorem1.3.Til队inawcasewehave车三OinQB川市三OinQcontradicts(4.23)andLemma4.2iSproved.. Stepfivj,Thus(4.12)and(4.13)yieId(4.3)asdesiredforT>TO with,say, 、.‘,,,d比1一ρUTAr-4Lxamhum- (4.26) TheproofOfproposition4.11Scomplete. . Remark4,l,IncaseofaradiaIfieId,asharpestimatefortheconstantk1in(4.11)maybeobtainedbyfoIIowingkomornjk,see [IJ0.1]-[IJ0.2].
ControllabilityandStabilizationofkirchqfYPlates 287 Remark4.l,WehavenotinvestigateddirectIyiftherequireduniquenesspropertyforthe中-problem(4.24)holdstrueif「。#中,「c「.AbovewefaIIintothe中-probIem(4.25)andappeaItothe1# 11IIiquenessrestIItin[L-L.1,p.27]whichrequiresinourcaser1=「·. Remark4.3.Inequality(4.3)with「1=rofproposition4.1impliesafortiorithefollowinguniquenessresult:If中satisfiesthekirchoffequation(1.2)=(2.19)and,moreover,tilethreeboundary COIlditions, 中|2=叫|z=弩但|z=0, (4.27) forO<t三T,Tsuffi山TMIylaMe,theninfact中EOinQByCOIltrast,astandarduniquenessrestIItrequiresa11fourboundaryconditionstobehomogeneous.. Remark4.4.When「=「,ineqllaIity(1.9)(tracerestlIarity)and1inequality(4.3)(COIltinllollsobservability)impIythatwecanintroduceanormontheinitiaIdataofthehomogeneousprobIem (2.19), ,νn飞ll〉llJnu内4飞,tl吁,,JAE呵。-rirtlEE、fk仆υ空白r、、rd噜,..Aψ』,nUAV'r41 兑wflichisequivalentotE(0),see(1.10),wllichinturniS中equivalentto||{中。,中1)il元兑.JD(A)×JD(A) TheproofofTheorem1.4ismuchmorecompIicatedandisgivenin[L-T.12].
288 LasieckaandTriggiani AppendixA Forhturemfemαtoregularity/exactCOruroliability/uniformstabilizationproblemsforequation(1.1a)withboundaryconditionsdiffemntfrom(1.1cd),weShanamtderiveageneralidentityfor@onlySOILIHonto(1.2a),withnouseofboundaryconditions(1.2c-d),intermsofageneralsmoothvectorMdh(x)=[h1(x),…hn(x)]overQ,seeEq(A5)below.Next,weShanspecializesuchm1dentity(A5)to中whichsaHsnesalsotheB.C.(1.2c-d)of由ispaper- Identityforowhichsolves(12a).Wemultiply(12a)byh·V(AO)andintegrateoverQ-Weobtain,seemspectively[LTA,Eq(A6)andEq(A7),AppendixA]JOtthmM=ωmwz-j川thvdz-L生jwot|2hvdzz ra中tra申tr+jτ丁h-vetdZ+jτ丁中tdivhdZ-jHWt-VOtdQzuvzV,Q -比JlWt|2川dQ-ptV(川 (A.l) ThAUVKU饨,-AVAra飞、v,ll'?"以"ZAUAmyAvhu瞥,|dz--nu-AUAVAJa‘、VKUAV句,&Ar--ny -jHWM)伽比J|mo)|2d川Q川 Finally,wecomputethenewtermbyintegradonbypansont yjMtth-V(忡)dQ=YKA叭,hV(忡))。]5Q -fi时hvdZ+fj时d川Q川Q afterusingalsotheidentity JhVVdQ=jvhvdr-JW川Q (A.4) withv=AOtandhreplacedbyA中thUsing(AJ)-(A3)inEq(12a)resultsinthc
ControllabilityandStabilizationofkirchq矿Plates 289 identityj嗖平fa@t工f2山V侧dZ+j百hWtdZ+2j(Mt)hV岱L生llV侧|2zuvzzz rqra中trh-vdZ-1AilV中tvh-vdZ+lτ丁中tdivhdZ-j中tA中th-vdzzzU,z =jHM)mm+JH川同tdQ +%j{Wt|2+Y(A中t)2-lV(A中)|2}dMdQ+j中tV(dM1)VOtdQQQ +jfhV(AO)dQ-[(叭,h-V(忡))Q+Y(AOt,h-V(忡))Q]J(A5)Q Specializationofleft-handsideof(A.5)t。中satis句ringalsotheboundawconditionsi13立尘二Recalling(12c-d),weobtain onz:中tEO;AOtsO;VOtir;V(AO)ir a中taoth-VOt=一一h-v;|17中tl==l一一|队Fva(A中)|ao主)|(AO)=-I-h-v;lV(A中)|=|1-| (A.6) Thus,theLehHandSide(L-H.s.)ofidentity(A5)becomesLmd(A5)=气[[等于+[蚓hvdZ(A7) Specializationoftherighthandsideof(A5)toradialvectoraddsh(x)=x,X0·UsingdivhEdimQ=n;H(x)sidentity,weobtainfortheRightHandSideof(A5) REfsof(A5)=jw(AO)|2+|问|2dQ+?jlWt|2-|V(崎)|2dQQQ
290 LasieckaandTriggiani +乎j(AOI)2dQ+jfhV(AO)dQ-[(叭,hV(崎))Q(A-8)"QQ +Y(A龟,h·V(A@))Q]J
ControllabilityandStabilizationofkirchQHPlates 29l AppendixB Again,weShanamobtainmidentity,(B.4)below,for中由atsolvesonly(12a)andformarbitraIysmoothvectorhldh(x)onQ.Next,weShanspecialize由isMenuty(B.4)tothecasewhere中satisks,inaddition,theB.C.(12c-d)and,momover,thevectorkldisradial. IdentityforOwhichsolves(1.2a).WeIIluluplyEq(1.2a)byAOdivhandintegrateoverQ.Weobtain,seerespectively[L-T.4,Eq(B.1),andEq(B2),AppendixB] j叫伽陀|jhdivhdr=jV中叫di川d正QJ2Q『UvQ」Ozu' +J|Wt|2d川Q+jetV(d川同dQ(B1) jpa(A中)f川川叫7Mdivhdz-jlV(AO)|2d川Q(B2)Q -jA中V(divh)V(AO)dQQ Finally,thenewtemis YJ俨俨A崎0川申d副仙川i忖vQ Using(B.1)-(B.3)inEq(12a)yie1ds||ra中tj{lVOthy(AOt)2-lRAO)l斗divhdQ=jτOtdivhdZQLJzuvfd(A中)r-j二专子A中divhdZ+lfAOdivhdQzuvQ +JM(dvh)阳)dQ-jV叫阿tdQQ (B.4)
292 LasteckaandT〉iggiani +iv争旧机j和1divhdrl: 丁01gillJ。、‘,,,huva--···AUAMYAAVAJEE飞ri--lLγ·+ Specializationof(B.4)toosatisfyinzalsotheB.C.(12cd).From(12c-d)weobtainforhItu陀陀femnce 比J{|Wt|2+叫)L|mo)|2}川dQ=比JM(川阳)dQ …比j申tV(divtOV命tdQQ (日.5) 十比jfA争divMQ÷均γ〈A龟,A争div盼。-〈熟,A争div挝、12Q Inpaniculaz,ifweuseznORS垣IFly出emul岳plierA争inaea§ovep113ced131毛,weO-iainj{iVW2÷Y(A取)2-iV〈A的i2}dQ=i〈YA争t-叭,A命〉alJ+jfA争dQ(B.6)Q Iobeusedin〈4.8〉azld〈623).
ControllabilityandStabilizationofkirchq矿Plates 293 References [G.1]P.Grisvard,caracterizationdequaIquesespacesd'interpoIation,Arch.RationaIMecflaniCsandAnaIysiS25(1967),pp.40-63. [Lag.1]J.Lagnese,Boundarystabilizationofthinplates,SIAMStudiesinAPPIiedMathematiCs. [L-L.1]J.LagneseandJ.L.Lions,ModelIing,analysis,andcontroIofthinp1ates,MassOI11988. [Li0.1]J.L.LionspExactcontrollability,stabiIizationandperturbatiOIls,sIAMReview30,1988,pp.1-68. [IaiO.2]J.L.Lions,ControIIabiIit6exactedesyst§IIlesdistribues,MaSSOIl,toappear. {Lit.1]W.Littman,NearoptimaItiIIleboundaryCOIltroIIabiIityforacIasSofhyperboIiCequationsppp.307一312,Sprjnger一verIagLectureNotesLNCIS#97,1987. [L-T.1]I.LasieckaandR.Triggiani,ReguIaritytheoryforaClassofEuler-Bern01111iequations:Acosineoperatorapproach,BoIIettinoUnioneMatematiCaItaIiana(7),3-B(1989),pp.199228. [L-T.2]I.LasieckaandR.TriggianiPExactCOIltrollabilityofEuler一Bern01111iequatiOIlswithboundarycontrolSfordispIacementandmomentS,J.Math.AnaI.andApp1.,toappear. [L-T.3]I.LasjeckaandR.Triggjani,ExactCOIltro11abiIityanduniformstabiIizationofEuler-BernouIIiequationswithonIyoneactivecontrolinAW|2· [L-T.4]I.LasieckaandR.TriggianiPExactCOIltroIIabiIityoftheEIIIer-Bern011111equatiOIlswithCOIltroIsintheDirichIetandNetlmarlnboundaryCOIlditions:Anon-conservativecase,SIAMJ.ControIandOptim.27(1989),pp.330-373. [L-T.5]I.LasieckaandR.Triggiani,AcosineoperatorapproachtomodelingL(OT·L(「))-boundaryinputhyperboliC2"2equatiOIls,App1.Math.andOptim.,vo1.7(1981),PP-35-83. [L-T.6]I.LasieckaandR.Triggiani,RegularityofhyperboliCeqllationsunderL2(0,T;L2(「)-boundaryterms,App1.Math-andOptim.,V01.10(1983),pp.275-286.
294LasieckaandTriggiani [L-T.7]I.LasieckaandR.Triggiani,UniformexponentialenergydecayOfthewaveequationinaboundedregionwith,JDiff.Eqns.,V01.66(1987),pp.340-390. [L-T.8]I.LasieckaandR.Triggiani,RiccatiequatiOIlsforhyperboliCpartiaIdifferentialequationswithL(0,T-L(「)-Dirichletboundaryterms,SIAMJ.COIltroiand2'2OptifI1.24(1986),pp.884-926. [L-T.9]I.LasieckaandR.TriggianipExactcontrollabilityOfthewaveequationwithNelImamIboundarycontr01,App1.Math.andOptim.19(1989),pp.243-290. [L-T.10]I.LasieckaandR.Triggiani,AlgebraiCRiccatiequationswithappIicatiOIlstoboundarypointcontrolprobIems:COIltinuoustheoryandapproxjmatiOIltheory,preprint,October1989. [L-T.11]I.LasjockaandR.Triggiani,AIiftingtheoremforthetiIIlereguIarityofsoIutiOIlstoabstractequatiOIlswithunboundedoperatorsandappiiCationstohyperboIiCeqllations,ProceedingsAmer.Math.Soc.,V01.104(1988),pp.745,755. [L-T.12]I.LasieckaandR.Triggiani,ExactcontrollabilityandtlniformStabilizationOfkirchoffpIates,preprint1989. p[S.1]J.Simon,CompactsetsinthespaceL(0,T;B),AnnalidiMatem.PtlraeAppIicata(iv),V01.CXLVI,pp.65-96. [T.1]R.Triggiani,AcosineoperatorapproachtomodeIingL2(0,T;L2(「})-boundaryinputproblemsforhyperboIicsystefns,LeeturesNotesCISSpringer-VerIagJr6(1978),pp.380一390;Proceedingsof8thIFIPConference,IJniversityOfwurzburg,W.Germany,JUIy1977. [T,2]ReTriggiani,ExactboundarycontroIIabiIityon'--1,"(Q)×HA(Q)forthewaveequationwithDirichIetCOIltrυIactingonaportiOIloftheboundary,andreIatedprobIeIIlSPAppl.Math.andoptiI11.18(1988),pp.241-277. [T.3]R.Triggiani,WaveequationonaboundeddomainwithboundarydiSSipation:Anoperatorapproach,J.Math.Ana]-andAppl.,V01.137(1989),PP-438-451.
C-ExistenceFamilies RALPHdeLAUBENFELSDepartmentofMathematics,OhioUniversity,Athens,Ohio LINTRODUCTION. Wewilldiscusstheoperator-theoreticapproachtotheαbstmctCGuchyproblem 兰也(t,z)=A(u(t,z))(t兰的,u(0,z)=z,dt 、..,,,'EA.嘈EA,,E飞 whereAisalinearoperator,withdomainD(ALonaBanachspace,X.ByasolutionwewillmeanUEC([0,∞),X)nc1([0,∞),[D(A)]),wherewedenotebylD(A)]theBamchspaceD(A)withthegraphnorm,satisfying(1.1).ByamildsolutionwewillmeanUEC([0,∞),X)s时thattHSu(s)dsEC([0,∞),[D(A)]),削is命ing nu>-eιJ'··、z+＼it/eu,α、1EJGVJ'E飞ur--o/FEEt＼A=、‘,,,''ι,,.‘、包 WewillwriteIAforthesetofailZforwhichauniquemildsolutionof(11)exists.Noproofswillbegiven.ProofsoftheresultsinSectionsIIIandWmaybefoundini17]-Thefollowingrelationshipbetween(1.1)andstronglycontinuoussemigroupsiswen.known;see,forexample,[1月,[25],[26],{39],[48lor[58]. 295
296 deLattbenjbl5 Proposition1.2.Supposeρ(A)ismmmpty.TIleIlthefollowingareequivalent.(a)IA=X.(b)Ageneratesastro昭lyco川nIJO山semigro叩· 叽与wouldliketocomderliowtoddICSS(1.l)whenIAislarge,butdoesIIothappentoequalallofX.Example1.3.LetA三一(jt)2,OIlaIlaIP仰〉吁叩Ip)mrOIP归川汀川Ti归at阳es叮I肌eOff1ulI川iO肌1T、ihlm(υ1l川.jI)川ib)川阳(川Qthebackwardheateq1uIatUiOInl.WhenX=LP(R)(I三IJ<∞),tileIIIAisdense,butAdoesnotgemrateastronglycontinuoussemigroup;tilebackwardsileatequationisanexampleofwhatisoftencalledmimpmper-lyposedorillI刷cdpM}lemhee[47]).Motivatedby(1.1),somePIle时MathmofstronglycontinuoussemigroupshavemeIItlyreceivedmudattention-DeanitioI11.4.Tilestmlglyco川nuomfamilyofboundedoperators{5(t)}t主oisaIlezpovzentiallgboundedn-hmcs171tqmtcdsUTIt-gmupgEFIEFUtcdbyAif31ty>Osuchtll川(队∞)豆p(A),||S(t)||is。(ctw),5盯(0创)=0aU1n1Fc卧(λ一川一12二人n/fMS(t)241t,JO VA>147,zEXf.ThiswuintroducediIll1];malso[21,问,[叫,[叫,[30],[3月,[38],[42],[43],[44la川[57lDe自nitioI11.5.SupposeCisboundedaIMiinjedive-TilestronglycontinuousfamilyofbollMedoperators{lV(t))t>oisa(',JZE1TItigrw叩if117(0)=CandIV(t)IV(s)=CIty(1十s),VAt主0.IfitexteMstoaIlentirefamiiyofboundedoperators{lV(z)}zεc,川(Jailitanmti陀C-group.ThegeMmtorof{lV(t))t>oisdennedbykCr俨-→lll!巴阳时时坦出UU州;斗山山:卡扫(υlW川川川V叭叩(υtAi比sclosedandD(以A)川isleftinvariantbyW(t),Vt主0.Thiswasintroduced,independently,in[9]ad[12];thegeneralityofDeanition1.5wasintroducedin[13];seealsoi10],[14],15],[16],[18],[20],[28],[29],[35],[36],[37],[381,[45],[46],[53],[54],[55]aad[561.Wethenhavethefollowingrelationshipto(1.1).Proposition1.6..Supposeρ(A)ismmmpty.ThenthefollowingareeqtuIiVm&dlen(归ωa叫)D(Anη)5IhA,and3tωtU1>0suchthatallsolutionsareO(etUF).(b)Ageneratesanexponentiallyboundedn-timesintegratedsemigroup- Proposition1.7..Supposeρ(A)isIIOMIIIptyandC(λ-A)-1=(λ-A)-IC,Vλερ(A)-ThentheibMowingareequivalent.(a)Im(C)gIA·(b)AgeneratesaC-semigroup- Example1.8:BackwardsHeatEquation.2川+Aμu叫呻(ο川t包(οt,5)=0(0t主0,EεθD)包(0,E)=f(5)(EεD), (1.9)
C-ExistenceFamilies 297 whereDisaboundedopensetinRn,withsmoothboundaryODandAistileLaplaciaIlonRn,立=1(否37)2LetA三-A,OIlLP(D)(1三p<∞),D(A)三IV2♂(D)nwJ,p(D).Thenitnl叮beshownthatthereexistsCsuchthatIm(C)isdenseandAgeneratesanentireC·.group-Thisproducesuniqueentiresolutionsof(1.9),forallinitialdatainadenseset.Example1.10:CauchyproblemfortheLaplaceequation,inaninfinitecylinder- Au(t,E)=0020,5εD)u(t,E)=0(t主0,zεθD)u(0,王)=f(5)(5εD)ZM=到附 、‘.,,v-A'EEA.·ES--,,..‘、 whereDisuinExample18andAistheLapkianonRn+1,(￡)2+立=1(￡)2(1.11)maybewrittenasasecondorderabstractCauchyproblem, 口可d一一、‘E,,,nυ,,..、、urId--、、..,/nUJ,,‘‘、u、‘IJ、‘.,,,4ι/,,‘飞u'''t、、A一一、、..,,,a''b,,..‘飞u WIleIVAisasiIIi议JMIll)lel.8.TIll18,uvcmaltetlIQUSualmatrixreductiontoafirstorderproblem,byωIMidering X×AD---AD、EEEEZEB--JYInUOA一一-A ItmaybeshownthatthereexistsCsuchthatIm(C)isdense,andAgeneratesanentireC-group.Thus,asintilepreviousexample,weobtainuniqueentiresolutions,forallinitialdatainadenseset.Example1.12.SupposeBisboundedandAgeneratesaboundedstronglycontinuousgroupthatcommuteswithB.Then3afamilyofboundedinjectiveoperators{CJtsuchthat(1)VE>0,BAgeneratesanentireCE-group.(2)UooCE(D(A))isdense.H||♂ω||tεRis0((1+|t|)n),thenBAgeneratesanexPO阳s盹eImInli地grOu叩P(C∞Or口mresPOnMdωi鸣toU,=BAu,也(0)=zhavingauniquesolution,VZεD(A(n+2))-Thismaybeshowntobebestpossible.Thisresultmaybeappliedto告(t,z,u)=h(u)35(tJ,u),u(0,z,u)=f(ZJ),onwmJ(R2).Example1.13.ForthehigherorderabstractCauchyproblem uh)(t)=A(u(t))(t主0)u{t)(0)=zt(0三t<n) tohaveanentiresolution,for41initialdatainadenseset,andn>larbitrary,itissumdentthatλAgenerateaboundedstronglycontinuoussemigroup,forsomecomplexλ,andhavedenserange.ThismaybeshownbymakingtheusualmatrixreductiontoaErstorderproblemd=Au,thenshowingthatAgeneratesanentireCf-group,VE>0,suchthatLJooCE(D(A))isdense.(see[20])Forn>2,itiswellknownthatAd伺snotgenerateastronglycontinuoussemigroupunlessAisbounded-(m[24])
298deLaubenfels Example1.14.Deheφ,onL(X),by φ(B)三AIB+BAL where/1laIldA2generatestronglycontinuoussemigroups-Thenφgeneratesanexponell'tidlyboundedA-semigroup,where A(B)三(T-AI)-1B(T-A2)-1(rEρ(A1)np(A2). ThisyieldsuniquesolutionsofU,=仇,u(0)=z,VzεIm(A2). IlINFORMALDISCUSSION. However,therearelimitationstobothC-semigroupsandintegratedsemigroups-Considertllefollowingsimpleexample.Example21.KtX三{continuousf:R→C|iim|zh∞f(z)J=0),aMlct||f||三叫zd|f(z)e升,A三d/dz,withmaximaldomain.TilenIAisdense,sinceitincludesailfunctionsofcompactsupport,butitmaybeshownthatthereexistsnoCsuchthatAgeneratesaC-semigroup,nornsuchthatAgeIieratm川口timesinteg川edsemigroup(see[17]).AnotherlargeclassofexampleswhereintegratedsemigroupsaIIdC-semigroupsseemdiiiiulttoapplydirectlyismatriEesofoperators,actingonproductsof(iP)O5附圳5剑i川山ωlbM}才lydiiT扣e阳l}h丛川CihiSiρ}川e5.Thisisanactiveareaofcure川research(see[21],[22l,[23],[40]aM[41])FroapplyC-semigroups,ithasbeennecessary,inpractice,toilavetileoperator。IItriesCOHIImte(m[14,[15j,[20iand[451);thisiscertainlyimpossiblewhentilemat川actsOIltileproductofdiEerentBanachspaces-Toapplyintegratedsemigroups,itisnece55arythattileresolventsetbenonempty;thisisrarelytilecasewilenthematrixhasIIondiagonaldomain-Besides,anoperatorthatgeneratesanintegratedsemigroupalsogeneratesaC.mdgro叩(see[14lor[38]).TherearealsoproblemsindevelopingaperturbationthωryforC-semigroupsorinte-gratedsemigroups.Similarly,whenAtgeneratesaCtsemigroup,fori=1,2,thenitisdimculttoobtainaC-semigroupgeneratedby(A1+A2),unlessC1commuteswithC2;inpractice,thislmdlymeansthMA1andA2mustcommute(S伺[14],foraresultofthisIlat盯e).TheproblemisthatCmustcommutewith(A1+A2),sothattheobviouschoice,C三CICbwillworkonlywhenC1C2=C2C1·Agoodgeneraiiutionofstronglycontinuoussemigroupsshouldsatisfythefollowing.WhenIAisnontrivial,itshouldbeaccessiblethroughthisfamilyofoperator飞withoutanyrenormingorconstructionsofnewBanachspaces-ThiswouldthenbringbeItentsanalogoustothoseofstronglycontinuoussemigroups,e.g.,continuousdependenceontheinitialdataandotherparameters,perturbationtheory,approximationtheory,asymptoticestimates,andageneraluIlineationmdsimplineation-Intuitively,thesolutionof(1.1)isu(t,z)=etAtHC(D(A))EIA,then{etAC}t20shouldbeastronglycontinuousfamilyofboundedoperators-IfW(t)三etAC=CetA,thenW(t)W(s)=CW(t+s);thisisthealgebraicpropertiesofaC·.semigroup-Toremovethe
C-ExistenceFamilies 299 mild叫utionof(1.1),withZ=0't山ihlmeCα2j卫Ju叫(仆T)MdT=IV2(tM(0)=0.飞VemustdefineW1(t)andW2(t)witlmtmfemlcetotlhhl陀eIP严〉陀附erlhl旧叫a叫Ip庐〉邓s川ystω阳e盯rm山eXPO川lεtA.TodefimtlIealgebraicpropertiesoftllcpair{(11710),11、(t)))吃。,itismcess川FtointertwinelvlaIldwhasfollows.11720)IV1(s)=CMtACSAC1=CM(t+s)ACl=C2lVI(t+s)=1的(t+s)C1.Thiswillbeourdefinitionofamild(C1,C2)-cziStenCCGnduniqtmlesshmilyinthemxtsection.Itis450clearllowto削巾veA,bydi阮renHatingIV2(t)att=0,tilenapplyingCF1;aswithstronglycontinuoussemigroupsandC-semigroups,thiswillbetilegeylemtor- III-EXISTENCEANDUNIQUENESSFAMILIES.Deanition3.1.TIlepair{(W1(t),IV2(t)}t〉oofstro吨ly-ωntinuomfamiliesofboundedoperatorsisamild(ChC2)ezistentEGMuniquenessfrImilyif(l)Wt(0)=Chfori=1,2.(2)C2isinjective.(3)W2(t)IV1(s)=C214气(t+s)=1飞(t+s)C1,怡,t主0.TlleoperatorAgolemtes(If气,If-)if 12叮1(jMF2(t)比二。) 阳)三{Z|轩(t)zexiMSJI川川(t)叫和(t)Z|t=o)川} Remark3.2.叽rileIICIequalsCuandcommuteswithIV1(t)aMW2(t),Vt三0,tileIIIVl(t)equalsIV2(t),andIV1(t)isaC1'semigro叩·Acomeque盯eof[13],TIleom1124,isthatAgeneratesH气(t).Theorem3.3.SupposeAgeneratesthemild(Chc2)existenceanduniquenesshIIlily(叭,W2).Then(a)Aisclosed.(b)Vt>hEX,KW1(收dsED(A),withA(j;W1(s)zh)=W1(加-Gz(c)D(A)2Im(C1).(d)H||Wt(t)||is。(et勺,tbrsome切>0,i=1,2,thenVT>切,Im(C1)EIm(T-A),and(TJ)isinjectimwith(T-A)-1CIZ=rrtW1(t)zdt,VZEX,C2Z=jJofTtW2(t)(T-A)zdt,VZED(A). Proposition3.4.SupposeAgeneratesamild(Cl,cdexistenceanduniquenesshmily-Thenallsolutions。ff1.ljareunique.DeanitioI13.5.SupposeqεL(X).Wewiil8aythat(1.1)isC1ωdl-posedifithasauniquemildsolution,VZEIm(C1),and3continuousg:[0,∞)→{0,∞)suchthat||u(t,C1Z)||三g(t)||z||,Vt主0,VzεX.Theorem3.6.HAgeneratesamild(ChcdexistenceanduniquenessibIdly,then(1.ljisC1well-posed.AslongasD(A)issumdentlylarge,itissumdentthatanextensionofAbesuchagenerator.Condition(b)ofthefollowingtheoremguaranteesthat(1.1)isqwell-posed.
3ω deLaubenjkls Theorem3.7.Suppose{W1(t)}t>oand{IV2(t)}t>oarestronglycontinuoωfamiliesofboMcdoperators,anddW叫叫1叫(μS巾tdιι1ih‘爪QεD叫(A4),VhZεXTlh如阳阳l阳阳川e凹mInI山岛ω创lHlO圳W川in昭ga盯川r刊C凹叫叩叩1μ刊Iu川Iμi(a叫)(υlW4V叽F气1,W2)isam川iυld(Cα1,Cα2)川i川5叫t忖阳e凹I盯θa川IMu川川IηliqtuI川川e创mIn1阳e臼S臼S白m川i叶lb抄Jy,,EeωI川}ο盯川r口ta川ite刷叫(d1lJYAII川ten-SionofA.(仙向b问)川川WtV川/andequalsW2(t)AzjzED(A),t主OIf||lvs(t)||isO(eut),ibrsomeω>0,t=1,2,AjsclosedandD(A)isdense,thenthesearebothequivalentto(c)Im(COEIm(T-A),and(r-A)isinjective,with(T-A)-1CIZ=rrtIV1(t)zdtVZEXJ22=rc-Ttlf包(t)(TJ)z川ZED(A),VT>川川(KW1(巾ds)isacontinuous。(euytjmapikoIII{0,∞)intoX,VZEX IV.EXPONENTIALLYBOUNDEDEXISTENCEFAMILIES. WearemotivatedbyTheorem3.71odefinetileexponentiallyboundedexistenceandunique-nessfamiliesseparatelywithtileLaplacet阳S阴阳e凹Inm川I口1i厄gm阳叩saMC-mIIIigmlIhandtheirgcM川0叭arededfnimdd川nm川1让i川la盯rlh〉y?i川n[口川l叫]L?[μ2lL?[口l2勾lan[4衍叫5叮]Forclwell-posedness,itisEtunicieIlttllJUallextensionofA,rMIllertilλIlJiihelf、lM、tllfgeHeratorofa(Cl,C2)以iStenceLintiuniquenessfamily,wlienU(A)issuiUrimtlyhIgt＼asiIITheorem3.7.飞飞fehavetherefore,iIIDefinitions42aIld4.3,removedtiledistinctionbetweenAbeingageneratoraIMiaIlexteIlsioliofAbeingagenerator;Proposition4.521Il(iTheorems4.7alld4.8demoIIStratetilerelevanceofthisto(].1).Intuitively,wearetakingtilestandardLaplacetransformforstronglycontinuoussemi-groups, (T一A)一1Z=leff-→vT~.JOandapplyingC2ontheleft,fortheuniquenessfamily,andCIontheright,fortheexistencefamily,onbothsidesoftheequation.Fortheexistencefamily,itisnotnecessarythatrερ(A),todehe(T-A)-IChandinfact,requiringthatp(A)bemmmptyturnsouttobetoorestrictiveahypothesis-GeneratorsofexponentiallyboundedC1·semigroupsmayhaveemptyresolventsets,ingeneral;theirC-resolventsetcontainsahalf-plane-Deanitiox24.1.ThecomplexnumberTisinpc(A),theC-resoltJOltsetofA,if(T-A)isinjectiveandIm(C)EI7710-ALDeanition4.2.SupposeAisclosable,{W1(t)}吃oisastronglycontinuousfamilyofboundedoperatorsand3tty〉Osuchthat||WI(t)||is。(etut),忡,∞)三ρCE(互),with 川)飞zf产 (ωa功)H川jd;阴W叭F气1(ω吵咐zd出sεD凤(A均),amnd们tHA(K叭(例S吵)Z白叫)川CωO川u川趴仆Oq(et俨俨‘wω叩U川t[归0,∞)intoX,VzεX,then{WI(t))isanαpOMntitilluboundedmildC·I-mste町efGFrailgforA.(b)IftHWI(t)zisacontinuousmapfrom[0,∞)into[D(A)],and||AW1(t)z||is0(ewt),vzεD(A),thenW1(t)isanezpOMMMlluboundedC1·αtStencefαmtlyforA.
C-ExistenceFamilies 30I Deanition4.3.Tilestronglycontinuousfamilyofboundedoperators{14%(t)}Qoisanez1307lentMlluboundedC2·uniquenessh771ilyforAifC2isinjectiveand3tu>Osuchthat||W2(t)||is。(eM),with C户fe-Ttl几(t)川MVM川r>ω Proposition4.4.SupposeAisclosed,D(A)isdenseandIV10)isanexpomntiallyboundedCl-existencefh灿aM川In川IexistencefamilyforA. Uniquenessofthesolutionsof(li)foilowsfromDehition4.3.Proposition4.5.IfthereexistsanexponentiallybolIIIdedC2-uniquenessihmilyforA,thenallsolutions。f(1.ljareunique. Ifwepermitonlyexpomntidlyboundedsolutionsof(1.1),thenaCrezistoleefamilyissufEdenttoguaranteeuniquenessandtllefollowingwell-posedness-Deanitiox14.6.SupposeClEL(X).Wewillsaythat(1.1)isC1·ezpOTIcnttdlutiyell-posedif3λf,tu>Osuchthatthereexistsauniqueexponentialiyboundedmildsolutionof(l.l)-VzεIm(C1),with vtf℃znu>-a''ιwvvzuC,I且川川<一、、E』,,rzca''LWU Theorem4.7.SupposeAisclosable,(T-A)isinjectiveandIm(CdEI771(r-A),forTlarge.Thenthefollowingareequivalent.(a)3anexpoMIltiallyboundedmildCl-existencehIIlilyibrA.(b)(1.ljisC1-exponentiallywell-posed. Theorem4.8.Suppose3anexponentiallyboundedC1-existencehmilyforA.Then3auniqueexpomntiallyboundedsolutionof(1.1j,VZECl(D(A)). WealsohaveTfille-YosidatypetheoremaTheorems4.1Oand4.12usethefollowingequivalentformofArendt'sintegratedversionofWidderktheorem.Lemma4.9(fromArendti2]Theorem1.landitspr∞f).SupposeGisaBamcllspace,f:(0,∞)→GandM<∞.ThentheibMowingareequivalent.(a)fisinErlitelydiHUrentiable,with||Tn+1fh)(T)||三Mn!,forr>0,n=0,1,2,.(b)3F:[0,∞)→Gsuchthat||F(t)-F(s)||三M|t-sLVs,t主0,F(0)=0,and 只T)=frfTtmt,bO ResultsanalogoustoTheorems4.lOand4.12,forC-semigroupsandintegratedsemi-groups,areinmostofthereferenceshrC-semigroupsandintegratedsemigroups-Theorem416isverysimilarto闷,Theorem32.Corollaries4.15and4.17,wherewegivesumdentconditionsfor(1.1)tohaveauniqueexponentialiyboundedsolution,foroneparticularRxedinitialvalue,asortof"pointwiseabstractCauchyproblem,"areputheretoillustratethewiderangeofapplicabilityofclexistencefamilies.Thesecomllamsallowonetoconsider(1.1)onepointatatime.
302 deLaubenfeIs Theorem4.10.SupposeAisclosedand3M,ω主OsuchthatIm(CdEI771(T…A户,VT>衍,nεN,(T-A)isinjectiwVT>衍,and Af||(T-A)-nc1!!三?一气立,VT>ω,7ZEN.T-w·川 (4.11) ThenVS>衍,thercexistsaftexponentialiyboundedmiid(s-A)-lCI吃xiStencciAIIIiiyibrA,thatisLipschitzCOIltiIIUOtlsonboundedintervals. TIleomxI1412.SupposeAisasiIITheorem4.Io,andinaddition, MjjAzii||A(r-A)-nclZ||三可丁Zt, (4.13) VT>ω,nεN,Zερ(A).TitezbVS>衍,thereexistsanexpomntiaiiybounded〈s…A)-ICE-existencehnzibFbrA?thatisLipschitzcontiIIIIOtISonbotIIldedintervals. Remark4.14.WhenCICommuteswith(T-A)…1CI(aswhenAgeneratesaCrsemigIfMIlo?Iim(413)followsauiomaticaiiyfmIl(4.11).Coronary4.15.SupposeAisclos叫,ZED(A)aIId3M、1tj>O.S>tsuchtll川(卜J1)J夺I711(T~A户,VT>tu,7tEN,(T-A)jsinjective,VT>仙,tilld AfiIT一均-n(s…A}zii三?一气二VT>队HN.T…tiJZ叫 Tlien(1.ljllasalJniqmexpo川ntiallyboundedmildsolution,t仙ha剖tiSILJi订l肮、lhlitZCωOI川ltυj川川川lωuOIlboundedintervals. ForTheorem4.16,weconstructa(k+1)-timesintegmtedC1·αiStencefamiigforAhee{17lLwhichimpliesthat3an(s…A)-(k+1)CI吃xiStencefamilyforA,Mfollows.LetrgOR+E),forsomer>c,anddeRne S(巾三jettu(ttJ-A)…ICZJ立一h114πzywH+1 fort;20,zEX.TEzeomm4.16.SupposeAisclosed,kEPfU{吟,M,c,ε>G,Sεc,with(a)Im(CdEIm((T-A)(s…A)(k+1}),(r…A),(s-A)injective,wheneverRe(叶>c,withthemaprM(T-A)→c1analytic.(b)ll(r…AY1C115MlTFK叫),wheneverReo-)>c.Thenthereexistsmexponentiaiiyboundedmild扫一A)什k÷E}C1·existencehmiiyibrA Corollary4.17.SupposeAisclosed,kεNU{0},M,qf〉0,Re(s)〉c,zED(AHILzand(s-A)k+1zEIm(r-ALand(r-A)isinjective,wheneverRe(r)>c,with ii(T-A)→(s-A)忡1zii三Mirik叫, TH(r…A)-1(s一A)k+1zanalytic.
C-ExistenceFamilies 303 Then(1.1)IIasauniqueexponentialjyboundedmildsoltItiOIl- V.HOLOMORPHICEXISTENCEFAMILIES.IIIorderthatthereexistaIlexponentiallyboundedholomorphicC-existenceforA,itissdidentthat||A(ω-A)-1Cl|beO(击),inasectorofanglegIeaterthan2,fol-sOnm1positivef叫(,Tr}hI阳meωOreImIn155.3a叫5.4付).If{ZED(AC)|ACZEI叫C))isdense,itis阳Ce5saryands1u1mCdien川tthat||(A(ω一A)-1CC'!川|bebO1um1口mn时1ded(,TT}h1eO阳1ProofsoftheseresultsandcorrespondingresultsforlhIOdlOImIn1Or叮IP抖〉才lhliCCC.-Se凹ImIn1i厄grmO1uIIp〉sandiMInlt怡e-grateds臼eImInIigroups,maybefoundin[19LWewillwritehfor{TESφ|T>0,|￠|<的,问for{resφlr>O?lol三。).DeanitioI15.1.Suppose?三。>OTheexpomltiallybo川dedmildC-existencefamily{W(t)}t主oisanezpOMntidly加uMedholomorphicC-czistc盯cfGmtlyofGnglcOforAifitextendstoafamilyofboundedope川ors{W(z)}zεsosatis乌Ping(l)Themapz←→IV(z),from50intoL(XLisilolomori)Ilk-(2)Whenever|￠|<0,{W(tesφ)}tEoisanexpomntiallyb∞川edmildC-existe盯efam-ilyforesφA.(3)Forallψ<0、{IV(z)}isstmlgiycontinuousonSUPIf||IV(z)||isboundedOIlSψ,forallψ<0,tileII{lVK))z已句lsλ{MYIi71drdlido771Orlyf山mildC.eziste71cεfomily.Tilefollowingtileorcmsilo叭'sthatthisgeneralizesllololllorphirfd川Iili民rollps(iIItrodlIEediII[10])Theorem5.2.5叩poseAisclosed,520>0,S(θ+号)仨ρfJ(J1).CisiIljcctiwalldcommuteswith(仙一A)-吧,VWEK(A)and{IV(z)}zuoisasωsftofL(X)TljelltllfibMowingareeqtJivalcIIt.(a){W(z)}zεseisanexpomntiallyboundedl10lomorIJllicC-mIIIigro叩ofaIJEleOgen-eratedbyanextensionofA.(b){W(z)}zεS@isanexpomdallyboundedholomorphicmildC-existencefamilyofangleOforA. Inthefollowingtwotheorems,wemayconstructtheC-existencefamily,aswithgeneratorsofholomorphicstronglycontinuoussemigroups,asfollows.ForT>0,letrT三{se土tψlS主T}U{Tete|-ψ三。三ψ},orientedcounterclockwise-DeRIle,forZES(ψ-苦), mz)三Iy叩-Am去 Theorem5.3.SupposeAisclosedand例如>ψ>?suchthat町,EK(A),and(b)wH(ω-A)-1,ikomvt,intoB(XLhholomorphic.(c)3ε>Osuchthat||A(仙一A)-1C||isboundedandO(|ω尸)jnVψThenthereexistsaboundedholomorphicmildC-existencefamilyofa略le(ψ-5)ibrA Theorem5.4.SupposeAisclosedand(a)3π>ψ>5,k>Osuchthat(k+Vψ)Eρc(ALand(b)WH(ω-AYIC,ihrII(k+Vψ)intoB(X),isholomorphic,and(c)Thereexistsf>Osuchthat|jA(ω-A)-1C||isO(|ω尸)in(k+比).
304 deLGttbenfels ThenthereexistsanexponentiallyboundedholomorphicmildC,existencefamilyofaIlgle(ψ-5)ibrA. Remark5.5.IIItileprecedingresults,iIlorderthattileIII叩ω,→(ω-A)-IC',froml匀,intoB(X),beholomorphic,itissuHicMIlttoihla飞{e?JA1EdlOforωεVψ,with||(UY{A)-1(T-A)-l(s-A)-1||locallybomded.Thism叮beshownwithtileide川ity(r-A)-1C一(s-A)-IC=(s-T)(T-A)一1(s-A)-1C.Theorem5.6..SupposeAisclosed,{ZED(AC)lACzεI叫C)}isdense,?三θ>()aIldS(f+@)三ρc(A).TlhleInltlhiefOlHlOWiIn】gareC{q伊机1严刊lu山Iμjh、V,alhh阳C凹yInl(a叫)Thereexistsalb}OtuI川I川ldedlh1OdlOIIωr叮lpμ〉对lhliCI川川I川川lυil(dj(fιr1丁7:丁'＼-ttθ叫:7川B叹XihSft忖阳1℃we凹InICOi￡h仙}泣川tυIηm川InI川j川lyofaIn1昭ElheθforA(b)Vψ<(?+0),||A(ω-A)-1C||jsbomdcdiIiSψ Theorem5.7.SupposeAiscloseddzED(AC)lACzεIm(C)}isdense,?主θ>0anVψ<(5+θ)L,t仙lh1陀町e盯reexistskψεR51uIClh1t仙lhla剖t(kψ+S4ψ,)EρCC.(A)}.Thentheibllowi昭areeqtIivalent-(a)ThereexistsaIIexponentiallyboundedholomorphicmildC-existencehIIlilyofaIIgltB。ibrA.(b)Vψ<(号+0),l!A(ω-A)』IC|lisbo川tiedjll(ktp+仇,). Remark5.8.LVIleIICAEAC(川飞viwlifllh川队、gflloratorof礼(77-semigrollI川、tlICIlD(A)E{2ED(AC)|ACZEIm(「)). VLEXAMPLES.叽7ewillwrite 句,"Al--nAA/Ill---11＼ ＼Illi--J/nn1··nAA actingonixhtomeanthatAsjmapsa511bspaceofXjintoXtforl三i,j三n.Example6.1.LetXandAbeMinExample21.Let(CJ)(z)三fJf(z),C2三Ch(WI(t)f)(z)三fMMf(z+t)(W2(t)f)(z)三ftf(z+t)·(Intuitively,WI(t)=etAC1,W2(t)=C2etA,whereetAistranslation,(etAf)(z)=f(z+t).)Thenitisstraightforwardtoshowthat(叭,W2)isa(C1,C2)existenceanduniquenessfamily,generatedbyA.Infact,theabstractCauchyproblem(1.1)hasauniquesolution,forailinitialdatainadenseset.Example6.2.SupposenENU{0}, A=(112), D(A)三D(G1)×[D(B)nD(G2)] whereD(G?)ED(B),Bisclosed,andGtgeneratesastrongly-continuoussemigro叩,fori=1,2.Then/Io飞(a)Ifn=1,thenthereexistsanexpomltidlybounded(o(←G2)-ljexi归ncefamilyforA,forssumdentlylarge.
C-ExistenceFamilies 305 (b)IftherepxistmεN,wEρ(GI)suchthat(ω-G1)-mBand(ω-G1)-mG2aleboundedjlienAibSCdlOω州lke川阳以创mihs"归阳t阳e创In1ωλ旧川川I盯mI口1i川ly川川'川Jj;a川1订I川aI川川e凹ω创In川l川tUhω川i血MaddlHl内1u1川InlK叫削(d巾削i扣Me叫d((huJ-飞寸?:T?1ο)γ-m!)-uniqueII叫aI11i队hIA.(c)If,inadditionto(bLetG.isholomorphic,fori=1,2,tdlhhl陀mefamilyisihlOdlOInmIn1Oωr叮Ipμ】才lhli比C. DetailsofthismaybcfoundiII[17]and[19].In[17],for(a)and(b),weEomtruettileexistenceanduniq1lonessfamiliesexplicitly,asfollows.FortJE0,let HW叫/＼0er.υ2(υS一G2才)-n/ acalculationshowsthalt ,dZ、、、.1,,yJ4,.E飞V,..Te∞f''1'。一-z＼tlf/n9·OGSJ'''飞rtnu/lt＼ATJ''1、马 forTS11日identlylargf＼2εXI×λ'2Fortilemiqllcmssfamily、definf.forz2iIID(G?),t主0,Z二(町,22),/(h1iοl)γ尸尸-→→?忖川川Vy川~A气4lHWfV/飞饥2纠(t)μ1.三lJ飞!飞()rt(,2/ SinceG2generatesas1rollglycontinuoussemigIOIlp,D(G?)isdense-01lIhypotilesobimplythatW2(t)isbom山d-FrillIS川、(t)maybeextendedtoastronglyωntinllO旧faIIIil〉ofbollIldedoperatorsOIlλ'1×λ'2·For(吟,weuseTileom1157,aMtilefactthat 1|G1BGTn|cγAC1=|oA;|sothat{zED(AC1)|ACIZEIm(C1)}isdense.Notethat(T-A)-11I。|-l(TJ1)-1(TJ1)叮B(s-G2)-nl(r-G)叮l(s-G2)-nj-lo(T-G2)-1(s-G2)-nj Remark6.3.Whenn=lJlmoperatorA,ofExample62,generatesaonce-integratedsemigro叩·ThisyieldsuniqueeXPO阳D(A2勺)=D(Gi)×D(G;幻).Example6.2(a)yieldsuniqueexponentiallyboundedsolutionsof(1.1),forinitialdatainD(G1)×D(Gi).Forn>1,Awill,ingeneral,haveemptyresolventset,hencewillnotgenerateanintegratedsemigroup-RemarkeA.Itisnotdimeulttogeneralizetheseexampleston×nmatrices,forarbitraryn,withthediagonalentriesgeneratingC-semigroupsorintegratedsemigroups-Hereisanexample.Example6.5.Supposeh+l,...Jn+1,N2,...,NmM1,...,Mn-IEN, GGOO/It--EIt--飞=A B13 B1.n＼......Bn-1月lOGn/
306 deLGtdbenjtls npuD(A)三D(GI)LD(GYLJAV、(1)D(G;y)ED(BtJ),for1三t<j三n.(2)3λfstichthat(r-Gs)一1exists,aM||(T-G‘)-1||is。(|r|k·),for1三t三叽lfr(T)>M(3)hεCsuchthat(s-GJN-BhJ(s-GJ)-NpisinL(XhXi),forl三t<J三?t.(Arl三0).(4)3sECsuchthat(s-Gs)-M-BhJ(s-Gj)Mpisbounded,forl<t<J三η,1TIr、ihkMl阳阳C创Inle1ih吕CdlOS川叫Jλ址tdihM)址ica叫tuihlereexistsaIIexpomltidlyboundedmild(s一王)42+艺LKJC-existencofamilyforA,where where C三iO(s-G2)-N20:|＼0(s-Gn)-Nm/ Tiici}roof(sw[17])i川OIvescalculating(T-A)-IC,showingthat||(r-J1)一IG||bourlE二二lk‘LfoIIK(r)811fucieIItl〉'large,thenappiyi吨TlleomII14.14.1TU、1lhl阳e5u川a川lI川a盯r咆吕E引1ul门川Inm11ul归川川5盯川i川I咔F1丁lr、lhl川叫l口《俨川mm、吁呐Inmlnl5.7iIlplaωofTIW旧时I14]4,givosmtilefollowi吨(mi19l).Example6.6..SIllyp(}fftBJ飞,...AL,Af1,...Mn-IEN, 「GIB13…B1月1.一·.a一|OG2·.:lti二=lIE·.·...…·u.1···Un-1.nlL...OGnJ D(A)三D(G1)j52D(G?), where(1)D(G?)ED(BU),for1<j三n(2)Gzgeneratesastronglycontinuousholomorphicsemigroup,forl三i三n.(3)ThereexistssECsuchthat(s-GJN·BtAs-Gj)-NzεB(Xhx‘),ibr1三i〈j三n.(NI三Oj(4)ThereexistsSECsuchthat(s-Gg)-M-Bu(s-Gj)Msisbounded,forI<t<j三n-ThenAisclosableandthereexistsanexponentiallyboundedholomorphic(s-A)-1C-existencefamilytbrA.Example6.7.InExample6.5,wecouldch∞seXj=LP(RN)(1三p三∞),GJequaltoaconstantcoemdentdiferentiaioperatorpj(D),whereD=(t(θ/θZI),...J(θ/δZN)),PJisanellipticmnconstantpolynomialsuchthat{Re(Pjh))|zεRN}isboundedaboveandBuequaltoalinearpartialdifemtialoperator,Bu=艺lα|主mJhαJJDα,ofarbitraryorder,wherehaJJisinhitelydiferentiable,withboundedderiva,thesisee[17]).Similarly,wemaychoose,inExample6.6,Gjequaltopj(D),wherepjisaneilipticno肌omtantpolynomialsuchthat{的(z)|zεRN}E(k-Vφ),forsomekεR,0三φ<?(suchasGjequaltotheLaplacian),XJ,Buasalreadystated(seei191).
C-ExistenceFamilies 307 REFERENCES lllw.Arendt,ResolventpositiveoperatomProc.LondonMUll-Soc-54(1987),321-349.i2iW.AreMLVectorvaluedLaplacetra川ihb∞r口mIn川I(1987η),327-352.[归川3叫lW.A加rme凹In1d们tandIHfi.IKb刷〈臼ωedlHlk阳e盯r口mImna圳n川mIn叽1h,I灿n川te喀grmatωCddS叫Odlu叫tυiOmOf门1V怡W/勺刨恤bωlhtω阳e盯rmrma川ih川In川Ite唔ψEFr卧ddii阮1岛U祉白r陀阳e创In川ItυiOInlsandapplications,in"IIltegro-DiferentialEquations,"Proc.Conf.Trento,1987,G.DaPrato,Ma.Ianmlli(edi),Pitman,toappear-[4lW.Arendt,F.NeubrandehandU.killotterbeck,InterpolationofS衍川e凹Iηm川Iη]咆i恒ErO叩Sani川I川ItωegratedS肥eImIη】i也grOtu1ps乌,SeImIn1e臼st阳erber川iCd}hltF1u1mInIlk〈tuiOInIalaIn1alwysihs,TUbinge11、Wintersemester,1988/89.[归5jhMf.Ba剖latb〉陆削a剖I肘andIHHfi.Emammi让rad,Snm】O∞Ot仙lh1ddiS"trdilb}u川tυiOInlS阳阳e凹Iη川ItυiOIn1i归nLP叮(Rnη),J.Math.An.Appl.70(1979),61一71.[6lA.V.BalakrishM叽Fraetiomlpowersofclosedoperatomandtllesemigroupsgemratedbytlle叽Pac.J.Math.10(1960),419437.[7]11.Beals,OntheabstractCauchyproblem,J.Func-An.10(1972),281299.[8lR.Beds,Semigro叩sandabstractGevreyspaces,J.Func-Al10(1972),300-308.[9jG.DaPrato,Semigr叩piresola巾zabili,RicercheMatt.15(1966),223-248.[10]G.DaPrato,12·semKruppiamliticiedeqmZionidjevoluziominLP,RicercheMatl6(1967),223249.[lljEI1.Davies,"One-ParameterSemigroupsJ'AcademicPros-、LoMOIl,1980.[l2lE.R.DaviesaIMiM.M.pa吨,TileCauchyproblemaIldageneralizatiolioffllθIIilltL31λ亏j山tlleoreI叽Proc.IJOMOIlMUll-Soc-55(l987),181-208.[l;ijIt.(ieLa111〉eIIfels,C-semigroupsaIldtlleCauchyproblem,J.VInic-All-,toappeal!14]I1.deLaubenfe15,Integratedsemigroups,C-semis-ro叩saIIdtlleabstractCauchylYIob-lcm,SemigroupForum,toappear-[15lR.deLaubeI巾ls,EntireSOl川OmoftlhleabstractCa剖1旧lh1JyFlmlb山lhh灿e创ωIη叽ItoaPPear[16lR.deLaubedels,C-semigro叩sandhigherorderabstractCauchyproblems,submittedtoJ.Math.An.andAPPI-[17lR.deLaubedelsExistenceanduniquenessibmiliesibrtheabstractCauchyproblem,Proc.LondonMath.Soc.,toappear-[18lR.deLaubedelsBoundedcommutingmultiplicativeperturbationsofgemratorsofstronglycontinuousgroups,submittedtoHoustonMath.J.[19lR.deIdaubenfelsHolomorphicC-existenceihrrlilies,submittedtoProc.LondonMUll-Soc-[20lR.deIdaubenklsandF.Neubrandehuh)=Au
308 deLatdbenfkis [28iJ.A.Goldstein,R.deIdaubeIIfeisandJ-T.saMefuhEquipartitionofemrgyonBa川cllspaces,pzepriz叫1§98).[29!K.ltASpectralmappingthωremsforeXpο阳spaces,SeznigroupForum38{1989〉?215·221.[301M.11iebehIntegratedsemigro叩5aMdiIIerentialoperatοmoIILP,DismtatioIhTub-ingen{1989).[31iM.HieberandH.kdlemanIhJutegratedsemigro叩s,jFmc.A184(1989),160180.i32jRilfugkes,SemzroψsofunboωdediimaroperatorsinBaMCiispace,'Yra瓜Am仁Math.Soc-230(1977),113一145.{33iTkato,"PerturhGOEIT}mzyfozLinearOpeMO队"Spmger,RewYork,1968.[34]J-L.Lions,Semi-groupesdistributions,Port.Matil-19(1960),141…164.i35jlMiy&d衍&,OnthegeneratorsofexponeMaiiyboundedC侧semigro叩s,Proc.JapanAcad-62Ser.A(1986),239一242.{36iLMiyaderAAgeneralization。fthelfille-Yosidaihωrem?prepriM(1988)·.[37!I.MWaderaandN.'ranal〈a,Exponentiallyboundedc·.semigroupsandgemrationofsemigroups,prepzin1(1988).[归38叫)Il.Mi均3yraderaandN.,TrhaM川nm1a1k矶a,EXlp川O阳groups,TokyoJ.Math.12,No.l(1989),99…115.[p39叫)R.Nag萨edl(μed.),J‘哈ne夺幡♂Parame创tω阳e盯rS轩切e凹Im川Iηli地g阳IIpμ}附sofP册ωiMtUiV刊eOpera川tOl184轧,1986,Springer唰Verlag-[40jR.Nagei,Towardsa"nIat盯theory"for川bmIMedοpe川ormtricmMathZ-201(1989),57-68.[41)R.Nagei,ThespectrumofuIIMIndedoperatormtricmwith川li-diagonaldomain,prepriM(1989).[42jF.Ne1uIb川problem,Pac-JMath.135(1988),111-155‘[μ43句)F.Ne1u1b川problems,SemigroupForum38(l989),233451.{44jF-PJeubranderAbstractellipticoperators,analyticinterpolatiοIIsemigroups,andLaplacetransibrmsofanalyticfunctionsFSemesterberichtFunktionalanalysis,Tubinger,Wintersemester1988/1989.[45!F.Ne1巾础derandR.deIA1巾nkls,LaplacetransformandregularityofC-semigroups,preprint(1990).{46!F-NeubranderandB.StratlbFractionalpowersofoperatorswithpolynomiallyb?mdedresoivent,SemesterberichtFUElMORalanalysis,TUbingexhWinteIJsemesterI§88/1§89.[47)LE.pay肘,"Improperlyposedproblemsinpartiddi仔erentialequations",SIAM,phiiadeiphia,Pa.1975.[48iA.pazy,"SemigroupsofLinearOperatorsandApplicationstoPartialDifrerentialEqua-tionsfSpringer,pfewYork,1983.[49lJ.T.sandefuhHigherorderabstractCauchyproblems,J.ofMath.An.aMAPPI-68(1§77),72§-742.[50iN.sam,LinearevolutionequationsintwoBaMChspa
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OnSomeSpectralPropertiesoftheStreamingOperatorwith岛follinedBoundaryConditions G.LAURODepartmentofAppliedMathematics,SchoolofEngineering,Florence,ItalyALDOBELLENLMORANTEDepartmentofCivilEngineering,SchoolofEngineer,ing,Florence,Italy 1.INTRODUCTION SOnm1eSpe倪Cthm川Ea川1dlp伊r叫O1p〉咒m川e创盯1r川. To=叮￡(fleestrmli川 (la) T=To-vEI(811It-ttIlli吨withcaptureoperator) 、‘../10·EEA4,,.、飞 subjecttoMaxwell'sboundaryconditions: O川 (2a) 3Il
3I2 LauroandBelleni,儿fOrante VWEQ、‘.,,vda,,,‘、、P'且VMir-JORAU9-+、‘..,,vea''l、、俨霄..or--、、..,,vda,,,.、俨I ,yE(斗,0) (2b) havebeenrecently吼叫ied[l],[2],[3].In(1)and(匀,f=f(XJ)istileparticledensity,V>Oistheparticlespeed,2>OisacapturecrosssectioIi,XE(-aA)isthepositionvariableandyE(-1,1)isthecosineoftheanglebetweenthevelocityaIKlthepositivex-axis.MoreoverJp1646益严bω〉OaretheaccomodationcoffnciclltsofthespecularrenectioIlandofthediffusiverenectioIhrespectively.IfweintroducetheBanttcllspace: X1二阶M)X(i,l)],|lfl|=jdxjlf(XJ)|dy antiifwedennetiledoIIlttillofTandToMfollows: D(T)={f:ftXl,Tikxl,fsatisfiesthebι(2)},D(To)=D(T) tlienitisknownthattIIltlcrtilehypotedsρ+6三l(dissipativeorconservativeboundaryconditions)Toistdlhl阳egF伊emI川[2勾].COr附lp〉泊川O创I叫i阳n鸣gUly,Tgeneratesapositivesemigmup{expoT)=exp(-vst)exp(Tot),t主0},[4],[51‘ SomedifficultiesttriseifjJ+6>lsinceitseemsthatonecaIlonlyprovethattheresolventoperatorR(λ,To)isbollli山diIIXUVλ>0.OntheotherilaIitl,Illit-ctJIltli1ioliρ+币>limpliesthattheboundarywallsmultiplyparticles.IIIfact,(2a)letuiδtotllefollowingexpressionfortilenetnuxofpartiElmatthewallx=-a: Ojvy川句=-[l-川]jvlf|W)dy' (3) Hellce,ifρ+6>l,IliltJIleti1llxispositivealldthismeansthatthewallatx=-aIIlultiplyparticles.Ofcounth,tisiItlilarresultholdsfortilewallatx=+a.Inamrlltpaper,[叫,ll川ll|lri仇ωMυ川』l川lhOwi川I川l8n川lK山川〈Oω}才lHlifnife盯叫!吃d【diMaXwedlHlr'、Slb〉OmdaryCωOMiutihtO}mICOInlsidered: +aO+a川川)=小一」JJ刷(仅旷川X扩均,丁)f叫(川X矿扎ιh,＼斗vw'「d呐'γ训y川)〈d!矿ι川+川叫2纠叶6.a田斗1-a
SpectralPropertiesoftheStreamingOperator3I3 +al+aq阳川a趴叫M'Jωy川)=小μ户J川x扩均,丁H)川f川)d川6jd叫~川扩仆ω川'JJ川yγ川,丁)dx矿,,J冽付叫y严叫阶川f叫仆咐(←μ-」1L附,-4aO.a whereh-,il+,k-,k+,aregivennon-negativefunctionsoftf(-A+a). Remarkl.ForiMaMe,wemayassumethath-(x')=k-(x')=OVxY(-a+AaA)andh+(x')=lt+(x')=OVXY(均十a-Aa),withAa《a.IfAaisoftileorderofIliagllitildeofsomemeanfreepaths,thenthespecularrenectionandthediffusionreflectionphenomenatakeplacewithintiletwoboundarylayers(-a,-a+Aa)aM(+a-Aa,+a).食 In[6],tilefollowingtlMortJIllwasproved- TlttOTEmLlfO三li一(XLit一(x),il+(x),k+(x)三ilVXE[-A+al andifthe crosssectionSischosen币1lrhihatE主(ρ+6)ll,thenthestreamingoperatorT,subjecttotileIIiollinedbolllldal-yconditions(4),generatesapositivecontractionsemigrouponxl{exp(tTLt主OLCOIlml〉OMillgly,Togemmtesapositivesemigroup{expoT。)=exp(vEt)expoT),t主O}. Remark2.BecauseoftlleresultsofTheoreml,semigrouptechniquescanbeusedtostudylineal-and110IIlineartransportproblemseveniftileboundarywallsareparticlemultiplying-,t IIlSection20fthislMl〉t汀,wetOIMidertilestreamingoperatorTsubjecttothemollifiedbo川dary盯Oliditiom(4),aMproveti川人=-vZisaneigenvalueofT.IIISectiom3aM4,weshowtl川tileinterval(-vZ,Oiandthestrip{A:-vE<ReA三0}ofthecomplexI〉la川lmevoidhItemctionwiththepointspectr11日1,Pσ(T),ofT.Finally,iIISertiolltweconsidertlitecaseinwhichtheboundarywallsmultiplyparticles. 2.CONSEItVATIVEItouNDARYCONDITIONS,PARTI InTIh1MrmtCω?泪Or阳tρ(T)〉{A:ReA>O},ρ(T。)3{入:ReA>vZ}.Consequently,σ(T)C{人:ReA50}andσ(To)C{A:lteA三vELwhereσ(17)istilespectrumofTandσ(T。)isthe
3I4 LauroandBelieniL儿fOranfe spectrumofTo,闷,[51.Inthissection,weshallstudysomepropertiesofthepointspectrumPσ(TLundertheassumi〉tionthattitleMIll}tlaryconditions(4)areconservative,see(9a).Withthis aiminmind,weconsidertiltEequation: (AI-T)f=O (5) wheretileunknownfIIillstbesoughtiIID(T)andwhere入isarealparameter-Notethat,fromIIOWOIl,tiledollltkilldFIisdefinedωfollows: 、、..,F吁A,,..、、D一一、‘-z''OFi,,.‘飞D、,‘,,、‘..,,A哇4,..、FiLν.be‘EUQdC俨uc36'uauup---'Axrtp'且IA电Axr,、俨...俨,..,-aEE、一一、‘,,,,T,,‘‘、D Thegeneralsolutionof(5)readsasfollows: λ(tL+x)f(XJ)=b+(y)exl〉[-一亏了一],yf(0,i) 人(a-x)f(XJ)=b一(yhBXi〉[+「亏「-l,yr(-l,0) 、‘,,,,nhu,,..‘、 wheretilefunctionsb+(y),l}一(y)tmtobedeterminedbyusingconditions(4)andwhereλ=A+vE,Areal-Ifwes山stit川e(6)into(4a)aM(4lE)aMifwedefinetilefollowingfunctions: 们(y)=vyl}+(y),jj2(y)=vyb一(-y) VYE(0,1), tileIlweget: 们(y)=川I1(入,yWY)+均叫j户IK州〈O yf(0,l) 州)=川、y)dlb)十均叫jhμKh〈乌2仙,丁vMWW)M叭W州3内州削1川山J(U旷川yγ,O (7) yf(0,l) where X,,、])-4-y17t··、''l飞刷-IA-[、,,XHi飞、、‘,,,x,ge--、lap--Jn咱二、l/VVM-A,,.‘‘、'AVEEE·冒FA
SpectralPropertiesoftheStreamingOperator 3I5 XBG、‘..,,,X一+-ya-v,,EE‘飞'一人一、,xe、‘.,,,x4,,,、+、-aaf|J4一一、、..,,,vu飞AJ,.,、、ndH (8a) XEG、‘..,,.XFa一VI,,...飞--A-nrxρ-L、‘.,,,x,,,.‘、'Ka「|ld4一一、、.,FVW、A,,-E飞唱AK (8b) XBGX一十一ya-v-A一Xρlvx+-Lha「|J4一一vd、A町,ιK withyE(0,l). NotethatkPσ(T),i.e.,人isaIleige川alueofTifandonlyifsystem(7)iωaωIItrivialsolution. Inwhatfollows,weshalloftenusetiletwoquantities: Xi=ρIIi+6Kl i=lJ (9) wherewrllJ叭'tytlriiυ1t飞|willlλhtil-tlievttlliestakenbyII1,II2,KUK2atA=-vE,i.e.,atA=0,sec(8). RETYlαrk3.TileconstantsxlaIltlX2haveaninterestingphysicalmeaning,(seealsothediscussionleadingto(:1).IIIfact,tilenetnuxofparticlesenteringtheslabatx=-acallbeobtaimdfrom(4a)ωfollows: lOj叩川 Oa+jd叫ρ4个jh户仆ih扎lL-川O-4a TIe肌e,ifil-aMlLalezrmoutsideoftlle"thin"bollMarylayer(-a,-a+Aa),wehave:
3l6 LauroandBelleni·儿fOrante WeCOIleludeLhatthewdlatx=-aIIlaybeconsideredconservativeifX1=l,dissipatliveifxl<l、tilltll川rtit-lelllliltiplyingifxl>l.Ofcourse,similarresultsholdfortilewallatx=+JL·Mort?iIIgeneral,tiletwowallscanbesaidgloballyconservativeifxlk2=l,globallydissipativeifxlX2〈l,andgloballymultiplyingifxlX2>i.,t Assumenowthattiletwobolllkittrywallsaregiobailyconservative: LIX2二l (9a) TileIIWtsllttve: Tftroy、εη12.人=-vEtIEσ(T)alldA=OEPσ(T。),ifaIKionlyifcondition(9a)issatisfied. Proof:IfwedeHIIC bfhi川〉i二l2O thcIhsy8tlfIIl(7),withλ二,vE,|)thtWIltJS: 81=X102 β2=14231 (lOa) antialso 81=(11飞2)Jj1 Jj2=(Xlk2)jj2 'EOO-- SinceWa)llol山问'川HIlllil川ioll,叶steIll(10loadmitsIIOIltrivialsolutions.Thus,λ=-vEElEσ(T)alitlA二OtlEσ(To).Convemly,i汀fWeaf怡忧州削、世悦训t吕剖甘讥lu川imntrivialsolutionf(XJ)tlulti讯tidiestilebollMarycOMitiom(4).Asaconseq时Ilce,system(7)ilωaIlolltriUJIlωliltioII(ih(y),β2(y)),andsystem(1013)admitsasolution(β1,β2)并0.Wcωlichith、thal(hb)ulustbeequaltol.Remark1.lfpkt)iioltl币,l|irIl入二OfiEσ(T。)aIId80tileevolutionproblemrelativetothefrwstINIHiligopt-I川。I'I。川tlitlMIIlollin川lb-t.(4a)and(4b)admitsastationarysoilltiOIl-食
3I8 LGur-oandBelleniL儿fOrante withI川m||||y: V6·dlp』、IFVdi,,‘、呵,ιaμ+vu,,..、'i叫μ,EdE·、lf|do--Y「llBEll-4-AnJ-4μ4μ「Illi--lL一一yaμ' antiweprovetliefollowiligtiiCOINIl. Theorem3.IfcoluluiiOIIWa)holds,thentileoperator(I-ρHλ;lisbounded. etum--,,.、li&ELWVUP--a-4飞VO--utp飞俨Hit--uolgJetuIι卜'oorp 寸-ll·-J唱A。r叮,‘「ll|iL-呵,&-咆ii-2-oy--、八。' (14) seetiledefinitionofiIA·wrila、efIOHI(14)aM(11) YJJW一问<-Y、AOY/,,‘飞 (15) wh川 Byusingtilc川lll180i'llit-tJlulll:1,wtfEtuiwriteequation(l2)asfollows: VMr·、VMJ,.‘、、叶μ、‘BaF‘险,町'J'-E飞IA,、、if--JO‘vν电、nvOF--EEA-、‘,,,,1AE,aoy曹··A,,..、一-dμ 、‘..,FPO--EA,,..、 Now,let、stit-「iIltB A=jIMY)州l〉O (17) tileI1,ifwcapplytllt-olwrJI10lkλtobothsidesof(16)andintegratewithrespecttoy,weobtain: i1=SAlif (18)
320 LGur-oandBellenL儿forante 1-ρII2(ρIil+6kl)>6K21,ρIIl(ρII2+6K2)>61〈1 atanyAE(-vEOl. Itfollowsthat (I-SA)11>(I-SA)21'(I-SA)22>(I事SA)12andSO φ(人)=tlet(l-5人)=(l-SA)11(I-S入)22-(I-S入)12(ImSA)21>OVAE(-vE,Ol Weco川llitivtli川叫ll儿li川 β(y)=OVyf(0,l) ρLIOμ飞-EIWHb飞电、)t}..、···-a飞-EIUL,,‘·冒LUult-4俨飞μ飞vv‘-udil--T TltforEm4.UMertllt-UHUlliptiou(Ob),tileoperatorTllasnoeigenvaluesbelongingtotileillterul(-vE,OL川ldTohMωeigenvaluesin(0,v21. REmark6. 'rlit-lWIll1h。l-FIllfol-eIIi4tireiIiagreementwiththephysicalintuition- Indeed,ii、tilthfIWWtviillilig叩flatoI1、ollatitileeigenvalue人oE(vZ,olthenthe"iliatioliTof-人。f二()川llltili川luttωlitrivialsoltltioIlftD(T。),incontrastwiththefactthatasttitioliJIryωlll1ioIICJKIlllotexistfortileparticletransportprobleminamaterial,whichc叫}tIll-ωl〉til-llich-8ttIitlwhichisboundedbytwoconservativewalls.,t VWLυir飞。飞&EUl--‘4ZLU--LQO'bu也UQU、B‘,,'bQU,,-E‘、俨'且·1,飞。-i‘EIUQUAHI、‘飞FI)(‘飞、ir、ptuJQH-E飞..4·飞-iI『iEKF'nu17ρLR X155l,121二l 食 (9c) 3.CON忖l5liVAIlVIli()UNOAltYUONDrHOlys,尸AItTIIl Inthissectionwedltilltl州lullftliAttilecondition(9b)holdsandweshalliIIves1igatewi川llerorI川tilt-6trip{人:-vEJ<Reλ三O}co川ains(coml山x)eigenvaluesofT.IfA二人1+iA2withAl=lit、人f(-vE,OLtiltj?IIWehavefmI1(8)(withA=人+vZ=(人1+vE)+i人2)
SpectralPropertiesoftheStreamingOperator 32I |Hi(入,y)|三IIi'|Ki(λ,y)|三I句,VReAE(-vz,0] (22) Since(9b)aM(22)implythat ρ2|Ihll2|<p2日1日2=(l<l、VReAE(-vz,oi (seeRemark5),C【l川Ilion(lG)Et山stillbederivedfrom(12).ComidermwtlmeleIUNItsSijoftllematrixSAdefiMiby(19).Wehave: ρH6l〈+6li|S11|+|S21|<21一-2=ll-ρ4H1112 (23a) jJH1币li2+6I〈1|S12|+|S22|<?一一=ll-ρ4H1H2 (23b) wheretlMHlllalith-sfυli川ViIOIIl(Ob).CO川eEllMIltiy, ||Sλi?||R2<||川||i12 (24) 勺ι-mj4,J+唱EAA、J一-叮4、1《JμeF··OL--nw WeCOMMetilthtβ=Ohtllflu川川I川川川I川川l川i问【q叩lμl川SOlh阳u川ltUiOnof(υl8创)and,comeflmntly,β(y)=Oistileuniquesolutionof(lG)-Thllf九westatetllcfollowingtheorem: TlttOTE川5.liwh)liolth,1llfllTli川IIO(complex)eigenvaluebelongingtothestrip{λ:-vE<ItrA三()). 5.PARTiCLEMIMJIH3IJYINGROUNDARYCONDITIONS InthisSectionweti忖SIllIlflllf111liefollowingrelationshold:
322 LauroandBelleni.儿forante ··A>叮,‘X'AV4'luMV‘、‘E···>一句,‘VA叫‘EE-->-唱AVA (90 Notethat(9c)implythatatleastoIleoftlletwoboundarywallsmultipliesparticles.飞VealsoSIll}imsethattilt-si〉ffll!ttrrentTtioIIphenomenondoesnotmultiplyparticles: ρiil<i‘ρil2<l (9d) NotethatWEl)implythat ρ2HIll2<lVλt(-v旦,ol (25) 、‘..,,内,,"l,,..‘、Baw--OFEP'··,飞ρ飞V·lFAeta飞}h-leet飞、lEFρυi(0·EU、毡,,‘-lr·、eOQU,dna RrYJIGTA-8.ifwr)JKIltlwt|)lioltitjlit-IIλ二·vEisIIotaIleige川、FailleofTaIIEiA=OisIlotalirigrlivtillit·υllOlbrrIlitvllhlll2):Illit-sttitioIiaryparticletransportproblemTof=0,ffD(1、。)felllliotilavealiontrivialsolutionbecauseatleastoneoftileboundarywallsmultil〉lit·目ptil-tit-ius.,t Let山IlOWCOIINKlelilwfUNCtionφ(人)=det(IESA)whereSAisgivenby(l!})andwhere人E{-vE,Ol.飞飞FVll川thhVIII(20)aM(9c): φ(-vE二)50 (26) OIItlubotherhaIiti,if入二()‘lllriiwrob1λilifroIII(8): 川1i(OJ)+风(OJ)三i-up(斗斗<1,vyf(0,l),i=l,2 where225(ρ+币)li,wrTilt-oITIlll. lt,,‘}}tt11)()(t,,‘)、-a,,dt叶,-(、vu·μ飞1,.、μ飞飞bloc φ(0)>() (28) 飞VeIIOWl〉rovetilefollo叭'iIigleIlulla:
SpectralPropertiesoftheStreamingOperator 323 LEmma1.φ(λ)=det(l-SA)isacoutimousfunctionofλELvz,OL ProofGiven人,人'ELvE,01,from(20)and(8),simplecalculationsshowthat: qFM·E·--一一.,u·l、λ、八···J守,<一···J)IA们乃Idt飞···J、、..,F、八户、uw··A,,..‘、 whereγijarepositiveCOIlsIJIlltJ亏,fol-exampleγ12=6E4a2.口 Relations(26),(28)JKll〈lLrIIlliltLltillowωtostatetilefollowingtheorem: TltEOTEm6.If(Oc)tLIltlWtl)hold,thenaAE(-vE,0)exists,suchthatφ(入)=0. SuchavalueofAisalleigrIIvalueofT. REYYIGTK9.l1lollo川ilt}IIl11lfOI川116that入。=入+vEE(0,vE)isaneigenvalueofTo-Thus,tlleHIll儿!ioIiTof,入。f==OllasamutrivialsolutionffD(T。).Weconcludethattilecorrespondillgptl11iclctrUISl〉ortproblemhEL弓astationarysolution,duetothebalancebetweentilthl}JIrtlich矗llllduplicationofthewallsandtileparticlecaptureofthematerial,withacaptllICt-losssectionEc=人。/v.食 6.COlyCLUDINUItEMAIiiis IftdIhlellIω,C∞Ome凹r川川.飞、V川ya川川川ttU,ji忖VWtfA(X11b2=l门)'thull-vSEPσ(T)aMOEPσ(T。).AsToisthefreestreamingυpeltLiol,1lieuilirstatiollaIyparticletransportproblem·vyθf/θx=0,ffD(To),iiasaIlolllriLPitti(l}ositivc)solution-ThisresultisinagreementwiththephysicalintuitionbNAust-l〉articlesareneitherproducednorcapturedbytheboundarywallsorbythematerialoftiltBhomogeneousslabunderconsideration-Ifeachoftllflwo认TIll币hEO川ervative(Xl=X2=l),thentherearenoeigenvaluesofTobelongingtoIlk-川lril】{人:O<ReA三vZ}oftllecomplexplane.Inparticular,Pσ(To)门(0,v汇]=OallElthiscallbceasilyuMeMoodoIIaphysicalgroundbyconsideringaglobJtlhtliJLUffoftllt、particlesilltheslab-IftllebollM川ywtillhlllululdyparticles(h主1,12主lwithUX2>l),thenTohasatleastjall咆rllullllt·lwlongingtotlleinterval(0,vELFIllei】hydralmeaningofthisresultwaSElicussmliIIRtPIlitilt49.
324 LauroandBellenL儿fOrante Alitlleprecedillgresultsholdtinderthe"boundarylayer"assumptionofRemark1.Inthiscase,ofcourse,tillecalculationsaremuchsimpler.WefinallyobservethatafurtherpaperwillbedevotedtocompletetheanalysisofthespectrumofthetransportoperatorT. AKNO飞VLEDGEMENTS Thispalm-waspIt、thlilt-tlJtlthr"2IIElIIlI,m-nationalConferenceonTrendsinSemigroup Theol-ytLIIEiEvollltioIilklllMioIIJDelft,September1989.Theauthorswouldliketothanktileorganizersoflluhmeeting.ThisworkwasperformedundertheauspicesofMinisterodell'IJIliversita'edellaRicercaScieIItineaeTeenologicAItaly. REFERENc-ES 1L.A.iB1如ted32.(;.Boruolialkl凡1、otttlO,AIIIVIIICongr.Naz.AIAfETA,tlol.1,3934398,Tortno(l986)3.W.Greenbel-get爪,Houndarv卫丛丛主Problems坦A草草旦EL旦旦且江工且旦旦,Btri-hauser,ljωrl(l987)4.T.kato,Peltllr1〉titioI1工监且工鱼zLU旦旦Operators,Springer-Verlag,BEThn(1966)5.A.Bclleni--Morallte,A且且且旦lsemigrollPS型ldEvolutionEquations,Clarendonprrss,017ford(lO79)6.A.Be叫li仆e盯川川In川l川i-MOωI川川tλ川川kuI川l1忖tr.'7T1￥r「dlht川川7川阳ilρjtlp』川f"tυ1r、f1.lNg.{'IIUlt?lJtrlItzr(l989)
OnaNonlinearHyperbolicIntegrodifrerentiaIEquationwithSingularKernel STIG.OLOFLONDENInstituteofMathematics,HelsinkiUniversityofTechnology,Esp00,Finland 1.INTRODUCTION Theequation咐,z)-fh川SJM=川,伦0,zd;也(0,z)=uo(功, (V) whereα(t)ispositiveinsomesense,comeetsnonlinearp町abolicmdnonline缸hyperbolicproblems.Ifα(t)EELthenEq.(V)isnonlineuhyperbolic;ifα(t)dtconsistsonlyofapointmassattheorigin,thenEq.(V)isnonlinearparabolic.Intheiatemediatecase,whereα(t)ispositivemd,say,decreasing,coavexandinsomesensesingularattheorigin,。nemayexpectsolutionscombizlingfeaturesofboththeextremecases.Inthelinearcase,whereσ(u)=Ku,thishasbeenestablishedingreatdetail.Themoresingularthekernelisattheorigin,themoresmoothingout 325
326 Londen ofinitialconditionsdoesthesolutionpresent-SeeDeschmdGrimmer[3],Erusa,NohelmdRenardy[7iandthereferencesmentionedtherein,ErmaandRenardyi8],pdss[15],andRenardyi161.Inthenonlinearcase(whereoneassumes,atleast,thatσissumdentlysmoothandsatidesd(z)〉0,zεR)theresultsonclassicalsoldoasuemainlyoftheetypes,i.e.,local(intime)dStenceforlargedata,globalexistenceforsm41dataandresultsonthedevelopmentofsingularitiesofsolutionsf。rcertaininitialdatainthecωewhereaissufEdentlyregularattheoriga-Theequation(V)was鱼rstcoadderedbyMacCmy[12laadlaterbyDaiermosmdNohelill.Theire对StenceresultswereimprovedbyStdms[19iwhodemonstratedthatsumdentconditionsonthekernelforlocal(largedata)andglobal(mddata)e对Stence町e,respectively, α"εLL(R+),α(0)〉0, and αofstrongpositivetype;α',α"εLZ(R+). Obviously,lessassumptionsonthesizeofthederivativesofαdowsetupsclosertotheparaboliccaseandshouldBotmaketheexistencequestionmoredimcult-However,themoresingularthekernelis,thegreaterarethetechnicalproblemsinvolvedintheproofs.Infact,evenlocalexistenceforlargedatainthecasewhereα(0+)=∞ord(0+)=-∞,ismintricatematter.Theequation(V)isap缸tic111町Cωeof r州,z))ffdh)ψ(uaMzdhfMt主0,(W) whichhasbeenstudiedinseveralrecentpapers-(Includedintheproble皿(W)areinitialconditionsand,ifzisrestrictedtoaboundedinterval,someboundaryconditions.)AmajormotivationforthestudyofEq.(W)isthefactthatthisequationoccursinviscoelasticity,seeErusa,NohelandRenardymmdbnardy,EnmandNohel[181.htheseapplications,both￠and飞baretakenmonotonestrictlyincreasingmdueassumedtobesu纽ciezitlysmooth.ThusEq.(V)maybeviewedasabstmodelofthetimebehaviorofanuaboundedbarofamaIerialwithmemory.
HyperbolicInfqrodub-entialEquation 327 ThestudiesonEq.(W)includeDdemosmdNohel[刻,ErusaandRe-nardyPLHOI-hi9l,wherebothlocalandglobale对Stenceresultsareob-tained,itisassumedthat q,"咱inu--.··nu〉-、‘.,,...,,‘、,。...、‘..,,咱i,,..、、、、.,,,+R,,..、、-ALι」α,。 and(forglobalexistence)that ￠'-b(0)ψ'〉0. Hereb(t)乞f-A中d(s)ds.Thusthislastconationrequres,ifonehasEq.(V)inmind(whereaisgivenad￠=ψ),thatα(∞)〉0.Ia[10],alocalexistenceresultisobtainedforthecase bεL1(R+),(-04bO)主0,i=0,1,2,3. IntherecentpaperRenardy[17ibothlocalandglobalexistenceresults缸eprovedforagenerdizationofEq.(W).Theseresultsdowd￠LL;i.e.,α(0+)=∞inEq.(V)isnotexcluded.Instead,thetransformconation阶b(ω)|;三C|Sb(ω)|,forωεRmdmmeconstantC,isimposed.(Again,b(t)=-L∞d(s)dsjFortheglobale对Stenceresult(inthenotationofEq.(V)),theconditionα(∞)〉Oappeustobeessential.Wehaverecentlybeenabletoshowthat α'εL1(R+),aisdstrongpositivetype, ares11伍dentconditionsonthekemdfor。btaiainggi。baiexistenceofsolutionsofEq.(V)f。rsmdldata-Weimposemc。nations。nα飞andd(0+)=一∞,α(∞)=Oarenotexcluded.Neitheristhekernelrequiredtobemonotoneinanysense.WithdεLI(R+)replacedbydELL(R+),wecmgivealocaiexistenceresultfordataofarbitrazysize.Ourmethoddproofuseskernels句,uniformlyofstmngpositivetype,thatapproximatethegivenkernel,andmodi负edversimsoftheenergyestimatesdevelopedinStafans[191.ItisofinteresttocompamthepresentTheorem2.lwitharesultconcern-ingghbalweaksolutionsof问叫咐(0队Mtι'u叫(0'z叫)=u均o(归z)εH;(仰0'J1盯). (Vb)
328 Londen Wehave,Loadeni11,Coronary斗, Theorem1.1.Let αεc2(0,∞)nc[0,∞),α(t)>0,(-lydOO)三0,i=0,13;t〉0,吨护fool号Ltd(T))=∞,(1.1)αsstunethdσiscontinuous,monotone7BondeCTeαsingpandletfoTsomecon-stα71tsλ1,入2 |σ(z)|三λ102|+1),M(z)三λ2(22-1),zεR. Finallusupposethαt fεACI。c(R+;L2(0,1)). ThentheTeezistsUsuchthat UεLZ二(R+;EJ(0,1)),也tεLZ二(R+;L2(OJ)),tLttεLL(R+;H-1(0,1)), α叫suchthatUsatidesEq.(民). Itisseenthat(1.1)(roughlyeqUvalenttod(0+)=一∞)givesusglobalexistenceofweaksolutionsforlugedata.hviewofTheorem2.landourlocalexistenceresult(aadoverlookingthefactthatinTheorem1.1,gε(0,1)whereasinourpresentanalysiszεR)itisachallengingproblemtoanalyzehowsmooththesolutionsofTheorem1.1缸e,饵,alternatively,whetherandhowthelocalsolutionsobtainedundertheassumptionsofTheorem2.1,butwithlargedata,bredd。wnif(1.1)holds.FurtherresultsonweaksolutionshavebeenobtainedbyEagler[41(onEq.(W))andbybhel,RogersandTZamas[141(onEq.(V)). 2.SUMMARYOFRESULTS OurmanresultconcernsEq.(V)inthecasewherethedataaresmdLWeshowthatifαisofstrongpositivetype,withdεL1(R+),thenEq.(V)hasasolutionthatexistsforallt主0.Noassumptionsaxemadeona",
HyperbolicIntqrodwerentialEquation 329 andd(0+)=一∞isnotexcluded.Thesolutionobtainedissmoothinthesensethat(2.10),(2.12)aresatiSEed-More。ver,thesecondandthirdorderderivativesaresmdlatinhityinthesenseUveaby(2.11),(2.12).Thesymbolu(L勺,pε[1,∞],归ndsfortheclassoffunctionsfOAdehedfort主0,zεR,satisfytng||f(t,·)||22=j二|f|2dz〈∞a.e.onR+,andsudthat||f(t,·)||22isintegablewithrespecttotoverR+. Theorem2.1.Let αεAChe(R+),α'εL1(R+), (2.1)(2.2) αndGSStATrzethat aisofstmzgpositimtupe. (2.3) Let σεcs(R),σ(0)=0,σ'(0)〉0, (2.4) assumethdtheinitidhTactionuosatides uomuon,也0222εL2(R), (2.5) mTiteUI(z)=f(0,z)andsupposethat 句,也12,也1"εL2(R). (2.6) Assumethatf=f1+f2+fspmhe俨e fIεL∞(L勺,hzε(LInL∞)(L勺,Assε(L2nL∞)(L勺,f1222εL2(L2),fu,fztz,f1tszεLL(L2), (2.7) f2,f2t,f22,f2tmf222,f2tzzεL2(L2),fs,fsz,f322εL∞(L2),fshfsu,fstmεLZ(L2). (2.8)(2.9) hdditto叽扩α(∞)=Opthenleth=0.lytheLP-nomsofuo,也1,元,f2,hmdthebdedmumslistedinfs.5/-fz.9jαTesumdentlusmαllpthenthemegistsaglobalsolutionuofEq.(V)suchthat uhUhUtz,也"'也tzmU222εL∞(L勺, (2.10)
330Londen 、‘,,J噜i咱iq,",,.‘飞、‘E,,向4L,,.‘、qaLζ」zzzuzzez'"uzzuzaz-u tLtt-ft,也ttz-ftuht-ftt-d(t)σ(uozhε(L2nL∞)(L2).(2.12) OurproofmaybeoutlinedUfollows.First,wereplaceαbyasmoothkernelαhhavtngthesa皿ecoastmtofstrongpositivity臼α,andsuchthat句→αinasUtablesenseash→∞.(SeeLemma3.1.)PreviousresultsallowustoconcludethattheequationwiththeapproximatingkernelαhhasasolutionUh-Next,weshowthattusa,tisitescertainbounds,uaiformlyinh.Thefactthat句hasthesameconstantofstrongpositivityasαiscrucialforthisstep-Toobtaintheseboundsweproceedωintheproofof[19,Theorem2!-However,certainchangeshavetobeintroducedsincewemakeaoassumptionsond.OnceudforZnboundsonuMJhJMz,tugs,uht"'tLKzgzhavebeenestablished,onemayletk→∞andobtain问→u,whereUsolvesEq.(V).Ofcou风仕om(2.10)一(2.12)andEq.(V)。nemayobtainfurtherresultsontheasymptoticsizeofthederivativesofu-Wereferthereadertoi19lforsuchstatements.Theprocedureoutlinedabovecmbeusedtoobtainalocalexistenceresultforlugedata.TheglobalconationdεL1(R+)isnowreplacedbydεLIoc(R+);ag回民noassumptionsaremadeondmdd(0+)=一∞isnotexcluded.Intheproofofthelocaiexistenceresultwereplacethegivenkernelαbysmoothappr创mahgkernelsαhhavtngthesameconstantofstrongpositivityasα.TheappmximatedequationhasauniquelocalsolutionUh-Next,weprovethatthesamederivativesoftuthatwelistedabovehaveuniformlyboundedL∞((OpT);L2)-normsforsomeT〉0.Toobtaintheseboundsweapplythesame(dthoughsomewhatmpmed)町gumeatsasintheproofofTheorem2.1.Lettingh→∞weobtainuh→钮,whereUsolvesEq.(V)on(or)·Fiadywesh。wthatiftheL∞((0,To);L2)-normoft￡remainboundedonthemuimdinte叫ofmtem(0,町,thenTb=∞.Theproofofthisrequiressomeadditionalaaaiysis-Thecompleteproofswinappearelsewhere.
HyperbolicInfqrodUKrentialEquation 33l 3.AUXILIARYLEMMAS TheproofofTheorem2.lreliesonmapproximationofthegivenkernelα(wMchisofstrongpositivetype)bykernelsαhthataresmoothuptotheoriginandareofstrongpositivetypewiththesameconstantqωα.Below,weformulateandprovethelemmaneeded.Thesecondlem皿aprovtdesacon-venientestimatefortheevaluationofintegralsofthetypej;￠(s)(hv)(s)ds-Theuseofthislemmaisakeystepinavoidingmyassumptiononα".Althoughitisusedonlyinthescalarcase,weformulatetheapproximationLemma3.1for(complexand)matrix-valuedkernels.TodotMs,wemedtoreedsomenotation.Let(·,·)denotesomeinnerproductoncn.Ann×nmatrixAissaidtobepositive,denotedA兰0,if(旬,Au)25Ofordvectors。εcn.DenotetheadjointofAbyA·.Thematrix捉A=j(A+A·)iscdledtheredputofA,andthematrixSA=去(A-A')theimaginarypartofAThusA=捉A+iSA.Amatrix-valuedmeasureαthatis鱼niteonJCRissaidtobepositiveifα(E)兰OforeveryBorelsetECJ.AfunctionaεLioc(R+;cn×π)issaidtobeofpositivetypeifforeveryvεP(R;cn)withcompactsuppodonehas nu〉-AEWJU』EWV*αa7·myrtIRm沉 (3.1) AfunctionaεLIoc(R+;cn×n)issaidt。be。fstrongpositivetypeiftheree对stsacomtmtq〉Oforwhichthefunctionα(t)-qe-tIisofpositivetype.LetαεLioc(R+;cnm)satisfyh+fdh(t)ldt〈∞forde〉0.Thenthefollowingc。aditionsueeq1dvBlent-(i)αisofpositivetype,(且)况&(z)兰Of。r提z〉0,(iii)Hminfz→衍,如〉。提&(z)兰OforTεRaadEmiaflz|→∞,如〉0捉&(z)兰0.Obviously,αisofstrongpositivetypewithconstantqif(ii)or(iii)holds时thm(z)即l叫by咋(z)-q(1+z)叮Forfurtherpropertiesofpositivematricesandfunctions(andmeasures)ofpodivetype,seeGripenberg,LOMenandStafans[5,Ch.16,Sections24.
332Londen Lemma3.1.Assumethd αεAClodR+;cnm), α'εL1(R+;cn×n), αndlet αbeofstmngpositityetupettyithconstαntq〉0. Thenthemezist{α叶江1sαtishing αhεC∞(R+;cnxn), suplMLl|L1(R+)<∞,α;εL1(R+;cn×勺,hαhisofstTongpositityetgpetuithconstαntq〉0, GTzdsuchthatfoTk→∞, 问(t)→α(t)mifomluonY, 公→dinL1(R+;cnxn). M07·eotye7·,α-ahisofpositityetupefoTallk. ProofofLemma3.1.WithoutlossofgeneEdity,takeα(∞)=Oandq=1.Noteth剖归cedεL1(R+),the(distribution)Fouriertransformaofαisafunction,dehedforω并0.Moreover,theconationα(∞)=OimpliesthattheFouriertransformofahasnopointmassattheorigin.Writeα(ω)=航(ω).Bythefactthatαisbounded,continuousmdofpositivetypeonR+,onehas,usingBocher'sTheorem[5,p.498], α(t)=1jeMα(ω)伽,tεR+nJR (3.2) Furthermore,αispositiveandintegable,i.e.,α兰OandαεLI(R).Letηbede负nedby 、LJnur4th11Rε.,aw4bnLguoc4ιEOoc-吨4····L-4b-z'-n..一臂,‘·二、自一一一一"υ0'η,wl
333 'rheaηεc∞(R),η(4)εL1(R)fori=0,1,2,...,aadAη(t)dt=1.Thetransform号saiis直es HyperbolicIntqrod{UbrentialEquation |ω|三1,1三|ω|三2,|ω|三:2. ω咱inLAU/EEEFtEEE飞一一、、..,,ω'''E‘、-n叮, Forh〉OandtεR,letm(t)=hη(ht).Cle缸1y, η;εL1(R). !!m|!L1(B)=||η||L1(Eh |ω|三h,k三|ω|三;2k,2h三|ω|. Inaddition,onehas仇(ω)=号(专)andso ω-h咱in4nu/EEEJ飞tEE飞=、‘,,,ω,,..飞'm-n叮, Define t〈0. b(t)=α(-0·, t220; b(t)=α(吟, (3.3) ftεL1(R+), Thenb=2α.Letfh=ηh*b.Therefollows s:pllf;||LZ(R)〈∞, fhεC∞(R;cnxn), |ω|三h,h三|ωlf二2h,2h三|ω|. 、‘..,,ω,,.‘飞α、‘..,,ω-Lm叫一,,,‘、‘,,αμhv'。4。tHnu--,飞lE飞=、‘EEJω,,..飞-'hv、‘.,,ω,,..、、-m'"吁,一一、‘,,,ω,,..飞'九 and NextdeheEO)=e-H|I,tεR,andgh=E-m*E.ThengKEC∞(R+)nc∞(R-),with g;εL1(R+). (3.4) s:P||gL||L1(R)〈∞, |ω|三k,h三|ω|:三2k,2h三|ω|. 7'品、、E,,,咱iω-Lhft飞ri!-…wl-ω':-+」+nu----rllJIB-一一ω,,,‘、IRZudm沉一一、、,,,,ω-K-n3 Obviously,
334 LOFtden Letb(t)=fh(t)+gh(t),tεR.Thush=fh+Shandhence,sinceαisofstrongpositivetypewithconstant1, {2α(ω),儿(ω)=(2(2一|专|)α(ω)+2(it|-1)市I兰布I,飞1+ω2I, |ω|三h,h三|ω|三2k,|ω|三:2h.(3.5) Consequently, -LτI纣h(ω)=玩儿(ω)主2α(ω)-i十ωa Finailydeane αh=bht220;αh=0,t〈0.Thenαkqg捉ah=38h;thuseachahisof毗ongpositivetypewithconstant1.Moreover,eachαhisboundedandcontinuousonR+,hencebyBochnetsTheorem[5,p.498ip句(们ije川h(ω)伽,tεR+(36)nJaUsing(32)-(3.6),itisnoth町dtodeckthatahhasdthedesiredproperties-. FortheestimationofthehigherderivativesintheproofofTheorem2.1thefollowinglem皿aiscrucial.ItisasimplereformulationfortheHilbertspacecaseof[5?Ch.17,Lemma4.2l-LetHbeacomplexHilbertspacewithscalarproduct(·,·)andnorm||·||H·ForvεLfoc(R+;E)a-ndαεLioc(R+;R),dehe,T,tQ(v,TJ)=j(ψ(吟,/α(t-s)v(s)doa.NotethathereaismiaMaimd,whereuvt让esvaluesinE.(Cf.(3.1).) Lemma8.2.LetT〉0,letψεP((01);H)beαbdut吻continuousmithvεL2((0,T);HLassumethatbεL1((0,T);R)andthatvεLL(R+;E)-Then 才fT(忡ψ(0t)L'(σbh川*叩伊州)dat三到剑仰剑州l川阶阳例|陆川川b叫州|川|k叫1气咐(仰队0叽'T叫)川||忡W￠圳仰(σT)川||阳H品(0仰阳均蚓Q(仰州川'JAtι'斗+|川|圳州驰胁|比bLυ1(仰0趴Mm叫'rmTη叫)泛(|川仰MM|忡M川ψ圳训|川|比叫川2气盯叩(αω(仰0,m)+||ψ'||LW,m识3(川,e)
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GeneralizedEvolutionOperatorsand(Generalized)C-Semigroups GUNTERLUMERMathematicsInstitute,UniversityofMons,Mons,Belgium 0.INTRODUCTION Integratedsemigroupshavebeenstudiedrecentlybyanumberofauthors·werefersimplyheretothepapersofArendt(1987),keHermannandHieber(1989)andNe由rader(1988).Cmsemigroups,alreadyconsideredbyDaPrato(1966),havebeenagainstudiedveryrecentlyratherextensivelybyseveralauthors,DaviesandPang(1987),Miyadem(1986),MWaderaandTamka(1987),Tanaka(1987),deLaubedels(1989ab).Both,integratedsemigro叩sandC-semigroups,haveniceapplicationsconcerningevolutionequations(Cauchyproblem)andturnouttobecloselyinterrelated.Ontheotherhand,veryrecentlyinLumer(1990a),byintroducingandstudying,besidestheclassicalsolutions,thegen-eralizedsolutionsofordernofhomogeneousevolutionequationsofthetypedu/dt=Au,u(0)=f,inaBmachspacex,mdthecorresponding(gener-allyunbounded)evoiutionoperatorsSn(t)on(generallynonclosed)subspamszn+1ofX,wedevelopedageneralmachineryquitesuitableforproblemofthetypedu/dt=Au+F(t),也(0)=f,whetherwell-posedintheusualsenseornot.Inparticularn-timesintegrated(non-degenerate)semigroupsofexpo-mMialgrowth(expomltidlyboundedintheterminologyofArendt(1987),keHermannandHieber(1989)),amspeciaiexamplesofsuch(familiesof)o?eratomSn(t).(1)Inthegeneralsetupjustmentioned,thebasicrolesareplayedbytheSn(t)andthespacesZn+1(orZn+1-kforappropriate。三k三n+1integers)(1)Applications。fthemethodsandresults。fLumer(1990a)aretoappearinLumer(1990b) 337
338 Ltdmer- (2).InthepresentpaperweinvestigatehowevertheroleplayedbythespamsZn+1nD(A勺,D((AZ叶1)k)whereAzn+1isthep町tofAinZn+1,whichcompletestheinformationneededforafullunderstandingofthemachinerydevelopedinLImer(19ωa)-andusingthisandanappropriategeneralizationofthenotionofC-semigroups(whicharenowfamiliesofgenerallyunboundedoperators)showthataverysimplereiationexistsbetweenany{Sn(t)}andacorrespondinggeneralizedC-semigroup,providedp(A)theresolventsetofAisnotempty.Onehasaveryneatformuiawhen王A-1εB(X)={aiiboundedeverywheredennedlinearoperatorsonX},andalesssimplerelation.butonethatcanstillbeexplicitelywrittendown-wheno￠p(A)butsome。并ωξp(A).Theseresultsextendfactsobtainedearlier,intheliteraturequotedabove,forn-timesintegratedsemigroupsandtheusualC-semigroups,underquitemorerestrictiveconditionssuchasexponentialgrowthconditions,andothers,usingtoalargeextentLaplacetransformargumentswhicharenotavailableinthegeneralsetup. 1.BASICCONTEXT,NOTATIONSANDTERMINOLOGY Thebasiccontext,assumptions,notations,andterminologyareindeedthoseofLumer(19ωa).Werecallnexttheessentialitemsneededinthispaper-XwillaiwaysdenoteaBanachspaze,2(X)denotesthesetofalllinearoperatorsinX(i.e.withdomainD()andrange(image)I()inX).Below,Aε￡(X)willbeassumedtobeclosedandsuchthatV=Au,也(0)=。"hωonlythesolution"u=。"·AsusualVλεCforwhich3(入一A)-1εB(X)(B(X)wasdeanedinsectionO)thatoperatorisdenotedR(λ,A)("λ←→R(λ,AYbeing"theresolventofA").WeconsidernowthegeneralinhomogeneousevolutionequationinX 包'=Au+F(t),u(0)=f,(食)(包'=du/d),FεLLC([0,+∞[,X).DEFINITION1.1Forninteger主1,weshallsaythat(*)hωastronggen-eralizedsolutionofordern(n-s.g-s.)tJniE3aclassicalsolutionun二Un(tJ)of u;=Atln+t1f+F(t)U(0)一0,(1)(n-1)!凡,n一where凡(t)=j;dTn-1fp-1...j;1F(To)dTo=(1/(n-1)!)j;(t-s)V1F(s)ds;inthatcaseweCalitYL=咋(tJ)themildgeneralizedsolutionofordern-1((n-1)-m.g品).Wesaythatuoisastronggeneralizedsolutionoforder。(CM.gs.)ofNiEitisaclassicalsolutionof(食).(2)ratherthanD(AH+1),D(A勺,槌inthecaseof(non-degenerate)n-timesintegratedsemi-groupsofexponentialgrowth.
GeneralizedEvolutionOperators 339 Forn=0,1,2,...,wesetZn={fεX:3an-s.g-s.h=%(t,f)of(食)wifhF=0,也(0)=f},andforn=0,1,2,...,t主0,wedeanethegeneralizedevolutionoperatorssn(t):Zn+1→X,bySn(t)f=吆+10,f).Oneshows(see1.3,1.5ofLurner(1990a))thatactuallyeachSn(t)operatesonZn抖,i.e-sn(t):Zn+1→Zn+1forn=0,1,2,...,t主0,ZoCZ1CZ2C…CZnc…,andonZn+1,飞二11Sn(t)Sn(s)=兔"(t+s)-LEi(刮2川(t)+tkhn-k(s)),(2)k=0 fors,t兰0, whereforn=Oweinterpret(2)sayingthatthesummationfromOto-1disappears(3).ForotherfactsconcerninggeneralizedsolutionsandevolutionoperatorsseeLurner(1990a).Toendthissection,letusrecallthatthespectrumof(theclosedoperator)A,sp(A),isdennedasbeing{λεC:βR(λ,A)},andp(A)theresolventsetofAisdeanedasCVp(A)thecomplementofsp(A)inC.Inthefollowingsectionsthegeneralcontextwillbetheoneofthepresentsection. 2.THESPACESA-k(Zn+1)ANDZn+1nD(AK).Inthissectionandthenextoneweassumethat3A-1εB(X),forthesimplicityofcomputations,results,andformulas;butaswesee!aterinsection4wecanreplaceA-1by(A一ω)-1=-R(ω,A)forωεp(A),providedp(A)弄。,toobtainsimilarresults(mdweshallalsoexaminetheextenttowhichtheseareindependentoftd,intermofAA+1,D((AAJK)).LEMMA2.1Forn=0,1,2,...,t主0,A-1:Zn+1→Zn+1nD(A), (3) Sn(t)A-1=A-lsn(t)onZn+1· Proof:Letfεzn+1,sothatwehaveaclassicalsolutionun+1for 4872buL1=AVn+1+二丁f,Un+1(0)=0.-7Bl (4) Now,applyingA-1tobothsidesof(4)andsettingω(t)=A-1tMl(t),wehavethatw(t)εD(A)andprovidesaclassicaisolutionof 叨'(t)=AW(t)+主A-1f,四(0)=0.n! (5) (3)Infact,usingwhatprmd锢'theformulain(2),and140fLumer(1990吟,onecanshowmorepreciselythatSn(t):Zn+l→ZnCZn+lhrallt主Oandn,exceptedn=0,i.e.f。rn=1,2,
340 Lumer HenceA-1fEZn+1,ω'(t)=Sn(t)A-1f=A-luL+1(t)=A-1Sn(t)f,whichproves(3).(4). LEMMA22.A-1:Zn+1→D(A)nclA+1=D(A)nZn-F)Pmof:Letg=A-1fJt三Zn+1·By2.1gεD(A)nzn+landSAT)gεD(A)forallT主0,SAT)g=A-lsn(T)f,sothatASAT)g=Sn(T)f,henceT叶ASAT)giscontinuouswhichsinceAisclosedyieldsVt220 AIt￡zM (6) Thensincegεzn+1, 队(t)g=AItsn(T)gdT+二g=Itsn(T)fdT+2g, (7) sotHSn(t)gisC1,henceg=A-1fεCIA+1;C1,叶1=Znby1.4ofLmer(1990a).-LEMMA2.3IfgεZnnD(A)thenAgεZn+1·(Forn=0,1,2,...).Proof:SincegεZ川D(A),gεCM+lnD(A)by1.4ofLumer(19ωa).Sowehaveaclassicalsolutionwfortu,=Aw+(tn/n!)g,ω(0)=0,withw'(.)=Sn(.)gbeingC1.ItfollowseasilysinceAisclosedthatw'。)εD(A),(Aw)'=Ad,andw"=Aw'+(tn-1/(n-1)!)g(1.4ofLumer(1990a))(6).Sinceg,叨'(t)εD(A)sodoesAw(t)and Aw'=(Aw)'=A(Aw)+(tn/n!)Ag. (8) Settingz(t)=Aw(t),wehavethusby(8)aclassicalsolutionofj=Az+(tn/n!)AgJ(0)=0,henceAgεzn+1··FromtheprecedinglemmasweconcludethatTHEOREM2.4Forn=0,1,2,...,t20, A-1(Zn+1)=ZnnD(A)=CM+1门D(A), (9) A(ZnnD(A))=Zn+1 A-1Sn(t)=Sn(t)A-1onZn+1 (10) ASn(t)=Sn(t)AonA-1(Z叶1). (4)Onecmalmaddin(3)thatA-1:Zo→Zo(samepmofwithminimalchanges),andsoO)A-1=A-ISo(t)anyway。nZoCZ1·(5)ForthenotationC1.n+1seeLumer(199Oa)(e)thmworksals。。fcourse(withtrivialmOdincation)forthecasen=0.
GeneralizedEvolutionOperators 34I Moreover,2.4andsimpleiterationargumentsyield:THEOREM2.5Forn=0,1,2,...,t主0,kinteger, A-k(Zn+1)=Zn+1-knD(A勺,forO三k三n+1, (11) AK(Zn+1-knD(AK))=Zn+1,for。三k三n+1. A-KSn(t)=Sn(t)A-konZn+1, (12) AKSn(t)=Sn(t)AKonA-k(Zn+1),k三0. 3.GENERALIZEDC-SEMIGROUPSANDREPRESENTATIONOFTHEEVOLUTIONOPERATORSSAt)WestartbyintroducingthefollowingdeanMOIlofgeneralizedC-semigroups-DEFINITION3.1LetYbealinearsubspaceofX,Cε￡(X)withD(C)3Y,Cbounded(onD(C))andiIEjective-Let9:I0,+∞[→￡(Y):tH90),D(90))=YW三0,and:(i)90)9(s)=C9(t+s)Vt,s主0;(ii)9(0)=C|Y(where"|"denotes"restrictiont。");(iii)tHg(t)isstronglycontinuous.Then9willbecalledagenralizedC-semigro叩(inX,operat-ingonY);byabuseoflangmge/notation,weshallalsooftenreferto9bythenotation{90)},or90)(forexampleweshalltalkaboutthegeneralizedC-semigro叩{90)}).WehavenowthefollowingrepresentationresultconnectingSn(t)withacor-respondinggeneralizedC-semigroup-(WeusethenotationNfor{0,1,2,...}.) THEOREM32(Recallthatwehaveassumed3A-1εB(X).)ConsidermynεN.DeaneonZn+hvt主0,9(t)=So(t)A-叮Zn+1.Then{90)}isageneralizedC-semigrow(ofunboundedoperatorsingenerai,withY=Zn+1andC=A-n)andonehastherepresentationformula(Vt主0): 以t)=9(t)-ZZAhonA+1' (13) whereitisunderstoodthatforn=0,thesummationin(13)disappears.Proof:Westartbyestablishing(13).LetfεZn+1·Hn=Othereisnothingtoprove,soassumen;21.Then ￡zW (14)
342 Lurner andapplyingA-1tobothsidesof(14),andusing2.1,儿(tWf=It￡BM (15) Thenby(15)(also22)and1.4ofLumer(1990a),A-1fεC1.n+1,S二(t)A-1f=ιdoff=钊)f+岗7444fHence: 队(t)f=(Srl(t)A-1-rt:.A-1)f (16) Hn=1weareanistled;证n主2,replacingin(16)nsuccessiveiybyn-kfork=1,...,n-1,andfbyA-kffork=1,...,n-1,weEnd Sn-1(t)A-lf =(hO)A-2一(;J→)f,,n-(k+1)=(Sn-(川)(t)A-(川)-t(k+旷 Sn-k(t)A-kf SI(t)A-(n-1)f=(So(t)A-n-A-n)f, andequatingtheseparatelyaddedlefthandandrighthandsidesoftheequalitiesin(17)and(16)weobtain Sn(t)f=(So(t)A-n-tA-1一…-A-n)f(n-1)! (18) =(的)A-n一汇2Ah)f, whichproves(13).Thereremainsnowonlytocheckthat90)=So(t)A-n|YdeanesageneralizedC-semigroupwithY=Zn+1,C=A-n-From2.5wehavethatA-n:Zn+1→ZlnD(An)andA-nSo(t)=So(t)A-nonZuhenceforfεZn+1=Y,A-nfεZhSo(s):ZI→ZlSOSo(s)A-nfεZI,and So(t)A-nso(s)A-nf=A-nso(t)So(s)A-nf=A-nSo(t+s)A-nf,(19)by1.5ofLumer(1990a).Thisshowsthat90)9(s)=C90+s)onYwithC=A-n;therestischar-.
GeneralizedEvolutionOperators 343 4.THEGENERALSITUATIONFOROPERATORSWITHp(A)并0.WeconsidernowthematterofrepresentationofSn(t)intermofgeneralizedc-semigroupsinthegeneralsituationwithp(A)#0.Indeedinthiss让uationitwillbepossibletoapply(13)tosf-ω(t)withωεp(A),andwithS♂-ω(t),(A一ω)k-n,ontherighthandside,recoveringthereaftersf(t)assr-ω)+ω(t)byperturbation,viaaperturbationfo口nulaextendingtheoneof31ofKeller-mannandHieber(1989)(7)totheSn(t)operators(1.1,(1)and(2),ofLuzner(1990b)),whichgivesinsymbolicnotation(explainedfollowingtheformUa): sf(t)=((1一ωI)neωsf-ω(J)(t) (20) where(Jg)(t)=j;g(T)dTforanygεC([0,+∞l,于),Y=Zn+1,andthenota-tionIisusedin(20)toactuallydenotetheoperatoronthespaceofstronglycontinuousmapsT:i0,+∞[→￡(Zn+hZI):t问T(t),D(T(t))=Y,dbEnedby(IT)(t)f=(J(T(·)f))(t),fεZn+1=Y,tε[0,+∞[.Inexplicitform(20)reads"follows 可川 (theintegralsbeingunderstoodinthestrongsenseofcoum).Now,computingsf-ω(t)via(13)wehave 俨(t)=广ω(t)一ε2(A一ω)k斗, (22) where9A-ω(t)=S♂-ω(t)(A一ω)-n=e-ωts♂(t)(A一ω)-nonZn+1·Settings♂(t)(A一ω)-n=M(t)=eωt9A-ω(t),(23) oneseesatoncethatN(t)isageneralizedC-semigroupwithC=(A一ω)-n,Y=zn+handintermofthegeneralizedC-semigroupM(t)werewrite(22)as 俨(t)=产N(t)一Z号(A一ω)bn (24) Combining(24)with(21)(or(20))oneobtainsasomewhatlaboriousthoughquiteexplicitformularepresentingSn(t)=sf(t)intermofthegeneralizedc-semigroupM(t)mdthe(A一ω)k-n.Asfortheresultsofsection2,theycarryoverindeedtothegeneralsitua-tionwithp(A)#",whereA-kisreplacedby(A一ω)-k,ωεp(A).Weshallnotgothroughthisindetailsincetheargumentsareessentiallythesameas(7)giventhemf。rintegratedsemigmu"。fexponentialgrowthh=1)
344 LUFF1er insection2.Letmmerelyconsideranexampleofthekindofsmallmodi-Ecationsarisinginthearguments:oneshowslikeinsection2thatR(ω,A):Zn+1→Zn+1,Sn(t)R(ω,A)=R(ω,A)Sn(t)onZn+1fort主0,andifgεR(ω,A)(Zn+1)theng=R(ω,A)fwithfεzn+1,soAg=AR(ω,A)f=ωR(ω,A)f-fεZn+1,sdt)Ag=Sn(t)AR(ω,A)f==Sn(t)(ωR(ω,A)-1)f=(ωR(ω,A)-1)Sn(t)f=AR(ω,A)Sn(t)f=ASn(t)R(ω,A)f=ASn(t)gwhichprov臼thatASn(t)=Sn(t)AonR(L印4)(Zn+1).hthepresentcontextitisofcoursealsoimportanttoseethatthesp缸esR(ω,A)(Zn+1),R(ω,A)k(Zn+1),donotreallydependonωandadmitasimpleintrinsicdescription(includingthecuewhen3A-1εB(X)i.e.whenO=ωερ(A)).Indeed,considerAZ叫,the"partofAinZ叶1",dennedby D(AZ叫)={fεD(A)nzn+1:AfεZn+1}, (25) Azn+1f=AfforfεD(Azn+E). Onecaninfactshow-weshallnotgivethedetails·thatforn=0,1,2,. D((AZ叫)k)=D(AK)nzn+1-K,。三k三n+1. (26) Soanally,ifp(A)并",2.5becomesinthisgeneralsituation:Forn=0,1,2,...,t兰0,kinteger,ωεp(A),R(ω,A)k(Zn+1)=D(AK)nzn+1-K=D(AK)nch+1=D((Azn+1)勺,for。三k三n+1,(27)(A一ω)k(Zn+1-knD(AK))=Zn+1,forO三k三n+1 R(ω,A)ksn(t)=Sn(t)R(ω,A)konZn+1, (28) AKSn(t)=Sn(t)AKonD((AZ叶1)勺,k主0,(whereD((AZEJ勺,D(A勺,areunderstoodtobeZn+1,X,fork=0)-Asaanalremark,noticethatif,forsomen,Zn+1工X,thenAgeneratesan-UmesintegratedsemgmupSJ)=Sn(t)onX(seeLmer(1则a))whichm叮notbeofexponentialgrowth(inthelattercaseAgemratesSn(t)intheextendedsenseexplainedinsection2ofIArner(1990a)),andofcourseAzn+E=Ainthementionedsituation,sowhethersdt)isofexponentialgrowth,ornot(whichmaywellhappen),wehavethenD(AK)cZn+1-k=:CK,n+1. REFERENCESAmdt,W.(1987).Vector-valuedLaplacetransformandCauchyproblem,IsraAlJ.Math.59,327.DaPrato,G.(1附).Semigrumregolarimabili,RichercheMat-15,223.
GeneralizedEvolutionOperators 345 Davies,E.B.,andPang,M.M.(1987).TheCauchyproblemandagenrealizationoftheHillbYosidatheorem,Proc.LondonMath.Soc-(3)55,181.deLaubedels,R.(1989功.C-semigroupsandtheCauchyproblem,preprint-(ToappeuinJ-dFmet.Analysis).deLa,由enfels,R.(1989b).Integratedsemigroups,C-semigroupsandtheab-strutCauchyproblem,preprint-(ToappearinSemigroupFonm).KbeiHh1怡e口mnamnmrnlH.andEHEi怆e伽b阳e町rM.(仰1ω98蚓9创),IntegratedS臼e叨r虹E咿84'160.Lum町,G.(19ωa),Solutionsg缸16ralidesetsemi-groupesintGgr缸,C.R.Acad-Sci.Paris,310ser-I,577.Lumer,G.(1990时,Applicationsdessolutionsg旬的,lidesetsemi-groupesintegres主desprobl也mesd'dvolution,forthcoming.Miyaderaj.(1986).OnthegeneratorsofexponentiallyboundedC-semigroups,Proc.JapanAcad-,62ser.A.MiyaderALmdTanah,N.(1987).SomeremarksonC-semigroupsandinte-gratedsemigroups,Proc.JapanAcad-,63ser.A.Ne由randeLF.(1988).IntegratedsemigroupsandtheirapplicationstotheCauchyproblem,Paz.J.Math.135,111.Tamka,,N.(1987).OntheexpomntiallyboundedC-semigroups-TokyoJ.Math.10Nol.
ExamplesandResultsConcerningtheBehaviorofGeneralizedSolutions,IntegratedSemigroups,andDissipativeEvolutionProblems GUNTERLUMERMathematicsInstitute,UniversityofMons,Mons,Belgium 0.INTRODUCTIONInrecentpapers,LImer(1990a,,时,wedevelopedthebasicmachineryandresultsconcerninggeneralizedevolutionoperators(quitesuitableforhandlingproblemofthetypedu/dt=Au十F(t),u(0)=f,inaBanachspaceX,whetherwell-posedintheusualsenseornot(1)),extendinginparticularmanynotionsandresultsconcerningintegratedsemigroups,andwealsotreatedsomespeciEcaspectsoflocallylipschitzintegratedsemigroups-ConcreteappplicationswillbegiveninLurner(19900.Inthepresentbriefarticleourpurposeistwofold:(i)giveexampleswhich(2)showorpermittoderiveeasilythat,roughlyspeaking,thediEerentbasicsituationsandobjectstreatedinthearticlesmentionedabove(spacesofgeneralizedsolutions,domainsoftherelatedoperators,fordibrentvaluesofn)doindeedalloccur(3);(ii)showthatfordissipative(clωed)operatorsA,roughlyspeakingagain,thingsareeitherquiteniceandsimple,orelseratherbad.Inparticularwesho啊that(inthiscontext)eitherthe(homogeneous)initial-valueproblemisnotsolvableforanydensesetofinitialvaluesinthesenseofstronggeneralizedsolutionsofordernforsomespeciEed(1)Abeinglinear,asareall。peratomc。nsideredinthisarticle(belowhperator"means"linearoperatof').(2)Mgetherwithandinadditiontowhatisalmadyknownfmmthearticlesmentionedabove(3)theseexampl倒givealsoagoodde剖。finsightintothevarietyofwaysinwhichsomeofthementionedsituationsandobjectscanarim- 347
348 Lumer nnomatterhowlargethatnhchosen,orelseAgeneratesatworstalocallyupsetlitzintegratedsemigroup;onewillinfactalwayshave(inthiscontext)A=Z2=ZZforailn》2,andatworstSn(t)=j;(127:"s)ds(againforn注2).Fornotionsandnotationsconcerninggeneralizedsolutionsandevolutionoperators,Zn,Sn(t),etc.,werefertoLumer(1990ah). 1.EXAMPLESWITHDISSIPATIVEOPERATORSTakeX=Co(l0,+∞[),AinXdeanedby:D(A)={fεxnc1(l0,+∞!):f'εXL 、‘..J'i,,,‘飞 Af=f'forfεD(A) ThenAhadissipative,closed,denseiydennedoperatorinX,notgeneratingasemigroup-(4).WeshallnowdeterminethespacesZnforA.LetfεX,n注1.ThenfεZnmeansthatwehaveaclωsicalsolutiontJn of UL=A%+tlff,un(0)=0·(n一1)-bornthefactthatunisC1,and(2),oneseesimmediatelythatthefunctiondenotedalsobyun,fromR+×R+→C,deanedbyun(tJ)=(un(t))(z)(fort注0,0《ZξR)bCIin(tJ);句,θun/θz,θ阳/θtMe(tJ)-continuous,(叫(t))(z)=(θun/θt)(tJ),(AUn(t))(z)=(θun/θz)(t,吟,andhence(2)impliesthat (2) θ"-A"一tn-1」=τ平+r,飞,f,比(OJ)=0,θtOz (3) onR+×R+· Nowusingthemethodofcharacteristicsi.e.makingthechangeofvariablesE=z-Lη=Z+t,theequation(3)istransformedinto 等(EJ)+(吉(71厂fG(5+牛 (4) where百(己,η)=Un(t,z).TheinitialconditionUn(0)=0,un(OJ)=OallZ》0,goesinto百(zo,η)=Oforη=zoandallzoεR+· (4)ClearlyhrA一likeisinfactthecaseforanydissipativelinearoperator(seeBedim3)-theuniquenesscondition句'=Au,u(0)=0,hasonlythesolutionU=0"consideredinLumer(1990a)四automaticallysatiSEed-
IntegratedsemigroupsandDiss伊afiveEvolutionProblems 349 Wecompute百from(4),integratingin5alongthelineη=constant=Z+t=zo(takenarbitrarilyhRd,startingfrom￡=zo=η.Takingintoaccountthat百(zo,η)=Owehave一'rez百(EJ)=ι晶川lo(η-x)n-1f(j(χ+η))句, (5) whichbythechangeofvariableofintegrationj(χ+η)=ggives 瓦(仅￡已hω叶,A叶η创←)←=%川川州(0队Mtι,SinceUn(t)εX=Co(l0,+∞l),un(t,o)=Oforailt注0,hence(6Mensthato=1fo(η-gr-1fk)dg=儿(ZO (6) (7) Vzo注0(wherefndenotesthen-thiteratedintegraloffinthewaydennedinLuner(1990a)),sofn=Oandthusf=0.Hence: Zn={0}forn=0,1,2,. (8) (actuallytheproofaboveisgoodforn=1,2,3,...,butonehasalsoZoCA-seeLumer(1990a)-so(8)holdsindeedforn=0,1,2,...).Theexamplejusttreated,withtheoutcome(8),revealsextremepossibilitiesinonedirection(badbehavior)inlinewithwhatwasmentionedin(ii)oftheintroduction,forAdissipative.Letusnextconsiderasimpletypicalexampleintheotherdirection(verygoodbehavior),forAdhsipative,andthenseehowoneobtains"intermediatebehavior"by"combining,,suchextremeexamples.LetY=C([0,1l),比=Co(l0,10.DeaneBinYbyD(B)={fε比nc2(l0,1l):f"εY}, (9) Bf=f"forfεD(B) Bisaclosed,non-denselydenned,dimipativeoperatorinY;BothepartofBinYb(seehzy(1983),p.39)is-asiswellknown-thegeneratorofacontractionsemigrouponEb(ifnecessaryseeLurner(1975)).Itisnotdime1山toseethatBgeneratesindeedalocallylipschitzintegratedsemigroupS(t)onY(usingforinstancefactsinLuzner(1975)andkeHermannandHieber(1989)),S(t)beinggivenbyS(t)=价值。一1)B-1(where3B-1εB(Y))-seeIAIner(1990a).Underthesecircumstancesitisnotdime1山toshowthatforB, Zo=D(Bo),zl=Yb=D(B), (10) Z2=Y.
350 LUFF1er Suchsituations,inasenseextremecasesfordissipativeoperatorsnotgen-eratingacontractionssemigroup,部exhibitedaboveforAandB,cannowbe"combined"toproduced"intermediatepossibilities",toobtain,forinstance,acloseddissipativeoperatorAinsomeXforwhichZi,say,isneithercontainednorcontainsD(A).Indeed,ifweconsiderthespacesX,Y,ofthepreviousexamplesandnowtakeJt=X@Ywith||.||立=||.||x+||.||yF),thenAdennedbydirectsumforf=(g,h)εXvia fεD(A)iEgεD(A),hεD(B), 、....,,'且'i,,,.‘‘、 Af=(Ag,Bh), satiSEesasiseasilyseen(pndconsideringX,Y,togetherwiththecorrespondings由spams,imbeddedinXintheobviousway): Zl(A)=ZdA)+ZI(B). (12) OfcourseD(A)=D(A)+D(B)soindeedfromwhatwehaveseenabove,Z1(A)￠D(A), (13) D(A)￠ZI(A), ipfactIZ(A)￠zl(A),Z1(A)nD(A)isnotdenseinD(A);onecanthencompareAwithAawhichis"quitesmaller".SinceforA,alln,Zn豆D(A),andforBandalln注1D(B)豆zlCZn,oneseesalsothatonehasinfactforailn》1theconclusionstatedin(13)forn=1. 2.EXAMPLESWITHNON-DISSIPATIVEOPERATORS.SOLUTIONSWITHLARGERTHANEXPONENTIALGROWTH.Inthissectionwetreatbdeny,inh)and(b)below,twotypesofsituations(theErstonerathergeneral,thesecondquitespecia儿withtheSrsttypegivingoftenrisetooperatorssn(t)whichcannothaveexponentialgrowth,andoper-atorsAwhichdonotgenerateanyn-timesintegratedsemigroup(whetherofexponentialgrowthornot),buthavingnontrivialZn,sn(t),foralln. (a).LetXdenotethespaceP=P(N+),N+={0,1,2,...},isometricallyisomorphictoanyseparableHilbertspace(ofMarlitedimension).ConsideronX (5)Forthenotionof"directsum"ofhitelymanyBanachspae钮,。riz由1itelymany(础laterin(b)ofsection2),weusethedenmtionofReedandSimon(1972)p78.
35I anidElitematrix(A)二(句·)withrealorcomplex-valuedentries;iJεN+,andtheassociatedoperatorAinX(thespaceofsequencesf=(fk)旱。=(fk)εP)dennedby IntegratedsemigroupsandDiss伊ativeEvolutionProblems {f=(儿)εX:((Af)4=汇向jh)创stsandεX}, D(A) (14) ((Af)4). SuchoperatorsAconstitutealargeandinterestingclassofHilbertspaceop-eratom,whichweshallofcoursenotinvestigatesystematicallyinthispaper(concerninggeneralizedsolutionsandSn(t)operators)but,ωwassaidintheintroduction,(i),onlyconsiderherefromthepointofviewofexamplesandcertainpropertiesrelatedtosuchexamples.Forourpresentpurpωes,letussupposehereafterthat Af (15) andfromhereonwriteforsimplicity向inlieuofa44·MoreoverwriteDo={f=(fk)εX:儿#OforonlyEnitelymanyk},soinparticularDoCD(A)-Thentheconditions(15)implyωiseasiiyseenthat3A-1εB(X)soAisclosed;andmoreoveronehasatoncethat"d=Au,u(0)=。"admitsonlythesoiution"u=。". fort并j,Reα4422α〉0,.lirnα44=+∞s-呻EX3 α4j=0 2.1.PROPOSITION.IfAisasabove,satiSEes(15),andgeneratessomen-timesintegratedsemigro叩Sn(t)intheextendedsenseofLumer(1990a)(andinparticularinthesenseofArendt(1987)),thensdt)caninfactneverbeofexponentialgrowth. prωf.IfAgeneratesann-timesintegratedsemigroupSn(t)andisofexponen-tialgrowth,||SJ)||ζMe"forallt注0,then3R(λ,A)=(λ-A)-1εB(X)forReλ>ω,where R(λ,A)=λnje-MSn(t)dt,JO(seeLuner(1990吟,Arendt(1987)).Butoneseesatoncethatail向areeigen-valuesofAandthereexist向withRe向〉ω,acontradiction.-NextobservethatforAasabove,f=(fk)εZn,n注1,meansthat3a川叫山Cd巾伽lha创ssik阳阳川Cωωaa,computationshowsthatforthecomponentsunkoftynonetm(forn》1,n=0,respectively): etαk-1-tι-it2α2-…-T屯τtn-1d-1比21k{n-1}!岛tr川SH儿,Bαk Unk(t)
352 Lumer (16) UOA(t)=eukfk· Considerthespecialexampleak=k+2k27ri(exampleconsideredinadiferentmannerandcontextbykenema-nnandHieber(1989),EXM叩le1.2).OneEndswithoutmuchdimeuitythatZ2=Xandonehωthusinthatcueanintegratedsemigroupwhichby2.1isnotofexponentialgrowth.Ingeneraloneneednothave,withAasconsideredin(15),mn-timesintegratedsemigro叩atall.Forinstanceifak=k+1thenfrom(16)itisclearthatZn并Xforalln.Moreover: 22.REMARKS.WhileforAasabove(in(15))onemayhaveZn+1=Xforsomen(andhenceAgeneratesanintegratedsemigroup),orZndifereMfromxforalln(seetheparagraphjustpreceding2功,onehasinanycaseDoCZnforanyn=0,1,2,...,andhencetheZn,Sn(t),arenevertrivial.Ontheotherhandusingasequence{fm}ofelementsinDo,wherefm=(fmdwithfm,k=Ofork并mandfmm=1,takingak=k十1,andmakingmeof(16),asimplecomputationshowsthatw〉oaxedlim||Sn(t)fml|/||fm||=∞(foranygivenn=0,1,2,...)soailtheseSn(t)areunbounded. (b).ForthenextPropositionweconsideranother(special)typeofsituationdiferentfromthetypeofsituationstreatedin(a). 2.3.PROPOSITION.Thereexist,inthecontextofLImer(19ωa),operatorsAforwhichailtheZn,n=0,1,2,...,arediEerent,anddiEerentfromX,i.e. Zocz,CZqC...CZC...ζX.u3ti平t三3t石tUTB (17) prωf(Indications).ByaresultofArendt,NeubranderandSchiotterbeck(1989),01ofthatpaper,(seealsoNeubrader(1988)),thereexistsforeachn注1anoperatorAnεZ(XJ(inaBanachspaceXn)whichgeneratesanexponen-tiallyboundedn-timesintegratedsemigroupsothatZn+1=Zn+1(An)=XmbutforwhichZn并Xn(whichonecanseefromresultsinArendt,NeubranderandSchlotterbeck(1989)andfromfactsininLumer(1990a)).Forn=03asemigroupetAoonXowithZo(Ao)=D(Ao)并Zl(Ao)=Xo·LetnowX=@ZLoXnwithf=(fn),||f||=艺||fn||<∞,forfεX,andAn=0bedeanedaccordingly,f=(fn)εD(A)ifalifnξD(An)and(AJn)εX,andofcourseAf=(Anfn).ThenonecancheckeasilythatAisclosedandhastheuniquesolutionOfor"u'=Au,u(0)=。",andbyconstructionitisimmediatethat(17)hoids.(OfcourseAisnOI叫is句"iveandindeed,asweshallseeinthenextsection,(17)couldnotholdforadissipativeA).
IntegrafedSemigroupsandDissibafiveEvolutionProblems 353 3.BEHAVIORINGENERALOFSn(t)ANDZnFORDISSIPATIVBA.LetAGZ(X)bedissipativeandclosed.(UndertheseconditionsonehasautomaticallytE阔、'=A吼叫G)=。"执asoniythesolution"U16势sinceforAdissipativeonemusthave||u(t)||ζ||u(0)||(see3.1below).) 3,1.LZMMA.ForAdissipativeinXonekas:||Sn(t)||ζtn/n!fort注0,nzO,1,2,..., (18) onZn- P机ωf.Fornt0,letfεZosowehave:矿工AU事也(0)=f.(19)Takeh〉0,t〉0.Wehaveu(t-h)="。)-hd(t)+o(h)口也(t)…hAu(t)+o(h)-Hencellu(t-h)||=liu(t)-hAu(t)||+o(h)》||u(t)||+o(h)bythedissipativemssofA.EZezlcewegetfo川keiefiuppezDinideyivuiveD…ofiiuO)ii?←||u(t)||一||u(t-h)||(D-(||u(.)||))(t)=lm《0.(20)hiohSincetHiiu(t)iiiscontinuousand(20)holdsfort〉0,iiu(jlhdecreasing(勺,so|lfl|》|问(t)||forallt注0.NowffZoCZhu;=A叫+fandu1(0)=0,withuiPJ)=550)f事where吨。)zj;:u(s)ds.sojlso(t)fij乙:!!叫(tJ)ii=iiu(t,f)iiζ|ifiiwhictlgives(18)fornt0.Forn;三1,letf巳Zn-Thenwehaveaclassicalsolutionunof UL=AUn令,:".f,以0)=0, (21) andacomputationsimilartotheaboveonegives,usingnow(21)toestimateD…(iju(jii)(.), (D-(||时)||))(t)坛,jd|f||, (22) ((沪、D-(||un(·)l卜写了llfll))(t)ζ0.Hence,tH||un(t)||…号||f||beingdecreasingandoattz0,wehaveijuJMJii凡(纠 whichholdsforn=0,1,2,...fεZnCZn抖,以七月1后Srl(7)fdTtL(t)f,soby(23)事(18)holds. . (6)SeeTitchmarsh(1975)p.354.
354 Lumer 32.LEMMA.Underthesameassumptionsasin3.1. ZnCZTB+1,forn=0,1,2,..., (24) Sn(t):Zn→Zn,forn注1 prωf-Wetreatthecωen注1(n=Oissimila吵.Let{fk}ζZn,fk→f.Then儿εZn+1,Sn(t)儿=uL+10,儿)=tJL+lh(t)(wherewehaveset%+10,儿)=Un+M(t)), nu--、‘..Fnv'κ·且+饵"VLhrJ俨-d+Lam--+"muA一一Lam-A+'nu (25) By31,叫+1,k(t)=Sn(t)fkconverges,uniformlyonbcompacta,tosayh(t),and%+LK(t)=j;sn(T)儿dT→fth(T)dT=ω(t).NowwisC1,ω'(t)=h(t),AUn+1,k(t)=Sn(t)fk一(tn/n!)儿→h(t)-(tn/n!)f,hencew(t)εD(A),ω'(t)=AW(t)+艺f,ω(0)二0.(26)n! ThusfεZn+1,Sn(t)f=limSn(t)儿,瓦cZn+1·Butitisageneralfeature,kfollowingeasilyfrom1.5and1.4ofLumer(1990功,thatifn》1Sn(t):Zn+1→Zn-Henceinour(dissipative)context: sdt):瓦→ZnforO1. (27) Thiscompletestheproofof3.2.-From3.2,Lumer(1990吟,and3.1,followsthatforn》1Sn(t)=Sn(t)|Znisan-timesintegratedsemigroup,ofexponentialgrowthontheBanachspazeZn-Thegtfm,torofSn(t)isA互Z,thepartofAinZ.Thiscanbeseenasfollows:VfεZnCZn+1,wehavefor阳+10,f)withtYL+1(t,f)=Sn(t)f,thatun+1ξD(A)nZsinceSn(t)fεAby(27)(and%+1(tJ)=j;sn(s)fds),whileAUn+1=ULl一(tn/n!)fεZnforthesamereMon,sotYVL+1εD(A瓦),A瓦Un+1=AUF-+1,henceoneMfεZn+1(A瓦)onZanda(t)=sfh(t)onz;二Zn+1(A五ζ),andhencea(t)hasAEZasgeneratorintheextendedandtheLaplacetransformsense(seeLurner(四9Oa)section2). 3.3.LEMMA.Undertheassumptionsof3.1onehas: (i)Z1=Z1, (28) (ii)Z2=Z2=Znforn》2.
IntegratedSemigroupsandDiss{pativeEvolutionProblems 355 prωf-Weantprove(ii).Bywhatwehavejustseenabove,AzzisthegeneratorofSn(t),R(λ,A瓦)=λnjfe-MSn(t)dtforA>0.SinceA瓦isdissipativewehave||R(λ,A五Z)||ζ1/Aforλ〉0,so 1||R(λ,AEZ)mkFforλ〉0,m=13,3,. (29) But(29)implies(seeArendt(1987),keHermannandHieber(1989)Theorem2.4)thatAEZgenmtmatwoMalocallylipschitzintegratedsemigrouponZ二(ora民migrouponZ)SOthat(seeAmndt(1987),Lmer(1990a))inmyc些the(homogeneous)initial-valueprobleminthe2·s.g.s.senseissolubleonZnintermofAzhenceintermofAwhichsaysthatZCZ2forn注2,whileZ2CZnforn》3by3.2,soZ2=Z2=瓦=Znforn注2.Wenowprove(i).Firstageneralfact:ifBinXgeneratesa1o侃Caa刮,1il坊ylHip庐sCd}h1山ibtuzintegratedsmeImni堪grmOu叩pS(0t)onX,thenb均yILd旧nme町r(1990aa叫,j)andperturbationB=(B←ω叫)十ωtωO陀阳Imnowvet}h1e附tbrikMCdtonD(B)L,henceD(B)CC12=ZuontheotherhandinthatsituationforfεZhS(t)f=词。,f)andu;=Bul+f,UI(0)=0,sof=的(0)=fnh-1Ul(h)εD(B),thuszlCD(B)·SoSnallyZl=万币;.Inoursituation,fromtheproofabovefor(ii)weknowthatAzgeneratesalocdlylipschitzintegratedsemigroupsdt)onZ,sobywhatwehaveseenD(AEZ)=Z1(Ar)andthelatterishenceclosed.Wethuscompletetheproofbyshowingthatzl(Az)=Z1(A)=Z1·Indeed,itisobviousthatZI(AT)CZ1·IffεZ1=Zl(A)Jthenfor。1=叫(t,fLul=A叫+f,whereui(t,f)=So(t)fεZ1,叫(tJ)=fJSo(s)fdsε瓦,MdAU1=ui-fεZ1,henceUl(tJ)εD(A)nZ,AU10,f)εZ1,andthereforeU1εD(AZ),AUI=AZUhsothatZ1CA(AZT),henceequalityholds.. Fromthepreviouslemmasweconclude,summarizing, 3.4.THEOREM.LetAbeaclosed(linear)dissipativeoperatorinX.Thenwehave Zoczl=Z1CZ2=Z2=Znforalln;22, (30) Sn(t)=(1/(n-2)!)兀(t-s)←2sds)dsforn注2(andofcoursethesitu-tionhwhichzhzhZ2aredistinctdoesoccurforlocallylipschitzintegratedsemigroupswithgeneratorhavingnon-densedomain).Onehasthenalsoimmediatelythefollowing,apriorisomewhatsurprisingconsequence-
356 Lumer- 3iTHEOREM.LetAbeaclosed(lineM)dNipativeoperatorinX.Theneithertheinitialvalueproblem u'=Au,u(0)=f (31) cannotbesolvedforadensesetofinitialvaluesinthestronggeneralizedsenseofordernwithsomespeciEedn,nomatterhowlargethelatternischosen(inotherwords,fornon注Ocanonesolvetheinitial-valueprobleminthen-s.g-s.senseforadensesetofinitialvalues),orelseAgeneratesalocallylipschitzintegratedsemigrouponX(ifD(A)isnotdense)andanordinarysemigroupiRD(A)isdeme.pmf.Ifforsomen,Znisdense,thenby(30)Z2=石=Xand(29)holdswithA亏-=A.Therestisnowclear.-02 REFERENCESArendt,W.(1987).Vector-valuedLaplacetransformsandCauchyproblem,Ism-lJ.Math.59,327.Arendt,W.,NeubrandehF.mdSchlotterbeck,U.(1989),Interpoiationofsemigroupsandintegratedsemigroups,SemesterbeI讪1tFunctionalanalysis,TGbingen,I5,1.kenema-rhH.andHi伽r,M.(1989).Integratedsemigroups,J.Fund-Anal.84,160.Luzner,G.(1975).ProblhmedeCauchypourop4rateursbeaux,et"changementdeternps",AnnalesInst.Fourier,E5,fasc.3and4,409.LumehG.(1990a).Solutionsgeneraheesetsemi-groupesintdgr缸,C.R.Acad-Sci.Paris,SIOser-I,577.Lumer,G.(1ωob).Generalizedevolutionoperatorsand(generalized)C-semi-groups,toappear-Lumer,G.(19900.Applicationsdessolutionsgdneralisdesetsemi-groupesintdgresadesproblhmesd'evolution,toappear-NeubrandehF.(1988).IntegratedsemigroupsandtheirapplicationstotheabstractCauchyproblem,PaciEcJ.Math.,135,111.pazy,A.(1983).Semigroupsoflinearoperatorsandapplicationstopartialdiferentiaiequations,SpringerVerlag,Berun-Reed,M.andSimon,B.(1972).MethodsofModernMathematicalPhysicsI:FunctionaIAnalysis,AcademicPress,NewYorkandLondon.Titchmarsch,EC.(1975).Thetheoryoffunctions,reprintedfromthe1968correctedreprintingofthe19392nded.,OxfordUniv.Press.
SomeRegularityResultsforLinearVariationaISecond-OrderParabolicEquations ALESSANDRALUNARDIDepartmentofMathematics,UniversityofCagliari,Cag-liari,Italy 1.INTRODUCTION Inthepaper[4lweshowedoptimalregularityresultsforaclassofparabolicequationsMaboundedopensetQ: 、‘.,,,4··且.喃自A/,‘、 u.=a(aJI)+a(au)+bU+Cu+fn+a-f,O三tgT.11XJ111XaU UndertheassumptionthatQhasC2boundaryaQ,weconsideredboththeCauchy-DMchletinitial-boundaryvalueproblem: ju(0,x)=u。(x),xeQ;(12)tlu(t,x)=g(t,x),05tgT,xeaQ andtheCauchy-NCumann(conormal)initial-boundaryvalueproblem:lu(0,x)=uAx),xεQ;(13)fU|a..(tA)V4(x)uv(t,x)+a(tA)V(x)u(tA)+f.(tA)VU)=g(t,x),OgtgT,xeaQ1111XJ11 (hemv(x)is由eexteriorunitnormalvectorto讪atthepohtx). 白1eauthorisamemberofG.N.A.F.A.ofCN.RTTmworkwasputidlysupponedbytheItalianNationalproj饵I4096M.PI.··EquaZionidiEvoltlZioneeApplicaZioniFisico-Maternatick--. 357
358 Lunardi Wemcall由atacontinuoushnCHonUwithLPspatialnrstorderderivativesuxiissaidtobea(generalized)solutionof(1.1)·(12)ifuOA)=g(tA)forogtgT,xeaQ,andrnoreover j:jQ川阳+a飞iμu飞x句J俨jJ严严ρ(0仇ω叫tL以,Ax..T.c(IA)u(tA)v(ti)]dxdt=!un(x)v(OA)dx+ll[ι(tA)v(tx)-40,x)VXOA)]dxdt,lzzdtotQ foreverysmoothVvanishingon[0,Tl×aQU{T}×Q.Similarly,Uissaidtobeasolutionof(1.1)-(13)if jf:jLQ川..T.c(tA)u(tA)v(tA)]dxdt=lun(x)v(OA)dx+ll[LOA)v(tA)-飞(tA)vx(ti)]dxdt+dQ-totQ'+flQg(tx)V(tX)ω foreveηsmoothhnCHonvvanishingon{T}×Q.wcprovedoptimalmgularityresultshtheclassescαnn([OJrl×Q)={中εC([0,T抖。):supI(t-OW2+|x,ylαr1|中(t,x)-0(s,y)l〈+∞}andCOA([0,T]×Q)={OeogsdgT,XJε口,x#yC([0,T]×Q):sup|X7|叫中(tA)-90,y)|〈+∞}fortheCauchy-DMIChletproblem,0515TλyεELMYandmtheclassC叫2A([0,T]×豆)fortheCauchy-NetImamproblem.Nowwerestricttothecasewherethecoefndentsaifai,bi,Cdonotdependontime,andw附eS由hO仰wO阳EHm刷f怕削6创lder叼u1h趾Ma盯mrd向ij=0矶,..叶,n叽,belongtωOCω2气([凹0,T]k;C(Q)》)={0εC(α[0,T]×。):su叩p(0t-S吵).ω2|忡申(0I,x对)-ogt〈sgT.xeE。(SA)|〈+∞}.Ifuoecl(Q)andgeC(1+α)/2([0,Tl;COQ))(Cα/2([0,T];C1(aQ)),uolMK=g(0,·),weshowthatthcsolutionofproblem(1.1)·(l.2)issuch由atUeC([0,T];W1.p(Q))foreveryp〉1,andUεC(α-M)/2([ε,Tl;CO(Q)),εel0,T[andOε[0,1[(oncemotem川at附)klO咿toc10if阳data1arcmerelycomuouswithrespecttox).SimilarmstiltsarcshownalsoforthcCauchy-Neummnproblem:mthiscaseitissufficientthatgeCW2([0,T];C(aQ)),andthecompatibilityconditionuij(0,·)auo/axjt)+ai(0,·)uo(·)+飞(Ov)]Vi(·)=g(Ov)onaQholds.Inbothcases,i证fbjalu~lhO/启ax~j+CαωluIO+乌(仰0,旷.))andaiJJ产alu』Od/ax飞j+a气iu~0+飞(份0,扩.))klongtωOCα气(Q)forcveηi===1L,v….川叫J.叶J,J凡Iforevery0ε[归0,1H[l.
LinearvariationalParabolicEquations 359 'ITmtechniqueusedhemis由esameash[句,由atis,weconsiderproblem(1.1)-(12)and(10·(13)aswolutimequaMM1thespamE040ε((w;-m))*:390',中nevp〉1-C(Q)such由atO=中。+Di申i}andE={中εf飞(W13(Q))*:300,..,中nεC(Q)suchp〉1由a挝t申忡叫=叫吗0h0+D州r陀e臼呻呻巾机S叩mWW叭p伊阳阳川e倪创创叫Cdωtu山i忖v川th1em〈Di占巳ζ协,QVOεW1·P(Q)).Then,weusethefact(showedM[4])由atellipticoperatorswithcontinuouscoefndentsgenerateanalyticsemigroupsMthespaαsEoandE,andweapplymSUItsaboutparabolicequationshgeneralBmachspaces-币1eadditionaldifflcultycausedbythenonhomogeneousboundaryconditionisovercomebymeansoftheBalaKrishnandeviceMthecaseoftheDMchlctboundarycondition,andbywritingtheboundaryintegrallg(t,对中(x)dσxasadomainintegral|[l。(t,对中(x)-li(t,对中x(x)]dxVtε[0,TLhthedaQdQcaseoftheNCumannboundarycondition. 2.PRELIMINARYRESULTS 'IT1roughout由epaper,QwilldenoteaboundedopensetMRn,withC2boundaryaQ.WeshallconsidertheBmachspacesE0,Edemdby (2.1) γQ(WT(Q)):刊Hω川川叫叫Cq"ω咱("伯仙￡Q岛》b)μs饥川l|川|f川E=iMn时f川{z川飞|川飞|L∞:f=f乌0+DiJfi} lE={fef寸,(W1·P(Q))丰:3乌,-4eqQ)suchthatf=fo+Difi}(22)1yanlHf||E=inf{zHfi||∞:f=fo+Difi}kizO Eachelementfofsuchspacesadmitsa(notunique)representationoftheform 〈协=jJWKX)·平均x)]dxwew川)(mspvoεw1叩门 with飞eLq(Q),q'=q/(q-1),fori=0,..AForshort,weshallwritef=乌+Difi·Toavoidconfusion,foreach巳MW14(Q)wedenoteby巳xiorauaxithederivativeof己withrespωtoxi,whemsfor创1己MLq(Q)wedm忧byDi巳the伽Ilentofw;,q(Q)(resp.w1气Q))*)definedby 〈D怜=-L己啕仰VM;飞Q)(WM1q(Q Weshallconsideralsothesubspacescα(Q),C穴Q)(O〈α〈1)delnedby
360 Lunardi (2.3) (Cα气叩喻协(ωω卧配Qω船町)忖川=斗{…fωεdC1气叩喃川。岛趴讪)N川su…un!lfilch=in川军。||飞licIWf=fo+Difi} |C了〈Q)={feE:3f0',fnec14(Q)Stf=fo+Difiandfivi=oonaQ}(ο2.4份)飞lHfH4=iMIn1f川{Ll|f|H|l:f=f+Df}IC.(Q)aiC气。) In[4]weshowed由atellipticvariationalopemtorswithcontinuouscoefficientsgenerateanalyticsemigroupsmEandhE0·Tobeprecise,weneedsomemorenotation.Ifa划,ai'bi,cεC(QLandtheellipticitycondition (25)ψ,成i飞28|巳l2VOA)ε[0,Tl×岳,已εRn holds,wedenncthescsquilinearform (2.6) du…L[ai川x巾 MW1·P(Q)×W1.pω),1〈p〈∞,p'=p/(p-1).Moreover,wcsetV=w1吨Q)orV=PUW1·P(Q)accordingasthcboundaryconditionisofDirichletorofNCIImamtype,andwedchemopemtorAP:VP→VJby (2.7)〈APu,v>=a(u,v)forUεVP,veVP·· WesetnowX=EoorX=EaccordingastheboundaryconditionisofDinehictorofNCumanntype-WedennemoperatorAmXby (2.8) D(A)={ue(V:AUεX};Au=AUVp〉1p〉1PP Westatenowthegenerationresult: 古fEORmf2.lTheoperatorA:D(A)→XgeneratesananalyticsemigroupMXForeveryp〉nthereareC,R〉0,θε]W2,π[,dependingonlyonQ,ontheellipticityconstant8,onthesupnormandthemodulusqfcompwiO14的eco吃FIdeFIts,suchthattheresolventrrofAcontainsthesecrorS={keC:|刘主R,|argλ|〈O},and(2.9)|λ|llUl|x+lλl1叫lulL+SURlAMP||U||W1,KEbB(XJMfZ))三C||λu-AUHxxe&a VUεD(A).· WecharacterizeddsotheinterpolationspacesDA(日,∞)forO<p〈1,n#1/2. PROPOSITION22Wehave--
LinearvariationalParabolicEquations 36l lc2β1(Q)for0〈P〈1/2DA(白,∞)={Alcf-1(。)={uecm(。):ui旷O}for1/2〈P〈1 inIhecafeqftheDirichletbour仰Fycondition,andic2P;1(Q)for0〈。〈1/2DA(白,∞)={.二lczls-I(。)for1/2〈p〈1 inthecaseqftheNetunamboundaηconditionJwithequivalenceqftherespectivenorms.Inbothcaseswehave C(Q)CDA(1/2,∞) andtheembeddingiscontinuous.4' 咀1efollowinginterPolatOIylemmaswillbeuseallhthesequel. LEMMA23LetAbetheoperatord41nedby(2.8).扩a〈b,O〈α<1,andUεc1吨n([a,b];X)ncαJ2([ab];D(A)),thenUeC(α-e+l)/2([ab];CO(。))jbreveηOe[0,1[. Proof:Firstweshow由atUbelongstoC(1+α)/2([a,b];C(Q)).Tothisaim,weneedaninterPolatoryinequality:(210)HO|LgCH中|lx1/2||中||D(A)1nvoεD(A)Initstum,(210)cmbeshownusingtheestimateHOH∞豆C|λ|-mHλ0·A中HX,whichho1dsforeveryλsufficientlylarge,sayk〉儿。([4,由.3.2]):actually,wehave||OtgC|λ|1β||0||x+C|俨||中||D(A);takingh(λ。+1)llO||Dd||中||x(so由atbko)weget(2.10)Nowwehave: ||u(t)+u(s)-h((t+S)/2)H问三|lu(t)+u(S)-2u((t+S)β)|lx1n.Hu(t)+u(S)-2u((t+S)/2)|lD(A)1ngconstiltdl1地nl|U|i1+qnfnltdl叫2llUHω11/25COIlst.(llUl|l+ωC(怡,b];X)Cqabl;D(A))CUAbkX)+|lUHM)|t-sl(1惜)β([a,b];D(A)) so由at由atUbelongstoC(1+α〉/2([a,b];C(。)).Moreover,thankstoIdemma1.2of[匀,UeCW2+1-P([ab];DA(P,∞))foreveηPel0,1[.ByProposition22,UεC(α-64)2([ab];co(Q))foreveryOε]0,1[.· LEMMA2.4扩O〈0,α〈1,then C(α-0+l)/2([0,T];CO(Q))nC(α+04)/2([0,T];CO(Q))CC(1+α)/2([0,Tl;C(Q)) andtheembeddingiscontinuous.
362 Lunardt Proof:ItissufficienttoshowthestatementMthecasewhereaQisofclassC3,thcgeneralcasebeingaconsequence-LetUbelongtoC(α-0+l)/2([0,Tl;CO(Q))(C(α+如l)f2([0,T];CO(Q)),andlctA:D(A)={Oef飞W1·P(Q):DOxεEo}→E0,AO=DPxp〉1·Abe由evaItatiorlalLaplaceoperator-Aisinvedible,andKILlbelongstoC(α-0+l)/2([0,T];c2+气。))thankstoSchalldehTheorem,whereasitbelongstoC(α+@+l)/2([0,T1;C20(Q))thanksto[1,th.llByinterpolation,wegetA-11leC(1+α)/2([0,Tl;C2(。)),so由atU=A(A-1u)belongstoC(1崎)β([0,T];C(Q)).. Ifanoperator-AgeneratesananalyticsemigroupinaBmachspaceX,therearcseveralexistenceandmgularityresultsforthesolutionofthcinitialvalueproblem jvyt)=Av(t)+申(t),OStgT;(2.11){lv(t。)=vo Herewementiononlythe陀SUItswhichwillbeusedinIhesequel. PRomsITION25LetXKGBamchspace,andletA:D(A)CX→XgenerateananalyticsemigroupctAinX.Then--(a)扩ObelongsωCO([0,Tl;X)fO〈O<ljandvoeD(A),Avo+0(to)εD(A),thenthejknction (212)V(加仇。+je(叫你)也O三t汀,Oistheuniquesolutionqf(2.11),bothv'andAvbelongωC({0,T];X)(CO([ε,T];XLforeνeηεel0,T[.Thereiscl〉Osuchthat (2.13)HVHelm-η;X)+||Av||C([0,TliX)gcl(llO||CQUO-Tl;X)+Hvo||D(A)) (b)扩ObelongstoCO([0,Tj;X)andvoeD(A),Avo+中(O)εDA(0,∞)thenbothv'andAvbelongtoCO([0,Tl;X),andvbelongstoB([0,Tl;DA(0,∞)).Thereisc2〉Osuchdωt (2.14)HvH+HVl|.+HAvHCO+I([0,TiiX)B([0.Tl,DA(0,∞))CO([0,Tl;X)-gc2(ll中l|ce([0,Tl;X)+HVollD(A)+||AVo+中(to)llDA(0,∞)) (c)扩中belongstoCO([0,Tl;DA(白,∞))withO+。〉1,GFld中(O)=0,thefunction (21臼5)均belongstωOC0如+P盹问(α[阴0,Tη];汉X)nC0如+n卡-η叫([阴0,Tηl;泪DAμ(η,∞)》)/户breVeηηε]p0,0阳],Z(0t)忏+0帜(tO)belongstωOD(A)foreveηt,and
LinearvariationalParabolicEquafions 363 (2.16)z'。)=A(z(t)+中(t)),OgtgT. Moreoverz+中belongstoCO+扣l([0,T];D(A))nC04-T1([0,T];DA(η,∞))foreveryηe邸,1[,andthereareC3,C4〉Osuchthat ll|zH〈C||OHlCF+0([0.Tl;X)-w3VCO([0,TKDA(P,∞))'(217)i川Z|川|CM忡川Tη叩1l川z+申|川|CnμMμ+州啡0ιh叫.叫η叫(u[川;D问A(η.F冲∞叫)川)豆C4H0|川|C町町叩I阴0川川.1Tlk;D问阳Aμ(R.P∞)川),0〈η豆Rβ. Proof:TEeproofsof(a)and(b)maybefoundM[5].Mostofstatement(c)wasshownh[31Herewehaveonlytoshow由atzbelongstoCO+萨η([0,T];DA(TI,∞))forO〈ηgp,andz+中belongstoCO+P-η([O,Tl;DA(η,∞))forP三η〈1.Forη=白,由isisaconsequenceofstatement(a)(withXreplacedbyDA(户,∞)).Byinterpolation,wcgetCO+R([0,T];X)(CO([0,T];DA(日,∞))CCO+P41([0,Tl;DA(η,∞))forcvcηηε]0,阳,andCO+扣1([0,T];D(A))(CO([0,Tl;DA(日,∞))CCO+萨叫[0,T];DA(η,∞))foreveryTIε]日,1[,withcontinuousembeddings,andthcstatementfollows.4' Weend由issectionwi由anotationdremark:lctU:[0,Tl→Z,wherea〈bandZisanyBmachspaceoffunctionsdefinedinQorMQ.Throughoutthcpaper,unlesssomeconfusionmayarise,weshallidentifythefunctionUWi由thcfunction(tA)→u(t)(x),dennedm[0,Tl×Q(resp-M[0,Tl×QLConversely,ifv:[0,Tl×Q→C(orv:[0,T]×Q→C)isanyfunctionsuchthatv(tv)belongstosomeBanachspaceZforeverytε[0,T],weshallidentifyvwi由thehIICHorl[0,T]→Z,t→v(tj. 3.THECAUCHY.DIRICHLETPROBLEM Wewritenowanabstractformulationofproblem(1.1)-O.D.Itisnotdimeu1ttosecthatifUbelongstoC1([0,Tl;EdnC([0,T]×Q)andv:[0,T]×Q→Risasmoothiunction,thenjTtfj〈11,vωdt=j心叫。,x)削+lm川川o(X)V(OA)]dxOOQdQ 111crefore,ahnCHonUec1([0,T];Eo)nC([0,T]×Q)isasolutionof(1.1)-(12)ifandonlyifUsatisfies(12)mdforcveηte[0,T]wehave u'。)=DKaij(t.讪XJ(t,·))+Di(ai(t,如(tv))+bi(t,讪Xi(tj+C(t,如(tj+food+Difi(tv) MtheEosense.Towdtearepresentationformulaforthesolutionofproblem(1.1)·(12)weneedasuitableextensionoperatorD.Thefollowingpropositioniswellknown.
364 Iunardi PROH〉SπION3.1ThereisamappingDsuchthat jεL(CO(aQ),C气。))VOe[0,2];(3.1){l(D巳)(x)=己(x)V己εC(aQLVxeaQ.. Undersomedifferentregularityassumptionsonthedata,weshowedm[4]由at由chnCHon (32)u(t)=♂vjdLS忖(s)+ADg(S)]ds-Ajc(14队Dg(S)也ogtgT。。 isthcsolutionofproblem(1.1)·(13).HcmetAisthesemigroupgeneratedbyAmEo'andf(s)=f00,·)+Difi(s,·).Wcshallshownowthefollowing: TMOREM32LetQbeaboundedopensetinRnwithOboundary,andlet (3.3) jbC(Q)Vijl;fo,飞εCW2([0,T];C(Q)); Letmoreover(25)hold.T刊hemFn1卢reveryu~0εC1飞(。岛)suchthatu~0创|阳a讪Q=g(仰0,.)),t仇Jh1ef户tu4FM1町Ctωtiω.UωOωFUd〈矿φf卢;FnM1阳edi川n(σ3.22匀)i"Sthetu4m川Fn川M1uUti.qlu4esolutionQ4f(υ1.1η).(υ1.22匀),iutbelongsωC1气([仰0,Tη];沮EOρ)nC(α[0,Tη];w1,P(Q))foreveηp>1,andωCI+αβ([ε,Tl;EdnC(α斗+1)/2([ε,T];CO(Q))ncα/2(忙,TI;W1♂(Q))foreveηεεl0,TLOε[0,l[.Thereisk1〉Osuchthatn(3.4)川U|lI+||UHlgk(||U||l二+乏JHfHωC([011;Iio)C([01liW,p(Q))lOC(Q)"1C(IMlle(Q))i=0+||glldl叫β([OJrliCOQ))+||g||cun([0,Tl;cl(aQ))) 扩,inaddition, (3.5) n'EEa--.....V、‘,,,一Q/,‘、αce、,,/nu,,.、、r11+nuuqa+VA飞σ/duu「09u、‘,,,nu/,、、nup'且+AUUHL+x「AU/σu「dvbJ thenUbelongsroC1+α/2([0,Tl;Edncα/2([0,T];W13(Q))(C(α-0+ly2([0,T];CO(Q))foreveηp〉l,Oe[0,lLandthereisk2〉Osuch的" (3.6)||U||cl+qn([OIl-马)+HUHcun(lOZl;Wl♂(Q))+HU||daO+lY2([011;CO(Q))三ngk(||b加Jax.+cu+f(0·)llα一+艺||a-aliax+aU+f(Ov)||α一+Jσ10O'C(Q)-121σJ1O1C(Q)n+||叭l旷主||飞||俨川C(Q))+||g||c叫 Proof:WesctU=U+U,with12
LinearVariationalParabolicEquaIions 365 U1(t)=etA(V卢以创+jc(tdM[f(s)+ADg(S)]ds,ogtgT;。 (3.7) U20)=-Aje(td)A[Dg(S)-Dg(WS+D酬,ogtgT。 Fir3tweconsiderthefunctionu1.WeshallapplyProposition25(a),withX=E。(m由iscase,D(A)isdenseMEohO=α/2.Theinitialvaluev。=u。-Dg(O)belongstoCI(OLandhencetoD(A)thankstothecompatibilityconditionuolao=g(0,·).Moreover,sinceS→Dg(s)belongstocα/2([0,T];Cl(Q)),thenS→中(s)=f(s)+ADg(s)belongstocα/2([0,T];Eo).ByProposition2.4(吟,wegetUIec1([0,T];EdnC([0,T];D(A))(c1+叫2(扰,T];Eo)nC叫2([ε,T];D(A)),andmomover||UH+|lAuH〈C1elm-T];Eo)1C([01];马)-(H中|lCF(I0.η;X)+HU0·Dg(O)l|D(八)),so由at (3.8)Hu1||c+||UHl([0,η;Eo)1C([01];w气Q))-ngc1(Huo||cI(Q)+zHPcω([0,η;C(0))+||g||cd(i0.TW10Q)))i=O Moreover,ulisthesolutionofproblem(2.11),so由at iu11)=Alll(t)+f(t)+ADg(趴05凶l(39)1u1(O)=u0·Dg(O)Lu1(t)(x)=OonaQ,05t三T. ByIdemma2.4wegetthatu1belongstoC(α-0+ly2([ε,T];C气。))foreachεε]0,T[andoε[0,1[.KtusconsidernowthehnCHon112·SincegbelongstoC(1+α)/2([0,T];COQ)),thenDgbelongstoC(1+α)/2([0,T];C(。)).ByProposition3.1,s→Dg(s)-Dg(O)belongsIoc(1+α)/2([0,T];DA(1/2,∞)),so由at,byProposition25(C)(withX=Eo,。=(1+α)/2,P=1/2)weobtainu2eCI+α/2([0,T];Eo),u2·Dgecα/2([0,Tl;D(A))nc1·η+α/2([0,T];DA(η,∞))foreachη21/2,andestimate(217)holds.ByProposition22,u2·DgbelongstoC1·η+叫2([0,T];C2T1·1(Q))foreachη21/2,thatisu2·DgeC(α-0+l)/2([0,T];co(0))foreachOε]0,11SinceDgbelongstoC(1+α)/2([0,T];C(Q))nCα/2([0,TKC1(Q)),then,byinterpolation,itbelongsalsotoC(α-0+l)/2([0,T];CO(0)),so由at112toobelongstoC(α-M)/2([0,T];CO(Q))forO〈O〈1.Moreover,U2belongstoC1·η+α/2([0,T];DA(η,∞))foreachηε]0,1/2[,SOthat-byProposition2.2,itbelongstoc(α+84)/2([0,T];CO(Q))foreveryoe]0,1[.ByLemma2.4,112belongsalsotoc(1+α)/2([0,T];C(Q)),andthereisC2〉Osuchthat
366 LunardI (310)llu2|lc1崎n([0,ηiEo)+llu2ll♂n([01l;wLP(Q))+!lu2|!dUEM川([0,ηco(Q-))三 55C2(llgl|cl阳qn([0.11iC(aQ))+Hgllcω([011;C10Q))) Moreover112'=A(u2·Dg),so由at juj(t)=A(u2(t)-Dg(t)),ogtgT;(3.11)iu2(O)=Dg(O);Lu20)(x)=(Dg(t))(x)=g(t,x)onao,ogt三T. Summingup(3.9)and(3.11),wefindthatU=u1+U2isasolutionto(1.l)-O.D.By(3.8)and(310)wegetUeC1([0,Tl;EdnC({0,T];W2,P(Q))foreveryp,andestimate(35)follows.Uniquenessofthcsolutionof(1.l)·(12)isasimpleconsequenceofuniquenessMhomogeneousequations.If,maddition,(3.6)holds,thenA(110-Dg(O))+中(O)=Auo+f(0,·)belongstoCM(Q)=DA(ω2,∞).1TmreforeProposition25(b)isapplicable,andwegetu1εC1+α/2([0,Tl;EdncαJ2([0,T];D(A)),and||UI||cl叫([0,ηω+HALI1||cd([0,η;马)gc1(||中HC棚,η;X)+HU0·Dg(O)|lD(A)+||A(U0·Dg(O))|lDA(ω2.∞)).币1isimplies由at(3.12)Hul||l叫+HUH+HU||(α品l归.ogC(HU||l←({OJlIEC)lcun([0.η,wl.p(Q)))1C([OJrl.C(Q))30C(Q)n+5时叭j+机。+fl(0,)Hhn+圣|H川|叫叩飞νmlH|俨叫([阴OTηlk阳附叫;汇炽呻Cq∞咱(伯向0白)沙)卢川川+川叶|H|uω叭g到μ叭|H凡|」C俨川ω叫(川Cdω巾1气怡(0ωa讪Q向))户) foreveryp>1.Summingup(3.10)and(3.12)we仙1dthat(36)holds.. 4.THECAIJCHY-NEUMANNPROBLEM Tointerpretproblem(1.l)·(13)asanevolutionequationinthespaceE,wehavetorcpmsentthefunctional中→lg(t,对中(x)dσxintheformlo(t)+Dili(t).η1crefore,wc.,atkstatethefollowinglemma: LEMMA4.lForeveηgεcα/2([0,T];C(aQ))thereexistljecα/2([0,T];C(Q)),j= 0,..,n,suchthat lgo,对中(x)dσx=![l00,对中(x)-lkM)叭(X)IdxV恒[O几VOew1·P(Q),p〉n,daQtQ andHljl|c耐,o(io,η心)gCl|g||CMρ([OZixatlywithCindependentqfg.
LinearVariationalParabolicEquations 367 Proof:1TmproofissimilartotheproofofProposition2.4of[4],howeverwewriteitdownforthereader'sconvenience.Bylocallynatteningtheboundary,itissufficienttosolveourproblemwithQmplacedbyBJR勺,aQmplacedbyz={yeRn:ly|至1,yn=0},andgecα/2([0,T];C(Z))such由atg(t,y')=Oforly121·ε.Inthiscase,forogt三Tandy=(y',yn)εB+(R勺,weset -Ea--n、‘,,,,-vlhJ'七1v''/,‘、浪=尺,yi,y-t'·trs·、、,,J'E飞σb,tσ。------、‘.,,、‘.,,、‘.,,yyv4·········E··,,E飞,,.飞,,.飞Bin---zal---EA---AfIl--L where巳eC∞([0,+∞[)isanyfunctionsuch由at巳(y)=1forogy三ε/2,and己(y)=Ofory三ε-TEenwehavel|lo||cα几0(IorlxB+(Rn))gH己'|lc([0,如[)-Hg||cq几O([0,T]XBdRγandHln||CMρ([01]×BJREbgH己Hem-+∞[)Hgl|CM([011;C(BJRn)))fori=1L,….H叮.吁J,JAra切/a均yn1Ln,SωO由a剖tforeachηεW1切'币叫p町(但B+(Rn勺)》),p〉n叽,weget -VE--』TnUFE--ke-E·E-vvdAUE··2、‘,/vd/,‘、xn叫、‘,,,vd..‘,,.、···A、‘,,,VJ/,‘、n叫、‘-sfvd..‘,,.‘、nur--E‘nR,,E飞品τB,.....,,--wvdAU、‘,/vd,,.、nH·、‘,/V6..‘,,.‘、σbz,····EEE-, WeremarkthatthechoiceldtA)=z(tA),lox)=z(t,x),wherezisthesolutionofUEXSAz(tv)=z(t,·)MQ,az(t,·)/av=g(t,·)MaQ,isnotsuitableforourpurpose,becauseifgεC的([0,Tl;CGQ)),thchnctionslidonotklongnecessarilytoC的([O,TkqQ)).ThankstoIJCmma41,wecanseeeasilythatahnCHonUec1([0,T];E)(C([0,T]×Q)isasolutionof(1.l)-(13)ifandonlyifu叭"州…(0ω加…tO们川)归川=斗D川t…)川训川…)μ川+叶D+Dif飞i(0t,.少)+DiJli(0t,.)),。三t三T; wherelv·-,lnamgivenbyIA3ITUna4.1.1Ylemfore,Uisgivenbythcvariationofconstantsformula (41)u(he飞 withhO)=fdt)+ldt)+Difibv)+Dili巾,·).Weshdlshownowthatthefunctiondennedin(4.1)isMfactthesolutionof(1.1)-(13). THEOREM4.1LetQbeaboundedopensetinrwithdboundaη,andler
368 Lunardi (4.2) jbiC(Q)Vij1;f。,飞εCW2([0,T];C(豆)); nzenjbreveηuoεcl(Q)suchthataij(Ov)audaxj(·)Vi(·)+ai(0,·)Vi(·)uoL)+飞(0,)Vi(·)=g(Ov)onaQ,thejunctionUdRflnedin(4.1)istheuniquesolutionqf(1.1)·(13),itbelongstoC1({0,T];E)(C([0,Tl;W1♂(Q))andtoc1+αf2([ε,T];E)(Cω(柱,Tl;W1·P(Q))nC(1+α-0)尼([ε,T];CO(Q))foreveryεε]0,TLP〉1,0ε[0,11Thereisk3〉OsuchdωI (43)HU||1+||UH1〈C([0,1丁;E)C(IOJ]iWS(Q))-ngk3(lluo||c1倍)+zH飞||C叫[aTliC(Q))+||gHc叫[(川CGQ)))1=0 扩,inaddition, (4.4) n4·EA--.....V、‘.,,,-QJS‘、αce、‘.,,,nu/,‘、rlI+nuu兔"+x气。/duu「dqu、‘.,Jnu,,‘、AUrl+nuunL+x飞。fduu飞。bJ thenUbelongstoC1+αβ([0,TI;E)nC(1+α-0)/2({0,Tl;CO(QDncα/2([0,T];W1·P(Q))foreνeηp〉1,Oe[0,1[,andthereisk4〉Osuchthar (45)HU||cl+qn+HU|ldu+|lU||(α-M归([0,T1iE)C(lo-Tl-w(a))C([0.T];C(Q)))-ngKA(||baudx+Cu+f(O)||α-+24||aallaxHiuo+f(0,)川α-+4【YJ00'C(Q).AEfEJσJEC(Q)n+||uo||cl旷王"。叫 Proof:WeapplyProposition25(a)(bLwithX=E,o=α/2,中=handv。=u0·白1ehnCHonhbelongstoC叫2([0,Tl;E)thankstoLemma4.1,uobelongstocl(。)CD(ALandAUo+h(O)=Di(aij(0,·)a/axJUo)+Di(ai(0,·)uo)+bi(Ov)a/axiuo+C(0,·)uo+Di(fi(0,·)· lI(0,·))+乌(0,·)-10(Ov)belongstoD(A)={中εE:300,..,。nεC(Q)such由at申IVi=oonaQ,中=中。+Dioi}becausethecompatibilityconditionaij(0,·)auo/axj(·)Vi(·)+ai(Ov)VI(·)u。(·)+fi(Of)Vi(·)=g(0,·)=1i(Ov)vi(·)onaQholds.App1yingProposition25(的,wefindthatUbelongstoC1([0,T];E)nC([0,TKD(A))(Chα/2([ε,Tl;E)ncαn([ε,T];D(A)),||U||l+HAuH三C(l|hHd+HU||),andC([0.Tl;E)C([0.η;E)lC([011;E)OD(A){u'(t)=Au(t)+h(t),ogtgT;(4.6)才lu(O)=uo
LinearvariationalParabolicEquations 369 Equdities(4.6)imply由atUisasolutionof(1.1)·(13).Momover,的mestimate(2.13)andIJemma4.1wegetestimate(43).ByapplyingLemma23,wcgetalsothatUbelongstoC(1+α-0)尼([ε,T];CO(。)).Itmaddition,(4.4)holds,thenAUo+h(O)εc:吨。)=DA(α/2,∞)byProposition22.ApplyingProposition25(b),wcgetUec1惜β([0,T];E)ncαn([0,T];D(A))CC叫2([0,T];w1·P(Q))(C(1+α-8)/2([0,T];CO(Q))foreveryp〉1,0ε[0,1[.Estimate(45)follows仕om(2.14)andIdemma23.. REFERENCES [llS.CAMPANATO:EquaZioniEllirrichedelIIOOrdineeSpaziL2.λ,Am.Mat.PLIraAPPI-68(1965),321·382 [2]A.LUNARDI:OntheEvolutionOperatorjbrAbstractParabolicEqωrions,IsraeliMa由.60(1987),281·314 [3]ALUNARDI,ESINESTRARI,W.VONWAHL:AsemigroupApproachtotheTimeDependentParabolicInitial-BoundaryValueProblem,PreprintDipart.Mat.Univ.Roma(July1989) [4lA.LUNARDLV.VESPRI:HdlderRegulariηinVariationalParabolicNonhomogeneousEquations,J.DifEEqns.(toappc盯)[5]E.SINESTRARI:OntheAbstractCauchyProbleminSpacesqfConttmotuFmctiorts,J.Math.Anal.APPI-107(1985),16.66
GeneralizationoftheEfille-YosidaTheorem ISAOMIYADERADepartmentofMathematics,SchoolofEducation,WasedaUni-versity,Tokyo,JapanNAOKITANAKADepartmentofMathematics,KochiUniversity,Kochi,Japan §1.INTRODUCTION-WeconsiderageneralizationoftheHille-Yosidatheorem-Throughoutthis paper,letXbeaBamchspaceandletusdemtebyB(X)thesetofailbounded linearoperatorsfromXintoitself,andletCbeaninjectiveoperatorinB(X). Aone-parameterfamily{S(t);t三0}inB(X)iscalledanexpomntially boundedC-semigro叩(hereafterabbreviatedtoC-semigro叩)if (11)S(s+t)C=S(s)S(t)fors,t主OandS(0)=C, (12)S(·)z:[0,∞)→XiscontinuousforZεX, nu>-a'tuFAO俨TAaPUM〈一、‘.,,,'tu,,,‘、sa'LWa-BEa-4bLUCucunu>一αZGnanu>-MeFAaere-ua'ιU、‘,,,,句。唱,..,,,‘飞 IfC=I(theidentity)theneveryC-semigro叩isasemigroupofelms (Co)introducedin[6land[14l.Let{S(t);t主0}beaC-semigro叩satisfyi鸣(1.3),anddenmLλεB(X)for入>αby (14) Lρ=ffMSMf…x SimHarlyasinthecasewheretilerangeR(C)isdenseinX(see[3l),wesee thatLλisinjectiveforλ>αandtheclosedlinearoperatorAdennedby (1.5)Az=(入一LγCMforZED(A)三{zεX;CzεR(Lλ)} isindependentofλ>α.TileoperatorAwillbecalledthegeneratorof{S(t);t主0}.Itisknownthat{D(A)={ZEX;lim(S(t)z-Cz)/tεR(C)}1t→0+(1.6)(lAz=c-IAZL(S(t)z-cz)/tforZεD(A) 37I
372 儿fvaderaandTanaKG (Seel13,Propositionlilor[4,Proposition32l.)Itshouldbemtedthataone-parameterfamilyinB(X)isaC-semigro叩ifandonlyifitisanIR2ι-S肥emmi厄grmOuofexponentialgrowthinthesenseofDaPratopl,and(1.6)showsthatthegeneratorofaC-semigroupcoincideswiththeinaIlitesimalgeneratorintroduced inl21. GenerationtheoryofC-semigroupsisveryimportantbecauseitprovidesuswithauniaedtreatmentforgenerationtheoremsofsemigroupsofthebasicclassessuchaselms(C(k))andgrowthorderαorn-timesintegratedsemigroupsbychoosingsuitableoperatorsC.(See[3l,间,[町,{9l,[12land[131.) LetusrememberthefollowinggenerationtheoremofC-semigroupsduetoDavies-Pangl3]: SupposethattherangeIZ(C)isdenseinX.Thentlmfollowingassertionsareequivalent: (I)AisthegeneratorofaC-semigro叩{S(t);t主0}satisfying(1.3). (II)Aismaximalwithrespecttothefoiiowingproperties; (Ai)Aisadenselydennedclosedlinearoperatorand入一Aisinjectivefor入〉α, (AdD((入-A)-m)3R(C)and||(入一A)-mc||三M/(入一α)mfor入>α and772221, (AOCzεD(A)andACz=CAzforZεD(A)-Later,Mi〉Faderal7lshowedthat(I),(II)andthefollowing(III)aremutu-allyequivalent:
HIlle-YosidaTheorem 373 (III)Asatisaes(A;),(A2)andtheconditionthatA=C-1AC. Inparticular,ifC=Ithentheresultaboveisthewell-knownHille,Yosidatheorem. Now,wedonotassumethatIZ(C)isdenseinX.Inthiscase,thegeneratorAofaC-semigro叩{S(t);t主0}satisfying(1.3)isnotnecessarilydemelydennedinX(see[4,Example620,butitisstillmaximalwithrespecttotheproperties(A2),(A3)and (AOAisaclosedlinearoperatorandλ-Aisinjectivefor入>α. Weareinterestedintheconverseproblem.Thatis,ourproblemisasfollows: SupposethatAismaximalwithrespecttotheproperties(A1)-(A3)-Then,doesAgenerateaC-semigroup? ToinvestigatethisweshalldealwithintegratedC-semigroupsin§2,whichhavebeenintroducedin闷,andtheresultsaregivenin53. §2.INTEGRATEDC-SEMIGROUPS- Aone-parameterfamiiy{U(t);t三0}inB(X)iscalledanintegratedC-semigroupif (21)U(·)z:[0,∞)→XiscontinuousforZεX,(2.2)U(t)z=Oforallt>OimpliesZ=0,(2.3)U(0)=0(thezerooperator)andU(t)C=CU(t)fort三0,(2.4)thereexistk主Oandb主Osuchthat||U(t)||三ICebtfort主0,
374 儿ft)FaderaandTanaka (25)U(s)U(t)z=fJ+tU(r)Czdr-fJU(T)Czdr-duo-)CzdrforZEXand s,t220, EveryintegratedLsemigroupisaonceintegratedsemigroupintroducedby Arendt[11. Let{U(t);t主O}beanintegratedC-semigro叩·For入>ωowe〈lean L(入)εB(X)by 川=入ffλt川dtf…x,whereω。=mαz{0,limt→∞(log||U(t)||)/t}. (2.6) ItisknownthatL(入)isiuectiveforλ>ωoandtheclosedlinearoperator Adennedby (2.7)Az=(入一L(λ)-IC)zforZ巳D(A)三{zεX;CzεR(L(λ))} isindependentof入>ωo(seel8l).TIleoperatorAiscalledthegeneratorof {U(t);t主O}.TIlegeneratorAhmthefollowingproperty{(入-A)L(入)z=CzforZεXαnd入>ω0,(2.8){lL(λ)(入-A)z=CzforZεD(A)α叫入>ω0·Wereferto[8lforfurtherinformationonintegratedC-semigroups- WenowgivetypicalexamplesofintegratedC-semigroups- Example1.LetAbethegeneratorofaCESmigroup{S(t);t三0}satis- fyi吨(1.3).IfwedeEmU(t)εB(X)fort主Oby川=fwdsf…x, then{U(t);t主0}isanintegratedC,semigro叩whosegeneratorisA,and ||U(t+lz)-U(t)||三MMa川的forkh三0.
HIlle-YosidaTheorem 375 Example2.LetAbeaclosedlinearoperatorsuchthatC-Aisinjective andD((c-A)-1)3R(C)forsomeconstantc.(NotetI时(c-A)-1CεB(X)bytheclosedgraphtheorem.)IfAisthegeneratorofa(c-A)-1C. semigroup{S(t);t主0}satisfying(1.3),thenitistlmgeneratorofmintegrated (2.9) C-semigroup{U(t);t主0}dennedbyU仰=川)f川ds=cf.S仰SforZεXandt〉0. Proof.WeErstnotethat (2.10)S(t)zεD(A)andAS(t)z=S(t)AzforZED(A)andt主0, (211)ds(收dsED(A)andS(t)z一(c-A)-1Cz=Ads(收dsforzεx andt220, (see[13,Proposition12l). By(210),S(t)Cz=(c-A)S(t)(c-A)-1Cz=(c-A)(c-A)-1CS(t)z=CS(t)zforzεXandt主0,andby(2.11),(c-A)ds(收ds=cds(收ds-S(t)z+(c-A)-1CzforZEXandt主0.TI阳efore{U(t);t主0}dennedby(2.9)satiSEes(2.1)-(2.4).Toseethat(2.5)isalsosatided,letZεXands,t主0.Itfollowsfrom(210)that叭印附S吟)川=川)才f川附(才fρρt〉〉S贝(ωωW川η川ψ例)Mμ灿Zd叫d&?O=(c一刊VS(己+川71)d己rsrt+￡re=l{(c-A)/S(η)Czdη一(c-A)/S(71)Czd什dz=f飞οczd己-JSU(υczdz
376 儿ft;yαderGandTanaka Consequently,{U(t);t主0}isanintegratedC-semigroup- ToshowthatAisthegeneratorof{U(t);t主0},letλ>mαz{α,ω。}-SinceAisthegeneratorof(c-A)-1C-semigroup{S(t);t主0}itfoilowsfrom [13,(1.5)lthat cz=(c一队(λ-A)z=川)fJtS州一机=fMt川-M=州)(入一均f…εD(A) HereLλmdL(λ)areboundedoperatorsdennedby(14)and(25)respectively. ThismeansthatD(A)CD(A)andAz=(人-L(人)-1C)z=AzforZεD(A), i.e.,ACA,whereAistilegeneratorof{U(t);t三0}.Next,letZεD(A)-TilenthereisazεXS旧llthatCZ=L(λ)z=ffM斗tU(t)zdt=(c-A)fffλtS(t)zdt=(c-A)LλAwhichimplies(c-A)一ICZεR(Lλ)μ., ZεD(A). Q.ED. AsaconverseofExample2weobtain Proposition.LetAbethegeneratorofanintegratedC-semigroup {U(t);t三0}andC>ω0·TilenAisthegeneratorofa(c-A)-1C-semigroup {S(t);t主O}dennedby (2.12) S(t)z=(d/dt)C一1U(t)(c-A)-1Cz=ere-csULH仙一川forJEεX叫主。 Proof.WeErstnoteby(2.8)that(c-A)-1C=L(c)εB(X).SinceW)(c一A)γ-→1CZ=才f俨∞飞C俨川υ=CJ二
HIlle-YosidaTheorem 377 forZεXandt之0,weseethatc-1U(·)(c-A)-1Cz:[0,∞)→XisdifereE tiableforZεXand 阳内附一川z=cffCSU(s+t)zds-mz forZεXaMt三0.LetusdeamS(吟,t主0,by(212).Clearly,(12)and(1.3)aresatidedandS(0)z=cjffCSU(s)zds=L(c)z=(c-A)-1CZforZεX,i.e.,S(0)=(c-A)-1C. Moreover,forZεXandt,822Oweobtain 才盯削削)X(才f俨ρ8V)S贝(价ω协川ηωψ)汩川Zdd悄=才f庐ρt、)S贝盯(α阳巳=f+t川 DifereMiati吨thiswithrespecttotandthens,wehavethatS(t)S(s)z= S(t+s)(c-A)-1CzforZEXandt,S主0.Therefore{S(t);t主0}isa (c-A)-1C-semigro叩· Next,toprovethatAisthegeneratorof{S(t);t兰0},let入>mαz{α,ω。}-WehaveforZEX (2.13) (c-A)-1CL(λ)z=L(入)L(c)z=cfh-M内。)川)叩t=cffMSM=ω HereLλistheboundedoperatordennedby(14).LetZbethegeneratorof {S(t);t三0}aMletZεD(Z).Then(c-A)-1Cz=LλUforsomeuεX,andhence(c-A)一1C-Cz=C(c-A)-1Cz=CLλU=(c-A)-1CL(入)uby(213).ThisimpliesthatCz=L(入)uεR(L(入))andhenceZεD(A)andAz=
378 儿fipFaderaandTanaka λz一L(λ)-1Cz=入z-U=(λ-L;1(c-A)-1C)z=Zz,i.e.,ZCA.ToshowthatD(A)CD(Z),letZεD(A).TIlellthereisauεXsuchthatCz=L(入)U-Itfollowsfrom(213)thatC(c-A)-1Cz=L(c)Cz=(c-A)-1CL(入)ν=CLλUwhichimplies(c-A)-1Cz=LλνεR(Lλ),i.e.,ZεD(Z).ThereforeA=Z-Q.E.D. 53.THEOREMS. Themainresultisstatedasfollows: Theoreml.TIlefollowingassertionsaremutuallyequivalent: (i)AistlmgeneratorofanintegratedC-semigroup{U(t);t主O}satisfyi吨 (31)||U(t+lt)-U(t)||三MKa(川)fort,la主0, whereAfandαareIIOIIIlegaiJiveconstants- (ii)Aismaximalwithrespecttothefollowingproperties(A1)一(A3); (A1)Aisaclosedlimaroperatorand入-Aisi时ectivefor人>α, (A2)D((入-A)-m)3R(C)and||(入-A)-mC||三M/(入一α)mfor人>α andm251, (A3)CzεD(A)aMACz=CAzforZED(A),i.e.,Acc-1AC.(iii)AsatisEes(AI),(Adaml(Ai)A=C-1AC.(iv)Aisthegeneratorofa(c-A)-1C,semigro叩{S(t);t三0}satisfying(3.2)||S(t+h)-S(t)||三MFlzea'仰的fort,h主0,
ffille-YosidaTheorem 379 whereCissomeconstaMsuchthatC-AisinjectiveandD((c-A)-1)3R(C), andαFaIIdλffareIKonnegativeconstants. Proof.Thefactthat(i)isequivalentto(ii)hasbeenprovedin[81.Supposethat(ii)issatiSEed-ThenACC-1ACmdc-IACsatides(A1)一(443)withAreplacedbyC一1AC.ThemaximalityofAwithrespectto(A1)一(A3)impliesA=C-1AC',andheme(ii)implies(iii).Conversely,letAsatisfy(A1ο),(A2υ)a缸In1(A;U).SupposethatACAFandAFsatiSEes(A1)一(AdwithAreplacedbyAF-Then,(λ-A)-lC(λ-AF)z=(入-AF)-1C(λ-AF)z=(λ-AF)-1(入一AOCz=Czforλ>αandZεD(AF),fromwhichitfollowsthatA'CC一1AC=AandheMeAF=A.ThereforeAismaximalwithrespecttotheproperties(A1)一(A3). Fhall〉飞theassertionthat(i)isequivalentto(iv)followsfromExample2 andProposition. Q.ED. [1,Example6.4lshowsthattheequivalentcOMitionsofTheoremldonotimplythatAgeneratesaC-semigroup-However,tllefollowiIlgisknown. Theorem2([80.IfAsatisaesoneoftheequivaleIItconditionsofTheorem1,thenthepartofAinD(A)isthegeneratorofaCrsemigro叩{S10);t三0}onD(A)satisfyiRg||S10)z||三Meat||zl|forZεD(A)aMt三0,whereCIistherestrictionofCtoD(A). FinallywecomderthecasewhereR(C)isdenseinX.ItiseasytoseethatifAisthegemratorofaC-semigro叩thenR(C)CD(A)(see[4,Theorem2.4l).Hence,ifR(C)isdenseinXthenAisdenselydeamdiBX.CombiIlhgthiswithExample1andTheoremsland2,weobtain Theorem3([3],[70.SupposethatR(C)isdemeinX.Thefollowin
380 儿fiyaderaandTanaka assertionsaremutuallyequivalent: (I)AisthegeneratorofaC-semigro叩{S(t);t三0}satisfying(1.3). (II)Aismaximalwithrespecttotheproperties(A;),(A2)and(A3). (III)AsatisEes(A;),(A2)and(Ai). AsadirectconsequeaceofTheorem1weobtain Corollary([8]).IfAsatiSEes(A1)一(443)thenC一1ACisthegeneratorof a口iMegratedC-semigro叩{U(t);t主0}satisfying(3.1). Proof.PutB1=C一1ACandB2=C一1B1C.TIleIIACB1CB2and Ba,i=1,2,satisfy(A1)一(443)withAreplacedbyBi-Similarlyashthe proofofn(iii)斗(ii)"inTheorem1,weobtainBzcc-IAC=B1.There- forewehaveB1=Bz=C一1BIC.ThecoachmioIIfollowsfromTheorem1. Q.E.D. Remark-[13,Theorem21lcanbeprovedfromthiscorollaryaMTheorem 2.OIltlleotherimld,theassertionthat(iii)implies(iv)inTheoremlcanbe provedbyusi吨tileargumentdevelopedinl13,§2]. REFERENCES 1.飞VArendt,vectcTvaluedLαplacetTMEshTmsaηdCαuchy17Toblems,IsraelJ.Math.59(1987),327-352. 2.G.DaPrato,SE77ugTUFFJtregolαTZZZαbzli,Ricerchedi肌ht.15(1966),223-248.3.EBDaviesandKIf-MIIPang,TheCauchyproblemandageneralzzαtzonofthefIzlie-YoszdαtftEOTEm,Proc.LondonMatil-Soc-55(1987),181-208.4.RdeLalit〉eIIfels,C-semigToupsandtheCαuchνPTobitm,J.FIIIICtAnal.(toappear).5.RtieLaubeIlilyls,IntqTatedsemzgTOUPSpC-semtgToupsαTLUitheabstractCαuchνpmbem1preprInt. 6.EHilleantiIZSPhillips,"FunctionalAnalysisandsemigroups,"Amer.Math.SocColioq.Publ.,1957
ffille-YosidaTheorem 38l 7.IMWadera,07sthegeneratorsofezponentiallνboundedC-semigToups,Proc.JapanAcad-Ser.A62(1986),239-242.8.IMWadeI飞AgeneTdizdionoftheHille-YosidαtheOTtm,Proc.JapanAcadSer.A64(1988),223-226. 9.IMWaderaandN.Tanaka,EzponenttallνboundedC-semtgToupsandgeneratzonofse7711·gToups,JMath.Anal.Appl.143(1989),358-378.10.FNeubrander,IMegTatedsemtgToupsandthezrappltcationstothedstTactCauchνprob.lem,PacincJ.Math.135(1988),11-155. 12.N.TanakaandIMiEFadeE飞SomeremuksonC-semzgToupsαndtntegmtedse7711groups,ProcJapanAcad-Ser.A63(1987),139一142.13.N.TanakaandI.MWadera,EzponentiallνboundedC-semtgToupsandintegratedsemz·gToups,TokyoJMath12(1989),99-115. 14.K.Yosida,"F飞IrldiorlalAnalysis,"Springez」Vedag,1980.
CertainSemigroupsonBaEEachFunctionSpacesandTheirAdjoin" J.M.A.M.VANNEERVENCentreforMathematicsandComputerScience,Am-sterdam,TheNetherlandsBENDEPAGTERDepartmentofMathematics,DelftUniversityofTechnology,Delft,TheNetherlands lnthisn。teC·O-semigr。ups。nBanachfuncti。nspacesarestudied.lnthenrstpartwearec。ncernedwiththepmblemunderwhat∞nditi。nsthesemigr。updualspaceisasubspace。ftheass。datespace-lnthesec。ndpartweinvestigatewhenamultiplteati。n。peratmdthef。rmApzf=hfgemrat臼aCvsemigmup.F。rth。"hhrwhichthisisthecasewegiveareprsentatbnhrthesemigr。updualspace. 1980MathematicsSuHectClassiHcation:47D05keywuds&phrases:Multiplicati。nsemigr。up.Banachfuncti。nspace-adj。intsemigr。up,sun-reflexivity. 1.Preliminaries Let(Q,2,μ)beaσ-RMtemeasurespaceandletLO(μ)denotethelinearspamofμ-measurablefunctionsonQwhichareRIliteaAAsusualμ-a.e.equalfunctionsareidentiaed.Alinearsubspa,ceXofLO(μ),eqUppedwithanorm||·||,iscailedaBMachfunctionspace(over仰,2,μ))ifXisaBmachspamwithrespectto||·||aadfELO(μ),gEXwith|f|三|g|a.e.impliesthatfεXMdl|f||三||g||.NotethateveryBamchfunctionsp缸eisaBamchlattice.Forthebasicthωryconcer旧ngBanadfunctionspaceswerefertothebooks[3i,[81,[9l.Wewillmailsomeoftherelevantfacts.WesaythatXiscarriedbyQifthereisnosubsetEofQofpositivemeasure呐ththepropertythatf=Oa.e.onEforailfεX,orequivdentlyifforeveryECQofpositivemeasurethereisasubsetFCEofpositivemeasuresuchthatthecharacteristicfunctionXFbelongstoX.QalwayscontainsasubsetQosuchthatXiscarriedbyQ＼Q0.ThereforewewillassumehenceforthwithoutlossofgenerditythatXiscarriedbyQ.Theassociatespace(sometimescailedthekatileduai)ofArisdennedby XF={gεf(μ):儿|均|dμ<∞,vfEX} XFisaBmachfunctionspacewithrespecttothenormevenby ||g||=品|儿fg句| 383
384 vanNeerenanddepagter EverygEX'deanesaboundedlinearfunctionalφgEX·viatheformula 川=儿川,vfEX Wehave||g|lx,=|!φg||x·.ThereforeX,canbeideMinedwithaclomdsubspamofx·.InfactX,isevenabandinX·.ThenormofXiscalledordercontinuousiffn↓OinXimplies||fnl|↓0.Xhasordercontinuousnormifandonlyifxr=X·.Alinearfunctionalφεx·iscalledordercontinuousiffn↓OinXimplies忡,fn)→0.OneCMshowthat4bEX·isordercontinuousifandoniyifoEXF.Finaily,apositivelinearoperatorT:X→Xiscdiedordercontinuousiffn↓OimpliesTfn↓0.Wewiudsoneedsometmmologonadjointmmigmups-S回[li,[51,[6iformoredet出ls.LetT(t)beaC·O-semigroupofoperatorsonaBanachspaceX.Theadjointsemigrouponx-isdennedbyT·(t)=(T(t))·.T·(t)neednotbestronglycontinuous.WedeRIle 、,,E,nu--.z-zd,..飞Tmwxr巳-z,,、.、--③Y4 X①isanorm-closed,weak--densesubspamofx·.Infact,ifAisthegeneratorofT(t),thenx@ispreciselythenorm-closureofD(A·).X9isinvariantunderT·(t),sotherestrictionsT①(t)ofT(t)toX③deheaCo-semigrouponX②.Applyingthesameconstructiontothissemigroup,wedeRIleX③①=(X9)@.Themapj:X→X③·, Uz,z①):=(z③,z) isacMailymembeddingwhichmapsXintoX③①.Incasejay=X⑦⑦wesaythatXissun-rebxivewithrespecttoT(t).Itiswell-knownthatthisisthecaseifmdonlyiftheresolventR(λ,A)isweaklycompact-HT(t)isaCrsemigrouponaBanachfunctionspaceX,thenonemayaskunderwhatconditionswehaveX①CXF.Triviaily,thisistruewhenerhasordercontinuousnorm.ReedthataBanadllatticeissaidtobeσ-Dedekindcompleteifeverycountablesubsetthatisboundedfromabovehasasupremum-EveryBanachfunctionspaceisσ-Dedekindcomplete.Lemma1.1.SupposeT(t)isaCrsemigrouponaBariachttnctioIlspaceX.ThenthebandgeneratedbyX①jsequaltoX·.Prwf:ByaresultofSellaefermabandintheduaiofaσ-DedekindcompleteBmachlatticeissequentidlyweak--dosed.LetYdenotethebandinx-generatedbyX⑦andtakeφEX-arbitrary.Since 人nR(λ",A)·φ→φwmk-foranysequenceAn→∞ing(剧,andsinceAηR(儿,A)'φε.Y⑦.itfollowsthatφeIrandhenceY=X·.////Theorem12.SupposeXisaCrsemigrouponaBanachfunctionspaceX.ThenX③cxrifandonlyifXhasordercontinuousnorm.Pmοf:IfXhuordercontinuousnorm.thenJX'=X·.SOtriviallyX⑦CXFholds-Conversely‘supposeX⑦CayF.SinceayFisabandinx·,byLemma1.iwehaveI'‘CaY'‘forcingX'=ey--////
aSemigroupsonBGFtaelVFunctionSpaces 385 Weremarkthatthes缸neresultholdsmutatismutandisformyσ-DedekindcompleteBanachlattice.TheequivalenthypothesesofTheorem1.2area-lwaysfunnedinthesun-reHexivecase-ThisisthecontentofTheorem1.4bdow-RecallthataBan缸hspaceiscailedweaklycompactlygenerated(WCG)ifitistheclosedlinearspmofoneofitsweaklycompactsubsets.Lemma13.SuppωeaBanadspaceXissun-rebxiveMthmspecttoaC·O-semigroup-ThenXdoesnotcontainasubspaceisomorphictol∞.proqf:SupposethecontraryandletYbeasubspaceofXwhichisisomorphictol∞.Sincel∞iscomplementedineveryBanachspaceCOMaiIUngitMasubpaceH,Prop.I.21.刻,itfollowsthatYiscomplementedinX.SincetheresolventR(λ,A)isweaklycompactandR(入,A)(X)=D(A)isdense,XisWCG.NowcomplementedsubspazesofWCGspacesMetriviailyWCGagain.Weconcludethatl∞isWCG,acontradiction.Infaεt,everyweaklycompactsetofl∞isseparable(e.g.notethatl∞embedsintoL∞[0,llandapplyp‘TilIII-VIIIA131).//// Aσ-DedekindcompleteBanachlatticenothavingordercontinuousnormcontainsasubspaceisomorphictol∞[4.Prop.ILIA-7].Hencethefollowingisanimmediateconsequenceofthepreviouslemma.Theorem1.4.SuppωeXisaσ-DedekindcompleteBanachlattice.IfXjsstIII-rebxivewith陀specttoaC0·semigro叩T(t),thenXhasordercontinuousnorm.InparticularthisresultappliestoBalladlfunctionspaces-Finallywewillconsiderpositivesemlgroups- Theorem15.SupposeT(t)isapositiveCrsemigrouponaBMachfunctionspaceIThenX①CX'ifaIIdonlyiffn↓Oimpliesl|R(λ,A)人||→0.pmf:SinceT(t)ispositive,邸,A)ispositiveforAlargeenough-SinceX,isdosedandX@istheclosureoflZ(入,A)·(X勺,itsumcestoprovethatforapositivelinearoperatorT:X→XwehaveT·(X·)CXFifandonlyiffn↓Oimplies||Tfn||→0.Firstweprovethey飞part.LetφEX·.ToprovethatT-OEX',letfn↓OinX.Byassumptionthisimplies||Tfn||→0.Inparticular,忡,Tfn)→0,SO(T-0,fn)→OandhenceT-oεXF.Conversely,assumeT-x-CXF.LetOEX·bepositiveandsupposefn↓OinX.SinceT·φεx'wehave忡,Tfn)=(T-0,fn)→0.SinceTispositiveweact114lyhave(φ,Tfn)l0.Sincethisholdsforailpositiveφ,from[9lwededuce||Tfn||→0.//// 2.Themultiplicationsemigroup LethELO(μ)beacomplex-v411edmeasurablefunctionanddeRIletheoperatorAhby D(Ah)={fEX:ftfEX};Ahf=fzf,fεD(AA). 、‘..,,,咱E··,,‘.‘、 NotethatAhisaclosedoperator.Put En={sEQ:|h(s)|三π}, (2)
386 vanNeerenanddepagter- letu.beitscharacteristicfunctionanddeRIlethebudprojections pn:X→X,Pnf=χιf. (3) Since|Pnf|三|f|forailf,pnindeedmapsXintoX.Infact,fromthelatticepropertyofthenormweseeimmediatelythatpnisacontraμionmapping.Ingenera-lD(Ah)neednotbedense,astheexampleX=L∞(0,1),h(s)=s-lshows.AsubsetBofLO(μ)iscailedsolidifthefollowingholds:whenever|/|三|g|andgEBthenaisofEB.Inparticular,ifBissolidandfεBthenalso|f|εB.Itiseasytoseethatthenorm-closureifasolidsetissolid.Anidealisasoiidlinearsubspace-NotethatbydeRMUoneveryBmachfunctionspamisanideaiinLO(μ).Proposition2.1.D(Ah)issolid.Moreover,D(AA)isdenseifarldonlyiflimn||Pnf-f||=oibrallfEX.Pmf:SupposegED(AJJandletfEXbeafunctionsatisfying|fl三|g|.ByassumptionhgEX,hencealsolhg|εXsinceXisanideal.But|hf|三|hgLSOhfEXwhichimpliesthatfED(Ah).ThisprovestheHrstassertion.Suppose||PJ-fll→OfordfEX.ToprovethatD(Ah)isdenseitsumcestoshowthatPJED(Ah)forailfEX.ButonEnwehave|h(s)|三n,SO |hpnf|:三|nPJ|三n|f| showingthathpnfEXandhencepnfED(AA).Conversely,supposeD(AA)isdense-Fir5tletfED(fih).Then FJKH呻Al-n一-rJL"1-n<-rJ-E、、、QV4=rJrJ凡P Hencebythelatticepropertyofthenorm, ||Pnf一f||才||Ahf|!→0,n→∞ SinceD(AA)isdensemd||Pn||三1fordin,thegemraicasefollowsfromadensityargument.//// ObservethatitisanimmediatecorollaryoftheabovepropositionthatontheBanachfunc-tionspaceX=L1(R)nL∞(E)equippedwiththenom||f||:=max{||f||LI(酌,||f|lL叫R)),everymultipiicationsemigroupisuniformlycontinuous.WewillnowcharacterizethosehELO(μ)whicheverisetoageneratorofaCo-semigroup-Theorem2.2.AhgeneratesaCrsemigrouponD(Ah)ifandonlyifReh三KforsomeconstaIltIY. PFWFSupposerlhgeneratesaCo-semigroupT(t)ontheclosureofD(fU).LetthesetsEnbedennedby(2).IfaconstantA'asabovedoesnotexist,thenforeverynthereisasetFnofpositivemeasuresuchthatReh>nonFn·SinceXiscarriedbyf1、therearesubsetsGnCFnofpositivemeasuresuchthatthecharacteristicfunctionsyG"belongt0.Y.SinceQ=lJKELthereisaknsuchthatEhnGnhaspositivemeasure.Since IEhnG‘三χG"
SemigroupsonBmachFunctionSpaces 387 itfollowsthatXEhnG.εX.Moreover,since|h|:三knonEhwehaveXEhnG‘εD(Ah),andXE--nG.isnotthezeroelementofXsinceμ(EhnGn)>0.Put ,XE--nG.'-一川-||XE--nG.|| Itisnotdimculttosee,e.g.fromtheexpo阳ltidfomUa(ct[1,p.791) 只t)f=J划。(;叫,Ah))V, fεD(Ah), thatforaimostailswehave T(t)fn(s)=etMS)fn(s)· NotethatthelatterformulamakessensesincefnED(Ah)andbyassumptionT(t)isdennedonD(Ah)·SinceReh>nonEh门Gnweget |T(t)fnl;三|entfn| implying |lT(t)||主||T(t)fn||主ent|lfn||=e时' acontradictionsincethiswouldmeanthattheoperatorT(t)isunboundedforeadt〉0.Conversely,supposeReh三Ifforsomek.DeHne -L凡--t圃,,..飞-nuζ」rJ、‘.,,,eurJSι"ρ』--eurJava-,,,‘飞T Thencle主lUiT(t)ikekt.wewillshowthatT(t)isaCH叫旦旦旦whosegematoriiiii二FixfED(Ah)andf>0.SinceD(Ah)issolid,SOisitsciosureD(Ah);inotherwords,D(rlh)isaBanazhfunctionspaceonitsownright.HencewemayapplyProposition2.ltoobtainannsuchthat||Pnf-f||〈ε.NowonEnwehave-n三|h|三n.ChooseO<to三1SOsmallthatforanyO三t三toand|α|三nwehave|eM-l|<ε.Thenforsucht. ||T(t)f-f||三||T(t)(f-Pnf)||+||f-PJ||+||T(t)Pnf-PJ||三(eIft+1)ε+||(eht-OXEJ||三(e|K|+l)ε+ε||χEJ||三(e|K|+l+||f||)ε. ThereforeT(t)isstronglycontinuousonD(Ah)andobviouslyAhisitsgenerator.////WeremarkthatthisresultcouldaisoeasilybederivedfromtheIfille-Yosidatheorem.ItisaneasyconsequenceofthedeRMUonthatXhasordercontinuousnormifandonlyifforailfεxanddecreasingsets几3F22...↓Owehave||/χ凡|!→0.UsingthisequivalentformulationtogetherwithProposition2.1andTheorem2.2weobtajn: Theorem2.3.XhasordercontinuousnormifandonlyifAhgmeratesarrSPITIigrouponxibreveryhwhoserealpartisboundedikornabove.
388 vanNeerenanddePagter- pmf:Suppo回Xhasordercontinuousnorm.TakehwithRehf二kanddeRIlethesetsEnandmapspnaccordingto(2)and(3).Since E1CE2C...↑Q, forailfεXweget ||PJ-f||=||fxn＼ι||→0. HencebyProposition2.1,D(AA)isdense.ThenThωrern2.2showsthatAhisageneratoronX.Conversely,letQ=几3F13F22...↓@.DeRIlehELO(μ)byh(s)=-n,sEFn＼Fn+1· Then En={SEQ:|h(s)|三n}=Q＼Fn+1· SincebyassumptionAhisageneratoronX,henceinparticularD(AA)isdense,wegetbyProposition2.1 ||fYF叫li!=||fYQ＼几H-f||=||PJ-f||一0.//// FromnowonweassumehtobeaxedwithRehboundedfromabove.HAJBisthegeneratorofasemigroupT(t)onX,thentheMjointT·(t)iswell-dennedonx·.InthefollowingtheoremwewillgvearepresentationforthesemigroupduaiX②.Let[PJ:-y-lzldenotetheclosedlinearspaninx-ofthesubspansIt:x·,n=1,2,...Theorem24X①=[P;x-lZL1·PF飞mfFirstnotethatx-isaBmachlattice,SOwhenever￠EX·,then|φ|isawell-dennedelementofx-ofnorm||φ||.WestartbyshowingthatD(A;)issolid.Supposelφ|三|叫withψεD(A:).Clearly, (hφ,f):=(φ,hf)deanesalinearfunctionaihφonD(Ah)andforfeD(Ah), (hφ,f)=(φ,hf)三(iφ|,|hf|)三(|ψ|,|hf|)=(|hψ|,|f|)三||A:ψ||||f||. ThereforehφisboundedonD(Ah).SinceD(Ah)isden肥,fzφextendstoaboundedlinearfumUonaionX.ThisprovesthatφεD(AP.Wewillnowprovetheinclusion[P;x-lzzlCX①.LetoEP;x·,sayφ=P;ψ.WehavetoshowthatφεX⑦.SinceD(APissolid,soisitsclosureX@.Thereforeitsumcestoshowthat|φ|εX①.Fixf>Oandchooseto>OSOSmailthatforanyO三t三toandiα|三nwehave|eM-1|<ε.Sincewehave|φ|=|P;ψ|==P;|ψ|,andhencefort三to, |(T·(t)|φ|-|φ|,f)|==|(|ψ|,Pn(ethf-f))|=|(|ψ|,XE.(eth-l)f)|三ε(|φ|,|f|)|三叫|φ||||fl|. Hence ||T·(t)|φ|-|￠|l!三ε||φ||
semigroupsonBanachFunctionSpaces 389 showingthat|￠|εxeandthereforealso￠εX①.SincexeisaclosedlinearspacethisimpHesthat[P;x-iZLICX①.Toconcludetheproofweshowthereverseinclusion.SinceD(Ai)=X①itsumcestoprovethatD(APC[P;x-lZL1·LetφεD(A;).SinceD(A:)issolid,wemaywithoutlossofgenerdityusumethatO主0.Itsumcestoprovethat||P;φ-创|→O槌n→∞.FormyfED(Ah)wehave rJAVA--n<-‘飞,,FJAMVL",,,‘飞1-n--1、,,FJL"AV,,,、、1-n〈-‘飞,,rJ-E‘、、αVAtAUV,,,‘、=飞JFrJAVAVV气,,,‘、 Thisshowsthat||P;φ-φ||三n-1||A:￠||→0. //// FinallywewiHconsiderthecasewhereQiscompactHallsdorfspaceandμisaBOrelmeasure.InthiscueitisnaturaitoseewhatimprovementscanbeobtainedwhenwerequirehELO(μ)tobecontinuous.Infactwewillasksomethingweaker,viz.that|h|isacontinuousfunctionQ→IR,theone-pointcompactiRcationofIR.ForsuchfunctionsweputE∞={SEQ:|h(s)|=∞}.SincehELO(μ),necessarilyμ(E∞)=0.WewillsaythatfεXiscompactlysupportedifthereisacompactkCQ＼E∞suchthatf=χKfa.e.andwedeRIlexctobethelinearsubspaceofayconsistingofallcompactlysupportedfunctions.Ofcoursearcdependsonh.Afunctionalφε.Y·issaidtobecompa,ctiysupPONedifthereisacompMtkCQ＼E∞suchthat(φ,f)=(φ-wf)forailfεX. Theorem2.5.AhgeneratesaCrsemigroupifandonlyifXcisdenseinX.Inthiscuex②jstheclosureofthecompactjysupportedfunctionals. PF-Emf:SupposeAhgeneratesaCo-semigroup-Since|h|iscontinuous,weseethatthesetsEnCQ＼E∞dennedby(2)areclosedinQ,hencecompaμ.NowtakefEXarbitrary.ByassumptionD(Ah)isdense,SObyProposition2.lwehavel|PJ-f||一0.SincePnfissupportedinthecompactsetEn,thisprovesthatXcisdenseinX.Fortheconverse,assumearctobedense.InviewofTheorem2.2wemustshowthatD(Ah)isdense(theconventionthatReh三KisstiHinhrce).InfactwewillshowthatxcCD(Ah).Indeed,letfEXcbesupportedinthecompactsetKCQ＼E旬·Since|h|iscontinuousasafunctionk→IR,weseethathisboundedonk.ThisimpliesthathED(Ah).TheassertiononX①isprovedinexactlythesameway,usingthecharazterizationfromTheorem2A////Example2.6.(i)LetX=LI(R),h(t)=t.LettingQ=ZEweconcludefromTheorem2.5thatX①istheclosedideaiinL∞generatedbyCo(IR).(ii)LetX=LI(D)呐thDtheclosedunitdiscinCSupposehiscontinuousinDwithurns-tlh(s)|==∞fordteθD.ThenX①istheclosedideaiinL∞(D)generatedbythesubpaceofcontinuousfunctionswhicharezeroonθD. FromTheorem2.40r2.5weimmediatelydeducethefollowing-Corollary2.7.LetXbeaBamchspacewithanunconditionalbasis{zn}注1·ThenAzn:=knz,zgeneratesaCo-semigroupifandonlyifRekn三kibrsomeconstantk.If|kn|→∞thenX①=[z川江1,theclosedlinearspanofthecoordinatefunctionals. Pr∞fJRegardXasaBmachfunctionspamonQ=Et ////
390 νanNeerenanddePagter- 3.References [llP.LButzer,H.Berms,Semigmupsofoperatomandapproximation,SpringerVedag,Berlin-Heidelberg-NewYork(1967).[2]J.Diestel,JiUh1,协ctormeasures,Math.Surveysnr.15,Amer.Math.Soc.,Providence,R.I.(1977).[31S.G.krein,Ju1.PetunirhEM.Semenov,InterpolationofLinearOperators,Transl.Math.Monogr.54,Amer.Math.SohProvidence(1982)[4iJ.LindenStrauss,L.Tzafriri,ClassicalBMachspae臼L汀,SpringerVedag,Berlin-Heidelberg-NewYork(1977,1979).[5lJ-LI-A.M.mNeem民ReH创叫ty,thedudRadon-Nikodymproperty,andcontinuityodadjointsemigroup,toappearin:Indag-Math.[6lB.dePagter,Acharacterizationofsun-reHexivity,Math.Ann.283,511·518(1989).[7iH.H.SchaefehWeakconvergenceofmeasures,Math.Ann.193,57·64(1971).[问8叫iA.C.Za汹a阳[问9叫lA.C.ZMIletRieszSpaces汀,Northtfoila时,Amsterdam(1983)
PhaseSpaceforann-thOrderDigerentiaIEquationinBaEEachSpace ENRICOOBRECHTDepartmentofMathematics,UniversityofBologna,Bologna,Italy 1.INTRODUCTION 工nrecentyearS,n-thorder(andespeciallysecondorder)differentialequationsinBanachspaceSbecameaverypopularsubject(see,e-9.,[3]-[6],[8]-[9],[11]-[14],[16]-[18]).ManyoftheSepaperstransformthen-thorderequationintoafirstordersystemeitherintheusualwayorbymeanSofSomeCleversubstituti。ns.However,the。peratorcoefficientoftheSOobtainedsystemmaynotbeclosedand,evenifclosed,maynotgenerateasemigroup.InordertOavoidthiSineonvenience,AqmonandNirenberq[1]requiredadifferentbehaviouroftheresolventinalltheproductspaceandinasuitablesubspace,ThiSdevicewasusedsuccessfullybyGrisvard[7]andLagnese[lO],too.InthiSpaper,weqiveadifferent,verySimpletechniqueofattackingtheCauchyproblemforann-thorder PartiallysupportedbyMURST{f。rldi40毛e60毛),Italy. 39I
392 obrecht equation:namely,wechooseFforthefirstorderSystem,a"phaSespace"FwhlchtakeScarefullylntOaccountailthecoeffiCientSOftheequationandthedominationrelationSexistinqbetweenthem,SOextendinqtotheqeηeralcasearlideawhichinspireSmuchoftheworkofFattorirli[5]ontheequationU"+Au=O.InSection2.,afterSomeprellITllrlarleSFweProveanatUralClOSedrleSSCorldltlononthefirstOrderSyStemcoeffiCientFwhichrequireSonlythatthephaSespaCebenottoolarqe.工nSection3.,westateandProvethemalnreSUltSOfthepaper,i.e.cor1ditionsformaximaldecreasinqofthereSOlventOfthematrixoperatOr.FirSt,weeStabliShaneCeSsaryconditionintermsOfthebehavioliratinfinityofp-l(λ)(theirlverseoftheoperatorpencilconnectedwiththen-thorderequationJseeDef工nit工on2.2)FthenweshowFbYaSimpleexample,thatthiSCorlditioniSnOtaSUffiCientone.Finally,inTheorem3.1.FweProvethatmaximaldecreasinqiSassuredifwerequireFbeSideSthealreadyeStablishedneceSSaryconditlon,aweakerformOfBreZiS-Fraenkelcondition.ToClarifythemeaninqOfthehypotheSeSofTheorem3.1rletUSconSidertheSLtuationwhenn=2.BythemethodsUsedintheAppendixof[6],itiSnotdifficulttoprovethathypotheSiSii)iSVerifiedifVl=(VOFV2)α,p,Vαε]OFl/2]andVpe[1,+∞];so,thiShypothesisrequireStochooseaphasespaceVl×X,wlthVlnottoolargeFWhilehypotheSiSi)requireStochooseVlnottoosmall.IfabalanCebetweentheSetWOrequirerTIerltSiSPoSSible,weshallhavemaximaldecreaSinq.ApplicationSOftheseresultSwillappearelSewhereJhoweverFletUSnotethathypotheSiSii)iSSliqhtlymoreqeneralthantheoriginalBreZiS-Fraenkelcondition[2],usedin[13]-[14}toproveexiStenceofclaSSicalsolutionsintheparaboliccase.HoweverFitSeemsverydiffiCulttOdeSCribeinterpolationspacesconnectedwiththematrixoperatorconSideredhereFSOthatitSeemspreferrabletOUsethetechniquesin[15]toqetmaximalregularityresultSintheparaboliccase.
393 PhaseSpaceforD{gerentialEquationinBanachSpace RESULTS PRELIMINARY SOME 2. neN,Vlr…rVnbe Set xand ln injection let space, contir111011S complexBanach xwith a be Of X SubspaCeSne=×Vii=l Let Set ir1X, operatorS CloSed linear are 啼,AnAJAFιT4 =lF…,n} 8(Ai-l),n-l=(b..,un,-ZAKUUl),k=0 i uie 49(q)={Uεe| 。U (ulv·-,un-l). U where the space" however "phase ClOSedJ the choOSe not we 。isif operatOr that Shows ,,V4awe-4ba←」.工quns-lwao14n1占·工ofe the reSUlt Usually quaranteed. 工S closedneSS and ViCs(Ai),alqebraically Suppose 2.1. PROPOSITION the Then X. VrI and n-1, lre. 工 if topoloqiCally, closedin lS →kurqωf队,ao、7亿。-VAirqp q that s(q),suchinViF ln IJK,i+l-→gics(Ai) sequenCe ftJkjkεNbein6.Hence,Vi a a gFThen Let bAU-- 1. =1,...,n ,i e vi+1 n-1. 1,..., nZAi-ltJLi→hi=lAsAi-11JK,i-4'Ai-lVilnn立Ai-lVii=2 Furthermore, X. ln that the follows ves(et),。V= -2,it n =2,..., X,i gand So -gn- tJK,l--' Ao closed. 工S q operator the 。f behaviour spectral the characterize tO order Zn followingdefinition.n-18(P)=(S(Ai),izO weneedthe Set 2.2. 已t, DEFINITION operator λε￠,xεsfpj. ZMAKX λnx+ pfλjx=
394 Obredu WeshallsaythatλεσbelonqstOtheresolventsetofthepencilp,andweshallwriteλε同Pj,if3p-l(λj三(P(λ)ylεεz(X). InthefollowinqFinordertobeabletowriteSimplerformulaSFweShallwriteAninSteadOfI,theidentityoperator. n-lPROPOSITION2.3.supposeViC(dpfAjj,alqebraiCally3=landtopoloqiCallYFifi=lr…rn-lFandVn=XJthenλεp(αjiffλεp(PjandFinthiscase,ifweset(λ-q)一1=||qkj(λ)||k,j=1,...,n wehave 。kjfλ)=λk寸1(P-lfλjZλhAf1·I)h=j ifjSK-1, n=λk寸lp-l(λjZλhAh,ifj主k.fI=jProof.Letfee.Ifues(巳),λε￠,(λ,。)u=f,wemusthave(U=(tJlr…rlJn),f=(flF··.,fn)) 嘈in74='K,i-7JKfe7Jqah?ιidηυ-『hFE=K『4.3吁IAU,4k句,儿--ku nf--nu‘,,,74nA+气人+kukA74i,嘈i二hp·=n咱dk (1) whenCe nnp从jtJl=ZZP-jAhfj;(2)jzlh=jnotethatfjes(Ah),hzjr...,n-l,SincefjeVj·Ifλεp(Pj,weqetthatthereexistSauniqueUleS(P),SatiSfyinq(2)Jthent12,...rtJrIareuniquelydeterminedand(U1,...Jn)es(q).SO(λ-qylexistSandiSeverywhere
PhaseSpaceforD{IYerentialEquationinBandchSpace 395 defirled.TheformulaforGLKj(λ)inthestatetTIer1tofthepropositiOI1thenfollowsandwehaveqkj(λ)εz(VjFVK), Krj=l,...rn.OntheotherhandFSupposeλepfPJ.IfpfλjiSnotinjective,thereexistSfneXFIJlJ,町,2εs(时,町,l乒肉,hsuchthatpfλjull=Pfλjt112Jthen(0,...,frl)εeandbothul,lrUl,2SatiSfy(2)withthiSCholCeOff.TT111Sλep(。).AηanalogousargumentshowsthatifpfλjiSnotorltoXFtherlλ-Ctisnotontoe.Finally,itiSobviousthat,ifpfλjhaSununboundedinverSeFthenitiSirTIPOSSiblethatλ-CLhaSaboundedinverse. 3.THEREDUCTIONTOAFIRSTORDERSYSTEM 工tiSfurldarner1talinSemigrouptheorytoeStabliShinwhichunboundedSetSofthecomplexplaneanoperatorhaSthereSOlventwithmaximaldeereasinqatinfirlity.Sinceqkn(λj=λn-lp-l(λj,Vλεp(PjF(k=l,...,n),itisobviollSthatanecessaryconditionformaximaldeereasirlqof(λ-q)-linanunboundedregionZisthefollowirlqone: 3MεR+,suchthat||λkp-1(λjx||kgM|lx||,VλεZ,k=l,...,n.(3) Hereandinthefollowinq,wedenoteby!|·|lkthenorminVK. However,thiSconditionisnotsufficierIt,asthefollowing exampleshows.Indeed,notethat 。n,n-1(λ)=p-1(λ)(λn+λn-lAn-1)-I,andletXbeaHilbertspace,n=2,AOaPoSitiveselfadjointoperatorinx,Al=0,vl=X.Thenetn,n-1(λ)=λ2(λ2+Ao)-1-I=Ao(λ2+Ao)-1;wehave||λ2(λ2+Ao)-1llgM,VλeR+andso(3)isverified,but||Ao(λ2+Ao)-1||doesnotdecayas|λl-latinfinity.Inordertogetmax工maldeereasing,weneedafurthercondition,whichiSVerySimilartoBreZiS-Fraenkel'Sone([2]JseealSO[5],[13]).
396 obrechI TheOreIT13.l.LetVCFV1,...,Vrl-lbeBanachSpaceS,SUchn-lthatVicf1s(Aj),izOF...,n-LalqebraiCallyandy=i topoloqiCallyJfurthezITloreFsetVn=X.SupposethereexiStS anunbourldedsetZOfthecomplexplanewhichiScontairled inptpjar1dsupposethereexistSroεR+,suchthat i)3MεR+,suchthat !|λkp-l(λ)x||kgM||x||n (4) VλεZ,|λ|〉ro,Vxεx,k=1,...rn; ii)lethefOr1,...rn-2J;VxeVf1,3KεR+,suchthat Lh(t,x)52K, (5) Vte]Orro-l[,where Lh(trX)=inf{Lh(trx,ψ)|φevo},hnLh(trXJ)=tM(Zt-ilWi|i+Zt-j||X叩||j).i=Oj=h+1 Then,3MεR+,suchthat||λ(λ-et)-1||:三M,VλεZ,|λ|;三ro· ProOf.UsinqthenotationSOfpropoSition2.3,wemuSt showthat3HεR+,suchthat .可JXH<--KX、‘EFqah.『JbA向U『句人 VxεVjrjrkε{lr…rnj.Supposej=nJthen,ifxεX,l!λαkrJ(λ)xllk=||λkp『l(λ)x|lngM|!xl|nr by(4).LetxεVjJbyHypothesiSii),ifλez,|λ|;Ero,3yλevo,suchthat Lh(|λl-1.x,yλ)〈K+1.Ifkzzj25n-1.weqet:
397 PhaseSPGcefor-D{gerentialEquafioninBanachSpace i|λαkj(λ)x||k=||λk-jp-l(λ)三儿叫x||k三 + +||λk-jyλ||k ZIm(λ)-m=l l|λk-jp-l(λ)立λiAifx-YK)||k俨 λiAiYK||k + < Zl|廿jAifx三叫工||ν-jfx-几)|lign叫fk||λk-jfx-Y儿)!|k+ we if qet, we 3-lF h with (6) Of arld (4) Of V工ew In choOSe -Yλ)||ng 三二M Il(λ) 1); + ||kk?jx||k ||x||KJ Ml(K+1). 〈 + 1 立||ki-jyλ K+ 〈 Ml 艺户hYAl|ng < gM 工2(λ) 工3(λ) n that hcuq】+RεXH『2问JbεX 25j5n-l,||λαkj(λ)x|lkuniform V k if Then, 〈 it prinCiple, bourldedneSS the .,,咱4e←」-amk·-tsse,7ddFetrs-lau---qe←』ra VλeZ,|λ|;三r0·BY the follows qet: kxx-iA-iqahn℃'但勺·i州人,4P Vj,we xε if Suppose, + 〈 xj|lk +||λk-j(YK- =l|λk-j k向人yx-AA--冽AhnT'句时州人嘈4P.守JK码儿<- ||λαkj(λ)x||k =立Jh(λ). +||沪jyl(λ)ZMAiYλllk -Yλ)||ig weqet: andby(6)withh=j-1,Jl似 {K+l), gnMl (4) By
3980bredf +KTiMn<--l代人y.iA·叮JmAL剖、,臼闻M<-n代人y-lA.工街AAlM℃/】刊+·.、4·iKM〈-<-hMλ叮ι『吨dTυγυ AfurtherappllcatlonOftheunlfOrmbOUr1dedReSSprirlcipleproveSthatFeveninthiSCaSe,λ巳tkj(k)iSuηiformlybOURdedfromVjtoVkrifλεZ,lλi三ro- REFERENCES 1.S.AqmonandL.Nirenberq,COIrlIrl.PureAppl.Math.,16:121(1963) 2.H.BreZiSandL.E.Fraenkel,J.Funct.Anal.,29:328(1978) 3.S.Cher1aRdR.TriqqianiFPaCifiCJ.Math.,136:15(1989) 4.Ph.ClemerltandJ.PrUSSFBoll.Ur1.Mat.Ital.,(7)3-B:623(1989) 5.H.O.FattOrini,SeCondOrderLinearDifferentialEquationsinBanachSpacesFNorthHolland(1985) 6.A.FaviniandE.obrecht,toappear 7.P.Grisvard,Ann.Sci.EcoleNorm.Sup.,(4)2:311(1969) 8.F.HIlanq,SIAMJ.ControlOptim.,26:714(1988) 9.F.Huanqandk.Lin,Ann.DifferentialEquationS,4:411(1988) lO.J.E.Lagnese,J.Math.Anal.Appl.,32:15(197O)
PhaseSpaceforDUKrentialEquationinBanachSpace 399 ll.F.NeubranderFTrans.AITIer.Math.soc.,295:257(1986) 12.F.Neubrander,semigroupForumF38:233(1989) l3.E.ObrechtFJ.Math.Anal.Appl.,125:5O8(1987) 14.E.ObrechtFSeIrlesterberichtFunktionalanalysiSTUbingenFSommersernester1986:191 15.E.ObrechtFiEpreparation 16.H.Tar1abe,J.FaC.Sci.UniV.TokyoF34:111(1987) 17.H.Tanabe,J.DifferentialEquationSF73:288(1988) l8.S.Ya.YakubOVFMat.Sb.fN.S.j,118(6O):252(1892)=Math.U-SSR-Sb.,46:255(l983)
QuasilinearParabolicVolterraEquationsinSpacesofIntegrableFunctions JANPRussDepartmentofMathematics,PaderbornUniversityofTechnology,Pad-erborn,Germany lINTRODUCTION Ir1flCHtNlx、oppll.bound刊lwithboundaryOQofrlass{气lr1tt(t)bt、ofboundtdvariation、aIKigε(VI(TT×IRJ飞jItNLL飞/eCOIlsidrr1llequasilimytIVolterrae(l1MtioII Ut(tz)u(t、工)tt(0、1·) Itdα(T){divg(υ1J圳、V川川v机机川Iu叫中t叫(0、tεJ『2、εδ07U011·)、2·ε0、 tεJ、rξQ『 、、..,F1i,EEE,,,..‘飞 40I
402 PrtE5 W川lhlυ川川lrr、lb)lktp川?1ImInl(l.l)ta川川tU甘Ir.才iSeSasa1mIn1Oded!l)川1r.刀(O)lb)i℃创Inm1口IiIn1Sθ叽兀V,飞呛刊(e创创?丸汀lr、孔址a川li且t扫扣te叶?才l(dlS'likethetheol-yofvisco-rlasticit〉'01·heatconductioniIImaterialswithmemory,andhasbeenstudl℃dbyII川〉a川ωIS;℃PNIac℃a叫·[lU5)‘lhfernlosamlhhell叶,E吨ler[8]、IJOIKielialkiNoiltBi[l21、LmaltiiaMSi附4trali[l3LStatrami23],thf?阳丁χ盯t守创?汀In川l1Im}门lO川1i(ο}g叫〉!hiIRt卡扑ht俨川川川、1旧刊l川川lμMta川lItisο1ui川Ir.p1uluIr.pOSfiIuitlhlHiSInlOtetOStIu1K川(diyE℃凹m?ηηJ〉X、〈d:iSte创InlCeOfStIr.OIn1gSOi1ultiOIn1fSiof(iL.l)iI川i川叮叩!p)泊￡alKCtrW7川刊(0}fiIlkgrabk?functionspithm-fol-smallTorfoISmalldatati仙f、aswellasnListellcpfo1·tilecorrespondingproblemontilelintki-t 1u叫t勺川t川(υ/I川)=儿ff沁气td山巾忡川lh归削忡{α叫忡州I叫巾川(忖例7叫){七d小iUih飞9(υl、V川v机机t"中i(l.2) tl(1、r)=0.l巳Ht‘」·εθfl ik』l‘吕malldata-('iloosillgtilt、spa(-wX=LIJ(.j;L叫。)))tISOIlrbask-spa(?一wllfrt7 JV2一+一<lqIY 、、EE'/、,、P...l,,tt飞 -OIll-rtFSUItsall-r(ii{fereIltfromthoseavailableil}tIIPliterature-Tll℃solutionswil门roi-tla8E411'lJ(J;lq(Q))nLP(J;lfrv24(Q)).0111·al}i〉roachreliesOIll℃川lltsoflhralxl＼fpIIIli[7LI〉liiwall(ihill[20、19lalld(jlulllmtalklP山兰日[310IilllutillltilI飞BRIlltutty‘tllkliIlialgiItal-ypowrr刊t)[trcolidoldrrellii〉ti(·。l}eratlorsalswthiltllf甘υ{飞bitlf11·tloptT1·礼tol--4illLPspa(·(11、ilfapplk、tktlionoftllfSPI℃SIlltsIYBqllirtBtlilt?川llltitiolliolrofp川abolit、1ypr、whirlilllt?tIllsthatytlr吨ν)isllIIiibHIll〉Yl)(}FSiiivf(lrjilli1t＼aIM!!litl11ilrkPIll俨irlrl(1)iIlvolvcdsatidimtlIiEiIlgirCOIlditioIloftliltBfol-lll 。U>、八ρL〉L霄··Aπ一仆9/】〈αAυ<一、λ气/.飞飞,J,.σο』仍U.ad (1.4) 叭'iicrt、tlilt;二haililldictulfsIAtli)lalcptransform-TIlt斗lttttlθ1·IIleaII日thaltltlMBKol-lipllIUl川llkivrti$1lifiUPlltlyNtlt}iigsillgldal-itya11lMB(}ligill、c.g.dtt(t)~俨-Vtasl→0十.Althoughmallyvi创刊rlastk-lIMliltTKIllsarrb〈flitTvrdtobeofthistyw句tllist-blissdonnoiliIicl1idctlk、pilyskllilyiIltprcstillgparabolic-llyprrboli(·llllutinguiM＼wiifl灯白(1)二tll(1)dt吨αl(()+)<∞‘bllt-ih(I)~t←lasl→0、silk-rtllPIlolriωOti=π/2.Ouapploachisilltir叩i山ofurwo山ofDaPratottIKIGrisval-d[4lwllol)ropost-飞itilisIIIFtllodfol-parabolicpartiai(lifeI℃IItiitltEquations-TilthOIliypa!〉℃1·
VolterraEquationsinSpacesofIrtIegrableFunctions 403 weKIlow。f、wherethemaximalregularityapproachisappliedtononlinearVol-terraeqmtionsisduetoLunardiandSiIlestrari[131.TheyusetheCαapproachwhichnecessarilyinvolvescompatibiiityconditions-ThemaindifereIICeexceptfol-thesettingisthattheyassume(inourterminology)thaltft(t)llasajumpatlt=0、whichleadstothepresenceoftiletermαodivg(1＼Vu(t、2))iII(l.l).Thenitissufficienttousetilewell-establishedIIMlximallregularityresultsforlinealrparab。licdiferentialequationsintileC飞setting-IIIdfpendentfromtileframp-1VOlthiIIourcasethisisnotenough、onehastoapplycorrespondingresultsfol-vector-valuedVolterraequations,sincewedonotamlmethatα(t)hasajumpatzero-ItshouldaisokmentionedthatunderslightlystrongerasSIHIll〉ti011SOIlthekernelα(t)(iIIHypothmis(H2)ofSectionIV、α(t)Illustbe2-regularillstcad)basedoIItilemaximal陀g川aritymuitpmvm!illPIiissIl8Lafu-tlle(川?for(ll)caIIbebuildup.IIOWO飞吧I＼herewellseLq-fl-theol-3·、siIK-r1llfllllOCOIllpailibili1lyconditionsal-einvolvedaIXilessregularityoftiO斗弘falxlα(1)isllwde(1.OUI-plaIIforthispaperisastilefollows-Scctlion2isdevotedtotileIle-cessarypreiiminarymaterialoniinearVolterraoperatorsinLP(J;Y)wherel'denotesaC-convexBanacIIspace,andboundednessOi-theirimaginal-ypowers-Section3dealswithimaginarypowersofsecondorderellipticdiferentialopera-tomi11(diUiV刊me创1r.毡ge创mInLCefOr1n川1taa川1tUlII川IFP气1SeCtiOInI4tlhle凹InlCOInltaiIn1Stlhlelnm1n1aihIn11r.eS1u1ltS'whiletheirproofsarrgivellillSrctioll5. Acknowledgement:Theauthorisindeptedtotilerefereeforanumberofvaluableremarksandsuggestionswhichleadtoseveralimprovementsintilepre-sentat1011. IILINEARVOLTERRAOPERATORSINLIP-SPA℃ES RecallthataBmachspaceYisfcontYEa-ifthereisafunction〈:Y×Y→R、(011vexiIIeachofitsvariables,suchthat〈(0,0)>OaIIdC(AU)三|31+u|forall二17、HεYwithlz|=|ν|=1.ABamciIspaceis〈-CO盯exifandonlyiftheHilbert traIIsforIn (H门口)=(1/方)pvl:f(t一斗tdR (2.l) isboundedinum;Y)forSOIlle(aIKlthenfolal!)IJε(l、∞).Hillm1spacesaIY
404 Prt4S5 〈-couvex(choose〈(1·,y)=1+Re(A二y)toseethis)aMLq(0,μ;γ)is〈L-CO盯e肘盯〉X;i让if.}}γ,y,ihlaωSt山ihlihSip〉Irω、飞刀(O)ip肘〉咒e创创创1r川叫.吃叮t勺3yvaInI(dli<p〈∞、Wihl忧阳削e创创1r.e(川Q'μ)川(d1扣he创InlOte创Sa创In1yII口-ImI〈-CωOIn1VeXiHt3yyhasbeenintroducedbyBurltholder;seetilt?surveyarticleBurlthol-der[iifol-tilePI-oofoftilem11ltsjustnlentiomdaswellasothers-WemTdtk、followinggeneralizationoftileMikilliIImultipliertlleoI℃111totlIevector-valuedcasrwhichisanextensionofamllltofMcConnell[16jd时toZimnm·111am[241 MultiplierTheorem:lidYKα〈-conum-BαnGcllspαctandh1171仨L$170(EZ＼{0})bεsuchthd |川(p)|+|ρ||1111(ρ)|三κmforαα.ρζ的 (2.2) DfjiltfαliIifα1·opεIUοI-ZLtbyIIlfαIi币。f 。υJT飞JEttHUζ」ρ'ρ'、rr1ργ一一ρ'~UF7t (2.;1) 川t阿fι(YF(HT＼{0}:?").Tlienjbrmeltpε(1,∞)1ltf陀18αCO川tt1711(丁1,>0.drlJflidiIl9011lyoηpfindY,suchtllαt |ZJJ|p三Cpκm|j|pjbrαlljεCF(EE＼{0}、Y):(2.4) 7;ytαd川itsα川tqMbo川仇dω Olbh〉M川8em凹mr川.飞飞VW7tlhle创肘1r.Y吐eddif.Olr.e(2.3)1ulIliq1uleιl〉y,de{iIlesILl-Actually,kicCOIlIIellaIKIZiIIlmermannalsoprovedmultidimensionalversionsofthisresult‘however,westateitone-dimensionalsiIIα-70IIlvthiscaseisusedbflow. AftertilesepreparationsweIiOWt111110IlI-MIteIltioIItolintTtil-VoiterratOi}Prato1·忖iIiIdly-spacps.AIIIaiIi叫θpiIiOIII-approachtol)IVlAPIII刊(l.l)ttIKi(l.2)(UIisistNilltileinversionOi-tileconvolutionoperator-Therefol-ewpconsidertik、equailioll 。(t)=μf白(T)[u(t-T)+j(t-T)|,tdR(25) inU(R;Y),whereYisa〈-convexBamchspaceaMpε(1、∞). Deanition:LetαεBILc(ET+)suchtlldfrrtklα(t)|<∞forεrtchε>0、α11ddα(入)并Ofol-RE入>0.
VolterraEquationsinSpacesofIntegrableFunctions 405 fijαiδcαllEdMct07、idwithα119lε0>Oif |argdα(入)|三Of01、αllRε入>0. (2.6) fiijαiscdledk-rεgulm、iftlM1、eisαc011δtti11tC>OsuchtfIM ((η)(|入ndα(入)|三Cn!|dα(人)|forαllRe入〉0,71三ι(2.7) Observethatαissectorialwithangle?ifandonlyiftlIemeasuredα(t)isofpositivetype-IIIapplicationstoviscoelasticityα(t)isoftheform 叫t)=αυ+αJ+Ith(5)仇I>O (2.8) whereα0,α∞主0、aIKlα1(t)主OisII011increasingaIXlOi-positivetype-SuchfmctiomarealW阳ayy'S阳tO创ωIrm.t山ha川tαisk-regularifα1巳CK-1(0、∞),(-1)ndn)(t)三Oforall11三k-1、and(-l)←lfiy-1)(t)ism11increasingandcO盯exOR(0、∞).Ifa(入)admitsamlyticextemolltoam!tor|arg入|<π/2+ε,fol-someε>0、aM(2.6)holdsforallsuch入,tileIIα(t)isit--regularforeverykε町.ThiscoversiIIparticulariter时lsoftheform(2.8)withα1(t)completelymomtomcWeshallneedthefollowing Lemma1:SupposeαεBV(EZ+)isbregulα1＼TllmdJl(i·)ε1417ο(EY＼{0})αnd ((n)(|pndα(tρ)|三Cη!|dα(ip)|forαllpεR、p#0、11三t ((n)P1·oqfJSincetlα(入)isboundedforRt入230、dα(入)isboundedOIlRe入>0、|入|三ε,foreveryη三kaMε>0.TlIelei-omtile山川a吨entiallin山s((n)fdp)=σhldtt(σ+iρ),ρεR,p手。咱 existalmosteverywhere,a11dweobtainfromtheidentity 品(n)(σ+仲 therelations 仰)-fn(7)二tffn+Ih)ds,扣laaρ7>O
406PrMS where11三k-1.Fromjb(ρ)=dα(ip)aIId(2.7)theassertionsfollows-口 IfαisSeCtOrmlwiμtlh1angle。<iaMl-mgularTtimn(2.5)behaves山plyiIilP(Et;11whereYisLconvex-kloreprecisely-wehavethefollowingresult.FOI-tileddiMioIiOi-complexPO附mot-closedlimaropeIMors,seeI4omatsll[llL TheoremtSttpposεYisα〈-corttyezBαTZGchspαctpε(l、∞LmdlrtαεBV(IR+)bεsectoridtuitftαnglε0〈π/2αηd/-regulαr.Thentht川iisαclοsetlliltmrdensdudeFMdopcIUtorAinLP(HZ;Y)suchthdU巳D(4)iifIt巳llf1,p(HZJrhlyltlthereMfεLP(HZJY)δMlttltαt d(I)=f山(7)f(川、td;1ltCIJ1户/(29) J1fol-torrtfihtlδthffollowiTtgpropε1·ties- fijM(44)={0}、冗(A均)=Lyp(倪ImR;才】/1(卜→∞、J0)Cp叫(JA均1川)句αm1ηl|川(μ+A)γ一1|厄三坐旦咱fordlμ>Oμ (2.lO) {Iijf1tltlIIIItsboundttliIIlilglItillylJOIUf11αIldtltfrri-HαCOIlstαltflbdrflJt1Id1119OItljvortlJtintfY,川trhthrlt 7「|Ji叫三lb(l+h|)exl〉((0+歹)|γ|)斗可ι的.(2.ii) {itij4COIn111utω1tyitlltliεgroupoft1、αrlslαtions- (it?jAIJItOIMinticilxltwy.1.t-hrcαcltμ>OU7fllit17f((μ+A)-lf)(I)三iOfrJYt<0矿f(t)=0卢rια.t<0. IJmofsinceαεBI/(Ei+)wellave|rh(入)|三Var(tltibrRe人主0.Byammlp-tiotαistllat 叫人+ltdα(人)|三|人|+μidα(人)|forallRe入主0,μ>0.(212) rIopbae、AVI--ae飞J、、IEYοrj,,..、飞ear-,,tUVEAOL--··Any&EU---A...、NHPL、,A''且ρhueD dG(1ρ)~(ρ)=(斗ρεR、ρ并0、μ>0.1.ρ+μdα(iρ)
VolterraEquafionsinSpacesofIntegrableFunctions 407 TheII |dα(ip)lC|rμ(p)|三C(<一、ρξR、p#0、μ>O|p|+μ|dα(iρ)|-一μhence~(p)isam山iplierforpm)andevenforPORJ)whenyisaHilbertspace-Forthecaseofgemmlpε(1,∞)and〈-convexYTweapplytileMultiplierTheol-em.SinceαisI-regular,by(2.12)weobtainbyasimplecalculationfromLemma1 C|rμ(ρ)|+|pr;(p)|三一,pεR,p并0,μ>O斗'μwithsomeconstantCindenpendentofμandp.Thereforc、thereareoperatorsRμ巳B(LPORJv))呼miquelyddIIedby RJ(ρ)=~(ρ)f(ρ),ρεR,ρ#0,fει了(R＼{0};Y)(2.13) andtileestimate lRμ|p三豆,μ>0,μholds‘withaldifferentconstantC.Sincetllerelation (2.14) ~(ρ)一几(p)=(ν一μ)~(p)1u(川、ρεR.ρ手0、 issatidedforallμ、ν>0、thefamily{Rμ}μ>OCB(LP(RJ'))isai附udo-molve川iIIthesemeot-Tfille-Phillips[10],p.185.Accordi吨toTlrom115.8.30f[10]、thekermlsk(Rμ)andtheranges冗(Rμ)ofRμareindependentofli、andtimeisaclosedlinearoperatorAiIILPm;γ)suchthat(it+A)-1=RAtiftilekernelM(Rμ)ofRμistrivial.TheoperatorA(ifitexists)isdenselydefinedif冗(Rμ)isdenseiIILPORJ);infact、44isgivenbyA=RJl一μl‘ieD(A)=冗(RJ.IJ1·omtile1111iquenesstheoremforfunctionsholomorphiciIlalhalfplanedllrtoFaMhf.RiesAseee.g.Priwalow[17l,itfollowsthatdα(iρ)并OforρεR、exceptforasetofmeasure0.Tih1mme臼1r.它甘edf扣Or屹e飞?a剖Sμ→∞we}h1aVel川t川7.〉μ川(p川)→1a.♂C.a创川I口叫1W附eO仙lb〉tu川aUiInlμR凡μf→fa汕1n1X叫(dlμμtR札μJf→t旷finLυ1(mmERRt飞I;;才】γ,卢,))、alldsoμRJ→falnl(d!μμtftμJf→t旷fim1n1Lι〈叮χ|MlμiR凡μ|bp三Ctilelatterimpliesidμ→lasμ→∞8trollgiyillLlqt;γ).IIlpaltiularM(Rμ)={0}aIIdD(A)=冗(ftJisdemeiIILPm:γ)sincetilesespacesareindependentOi-μ.
408 Prt45S ThusthereisaclosedlineaIoperatorAiIILPm;Y)withdemedomainDM)suchthat(一∞,0)cp(A)andRμ=(μ+A)-1;(2.14)impliesestimate(210)-SinceyaswellasLPOR;Y)amrenexive,theErgodicTheol-em18.7.3inHilleandPhillips[10limpliesμ凡→Pstronglyinum;Y)asμ→0'Wihk1町阳me创创Irr.B(仙Lp(但R;才}y,y丁F))川)istheprojectioIIWith冗(P)=λf(A)andM(P)=冗(A).However、sinceμ~(ρ)→Oforallp并0、asabovefollowsP=0、i时.足eλK/(川4A4)=0「ih}陀阳削e创InlC冗阳(A均)=L叮RI;才}}y,r丁F))忡by户Idee创川!X叮X川ih忖M~V川,才i问t叮yr11T;IIIlIr-飞1卫飞1卫lhMlU山iSCωOωInm叫Iu叫l刊ψ叭lpμ〉才let阳e臼StIhk1陀eP扒IOωOfof(0i)(iii)isobvious广foprove(23)observethatfol-μ>0. 1prμ(ρ)=dα(tρ)(1-μ~(p)),pεR,p#0 heImforfεcrmuo};γ)weobtain (JLfY=dα*f-μRμ(dα*f)=dα*f-litlα*ftJ(2l5) ThisimpiiesbyclosedIlessOi-(dli证{fe创凹Ir.它刊e创lnltiati(O)InlaIulK(d1叭WF才itlhl(dle创lnlS叮ityOif、8队1ulClh1if.1ulu川I川l(CJti(0)ll:S-fi山1口lLνp(但lERtR飞;J}γ,f1,Lνp(但ER飞;J}γ,y丁1,))k.ReplacingRJbytiε17(A)iII(215)附obtain(2.9);iIIi〉articularitisωwevide川thatthesymbolofAisgivenby,ip/dG(ip).Toprove(iv)notethatthereisafunctionbμ巳LjJR+)suchthat A(入)dα(人)bμ(入)=二(吨Re人>0.μ1+μ&(入)入+μdα(入) IFOIfεcr(R+)wetilerdoleobtain dα(入)bμ*f(入)=bμ(入)f(入)二(f(入)『Re人>0;人+μdα(入) passingtothehorizontallimit入=ε+ψ→ψthisyields bμ*f(iρ)=γμ(ρ)f(ip)、foraaρεR、 hencewetlavetherepresentationftμf=扎*fi-Ol-allfε(vi?(IR+)‘alld(iv)follows.Villally、weprove(ii).LetAμ=(li+A)(I+μA)-1、O<μ<l.ThenfLisbomdedaMρ(Aμ)〉(一∞叫,hencetheimaginalypowersAyofJlμexistandarebomded.Accordi吨tokomatsu[1月,for(2.11)itiseω1lghtoestimatt?|AY|andpasstotheiiIIIitμ→0.Thesebounds叭JillbeobtainedbytheMultiplier
VolterraEquationsinSpacesofIntegrableFunctions 409 Theorem-ThemultipliercorrespondingtoAistyD(tp)=iρ/dα(ip),hencethatofAμisgivenbymμ(p)=thv(ρ).Sincethekemeldα(t)isbyωsumptionasectorial,0<π/2、V阿have|argψ(ρ)|三0+π/2forallγεIR.Themtiomlfullction(it+z)(l+μz)-1preserveseachsector|algzl三功,lmIm|argm(ρ)|三。+π/2foraHγεR,0<μ<1.Fromthisoneseesthatthemultiplier(mAp))nwhichcorrespondstoA;?satis自es |mμ(p)叫三e|7||argmμ(ρ)|三ε|守|(0+π/2),pεR,p并0, and 1(p)m;(ψ)pij(p)η(mμ(ρ))t?·一一一mμ(ψ)ρ(p)(l一μ2)ψ(ρ)dJ(tρ)1η(711μ(ρ))门·(1-zρτ÷÷)μ(1+μψ(ρ))(μ+ψ(p))'thellimplieswithl-legularityofα(t)andCfromEstimate(2.12) 、‘‘,,,,~yρ'i,..‘飞μηd一句ρ' |A(m川7)|三|γ|ε|7|(川2)J(l+C)「ρεRP并0αpTheMultiplierTheoremtilenyieldstheestimate |A7|三I1'p(l+|η|)户|(0+π/2)、可εR、 whereI乌dependsonlyoIlpε(1,∞)aIKlY.TileproofofTheol-eIIIlisCOIN-piete.口 III-IMAGINARYPOWERSOFELLIPTICDIFFERENTIALOPERATORSINDIVERGENCEFORM LetQCRNbeaboundeddomainwithδflofclassC气bεlfJFlJ(Q;RNXNLsymmetrica11duniformlypositivede且Ilite,aIXiconsidertheoperator (Bou)(z)=-div(b(z)Vu(z)),zεQ (3.1) iIILq(0)withdomainD(Bo)=W2,q(Q)nwJA(Q),whe陀qε(l?∞)ands〉ATJ;三q.ItiswellknownthatBoisaclosedlineardenselyde自IIedoperatoriRLq(0)withcompactmsolveMaMspectrumσ(Bo)=巧(Bo)C(0‘∞)whichsatisaestileresolvelltestimate 1CM)|(入+Bo)-I|<一一-fol-allhrg入|三π-0(3.2)一l+|入|
4IO Prtus whereO〉Oisarbitrary;fol-q=2、Boisevenselfadjoint-TobeabletoapplytheDOIPVminiTheoreminSectioli5wrneedtileboundedliessOi-theiINagual-ypowersofBo-Tileresuitisasfollows- Theorem2:LetthεαssumpttortssttztedαbouebεsαtisjZed-Thε71f01、"chε〉Otltf1·εiδαcortst411ltI毛主lsuchtftd |BF|三IIFε♂|守|fm-fill寸εm. )?dqJ,EEE‘飞 i勺1·tlwdeiinitioIlofgeneralcomplexpowersofaclosedlinearoperatorSem(‘IK〈OEIuI川f怡川川4讥训1ulllilL.F1I、1lhi阳(e叽7又(川?汀InmIHl2W川aS{ihm川.3Stlp〉川lrω.ib)OluluIuiK(dla剖1r.γ3y,~V,a川lh1ule}p〉1r.OlbM)才lelnm1口1SoftaLu町1r.七lbM〉九itU1r.a1ryO创Ir.dde创盯1r飞.＼、ihlOWeVe创盯1r.＼、叽WviMtlhl(?∞-coetikicIltsaIKiOIlboundeddomainswith(vq己-boundary-FOI-applicationstononlinearproiJelll忖iikt》tiltOIifWCCOIlsidol-iIithispaper、(fmcoeiTlcie凹Inl此tSa川lr‘℃etooIr.创th1r.ik(Cd.ttUiV盯it伫-》飞、＼.IhInil川Eη1r.iaiS出吕t礼tttl((0)1tJlhlhlr℃(C.la吕臼8(7飞k山ha川lII川lK川〈ditihl凶a川ltttυ}川i[.tlhl{ο7lb}O1ulu川lnlX川(dlhalIr1r.}y'to(wl+山.fol-seCOIldoldr1·l}rohltBIllsaltlfast;alllclemPHial-yproofofrrilfOrell12callibrf01lIKithere-(‘OIlsidp1·llowallintervalJCRAIKldellotobyBtlk主point飞viseextellf4i(}liofffotoLP(J;Lq(fl))ddill叭lby (Btt)(t)二Bott(t)fol-a.a.tεJT (:1.4) withD(H)=LP(J;WM(fl)门LLjq(0)).Bis(losedliIlealdemeiyddimdagull、tilldweIIU℃ ((入十B)一1f)(1)二(λ+fjυ)一lf(t)fωa-a.1εJ叫人￠σ(β。)(35) as飞vrllas (Bnf)(t)=BFf(t)fora-a.tξJ-aIKi7ξR.(3.6) Thereforeσ(B)Cσ(Bo)allfiEstillIat创(3.2)alld(3.3)carl-yovertoBdirtytiJ‘tiltBCOIititantsC(0)aIKlII'EIYmailiunchanged- I＼lfy-LIAiNRESljLrIS ifol-COIlveIlimlcewe{irststailtBOIll-stallldinghypothesesOIlfl‘α、9叫"0、paIKlri·
VolterraEquafionsinSpacesofIntegrableFunctions 4ll fHljQCRNMαbomdctldomαillwithbouMGl-yOOofclαω〔VI. fHSjαεBV(HZ十)isMctoriαlmtllα叼hO<π/2αMl-1、叨ulα1＼ (H3)gεC2(TT×EZN;EZ勺,uoεW2,q(Q)nl刊,q(Q),mdtllEmα们、izbLr)gw(z,Vuo(z))issyTYmetMαndpositiMdeFnitc,uniformly0110. fHjjf+7〈l WebeginwithPIoblem(1.川、theequationontileline- Theorem3:Let(Hl)~(H4)besαtisFed.αIlrlldfo(1·)=divg(JI＼Vtto(1·))-Tlulltltoεexist7＼ρ>OstichtlIdformcllf+元εX=L13(EE;Lq(。))、tpith|f+jUx<γthwfiδαmiq盯80l山071ti。ffl.3jδucl11liftitiItoξZ二i/俨'J♂叮1p叮y气(值Ht七;Lυq(川Q)川)nLp叮(HEZ;Jl川/f斗俨1Lff+‘Aι←H→ti一1圳i句Oiωδ〔Fv叮1fρl、OIm7ηlXtωOZ Olb)S优阳ee创!1Ir、Vetlh1atjf.O叫(z叫)=01m丑e创aIn1St}hlat1ul句O叫(2Z叫7才)isaES4tatUiOIn1a剖Ir、3y7SOωlh1u1tUikhm(Oω〉川Inlof(υ1.2)ifhhh胃Lο山tOω)MIf三0.TtIllsTheorem3caIIbeviewedasastabilityresultwithrmpwttotirclassX=Lyp叮(mR;注lLiyq叫(Q)门)offbO创I盯n吨gμfumCtUiOnrTflhle创1r、℃ee!isaS剖切ihInmI丑1ilaIr、Ir、eS1uIltfortlh1elp〉eriOdi(C?Ip〉lr.刀叶(O〉才l〉lemInmI口1(1.2)i.广iIr:丁飞(O〉bοalb〉itf、ttl((O)S叫川1tl￡a川l川t(?Btlhl川ifS兰、W飞VyeiInltroduce LC(IR;Y)={1tεLLCOR;Y):Uisupem normedby |u|LE=(ω川|u(t)|Cdt)l/气wherel/denotesagem Theorem4:Ld(Hl)~(H4)bεδαtMfed,αMletfo(r)=divgh、VI川1·)).TllflitlM川口istr,p>Ostichthdf01、"chfεXω=LC(EZ;Lq(0))、mill|f十/U|入'临<γthuy1isα1』71iq1MUJ-IJCl、iodicsolutionU巳Zω二日,γ(的;Lq(0))nLC(的;11/24(II)nLPIq(Q))stichtflαth-uoh<ρ.TllesolutionmαpfHtiisofclαssCIfrο川XωloZω
4I2Prt45S Nextwpconsidertheinitialvaluei〉roble111(1.l)OIltllcllalaine-GivenfεLLCOR+;υ(Q))suchtI川f+jUεlym+;Lqm)),以telld‘ftoRbyf(1)=-jufol-I<O斗extendtitoRby-叫t)=tiofol-t<0;thelltiisasolutionof(1.l)OIlR+、ifaIKloIIiyifitutides(1.2)onR.ThusasacorollarytoTileoreI113wcobtaintllefollowingresultfor(1.l)OIIR+· Coroliary1:lift(iIi)~(H4)bf划tMFed.α川lltljbLI)=div9(1:Vtio(工))T/tfIIIlttl-ttIIYJ11ρ>Oslidilhαtjbrtαchf+jUεX二IY(Ht+;lq(0))『tuw川川iU川川)川川Ii.|lUf/.+jhU|1入X飞,<rt/I川日tl川Iq肌MlItlionliof{lljhitchtltMti-tit)ξ万二iivlJ(Ht+;Lq(0))门liFJ(H?十;IJIV24(Q)nl1?q(II))α叫|ti-uoU<ρ.71、/扣tH{川川川川川a1(刀ttο山:7』11川tψρf+jAiU)←H→ti-tioisofclub-sC1jLOINXtoZ. Asmultillesultholdsfora{且iXC叽叫(dliGi川《《ο;??i川e创盯Ir、S叮川川ih川In肌lNKt(Cω;:丁E飞+ Soi-al-飞velitiIVt:2been(-olICP1·liftlwith"!Tilll礼lldatan.IliM}lull1、liOWTVtBl‘alst)givfsrrsuitsOIllocalpxif41(?ll〈-p(iIltilll℃)fol-arbitral-ydata-rIbs(Ttilif丸。lltjittstoobservrthat(l.l)iscausal吁whichlIleallsthatltllesolutionti(t)Oi-(l.l)1ulip}t0tiImI口1eTisOInllb3y?ih1l让叫卅{in:1hlgiVe创mInlfεLLC但但飞+;Lq叫(【Ql)川)‘叫山cefby-jUfort:>TtoseethailtlrI川llloff+‘foiIIUm+:Lq(fl))calllrllladearbitral-ily→mallbyClioosingI刊mallellough-1111lsasacorollary10(101·oHall-yl飞vrobtaintllefoliυwillgIN11110liltx-tillexistpIi(-pi-01·(l.l)witharbitral--vdata- Corollary2:IAI(fIi)~(Ill)btMtlidittlT/it1191mtfεX=lifJHf+;υ(Q))tht阿isI二T(tto、f)>05Iichtlldfl.ljdm118α川iqtiflsol仙。Itttwflu-hb卜lο叼δIolivlJ(IO斗Tl;Lq(Q))nLP(lo-Tl;liftd(Q)nliftd(fl)). SeveralremarksareIIOWinolder Remarks:(i)('。II刊川Ilgllyl川 "(l)=tlo寸tttl(δ)山t〉O (1.i) wlwwα。三0咱们三Ois110IIi盯realsi吨ofpositivetype-aIIdα1εLl(Ei+);ili
VoHerraEquafionsinSpacesofIntegrableFunctions 4l3 viscoelasticitythisisthecaseofnmis.IhIn1(C!!d白创InImt山ha川tαiskι-寸1r‘eqglu川ldla剖1r.fO创1r.aIn盯1η3y7kεE凹＼VT亏ifα1(t)iscompletelyINOIlotolli町however、fol-α(t)tobe1·regularitissumcKIltthatα1(t)isiIIadditioncoIIvex吨asisshownillPI-ass[18!-Theanglecondition p、π|argdα(入)|<O<EforRe人>O (4.2) iseasilyseentobeequivalentto -ImaI(ip)三C(α。+Readip))forρ>O aIIdsomeC>0;apply-themaximumprillcipleforharmonicfunctionsillalhalfi〉lanetoseethis-Thus-illcaseαo>0、(4.2)isalwayssatisfied-I了。1·tlrmoleintemti吨caseα。=0、itllasbeensiloWRiIIPlfISSaMSohrl20]thath(1)completelymoωtonicandii叫→o-td1(t)/α10)>Oaresumcie川.Tih1iSCωOV刊me创1r.lk〈优mωe创1r、肘lhSα向1(归t)oftlh1efO创rm、mt俨α一」1εf一→εt飞'whereαε(0,1)aMε〉0.FOI-α10)CO盯exo吨飞amcesmryconditionfor(42)tobefunnedisα1(0+)=∞;thisnm1e侃maUInlt山ha川taS饥刊1ul日伍iCdi止阳e臼Inl此tl忖3yylS§thIr.乃uO创In1gi山In1S叫ta剖川Inl此tuuaalIInleO1uISViSCωOS盯it勺3y,I口m1丑11uIS川川ttlib)elp归〉川1r.eSe创mI川川1刊t.OIltlleotherha川、byImamOi-tlrestimatesobtaimdill('a1·1aMHallmge11[21、itisliotdiificulttosl川、vthatlli川→o-th1(I)/α1(t)>Oimpliesti-lea吨lrcOMitlion(4.2)alsoiIlcaseα10)aIId-G1(t)areconvex?only-(0ii斗)(C!OIn1Ce臼臼mr口.Impe创mIn阪川1X叫(dl扣he臼mIn1CeOIn11uiasWeddiHlL.Mu旧Cih1mO阳gem1n肘1e白rm、adlfO创r.mSO"f(υlL川.11门)aI口1d(υlL.22幻)Ca1n1betreatedbytllemethodpresentedhere,however,forthesakeofclarityaIKlsimplicity＼toavoidtechnicaldimculties,werestricedattentiontotilepresentformof(1.l)alkl(l.2).SymmetryalxipositivedefiniteIlessofMI)二gE(1＼Vuo(J·))isrestrictivebuseemstobephysically-1·eaSOIlable-FOI-example-ifφ:Q×RN→Rispositivedeimiteandot-classC3、theng(29ω)=φω(29W)satisaes(H3).ItslmIldalsobementionedthatincaseaNewtonianviscosityαo>Oispresent、symmetryisωtmededatall『onlytlIesymmtriepartofMz)mMbepositivedeallite-(iii)Assumeagainthatα(t)isoftheform(4.1).Thellifαo>0、bytlrPaley-WienerLemmaollecaIlsilowD(A)二IH忡/1+扩1FV//川vαO二0亏Inm1丑1O1r、ett(ii1nm1丑1e1r、eg1u川1dlha创lIrIr、才iHtyisactuallyiIn1VOlh兀VFed.rlfoseethisSIll}powthereall-econstantsQξ(0、ILC〉Osuchthat C三ldl(ip)(l+ρ2)?!三c-Ifol-allρεR; (4.3)
4I4 Prtus tile1117(4)=Hl+α,p(IR;31,t山lhieCωOImI口1可甲Ip}l扣te?X川iIn川l川tk阳e创盯叫1r盯.1lp仰〉川Olha川tUiOInlSpaCebe创tWee臼Inl讥Wy叫1,pm;J}γ,y,丁)ta川川Lu川l口lK(di川川lL/1+俨lL/tSι川兰升巾ihInlCeα(υt)isaS!S叭川-孔训1ulu山Inm1n1e仅叫(dltobe1-寸Ir.℃(eT宅g1ullaIr.＼.I盯f(4.3)holds、ifypothmis(H4)CELIIberdalxedtosomeextent;theembedding HI+α十叩但;Y)L→W1+α,p(R;Y) showsthatthen (TI5)卡tvl isslliiiciellt.Asaconcreteexamplewhere(4.3)holdsletusmentionagaintheh-BIndstt1(t)二thltiF-Et. v-PItoolfSOIJTHEMAINRIESULrIS ribproverfilm-eIII3、letX=LP(R;Lq(0)MIlddefillelineal-operatorsAaIKlβlIIXasfollowsSill刊tilt?!k〈优削川e创创Ir川‘ib}3yvHypotllmis(i口)、weddilltJluillTheol-tB111l、whichIIleall8 4tl=/iiIMI)二f的l(T)/(t-TJli巳HL(51) aIKlD(Ji)consistsoiallfm叫omuεIVLP(Et;Lq(II))suchthattheIeisjεxwithU=dα*fByTheoremLAisclosediiIleardenselydefinedinjectiv飞到4)clj川 ,AJ|(μ+A)-IK-fω (5.2) ttswellas π|4叫三li(i+|气,|)exp([0十歹lh|).可εEt. (3.:1) IJt吨Iβ(idinedbv (Bu)(t、2)=-div(b(z)Vu(tTZ)),tεR吨I巴。(5.4) withD(B)=LP(Et;W2,p(Q)nwjq(0)),wheleMI)=g川‘tVttυ(z))、‘I'巳0. (5.5) Silkttil℃ω凶时alityyis。{(lass(VIalkiuυ巳IUZJ(fl)nlf/14(il)b3Hypoill川is(H川、birlo吨8toIVIJl(ILIRNVN)astlrhbol叭可mbeddingwh(Q)」→fl(口)
VofferraEquationsinSpacesoffnfegrableFunctions 4f5 silows:oUsel-vetiiatq〉iVby(H4)-11}111STileolmη2applies、ilellccσ(β)C(0,∞)aIIdtheestimates CW)!但十B)-li〈一一一一,iargAi三万…8…1+iAi (5,G) aIId iBη;三Kεri句:专?在R. (5.7) hold.Let h(2,w)=g(tVuo(x)十ω)-g(AVuo(r))-gw(2、Vuo(z))tu(5.8) foi-rξ在zcGR八＼and H(υ)=divh(2,Vu(t,z))7tεR9ZGTT, (5.9) fo117εV:WMm;Lqm))nLPm;W23(mnif才气。)).Tim(12)caIIbe 1·ewrltteIlas A17十Bu=H(17)÷(f÷foL (3.iG} 叭W川FihM阳mθIe们盯lU)=u卜…1问iωO'anmM(di削lbk)咒eddifbjbdO创r阿、ejh趴b川(I叫)=『斗(diiVg纠(‘王?守(仔f}盯)、itisci。arfromiile1、emMKsatiflebeginzlingofscctiOI121ilaiYalldiilfilalsoAXareLconvex-ThepropertiesofAaIKiBderivedaboveandtilepa-川〉0!icitycOMitiomO〈7「/2showthatthetimomIIofDoreaMVemil?!illthevel-SionprovediIIPI-assandSoiIrpoiappiies.NotethatAaMBarerSOlVemln1tCOInmInm1UIIm{INIn11u1timIn1gS盯i1n1CeAactsonlyih川1n1tι-dih1r.eCtUiO1口1whileBdoessoOInllb3y'i川l川}」rι.＼.r1丁ir飞i』1淀号创创r时.eddif.αO1.εA十Bi古SCdiOSeddwi江i山』dOm&出i1n11万〉(A)η1D〉{B)&时bOmde付di鸟3y?i扫:i盯飞V7吧芒岱创I扎川.吃i1ii池』弘i仨e巳.S盯川ih川I口盯1!卡叫Lυ?斗iW=!ω叫lkX+!Aω叫lkX+lB叫dXXU;iu!h1田e1n1A+βisanisomorphismbetwemtile8i〉aces111'tillidX.observethati/l/」→ViIoids.11leoreIII3followsfromtileinversefunctiontheorem,providedwecallshowii时H:V→XisofciassCland毛主aiH(G)=HF(C)=Giloids.Si盯eif/ω17、tlIej川ersefunctiontheoremthenyieldsaunique叫utioIIUε14:|ν|liy<ρ、wilelleveff+jcεxissldl址latif÷foix<飞adiirsoilitpionmapf+AHU二U-IioisofciassClbetweenthespacesXalkilV〕l:rlbprovethatHiscontinuouslydiffermtiablt》flullll/toX.weneedSOIllfresuitsconcerniilgmllbeddiIIgsoffractionaiSoboim?spaces-
4I6 Prμss Le111111a2:lJdJcftbtttltyilttuuttl,YtIBα1titchδyαct,GthfgcIIUYltο1·οfαbo川dtilα1tαlytirco-sf1Itig1圳ilJinY,tlnfllet176·(t人p)dωοtttltt1·tttlililt1·lJοltitioli8ptlctJsbdtuffIIIU=D(G)fquillJIJdtuitltilit沪、αphnormofGα11d}＼TlIfIl fUWαJJ(J;Y)」→Cb(J;Y)矿αp>l; {iijW1,p(J;1')nLP(JJU)〕WW(J;DG(疗,p))formeγμ-dε(0、l)ωitliα+if<l. FOIthepmot-Oi-IJIIIIm2we时ere.g.toDiBlasiol6l『whodealswithtlrcaseJ=[0、Tl;tileprooigiventilelecamesoverdirectlytomb01肌 ('IloosillgY'=Lq(OLG=Awith17(G)=W24(0)门l414(QLitisweliltliOWlithatlDG(0‘q)ι→W圳(川、foIO#iatleast;seeegD趴iBlhaωSm[问6叫iNO创t肘川ttl}hthhl凶川attileem1mI口Ibe仅(di(d1iIn1gS DG(OITP)〕17G(0、q)、。r>0.}人qι(l、∞) alralwayssaudic(1.Tllelf-iolt一＼Clioosillgo>l/lyalidlij〉JV/2q+l/2SIKhthato+Id〈l、LeIIIIIla2飞Yields VL→Gm;CI(IT))门LFJ(IRf1(TT)) (51l) E6叫叫川4斗山叩ih川lnlCe川Wv队q叫(Qm)〕(σ丁叮1(而5)川i汀f2M3〉1+N/Mq、lb问〉巧〉yfS出川OωlbωMX汕川(O)址lhf们们飞VFIH崎1打ylP〉沟川Otlh怡町附eSmihS(川THI4川)刊d巾lhhlω川OtlhlataClhIOi(CjeofS悦川lulCIhlαaInlK(dl『ddisip〉OSSiib〉le巳.(5.ll)istile(C丁1·1lcialembeddiIlgwhichlIlakesH(t7)a(71-function-I08eethisobwrvrtiltulI(·、tlqbfloligstoiffyl4(IKIltJV)fol-eachiixmiti'已Et入' vi/t(l＼W)=g儿1·、Vuo(I)+w)-91(工,Vuo(2·))-tftAfi(1·-vuo(‘I))tu++(gutL1·呼Vh(I)+叫-gw(XTVuo(z))-gwJLVIto(31))ω)Vhω(‘I) holds-Thisrelationshows、thatthereisacontinuousIlolldecreasingfullttioIiv:Et+→R+withρ(0)=Osuchthat |Vi/I(LW)|三(l+lv~ -lfε5、tuεEtN (,1.l2) ,,、电、.·IAELHd、、,-4El,‘飞叫川-tu'E··a''飞··Aa、、,,,F)n、、J‘、HH且、.,,hwi飞咱J''1飞I11陀、、J‘.、HH,d''aEE--B‘、马/LC」.,,,,ρ飞飞、,、,也.,.、i4飞Z飞,‘、,'ιti、EFrd飞。.l‘EUr·飞IIl.,田、ia-ua、电、VLrtbe、EA飞、,AOρLev--1)gr·、,EEAKL'HU[·l 、、l,/'ι、,飞(F'n 9up(＼Vtto(·)+叫-guv(-LVFIlo(·))
VolterraEquationsinSpacesofIFTIegrableFunctions 4l7 vxlf(·,w)=g川(·、Vt川·)+ω)-g川(·吁Vuo(-))+(gww(·、V1以·)+tu)-guJ＼Vuo(·)))V2t川·). Thusenlargingψifnecessaryweobtain |Vzhr(2、ω)|三(l+|V2uo(z)|)v(|ω|) zε0,u?巳RN、 (5.l3) alld |lfh-w)|三ψ(|ω|),I巳TltuξRN. (5.l4) Sinceforeachuεl/(iIISomewhatsymbolicmtation) H(u)=Vz·h(·,Vυ)+hf(·亏V叫·V2t7 weobtainwithtileaidof(512)aM(514) |fI(叫|x三ψ(|u|OAJ;lA)(|t7loJ;1,q+|Iio|24|巾,队 where附medtileωtatioII|·|川aIKl|·|乱,s;川if扣h:b趴O创ltlhl(e干ln}凶川a川tu川allnlω川川(O创ω川)川汀Ir口.1I川iI口1lH11川v叮呐7盯川ya讪I川i(dii川Inll川/fι1LJJ川v叮7nE'Js0R;汁l俨川(Q)川)吨Ir、℃meSlp)eCdtih飞V7吧edly.ThustllcEmbeddillg(5.ll)SIlo川tlIMlfliswellde{iIledoI11/EIIIIdsatisflesaIlestimateOi-tile!form |H(υ)|x三Cv(C|υ|v)|υ|v、17ξ1,二 iIiparticularHisFrechet-dijTermtiabieatu=OaIldH(0)二Hf(0)=0、since手(0)=0.I切 HI(t70)17=Vz-lf(·、Vυo)V17+hf(·呼Vuo)V2υ+llff(·「Vuo)VIL-oV17 SiIK-e fr(·、叫=9ω(·,Vuo(·)+ω)、 byenlarginglTDif11οcessary,wealsohave VIII(1·JU)|三Co+v(|叫)?‘rε111t,εRAN. Therefore「tiltestimate |Hf(υ。如|入'三ICo+v(|uo|om;1,∞)l-l|叫。♂:ld+|tto|24|ν|OJhlm++|ν|0,p;h+|uo|OJ;24|u|ω;1.J<C[co+v(C|uo|v)l-[l+|υohJ|17仆,17、t70ξ1/
4I8 Prt455 刊ilowsthailHf(t70)isaboundtdliIleal-operatorfroml'1oX、fol-everytbξl/.Silk-egisofclassf29similarestimatesinvolvingtilemodulusofCOIltilluityo[guJ』alldgujtuonboundedsubsetsofTT×IRNshowthatHf(uo)ddimdaboveIeallyistheFrdchet-derivativeofHat1709aIIdthatfIFisCOIlti111101lsiIIoperator-uorIII-TilePI-oofofTheol-em3isthereforecomplete.Theorem4isprovedalongsimiiarlines.However,oneilastoobse1·飞℃thatM(4)并{0}intileperiodk-ωe;lq〉laωJlbyA+jtaMHbyB-il、wimpii>OisslliIicielullysmall、toci11lIII飞FelltthisdifkuityfrileiIIlagillarypo飞川、1·soIJl+liaikiB-lia山littilestulleboundsasabove-i.e.15.2)、(5.3)aIIEl(5.6)向(5.7)holdiIItlleperiodiccaseaswell(probablywithdiftBrentconstantsLprovidedμ>Oissosmallthatσ(B)C(μ、∞).TileproofsOi-(:10rollariesiaIld2havealreadybeengiveniIISectiolii-口 itElfitiiICNCES l.l).L.BIll-kholder-kh1·tillgalesaIKll怡Urim-analysisillBallacilspat-rtiliG.IJettaaIKlkIPratelli、Editors、Probαbilityα11dAnαlu813.SpriIlgrrLVPItag、1986. 2.R.飞VCarl-aIldk.B.Itamisgen.AIIOIIhomogeneousiIItegrodiirel-clltitil"l1latioIIillHi灿烂ltspace-SIJiMJ.Ahtlt.fiItttl『10:96l984(1979) :iIEiLfldINPUtaIidJ.PrihN.(bill!)lptdylXMitivrIIleaSIllyf日Jilldift-BUrl-MElul-groups-JILIth.41t11.、287:73105(l990). l.(;.DalPratotillldP.(;1·is飞Fall-d.EqualtiOIlsd飞vollltioIialbstraitcslionliI}(;allI℃drtypepal-aboliq1lt＼J11III-J1;fd.PUT-tiJippl.IL/;120:329356(l979). 5.CKJ1.DafernlosaIKiJ.A.Nolle1.Energymethodsfol-IionlinetIII-hypTlboli(VoiterrailltegrodiiItyelltialtB(illations.(-v011i17t.lJttIf.Dd:lfqnδ'·.4:川1}27以(i971}). 6.(;.l)itiltuio.Liliftlri)ttrtLboli〈-cvolutioneqllatiOIlsiliLP-spaces-fll111.Jl1fαi. 、、l'/.,主40凡UA可d''EA(ι丛IAυ1ivhJUW气。、uo心。‘JY曹··E,It'''''‘,,1,,a川ya4吨l,,‘-tt、J,a,. 7.(;.Dort?alklA.Venni.OIltileclosedllessofthes1lINoftwoclosedoPPI-ttto11.MilthZ、196:18920l(1987).
VolterraEquationsinSpacesofIntegrableFunctions 4l9 8.H.EIIglel--011thedynamicshearnowproblemforviscoelasticliquids-SIAJLIJ.Mαth.Aml.,18:972-996(1987). 9.D.Fujiwa11.Fractionalpowersofsecondorderellipticoperators-J.Afαth.Soc-Jαpα凡21:481522(1969) 10.EHilleaIIdR.S.Phillips-Sε们1mIηI,ιt咿gIm1ηο1uipSαTηldlFL二气、寸}hb1uuiillIn1kμtiωO1ηtαdl4ofAIm7ηlC1＼」/A11飞VlLMiffαtlhi.Soc-Colloq.Publ.Amel--Math.Soc.,Providence、RilodrIsland,1957. 11.HKomats11.Fractionalpowersofoperators.PGcijIcJ.MMlL、1:285-346(1966). l2.S.o.LolldellaIKiJ.A.Nollel.NonlinearVolterraliIitegrodiiferelltiaImlIla-tionoccuringinileatflow-J.I111叨1、αlEquαiiOIls、6:ll50(1984). l3.A.LunardiaIKlESillestrari.FullynonlineariIItegrodiferentialeq1latiollNillge时川Bamchspace凡,fαth.Z.、190:225318(1985). l4.R.MacCully-AIlintegrodiEel-entiale(illationwithapplicationsiIIilealtlHow-QMlf.Alypl.Mdli叫35:1-20(1977). 15.R-LfacCallIY-Amodelforone-dimensionalnonlinearviscoelasticity.QIif11、1.AP131JLIα1fl·-35:2133(l977). i6.1＼It.hfcCOIlllell-011FouriermultipliertransformationsofBa11aichvalumifmctiom.Trα川ATyler-Afdll.Soc-9285:739757(1984). l7.IiPriwalow-Rα11dcigmsellαftenαnαlytiscllC1、Funkti071川、volume250fffoclist-httlbdcllE1、jUrJVdlumdikDmltsCherVedagderWissellschaftcIl龟BerlinT1956. l8.J.Prass-MaximalregularityOi-lineal-vectorvaiuedpal-aboli℃＼YOKEB11·tllequaltiOIls-J.hltegrdEqη6.4IJpls-Ttoappear l9.J.PriissaIKiII.Soill--BoundedIlessofimaginarypowersofseCOIldoIYKBrellipticdi{ferentialoperatorsiIILP.PrepriIIt. 20.J.PriissaIKiH.Soilr.OIIoperatorswithboundedimaginal-ypo飞vel-siILBallaclIspaces-」thtl1.Z.吁203:429452(1990).
420 Prius 21.FVI.Renal-dy,WJ.Hrusa-a11dJ.A.NoIlei-MMfleyTIdiedProblemsiIIVts-coehsticity,volume35ofpttmαTtMOTtogmpfzsαηdSω、ueysinPω、tαndApplied几fdflunαtics.LongmanSdenti自candTechnical,Hariow、Essex,l987. 22.R.Seeley-NormsaIKldomainsofthecomplexpowersAb-4771tr.J.itfritjl川93:299-3ω(1971). 23.0.StaffaIIS.OIlanonlinearhyperbolicVolterraequation-SIA儿fJ-llfαtfl.Jlnαl.,11:793-812(i980). 24.FZimmermanII.Onvector-valuedFouriermultipliertheorems.Sitidirt JVM/1.、93:20122(1989).
AlmostPeriodicityPropertiesofSolutionstotheNonlinearCauchyProbleminBaEEachSpaces W.M.RUESSDepartmentofMathematics,UniversityofEssen,Essen,Germany 工NTRODUCTION Theobjectofthispaperistopresentseveralrecentresultsonal-mostperiodicitypropertieSofsolutionstononlinearev0111tionequa-tiORS(basedonjointworkwithw.H.Summers[61,64,67]).Wecomparetheresultsforthecompactandtheweaklycompactcase,anddiscusstheCOReeptsandtechrliqliesofproofandpossibleextensionsofthere-sultsaswellasproblemsrelatedtothem.Throuqhoutthepaper,AcxxxisanIII-aceretiveoperator(qeneral-ly,nonlineararldmultivalued)inaBanachspacex,qerleratingastrong-lycontinuoussemigroup(S(t))〉ofnonexpansivemapsS(t):C→Ct一O(Crandall-Liqqett[23]),whereC=ClD(A)issupposedtobe(closed arld)convex.WeconsiderboththehomogeneousandtheinhomogeneousCauchyproblemassociatedwithAandhEL1(E+,x) U(t)+Au(t)3h(t),t〉O(CP){andU(0)=11OEC, U(t)+Au(t)3O,t〉O(CP.){…U(O)=110EC Asiswell-known(Cf.[10,14,15,39,52]),processesirlphysics,chemistry,bioloqy...,oriqinallymodeledintheformofpartialdif- ThisworkwassupportedinpartbytheDeutscheForschunqsqemeiηschaft(DFG) 42l
422 Ruess ferentialequations,oftencanbesetupinthisqeneralformat.Then,xiSusuallyaspaceoffunctionslikeC(Q)orLP(Q),1〈p〈∞,QopeninRn,andDU)consistsOfsmoothelementsofxlike庐山orwk,p(Q),KEm,satisfyingcertainboundaryconditions.Also,itiswell-knownthata)stronqsolutionsto(CPh)aregiverlbytheInotionU=S(·)11oof(S(t))〉throllqhtheinitialvalue110,andthat,t一Ob)sincehEL1(EZ+,X),theinteqralsolutionto(CP)-ir1thesenseofB色rlilan[11]-isanalmost-orbitof(S(t))([50,Prop.7.1]).tELO DEFIN工TIONAcontinuousfunctionU:卫+→CissaidtobeanαZrnost- opb￡tof(S{t))、t二O liIIISUPHU(t+r)-S(x)U{t)||=O-t→∞rEE+ 工nvestiqatingthelong-termbehaviorofsolutionsto(CP)andto(CPh)thusamountstoadiscusSionoftheasymptoticbehaviorOfal-most-orbitsof(S(t))、tfOIncasetheranqeofthesolutionUis(norm-)relativelycompact,resultsinthiSdirectionarequitesatiSfactory,well-knownandeasilyderivedfromtheconceptOfasymptoticalmostpeziodicity('TheoremsA1andB1insection1).IncasetheranqeofUisonlyweaklyrelativelycompact,themaingeneralresultswereBaillor1.S[2-6]rlonliRearergodictheoremanditsextensions,concerninqtheexistenceofthenorm-limitofthetime-meansofthesolution(NonlinearMETinsection2below).In[61,64,67],wetookupEberlei口'S[29]classicalconceptofweakly alITlostperiodicfur1ctiorlstoderivearlaloqueSoftheresultsir1therl。rTTI-COInpactcaseontheasymptoticbehaviorofUitself-TheoremsA2andB2iIlsection2. 工ncontrastwithaLInostperiodicity(onE)andasymptoticalmostperiodicity(onE+),weakalmostperiodicityirltheserlseofEberleinSOfarhadnotbeenafactorinthecontextOfdifferentialeqIlatior1S-ItisoneofthemainobjectsofthiSpapertomakethisclassicalcon-cept,theresultsontheasymptoticbehaviorOfsolutionsto(CP)re-latedtoit,andthecorrespondingtechr1iqLIesOfprooftransparenti口thecontextOftheaboveCauchyproblem.
NonlinearCauchyProbleminBanachSpaces 423 NOTAT工ONANDTERM工NOLOGY (a)IfU:卫+→CisaRalmost-orbitof(S(t))〉tfOY(u)={u(t)|t〉O}denotesitsranqe,ω(u)={yEC|U(t)→YinnormforsomesequenceO三t→∞}itsnQ-limitset,andω(u)={yEClU(t)→yweaklyforsomesequenceO三tn→∞}itsnweakQ-limitset. IncaseU=S(·)xisactuallyamotionof(S(t))、,thesesetst乙owill,asusual,bedenotedbyy(x),ω{x)andω{x),respectively.W(b)ForasubsetDofx,CODdenotesitsconvexhull,!lll-clD(resp.w-clD)itsnorm-(resp.weak-)closure,aI1dextDitssetofextremepoiRts.BXdenotestheclosedunitballOfx.(c)ForJE{R,R+),(C(J,X),|ll|)denotesthespaceofboundedb∞continuousfurlctiorISf:J→x,equippedwiththesup-norm,andCO(J,X)itssubspaceoffunctionsvaRishinga℃infinity.ForfECb(J,x),andtEJ,fisthet-trar1slateoff:f(S)=f(s+t),tSEJ,andH(f)={f|tEJ}itssetoftranslates.tThespaceofallαZI770stpe240d￡σfunctionsf:R→X(Bohl/Bohr/Boch-ner[13])willbedenotedbyAPfEsXJ.Recall[13]thatoneofthe equivalerltdescriptionsofalmostperiodicfunctionsisthatofallfECb(卫,x)withtheirsetH(f)oftranslatesrelativelycompactlnsup-norm. 1THECOMPACTCASE-ASYMP'IOTICALMOSTPERIODICITY Forreasonsofcomparisonwiththeweaklycompactcase,webrieflyre-calltwoofthe-well-k口。wn-rnainxesultsontheasymptoticbehaviorofalmost-orbitswith(norm-)relativelycompactrangeinaqeneralBar1achspacex.Theconceptgoverninqthecompactcaselsthatofasymptotical-mostperiodicity. CZαss句αZσomept(Fr毛chet[30,31],1941):AfunctionfECb(卫+,x)issaidtobeαSH771ptot￡σαZZUαZmostpep￡od4σ(α.α.p.)ifitsset日(f)={ft|t兰0}oftranslatesisrelativelycompactin(Cb(卫+,x),|||ω. Thespaceofallasymptoticallyalmostperiodicfunctionswillbe
424Ruess denotedbyAAPfE飞xj. CZαss王σαZr'esuZt(Frechet[30,31],1941;DeLeeuw/Glicksberg[25,26],1961):AfunctionfEC(R+,X)isasymptoticallyalmostperiodicifandonlybifthereexistuniquefunctionsqEAP(IRA)arld。ECo(EZ+,X)suchthatf=q|EZ++中· TERMImLOGYInthefollowinq,wecallafunctionfEC(卫+,X)almostb+periodicifitiStherestrietiontoEofanalmostperiodicfurlctiorlq:R→X THEOREMA1ForageneralBanachspacexandanyalmost-orbitU:1R+→Cof(S(t))、,℃hefollowingassertionsareequivalent:tfO(a)Theranqey(U)ofUisrelativelycompactinx(b)UisasFnptoticallyalmostperiodic(C)ThereexistuniqueelementsYEω(U)and中ECO(EZ+,X)suchthat(i)U=S(·)y+中,and(ii)S(·)yisalmostperiodic THEOREMB1ForageneralBanachspacex,assumethatU:卫+→CiSanalmost-orbitof(S(t))、withrelativelyc。ITlpactrangeinxt三OThenω(U)isacompactcommutativegroup,and,ifl」denotesnorma-lizedHaarmeasureonω(u), lrTf||!i-liIIl￡jf(U(t+s))dt=jfdliT→∞-vω(u) 11rliformlyoverSER+,forallcontinuousfunctionsffrom||||-ClY(U)intoY,YanyBanachspace. NOTESInparticular,inteqralsolutionsUof(CP)withrelatively compactrange(A)alwaysaSFnptoticallyuniformlyapproachantZZmostperdod40(weak)solutionS(·)yof(CPL):1|||-lim(u(t)-S(t)y)=O,and-t→∞ (B)Uaswellascompositionswith"observablesHf-asspecifiedinTheoremB1-,areergodicinthesensethatthelimitofthetime-rnearlsalong(andstartingfromanypointof)thetrajectoryY(U)existsandisequaltothespacemeanoffovertheQ-lirnitsetω(u)
NonlinearCauchyProbleminBanachSpaces 425 Proofs.Bothres111tsare-moreorless-iITInediateconsequeRcesofthefactthatalmost-orbitsofnonexpansivesemiqroupswithrelative-lycompactrangeareasymptoticallyalmostperiodic,which,inturn,isatrivialobservation:Givenasequence(t)CE+,if(t)isRnnn+bounded,wecanassumet→tEE灵,andcor1eludethat11+→u+-innO」口」。sup-norm,forUisurliformlycontinuous.If(t)isurlbourlded,wenncar1assumethatO〈t个∞,arld(u(t))isnorm-Cauchy(y(u)beir19-nnnrelativelycompact).Thus,wehave ||u+(t)-U+(t)川〈川U(t+t)-S(t)U(t)||+川U(t)-u(t)||」n」m-nn+HS(t)u(tm)-u(tm+t)|| arbitrarilysITlall,urliformlyovertEE+,fornaRdmlargeenough-TherestofTheoremA1nowfollowsbyanobviousapplicatior1oftheFr色chet-DeLeeuw/Glicksberqresultto吐1egivencontext,andTheoremB1fromthedecompositionU=S(·)y+中ofTheoremA1(c),combinedwithseveralclassicalresultsondynamicalsystems(seeG1lunarl/pazy[37]):ω(u)=ω(y)isaminimalsetofalmostperiodicmotions([51,Part2,CT1.v,sections8ard9]),andthusisacompactcommutativegroup([51,Part2,CT1.v,Thm.8.16]and[27]).Also, 门vdp+ιyf以=tdytquf牛Tor-J1-Tn∞-u→lT ([51,Part2,ch.VLsections5arld6])-Finally,from中ECO(EE+,x),1rT1/I'||!|-lirn击jOf(U(t))dt=||||-lim子jof(S(t)Y)dt,T→∞T→∞ thuscompletinqtheproof. REMARKS1.Asmentionedearlier,bothoftheaboveresultsarewell-KRowrl.TheoremA1appearsinvariousformslnBhatia/chow[12,Thm.6.9],Dafermos/sleInrod[24],andHaraux[38,39,41,42](thislistmaynotbecomplete).ETaxaux[40]andzshii[43]extendedittothecaseofperio-dicevolutionsystems.comparealso[59,60,62,63],and,forthediscretecaseofiteratesofanonexpansivemap,Baillon/Bruck/Reich[9,ThII1.3.2].TheoremB1-forU=S(·)xamotionof(S(t))-isTheoremt〉OinGutmarl/pazy[37]. 2.Muchmorecanbesaid.Interestinglyenough,bothresults
426 Ruess essentiallydatebacktotheworkofM.Freerletintheyear1941.Hispapers[30-35],thoughforthefinite-dimensionalcase,alreadycor1-Minalltheresultsandideas.TheoremA1canbereadfrom[30,31,35],aMTheoremB1iscontainedin[32-34].Unfortunately,thesepapersarerelativelyir1accessiblenowadays,andthushavelarqelybeenover-looked.Nevertheless,itseemsitisFr邑chetwhodeservesthecredit.NotealSOthat,SiIIlilarly,TheoremA1arldaformOfTheoremB1forthediscretecaseofiteratesOfamaphadalreadybE-entakenupbykyFan[48]in1943. 2THEWEAKLYCOMPACTCASE-WEAKALFCSTPERIOD工CZTY WepresenttheconceptsandtechniqueSOfprooff-orthearla10911esOfTheoremsA1andB1forthecasethaty(u)isonlysupposedtobeweaklyrelativelycompact[64,68].Theresultsformallycorrespondexactlytotheonesfory(U)beingnorm-relativelycompactexceptthat-eSsentially-asymptoticalmostperiodicityhastobereplacedbyEberleirfS[29]conceptofweakalmostperiodicity-However,therewillaswe11betwoeSSentialdifferences 1.AsiSthecaseforfixedpoir1ttheorems,passirlqfromnorm-toweakrelativecompactneSSseems-SOfar-onlypossibleattheexpeηseOfimposinqrestrictiOIlsonthegeometryoftheunderlyinqBanachspace-uniformconvexity,attheveryleast.(Onthispoint,compareBruck'Sexcellentsurvey[22]-amustforreadingforthoseinterestedintheasymptoticbehaviorofnonexpansivemappinqsanyway.)2.WhileitwastrivialtoderiveasymptoticaImostperiodicityOfal-ITiost-orbitsUfromrelativecompactnessofy(11)(inageneralBanachspace),thecorrespondingstepfromweakrelativecompactnessofy(u)(inaur1iformlyconvexBanachspace-plussomefurtherextraconditiononU)toweakalmostperiodicityofUhereisthedecisivestepofproof,requiringspecialtechniques.First,werecalloneofthemaingeneralresultsknown工nourcorl-texthere:Baillor1.SarldBruck'Smeanergodictheoremforsemiqroupsofnonexpansivemappings(nonlirlearMET,forshort).RightafterBaillor1[1-6],in1975-78,hadbeenthefirsttosettlear1onlinearMETforrea1Hilbertspaceandspacesd,1〈p〈∞,quiteallimberofauthorswerlt
NonlinearCauchyProbleminBanachSpaces 427 ontoextendandrefinehistechniquesandresults;apartiallist(only)comprisesBaillon/Br色zis[8],Brezis/Browder[16,17],Bruck[18-21],Kobayasi/Miyadera[46,47],pazy[53,54]andReich[55-57].FOXacompletelistandsurveyof出isdevelopment,aqaihBruck[22]isobliqatoryreadir19·Initsfinalform,basedonBruckV[20,21]re-finementsofBaillor1.Stechniques,arldstatedexplicitlyinMiyadera/Kobayasi[50],itreadsasfollows. NONLINEARMETAssumethatxiSauniformlycor1vexBanachspace,andU:卫+→Cisanalmost-orbitof(S(t))kthatist二oi)bounded,andii)asymptoticallyisometric(a.i.),i.e. liIn川U(t+s)-U(s)||=P(s)existsuniformlyoverSER+-t→∞ Then 1rT||||-lirIl二IU(t+S)dt=ZEXT→∞l'O existsur1iformlyoverSEEZ+, andisafixedpoi口tof(S(t))、t三0 NotethattheassertiORiSthatofTheoremB1forthespecialcaseOffbeingtheiderltitymap.TheoremA2belowwi1lallowformoreqe-TIeralmapsf. 旦旦旦旦王EEonUbeinqasymptoticallyisometric(a.i.).1.Theexister1ceofthelimitinnormmayfailwithoutthiSextraas-sumptiononU:Baillon[7].2.a)工fω(u)半φ,thenUis(a.i.)[50,Prop.5.5].b)IfxiSrealHilbertspace,OEC,andthereexistsL主Osuchthat||S(t)X+S(t)Y||2三川x+y川2+L{|lx川2-HS(t)x川2+HY川2-||S(℃)y川2} foralltER+andx,yEC,theneveryalmost-orbitof(S(t))t主ois(a.i.)[50,Cor.6.2]-Inparticular,thisistrueifC=-candeachS(t)isodd.(Forapplications,thidt,forexample,ofAbeingthesubdifferentialB中ofanevenconvexl.S.C.functiononrealHilbertspace.) Wenowfollowformallytheformatsetforthinsectior11topresenttheresultsof[64,68].
428 Ruess TheconceptgoverninqtheweaklycompactcaseiSthatofweakal- mostperiodicity. CZαssfσαZσonσept(Eber1eirl[29],1949):AfunctionfECb(R飞X)issaidtobetJGαKZUαhustpeptod￡σf4nthesenseofEbepZe4713E.-ω.α.p.jifitssetH(f)={f|t〉O}isweaklyrelativelycompactt-in(Cb(IR+,x),||||缸,)· (Eberlein[29]consideredthescalarcase;forthevector-valuedcase,seeMilms[49],and[65].) ThespaceOfallweaklyalInostperiodicfunctionswillbedenotedbyW(R+,X),and吐1e(closedlinear)subspaceofthose中EW(IR+,X)forwhichthezerofunctioniscontairledintheweal〈closureofH(中)-somesequence(中f)oftrarlslatesof中converqestothezerofurlc-」nntiorlwithrespecttotheweaktopoloqyof(C(R+,X),||||∞)-bybWO(R+,x) CZαss4σαZPGSUZt(DeLeeuw/Glicksberg[25,26],1961;Jacobs[44,45],1956/57):AfunctiorlfE飞(R+,x)isweaklyalmostperiodic(irlthesenseofEberlein)ifandonlyifthereexistuIliquefurlctionsqEAP(R,X)arldOEW(卫+,X)suchthatf=q+|EE+ THEOREMA2([64,Thm.1.4])AssumethatxisauniformlyconvexBanachspace,andU:R+→Cisanalmost-orbitof(S(t))thatisboun-tELOdedandasymptoticallyisometrie.Then(a)UisweaklyalmostperiodicintheserlseofEberlein,and(b)thereexistuniqueelementsyEω(U)and4IEW(卫+,X)suchthatWO(l)U=S(·)y+中,and(ii)S(·)yisalmostperiodic THEOREMB2([68,ThIII-3.1])AssumethatXisamiformlyconvexBanachspace,andU:IR+→Caboundedandasymptotic-allyisornetriealrnost-orbitof(S(t))℃主owithadecompositior111=S(·)y+4;asinTheoremA2.Thenω(y)isacompactcommutativegroup,and,ifμdenotesrlorIria-lizedHaarmeasureorlω(y), 1rT||||-limz!f{U(t+s))dtlJ'OT→∞ =ffdlJ,ω(y) urliformlyoverSEE主+,for
NonlinearCauchyProbleminBonachSpaces 429 (i)alLboundedlinearoperat。Esf:X呻Y,and(ii)allcontir111011sfunctionsffromw-Cly(u)wiU1吐1einducedweaktop。loqyintoywiU1吐1enormtopoloqy,YanyBanachspace. NOTES1.Comparedto吐1enonlinearMETabove,theassumptionsarethesamebuttheconclusionOfTheoremA2is。rIU1easymp℃oticbehaviorofUi℃selfraUMZU1anonU1easympt。ticbehaviorofjustthetime-meansofU.(ThenonlinearMET,inturn,followsfromthismorespecificre-sul℃asaspecialcase。EUMclassicallinearMET,seeSteplinU1eprooEsbeLow.)2.znparticular,integralsolutionsU。f(CP)(wlU1Xunl-fortnlyconvex)thatareboundedand(a.i.)(λ)asympto℃icallyweaklyappx。achaweakSOLIltiORS(·)yof(CPh)吐1atisGhostpeptod40inthefoll。winqsense:thereexiStsasequenceO三℃n→∞suchthatw-liIE(U℃r1-S(·)Y)=O(吐1eliIEi℃withre-spectt。theweaktopoloqyof{Cb(卫+,x),|!||cn)).Theextenttowhichthisweakas-ymptoticapproximationworksisthepointofdiscussi。rI。fProblem1insection3.Moreover,(B)Uaswellascompositionswith"。bservablesHf-asspecifiedinThe。reIEB2-axeergodicinthesamesenseasinthecompactcase above(excep℃withω(u)beinqreplacedbyω(y)).3.PairedwiU1fur吐1erres℃ric℃ions。nX,A,。r(S(t))、"t二OThe。reIEA2canbeusedtoderivestx。nqerforms。fasympto℃icbehaviorf。rspecialcases,byprovinq,f。rinstance,thatS(·)yisconstant,and中isevenweak-。znorm-C.Thefadthu,hrA=忡wi由中。even,proper,convexandl.s.C.。rlrealHilber℃space,||||-limU(t)t.·CDexistsforeveryboundedwe出s。LIlti。nUto(σ)(Cf.Bzuck〔18,'I·M.5]),isoneoftheexamples℃。由iseffect,see[69].4.Theassertion。fThe。reIIlB2mayfailforHobservables"fthataxe。nlysupposedt。benorm-t。-rmmcmtinuOIls([68,Example3.4]). pr。。fs.B。thresultsaxe-rIl。ze。rless『immedla℃econsequences。fthefactthatboundedandasymptoticallyis。metzicalmost-。rbi℃s。fnonexpa口sivesemiqr。upsinaunihmlycmvexBanachspaceareE.-w.a.p.(pz。p。si℃土。n(a)inThe。reIIIA2),whichisthemainpart(Step2below)。f吐1epr。。f.
430 Ruess STEP1:(a)implies(b)inTheoremA2AccordingtoDeLeeuw/Glicksberq-Jacobscitedabove,U=q|EZ++。withqEAP饵,X)and中EW(E+,X).EleITIerltaryarqmentsshowthatothereexistsasequenceO三tn→∞suchthatboth||qtrl-q川∞→Oand中tn→Oweakly.AsY(U)isweaklyrelativelycompact,wecanaswellassumethatw-liIIlU(trl)=YEωw(u)Thus,fortE巳0,andx*EBX*, )+」+n+』(u)n卡』(u)ot(→cu|)牢+X,)川川t*(XGJ,、J-nt)(tur、-JnttrtAVCM+)yt)(tnf、tsqd((三+)*x,,)+L(qy)t(qM( asn→∞{wherewehaveusedLemma2irlSTEP2below). Hence,9|:阪+=S(·)y,and(b)isestablished. ProofofTheoremB2:Here,weo口lyneedtoshowtheexistenceoftheergodiclimit(innorm).TherestoftheproofthenfollowsalongsimilarlinestotheoneforTheoremB1,see[68].First,ifUEW(EZ+,X)andf:w-Cly(u)→Yisweak-to-normCORtirl11011s,thenf(u(·))EW(R+,X)([68,LeIIlm3.2]).Thenweusethefollowingqe口eral ERG3DICTHEOREMforweaklyaLInostperiodicfunctions2IffEW(R+,X), then ||!|-limlfTf(t+s)dt=Zτ飞→∞LU'O existsur1iformlyoverSEE+ (SeeEberlein[29]-scalarcase-and[68,ThIII-2.2].) Proof.ThisisaspecialcaseoftheclassicalIIlearlergodictheo-rembykakutani,vonNeumarln,Yosidaandothers(priorto1939)forZ￡-neGPC。-semiqroups:First,notethatanyfEW(卫王+,x)isuniformlycontinuous([68,prop.2.1]).Then,consideringthetranslatiorlsemiqroup(T(t))t注Oon(W(卫+,X),||||∞),T(t)f=ft,givenfEW(R+,x),thelinearMET([28,VIII.7.1,ThIn.1])appliestoshowthatthereexistsafixedpointZEW(R+X)of(T(t))Le.aconstantfunction't三0'Z(t)=ZEX,suchthat1rTliInsupHzjhf(t+s)dt-ZH=O,F+∞sEEZ+工'υ whichisthedesired(nonlir1ear)result.
NonlinearCauchyProbleminBanachSpaces 43l STEP2zpr。。f。fpr。p。siti。n{a)。fThe。reEIlA2Weshalln。tgiveadetailedpx。。王(see[64]),butrathereE吃plainU1emetrmdsandtechniquesin。rdert。beable-insecti。rl3-t。discusstheirlimitati。ns. STEP2.1:Characterizati。nofweakalmostperi。dicity LEMMA1(Gr。thendiedt[36],1952;Milnes[49],and[65])Afuncti。nfEcb{卫+,x)isinW(卫+,X)ifandonlyiff。ranyse-quences(tm,也)mC卫+xextBfand(ωn}nC卫+,α:=liznliIn{f{t+40),x*)=limlizn{f(t制),x*)=:8,nmInnmnpr。videdtheiteratedlimitsexist. 旦旦Lzc。rlsider出eimmetricembedding(Cb(及+,x),||||∞)→(C(EZ+x(extB*,w*)),||||∞)bfH{(t,x*)H(f(thx*)} andapplyGrotherldiedt'sclassicalscalarresult[36,Th§。reme6]. STEP2.22The。reIIIA2(a)f。XU=S(·)xam。ti。n。fac。-semiqr。11P{S(t))t〉O。fb。12rldedlinear。Perat。rsonaqeneralBanachspacex,withy(x}weaklyrelativelyc。mpactinx. Acc。rdingt。Idemna1,wehavet。c。11siderd。11blesequencesofthef。rm(S(t钊β}x,x*).Sincey{x)isweaklyrelativelyc。mpact,andn,m(S(tm)*唁)misMunchd,wecanassmethatS(ω)x→xweaklyinx,nand{S(t)*x*)clustexsw*atx*εB*-Thus,txivially,m。xα=liznliIE(S{t钊β)x,x*)=liznlim{S(ω}x,S(t)*x*}=(x,x*)mnmnmm。。nmnm=liznlim(S{ω-}x,S{t}*x±)=B.mnUU刷 Thep。int。fthisdeviati。nist。sh。wthatthepr。。fw。rkseasilyifallS(tysaxelinear(and{S{t}}t三。isurlif。rmlyb。11rlded)·This。bsexvati。nisthestartingp。intf。rthef。ll。wingextendedversi。rl。fThe。xeIBA2
432Ruess STEP2.3:ExtensionofTheoremA2toaqeneralBanachspace DEFINITION(Bruck[20,21])AmapS:C→C,Cclosedconvexinx,xaBarlachspace,issaidtobeε-αppmMmteZuαff4monasubsetkcCif||S(E入.x.)-ZA.sx.川〈εforall(fiRite)convexcombi-llllnationsZA.x.ofeleIIlentsx.EKll工 THEOREMA2.1AssumethatU:R+→CiSanalITlost-orbitOfastrorlq-lycontir111OIlssemiqroup(S(t))ofnonexpar1sivemapsS(℃):C→Ct注OonaclosedconvexsubsetCofageneralBanachspacex. Moreover,assumethat (i)Y(U)isweaklyrelativelycompactiIIX,and(ii)givenε〉O,thereexistsT=T(ε)主OsuchthatS(h)isε-approximatelyaffineonco{U(t)|t主T}forallh主0.Thenpropositions(a)and(b)ofTheoremA2hold. withtheproofforthelinearcaseaboveir1rnirld,thisresultshouldnotcomeastoomuchofasurprise.Indeed,thedetailsofproofcanbereadfromtheproofofTheoremA2[64,Thm.1.4].Alongtheway,thefollowir19LemmaisrequiredthatwealreadyusedinSTEP1above. LEMMA2GiventheassumptionsofTheoremA2.1,S(h):w-Cly(u)→Xisweak-to-weakcontir111011sforallh〉O (Lemma1.7in[64],togetherwith[66,Remark2.5].) WhatiSleftfortheproofofTheoremA2(a)istoshowthattheassumptionsofTheoremA2implythoseofTheoremA2.1.ThiSiSwhereBaillor1.S[6]andBruck'S[20,21]ideascomeintoplay.Asanintroduc-tionintotheseldeas,weStartwiththeHilbertspacecase. STEP2.4:TheoremA2(a)forX=H=realHiIbertspace 工nthiscase,wecanusezarantonelldSinequality([70,PartI,cor.toLemma1.7,p.248]): )勺品·叮JX)hcu-lx)h(mb「4.、JX-lx(-1」、人-1、A.、Jγ』〈-l〈-内4.lx)-n(CU·工、八z)-lx-L、λz()h(S
NonlinearCauchyProbleminBanachSpaces 433 forallh三0,andallconvexcombinationsZA.x.,x-EC.Thisine-lllqualityholdsforqeneralnonexpansivemaps,andfollowsbydirectcom-putationfromthepropertiesofscalarproducts.Obviously,ifU:IR+→Cisasymptoticallyisometric,thisinequalityyieldsassumption(ii)ofTheoremA2.1,andwearedone. STEP2.5:TheoremA2(a)forXURiformlyconvex BuildingonBaillorl'S[6]techniques,Bruck[20,21]extendedzaranto-ne110'Sinequalityinthefollowingform. NOTAT工ONANDTERMINOLOGY(a)Tdenotesthesetofallcontir111011s,corlvexandstrictlyincreasingmapsy:",∞)→[o,∞)withY(0)=O.(b)AnorlexpansivemapT:C→x,CclosedcorlvexirlaBarlachspacex,issaidtobeoftHpey,withyEr,if Y(川T(ZA.x.)-ZA.TX.H)主Iriax(川x.-x.川-川Tx.-Tx.川)lli,jl〕 forallconvexCOInbirlatiorISZA.x.,x.EC工工'工 LEMMA3(Bruck[21,ThITl.2.1])IfCisaclosed,bourlded,andconvexsubsetofaumiforInlyconvexBanachspacex,thenthereexistsyEr(dependingonlyondiamC)suchthatevery口oneE号aRSivemapT:C→xisoftypey Obviously,thisHmeasureOfnon-affirleness"fornonexpaRSivemaps inuniformlycORVexBar1achspacesworksjustaswellaszarantonello'Sinequalitytoderiveassumption(ii)ofTheoremA2.1fromUbeinqasymptoticallyisometric. Thesecorldrestrietiorl,liniformconvexityofx,iSneededforthe proofofLemma3attwostages.1.xisuniformlyconvexifandorllyif6(ε)〉Oforallεξ(0,2],where6(ε)=inf{1-1/2||x+y!||||刘|=|lYH=1,||x-Y||三ε}isthemodulusofconvexityofx.Bruck[20,21]usesthispropertyofS(·)tofirstconstructaY2Er(basically,byirlteqrating占(·))thatworksforconvexcombinationsoflengthtwo,andthenqoesontopro-duce,byirIductiorl,YpErthat-worksforconvexcombinationsoflength
434 Ruess p,pEL2.2.uniformlycor1vexBanachspaceser1joythecorlvexapproximationpro『perty,i.e.,givenMCBandε〉O,thereexistspENsuchthatxcoMCCOnM+εB,wherecoMdenoteSallconvexCOInbinationsofyxepeleIner1tsinMoflerlqthatmostP(Cf.Bruck[21]).Thisallowstoconstruct,outofthe(Yp)p三2,onesingleYErthatworksforcon-vexcombinationsofazvlenqth([21]). Thisconceptualinsightintothetechniquesofproofwillberlee-dedfoxthediscussionoftheirverylimitationsiRthenextsectior1. 3PROBLEMS TheassertionOfTheoremA2raisestwoimmediatequestions. +PROBLEM1If中εw(R,X),howCLoseis中toarlorIII-(orweak-)O Co-function? withzeqardtothedecompositionU=S(·)y+中ofU(TheoremA2),thiSamountstoaskingtowhatexterlttheweaksolutionUof(CP)asymptoticallyapproachesthealInostperiodic(weak)solutionS(·)yof (CPh) PROBLEM2Howmuchqeometry(fortheunderlyinqBanachspaceX)iSactuallyneededfoxtheassertionofTheoremA2? Problem1.If中EE=(W(R+,X),||||∞),由enOOa)℃hereexistsO三tr14∞suchthat中tn→OEEoweaklyinE。,arld1rTb)limsup{否f。|(中(t十h),沪)|dt|x*ξBJ,hEE+}=O([68,prop.2.3])·T→∞ThefirstquestionwithregardtothedecompositionU=S(·)y+。iswhetherhJOEEoweaklyinEofor兰主sequencesO三tJ∞,i.e.whether中isalsoaweak-CO-function.Thisisnottruein9eneral,seetheexamplesin[65,section3].Letusdefinetheweak-Q-limitsetomof中(withrespecttoWEO)tobethesetofallOEEnsuchthat中+→中weaklyinEoforULnsomesequenceO三tn→∞,Thenextquestiorl,吐len,is:工f中EWo(职+,x),isQW(中)CCo(卫+,xw)=weak-Co-functionsincb(E飞X)?
NonlinearCauchyProbleminBanachSpaces 435 Unfortunately,ingeneral,出isisnottrueeither.In[58],aclass+offunctionsFEW(R)isbeirlqconstructedwiththefollowingpro-perties(i)FEWo(卫+)＼Co(卫+),but(ii)Fr→FeverliRsup-normon」n 卫+,forsomesequeRceO三tn→∞. Atfirstsight,withreqardtoU1edecompositionU=S(·)y+中ofTheoremA2,thisisafairlyrleqativeresult.However,takingintoaccountrealprocessesqoverrledbysolutionsto(CP),thiskindofir-reqularityshouldcomeasnosurprise.ThoseprocessesarecomposedofareqularstateS(·)Y-thatisevenalmostperiodic-whichisbeinqattair1edasymptoticallyforaparticular(time-)approachtoinfir1ity,andofacriticalstate中whichisresponsibleforallirregularlong-termbehavior,andwhich,i口particular,foradifferent(time-)approachtoinfinity,mayevenleadbacktotheoriginalState. Thisisofpracticalimportaηce:While,inthecompactcase,theasymptoticbehavioroftheprocessUiscompletelydeterminedbyitsregularpartS(·)y-itscriticalpart中(ir1TheoremA1)bei口9 norm-co-,thisis,inqeneral,bynomeanstrueintheweaklycom-pactcase.Moreover,just"Ifleasurirlq"theliIIlitofthe(time-)Inear1SoftheprocessorofobservablesU1ereof-which,forallpracticalpurposes,usuallyisbeingconsideredsufficient-mayevenleadtofalseconclusiorls,asthiSliInit,accordingtoTheoremB2,complete-lydisreqardsthecriticalaRd,inqeneral,decisiveperturbation中EWo(卫+,x)· +Atanyrate,characterizationsofw。(卫,X)-functionsinCb(卫+,x)areofqreatinterest(seealso[65,FinalRemarks]),butseemtobeverydifficult,see[58]. Problem2.Oneofthebasictο。lsintheproofofTheoremA2isBruck'sHmeasureYErofnon-affinenessHf。rnonexpansivemapsin 11rliformlyconvexBanachspaces(Lemma3instep2.5above).Thecon-structionofymakesessentialuseofthefactthat,foxxuniformlyconvex,themodulusofconvexity6(ε)isstrictlypositiveforallX一一εE(0,2],forthisiswhatmakesYstrictlyincreasir19.moiIImediate questiOIlsaxea)工sitpossibletojustallowforunifo口nlycorlvexrenorminqs,and
436 Ruess b)whathappensifoneleavesoutuniformconvexItyaltoqether,likeX=L1[0,1]? withregardt。theexistenceofameasureyεrofrIon-affinenessforrlOIlexpansivemapsinthesenseofLemma3above,theanswerstobothquestionsarenegative.(TheauthorisgratefultoA.pelczynskiandw.Schachermayexforadiscussionofthesubsequentexamples.)Note吐1at,if,foragivenBanachspacex,suchameasureyErexists,thenanynonexpansivemapTisnecessarilyaffineonthelineseq-Inerlt[x,y]whenever||Tx-Ty||=||x-y||.This,however,isnottrueforAlspach's[1]exampleTinL1[0,1](baker'Stransformation)11'Tisactuallyanisometry,butT(-(r+r)+1)丰-T(r+1)+土T{r+1)2122122r.=thei-thRademacherfunction,iE{1,2}-lMoreover,adiscretea叫oqofbaker'stra时OXT川iorlon℃hespace21(R4withZi-norm)showsthatitisnoteventrueinfinite-dimerlsio-nalBanachspaces.Thus,吐1edesiredmeasuresγErdonotexisti口generalinbothclassesofBarlachspacesalludedtoirlquestior1sa)andb)above.Note,however,thatthisdoesnotnecessarilyrulecutthevalidityofaversionofTheoremA2inthesespaces.(Inparticular,4inthefirlite-dimensionalexampleZ1,everyalmost-orbitisevenasymptoticallyalmostperiodicbyTheoremA1.) Thus,onepossibleliReoffutureresearchcouldbeadiscussion ofBanachspacesxandalmost-orbitsUofsemigroups(S(t))t2LOforwhichtheassumptionsOfTheoremA2.1aref111filled. REFERENCES 1.D.E.AlspacthAfixedpointfreenomEpansivemap,proc.Mer.Math.soc.82(1981),423-424. 2.J.B.Baillon,IJntTIE-oxEEIIledetypeerqodiquepourlescontractionsnonlirl在airesdansurlespacedeHilbert,c.R.Acad.sci.PariS280(1975),A1511-A1514. 3.,Quelquespropxiet§sdeconvergenceasyITlptotiquepourlessemi-groupesdecontractionsimpaires,C.R.Acad.sci-Paris283(1976),A75-A78. 4.,QuelquesproprietesdeconverqenceasymptotiquepourlescontractiOIlsimpaires,C.R.Acad.sci.Paris283(1976),A587-A590.
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AsymptoticBehaviorof-somePerturbedC-Semigroups W.SCHAPPACHERInstituteforMathematics,UniversityofGraz,Graz,Austria ManyinterestingproblemsarisinginapplicationscanberegardedasdynamicalsystemsactingiIlacertainfunctionspace-Thetheoryofsuchlinearevolutionequtionsisratherwell-developped(seeforinstance,Amam(1984),Goldstein(1985),Pazy(1983).Whenwewanttoapplytheabstracttheoryto‘concrete'situations,however,weoftenfacethefactthatitisquitedimcultoratleasttedioustoverifytheassumptionsoftheabstracttheory.Ontheotherhand,weoftenhavesomeadditiorlalaprioriinformationfromthespeciEcproblemunderconsideration.Hencethereisanincreasinginterestinderivinggenerationresultsforevolutionequationswhicharemoreadequatetospeciaicsituationsaadareeasiertobechecked.Inparticular,wementiontheworkbyAmam(1986),ClementetaL(1987,1989),DeschaMSchappaCher(1985,1989)Descheta1.(1989),Greimr(19871989).TheobjectiveofthispaperistoinvestigatetheasymptoticbehaviorofII111ltiplicativelyperturbedlinearsemigroups-(Makinguseoftheprincipleoflinearizedstabilitylocalstabilityresultscanbethenobtamedfornonlinearsemigroups,too-)Forthereader'sconvenience,wepresentashortsurveyofthemotivationandthemostimportantgenerationresultsformultiplicativeperturbations.AdetaileddiscussioncanbefoundinDesch-SchappaCher(1989):LetQbeaboundeddomaininRnwithsumcKIltlysmoothboundaryr.WeconsidertheiIIMal-b011日daryvalueproblem 、、lf咱EAJ'z·飞、、 2川=AAu0u(仰0,Z)=u0叫(归例Z叫)L'Bu(t,z)=υ(t,z), t>0,Zε0,ZEQ,t>0,Zεr. HereAstandsforalinearpartialdiferentialoperator,andBrepresentssomekindoftraceoperator(usudly,Dirichlet一,Ne1Imam-ormixedboundaryoperators).Inordertoattackthisproblemwithfunctionalanalytictools?letXbeaBanachspaceoffunctionsmappingOintoRm,andletYbeaBanachspaceconsistingoffunctionsmappingrintoRn.Formallv,wemayproceedasfollows: ThisworkwaspartiallysupportedbyFondszurF&rderungderwissensehaftlichenFOE-sChurIg,Austria,S3206. 44I
442 SchappaCher 缸里立44ConsiderthehomogeneousproblemaadtheassociatedoperatorAdearledbyA=A|kerB'ie D(A)={uεX|AUEXand814=0},Au=Au. 缸主立2iConsidertheassociated‘Dirichlet'problem (λI-A)tA=0,Bu=-u. Ifthisproblemiswell-posed,inthesensethatthereexistsa112iquesolutionu=P(A)υ,andthisDirichletoperatorP(λ)isacontinuouslinearoperatorfromYintoX,itisnaturaltorewrite(1)asanabstractCauchyprobleminX: 去。)=川)+贝λ)υ(t))一λP队州u(0)=U0· Inmostapplicationstheboundarydataareobtainedbya‘feedback'oftheState,i.e. υ(t)=Qu(t), withacontinuous(ingeneralnonlinear)operatorQ.Consequently,wearriveattileCauchyproblem 生(t)=A(I+P(λ)Q)u(t)一人P(λ)Qu(t),t>0,dtorslightlymoregeneral 去(t)=A(I+P伽(川仰(t) withcontinuouslinearoperatorsHandPmappingYintoX,andacontinuous(maybenonlinear)operatorQmappingXintoYInordertoderivegenerationresults,itturnsouttobeusefultoformulatethesecondi-tionsasrestrictionsontherangeoftheperturbationPQ.Forthesakeofsimplicity,weshallassumeinthesequelthatQislinear. COMitiozl(Z):LetAgenerateaCo-semigro叩S(t)inaBamchspace(X,||||).Let(Z,lb)beaBmachspacecontimouslyembeddediaX.Wesaythat(Z,|-|z)satidesCondition(Z)withrespecttoAifthereexistsacOI1tiII11011s,IIondecreasingfunction7:[0,∞)→IO?∞)withγ(0)=0,suchthatforallcontimousfunctions￠:[0,Tl→Zwehave js(t-sM(s)dsεDM)
AsympfoticBehaviorofPerturbedC。-Semigroup 443 and ||AlS(t-s)￠(s)ds||三70)sup|￠(s)|zfor。三t三T.JO<s〈tCondition(Z)seemstobeverytechnicalaMartiEcialHowever,thereareseveralapproachestoverifyitquiteeasily.Inmostapplications,actuallyastrongerconditionissatisEed: Condition(ZnkLetl三p<∞andletAbetheinEIlitesimaigeneratorofaCo-semigroupS(-)onX.ABmachspace(Z,|·|z)thatiscontinuouslyembeddedinXissaidtosatisfyCondition(Zp)withrespecttoAifthereexistssomeT>OsuchthatforanyT￠ELP(0,T;Z)jS(T-s)￠(s)dsisinD(ALOurmaingenerationresultis:IDeschandSchappaCher19891THEOREM.LetAbetheidnitesimalgenerator。faCo-semigro叩S(,)onX.IfRange(PQ)iscontainedinaBamchspaceZsatisfyi昭condition(Z)andif(PQ)isacontinuouslinearoperator企omXintoZ,thenA(I+PQ)istheindIlitesimalgeneratorofaCo-semigroupT(·)onXwhichcmberepresentedas T仰=S仰+中。一树T(s)Z TheobjectiveofthispaperistocomparethebehaviorofthetwosemigroupS(-)amdT().Tobeginwith,werecallsomebasicfacts:LetU(-)beaCrsemigro叩onXwithinErlitesimalgeneratorC.Thegrowthratetiyo(C)ofU()isde岳阳dby 问(C)=出:ln|lU(t)llIIIgeneral,wecannotestimatetuo(C)intermsofthespectrumofC.Infact,ifweset s(C)=sup{Reλ1人εσ(C)} thentuo(C)主s(C).LetαbeameasureofnomompactnessinX.Wedeanetheesssentialgrowthrate叫(C)ofU(-)by 叫(C)=比;ln州))Sinceα(u(t))三|u(t)|,weclearlyhaveω1(c)三四o(c) ThesigIIi五caneeofthisessentialgrowthrateisbasedonthefollowingresult(seeDeimling(1984)).
444 SchappaCher THEOREMLetU(-)beaCoferI啤ro叩withiIIEnitesimalgeneratorCHκ>ωl(C)thenthereexistsadecompositionX=XI@X20fXintoU(t)-invariantclosedsubspacessuchthatXIishite-dimensionalandthegrowthrateoftherestrictionofU(-)toX2isboundedbyκ. Inparticular,wehavei)IfuJo(C)exceedsuJ1(C),thenChasaneigenvalueλwithReλ=ωoandEElitedimensionalgeneraiizedeigenspaceu;二Iker(λI一C)k ωo(C)=max(ω1(CLs(C)). ii)Ifω1(C)<κ,thenthereexistsadecompositionX=Xlex2@…@X7neyintoEIlitelyma町U(t)invariantsubspacessuchthateachXiisahitedi口lemonalgeneralizedeigenspaceofCandthegrowthrateoftherestrictioIlofU(t)toYisboundedbyκ.iii)IfRe入>ω1(CLthen入belongstothespectrumofCifλisaneigenvalueofamtemultiplicity-Moreover,theFredholmalternativeholds:(λI-C)isoIitoifitislnJectlve.Aftertheseprerequisitesletusreturntotheperturbationproblemwewanttodealwith-ConsidertheinaElitesimalgeneratorAofalinearCo-semigrouponXandletBandGbecontinuouslinearoperatorsinXsuchthatRange(B)satisaesCondition(Z)withrespecttoAConsequentlyA(I+B)+GistheidhtesimaigeIleratorofasemigro叩T()onX、whichsatidesthevariationofparametersformula T仰=S(t川户(t-s)BT(s灿+f机8)GT(s)zds Probably、thebestintuitivewaytolookatthisproblemistoviewitasaCOIltrolsystem 兰u(t)=A(u(t)十PU(t))+Hυ(t),dtu(0)=Z withsomefeedbackfromthestateUεXtothecontrolfunctionuEC(0,∞;Y): u(t)=Qu(t). Here,PandHarecontinuouslinearoperatorsY→XandQisacontinuouslinearoperatorX→Y.Moreover,Ra吨e(P)satisaescondition(Z)withrespecttoA.TherespORseoftheoutputQu(t)totheinputυ(t)isgiveILbythefollowingiIIput-outputoperatorthatmapsC(0,T;Y)intoitself: Qu(t)=QS(t)z+Q(Fu)(t)
AsymptoticBehaviorofPerturbedC0·Semigroup 445 with 叫)=A户。-s)PU(s)ds+户。一s)HU(S)dsREMARK:Obviously,theoperatorsPandHarenotuniquelydeterminedbythesemi-groupsS(·)aMT(·).IIIfact,ifRisacontinuouslinearoperatorsuchthat(P-P1)mapsyhtoD(ALwemayreplacePbyRandHbyH1=A(P-P1)+HtoobtainthesamegeIlerator A(I+PlQ)+HlQ=A(I+PQ)+HQ AstraightforwardcalculationshowsthatthischangehasBOiIlf111enceontheoperatorQF. REMARK:Iftheoperator(I十PQ)hasacontinuousinverse,wemayrecoverAasamultiplicativeperturbationofA(I+PQ)+HQbyA=(A(I+PQ)+HQ)(I一(I十PQ)-IPQ)+H(Q(I十PQ)一lp-I)QThisimpliesthatanyestimateonthegrowthoronthespectrumofT(·)iatermsofst)alsoyieldsaconverseestimateonS()intermsofT() PROPOSITIONl.LetAbetheinhitesimalgeneratorofalinearCo-semigro叩S(·)onxandletP,HbecontinuouslinearoperatorstkomYintoXandletQbeacontinuouslinearoperatorX→YandassumethatRange(P)satiSEescondition(Z)withrespecttoA.ThenthereexistsanoperatorPIfromYintoXSUchthatP-PlmapsYintoD(A)and(I+PIQ)isinmtible.Moreover,Range(P1)alsosatiSEescondition(Z)MthrespecttoAwhereasRange((I+PIA)-lpl)satidescondition(Z)withrespecttotheperturbedoperatorA(I+PIQ)+HIQwithH1=A(P-R)+H. PROOF:As(I+PQ)AisanidIlitesimalgenerator,theoperator(人I-A-PQA)一lexistsforsumdentlylargeλ>0.WesetP1=(I一λ(λI-A)-1)P.Clearly,P1-PmapsYintoD(ALaMIleImrange(P1)satisaescOMitioII(Z)withrespecttoAasPdoesbyassumption.Notethat(人I-A)(人I-A-PQA)一IisaC∞O口川tUimOulinearOpe盯ra时tOωr.AaeasycomputationshowsthatthefollowingequatiOIlsarevalid:(人I-A)(人I-A-PQA)-1=(I+PQ(I一λ(λI-A)-1)-1 aIId (I+(I一人I-A)-1PQ)-1=I←(I一人(λI-A)-1)(I+PQ(I一λ(λI-A)-1)一1pItiseasilyveHEedthatRange(P1)satiSEescondition(Z)alsowithrespectto(I+PIQ)A、andhencebyasimilarityargument,range((I+PIQ)-1PI)satisaescondition(Z)withrespectto(I+P1Q)-1(I+PIQ)A(I+PlQ)=A(I+PIQ),aMhencealsowithrespecttoA(I+PIQ)+H1Q.. Ourmainresultis
446 SchappaCher THEOREM2.LetAbetheinbitesimalgeneratorofalinearCo--semigroupS(-)onX,letpandHbecontinuouslinearoperators企omYintoXandletQbeacontinuouslinearoperatortkomXintoY.Moreover,assumethatrange(P)satidescondition(Z)withrespecttoALetT(·)denotetheCo-semigroupgeneratedbyA(I+PQ)+HQAssumethattheinput-outputoperator 仙。)=叫so-s川 iscompact企omC(0,T;Y)intoC(0,T;X)foranyT>0.ThenLAJI(S(t))=ω1(T(t)) PROOF:ForanyZεX,weconsiderthe"controlfunction,,υ(t;2):=QT(t)zForanyaxedt三0,T(t)zcaIlbedeterminedfromUandthusweconcludeinparticularthatT(t)zdependscontinuouslyonU.Next,weinvestigatetheoperatorν:X→C(0,t;X)givenby (Vz)(t):=υ(t,z). Evidently,theoperatorQ:X→C(0,t;X),(Qz)(t):=QS(t)ziscontinuousAsu(、z)=Qz+QFu(-,z),wededucethat ν-Q=QFV andconsequentlyV-Qiscompact.Thereforealsotheoperator z→A户。-s)附(川心肌 lscompact- SettingU(t)z:=AfS(t-s)PQS(s)ds+jS(t-s)HQS(s)zdsthismeansthatT(t)-U(t)-S(t)isacompactoperator-Inparticular,weobtaiIlα(T(t))=α(UO)十50))ItiseasytoverifythatforallBOIlnegativeSandtwehave S(t)U(s)+U(t)S(s)=U(t+s). Letμ>ω1(S(t)).Forsumdentlylargetwehaveα(S(t))<jeμtComeqmMly、 α(U(20)三α(U(t)S(t)+S(t)U(t))三2α(S(t))α(U(t))三εμtα(U(t)). Byinduction,weobtainα(U(2nt))三eμ2ntα(U(t)),sothat ωI(T(t))=lim土lm(T(2nt))=lim土lnα(U(2nt)+S(川))n→∞2ntn→∞2nt<lim土ln(川t(α(U(t))+1))=1·-n→∞2nt
AsymptoticBehaviorofPerturbedC0·Semigroup 447 Thuswegetω1(T(t))三ω1(S(t)).Thusitremainstoprovetheconverseinequality.ByPropositionlwemayassumethat(I+PQ)hasacontinuousinverseandthatthemageof(I+PQ)一1P则idescondition(Z)withrespecttoA(I+PQ).FormyzED(A)thefunctionω(t):=S(t)zsMisses主ω(t)=Aω(t)=A(I+PQ)(ω(t)一(I+PQ)-1PQS(t)z)+HQω(t)-HQS(t)zdt=(A(I+PQ)+HQ)(ω(t)一(I+PQ)一1P(?z)(t))+(HQ(I+PQ)-1P-H)(Qz)(t) andthereforeweinferthatforanyzεXmdt220, S(巾=T仰+叫一PQ)+HQ)fT(t一功(I+即lm)(s)ds +fm-s)(HQ(I+PQ内-H阳s)dsAsQ-νiscompact,theoperator Z→S仰-m)z一问+PQ)-HQ)fT(t一吵(I+PQ内川s)ds -fm-s)(HQ(I+网内-H川忡 iscompact,SOthatα(S(t))=α(T(t)+W(t))withW(t)beingdeanedby W(巾=川+PQ)+H州m一加即1问川ds +/m-s)(HQ(I+叼町γ内一H帆5Again,itisaneasyexercisetoverifythatforallnonnegatives,t T(t)W(s)+W(t)T(s)=W(t+s) andSOwemayutilizethesameprocedureasabovetoobtain ω1(S(t))三ω1(T(t)).. Thistheoremcoversinparticularthecasewhenrank(Q)isarlite,i.e.wemayputY=Rnandrange(P)satiSEescondition(Zp)withrespecttoAMoregemraly,wehave
448 SchappaCher- THEOREM3.LetAbethemarlitesimalgeneratorofalinearCo-semigroupS(-)onX,letpmdHbecontinuouslinearoperatorsfromarebxiveBmachspaceYintoX,andletQbeacontinuouslinearoperatorX→Y.Moreover,supposethatthereissomepwithl三p<∞suchthatrange(P)satidescondition(Zp)withrespecttoAandthatQiscompact-Thentheassociatedinput-outputoperatorQFiscompactforanypositiveTandcon-sequentlythesemigroupsgeneratedbyAaMA(I+PQ)+HQhavethesameessentialgrowthbound. PROOF:LetBbeaboundedsubsetofC(0,T;Y).Astheset{(Fu)(t)|11ξB,0三t三T}isbounded,theset{(QFu)(t)|UEB,。三t三T}iscompact-Itremainstovmfythatthefamily{QF叫tYEB}isequicontimous,ie.thatthefamilyofrealvaluedfunctions{fQF叫rEY飞||f||=1,UEB}isequicontimom-Astheoperator衍1':=(Fu)(T)iscontinuousfromLP(0,T;Y)intoXtheoperator衍*Q淑isacompactoperatorfromY*intoLq(0,T;Y*),whereqistheconjugateindextop,i.e.ll一+一=1.pqForanyfεY*wesetrQV‘:=叫·,U-e)and,inordertoavoidcomplicationswiththedomain,wesetu(t,ν*)=Ofort￠{0,TlandcontinueanyUεBbyu(t)=Ofort￠[0,TlForanyυε8,U*εY-andO三t三Tweobtain 问(FIY)(t)=fQ(Afs(t一S叫川抽州)归酌阿P托叫υ叫叫(=fQ45(T一s)P巾 =fuM Therefore, ||fQ(Fu)(t+h)-fQ(Fu)(t)||T=|川|f户μ仆u叫巾(υS俨川川)忡训υ叫(川+刊h一m一|川|小f户μ仆u叫巾川(υST三/||u川一u(s+时)||||巾+t+h一叩s +jllu(sj)|lll巾+t一叩s
AsymptoticBehaviorofPerturbedCo-Semigroup 449 AsBisbomdedandtheset{u(·,俨)|||f||三1)isrelativelycompactinLq(0,T;Y门CL1(0,T;Y仆,weobtainthattheexpressionontherighthandsideconvergestoOash→0,UBiformiyforuεBand||u叫|三1.· Asinallourapplicationscondition(Zp)issatisaed,itseemstobeatemptingtosuggestthatinTheorem3thecondition(Zp)canbereplacedsimplybycondition(ZLItturnsoutthatcOMiti02(Z)togetherwithdimYbeing丑niteisnotsumcieMtoobtaiI1acompactinput-outputoperator-LetX=Cub(R,R)betheBamchspaceofallbounded,uniformlycontinuousfunctionsZ:R→Requippedwiththeusualsupnorm||-||∞AsunperturbedseH鸣roupS(·)wechoosetheshiftgroup (S(t)z)(0)=z(t+9) withinaIlitesimalgeneratorAgivenby D(A)=CA(R,R),(Az)(8)=21(8). Next,weselectandaxsomenondecreasing,singularfunctionzεX.WesetY=RanddeEIleoperatorsPandQby Pu:=UZQz:=z(0),respectively WeclaimthatthecorrespondinginputoutputoperatorWisnotcompact-Tobeginwith?weverifythatra吨e(P)satisaescondition(Z)withrespecttoA:InthesequelaxsomeT>OIIIfact,weobtainforany￠εC(0,T;R) andtherefore |;(S(h)一旷S(t一吟P叫s)(8)|=|;jz(t+h+8-h(t+仁机)ds|斗(fz川一吟出一fz川一吟出)||￠||∞ andcomeqmntly,raage(P)satisaescondition(Z)withrespecttoAwithγ(t)SUP8εR(z(t+8)-z(8)).Foranyintegern,weset TEn:={机WUMRMM
450 SchappaCher Thenweobtainforany￠εEntheequality 4Sh)归阳P问叫￠圳仲川(υωws吵圳忡)川灿d出S仲多刮f介户μz抖州(归川t=fz(t-s)川的ds-z(OM(t)们川制0) andSOtheinput-outputoperatorWisgivenby (附)(t)=对)￠(0)一村附 fψ川)(WMt=fu(T-吟￠(t讪 whereweputu(t):=z(t)￠(0)一中(t)z(0)+jz(t-s)V(s)ds.Consequently,thereisaconstantC(ψ)dependingonvsuchthatforall￠εEn,wehavetheestimate |fψ川)川(t)dt|三C(非)fMS三仨(ψ) AssumenowthatWiscompactandletXEnz二1W(En).TfleaweconcludethatforanyTψεC1(0,t;R)|jψ(T-t)χ(t)dt|==OaMthusx=0.BythecompactnessofW(EILowededucethatforanyε>OthereexistssomeintegernsuchthatWF(En)iscontainedintheopenεbailcenteredattheorigin.Inparticular,wehavefor￠εEn T|川T)|=|z叫 Now?letO<SI<tl<82<t2<<Sm<tm<Tbeapartitionof[0、Tlsu(tithat 艺(tJ-w: andpickηsumcieMlysmallsothat2η<minj=1,.,m(tJ-sj).
AsymptoticBehaviorofPerturbedC。-Semigroup 45I Moreover,wedeaneafunction￠by ￠(T-t)=〈)i(t-Si)lj(tJ-t)TheII￠εEn,￠(仰0)=￠(T)=0a丑￠仰(σF川川Tι川一斗t ift<Sht>tmortε[勺,SJ+11,iftEISj十吁,tJ一训,iftε[s川J+时,iftEltj一η,tjl. iftεltJ一η,tjl,iftε[SJJJ十训,otherwise, andhencetheaboveestimateshowsthat uduzn-vfj句u,GUZ气ilJ叫14-n-m艺同一-JUTAVZTfIjo--TAVW>-F】 Takingthelimitasη→肘,weinferthatE二;二1tJ-sj三timplies F」<-QUZZmZ同 whichmeansthatzisabsolutelycontinuouscontradictingtheassumptionthatzissingular.ODenProblem.WedonotKIlowanexampleofamultiplicativeperturbationproblemA(I+B)withaCOBtimomlinearoperatorBhavi吨hitedimemiomlra吨eaMsatisfyingcondition(Z)withrespecttoAsuchthatessentialdestabilizationoccurs.Thefollowingsimpleexample,however,suggeststhatthecompactnessoftheinput-outputoperatormaybecrucial.TheexampledoesnotEtintoourframework?astheoperatorQisaotcontinuous:LetX=L2(0,1;R)anddeanetheCo--semigro叩S(·)onXby 、‘.,,,a'tuAσ,,..、、zor--〈1,、一-、、,,,,AVJ't飞、、,,,,2、、,,,,,,?ι,,..飞S,,..‘‘、 ifO-t>0,ifO-t<0. Clearly,S(·)isnilpotentandthusω。(S(t))=一∞.S(t)canberegardedasthesolutionsemigrouptotheCauchyproblem 兰u(tJ)=一二Lu(tJ),dtdou(t,0)=u(t)withu(t)=0,u(0,8)=z(8).
452 SchappaCher Thisinitialboundaryvalueproblemiswell-posedforUELL(0,∞,R)Settinguptheunboundedfeedbackofrankone u(t):=Qu(t,·):=u(t,1), weobtaintheinput-outputoperator 、‘,,,,咱E·-AVau,,..、、unurt『41lk一一、、..,,·φ'b,,,,‘飞、、-E,,UF门M币,,,.‘、 fort>1,fort<1 whichisevidentlynotcompact-TheresultingsemigroupT(t)isgivenby 、‘,/吨EE晶+、1'''a?ba?b一-AHVAσ,,,,、,,z1、zzrI,,tIL一一、、E,,,,AO'''E‘、、1,JZ、、..,,a,aw,,t飞T,,..‘、 ifO-t>0,if8-t<O isactuallyagroupofisometries,andhenceω1(T(t))=1. Wenowproceedtothesecondproblemofthissection,namelytodeterminethespectrumoftheperturbedgeneratorA(I+PQ):Inordertobuildupacharacteristicequation,weintroduceanoperator饵Y(入)by ptr(人):=QA(λI-A)-1P+Q(λl-A)-IH whichiswell--deanedforallλbelongingtotheresolventsetofAandisacontinuouslinearoperatorinyItiseasilyveriaedthatW(λ)istheLaplacetransformoftheinputoutputoperatorQF,andhenceW(λ)maybeviewedasthetransferfunctionofthecontrolsystem 去。)=州)+PQU仙HQU(t withinputUandoutputQu.FOI-technicalreasons,wealsointroducetheoperatorF(人)by F(人)=A(入I-A)一1P+(λI-A)-IH SOthat W(λ)=QF(λ). Tobeginwith,wediscussthepointspectrumoftheperturbedgenerator:
AsymptoticBehaviorofPerturbedC0·semigroup 453 PROPOSITION4.LetλbelongtotheresolventsetofA,andletAdenotetheperturbedgenerator:A=A(I+PQ)+HQ.ThenλliesinthepointspectrumofAiftheoperatorI-W(λ)isnotinjective.Moreover,thereismisomorphismj企ornthegenerdizedeigenspaceJEer(入I-A)mtothekernelKmoftheoperatorWmthatmapsymintoymandisddnedby/I-W(λ)一击WF(λ)…一百仨?W(m-1)(λ)＼w-lOI-W(入)-t布W(m-m)|vvtwz--EE＼OOI-W(λ)/ Thisisomorphismisdeterminedby 32=(Qz,Q(4一λI)z,...,Q(4一人I)m一12) f1(仇,...,umY=了一土-TF(J-1)(λ)UJ主:飞J品/whmmusual,F(j)(入)阳 PROOF:mfeshallfrequentlyusethewell-knownidentity (入I-A)-1)(j)=j!(-lY(入I-A)-J-1.WeErstassumethatZεker(入I一A)m.SettingUJ=Q(A一人I)J-b,weobtainfork=1,...,m 号if川(λ)Uj=寻(一川(人I一俨PQ+(人I一俨1HQ)的红飞J川江 =汇(-1)j-k(λI一A)kf =汇(-l)J-k((λI一A)k-1-1(L入I)+(入I-A)k-J)(4一人I)J-12 =艺(-1)J-k(入I-A)k叮叮A一λI)jz一汇(-1)j-k-1(λI一A)k-J(A一川一12=(A一λI)k-12· ApplyingQtobothsides,weseethat(U1,...Jm)tεkerWm,whereasthecasek=lyields =了一土寸F(J二1)(入)的-2UA/
454 SchappaCher Nowassumethat(U1,...,νm)tEkerWm·Thismeansthat 号」寸F叫人)Uj=户江k飞uJ,川ψ叮/J Consequently,weobtain (A一刀)兰市川=川Q)+卧λI)兰卢川J=A(了一L寸F(J一~i+PUK)+HUK一入γ-L寸F(J-k)UJIU川J/立1飞Jh/ =(A一λI)(汇(一ly-k(A(入I一A)k-J-lPUJ+(人I一A)k-HHUK)+PUK)+HUHλPUK =主1(j-J-1)!俨1)的 Byiteratingthisprocedure,weinferthat (A一人I)mE二F(川UJ工O and Q(A一λI)卜1了一上寸F(J-1)u=了三丁飞JA/J么(j-k)!J一 whichprovestheclaim.' WenextinvestigatetheresidualspectrumofA:PROPOSITION5.Let入belongtotheresolventsetofAThenrange(入I-A)isdenseinxifm昭e(I-W(入))isdenseiny PROOF:ByPropositionlwemayassumethat(I+PQ)isi盯edible-Moreover,weshallusethesimplefactthatQ(I+PQ)-1=(I+QP)-lQ.Assumearstthatrange(人I-A)isnotdense,i.e.thereexistsanonzero♂εx-suchthMforallZεD(A) f((λI-A)(I+PQ)-HQ一λPQ)z)=0 PuttingZ:=(I+PQ)-lz,weobtainforalizED(A) f(λI-A)z=z*(H+入P)(I+PQ)-IQz.
AsymptoticBehaviorofPerturbedCo-Semigroup 455 Next,weset俨:=((I+PQ)-1)气H*+λp*)f.As入belongstotheresolventsetofA,thereexistssomeZεXsuchthatO并f(λI-A)z=俨QAandconsequently,f#0.Now,weobtainforalluεY f(I-EU(λ))u=f(I-Q入(λI-A)一1P+QP-Q(入I-A)-lH)ν=f(I+PQ-Q(λI-A)-1(入P+H))ν=♂(H+入P)ν一♂(λI}A)(入I-A)-1(H十人P)u=0 andhencerange(I-W(人))isnotdenseinY.Conversely,supposethatthereexistssomenOIIzerofεY*sothat f(I-W(入))u=OforallUEY.Wesetz*:=((λI-A)-1)叩γandobtainforailzED(A) f(入I-A)z=fQ(入I-A)-1((入I-A)(I+PQ)z一(λPQ+HQ)z)=f(I+PQ-Q入(入I-A)-1P-Q(λI-A)-IH)Qz=f(I-W(入))Qz=0.. SoitremainstolookatthecontinuousspectrumofA:PROPOSITION6.LetλbelongtotheresolventsetofAThenrange(人I-A)isclosedifrange(I-W(入))is. PROOF:Let7rbethecanonicalprojectionofXontoX/ker(入一A),andlet户deBotetheprojectionofYontoY/ker(I-W(入)).Weshallusethefollowingcharacterizationofoperatorshavingclosedrange:(λI-A)hasclosedrangeif(λI-AMn→0implies7Tobeginwith,assumethat(I-W(入))hasclosedrangeinY.Let(Zn)beasequenceinD(A)suchthat(λI-A)zn→0.As F(入)QZn+(λI-A)-1(人I-A)zn=A(人I-A)一1PQzn+(λI-A)-IHQZn一(λI-A)-1A(I+PQ)zn-(λI-A)-IHQZn+入(人I-A)-12n=λ(λI-A)-1zn一A(λI-A)-12n=Zn, weseethat (I-W(λ))Qzn=Q(入I-A)-1(λI-A)zn→0.Since(I-W(λ))hasclosedrangethisimpliesthat什Qzn→OaMhencethereexistsasequence(Un)inker(I-W(λ))suchthat||Un-QZn||→0.
456 SchappaCher- FromTheorem3(forthecasem=1),weknowthatF(入)UnEker(入I-A).AsZn-F(人)Un=Jt(λ)(Qzn-un)+(入I-A)一1(λI-A)zn→0,wededucethat门,1→0,andthus(入I-A)hmdosedrange.Conversely,assumethat(人I-A)hasclosedrangeLet(un)beasequenceinYsuchIhat(I-W(λ))Un→0.Again,wemayassumethat(I+PQ)一1exists,andsetZn:=(I+PQ)一l(入I-A)-1(λP-H)作(入)Un·Thenwehave Zn=(I+PQ)-1(A(λI-A)-IP+P一(入I-A)-lH)仿(入)Un=(I+PQ)-1(F(入)+P)W(λ)Un· Inparticular, Qzn=Q(F(λ)+P)W(λ)Un=(I+QP)-1(IU(λ)2+QPW(λ))Un,aMthereforeW(人)h-un→OimpliesthatQZn-un→0Now, (入I-A)zn=(入I-A)(I+PQ)zn一(入PQ+HQ)zn=(入P+H)W(入)Un一(人P+H)QZn showsthat(λI-A)znandhencealsoMnconvergeto0,i.e.thereexistsasequence(zn)cker(入I-A)with||zn-zn||→0.Again,wemakeuseofTheorem3toinferthatQznEker(I-W(入))andm||QZn-h||三||Qzn-QZn||+|lQzn-b||→0,wtfconcludethat什Un→Oandhencetheclaimholds.' Collectingthesethreepropositions,weobtainthecharacterizationofthespectrumofAintermsofthetransferfunctionW(λ):THEOREM7.LetAbetheinEnitesimalgeneratorofalinearCo-semigroupS(·)onX,andletPandQbecontinuouslinearoperatorsYintoX,andletQbeacontinuouslinearoperatorX→Y.Moreover,supposethatrange(P)satiSEescondition(Z)withrespecttoA.LetA=A(I+PQ)+HQadformy入belongingtotheresolventsetofAwesetW(入)=QA(λI-A)-1P+Q(λI-A)-IH.Thenwehave人isthepointspectrumofAiflisinthepointspectrumofW(入),λistheresidualspectrumofAiflisintheresidualspectrumofwr(λ),λisthecontinuousspectrumofAifIisinthecontinuousspectrumofW(λ). Forthecaseofftrlitedimensionalmultiplicativeperturbations,weobtain:
AsymptoticBehaviorofPerturbedCo-Semigroup 457 THEROEM8.LettheassumptionsofTheorem7holdandassumeinadditionthatY=Rn.ThenW(λ)ismn×nmatrixandsomeλbelongingtotheresolventsetofAliesinthespectrumofAifdet(I-W(人))=0. ThecriterionabovehasthedisadvantagethatitcanmotbeappliedtoAlyinginthespectrumofA.Inparticular,wedonotgetanyinformationwhetheraperturbationstabilizestheproblemiftheunperturbedproblemisunstable,becausewewouldneedtodeterminewhethertheunstableeigenvaluesofAcanberemovedbytheperturbation.RecallthatforanyλobelongingtothespectrumofAI-W(人。)isnotdenned.Inmostcasesofinterest,人owillbeapoleofwr(·).However,onemightexpectthatλocanberemovedfromthespectrumofAbyaperturbationif(I-W(λ))-1caI1beextendedcontinuouslytoh.Thishopeisinfacttrue,providedthat入ocanberemovedatallbyafeedbackbasedontheobservationQandthecontrolconstructedbyPandH:THEOREM9.LetAbetheinEnitesimalgeneratorofaCo-semigrouponX,letPMdHbecontinuouslinearoperatorsY→XMdletQbeacontinuouslinearoperatorX→Y.Supposethatrange(P)satidescondition(Z)withrespecttoAmdletA=A(I+PQ)+HQ.Let入obeintheclosureoftheresolventsetofA.Assumethatthereexistsacontinuouslinearoperatork:X→YsuchthatAliesintheresolventsetoftheoperatorAdeanedby A:=A(I+PKQ)+HIfQ. ThenhbelongstotheresolventsetofAifthereexistsasequence(人n)suchthat入n→人oand(I一仿(λn))一1isbounded. PROOF:AccordingtoProposition1wemayassumethatboth(I+PQ)-land(I+PKQ)-lexist.Otherwise,wereplacePby(I一κ(κI-A)-1)Pwhereκischosensuchthatboth(κI-A-PQA)-Iand(κI-A-PKQA)-lexist-WeshallwriteAasaperturbationoftheoperatorAandcomputethecorrespondingtransferfunctionW(λ):Itisasimplecomputationtoverifythat A(I+PQ)+HQ=(A(I+PKQ)+HKQ)(I+P(I+KQP)-1(I-K)Q)+H(I+KQP)-1(I-K)Q. Formy入belongklgtotheresolventsetofA,thetransferfunctionsforAregardedasaperturbationofAisgivenby W(入)=Q(有人I-A)-1P+(入I-4)一1H)(I+KQP)-1(I-K)=(-QP+Q(λI-A)一1(λP+H))(I+KQP)-1(I-K). TocomputeQ(入I-A)-1,supposethatZ=(入I-A)-1z,i.e.wehave Z=(λI-A(I+PHQ)-HKQ)z
458 Schappacher- ApplyingQyields Qz-W(λ)KQz=Q(λI-A)-lz.Forλsumdentlydosetoλ0,weknowthat(I-W(入)K)一lexistssinceW(入)KisthetransferfundioforAregardedasaperturbationofAandλ。liesintheresolventsetofATherefore,Q(入I-A)-1=(I-W(λ)K)一IQ(入I-A)-1.Now,仿(人)=(-QP+(I-W(入)K)-IQ(人I-A)-1(入P+H))(I+KQP)-1(I-K)=(-QP+(I-W(入)K)-l(QP+W(人))(I+KQP)一1(I-K)=(I-W(入)K-IW(入)(I+KQP)(I+KQP)-1(I-K)=(I-W(入)K)一11归入)(I-K). AsλobelongstotheresolventsetofA,wemayuseW(λ。)todeterminewhetherλoliesintheresolventsetofA(I-W(λo))-IexistsifforsomesequenceλniatheresolventsetofA,λn→λoand(I-W(λπ))-1isbounded.Now, (I-W(入n))=I一(I-W(人n)K)-1(W(λn)-W(人n)K)(I-W(λn)K)一1(I-W(An)) andhenceitsinverseis (I-W(λn))-1=(I-W(入n))→(I-W(λn)K)=(I-W(入n))一1(I-K)+K Thereforethebomdedmssof(I-W(λn))-limpliesthatof(I-W(入71))-1andthushliesintheresolventsetofA.Conversely,ifhbelongstotheresolventsetofA,wecanchoosesomeA'SLImeieIItlyclosetoIsuchthathalsobelongstotheresolventsetofAand(I-K)-1exists.TheIIitfollowsthat(I-W(λn))一lisbounded.· Thecriteriongivenaboverequiresthatthefeedbackremovinghfromthespectrumof(A)isofthespeciacformKQByaformaltrick,wemayextendthistoageneralcontinuouslinearoperatorQ1:X→Y:COROLLARY.LetA,P,H,QandAbeasinTheorem9andassumealsothathliesontheboundaryoftheresolventsetofA.Moreover,supposethatthereisacontinuouslinearoperatorQImappingXintoYsuchthat入obelongstotheresolventsetofA:=A(Q+PQI)+HQ1DeEIiingW(λ):=QlF(入),weobtainthat入oisinthcresolventsetofAifthereexistsasequence(入n)suchthat儿→入oandboth,(I-W(入n))-1andWI(儿)(I-W(人n))-larebounded. PROOF:Wesety=Y×Y,anddearlelinearoperatorsP,坷,Qandby?(的?的)=PUo,对(ν。,的)=HUoandQz=(Qz,Q12),respectively.ThenA=A(I+PQ)+对Q
AsymptoticBehaviorofPerturbedCFSemigroup 459 and A=A(I+?κQ)+πκQwhereweput瓦(仰,U1)=(的,0)ThetransferfunctionforAinthissettingisinmatrixnotation /W(入)飞W(入)=l《)＼W1(入)//(I-W(入))一1(I-W(入))-l=(＼W1(入)(I-W(入))一1NowtheclaimfollowsimmediatelyfromTheorem9.' andhence ＼飞Il--/nuri TheexistenceofafeedbackremovinghfromthespectrumofAisessentialfortheabovecriterion,althoughtheoperatorkdoesnotshowupexplicitelyinTheorem9.Thiscaabessenfromthefollowingconstruction:LetX=Xlex2wheretheXiareclosed,A-invariantsubspacessuchthatXICker(Q)andboth,ra鸣e(P)andra吨e(H)arecontainedinX2·Clearly,thebehaviorofthesemigro叩onX1is口otafeetedbytheperturbation,mrcanW(入)tellusanythingaboutthespectrumofAonX1·Considerationsofthiskindleadtoconceptsofcontroliabilityandobservabilityforsuchproblems,aIIdwewillnotiIIVestigatethem.AsW(入)playsakeyroleforthedeterminationofthespectrumoftheperturbedgen-erator,wenextpresentapossibilitytocomputeit,iftheproblemarisesfromanabstractinitialboundaryvalueproblem:THEOREM10.Assumethattheabstractinitial-boundaryvalueproblem 兰u(t)=A叫t),t三0,dtBu(t)=υ(t),t主0,u(0)=Z iswell-posedinXforboundarydataUELL(0,∞;Y).Moreover,letQ:X→YbeacontinuouslinearoperatorandletAdenotetheinEnitesimalgeneratorofthesemigrouponXassociatedtotheboundaryfeedbackproblemu(t)=Qu(t).Finally}letAbethegeneratorcorrespondingtothecase叫t)=0.ThenthespectrumofAisdeterminedbythetransferfunction W(λ)=-QP(λ), whereP(λ)uisthesolutionto(λI-A)z=0,Bz=-u PROOF:ByTheorem3wemayrewriteAas A=A(I+P(μ)Q)一μP(μ)Qforsomeaxedμ.
460 Schappocher- Then W(λ)=Q(A一μI)(λI-A)-IP(μ)=Q(-P(μ)+(λ一μ)(入I-A)-lp(μ))=-QP(入).. Throughoutthischapter,wehavealwaystreatedperturbatiOIlsbymultiplicationfromtheright.Ofcourse,ailtheseresultscanalsobeappliedtothecaseofadditiveA-boundedperturbations. THEOREM11.LetAbetheinEIIitesimalgeneratorofalinearCo-semigrouponX,letpbeacontinuouslinearoperatorY→Xsuchthatrange(P)satidescondition(Z)withrespecttoA,aIIdletFbeanA-boundedlinearoperatorX→γThenthereexistssomeμ>OandmisoIIIorophismLonXsuchthat A+PF=L-IAL withA=A(I-PF(μI-A)-1+μPF(μI-A)-1.Inparticular,allspectralandstabilityresultsofthissectionalsoapplytoA+PFwiththetransferfunctionW(λ)=F(λI-A)-1p PROOF:AsA+PFgeneratesaCo-semigroup,wemaychooseμSumerntlylargesuchthat(μI-A-PF)-lexists.Thisimpliesthatalso (I-PF(μI-A)-1)-1=(μI-A)(μI-A-PF)-1 exists.(Notethat(μI一A-PF)-lmapsXintodom(A十PF)=dom(A))Nowweget (I-PF(μI-A)-1)(A(I-PF(μI-A)-l)+μPF(μI→A)一l)(I-PF(μI-A)-1)一1=(I-PF(μI-A)-1)(A+μPF(μI-A)一1(I-PF(μI-A)一1)-1)=A一μPF(μI-A)-1+PF+μPF(μI-A)-1=A+PF HencetheisomorphismLisgivenbyL=(I-PF(μI一A)-1)-1,andwealsohave W(λ)=-F(μI-A)一1A(入I-A)-1P+μF(μI-A)-l(入I-A)-Ip=F(λI-A)一1p asclaimed.'
AsymptoticBehaviorofPerturbedCFSemigroup 46l REFERENCES [llAmannH,"Gew&ω}叫ikCheDiπe盯r陀阳en川tUia剖lgl怡Me创ikc}h1ur吨en旷J,DeGruyter,1984[2]AmannH,pamboltceuoluttonequattonsmthn07BUntaTbouπdaTVC07zdittons,,Proc.Symp-PureMath.45(1986),17-27[3]ClementPh,0.Didmam,MGyllenberg,HHeijmans,HThieme,PeTturbattoηthEOTVfordualsemtgroups,MathemMischeAnmler1277(1987),709-725[4]ClementPh,0.Diekman凡M.Gyllenberg,HHeijmans,HThieme,AHtileYoszdαTheoremfOTαclαssoftueaklfcontmuoussemtgToups,SemigroupForum38(1989),157-178.[5lDeimling,K,"NonlinearhnctiomlAnalys凰"Springer,1984[6]Desch,W.吁,W.SCd}h1a叩pp阳aCdth1e矶Ir飞.＼'S句peCt衍TαdlPTmop严eT巾tuze臼sf扣OTβnmttE一d"tm7仰7nZe旧阳Difπrer陀en川thI陆Ma叫lEquations59(1985)'80一102.[7lDesch,W.,W.SchappaCher,GeMmuonTesuItsforpeTtwbedsemigToups,in"Semigrouptheoryandapplications,"PhClement,SInvernizzi,EMitidierl,IVrabie(eds),MarcelDekker,1989,pp125-152[8lDesch,W,WSchappaCher,kangPeiZhang,SemtiiMaTEml山071Equattons,HoustonJMath.15(1989),527-552[9]Goldstein,JA,SemzgToupsoflineαTOPETatoTSαndαpplimttons,OxfordUnivPress(1985)[10]Greimr,G.,PeTturbingtheboundα叩Condzttonsofagenemtor,HoustonJMath13(1987),213-229[11lGminer,G.,SemtuneaTboundaTVcondttiomfOTetMuttonequαtzons,SemigroupForum38(1989),203-214.[12lpazy,A.,"SemigroupsoflimaropemorsandapplicationstopartialdifemltialequatmsJ'Spri吨er,1983
SemigroupsDennedbyAdditiveProcesses LSMITSDepartmentofMathematicsandComputerScience,UniversityofAntwerp(UIALAntwerp/Wilrijk,Belgium JANA.VANCASTERENDepartmentofMathematicsandComputerScience,Uni-versityofAntwerp(UIALWilrijk/Antwerp,Belgium IADDITIVEFUNCTIONALSOFMARKOVPROCESSES. 1.1Motivation.Inthispaperwestudypropertiesofasemigro叩{P(t):t主0}ofoperatorsinaBamchspaceoffunctionsf:E→Cwhicharisesfromtheformula[PO)fl(z)=Ez[Flf(Xt)].HereEzdenotestheexpectationwithrespecttoagivenMarkovprocess{(Q,F,Pz),(Xt:t兰的,(dt:t主0),(E,E)}conditionedsothatXo=2,pfalmostsurely,and{Fl:t主O}issomesuitableprocessadaptedtothehistoryoftheMarkovprocess-ToExtheideas,letfbelongtoCo(E),theBanachspaceofcontinuouscomplex-vdmdfundiomonEvanishingatinftnity,equippedwiththeuniformnorm.LaterwewillalsolookatLPfunctionswithrespecttosomeregularBOrelmeasureson(E、E).InwhatfollowsEisasecondcountablelocallycompactHausdorfspacewithBOrelfieldEandwithone-pointcompactiftcationEA.Qistlmspaceofallrightcontinuousfunctionsωfrom[0,∞)toEAwhichhaveleftlimitsandwithtllepropertyVAt:(8三tandω(s)=A)=争ω(t)=A.Inotherwords,thereisneSCaPefl-Omthepointatinanity.Thelifetime〈ofsuchapathisdeanedby〈(ω)=inf{8〉0;ω(8)=A}.Ahstexampleofthekindofsemigroupunderstudyisprm讯edbyputtingF(t)三1.Undersuitableregularityhypothesesonthehfarlt01'process,tllesemigroup{PoO):t兰时,givenby[凡(t)fl(z)=Ez[f(Xt)],isapositivitypremvingcontractionsemigr0吨。nCo(E).See,foriasta町e,[23;chapter2lInaveryloosesense,allsemigroupsinthispaper 463
464 SHItf5undvtJFICOSIer-rn canbeconsideredaspert旧bationsofPo(t).Itisnownaturaltoask飞w、V?lhu;a川Ltlk￡d山山ih山Inloftime-dependentdensitiescouldbeinsertedundertheexpectationS1u阳1K(.tIhltlhla川tanotherSe臼mI口mI丑曰1让i达grO1u1Pa1HtSeS.Inotherwords7forwhichcomplex,valuedprocesses{31:t主0}doesthepreseEiption[P(t)fl(z)=巳[Yif(Xt)],f主0,fεCo(ELZεE,dieaIleastronglycontinmmsemigrouponCo(E)?Itiseasily阳'11thattllealgebraicsemigroupconditionisinthiscaseequivalentto: Ez[几+tf(Xs+t)]==Ez[凡·(31O178)f(Xs+t)] 、、,,,F咱』A唱EEA,,,..、、 whenevertheseexpressionsmakesenseandWIleEe179istheusualMarkovtimeshiftoperator.ClearlythisissatisfiedifforeachZξEandforPralmosteverypathω:凡+t=巳·(31008),s;三0,t主0.Weexpressthisbysayingthat(U}tismultiplicαtitye.InappendixAwegiveac011IIterexamplestmvi吨thatmultiplicativityisnotanecessarycondition.1.2.Deanition-Inspiteoftilecounterexample,weshallrestrictoursrhTSevenIIlore,req11idngnotonlymultiplicativityofyjbutevenadditivityofitslogarithm:37=exp(Zt),Zs+t=Zs+ZtO仇.However,assoonaswehaveimposedregularityconditionsGIl|log31|forsmallt,wecanalwaysaddmultiplesof2MtoZtiaordertoensureadditivity.Anadaptedprocess(Zt)tsatisfyingtheaboveconditionwillbecalledanαdditiueprocess-1.3.Examples.InthissectionwewanttogivesomeexamplesofadditiveprocessesZdeanedonE,wherepossiblyE=Ru,whichbelongtokato『sclassforcertainFellersemigroupsoriothecorrespondingdifusions.IIIwhatfollowsweshalla.0.beinterestedinkat0'sclasskuandIfuloc-Af1Inc-tioIIIV230issaidtobelongt-OKa-t0'sclassICνifitVedaestilefollowingffl/|z-u|21identity:limsupljexpl-lV(ν)dsdu=0.IRHiAiZen-t↓Ozdvjo(JZZ)ν飞28/IIlazlandSimonprovethatthehtterisequivalenttothefollowingcondition:1imα↓OSUPzεrf|俨z|三uPu(|ν-z|)V(u)du=0,whereuu(T)=r2-u,ifν主3orifv=landu2(俨)=-logr.ThefunctionV兰ObelongstoI飞、lociflAVbelongstoIfufol-aucompactsubsetsIfofRU.NextwedescItbesomeexam-PIes-UnlessspeciaedotherwiseVisaIIOIl-aegativeBOrelmeasurablef1111ctioII,whichmaytaketllevalue∞.Example1.LetIV=IV+-IV-beaBol-elmeasurablefunctiondeanedoII[0,∞)×EandputZJ(t)=兀W(T+8,X(8))dsHereweassumetllatZfbelongstok(E)andt-hat-ZJbelongstoI飞floc(E):seeDeamuon2.4.nrealsodeanethef扣Or飞巩m飞UsingtlhleSetwoprocesseswemaydeamoperatorevolutiom{P1(8,t):s三t}aM{P2(t,8):8三t}asfollows: [PI忡,t)fl(z)=Ez(exp(Zf(t-s))f(X(t-8))),(12)[P2(t,s)f](z)=EAexp(z;但-s))f(X(t-s))),(1.3)
seFF1号roupsD旷InedbyAridifiveProt-e曰" 465 WIleEeforexamplefbelongstoCo(E)ortoLP(E,m).Theseoperator们叶11-tionsmaybeemployedt。"solve"Cauchyproblemslike θuu(s)=f;(1.4)=A011+IVtht228,θt7θuu(t)=f.θ=Aott+IVthsf二t,8(1.5) InOl-dertosolveequation(1.4)wewriteu(t,z)=[P1(8,t)fl(z)andto叫veequation(1.5)wewriteu(8,z)=[P2(t?8)fl(z).Example2.Let{b(8):8主0}be护dimeaSionalBrownianmotionandletC:[0,∞)×Ru→RUbeavectoraeldonWwhichmightpossesssomesingularities-MoreoverletV:[0,∞)×Ru→RbeaBOrelmeasurablefunction-Put Zr(t)=jtc(T+Ab(8))odb(8)-llt|c(T+Ab(8))|2ds-jtV(r+89b(8))ds.h2joh(1.6)Thisadditiveprocesscaabeusedto"solve"tllefollowinginitialvalueprobleIn: 22(t,z)=1AU(t,z)+[巾,z).引](12)-V(t,z)州,叫,t主AU(892)=f(r)·θt2(1.7)InfactagoodcandidatefortllefunctionUin(1.7)isgivenby u(t,z)=IEz(exp(ZS(t-8))f(X(t-8))). Heretlle自rstintegralonthe吨IIt-handsideof(1.6)istobeinterpretedastlrforwardItointegral-Example3.ThetheoryisapplicableifthefunctionV-belongstoICujf14Lbelongstolyu,locandifweconsiderV=V+-V-asanperturbationoftllempertubedquantummechanicalHamiltonian-jAThisispartoftlleresultsofAimmnandSimon[1landofSimon[221.HereZT(t)=-j;v(b(8))dsExample4.In[24lwehavelookedatperturbationsofgeneratorsoftlromstein-UhKIll〉eckprocessoftlleformLo+V,whereLoisgivenbyLo=-jA+c(z)VaMwhereVisanappropriatepote叫alfmCHonFordetailstllereaderisreferredto[24]-TllefollowingexampleisinfactageneralizationofthepreviousOIr.Example5.Letα(z)=σ(z)Tσ(z)beaPOSdiHttJiiV陀eS巧3y,1咀I口mIvaluedfunctiondeda肘donRuandlet{Xt:t兰0}betlhle(们1um1amique叫)StrOnsolutionofthestochasticdifemtialequation:X(t)=X(0)+j;σ(X(s))db(s)Illadditionietc(z)beanRu-valuedso-calleddTiβtermdeanedonRU-PutlZ(t)=kc(X(s))db(s)-iK|c(X(s))|2dsAg出11wemaypro附dasin[24]HoweverthusfarwehaveIIotyetworkedouttlledetails.RelevaIItsourcesofinformationareprobably:Aze盯ott[3l,Azencottetal[剖,DaviesandTruman[12],MolellanovpoLBismlt[6landBismut[7]
466 SFFMISGFIdvaFICU5fereFI 飞飞TeIIowintroduceaphysicallyrelevantexamplewheresamplepathcontinuityisviolated-1.4.PaulitypeHamiltonians[13].AsahststepfromnO盯elath-istictorelatiusticquantummechanics,onemightreplacetheSchr凸dingerequationbythePauliequation-ThiscomesdowntoreplacingthestatespaceR3byaproductoftwocopiesofit,andtoconsiderthecross-productofthecontinuoushhrkovprocesswithajumpprocessbetweenspincomponents-TheimaginarytiImPauliequationforspinj,puttinga=m=C=1,lookslikethis: θ￠t贮;(iV一仙一V仇+;-H叫 whereαisalocallysquareintegablevectorpotential,Visalocallyintegral3lescalarpotential,HistIrmagneticikldandσisthethree斗vectorofstandardPaulimatrices: ＼、EE』,f/咱io--iAU/Ilt＼、一-zσ＼1111/io..,,。一/tlI飞、一一"'σ＼飞111/-inunu咱i/l＼一-zσ wheretlmspinismeasuredconventionallyinthedirectionoftheZ-axis.Theequationaboveissolvedbythefollowingexpression,whichcanbeconsideredasageneralizationofthebynman-kacformula: h巾υ队,Jσ…叩(μ例t吟)一i才α叫仙州(仙仇刷bιωsJ)d品仇bιs一j划才(仙川αωd8+jjtHzMdsJ吨川s)一川ν(bs)忡s)l where(bs九三0isBrownianmotioninR3startingatz,(Ns)s>oisastadal-dcomti吨processiMepedentofit,andσs=(-1)N,σ.Atarstsight,theaddi-tivefunctionalintlleexponentialmightbehavesimilarlytothef1111CHonalse11-C0111lteredinthef01mulaeofcaIIlerOIl-MartinandFeynman-kabIto-飞Veshallsee,llowe飞吧1·、thattherearetechnicaldimcultiesarisingfromthenoneontinu-ityoftllesamplepaths-矶、inseNtlle(classical)probabilisticJohn-Nirnberginequality.1.5.TheJohn-Nirenberginequalityformartingales[14][15].LetXbeαcαdlαgαdαpted771αTti7lgαlesuchthαtyfoTsomeconstαMC>0αndfoTαnytωostoppi叼timesS三TPEUXT-Xs一|:Fslf二cyP-almoststL时y.ThmfoTm町λ<(4c)-19E|exp(λsupXsl|<-L一·|＼s>o/|-1-4c
semigroupsD旷InedbyAddHiveProcesses 467 IIBOUNDSFORSUBADDITIVEPROCESSES.Supposethat,foreachT主0,wearegivenacadlagadaptedprocess{Zr(s);s主0}.Weassumetl刚thisfamilyofprocesseshastlrfollowingsubadditivityproperty:VT主0,VzεE,zr(t+8)三ZT(s)+Zr+80)odhzr(0)=0,Pra.s.呼(2.1)wheredsistheusualtimetranslationoperatorontllepathspaceQ:[ds(ω)](t)=ω(8+t).Ifin(21)三isreplacedwithEthenthefamilyissaidtokbαckmTtlαdditiveandif三isreplacedwith22,thenthisfamilyissaidtobebαchyαTd川pemdditi侃Ifintheright-handsideof(21)T+8isreplacedwithr-S句thenwesaythatthefamilyisfoTWαTdsubαdditimandsimilarconventionscanbeemployedtodeanefoTTiyαTdαdditiueandfoTTIYαTdsTApemddititye.MTealsoassumethatZT(8,ω)isjointlymeasurableinT78aMω.ForageneralvemoIlandaprofounddiscussionoftheJohn-NirenberginequalityseeDellaCherieaIKlhfeyer[14landalsoDurett[151.ForanoMrivialapplicationseeCramtlOIlandzhao[Ill.AlsoofinterestistheorigindpaperofIGlashinskii[19laMSimon[22].TIlefollowi吨res1山isslightlymoregeneralthanTheorem1.5in[24.AtemindstoppingtimeTpossesesthepropertytl时ontheevent(T>t),tlleequalityT=t+TodtholdsPra.s.foraIlzεE.HittingtimesaretermiIIalMOPPIngtunes-2.1.Theorem.(khashinskii,John-Nirenberg)Let{Zr(t):t主0}beαTeal-tyαltLedbαckωαTds包bαdditiMhmilu.LetTbeαteTmindstoppmgtimeα71ddenotezr(t)=supo<s三min(t,T)ZT(8).FUT主OmdFzt主0.S叩pOSEtUlbM川C们7T、eαZt白8dtPOMi倪阿αl川7myn川ZbbeTη‘S9λ0仇Fγ饥1yγη2αmηdt句0=t叫(λ0υ)tωOget伪heTωt伪lhbαmeωα川mblhCfαmtl切yOf8tOPpimn叼gtimes{SS'r+tUy:t主U兰OLsr+tJ<TFfoTwhich,foTGllrεEαndfoTαllt;EU;三0αnd05二ttyf二toytuithU+w三ty。三8三S叶ty〈w三mn(t,to)impliesλoZT+u(8)三γ1,Pz-dmostωTelu;(2.2a)αndfoTtuhichyαgαtnf07·αllzεEαndfoTallt2EU25OandO三ttYEEtoytiyithu+wf二tpPz(sr+tY〈w)三exp(一γ172).(22b)ChoosefoTO<λ<λoconstantsω(λ)SMhthd1+一土-exp(γ172)<exp(ω(λ)to)·(2.3)λ。一λ一The7zforO〈λ1<λ2〈λoyO<λ<λothefollotDingineqtLalitiesaTemlidfoTαllZεEPfOTαllt2U22OPfoTαll05二w三t-UαTBdfoTGll《〉0.·PzFK)>己:T〉w)三优P(m归p(-M)优P(ω(λ)凡(24)//-r+ν＼＼λ1Ez(exp(λ172Z(w)):T〉w)三1+γ-Texp(ω(λ2)t)exp(7172).(25) Ifforany05;UfEtandforany05二wfEto,withU+tu三tontlleeventsrH:=inf{T〉8>0:λoZTH(8)>γ1}<w三mn(t,to),tlleinequal-ityzr+u(ST+u)三Zr+u(sr+L)isvalid,thenS叶tYVedaes(2.2a).
468 SFFtifsandvanfusferert Proof.飞Veitxλo〉λ>0,λo〉λ2〉λ1〉OandftrstsupposethatO<t三to=t(λo).Dehefort主U主Othesequenceofstoppingtimes{s;+ty:kεN)asfollows:(seee.g.PortandStone[21,p.23])s;+u=Oand,fork三1、 u14·-TKSAUOν1+-TKS+U+rcu+u1+-rLhn3一-U+rLhcu (2.6) ontheevent{s;2〈t-u)andsfty=∞el民whereThen,Pralmostsue13飞。三8三sru〈w三mn(t,to)impliesλoZr+气8)三hlandhence,ontheevent{sru〈叫,withO<w三mn(tjo), s;+U三inf(s〉0:λJ~)>h) (2.7) Sothat,by(2.7)andbythestrongtimedependentMarkovpropertywesee、for(k+1)γ1/λ。>《主k71/λ。,。三17,tu,w<t,U+w三t, IPz(T~?)>￡)三Pz(λfh)>h)三Pz(sfu<w)/血'一自CT+ν＼三Pz(s;1:+sihJTLJb1ods;丁:<町,s;一1<叫//r+u+s;2:(ω')＼l=EzlJHPx(s;2:)(的(ω:S1(ω)<w-s;11(J))1[OJ)(stkJ)))三supsupPu(s;+u+u<w-u)Pz(s;1:〈w)uεEUJ>u250三SupSuPPν(S巾+u〈即一uνεEtuωAUy>tuA;主EO三exp(-71寸2k)三exp(+7172)exp(一λ720.(2.8) Thisshows(2.4)forO<t三t0·WealsohaveforsuchO〈u+w三t三to,。三tu,u, Ez(优P(MZ叫w)))=1+MJf∞Pz(伊玄7,(μ巾)〉小P州川(从仙λh1γ节试叫2必州￡C)三1+λ1寸∞exp川exp(-bγ2川(入1725)dz=1+告γ均川 In(2.9)wesetλ1=λandweletλ2tendtoλotoobtain 叫expoJK)):T〉w)51+泸γxp(川) (2.10)
semigroupsD旷InedbLVAddHiveProcesses 469 Nextwepickt〉Oandwewriteto=t1=…=队,tn+1=t-nto,1vherenεNischoseninsuchawaythatnt。三t<(n+1)toPutsk=ZL1tJByinductionweshallprovethat,for15二kfEn+1andforO三UEEt-tU吨。三ω558k, 叫fk)>￡:T〉们时1%)(1+泸γxp(γ172)yexp(-M) Suppose(211)isvalidforsomek三n.Thenweobtain,forO三U三t一町,SK<w三sk+19withOEEU:=tu-SK三tk+1and, Pz伊庐旷r叶+~U三Ez(…PX(u)(ω)(zrMU(8k)+7+U(u)(ω)>己:T〉801{T>u}(ω))三优叫P川(1+泸古Xγ产e优川x×Ez(优P(h27+U(u)):T〉u)三exp(γ172)(1+一土-expb172)lkexp(-h25)·(212)＼O一λ/ Thisisthesameas(211)withk+1replacingk.Hence,forO三飞仙,u+w三t, /-r+27＼/λ＼npz(Z'川(w)〉已:T〉w)三exp(71%)(1+一一-exp(γ172)lexp(-h必)＼/＼λ。一λ/三exp(γ172)exp(ω(λ)t)exp(一λ720.(213)Noticethat(213)isthesameas(2.4).Inequality(2.5)followsbecauseforλ。>λ2>λ1〉Owehave,againfor05二ty,tu,U+w三t://-r+ty飞飞lEz(exp(λ172Z(w)):T>w)三1+λ1寸∞pz(玄巾(W)〉￡:T〉w)优叫蚓P(从M1门γη2ο三1+MJf俨∞飞e优x时1门mωγη划2ρ)叫=41+」瓦γ7优叫xP(μ以川ω叫( . Nextsupposethatforsomestrictlypositiverealnumbers机,α,bandfo:supsuppz|λosupZT(8)>α|三exp(一αb).(215)ZEET20LO<s<toj
470 Smif5αFtdvanCasfer-en Fixt>Oandconsidertheexponentialboundsfromthepreviousprop0·sitionforthestochasticvariableszr(t)=supo<s<tZT(8)indexedbyrεR+飞飞7edeanetheexponentialboundsonthedistributionoftheprocessZ: ~(t)=(lirr出工y咐??俨严川巳叫呻z」[PF忡灿玄T扪盯mr飞VM(0例川t吟)A~川2纣州(0t归~川州(0t)=一叫哈工PYTPzpr(t)〉711/丁AAt)=(JELYEzpr(t)呐!l1/γh(t)=一叫15lyPzpr(t)>?l1/丁 2.2.Theorem.FOTthetyαTidlesjustdejtnedmdwiththemtαti07tsofabovth(t)=A2(t)=h(t)=A40)=A5(t)yPToutdedthePTocesstHZr(t)ispz-dmoststLTelucontinuoωfoTαllZεE.AZOTeotYETtheseqtLαntitiesdonotdepeηdont.Proof.The(iL)equalitiesA1=A2=h三min(A4,A5)andA4三eA5areelementaryaaddonotdependonthesubadditivityproperty-Aproofislefttothereader.Theremaininginequalities,whichwestillhavetoprove,areA50)三A2(t)andA4(t)三A2(t).Wehaveprovenearlier(seevanCasteren[24landTheorem2.1.)thatforbackwardsubadditivefamiliesofprocesseswithproperty(215)andforλε(0,λo),ω(λ)=tfln(1+λ(λ。一λ)-1exp(αb)),t三OandZ>0:supzεES叩rεmPz[s吨。<s<tzr(s)>《]:三exp(αb一λb《+ω(λ)t).Noticethatifλ<A5(t),thenthehypothesis(215)issatisaedwithα=7,λ0=1aω=λ们SO吐山ω怡ωl让山川i扫川SUiI叫1maλf(μ例t忖)exP以(一λ￡ο)fromwhichweseethatλ三it2(t)andtheinequalityAHt)三A2(t)follows.NextweshowthatA40)三A2(t).Supposethatλ主OandntεNa川…liII叫a叫刷,f…mF<1,sh川OEz((叮门三俨Fixn=km+j,k;三0,0三j三m-1.Then E((λZJt斗才<σ<…fl/dσ1仇Pz(λ玄Tr气V(μt们)>σ叫叫nJ)(TJ仨扫=习i叫I<才ι<0σ以<盯…<ωJ∞fd命σ叫1伽Pz(0λz矿玄Tr(刀MJ)+λz玄γr叫m(t-TJh川
SemigroupsD旷IFtedbyAddifiveProcesses 47l 〈才ι<0σ以<…f户μμd伽向σ叫1伽E巳z(伊h机P盯hX川(盯叫ηkMm)(λ玄叫km(t一勾km)〉σn-σkm:λ玄T(t)〉叫)三jJO<…Jdσ1dσkm咔叫m)((λZr+巧km(0t一巧孔ωMJ))j丁):λ矿玄γr(t川k三寸;才Lι<0σ<<σ川m<∞Jfdσ叫1dσ叽~k阳m叽叫(0λz矿玄T盯r气V(仪例t吟巾)>叫ν"♂εz2?巳V沙JrFE川玄Ts(旷 (byinductionwithrespecttok)〈sD俨ν(λ70))my叫(矿。))J)1.lsup,"E,沱rImJlME,往rJ:(E18ldehinequality) //-r飞＼βItfollowsthatEz(exp(λZO)))<1+m!一-.Consequentlyλ三A20)and飞飞//一1-FhenceA40)三it20).FinallyweprovethatA20)doesnotdependont.Fromthedeamuonof(backward)subadditiveitfollowsthatz-(8+t)三Z(s)+zr+s(t)。仇,PralmostsurelyforzεEThequantityA20)doesROtdependont,becausebytheMaxkovpropertywehave$forλ>0, 叫优P(λ玄T(s+州三Ez(e叫λ玄T(8)+λZ叫吟。仇))=Ez(呵(λT(s))Exω(呵(λZ叫t))))三Ez(优P(λ玄T(8)))sZEU←xp(λT叩))) . Nextwedescribesomerelatedresults.FromnowonwedenoteZV)=SUPOO〈tZ气功.Foranybackwardsubadditivefamily{Zr(t):t主0}.Ddnetlmqua叫到Z)by:以Z)=叫λ三0:叫,rEz(仰伊(t)/λ))<∞)Heret>Oand,bytllepreviousresult,p(Z)doesnotdependont.飞飞与also
472 SmitsandvartCasteren wdteZ={Zr(t):t主OLzεE,TεR+.Thefollowingresultsaystl刚thefunctionalZHp(Z)issubadditive-2.3.Proposition.IfzlandZ2arebackwardsubadditiveprocesses,thenp(ZI+Z2)三p(Z1)+p(Z2).Inadditionp(λZ)=λp(Z),WIlemverλ>OandforZabackwardsubadditiveprocess-Proof.Thepositivehomogeneityistrivial.Thesubadditivitywillbeprovedasfollows-Pickλ1〉p(Z1)andλ2〉p(Z2).Putλ=λ1+λ2andExε>0.ChooseNeinsuchaway阳n主几impliesEz(70)")三dh!andEz(T(叫三忡!,forallZεEaIIdforrεRFOI-n兰川川λN「Ew附…eωObt叽叫EH1凸况lde臼r,、Sinequality, EF(?吟去)毛(;)去(EzO(仲(川去三主(;)去(叫玄T扪盯r飞V(仅旷t吟)三去主(;)川 FromTheorem2.2.togetherwiththedeanitionofAI(t)thes山additivityoftlmfunctionalZHp(Z)follows.-2.4.Deanition-(a)AnadditiveprocessZ={zr(t):t主0}r>oissaidtωOhM协lO叩O盯)川l川SUI-ablefamilyofstoppingtimes{sr:俨主0}suchthatsupr>OSUPOGTZU)isaboundedrandomvambleandsuchthatlimtiosupzεESUPr>OPz(Sr〈t)=0.(b)ItissaidtobelongtoIfloc(E)ifforallexittimesToftheformT=inf{8〉0:X(s)εEA＼int(K)),wherekva时soverallcompacts山时tsofr--r飞E,EmtiOS11IUrEz(Z(min(t?T)))=Oandifthereexistsameasurablefam-ilyofstoppingtimes{Sr:俨主0}suchthats叩r主OSUPO<s三FZT(8)1{T>s}isaboundedrandomvaItableandsuchthatlimtiosupr>osupzεEPAST<t)=0.Iftheprocesses{Zr(t):t主0}r>0arecontinuous,thentheass旧IIptiomonthestoppi吨timesSrfollowfromtheothers.IfZbelongstok(E),thenitfollowsfromTheorem2.2.thatp(Z)=0.LetZ:={Zr(t):t主0}r>obeaforward(backward)additiveprocess-ThentheprocessMdeanmlEyλfr(t)=exp(zr(t))isaforward(backwax-d)multiplicativeprocessinthesemethatMr(8+t)=Mr(8)M…(t),pz-as.,forallzεE,forallT主OaMforallAt;20.LetTbeaterminalstoppingtimeintllesensethatPralmostsurelyontheeveM{T〉s}theequalityT=8+T01?sholdsfor8主0.IfAfisabackwardmultiplicativeprocessandifTisaterminalstoppingtime,tlrutlleprocesstHJVr(t)1{T〉t}isabackwardm山iplicativeprocess-
SemigroupsDR/InedbyAdditiveProcesses 473 III-INTEGRALKERNELSANDLPPROPERTIES.3.OAssumptions.Inthissectionwesupposethat{PO):t主0}isaso-calledFellersemigroupinCo(E)oftheformiP(t)fl(z)=fp(tl,u)f(u)dm(u),fεCo(ELzεE,t>0,wherep(t,AU)isanonnegativefmctionon(0?∞)×E×E,whichposssessesthefollowingproperties:jp(t,ZJ)dm(u)三1,zεE,t>0,andlimz→AS呻民KP(t,ZJ)=limν→AS吨zεKP(t,2,ν)=0,t〉0,forallcompactsubsetsKofE.ThemeasuremisastrictlypositiveRadonmeasureonEInsteadofdm(z)weusuallyjustwritedtFromnowonweassumethattheunperturbedMarkovprocesshasasumwzetTictransitionkernelp(t,2,u)whichsatisaesthefollowingconditions:(1)Thefunctionp(t,AU)isjointlycontinuousinitsargumeMst〉0,zεEandUεE;(2)ThefunctionzHjp(tl,u)duiscontinmus;(3)ForeverycompactkcEandeverycompactTC(0,∞),theexpressionS叩zεES叩UUζSUPsεTP(8,2,u)is缸lite-3.1Mirrorimageofanadditiveprocess-Let仰,A,μ)betllefullWienermeasurespace,i.e.,theintegralofthemeasuresPzoverallthestartingpoiIItszεE.Infacttheddzlingpropertyofμistheequality fF川dμ=ffp(川叫斗,uj)FM forall71·tuplesofboundedBOrelfunctions元,...,fndeanedonEandforallincreasing71·tuplesoftimesh,...,tn·Inwhatfollowsweshallneedtllerdectionoperators(Rt)t>0,whichareμ-preservingtransformationsofofortlMFalgebrageneratedby(Xs:0三8三t):(Rtω)(8)=ω(t-s),ωε0,0三8三t.Let(Zt)t>obeanadditiveprocess-PutZKUJ)=Zt(RtUJ).Then(ZJ)t>oisadditive:Z;+t(ω)=Zs+t(Rs+tω)=Zs(Rs+tω)+ZtwsRs+tω)=Zs(RJtω)+Zt(Rtω)=Z;(8tω)+Z;(ω).Moreover,(ZUF=Zt.TwobasicequalitiesforμarejyoRtdμ=jYdμ,wlmeverYisaboundedrandomvariablethatismeasurablewithrespecttoFhandEz(Yods)=jp(8,2,X(8))Yo仇ditforanyboundedrandomvariableY.Weshallfromnowoninterpretthedefinitionofanadditiveprocessasanequalityalmosteverywherewithrespecttoμ.3.2Lemma.Letz,UεEFO<σ<t-T<tmdletMbeaTαndommTMblewhichismeω包TdlewithmFeetto{X(8):σ〈8〈t-T}.Then EzIMp(7,X(t-T),ul=Eu[MoRd(σ,21(t一σ)l tρhe7letyeTtheintegTalsezist.Proof.ItissumcieMtoconsiderthespecialcaseM(ω)=口二114(X(stTω))foraxedAtεB(E),1三i三n,andσ<81〈...<sn<t-7.ThenM(Rtω)=H二11At(X(t一句,ω))andweobtain Eν[MORd(σ,2,X(t一σ))]
474 SmtfsandvanCasIeren =f户μ以(川X(μt一矶川叫ω叫)川=f户μP以(仰σ…'JA2矶川?J2句叫州ωOω伽喇)》以忡p以仰(μ81一川Z1)HUJZi)148t+1一川川+1)p(t-T-Sn,Zn,zn+1)dzo…dzn+1=巳[Mp(t-7,X(t-T),u)l. -NotethatthereexistsageneralizationofthisresultJtonon-symmetrictransi-tiondensities,providedthe"adjoint"kernelp(8,y,z)ddmsasUtableMarkovtransitiondensityaswell-3.3.Deanition-Wesaythatanadditiveprocess(Zt)tEoissymmetmifitisequaltoitsmirrorimage(ZOGo,andαMisvrzmctTicifZt+Z;=OSincetlmmirror(dash)operationisline町,wecansplitZtintoaS巧3y,丁1I日mmInl江IIn配lr削eturd忧.tikCa剖In阳川IK(d1ana丑旧t"iS巧3y,丁咀I口mIPrOCeSSZwithtlmpropertythatZaswellasZ,belongstok(E)iscalledElKMOPTocess-IfZandztbothbelongtoIIfloc(ELthenZiscalledaloCGlkdoprocess-AIlycomplexvaluedadditiveprocessZwiththepropertythat5日Zas附llasSZare(local)katoprocessesiscalledacomplezmluedflocαljkdoPTOCESS-IfZisacadlag(local)katoprocess,tImltheresultsofsectioI12areavailable.3.4.Examples.Let(X(8))spbestandardz/-dimensionalBrownianmotion.IfVεLLJRU,C)isapotentialinIUt0'sclassIfuasdescribedbyAizeIman-Simoni1],thenSt=j;v(X(8))dsisasymmetricadditivekatoprocessIf3εLLc(RU,CU)削isaes|5|2εIfνanddivaεItfu,thenAt=兀δ(X(8))dX(8)+ijkdiv3)(X(8))dsisanantisymmetdcaddtivebtowmssOneBLightstartbelievingthatallkatoprocesseswithrespecttoBrownianIIlotionres1dtfromalinearcombinationofthetwotypesdescribedabove.How-ever,Simon[22lhaspointedoutthatkat0'sclassofpotentialsisnotcompletewithrespecttothenorm||Vl|kv三SUPzf|z-u|至1|z-u|2-u|V(U)|du,(ν主3)Indeed,tllecompletionoftknormedspacekucontainssomenoncontinuousmeasuresaswellasallthedistributionalLapladansofcompactlysupportedcontinmmf1mctiom.Notethatthenorm||-||儿iseq甲1u1山ii|lV卢|川|S三|川|S乌t||S三S1u1PzEz[K(飞V7、arS)(t)landwhere(VarS)(t)isthetotalvariatioIlofSSontheinterval[0,tl.If(Zr:t主0)叫NisaCmchysequenceofadditiveprocessesinthelatternorm,thenitmusthavealimitlstochasticprocess-Be-sidesadditive,thislimitisdsokatoandsymIIlddciftheZfare.Hence,tlleexamplewhichSimon[22lgivesgoesthroughtocomtruetasymmetricI臼toprocesswhichisnottlmintegralofafunction.Finally,ifICCEisaIlollemptysetsuchthatforallzgfILICllashittillgprobabilityzero,thenUAυ(0))isanexampleofanalmosteverywhereb01肌lmladditiveprocesswhichisnotkat0.
SemigroupsD旷InedbyAddifiveProcesses 475 3.5.Continuityofintegralkernels:motivation.Weliketostudyli11mroperatorsoftheformfHZf,wherefisinsomeclassofmeasurablefunctionsfromEtoC(e.g.L1),and[Ttf](z)=lLIexp(Zt)f(Xt)],2εE.Theadditivityof(Zt)t>othentranslatesintothesemigrouppropertyof(TI)t>0·ToobtainanintegralkernelforTl,weshallstudythelimit 汁。lEz[exp(Z川一Zσ)p(7,Xt-TJ)] (2,UξE). 3.6.Lemma.Let(Zt)t>oω肌llω(-Zt)t>obeTedmhedddtti时KMOPTocessesαηdsupposethαtt/2<to<t.Thc7Z :131S叩Ez[|exp(-Z川-exp(-Zσ2)|exp(ZtJ)p(71t-4)]=0 叫ε陀thes包PTemumistαkeηOMTαllσ1ε(0,t/4)?lσ2一σ1|<6?2,νεE?Tε(0,t-t01.Proof.Fix町,吨,z,uandTinsuchawaythatO<σ1〈σ2〈t/2andthatO〈7三t-t0·Theexpectationunderstudyisboundedby lEz[lZσl-Zη|exp(max(Z叭,Zσ2)+Zt-T)p(7,X(t-7),u)l三IEz[|Zσ1-Zσ2|exp(max(Z叭,Zσ2)+Zt/2)exp(Zt-T-Zt/2)13(TJ(t-TLu)]三lEz[|Zσ1-Zσ2|exp(4max(Zσ17Zσ2)+4Zt/2)p(7,X(t-TLU)]1/4×Ez[lZσ1-Zσ2|exp(4Zt-T-4Zt/2)p(7,X(t-TLu)]1/4×Ez[lZσ1-Zσ2|p(7,X(t-T),u)11/2三Ez[exp(8玄t/2)]1/4supp(t/2,只W)1/4 ×Ez[|Zσ2一σ1O8σ1|]1/4supEz[exp(4Zt-T-σ2-4Zt/2-σ2)p(7,X(t-7一σ2),u)]1/4 ×Ez[|Zσ2-σ106σ1|lI/2supp(t/2,AW)1/2. Thearstfactorisboundeduniformlyinz.ThesecondaswellastllesixthfactorisEIlite-TilethirdandalsothettfthfactorconvergestoOasσ2→σ1uniformlyinZandσ1·Finally,tothefourthfactorweapplyLemma32: lEz[exp(4Zt一←σ2-4Zt/2-η)p(7,X(t-7一σ2)J)]=Ey[exp(叫/2一叫)P(t/2一σ2,zl(t/2))]三γEωlexp(的/2))lr?P(t/2一句,uv) -IIIthefollowinglmmawewdteE=UL儿,wllmIIfrngint(Ifm+1)amdeveryIIfmisacompacts山setofE.WealsowriteTm=inf{s>0:X(s)ε
476 SmiI5andvanCasferen EA＼int(ICm)}.Moreoverpm(t,z,ν)isdeanedbypm(t,z,ν)=p(t、ιν)-Ez(p(t-Tm,X(TmLu):Tm〈t).AproofcanbepatternedaftertllepmofofLemma3.6.3.7.Lemma.Let(ZOQobeαηαddtttMcomplezmlMdlocdkatopTOcmsOTlet(Zt)t>0αswellas(-Zt)t主0beα陀αlwluedlocαlkαtopTocessα叫supposetlzdO<σ1<t/2<to<t.The凡foTαllmεNpmdtmifomzlyfoT(11y)εE×E×(0吨t-to)pli1IIlEz[|exp(-Zm)-exp(-Zσ2)|exp(Zt-T)pm(hXt-T,y):Tm>t-Tj=0.σ2iσ1 3.8.Proposition.Ftzσ>OyT>Oandlet(Zt)tbemαddittMKMOPTOCESS-Supposep(t929U)sαtisFes(1),(2)αηd(3).The71theftLMtionf(t92.y),dEFMdbyf(t,z,ν)=Ez[exp(Zt-T-Zσ)p(7,Xt-T,u)],i归3jointlyCωOTnZt"t川O包i叩7ηL(υt、且趴.＼、Uω)ε(伊σ+T飞、∞)×E×E.stmilα7·luthefunctionsfm(1ι训,mεN,deFMdbufm(t,2,u)=lEz[exp(Zt-T-Zσ)pm(7,Xt-T,ν):Tm>t一汁,tαZTjointlyCωO7ηLdt刮t川Oωt川n(υt,2趴?Uω)ε(伊σ+T飞'∞)×imnt叫(IK{~m川)×imIn川lt叫(IK1f了7m川nJ).Proof.Choose(t1,zhU1)andaxε〉0.飞Veshallprovethatthistuplehasa时ighborhoodoftuPIes(t2,22,U2)suchthat|f(t2,r2,U2)-f(tuZ1·U1)!<巳bysplittingthediference:|f(句,ZLU2)-f(t1,rhUl)|三A+B+C',叫m-eA=|f(t2,zuU2)-f(t2,zuU2)|,B=|f(t2,zhU2)-f(t2921,UI)|andc=|f(f2句川、Ul)-f(tIll-U1)|-WeestimatetileterImA、BandCseparatt灯 A=|Eh[EaYσ[exp(Zh-T一σ)13(TJh一T一σ,U2)ll-EZ1[ELlexp(Zt2-T一σ)p(71t2-T一σ?U2)li|. ChoosecompactneighborhoodsCofhandDofu1inE.Thentheboundλf.deanedby Iσ+T十t1iλJ三sup{lEz[exp(Zh-T一σ)p(7,xh-T一σ,u)];zεc,νεD,今1三t2三2t17 isaIlite-FOI-anycompactIfcE, 4三刊时|P川 ForeveryzεC,chooseacompactsetIIfzCES盯llthatlfE＼儿P(σAZ)ilz<ε/(12λf).TIlefunctionzHfp(σ,AZ)dzbeingcoatimousina＼tlr时existsaIIopenneighbodood1年ofZiIIES町hthatforallUε1二:fE＼IU12(σ、弘Z)dz<ε/(12M).SiMe(飞)zεcisanopencoverofthecomI肌tsetC,tlm‘eexistsaaIlitesubcover-Itfollowsthatthe1·eisaaIlitefamily{If19··,kn}ofcompactsS旧llthatfE＼UsIUP(σ,u,z)dz<ε/(12MLVyεc.SinceLJtIfziscompact,themexistsamghborhoodwof21suchthatifz2ε1亿
SemigroupsD听nedbyAddiriveProcesses 477 p(σ,22,z)|dz<ε/(6M).Ontheotherhand,ifz2εwncpfE＼Utkt(phrhz)+p(σ,Z27Z))dz〈ε/(6M).飞机concludethatf灿Oωrz叼2ε1WVnC'U仇2εDamI丑1(σ+T+th1)ν/2三t句2三2th1,wehaveA三ε/3.TheestimateforBissimilar.SelectcompactneighborhoodsTofhandYofU1andput N=supEz[exp(2ZM1)(p(Tlh-T-σ,的)+p(7,xh-T-σ,U1))],zεE,U2εY,t2ξT whichisnnitebythekatohypothesisandbyassumption(3).Then B=|EZ1[EXσ[exp(zh-T-σ)(p(7,xh-T一σ,的)-p(77xh-T-σ,的))ii|三N1/2EZ1[|p(7,xh-hU2)-p(7,xh-TJ1)|l1/2三N1/2EZ1[|p(Tvrh-TJ2)-p(7,Xt2-hY1)|:Xt2-TεK]1/2+N1/2Ezl[|p(7,xh-hb)-p(7,Xh-hU1)|:Xt2-T￠Kl1/2 foranymeasurableIfCE.Again,puttingsuitablerestrictionsonbandbTweobtainB三ε/3.Finally,forC, C三|EZ1[exp(Zh-T一σ)p(7,Xt2-T-σ,U1)]一|EZ1[exp(Ztl→-σ)p(7111-T→,的)l|三lEZ1[exp(Zt2)|Zh-T-σ-Zt1-T-σlp(7,xh-T-σ,U1)]+|EZI[exp(Zh-T一σ)(p(7,xh-T-σ,U1)-p(7,Xt1-T-σ,U1))]|三supEz[exp(3Zb)]1/2S叩Ez[|Z|t2-h||]1/2supp(7,zJ1) +EZ1[exp(Ztl-T-σ)|p(T+t2-t1,Xt1-T-σ,Ul)-p(7,Xt1-T一σ,U1)|l Thefirsttermconvergestozeroasb→t1·Thesecondtermisboundedby s1PEzMZt1-J咐时+t2一川1)+…川1))/2×(EZI[|p(7+t2-t1,xh-T一σ,U1)-p(7,Xt1-T一σ,U1)|])1/2. Observethatlimb→t1P(T+t2-t1,xh-T-σ,U1)=p(7,xh-T-σ,U1),PZ1·almostsurely-Alsonoticethat tJ!211EzJ(T+t2-t1,Xt1-T一σJ1)=JM1fp(t1一T一仇Z)P川2一川1-J1)dz=t:1311p(t2-T一σAJ1)=EZ1忡,X(t1-T一σ)JI)
478 SmiIsandvanCasteren FromSchefd'stheorem(seeBauer[5])itthenfollowsthatthefamily{p(T+tn-t11(t1-T一σ),U1):nεN}isuniformlyintegrable,whenever(tn:71εN)isasequencetendingtot1.Consequentlyweinferthat JMllEZ1[|p(7+t2-t1,X(t1-T一σ)J1)-p(TJ(t1-T一σ)J1)|]=0 HencetheconclusionProposition3.8follows-Asmdlarargumentcanbeem-ployedtoprovethesecondstatementintheProposition3.8.HereweusetllefactthatforZ三Otheresultisknown.3.9.Proposition.Let(Zt)tαnd(-Zt)tbe陀αlkαtopTOcesses.The7lthelimitlinhrioEz(exp(ZtE,T-Zσ)p(7,X(t-7),u))czistsmdisjomtlyCOMm-uoustnZα叫y.Let(Zt)tαηd(Zt)tbelocdkdopTocesses.Thenthelimitslimσ,TioIEz(exp(Zt-T-Zσ)pm(T,X(t-T),u):Tm〉t-7)ymεN,α阿joi71tlycontinuoωfoTZα叫UiniM(Ifm).Proof.LetO<σ1〈σ2〈t/2<t一巧<t-71<t.Thenwehave |Ez[exp(Zt-Tl-Zm)p(71,Xt-T1,u)l-lEzlexp(Zt-T2-Zη)p(72,Xt一町,y]|=|lEz[(exp(Zt-Tl-Zσ1)-exp(Zt-T2-Z的))p(Tl,Xt-T1,u)]|三|lEz[exp(Zt-T2)(exp(-Zσ1)-exp(-Zσ2))p(72,Xt-巧,u)]|+|lEzkexp(Zt-T1)-exp(Zt一句))exp(-Zσ1)p(71,几-T1,u)]|. ByvirtueofLemma32,thelasttermcanberewdttenas |Eν[(exp(-ZL)-exp(-ZL))exp(ZLσ1)IY(σ1,2,Xt一σ1)]| TIleresultfollowsbyapplyinglemma3.6twice-IIIordertoproveasimil;1rresultfol-thelocaldensitiesweinvokeLemma37.insteadofLemma3.6.TIlemainresultofthispaperistllefollowingtheorem-Thesymbolsk(E)aIKlIfloc(E)foradditiveprocessesaredeamdinDeamtion24.TIleres1山isknowniftlmpIocessZisgivenbyZ(t)=-kvms))缸,whereVisanaPIx-oIXiatcpoteMialfmction:seee.g.[24].3.10.Theorem.S叩posethdfoTαllyzεE,{Z(t):t主0}Teul'υGl-MdαdαptedαddtttMPTocess川chthdZα叫ZIbelongtok(E)mds'Itchthd-Zmd-ZtbelongtoIQoc(E).DeFMthesemigTO叩{Pz(t):t兰0}by[Pz(t)f](z)=lEz(exp(ZO))f(X(t))).U叫eTtheαmLmpti071thdp(t,且＼U)issymmetTic-t.e.p(t,AU)=p(t,y,z)foTGllt>OyfOTαllzpUmE,mdCOMm-wωthesemigTO叩{Pz(t):t主0}tsαstTO叼lycontimoωsemigTO叩tnCo(E)αηdαlsomLP(E,m),1三p<∞.(a)E阳、yopemtorPz(t)tsofthefomzIPz(t)fl(z)=fpz(t,2,u)f(u)d川(u),fεCo(ELti加Tepz(t,2,y)isαcoηtinuousftLMUontDhichMTiFesthcthT山tivofclbαpmαη-kolmogOTorIYz(8+12?,u)=fpz(8,ιz)pz(t‘AU)dm(z).t〉0.41U巳E.
SemtgroupsD旷InedbyAdditiveProcesses 479 (b)ThesemigTO叩{Pz(t):t主0}αlsoαCt8αω8α8tTOη叼gl切νCωOπtuiη川包Ouω8Semmi叨gmm包LtmηLp叮(E,m)F1三p〈∞.ThedudPz(t)*tsgiMnbuPz(t)*=Pz,(t).(c)Lf凡(t)mαpsL1(E,m)intoL∞(E,m)foTGllt〉Ofi.e.ifsup{po(t,29U):2,UεE}〈∞foTdlt>OjythenPz(t)yt>OymαpsLP(E,m)intoLq(E,mLf071三p52qf二∞.(d)IηL2(E,m)thefαmilu{Pz(t):t主0}isαSe叫{vf.αd句jOωtη时tPOSt此tt切η仇州iutyPTeSeT川η阳i切7ηtStTmO叼l句yCωO犯Mtuimη川包ωO包ωssemigTO叩withαse扩-αdjomtgeMmtoTFPTO仇dedZF(t)=Z(tLt〉0.Remark.FromtheoutlineoftheproofbelowitwillfollowthatthesameresultistrueforacomplexvaluedadditiveprocessZ,provideditisalocalkatoprocesswiththepropertythatbothReZandbZFbelongtoIζ(E):seeDefinition2.4.Outlineofaproof.DeaneformεNthedensitiespz,m(t,2,u)andpz(t,2,u)respectivelyby pzAMJ)=悍Ez(叫(Z(讨)pm(t一71(TM):Tm>T)pz( TileILbythetimedependentstrongMarkovpropertyweobtain: 12z(t,且＼U)-pz,m(t,27U)=悍(Ez(吨(Z(T))p(t-TJ(7)J))-Ez(exp(Z(T))pm(t-TJ(TLU):Zn〉T))=可行(Ez(exp(Z(T))p(t-7,X(T),u):Tm三T)-lEz(exp(Z(7))EX(T)(p(t-T-Tm,X(Tm)J):Tm<t-7):Tm〉T))=WEz(叫(Z(7))p(t-71(T)J):Tm三T)+hEz(优P(Z(T))p(t-Tml(Tm)J):7〈Tm<t)=IEz(exp(Z(Tm))pzO-Tm,X(Tm),u):Tm<t). 飞飞恒alsoeasilyseetl刚pz(8,z,u)2三P2z(8,AU)p(8,AU)andhence pz(t,ZJ)-pz,m(t,2,u)三Ez(exp(Z(Tm))(P2Z(t-TJ(TmM))1/2(P(t-TJ(TmLU))1/2:Tm<f)三(Ez(exp(2Z(Tm))P2ZO-Tm,X(TmLu):Tm<t))1/2×(Ez(p(t-Tm,X(TmLu):Tm<t))1/2=(P2Z(t,2,u)-P2Z,m(t,2,u))1/2.(Ez(p(t-Tm,X(TmLu):Tm<t))1/2三(132Z(12,u))1/2.(lEz(p(t-Tm,X(TmLU):Tm<t))1/2.
480 SmitsandvanCasferen Forz,uinacompactsubsetKofEwehavethefollowingestimate P2z(t,2,u)三||lkP2z1I1'||1∞三||1KP2Z(t/2)P2z(t/2)1K||1∞三||1I飞,p2z(t/2)||2∞||P2z(t/2)1Idi12=||1KP2z(t/2)||2∞||1Itp2Z'(t/2)||2∞三(||lkh(t/2)||∞,∞)1/2(||队r(t/几 beca1mtheprocessesZaswellasZ,belongtoIC(E).Inadditionwealsolm'ethattheseq时nceoffunctionslL(p(t-Tm,X(TmLu):Tm<t)、mξN、decreasestothezero-function.Sincethesefunctionsarecontinuous02(0,∞)×int(IIh)×int(km),itfollowsthatthisconverge盯eisuniformoncompactsubsetsof(0,∞)×E×E.TIleconclusionsinTheorem310thenfo1lowasiathecasewhereZisoftileformZ(t)=一兀V(X(8))ds Toconcludethissectionwewanttomentionsomeinterestingproblemsandpossibleapplications.(i)ObtainaresultlikeTheorem3.10.incasethesemigroup{Pz(t):t主0}isreplacedwiththeoperatorevolutionsasdeanedby: [P1(89t)fl(z)=Ez(exp(Zf(t-s))f(X(t-8))),(31)[P2(18)fkz)=lEAexp(z;(t-8))f(X(t-8))),(32) wherein(31)ZJ(吟,t主07isbackwardadditiveELMwherein(32)ZJ(门,t〉0,isforw町dadditive-(ii)Letthedifusioncoemdentα=σTσbesuchthatthestochasticdifez-entialequationdXO)=σ(X(t))db(吟,X(0)=2,possessesuniqueweaksolutionsforeveryZξRU.LetCbeasuitablevectorfield,definedonRu.PutZ(t)=兀c(X(8))db(s)-i兀|c(X(8))|2dsTIlepromsZisad〈li山eHereb(t)is川iimemonalBrownianmotion.TreattIds(kind)ofexampleindetail.Pz(t)-I(iii)LetZbeanadditiveprocessandletA=s-limbetllet↓otgeneratorofthesemigroup{Pz(t):t主0}definedby[Pz(t)fl(z)=Ez(exp(Z(t))f(X(t))).LetψbeaneigenfunctionofA,i.e.Aψ=一λψ,Studythedecayofψ.Itisperhapsusefultoobservethat,asinthecaseofapotentialfunction,theprocessMf:=￠(X(t))exp(λt+Z(t))句t主O句isamartingale:seeCarImna,MastersandSimonHoi-(iv)Howdoqmdraticforms、Q(fJ)=-fAofMm+ffEdμ,atintoourtheory?Hereμtakestheroleofpotentialfunction.Forresultsinthisdirectionseee.g-JUbeverbandhfa[21,BlancImdandMa间,FeyelanddeLaPmdelle[16],FuKushima[17landVoigt[25].(v)T时atinmoredetailtllePaulitypeHamiltoniamofsection1.
SemigroupsD旷InedbyAdditiveProcesses 48l AppendixA.Exampleofamultiplicativeandanonmultiplicativeprocessyieldingthesamesemigroup-Let(Xt:t兰0创)bestandardBrmmOw怀.IlmImIn1O创tihOInl川=R附ν飞υ,J1伦川eaClh1pathω'letMtωbeitsreHectionwithrespecttotheaxisω(0)ω(t): (Mtω)(8)+ω(s)||ω(0)一ω(t);(Mtω)(8)一ω(s)上ω(0)一ω(t)句 tlmpointbeingthatliftleavesBrownianbridgeinvariant:Ez川[F贝(ω叫)f刀(X几t)川l=E巳z[F刊ftLμ叫岛tdlhl町ef扣Orn1mI丑le臼risobviouslyImIn11u吐1dlt"iP抖1ikCa川ti忖飞Vye飞,weshalldeImIn1OIn1StratetlhlaMtlthelatterisIlot-Toseethis,ttxt>OandtwopointsZ,yinRUsuchthatz『Uandtheoriginformanequilateraltriulglewhosesideshavelength10.ConsidertllesetAofallpathsωεQwhichsatisfy:1.|ω(0)一刘<1;2.|ω(t)-u|<1;3.|ω(t/2)|<1;4.t<inf{s主0;|(Mtω)(s)|三1}ThenAlmnont时ialfPzdbm (31OMt)(ω)=1>(31/2OMt/2)(ω)·(31/2OMt/2)(8t/2ω). Acknowledgement.TheauthorsareobligedtotlleUIliversityofAnt飞verp(UIA)andtotlleBelgianNationalFundforScientiacResearch(NF飞VO)fortheirmaterialsupportadtotherefereefordere盯e[161.Referexlees.[1]AizenmanM.andSimonB.,BrownianmotionaMHamacki时qmlityforS仕Clhl时d心in吨ge盯rOperators,Comm.PtLTEAppl.Mαth.35(1982),209·273.[2lAlbeverioS.andhfaZ,Additivefunctiomls,mwIm'ereRadonadkatoclasssmoothmeasuresassociatedwithDirichletforms,PrepdntNr.66(OFtober1989),SOMerforschmgsbereich237?IEMit旧ff1rMatheIIlatikRuhr-171liversitatBochum,GDR.[3lAze盯ottR.,GradesDAviationsetapplications,Ecoled'dtddePTobαbilitdsdeSaint-FlotLTVIII1978,SpringerLNhf774.[4lAzencottR.eta1.,GdωOddι臼‘S叫98创4-8创5,SocidtAmathAmatiquedeFrance,1981.[5lBauerH.,PTobαbilitytheoTyαndelementsofmeαstL陀theOTU,Holt,Re!i111川、tandnTiIIMOIl,NewYork1972.[6lBism1tJ..1ιLαTgedmidiomα叫theMdlmmcαlcdus,Bih1r此ω.业讪kMl山BaSe11984.[7lBismutJ..孔19ProbabilityaMGeomet毗inPTohbuttymdmαlysis‘Spri且gerLNKI1206,1.60.[8lBlanclmdPl1.aMhfaZ.,S如emmiggrO佣1u1pofS仕创Cdlhl时din鸣geroperatorswithpo-teIdialsgivenbyRadonmeasures,BiBoSNr.262919879toappearintlrproceeElings2MAscom-Locarno-Comomeeti吨(Internati01叫confere町e
482 SmitsandvanCasferen onStochasticProcesses·GeometryandPhysics),Albevems.,CasatiS-andCatuneoU,MerliniD,MoresiR(editors),WorldScimtiac,SiRga-pore1990.[9lBlumenthalRM.andGdoorR.K,MαTKotYPTocessesmdpotentMltlMOTY,AcademicPress,NewYork1968.[口10叫lCC!a剖r口m、1咀InImnmaR.,孔MIa臼St优er臼S飞W飞V7.C.andSimonBB.,RelativisticS&创CdIhl时d占i吨erOPe盯rma-t013:asymptoticbehaviouroftheeigenfunctions,toappearinJ.。fFmet.Aηαlysis-[11lcramtonM.andZhaoZ,ConditionaltramformationofdriftforuMaandpotentialtheoryforjA+b(·)-V?CommMdhphys112(1987),613·625[12iDaviesIandTrumanA.,OntheLaplaceasymptoticexpansionofCOIl-ditionalnTienerintegralsandtheBender-飞再711formulaforz2N-anhanno11icoscillatorsj.Mαth.Physics,24(1983),255,266.[13!DeA吨elisG.,Jona-LasiImG.andSir吨mhf,ProbabilisticωlutionofPaulitypeequationsj.phys.A.·Mαth.Gen.16(1983),2433.2444.[14lDellaCherieC.andhfeyerP.,PTobabuttdsetpotentielV-VIII,Hernm111、Pads1980.[15lDumtR.,BTownimmottoηαndmαTtingdestnmαlysis?Wadsworth,BefnlontCA1984.[16lF巧'elDanddeLaPraddleA,Etdede196quationjAtt一叩=0‘oilμestunemes旧epositive,AmαlesImtitutFotmeht.38(1988).f3、199.218.[17lF此时himahf,DirichletformsandMarkovprocesses,NOωrt山Ihl-THtOdlHlha且hMlfhfaMttllhlhleImIn1a剖tikCadlLibrary23'Amsterdam1980.[18lJohnFandNirenbergL.,Onfmctiomofboundedmeanoscillation、Comm.PU陀Appl.Math.14(1961),415426.[19lkl邸'mmltiiR.,Onpositivesolutionsofuu+Vu=0,TheOTUPTob.ApplvoLIV,nr.4,(1959),309·318.[20]MolellanovS.A?D趴iE伍仇1u1山S的i扣OnP巳rm.OCeSSeSandIRm飞Uie臼m1口mI1anmmIn川liamnGemmO创Inm11e叫巾tthlIIr坏733MddlhL.S盯My30,1(1975),1.63.[21lPortS.CaMStoneC-J.,BmtMZMηmotionα叫pote7btidtheOTY、(1978),AcademicPress,NewYork.[22lSimonB,Sell凶di吨ersemgroups,Bull.Am.Mαth.Soc-7(1982)i447-526.[23]飞TallCasterenJ、GeMmtomofstTO叼lucontt川0ωsemigTO叩3.Pitmanresearchnotesinmathematics115,London1985.[24ivanCasterenJ.,OIlnωOMymmmI1e创tr町ikCgeneralizedSClhl时d也i吨erS肘p山g职rm、O1u1P严S趴‘stocltωticαmlysisα叫αppltcαtiom8(2)(1990),225.262.[25lvoigtJ,Leet旧eatthe2n叫dInternationalCOInlfemgroupTlh1eO创1r.〉y,andEVOdlhu川1tUiO1nlEqmtUiO川'Delft25/9弘-29/9/1989:Sellriidi吨PIoperatorswithsingularpotentials-
UniformlyBoundedSolutionsofQuasilinearParabolicSystems BRUNELLOTERRENIDepartmentofMathematics,"F.Enriques,"UniversityofMilaIUMilan,Italy 1.INTRODUCTION 3LetQCIRnbeabounded。pensetwithCboundary.Considerthef011owinginitiaIboundaryvalueproblemofparabolictype: DU(t,x)-a(t,X,U,vU)DDU(t,x)=f(t,x,U,vU)inR.×Q,'ijEE'XXX""x+'j1 δ-lb.(t,x,u)·DU(t,x)|+(I-δ)-u=δ-g(t,x,U)onR×δQ,(1.1)L1x」+U(0,x)=￠(x)inQ, NwhereU,f,g,￠areC-valuedfunctions(N主1),a,bareN×NijjmatriceswithC-entries(i,j=1,...,n)and δisaN×N-diagonalmatrixwithδhheC(δQ,{0,1}). (1.2) Inotherwords,onsomec。nnectedsubsetsofθQtheconditionsattheboundarymaybeofhomogeneousDirichlettypeonsomecomponentsofthevectorU. Thegoalofthispaperistopresentan,eeasy-,proofofthefollowing THEOREM1.1-SupposethatHypotheses2.1,2.2,2.3beIow(ofregularity,parabolicityandcompatibility)hold.IfUisaSOlutionofproblem(1.1)suchthat,forsomep〉n, 483
484 Terrent 、‘,,N、...,QFE飞nFT」,E···‘+R,,..、'且-huFUn、...,MH、...,、...,Q,..、nr2δ口",.,,‘+D川,SE飞hupueu (1.3〕 with 、..,,au14EEA,,..、∞+〈、,,、..,,。,,..、p『L、..,,&L,,..、u··』nu+、..,,Q,...、DET」、..,&L,,.、u2xnu+、‘..,Q,s-·、nFYL、‘,,,争L,,..、u,41+PRuest--、EEJur·、nFM dna、...,MH、...,、...,-Q,s-·、2δ户EM,...‘+RU,,..、huFUεuy'EE--,.品rassecenneh&L 、...,、..,,u,...、PM,,..、CJ飞、..,,一Q,s-·、呵,"fu、..,+-险,...、u+nrp川uεst (1.5) withCdepending。nIyonM(u).p REMARK1.2InwritingnormsweomitthesuperscriptN;H2,pcαδ'δrespectivelytheusualSobolevandHolderspaces,andthesubscriptδmeansthat(I-δ)U=0。nδQ. ThereSIlltexpressedbyTheorem1.1isinsomesenseexpected,althoughithasneverbeenstatedexplicitlyinthepresentform;butatpresentitlacksanexplicitandcompletepr。。f.Letusbrieflyexplainwhy,inouropinion,thisiss。-MeobservethatTheorem1.1claimstwothings,namely:(a)thes0111tionUismoreregularthan(1.3)says;inparticular '-U(t)ec-(Q)VteR+; (b)moreover,theuniformestimate(1.5)holds.Theadditionalregularityproperty(a)canbeobtainedusingSolonnikov'stechnique(s。lonnikov1965andrelatedpapers):namely,J-foreveryT〉OthereiSanestimatefortheC([0,T];C』(Q))normofU.ConcerningClaim(b),。nehastoverifythatsuchestimatesdonotdependonT;butthisfactdoesnotseemtohavebeenprovedinthat papers.Inordertoobtainaa,shortHproof。fTheorem1.1weprefertoproceedinanalternativeway:weusesomeargumentsfromanalytiCsemigrouptheorytoshowthatUmustsolveasuitableintegFalequation,andusingthiSinformationwederivefirstly(a)and
QuasilinearPambolicSystems 485 consequently(b).Thereisanotherreasont。provideanewproofofTheorem1.1:weareinterestedintestingthecapabilityofourtheoryofquasilinearparabolicequations(Acq1listapace--Terrer111988),inordert。getinformationsnotonly。nexistence,uniquenessandregularityoftheSO111ti。nbutalsoonapri。riestimates,whichofcourseareessentialinseveralsituationst。deriveglobalexistenceandstability.Thispaperisafirststepinthisdirection.In(AIrtann1990)problemsoftype(1.1)areconsideredandinmanycasesheisabletoshowexistence。faglobals0111tionwiththeproperty(1.3).Theorem1.1allowsustosaythatinthatcasesthes。lutionisClassicalandisuniformlyboundedintheC2norm.Thispaperisorganizedasfollows:section2c。ntainsassumptionsandsomepreliminaries;insection3wewritedowntheintegralequationtobesatisfiedbyU,andgivetheproofofTheorem1.1.FinallyintheappendixwediscusssometechnicaldetaiIs.Acknowledgment:IthankProf.HerbertAmannwhointerestedmeinthiSquestion. 2.PRELIMINARIESANDASSUMPTIONS Foreachk〉Oset 、SJVA〈-NUNcεu,,、‘=VAA 、,JK〈-MHnPAnNcep,41='vhA 7:=R×A×A'.K+KK Merequirethefollowing HYPOTHESIS2.1(Regularity。fthedata〕 (i) 一NIENN∞-NnNa1jeC(R+×Q×C×C;Z(C)),feC(R×Q×C×C;C),andVK>O+…-a,Da,Da,Da;f.Df,Df.DfeLW(R×Q×A×A').1jxijU1j'pij"X'U'p+KK (ii) -NN∞-NbjeC(R+×Q×CH;JE(C)),geC(R×Q×C;c),andVK〉O+…-b,Db,Db;g,Dg,DgεLW(R×Q×A).jxjUjXU (iii) ,....‘-EAnu咱···AeσemosroriN噜···』、...,-Q,s-·、σ句,"+2δnEU,···LeAY HYPOTHESIS2.2(Parabolicity)Foreachk〉Oand(t,U,p)egthepairofdifferentialoperators(d,2)K
486 Terreni definedby d(.,D)Mw:=a‘‘j{t,11xx』11iSO-regularellipticinthesense。fAmann1985,uniforInlywithrespectto(t,U,p)ε7.{SeeAmann1984andGeym。nat-Grisvard1967).K Moreprecisely,theN×NIIlatrices A(0;t,x,u,p;已ρ):=a(t,x,u,p)ee+e10ρ2I,sjBδ(t,x,u;写):=δb(t,x,11)毫+(I-δ),jnwhereoεR,￡eR,ρeR,(t,U,p)e7mustsatisfythefollowingKconditions: π(i)thereexistoe]-,π[,X〉OsuchthatK2|detA(0;t,x,u,p;笔,ρ)|主χK(|￡|2+|ρ|2)N,vxe页,voe[-OE,。],veeRn,vρeR;KK n(ii)foreachxeδQ,Oe[-0,o],ee!R,ρeRwith(￡,ρ)笋(0,0)andKK￡-v(x)=0,thepolynomial τ→detA(0;t,x,u,p;笔+τv(x),ρ)+haspreciselyNrootsτj(0;t,x,u,p;笔,ρ)withpositiveimaginarypart. (iii)(Complementarity)ifk〉0,xeδQ,(t,u,p)e7,oe[-0,Ow]EeRn,KKρεRwith(笔,p)#(0,0)ande-v(x)=0,therowsofthematrix 血市电···4、...,p、..,x,,..、ντ+C飞P-ux&LO,,.‘、A,....、...,、...,x,..、vτ+产气yx&L,...、XURU arelinearlyindependentmodulothepolynomial Nτ→n(τ-τj(0;t,x,u,p;已p));i=1E 'hereMiSthematrixalgebraicallyadjointofthematrixM. (iv)(Uniformcomplemerztarity)FixnowaconnectedcomponentrofδQ,randdenotebyBthematrixBδwithxrestrictedtostayinr;write,accordingto(iii),
QuasilinearParabolicSystems 487 .Br(t,x,y;言+τν(X))·[A{0;t,x,u,p;吃+τv{x),ρ)]= N-1-Nj+zQ;(0;t,x,u,p;已ρ)τ(modn(τ-τ‘(0;t,x,u,p;吃,p)))j=O'1=1'rrrranddefinethe(N×N)-matrixQbyQ:=[Q,...,Q].ThenwedefineoN-1theminorconstantn〔onrby ArβK:=infrnaxidetQ(0;t,x,u,p;笔,ρ)|,σσrwhereQσdenotethevarious(N×N)-submatricesofQandwheretheinfimumistaken。verallxer,(0,t,U,p,p)ε[-0,49]×7×RsuchthatKK22￡-v(x)=Oandi￡|+ρ=1.Finally,weassumethattheminorconstant凡:=川曰:|「connectedcomponentofm}satisfiesHJom HYPOTHESIS2.3(CompatibilityattzO).Thereexistsσe]]0,1[such2+2σ-thatthefuncti。r14bbelongst。C(Q)andsatisfiesthef。11owingδcompatibilityconditions: nMnuexv、..,,、‘,,,x,,.、AVxnu,,..、σ。采V=、..JX,,.、AWY、..,,XυTEE.,,.、+『BEad、..Jxnu,..、AV···Jxnu、..,,、...,x,,..、Awyxnu,,.、aEdkurEELXU '…-x→a..(0,x,￠(x),v￠(x))DD￠(x)+f(0,x,￠(x),v￠(x))εc:v(Q)11xxxxo Beforetoillustrates。meconsequencesofourassumptionsweneedthefollowinginclusionpr。perties: PROPOSITION2.4Letp〉n.IfveC(R·[H2,p(Q)])nC1(R;[LP(Q)]),thenb+'δb+foreachαe]0,1/2-n/2p[andT〉Owehave 2α-α(i)vveC([OT]·C(Q))nC([0,T];C(Q))x"δα+1/2(ii)vεC{[0,T];Cδ(Q)) andinaddition +、..,、...,一Q,,..、xu「M、...JTinur--··,,..、αpuvxnv+、..,,、..,,-Q,s-·、α2δ户-u噜···aTAnu,.....,,..、产UVxwv,4·‘+PRuεST (2.1) +||v||川'J9一}〈C(α,MR(v〕).CFU』{[0,T];Cδ(Q))VEE豆豆主ιParts(i),(ii)andestimate(2.1)。naboundedinterval[0,T]
488 Terrent follOWfromthewellknownincIusi。n ,Ez--‘4,4nu咱E·E-aptAV、-za,、..,,Q,,..、pavq6?"曰"‘....aTAnu,···hr·飞。FUC、...,、...,、...,Q,..、nFI』、...aTAnu,.....,,..、4Apun、..,,、...,Q,,EE、nF叮'』口UTinu,...‘''E‘、FU andfromSobolevimbeddingtheorems(p〉n).Byhomotheticargumentswe canseethatindeedtheconstantCdoesnotdependonT. 口 PROPOSITION2.5LetuεC(R;[H2,p(Q)])nC1(R;[LP(Q)])(p>n);thenforb+b+eachT>0,αε]0,1/2-n/2p[thefurlctions 、..,、..JX&L,,.、uxwv、-ZE,x+L,,..、ux&L,,..、··J··Aa→、‘,,,xι··",,a飞 (t,x)-→f(t,x,U(t,x),vU(t,x))x belongtothespace 、EZ,,、..,,一Q,,EZ、FEM--··JTAnu,...‘,,..、αFUn、‘..,、..,一Q,...、αq6「MTAnu,.....,,..、户U (2.4) andthefurlcti。ns (t,x)→bj(t,x,u(t,x)), (t,x)→g(t,x,u(t,x)) belongtothespace 、-Ea'RJ2,,..、、..,、..,,-Q,,..、「U嘈···aTAnu,aa·‘,,..、叮,"J'..‘+α户-vn、..,、..,-Q,s-·、··a·户UTAnu,..,‘,,..、αFLVn、‘..,、...,-Q,s-·、α叮4+-AFL'···aTinur--··‘rE飞户」 Inadditiontheirnorms,intherespectivespaces,areboundedwithaconstantdependingonlyonM(U),uniformlywithrespecttoT>0.PEζ22主」.ItisaiIIlInediateconsequenceofProposition2.4andtheregularityassumptionsonthedata(HypothesiS2.1). 口 Letp>n,letueC(R-H2,p(Q))nC1(R;LP(Q))betheSOILItionofb+'δb+problem(1.1),andconsiderthefollowingoperators: A(t,x,D):=a(t,x,U(t,x),vU(t,x))DD,(t,x)εR+×Q,1jx1j B。(t,x,D):=δb(t,x,U(t,x))宫D+(I-δ)歹,(t,x)εR.×δQ,。jojO+wherezodenotesthetraceoperator.AsaconsequenceoftheprecedingpropositionsandassumptionswecanmakethefollowingREMARK2.6(i)ForeveryteRthepairofdifferential。perat。rs+ (A{t,·,D),Bδ(t,·,D)) isO-regulare111pticuniformlywithrespectt。teR.Moreprecisely,+choosingk:=C(M(U)),thec。stantsO:=49,X:=;E,β:=β,appearingpoKOKO
Quasilinear-ParabolicSysfems 489 intheHypothesis2.2(i)-(iv),dependonlyonk;In。reoverthecoefficients。fA(t,x,D)andB((t,x,D)areequic。ntinuousinxeQδwithrespecttoteR+·(ii)UsingTheorem(12.2)ofAmann1984andTheorem(5.3)ofGeyIIIonat-Grisvard1967wecansaythatthereexistsλEE0,dependingOonlyonM(u),suchthatforeachtEE0,q>n,andλinthesectorPZ(0):={λeC||argλ|<0},themap{λ-(A(t,·,D)-λ),BE(t,·,D))isan'oooisomorphismbetweenH2,q(Q)andLq(Q)×H1,q(Q).δ(iii)Anyso111tionuofproblem(1.1)s。Ivesalsothefollowingnon-autonom。uslinearproblem: ut-A(t,x,D)U=f(t,x),Bδ(t,x,D)U=G(t,x),U(0,x)=φ, 、EEtl--FEElJ'QQθ××++pmm川εε、...,、...,.,xx-Q''εttxFE飞,『飞 (2.6) where: A(t,x,D):=A{t,x,D)-λOI, (t,x)εR+×Q, F(t,x):=f(t,x,U(t,x),vU(t,x))+λ-u(t,x),xυ (t,x)εR+×Q (t,x)εR+×θQ.--'"-(iv)Thefunctiont→G{t,·)behngst。cbm+;CJm(Q). G(t,x):=δg(t,x,U(t,x)), (v)Mecanrewritecompatibilityconditionsatt=Ointhisway: 、..,,nυ,,..、G--AWY、..Jnunu,,-E、XURU、..,,一Q,..飞σ?-+2δ户UrtAWY 、‘..,一Q,s-·、σ2δ「ve、‘..,nu,,E·、nr+AWY、EJnunur--、A FollowingAcquistapace-Terreni1988,wecanconstructtwofamiliesofHresolventHoperators RU,t):LP(Q)→H2,p(Q)NU,t):H1,p(Q)→H2,p(Q),δ'δδwhicharedefinedforeveryteR,λeZ{0):={λεC||argλ|<0}by:+ol r且nu----Hunuvvpu--=、J--nUHunuv··,、..,,.,、...,nu·nu&L·+L·,,、,,t、,A+LAt-'1-'1HU2uvv采u飞ARU飞八RUrts吨E飞ridtt、白白r&户U、E.,,、‘..,&L+L、八飞八,,..、,,,‘、采υ,ORHM川----HUVV inQ onδQ, inQ 。nδQ. NowwelistsomepropertiesofRδ(λ,t),Nδ(λ,t):PROPOSITION2.7Foreachεe]0,1/2[,q主p〉nand(λ,t)eZ(。。)×R+wehave:
490Ter-rent ||Rδ(λ,t)F||qsffcz+nnq||F||p;(27)L『(Q)|λ|F『LF(Q) C(ε)|lR采(λ,t)F||9P-三,-F||F||一;(2.8)vc--(Q)|λ|‘-C(Q) C(ε,p)||Rδ(λ,t)F||2ε一三1-ε-n/2P||F||p;(2.9)C(Q)|λ|FLF(Q) C(q,p)1/2||NX(λ,t)F||三{||VG||+|λ|||Gl};(2.10)-n/2p+n/2qxpoLq(Q)|λ|L(Q)LP(Q) C(ε)llNx(λ,t)Flqp三嘈p{iivG||+|λ|121Gll-};(2.11)odkJ-l-KJX-vh』(Q)|λ|品』C(Q)C(Q) C(ε,p)1/2||Nδ(λ,t)F||2εs{||VG||+|λ|||G||}(2.12)一1-ε-n/2pxpC(Q)|λ|L(Q)LP(Q) ||[Rδ(λ,t)-Rδ(λ,r)]h||2ε-gck)it-r|α|lh||一(2.13)C(Q)C(Q)||[Nδ(λ,t)-Nδ(λ,r)]h||2ε-三C(ε)|t-r|α{||vxh||-+|λ|1/2||hl一}(2.14)C(Q}C(Q)C(Q) EE22主ιTheproofiScontainedintheunionoftheworksofAInann1984,Acquistapace-Terreni1988,Acqllistapace-Terrer111987andTerreni1989;howeversomeadjustementsandremarksarenecessary.Estimates(2.7),(2.9),(2.10〕and(2.12)followfromTheorem(12.2)ofAmann(1984)withsomeinterPolationarguments.MeInustremarkthatifδrr=Oforsomere{1,....N}inaconnectedsubsetofδQ,thenther-thboundaryconditionisofhomogeneousDirichlettypeandthecorrespondingboundarydatumiSzero(seeRemark2.6(iv));thusnocontributionofhighernormsofGiSnecessaryintheright-handsideoftheestimates(seealsoRemark1.5and§2.AofTerrer111989).Estimates(2.8),(2.11),(2.13)and(2.14)followfromTheorem1.1andRemark5,1ofAcq1listapace-Terreni1987withsomeinterpolationarguments,takingintoaccounttheaboveremarkonthematrixfunctionδ.FinallyrecallingRemark2.6(i)wecanseethatallconstantsappearingin(2.8)-(2.12)donotdependont.口
Quasilinear-porGbolicSysfeF715 49I 3.PROOFOF四fEOREM11 Letusdescribehowwecancostructarepresentationformulaforthesolutionofproblem(2.6).Hereandfromnowon,weSimplywriteA(t,D),B(t,D)insteadofA(t,·,D),B(t,·,D).Supposeforsimplicityδthatp>2n,andfixαε]n/2p,1/2-n/2p[(wewillseeintheAppendixthattheassumptionp>2ncanbedroppedbymeansofabootstrapargument).FollowingAcqllistapace-Terreni1988(formu1as(4.10),(4.12),(4.15),(4.19))thesolutionUofproblem(2.6)mustsatisfythefollowingintegralequation.tU(t)=lk会(t,S)11(s)ds+L(φ,F,G)(t),te[0,T],(3.1)lδδwowherefor025s<twehaveset 、八-q、,,『...」、...,nυ+L,,.、RU、..a'nu&L,...、RUFEEE』、1'+L飞八,,..、。OM川+『IEJ、..,nus,,..、AA、..,nu+L,,..、AAFEEE』、...,+L、八,,.‘、军UR,4飞飞八、‘,,s'Le甲or'十Id--、..,,S&L(d串onka 「-1tλLδ(φ,FJ)(t):=￠+十λe{Rδ(λ,t)A(t,D)￠-Nδ(λ,t)B(t,D)￠}dλ+'ir+j宫λ-1州v,t)F(t)+V,t)以t)}dλ++jtj(t-s}λ十e{Rδ(λ,t)[F{t)-F(s)]+Nδ(λ,t)[G(s)-G(t)]}dλds.O':r-149ioThepath万liesinz(0)andjoins+∞eto+∞e,。ε]π/2,o[,ando九means(2πi)1九Weremarkthatby(4.13)andPr。position4.3。fAcquistapace-Terrer111989wehave α-i、..,,S&L,,.‘、cs、...,、..,,Q,..‘、p2δ曰",,..、￡、‘..,S&'iw,,EE、XUK Vt>SEE0, and C三、..,,、‘.,,nM,...、p2δ口u嘈···aminu,..E··,,..、户U、..,,GFAWY,,.‘、XU7L andtheconstantsCdonotdepend。nT;SOwecanapplytheoperatorA(t,D)tobothsidesof(3.1),obtainingtA(t,川t)=jkδ〔tdus)ds+kw,G)(t),比[0,T](32)O where
492 Terrent +、..,,Dλ'AUsrt、,,A]-nu、EJ'DS.,,,..、+Lnprt-AA、l'rthu、..,,.,&L&L,,,、λB,,.飞,...‘又u、JRt,,飞'飞八、八飞Aft、,XUSM川t+e叩orilu=、...,S&L,,..、采UVA、..,nu&L,,,‘、AA--、..,,S+L,,..、XU~VA (3.3) and Lδ(φ,F,G)(t):=A(t,D)Lδ(φ,F,G)(t)=「tλ=A(t,D)φ+十e{Rδ(λ,t)[A(t,D)φ+F(t)]}dλ+':r-F(t)+letλ{Vλ,t)[G(t)-B(t,D)￠]}dλ+':vat.、+IJie(t-s)Aλ{Rδ(λ,t)[F(S)-F(t)]+飞(λ,t)[G(S)-G(t)]dλds(34)'O':r ApreliminarystepintheproofofTheorem1.1iSthefollowing PROPOSITION3.1Foreachεε]0,(α-n/2p)〈σ[andT>Owehave'-,,-ueC([0,T];CFU』(Q)). (3.5) ELgg￡4,Theres111twillf011owbystandardargumentsofinterpolationtheoryprovidedweshowthat 、EEJ、...,-QrE飞ε「J-FEV咱···aTAnu,...、∞7」ε,,u、...,、...,-Q,,..、ε2+2户L咱···aTιnur---‘,...、∞冒Lεu (3.6) Tostartwith,wweprovethat ∞2ε-t→A(t,D)U(t)eL([0,T];CU』(Q)). (3.7) By(3.2)wehave =、...,-Q,s-·、ε叮LFU、..,,&L,,..、、...,GFrAV,...、XUNIL+SAU、..,,一Q,...、ε2FU、..,,S,...、u、..,,SφL,...、XU~VA--unu'EEEEEEd〈甲、..,,-Q,,..、ε叫J』FU、...,+L,,..、u、..Jnu&L,,..、AA =:I+I12 Using(3.3),(3.4),estimates(2.9),(2.12)andastandardchangeofcomplexVariables,asthecoefficier1tofA(t,D),B(t,D)belongtothespaces(2.4),(2.5)weobtain .t.I1=IIIeμ士{Rδ(士,t)[A(t,D)-A(s,D)]u(S)+vOwir+飞(击,t)[B(t,D)-B(s,川S)}艺||2ε一三C(Q)
QuasilinearParabolicsysferns 493 .t三I(t-s)-2{(t-s)1+n/2P{||[A(t,D)-A(s,D)]u(s)||n+JOLF(Q)+(t-s)-1/2||[B(t,D)-B(SJ]u(s)||+lV([B(t,D)-B(s,D]uh))||n}dsLP(Q)xLP(Q)at三I(t-s)α-1-ε-n/2P||u(S)||qmds三c·Tα+n/2P-MP(u)-JoHu,F(Q) +S)d-Q)(-Qεf、2εC2HHHFlvλHH1、八、'd、EJ、z'、...,,...』&L、,Jrt&LF(+GAwv-、..,飞,,,DS',E飞+LFU,...‘,,..、A+j,...‘...‘、E,,、..,,t-Q)'(|、代、,。。(剖NXU【】RHHH1+,1λhd]&-uh‘,,、EJe]{tγurtptTJAVF·IB--、...,.+D)飞J's-Qt(rtrtram-~B[;吨'-、Jrt、』'&L;'、Jft、八....·ZFE、吁thuλuF)R1叶fl&L,414'λ、...,因‘..、50.L、i飞δ〕-MNt叫,1{去λeHHeter--J~dAMY叩OHHHHVf+Jto〈-HHHP-EIIJ?』I++ ThelastintegraIofthiSsumcanbeevaluatedinthesamewayofI1takingintoaccounttheregularityofFandG.IntheothertwointegralSweaddandsubtractsuitabletermstakingintoaccountonRemark2.6(v),wehave 「tλI乒C(札T,MR(u))+||十e{Rλ(λ,t){[A(t,D)￠+F(t)]-[A(0,D)￠+F(O)]}dλ||"一-vJZVC』』(Q)+||十etλ{Rδ(λ,t)-Rδ(λ,0)}[A(0,D)￠+F(O)]dλ||2ε一JγC(Q)+||十etλRλ(λ,t)[A(0,D)￠+F(O)]dλ1"一J万vC』』(Q)+||十etλNλ(λ,t)[G(t)-G(O)]dλ||qp一JYVCh』(Q)+||十etλNλ(λ,t)[B(0,D)-B(t,D)]φdλ||qp一J古vC"』(Q) UsingtheestiInatesofProposition2.7andtheregularitypropertiesofthedataweget εn4C飞八AU、...,、..Jnu,,..、rA+AV、..Jnunu,,E、AA,...‘、...,&Lnu,,..、XUR飞八··ueTO俨l+IJ+、..,、..JU,,E·、nrMUTAAV''E·、C〈-?-TE-- Finally,takingintoaccountthecompatibilityconditions(Remark2.6(v)),(3.7)follows.Weremarknowthatproperty(3.6)forU,followsdirectlyfrom
494 Terrent equation(2.6).RecallingRemark2.6(iv),(3.7)iITIPIies,viathestandardschauderestinlateforelliptiCsystems,therequiredproperty(3.6)forU.口 LetusproveThe。rem1.1.TakingintoaccountProposition3.1wecan'-…-saythatuεC(R;C』·』』{Q))and+sup||u(t)190F-三C(￠,MR(u)).(33)tε[O,2]Cu'"』(Q)FThustheonlythingtoshowiSthatsupsuplu(t)||90F-SC(φ,Mn(u)).T主2te[2,TICV』』(Q)FFixTEE2andtake225tzET;applyingagain(3.2)wehave +sd、..,,-Q,..‘、ε叮'』户-u、...,S,,.‘、u、..FS&L,..、2υNVAMHHHH-A·、-&LPEEl-J+sd、..,-Q,··E、ε?"【L、...,sr·飞u、..,,s+L,,..、XUNVA-A"HHHH'--nu'-EEE-zd三、-E,-Q,,..、ε?-FU、...,&L,,..、u、...,nυ+L,,..、AA +、-E,,-Q,,..、ε叮ιFU飞AAU飞,AV'、..Fnu&L,,E··、RU、..,,&L飞八,...、XUM川+AV、...,nu&L,,..、AA、..,,&L飞八,,...、XUR,4、飞八··ueTO俨ITld+、..,,一Q,,.、ε?MFEVAV-、..Jnu&L,,..、A+ +、...,一Q,s-、ε?HFLV飞八d、,,、12&L,,.、户U、..,,&L飞八,...、采UM川+、..,,&L,,.‘、rA、..,&L飞八,...、eOR''1飞八.."eγur--+、..,,一Q,s-E、εq4「M、...,&L,,..、rA+ +|||1十e{h}λλ{Rδ(λ,t)[F(S)-F(t)]+Nδ(λ,t)[G(S)-G(t)]}dλds||2ε一+JOJ歹vC(Q) +|||十e{h}λλ{RA(λ,t)[F(s)-F(t)]+NA(λ,t)[G(S}-G(t)]}dλds19F一Jt-1J古vvc-』(Q) RUT-A+7'TEA+瓦UYEA+5TEA+A噜VEA+qJVTEA+?-YEA+··且'YEA ItiseasilyseenthatI+I三C(φ,M(u)).35pApplying(2.8),(2.11),theusualchangeofcomplexvariableandrecallingtheregularitypropertiesofA(t,D),B(t,D)wehaveI1三IIJe(问)λλ{Rδ(λ,t)[A(t,D)-A(s,D)]'t-1·:r+Nδ(λ,t)[B(t,D)-B(s,D]u(s)dλ||qp-ds主cu、,(Q)三C(MR(u))I(t-s)-1气||[A(t,D)-A(s,D)]u(S)||一+FJ-1C{Q} +(t-s)-1/21[B(t,D)-B(s,D)]u(s)||-+||VV([B(t,D)-B(s,D)]u(s))||一}dsC(Q)AC(Q)
QuasilinearPGrabo/icSysferns 495 、..,,,-Q,,..、FU、..,,&L,,..、uTAP-,uoSIG&L、...,、..,u,,..、PM,,..、CSSAU、..J-Q,s-·、FUUα+ε4且、..,,s+L''14A&·.&LFIIJ、...,、..a'u,...、PM,,.、cs Usingestimates(2.9)and(2.12)weget +、...,Q,,..、nrYL、..JS,,..、u、...,、..,nus,,..、A、..,,nu&L''E飞AA,....‘,ekD·句dJ'nε..‘、..,,S+L-A,,.、tor--J、..,,、..Fur·、nrMU,,.‘、C〈-?"?EA +||VX([B(t,D)-B(s,D)]u(S))||n+(t-s丁1/2||[B(t,D)-B(s,D)]u(S)||n}dssLV(Q)LF(Q).t-1三C(MR(u))I(t-s)-1-ε-n/2pl|u||qndssc(Mn(u))VJOHHV(Q)v Recallingthatt注2andapplying(2.8)and(2.11)wehaveIa+I,三C(φ,M(u))and4bp +、..,,-Q,,..、户U唱...,、...,S,,.、G、..,&L,,..、GFaEEKxwv+、‘..,-Q,,..、产U、..z's,,.‘、F、..,&L,,..、rA,‘.、ε....、...,S&L''1·A·L·-wPEllJ、..,,、...,u,,..、PMFE飞C〈-RMTEA 、..J、..Jur·-、nrMU,..、CSSAU、坠,、‘..,-Qr·-、户-M、..,,S,..、G、..,,+L,...、户U?』/eA、..,,s&L,..、+ FinalIyusing(2.9)and(2.12)weget.t-1I守主C(MR(u))|(t-s)+ε-n/2Psup{||F(t)||n+||G(t)||,n}ds三C(Mn(u))-eFJot主oLF(Q)H‘'F(Q)F Summingupandtakingaccounton(3.8)wehave 、..,,、..,,u,,.、PM,,..、C+、‘,,一Q,s-、句ι户L、...,&L,..、uTPou[sε&L、..,,、..,,u,..、PM,,.‘、C〈-、..,,-Q,,..、ε?-FU、‘..,&L,,.、u、...,nu&L,,..、An甲且P·,uos[ε&-" Next,usingagainSchallder'Sestimate,theuniformregularityofthecoefficientsofA(t,D),B(t,D)andtheuniformO-regularellipticity wehave 、..,,QJnJ,,a‘、、.,,,、..,u,,.‘、PM,,..、C+、...,一口,..、吨,』FU、...,&L,,..、uTPou-se&-h、-Ea'、...,u,,.‘、D-M,,.‘、cs、..,,一Q,,.‘、ε呵岳+q4户-v、..,,+L,,..、uTP'uos[e··· Nowweremarkthatthereexistsρ=p(Q)suchthatforeachρ<ρandoo牛'·-weC』'"-(Q)wehave 一口ε吨,-『,--dw2XWVF··'』εq6n俨〈--Q,,.‘、户Lw2xwv +ρ-{2+2ε}||w||一'C(Q) and
496Terrent 、...,一口,『飞FUW、.,εq4+'Ap+-nuε句6『...」w2xwvrsLε句6+-Anr〈-、..,,一Q''ZE、【LWxnv SOwecanchooseρsuffiCientlysma11,dependingonlyonM(U),inp2ordertodroptheCnormofUintheright-handSideof(3.9):weget 、...,、..,u,,..、PM,..飞,,C〈-、..,-Q,s-·、ε句6+叮6FU、...,+L,,..‘、uTP'uos[ε‘.、 ObservingthatC'(M(U))doesnotdependsonT,(1.5)follows.口p 4.APPENDIX Wewillseethattheassumptionp>2nintheproofofTheorem1.1iSnotrestriCtive.Indeedwehavethef。11owing PROPOSITION4.1LetUbethesolutionofprob1em(1.1)andsupposen〈p<2n.Thenforeachqe[p,np/(2n-p)[M(u)<+∞斗M(u)〈+∞.pq Proof.Ifweshowthat t→(A(t,D)u(t),B(tJMt))e山R+;山俨(川 theresultwiIlfollowbytheRemark2.6(i)andthestandardaprioriestiInatesinLqspacesforellipticproblems(seeTheorem12.2ofAmann1984).NowrecallingRemark2.6(iv),wehaveonlytoshowthat 、..,,、‘..,u,,.‘、nFMAVq‘,..‘、C〈-q-VL、‘..,&L,,E·、u、..,,nu&L,,..、AA+PRUεst Vqε[p,np/(2n-p)[. (4.1) Fixqe[p,np/(2n-p)[andchooseαe]n/2p-n/2q,1/2-n/2p[.Using(3.2),estimates(2.7),(2.10)andRemark2.6(v)weeasiIyseethat sup||A(t,D)u(t)||sc(q,￠,M(u)).,qte[O,2]LNextassumet注2andusingagain(3.2)weget +sd、...,Q,,E‘、qBYL、..,,S,,.、u、...,s&··",,..、,ONVR屿'A+···LFEEl--d+SAU、‘..,Q,,.、n『IL、..,,s,,..、u、..,,S+L,,..、eo~VA-AρlJO〈-、..,,Q,s-·、nr冒L、..,,&L,,..、u、...,nu&L,,..、An qJET-A+q4TEA+....T'A=n『T」、..,,&L,,自‘、、E.,,GFAAWY,··E、XUNIL+ Takingintoaccount(3.3),(3.4)and(2.7),(2.10)wehave
Qt405IllnearParaboliclivsIeFF15 497 、...,、...,u,,.、nrMqp,,.‘、C〈-SAU、..JQ,s-z、D·q4口u、...,S,,..、unH.「LJ''n+nr「d/'n咽A、..JSιL,,..、4,..+凰"nur--JC〈-咽A?EA +、..,,nu,..、DEYL、‘.,,S,,.‘、u、EEE-4、...,nus,,..、AA、...,nu&L,...、AA,EEE·‘rekα+qJ'n+p叫dJ''n4A、...,S&L,,..、·····L&LFllJC〈-2TEA (t-s71/2||[B(t,D)-B(s,D)]u(s)||n+||vv([B(t,D)-B(s,D)]u(s))||n}ds三LV(Q)ALV(Q)at三c|(t-s)-m-n/2p+n/2q似||u(s)||qm三C(p,q,MR(u));J-1H叮俨(Q)F IA三C(￠,M(u)-JP Thus(4.1)holdsandProposition4.1isproved. 口 Bytheabovepropositionweseethatifp>(4/3)nwecanselectaq>2nsuchthatM(U)<+∞.Fixnowanyp>nandchoosemeNsuchthatqm-1m-1p>2n/(2-1);thennp/(2n-p)>2·n/(2-1),SOthatwecanselectaq>2m-1n/(2m-1-1)suchthatM(U)<+∞·Aftermstepswewillfindaqq>2nsuchthatM(U)<∞.q REFERENCES 1.P.AcquistapaceandB.Terreni(1987).Hath.Z.191,451-471.2.P.AcquistapaceandB.Terreni(1988).Hath.Ann.282,315-335.3.H.AInann(1984).AnnaliScuolaNorm.SUP-PisaIV11,593-676.4.H.AIIlann(1985.J.reineU.arlsew-Hath.366,47-84.5.H.AInann(1990),Diff.Int.Eq.3,13-75.6.G.GeyIIIonatandP.Grisvard(1967).Rend.Sem.Hat.Urliv.Padova38,121-173.7.V.A.Solonnikov(1965).BoundaryvalueproblemsofIIlathematicalphysiCsIII,ProceedingsofthesteklovInstituteofNathematiCs,A.M.S.,Providence,1967.8.B.Terreni(1989).StudiaHath.92,141-175.
SemigroupsGeneratedbyFirstOrderDifferentialOperatorsonN-DimensionalDomains MARLENEGABRIELEULMETMathematicsInstitute,UniversityofTUbingeILTUbingeruGermany ItisthepurposeofthispapertosummarizesomeresultsofmythesisconcerningsemigroupsgeneratedbyErstorderdiferentialoperators02日-dimensionaldomains.Weonlysketchsomeoftheproofs;furtherdetailswillbepublishedlater.GivenaSrstorderdiferentialoperatorA,ourcentralproblemistocharacterizethosedomainsD(A)CC(Q)forwhichAgeneratesastronglycontinuous,usuallypositive,semigrouponC(Q).(Qdenotesaboundeddo-mainofRnwithsmoothboundary.)Weandoutwhetherandwhatkindofboundaryconditionsareneeded.Firstietusfixthegeneralframework:InthesequelweusethefollowingIlotationsandsupposethattheassumptionsbelowaresatisfied:a)LetQbeaboundeddomainofRn.WedenotebyθQtheboundaryofQ,andbyQitsclosure.Moreover,weassumethatQhassmooth 499
500 Ulmef boundary,i.e.,foreachZ巳θQthereisaneighbourhoodWandafunctionηεC1(IVLsuchthati)QnW={zεW:η(z)<0},ii)WW={zEW:η(z)〉O},iii)forzεWnθQwehavegmdMz)并0.ThismeansthatθQislocallyaC1manifoldofdimensionn-1.Moreover,97'αdTIhasthesamedirectionandorientationasthenormalizedoutwardIIormalntoθQ,i.e.,gmdη=n||grαdη||.b)TheunderlyingBanachspaceXisC(Q),thespaceofallrealvaluedcontinuousfunctionsdeanedonQ,equippedwiththesupremumnorm.c)ThesetofallC1functionsdeanedontheclosedsetQisC1(Q):={gb:gεC1(U),UaneighbouhoodofQ}.d)LetF:Q→RnbeaC1vectoraeld-Acorrespondingarstorderdif-ferentialoperatorAonC(Q)isdennedasthederivati飞吧alongthevectoraeldF,i.e.,Af:=(g71df|F)withdomainD(A).Here(gmdf(z)|F(z))denotesthesumz二1苦于瓦(z)ItisoneceMraltaskofthispapertodescribethedomainD(A)ofthegeneratorA.ItturnsoutthatthedomainisdeterminedbyasmoothnessconditionaMaboundarycondition(dependingonthepositionofthevectoraeldFrelativetoθOLThereforewedistinguishthefollowingsituatiOIls:α)ThevectorEeldFonmpointsinwards.Wedescribethissituationanalyticallyby(F(z)|n(z))〈OforeveryZεθQ,wherenistheoutwardnormaltoθQ.3)ThevectoraeldFonθQisoutwardsoriented,i.e.,(F(z)|n(z))>OforallZεθQ-7)ThereexistzoεθQsuchthatthevectoraeldFistangmttoθQ;thismeans(F(zo)|n(zo))=0.Itturnsoutthat,roughlyspeaking,forpointszεθQlikethosedescribedinα)wedonotneedanyboundaryconditions;forZξθQsatisfyingd)wedoneedboundaryconditionsandthecaseγ)leadstocriticalpoints,whereweneedadditionalassumptions.InthesequelwedescribetheErsttwosituationsα)aIldPLaIlddiscussfurtherpropertiesofthecorrespondingsemigroups- NotethattheErstorderdiferentialoperatorAdennedonaxwithdo-mainDm(A)=C1(Q)isclosable-HenceanyrestrictionofAtoD(A)CDm(A)isclosableaswell. §1Thevectoraeldattheboundarypointsinwards Amlytically,thisisthesituationwhen(F(z)|η(z))<OforeveryZ巳θQ,andduetoassumptiona)iii)thisislocallyequivalentto(F(z)|grαdη(z))<O
semigroupsGeneratedbyDWbrentialOperators 50I forZεθQ.Inthiscasethefollowingtheoremholds(seealsolNal,p.198,wherethetheoremisstatedwithoutproof). 1.1Theorem.LetFεC1(0,Rn)satisfytheaboveconditionsandcon-sidertheoperatorAdeβIIedonX=C(Q)withdomainD(A)=C1(Q).ThentheclosureofAgeneratesapositivestronglycontinuoussemigrouponX.Infactthesemigroupconsistsoflatticehomomorphisms. Theabovetheoremandallsimilarstatementstofollowareprovedper-formingthefollowingsteps:1)Weconstructthesemigro叩(T(t))Qoexplicitly.2)WeprovethatD(A)is(T(t))吃oinvariant.3)Weshowthatthegeneratorcorrespondingto(T(t))位。coincideswithAonD(A).4)WeobservethatD(A)isadensesubspaceofX.Theorem1.34from[Na]assures,ttmif1)-4)aresatisaed,thenD(A)isacoreofA,andtheclosureofAonD(A)generates(TO))t主0·OneoftheparticularaspectsoftheseproofsisthatwesucceedtoCOIlstruetthesemigroupexplicitly.IntheabovesituationthevectoraeldFinducesaglobalnowψ(t,z):E十×Q→Q,andwedeEmthesemigroup(TO)fkz):=f(ψ(t92))foreveryt主0,ZξQandfεC1(Q).Moreover,wecaIIdescribethegeneratormentionedinTheorem1.lexplicitly. 1.2Proposition.ConsidertheoperatorAf=(grαdf|F)onXwithdomainD(A)=C1(Q)αsαboue.ThenitsclosureAisgivenbythederivativedongthetrajectoryofψ(t,z)throughZεQattimet=0,anditsdomainD(A)isthesetofallfunctionsfεXsuchthatthederivativeoffalongthetrajectoryofψ(t72)throughZattimet=OexistsineverypointZεQanddeftnesacontinuousfunctiononQ. NotethatiffεC1(Q)thenthederivativealongthevectorEeldFcoincideswiththederivativealongthetrajectoryoftp.InthiscaseORecaaequivalentlydescribeD(A)asthesetofallfunctionsfεXhavingcoMim-011sdirectionalderivativeinthedirectionF.
502Ulmet §2Thevectoraeldontheboundarypointsoutwards Thisisthecasewhen(F(z)in(z))>OforallZεθQ.AtrajectoryofthenowinducedbythevectorgeldF,whichleavesQwillneverreturnintothedomain.InthissituationwehavetoimposeboundaryconditionsforallZεθQ.Inthefollowingweshalldiscussthemostcommontypesofboundaryconditionsleadingtosemigroupsgenerators- 2.1Ventcelboundaryconditions Themostgeneral飞rentedboundaryCOIlditioIIisgivenforageneratorA,whichisa(weakly)ellipticdiferentialoperatoroforderatmosttwqseeivel,[GLl.Thisboundaryconditionhastheform ?θu(z)Au(z)+〉;αz(z)一τ一+α。(z)u(z)=OforZε侃,』-dozt andarisese.g.inapproximationtheory,seelTil-Itisoftenusedwithαz=αo=0,i.e.,Au|θQ=0.Oursituationcorrespondstothecase,whenAre咱ducestoaErstorderoperator.InthiscasewecalltheboundaryconditionAu|θQ=OaVentcelbotmdaTycondition,althoughthisisaslightlyn∞OmtandaIr叫dduseoftheterm. Ifweimpose飞rentedboundaryconditionsforallZεθQ,thenthefollowingtheoremholds. 2.1Theorem.TheCdlOS盯eOft伪lh1eOpe叮1r?a剖tO叮zr.AonCC,(Q)withdomainD(A)={fεC1气(Q)伫:Af|MθQ=0}ge臼I肘mteSaStrmO鸣l抄yC∞On川t"inmO山pOSitiVeS町me臼ImmIη1i号grmOuO∞nX. InthefollowingweexplainhowtoconstructthesemigroupmentionedinTheorem21.LetU:={zεEn:thetrajectorythroughZintersectsθQ},anddeanethefunctionS:U→Rbys(z):=thetimeneededbythetrajectoryofvstartinginZtoreachtheboundaryθQ.OnecanshowthatthesetUisopen,thefunction8isuniqueiydeanedandcontinuouslydifferentiable-
SemigroupsGeneratedbyDUKrentialOperators 503 Bydeanitionwehavethats(z)〉OforZεQ,s(z)=OforZεθQands(z)<OforZ￠Q,andwesets(z)=∞(s(z)=-∞)ifthetrajectorythroughZεQ(z￠Q)doesnotintersecttheboundaryθQ.Furtheron,weddmthemappingσ:U→θQbyσ(z):=ψ(s(功,z).TheassumptionconcerningtheorientationofthevectortkldFontheboundaryimpliesthatthenowv,inducedbyF,isdennedonLJzd(一∞,s(z))×{z},(fordetailssee[Aml,p.134E).LetZεQandt主0.TIleaeitherψ(t,z)εQforallt主0(ifs(z)=∞),orthereexistsauniques(z)<∞withO三s(z)<t+(z)suchthatv(s(z),z)εθ0.Wenowdeaneaglobalsemiaowφ:m+×Q→Qas 飞EIZJ,,‘、、s'、,,/<-zJ'''飞4'UQU〈一〉-nU4ιF?AF,..、‘,,,FZ、‘,,/、‘.,/ZZ‘,,,..飞AEKVQU/,E‘飞J'··飞ψψ-rt』411一一、、.,,,Z''ιJ,,‘‘、φ andthesemigrouponXgeneratedbytheoperatorAwithdomainD(A)={fεC1(。):Af|θQ=0}is(T(t)f)(z):=f(φ(t,z))forallZεQ,t主0.(1) Relation(1)ddmsalatticesemigro叩,seeTheorem3.4of[Nal,p.144,so(T(t))t20ispositive. 2.2BoundaryconditionsofNeumanntype LetusreplacetheVentcelboundaryconditionsinTheorem2.lbyaθfNemanntypeboundarycondition-一(z)=Oforallzεθ51.Hereνdemtes.θνaIIOIItangentC1vectoraeldonθQ.ForthissituatioaweobtainaresultsimilartoTheorem21. 2.2Theorem.LetFεC2(0,Rn)andconsiderthehstorderdditfre叮rMme臼mInIt"i归adloperatorAf=(g俨αdf|F)onXwithdomainD(A)={fεC1(Q):θf、否(z)=OforallzεθQ}-ThentheclosureofAgeneratesastronglycontinuouspositivesemigrouponX. Forsimplicitiywerestricttothecasewhere叫z)=n(z)forZεθQ,71beingtheoutwardnormaltotheboundaryofthedomain.AgaiIlwepoilltouttheconstructionofthesemigroup,whichismoreiavolvedthaatheoneinSection2.1.ThemainideaistoconstructanewvectoraeldGiIlaBeighbourhoodoftheboundary.引ferequirethatontheboundaryθQthisaewvector自eld
504 Ulmet Gcoincideswiththeoutwardnormalt00,andoutsidethedomainweshiftthevector叫z)alongthetrajectorythroughz.Infact,wedeane G(z):=n(z)G(v(t;z)):=Dv(t,z)-n(z) ZεθQZεθQ,t兰0,v(t,z)εU＼Q. HereDv(t,z)isthederivativewithrespecttotTheformaldeamtionofthenewvectorfteldGwouldbethen G(z):=[Dv(-s(z),σ(z))ln(σ(z)), forallZεU. ThisnewvectoraeldGinducesforitspartanowO(t,z)andweshallextendthefunctionsfoutsideofQ,(inaneighbourhoodofOLeoI川aMa10吨thesenewtrajectories.SimilartωOS叫(Z叫)wedeanT叫(Z叫):=thetimeneededbythetrajectoryof￠startinginZtoreachtheboundaryθQ.IfVdenotesthesetV:={zεU:|r(z)|<∞}thenVCUandweιdled且nthemapρ:V→θQbyp(z):=￠(T(z),z).Thesemigro叩ofTheorem22isconstructedas jf(v(t,z))ifψ(t,z)εQ(T(t)f)(z):={lf(ρ(ψ(t,z)))ifψ(t,z)εV＼Q. Finally?onecanshowthattheconstructiondoesnotdependOIltheextensionofF. 53FurtherpropertiesofthecorrespondingSemigroups Thesubsequentpropertiesturnouttobeusefulforthequalitativestudyofsemigroups-WeassumethroughoutthissectionthatthevectorfieldF0日theboundaryiseitherorientedoutward,orpoiIItsinwards. 3.1TheTranslationProperty Consideracontinuouslydifez-entiablevectoraeldFdeanedonQandthecorrespondingdowv(t,z)fort>OandZεQ.IRthecasewherethevectoraeldattheboundarypointsinwardsisparticularlysimple.WealwaysobtainT(t)f(z)=f(ψ(t,z))forrεQaIKit〉0,i.e.,weobtainatranslationalongthetrajectorythroughz.
SemigroupsGeneratedbyDWKrentialOperators 505 InthecasewherethevectoriteldattheboundarypointsoutwardsthekeypropertyisthatifthetrajectoryofvstartingiIIZisoutsideofQthenthevalueofthesemigroupattimetisobtainedbyconsideringthevalueinsomeboundarypointsometimeago.Wedeanethetranslationpropertyasfollows(thenotationsarefromtheprevioussections). 3.1Deanition-Thesemigro叩(T(t))GohasthetmnSuttonpTope均if Jf(v(t,z))ift三s(z)T(t)f(z)={lT(t-s(z))f(σ(z))ift三s(z), (T) forallt主0,ZεQnU,fεX.InthecasewherethevectoraeldattheboundarypointsiIIWardsthetrajectoryoftpstartingiIIZwillneverleavethedomam-ForthissituatiOIlweformallysets(z)=∞,sowehavetoconsiderOII1ythe自rstlimeofthedefinitionabove.IfweassumethatontheBamchspaceC(Q)theclosureoftheErstorderdifemltialoperatorAf=(grαdf|F),togetherwithasuitabledomainD(A)generatesastr0吨lycontinuoussemigro叩(TO))t>09(forexampledomainsthatiMIMeVeMcelorNeumamboundarycOMitions),theIIthef扣OlHlOWimI且lholds. 3.2Proposition.Thesemigro叩T(t))Qohasthetranslationproperty(T). 3.3Remark.Thetranslationpropertyisindepmdentoftheb0112daryconditions. 3.2Thenon-compactnessproperty ThequestionwhetherastrOIlgiycontinuoussemigroupofoperatorsiscompact(eventuallycompact)isimportantforspectralandasymptoticthe-ory(e.g.see[Nal,p.87).LetuεθQ,anddeanezo:=ψ(-t-1,ν).Thenforeveryt三OthereisazoεQsuchthatt〈s(zo),andduetothetranslationproperty(T)wehave (TO)f)(zo)=f(v(t,20)). Wepasstotheadjointsemigro叩(TO)')t沪'andconstructabomdedse-quence{μn},μnεM(Q),suchthattheimage{T(t)Flin}hasmcO盯ergentsubsequence.Alongtheselines,onecanprovethattheadjoiatsemigroupis
506 Ulmer notcompact,thusthesemigro叩(T(t))Goisalsonotcompact-Sincetheconsiderationscanbedoneforallt三0,weobtainthat(T(t))t>oisnoteventuallycompact- 3.4Proposition.AssumethattheErstorderdiEKrentialoperatorAto-getherwithasuitabledomaingeneratesastronglycontinuoussemigrouponX. Thenthissen鸣ro叩isnot(eventually)compact. 54PerturbationofVentcelboundaryconditionsAcharacterizationofpositivity Throughoutthissectiontheframeworkofpisessential.ThusweassumethatthevectorEeldFonθQisorientedoutward.AspointofdepartureweuseTheorem2.1.Sinceouraimistoconsidervariousclassesofboundaryconditions,wetrytoperturbthemsuchthatwedonotlosethegeneratorproperty.In[Grlthisproblemissolvedinageneraisetting.UnfortuIlatelywecannotapplythistheorydirectlysincewedoknowonlyacoreofD(A). 4.1Theorem.LetX=C(Q),Y=C(θQ),φεζ(X,Y)andddmBtheoperatorAφonXbyAφf:=AfwithdomainD(Aφ):={fεC1(Q):Afho=φf}.Thentheclos盯eofAφgeneratesastronglycontinmmsemigro叩(Tφ(t))t>0· InthespecialcaseofVentceiboundaryconditionswehaveequalitybetweenthedomainsD(Aφ)={fεD(A):Af|δQ=φf}andD(Aφ)?ie.吨 D(Aφ)=D(Aφ). (2) Thisrelationisessentialfortheproofoftheabovetheorem.DuetoI-elation(2)itispossibletodescribeexactlytheclosureofAφ,thegeneratorofthesemigro叩(Tφ(t))t主0· 4.2Remark.ThedomainoftheoperatorAφisthesetofallfunctionsfεXsuchthatthederivativeoffalongthetrajectoryofψ(t,z)atUmet=OexistsforallZε0、isacontinuousfunction,andsatisaesAf|θQ=φf.
-SemigroupsGeneratedbyDWKrentialOperators 507 Byperturbingtheboundaryconditions,wemightlosethepositivityofthesemigroup(几(t))t主0·Wethereforelookfornecessaryandsumdentconditionsonφεζ(X,Y)suchthatthesemigro叩(Tφ(t))t主oispositive. LetusErstassumethatthesemigro叩(Tφ(t))t主oispositive.Weneedthefollowingtwolemmas. 4.3Lemma.LetAbethegeneratorofastronglycontinuoussemigrouponC(KLKcompactandA'theadjointoperatorofA.ThenforO三fεC(K),0三μεD(AOand〈f,μ>=0=〉<f,AFμ〉主0.FortheproofseeiNal,p133. 4.4Lemma.IfνisaregularboundedBOrelmeasureonQandzoεQsuchthatf(zo)=0,f主O=今ν(f)主O,then1/十c仇。isapositiveboundedmeasureforeveryC〉||ν||. SincetheDiracmeasure6zatZ巳θQbelongstoD(AUwecanapplyLemma4.3withμ=儿,andobtainthatZεθQ,fεC(Q)+,andfh)=0斗〈f,V儿〉主0.ForaaxedzoεθQaIIdν=守儿。,C主||φ||,Lemma4.4yieldsthatφ'6zo+c仇。isapositivemeasure.UsingtherepresentationofboundedlinearoperatorsonspacesofcontinuousfunctiOIlsdescribedin[D旬,p.490,weobtainthefollowingdecompositionfor||φ||: φ=一||φ||φ。+φ1, whereφ1主OaIIdφ。:C(Q)→C(θQ)istherestrictionmappiBgf→fho.Conversely,wecanverifyusingtheArendbCherIIoff-katoCharacteriza-tiOIITheorem(seelACKl),thatifφε￡(X,Y)isoftheformφ=kφ。+φ19forkε鼠,φ1主Oandφoistherestrictedmappingf→f|侃,mdtheclosureofAφgeneratesastronglycontinuoussemigrouponX,tllenthesemigroupispositive.Theseconsiderationsleadtothefollowingtheorem. 4.5Theorem.LetFεC1(Q,R勺,争εζ(C(Q),C(δQ)),anddd肘thearstorderdig-rentialoperatorAφonXbyAφf=(grdf|F)withdomainD(Aφ):={fεC1(Q):Af(z)=φf(z)forallZεθQ}-ThentheclosureofAφgeneratesastronglycontinuouspositivesemigrouponC(Q)ifaIIdonlyifφhastherepresentationφ=胁。+φ1withφ1主0,φ。:fHf|θQandkεEI.
508Ulmet 4.6Remark.TheonedimensionalcaseisinvestigatediniNaLp.133. Someexamples- 1)IftheperturbationφoftheVentcelboundaryCOILditionisoftheform (φf)(Z)=LKM withk:θQ×Q→Racontinuouskernel,thenthesemigro叩(T创刊)t>oispositiveifandonlyifk(2,ν)主0(φ。=0,φ1=φ).2)Forα:θQ→Q,andc(z)continuous,let (φf)(z)=c(z)f(α(z)). Ifα(z)=zforallzεθfl,theaboveperturbationleadstobouIIdt113't0日dLtionsofmixedtypeaIIEithecorrespondingsemigroupispositiveforarbitraryc(z).Ifα(z)并ZforsomeZεθQ,weobtainnonlocalbouMarycoaditiom,aIldthecorrespondingsemigroupispositiveifandonlyifc(z)三OforzεθQS旧IIthatα(z)并z.ThiskindofbomdarycOMitiomhavebeeII(Tailed"lateralconditions"byW-Feller,m[叫. Acknowledgment ITDishtothαnkPTofessOTG.GTeineTfoThissTL99estionsαηdpαttentguidαηCe. RefereIlees [ACKIAreMtW.,P.ChermtTKatqAgeMrah-zatioIIofdissipativityamlpositivesemigroups,JOperatorTheory8(1982),167-180.[AmiAmann,日,Gewd1111licheDiferentialddcflung℃凡WalterdeGI叮teIVerlag,Berlin-NewYork1983.[DSlDunford,N.,J.T.Schwartz,LinearOperators、PartLIIIterseie町CPub-lishersINC,New-York1967.[blFeller,W.,Gem-alizedsecodorderdiferentialoperatorsaMtlieu-lat-eralconditions,IllinoisJMath.1(1957),459-504.iGLIGoldstein?J.A.,C.Lin,Highlydegenerateparabolicboundaryvalueproblems、DiferentitllandIntegralEquations2(1989),216-227.
&migroupsGeneratedbyDWKrentialOperators 509 [GI-lGreim·,G.,Pert盯bi略tlIebomdarycorditiomofagem个ator,HomtOIIJ.Math.13(1987),213-229.[NalNageLR,(州,One-ParameterSmigroupsofPositiveOperators,Lect.NotesMath.1184,Springer-Verlag,Berlin-Heidelberg-NewYork-Tokyo1986.[TilTimmermam,C.A.,Semigro叩sofOperators?ApproximationandSat-urationinBanaclzSpaces,Ph-D-Thesis,DelftUniversityPress1987.iVelveMce1,A.D.,Onboundaryconditionsformultidimemomldifusionprocesses,TheoryProb.APPI-4(1959),164-177.
ANonhomogeneousDirichletProblemforaDelayDifferentialEquation PAOLAVERNOLEDepartmentofMathematics,UniversityofRome,"LaSapi-enza,"Rome,Italy lINTRODUCTION TheobjectofthispaperisthestudyofthesolutionofadifferentialparabolicequationwithadiscreteandacontinuousdelaywithanonhomogeneousDIRICHLETboundarycondition.LetQbeaboundedopensetofRn,withasmoothboundaryr.We considerasecondorderlineardifferentialoperator- nnAu(x)tEfij(x)uxiXj(川三bi(x)uxi(x)+c{x)u(x)(1)uniformlyellipticinQwithcoefficientsau,bi,cεcα(。);c(x)三OVXEO.Givenr,TpositivenumbersandAf,g,kfunctionsbelogingtosuitable Banachspaces, weareconcernedwiththefollowingparabolic problem: workdoneasamemberofG-N.A.F.A.ofC-N.R. 5II
5I2 VerFtole (t(tXMU(t)叫对中(σ)A(t叫+f())也(tx)ε[OTKOu(tA)=k(t,x)for(tA)ε[-r,OlxQ(2)u(tA)=g(t,x)for(tA)ε{-r,T]xrInpreviouspapersasimilarproblemwasconsideredwiththefunctiong(t,x)三ObySinestrui(1984)andVenlole(1989).ThehomogeneousDirichletboundaryconditionlettheauthorsusetheresultsofmaximalregularityforevolutionequationsconnectedwiththeinfinitesimalgeneratorofananalyticsemigroupinasuitableBanachspacee.g.C(Q).Inthiscasewecannotuse,directly,theseresults,becauseanellipticoperatorwithIIonhomogeneousboundaryconditioninthespaceofcontinuousfunctionsdoesnotgenerateasemigroupoflinearoperators-Webrief1ydescribenowthecontentofourpaper:insection2wegivethebasicassumptionsontheoperatorAtogetherwiththedefinitionsandtheresultsweneedlater.Section3isdevotedtothestudyoftheexistenceandtheurlicityofthe IlPRELIMINARIES LetEbeaBamchspacewithnorm川·||andA:DACE→Ealinearoperatorverifyingthefollowingassumption: (HA)Thereexistpε]7r/2,叫andM>Osuchthat ZA={zεc,z#0;|argzl三9}cp(A)and|AR(入,A)li(E)三M;ρ(A)istheresolventsetofAandR(入,A)=(九A)-1ItwasprovedbySinestrari(1985)that(evenifDA#E).Ageneratesan analyticsemigroupt→T(t)t>OandthereexistsMKsuchthatforeacht〉O ||tkAKT(t)||￡(E)三Mkandlim+T(t)x=xifandonlyifxEOA·t一+o-DAisaBanachspacewiththegraphnorm-wedenotebyDA(α)andDA忡,∞) forO<α<1thefollowingtwointermediatespacesbetweenDAandE: DA(α)={xεE:limtI-αAT(t)x=0}问t→oDA(α,∞)={xεE:||x||α=sup||tldAT(t)x||<∞}t〉ODACDA(α)CDA(α,∞)CEWesetCα/2,α([01]xO)=Cα/2([011,C(Q))nB([03];C气。))andhα/2,α([0,TlxO)=hα/2([0,Tl;C(0))nC([0,T];hα(0))
DinchlcfPfoblcmftJroDdawrIfqtddfIOFI 5l3 Inthenextsectionwewilluseatheoremoftimeregularity、theproofof-whichcanbefoundinSinestrari(1985)、whichwerecallforsemplicity THEOREMI LNlusconsidertheproblem (…)u叫(0创)=x tε[0,Tl (3) whereAverifiestheassumptionHA-Iffεcβ(0,T:E)(O<3<1),xEDA,Ax+f(0)ξDA(A∞),thenthefunction u叫(们川t付)=T(们例t吟)民x+(T申叫fη)(们t)whh阳阳叫e盯盯叫r陀叫e叫(叫t付巾叫)忡叶=斗jTm川川(υ仆h川tιM叫-4叫S咛圳忡)芦贝阳附f町巾(hSOistheuniqueUECl([0,Tl;E)nC([0,T];DA)whichverifies(3). 0<t<T (4) βMoreoveruεC([0,Tl;E)nC([0,T];DA)andu飞B(0,T;DAA∞)) Thefollowingestimateshold: ||u||c1+3(0,TZ)三Cl(T)[||f||cd+||Ax+f(0)||DA(A∞)+||X||](5)||u||CF(0,T,E)三C2(T)[||f||c3+||Ax+f(0)||DA(A∞)+||X||](6)||u'||B(0,T;DA(β,∞))三C3(T)[||f||cd+||Ax+f(0)||DAA∞)](7) Whenfξhβ([0,Ti;同,xξDA,Ax+f(0)εDAF),thestrictsolutionof(3)叫hl+α([0,TLE)nhα([0,TLDA)andu'εC([0,T];DA(F))· III-GLOBALSOLUTIONOFTHEDELAYEQUATION InordertousethemaximalregularityresultsoftheprevioussectionweemployawellFknowndeviceduetoBaiacKrishnan(1976)toreduceproblem(l)toahomogeneousboundaryvalueprobleII1.叭feconsiderthelinearoperatorD、suchthatz=Dgisthesolutionof theproblem
5I4 Ver-Ftole nu--)xt(zAi--JI--飞ln[,rrlxQ(8)z(t,x)=g(t,x)ln[-r,TlxrByVII-tlieofClassicaleSUITlabesforelliptiCequatlonSand interpolationtheoremsit'seasytoseethat DEl(C(F),C(。))ni(C2+α(F),c2+α(。))andDghasthesameregularityintasgandADg=0.Ifwesetv(tA)=u(ti)··Dg(tA),thenUisasolutionofproblem(1),ifandonlyifvisasolutionofthefollowingproblem: O飞川斗V(tA)+b(tM)+ja(σ)AV(…阳Htx)(Dg川) for(tl)ε[0,Tlxo(9)v(ti)=k(t,x)-Dg(t,x)for(ti)εLr,OlxQv(t,x)=0for(tA)ε[,rZlxFWemakethefollowingassumptions:either…..…-fεcufhu([0,TlxQ)l+α/2,2+α-KEC([-110]xO)(10)l+α/2,2+αgEC([-hTlxF)aELl(-r,0)or…,'…-fEhur…([0,TKQ)l+α/2,2+α-KEh([-r,TlxQ))噜EA'EAJ,,‘、l+α/2,2+αgEh([-hTKF)aεLI(-r,0) withthecompatibilityconditions
DiFjchleffhblem户raDdayEquation 5l5 g(t,x)=k(tl) for(tJOELr,OixF (12) gt(t,x)=kt(tA) for(tJOELr,Olxr Wewritetheabstractformofproblem(9)ov'(MmAV(tf)+ja(仙川阳仙(闹'(t)fortε[0 (13) v(t)=k(t)-Dg(t) fortε[-r,Ol whereE=C(Q)A:DACE→E DA={vεw23(Q)withp>hAvεC(Q)v/F=0}Av=AV ItwasprovedbyStewart(1974),thatchoosingAoasufficientlylargenumber,(A-An)t(A-λ。)satisfiesHAandSOgeneratesananalyticsemigroupeUnotstronglyCOIltinLlollsint=0. HoweverthesameistrueforanytranslationofA-Ao,inparticularforA;wehaveλot(A,λ。)tT(t)=et=Wehavethefollowingcharacterizationoftheintermediatespacesforβε]0,1[,2β233#l/2DAS)=hoandDA(β/2,∞)=Co(Q)seeLmardi(1985). NowweareabletoproveOUTmainresult. THEOREM2 Ifassumptions(10)and(12)hold,thenproblem(13)hasauniquestrictsolutionvεcl+α([0,Tl;E)ncα([0,TLDA)andv'EB(0,T;C02α(。))·Ifassumptions(11)and(12)hold,thentheuniquestrictsolutionof(13)VEh1+α([01l;E)nhα([0,Tl;DA)andv'εC([OJrl;hα(。))· PROOF Wesolvetheproblem(13)byastepbystepmethod.WeGrstsolvetheproblemintheinterval[OJLthen,usingthesolutionfhOu川Inlaωsanewinitialdatum,wesolvetheproblemintheintervallr,2r]andSOon. LetusfixtlE[OAsuchthat
5I6 Ver-nole p(t1)=C20)||A||||a||<112ll(DAE)Ll(-tl,0)whereC2(tl)istheconstantofestimate(6).WR-denotebyXthefollowingsubsetofCα/2(0,tl:DA)X={VECα/2(0,tl:DAKV(0)=k(0)-Dg(0)}(15)αI2Wecanassociatetoeach亏εXthefunctionvEC'(-r,tl;DA)dennedωfollows (14) 合田"σ。DφL&EW-vfl〈1、一一v tε{0,tll (16) tε[-r,ol ForeachVEXwedefine 臼(t)=叩)[k(0)问(0)]+jmm川-4叫s叫)川r忖)十H巾制S吟训川)川(+jT(们仆阳t仁M.4S什a叫巾(μσ)叫叫σds Fromtheassumptions(10)itfollowsth创 ORSJ4kh)+f(队)(DU(0,)+ja(σ)AV(机)dσ belongstoCα/2(0,tuE);k(0)-Dg(0)εDAMoreoverA[k(0,.)-Dg(0,.)]+F(Ov)εcα(。)aMbyvirtueofthe compatibilityconditiongt(OA)=kt(OA)ifxεr oAK川-ADg川材k(以)到州(圳0,对中(MW)dσ=oi.eA[k(0)-Dg(0)]+F(0)εDA(α/2、∞) Hencefromtheorem1wededucethatSVεXandproblem(13)hasasolution,ifandonlyifthereexistsVEXsuchthatSV=Vandinthiscase, alwaysusingtheoreml 、BE,,、‘.,J∞9"//α,,,.、AD唱,..+Lnu,,t飞Bζ」-VJuna、‘,,,E电·&φLAU''1、呵,ι,fα+Cε-V ThereforetocompletethePI-oofoftheErstpartoftheorem2is sufficienttoprovethatSisacontractiononX.to衍(问们)=jm-叫a(σ)A(叫叶σ附s
DirichlcrpmblemjiJraDdayt唱quafion 5I7 oSet川andaεILJl气(-rh,A0)thenlK(u叫)εCα([归0,Tlh;E)andi江fTε[归0,♂r叶l ||l(u)||<川A||[||a川1||u||cα(0,T,E)-i(DA,E)L(-IVO)Cα(-r,0,DA)+||al|||u||Ll(-T,0)Cα(0,T;DA)Soweget||SV-SW||<C(tl)||l(v-w)||x-2cα(OA1,E)一<C(t)||A||||!||忻-W||x=p(tl)||亨,击||x￡(DA,E)aLl(-tl,0)Thisimpliestheexistenceofasolutionof(13)intheinterval[0,tll.Iftl三rweconsiderthesameproblemwithinitialpointtlinsteadofOand,inthesameway,wecanprovetheexistenceofasolutionin[0,tl+t2lwitht2=min(tlJ-tl).Theproofofthetheoremisobtainedrepeatingthesameargumentuntiltl+与+t3+-H·H··>r.Oncewehaveasolutionofproblem(13)thesolutionUofourproblemisU(t,x)=v(t,x)+Dg(t,x). ToprovethesecondpartofthethesiswecanrepeatthepreviousproofsubstitutingC气DA(α,∞)andBbyhα'DA(α)andCrespectively. AKNOWLEDGMENT Theauthorthankstherefereeforusefulsuggestions.
5I8 VerFTole REFERENCES 1.A.V.BalaKrishnan:AppliedFunctionalAnalysis,SpringerVedag,NewYork(1976) 2.A.Lunardi:InterpolationSpacesbetweenDomainsofEllipticOperatorsandSpacesofContinuousFunctionswithApplicationstoNonlinearParabolicEquations,Math.NaChr.121(l985),295·318 3.A.Lunardi:MaximalSpaceRegularityinNOIlhomogeneousInitialBoundaryValueParabolicProblem,Numer-FUIlet-Anal.Optim.10(1989),323-349 4.E.Sinestrari:OnaclassofRetardedPartialDifferentialEquations,Math.Z186(1984L223·246 5.E.Sinestrari:OntheAbstractCauchyProblemofParabolicTypeinSpace8ofContinuousFunctions,J.Math.Ami-Appl.,66(1985),1666 6.H.B.Stewart:GenerationofAnalyticSemigroupbyStronglyEllipticO-peratorsunderGeneralBoundaryConditions,Trans.Amer.Math.Soc-259(1980),229-3107.P.Venlole:LinearDifferentialProblemwithDiscreteandDistributedDelayTerms,Boll.Un-Mat.Ital.3·B(1989)601·621.
LocalExistenceforaParabolicProblemwithFullyNonlinearBoundaryConditionArisinginNonlinearHeatConduction:AnI47·Approach P.WEIDEMAIERFacultyofMathematicsandPhysics,UniversityofBayreuth,Bay-reuth,Germany Wepresentthemainresultsofaforthcomingpaper(WEIDEMAIER(1990));theretheinterestedreadercanandalldetailsoftheproofsnotgivenhere,aswellasamoreextendedbibliography. Theorem1.Assume (Q)QCRnαboundedd0771αtnyδQεC2.(A)叫=αi(27tJ,p)εC3(百T×R×Rn)(1三i三n);Dpα(·)symmetmαηdpositityedeFTliteuniformluonQT×BforeveruboundedsetBinR×Rn;(Dpα:=(OPJαi)13,总n).(F)f=f(2,tJ,p)sdsFesthefollowingCbmthdodoTyconditions(cf.FUCIketd-f197刀/:VBαsin(A)3hB,LBεLAQT)Stichthdforα.ι(z,t)εQuV(uJ),(岳,户)εBijf(-,-AP)ismeαsurdleinQTiO|f(z,t,uJ)|三hB(凯t)iiO|f(2,t,uJ)-f(z,t,岳,P)|三LB(zj)·|(UJ)一(岳,P)|.(ψ)ψ=￠(Z?ku)εC2(51T×R).(φ)vεWJ(Q),p>n+2. Lfthecompαtibtlitycondition 5I9
520 Weidemaier (α(-,0,v(-),Vv(-)),ν(-))三￠(-,0,v(-))onθQ issdisjted,theproblemUt(zj)-diuzα(zj?u(zj),Vz叫z,t))=f(z,t,u(zj),Vzu(zj))(α(己,t,u(己,t),Vzu(已t)),ν(己))=ψ(己,t,u(己,t))u(0)=v 1ηQTonOflT111Q 、、..,/、J育,.,A,,EEE飞 fuoute俨normdtoOQjhαsαuniquelocdfintiTMjsolutioni11117·1(07):={uεLAQT)|δ;u,δttifdistr.smsejELp(OT)Vh|三2}?P>η+2. Example:BVP'softhistypearisenaturallyinnonlinearheat-conduction、wheretheusualFOURIER'slaw5=-K(d)-Vzdbetweenthegradientofthetemperaturedandtheheat-nuxvectorSisreplaced1337thenonlinearconstitutiverelation5=5(zj,d,Vzd)(cp.POTIER-FERRY(1981)). Remark2. (a)ThesolutiondescribedinTheorem1isalsogloballymimrandcoI卜tiI111011sdependenceontheinitialvaluesholds.(b)Let[0,T+[denotethemaximalintervalofexistence;ifthereisaSQ-quence(tn)↑T十suchthat(||u(·,tn)||wJ(fl))isbounded,thenT+=∞,i.e.globalexistenceholds. Problem(σP)(andamlhOgOmIp〉rmObMiemmSfOrSySt忱阳e臼ImIbyACQ1UjI巴STAPACE/TERRENI(l987)门)inmixedHG创lde臼Ir卜.飞-SOlb〉Olke肝VSIp〉aC俨何S(Oi.e.Uεcα([0,TLLV;(fl))andUrεCα([0,Tl,Lp(Q)),αε(0,钉,p〉η).Intheirtreatmenttheinitial-valuevnecessarilyhastosatisfyasecondCOIN-patibilitycondition,whichisabsentinmyapproach-MoreovertheirdiEP-rentiabilityassumptionsforf,ψaremorerestrictive-LUNARDI(1989)alsoconsideredproblemswithfullynonlinearb.c-ksheworkedinlittle-T16lder-Spaces.Mostpapers(AMANN(1985),(1986),GIAQUINTA/hIODICA气(1987))however,donotcoverthefdynonlinearcase,0吨7the〈lmsiii时arone.Theoremlisprovedbyconstructingaaxedpointi-orthemapS:1U叶飞dennedbythefollowingiinearproblem:ut一仇Jαyvi句ti=f(ω)一[θpAψ)-ajdw)WAω(Dpα(ψ)Vzu,ν)=47(w)+(lDpα(ψ)Vztu一α(ω)],ν)u(0)=v,whereθPJdω)(zj):=OPJ向(z,t,ω(z,t),Vzω(zj))j(ω)(zj):=f(zj,ω(z,t),Vzω(2,t)),wheref(2,t,ω(2,t)?Vzω(27t)):=
ParabolicProblemArisinginNonlinearHeatConduction 52I f(队t,ω(zj)Yzω(zj))+diυzα(zj,ω(zj)Yzw(zj))-aJα斗jr尸w叫)(μZ'jt忖)θtJ冯θ句Juω?θ仇p3αjψ)(己):=OPJdψ)(己,0),α(ω)(己,t):=α(己,tJ(己,t)Yzw(己,t)),ψ(ω)(己,t):=￠(己,仁tIJ(己,t).InordertoshowthatSisaself-mapofanappropriateset?weneedresultsof771αzimdregulαrituforthefollowinglinearproblemwithinhomogmmusconormalb.c.: Ut-ctj(2,t)aθju=F(zj)Cij(己,t)νj(己)θtu=G(已t)u(0)=ψ inQTonθQTinQ; suchresultswereprovedbySOLONNIKOV(1967)andGARRONI/SOLONNLKOV(1984):||忡 (1<p〈∞);however,inourcontext,twoadditionaldimcultiesoccur:FirstlyweneedSOLONNIKOV'sestimateforonlyminimalcontinuityof的,的一1-ii(1-i)thecoemcKIlts:ctjεczt(QT)「114=(θQT)forsomesmallα1,α2ε1-抖(1-3)一(0,1)insteadofctuεczt(QT),asinGARRONI/SOLONNIKOV(1984);itturnsoutthatthenecessarymOdincatiomcanbecarriedout,ifp>n+2.SecondlywehavetomakesurethattheconstantcoinSOLONNIKOV'sestimateisindependentofT(forsmallTLUnfortunatelythiswasnotguaranteedinSOLONNIKOV'soriginalversion(cf.SOLONNIKOV(1984))-ThisdimcultywasovercomeinWEIDEMAIER(1988).OnthebasisoftheIIIodinedSOLONNIKOV-estimateitistheneasytoapplyBANACH'snxed-pointtheoremintheset MZ(ψ):={UEWF1(QT),p〉n+2|u(0)=件||u-v||时'1(QT)三K}(forK=If(ψ)largeandT=T(If(ψ))small).Finallyweconsiderthecase,whentheellipticpartisinnon-divergenceform: udz,t)-ctj(z,Lu(2,t),Vzu(zj))δJju(z,t)=f(z,ku(zj),Vz叫zj))inQT(P)Cij(己,t,u(己,t),Vzu(己,t))η(cau(己,t)=ψ(己,t,u(己,t))onOOTu(0)=vinQ. HeretheresultofTheomII1(undertheanalogousassumptionsofthisTheo-rem)cannotbeexpected;comparethecounterexampleinDEURING(1989);however,undertheadditionalsmallnessassumption
522 Weidernaifer co-C4·.切怀||仇,cil(-,0,以-LVv(·))θlv(-)||悦,m〈1(0.1)i三t,J三n'(句fromSOLONNIKOV'sestimte,cltheembeddingconstantof143117,1(QT)1-t,3(1-t)HVztiεIYp(δQT)),weagaingettheresultofTheorem1.Remark3.Theoremlremainsvalidforalargeclassofparabolicsystems、sincethecorrespondingsharpLP-estimatesforthelinearproblemsareknown(cf.SOLONNIKOV(1965)).References [1]P.ACQUISTAPACEandB.TERRENI,"QuαsilinmrpGmbolicintegrodilTε陀ntidsustemsIiyithfullunonlineαTbouηdαruconditions",Proceedingsof"VolterraIntegrodiferentialEquationsinBanachSpacesandApplications",Trento1987,Pitman- [2lH.AMANN,J-rdMU.αng.MdlLy360(1985),pp.47-83.[3iI1.AMANN,A叫.Rd.Mech.Aml.,92(1986),pp.155192.HlP.DEURING,Boll.U-Af-I.3·BP向(1989),pp.955982.[5iS.FUCIK,J.NECASandV.SOUCEItEmfd-hruηgindieIU川-tioTIS陀cll111i719?Leipzig,Teubner,1977. [6lM.GARRONIandV.A.SOLONNIKOV,Comm.PDE,9flJj(1984),pp.13251372.[7lM.GIAQUINTAandG.MODICA,Ann.TnGt.pumαppl.,sεr.IV,1J9(1987),pp.41-59. [8lA.LUNARDLlVm们tdFttMtiomlATltzlpisαndOptimizdiorl10ω(1989),pp.323-349.[9lM.POTIER-FERRY,AM.Rd.Mech.AMl.77(1981),pp.301-321.[10lV.A.SOLONNIKOV,TrdyAfd.Inst.Stdl01770(1964),pp.133-212,(TmmlAMS65(1967),pp.31-137).[11lV.A.SOLONNIKOV,Proc.StdlouInst.凡LIth.83(1965),pp.1-184.[l2lP.WEIDEMAIER,Mαth.Z-199(1988),pp.589-604.[13lP.WEIDEMAIER,Locαleziste盯eforpαmbolicproblemswithfullu7107llinmrboundαryconditto71;αLfαpp70αch,Ann.mIGt.pttrααppl,,toappear1990.
Page523 IndexAAccrctivc,43AdditivefunctionalsofMarkovprocesses-4益主Algebraicmultiplicity,lP2Almostperiodicity,421Analyticsemigroup-193益。Asymptoticalmostperiodicfunctions,4229422Asymptoticbehaviour,1539441BBackwardheatequation,29重Banachfunctionspaces-382Bcurling-Denycriterion,35Bochncfstheorem,334Boundaryconditions,192Boundarycontrol,2重1Boundedimaginarypowers-4Q鱼,4Q9CCauchy-Dirichletinitial-boundaryvalueproblem,3579363Cauchy-NCumanninitial-boundaryvalueproblem,35193豆豆C-existencefamilies,295Characteristicequation,lmCompletecontraction,43948962965Completelyaccrdive-42950952952Completelyaccrctivcsubdifferentials-68Completelymonotonic,43Complexinterplationfunctor-122
ConditionF,263Conservativeboundaryconditions,313Corners,213Cosinefunctions,263Cracks,213C-semigroups-88929鱼,刀1DDelaydifferentialequations,511Differentiablesemigroups-21Diracfsequation,25992重lDirichletcontrol,214Dissipativcoperator,3479348Dissipativcquantummechanics,254Dualsemigroups-m2,m8EEntireC-semigroups-29重Evolutionoperator,69342Exactcontrollability-21292重工2109280FFeedbacksystem,272Fcynman-kacformula,口Q,口2Fitnessfunctions,11Fullynonlinearboundaryconditions,519GGeneralizedC-semigroups-341Generalizedsolutionsofordern,P79PB
Page524 HHilbertUniquenessMethod,2111Hillc-Yosidatheorem,371Hδldcrregularity,352Hyperbolicintcgrodifferentialequation,325IInitialboundaryvalueproblems,口7Integratedsemigroups-29911091169112914125993529312Intcgrodifferentialequation,119322Interpolationofsemigroups-30JJohn-Nircnberginequalityformartingale,4豆豆KKirchoffplates-2鱼?7k-timesintegratedsemigroup-3930931932939LLangcvinalgorithm,1豆豆Laplacetransform,?Linearelasticityoperator,149Linearquadraticproblem,2Lipschitzkernels,UIRfy714ccrcti、吃439145Maximalmonotone,18Maximalregularity,122m-completelyaccrcti、吃4394495益,58959965Mixedhyperbolic-parabolicproblem,23Multiplicationsemigroups-19982985
此411ltiplicativcperturbation,44lMultipliertheorem,组4h411tationandselection,77NN-contractions,46NonautonomousRiccatiequation,2Nonhomogeneousboundaryconditions,227Nonlinearergodictheorem,422nth-orderdifferentialequations,39lNormalBanachspaces-4495496997QO Order-continuousnorm,384Order-preservingcontractions,42942 PParticulcmultiplyingboundaryconditions,321Pauli-typeHamiltonians,4豆豆Phasespace-沙1Populationgenetics,11Potentialwell,1559152Power-bounded,95 QQuasilinearparabolicequations,482QuasilinearparabolicVolterraequations,4Q1RRamifiedspaces-111Realinterpolationspaces-12重Regularizingsymmetricsubmarkoviansemigroups-91Retardedfunctionaldifferentialequation,lP1
R-semigroups-372SSchI-6dingerequation,30,3593292439245SchI-6dingeroperators,14891"SemigroupsinFItchctspaces-12Semipermeableinterface,21SimulatedannealiI毡,些重,口重Singularkernel,322Singularsolutions,219Smoothdistribution(semi)-group-30,H3Sobolevimbeddings,29935Sobolev-Slobodeckiispaces-12fiSolonnikovfstechnique,484Spectraldistributions,143,H49H5StochasticLyapunovtheorem,l重!1Stonefstheorem,146Streamingoperators,311Strongposmvctype,32793299Plsubdifferential-16Sun-reflexive,322Symmetrichyperbolicsystems,2重QSymmetricsubmarkoviansemigroups-92 TTopologicalnetworks,20921Transferfunction,452Transmissionproblems,15Transportoperators,14重
Page525 UUltracontractivebounds,1709171Unconditionalbasis,389UniformLegendrecondition,154Uniformstabilization,272VVariationalinequalities,20Variationalsecond-orderparabolicequations,351Variationofconstantsformula,10891099119V二coercive-17Viscoelasticity,32鱼,4Q2明fWaveequation,214Weaklyalmostperiodicfunctions,422Weaklycompactlygenerated,325认rentzclboundarycondition,1211229主Q2Widdcrfstheorem,2Q1Wienermeasure,167