Die Grundlehren der
mathematischen Wissenschaften in Einzeldarstellungen
Band 182
J. L. Lions E. Magenes
Non-Homogeneous
Boundary Value Problems
and Applications
II

Die Grundlehren der
mathematischen Wissenschaften
in Einzeldarstellungen
mit besonderer Berucksichtigung
der Anwendungsgebiete
Band 182
Herausgegeben von
J. L. Doob • A. Grothendieck • E. Heinz • F. Hirzebruch
E. Hopf • W. Maak • S. MacLane • W. Magnus -J. K. Moser
M. M. Postnikov . F. K. Schmidt. D. S. Scott. K. Stein
Geschdftsfiihrende Herausgeber
B. Eckmann und B. L. van der Waerden

J. L. Lions • E.Magenes
Non-Homogeneous
Boundary Value Problems
and Applications
Translated from the French by
P. Kenneth
Volume II
Springer-Verlag Berlin Heidelberg New York 1972

J. L. Lions E. Magenes
University of Paris University of Pavia
Title of the French Original Edition:
Problemes aux limites non homogenes et applications (tome II)
Publisher: S. A. Dunod, Paris 1968
Translator:
P. Kenneth
Paris
Geschaftsfiihrende Herausgeber:
B.Eckmann
Eidgenossische Technische Hochschule Zurich
B. L. van der Waerden
Mathematisches Institut der Universitat Zurich
AMS Subject Classifications (1970)
Primary 35J20,35J25, 35J30, 35J35, 35J40, 35K20,35K35,35L20,
Secondary 46E35
ISBN 3-540-05444-8 Springer-Verlag Berlin Heidelberg New York
ISBN 0-387-05444-8 Springer-Verlag New York Heidelberg Berlin
This work is subject to copyright. All rights are reserved, whether the whole or part of the material
is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction
by photocopying machine or similar means, and storage in data banks. Under § 54 of the German
Copyright Law where copies are made for other than private use, a fee is payable to the publisher, the amount
of the fee to be determined by agreement with the publisher.
© by Springer-Verlag, Berlin • Heidelberg 1972. Printed in Germany.
Library of Congress Catalog Card Number 71-151407

Introduction
1. In this second volume, we continue at first the study of non-
homogeneous boundary value problems for particular classes of
evolution equations.
In Chapter 41, we study parabolic operators by the method of
Agranovitch-Vishik [1]; this is step (i) (Introduction to Volume I,
Section 4), i.e. the study of regularity. The next steps: (ii) transposition,
(iii) interpolation, are similar in principle to those of Chapter 2, but
involve rather considerable additional technical difficulties.
In Chapter 5, we study hyperbolic operators or operators well-
defined in thesense of Petrowski or Schroedinger. Our regularity results
(step (i)) seem to be new. Steps (ii) and (iii) are analogous to those of
the parabolic case, except for certain technical differences.
In Chapter 6, the results of Chapters 4 and 5 are applied to the
study of optimal control problems for systems governed by evolution
equations, when the control appears in the boundary conditions (so that
non-homogeneous boundary value problems are the basic tool of this
theory). Another type of application, to the characterization of "air'
well-posed problems for the operators in question, is given in the
Appendix. Still other applications, for example to numerical analysis, will
be given in Volume 3.
2. For the same systems {P, Qj), Volume 3 will proceed in an
analogous way, but starting from regularity results in real analytic or
Gevrey classes. In this way, we shall be able to reach the case in which
the data /, gj are Gevrey functionals or analytic functionals.
3. The organization of the chapters is the same as for Volume 1.
In particular, the open problems are assembled in the last section of
each chapter.
J. L. Lions E. Magenes
1 The numbering of the chapters follows that of Volume I.

Contents
Chapter 4
Parabolic Evolution Operators. Hilbert Theory
1. Notation and Hypotheses. First Regularity Theorem 1
1.1 Notation 1
1.2 Statement of the Problems 3
1.3 (Formal) Green's Formulas 3
1.4 First Existence and Uniqueness Theorem (Statement) 5
1.5 Orientation 5
2. The Spaces Hr's(Q). Trace Theorems. Compatibility Relations. ... 6
2.1 Hr*s-Spaces 6
2.2 First Trace Theorem 9
2.3 Local Compatibility Relations 10
2.4 Global Compatibility Relations for a Particular Case 12
2.5 General Compatibility Relations 17
3. Evolution Equations and the Laplace Transform 20
3.1 Vector Distribution Solutions 20
3.2 Z.2-Solutions 22
4. The Case of Operators Independent of t 24
4.1 Hypotheses 24
4.2 Basic Inequalities 25
4.3 Solution of the Problem 27
5. Regularity 28
5.1 Preliminaries 28
5.2 Basic Inequalities 29
5.3 An Abstract Result 30
5.4 Solution of the Boundary Value Problem 32
6. Case of Time-Dependent Operators. Existence of Solutions in the Spaces
H2rm-m(Q), Realr^l 33
6.1 Hypotheses. Statement of the Result 33
6.2 Local Result in t 34
6.3 Proof of Theorem 6.1 36
6.4 Regular Non-Homogeneous Problems ....'. 36
7. Adjoint Isomorphism of Order r 39
7.1 The Adjoint Problem 39
7.2 Adjoint.Isomorphism of Order r 40

Contents VII
8. Transposition of the Adjoint Isomorphism of Order r. (I): Generalities 40
8.1 Transposition 40
8.2 Orientation 41
8.3 The Spaces #-*•-£ (Q), #-*•-£ (27), <x, 0 ^0 41
8.4 (Formal) Choice of L 42
9. Choice of /. The Spaces E2rm*r(Q) 43
9.1 The Space S2rm'r{Q) 43
9.2 The Space E-2rm>-r{Q) 43
9.3 Choice of /. The Space D^r~Vi(Q) 45
10. Trace Theorems for the Spaces Dp{r-l){Q), r ^ 1 46
10.1 Density Theorem 46
10.2 Trace Theorem on U 49
10.3 Continuity of the Trace on Surfaces Neighbouring £ 53
10.4 Trace Theorem on Q0 55
10.5 Continuity of the Trace on Sections Neighbouring Q 57
11. Choice of gj and u0. The Spaces H2<xm E* (27) 57
11.1 The Spaces H2«m S«{Z) 57
11.2 Choice of gj 59
11.3 Choice of u0 59
12. Transposition of the Adjoint Isomorphism of Order r. (II): Results;
Existence of Solutions in H2mr*r(Q)-Spaces, Real r <: 0 59
12.1 Final Choice of L 59
12.2 Results 60
12.3 Complements 62
13. State of the Problem. Complements on the Transposition of the Adjoint
Isomorphism of Order 1 65
13.1 State of the Problem 65
13.2 Complements on the Transposition of the Adjoint Isomorphism
of Order 1 66
13-3 Orientation 67
14. Some Interpolation Theorems 68
14.1 Notation. Statement of the Main Result 68
14.2 Outline of the Proof 69
14.3 First Auxiliary Interpolation Theorem 70
14.4 Second Auxiliary Interpolation Theorem 71
14.5 Third Auxiliary Interpolation Theorem 75
14.6 Proof of Theorem 14.1 78
15. Final Results; Existence of Solutions in the Spaces H2mr*r (Q), 0 < r < 1.
Applications 78
15.1 Application of the Results of Section 14 78
15.2 Examples; Generalities 80
15.3 Examples (I) 81
15.4 Examples (II) 82
15.5 Some Complements on the Dirichlet Problem 83
16. Comments 85
17. Problems 87

VIII
Contents
Chapter 5
Hyperbolic Evolution Operators, of Petrowski and of Schroedinger. Hilbert Theory
1. Application of the Results of Chapter 3 and General Remarks ... 91
1.1 Notation. Hypotheses 91
1.2 Application of the Results of Chapter 3 93
1.3 A Counter-Example 93
2. A Regularity Theorem (I) 95
3. Regular Non-Homogeneous Problems 99
3.1 Statement of the Problem 99
3.2 The Compatibility Relations 100
3.3 The Case of the Dirichlet Problem 104
4. Transposition 105
4.1 Adjoint Isomorphism 105
4.2 Transposition 106
4.3 Choice of L 107
4.4 Conclusion 107
5. Interpolation 108
5.1 Statement of the Problem 108
5.2 Some Interpolation Results 109
5.3 Consequences 114
5.4 The Case of the Dirichlet Problem 115
6. Applications and Examples 117
■ 6.1 General Results 117
6.2 Examples 119
7. Regularity Theorem (II) 122
7.1 Statement 122
7.2 Proof of Theorem 7.1 123
8. Non-Integer Order Regularity Theorem 127
8.1 Orientation 127
8.2 Interpolation in y 127
8.3 Interpretation of the Space v<2r-»m'2r (Q), r^l 128
9. Adjoint Isomorphism of Order y and Transposition 129
9.1 Adjoint Isomorphism of Order y 129
9-2 Transposition 130
9.3 Formal Choice of L 131
10. Choice of /, g , u0, ux 130
10.1 Choice of / 131
10.2 The Space DaTd^ (0) 131
10.3 Choice of gj 131
10.4 Choice of %, % 132
10.5 Conclusion 132

Contents IX
11. Trace Theorems in the Space DA+iyi. (Q) 133
11.1 Density Theorem 133
11.2 Traces on U 137
11.3 Continuity of the Trace on Neighbouring Surfaces 140
11.4 Traces on ^0 140
11.5 Continuity of the Trace on Sections Neighbouring Q0 .... 141
11.6 Remark 142
12. Schroedinger Type Equations 142
12.1 Notation 142
12.2 First Regularity Theorem. Parabolic Regularization 143
12.3 Second Regularity Theorem 145
12.4 r-Isomorphism Theorem 148
12.5 Choice of L 149
12.6 Trace Theorem 149
13. Comments 150
14. Problems 152
Chapter 6
Applications to Optimal Control Problems
1.. Statement of the Problems for the Linear Parabolic Case 157
1.1 Notation 157
1.2 Optimization Problems 159
2. Choice of the Norms in the Cost Function 160
2.1 Reminder. Condition on ^! (Q) 160
2.2 Space Described by Sy. Conditions on K2{Z) 161
2.3 Space Described by y(x, T; u). Condition on K3(Q) 162
3. Optimality Condition for Quadratic Cost Functions 163
3.1 Notation 163
3.2 Optimality Condition 163
4. Optimality Condition and Green's Formula 164
4.1 Optimality Condition. Application of Section 3.2 164
4.2 The Isomorphisms At 165
4.3 The "Adjoint" Problem 165
4.4 New Form of the Optimality Condition 166
5. The Particular Case // = m -f ^, E3 4= 0 168
5.1 Properties of y 168
5.2 Choice of Kx (Q) 168
5.3 Choice of K2 (U) and K3 (Q) 168
5.4 Adjoint Problem and Optimality Condition 168
6. Consequences of the Optimality Condition (I) 169
6.1 Generalities 169
6.2 Consequences of Theorem 6.1 169
7. Consequences of the Optimality Condition (II) 171
7.1 Additional Hypotheses 171
7.2 Optimality Condition 171

X Contents
8. Complements on the Choice of the Spaces Kt 173
8.1 Orientation 173
8.2 Choice of Kx (Q) 173
8.3 Choice of K2(Z) 173
8.4 Choice of K3 (Q) 174
9. Examples 174
10. Non-Parabolic Cases. Statement of the Problems. Generalities . . . .178
10.1 Notation 178
10.2 Cost Function 179
10.3 Optimality Condition (I) 180
10.4 Adjoint Problem 181
10.5 Green's Formula 182
10.6 Optimality Condition (II) 183
10.7 Consequences 183
11. Applications. Examples 183
11.1 Control in the Boundary Conditions 183
11.2 Choice of Kx 183
11.3 Choice of K2 , 185
11.4 Examples 189
12. Comments 191
13. Problems 192
Appendix
Boundary Value Problems and Operator Extensions
1. Statement of the Problem. Well-Posed Spaces 193
1.1 Notation 193
1.2 First Condition for U to be Well-Posed 194
1.3 The Space D0 195
2. Abstract Boundary Conditions 196
2.1 Boundary Spaces and Operators 196
2.2 Characterization of Well-Posed Spaces 196
3. Example 1. Elliptic Operators 198
3.1 Notation 198
3.2 The Boundary Operators and Spaces 199
3.3 Consequences 199
3.4 Various Remarks 200
4. Example 2. Parabolic Operators 202
4.1 Notation 202
4.2 The Boundary Operators and Spaces 203
4.3 Consequences 205
5. Example 3. Evolution Operators of the Second Order in / 205
5.1 Notation 205
5.2 Formal Results 206
6. Comments and Problems 206
Bibliography 208

Contents of Volume I
(Published 1972)
Chapter 1 Hilbert Theory of Trace and Interpolation Spaces
Chapter 2 Elliptic Operators. Hilbert Theory
Chapter 3 Variational Evolution Equations
Contents of Volume III
(In preparation)
Chapter 7 Scalar and Vector Ultra-Distributions
Chapter 8 Elliptic Boundary Value Problems in Spaces of Distributions and
Ultra-Distributions
Chapter 9 Evolution Equations in Spaces of Distributions and
Ultra-Distributions
Chapter 10 Parabolic Boundary Value Problems in Spaces of Ultra-Distributions
Chapter 11 Evolution Equations of the Second Order in t and Schroedinger Type
Equations
. Appendix Calculus of Variations in Gevrey-Type Spaces.

Chapter 4
Parabolic Evolution Operators. Hilbert Theory
In this chapter we develop the theory of boundary value problems
for partial differential equations of the parabolic type; this theory
is in a way the analogue of the non-variational elliptic theory developed
in Sections 1 — 8, of Chapter 21.
We shall not go back over the concrete applications of the results
of Chapter 3 to parabolic equations, for which we have already given
examples (see Section 5.2, Chapter 3). This chapter (except for
Section 17.2) does not require the knowledge of Chapter 3.
1. Notation and Hypotheses. First Regularity Theorem
1.1 Notation
Let Q be an open set in R"; we assume Q to be bounded and to have
an (n — 1) dimensional, infinitely differentiable boundary r such that
Q is locally on only one side of r (see Chapter 1, Section 7).
Remark 1.1. The following results can be extended to the case where
Q is not bounded but where r has a "suitable" behavior at infinity.
We have not tried to extend what is to follow (as was not done in
Chapters 2 and 3 either) to the case r = [j TJ} where the dimension ns
j
of .Tj may be strictly less than (n — 1); see Problem 17.7. D
Let T < oo and
Q = Qx]0,T[,
S = r x ]0, T[, lateral boundary of Q
Q0 = Q x {t = 0}
QT = Q x {t = T}.
1 The references are given according to chapter; thus, for the first three
chapters we refer to Volume I.

2 1. Notation and Hypotheses. First Regularity Theorem
In Q, we consider the operator
(i.i)
where A is given by
d \ d d
P = A [x,t, + = A +
1 dx dt dt
(1.2)
A<p= I {-\)^Dl{apq{xyt)Dl9)i
and Dpx is defined in Chapter 2, Section 1.
The coefficients apq satisfy
(1.3)
«*e#(C). □
Remark 1.2. The differentiability hypotheses (1.3) may be weakened
in each of the following statements. However, in each statement, the
"optimal" differentiability hypotheses are unknown; in fact, we recall
— see Chapter 2, Remark 8.2 — that the systematic application of
interpolation theory cannot yield optimal differentiability hypotheses. D
The boundary operators
We denote by Bj, 0 ^ / ^ m — \,m boundary operators defined by
(1.4) Bj(p= £ bJhDhxip,
\h\Zmj
where the functions bJh — bJh(x,t) satisfy
(1.5) bJkeS{£)
(Remark 1.2 applies to the. regularity hypotheses on bJh).
We assume that
[ 0 ^ ntj = order of Bj ^ 2m — 1;
(1.6)
for each t0 e [0, T], the system \Bj I x} t0,
dx
j=o
is normal on r (see Chapter 2, Section 1.4)
and that (this is the fundamental hypothesis):
for every 6 e
A lx,t09-
[71 7l~\
-T'TJ
and Vi0 6 (0, T\
+ (-1)- et9D2ym,BAx,t0,--
OX J \ OX
is a "regular elliptic'' system in D x Ry; in other words, for
, — and every t0 e [0, T], the operator

(1.7)
1.3 (Formal) Green's Formulas
d
A [x,t0,
dx
id n2m
+ (-lJ-e^Z);
is properly elliptic in D x Ry and the system of boundary
operators
I m-l
(which, according to hypothesis (1.6), is normal on r x Ry)
covers the operator
d
Alx, t0f-
dx
+ (-\)meleD;m on rxRy.
In (1.7) the proper ellipticity and covering hypotheses are to be
understood in the sense of Chapter 2, Section 1 (in an obvious formulation for
the case in question).
Hypotheses (1.6) and (1.7) are very general; for instance, we see
immediately that if for every t0 e [0, T] the operator A\x,t0, 1
dJ
dx
is strongly elliptic in D and if Bj = r (derivate of order / along the
dvJ
normal v to E), j = 0, . . ., m — 1 (which corresponds to the conditions
of Dirichlet on Z), then (1.6) and (1.7) hold. D
1.2 Statement of the Problems
In this chapter we shall study — always under the hypotheses of
Section 1.1 — the problems (mixed in the sense of J. Hadamard [1]):
(1.8)
Pu = Au + u'=f in Q \u' = ),
(1.9) BjU = gj on Z, 0 ^ / ^ m - 1,
(1.10) u(x,0)=uo on Q0)
where the "functions" /, gd and u0 belong to various Hilbert classes. D
1.3 (Formal) Green's Formulas
The following Green's formulas should be compared to those in
Chapter 2, Section 2.
The functions in the following formulas are always assumed to be
"sufficiently regular"; one of the problems of the present chapter is in
fact to justify these formulas under "weak" hypotheses. D

4 1. Notation and Hypotheses. First Regularity Theorem
{Bj}™^1 is completed by a system {Sj}]1^ — which is not unique —
in such a way as to make the system {Bj} Sj]J~^ "normal" and "of
Dirichlet" on r for each t (Chapter 2, Section 1). Then there exist 2m
boundary operators CJt TJt j = 0, . . ., m — 1, with the following
properties:
the coefficients of Cjt Tj belong to @(E),
order Cj + order Sj = order B) + order Tj = 2 m — 1,
m—1 m—1
(1.11) (A9,V) -(<p,A*y>) = £ Sj<pCjVdZ-Z \Bj9T-jVdi:
where q> and y> are "regular" functions in Q, which, for example, belong
to 3>{Q), where ( , ) denotes the scalar product in L2(Q) and where
A*y = E (-l)WB*(a„(x,f)DlV).
|p|.|4|£m
The adjoint (in the sense of distributions on Q) of P is
d
dt t
P* = A*
We have
(1.12)
m-1
(P <p, v) - (<p, P* v) = £ f Sj 9) Cj ipdE -
m-1
- Tj \ Bj(pTjipd U + <p(xtT)ip(x,T) dx —
1 = 0 J •*
- f <p{%>0)y>(x,0) dx. U
Do
In order to simplify the notation, we set:
and we write (1.11) and (1.12) respectively in the form:
(1.11a) (A <p,y) - (<p,A*y>) = (S <p,Cf)z ~ (B(p,Tip)z,
(1.12a) (P <p, y) - (<p, P* W) = ($<p,€ v)z -(E<p,T y)s +
+ (<P(T),v(T))a-(<p{0),v(0))o,
where the index E (resp. Q) on the parentheses means integration on E
(resp. Q) and where for example cp (T) denotes the function
x -> <p(x, T). D

1.5 Orientation 5
1.4 First Existence and Uniqueness Theorem (Statement)
Our first aim in this chapter is to prove the following theorem (the
proof, as well as some additional results, are given in Sections 2 through 5
below):
Theorem 1.1. Let the hypotheses of Section 1.1 be satisfied. Let f be
given in L2(Q) = H° (Q). Then there exists one and only one function u
such that
( du
(1.13) Pu = Au + u'=f in Q \u' =
\ dt
(1.14) BjU = 0, Ogj^w-1,
(1.15) u(x,0) =0,
(1.16) ueL2(0,T;H2m(Q)),
(1.17) u' eL2(0, T;H°(Q)),
(we summarize (1.16), (1.17); by: u e H2mtl (Q) see Section! below).
Remark 1.3. In comparison with problem (1.9), (1.10), we have taken
the functions gj and u0 to be zero. Later on, we shall consider the case
of non-zero gd and u0. D
Remark 1.4. The space L2(0, T; H2m(i2)) (defined as a particular
case of L2(0, T; X), see Chapter 1, Section 1) coincides with the space
of functions v such that
veL2(0,T;H°(Q)) = L2(Q) = H°(Q),
DpxveH°(Q) for \p\£2m. U
du
Remark 1.5. In (1.7), u' = is defined as in Proposition 1.1,
Chapter 1. □ dt
Remark 1.6. Theorem 1.1 may be stated in the form of an isomorphism
between two Hilbert spaces. We shall do this, in a more general setting,
in Section 6. In any case, there exists a constant c such that
llWllL2(0,r;H2»»(a)) + II U' IIL2(0,T;HO(Q)) ^ C ||/||l2(Q). E
1.5 Orientation
The proof oi Theorem 1.1 (with numerous additional results) will
be given in Sections 3, 4 and 5.
For the sake of convenience we start with a brief study of certain
"Sobolev type" spaces which have a more complicated structure than
those of Chapter 1; this is the aim of Section 2.

6 2. The Spaces Hr*s(Q). Trace Theorems. Compatibility Relations
In Section 3 we shall make some general remarks about the Laplace
transform and evolution equations; these remarks may be useful in
other contexts as well.
In Section 4 we shall examine the case where A has coefficients
independent of t. In Section 5 we shall obtain a regularity result via
the method of Section 4. Finally in Section 6 we shall examine the general
case of coefficients which depend on x and t.
2. The Spaces Hrs(Q). Trace Theorems. Compatibility Relations
2.1 #rs-Spaces
Let r and s be two non-negative real numbers. For Q an open set
in R", we define:
(2.1) Hr's(Q)=Ho(0,T;Hr(Q))nHs(0,T',H°(Q)), (Q = Qx]0,T[),
which is a Hilbert space with the norm
ij\\u(t) \\HriQ) dt + ||
In (2.1), H°{QtT')Hr{Q)) = L2{QtT')Hr{Qj)) where Hr(Q) is
defined in Chapter 1, Section 9, and Hs(0, T; H° (Q)) is defined in
the same way as Hs (0, T)\ see Chapter 1, Section 9; for example, if X
is a Hilbert space,
Hs(0,T]X) = [^w(0,r;Z),^°(0,r;Z)]e, (1 -d)m = s,
integer m > s, and
Hm(0,T;X) = {v\v,v',. . ., y(w) eL2(0, T;X)}. D
Remark 2.1. The space Hr,s(Q) is the space of restrictions to Q = Q x
x]0,T[ of functions in Hr's(QxR); of course, according to (2.1):
Hr's(Q x R) = #°(R; Hr(Q)) n HS(R; H°(Q))
and this time we may define #S(R; H° (Q)) by a Fourier transform in t:
veHs(R; ff°(fi))o (1 + |r|2)s/2 veL2(Rx; H°(Q)) = L2(Qx RT)
if v(r) = Fourier transform in t of v(t). D
Remark 2.2. We have assumed that Q is a bounded open set in R",
whose boundary r is an (n — 1)-dimensional, infinitely differentiate
variety such that Q is locally on only one side of r (see Section 1.1).
J./Z

2.1 H r's-Spaces 7
Then
(2.2) Hr's(Q) = space of restrictions to Q of functions
in Hr's(RnxxRt)
and when i2 and ]0, T[ are replaced by R" and Rt, definition (2.1) is
equivalent to (see Chapter 1, Section 13.2)
(2.3) H'-*(RZ x R,) = {v\ [(1 + HI2)"2 + (1 + |t|2)"2] veL2(Rn^ x R,)}
where v (£, t) = Fourier transform in x and t of v(x, t).
Under these hypotheses, @(Q) is dense in Hr,s{Q) (proof analogous
to the density theorems of Chapter 1, Sections 8 and 9). D
An interpolation property:
Proposition 2.1. For r, s ^ 0 and 0 e]0, l[, we have
(2.4) [Hr's(Q)f L2md = H"-9™1-'*®).
More generally, .if r, s, q , a ^ 0, we have:
(2.5) [Hr's(Q)f HQ>a{Q)-]Q = Hil'eyr+eQA1''e^+ea(Q).
Proof. We restrict the proof to (2.4); the proof of (2.5) follows the
same principle.
1) We shaU use Chapter 1, Section 13.1. We note: L2(0,T,Hr(Q))
is the domain in L2 (Q) of a positive self-adjoint operator R (with norm
equivalent to the norm of the graph) and Hs(0, T\ H° (£})) is the domain
in L2(Q) of a positive self-adjoint operator S — and the operators R
and S commute (one is an operator in x, independent of t and the other
an operator in t, independent of x). Then
H"°(Q)=D(R)nD(S)
and, according to Chapter 1, Section 13.1, we therefore have
[Hr-s(Q),H°(Q)le = XnY,
where
X = [L2(0, T; Hr{Q)) = D (R), H° (0]9
and
Y = [ff'(0, T; H°(Q)) = D (S), H° (0],.
2) Then (2.4) follows from the two identities:
(2.6) [L2(0, T; H'{Q)), L2(0, T; H°(Q)J]e = L2(0, T; H«-e»(Q))

8 2. The Spaces Hr's(Q). Trace Theorems. Compatibility Relations
and
(2.7) [Hs(0, T; H° {£))), L2(0, T\ H°(Q)J\e = ^"^(O, T\ H°(£})).
Identity (2.6) is a particular case of the general identity (C0 and Cx
are Hilbert spaces):
[L2(0, T; C0), L2(0, T; CJ], = L2(0,T; [C0, Cx]a)
(see Chapter 1, Section 14.2)
and of
[Hr(Q),H°(Q)]e = H«-<»r(Q)
(Chapter 1, Section 9).
Identity (2.7) is the analogue (with values in a Hilbert space) of
Chapter 1, Section 9. □
Some subspaces of Hr,s(Q).
We shall write:
(2.8) H£(Q) = L2(0, T; Hr0(i2)) n H'(0, T; H°(Q)):
and it can be verified that this space coincides with the closure in Hr,s (Q)
of the subspace of functions which vanish in a (variable) neighborhood
of Z = Tx]0,T[;
(2.9) Hr;0s(Q) =L2(0,T;If (£})) nHso(0,T; H° (Q)),
which is the closure in HTtS (Q) of the subspace of functions which vanish
in the neighborhood of t = 0 and of t = T\
(2.10) m>M = f*:(q) nirjs(Q),
which is the closure of ®(Q) in Hr's (Q). D
The space Hr>s{Z).
Definition (2.1) extends without difficulty to the case where Q is
replaced by its boundary 27; it suffices to use the spaces Hr(T) instead
of Hr (Q) (see Chapter 1, Section 7); we obtain:
(2.11) iT's(27) = H°(0, T) Hr(T)) n Hs(0, T\ H0(T)).
A result analogous to Proposition 2.1 holds for these spaces. We
recall that we have already introduced these spaces in Chapter 1,
Section 13.3 with a definition equivalent to (2.11).
Since Hr0 (T) = Hr(T) (Q) (T) is dense in Hr{F)), there is no distinction
between #£s(27) and #r'5(27); but we may set (by analogy with (2.9)):
(2.12) Hr;*(Z) = H°(0, T\ Hr(T)) n H'o(0, T\ H°(r)). U

2.2 First Trace Theorem
2.2 First Trace Theorem
Theorem 2.1. For u eHr,s(Q) with r > \y s ^ 0, we may define:
dJu j 1
— on 2 if 7 < r (integeY j ^ 0),
dvJ 2
(2.13)
wheYe
(2.14)
5J^
~s7
e#""v'(i;)
3 normal derivative on Z, \
dv oriented toward the interior of Q)
Y
r - / - i
(Vj = 0 if s = 0),
dJu
dvJ
are continuous linear mappings of
Hr's(Q) -#"""' (I)al)\
We may also define, joy s > \y y ^ 0:
dku
■ (x, 0) ow i2 # & < s
(2.15)
wheYe
(2.16)
5**
(integeY k ^ 0),
3*^
S*J
^>fc = I S — k ■
ik (x,0)eHp*(i2),
dku
(x,0) aYe continuous mappings of Hr*s(Q)
Before proving the theorem, let us point out that the mapping
\dJu dku } 1 1
(2-17) H^'^~(*'°r /<*-T.*<'-T.
of
;T'S(<2) - nHvv>{Z) x IIHPk{i2)
j k
is not suYJective (see the sections immediately following this one).
Pyooj of TheoYem 2.1. 1) Pyooj of (2.13) a^ (2.14). All spaces under
consideration are local (see Chapter 1, Section 7); we may therefore
((D) The traces are always understood by extension by continuity from @{Q),
wich is dejise in Hr,s(Q) (see Remark 2.2).

10 2. The Spaces Hr's(Q). Trace Theorems. Compatibility Relations
reduce the problem to the case
Q = {x\xn/>0}.
Let
X = \X , X„} , X = \X± , . . ., Xn_ ! ) .
Then u e Hr,s[Q x ]0, T[) is equivalent to:
ueH'iR+^-.HOiRlT1 x]0,T[)),
ueH°(R+>Xa;Hx-°t),
ueH°(R+,Xn;H0x::t),
or
(2.18) m e H° (]0, oo[; Hr£t) n Hr (]0, oo[; tf °;°)
(we have set: H% = #°(]0, T[; ^(BJ-1)) n #s(]0, J[; fl0^.-1))).
Now we apply Theorem 4.2 of Chapter 1, with
(2.19) X = fl£„ y = *#.°
(and y instead of s; Theorem 4.2 remains unchanged if R is replaced
by the half-line ]0, oo[); we obtain:
^a)(0) =-Lu{x', 0,t)e [X, Y]j+1/2
which, together with (2.19) and Proposition 2.1, yields (2.13) and (2.14);
indeed , j+i/2\ /_ J+i/2\
[X, Y]j+1/2 = H\,tt r I ^
r
whence (2.14).
2) Proo/ o/ (2.15) #w^ (2.16). This is an immediate consequence of
Theorem 4.2 of Chapter 1: since
ueH°(0, T\ Hr{Q)) n HS(0, T; H°{Q)),
we apply Theorem 4.2 (having replaced' R — which is permissible —
with ]0, T[) with X = Hr{Q)t Y = H°{Q)\ then
dku r(i-J+1/2\
«(*)(0) = —- {x,0)e [X} Y]j+1/2 =HK s j{Q)
dtk —i—
(according to Chapter 1, Section 9). D
2.3 Local Compatibility Relations
We shall see that the mapping (2.17) is not suYJective; the functions
a j 4/ 3 u
_ and —— (x, 0) satisfy additional relations, called compatibility
dvJ dr
Yelations (&.&.).

2.3 Local Compatibility Relations 11
Formally, it is clear that
dk I dJu
(2.20)
dtk \ dvJ
27 t = 0
(yj = normal derivative on r, of order /).
The {01. &.) express (2.20) — whenever this has meaning — with
a "limit case" which will be investigated in the following sections.
We shall prove
Proposition 2.2. Let r,s > 0 with 1 (— + —)> 0 ((1));
relations (2.20) are valid for all integer couples j, k such that
1 k ! / 1 1 \
(2.21) - + -< 1 -_-_ + —.
r s 2 \ r s J
Proof. Again we reduce the problem to the case Q = [x \ xn > 0}.
We have to show that
dk dJu dJ dku
According to (2.13):
■ (*', 0,0 6 #*-"v' (RJ71 x ]0, T[)((2))
and again applying Theorem 2.1 (but this time (2.15)), we have:
dk dJu J±.(VJ-k-i/2)
if k < vj - i, i.e. (2.21) (applying (2.14)) and
dk dJu
dtk dxjn
is a continuous mapping of
(2.23) „-(*, o.O)
JH-(VJ-k- 1/2)
Hr-s(Q)-+H*J (RJ71).
Similarly, according to (2.15),
dku
~Jik~{
x,0)eHPk(Q),
((1» Note that in this case; r > ±f s > £.
((2)) Ncrte that in this case: /j,j > -J-, Vj >• -J-.

12 2. The Spaces Hr,s(Q). Trace Theorems. Compatibility Relations
therefore
dku
yj —r (x, 0) e #'"-•'-1/2 (J1) (Chapter 1, Section 9)
if 1 <Pk- i,
i.e. (2.21) (and £fc - / - — = —Ivj - k J is satisfied].
Furthermore
dJ dku
(2.24) u-+ (*',0,0)
is a continuous mapping of
JU-ivj-h-l/2)
Hr's{Q) -#v' (Rx" )•
Since the mappings (2.23) and (2.24) coincide on the dense subspace
Q>{D)t the result follows. □
Remark 2.3. In the proof of Proposition 2.2 we may also use the
following result:
Proposition 2.3. If ueHr,s(Q), r and s > 0, /, k integers such
that 1 — (— + —) ^ 0, we have
\r s)
(2.25)
. dk
DJX—-ueHflfV(Q) (DJX = ?th order derivative with respect to x),
dtk
with
U V I j k
(2.26) -iL = _=i_jL + _
r s \ r s
Proof. We reduce the problem to the case Q = R£ x Rt and use the
Fourier transform in x and t. Then the result follows from
If |' |r|*[(l + |f \2r2 + (1 + r2)v/2] ^ c[(l + If \2Y'2
+ (1 + r2)s/2] D
State of the problem
We shall now see that when the equality holds in (2.21) we again
have a compatibility relation, but this time a non-local one. The relations
are due to P. Grisvard [4] and [8]; we shall specify them first for a
(fundamental) particular case. This is the aim of the next section.
2.4 Global Compatibility Relations for a Particular Case
We consider a Hilbert space X and the space
iT's(R2+;X), R+ =]0, oo[, {:M}eR2+.

2.4 Global Compatibility Relations for a Particular Case
Theorem 2.2. Assume that
13
(2.27)
1 1
— + — = 2, r,s>0.
r s
Then (since r > \, i > %) we may define
\u(x,0) =f{x)eH1/2(R+;X)«1»
l«(0,<) = g(<)e#1/2(R+;X).
We Aaae (global compatibility relation):
(2.28)
(2.29)
0
I
\\f(as)-g(ar)\\2x <oo.
a
Proof. 1) To u, we associate the function v = v(x,t) (for reasons
which will become clear later on) defined by
(2.30)
where
(2.31)
v(x,t) = £ 'ZockPiu(kx,lt),
fc=l 1=1
m = [r] + 1, w = [s] + 1 (M = integer part of a)
X*1** = 0, t = 1,2, . . .,m - 1,
£/'/?, = 0, /=l,...,n-l.
The function t; also belongs to #r,s (R+; X) and
*>(*,0) = X>fc/(&*) = 9?W,
»(<M=L/W) = v(<)-
(2.32)
But the linear mapping
(2.33) / - ? - /
is a continuous mapping of
H1 (R+ ; X) - #* (R+; X) (since J>k = 1)
and of
iP(R+;X)->#°(R+;X);
K1)) Apply Theorem 2.1 (valid without change for functions taking their
values in a Hilbert space.)

14 2. The Spaces Hr*s(Q). Trace Theorems. Compatibility Relations
therefore by interpolation and application of the results of Chapter 1,
Sections 9 and 11.5, Theorem 11.7 (valid without change for functions
with values in a Hilbert space):
(2.33) is a continuous linear mapping of
^2(R+;Z)^^/2(R+;Z),
(2.34)
where
(2.35) Htf (R+; X) = \v \ v e H"* (R+; X), f i- || v (a) fxda < oo
0
The same remarks hold for the mapping g -> ip — g.
Therefore, in the notation of (2.28) and (2.32):
DO " DO
f 7 f 7
I ll/M - y(o)\\2x-?-<<x>, I ||g(cr) -v(o)fcJL
<00
or, equivalently
00 00
I ||/(as) -cp{<f)\\x — < oo, I ||g((0 - V(<0llx—<«.
0 0
Therefore we shall have obtained (2.29) if we can show:
oo
/» 7
(2-36) , j ||9,((TS)_v(<T')||2_^<oo.
0
2) According to (2.32), we may write
<p{(f) - ^(orr) = v(<f, 0) - v(ors, ar) + v(ors, of) - v(0, Or)
and since the variables x and £ play analogous roles, we shall have
(2.36) if we can show that
oo
J ll»(«V
(2.37) | || v (<?, 0) - v (a5, of) ||J ^- < oo.
0
We shall consider the two cases s ^ 1 and s > 1.
3) The case s ^ 1.
We shall need the following lemma (which we shall prove further on):
Lemma 2.1. Let \ < s < §. Then, if u e HS(R+] X), there exists a
constant c such that
(2.38) ||«(f+ A)-«(*)||,^eA-1/2> *.f + AeR+.

2.4 Global Compatibility Relations for a Particular Case 15
Since v e Hr's(R2+; X), we have:
veL2(R+;Hs(R+;X))f
and therefore, according to (2.38),
veL2(R+;Ups_U2(R+;X))ai)\
therefore
\\v(x9t) - v(x9t')\\x ^ w(x) \t - t'\M-112, weL2(R+).
Then
|| V(<f, 0) - V(<f, <Tr) \\x ^ w{<?) (Tr(s-1/2>
= w{(f) <f12 (according to (2.27)].
and
oo oo
||v(<f, 0) - v(as, <f) \\x— = ™{<y)2 da < + oo.
0 0
4) The case s > 1.
Thanks to (2.31), we have
(2.39) lil(*,o)=0, / = 1,. . ., [s] - 1
and therefore, setting ju, = [s] = n — 1:
(2.40) v(x,t) - v(x,0) =—J-—[(t-ay-i^(x>(T)da.
{fi - i)! J a^
Consequently °
t
f II 3Mv
(2.41) || »(*, 0 - *(*, 0) ||, <cr-i\\ — (x,
o
where c denotes a constant.
But according to (2.30):
lSr(*.') =IJlflPih(x,lt)f
01 i
where
*(*,*) = X*k-^-(£*,*)•
If ft' = /"ft and if we set
X(xj) =liP'lh(x,lt),
da,
\x
(d
(1)) Lip0 (R+; X) = space of continuous functions /, mapping t^O -+ X,
such that sup h~d ||/(* + A) — f(t)\\x < °°.
/.a>o

16 2. The Spaces Hr*s(Q). Trace Theorems. Compatibility Relations
we have
(2.42)
But
h- x = -
IF
heL2(R+y,H°-»(R+;X))
and since £# = 1, we see (as for (2.34), (2.35)) that
h-XeL2(R+tX'tHs0-»(R+',X)y,
applying Chapter 1, Section 11 and (2.42), it follows that
oo oo
I Wv I12
(2.43) dx\ r*<*-"> |
0 0
But then
(x,t)
At < oo.
ar/s
{°>y)
,2(s-M)
oo oo
J \\v(<f, 0) - v{<?t </) ||^ = f || V(or, 0) - v(a,o"<) \\2X^- ^
0 0
oo / W
g (according to (2,41)) ^ c\ a"1^ ( j
o \o
oo / ar/s
f 2-L(M-l) / f II g/«v
"J \ J \\^
o \o
oo / ar/s
a
dy J da
(applying (2.27),
o \o
which is finite according to (2.43). D
Proof of Lemma 2.1. Of course it is equivalent to establish an
analogous result for
ueHs{R;X).
We apply the Fourier transform; if for example
+ oo
0(f)- J-
we have:
-2nix£
+ oo
u(x) dx,
|«(* + A)-«(*)||xg j |e2*^-l|||*(f)||x«.

2.5 General Compatibility Relations* 17
Choosing c0 such that |e" — 11 ^ 2 11| for \t\ ^ c0, we obtain
||„(s + *) _ „(*) ||x ^ 4w |A| J ||f rf(f) ||x if +
|*|£*o/2*|*|
+ 2 J ||rf(f)||xrff £4*|A| f |||fr«||,|f|1-^ +
|*|>c0/2*|*| m^Co/2n\h\
+ 2 J Ifl-lllfl'tfllx^^
|5|>c0/2ic|*|
^47t\h\( j ifi2(i-'>ify/2/jiiifi-rfiii«y/2
\|^|^^o/27c|fc| / \R /
+
+ 2/ i ifr2*^r71 iiifi^iu^172^^*!-^. n
j \e\-*de\1,2(j\\\e\'4iiide
■|*c0/2*|*| / \R
2.5 General Compatibility Relations
We are now ready to specify all compatibility relations satisfied
by the traces of u e Hr,s(Q). With the aid of local maps, we can establish
a correspondence between each u e Hr,s (Q) and a finite number of
functions, denoted by
ult u2> . . ., uN,
which define u uniquely and are such that
(2.44) ut eHr's(Rn+ x ]0, T[), Rn+ = {% \ xn > 0}.
On the other hand, there exist compatibility relations of the same
kind on the hypersurfaces {x e T, t = 0} and [x er,t = T}. Therefore,
we do not lose any generality by assuming T = + oo and consequently
it is sufficient to specify all (&.<£.) for the case Rn+ x ]0, oo[. Therefore,
let
ueHr's(Rn+ x ]0,oo[), r,s>0, 1 - — (— + —) > 0.
2 \r s J
(*.0) =/*(*),
Set:
dku
(2.45)
dtk
dJu
(x',0,t) =gj{x',t).
dxi
According to Theorem 2.1, we have:
(2.46)
{fk,Sj}eF= n H"k(K) x II HV{Z),
k<s- 1/2 j<r- 1/2
Piofa'Vj given by (2.16) and (2.14).

18 2. The Spaces HTtS(Q). Trace Theorems. Compatibility Relations
According to (2.20),. we know that
dkgf dJfk j k 1/11
(2.47) _|L(*f0)--^(*',O) if 1+-<1 -_(-- + .
According to Theorem 2.2, we have:
2 \r
(2.48) \
0 R»-*
/ * 1/11
r s 2 \r s
2 da
dx' < oo
a
Indeed, according to Proposition 2.3,
and from (2.26) (with (2.48)) we deduce that
The function
1 1
- + - = 2.
JLt V
xn,t-+ "x' -+u{x',xnttn)"
thus belongs to H^V{R\\ ^(RJ"1)) and therefore (2.48) is just the
interpretation of (2.29). □
We denote by F0 the vector subspace of the product space F of
elements {fk, gj} which satisfy (2.47) and (2.48); F0 is a Hilbert space for
the norm whose square is given by
ll{/,
00
skgj,,
dtl
, (x',Gr)\ dx' -
da
o r*-i
Then the fundamental result is:
Theorem 2.3. u -► {fk,gj] {defined by (2.45)) is a continuous linear
surjection of
Hr's(Rn+ x ]0, ooD-*F0> */ r>0, s>0, 1 —— ( 1 j > 0.
Proof. Only the surjectivity remains to be shown.

2.5 General Compatibility Relations
19
According to Theorem 4.2 of Chapter 1 (applied as in 1) of the proof
of Theorem 2.1) there exists a function v e Hr*s(Rn+ x ]0, oo[) such
that
(2-49) ^j (*', (U) = gj (*', t), 0 ^ / < r - I
d*n 2
It remains to construct weHr*s(Rn+ x ]0, oo[) such that
dJw
(2.50)
where
(2.51)
^(•.o.o-o.
3*^ 1
-t— (*.0) = /*-K, ozk<s--,
dtk 2
Ak
But since by hypothesis {fk, g^} e jF0 and since {/&fc, gj} e F0, we
obtain:
{/*-A*,0}eFo.
Thus, if we set
(2.52) <pk = fk-hkeH>*(Rn+),
we have {<pkt0} eF0, that is
(2.53)
and
dJwk 1
^-(,■.0,-0 if i<h--
U\fr"
da
dx' < oo if j = pk •
a 2
(2-54) | | | "^f (*'>*)
d r»"-i
Therefore (see Theorem 11.7 of Chapter 1):
<pk e H%% [RD = [Hr0 (Rn+), H° (Rn+)]ik+ 1/2)/s (r > i)
(replace #£ (R+) by i/Jo (R+) if r = integer + £) and consequently,
according to Theorem 4.2, Chapter 1, there exists a weL2(0, oo;#J(R+))n
n #s(0, oo; L2 (Rn+)) such that
z£/<*>(0) = 9?fc;
we can choose the desired function w in this way. □

20 3. Evolution Equations and the Laplace Transform
Remark 2A. In the sequel, we shall constantly use the spaces
H2rm'r(Q), r ^ 0, integer m ^ 1.
Then, for r > \t Theorem 2.1 yields:
dJu 1
(2.55) e H2rm-J- U2.<2rm-j-1/2>/2« (^, / < 2r m
5W 2
dku 1
(2.56) -z-r(*> 0)eH2m^-k~1^(Q)t k < r
or 2
3. Evolution Equations and the Laplace Transform
3.1 Vector Distribution Solutions
Let H be a Banach space (later on we shall assume that H is a Hilbert
space) and let A be a closed unbounded operator in # such that
(3.1)
A + p is an isomorphism of D(A) (norm of the graph) -+H,
for p = g + irj, | > f0, jjeR, swc& ^^
II (^ + ^)_1 ||*(h:h) ^ polynomial in \p\. D
The space @'+ (H)
In Chapter 1, Section 1.3, we introduced the space 2' (H) of
distributions on Rt taking their values in H (let a = —oo, b = +oo in
(1.17), Chapter 1), where H is a Hilbert space or a Banach space.
We denote by @'+ (H) the subspace of &' (H) of distributions /
which vanish for t < tf (where tf depends on /), that is such that
</,<p> = 0 V(pe@ = 0(Rt),
where cp has compact support on t < tf.
It can be shown (see L.Schwartz [6]) that 2'+(H) = J^(^_;#)
= space of continuous linear mappings of Q)_ into H, where Q)_ = space
of functions cp of class C00 on Rt, with support restricted to the right
(i.e. cp (t) = 0 for t > tv> where t9 depends on cp). The space @_ is provided
with the inductive limit topology of ^ln) = {cp \ cp e @- ,q>(t) = 0 for
t > tn, tn fixed} {the space &ln> is provided with the topology of compact
convergence of the functions and of each of their derivatives). In general,
&+{H) = &(&_\H)
is provided with the topology of uniform convergence on the bounded
sets of £&_. □

3.1 Vector Distribution Solutions 21
We may now state1
Theorem 3.1. Let A be given with (2.1). Then, for f given in <$'+ (H),
there exists a unique u e @'+(D(A)) such that
(3.2)
/ du \
Au + u'=f in @+{H) lu' = •
[Furthermore, / -> u is a continuous linear mapping of &+ (H) -►
-+&+(D(A))].
Proof. 1) We shall show the existence of a distribution 9 with the
following properties:
(3.3) 9e9+(&(H\D{A)))9
(3.4) j \A + — ) * 9 = d ® IH, IH = identity in H9
\ Ot J (t)
(3.4)2 9 (* U + — j = <5 ® /DU), IDU) = identity in D {A)
(where * denotes convolution in t\ for convolution of vector-valued
distributions we refer the reader to L. Schwartz [6]; we shall only need
the elements of the theory here); furthermore
!<& admits a Laplace transform in t (in the sense of distributions;
see L. Schwartz [2]) and 9 has support in t on t ^ 0.
The theorem follows immediately from these properties: the unique
solution to the problem is
(3.6) u = <$ * f.
2) It remains therefore to construct 9 with properties (3.3) to (3.5).
Let 9 (p) be the Laplace transform of 9 (^t = Fourier transform in t):
/ — 00
(3.7) &(p) = ^ (e-* 9) = J e"l,|f(e"* #(*)) dt (formally).
+ 00
Then — assuming the existence of 9 (p) for f > f x with a suitable f x
— equations (3.4) x and (3.4)2 become
(A+fi)9{p) =IH
<£(j>) {A+p)= ID(A).
1 We shall obtain more general results in Chapter 9 of Volume III.

22 3. Evolution Equations and the Laplace Transform
Therefore, we may take
(3.8) 9(p) = (A + p)'1, I > So (which exists according to (3.11)),
if the inverse Laplace tiansform of (^4 + p)'1 exists and has properties
(3.3) and (3.5).
But existence and properties (3.3) and (3.5) follow from (3.1), by
a result on supports in Laplace transformation (Lions [1]), and from the
property:
p -> {A + p)'1 is a holomorphic mapping of f > |0 -> 3?{H\ D(A)).
Indeed, this property is a consequence of the resolvent identity:
if p and p' satisfy Rep > |0, Rep' > |0, then
(p,-p)-1\.(A+pri-{A+p)-^ = -{A+pri{A+p)-\
which shows that {A + p)"1 exists (in J?(H;D(A))) and is equal
dp
to — (A + p)~2. Whence the theorem. D
3.2 Z,2-Solutions
Theorem 3.2. Let A be given with (3.1) and furthermore satisfy
(3.9) || (A + p)'1 U^(h.h) S ,* , for | > |0> c = constant.
I Pi + 1
Assume that H is a Hilbert space.
Then, for f given in L2 (0, T\ H), there exists a unique function u
satisfying
(3.10) ueL2(0,T;D(A)),
(3.11) Au + u' = f,
(3.12) u(0) = u0, u0 given in [D(A),H]1/2
(ndtation of Chapter 1, Section 2; u (0) is well-defined since u'eL2(0,T;H),
thanks to (3.11)).
Proof. 1) We can reduce the problem to the case "u0 = 0" (which
is not essential); indeed, since u0 e[D(A), H]1/2, there exists (see
Chapter 1, Section 3) a w satisfying
(3.13)
weL2(0,T;D(A)),
w' eLz(0,T;H),
[ w(0) = u0

3.2 L2-Solutions 23
(and which depends continuously on u0). Then
U — W = 0
must satisfy
A 0 + 0' = / - (A w + w') e L2(0, T; H),
0eI2(OJ;D(i)),
0>(0) = 0.
2) Let u be the solution of (3.10) to (3.12) with / = 0, u0 = 0.
Let « = extension by 0 of u outside ]0, T[. Then
A u + («)' = «(T) ®d(t - T).
and according to Theorem 3.1,
u = ^ * («(r) ® d(* - r)) = #(* - T)u(T).
Since #(*) = 0 for * < 0, we see that 0(* - T) = 0 for t < T and
therefore
u = restriction of « to ]0, T[ = 0;
this shows the uniqueness.
3) Now we consider
/ = extension by 0 of / outside]0,T[.
We shall verify that
(3.15) e~|f UeL2(Rt;D(A)) Vf > £0>
(3.16) AU + U' = f on R.
Then
U' =f - A UeL2(Rt',H)
and since C7 = 0 for £ < 0, we shall have
(3.17) £7(0) = 0.
Then the restriction u of U to ]0, T[ is a solution to the problem.
(3! 14)

24
4. The Case of Operators Independent of t
Since (3.16) follows from (3.14) and Theorem 3.1, it just remains to
verify (3.15). But
e-^=(e-*»ar)(?)(e"«»/)f
therefore by Fourier transform in t:
JMe-*t/)=a^)JMe-*7).
Applying (3.9), which is equivalent to
(3.18) \\{A+p)-'
we deduce from it, thanks to (3.8):
IIJMe"* U) (r])\\2DU) ^ c\\\&t{z-*l) fo) |||.
Integrating in rj and applying Plancherel's theorem (which assumes
H to be a Hilbert space) it follows that, for £ > |0:
/ lie"** U(t) fD(A) dtZclf \\e-(,f(t) ||| it < oo;
whence (3.15). □
4. The Case of Operators Independent of t
4.1 Hypotheses
We shall again consider the operator P given in (1.1) and (1.2), and
assume that the coefficients apq do not depend on t; thus apq = apq (x) e
We introduce the auxiliary variable yeRy and the operators
71 71
T'T
(4.1) As = A(x,Dx) + eid(-\)mD2ym, 6
The formal adjoint of AQ in Q x Ry is
(4.2) 4* = i4*(*f Z>x) + e-i0(-l)mZ)f\
We again consider the boundary operators {B/J/lV given by (1.4)
and assume that the coefficients bjh do not depend on t\ thus bjh = bjh (x) e
e®(r).
Finally, we assume that the system {Ad, Bj}, for all 6 e
71 71
T'T
is a "regular elliptic'' system in Q x Ry; more precisely, we assume that
7C 7C
(i) /I0 is properly elliptic in D xRy V0 e
2' 2
(ii) {-B/lJfLo1 is normal on r and of order mj} with O^mj^lm— 1,
(iii) {B/JJTo1 C(WS ^^TxRy V0 e - —, —

4.2 Basic Inequalities
25
Then, applying the elliptic theory of Chapter 2, it follows that the
formal adjoint system {/I*, C,} is also a "regular elliptic" system (see
Chapter 2, Section 2) and that we have the following estimates:
(4.3)
theie exists a constant c, independent of 0 e
71 71
, such
that Vco e #2m (12 x Ry) with support in y on ] — 1, 1 [ and such
that Bj co = 0, 0 :g / ^ w — 1 (respectively C,- co = 0, 0 rg / rg
^ m — 1) on fxRr we have
|||/l0co||o + |||tt)|||0 ^ c |||co|||2m
(respectively ||M*co|||0 + |||co|||0 ^ c |H||2m),
where we have used the notation
III <w 111ft = norm of co in Hk{Q x Ry).
(See Chapter 2, Sections 4 and 5; note that, by continuity, we easily
I 7€ 7€
see that the constant c is independent of the parameter d e
see also Chapter 2, Remark 4.1.) D
2 ' 2
4.2 Basic Inequalities
We define A (resp. ^4*) to be an unbounded operator in H = L2 (Q)
with
(4.4)
D(A) = \v\veH2m(f2),Bj(x, )v = 0 on r, 0 ^ / ^ m - 1
and
(4.5)
D(4*) = H^e#2m(£),C,. *,
dx
v = 0 on r, 0 ^ / ^ m - 1
Theorem 4.1. Under the hypotheses of Section 4.1, tf/ters owsfc a |0eR
#w^ aw «>0 s^c& £/^, V^> = I + i ?7, I > lo» ^ ^a*^
(4.6) ||(4 +#)i;||o^*||w||2- Vi/6£>(4),
(4.7) \\{A* + -p)v\\0>oc\\v\\2m VveD(A*)
(where || \\k = norm in Hk(Q)).

26 4. The Case of Operators Independent of /
Proof. We apply a method of Agmon and Nirenberg.
We consider veD(A) (the proof of (4.7) is analogous to the one
which follows) and we introduce
w(x, y) = z(y) e1/J,yv(x)} ju, e R,
ze@(R) with support in ] — 1, 1[.
Since veD(A), the function w satisfies the conditions of (4.3).
Denote by |M|2m,o (resP- II w\\0,2m) the norm in L2{Ry\H2m{Q))
(resp. H2m(Ry;L2(Q))); since in H2m{QxRy) we have:
|HI|2m~IMl2>«.0+ lkllo,2m((1)>,
the inequality for AQ in (4.3) is equivalent to
(4.9) IMHIlo + ||k 10 ^ MIMl2m.O + ||W||0.2»).
But
AQw = z{y)el*"(Av + (-l)w ei0(i p)2"1 v) +
/2m-l f2m\ \ .
from which, since \eifiy\ = 1:
IMHHo ^ c2 \\(A + e'VVIIo + e3(l + l^l2"-1) IMIo
and since
IHlo ;S c2 iMio
we see that (4.9) implies
(4.10)
|| (A +e'V"Vllo ^adMUo + IMI0.2J -eA(l + I ^* I2"*"x) II ^ II o >
where the constants cit here as well as in the sequel, are independent
of [i, 0, v.
But
II ** II 2m,0 = C2 lb II 2m
and
ll^llo.2m-||h|o+||^2m^|||o;
from which it follows that
lkl|0>2m ^ c5 \fi\2m \\v0\\0 - c6{\ + \fi\2"-1) \\v\\0.
((D) xhe symbol ~ denotes equivalent norms.
(4.8)

4.3 Solution of the Problem
Introducing this inequality in (4.10) we obtain:
\\(A + e'V-)»||o ^ MIMI2. + IH2m IMIo) -
-c8(l + \fi\2"-1)\\v\\0
27
and therefore
(4.11)
(A +eiV2W)^llo^^9(bll2m+ \f*\2m\\v\
for ju, ^ |0 sufficiently large.
But since this is valid for VS e
71 71
T'T
, we can choose [i and d, if
P = £ + ir), !>!o>
so that eie jbt2m = p and therefore
(4.12) ll(4+«w||o^c9(||w||2. + |^|||i;||o),
from which (4.6) follows as a particular case.
The proof of (4.7) follows the same lines. □
4.3 Solution of the Problem
Theorem 4.2. Under the hypotheses of Section 4.1, there exists a unique
function u e H2mtl (Q) satisfying
(4.13)
A u + «' = /, / given in L2 (Q)
BjU = 0 on Z = rx]0,T[, 0^/^w-l,
u{x,0) = 0.
Proof. Apply Theorem 3.2 with Z)(^4) given by (4.4) (and u0 = 0).
It is sufficient to show that (3.9) holds.
But according to Theorem 4.1 and the elliptic theory (Chapter 2,
Section 8) it follows from (4.6) and (4.7) that A*, defined with D (A*)
given by (4.5), is the adjoint of A in the sense of unbounded operators
in H and that (A + p) is an isomorphism of D(A) -► H for | > |0.
Furthermore, according to (4.6):
II (A + P)'1 II^(h;du)) ^ constant,
(which is equivalent to (3.9)). D
Remark 4.1. Non-homogeneous case. Let the problem be
Au + u' = f9 feL2{Q),
gj given in H2m-mJ-1/2'<2m-mJ-1,2>,2m(Z)
[ u(x,0) = u0, u0 given in Hm(Q),
(4.14)

(4.15)
28 5. Regularity
where gjt u0 satisfy the compatibility relations {M.^.):
gj{x,0) =Bjlx,—\u0 on T
for all j such that nij _^ m — 1.
Thanks to the (#. «\) (4.15), there exists a w e H2mtl (Q) with
{#/ w = £/» 0 < j < m — 1,
w(x, 0) = u0[x)
(in fact the compatibility relations are precisely the necessary and
sufficient conditions for the existence of weH2m,1{Q), with (4.16); see
Section 2.5).
Then (u — w) must satisfy
A (u - w) + (u - w)' = f - (A w + w') e L2 [Q),
Bj(u - w) = 0,
m(0) - w(0) = 0.
Therefore the problem is reduced to (4.13) and we have
Theorem 4.3. Under the hypotheses of Section 4.1, if gj and u0 are
given with the {M. *&.) (4.15), then problem (4.14) admits a unique solution
in H2m^{Q).
5. Regularity
5.1 Preliminaries
In Section 4, we started with unbounded operators in H = L2 (Q).
Now, we may consider (having assumed that the coefficients of A
and Bj are sufficiently regular, see Section 4.1) A to be an unbounded
operator in Hk(Q), arbitrary k §: 0, integer or not. However, we shall
assume k to be an integer in order to simplify some technical points.
Thus, we take
(5.1) H = Hk(Q),
(5.2) D(A) = {v\veHk+2m{Q),BjV = 0,0 ^ j£m - 1},
(5.3) D[A*) = {v\ veHk+2m{Q), Cj v = 0, 0 = / = m - 1}.
In order to arrive at the inequalities which will replace (4.6) and (4.7),
we shall use the fact that, under the hypotheses of Section 4.1, we also

5.2 Basic Inequalities
have (see Chapter 2, Section 4.5) the estimates:
29
(5.4)
for fixed integer k, there exists a constant c, independent of 6,
such that for all w e Hk+2m (Q x Ry) with support in y on
] — 1, 1[, and with Bjw = Oy0^j^m— 1 (resp. Cs w = 0,
0 ^ / ^ m — 1), onTxRy, we have:
IIMfl^l* + lll^ll* ^ C ||MI*+2»
(resp. Ill-will* + I w I* ^ c I w I
k+2m) •
(in the same notation as in (4.3)).
5.2 Basic Inequalities
Theorem 5.1. Under the hypotheses of Section 4.1, there exists a £0eR
and an oc > 0 such that Vp = £ + if], g > f0> ^ have
(5.5) ||(4 +0»||lk + (l + |£|W2")||(4 +P)v\\0^
£*(IMI*+2» + l£l1+W2"IMIo) VveD(A)
(resp.
(5.6) ||(4* + p)v\\k + (1 + l^r2"") 11(4* + p) v\\0 ^
^«(ll»l
fc + 2m
+ I*
|l+fc/2m
V^ eD(i*)).
Proof. The principle of the proof is analogous to the one of
Theorem 4.1. Let v e D(A) (defined by (5.2)). Introduce w by (4.8). In the
notation of the proof of Theorem 4.1, (5.4) may be written:
(5.7)
M«Hlo.fc + Me^lko + IMko + IMIo,* ^
^ Cl(IMI*+2m,0 + II » II0.*+ 2m)-
But using the explicit form of Agw, we have:
Ma"1*.o ^ o2 ||(4 + e'VHI* + c2(l + Ia*!2-1) ||w||*
and *
\\Aew\\0,kZc3(l + |,u|*)||(4+eV>llo + c3(l + \/t\k+2m-1)Mo-
On the other hand:
\w\
k + 2m,0
= C±\\V\
and
iMio.i+2.. ^ c5 i^r2- iit»n0 -c6(i + i^r2-"-1) iiw||0

30 5. Regularity
and therefore (5.7) yields
(5.8) || (A + e'V") »ll* + (1 + \fi\*) || (A + e'V*) «||0 ^
i^7[IMI*+2m + br2mIMIo-
-(i + \Mk+2m-1)Mo-(i + \/*\2m-1)Mii-
But (see Chapter 1, Remark 9.1):
^ c* II v
.fc/(fc + 2m)
\k + 2m
.l-fc/(fc + 2m)
therefore
l^l21""1 IMI* ^ i^7 |M|fc+2m + c9 i^ia-DW+a-)^- ||V||0
and consequently (5.8) yields:
| || (4 +eiVmM|fc + (l + l^lfc)ll(^ +e»V2w)^llo^
I ^ioCIMU+2* + Hfc+2wIM|0] for ii ^ fi0 sufficiently large.
Letting, as in Theorem 2.1, p = eid jbt2m, (3.5) follows. □
5.3 An Abstract Result
In order to clarify the use of the "basic inequalities", it may be
useful to first state an abstract result — a very simple variant of
Theorem 3.2.
Let H be a Hilbert space contained in another Hilbert space Jjf:
(5.10) H c £F with continuous injection.
Let the given operator A be unbounded in H. We assume:
(2.2) holds and furthermore, VueD(A), we have:
||(4 +p)u\\H+(l + \P\')U* +*)«Lr £
(5.11)
^c(\\u\
D(A)
+ IJI
1 + 0
l V p with f > f0, where /? > 0 is given.
Theorem 5.2. Let A he given and satisfy (5.11). Let f he given with
(5.12)
feL2(0tT;H) nHfi(0, T;Jf)} /<'>(0) = 0
if j < P - i, /? 4= integer + \\
——/<*> (t) eL2(0, T; JT) if p = ^ + —, ^ = integer.
V< 2

5.3 An Abstact Result 31
Then there exists a unique u, with
(5.13) ^eL2(OJ;D(i))n^+1(OJ;^),
(5.14) Au + u' = /,
(5.15) u(0) = 0.
Proof. Since (3.2) holds, uniqueness follows as in Theorem 4.1.
We extend f to /on R by 0 for t < 0 in such a way that / has compact
support in t and satisfies
(5.16) feHfi(R; Jt) n L2(R; H).
(Extension "by reflection" about t = T; see Chapter 1, Section 2.) As
in Theorem 3.1, we solve
dU
(5.17) A U + —- = f on R.
dt
By Laplace transform followed by inversion, we have
(5.18) e"** U = &;l[(A + p)-* Srt(e-*'?)]
for f > |0.
But since / has compact support and satisfies (5.16), we have:
(5.19) e-(rfeH^(R;je)nL2(R;H).
Let
Then, according to (5.11):
(5.20) || (A + P)-1 F(p) ||DU) g c \\F(p) \\H + c(l + \p\») || F(P) \U
and according to (5.19):
(5.21) (l + |i?|')2?(£ + ii?)e£*(Bii;Jr) Vf.
Therefore (5.20) shows that
rj-+(A +p)-iF{p), f >f0f
belongs to L2(R^;Z)(^)) and then (5.18) shows that
e't* UeL2(Rt;D(A)).
Therefore, if u = restriction of U to ]0,T[, we have:
(5.22) ueL2(0,T;D(A)).
But according to (5.11), we have an analogous bound for
\p\^(A+p)-lF(p),

32 5. Regularity
from which we obtain that
(5.23) e-^[/e^+1(Rt;Jf)
and therefore that
ueHfi+1(0, T]JT),
which, together with (5.22), yields the theorem. D
5.4 Solution of the Boundary Value Problem
Theorem 5.3. Under the hypotheses of Section 4.1, let f, gjy u0 be given
with
(5.24) / e Hk>k/2m (Q), integer k > 0,
(5 25) g e JJk+2m-mj-1/2,(k+2m-mj-1/2)/2m/2J)
(5.26) u0eHk+m(Q),
and satisfy the compatibility relations (5.31) below. Then there exists a
unique function u satisfying
(5.27) ueHk+2m>k/2m+1(Q),
(5.28) A u + W = /,
(5.29) BjU = gJt 0 g / g m - 1,
(5.30) u(0) = u0.
Proof. 1) The compatibility relations are the "relations needed"
(see Section 2) for the existence of w e Hk+2m>k/2m+1 (Q), with
\ Bjw = gJt w(0) = u0
k 1
(5.31)
I (A W + W')"'(U) = PJ'(V), USf <
Then u — w = 0 satisfies
A 0 + 0' = / - (A w + W) e Hk>k/2m(Q),
Bj0 = 0, 0^'gw-l,
<Z>(0) = 0,
and
k 1
(/_(,!«, + W'))u>(0) =0, 0 ^ 1 <
2m 2
2) Thus we are led back to Theorem 5.2 with
k
H = Hk(Q), 3f = H°(Q), p=
2m
and (5.11) following from (5.5); whence the theorem. D

6.1 Hypotheses. Statement of the Result 33
6. Case of T:r.:e-Drprndent Operators.
Existence of Solutions in the Spaces H2mrr(Q)9 Real r ^ 1
6.1 Hypotheses. Statement of the Result
We now consider the operators:
(6.1) A(t) = A(x,t,Dx) = £ (-I)'-' Dpx(apq(x,t)Dl),
(6.2) *,(*) = Bj(x, t, Dx) = X bJh(x, t) Dx.
We shall make the following hypotheses:
I for every t = toe[0,T], system {A (t0), Bj(t0)} satisfies the
I hypotheses of Section 4.1;
(therefore we shall have:
|||(il(*0) + zie{-\)mD2ym)w\\0 + I w 10 ^ ct0 \\\w\\\2m
\fweH2m(Qx Ry), with Bs (t0) w = 0
and the analogous "adjoint" conditions).
We then make the following regularity hypotheses on the dependence
on t (these hypotheses can be weakened, as we shall see in Remark 6.1):
(6.4) a„ e ^([O.T];#(£)) Vp.q,
(6.5) bJh S Ca»—^([O, T\;9(r)) Mj, h.
Theorem 6.1. Assume that (6.3), (6.4) and (6.5) hold. Let f, gd and u0
be given with
(6.6) feL2(Q),
(6.7) gjeH2m-mj-l/2, (2m-m,-l/2)/2m^ 0 ^ / ^ W - 1,
(6.8) u0eHm(G),
and let the following compatibility relations hold:
(6.9)
3<{X>°'lL)Uo
= gj(x> 0) if nij < m .
r 2
Then there exists a unique function u in H2m,1(Q) satisfying
(6.10) A u + u' = f,
(6.11) Bju = gj, 0^/^m-l,
(6.12) «(0) -«0.
The proof of this theorem (which contains Theorem 1.1) will be given
in two steps.

34 6. Case of Time-Dependent Operators
6.2 Local Result in t
Let At be an interval to be chosen "sufficiently small" later on.
Set •
QAt = Qx]0,At[, ZAt = rx]0,At[.
Define
(6.13)
KAt = \l2{Qm) x mf[H2m-m>-1/2-<2m-mi-1/2y/2m(i;jt) x Hm(Q),@.<i?\
that is
KAt = {{/,g,,M>/e£2(&,),^^
u0eHm(Q) and (6.9) is satisfied}.
Let R and R0 be the operators
(6.14) R:u-> {A(t)u + «', Bj(t) u, u(x,0)},
(6.15) R0:u-+ {A(0)u + u', Bj(0) u,u(x,0)},
which both satisfy
(6.16) R,R0e^(H2^(QAt);KAt).
In the cylinder QAt ("local" case in t), problem (6.10), (6.11), (6.12)
is equivalent to
(6.17) Ru = {/, gJt u0} = x given in KAt.
But thanks to Theorem 4.3, R0 is an isomorphism of H2mA (QAt)
onto KAt for all At, and furthermore, according to the estimates,
(6.18) WRo1 \\j?(KAt;H2m.HQAt» ^ (constant independent of At) = cti{1))
Problem (6.17) is equivalent to
(R0 + (R-R0))u = x>
that is to
(6.19) (1 + Ro'(R - «o))« = Ro'x-
From this fact we shall deduce:
Lemma 6.1. Hypotheses of Theorem 6.1. There exists a At such that
problem (6.10), (6.11), (6.12) has a unique solution in Q x ]0, A t[. Further-
(CD) Note that the bounds in the case of time-independent coefficients hold
with constants independent of the length of the interval in t.

6.2 Local Result in t 35
more At does not depend on the origin of t, in other words, problem
x A(t)w + w'=f in Q x]t0tt0 + At[,
Bjw==gj on rx]t0,t0 + At[,
w(x, t0) = uto{x),
with
x J 2
admits a unique solution for every t0 e [0, T].
Proof. We need to find an estimate for
<p{At) = \\Rq (R — Ro)\\<e{H2rn.l(QAty;H2m.HQAt»'
According to (6.18), we have:
(6.20) <p{At) ^ c, \\R - R0\\x<B2>n.iiQAt>;KAt) = *i y>(At).
Let
B1 = {u\uGH2^(QAt)f \\u\\H2m.HQAt) ^ 1}.
Then
y>{At) = sup||jR^ - R0u\\KAt
ueBt
= sup ||(il (t) -A(0))u\\HOiQAt} +
m-1
(where fjtj = 2m — mj — ^-).
But thanks to (6.4) and (6.5) we have:
(6.21) \\(A(t) -A(0))u\\HoiQAty^c2At\\u\\H2m.HQAt) if te[0fAf]
and
(6.22) 11(5,(0 -£,((>))«||
if *e[0,Zl*], V/.
Remark 6.1. Hypotheses (6.4) and (6.5) are only simple sufficient
conditions implying (6.21) and (6.22); they may be weakened by using
the spaces of multipliers on the Sobolev spaces. D
Remark 6.2. All the constants are independent of the origin in t. D
From (6.20), (6.21), (6.22) we deduce:
<p{At) ^ c^At
and therefore
<p(At) < 1 for At < —•

36 6. Case of Time-Dependent Operators
Then 1 + R'1 (R - R0) is invertible in H2m'1{QAt) and Lemma 6.1
follows.
6.3 Proof of Theorem 6.1
The theorem is an immediate consequence of Lemma 6.1. Choose A t
as in the lemma. Solve the problem in Q x ]0, A t[) let u1 be the solution.
Then consider u2, solution (which exists according to Lemma 6.1) of
Au2 + (u2)' =/ in Qx]At,2At[,
Bju2 = gj on rx]At,2At[,
u2(x,At) = u1 (x,At)\
the compatibility relations on r x {t = At} hold since u1 is the solution
of the problem in Q x ]0, At[; thus u2 exists and so on. Taking u = u1
in Q x]0,At[, u2 in Q x]At,2At[, etc. we obtain the existence of
a solution.
Uniqueness also results from Lemma 6.1; if u satisfies (6.10), (6.11),
(6.12) with / = 0, gj = 0, «0 = 0, then u = 0 in Q x ]0, Zl*[ according
to Lemma 6.1; then u(x,At)=0 implies u = 0 in Q x]At,2At[,
etc. D
6.4 Regular Non-Homogeneous Problems
The same method as in Section 6.3, together with Theorem 5.3,
yields the following result. Under the hypotheses of Section 1.1, let
(6.23) r ^ 0 be given with 2r m = integer]
let gj, u0 and / be given with
(6.24) gj e H2(r+l)m-mj-l/2t(r+l)-(mJ+H2)!2m ^^ 0 ^ / ^ W - 1,
(6.25) ^0e#2(r+1/2)w(£),
(6.26) feH2™-'(Q),
and with the compatibility relation (&. ft.), which may be expressed by:
there exists a w e H2<r+1)m-r+1(Q) with B5w = g3
{Bj = Bj (t) now depends on t), 0 ^ / ^ m — 1,
(6.27) | zc(*,0) = «oW» *e*2,
Z>*[>4 (*, *, Z)J w + Z>,w] 11=0 = £*/(*, 0),
0 ^ * <r - i.(1)
Then problem (6.10), (6.11), (6.12) admits a unique solution in the
space #2(r+1>m.r+1 (^) (solution which depends continuously — in the
(1) Of course, with the help of Theorem 2.3, we can write these relations
explicitly.

6.4 Regular Non-Homogeneous Problems 37
obvious topologies — on the data {gj, u0, /}). We shall now extend
this to the case "non-integer 2r m" (with a condition on r):
Theorem 6.2. Let real r ^ 0, with
(6.28) 2r m and r 4= integer + i-
^4ss^m£ tf/^ £&£ hypotheses of Section 1.1 a^ satisfied. Let {/, gj, ^0}
6* given with (6.24), (6.25), (6.26) and satisfy the {<%.<#.) (6.27). Then
problem (6.10), 6.11), (6.12) admits a unique solution u in H2ir+ 1)m,r+ * (Q).
The mapping {/, gj} u0] -+ u is a continuous mapping of
m- 1
Jj2rm,r/Q\ x FT Jj2(r + Dm-mj- 1/2,(r+ l)-(mj+ l/2)/2m /™ x
i = 0
X #2(r+ 1/2)m (fl) =&r-+ #2(r+ 1)w'r+ * (<?) .
Remark 6.3. We may also state:
the mapping u -+ I + Au,Bju>u(x>0)\ is an isomorphism of
^r2(r+i)m,r+i ^ Qnio ^ sufrspace 0j foe product space &\ of elements
{f,gj,u0} which satisfy the (^.^.). U
Proof of Theorem 6.2. Since, by hypotheses, there exists a w with
(6.27), it all comes down to solving (setting 0 = u — w):
80
— + A0 = v,
ot
Bj0 = 0, 0 ^ j <; m - 1,
<2>(*,0) =0
the function ip I = / — ( h A w J J satisfies
(6.30) ipeH2™-r{Q), wf\xt 0) = 0, 0^i<f-f
We have to show that 0 e H2Cr+1)m'r+1 (Q).
But this result is known for integer 2r m\ so let us take r1 with
rx > r, 2r1 m = integer and let us call Gr the space defined by (6.30).
Thus we know that the mapping
ip -+ 0
is a continuous linear mapping of G° -+ H2"1*1 (Q) and of Gri -+
^ H2{ri + 1)m,ri + 1 (Q). Therefore, by interpolation, it is a continuous
mapping of
(6.29)

38 6. Case of Time-Dependent Operators
According to Proposition 2.1, the second space is equivalent to
jjlUl-dfrt + l)m,(l -0)1-! + 1 ,0v
Choose 6 with (1 — 0) rt = r. It remains to see that [Gr\ G°]d = Gry
2r m and r 4= integer + -£. The proof of this is analogous to the proof
of Theorem 11.6. of Chapter 1. D
Remark 6.4. We can also deal with the exceptional values of the
parameter r by considering the integral compatibility relations. If we
do not introduce these integral relations, Theorem 6.2 is no longer
valid for r exceptional. Indeed, let us take / = 0 in the statement of
Theorem 6.2, and in order to simplify the notation, let us replace r + 1
by r. Then if ,. n
2(r — j-) m = integer + i,
or if
2rm — m* — \ 1
= integer -\ ,
2m 6 2
we shall see that the following inequality does not hold (see Solon-
nikov [2]):
(m-l
2j llgrjllH2rm-mJ-l/2.(2rm-mj.-l/2)/2m(i7) +
j = o
+ H^0llH2(r-l/2)m(£)J
for gj e @{Z) and u0 e@(Q).
In order to simplify somewhat, assume S = r x ]0, oo[,
Q = £x]0, oo[.
First we consider u0 = 0. Let gj (resp. u) be the extension of gj
(resp. u) tofxR (resp. Q x R) by 0 for t < 0. According to the trace
theorem of Section 2, we have:
X \\£j \\ Hlrm-mj-1/2. iirm-mj-l/2)/2mirxR) ^ C2 || U || fl2rm,r(fixR)
J
and since r =J= integer + i> we have
II ^llH2rm,r(£xR) S C3 II W II H2rm,r(Q)
and therefore if the inequality (6.31) would hold, we would have
m-l
s
j = o
X llfjllH2rm-mJ-l/2.(2rm-mj.-l/2)/2m(rxR) ^
m-l
^ C3 X ll£jllH2rm-mJ-l/2.<2rm-mJ-l/2>/2ma;)
J = 0
w/wc/& is false (see Chapter 1, Section 11.3) when
2r m — m, — \ 1
= integer -\
2m 5 2

7.1 The Adjoint Problem 39
The same proof applies when gj = 0 and u0 =J= 0, by considering
the extension by 0 outside Q. D
We make the following additional remarks:
(i) the preceding counter-example extends (same type of proof) to
the case of Lpy p 4= 2;
(ii) the "explanation" for the preceding counter-example is the
following: for the exceptional values of the parameter, the natural
topology on the space of elements {gjfu0} satisfying the {&. <&.) is
strictly finer than the one induced by
m-l
TT Jj2rm-mj-l/2(2rm-mj-l/2)/2m/2J) x H2ir~ 1/2)m (Q)
j = 0
(See Chapter 1, Section 11 and Section 2 of this chapter.)
Thus if we work in
Tj2rm — mj — 1/2,(2rm — mj— l/2)/2m / y-i\
11 0,0 \^)
(space obtained by interpolation) or in H^r~1/2)m(Q), the preceding
considerations yield no counter-example. D
7. Adjoint Isomorphism of Order r
7.1 The Adjoint Problem
We shall always assume the hypotheses of Section 1.1 to be satisfied.
Replacing P by P* and the boundary conditions pertaining to Bj
by the "adjoint'' conditions pertaining to Cj, we deduce the following
result from Theorem 6.2: let cp be given in H2ir~iym0\l" 1 (Q), real r ^ 1,
2r m and r 4= integer + \\ let v be the solution of
(7.1) A*v - v' = <p in Q,
(7.2) Cj v = 0 onl, 0^'gw-l,
(7.3) v(x,T) = 0, xeQ.
Then
(7.4) VEH2rm*r(Q).
In fact, as cp describes i/20""1*™^-1 (Q), v describes a subspace of
Definition 7.1. 7/ ra*/ r ^ 1 with 2r m and r =J= integer + ^, w
denote by Xr(Q) the space described by the solutions v of (7.1), (7.2), (7.3)
as cp describes ^2(r"1)"-J"1(G)-

40 8. Transposition of the Adjoint Isomorphism of Order r. (I): Generalities
Therefore, by (7.4), we have
(7.5) X'(Q)<=H*""(Q).
Equivalently, we can define:
Xr{Q) = {v\veH2™>*{Q)iCjv = 0, / = 0, . . ., m - 1, v{x9 T) = 0,
7.2 Adjoint Isomorphism of Order r
Providing Xr(Q) with the norm of the graph
IMIxr(Q) = (\\v\\2H2rm.r(Q) + || P* V || ^(r- l)«.r- l(Q)) 1/2,
we have
(7.6)
3 \
for r ^ 1 #w^ ze;#& 2^ m #w^ r =J= integer + ^, P* [ = ^4*
« an isomorphism of Xr(Q) onto H2^'1^'1 (<?).
This is what we call the "adjoint isomorphism of order r'\ D
Remark 7.1. The choice of the space ^2(r"1)";J"1 (0 (as "domain"
of <p) is, of course, completely arbitrary. We have considered this case
for the following reasons:
(i) #2(r-1)^o-1 (Q) is a space for which we have a "natural"
regularity theorem;
(ii) &{Q) is dense in H2ir~ 1)m0-yx (<?).
In fact, in Chapter 2, Section 6 (see (6.2)), we introduced X* (£2)
with A* v e Hl(Q) for analogous reasons; here, the space #2(r~"1),^S~"1(())
"replaces" the space HrQ{Q) of the elliptic theory. □
We shall now transpose the isomorphism (7.6).
8. Transposition of the Adjoint Isomorphism of Order r.
(I): Generalities
8.1 Transposition
We still assume that the hypotheses of Section 1.1 are satisfied.
By transposition we deduce from (7.6):
f let v -► L (v) be a continuous antilinear form on Xr (Q); there
(8.1) ] exists a unique u e (#2(r~"1)£o~1 ((?))' such that <«, A*v-v'y
= L(v) VveX'(Q)t

8.3 The Spaces H-"'-P(Q), H'*^(27), a, $ ^ 0 41
where the brackets < , > denote the duality between
(tf2*-1^-1^))' and ^2(r-»^-1(«- □
8.2 Orientation
We shall now — until Section 12 inclusively — specify and
interpret (8.1).
For this purpose, we shall:
(i) interpret the space (H2^'1^"1 (Q))'\ Section 8.3;
(ii) choose L; Sections 8.4, 9 and 11;
(iii) state trace theorems; Section 10;
(iv) interpret (8.1); Section 12. □
8.3 The Spaces #""'""'(0, H~"-?(£), <*,$ ^ 0
Definition 8.1. Let <x, /? e R and oc, fi ^ 0. We define:
(8.2) H-"--'(Q) =dual of H<0%(Q)
(where we recall that H^(Q) = closure of ®(Q) in H«-p(Q)).
If ft is an integer, we obtain from (2.1) and the Hahn-Banach theorem
that every continuous antilinear form on H^(Q x Rt) may be written
(non-uniquely) as:
(8.3) (ymO = (y>0,v) + (tpl9D'tv),
where
y0 eI2(Rt; H-«(Q)) (H~«(Q) = dual of Ha0(Q)),
(see Chapter 1, Section 12) and where
?16l2(Rr;ff°(fl)).
Then yj e H~"'~P(Q) may be represented by:
[ y = Vo + £>?v>2>
(8.4) < y)0eL2(0}T;H-«{Q)),
[ W2eL2(0fT;H°(Q)). D
If /? w watf aw integer, we may still write a formula of type (8.4) by
using fractional derivatives in t. D
In the same way, we define #■*•■"* (27) = dual of H*\%(Z) and, for
integer /? (with fractional derivatives for /? =J= integer), every y e
e jj-oc,-p ^) may be represented, non-uniquely, by
(8.5) j ^0eI2(OJ;^«(r)),
y>2eL2(0tT;H°(r)) {= H«{Z)).

42 8. Transposition of the Adjoint Isomorphism of Order y. (I): Generalities
8.4 (Formal) Choice of L
With the notions of Section 8.3, we may state (8.1) in the form:
let L be a continuous antilinear form on Xr(Q); there exists
a unique
«e#-2(r-1)m--(r-1)((?),
such that
(8.6)
(u,A*v - vfy = L(v) VveXr{Q).
Furthermore
!L -> u is a continuous linear mapping of (Xr (()))' (antidual of
X'{Q)) - fir-2(r-l)w.-(r-l)(g)B D
We now formally choose (in the sequel we shall be concerned with
giving meaning to this!):
r V-1 /• /•
(8.8) L(u) = Xfvdxdt + J] ^1^^+ Uoy(^,0)^,
Q Z Q
where /, gj and u0 are "functions" given in Q, on 27 and in Q.
We shall "show" (this will also be justified in the sequel) that
(8.9) A u + u' = f in Q,
(8.10) Bju = gJt 0 ^ / ^ w - 1,
(8.11) u(x,0) = u0.
Indeed, we first apply (8.6) and (8.8) with v e@[Q). Then the
integrals on S and Q vanish and there remains
<utA*v - v'} = [jvdxdt Vv e@{Q)
Q
therefore (8.9) in the sense of distributions on Q.
Then, assuming that Green s formula holds (which will be shown
later on), we deduce from (1.12a) that
(Pu,v) | = j fvdxdt\ = {u,P*v) + (5«, €v)j, - {Bu.Tv)^ +
+.(u[T),v(T))Q-(u(0),v(0))Q;
but if veXr(Q), we have:
Cv = 0 and v[0) = 0,

9.1 The Space E2rm-r (Q) 43
therefore
(8.12) [u,P*v) = j fvdxdt + {Bu.Tv)^ + («(0), u (0))fi
Q
and comparing with (8.6)-(8.8), we obtain (8.10) and (8.11). D
Now the problem is to choose f, gj and u0, and then to justify the
preceding formal considerations.
9. Choice off. The Spaces E2rm>r(Q)
Compare this section with Chapter 2, Section 6.3.
9.1 The Space E2rm>r(Q)
Let q be the function introduced in Chapter 2, Section 6.3 (function
"equivalent" to the distance d(x,r) from ^ to T).
We recall (Chapter 2, Section 6.3) the definition of S**(£2), integer
ix ^ 1:
(9.1) E*{Q\ = {v\qMD*veL2(£i)y\(x\ £/*},
which is a Hilbert space with the norm
(9.2) IMI*Kfl> = ( E Ile|a|w2
1/2
L2(Q) I
Next, let d (t) be a fixed infinitely differentiable function on [0, T]
such that
f t if t < t0
(9.3) d(t) = \
[T - t if T - t0 £ t £ T,
t0 fixed with 0 < t0 < T — t0.
Let us start with the case "integer r". We define
(9.4) E2rm-r{Q) = {v | dJ(t) ^>el2(0, T;S2<r-J>m{Q)), O^j^r}.
Provided with the norm
(9.5) || v ||32rm.riQy = ^£ II ^»U)IIw(0.r;Sa(-i)»(O)))1/2.
it is a Hilbert space. D
Proposition 9.1. The space @(Q) is dense in E2rm,r{Q).
Proof. 1) It is sufficient to show that the space of elements v eE2rm,r (Q)
and having compact support in Q is dense in S2rm,r (Q) (then we regularize).

44 9. Choice of /. The Spaces E2rm*r{Q)
2) Let de(x) be a sequence of functions belonging to @{Q) and
cpe(t) a sequence of functions belonging to £&(]0, T[) such that
de(x) = 0 if ^(*,T) ^ e,
<J,(*) = 1 if <Z(*,r) ^ 2e,
y8(0 = 0 if t <Le or t^T-e,
<pe(t) = 1 if * ^ 2e and t ^T - 2s,
and such that
f |^(*,r)w Z)"(5e(*)| ^ constant, |a| g 2m,
1 I # yV) (0 I + I (^ - *)' 9^ W I ^ constant, £ ^ r.
Such sequences exist.
Let ueS2rm'r(Q)] we introduce:
(9.7) ue = de{x)cpe{t)u.
It is sufficient to show that ue -> u in S'2rm'r(^) as's -> 0, therefore
that
dW-W"dW~W in ia(o.r;S2fr--»-(«).
therefore finally that
e(*)w d(t)JDiD$ue -> e(*)M d{t)3D3tD%u
(9-8)
in L2(fi x ]0, T[) as e -> 0, |a| ^ 2(r - /) m.
Replacing ue with its value (9.7), we see that we shall have (9.8) if
(9 9) ! pH d{t)J ^ {D* 6e) {D''k Dlu)^° in L2{Q)•
But the functions appearing in (9.9) may be written:
(QW Di de) Wf <pf) (d(ty->< eM DJt~k Dlu);
according to (9.6), the first two terms are bounded and furthermore
tend towards zero a.e. as e -> 0, from which (9.9) follows by an
application of the Lebesgue theorem. D
For the case "non-integer r"', we define S2rm'r(Q) as follows:
(if r = k + \ - d, 0 < 0 < 1, integer k > 0,
(9.10)
[ S2m>r(Q) = [52(*+l)m,*+l(g) £2fcm.fc(0] Q

9.3 Choice of /. The Space Dpir~l)(Q) 45
Proposition 9.1 is still true for non-integer r, thanks to the properties
of the interpolation spaces [ , ]0 (see Chapter 1, Section 2). D
Proposition 9.2.
Xr(Q) c S2rm'r(Q) for r ^ 1, 2r m and r 4= integer + \.
Proof. FoUows immediately, since (see Section 7) Xr (Q) cz H2rm>r (Q)
and obviously H2rm'r{Q) cz S2rm'r (Q). D
Remark 9.1. (Compare with Chapter 1, Section 6).
We have sought "the smallest possible" space which has the two
properties of Propositions 9.1 and 9.2 and which furthermore is
independent of the boundary operators Bj (in order not to increase the
technical difficulties). This has led us to the spaces S2rm,r(Q). D
9.2 The Space 3'2lm^r(Q)
By definition, we set:
(9.11) E-2™-~r{Q) = (S2rm'r{Q)y, r^O.
Proposition 9.3. The space E~2rm,~r(Q), for integer r ^ 0, coincides
with the space of distributions on Q which may be written (non-uniquely):
(9.12)
where
f = ZD«xDJt(QW(x)dJ(t)faJ(x,t)),
<X,J
|*| g 2(r - /) m, 0 g / ^ rt
(9-13) LjeL2(Q).
Proof. By the definition of S2rm,r (Q) and the Hahn-Banach theorem,
e^ery continuous linear form on this space may be written
M
(?) = I / QM & »JtD%<p- gaJ dx it, gxJ e L> (Q).
By Proposition 9.2, M is defined by its values for <pe@{Q), from
which (9.12) foUows (setting faJ = (- l)"+j gaJ). D
9.3 Choice of/. The Space D^'1*(Q)
From Propositions 9.1 and 9.2 we obtain:
Proposition 9.4. For f given in E~2rm,~r(Q),
(9.i4) *-►</;»>
is a continuous antilinear form on Xr(Q). D

46 10. Trace Theorems for the Spaces JDp(r~1)(0), r ^> h
In (8.8), we shall take </, v} under the preceding conditions. Thus
we shall obtain a solution s/e #~2(r~1)m'~(r~1)((?) such that Au +
+ u' = /, therefore
Pu = A u + W eE-2rm'-r{Q),
which leads to the following definition:
(9.15) £>p(r_1)((?) = {u\ueH-2^-l^-^-l^{Q)t P u eS-2rm^r{Q)}
(r ^ 1, 2rm and r =J= integer + -J).
Provided with the worm o/ the graph
(9.16) IIwIIdp^"1^) = (||«||H-2(r-l)«.-(r-l)(Q) + || P « || J-2rm.-r(Q))1/2
Z)p(r_1)((?) is a Hilbert space. D
We shall now obtain trace theorems for Dpir~ly(Q), which will
show how (8.6)-(8.8) "contain" (8.10)-(8.11).
We shall follow the methods already used in this book (in particular,
see Section 6 of Chapter 2). But we call attention to the fact that we
shall not obtain "optimal" results as we did for the elliptic case. Indeed,
as will be explained in detail in Section 12.3, we can globally, by
extension by continuity define the operator a u = {BjU, u(x, 0)} for the u's
in Dpir~1)(Q); but we face rather substantial technical difficulties if
we seek to "separate" the operators u(x, 0) on Q and B u on Z in a u
and to obtain the "optimal" spaces for u(x,0) and B u separately.
Nevertheless, we believe that the results we shall obtain are already
satisfactory from the point of view of applications. D
10. Trace Theorems for the Spaces Dp0""1*©), r £ 1
10.1 Density Theorem
Theorem 10.1. Under the hypotheses (1.1), (1.2), (1.3) of Section!.I
and if for every 0 e
and t0 e [0, T] the operator Alx,t0, ) +
dx )
71 71
+ {-l)m eie D2m is properly elliptic in D x R, the space Qj{Q) is dense
in Dp^r~1)(Q) for every r ^ 1 with 2rm and r 4= integer + \.
Proof. 1) Let u -> N (u) be a continuous antilinear form on Dp (r~*} (Q);
according to (9.15), (9.16) and the Hahn-Banach theorem, we may
write:
(10.1) N{u) =(ip0,u> + <>!,i^>,

10.1 Density Theorem 47
where
„,. r- rj2(r-l)m,r-l (n\
Wo^H 0,0 (v)>
WleS2rm'r(Q)t
and where the first (resp. second) bracket denotes the duality between
^(r-Dm.r-l^ ^ R_ 2(r_ 1)m,-(r- i) ((?)
(resp.£'2rm'r((2) and S'-2rm'-r ((?)).
Assume that
(10.2) N((p)=0 V<pe@{Q).
We shall show that this implies
(10.3) N(u)=0 VueD;ir~iy{Q)
(which will prove the theorem).
2) Let y)0, ^x be the extensions of the functions ipt to R" x Rt by 0
outside Q. The function y)x belongs, in particular, to L2 (Q), therefore
Let 0 be arbitrary in ^(R"*1); if cp denotes its restriction to Q,
we have:
(10.4) <^0, ®> + <Vi, P~®> = <Wo > $> + <Vi > Tyy
where P = A -\ is an "extension" of P, A = A lx,t, ) being
dt \ dx]
an operator defined in R"+x, with coefficients of class C00, which coincides
with A on Q and is such that the operator
X(x,t0, J + (-l)mei9Dym is, for every d e
71 71
T'T
and every t0 e [—e, T + e], properly elliptic in (PxRy, where (9 is a
suitable neighbourhood of D and £ is a suitable positive real number;
the first (resp. second) bracket on the left-hand side of (10.4) denotes,,
for example, the duality between £r2(r-1)J;r0"1 (R^1) and its dual (resp.
between L2(R^X) and itself). According to (10.1) and (10.2), we have:
<Vo.#> +<Vi,-P"#> =0,
from which we obtain
(io.5) n + (P)*fi = o
or
(10.5 a) yj0 + {A)*y)l--ZL = o.
ot

48
10. Trace Theorems for the Spaces Dpir~ly(Q), r ^ 1
The function ip1 belongs to L2((P x ]— e, T + e[), vanishes outside Q
and satisfies:
(10.5b) (Z)*^ --^-=Vo in <Px]-e,r + «[.
But there exists a unique function w (see Theorems 5.3 and 6.2,
f 3J lm_1
which apply since the system |—-\ covers every properly elliptic
operator) such that \dv W-°
dw
(10.6)
(A)*w = — yx0 in (P x]—e,T + e[,
dt
d3w
= 0 on d(9x]-e,T + e[, Og/^w-1,
dvJ
[ w(x, T + e) = 0,
where, since (as r and 2rw + integer + i) y>0 e ^2(r'"1)wJ'o~1 (# x
x]—e, r + e[), we have:
(10.7) ' w e H2rm-r((9 x] -e, r + e[).
We shall show that
(10.8) ^ = w.
Indeed, let <p be arbitrary in @(Q)\ we introduce v, solution of
A v + v' = <p in 0
(10.9)] —- = 0 on d(9x]-e,T + e[, 0^/^m-l,
v(#, — e) = 0.
We have
- [ <((^)* -A)yi.g> = <Vo,«> (by (10.5b))
(10.10) «! = <^1, (^ + jDf)v> (since %p1 has compact support in
[ 0x]-e,T + e[) =<Vi,^>.
On the other hand, since w satisfies (10.6) and (10.7) and since v
is infinitely differentiable in 0 x [—e, T + e] (since (°| Hktk/2m((P x
_ fc
x ]— e, T + e[) = C°°-functions in 0 x [ —e, T + e]), we have:
<((*)* -D,)w,f>> = -<y>0^>
= <w, (^ + Dt) v} (by Greenrs formula, Section 1)

10.2 Trace Theorem on E 49
which, upon comparison with (10.10) shows that
<Vi>P> = <J*>,f> V<pe^(<2);
whence (10.8).
Therefore, according to (10.7), y}1 e H2rm'r{(P x]-e, T + e[) and
since <p1 vanishes outside Q, it follows that ipx and all its derivatives
with well-defined traces vanish on the boundary dQ of Q. Therefore
(10.11) ^eC(^ ]0,r[)
and (by restriction of (10.5) to 0:
(10.12) v>o + **Vi =0 in Q.
3) Now, if ueDpir-iy(Q), then we have:
<Pf*",V>i> = {u,P*y)iy
and therefore
N{u) = <Vo'+ P*y)ltu};
from which we obtain (10.3) with the help of (10.12), which proves the
theorem. D
10.2 Trace Theorem on £
Theorem 10.2. Under the hypotheses of Theorem 10.1 and (1.6) of
Section 1.1, the mapping u -+ BjU of @{Q) -+ @{E) extends by continuity
to a continuous linear mapping, still denoted by u -+ Bj u, of
£)-(r-l)/0\ _^ JJ-2rm+2m-mj—l/2,-r+(2m-mj-l/2)/2m/2J\
for every r ^ 1 with 2r m and r 4= integer + •£.
Proof. 1) We shall define all BjU simultaneously, i.e. B u (notation
of Section 1). Let us for the moment accept the following lemma without
proof.
Lemma 10.1. There exists a right inverse
(10.13) g={g}7=o1-R(g)=v,
a continuous linear mapping of
m-l
(10.14) yi jj2rm-(2m-mj-l/2y,r-{2m-mj-l/2)/2m,£\ _> -f~
j = 0
where
(10.15) TT = {v\ve H2r% {Q), P* v e H2'-1^'1 (Q)}
such that
(10.16) €v = 0, Tv=g.

50 10. Trace Theorems for the Spaces £>J(r_1)(0)f r ^ 1
m-l
2) For given ueDpir~1}(Q) and given g e fj (which here denotes
the first space in (10.14)), set J = 0
(10.17) Z(g) = -<Pu,v) + <^,P**;>,
where the first (resp. second) bracket denotes the duality between
E-2rm>-r{Q) and E2rm'r{Q) which contains H2rm>r(Q) (reps, between
#-2(r-l)m.-(r-l)((?) and #2(r- l)m.r-1 ^ and where y = Rgy
The (antilinear) form Z (g) is independent of the right-inverse (which
of course is not unique); indeed, if vx is another function of Y° (defined
by (10.15)) satisfying (10.16), then v — vt = w satisfies:
€ w = 0, T w = 0
and therefore, {C, T} being a Dirichlet system of order 2m:
dkw
(10.18) = 0 on 27, 0 ^ k ^2m - \.
dvk
But
P*weH2«-l)Z;r0-l(Q), i.e. A* w - —e H2^1^1 (0
or
and therefore
I dw\ 1
£>£L4*w =0 on 27, V/> with \f\^2(r - \)m .
dw
Writing first that A* w — = 0 on 27 and since, by (10.18),
= 0 on 27, we see that,^4* z# = 0 on 27, which, thanks to the el-
dt d2mw
Hpticity of A*, implies 2m = 0 on 27. Next, we take \p\ = 1, .
and we obtain
dv2m
dkw 1
(10.19) - = 0 on 27 for 0 ^ k < 2r m ,
dvk 2
by recurrence.
Furthermore, vu>(0) = vu>(T) =0 if 0 ^ / < r - -J, with an
analogous condition for z^, therefore for w, which together with (10.19)
shows that
(10.20) wefl2rJr0(e)((1,).
((1)) Indeed, we verify, as in Chapter 1, Section 2, that if w satisfies (10.19)
and
«/C/> (0) = wU) (T) =0, 0 ^ / < y — i,
then w belongs to the closure of 9{Q) in H2rm'r(Q), i. e. in H*%£(Q).

10.2 Trace Theorem on 2J 51
Then
<u, P* w) = (P uy w},
which shows that
~(Pu,v} + <u,P*vy = - (Puyv{) + <u,P*v{)'y
whence our assertion.
It follows from Lemma 10.1 that the form g -+ Z(g) is continuous
m- 1
on Y[ > therefore it may be written
j=o
(10.21) *d) = <&.i>, &e("n)'.
In this way, we define a linear mapping
«-& of ^^"(^^fnY,
which is continuous: if u -> 0 in Z)p (r~1} (Q) and if g belongs to a bounded
m-l
set of Y[ y then R (g) belongs to a bounded set of ir9 so that, according
j=o
to the definition (10.17), Z(g) -> 0 uniformly in g and therefore ftu -> 0
3) Now, assume that ue@{Q). Then Green's formula (Section 1),
which we are now allowed to usef shows that
Z(g) = (Bu,T w)s = <5«f|>
and therefore:
(10.22)
| for *ue@(Q), we have <$, — Buu,g} = 0
m-l
Vg e f|, therefore f$u = E u.
j=o
This proves the theorem if we can prove Lemma 10.1. D
Proof of Lemma 10.1. Let us write all boundary conditions to be
satisfied by v.
First, we must have (10.16), which, since {€, T} is a Dirichlet system,
is equivalent to (see Chapter 2, Section 2):
(10.23) —Vr = hk> 0 ^ ft ^ 2m - 1,
with
(10.24) hk given in ^-(fc+i/2);r-(Hi/2)/2W(i:)i
Next, we must have
v e H2rm:r0 (Q) and P* w = A* v - v' e H2*'1^' * (Q),

52 10. Trace Theorems for the Spaces Dp{3r"1)(Q)t r ^ 1
which entails two types of boundary conditions:
(i) on U:
(10.25)
dkA*v dkv' 1
17-17' 0SA<2(r-l)*-T;
(ii) on Q0 and QT:
dJv dJv 1
(10.26) -^j-(*,0) = —T(x,T) =0, 0^/<r- —
Apply (10.25) with ft = 0; then v' \s = h'0> therefore
A* v \s = % e^2rw-1/2;r0"(2w+1/2)/2w(^).
and this together with (10.23) yields
d2mv
d'v2
. 7L e rT2rm-(2m+l/2),r-(2m+l/2)/2m/y.x
By induction, assume that (10.25) implies
(10.27)
7, _ tj2rm-(2m + fc+l/2),r-(2m+fc+l/2)/2m/y\
= hlm + k tU ,0 U<)
dv2m+k
for ft <^ ft0 — 1; then, for ft = ft0, (10.25) yields
dv
,*o
5v
fco
. TT2rm-(fc0+l/2),r-(2m+k0 + l/2)/2m/y.x
whence (10.27) for ft = ft0.
Finally, £&£ boundary conditions are equivalent to
(10.28)
dkv
hk given in ^-(Hi/2);r-(ft + i/2)/2m(i:))
0^ft<2rm-£
#w^ to (10.26) (note that we do not have global compatibility conditions,
since 2r m and r + integer + ■£).
Remark 10.1. Of course, we define S, * for ueDpir~iy(Q) in the
same way, we have:
the mapping u -> Ssu of @(Q) -> ^(£) extends by continuity
to a continuous linear mapping, still denoted by u -> SjU, of
(10.29)
D;«-ly[Q)
tt-2(r-l)m-(orders^)-1/2,-(r-l)-(order Sj + l/2)/2m/y.\

10.3 Continuity of the Trace on Surfaces Neighbouring 2 53
It follows that
(10.30)
dku _ _
the mapping u -> of @(Q) -> @(Z) extends by continuity
dvk
dku
to a continuous linear mapping, still denoted by u -> ——, of
dvk
Z>""(r-1)(0) _> #-2(r-l)m-fc-l/2,-(r-l)-(fc+l/2)/2m/JT\ Q
Remark 10.2. Every distributions u e H~ °° (Q) = [j H~k{Q) belongs
fc = 0
to a space H~2ir-1)m^ir-1)(Q) for a suitable integer r. Theorem 10.2
therefore implies that, for every distribution u in H~ °° (Q) such that
P u e (J S~ 2rm'r (Q), we may define Bj uonZ as a distribution in H~ °° (Z).
r
In a very approximate manner, we may say that we shall see in Volume 3
how this can be extended when u is an "arbitrary distribution" on Q
such that P u" does not grow too rapidly" in the neighbourhood of Z;
then we can still define BjU on Z, but they are no longer distributions
in the usual sense. Q
Remark 10.3. We now have the following partial Green's formula
(see also Section 12.3) at our disposal:
<P u, v} - <u, P*~v} = -(Bu, T v}
for
« e Dpir~1}(Q), ve H2r%(Q), P* v e #2(r"U'o"x (Q)
CjV = 0, / = 0, . . ., m - 1. D
10.3 Continuity of the Trace on Surfaces Neighbouring Z
Let re be a family of surfaces which are "parallel" to r and "tend
towards i"1" as q -* 0 (see Chapter 2, Section 8.1).
We assume that a correspondence
(io.3i) x-*v(*ie) of re-*r>
which is infinitely differentiable both ways, has been established.
Let
(io.32) re=rex]o,r[.
We assume, as we are allowed to do, that the coefficients of the operators
{Bj}, {Cj}, {Sj} and {Tj} are defined and infinitely differentiable not
only on Z but in Q - QQQ, so that the systems {Bj}, {Cj}, {Sj} and {T3}
are normal on re for 0 ^ q ^ q0, 0 ^ t <Z T. We introduce the sub-
space of Dpir~1){Q) defined by
(10.33) Dpir~1\Q) = {«|«6ff-2fr-1)m--°,-1)(0), P^eL2(<?)}

54 10. Trace Theorems for the Spaces £>p(r_1)(0), r > 1
and in analogous fashion
(10.34) D;ir-1}(Qe) = {u\ue fl-ao-i)..-*-!)^, Pue L*{Qe)},
with the norms of the graph. We note that the restrictions to QQ of the u's
in Bpir'^(Q) belong to £p(r_1)((?e)- By the _same type of proof as
in Sections 10.1 and 10.2, we can show that @{Q) (resp. @{QQ)) is dense
in i5p(r-1)(()) (resp. ^p(r_1)(^e)) and that we have a trace theorem for
the u's in i5p(r_1)(<2) (resp. Bpir~iy(QQ)) which is the analogue of
Theorem 10.2. Therefore, we can define BjU \z and BjU \E for the u's
in B^'^(Q).
With the aid of transformation (10.31), we can — tranfer of
structure — define the image of BjU \s on S\ we set:
(10.35) Bfu = image of BjU\Eq under (10.31).
By the trace theorem (and the fact that (10.31) is infinitely
differentiate both ways) we have:
(10.36) S5?)WeJHr-2(r-l)m-(mJ+l/2).-(r-l)-(m</+l/2)/2m(2^.
Theorem 10.3. Hypotheses and notation of Theorem 10.2. If
u e Dpir~1)(Q) and the operators Bje) are defined by (10.35), then
(10.37) Bf U-+ BjU in #-2(r-Dm-(m^ 1/2), -<i-l)-<mj+l/2)/2iii(£)
as q -> 0.
Proof. If ue@(Q), then (10.37) is obvious — with convergence
in £&(E). Since £&(Q) is dense in Dpir~1)(Q), it is sufficient to show that
\Bf remains in a bounded set of
J^Yi5~(r~1)(0)* /f-2(r-1)m-(mJ+1/2>»-(r-1)-(mJ+1/2)/2m/^\\
as q -+ 0.
(10.38)
Set
TT(e) __ ]~[ ^-2(r-l)m-(mJ+l/2),-(r-l)-(mJ+l/2)/2m/^
J J = 0
and (compare with (10.15)):
rt = {v\ve H2'% (Qe) ,P*ve H^~ ȣ-1 (0,)}.
For g fixed in
m-l
FT /f-2(r-1>m~(mi+1/2).-(r-l)-(mJ+l/2)/2m/2^\
J = 0
we have to show that \(BiQ)u,g}\ ^ constant (independent of q).
After a transfer of structure by (10.31), this amounts to showing
that
K5wU,»Sc>l ^ constant,

10.4 Trace Theorem on Q0 55
/
where (in particular) g6 belongs to a bounded set of fj(e). But (by the
, definition of B u j^-J: J
<Su \St, q = <«, p^e>Qp - <p u, veyQe,
where
vQ = Re(ge)
and 7^e = right-inverse analogous to the one in Lemma 6.1 but passing
from ZQ to QQ (instead of Z to Q).
It suffices now to verify that we can choose RQ in such a way that
R6 (g6) remains in a ball of i^Q with radius independent of q , as q -* 0.
However, this follows from the proof of Lemma 10.1 and from
Theorem 2.2, since all local maps take place with functions which, together
with all their derivatives, are bounded in q. D
Remark 10.4. Assume that u is given with
(10.39) ueH-°>{Q)t Pu = 0.
Then (see also Remark 10.2) u e #-20-i)m.-o-D(<2) for a suitable r
and on the other hand, since P is hypoelliptic (see S. Mizohata [2]; we
shall prove this in Volume 3), we have:
(10.40) ue£(Q) =C°°((?).
Therefore
(10.41) Bju\s,e*(ZQ).
Therefore in this case
(10.42) B(B) W6jff-2(r-l)»-(»J+l/2).-(r-l)-(»J+l/2)/2»^ n ^ (£) ,
(but this does not alter the nature of the convergence in (10.37)). D
10.4 Trace Theorem on QQ
Theorem 10.4. Under the hypotheses of Theorem 10.2, the mapping
u -» u (x, 0) of @{Q) -» @ (D) extends by continuity to a continuous linear
mapping, still denoted by u -» u{x, 0), of Dpir~1){Q) -» H-2ir-ll2)m(Q),
for all r ^ 1 with 2r m and r =j= integer + \.
Proof. Let q) be given in H2ir~1/2)m(Q). For the moment we assume
Lemma 10.2. There exists a right-inverse
<p-*Ri(<p) = v>
which is a continuous linear mapping of Hlir~1,2)m[Q) -» irl, where
(10.43) r± = {v\veH2^r(Q)}P*vEH2ir-1Xo~1(Q)}>

56 10. Trace Theorems for the Spaces £>p(r-1)(0), f ^ 1
such that
(10.44) Cv1=0i Tv1=0
and
(10.45) vx(x, T) = 0, v±(x,0) = <p(x).
Then if we set
(10.46) Zi(<p) = -<i>w,0i> + <«,i>*t>i>,
we note (as in the proof of Theorem 10.2) that Zx is independent of
the choice of the right-inverse (satisfying (10.44) and (10.45)), therefore
depends only on q>. The antilinear form <p -* Zx (<p) is continuous on
#o(r~1/2)m(£) and therefore of the form
(10.47) Z±(9) = <ru,<p(x,0)>, rtte^-2(-1/2>-(D).
The linear mapping u -» ru maps Dpir~1\Q) -» i7-2(r-1/2)m(D)
continuously.
If u e 3(Q), then (we have done all that is necessary for this)
ru = u(x,0),
which proves the theorem if we can prove Lemma 10.2. D
Proof of Lemma 10.2. First of all, conditions (10.44) are equivalent to
3k
(10.48) —7 = 0 on St 0^H2»-1,
and (see the proof of Theorem 10.2) the other boundary conditions will
be satisfied if we (bluntly) take
(10.49) —- = 0 on 2", 0 = * < 2rm .
dvk 2
Choose vx such that
(10.50) v[J){x, T) = 0, 0 = / < r - \.
Next, we must have
M*,0) = <p{%),
P*v1(x,0) = 0,
i.e.
vi(*,0) =A*(peH20ir-1-1/2)m(Q)
and so on, whence
f v[J)(x,0) = (pj given in #2(r-'-1/2)m(i2),
(10.51)
! 0 ^ 7 < r - \, 9?0 = 9?.

11.1 The Spaces H2"" E*(E) 57
According to the trace theorem, Chapter 1, Sections 3 and 4 and
according to Chapter 1, Section 11, there exists a continuous linear
mapping
{yjYjZh-tv, of n^(r--/-1/2)m(fl)->ff2""-r(e),
J = o
which satisfies (10.49), (10.50) and (10.51), whence the lemma. D
Remark 10.5 (Partial Green's formula). We now have the following
"partial Green's formula" at our disposal (compare with Remark 10.3):
(10.52)
<Pu,vy - (u,P*v> = -<u(x,0),v(x,0)>,
u e D?r-"(Q), v e H2""-'(Q), P* v e H2ir'1)m0'fr0-x (Q),
v(x, T) = 0,
Cv = T v = 0. D
10.5 Continuity of the Trace on Sections Neighbouring Q
Denote by QtQ the section
fi*o =(?n{* = *0}, 0<to<T.
By restriction to <2*0 = !2*0 x ]t0, T[, if w e Bpir~iy(Q), we have
w eSp^'^fftJ (definition analogous to (10.34)) and therefore, by the
analogue of Theorem 10.4 for Dpir~1)(Q)t we may define
u(x, t0) e#-2(r-1/2)m(!2,0) « i7"2(r-1/2)m(D).
Then, with the same type of method as for Theorem 10.3, we can
prove
Theorem 10.5. Hypotheses and notation of Theorem 10.4. For u given
in Dpir~X){Q)} we have
(10.53) u(x, t0) -» «(*, 0) in H-2<r-1/2)m{Q) as t0 -» 0.
11. Choice of gj and w0. The Spaces H2«m B"(£)
11.1 The Spaces H1(xm3"(2:)
We start with the case integer oc (^ 0).
Let d(t) be the function defined in (9.3). We define:
(11.1) H2«m3«{E) = {v | dJ(t) vU) e L2(0, T\ H2<"-"">m(r)),
0 ^ / ^ oc};

58 11. Choice of gj and u0. The Spaces H^E^iE)
provided with the norm
1/2
this space is a Hilbert space.
For the case non-integer oc we use interpolation and intermediate
spaces; for oc = k + 1 — 0 and 0 < 6 < 1, we define
(11.3) H2«m3«(Z) = [H2^^m3k+1{E)tH2km3k{E)]d.
Proposition 11.1. The space 2{S) is dense in H2"m3"(Z).
Proof. 1) Since X is dense in [X, Y]0, it is sufficient to prove the
proposition for integer oc.
2) Introduce <pe(t) as in the proof of Proposition 9.1 and, for u
given in H2"m 5" (27), introduce the functions
(11.4) ue = <pe{t)u.
It is sufficient to show (J1 being a compact variety) that
ue -» u in H2<xm3*(Z) as s -» 0,
therefore that
dJ(t)u™->dJ(t)uW in L2(0, T; ff2<"-''*>",(J1))> O^j^oc,
and therefore, replacing ue by its value, that
(11.5) dJ{t)<p?)uu-V{t)-*0 in L2(0fT;H2^-J^m{r))} l^k^j;
but this can be written
which tends towards zero in L2(0, T; tf^-u-w/"^/')) and therefore
in L2(0, T;H2("-j/")m(r)); whence (11.5) and the proposition. D
Now, we define:
(11.6) H-^S-'iZ) = 'dual of #2"w2*(27), real * ^ 0.
Thanks to Proposition 11.1, we have:
(11.7) H-2<xmE-<x{Z) c 2'(Z).
For integer oc, the structure of H~2am3~*{E) is given by:
every distribution f e H~20lm 3~*(Z!) may be represented by
(11.8)
f = ZDi(d>(t)fj),
J=o
where fj e L2(0, T; tf-w-.//*)™^)). D

12.1 Final Choice of L 59
11.2 Choice of gj
Proposition 11.2. Let gj be given with
(11 9) „ G fl-2(r-l)m-(mj+1/2) &-(r- l)-(/n,+ l/2)/2m / y\
for 0 ^ / ^ m — 1. 77&0n £/&0 antilinear form
(n.io) w-I<gj,r,w>
J = 0
?.'s continuous on Xr(Q), r ^ 1, 2rw #?z^ r =j= integer + ^.
Proof. If ^elr (5), it follows from Theorem 2.1 and the fact that
the order of Ti = 2m —ms — 1 that
T i) c. ZT2rm-(2m-mj- l/2),r-(2m-mj- l/2)/2m /y\
_ Tj2(r- l)m + mj+ l/2,r- 1 +(mj+ l/2)/2m / y\
Tj2(r-l)m + mj+l/2 ^r- l+(mi+ l/2)/2m / v-»\
the mapping v-+TjV being continuous; whence the proposition. D
11.3 Choice of «0
Here, we use the spaces E" (Q), S~" (Q) introduced in Chapter 2,
Section 6.
Proposition 11.3. Let u0 be given in S~2(y~ll2)m{Q). Then the anti-
linear form
(11.11) v-+(uo,v(x}0)}
is continuous on Xr(Q), r ^ 1, 2r m and r =j= integer + \.
Proof. Since X**(Q) a H2rm'r(Q), v -* v(x,0) is a continuous linear
mapping of ^ ^ _^ H2rm-"(Q) c i5'2(|-1/2)m(D),
from which the result follows.
12. Transposition of the Adjoint Isomorphism of Order i*.
(II): Results; Existence of Solutions in H2mr>r(0-Spaces,
Real r g 0
12.1 Final Choice of L
Let /, gJt and u0 be given with
(12.1) feS-2rm>r(Q) (r ^ 1, 2r m and r * integer + ±),
(12.2) &er2(r-l)m-(m,+ l/2)5-(r-l)-(m,+ l/2)/2m(r)) Og/^W-1,
(12.3) ^0eS,-2(|-1/2>m(D).

60 12. Transposition of the Adjoint Isomorphism of Order r. (II): Results
Then, according to Propositions 9.4, 11.2 and 11.3 the antilinear
form
m-l
(12.4) L(v) = </,«;> + £ <gj.TjV>+<uo,v(x,0)>
j = o
is continuous on Xr(Q). This is the form L which we choose for (8.1).
In this manner, we have justified Section 8.4. D
Remark 12.1. The choice of L we have just made is not the most
general one possible; see Sections 12.3 and 13. D
12.2 Results
Theorem 12.1. Hypotheses of Section 1.1; r is real and r ^ 1, with
2r m and r =j= integer + \. Let f,gj, u0 be given with (12.1), (12.2) and
(12.3). There exists a unique distribution u e Dpir~'1)'(Q) satisfying
(12.5) <«,P*v> = L(v) VveXr(Q) (see Definition 2.1),
where L is given by (12.4). Furthermore
(12.6) {f.gj.Ho}-*"
is a continuous linear mapping of
m-l
S~2rm-r (Q\ x TT #~2(r- l)m-(mj+ 1 /2) 77-(r- l)-(mj+ II2)12m iy\ x
J = 0
x S'2^-1/2^(Q) -+ Dp*'1*®).
The distribution u satisfies
(12.7) A u + u' = f in Q,
(12.8) Bju = gj on 2J, 0 ^ / ^ m - 1
(BjU taken in the sense of Theorem 10.2),
(12.9) u(x,0)=uo in Q
(u(x, 0) taken in the sense of Theorem 10.4).
Remark 12.2. We must be very careful about the following point:
the distribution u is not necessarily unique in satisfying (12.7), (12.8)
and (12.9); uniqueness occurs only if the boundary conditions (12.8)
and (12.9) are interpreted in a stronger sense than in Theorems 10.2
and 10.4, as we shall indicate in Section 12.3. D
Proof. 1) By (8.1), there exists a unique u in

12.2 Results 61
such that
(12.10) <u,A*v- v'y = L{v) VveXr(Q),
L given by (12.4).
Furthermore u depends continuously on L, which will prove the
continuity of (12.6).
2) In particular, we have (12.10) for v = cp e @{Q), therefore (12.7),
which entails that u e £>p(r_1) (Q).
3) Now we take
v e H2rm% (Q), PtveH2*-1 ^o * (Q), Cv = 0
(thus v eXr(Q)) and apply Remark 10.2; we have:
(12.11) <Pu,v} - <w,P*v> = -<S^,F7>;
but
(Pu,v} = </,£>
and
<«, P*7> = I(v) = </, v> + <g, 7^> (since v{x, 0) = 0),
so that (12.11) becomes
(12.12) <g,f^> =<5^,f7>.
Since, according to Lemma 10.1, we may take Tv in (12.11)
arbitrary in
m-l
rr Tj2rm-(2m-mj- l/2),r-(2m-mJ- l/2)/2m / v-.\
J = 0
(12.8) foUows from (12.12).
4) Finally, we take
veH2rm>r{Q), PtveH2^-1^-1®), v{x, T) = 0, Cz; = T v = 0.
Then (10.52) yields
(12.13) <Pu,v} - <uyP*~vy = -<«(*, 0),v(*70)>.
But here again
and
<«, P*~v> = I (v) = </, £> + (uo,v(x,0)y (since f v = 0),
which, together with (12.13), yields
(u0,v(x,o)y = (u(x,o),v(x,o)y,
from which, thanks to Lemma 10.2, we obtain (12.9). D

62 12. Transposition of the Adjoint Isomorphism of Order v. (II): Results
12.3 Complements
We shall complete the statements made in Section 9.3 and in
Remarks 10.3, 10.4, .12.1 and 12.2, relative to the fact that the results we
have obtained are not the best possible ones. Indeed, for example, in
the existence theorem (Theorem 12.1) we have taken the data gj and u0
in smaller spaces than the ones in which we have given trace theorems
(Theorems, 10.2 and 10.4) for BjU and u{x,0), u e Z>p(r_1)(<?).
Thus, we have not obtained a characterization of traces of u's
belonging to D ■
-(r-l)
(Q), in opposition to what we have been able to do in the
elliptic case (see Chapter 2, Section 6, Theorems 6.5 and 6.6).
Let us take a closer look at the difficulties encountered in this problem.
As in the elliptic case (Chapter 2, Section 6.2), in order to obtain optimal
trace theorems we must study the image ^'(Xr(Q)) of Xr' (Q) under the
mapping ^ defined by
v-*rv = {Tov,...,Tm_1v,v(x,0)}, veXr(Q).
But, thanks to Theorem 2.3, we see that ^(Xr(Q)) coincides with
the subspace of the product space
m-l
(12 14") FT #2(r~1>m+mJ+1/2'r~1+(mj + 1/2V2m/T) x #2(r_1/2)m(iQ\
j = o
formed by the functions {<p0, . . ., (pm
compatibility relations1
(12.15). ~(fc)
%p} = {0,ip} which satisfy the
<p)K){x,T) =0, Og * <r-f, xeT,
(12.16)
(12.17)
Tj[x>0>j^jV> = <Pj{%>0) on T
for all / such that 2m — m3 < 2(r — %) m + \
(so that Tjip is well-defined),
d
Cjlx.O,
dx
ip = 0
on
for all / such that 2m — fij < 2(r — £) m + \.
We denote this subspace by yr.
According to Theorem 2.3, there exists a right-inverse
(12.18)
{$> v>} -* R($>v>) = v
1 Note that in the case of the Dirichlet problem [ Bj = Cj = —j 1 and if
r = 1, the compatibility relations are reduced to ip € H™ (G); see Remark 12.3
on this point.

12.3 Complements 63
which is a continuous linear mapping of
-Tr -» {v | v e H2rm>r(Q),P* v e ^2(r"1)o,o"1 ((?)}
such that
(12.19) v<'>(*, T) = 0, 0^/<r--f,
(12.20) Cfl = 0, Tv=y,
(12.21) v(*,0) = y>(*)-
Thus velr(5) and jR is a continuous linear right-inverse of the
mapping y.
For u given in £>p(r_1)((?), we consider the antilinear form
(12.22) {y,ip} -*Z{$,y) = -(Pu,v} + <^,P*z;>, v = R($,ip).
As for Theorems 10.2 and 10.4, we verify that Z(<p, ip) is independent
of the choice of the right-inverse and continuous. It follows that
(12.23) z@,v) =<<yu>{<p>w}y,
where
(12.24) GUEi^r,
the mapping u -* a u being a continuous linear mapping of DP~ir~1)(Q)
into this space.
According to the Green's formula of Section 1 and (7.6), we easily
see that if u e @(Q), then we can identify a u with {E u,u(x,0)} and
therefore, thanks to Theorem 10.1, au is the extension by continuity
of the operator {Bu,u(x,Qi)} from 3(Q) to £p(r_1)((?).
Thus, we habe obtained a new trace theorem for the u's belonging
[ the mapping u-* au = {Bu, u(x, 0)} of £&(Q) into [S(2T)]m x
x S(£?) extends by continuity to a continuous linear mapping,
[ still denoted by u -> a u,
We also have the Green's formula'.
(12.25)
(12.26) X > / \ • / \ /
VueDp^-^iQ) and VveXr{Q).
Finally, we have
Theorem 12.2. Under the hypotheses of Theorem 12.1, if f is given
in S~2rm*~r(Q) and g* in 1^'T, then there exists a unique u in DP~ir~1)(Q)

64 12. Transposition of the Adjoint Isomorphism of Order /. (II): Results
satisfying
(12.27) A u + u' = f,
(12.28) ou = g*,
where (12.27) is understood in the sense of & (Q) and (12.28) in the sense
of (12.25).
But now the problem is to separate the operators B u on S and u (x, 0)
in Q in au.
Now, if'T is a quotient space of
m-l i,
rT ^2(r-l)m+mj+l/2,r-l + (mj+l/2)/2m/JT\ I X (H2ir~ 1/2)m (Q)Y
.J = ° J
which is the dual space of (12.14).
By suitable identifications, this space can be interpreted as a
product of spaces of distributions in the usual sense on Q x S x J"; the
introduction of J" is necessary in order to account for the quotient;
therefore for the case of generalized solutions, the boundary data — if
we want to interpret them in the usual form — do not appear as in the
case of ordinary solutions, see Baiocchi [2] (where this author restricts
his arguments to second order operators, but his method is general);
see also Lions/Magenes [2].
In order to avoid the technical difficulties inherent to the optimal
case, we have, via Theorems 10.2 and 10.4, given meaning to the
operators B u and u(x,0) separately in spaces which are in a certain
sense larger than the restrictions toZ and Q of the "optimal" space1 i^'r.
(Note that Sections 10.2 and 10.4 served to insure that the relations
(12.15) and (12.16) hold by taking all terms of (12.16) to be zero.)
And, in (existence) Theorem 12.1, the data gj and u0 belong to
spaces smaller than the "optimal" spaces. D
Remark 12.3. For the Dirichlet problem with r = 1, i^r coincides
with the product space
m-l
f] HJ+lf2-u+1/2>f2m(Z) xH%(Q)
j = o
of which the dual is (since (/ + %)j2m < \, see Chapter 1, Section 11)
m-l
]-j jff-j-i/2.-t/+i/2)/2»(2^ xH-m{Q).
j = o
Then the "separation" is possible and condition (12.18) can be
written in the form
(12.29) 7jU = gj> j = 0,...,m-l,
(12.30) u{x,0) = u0,

13.1 State of the Problem
65
where gjEH'^1^'u^1'2^2m(Z)9 u0EH~m(Q) and
[ the mapping u -> {A u + u'; y3 u, j = 0, . . ., m — 1;
u(x,0)} is an isomorphism of Dp(Q) onto
(12.31)
3-2m,-i(Q) x YlH-J-1/2>-u+1/2)/2m(Z) xH~m(Q).
J = o
Furthermore, for this case, the space S2mA (Q) can be replaced by
the smaller space
{v | v eL2(0, T; H^{Q))} dv' eL2(0, T\ L2{Q))}}
where d is defined in Section 9.1, and then A u + u' can be taken in
the dual space.
13. State of the Problem. Complements on the Transposition
of the Adjoint Isomorphism of Order 1
13.1 State of the Problem
Let us briefly recall Theorems 6.2 and 12.1; the problem
[ A u + u' = f,
(13.1) i Bju =gj, j = 0,...,m - 1,
I u (x, 0) = u0,
is solvable with:
\ ueH2n*s+1)>s+1{Q), if feH2ms-s{Q),
gj EH2mis+1)-mJ--1/2>s+1-imJ+1/2)/2m(Z), u0 eH2is+1/2)m(Q)
and satisfy the compatibility relations; and this for s ^ 0, with
2s m and s =j= integer + \\
[ ueH2n«s+1^s+1(Q), if fe52m>s{Q),
(ii) \ gj eH2mis+1)-mJ-1/2's+1-imJ+1/2)/2m(Z)tu0 eS2is+1/2)m(Q);
[ and this for s g —1, with 2 s m and s =j= integer + %.
Note that in (i) and (ii), u is a solution of (13.1) in the sense that
A u. + u' = f is satisfied in Of' (Q) and the conditions Bj u = gj and
u (x, 0) = u0 are satisfied in the sense of Theorem 2.1 for (i) and
Theorems 10.2 and 10.4 for (ii).
In summary form, we say that the data are of "regularity s", seR
(s of arbitrary sign), if feH2mStS(Q) or 5'2ms,s(^), and gj and u0 are
'' suitably'' associated.
(i)

66 13. Transposition of the Adjoint Isomorphism of Order 1
Then (i) and (ii) mean this:
if the data are of <f regularity s", with s ^ 0 or s ^ —1, then
the solution u belongs to #2m(s+ *>•s+1 (Q).
Therefore it remains to study the case in which the data are of
"regularity s", with —1 < s < 0.
Following the "plan" of this book, it would be natural to interpolate
between (i) and (ii), in particular between the case s = 0 of (i) and s = — 1
of (ii). But it must be pointed out that we are not in the same situation
as, for example, in the elliptic case studied in Section 7 of Chapter 2,
because, whereas Theorem 6.2 yields an "optimal" isomorphism theorem
for s ^ 0, the situation is different for s ^ — 1 since the result of (ii)
is not an "optimal" isomorphism theorem.
Furthermore, interpolation between the spaces of data which appear
in (i) and (ii) involves considerable technical difficulties.
Thus we shall reconsider the problem without using (ii); we shall
start directly with the transposition of the adjoint isomorphism of
order 1, according to the notation of Section 7 and without making
any simplification; we shall seek the regularity properties of the solution
of (8.1) when the form L.(v) is defined by data /, gj} and u0 of
"regularity s", with -1 < s < 0. D
13.2 Complements on the Transposition of the Adjoint Isomorphism
of Order 1
We again consider (8.1) with r = 1; there exists a unique u e H° (Q)
= L2[Q) with
(13.2) <w,1*v - »'> = L(v) VveX1®),
where L is a continuous antilinear form on X1 (Q).
We again choose L in the form
(13.3) L(v) = </, v> + <g, T7> + <uo,v(x,0)>.
On the other hand the choice of /, g = {g0, . . ., gm_!} and u0 shall
be different from Section 11: we no longer require /, gj and u0 to be
distributions on Q, S and Q. D
Thus we take
(13.4) fe(H2m^(Q)y. U
Since, according to Section 2, v -» T v is a continuous linear mapping
of
m-l
■ x1 (Q) -»ii HmJ+1/2><mJ+1/2)/2m(i;)
j = o

13.3 Orientation 67
we may take
m-l
(13.5) g e n (HmJ+lf2-<mJ+lf2>f2m{Z))'
in (13.3) (the spaces (HmJ+1/2'imJ+1/2y2m(Z))' not being spaces of
distributions on 27). D
Similarly, since v -* v(x,0) is a continuous linear mapping of
X1 (Q) -» Hm(Q), we may take
(13.6) u0e(Hm{Q)y
in (13.3) ((Hm(Q)Y not being a space of distributions on Q). D
In short:
Theorem 13.1. Let /, g, u0 be given with (13.4), (13.5) and (13.6),
under the hypotheses of Section 1. There exists a unique u e H° (Q) such
that (13.2) holds with (13.3). D
13.3 Orientation
We shall seek suitable hypotheses, which depend on s (— 1 < s < 0),
on /, g and u0 such that we can show that the solution u of Theorem 13.1
belongs to H2mis+1)'s+1 (Q). We shall apply interpolation theory in the
following way.
If we consider the mapping ^ (Green's operator for problem (13.1))
(13.7) 9'.{f,g,u0}->» = 9V>g,"o),
then Section 13.1 and Theorem 13.1 tell us that it is a continuous
mapping of
J H°(Q) x I Y\H2m-^J+1^2^1-^J+1^2m(i:) xHm(Q), 0t<g\
and of
m-l
(13.9) (#2m'1((?))' x n (HmJ+1/2^mJ+1^2m(2))f x (Hm(Q))' -» H°(Q).
j = o
We recall that
| n#2m-(mi+l/2),l-(mi+l/2)/2m(2;) X Hm (Q) , 0t%\\
denotes the subspace of the product space whose elements satisfy the
compatibility relations.
We shall now interpolate between (13.8) and (13.9). D
Remark 13.1. The mapping ^ is not injective from
m-l
(H^-HQ))' x n (HmJ+U2.(mrn/2V2m^y x (Hm(Q))' -* H°(Q),
j=o

68
14. Some Interpolation Theorems
since we can choose {/, g, u0} =)= 0 with (13.4), (13.5) and (13.6) such
that the form L(v) defined in (13.3) = 0 on X1 (Q) (since /, g and uQ
may have support onf). D
14. Some Interpolation Theorems
14.1 Notation. Statement of the Main Result
We set (see (13.8) and (13.9)):
(14.1) A0 = j"n H2m-^+1/2hl-^+lf2)f2m(Z) x Hm{Q)tm^\
m-l
(14.2) A, = n (fl-'+1/2'<"'+1/2)/2"(i))' x (Hm{Q))'.
j=o
We also set , .
\H?(Q) if B > 0
(14.3) je"(Q) = \ \
\H"-«/2m(Z) ii «^0
(14.4) Jf*B/2"(2^ = „
\(H-*-"'2m(Z)y if a, ^ 0.
Theorem 14.1. Notation of (14.1) to (14.4). irf m} be the order of B]t
0 ^ / ^ m — 1, ««rf assume (1.6) to 6e satisfied. If
1 1 rninw,
(14.5) 0> i = «
2 4w 2m
then
m-l
(14.6) [^0> ^i)]e = J! je2m^-e)-^+1^1-e'^+1^2m(i;) x
xJeil-2d)m(Q).
Remark 14.1. In (14.5) we always take 0 < 6 < 1.
Condition (14.5) is therefore always satisfied iirj ^ 0, i.e. if minms ^
^ m — %, therefore if
(14.7) Mj ^ m V/.
Then (14.6) Aofcfe V0e]O, 1[. D
Remark 14.2. The compatibility relations do not appear in [yl0, A{\e.
In fact these relations would bring in the expressions Bj u0 \r for
the element {gJfu0}'f if u0 e Ha~2d)m(Q), then £,-^0 |r has meaning
only if the order of Bj = ms < (1 — 26) m — \, therefore if
1 1 m*
(14.8) fl<___ L.
2 4m 2w

14.2 Outline of the Proof 69
Thus the compatibility relations are not defined if (excluding the
ambiguous case of equality)
1 1 mi
6>--- - V/,
2 Am 2m
that is if (14.5) is satisfied. D
Remark 14.3. It seems (very likely) that, for arbitrary 0e]O, 1[,
[AQyA^Q is given by the subspace of
m-1
•pr ^2m(l-0)-(mj+l/2),l-0-(mj+l/2)2m/™ ^(1 -20)m /q\
J = 0
defined by those compatibility relations which have meaning on this space
(with in addition the exceptional cases
i-e-mj + i =±
2m 2
where integral relations appear).
14.2 Outline of the Proof
The compatibility relations- for {gJf u0} can be expressed by
(14.9) Bj tx, 0,—\ u0 = gj(x, 0), xeR
Writing Bj(0) = Bj(x,0, 1, we introduce
\ 8xJ
(14.10) V = {v | v EHm (Q) ,Bj{0)v = 0 for all / such that w^w-l}.D
Remark 14.4. \ims^m V/, then V = Hm(Q). D
Then we introduce
(14.11)
^2m-(mi+l/2),l-(mJ+l/2)/2m/^\ _ \y . y rj2m-(mj+ 1/2), 1 -(mj+ l/2)/2m / y\
v(x, 0) = 0 for x e r if 1 - UlLtl. > __
2m 2
(space which differs from #";§ (27) as defined in Section 2, except if
2w

70 14. Some Interpolation Theorems
We set
f m-l
C0 = n 0H2m-imJ+lf2)>l-imJ+1/2)/2m(Z) x V,
(14.12)
j=o
m-l
Do = n H2m-imj+1/2)tl-imj+1/2)/2m{i:) x Hm{Q).
j = 0
Remark 14.5. In C0 the compatibility relations are trivially satisfied,
each well-defined term of (14.9) being zero. There are no compatibility
relations in D0. D
Evidently C0 cz A0 cz D0 and hence
(14.13) [C0, A{\9 c= [A0> Ax]d cz [D0> Ax]9.
In the sequel we shall specify the spaces [C0, Ax]e and [D0, A^,
spaces which coincide when (14.5) holds, from which Theorem 14.1
will follow. D
14.3 First Auxiliary Interpolation Theorem
Theorem 14.2. We have (in the notation of (14.4)):
(14.14) [ff2"-*'1-*'2"^, (H«-«'2m{Z)y]e
= <?r(i-e)2»-«.i-e-«/2»(2^> 0<oc<2m.
Proof. (This proof is very similar to the proof of Theorem 12.5 of
Chapter 1.)
1) In general, let y and 2tf be two Hilbert spaces, y cz ^, y dense
in £F\ if £F is identified with its antidual and if y is the antidual of y,
then we have: ^ = ^ = ^
IfWCSet ^- = [^„
then
y1-20 if i - id > o
(14.15) [y,y],=
1 (y2*-*)' if i - 20^ o.
This is a consequence of the reiteration theorem (Chapter 1, Section 6.1),
since [y, ir']lf2 = ^ (or else can be easily verified directly).
2) Choose an integer ji such that
(14.16) ji > 2m — oc, /Lt > oc.
By 1) and Section 2.1, we obtain
#2m-«.l-«/2m(£) = ^JJH.H/2m (£) y ((#7^/2m (27))']^ ,
where
(1 - 20!) [i = 2m - oc
and
(ff«.«/2»(23)' = [^•^2m(r),(^''t/2m(i:)),]02,

14.4 Second Auxiliary Interpolation Theorem
71
where
(202 - 1) ji = oc.
In order to simplify the notation, set Hll*ll,2m(Z) = E; then
[#2m-a'1-a/2'm {H«>«i2m(E))f\ = [[E,E']ei, [E,E']d2-]d = [E,E']d3
(by the reiteration theorem), where
e3 = (i - 0)6, +ee2.
Applying (14.15) we find
[E,E']03 = ^<l-2«3).M<l-2«3)/2«(£);
but
(1 - 263) p = (1 - 6) 2m - oc,
from which (14.14) follows. D
14.4 Second Auxiliary Interpolation Theorem
Theorem 14.3. Let 0 < oc < 2m. If
1 - oc\2m =j= \, 1 - 0 - oc\2m =j= ^,
we have:
[H*m-"\%m-"},:im{Z), (H"'*/2m{i;)y]d =
2m-txA2m-tx)/2m ( y\ fjjtx,tx/2m t
_ rr(2m-a)(l-0)-a0,[(2m-a)(l-0)-a0]/2m/v-»x
(14.17) ] */ (2m - oc) (1 - 0) - oc 0 ^ 0,
= (77-[(2m-")(l-0)-"0L-[(2m-a)(l-0)-a0]/2m^\y
if (2m - oc) (1 - 0) - ocd ^0.
(For the case 1 — 0 — <x/2w = |, we have a result of the type
given in Theorem 12.6 of Chapter 1.)
Proof. The plan of the proof is analogous to the one for the proof of
Theorem 12.6 of Chapter 1.
In order to simplify the notation, we set
E0 = H2m-*i2m-ay2m{Z),
El = (Ha^2m(2)y.
1) For the time being we assume
Lemma 14.1. Let ju be an integer > 0 such that ii\2m is an integer.
Then
(14.18)
[iF;g/2m(2;), (#^/2m (£))'] i/2= H°(S).

72 14. Some Interpolation Theorems
2) Let us now verify that
(14.19)
Indeed
(14.20)
11-6) u 1
if =f= h integer.
2m 2
| H";g/2m(i:) = #g/2m(0, T; #°(r)) nH°(0,T; H"{r))
j H°(Z) = H°(0,T;H°(r))
and we may apply the result pertaining to "interpolation of
intersections" (Chapter 1, Section 12.1); we obtain
(14.21)
[H»tf2m(Z), #°(2% = [Hg/2m(0, T; H°(r)), H°(0, T;H°(r))}e n
n [ff°(0, T; ^(i1)), tf°(0, T; H0(DJ]e.
But the first space on the right-hand side of (14.21) is equivalent to
(according to Theorem 11.6 of Chapter 1, a result which is valid without
change for functions taking their values in the same Hilbert space)
Hu-6W*(piT.>Ho(r)) if (1 ~6)lJl * integer +— •
2m 2
The second space on the right-hand side of (14.21) is equivalent to
#°(0, T; #(1-0)"(r))(by (14.17), Chapter 1) and therefore (14.21) shows
that
[H^/2m{Z)t H°(Z)\ = i#~e)"/2m(0, T; H°{r)) n H°(0, T\ IF1-0*{I7))
from which (14.19) follows.
3) Choose [i such that ii\2m = integer ^ 1. Set:
H^l2m{E) = X, H^/2m(Z) = Y
Then, by (14.19), we have:
(14.22) E0 = [X, H°{Z)\, (1 - ex) p = 2m - *,
and on the other hand
(14.23) H^2m(Z) = [H^'2m{Z)fH°{Z)]Q2i (1 - 62) [i = oc.
Then, according to the duality theorem (Chapter 1, Section 6.2),
we have
and by (14.18):
that is, by reiteration (Chapter 1, Section 6.1),
(14.24) EA = [X,Y']1.ta/2.

14-4 Second Auxiliary Interpolation Theorem 73
In the same way, with (14.18), (14.22) yields
(14.25) E0 = [X,Y']ei/2.
Then the space [E0, Et]e is equivalent to
[x, y](1_e)ei/2+e<i-e2/2) = \x, y\,
(14.26)
01 / #2
according to (14.24), (14.25) and the reiteration result.
If /? ^ \, this is equivalent to
\X, [X, Y']1/2]2, = [X, H°(i7)]2/1 = (by (14.19))
from which we obtain the first formula (14.17), since
(1 - 2/3)// = (1 -6) (2m -a)-6«.
If /? > ^, this is equivalent to
[[X, Y']1/2, Y']2/l_1 = [#0(2;), Y'k,., = (duality theorem)
= [Y,tf°(Z)]2_2/J,
from which we obtain the second formula (14.17), since
(1 - (2 - 2P)) p = - [{2m - a) (1 - 9) - <x 0]. D
Before giving the proof of Lemma 14.1, let us state the following
corollary which is fundamental to our present discussion.
Corollary 14.1. Let 0 < <x < 2m. Then if
2m (1 - 6) - oc 1
2m 2
we have (in the notation of (14.11))
r rj2m-a,(2m-a)/2m/rrv (ZJ<x,<x/2m (Y\\'l _ ^2m(l -fl)-*,(2m(l -6)-*)/2m / n\
Proof. We always have
^2m-«,(2m-«)/2w/2'\ c ^2ffl-«,(2m-*)/2m /JT\ «-- ^2m-*,(2m-«)/2m /£»\
and therefore, if is denotes the desired interpolation space,
c [#2»-«,(2m-«>/*» (2^ (#«.«/*» (2^],.
Now the first space is given by (14.47), the second is given (as in
Chapter 1, Theorem 12.5, replacing Q by Z) by
^?2m(l -6)-*,[2m(l-0)-«]/2ffl/r.\

74 14. Some Interpolation Theorems
from which the result follows, since these two spaces are identical for
2w(l - 6) - <x 1
- - < —• D
2m 2
Proof of Lemma 14.1. 1) We always have
H^/2m(Z) c H^2m{Z),
therefore, if we denote the interpolation space on the left-hand side of
(14.18) by F, we have:
F c [H^/2m{Z), (#M/2m (£))'] 1/2
and this last space is H°(Z).
Therefore F cz H°(Z) and, by duality
H°(Z) c [H^/2m{Z)tH-^-^2m{Z)]1/2.
2) Therefore, we shall have (14.18) if we can show the inverse
inclusion. For this purpose we apply (compare with Chapter 1, Lemma 12.2)
Lemma 14.2. There exists an extension operator P with the following
properties:
P e &(HW2m {Z); H^2m {ZJ), where Z^ = T x R,,
P e^(H-^^2m(Z);H"^'^2m(ZJ)>
rEP u = u \fu e H^^"^2m(Z)J where rz = operator
of restriction to Z.
The existence of this operator P can be shown, by " reflection in t"
about t = 0 and t = T (the variables corresponding to r playing the
role of parameters), exactly as in the proof of Lemma 12.2 of Chapter 1. D
Then, by interpolation, we have:
P e Se{\_R^l2m(Z) > H-»--**"(Z)]1/2; [H"-"'2"(ZJ, H-"--"2*(ZJ] 1/2).
But
[H^2m(ZJ,H-^-^2m(ZJ]1/2 = H°(ZJ
and therefore, if
u e [H^2m(Z), H-»>-v2m(Z)]1/2>
we have:
rzP u = u e H° (Z), whence the desired result. D

14.5 Third Auxiliary Interpolation Theorem 75
14.5 Third Auxiliary Interpolation Theorem
Theorem 14.4. Let the space V be defined by (14.10) and let 6 satisfy
(14.5). We have
(14.27) [7, Hm(Q)']d = je^-2e^{Q)
(where ^(Q) is defined by (14.3)).
Proof. 1) Since 7 c Hm(Q), we always have
(14.28) [7, Hm{Q)']e c [Hm{Q),Hm{Q)']e = je(1-20)m{Q)
(Chapter 1, Theorem 12.5).
2) On the other hand
H'SiQ) c 7,
therefore
(14.29) [7, ff»(fl)'], = [#S(fl), ff»(fl)'],
and (Chapter 1, Theorem 12.6 and the end of Remark 12.5)
f [#JJ (13), #m (13)']- = tfi1"2*)m (R") (space of restrictions to
(14.30) Q
[ 5 of distributions in H(1~2d)m(Rn) with support in D).
But if
(14.31) -i < (1 - 26) m<\,
this last space coincides with ^"^"(D) and (14.29), (14.30) yield
the inverse inclusion to (14.28). Therefore we have already shown the
theorem for 8 satisfying (14.31).
3) In particular, taking 0 = \, we have shown that
[F.fl* (£>)'] 1/2=ff <>(£>).
For 6 < i, the reiteration theorem (Chapter 1, Section 6.1) yields
[7,J?-(fln, = [F;[7,fl»(fl)']1/2]2,
and the theorem then follows from the following identity (which will
be verified in step 4) below):
(14.32)
[V,H°{Q)]tl =tf(1-<")m(£)
1 minw/
if 0! > 1 -
2m m
For 0 > \, reiteration yields
[V, H-(Q)'], = UV, H">(Q)>]1/2, H«>(Q)%e
= [H°(Q),Hm(Qy]e,

76
14. Some Interpolation Theorems
from which the desired result follows upon application of Theorem 12.5
of Chapter 1.
4) Proof of (14.32). j m[nm
Note that the relation 6t > 1 — — - may be written
2m m
(14.33) mj > (1 - 0i) m - ±, V/.
Of course, we have
[V,H°(Q)]9l cz [H»(Q),H°(Q)]ei = ^(1-ei)m(D),
and therefore we need to show the inverse inclusion. Thus let / be given in
#a-*i>»(fl). By Theorem4.2 of Chapter 1, to show that fe[V,H°(Q)]dl
is equivalent to showing that there exists a function u satisfying
(14.34)
ueL2(0,oo; V) nP(0, oo; H°(Q)), — = 6l9
2s
[u(x,0) = /(*),
or, in the notation of Section 2 and setting
(14.35)
dv J 0 = 0 0VP
bjp = tangential differential operator on r
(with regular coefficients) of order ^ mi — j3:
[ ueHm>s{Qx]0, oo[),
(14.36) Bju = 0 on Tx]0, oo[,
[ u{x,0) = /(%).
Since / is given in H(1~d0m(Q), we can define
dJf
(14.37) -±eH(1-60m-J-1/2(r) for / < (1 - d,
dvJ
1
m .
2
We define gj to satisfy
a* Vi m — j — 4
?Jeff«"'(I),^ = -i = i 1, (27 = rx]0,coD
(14.38)
m
dJf 1
oj'-' 2

14-5 Third Auxiliary Interpolation Theorem 77
which is. possible (in an infinity of ways) according to Chapter 1,
Theorem 4.2.
Next we define,
(14.39) gj = 0 if /^(l-fli)**-*, j±mj
and finally we define gmj by the system (note that m3 > (1 — 0i) m — ^):
(14.40) gmj+'lbjiigfi-O.
p = o
Let us verify that
(14.41) gj e #"'■'' (E) V/ (£ max tnj).
It only remains to be verified for gm ; we proceed by induction on
increasing m3\ we may then assume that gp e H/Xp,vii (U) in (14.40) and
according to Proposition 2.3, bJpgpeHll*,v*{E), where
Pa V<x mj ~ P
= — =1 , whence the result.
ft* vp ftp
Thus we have constructed functions gj satisfying (14.41), (14.40)
and (14.38). Then, according to Theorem 2.3 (in the "easy" case in which
there are no global compatibility relations), there exists awe Hm's(Q x
x]0, oo [) with
(14.42) 44 (*''*) =£/(*''*), *'er,
and
u(x,0) = /(*).
But (14.42) and (14.40) imply (we have done what is necessary for
that): BjU = 0 on r x]0, oo[, and therefore we have (14.36). D
Remark 14.5. Result (14.32) can be completed for all values of the
parameter d1 (see Grisvard [8]): the space [V, H° (i3)]01 coincides with
the subs ft ace of elements u e Hil~6l)m(Q) such that
(14.43) BjU = 0 for m3 < (1 - Bx) m - \,
(14.44) -~rBj ueL2 (£) for mj = (1 ~ ei) m >
(q as in (11.18) of Chapter 1).
The case (14.43) can be shown by a simple modification (in the
way of choosing gj) of the preceeding proof; the case (14.44) can be
attributed to the same principle, but with much more delicate technical
details (and the use of Theorem 2.2). D

78 15. Existence of Solutions in the Spaces H2mr'r(Q), 0 < r < 1
14.6 Proof of Theorem 14.1
In the notation of (14.2) and (14.11), we have:
m-l
[C0,Ai]e= 11 [o#2w~ (w'+1/2)'C2w- c»i+i/W2»(r)#
^wy+l/2.(wy+l/2)/2w(r)y-|fl x ^ #™(D)']0
and applying Corollary 14.1 and Theorem 14.4, this is equivalent to
"TT ^7(l-0)2m-(mi+l/2),l-0-(mJ+l/2)/2m/v-»x ^?i
j = o
(l-20)m
W
if 0 satisfies (14.5).
But [£>0>^i]e is always (for any 0) equal to the preceding space,
from which the theorem follows by (14.13). D
15. Final Results; Existence of Solutions
in the Spaces H2mr>r(Q), 0 < r < 1. Applications
15.1 Application of the Results of Section 14
We reconsider Section 13.3 with the notation of Section 14.1; then ^,
defined by (13.7), is a continuous linear mapping of
IH»(Q),{H^(Q)Y\9 x [A0, Ax]6 -> [H2^ (Q),H°(Q)]9.
Set 34r-26m>-6(Q) = (H2m6>6(Q))'.
Applying Theorem 14.1, we obtain:
for 0e]O, 1[ and satisfying (14.5) (see also Remark 14.1),
^ defined by (13.7) is a continuous linear mapping of
(15.1)
-26m,-6
(0) X FT ^f72(1-0)m~(mi+1/2), 1-0-(mj+ l/2)/2m/j£\
j = 0
x ^-^(Q) -►#(1-e>2w'1-(<2).
It suffices to note that [H° (Q), (H2"1-1 (Q))']d = J$r-26m>-6(Q) (same
proof as for Theorem 12.5). D
We can now complete the results of (i) and (ii) of Section 13.1 with
the following result:
f if fe3f2ms's{Q), gje^2m(s+1)'mj'1/2's+1'(mj+1/2)/2m(i:),
u0 e^2(s+1/2)w(2;), then the solution of problem (13.1) belongs
(iii) I to H2n*s+1>'s+1(Q); and this for
— 1 < s < —max
1 1 minw/
0,
2 Am 2m

15.1 Application of the Results of Section 14
79
Of course, we must point out that in (iii), u is a solution of problem
(13.1) in the sense of Theorem 13.1, i.e. u satisfies
<uyA*v -v'> = </,£> + (g,Tv>z + <uo,v(x,0)>Q
VveX1®).
If
1
Am
2m
-<0,
that is if
(15.2) mj ^ m V/,
then we have obtained all possible orders of regularity for — 1 < s < 0.
If, on the other hand, — — — > 0, then, in the
2 Am 2m
terminology of Section 13.1, we do not have the optimal result for the
order of regularity s, with
(15.3)
1
mmm,
Am
2m
< s < 0.
For the case (15.3), there are compatibility relations between the data:
we shall not specify them here. D
Remark 15.1. We can also give another application of the interpolation
results of Section 14; a ''parabolic" adaptation of Remark 7.3 of
Chapter 2.
We introduce the following spaces:
H2mB\1o(Q) = {v\ve H2^ (Q),BjV = 0 onZ,
0 <; / ^ m - l,v(#,0) =0}
H2c:t(Q) = {v\ve H2*»-i (Q),Cj v = 0 on 27,
0 ^ / ^ m - l,v{x, T) = 0}.
(15.4)
Then
(15.5) \
+ A (resp. h ^4*) is an isomorphism of H2^\q(Q)
dt \ dt
(resp. H2^T(Q)) onto L2(Q).
By transposition
(15.6) ( + A*\ is an isomorphism of L2(Q) onto (#2*;i ((?))'

80 15. Existence of Solutions in the Spaces H2mr>r(Q), 0 < r < 1
Id \*
and with the aid of Green's formula we easily verify that \- A*\
rs rs
is an extension of h A, which we shall still denote by h A.
dt J dt
Then, interpolating between (15.5) and (15.6), we obtain:
d
(15.7)
— + A is an isomorphism of [H2mBAQ{Q), L2(Q)]e
dt '
onto[£2(e),(tf2^(<2))']e.
The second space in (15.7) coincides (according to Chapter 1,
Section 6.2) with
QLnPS'MQ).!-2®)]!-,)'.
By variants of the preceding methods, it can be shown that
(15.8) [H2mcMQ), L*(0)]i-, = H2mM(Q),
liin particular 2m 0 < \ (in which case there are no more compatibility
relations). In this case the space 2f{Q) is dense in H2mdfd(Q) and we have
(see Section 8.3)
(15.9) (H2md'd{Q))' = H-2md'-d(Q);
furthermore, since 2m(I —6) > 2m — -J-, we can verify that
(15.10) I ={v\veH2mil-d)'1-d(Q)>BjV = 0 on T,
[ 0 ^ ; g m - 1, v(x,0) = 0}
Therefore:
Theorem 15.1. Under the hypotheses of Section 2.1, the operator
rs
h A is an isomorphism of H2sr%sQ(Q) onto H-2ms'~s(Q), for 0 < s <
dt
<4L' D
Am
15.2 Examples; Generalities
Suppose that we have to solve problem (13.1) with
(15.11) gjeL2(Z),j = 0,...,m-l
(this situation will come up in an essential way in Chapter 6). Then we
have to apply (iii) of Section 15.1. Choose s such that (if possible!)
(15 12) L2IE) <z 2^2(s+1>m~(mj + 1/2)'s+1~(mj + 1/2V2m/r\

15.3 Examples (I)
81
that is such that
1 +s iZJL^o
2m
with s taken as large as possible. Therefore
1 min Mj
(15.13) s
We then obtain:
optimum
4m 2m
(<0).
Theorem 15.2. Assume that the g/s are given so as to satisfy (15.12).
// we take:
i jp> — 2m+l/2+minmJt - 1 + l/4m-f min(mJ-)/2m /^\
(15.14)
(15.15)
U0 6.
-m+ l/2+minmj
(O),
then the solution u of problem (13.1) (in the sense of Theorem 13.1)
satisfies
(15.16) u e H1/2+minmJ'a/2+minmJ)/2m(Q). D
Example 1.
Consider
(15.17)
\ du
~dt
15.3 Examples (I)
- Au = f,
u\z = go> u(x> 0) = u0.
Then m5 = m0 = 0, m = 1 and therefore: if g0 e L2 (27), we may-
take /eJT-3/2'-3/4(<2), u0eJf-1/2{Q) and so
(15.18)
UE #1/2.1/4(0) g
Example 2.
Now consider
(15.19)
du
du
dv
- Au = f,
= go, u(x,0) = u0.
Then m = 1, m5 = m0 = 1 and therefore if g0 e L2 (27), we may
take /eJT-1/2--1/4(<2), ^0eJr1/2(i2) and so
(15.20)
ueH3f2'3f*(Q). D

82
15. Existence of Solutions in the Spaces H2mr*r(Q), 0 < r < 1
Example 3.
Consider
(15.21)
dt
+ A2u = f,
u\z = go>
du
dvy
= glt u(x,0) = u0.
Then m = 2, m0 = 0, m1 = 1, minwj = 0. Therefore if g0, g1 e
eL2{Z), we may take / g ^-4+1/2'-1 + 1/8(<?), % e ^"3/2(i2) and so
(15.22)
ueH1'2'
1/8 (Q).
Example 4.
Finally consider
(15.23)
du
IT
du
dv
+ A2u = f,
dAu
27 dv
= Si,
D
u(x, 0)
= «0-
Then m = 2, m0 = \, m1 =3, minm,- = 1. Therefore if gt e L2 (Z),
we may take / g tf-5'2--5'* {Q), u0e3f-1/2{Q) and so
(15.24) ueH3l2'3/8{Q). U
15.4 Examples (II)
Consider the (homogeneous) boundary value problem already studied
in Chapter 3, Section 4.7.5 in the setting of variational evolution
equations:
(15.25)
(15.26)
du
~Jt
+ A2u = f in Q
u= 0,
dAu
dv
= 0 on Z,
(15.27) u(x,0) = u0(x), u0 given in Q.
Example (12.25) —(15.27) also fits the theory studied in the present
chapter.
It is required to verify (1.7). Thus we consider the operator
(15.28) A2x + eidD$ in {xn > 0 | x e R", y e R},
with the "boundary operators"
d M
(15.29)
Set
B0 = identity, Bt =
dxn
f=il +
+ £_i, q>0;

15.5 Some Complements on the Dirichlet Problem 83
the polynomial A0 (x, £ + r f'), already considered in the elliptic theory,
corresponding to (15.28) (see Chapter 2, Section 1), becomes
[71 7l~\
-t-tJ
(15.30) (q2 +t2)2 + e'V« 6e
and the polynomials corresponding to (15.29) are
(15.31) i, T(e2+r2).
For r\ = 0, we have seen (Chapter 2, Section 9.4) that the polynomials
(15.31) are independent modulus (t — t^) (t — t£) , where
t£ = root of (15.30) with imaginary part >0, k = 1,2.
It remains to investigate the case r\ =j= 0. Then x\ =j= r£ . We must
have that
a + brt(g2,+ (r2+)2) = 0,
a + brt{Q2 + (r2+)2) =0,
imply that a = b = 0, theiefore that
T2+(<?2 + (T2+)2) - T^e2 + (T + )2) + 0,
that is
(15.32) q2 + (rt)2 + rt r2+ + (r2+)2 * 0,
which is easily verified. D
15.5 Some Complements on the Dirichlet Problem
Note that for each particular problem, we can obtain new results
by applying the same types of methods as in Chapters 3 and 4 and
choosing the spaces as a function of the boundary conditions as well.
For example, consider the Dirichlet problem. Assume the operator
given by (1.2), with (1.3), to be strongly elliptic in D uniformly in t e [0, T].
Then, according to the results of Chapter 3 and the trace theorems of
this chapter, we find that the problem
(15.33)
(15.34)
(15.35)
du
Au + —=f,
ot
dJU
-z~r = gj. / = o,.
OVJ
u(x, 0) = u0,

84 15. Existence of Solutions in the Spaces H2mr*r (Q), 0 < r < 1
admits a unique solution in the space
(15.36) W = |i;|i;eL2(0, T;Hm(Q)), — eL2(0}T]H'm(Q))\
( /eL2(0, T;H-m(Q)), gj e Hm^-1^m-J-1^2m(Z)9
for all
(15.37)
[ u0 e L2 (Q).
More precisely
the mapping u ->
(15.38)
Idu dJu
A u + -—; —-r, j = 0, . . ., m - l;u(x, 0)
dt dv3
is an isomorphism of W onto
L2(0,T;H-m{Q)) x n Hm7J-1/2>im-J-1/2y2m(Z) x L2(Q).
j = o
We also recall that, according to Theorem 6.1, we have
the mapping u ->
(15.39)
Idu d3u |
is an isomorphism of R2m>* (Q) onto
lL2(Q)x\\H2m-mJ-1/2'i2m-m"1/2)I2m
and that, according to Remark 12.3, we also have
the mapping u -»
idu dJu
Au + —-, —— / = 0, . . ., m - l',u{x,0)
ot ovJ
is an isomorphism of D%(Q) onto
m-l
3-2m.~HQ) x ]-J jff-J-l/2.-C/+l/2)/2»(2^ xH~m(Q).
j = o
Of course, we can now interpolate between these three isomorphisms,
which leads to new interpolation problems such as, for example, the
interpretation of the space
IL2(0,T; H->(Q)),3-2m'-1(Q)-]9.
(15.40)

16. Gomments
85
16. Comments
Boundary value problems for second order parabolic equations
have since long ago led to a great number of works; for these equations,
we refer the reader to the books of Friedman [1] and Ladyzenskaya
Uraltzeva-Solonnikov [1], to irin-Kalashnikov-Oleinik [1] and Solon-
nikov [2] and to the bibliographies of these works.
The boundary value problems for parabolic operators studied in
this chapter (arid which do not necessarily fit the variational theory of
Chapter 3) have been solved by Agranovich and Vishik [1] in the Hilbert
setting of the spaces H2mr'r (Q) with r g; 1.
The method followed here (Sections 1—6) is, in our belief, somewhat
simpler than the one adopted by these authors. We apply, on the one
hand, rather general abstract results on the solution of the equations
by Laplace transform (Sections 3 and 5.3) (variants of Doetsch [1],
Garnir [1], Lions [1] and [13], Chapter 11) and, on the other hand, an
idea of Agmon and Nirenberg (see Agmon [6], Agmon-Nirenberg [1])
to prove the "basic inequalities" (Sections 4 and 5) (Agranovich and
Vishik, loc. cit., prove the inequalities directly, i.e. by reconsidering
the "usual" proofs of the elliptic case (one could also follow the method
of Chapter 2, but being careful to examine the contribution of terms
containing the variable "Laplace transform")).
Section 2 gives trace theorems in the non-isotropic Sobolev spaces
Hr's[Q). Theorems 2.2 and 2.3 are due to Grisvard [4, 8] (and we have
adopted Grisvard's method here). Particular cases were obtained
independently by Fuji vara [2]. For Theorem 2.1, see also Pagni [1].
In Grisvard [8], one also finds trace theorems in Sobolev spaces of
Hr's-type, but constructed on Lp, p =j= 2; for other bibliographical
references on these spaces and on non-isotropic Sobolev spaces, see the
Comments to Chapter 1. For non-isotropic Morrey-type spaces see
Barozzi [1], Da Prato [3].
Coming back to the boundary value problem, the Hilbert setting
(that is the setting of Sobolev spaces H2mr,r (Q) — see Section 2 —
constructed on L2) is, of course, not the only one:
i) We can replace L2 with Lp, p=^2,l<p<oo; the corresponding
theory applies, see Grisvard [6 — 8], and Solonnikov [2 — 4]; the first of
these authors uses techniques of functional analysis (Dunford integrals
to respresent the "inverse operators", furthermore taken in interpolation
spaces); the second author uses a more classical method, starting from
the analogues of the Poisson kernels, see also Sobolevskii [3, 4].
We also call attention to the studies on parabolic potentials of
Arnese [1, 2], Jones [1—4], followed by Fabes [1], Fabes-Jodeit [1],
Fabes-Riviere [1], Jodeit [1], Kree [1—4].

86
16. Comments
ii) We can also consider the setting of spaces of differentiable
functions in the usual sense constructed on Lipa; see Arima [1].
Once in possession of a "good" theory of regularity of boundary
value problems, homogeneous or non-homogeneous (the later ones thanks
to suitable trace theorems, given in Section 2) we go over to the
"irregular" cases (indispensable for the applications to optimal control problems
among others) by the general method: transposition and interpolation.
We start solely from L2-results; the technical difficulties already
encountered in this chapter with L2 indicate how considerable the
"technical" difficulties are for the cases Lp, p =# 2. Transposition and
interpolation are presented in Sections 7—17, the results of which are
published here for the first time. In connection with the results of
Sections 10.3 and 10.5, see also Pini [1].
Questions connected to regularity in Gevrey-type spaces and to the
study of boundary value problems of parabolic type in spaces of ultra-
distributions will be taken up in Volume III.
Let us call attention to some further questions pertaining to
parabolic boundary value problems which we have not treated in this book:
a) Green's functions of parabolic problems; see Dressel [1], Eidel-
man [1] and the bibliography of this work and Arima [1].<
b) Equations with discontinuous coefficients, Harnack's inequalities
etc.; see Aronson [1], Aronson-Serrin [1, 2], Guglielmino [1, 2], Kroujkov-
Oleinik [1], Kroujkov [1, 2], Ladyzenskaya-Uraltzeva [2], Ladyzen-
skaya-Solonnikov-Uraltzeva [1], Moser [1].
c) Behavior of solutions at infinity in the time variable (see
Comments to Chapter 3) and in x, when Q is unbounded; see Barozzi [2, 3],
Friedman [2], Juberg [1, 2], Tacklind [1], Tikonov [1, 2].
d) Uniqueness and unique extension problems: Agmon-Nirenberg
[1, 2], Ito-Yamabe [1], S. Krein [1], Landis [1], Lions-Malgrange [1],
Mizohata [5].
e) Study of boundary value problems using the theory of pseudo-
differential operators (see Vishik [5], Vishik-Eskin [3]), which
furthermore allows the treatment of problems with mixed conditions on E
(for particular cases, see also the examples 4.7 of Chapter 3) or of oblique
derivative problems.
f) Boundary value problems for parabolic systems] see the books
of Friedman [1] and Eidelman [2], the work of Solonnikov [3] and the
bibliography of these publications.
g) Boundary value problems in non-cylindrical open sets; see
Kaminin-Maslennikova [1], Lions [5, 6], Vishik-Eskin [3]; a method

17. Problems
87
using the theory of semi-groups (results of Kato and Tanabe) and the
theory of interpolation has been developed by Krein-Laptiev [1].
h) Questions pertaining to approximation of the solution by finite-
difference methods; there is a vast literature on this subject, but the
study of the general case still poses numerous problems (see
Problem 17.19); for a possible way of approximating solutions of non-
homogeneous problems, see Lions [29].
i) Questions pertaining to "regular'' and "irregular" points of the
boundary and to "parabolic" capacity for equations of the second order,
"isolated" and "removable" singularities of the solution: see Aron-
son [2, 3], Eidelman [2], Pini [13, 14].
17. Problems
17.1 In the text we have left aside the complete interpretation of
the spaces 32rm,r(Q), H2rmSr(U) for non-integer r.
This problem is not a fortiori solved for spaces of this type constructed
on Lp, p =j= 2, instead of L2 (see also Chapter 1, Problem 18.4).
17.2 More complete study of traces of u's belonging to.Dp (r~1) (Q),
in particular the problem, noted in Section 12.3, of separating the
operators E u and u (x, 0) in au •
17.3 Complete the results of "regularity s" with — l<s<0,
obtained in Sections 13 — 15; this will pose many interesting problems
of interpolation between the various spaces under consideration (see
also Section 15.5).
17.4 In this whole chapter the question of'' exceptional parameters''
is not treated in optimal fashion (as was attempted in Chapter 2); as
a matter of fact, it seemed to us that the technical difficulties encountered
in the study of all exceptional cases (in particular with the global
compatibility relations) were out of proportion with the interest of the
results one might obtain, at least with the methods used in this chapter.
It would be interesting to see if other methods (for example
transforming the problems to equations for pseudo-differential operators on the
boundary — but here the boundary is Q x S . . .) allow for a less difficult
study of these exceptional parameters.
17.5 We have only used L2-theory; one could also use the regularity
theorems in the spaces Lp, p =j= 2, 1 < p < oo (see the Comments).
Similarly, one could use the regularity theorems in the spaces Lipa (Q)
(see Arima [1]).

88
17. Problems
Then the technical details are very complicated and we have not
undertaken any write-up on this subject.
17.6 It would be of interest to systematically extend all the
preceding theory to general parabolic operators (see Agranovich-Vishik [1] for
the L2-case and the works cited in the Comments for Lp, p =# 2). The
Z,2-case probably does not present any essentially new difficulties in
relation to what has been done in this chapter.
17.7 A systematic study of the case in which the open set Q has
r
a boundary r = (J rt with r/s of different dimensions remains to be
done. i = 1
17.8 Also the study of non-homogeneous problems in non-cylindrical
open sets remains to be done (see Comments).
17.9 It would be of interest to study non-homogeneous problems
for "elliptic-parabolic" operators; see Fichera [3], Oleinik [1, 2], Kohn-
Nirenberg [1, 3], Hormander [11].
17.10 In this text, we have not considered non-homogeneous
problems with "periodicity conditions" instead of "initial conditions":
A u + u' = /,
Bu = g,
u(0) = u{T)
(or w(0) = u(T) + x» X given)
(see Chapter 3, Section 6).
17.11 One can consider "coupled" problems (elliptic-parabolic in
a different sense than in 17.9) as follows:
let Q = Qx n Q2> A = common part of the boundary; find ut (x,t),
i = 1,2, defined in Qt = Qt x ]0, T[ with for example
-Aux = fx{x, t) in Qlf
du2
-A u2 + —— = f2 (x, t) in Q2,
ot
transmission conditions on 7\ x ]0, T[ and the "usual" conditions on
the other parts of the boundary (and an initial condition on u2). As was
pointed out to us by O. A. Oleinik, this case fits the general study of
elliptic-parabolic equations (see Problem 17.9) with piecewise continuous
coefficients.

17. Problems 89
17.12 Same question for the problem
-Au = f in Q,
the time appearing only in the boundary conditions:
du du
-T- + —= £ on 27,
dv at
with u(x,0) given for x eT (see Lions [13]).
Here is a method of solution. We bring the problem back to the case
••/ = 0".
Denote by 38 the continuous linear operator, mapping H1I2[T)-+
-» H~1/2(r), defined as follows: if h is given in H1/2(r), denote bv co
the solution in H1 (Q) of
Aco = 0,
co \r = h,
and set
dco
38h =
dv
\-
Then the problem to be solved is equivalent to the following one:
if ux denotes the trace of u on 27, find ul9 solution of
du-,
&ut +
dt
ui (0) given.
It is easy to verify that & is coercive on V = H1/2(r) in the sense
[&v, v) + X \\v\\2LHn ^ p |M|Hi/2(r), P > 0, X > 0, VveV.
Now, the general theory applies, either as in Chapter 3 or as in the
present chapter (for we have regularity theorems for the first-order
elliptic operator 38) \ see also Friedman-Shinbrot [1].
In particular, this yields the results of Garipov [1].
This method extends to the case in which — A is replaced by a
second-order elliptic operator A (t) with time-dependent coefficients.
The method also extends to the problem of the second order in t (see
Chapter 3):
-Au = 0 in Q,
du d2u
1 r- = g on 27,
dv dt2 6
u(x,0), ut(x, 0) given for x er.
For this type of problem, also consult Friedman-Shinbrot [1],
Odhnoff [2], Pleijel [1].

90
17. Problems
17.13 We most probably have "stability of traces'5 upon passing
from the operator A + djdt to "ellifitically regularized" operators:
d d2m
(17.1) A + — + (-l)me-
dt v ' dt2m
The process of elliptic regularization (17.1) is justified when A and Bs
correspond to a variational problem (see Chapter 2, Section 9 and
Chapter 3, Section 4). It is very likely — but not proven, it seems — that
the process (17.1) leads to solutions ue which converge towards the
solution u of the evolution problem as e -» 0, when the hypotheses on
A, Bj are as in Section 1.1.
It would not be without interest to obtain the non-homogeneous
boundary value problems for the evolution case by passage to the limit
(e -» 0) starting from suitable elliptic boundary value problems. (Can
one in this way reach the LMheory, p =j= 2?)
It would be of equal interest to extend elliptic regularization to the
general setting of Agranovich-Vishik [1]; probably, we would have to
consider
P{x>t>DX}ezD2tml1 + £>,), e> 0, lz = eie,de
71 ' 71
~2' ~2
with the boundary operators (in x)
Bs{xytyDXiEzD]ml1 + Dt).
17.14 It is certainly technically very difficult to extend the results
of this chapter to operators A + djdt considered in the spaces
L>(0,T;L*(Q)), p * q.
(There are some results in this direction by Da Prato and by Gris-
vard.)
However, such a study might be useful for nonlinear problems.
17.15 Problems analogous to the ones studied in this chapter for
the operators A + djdt, where A has unbounded coefficients; for
example with coefficients in Lp (Q) (some results for these operators are
given in Baiocchi [7], Guglielmino [2], Lions-Raviart [2]).
17.16 Non-homogeneous problems with "mixed" conditions on S
or with oblique derivative conditions on S (see Comments).
17.17 Non-homogeneous parabolic problems in "weighted" spaces.
17.18 Non-homogeneous problems for parabolic systems.
17.19 Questions pertaining to approximation by finite difference
methods for the non-homogeneous problems studied in this chapter are
still open in general; see Lions [29] for a partial result.

Chapter 5
Hyperbolic Evolution Operators, of Petrowski and of
Schroedinger. Hilbert Theory.
The reading of this chapter does not require the knowledge of
Chapter 4. It rests essentially on Chapters 1 and 2 and on the elements of
Chapter 3.
We call attention to the fact that transposition and interpolation
are used in a slightly different order than in Chapters 3 and 4.
1. Application of the Results of Chapter 3
and General Remarks
1.1 Notation. Hypotheses
We shall study the equations
/ d \ d2u
(u) *(*'''j;r + -3F-f in Q'
or
(1.1a) A u + u" = /,
with the boundary conditions
(1.2) BjU = gj} 0 ^ / ^ m - 1, onfl
and the initial conditions
(1.3) u(x,0) = u0(x) in 27,
/ du \
(1.4) u'(x,0) = u1(x) in Q lu'(x,0)= (^,0)1
(in the notation of Chapter 4, Section 1 for Q, Q, E).
The hypotheses on the operator A and on the boundary operators Bs
are the following: A is defined by
(1.5) Acp= £ (-\)\>\D>x{am{x,t)D*xcp),
|p|.|«|Sm

92 1. Application of the Results of Chapter 3 and General Remarks
where the functions apq are given in @(Q).
The operator A is assumed to be symmetric:
(1.6) A* = A (i.e. am = a~^).
Remark 1.1. In fact, for the sequel, it is sufficient that the principal
part of A be symmetric. For this reason, we shall keep both notations A
and A*. Q
Let a (t; u, v) be the continuous sesquilinear form on Hm (Q):
(1.7) a{f,u,v)= £ ( apq(x>t)D^uDxrvdx.
japa(x>
(1-8)
Q
We assume the operators Bj to be defined by:
BJ = BJ(x,t;Dx) = X M*>')£*>
\h\zmj
where the functions bJh are given in @(E) , 0 ^ Wj < 2w, and,
for every toe[0,T], the system {Bj(x> t0] Dx}™~o
is normal on r.
Furthermore, we assume the boundary conditions (1.2) to correspond
(see Chapter 2, Section 9.4) to (1.7) and a closed vector subspace V
of Hm(Q):
V H%(Q) czV c #m(i2),
the form a(t;u,v) being V-coercive, i.e.
(1.9) a(t;v,v) ^oc\\v\\Hm(Q)> <x > 0, Vv e 7. D
Remark 1.2. It would be sufficient to assume
principal part of a(t\ v, u) ^ oc IMIjUuw _ ^ IMIh°(g) VveF
and for suitable X. We shall assume (1.9) in order to simplify some
technical points. D
Greens formula (compare with Chapter 4, Section 1) may be written
(assuming all functions to be regular):
(1.10) (A<p + (p'\y))-(<p>A*tp + y)'') = (3(p,Cy))i:-{B<p,Ty))z +
+ (<p'(T),y>(T))n - (9/ (0),y (0))fl - (<p(T)iW> (T))Q + (<p(0),W' (0))fl. D
Variational formulation of the problem.
Assume gj = 0, 0 ^ / ^ m — 1, in (1.2).
Then problem (1.1), . . ., (1.4) may be stated as follows (see Chapter 3):
(1.11) a(t; u(t),v) + (u"(t),v)Q = (f(t),v)Q Vv e V,
(1.12) u(0) = u0> u'(0) = ux.
(u(t) denotes the function x -* u(x, t); similarly for f(t)). D

1.3 A Counter-Example
93
1.2 Application of the Results of Chapter 3
The application of Theorem 8.2 of Chapter 3 to the present setting
yields:
if feL2(0,T;H°(Q)) = L2(Q), if u0eV and if u1eH(=L2{Q)),
then u is (after a possible modification in t on a set of measure zero) a
continuous mapping of [0, T~\ -* V and its derivative du/dtis a continuous
mapping of [0, T] -* H.
Our tool is again the transposition of a regularity theorem, but as
always in this text, we have the choice of the regularity theorem, and
this choice is dictated by considerations of (relative . . .) simplicity of
the final results.
In the present case, we shall consider (see Section 4 for a precise
statement) the adjoint problem:
(1.13) A*v + v" = <p in Q = Qx]0,T[
(1.14) Cjv = 0 on Z, 0^j^m-l,
(1.15) v{x,T) = 0, v'[x,T) = 0.
//, by an application of the results of Chapter 3 which we have just
recalled, we take q> e L2 (Q), then v describes a space — say X — and
(1.16) Ic C°{[0,T]; V) nC^tOJ];^).
For transposition, we need the space described by T v = [Tj v}™^1
as v describes X. Now, the result (1.16) is not on general (except if the
order oiTj^m— 1, V/) sufficient to define {Tj v}; we could still define
Tj v by using the additional fact that A* v + v" e L2 (Q), but this
would complicate matters unnecessarily. D
The situation is as follows: either we must improve the regularity
theorem (1.16), or we must establish and apply other regularity theorems.
In the next section we shall show that an improvement of (1.16) is
essentially impossible, and thus we shall be led to establish other
regularity theorems in the sequel.
1.3 A Counter-Example
Assume the operator A to be independent of t. We show that for feL2 (Q),
u0 = 0, u± = 0, the solution u of (1.11)— (1.12) does not in general
belong to L2(0, T;H2m{Q)) (so that, in the notation of (1.16), X is
not contained in L2(0, T;H2m(Q)) and there is no simple "natural''
definition of TjV, V/, v e X).

94 1. Application of the Results of Chapter 3 and General Remarks
We first consider w to be the solution of
(1.17)
with
(1.18)
a{w{t),v) + {w"{t),v)Q = 0
w(0) = 0,
w'(0) = wlt
wlE[D(A),H]3
/4-
We shall show that in general w$L2(0, T;H2m(Q)). This will in
turn imply the truth of our assertion, for we can construct 0 with
(1.19)
[ 0eL2(O,T;H2m(Q)), <Z>" e L2(0, T; H°{Q))y
| 0(0) = 0, 0'{O) = wx
and w — 0 then satisfies
(1.20) A(w - 0) + (w - 0)" = - {A 0 + 0"),
(w - 0) (0) = (w - 0)\(O) = 0
and therefore
(1.21)
w — 0 = u if / = — (A 0 + 0").
Now / e L2 (Q), and if we had u e L2(0, T; H2m(Q)), then we would
have z£>eL2(0, T;H2m(Q)) and we shall show that this is not true.
The simplest way is to use the spectral decomposition of A. Let
e
Ij = \ f)(X) dfi(X) be a measurable Hilbert sum (see Dixmier [1] and
Chapter 1, Section 2.3) and °H a unitary operator mapping H = L2(Q)
onto I) such that °ll diagonalizes A:
fAv = I
VveD(A).
(1.22)
If we set
(1.23) w{Xj) = (®w{t)){X), iMA).= (*»i) W),
then (1.17) is equivalent to
(1.24)
rf2 *.
**(M + -73-*(*,*) =0,
tf (*,<>) =0,
<*2
rf*
■(A(0) = ^W;

2. A Regularity Theorem (I) 95
therefore
(1.25) W{XJ) =-^=3^1 (tyffyw^X).
Since w1 satisfies (1.18), we have. (Chapter 1, Section 2.1)
(1.26) XllAw{X)e^.
If w did belong to L2 (0, T; H2m (Q)), then, since w satisfies the
boundary conditions, we would have: w e L2(0, T; D(A)) and therefore
Aze>eL2(0,r;f)).
But
T T
\ \\Xw{t)^dt = I \\Xll2w1[X)\\sm2{tJ~X)dt
0 0
and this, under just the condition (1.26), is generally infinite; which
proves our assertion. D
Remark 1.3. Let XA be the space described by the solution v of
(1.13), (1.14), (1.15) as cp describes L2(Q). This space depends on A (here
we see an essential difference with the parabolic case). Indeed, let A0
be fixed and u0 e XAo, with
u0$L2(0, T\H2m[Q))
(which is possible according to what we have just seen); therefore, for
a suitable index of differentiation oc, \oc\ = m, we have:
D2x«u0$L2(Q)
(since if D2X« u0 e L2 (Q), V*, \oc\ = m, then u0 e L2(0, T; H2m(Q))).
So let A = A0 + A £>*"(-l)w; for sufficiently small real A, A is
still a (self-adjoint) elliptic operator and u0$XA, since
Au0 + < = (^o^0 + <) + X(-\)mD2x«u0$L2{Q). D
2. A Regularity Theorem (I)
Theorem 2.1. Assume that the hypotheses of Section 1.1 are satisfied.
Assume that f is given with
(2.1)
f<=L2(0,T;Ho{Q)) = L2(Q)
J' =^-eL2(0,T;H<>(Q))
ot
(i.e. feH^iQ))

96 2. A Regularity Theorem (I)
Assume that u0tu1 are given with
(2.2)
u0 e domain
oiA(*.0,±)
(defined by a(0;u,v) and V)
(therefore, in particular, u0eH2m(Q)),
ux eHm{Q).
Then the solution u of problem (1.11), (1.12) {which exists and is unique)
(see Chapter 3, Section 8 and the preceding Section) satisfies
(2.3)
that is
(2.3 a)
' ueL2(0,T;H2m{Q)),
d2u
ueH2m>2(Q).
Proof. 1) Let us (formally) differentiate (1.11) with respect to t\
we obtain:
(2.4) a(t; u'(t), v) + (W"(t), v)D + a'(t; u{t), v) = (/'(*), v)D
where
a'(t;u,v)= Y \ l—apq(x,t))DluD*vdx.
W.|«I^»J \dt I
But (1.1) yields
(2.5) w"(0) = /(0) - A (x, 0, — )u0 = u2eH°{Q).
We then consider the a priori resolution of the problem: find u
satisfying (2.4) Vfl e V, with (1.12) and (2.5) (u2 given).
We shall establish the a priori estimates which (by, for example,
the method of Faedo/Galerkin) will show that
(2.6)
there exists a (unique) u satisfying (2.4), (2.5) and
u'eL2{0,T;V), u" eZ,2(0, T; #°(D)).
For this purpose, we replace v by u" (t) in (2.4); taking twice the
real part of the result thus obtained, we get:
dt
[a(t; u'{t)tu'{t)) + ||.«"(*) llJoo,,] + 2Rea'(*; u{t),u"(t)) -
- a'(t; u' (t), u' (t)) = 2Re(/' (t), u" (t))Q

2. A Regularity Theorem (I) 97
from which, by integration over t:
a(t;u'(t),u'(t))+ \\u"(t)2HQiQ)\\ =
t
= a(0',u1,u2) + ||«2||ho(0) — 2Re \ a,(a;u(a), u"(a))da +
(2.7)
+ [ a'(a\ u' {a),«' (a)) da + 2Re j (/' (a), u" (a))Q da.
But
I a'(a; u(a), u"(&)) da = a'(t; u(t), u'(t)) — a'(0; u0, ut) —
o
t t
— a' (a\ u', u') da — a" (a; u, u') da
O 0
and (2.7) becomes:
«(*;«'(*),«'(*)) + II«"(*)IIh0(O) = «(0; «!,«!) + ||«2|lHO(fi) +
t t
+ 3 \ a' (a; u', u') da + 2Re a" (a, u, u') da —
o o
- 2Rea'(2; u(t), u'(t)) + 2Rea'(0;^0> ui) +
t
+ 2Ref (/', u")ada.
o
Using (1.9), we find (the c's denoting various constants and setting
Ml = IMIfl"»«3), \v\ = II"IIho(O)):
(2.8)
(2.9)
l«'WII2 + l«"WI2 £«
kill2 + Kl2 + 11% II IK II +
i«wiiii«'wn + J"(ii«t + b«iiii«'ii)^ +
JVIK'I
^0*
But ^ (£) = u0 + I ^' (a) da yields
o
t
H«W1 ^ll«oll+ f fl«»M*

98 2. A Regularity Theorem (I)
and so we easily deduce from (2.9) that
\\u'(t)\\2 + W'{t)\2^o
"oilman + IKP+ / l/'(ff)l2^ +
0
t
+ J(ll«»||2 + |«"(<r)|2)<*<r
from which we obtain the desired inequality by an application of Gron-
walTs lemma:
(2.10)
\\u'(t)p + \u"(t)\2^c
0 < t £ T.
*0llH2m(fl)
l%ll2+ j\f'(°)\2do
(2.6) follows, as in Chapter 3, or by the method of Faedo-Galerkin
(see Chapter 3, Section 8.2).
2) Next, we note that (2.4) can be written (we have done exactly
what was necessary for this):
it
[a(t; u(t),v) + (u"(t),v)Q - (/(*), v)!, = 0
and since u2 has been chosen such that
a(0,uOfv) + {u2tv)a - (/(0),v) = 0,
we see that u satisfies (1.11).
But then
(2.11) a(t; u(t),v) = (y(t),v)Q, \tv e V,
where
(2.12) W = f-U" eL2(0, T; H°(Q)) = L2(Q).
Let us, for the moment assume
Lemma 2.1. Under the hypothesis: apqe@(Q) and if (1.9) holds
(hypothesis (1.6) is not necessary here), if u satisfies (2.11) with (2.12),
then
(2.13) ueL2(0,T;H2m(Q)).
Then (2.13) together with (2.6) yields the desired result. D
Proof of Lemma 2.1. Let A'^-^f be the solution of
a{t\ A1- (t) /, v) = (/, v)Q, VveV, f given in L2 (Q).
We know that under these conditions (see Chapter 2)
A-1(t)feH2m(Q).

3.1 Statement of the Problem
99
and according to Remark 4.1 of Chapter 2, || A'1 (t) fWm^n) is bounded
in t on every compact set. Therefore, according to the formula
A-1 (t + h) - A'x{t) = -A-1 (t + h)(A(t + h)-A (t))-lA (t),
the function t -* A"1 (i) f is a continuous (and even C1) mapping of
[0,T]-+H2m{Q), VfeL2(Q).
Then the function u, which is given by
t-+u(t) = A'1[t)fp[t)>
is measurable from [0, T] -* H2m{Q) and
ll«Wllfl2m(u) ^ C ||yWllL2(u);
therefore, we have (2.13). D
3. Regular Non-Homogeneous Problems
3.1 Statement of the Problem
We now consider problem (1.1), . . ., (1.4) with gj =j= 0; Theorem 2.1
yielding one solution for the case "gj = 0". We reduce the problem to
Theorem 2.1 by the usual method:
assume u0> ult gj (0 ^ / ^ m — 1) to be given such that there
exists a function w satisfying
Aw + w" = (peL2(0,T;H°(Q))
and <p' eL2(0,T;H°(Q)),
(3.1) \ and the conditions
BjW = gj, 0 ^ / :g m — 1,
w(0) = u0, w' (0) = Wj.
Then setting
(3.2)
problem (1.1), . .
u — w = &,
(1.4) is equivalent to
A 0 + &" = / - <p,
(3.3) j By 0 = 0, 0£/£ m - 1,
0(0) =0, 0'(O) = 0,
and we apply Theorem 2.1 to problem (3.3). D

*fQO 3. Regular Non-Homogeneous Problems
3.2 The Compatibility Relations
We now have to specify (3.1). In order to realize conditions
"(p, <p' eL2(0, T; H°(Q))" in a "practical" way, we shall require w
to satisfy
(3.4) weL2(0>T\H2m{Q))> ^eL2(0,r;i72m(D)), w"'eL2(0,T;H0(Q)).
Proposition 3.1. The necessary and sufficient condition for w to
satisfy (3.4) is that
(3-5)
w
(t) = w0 + J \p{<y) da,
y)eH2m,2^} w0eH2m{Q).
ProofAifEH2m'2{Q)(le.y)EL2{0>TiH2m(Q))>y)ffeL2(0>T'>H0(Q)))>
then w, given by (3.5), satisfies (3.4).
Conversely, if w satisfies (3.4), then w' = ip e H2m'2 (Q) and w is a
continuous mapping of [0, T] -* H2m(Q)', therefore w(0) = w0 eH2m(Q)
and we have (3.5). D
Proposition 3.2. If w satisfies (3.4), then we have:
(3-6)
BjW = gjeH1(0,T;H2m-m^ll2(D)nHl3m-im'+1/mim(0,T;H°(r))>
(3.7) w(x,0) = w0eH2m{Q),
(3.8) w'{x,0) = w1eH3ml2(Q),
and the compatibility relations
(3-9)
BJ[x,0, — )wo = gJ(x,0), xeT,
m — 1
(3.10)
for all j such that m5 =j=
Bj[x,0, )w1 + ( Bj\x,0, ))w0
J[ dx x \dt J\ dx " °
= —-gj{x,0),xer>
ot
for all j such that ms g
3m — 1
to which global relations (see below) must eventually be added when m is
odd.

3.2 The Compatibility Relations
Proof. By (3.5), we obtain
(3.11)
101
Bjw(t) = Bjw0 + Bj(jf(a)da\.
and
Since w0eH2m(Q), we have (Chapter 1, Section 9):
BjweH2"-mJ-1/2(r)
t -> Bj ix, t 1 w
OX
belongs to the space defined in (3.6).
On the other hand
Bjl \y{o)dc\= £ bjh{xJ)\DhMxta)da (xeT).
\0 / \k\^mJ 0
But xp e H2m'2 (Q) and therefore, according to Theorem 2.1 of
Chapter 4, we have
't^_eff2m-j-l/2,(2m-j-l/2)/m/2J\ _
dvJ
= #°(0, T; H2m-J~1/2(r)) n #(2m--/-1/2)/w(0, T; H°(r)),
so that
t
J y,. y>(cr) rfo-e tf1 (0, T; #2m^^^^
o
from which (3.6) follows.
We have already seen (3.7); (3.8) follows from trace theorems for y>.
It remains to show the compatibility relations. For this purpose, we
apply Theorem 2.3 of Chapter 4. We obtain:
dkw . m — \
|,=0 =yk(y>{0)), 0 ^ * ^2m- i, a #—-—,
(3.12)
and
(3.13)
dvk
d dky)
dt dvk
3m — 1
= y*(y('0)), o ^ k < —-—
t = 0 ^
(relations which imply (3.9) and (3.10)), to which we must add the
global relations (assuming, after local maps, that Q = {x \ xn > 0}):
(3.14)
ou
II.
d"f ,
&y f
(x',a2,0) r(*'.0'ff )
dx"n ' ' dx*
da
dx' < oo
a
if k =
m — \

102 3. Regular Non-Homogeneous Problems
and
(3.15)
II
d dk
k ip' (*', cr2, 0) - —-rry^, 0, a2
Rn-A d%n St dx{n
da
dx' < oo
a
3m — 1
if k =
relations which imply global relations for BjW0, BJw1> gJ} if there
m — 1 3m — 1
exists mj such that
2
Conversely, we have:
Proposition 3.3. Let gj, w0, w1 be given and satisfy (3.6), (3.7),
(3.8), (3.9), (3.10) and the eventual global relations of Proposition 3.2. Then
there exists a w satisfying (3.4) and
Bjw = gj> w{x,0) = wo, w'(x,0)=w1.
Proof. 1) We first consider 6eH2m-2(Q) with
Bjd=g;-B'j(xJ,^-\w0,
d(xf0) = wx.
Such a function exists, according to the compatibility relations and
Chapter 4 (Section 2 and pait4 of the Proof of Theorem 14.4).
We shall seek w in the form
t
w = F + w0 + \ 6(a) da,
o
with
FeL2(0, T;H2m(Q)), F' eH2m>2{Q),
F(x,0) =0, F'(x,0) =0,
BjF= - fj^U^-^-) (oiaJdaAdo^hj.
Then we shall have:
w[xy 0) = w0, w'(x, 0) = 6{x, 0) = wx

3.2 The Compatibility Relations 103
and , , x
(Bjw)' = (BjF)' + Bjd + B'jljd'da] + B'jW0 = g'Jt
therefore BjW-gj = Bj(0) w0 - gj(0) = 0.
2) Construction of F. Let us order the B/s in increasing order of m5.
Thus, we replace the conditions "BjF = hj" with:
dJF
= 0 if 7 ^ m0 — 1,
amoF
■ = *<>;
fimi mi-l g|8
next, if Bx = h V &lj8 —^, &lj8 = differential operator of
order ^ m1 — ^ on 2, we take
and so on. Therefore
dJF
where the &/s belong to the suitable spaces on Z and the compatibility
relations hold:
kj\t = o = 0, k'j\t = 0 =0.
In order to construct J7, we start with the construction of G e
eH2m>2(Q) with dJG
G(x90) = 0, —r= *;
dvJ
(which is possible by Theorem 2.3 of Chapter 4), and then we take
t
F=JG(c)do. D
o
From the preceding results we deduce
Theorem 3.1. Assume (1.6) and (1.9) to be satisfied. Let f, gj} u0, ux
be given with
(3.16) feH^iQ),
(3.17) gJeH1(0,T;H2m-mi-1l:i{r))nHlim-im^ll2nim{0,T;Ho{r)),
(3.18) u0eH2m(Q),
(3.19) «! e H3"'2 {Q),

104 3. Regular Non-Homogeneous Problems
and the compatibility relations of Proposition 3.2. Then the solution u
of problem (1.1), . . ., (1.4) satisfies
(3.20) ueH2m-2(Q). D
Remark 3.1. We can verify that:
c #!(0, T; H2m-mJ-1/2{D) n tf3m-^ + 1/2)]/«(0j T; H°(r)).
We therefore a fortiori have (3.20) if we replace (3.17) with the
condition
(3 22) E jj3rn-mj-l/2,l3m-(mj+l/2)l/m,2J)t Q
3.3 The Case of the Dirichlet Problem
We consider the Dirichlet problem, that is
(3.23) Au + u" =f in <?,
dJu
(3.24) -— = g,-, 0^'gw-l, on 27,
(3.25)
^ (%, 0) = u0 in i2
du
(x, 0) = ut in !2.
We have
Theorem 3.2. Assume that (1.6) and (1.9) &0&Z. Letf /, gJf u0, u1 be
given with
(3.26) feL*(Q),
(3.27) g,6ffta-|-1'2i(J"-J-1'1,*(20I 0 g / £ » - 1,
(3.28)
| «iefl*/2(0),
zotW £&<? compatibility relations such that there exists awe H2m'2 (Q)
satisfying
dsw
dvJ
gJt 0 ^ / ^ m — 1, w(#,0) = «0(#), w'(#,0) = «!(#)
(see Chapter 4, Section 2, Theorem 2.3).
Then the solution of problem (3.23), (3.24), (3.25) satisfies
(3.29) umH^iQ).

4.1 Adjoint Isomorphism
105
Remark 3.2. Here the compatibility relations are (see Theorem 2.3,
Chapter 4):
(3.30) gj \t = 0 =yju0 for 0 <; / ^ m - 1
d m — \
(3.31) — gj\t = 0 =yJu1 for 0^/^ —
ot 2
and the following global relation when
m — 1
(3.32)
(3.33)
1
-, m odd:
I
0 R»-l
^«i . , ,. Sgj
dxi
x',«2) -^r(%',e2m)
01
do
2 ax' < oo,
(where we assume that Q = {% \ xn > 0}, to which we are led by local
maps).
Proof of Theorem 3.2. The proof is immediate; having chosen a
function w as in the statement of the theorem, the function & = u — v
must satisfy:
A 0 + &" = f - [A w + w"),
dJ0
— = 0, O^^-l,
3vJ
<P(*,0) = 0,
■(*,0) = 0.
Since w e H2m'2 (Q), the function f — [Aw — w") belongs to L2 (Q)
and therefore Theorem 8.2 of Chapter 3 applies. Q
4. Transposition
4.1 Adjoint Isomorphism
Consider the "adjoint problem":
(4.1) A* v + v" = q>
with
(4.2) <p e #o!o ((?) = closure of S(<?) in H0'1^),
and the homogeneous conditions
(4.3) C> = 0, 0 ^ / ^ w - 1,
(4.4) v{T)=0, v'(T)=0. U

(4-5) , , ., TJ0,
106 4. Transposition
Remark 4.1. For 9?, element of H0-1 (Q), to belong to H°0$(Q), it
is necessary and sufficient that
<p(x,0) = <p(x, T) = 0. D
If we define
X = space described by v, solution of (4.1),. . ., (4.4), as q>
describes Hq[1(Q), provided with the translated topology,
then we see that
(4.6) A* + D2 is an isomorphism of X onto H%\q{Q) .
It is this "adjoint isomorphism" which we shall now transpose. D
By Theorem 1.1, we have:
(4.7) XaH2m-2(Q). U
4.2 Transposition
From (4.6) follows (compare with Section 13 of Chapter 4, for
instance) :
if v -» L (v) is a continuous antilinear form on X, then there
exists a unique u e (Hq[q (Q))' satisfying
(4.8)
<«, {A* + D2t) v>=L(v), VveX,
where < , > denotes the duality between (.ffo.'o ((?))' and
h°oMq).
If, as we have already done previously (Chapter 4, Section 8.3), we
define
(4.9) H°--1(Q) = (H°0'MQ)y,
then we can state (4.8) in the equivalent form:
[ for every continuous antilinear form v -> L (v) on X, there exists
(4.8 a) la unique u eH°'~1(Q) satisfying
<uy (A* + D2)v} =L(v), VveX. D
Remark 4.2. We recall that every element h of H0t~1(Q) may be
represented (non-uniquely) by
(4.10) a = a0+JLAi> hteL2(Q). D
ot
As we have already seen in Chapter 4, the problem now is to
choose L. D

4.4 Conclusion 107
4.3 Choice of L
Formally at first
(4.11) L(v) = </, v} + <g, fT> + <%, t7(0)> - (u0y t7W>,
where the brackets are taken according to suitable dualities.
The choice of /, g, ult u0 follows from (4.7).
First v -» </, v} is evidently, according to (4.7), a continuous anti-
linear form on X, if we take
(4.12) fe(H2m>2(Q))'. U
Remark 4.3 (Compare with Section 13.2, Chapter 4). Thus / is not
a distribution on Q. D
Next we note that:
v -> T v is a continuous linear mapping of
m-l
TJ2m,2 IQ\ . TT rj2m-(2m-mj-l)-l/2,(2m-(2m-mj-j)-l/2)/m/ v-»\
(4.13)
m-l
= 77 /fmi+1/2»(mJ+1/2>/m(27)
J = o
and consequently, we may take
m-l
(4.14) gejl (/r'+1/2'<^+1/2>/"(D)'
j = o
in (4.11).
(As in Remark 4.3, g is not a distribution on 27.) D
Finally, v -* v(0) (resp. v -* v'(0)) is a continuous linear mapping
of H2m>2{Q) -» H3m/2{Q) (resp. Hm/2{Q)) and therefore, according to
(4.7), we may take:
(4.15) ^o e (#m/2 (£))', «! e (#3m/2 (fl))'
in (4.11) (and here again, the u/s are not distributions on Q).
4.4 Conclusion
The above results may be summarized in
Theorem 4.1. Assume that (1.6), (1.8) and (1.9) are satisfied. Let
f, g, u-o> ui be given with (4.12), (4.14), (4.15). There exists a unique
ueH°--1{Q) such that:
(4.16) <«, (A* + D2) v> = </, v> + <g, f v> +
+ <«i,w(0)>-<«o,»7(0)> VveZ. D

108
5. Interpolation
Remark 4.4. In (4.16), the term on the left of the equality sign is
taken in the duality between tf0'"1^) and H°0$(Q). On the right of
the equality sign, the first term is taken in the duality between [H2mA (Q))r
and H2m>1 (Q), the second between
m-l m-l
FT (Hmj+1,2'(mj+1/2)/m(U)y and Y] HmJ+1/2>(mJ + 1/2Vm(£),
j=0 j=0
the third between (H3m/2 {Q))f and H3m/2(D) and the last between
(Hm/2(Q))' and Hm/2{D). D
Remark 4.5. The interpretation of (4.16) in terms of the "usual"
boundary value problems is not considered here (see Problem 14.1).
Under different hypotheses on /, g, u0, ult this problem is studied in
Section 11. D
Remark 4.6. Taking account of (1.10), the signs in (4.16) are chosen
uch that, formally, the solution u of (3.16) satisfies (4.1), . . ., (4.4). D
5. Interpolation
5.1 Statement of the Problem
Consider the mapping 0((1)):
(5.1) #:/,g,«o»«i -> «.
In order to simplify the notation, set
{m-l
7t . i 7Z — TT rjSm—m,-—1/2,(3m—fit,-—l/2)/m / v-i\
j = o
u0 e H2m(Q), %e H3m/2 (Q), giu0iu1 satisfy the
(5.2)
m-l
(5.3) Ax = n (Hm'+1,2-im>+1,2>,m(Z)y x (ff»'2(fi))' x (H3m/2{Q))'t
j = o
where each space is provided with the norm (||g||2 + \\u0 \\2 + ||^i||2)1/2,
and where each norm for g, u0, ux is taken in the appropriate space.
Then, according to Theorems 3.1 and 4.1, the linear mapping (5.1)
has the following properties:
{^ is a continuous mapping of H°A(Q) x A0 -* H2m'2 (Q),
& is a continuous mapping of (H2m'2(Q))' x Ax -> H°^1{Q).
((1)) As in Remark 13.1 of Chapter 4, we must be careful to remember that
this mapping is not injective.

5.2 Some Interpolation Results 109
Therefore by interpolation of (5.4), we have:
(^ is a continuous mapping of
lH°*(Q),(H**>*(Q)y]9 x [A0, A J, - [&•»■> (Q),H°--i(Q)]e
We need to make (5.5) explicit, and we shall partially accomplish
this.
5.2 Some Interpolation Results
We start by giving an interpretation of the space [H2m>2 (Q),
But we must be careful: it is not true that this space coincides with
the space of restrictions to Q of the v's eL2(Rt; H2mil-0)(Q)) such that
T-L-r,eL»(*,;2m),
v = Fourier transform in t of v. The correct interpretation is:
Theorem 5.1. The following identities hold:
[H2m>2(Q),H°--1(Q)]e = H-e(0, T; H2m<l~9>(Q)) o
(5.6)
nfl2-3,(0, T:,H°(D)), 0*- and H-
2 6
(5.7) [H2^2{Q)} H°>-i(Q)]1/2 = #oo/2(0, T; Hm{Q)) n
n#1/2(0, T;H°{Q))
( where Hq01/2(0, T; X) = (#S£2(0, T\ X))' if X is a Hilbert space and
#j^2(0, T\X) = [#<5(0, T;X),H°(0, T;X)]1/2] see Chapter 1,
Section 11);
(5.8) [H2™>2(Q), H°>-i(Q)]5/6 = #"5/6(0, T\ H«*(Q)) n
n^0-01/2(0,r;^°(D)).
Proof. We first replace the cylinder Q = Q x ]0, T[ with Qo0=Q xRt',
we already assume that Q = Q x ]0, oo[ (otherwise make two extensions
by symmetry, instead of one as below, and use the truncations).
Set \u{t) if *>0
pu(t) =
[<x1u( — t)-\-<x2u( — 2t)+oc3u( — 3t) if t < 0,
with
«1 + ^2 + ^3 = 1
—<*i — 2oc2 — 3oc3 = 1
<*2 ^3 .
— 0C± = 1 •
2 3

110 5. Interpolation
u -* p u is a continuous linear mapping of
H2»>>2(Q)-+H2»>>2(QJ
and of
H°>-HQ)=H-i(0,T;H°(Q))^H°>-i(QJ=H-i(Rt',H<>(Q))
(if we note that the transpose of p maps
W(Rt; H\Q)) -* #£(]0, T[; H°(Q))).
Therefore u -* p u is a continuous linear mapping of
[&">■>(Q), H°--i(Q)]e -> [&»>•>(QJ, H°--i(Qx)]e
and since the restriction of (^ to ^ is a continuous linear mapping of
tf2ra-2(CJ (resp.tf°--1(ej)-*#2'»-2(e) (resp.tf0-"1^))
and satisfies r p u = u, we finally obtain:
[#2m-2(<2), tf0'-1^)]* = space of restrictions to Q of
[^-■'(OJ,*0--1 Woo)],.
But using the extensions of Q -* Rw (see Chapter 1, Sections 8 and 9),
this last space coincides with the space of restrictions to Q^ of
[H^(Rl x Rf), ff«>.-i (K x R,)]e = Xe.
2) Now, by an application of Fourier transformation, the
characterization of Xe is immediate. If tJ denotes the Fourier transform in %
and ^ of (7, we have:
UeX9o(l + l^l2" + \r\2)l-e n /, .. ffeL2(R^xRT),
(1 + 1*1)
and thus
Xe = H~d(Rt; H2*1-*^)) n #2"3"(Rr; #°(R*)).
Since the space of restrictions to Q of Hs (R"), s > 0, is Hs (Q), it follows
that:
[H2m>2(Q), H°*-1(Q)]9 = space of restrictions to ]0, T[ of
H~e(Rt; H2m^-^(Q)) n #2"3"(Rr; H°(Q)).
3) In order to complete the proof of the theorem, it only remains
to verify:
if Jf is a Hilbert space, then the space of restrictions to ]0, T[ of
Hs(Rt, Jf), -1 < s < 2, coincides with Hs{0, T\ X) for s + -£, and
with
^oo1/2G0,r[;jT) for s= -±-

5.2 Some Interpolation Results 111
For s ^ 0, this is Theorem 9.1 of Chapter 1 (for functions with values
in a Hilbert space). It remains to study the case — 1 < s < 0. First
of all, if U e Hs(Rt; X) and if (9 = ]0, T[, its restriction U0 to (9 is
defined by
(5.9) <U0>(p> = <U><py, q>e9(0),
<p = extension of (p by 0 outside (9.
Since cp -» <p is a continuous mapping of
#o'(0)->ff-'(R), 5+ -±,
^2(^)->^/2(R),
it follows that
U€eH'{0, T\jf), s * -i
^6ffoo1/2G0,r[;jT) if s= -i.
We still have to show that the mapping U -» C7^ is surjective. We know
(Chapter 1, Section 12.2) * that there exists a continuous linear mapping
u -+ P u
of
#°(0, T]Jf) -+H°(Rt;jf) and of ff-^O, T\X) -» i?"1 (Rr; jf)
such that (P tt)^ = m . By interpolation, P is a continuous linear mapping
of
[#°(0, T\ X), H-1 (0, T; X)\ -> ff-'(Rt; jf)
and the first space is (Chapter 1, Section 12)
#-"(0,r;jO, d *±,
^oo1/2(0,r;jT), if 0 = ±.
Whence the desired result. D
We have not solved the problem of the complete characterization
(different from the definition!) of \A0iA^\e\ see Problem 14.1. Here,
we give a particular case (which yields interesting applications to
optimal control theory, Chapter 6, Section 11). We consider the case:
u0 = 0i ux = 0.
We introduce
f Ej[S) = space described by gj when {g, 0, 0} belongs to A0t
(5.10) \ with g = {0, . . ., 0, gjt 0, . . ., 0}, provided with the topo-
I logy induced by A0]
1 Lemma 12.2 of Chapter 1 is valid for vector-valued functions.

112 5. Interpolation
and
m-l
(5.11) A00 = EI Ej(Z).
j = o
Next, we set
m-l
(5.12) ^ A10 = n (Hm'+lf2-im'+1,2>/m(Z)y.
j = 0
Then, we have
m-l
(5.13) [A00,A10]e= [I {.Ej(Z),(Hm>+1>2'«"+1'2>""(Z)y]t.
J = o
Replace u0, ux with 0 in (5.1); thus, we consider
(5.14) /fgj0>0-=Ui#,
which is a continuous linear mapping of (we do not write the
multiplication by {0} x {0})
H^(Q)xA00-^H2^(Q)
and
therefore of
(5.15) [#°-1(e),(tf2ra-2(0)']»x [^oo. ^io].- [H2"'2^).^0'-1^)]..
where the last space is characterized by Theorem 5.1.
Thanks to (5.13), the space [^oo^iole is characterized with the
help of the following theorem:
Theorem 5.2. We have:
[Ej(Z), (H"'+1'2'im>+1,2>,m(Z)yiie
_. ^3m(l-0)-(mj+l/2),[3m(l-0)-(mj+l/2)]/m/v-»x
if
3m(l - 0) - (mj + i) 1
(5.16) y- '- ^ 2-L < - •
m 2
(For the definition of the spaces 34?">"Im(Z!) see Chapter 4, Section 14.1).
Proof. 1) We first show that
(5.17) [Ej(Z),(Hmj+1/2'imj+1/2)/m(Z)yieQ = L2(Z)
if 3m(1 - 0O) - (mj+-i) = 0.
Indeed ^3m-(mj+l/2),[3m-(mj+l/2)]/m/™ <_ £. (JT\ cz
c: /73m~(mJ+1/2)'[3m~(mJ+1/2)]/m/r\
so that [£'</(2,),(fiPmi+1/2'(mi+1/2)/m(2,))/]e0 contains and is contained in
L2(Z) (apply Theorems 14.2 and 14.3 of Chapter 4), from which (5.17)
follows.

5.2 Some Interpolation Results 113
2) Next, we apply the reiteration theorem (Chapter 1, Section 6). If
6 > 60, then:
lEj(Z), (fr^+1/2-(^+1/2)/m(^)']6
= l[Ej(Z), (Hmi+ll2'imt+ll2)lm(Z))']QQi (Hm>+lf2'im'+lf2Vmffl)'-]ei
where
6 = 60 + 6,(1 -60);
according to (5.17), this space is equivalent to
\p (2), (HmJ+ ^2^j+ "»»* (Z))']6l,
which, by duality, reduces to (Chapter 1, Section 6.2)
[L2(Z),HmJ+1/2'imJ+1/2)/m(Z)]dl,
from which the result follows (by an application of Chapter 4, Section 2).
Now if 6 < 60 (with or without (5.16)), then:
\Ej(E), (Hm>+ll2'imt+ll2)lm(E))']Q
= [£,(23, [Ej(Z), (^+1/2'(^+1/2)/w(r))10o]02,
where
6 = 60 62
and, by (5.17), this is equivalent to:
(5.18) [Ej(Z),L>(Z)]e2.
It remains, therefore, to characterize this space, in particular when
(5.16) holds.
3) The space Ej(Z) appears as an intersection: define
(5.19) Fj{Z) = L2(0, T; H3m'(m^ll2\r)),
(5.20) Gj(2) = \g\ge #»—<-*+V2>v»(0> T; L2(D),
S(0) = 0,
and
3m — (m1 + 4) 3
e' (0) = o, if ^—2± > -,
m 2
" ' rw.L.(p.ri£.m). a 3"-^ + «-2|""'.
Then
(5.21) £^=^(2)0^.
<(1)> See (11.52), Chapter 1.

114 5. Interpolation
But
Fj(£)c:L*(Z), G,(27)c=L2(r),
and therefore
Fj(2) = domain in L2 (Z) of a positive self-adjoint operator, say @j = 0
and
Gj(E) = domain in L2 (27) of a positive self-ad joint operator, say Wj = S7.
Furthermore 0 and W commute. (Indeed, after local maps and Fourier
transformation, 0 amounts to multiplication by
(1 + |f|)3—(-,+ l/2>f |eR»-l
and W to multiplication by
(1 + |T|)t3ra-(ra'+1/2)3/m)TeR).
Therfore (Chapter 1, Section 13):
(5.22) [^BlI2W]fc-[FJ(25lIa(%n[CJ(23lL2^
But
(5.23) [Fj(Z),L2(Z)]d2 = L2(0, T\ ^-Ci+i/2))(i-w(f))
and therefore the desired result is a consequence of the identity (see
Chapter 4, (14.32)):
(5.24) [Gj(Z), L2(2)]d2 = ^(3m-(m,+ l/2))(l-02)/m(r)
if
v J ^™ \(i _02) <_.
ml 2
(l 3m — (mT- + +) \
Note that ^ —J (1 - d2)
_ 3w - (wj + i) / (9 \ _ 3w(l - 0) - (m, + i) \
w \ 0O / w /
5.3 Consequences
From (5.15) and Theorems 5.1 and 5.2, we deduce:
Theorem 5.3. Assume that 0 < 6 < 1 and that (5.16) M^s for
j = 0, . . ., m — 1.
Let f,gbe given with
(5.25) /e[ff°-1(e),(#2ra-2(e))']e>
m-1
(5 26) £ e TT 2f3m(1~0)~(mi+1/2>'(3m(1-0)~(mi+1/2»/m/r\

5.4 The Case of the Dirichlet Problem
Then, the solution u of (4.16) (with u0 = 0, ux = 0)
belongs to the space described in (5.6), or (5.7)
115
(5.27)
*/ 0 = —, or (5.8) */ 0 = —.
2 6
Remark 5.1. The condition on 0 is equivalent to
5 1 min m f
(5.28)
e>
6 6w
3w
Remark 5.2. Formally, u is a solution of
f A u + u" = /,
(5.29) | 5^ = g
[ u(x,0) = 0, u'(x,0) = 0. D
5.4 The Case of the Dirichlet Problem
For the Dirichlet problem, we may use Theorem 3.2 instead of the
regularity Theorem 3.1.
We introduce the space &/0 (which "replaces" the space A0 defined
in (5.2)):
{m-1
z ». ». \ X r- TT rj2m-j- 1/2,(2m—j- l/2)/m / \i\
g,«o* wi \g e [I H ' ' )l (2/),
j=o
u0eH3m'2(i;), ux eHm'2(Q),
g,u0iu1 satisfying the &.<£. defined by (3.30) to (3.33) .
If ^ denotes the mapping
(5.30)
where u is a solution of
f A u + u" = /,
(5.31)
dJu
dv-
j = gJt 0 ^ j ^ m - 1,
du
u(x,0) = u0(x), —— (x,0) = ux (x),
ot
we now have:
is a continuous mapping of L2 (Q) x £?0 -► H™*1 (Q),
is a continuous mapping of (H2m'2(Q))' x Ax -► iP*"1^).
(5.32)

(5.33)
(5.34)
116 5. Interpolation
Therefore, by interpolation of (5.32):
^ is a continuous mapping of [L2 (Q), ((H2m>2 [Q)y]e x
x [s'o,Ai]e'+lH"-i(Q),H<>--*(Q)]e. D
As for Theorem 5.1, we can show:
Theorem 5.4. We have the identities:
[H*»>i(Q),H°>-i(Q)]d = H'e(OtT;H^-e^(Q))n
nHi-MQ.TiHOiQ)),
0*±, 0*1;
(5.35) lH».*(Q),H°--*(Q)]1/2 = H^2(0, T; H«*(Q)) n H°(Q);
(5.36) [H"-i(Q),H°>-i(Q)]3/4 = H-3!\0,T;Hm/*(Q))n
n^oo1/2(0,r;^°(i3)). □
By the same arguments as for Theorem 5.3 (introduce <stf00 = space
of g's, with u0 = ux = 0,
m-l
J = 0
m — \ m — 1
gj(0)=0 V/, g'(0)=0 for />— and, for / = — ,
00
[ flg'o-l)/2(*',<T2-)|2i*'-^-<00 if £ = {*|*,,>0})
we obtain:
Theorem 5.5. Let f be given with
(5.37) /6[£2(0),(ff2--2(0))'],
m-l
(5.38) | erTe^72m(l-0)-(j+l/2),(2w(l-0)-(j+l/2))/m/2T\
J=0
Assume that u0 = 0 *wmZ % = 0.
Assume that
3m — 1
(5.39) 0> •
Am
Then
(5.40) if 6 * J, w belongs to the space described in (5.34);
(5.41) if 6 = J, u belongs to the space described in (5.36). D

6.1 General Results 117
Remark 5.3. It is interesting to compare Theorem 5.5 with Theorem 5.3
applied to the Dirichlet problem, that is with ms = /, V/. Let us just
concern ourselves with the regularity of the g/s. Take 0 = 61 in
Theorem 5.3 and 0 = 02 in Theorem 5.5. Then, we have the same regularity
for the g/s if
3(1 -00 =2(1 -02).
Then 2 — 30j = 1 — 202. Therefore, Theorems 5.3 and 5.5 yield
respectively :
u e H-ei(0, T; H2m^-ei){Q)) n H2-3dl(0, T\ H°{Q))
and
u e H-2d2(0, T\ H^-^iQ)) n H2~^{0y T\ H°{Q)). D
Remark 5.4. For the Dirichlet problem, we can also transpose the
isomorphism q> -> v, wehre v is the solution of
A* v + v" = <p,
v{x,T) =0, v'(x,T) =0,
= 0 on 27, 0 ^ 7 ^ w — 1,
for <peL2(Q). U
6. Applications and Examples
6.1 General Results
We consider problem (5.29), assuming that the data gj satisfy
(6.1) gjeL2(Z), / = 0, l,...,m- 1.
In order to simplify, we choose
(6.2) u0 = 0, ut = 0.
Our aim is to investigate [assuming f to be chosen in an "appropriate"
manner) the corresponding order of regularity for the solution u. D
Case in which the data are not of Dirichlet type.
We apply Theorem 5.3. This is permissible for 0 such that
^3m(l-0)-(mJ+l/2),(3m(l-0)-(mJ-+l/2))/m/v-»N j 2 / Y) y7'
that is if
(6.3) 3m(l - 0) - (tnj + ±) = 0, V/.
The optimal choice of 0 (i.e. which yields the maximum information
about the regularity of the corresponding solution u) is the least 0 such

118 6. Applications and Examples
that (6.3) is satisfied, that is
1 min mt
(6.4) 8 = 1 J
6m 3m
Let us examine the particular cases:
(i) the case 6 = ^ yields: 2minmj = 3m — 1; but minm,- _ m and
therefore this is possible only if m = 1, m0 = 1;
(ii) the case 6 = — yields: minm,- = , which is possible if m
is odd. 6 2
In conclusion:
Theorem 6.1. Let the g/s be given in L2(E) and f be given by (5.25),
with 6 given by (6.4). Assume that u0 = 0, ux =,0. Then, unless
(6.5) m = 1, tn0 = I,
or
m — 1
(6.6) w *s o^, minmj = ,
w have
(6.7) « e tf-*(0, r; H2m^-9^(Q)) n tf2-3'(0, T; #°(£)).
For the case (6.5), Z0£ have
(6.8) « e Hoo/2(0, T; H1 .(£))) n tf1/2(0, T; H°(£))).
For the case (6.6), we have
(6.9) ft e tf-5/6(0, T; Hm'*(Q)) n #oo1/2(0, T;:H°{£))). U
The case in which the data are of Dirichlet type.
Here, we apply Theorem 5.5. This is legitimate if
t^2m(.l-0)-U+l/2K2mCl-O)^U+l/2))/m/2J) Z> L2 (2) , V/,
that is, if
2m (I -0) -i = 0
and therefore the optimal 6 is
(6.10) (9=1
Am
The only singular case, 6 = f, corresponds to w= 1. Therefore:
Theorem 6.2. Let the g/s be given'in L2(Z) and f be given by (5.37),
with 6 given by (6.10). Then, unless
(6.11) m = 1,

6.2 Examples 119
we have
(6.12) ueH~0(O, T; H^~0^(Q)) n H1"29^, T\ H°(Q)).
For the case (6.11), we have
(6.13) u e#"3/4(0, T; W*(Q)) n H^'2(0, T; H°(Q)). U
6.2 Examples
Example 6.1
Consider the problem
-Au + u" = 0
(6.14)
du
—- = g on 27,
u{x,0) = 0, «'(*, 0) = 0.
Then m = 1, m0 = 1; therefore we have the case (6.5) of Theorem 6.1;
we obtain:
{if geL2(E), the solution u of (6.14) satisfies
u e #oo '(0, T; # * (fi)) n #1/2(0, T; #° (fl)).
Remark 6.1. We shall see how result (6.15) compares to the result
given by Theorem 9.5 of Chapter 3.
First of all, let us see how the preceding problem may be stated in
variational form. Introduce
V = H1(Q), H = L2(Q)
and V = V~x = antidual of V (it is not a space of distributions on Q).
For v e H1/2+e(Q), arbitrary e > 0, we know (Chapter 1, Section 8)
that y0 v = trace of v on r e He(r). Consequently, for g given in L2 (27)
= L2(0, T\ L2(r)), we may define / by:
(6.16) (/(t),v)=fg{x91) y0v(x) dr
r
and thus
(6.17) /eL2(0, T;(H^2+E(Q)y)t Ve > 0.
But (6.17) is equivalent (by Chapter 1, Section 9) to
f feL2(0,T;V-1/2-E)
(6-18)
[V-e = [H, V']0.

120 6. Applications and Examples
Thus, the problem is equivalent to the search for the solution u of
(6.19) a(u(t),v) + (u"(t),v) = (f(t),v), Vv e V,
where
- r du Tv
a[u,v) = X ~—dx
and where ( , ) denotes the scalar product in H (or in the antiduality
between V and V).
Therefore, Theorem 9.5 of Chapter 3 applies, with
6 = i + e, arbitrary e > 0.
We obtain:
(6.20) ueC° ([0, T]; F1/2-£), Ve > 0,
(6.21) «'eC°([0, T]; V'1*2-*), Ve > 0.
i
Therefore, in particular:
ueL2(0,T;V1/2-E)f u' eL2(0,T; V~1/2-E)
and therefore, according to Chapter 1, Section 4:
Df/2u (^-order derivative of u with resprect to t) eL2(0, T; V"e).
In summary:
(6.22) ueL2(0, T\ H^2-E(Q)) n H1/2(0, T; V~E), Ve > 0.
It can be shown that (6.15) implies
, u e L2(0, T; [H1 (Q), H° (Q)]l/2) = L2(0, T\ H^2(Q))t
so that we may take e = 0 in (6.22) — a result which does not follow
from Theorem 9.5, Chapter 3.
On the other hand, the results of this chapter, which are all of a
"Hilbert" nature, do not yield (6.20) and (6.21). □
Remark 6.2. In all this chapter, no regularity result is optimal, since
from the start we do not have optimal regularity results. This may
be verified with a trivial example; take (6.14) in one dimension, with
Q = ]0, oo [ (the fact that Q is unbounded is of course without
significance for the study of a local regularity result). Then
u(x,t) = \j g(o)da, if t°Z x; 0, if t^ x ;
the solution is locally in H1 in the neighborhood of E\ which is of course
better than the result given by (6.15). D

6.2 Examples
121
Example 6.2
Now, consider
f -Au + u" = 0
(6.23) | u = g on r,
[ u[x,0) = 0, «'(*, 0) = 0.
We can apply:
Theorem 6.1, with m = 1, case (6.6); Theorem 6.2, with m = 1.
Then:
if geL2 (E), the solution u of (6.23) satisfies
(6.24) u e H-5'6(0, T; H^(Q)) n tf-3/4(0, T; #1/4(fi)) n
n#oo1/2(0, T;H°(Q)). D
Remark 6.3. Same remark as Remark 6.2. Taking Q = ]0, oo[, this
time we obtain
u(x,t) = {g(t — x) if t > x; 0 if t < x},
a function which is locally in L2 in the neighborhood of E (which is
better than (6.24)). □
Example 6.3
Consider the problem
(6.25)
A2u + u" = 0,
Au I
dAu
£o>
dv
u(x,0) = 0, u'{xfl) = 0.
Then, m = 2, m0 = 2, mx = 3, minrn^- = 2.
Formula (6.4) yields 6 = J and we obtain:
(6.26) m e #"3/4(0, T; tf1^)) n #5/4(0, T; H°(Q)). D
Example 6.4
Consider the problem
' A2u + u" = 0,
dv
(6.27)
«ta = ^o,
«(tf,0) = 0, u'(x,0) = 0.

122 7. Regularity Theorem (II)
Then m = 2, minm^ = 0 and Theorem 6.1 yields
(6.28) u e #-11/12(0, T; H^3(D)) n #"3/4(0, T; H°(Q))
and Theorem 6.2 yields:
(6.29) u e#"7/8(0, T; H^{Q)) n #"3/4(0, T; tf0(i2)). D
7. Regularity Theorem (II)
7.1 Statement
We again consider the problem:
(7.1) Au + u" = f
(7.2) Bu = 0,
(7.3) u(xt0) = 0, u'{xt0) = 0.
We shall prove the following regularity theorem:
Theorem 7.1. Let r be an integer §: 0. Let f be given with
(7.4)
and
(7.5)
feL2(0,T;H^-"m(Q)),
/(2r> = ~£kf e L2(°* T'' H°(Q)) = L2 (Q)'
f(x, 0) = /'(*, 0) = • • • = P'-^^.O) = 0
(no condition if r = 0.)
Then the solution u of (7.1), (7.2), (7.3) satisfies
(7.6) M6H(2r+1)m-2r+1(<2)
(that is:
ueL2(0, T; H<2r+ 1)m(£)), «<2'+1)6l2(0, J; H°(Q))).
Remark 7.1. If r = 0, (7.4) reduces to
/eL2(C).
Then (7.6) holds (Chapter 3). D

7.2 Proof of Theorem 7.1 123
7.2 Proof of Theorem 7.1
1) By formal differentiation in t, we have
(7.7) A «<2'> + u<2r+2> + £ (2r) i4^-»««) = /(2D
fc = 0 \ k )
(where A<* <p = £ (-1)1-1 Z^Df *„(*, *)) Z)^)
|j>|.|«|£m
and
(7.8) «<■»(()) =0, 0 ^ / ^ 2r + 1.
Multiplying (7.7) with w(2r+1)(£) and applying the appropriate
integrations by parts (as in Section 1), we obtain
iN(2r+1)e)iiHo(fi) + ii«(2r)wni-a»^
r
^ c j [||«(2r)(ff) II™ + IK2'+1)(a) H^aw + ||/(2'>(a) fl2^,] da
0
from which, by an application of GronwalTs lemma, we obtain an a
priori inequality of the form
T
(7.9) || «<*■+ »(*) ||S«a» + II «(2r) (') Uh"<«) ^cj\\ /»'> (or) ||20(fi) icr.
0
Zw particular, it follows (using, for instance, the method of Faedo-
Galerkin) that
(7.10) u^eL2(0tT;Hm(Q)),
(7.11) «<2r+1)eL2(0,:r; tf°(0)).
2) We shall prove the theorem by induction onr It holds for r = 0
(see Remark 7.1). If we admit that it holds up to (r — 1), then we obtain
(7.12) ueL2(0tT',H^2r-^m(Q)) (and also ui2r'^ eZ,2(0, T; H°{Q))),
which, together with (7.11) and the intermediate derivative theorem
(Chapter 1, Section 2), yields
(7.13) u^eL2(0, T\ tfur-nmu-j/ur+i))^)^ 0 ^ / ^ 2r + 1.
We shall, as a first step, show that
(7.14) u e Z,2(0, T\ ^(2r+1)m"£(i3)) for all e > 0.
Set
(7.15) So = (2r - 1) m, f, = Sn-i (l - -^-r) + 2m, * ^ 1;
\ 2r + 1/

124 7. Regularity Theorem (II)
according to (7.13):
W'eL2(0J;^o(1'2/(2r+1))W);
and since / satisfies (7.4), we have:
/eL2(0,r;^o(1-2/(2r+1))(i3)),
since
Therefore
A u = f - u" eL2(0, r;#lo(1-2/(2r+1))(fi)),
Bu = 0;
from which, as in Lemma 2.1, we deduce:
ueL2(0, T;H^(Q)).
Admitting, by induction on nt that
weL^OJ;^-1^)),
we obtain (with (7.11) and the intermediate derivative theorem):
^eL2(b,r;//^-l(1-2/(2r+1))(i3)).
But
therefore x
A u = f - u" eL2(0, T; ^-l(1"2/(2r+1))(i3)),
Bu = 0;
therefore
(7.16) ueL2(0, T]H*n(Q)).
We therefore have (7.16) for all n. But £„-►£ = (2 r+l)m, whence
(7.14).
3) We shall now show, by induction on k, integer k increasing, that
(7.17) «<2<r-*» eL2(0, T\ #(2fc+1)m(fi)), k£r.
The result is known for k = 0. We assume it to hold up to (k — 1),
that is
(7.18) ^-fc)+2)6I2(0, T\ ^(2fc-1)m(i3)).

7.2 Proof of Theorem 7.1 125
Differentiate (7.1) 2(r — k) times with respect to t:
2(r-fc)-l
(7.19) A ui2ir~k» + uW-k)~r) + £ Ai2ir~k)-J)uU) = /<2<'-fc».
j=o
According to (7.14) and (7.11),
^eL2(0, T; Hi2r+1)mil-J/i2r+1))-^(Q))t
arbitrary £j > 0, and
(2r+l)w(l )>{2k+l)m if ;^2(r-*)-l,
\ 2^+1/
therefore, in (7.19):
(7.20) 4«2r-*)-./> UU) e £2^ J. ^(2fc-Dm(i3))
Furthermore
/<2(r-*)) GL2(0, T; #<2r-l)m<l-2<r-*)/2r)(£))
and
2(r - A)\
(2r- l)m(l -) ^ (2A - 1) w,
which, together with (7.18), (7.20), and (7.19) yields
A «<2<'-*» eL2(0, T; ^2fc"1)m(i3)),
and therefore we have (7.17), which proves the theorem. □
Remark 7.2. If A is independent of t and if we make somewhat
stronger hypotheses on / (see below), we can give another proof of the
preceding theorem — proof which shows that (7.6) is essentially the
best possible result (at least in the Hilbert space setting).
We assume the hypotheses of Section 1.3 to be satisfied and use the
same notation. We introduce
(7.21) u{Xt t) = (#«(*)) (A), /(A, t) = («/(*)) (A)
and assume
' r1/2/eI2(0,T;I)),
d2r
(7.22)
and that (7.5) holds.
dt2r-?emo,T;y,

126 7. Regularity Theorem (II)
The first condition in (7.22) is more restrictive than the first condition
in (7.4): it means that feL2(0, T\ D(Ar~1,2))J which implies that
feL2(03T'9H^2r"^m{Q))
and furthermore that / satisfies certain boundary conditions on 27.
We shall recover (7.6) by applying the explicit formula:
(7.23) u = E*f, E(k,t) =
From the formula
^=-sin(* Va), t>0
Va
0,
t < 0.
d2E
dt2
+ XE = d(= d(t)),
we deduce that
(7.24) £=|
Ld»+±.di*> + ... + (_iri_J_a(2r-2, +
A2
1 d2r
X-1 dt2r
Since (7.5) holds, the derivatives of / on R, do not introduce any mass
at the ^-origin and therefore, introducing (7.24) into (7.23), we obtain
(7.25) * = jf~ jit" + ■'• + (-ir1^r/-(2,-2) +
+ (-l)ri-(£./t2r)).
We must show that
(7.26) X<r+1^ueL2(0,T;fy
(which implies (7.6) and in fact, in this case, is equivalent to (7.6), since
the boundary conditions for u are automatically satisfied).
But A(r+1/2)(—/) belongs to L2(0, T; f)) according to the first
hypothesis in (7.22) and
Ar+1/2 — (£*/<2'>) = A1/* (E */<*'>) = (sin*>/*) */<*'>
A (t)
belongs to L2(0, T\ f)) according to the second hypothesis in (7.22) -
and this cannot be improved upon.

8.2 Interpolation in y 127
It remains to show that
(7.27) Ar+1/2_L_/<2>6L2(0):r.W
for 1 <! / ^ r — 1.
Now, from (7.22) and the result on intermediate derivatives
(Chapter 1, Section 3), we deduce that
i.e.
^(r-l/2)(l-2i/2r)/(2J) eL2(0> T', f)) ,
fc-i/2-J+J/2rf<2J)eL2(0,T;fy,
which, in particular, implies (7.27) [thus, we could improve the result
on the "intermediate'' terms in (7.25), but not on the extreme terms].
Hence, our assertion follows.
8. Non-Integer Order Regularity Theorem
8.1 Orientation
As we have seen in Chapter 4, Sections 8, 9 and 10, we can either
transpose the situation of Section 6 and then interpolate, or interpolate
first — and this second possibility is probably more favorable here.
We therefore briefly indicate the "interpolation in r" of Theorem7.1.
8.2 Interpolation in r
Conditions (7.4) may be summarized by:
(8.1) /eF(2r-l)m,2r((?)j
where
V(2r-l)m,2r(Q) = y | f £ #(2r- l)».2r (£) •
f(x,0) = ...=/<*-i>(*,0) =0}.
(the space H2ir~1)m'2r [Q) (see Chapter 4, Section 2) consists of the
elements /eF(2r-1)m»2r(0) such that, furthermore, f(x,T) = ••• = Pr-^{x,T)
= 0).
According to Theorem 7.1,
(8.3) / -► u = "solution of (7.1), (7.2), (7.3)"
is a continuous linear mapping of
v<2r0-l>m.2r0{Q) ^ H<2r0 + l)m.2,.0 + 1 {Q) ^ -^^ ^ > j ^
Vm-2(Q)^H3m-3(Q).
(8.2)

128 8. Non-Integer Order Regularity Theorem
Define:
(8.4)
where
(8.5) r = (1 -6)r0 + 6.
V(2r-l)m.2r(Q) = [y(2r0-l)m,2r0 ^^ ^,2(0)]^
r^l and arbitrary (<r0, but r0 is arbitrary!),
(Proposition 8.1 will show that, up to an equivalence in norms,
definition (8.4) depends only on r and not on r0 and 6.)
Since (see Chapter 4, Section 2), with (8.5):
[Hi2ro+iym'2r°+1 (Q) , H3m'3 (Q)]0 = #<2r+l)».2r+l(0)
it follows that we have:
Theorem 8.1. Let r be arbitrary, r 2> 1. Then, if f e Vi2r~ 1)m>2r (Q)
(defined by (8.4) and (8.2)), the solution u of (7.1), (7.2), (7.3) belongs to
H(2*+l)m.2r+l(Q) []
Remark 8.1. The case 0 :g r ^ 1 can be treated in the same way,
using the fact that / -> u is a continuous mapping of H° (Q) -> Hm' 1(Q). U
8.3 Interpretation of the Space V^2r'^m'2r{Q)9 r^l
Proposition 8.1. For 2r 4= integer + \, we have:
Vi2r-l)m,2r(Q) = y | f £ #(2r- l)m.2r ((?) ^ /">(*, 0) = 0
/or 0 ^ / < 2r - ^}
(therefore definition (8.2) for integer r extends to the non-integer case).
Remark 8.2. If 2r = integer + \, then we have an integral relation
(see Chapter 4, Section 2.5). □
Proof. We have:
(8.6) F(2r0-l)m,2r0((?) = £2^ j. #<2r0-D»(£j) n ^
where
(8.7) Gro = {/ | / e #2r°(0, T; H°(Q)), /"> (0) = 0, 0 ^ / ^ 2r0 - 1}
and similarly
(8.8) Vm>2(Q) = L2(0, T; ff"(fl)) n G1.
But L2(0, T; ^(2r°-1)m(i3)) is the domain in L2{Q) of a positive self-
adjoint operator A0 and L2(0, T; #m(i2)) = D(AS°) with:
(2r0 - l)*o = 1-
On the other hand, Gr° is the domain in L2(Q) of a positive self-
adjoint operator Ax and G1 = D (A*1) with f0 ocx = 1 (this can be shown

9.1 Adjoint Isomorphism of Order r 129
with the same type of technique as for the interpolation between H™ (Q)
and Hq (Q); see Chapter 1). Therefore
Vi2r-l)m,2r(Q) = [D(Aq) n D (Ax) , D (A$) O D (A?)]$
and since the operators A0 and Ax commute, this last space coincides
with
[D (A0), D {A%)]e n [D (A,), D (A?)]„
therefore:
(8.9) V<<2r-1)m>2r{Q) = [L2(0, T\ Hi2r°-1)m(Q)),
L2(O,T;H>»(Q))]0n[Gr°,Gi]9.
The first space on the right side of the equality sign is L2(0, T;
Hi2r~1)m(Q)) and, as in Chapter 1, Section 11.5, the second space
coincides with
Gr = {/ | / e H2r(0, T; H°(Q)), /<■»(()) = 0, 0 ^ / < 2r - ±}
from which the result follows. D
9. Adjoint Isomorphism of Order r and Transposition
9.1 Adjoint Isomorphism of Order r
We shall use only Theorem 7.1, that is the case integer r — and
furthermore r g: 1. The adaption to "non-integer r" corresponding to
Section 8, is not given here. D
As usual (see Section 4.1), we consider the adjoint problem:
(9.1)
A* v + v" = q>,
Cv = 0,
I v(T) = v'(T) = 0,
and we assume that
(9.2) (peHi2r-^r{Q) = closure of ®(Q) in H^~v>»-2r(Q).
Define:
(9.3) Xr(Q) = space described by v as <p describes ^2(r"1)o,'or((?)
and then (Xr(Q) being provided with the "translated" topology) we
have (done what was necessary to have):
(9.4) v -► A* v + v" is an isomorphism of Xr(Q) -► ^(2r'1)J;02r((?).
This is what we shall call the adjoint isomorphism of order r.

130 10. Choice of /, g, u0, ux
9.2 Transposition
Let us transform (9.4). Note that (see Chapter 4, Section 8.3)
(9.5) (^(2r~1)o;or(0y = ff-<*-i>».-*(0);
and recall that every element g of this space may be written (non-
uniquely):
\g~ I DigP+ I Dkthk.
(9.6) J |j>|£(2i—l)m 0ZkZ2r
[gp.hteL2®). U
Then:
Theorem 9.1. Let v -> L(v) be a continuous antilinear form on Xr{Q)\
there exists a unique u in H~i2r~1)m,~2r (Q) such that
(9.7) (u,A*v + v"> = L(v), VveXr{Q).
9.3 Formal Choice of L
As in (4.11), we choose
(9.8) L(v) =<f,v> + (g,Tvy + <%,^(0j"> -<^0,^0)>,
but (which we have not done in Section 4), we choose /, g, ui to be
(suitable) distributions on Q, E and Q.
Thus, we shall follow a procedure analogous to Chapter 4, Sections 8
to 12. □
10. Choice of/, g, w0, Hi
10.1 Choice of/
According to Theorem 7.1, we have:
(10.1) Xr(Q) C #(2r+l)m,2r+l(0
Introduce (as in Chapter 4, Section 9) the space:
(10.2) 5(2r+l)m,2r+l(0 = ^ | ^) W(J) e£2(Q> J'• £(2r4 l-7>»(fl))f
0 ^ ? ^ 2*- + 1},
where
<*(*) is defined by (9.3) of Chapter 4,
3^(0) is defined in Chapter 2, Section 6.3.
As for Proposition 9.1 of Chapter 4, we verify that
(10.3) 9(Q) is dense in S^r+i)m,2r + i ^

10.3 Choice of gj 131
Therefore
(10.4) 5-(2r+l)m,-(2r+l)(Q) = (£(2r+ l)m.(2r+ 1) (£))' ^ 0'(0) .
For this space, we have a structure result analogous to Proposition9.3
of Chapter 4 and, thanks to (10.1), we have:
(10.5) if /e£,-<2r+1>"l--(2r+1>(0), then z,-► </, v>
is continuous antilinear on Xr (Q).
10.2 The Space D^*" (Q)
Taking, in (9.7), v = <p e @{Q), we have L(<p) = (/, <p)Q and
therefore, in the sense of 2'(Q):
(10.6) A u + u" = /.
This leads to the following definition:
(107) ! Z):^?"1)(<?) = {«l«eff-<a'-""--am
\ Au + u" es-(2r+1>M-(2r+1)(g)}, integer r ^ 1;
provided with the norm of the graph, this is a Hilbert space.
77^w ^ solution u of (9.7) belongs to Da+d? ((?) •
10.3 Choice of £,
From (10.1) and the trace theorems of Chapter 1 (see also Chapter 4,
Section 2), we deduce that:
[ v -> T v is a continuous linear mapping of
m-l
Xr(0) -> T~T Hi2r+1)m~i2m~mJ~1,2)'ii2r+1)m~i2m~mJ~1/2))/m(Z). D
j = o
(10.8)
We introduce (compare with Chapter 4, Section 11.1) the spaces
H*mS*(Z), real oc ^ 0;
first for integer oc ^ 0:
(10.9) H«mE«{Z) = {v | <**(*) ^> e L2(0, T; #<«-■"«>» (J1)), 0 ^ / ^*}
and then for non-integer oceR+, we define by interpolation:
f H«mB*(Z) = [H«°m Sao (27), H°'° (27)1-
(10.10)
I integer #0, (1 — 0) a0 = a.

(10.11)
132 10. Choice of /, g, u0, u±
The space defined in this way depends only on oc. Then:
rr(2r-l)m + mJ+l/2,((2r-l)m + mJ+l/2)/m/v-.\
C #(2r-1)m + m./+1/2 &«2r- l)m + mj+ l/2)/m /y\ r-.
As for Proposition 11.1, Chapter 4, we verify that
(10.12) @(Z) is dense in H«mS«(Z)
and therefore
H-«mE-«(Z) = (H"mS"(Z))' c ®'{Z).
According to (10.11), we have:
(10.13)
if £ e FT ^-((2r-1)m+mJ+1/2)45,"'((2r-1)m+mJ+1/2)/»»/27)
J=0
then v -+ (g,T vy is continuous on Xr (()).
10.4 Choice of u0 and ux
v -+ {v(0), v' (0)} is a continuous mapping of
^(2r+l)m,(2r+l)//}\ _^ £[(2r+l)m-m/2/Q\ x #(2r+ l)m-3m/2 /Q\
Using the spaces 3*(Q), S~"(Q) introduced in Chapter 2, Section 6, it
follows that:
(10.14)
if {u0f «J e5-(2r-1/2)m(i3) x S'-(2r+1/2)m(i3),
then the form v -+ <ux, v(0)} — <uQ} v' (0)>
[ is continuous on Xr (Q). □
10.5 Conclusion
The preceding results may be summarized by
Theorem 10.1. Let r be an integer, r ^ 1. Take
(10.15) feS-(2r+l)m,-(2r+l)(Q)f
m-1
(10.16) g eY] H~a2r~1)m+mj+1/2)S~a2r~1)m+mj+1/2)/m(Z),
j=o
(10.17) u0eE-(2r-ll2>>m{Q)}
ux eS-<2r+1/2>m(Q);
(10.18) <*!
£/&m tffo solution u of (9.7) satisfies
(10.19)
ueD^Z^iQ) (see (10.7)). D

11.1 Density Theorem 133
It remains to specify (which we shall only do in a partial way) in
what sense the solution u of (9.7) satisfies the relations
(10.20) A u + u" = /, Bu = g, u(0) = u0, u'(0) = ut.
This is the objective of the next section.
11. Trace Theorems in the Space Da%^(Q)
11.1 Density Theorem
Theorem 11.1. Assume that hypotheses (1.6), (1.8), (1.9) are satisfied.
Let u be a solution of (9.7) with the choices (10.15), . . ., (10.18). Then u
belongs to the closure T>a + dI ((?) °f ®(Q) ^n Da+dI (Q) •
Before proving this theorem, we make
Remark ILL We do not know whether @(Q) is dense in DA + rD\ (Q);
see Problem 14.5 and Baiocchi [8], Hormander [4, 5]. The application
of the method of Theorem 10.1, Chapter 4, leads to a difficulty which
we believe to be of sufficient interest to be discussed in some detail.
Let u-+N(u) be a continuous antilinear form on DA + rD\ (Q),
such that N((p)=0, V<pe@{Q). According to the Hahn-Banach
theorem and definition (10.7), N(u) may be represented by
(11.1) N(u) = <ro, u} + <Vl> (A + Dt) u},
where
y)x e3i2r+l)m.2r+HQy
Let ip0, ip1 be the extensions of ip0 and tpx to R^1 by 0 outside Q.
Let 0 be a bounded open set in R", with regular boundary d@, such
that D c 0 and such that the operator A can be extended to an
operator A on
0x] -t0,T + t0[,
for a suitable t0 > 0, which is elliptic in x, and has regular coefficients
on Ox [-t0,T + t0].
For arbitrary 0 in @((9 x] — t0,T + t0[), its restriction 9? to Q
belongs to @{Q), and therefore
(11.2) <fO)0> + <ylf (J+ £»,2) <P> = iV(9,) = 0,

134
-(2r-l)
11. Trace Theorems in the Space DA+D2 (Q)
where the brackets denote, for example, the duality between Of' (0 x
x]-t0, T + *0[) and 3(0 x]-t0, T + *0[). Therefore
(11-3)
{A* + Df)y}± = -y0 in 0 x]-t0,T + t0[.
Note that, in particular, ip1 e L2 (d) x ] — t0, T + t0[).
Now let w be the solution of
(11.4)
(J* + D2) w = -y>0 in Ox]-t0,T + t0[,
w(x, T + t0) = 0, \ w'(*, T + t0) = 0,
3W
= 0, 0^/^m-l, on d0x]-*o, ^ + *0[,
(or the same conditions "corresponding to B/': BjW = 0).
Since y>0 e Hi2r~ 1)m0'2r(Q), we have
y0e^(2r"1)?:2r(»x]^o.r + ^[)
and Theorem 7.1 yields
(11.5) V g tf<2r+l>».2r+l(0 X ]-*0, T + t0[)
(a result which cannot be improved in an essential way, see Section 1).
Let us now verify that
(11.6) w = %p1.
To this end, let 6 be arbitrary in @(& x] — t0,T + t0[) and let v
be the solution of
(A + D2) v = 0
v(x, -t0) = 0, v'{x, -t0) = 0,
dJv
0 on d<9x]-t0,T + t0[, 0^/^m-l.
(11.7)
dv>
Since <px has compact support in^ x ] —10, T + t0[, we have:
<(I* + D2t) f1,v}= <fc, (J + £>?) v} = <fc, 0>,
and according to (11.3), we therefore have:
(11.8) <Vi,0>= -<^o^>.
On the other hand, Green's formula, valid "between" w and v, yields
<{I* + D2)w,vy = -~(ip0tvy = <wt (A + D?)vy = <w,ey,
which, by comparison with (11.8), shows (11.6).

11.1 Density Theorem 135
But then, according to (11.5), ipl belongs to
Hi2r+1)m'2r+1(t)x]-t0,T + t0[)
and since <px vanishes outside Q we finally obtain
(11.9) V>ie#(2r+1)£or+1(e),
and this regularity result cannot be improved in an essential way.
But it is not sufficient to conclude that
<Vl, (A + Z)2) u> = <(,4* + Z)2) Vl, «>,
so that it does not follow that N (u) = 0, Vw.
Here we see the essential difference between the present case and
the parabolic, therefore hypoelliptic, case of Chapter 4, Theorem 10.1.
Of course, the preceding discussion does not constitute a
counterexample ! D
Remark 11.2 (due to C. Baiocchi). If we denote by Da+dI(Q) tne
space of u's belonging to &'(Q) such that
Au + u" eE-(2r+1>m'-(2r+1>(Q),
then the preceding arguments, this time, yield the fact that @(Q) is
dense in Da*+d\(Q). D
Proof of Theorem ILL Since Qf\Q) (resp. @(Z)m, resp. @(Q), resp.
@(Q)) is dense in
g-(2r+ l)m.-(2r+ D (Q\
(m— 1
resp FT H~a2r~1)m+mj+ll2)S~a2r~1)m+mj+ll2)/m(Z)
j=o
resp. B-<2r-1'2>m(Q), resp.E-<2r+1/2>m(Qy
we can find
(11.10) fne9(Q), gne®(Z)m, u0ne®(Q), ulHe9(Q),
with
(11.11) /„ "►/ in C'-Ur+Dm.-Ur+l)^
m-1
(11.12) g„ -> g in 7T ^-((2r-Dm + mJ + l/2)^-((2r-Din + inJ+l/2)/in/2'^
J = 0
(11.13) «0i,->*o in 45'-(2r-1/2)lw(i3),
(11.14) uln^ux in £,-(2r+1/2)w,(i3).

-(2r-l).
136 11. Trace Theorems in the Space DA+'D2"(Q)
Set (compare with (9.8)):
(n.15) LM=<fn,v> + <gn,Tvy + <uln,v(o)y-<u0n,v'(o)y.
Let (applying Theorem 9.1) un be the solution in H"i2r"1)m'"2r(Q) of
(11.16) (*n,A*v + v") = Ln(v)> VveX'(Q).
As n -> oo, Ln (v) -> L (v) uniformly for v in a bounded set of (XrQ)
and therefore
(11.17) «„-►« in #-(2r-1)m--2r(0).
Since, on the other hand-:
Aun + < = /„,
we have, according to (11.11):
(11.18) Aun + U'H' ^AU + U" in tf-Ur+Dm.-Ur+l)^)
and from (11.17)^(11.18) foUows that
(11.19) «»-« in D~aZV\Q)>
But let us a priori consider un as a solution of
(A +D2)un = fn,
Bun =gn>
(11.20)
un(x,0) =,u0n> -rr^'°) = Wl»*
Then, according to the regularity Theorem 7.1 (the problem can be
easily reduced to the "homogeneous'' case: gn = 0, u0n = 0, uln = 0)
it follows that
(11-21) *.e9(Q).
But then the usual Green's formula yields
(u„, (A* + D2t) v} = L„(v), WeX'(Q)
and since (11.16) admits a unique solution, we have
(11.22) un = un.
The desired result follows from (11.21), (11.22) and (11.19).

11.2 Traces on 2 137
11.2 Traces on £
Theorem 11.2. Let r be an integer, r g; 1 and
^ (2r + 2\ lk + ±\ .
2r + 2 - 4= integer + \,
\2r + 1/ \ m /
/or 0 ^'& ^ (2r + 1) m - 1.
The mapping u ^> BjU of @ (Q) into Q) (27) extends by continuity to a
continuous linear mapping, still denoted by u -> BjU, of Da+d^ ((?) (see
Theorem 11.1) -+H \^+i)\ m ) (jjy
Proof. Let us, for the time being, admit
Lemma 11.1. Let r be an integer ^ 1 with
I2r + 2\ Ik + i\
2r + 2 - 4= integer + 4,
\2r + 1/ \ w /
for 0 ^ k ^ (2r + l)m - I;
then there exists a right inverse
(n.23) g = {gj}7:Z^R® = v,
which is a continuous linear mapping of
m— 1 / 2r + 2 \ / 2m— m*— 1/2 \
(11.24) nflw,+ 1,"'<2"'"'' 1/2)',2or+2-(^r)( ^(S)-*
^#(2r+1)m;2r+2(0
(11.25) (4* + £>,2) v e Hi2r-1)m0f(Q),
(11.26) £^ = 0,
(11.27) Tv=g. D
For given m in D^+d? (0, set
(11.28) Z{g) = -<4 m + Dfu, v} + <«, i*» + 5fi>>,
where the first (resp. second) bracket denotes the duality between
£'-(2r+l)in,-(2r+l)^\ an(j £&r+ l)m,2r+l /Q\ _, #(2r+ l)ro,2i-+ 1 /fl\ _,
=»Jff(2r+1)m:^+2(0 (resp.tf-^-1*-'-2'^) and H*2'-1^;2,'^)).

138
11. Trace Theorems in the Space DA^D2 (Q)
We verify that Z (g) depends only on g and not on the choice of the
(non-unique) right inverse satisfying the conditions of Lemma 11.1.
Indeed, if vt is a second function satisfying these conditions, then
v — vt = w satisfies (since {€, T} is a Dirichlet system; Chapter 2,
Section 1)
dkw
= 0 on E, 0 ^ k ^2m - I.
dvk
Next, A*w +D]weHi2r-1)m$(Q) implies (as in Theorem 10.2
of Chapter 4) that
dkw
~dvir
= 0 on E for 0 g k ^ (2r + 1) m - 1, and
finally w e #(2r+1)£02r+2 (Q), from which the result follows.
The antilinear form g -► Z(g) is continuous in g, therefore
(11.29)
where
(11.30)
2r + 2 \ / 2m-mj-l/2
■><*>)'
= '»-^-c2r+1)ra+2ra-raj-1/2.-(2r+2>+(illl)(ifiz^Zi)
n
J = 0
(£)
In this way, we have defined a linear mapping
«-& of Dl^r1^)^^
which is continuous (as in Theorem 10.2, Chapter 4).
Finally, we verify (starting from Green's formula) that.
#, = Bu if ue9(Q),
which, thanks to Theorem 11.1, proves the theorem, if we can prove
Lemma 11.1. D
Proof of Lemma 11.1. We use the same principle as for the proof of
Lemma 10.1, Chapter 4. Conditions (11.26) —(11.27) are equivalent to
the data
dkv
\ = hk, 0 ^ k ^ 2m - 1,
(11.31)
dvk
hk eH ,o \2r + iM « / (E).

11.2 Traces on E
139
Next, we write the boundary conditions corresponding to (11.24) and
(11.25):
8"A* v dkv"
(11.32) —+ —— =0 on E, 0^ k ^ (2r - I) m - 1,
(11.33)
dvk dvk
dJv(x,0) dJv(x,T)
dtJ
dtJ
0, xeQ, 0 g / g 2r + 1.
We apply (11.32) successively: first for 0 tk k g 2m — 1, we may
use (11.31) and therefore
*£l = -k e *<—<-/2,r m) m
(£)
dvk
therefore, in particular:
l 2r+l \ /k + 2m+ 1/2 \
^ 6^+1)._tt+1/2).^+2)_(__)(___L.) (r)
and finally it follows that the boundary conditions are equivalent to
(11.34)
dkV „<2r+!),-»+l/2).(2, + 2)- (g + I) ( "+1/2 )
dvk
for 0 ^ A ^ (2r + 1) m - 1.
W
The lemma then follows from Theorem 2.3, Chapter 41. D
Remark 11.3. If
'2r + 2\ jj + iN
Vj = 2r + 2 —
2r + 1
equals an integer + \, then Lemma 11.1 contains integral compatibility
relations. Indeed if u e Hi2r.+1)m>*r+2 (Q), then
dJu dhu
dvJ
df
and, assuming that Q is the half-space %„ > 0, we must have (see
Theorem 2.3, Chapter 4)
f f I 3*
0 R"-*
da
dx' < oo
a
1 Applied with fk = 0 and gj = fy.

140 11. Trace Theorems in the Space DA+D2 (Q)
if
/ k 1/1 1
■ + -= 1 -- \-tl r.— + ■
{2r+l)m 2r + 2 2\{2r+l)m 2r + 2]
which is equivalent to Vj = k + \.
Of course, this corresponds to complicated integral relations on the
g/s in Lemma 11.1. D
11.3 Continuity of the Trace on Neighbouring Surfaces
The notation is the same as in Chapter 4.
For ueDA+Di^iQ) (= closure of ®(Q) in the space D^i^iQ)
defined by {u \ u e#-(2r-1)m'-2r((?), A u + D? ueL2{Q)}), we define
/ 2r+2 \ / 2m-my—1/2 \
(n.35) fi««6J/-«2'+i"+2-"-"!'-(2'+2,ty(—h-H
(2)
("trace of Bju" on "neighbouring surfaces" of 27).
By the same method as for Theorem 10.3, Chapter 4, we show
Theorem 11.3. Under the hypotheses of Theorem 11.2, ifuet)A+D2t (Q),
we have
[ B^ u -> BjU in
/2r + 2\ /2m— m*— l/2\
J?-<2r+l)»+2m-m,-l/2.-(2, + 2)+(-5_n-)( L—'-)
(11.36)
W.
as, q -> 0. D
11.4 Traces on Q0
Theorem 11.4. Let r be an integer, r ^ 1, with
(2r + 1) m - I ] I J 4= integer + —, * = 0, 1((1)).
The mapping
u^> {u(x,0),u'(x,0)} of @(Q) -+@(D) x@(D)
extends by continuity to a continuous linear mapping, still denoted by
u -> {u(x, 0), u'(x, 0)},
of
D^^ffl -+ H-t2r+1}m*V2G&) (Q) x H-i2r+iym+ll2(^) (O).
((i)) For the exceptional cases, a remark similar to Remark 11.3 holds.

11.5 Continuity of the Trace on Sections Neighbouring QQ 141
Proof. As for Lemma 11.1, we verify that there exists a right inverse
{<Po><Pi} = <P^> Ri{<f>) = i>i,
which is a continuous linear mapping of
H^r+1)n- (w)t {q) x ^2r+1)m- @nr)r {Q) _ i^1^2^)
such that
^,1+Z)r2,1e//(2r-1)^((?),
i^*, r) = o, i>i(*, r) =o.
For w given in D^+^f (@), we define:
(11.37) Zx($) = -(Au + «",«?!> + <«,4*i>i + *//>,
the duality being taken as in (11.28).
We verify that Zx (0) is independent of the choice of the right inverse
and consequently that
Zl($) = <*u> {<P0> <Pl }>
the mapping u -> ru being linear and continuous.
Next, we verify that if ru = {x°u, rj},
then r2 = —u'(x,0), x\ = u(x,0) if ue@(Q). Whence the theorem.
11.5 Continuity of the Trace on Sections Neighbouring Q0
In the notation of Chapter 4, we have:
Theorem 11.5. Assume (1.6), (1.8) and (1.9) to be satisfied. Let integer
r ^ 1 with
'2r + 1\ li + 1\ . 1
(2r + 1) m - [2r + 2J [—j—) * inUger + ~2' i = 0>1'
For u given in i}A + D2t (Q) > we have:
/2r+l \ 3
(11.38) u(x,to)^>u(x,0) in H (2r+1)m+ [ir+T) T (fi),
/2r+l \ 1
(11.39) u'(x,to)-+u'(x,0) in H~(2r+1)m+l^+TJT (£),
as i0 -»• 0. D

142
12. Schroedinger Type Equations
11.6 Remark
The spaces appearing in (10.16), (10.17) and (10.18) are contained in
the spaces appearing in the trace Theorems 11.2 and 11.4. Theorems 11.2
and 11.4 give meaning to (10.20). But we must be careful of the fact
that we do not have "optimal" results. Formulation (10.20) is only a
weakened (but intuitive) formulation of Theorem 9.1. As we have done
for the parabolic case in Section 12.3, Chapter 4, we could seek a
characterization of the traces of u's belonging to D^+i? ((?), but we would
find the same type of difficulties. Indeed, we can globally, by extension
by continuity, define the operator u^ru= {Bu, u(x, 0), Dtu(x, 0)}
for all u e DA + d? ((?) and characterize the space y°'r described by r u
as u describes DA+D2t (Q), so that the problem
A u + D2u = /; r u = g*
admits a unique solution u eDa+d^ (Q) for all / eE-i2r+1)m'~i2r+1>(Q)
and g* e y,. But then we meet the same difficulties as in Section 12.3,
Chapter 4, if we want to separate the operators B u,u(x,0) and Dtu(x,0)
in ru. We shall not insist on this point (see Problem 14.1). □
12. Schroedinger Type Equations
12.1 Notation
The operator A and the boundary operators Bj have the same
properties as in Section 1. For Theorem 12.2, we shall also require the
hypothesis
(12.1) Au\(t) e^(H°(Q)'>H°(Q)), V/ ^ 1
(that is, the functions apq(x,t) are independent of t, except a00(x,t}).
We shall consider the (non-homogeneous Schroedinger type)
problems :
\iAu + u'=f,
(12.2) J Bu =g
[ u(x, 0) = u0. D
We shall give two regularity theorems. And we shall introduce the
so-called method of parabolic regularization (see also Chapter 3,
Section 8.5).

12.2 First Regularity Theorem. Parabolic Regularization 143
12.2 First Regularity Theorem. Parabolic Regularization
Theorem 12.1. Assume the hypotheses of Section 1 to be satisfied. Let f
be given with
(12.3) f,f'eL2(Q) (i.e./e^'(?))
and u0 be given with
(12.4) u0eH2m(Q), Bj(x,0, J u0 = 0, Og/gm-l.
Then the solution u of problem (12.2), with g = 0, satisfies
(12.5) ueH2m^{Q)
(i.e.: ueL2(0,T;H2m(Q)), u' eL2(0, T\ H°(Q))).
Proof.
1) Variational formulation.
In the notations of Section 1 and Chapter 3, problem (12.2) is
equivalent to:
( i a(t\ u(t), v) + (u' (t), v) = (f(t),v) Vve V,
(12.6) {
| ueL2(0,T; V), «(0) = u0.
2) Parabolic regularization (see also Chapter 3, Section 8.5).
Let e > 0 be arbitrary. Denote by ue the solution of the problem
(which we shall call the parabolic regularization of (12.5)):
(12.7) (c + i) a(t\ uE, v) + «, v) = (/, v), V^e V.
Letting v = ue in (12.7) and taking twice the real part of the result,
we obtain:
d
2e a(t) uE, uE) + — \uE{t)\2 = (f(t),uE(t))t
at
and therefore
and consequently
(12.8) uB remains in a bounded set of L2(0, T; H°(Q)), as e -* 0.
3) According to Chapter 4, we know that we can differentiate (12.7)
with respect to t:
u'EeL2{0,T\V)

144
satisfies
(12.9)
12. Schroedinger Type Equations
{e + i)a{t;uE,v) + (e + i)a' (t;uE,v) + (u'e' ,v) = (f' ,v) MveV,
uE(0) = u0, <(0) = /(0) - (e + i)A (0)«0 = uueH°[Q).
If we replace v with uE(t) in (12.9) and take the real part, we obtain
(12.10)
ea(t;uE(t),u'E(t)) + —^\u'E(t)\2
(12.11)
= Re(/', <(*)) - Re(e + i) a'(t; uE> <).
But, thanks to (12.1):
\a'{t)uE,u'E)\ £c\ua(t)\\u'B(t)\
(where | | denotes the norm in H°(Q)) and thus it follows from (12.8)
and (12.10) that
[ uE remains (in particular) in a bounded set
j of L2(0,T',H°(Q)), as e-► 0.
On the other hand \]e u'e remains in a bounded set of L2 (0, T\ V).
4) Letting v = ue in (12.7) and, this time, taking the imaginary
parts, we obtain:
a{t\ uE(t),uE(t)) = Im(/, uE(t)) - Im«, «a)
and thanks to (12.11) it follows that
' (12.12) uE remains in a bounded set of L2(0, T\ V), as e -> 0.
5) We can therefore extract a sequence — still denoted by ue —
from u£, such that
u weakly in L2(0,T;F),
u weakly in L2(0,T;H), H = H°(Q),
and necessarily u = solution of (12.6). By uniqueness, it is unnecessary
to extract a "subsequence'' from u£.
Thus we have shown the (weak; see Remark 12.1) convergence of
ue to u and that the solution u of (12.6) satisfies
But
ueL2{0,T\V), u' eL2(0, T;H).
i a(t\ u(t), v) = (f(t) - u'(t), v), Vv e V,
f-u'eL2{0,T;H)

12.3 Second Regularity Theorem 145
implies {elliptic regularization) that
ueL2(0,T;H2m(Q))
and this completes the proof of the theorem. D
Remark 12.1. We have:
i a (t; u — uE, u — uE) + (uf — u'E, u — ue) = e a(t; uE, u — uE)
and hence
d
— \u(t) - uE(t)\2 = 2eRea(t;uE,u - uE)
at
and therefore in particular ue -► u strongly in L2(0, T\ H°(Q)), as
£-►0. D
12.3 Second Regularity Theorem
Theorem 12.2. Assume the hypotheses of Section 1 and (12.1) to be
satisfied. Let f be given with
(12.13) /e#(2r~1)m:ro((?)> inteZer * ^ 1-
Let
(12.14) i = 0, uo = 0.
Then the solution u of problem (12.2) satisfies
(12.15) ueH2rm>r(Q).
Proof. The theorem is valid for r = 1 (consequence of Theorem 12.1).
We assume the theorem to hold up to r — 1, that is
^e#2(r-1)m'r-1((?),
and we show (12.15).
Again, let ue be the parabolic regularization of u (see Section 12.2),
that is the solution of
(e + i)a(t;ue,v) + (u'E,v) = (/, v).
Differentiating r times with respect to t:
(12.16) (e + i) a(t; uEr\ v) + (e + i) £ M *<-■'>(*; u\j\ v) +
+ (^r+1), V) = (/oo, W); ^>(0) =0, 0 ^ / ^ r.

146 12. Schroedinger Type Equations
Replacing v with u^r)(t) in (12.16) and taking the real part, we obtain
1 d
(12.17)
ea(t;u?(t),ulr\t))+-—\u?(t)\2
2 at
It follows that
(12.18)
| V e uKp remains in a bounded set of L2(0, T\ V),
as e -> 0 and «*r) remains in a bounded set of
L2(0,r;tf°(fi)).
The derivative of order (r — 1), instead of (12.16), and after having
replaced v with u%~l), leads to
(12.19)
(e + i)a(t;u?-1,,u(;-1>) +
r~2 (r - 1
/
r^2 It
+ (e + i) Z
^-1-J>(^;^),<-1)) +
and therefore, taking the imaginary part of (12.19) and using (12.18),
it follows that
(12.20) ^r_1) remains in a bounded set of L2(0, T; V), as e -► 0.
Extracting a suitable sequence and letting e -> 0, it follows that
j ^r)GL2(0,r;^°(i3)) = L2((2),
J u<r-1>eL2(0,T;Hm{Q)),
(12.21)
(12.22)
r-2
iap;*0"-1*, v) + i£
r - 1
7(r-l-J)//.^U)
(*; wU), v) +
+ (u<r\v) = {p-^.v) \fveV.
But according to the induction hypothesis and the intermediate
derivatives theorem (Chapter 1, Section 3), we already know that
#6l2(0J;^2(p-1)ffl(1
-J/(r-D)
(Q))
and therefore
Air-i-j) u^eL2(0, T; H2m«-l->-V{Q)) e L2(0,T; H°(Q)),

12.3 Second Regularity Theorem 147
since
Therefore it follows from (12.22) (the uuys satisfying BkuU) = 0)
that
iA{t) u<r-»(t) = Z0-1* - u<* - r£ (""^x
x Air-J-»uW eL2(0, T;H°{Q))
and since Bk w(r""1) = 0, O^&^m— 1, it follows that
(12.23) ^(r-1) eL2(0, T; H2m(Q)).
But, differentiating the equation
ia(t; u, v) + (u', v) = (/, v)
(r — 2) times with respect to t (which is now permissible), we obtain
'-3 iY _ 2\
(12.24) ia(*; u^~2\ v) + i £ I . I a«-2-»(t\ uu\ v) +
+ («o-d,i;) = (f«-2\v) VveV.
Now, we have (12.23). Furthermore
and (12.13), together with the intermediate derivatives theorem, yields
p-2)el2(0, T; ^2(r-l)m(l-(r-2)/r)(i3)) c p(0) J. ^2m(fi))
(we see that here we could weaken the hypotheses on /) and (12.24) yields
wo-2)gL2(0, T;H*m(Q)).
And so on, taking the derivative with respect to t, of order r — 3,
r - 4, . . . □
Remark 12.2. As in Remark 7.3, we see that Theorem 12.2 is
essentially the best possible result. In the notation of Remark 7.3, we have:
u = G* /,
(O

148
where
12. Schroedinger Type Equations
f exp(—i Xt) if t > 0
G = G(X,t) =
0
From the formula
1
G =
iA
6 + -±^6'+ l
(_iA) (-U)2
1
if * < 0.
1
JO-1) +
(-iA)
ir-1
G(r)
<P
(where G(r) = G), we deduce that
df
1 . 1
&=^rf + V +
1
iA iA(-iA)
(iA)(-iA)'
-/<-!> +
+
iA(-iA)'
-G*/
(r)
Therefore if we assume slightly more than (12.13) (see Remark 7.3),
that is:
Ar-1/eL2(0,r;t)) and /<'> eL2(0, T\ f)),
then we verify that
ArfieL2(0,r;t))
and with no possibility of improvement (in the Hilbert setting). D
12.4 r-Isomorphism Theorem
We shall give only the steps corresponding to Sections 6 to 10. (The
point of view corresponding to Sections 1 to 5 is not specified here.)
We consider the adjoint problem:
(12.25)
and we assume that
(12.26)
— i A* v — v' = <p,
Cv = 0,
v{x,T) = 0,
integer r ^ 1 (by interpolation, we could also consider the case "non-
integer r").

12.6 Trace Theorem
149
Define
(12.27)
Xr(Q) = space described by the solution v of (12.25)
as <p describes #2(r~1)J;S((?), provided with the
"translated" topology.
(We use the same notation as in (9.3), although we evidently deal with
different spaces here!).
Then:
(12.28) -i4* + Dt is an isomorphism of Xr(Q) -> ^2(P"1)JJ((?)
and therefore
for every continuous antilinear form v -> L (v) on Xr (Q),
there exists a unique u e H~2(r~1)m,~r(Q)
which is a solution of
(12.29)
(uy (_i,4* - Dt)v} =L(v), VveXr(Q).
12.5 Choice of L
According to Theorem 12.2, we have
(12.30) X*(Q)<zH*™''(Q).
Therefore, if we take
(12.31) L(v) = <j,vy + i<g,fV> + <%,^0)>,
we see that we can choose (notation of Section 10):
(12.32) /eS-2""--'(0),
m-l
(12.33) g e n H~2rm+i2m~mj~ll2) S~r+i2m~mj~ll2)l2m(Z),
(12.34)
UqEZ
;7-(2r-l)m
(Q). U
12.6 Trace Theorem
We can (however, keeping in mind the remarks made in Section 11.6
and in Section 12.3 of Chapter 4) interpret problem (12.29) in the
"usual" form:
I i A u -{- u' = /
(12.35) J Su = g
u(0) = u0,

150
13. Comments
with the following "trace theorems" (proofs analogous to those of
Section 11); first of all, let:
(12.36) Dj?»»(Q) = {ueH-2«-v*»'-r(Q),iAu + u' e S~ 2rm>-2r (<?)};
note that
if u satisfies (12.29), then u belongs to the closure
(12.37)
Then
D{
-(2r-l)
"\A + Dt
(Q)otS(Q) in DrA%-»(Q).
(12.38)
and
r + 1
k + i
if r is an integer, r g: 1, with r + 1 —
r I \ 2m
4= integer + \, for 0 ^ k ^ 2r m — 1, then the mapping
u^Bu of @(Q)^>@(E)m
extends by continuity to a continuous linear mapping, still
denoted by u-+Bu, of D^i + rJ^1)'((?) into
m-l
T~T rr-(2rm + (2m-mj-l/2),
j = 0
/ r+1 W 2m-mj--l/2 \
,-(r+l))+ (~ J( 2m j /£\
z'/ f is an integer, r §; 1, z#^/& 2rw -
\ 1
4= integer +
r + 1 ' " 2
(12.39) I then the mapping u -> ^(0) o/ i^($) -> ^(i3) extends by
continuity to a continuous linear mapping, still denoted by u -+u(0),
of D^?D-iy(Q) into H2rm+r"r+1>(Q). U
Remark 12.3. It is possible to develop a remark entirely analogous
to Section 11.6, in connection with the trace theorems for Schroedinger's
equation.
13. Comments
The operators studied in Sections 1 to 11 of this chapter are
hyperbolic if A is of order 2 and well-posed in the sense of Petrowski if A is
of order 2m, m > 1.
Our objective is in no way the study of the particular properties due
to the hyperbolic nature of the problems. For the study of mixed
(homogeneous) problems pertaining to second order hyperbolic equations, we
refer the reader to Hadamard [1], Krzyzanski-Schauder [1], Schauder [1]
and Ladyzenskaya [1]. For the Cauchy problem for hyperbolic systems,
see Garding [3], Leray [1], Petrowski [2], the works of Garding and
Leray, as well as (application of singular integral methods) Calderon [6],
Mizohata [1, 3, 4].

13. Comments
151
For the case of constant coefficients and the use of the Fourier
transform, see Schwartz [3], Gelfand-Shilov [1], Volume III, and for the
use of the Laplace transform, see Garnir [1].
For mixed hyperbolic problems of order >2, for which many open
questions still remain, see Agmon [5], Campbell [1],
Campbell-Robinson [1], Duff [1], Thomee [1] and Peyser [1].
For first-order hyperbolic systems, see Agranovich [2, 3], De Prima
[1], Friedrichs [3], Friedrichs-Lax [1], Lax-Phillips [1], Phillips [1],
Phillips-Sarason [1], Sarason [1, 2], Wilcox [1, 2].
Curiously enough, the non-homogeneous boundary value problems
do not seem to have undergone any systematic study for the cases
considered in this chapter, even for second-order hyperbolic operators
(except, of course, where only "regular'' data on the boundary are
considered, and where a "loss of regularity'' in the result is accepted;
see V. A. Il'in [1], Vaghi [1]).
It was therefore necessary to take up this question from the
beginning; we start with the proof of regularity results, which (although far
from optimal, see the Remarks of Section 6) seem to be new.
Once the regularity theorems have been established (Sections 2
and 7), we apply the general procedure followed in this text: we transpose
and interpolate (or vice-versa).
Unfortunately, here again, "exceptional parameters" leading to
serious technical difficulties appear.
We consider the variational setting; here, we can make a remark
analogous to Chapter 2, Section 9 and to Chapter 4, Section 17; if we
consider the problem
r d2u
a) A2u + i^ = f>
{uix, 0) = u0,
U' (X, 0) = %,
dAu
(3) «|* = 0,
= 0,
E
dv
then we can treat it by the variational method by taking
H = Hl0(Q)t V = \v\veHl{Q), Av eL2(Q)\,
[ dxt J
" / d d \
a(t;u,v) = a(u,v) = V A u, A v
i=i \ dxt dxt J
' L2(#)
The corresponding non-homogeneous boundary value problems have
not been studied here. For other boundary value problems in connection
with (1), see Prouse [1].

152
14. Problems
r\td
An analogous remark holds if we replace (1) with iA2u+ = /,
(2) with u(x, 0) = u0 and leave (3) as it stands.
In the preceding Chapters (as we have already pointed out) we
could have taken regularity theorems in Lp, p =j= 1, 2, oo, as a starting
point. No doubt, this would have led to rather considerable additional
technical difficulties and to certain open problems (which we have
pointed out), but the starting point (regularity theorems) was accessible.
This is no longer true for the present case; the "LMnequalities", which
one could formally expect, are Inaccurate (see Littman [1]); substitutes
can been found (see Giusti-Da Prato [1], Da Prato [6]), but much work
remains to be done in this direction, and we have not undertaken it.
We have purposely left aside a certain number of cases which are
probably accessible, but which would have considerably complicated
the presentation: transmission problems, coupled problems, etc. . . .;
we refer to these cases in some of the problems which follow.
For almost-periodic solutions in t, see — as was already pointed out
in the Comments to Chapter 3 — the text of Amerio-Prouse [1] and
its bibliography (it is likely that the almost-periodic properties of the
solution extend to the various classes given in this chapter, in particular
when the boundary data g are almost-periodic in t in suitable spaces
H~s(r); but it seems that this point has not been studied to date).
d2
For hyperbolic operators of the form h A, where A is an
dt2
elliptic operator of the second order which can degenerate, see Oleinik
[3'4]- 82 k d
For hyperbolic operators of the form h h A, see
dt2 t dt
Weinstein [2] and also Diaz [1], Walter [1] and the bibliographies of
these works.
14. Problems
14.1 More complete study of the traces of u's belonging to
This problem is more difficult than for the parabolic case (see
Section 12.3 and the Problems for Chapter 4), since, here, we do not have
"optimal" estimates (see also Section 11.6 and Problem 14.5 below).
14.2 Consider the spaces At described in (5.2) and (5.3). It seems
likely to us, but it has not been demonstrated ,that
m-1
[A A] _ TT ^3m(l-e)-(inJ+l/2),(3in(l-e)-(inJ+l/2))/in/2'\ x
*=0

14. Problems 153
if
3m(l - 0) - (mj + j) 1
U4-l) < ~ '
m 2
and with the ^. %>. being well-defined if (14.1) is not satisfied.
14.3 Characterize (in a different way than by the definition!) the
space
[tf°.i(0,(ff2m-2 (Q)n
(which appears in (5.5), for instance).
14.4 Because we wanted to remain within a Hilbert theory, our
regularity results are never optimal. For example, in Theorem 2.1, we
may conclude that
ueL"(0,T;H2m(Q)),
u"eL"(0,T;H°{Q))9
(instead of L2(0, T; H2m(Q)) and L2(0, T\ H°(Q))). Therefore, by
transposition more precise results can be obtained by using a non-
Hilbert theory.
In this way, problems of interpolation between subspaces of Sobolev
spaces with different exponents arise.
The following is an example of such problems: consider an open set
Qx®, QeR", tfeR™,
and first let
Wi(Q\ Wva(0)) = {v | D*xveL>(Q\ W\(G))9 |*| £ //},
where
Wl(0) = {w\I%weL*(Cf),\p\ ^ v};
next, let
V = {v | v e W%(Q; W\ (<P)), Bj v = 0 on all (or a subset) of the boundary
of Q x 0y the B/s being suitable boundary operators}.
Can the spaces (V, Lp{Q\L*(C))))Q,r (in the notation of Peetre [8]) be
characterized?
14.5 Is @(Q) dense in D^ft*(Q) (Theorem 11.1)? In this
connection, see Baiocchi [8] and Hormander [4, 5].
14.6 It might be useful to study the "non-integer r" cases and,
in particular, the exceptional values of r\ a study which we have not
attempted.

154
14. Problems
14.7 Non-homogeneous, general hyperbolic problems; the theory of
mixed problems (in the sense of Hadamard) for these cases is still wide
open; see the Comments.
14.8 Consider the operators
(14.2) A^.^+A^x^^ + A^.t.J-)^.
under various hypotheses on At (see Lions [13]; Baiocchi [6, 7]; see also
Chapter 3).
It should be possible, without too much difficulty, to extend the
results of this chapter to the operators (14.2). See also the study of
operators of type (14.2) in Albertoni/Cercignani [1] (see also Scarpini [1]).
14.9 Similarly, it should be possible to extend the results of this
chapter to coupled problems: on an open set Qt xQ2, we have an
operator of a given type — for example parabolic — on Qt x ]0, T[
and of another type, for example hyperbolic, etc., on Q2 x ]0, T[, with
transmission conditions on rx x ]0, T[. where rx is a common part of
the boundaries of Qx and Q2.
For homogeneous problems of this type, see Gelfand [1], Hersch [1],
Lions [18], Lions-Raviart [1].
14.10 We have always assumed that the operators' A (or At in
(14.2)) are "regular". Study the non-homogeneous boundary value
d2
problems for the operators A + , where A has singularities. For
k 3 8t
example: A = — A , where Q is contained in {xn > 0}, with
xn dxn
part of the boundary on the (singular) hyperplane xn = 0. Or, where
A can degenerate; see Baouendi [1] and studies by O. A. Oleinik which
are still to be published.
14.11 Regularity theorems and non-homogeneous problems for
the evolution operators corresponding to the semi-groups of order n
of Da Prato [1, 2].
14.12 Non-homogeneous problems for, for example,
d2
on Q x ]0, T[, where Q has various parts of the boundary rt of different
dimensions (analogue to Problem 17.7, Chapter 4).
14.13 Analogue to Problem 17.13, Chapter 4, with the "elliptic
d2
regularization,, connected with A + -—- (see Strauss [1]).

14. Problems
155
14.14 Analogue to the preceding problem with the " parabolic
regularization'' introduced in Section 12.2, and the one of Chapter 3,
Section 8. (In this connection, note that, having applied the " parabolic
regularization''
rs rs
\A + —^(£ + i)A + —-,
ot ot
we can apply the "elliptic regularization to the parabolically regularized
operator", to obtain
^ + iM+__j_£l_, £l>0,
or, if A is of order 2m:
3 d2m
(14.3) le + i)A + + £1(-l)m——.
v > v t ) dt -r iv i dt2m
In this way, we are led back to solving non-homogeneous problems for
the elliptic operators (14.3), followed by results of the type discussed
in Section 12, as eJel -> 0).
14.15 Problems of the preceding type in non-cylindrical domains;
see Rogak [1] and Lions [31].
14.16 Which non-homogeneous problems for the operators
considered in this chapter can be solved in Lp, p =J= 2? There are essential
differences between the cases p — 2 and p 4= 2 (see Littman [1]), but
the results of Da Prato-Giusti [1] and Da Prato [6] should serve as a
"starting point'' for a study of these problems.
14.17 Study of coupled problems (/ and g being taken from very
general classes) of the type
-Axu = f(x,t) in Q;
du d2u
dv ot2
u(x, 0) = u0 for x e Q,
du
(x, 0) = ux for x e J7,
dt
(see Problem 17.12 of Chapter 4); see Friedman-Shimbrot [1], Lions [13].
14.18 The problem of oblique derivatives for the wave operator seems
to be unsolved. Is it well-posed?

156
14. Problems
14.19 The problems of this chapter can be studied in weighted
iP'-spaces (for the elliptic case, see, in particular, Geymonat-Gris-
vard [2]).
More generally, can these problems be studied in Morrey type spaces?
As we have already pointed out, the "natural" inequalities are inaccurate
in Lp, but do there exist (non-hilbertisable) Banach spaces where the
"natural'' inequalities hold?
14.20 The questions pertaining to approximation by finite difference
methods for the non-homogeneous problems studied in this chapter lead
to numerous, still unsolved problems.

Chapter 6
Applications to Optimal Control Problems
Note. This chapter is of a somewhat different nature than the
others: its sole objective is to serve as an introduction to the (enormous)
subject of control problems for systems governed by partial differential
equations, with the controls appearing (mostly) in the boundary
conditions. Sections 1 through 9 rely in an essential way on Chapter 4,
and the remaining sections on Chapters1.
We call attention to the notation, as it is different than for Chapters4
and 5; we have, whenever possible, followed the usual notation of
control theory.
1. Statement of the Problems for the Linear Parabolic Case
1.1 Notation
In this chapter, we adopt, as much as possible, the usual notation
of control theory, which is different than the notation for the theory of
boundary value problems (and thus from the one used up to this point).
For the geometric concepts the notation remains unchanged: Q,
r, E, etc.
Similarly, we still have: x e Q, t = time, etc.
The state of the system is represented by y (x, t), a function taking
its values in R (or in RN for "differential systems").
Every function under consideration takes its values in R; thus the scalar
products are real, and denoted by < , > or ( , ), without distinction. It is
assumed that the state of the system y is provided by the solution of
the non-homogeneous boundary value problem:
(1.1) Ay + y' = Exux in Q = Q x ]0, T[,
(1.2) By = E2u2,
(1.3) y(x,0) =E3u3,
1 Nevertheless, Section 1 may be read independently from the rest of the text.
The fractional Sobolev spaces intervene (as trace spaces) in the following sections.

158 1. Statement of the Problems for the Linear Parabolic Case
where
(i) A H is a parabolic operator satisfying the hypotheses of Sec-
dt
tion 1, Chapter 4, for the operators {A, Bj}, with B = {#,};
(ii) ult u2t u3 denote the controls. D
More precisely, we assume that
(1.4) ut e J^t = Hilbert space on R, * = 1, 2, 3
(with scalar product ( , )Jfi) and that
(1.5)
E2e&(tf>2\L2{E)m)i
E3e&(Jf3]L2(Q)).
Then, for u = {ult u2, u3} given in Y[ «^»> problem (1.1), (1.2), (1.3)
i=i
admits a unique solution (see Chapter 4; we shall come back to this
point):
(1.6) y(x, t) = y(xtt\u), u = {ult u2i u3}.
Remark 1.1. Oi course, we could consider an infinity of other situations:
the operators Et mapping J^t into spaces different from L2 (Q), etc.
(for example, H2rm-r(Q)t 5-2rm--r(Q) instead of L2(Q), etc.); note that:
(i) setting (1.5) seems to be the most important one for applications;
(ii) the method developed below is general. D
Remark 1.2. We could also consider
(1.7) Ay + y' = E1u1+f,
(1.8) By =E2u2+g,
(1.9) y(x,0) =E3u3 + y0{x),
with given f, g, y0. Everything that follows can be adapted to this
situation, without difficulty. D
An important case for the applications is when /, g, y0 are random
functions. We shall not develop this point of view here; see Bensous-
san [1]. D
Remark 1.3. We insist on the fact that T is fixed] we do not consider
optimal time problems here. D

1.2 Optimization Problems 159
1.2 Optimization Problems
1.2.1 The constraints on the control
The controls ut do not always describe the entire space 3^it but
3
(1.10) u = {ultu2t %} describes <% c 3tf = f] #i%
i = i
We shall assume that
(1.11) ^ is a closed, convex subset of 3tf'.
1.2.2 The cost function J(u)
Let us begin with a "vague" statement of the problem: as u describes
<%, y(x,t;u) describes a set of functions; we seek u (or the u's) (if it
exists) such that y(x,t;u) is "as close as possible" to a desired state
y*(%,t).
It is not necessary, but convenient (and useful for the applications)
to consider y(x,t;u):
1) in Q;
2) on 27, via S y (see Chapter 4, Section 1; it is natural to work
with 3 y, since B y is given and since {B, 5} is a Dirichlet system);
3) on QT, that is y(#, T; u).
We thus seek (and this is still not a precise statement) u in °tt such
that y(#, 2; u) is "as close a possible" to y\ in (), 5 y is "as close as
possible" to y% on Z, y(x, T; u) is "as close as possible" to y\ in £?r. □
Now, let us make the sense of this approximation precise.
Let Ki, i = 1,2,3, be three Hilbert spaces of functions or
distributions on Q, 27, £?, respectively. For the sake of precision, we shall
sometimes write
K1(Q), K2{Z), K3(Q).
We assume:
(1.12) yj given in Ki9 i= 1,2,3.
77^ essential hypothesis is:
f iw y(#, 2; w), solution of (1.1), (1.2), (1.3), ^ spaces K( must
(1.13)
( be chosen such that y e K1(Q)t S y e K2(U), y(x,T) e K3 (Q).
(This shall be made precise in Section 2.) □
It is then natural to seek a solution "as close as possible" to yld in
the sense of Kt. From which we obtain the following precise definitions.

160 2. Choice of the Norms in the Cost Function
Definition 1.1. The cost function is the functional:
(1.14) /(«) = & \\y(x,t;u) - yjr-li, + fi2 \\Sy - y\\\^ +
+ p3\\y(x,T;u)-y3a\\2Ki,
where
ft ^ 0, i»l + i»2 + i»3 > 0.
The control (or optimization) problem is to seek the u° (or w0's)
in °ll (if it exists) such that
(1.15) J{u°)^J(u), Vue<%.
Definition 1.2. Every control u° satisfying (1.15) is said to be an
"optimal control".
Remark 1.4. Up to a certain point, we are free to choose the spaces Kt
(see below); the essential point is that these spaces cannot be chosen
"arbitrarily small". For example, we may think that we can seek
an "optimal control'' for /
P3j\y(x,T',u) -y3d\2dx,
with y\ given in L2(Q). But under hypotheses (1.5), this problem is
meaningless (except if inf.m5 = m)\ we must take (except if inf.mi — m)
K3 (Q) larger than L2 (Q). U
Remark 1.5. We restrict our study to quadratic functional of type
(1.14). Of course, one could state the same kind of problem, replacing
the spaces L2 (Q) with LP(Q), 1 ^ p ^ oo, L2(Z) with LP(Z), etc. and
using other, non-quadratic functional. D
Remark 1.6. The cost function is generally written in terms of
observations, which are themselves linear functions of the state y. Up to
certain technical details, the following methods remain valid for this
case. □
2. Choice of the Norms in the Cost Function
2.1 Reminder. Condition on KX(Q)
According to Theorem 15.2, Chapter4, we have:
(2.1) yeH^2m(Q),
where
(2.2) /u = \ + min mjt
J

2.2 Space Described by S y. Condition on K2(£)
161
except if ju = m + \t i.e. if minm^ = m. (Indeed, for this case,
Theorem 15.2, Chapter 4, assumes that y(x,0) is given in H1/2(Q) and
not only in L2{Q). But (2.1) is still valid if E3 = 0 {or if E3 e
eJ£?(^3;#1/2(£))).)
In the sequel, we shall assume that ju < m + -J and that E3 = 0 if
[A = m + \. (The case <{ ju = m + ^-, E3 4= 0" is studied in Section 5.)
Under these conditions (and assuming that u describes the entire space
3 \ ^
& = U^t)' we have:
Proposition 2.1. Kt(Q) (see 1.2.2) must be chosen to satisfy
(2.3) Kl(Q)iDH^2m(Q). D
Furthermore, we shall assume that
(2.4) H^/2m(Q) is dense in KX{Q).
2.2 Space Described by Sy. Condition on K2(£)
We know (Chapter 4, Section 2) that, if jUy denotes the order of SJt
y -> 5 y is a continuous linear mapping of
(2.5)
JJ2mA(Q\ __> T~T Jj2m- ftj- ll2,(2m- ftj- 112)l2m ,™
J = 0
and that (Chapter 4, Section 10; Remark 10.1)
(2.6)
y -> 5 y is a continuous linear mapping of
m-l
D°p(Q) -> ["J ^-^-l/2.(-^-l/2)/2m(r)>
J=0
But y e H^/2m(Q) and P y = A y + y' e L2(Q), so that, by
interpolation of (2.5), (2.6), we obtain:
ii ye H^2m (Q) and Pye L2 (Q), then
m-l
5 y e FT iy^~(2m"^"•1)~1/2•(^"(2m"^"1)"•1/2)/2m(2,).
(2.7)
Therefore:
Proposition 2.2. K2 (27) rawstf fo chosen to satisfy
(2.8)
#2 (2?) => n jy"w"i/2,(Ai"w"i/2)/2m(2^.
Furthermore, as in (2.4), we shall assume that
(2.9) w (2.8), the second space is dense in the first.

162 2. Choice of the Norms in the Cost Function
2.3 Space Described by y(x,T;u). Condition on K3(Q)
We have (2.1), therefore
yeL2(0,T;H»(Q)).
According to Theorem 12.7, Chapter 1, we therefore have (since
ju =■ integer + \)
AyeL2(0,T;(H2orfimf)
and therefore
y' = ElUl-A yeL2(0, T; (#*?" " (£>))').
Then (Chapter 1, Sections 3 and 9)
y(., T; u) e [#"(£>), (H^-"(Q)Y]1/2 = X;
by the duality theorem, Chapter 1, Section 6.2:
X' = [# oo-" (#).(# "(£))'] 1/2
and according to Remark 12.6, Chapter 1, we have
\H^"(Q) if lx^m-\,
~ J Hm-"(Q) if n = m + i.
Therefore
(2.10) y(., T\u) \
Therefore
Proposition 2.3. i£3 (Q) must be chosen to satisfy
[K3(Q) = (HZo"m if (*^™-i,
J K3{Q) z>#1/2(fi) if fi = fn + i.
And, as in (2.4) and (2.9), we add
(2.12) (tfoo^CG))' (or H^2(Q)) is dense in K3(Q). D
Remark 2.1. Of course, conditions (2.3), (2.8) and (2.11) do not
uniquely define the i£/s! There exists an infinity of possible choices. □
. Remark 2.2. If
m—1 m—1
^2 = n^2.„ With ||*||£2 = £||*,||Lr
than we can consider
m-l
j = 0
instead of /S2 |5y - y^|||2, in (1.14). Q

3.2 Optimality Condition 163
3. Optimality Condition for Quadratic Cost Functions
In this section, we recall a well-known result.
3.1 Notation
Let n (u, v) be a continuous bilinear form on a real Hilbert space $f
and let v -> M (v) be a continuous linear form on £F.
Let ^ be a closed convex subset of 3tf.
Set
(3.1) J(u) = n{u,u) - 2M{u).
We seek the u° (or u°'s) in °tt (if it exists) such that
(3.2) J(u°)^J(u), \fue<%.
3.2 Optimality Condition
Proposition 3.1. Assume that 7t(ufv) satisfies:
(3.3) 7t(u, v) = 7i(v, u), V^,J/e/,
and
(3.4) 7t[u,u) ^ 0, Vwe^f.
Then, a necessary and sufficient condition for (3.2) to be satisfied is
that
(3.5) 7i'[u°, u - u°) ^ M(u - u°)t Vue<%.
Proof. 1) (3.2) => (3.5).
Let u be arbitrary In °U\ then (1 - 0) u° + 6 u e °U for 0 < 0 < 1
and therefore (3.2) implies
J(u°) ^ J((\ -6)u° + du)t
that is, expanding (and using the symmetry (3.3)),
0 ^ 26[ti(u°tu -U°)-M(u- u0)] + d2n(u -u°,u- u°).
Dividing by 0 and letting 0 -> 0, we obtain (3.5).
2) (3.5) => (3.2).
For u (and u°) arbitrary in °U, we have:
(3.6) J(u) -J(u°) =2[n(u0,u -U°)-M(u- u0)] +n(u- u°tu - u°),
from which the result follows, since n (u — u°f u — u°) g: 0, by (3.4). D

164 4. Optimality Condition and Green's Formula
4. Optimality Condition and Green's Formula
4.1 Optimality Condition. Application of Section 3.2
We apply the remarks of Section 3 to the problem stated in
Sections 1 and 2. Expanding (1.14), we may write
(4.1)
where
J(u) = n{ut u) — 2M(u) + constant,
(4.2)
(4.3)
n{u,u) = P1(y{x,f,u),y(x,f,u))Kl + fi2(Sy(x, t; u),
I Sy(x,t;u))K2 + p3(y(x, T;u),y(x, T;u))Ki,
M(u) = ^(y(x, t; u),yld)Kl + p2($y(x, t; u),yl)K2 +
+ ^(y{x.,T\u),yl)K3
and where the constant equals
Since
u = {ultu2,u3} -► {y[x,t;u)fSy,y(x, T;u)}
is a continuous linear mapping of
tf> = J"] #% _> Kx x K2 x #3
(see Propositions 2.1, 2.2. and 2.3) we may apply Proposition 3.1 to
J(u). We thus obtain:
Proposition 4.1. A necessary and sufficient condition for u° to be
an optimal control is that (we have written y(u) for y(x, t; u))\
[j8i(y(«°) -yi.y(«)-y(«0))zt +
+ j82(5y(«°) - yl$y(u) -Sy(«%2 +
+ 03(y(*. T; «°) - yj, y(x, T; u) - y(*, T; u°))K3 ^ 0.
\/ue<%. D
(4.4)
We shall now transform this condition by using Green's formula.
To this end, we need some preparation.

4.3 The "Adjoint" Problem 165
4.2 The Isomorphisms At
We distinguish two cases for the choice of Kt\
First case: p ^ m — \. Then, we use (2.3), (2.8) and (2.11), to obtain:
(4.5) x K[(Q)c:(H^^(Q)yt
m-l
(4.6)i K'2(Z) c Y\ (ff"-«-1/2-("-«-1/2>/2-(Z))',
j=o
(4.7)! ^(fl)cC(fl).
Second case: fi = m + -J-. Then, so as to avoid having to use the
boundary value problems for exceptional values of the parameter, we
choose e > 0, fixed arbitrarily small, and replace (2.3), (2.8) and (2.11)
with
KX(Q) 3#^e'(^-e)/2m(<2),
* m-l
K2{E) => Y\HfX~e~fXi~ll2titl~e~tl3~ll2)l2m{E)t
j=o
K3{Q) r> Hll2'e(Q).
Then
(4.5)2 ^(^d^-.c-^-^y,
m-l
(4.6)2 ^(Z) C fl (Jffl«-«-«-1/2.(|.-.-W-l/2)/2»^y>
J = 0
(4.7)2 XJ(fi) c (H^2-E(Q))f = H-^2+E(Q). U
Definition 4.1. PFe denote by At the canonical isomorphism of Kt
onto K'i.
4.3 The "Adjoint" Problem
The notation for the operators A*, C, T is the same as in Section 1
of Chapter 4.
->
For we/,we denote by p (x, t; u) = p (u) the solution of the
following ("adjoint") problem:
(4.8) (a*--^\p(u) =i81^l1(y(«) -yi),
(4.9) Cp(u) =P2A2($y(u)-y2d),
(4.10) fi(x,T;u) = P3A3(y(x,T;u)-y*).

166 4. Optimality Condition and Green's Formula
Proposition 4.2. Problem (4.8), (4.9), (4.10) admits a unique
solution, which satisfies:
(4.11)! p(u)eH2m-^1~tli2m(Q)t if ft^m-i,
(4.11)2 p(u)eH2m-^^-^^2m(Q)t if Jbt = m + i.
Proof1. We use Chapter 4, Section 15.1, (iii), with 26 m = fi. If
y(u) — y1deKl(Q), then A1(y(u) — yd)eK[(Q) and (we have done
what was necessary for that) A1(y(u) — yd) therefore belongs to
^-^-"/2m(@), with the notation of Chapter 4, Sections 14 and 15.
Next
m- 1
A2(5y(u) - y2d) eK'2(E) c: Y\jf+"+1l2A-»+"+1l2)2m(Z)
j = o
and
A3(y(x,T;u)-y3d)eK3(Q).
Thus Chapter 4, Section 15,1. (iii) applies and we obtain (4.11). □
4.4 New Form of the Optimality Condition
Theorem 4.1. Assume that minrnj < m or, if minrnj = m, that
E3 = 0. A necessary and sufficient condition for u° to be an optimal
control is that:
f ip (u°), e, (Ul - u°)y + <Tp (u°), e2 (u2 - u°2)y +
(4.12) + <p(x,0',u°),E3(u3 - u°3)} ^ 0
[ "iu — {ult u2, u3) e °tt,
where p(u°) is defined by (4.8), (4.9), (4.10) (with u = u°), and where
the brackets denote the scalar products in L2 (Q), (L2 (Z))m and L2 (Q),
respectively, since2
P(u0)eH2m-^1~^2m(Q) c L2(Q),
m-l
Tp(u°) e n Hm^+ll2-^m^ll2-^2m(Z) c {L2(Z))m
j = o
and p(x,0',u°) eHm~fl(Q) c L2(Q), except if minm, = m, in which
case the last scalar product in (4.12) drops out (by hypothesis!).
Proof3. Set 0 = y(u) — y(u°) and take the scalar product of (4.8)
(with u = u°) with 0 (in the duality between K\(Q) c (^^/2m(Q))'
1 If fi = m + \, replace ^ with p - e and H™~I*(Q) with H-1/2+8{Q) in the
proof [see (4.7)2]-
2 Here and in the sequel, replace ft with ju, — e, if ^ = w + J-
3 See the footnote to theVproof of Proposition 4.2.

4.4 New Form of the Optimality Condition 167
and H^l2m(Q)). We obtain
(4.13) (L* - -^W°)> A = Pi(y(»°) - y1*,*)*
(since Ax is the canonical isomorphism of Kx onto K[).
But, according to Green's formula, valid here by extension by
continuity (see Chapter 4,) the first term in (4.13) is
<f j>(u°), B 0} - (C j>(u°),S 0} -ip(T;uQ)}0(T)y +
+ <£(0; u°), 0(0)} + <fi(u°), (A + dldt) 0},
where the first bracket denotes the duality between
m-l
J-J Hmj+l/2-r.imj+l/2-M2m(Z) and (L2 (Z))m,
J = 0
the second between K'2 and i£2> tne third between i^3 and K3 for
example, the fourth between Hm-/X(Q) c Z,2 (12) and L2 (12) (this bracket
vanishing for ^ = m + -J-) and the last between Z,2 (@) and itself.
Using (4.9)-(4.10), it follows that:
Pi(y(u°) - yl *)Kl + j82(3y(«°) - y2,3 <z>)*2 +
+ p3(y(T; u°) - y3d>0(T))K3 = (p(u°),(a +-^j0) +
+ <f p(u°), B0 > + <£(0; «°), <Z>(0)>.
(4.14)
Replacing y(«) — y(«°) with its value, we see that the first term
in (4.4) equals
P{u°), IA + -^) <Z>\ + < Tp(u°),B0 > + <£(0; «°), <Z>(0)>.
But
(^ + 4r)0 = £iK "w?) (if w°= {u°itu°2'u°^'
B0 = E2(u2 - u°2),
0(0) =E3(u3-u°3),
and the theorem follows. □
In the following sections (6 and 7), we shall give some applications
of this theorem. Now, we shall briefly investigate the particular case
^ = m + \y E3 4= 0.

168 5. The Particular Case fx == m + \, E3 =j= 0
5. The Particular Case §i = m + i, £3 # 0
5.1 Properties of .y
We no longer apply Theorem 15.2, Chapter 4, but instead Section 15.1
(iii) of Chapter 4, with s = — \\ then we must have
E1u1 (which replaces /) e ^-m'-1/2((?),
which is evidently satisfied, and we must also have E2 u2 (which
replaces g = {gj})
m-l
_ T~T j^m-mj-1/2,(m-mj-l/2)2m/v-t\
j = 0
which is satisfied since rnj §: m. In this manner, we obtain:
Proposition 5.1. // /u, = m + \, then
(5.1) yeHm^^2(Q). U
Remark 5.1. This is indeed different than (2.1). □
5.2 Choice of Kt(Q)
\
Following (5.1), we take (compare with Section 2):
(5.2) Kt (Q) z> Hm>1/2(Q), the second space being dense in the first.
5.3 Choice of K2(£) and KS(Q)
The remarks of Sections 2.2 and 2.3 hold if we replace jbt with m + \\
(5.3)
m-l
K2{E) => J] Hm+wim+'"M2m{Z),
j=o
the second space being dense in the first.
According to Proposition 2.3, we must take
(5.4) K3(Q)zdH^2(Q),
the second space being dense in the first.
In this case, a simple choice is
(5.5) K3(Q) =L2(Q). D
5.4 Adjoint Problem and Optimality Condition
We obtain results analogous to those of Section 4, with the new
choices (5.2), (5.3) and (5.4).

6.2 Consequence of Theorem 6.1 169
6. Consequences of the Optimality Condition (I)
6.1 Generalities
Let us consider the operator Ef, adjoint oiEifi=l,2,3', with the
appropriate identifications, we have:
f Ete&tL2®);^),
(6.1) E*e&{(L*{Z)y',3#>2),
{E*eJ?(L2(Q)',jr3).
Then, if we introduce
(6.2) C(«°) = {Etp{u°),E*Tp{u°),Elp(x,0;u0)},
element of Jf = f] Jf ^, Theorem 4.1 yields the following equivalent
result: i = 1
Theorem 6.1. Hypotheses of Theorem 4.1. ^4 necessary and sufficient
condition for u° to be an optimal control is that u° e °ll satisfies
(6.3) (C(«°),« - «°)j? ^ 0, Vue<%,
where £(«°) is gww fry (6.2), ^>(^°) being given by (4.8), (4.9), (4.10).
6.2 Consequence of Theorem 6.1
The set of inequalities (6.3) may also be written
(6.4) (C(«0),«0)^ = inf(C(«°),«)^;
consequently, if the set °tt is strictly convex (in the sense: every
supporting hyperplane of °ll cuts f at a unique point) and if £(u°) =J= 0, the
vector u° is uniquely determined by C(w°): we have a relation of the
form
(6.5) u° = -<Z>(C(^0)). D
Example 6.1
->
If % is the ball 3tf with radius M and center at the origin, relation
(6.5) may be'written (still if f («°) =t= 0):
M
u° = f(«°). D

170 6. Consequences of the Optimality Condition (I)
We can now write the system of partial differential equations satisfied
by y(u°) and p(u°) corresponding to an optimal control u°; thanks
to (6.5), u° no longer appears; let us write (6.5) more explicitly:
(6.5 a)
f „;= -01(E*1p(u<>),E*2 Tp(u°),E$p(x, 0; u0)),
u°2= -02(...),
u°3= -<P3(..,).
Substituting (6.5a) into (1.1), (1.2), (1.3) and adding (4.8), (4.9),
(4.10), we obtain:
Theorem 6.2. To every optimal control u° (if it exists) corresponds a
couple {y,p}> solution of the (nonlinear) system of partial differential
equations
(6.6) Ay + ^f + E, 0^* p, E* Tp, E%p{x, 0)) = 0,
ot
(6.7) By + E2 02(E* p, E*Tp, E*p(x, 0)) = 0,
(6.8) y(x, 0) + E3 03(E* p, EtTp, E*p(x, 0)) = 0,
(6.9) A*p-?£-p1A1<y-yl) = 0,
ot
(6.10) Cp-p2A2($y-y2d)=0,
(6.11) P(x, T) - 03 A3(y(x, T) - y3d) = 0,
as long as
{EXp.EtTp.Etpix.O)}**).
If {y,p} is a solution, then the corresponding optimal control u° is
given by:
(6.12) u° = -0(E*lp,EtTp,E*p(x,O)). U
Remark 6.1. For systems governed by ordinary differential equations,
the operators Ax and A3 reduce to the identity, and the equations (6.7)
and (6.10), corresponding to the boundary conditions on Z, no longer
appear. D
Remark 6.2. We have the existence of (at least) one optimal control,
if % is bounded (easy verification). D

7.2 Optimality Condition 171
7. Consequences of the Optimality Condition (II)
\ 7.1 Additional Hypotheses
Let:
0 = an open set in RN, £ e 0,
d£ = Radon measure ^> 0 on 0,
t) = Hilbert space on R.
Assume that
(7.1) ^ = Z,2(0;i))
and that
(7.2) W = {u\ueJP,u(g) eW^, <Jf-almost everywhere},
where
(7.3) ^l} = closed convex set in t).
Lemma 7.1. 77^ set <% defined by (7.2), (7.3) is closed and convex
in J?'.
Proof. Let une W, un -> u in Jtf. We can extract a subsequence uv
from ww, such that uv (£) -> w (£) in t) for almost every £: thus since
«„(£) belongs to the set °U^ which is closed in t), it follows that w(£) e ^
a.e., therefore that u e <%, whence the lemma. D
We shall now apply Theorem 6.1 to this situation; then £ (u°) is a
function £ -> £(£, «°) of 0 -> t). D
7.2 Optimality Condition
Theorem 7.1. Under the hypotheses of Section 7.1, if u° is an optimal
control, then, except possibly for £ belonging to a set N a 0 of measure
zero [for d£):
(7.4) (f (£; «»), u°(e>\ = inf (f(£; ««),«),,
£ fremg defined as in Theorem 6.1, and conversely.
Proof. 1) Let u° be an optimal control. Denote by J£?t the Lebesgue
set of the (vector) function £ -> £(£; w°) and by J£?2 the Lebesgue set
of the (scalar) function
f-»(C(f;«°).«p(f))i.
We shall prove (7.4) for ^ e^>l nJ?2> the complement of which
has measure zero.

172 7. Consequenses of the Optimality Condition (II)
2) Let f0 e J2\ n J^2 and let 0,, / = 1, 2, . . ., be a family of cubes
with center £0, with
^ d ^2 D " ' and measure (0y) -> 0 as / -> + oo.
Then thanks to the choice 1),
0)<*f-C(f0;«°)M,
(7.5)
measure
measure
ire(^) J ^'
l—z- [ (C(f;«°),«°(f))^f->(C(fo; «°),«°(fo))6-
ire(0,) J
3) Let w be arbitrary in ^. Define
f «°(f) outside 0,,
(7.6) «,tf) = ;
[ w in Oj.
In this manner we define an "admissible control", i.e. Uj e °U (see
(7.2)) and therefore, according to Theorem 6.1:
(7.7) (t(«°),«j-«°)*£0,
or
(7.8) J(C(f;«°),«-«0(f),j«^o.
Dividing by measure (0,.), it follows that
measure
^-|(C(«:«»),«»(«MfS0
measure
and as /-> oo, we obtain, with (7.5):
(7.9) (C(f0;«°). «)* ^ (f(fo;«°).«°(fo))j,
whence (7.4).
Conversely, if we have (7.4), then we have (7.9), or
(C(f;«°),«(D),^(f(f;«0),«0(D)
for almost all £, if u e <%, and therefore we have (6.3) (after integration
in £ over 0). D

8.3 Choice of K2(Z) 173
Remark 7.1. If t) = WL2^1]^)} 0l = open sets in RNi provided
i = i
with the Lebesgue measure, and if
(7.io) % = n %,.
i = 1
where
(7.11) °U^ = closed convex set in \,
then (7.4) "decomposes" into i0 conditions (1 ^ i ^ i0):
(7.12)
Uie®f)i
except possibly for £* in a set of measure zero on 0\ D
8. Complements on the Choice of the Spaces K(
8.1 Orientation
The nonlinear system of partial differential equations (6.6), . . ., (6.12)
contains the isomorphisms Ait which we must now specify completely.
We recall (Definition 4.1) that At is the canonical isomorphism of Kt
onto K't.
In the examples, the K/s are (integer or fractional order) Sobolev
spaces and the At's are therefore differential or integro-differential
operators, or Green operators, according to whether the order of Kt is
> 0 or <0 (if Kt has order 0, therefore is an L2-space, A{ is the identity).
8.2 Choice of KX(Q)
The simplest choice for Kx (Q) (but this choice is not mandatory and
for this reason we have preserved the general notation Kx (Q)) is
(8-1) Kt(Q)=L*(Q)
and thus
(8.2) Ax = identity. □
8.3 Choice of K2 (Z)
We seek K2(Z) in the form
m-l
(8.3) K2 (Z) = J] E"J (IT), integer a,.
j=o
If
(8.4) fi - Hj - i = (minw,) - fi} ^ 0,

174 9. Examples
we take
(8.5) Haj(Z) = L2{S) (and then the corresponding operator A2j is
the identity).
If
(8.6) fi - pj - i < 0,
we take
(8.7) ocj = (minm,-) — fij (<0).
In this case A2j is given by
A2j(o = {-A£ + I)ajco ((1)), with Dirichlet
(8.8)
conditions on I, w e //"'(<£). D
8:4 Choice of KZ(Q)
We may take
(8.9) K3(Q) = HminmJ-m(Q).
Therefore
(8.9)! A3 = identity, if rninw^ = m,
and
| ^13(0 = (-A + 7)minm^~mco, with Dirichlet boundary
[ conditions, if-minwj < mt co e K3(Q). D
(8.9)2
9. Examples
Example 9.1
Consider the following very particular case of (1.1), (1.2), (1.3):
(9.1)
^Ay +4^ = 0,
dt
y Is = ^2 =
ly(*.0) = o.
(Therefore Ex = E3 = 0, JT2 = L2^), £2 = identity.)
Then w = 1, m0 = 0 and, according to Section 8.3, we take
K2{S)=H-i{Z).
((1)) Zlj; = Laplace-Beltrami operator on L.

9. Examples
Thus the adjoint system (4.8), (4.9), (4.10) is
175
(9.2)
(9.3)
(9.4)
(9.5)
dp
-Ap - = 0,
e dt
(-Az + i)(p\*) = p2(4-y
\ dv
,-*)
p\x = 0 for t = 0 and t = T,
P(x,T) = 0.
-Ay + ——
dt
dy
dv
. y{%>
= u2 ■
z
0) = 0.
= 0,
= u,
[Equations (9.3), (9.4) define p\s, which, together with (9.2) and
(9.5), determines p in Q]. D
Example 9.2
Now consider the system
(9.6)
Therefore Ex = E3 = 0, J^2 = L2{Z), E2 = identity. Then we
have ^0 = 0, m =■ 1, m0 = 1 and therefore
minmj — ju0 = 1
and therefore (see (8.4), (8.5)) we take
K2(Z) = L2(Z).
Thus the adjoint system is
(9.7)
dp
(9.8)
dp
dv
P2.(y\z-ri),
(9.9) p(x,T) = 0. U
Remark 9.1. For the case in point, we could have chosen
K2(Z) = H^'2(Z).

176
9. Examples
Then A2 is an integro-differential operator. System (9.7), (9.8), (9.9)
is replaced by
(9.10)
dp
-At- — -*,
dp
dv
= P2A2(y\i-y2d)i
[p(x,T) = 0%
The cost function, in the first case of system (9.7), (9.8), (9.9), is
(9.11) Pij(y(x,t;u)-y*yd£
z
and in the second case (of (9.10)) is
(9.12) p2\\y(x,f,u)-y2*\\2HuuH».
Thus, if we seek an optimal approximation in a larger norm (i.e. a
finer topology), the adjoint problem is more complicated (as one would
expect). U
Example 9.3
Consider
(9.13)
dy
-Jy + -f = 0,
at
dv
0,
l y(%> o) = u3 = u.
Then Ex = E2 = 0, J^3 = L2(Q), E3 = identity. We have
m = 1, m0 = 1,
and therefore we have the particular case of Section 5.
Thus we take (see (5.5)) K3(Q) = L2(Q) and obtain the adjoint
system:
dp
(9.14)
dp
dv
0,
{p(x,T)=p3(y(x,T)-y3a). D

9. Examples
177
Example 9.4
Consider
(9.15)
d.y
01
y(%, 0) = ^3 = U.
Therefore Et =± £2 = 0, ^3 = L2(£), £3 = identity. We have
m = 1, w0 = 0 and therefore (see (8.9)):
K3{Q) = H'^Q).
Then the adjoint system is
(9.16)
-Ap - = 0,
y dt
(9.17)
(9-18)
(-A + I)p(x,T)=p3(y(x,T)-yl),
p(x,T) = 0 for xeT.
In this case, (9.18) determines p(x,T) (Dirichlet problem!), which
together with (9.16), (9.17) defines p in Q. D
Example 9.5
Finally, we consider an example with m > 1. Let the system be
(9.19)
(9.20)
(9.21)
dy
01
dy
dv
y{x,0) = 0.
Then Ex = E3 = 0, Jf2 = (L2(Z))2, E2 = identity.
We have: m = 2, m0 = 0, mx = 1, ^0 = 3, /^x =2, minm^ = 0
and, according to (8.4) and (8.5), we therefore take:
(9.22)
K2(S) = H-*(Z)xH-2{Z),

178 10. Non-Parabolic Cases. Statement of the Problems. Generalities
and the adjoint system becomes:
(9.23)
(9.24) \
dp
A2p -_L = 0,
(-Az + I)3fi\z = p2
dt
dAy
dv
-y^*
dp
82p
IF
0, for t = 0 and t = T,
(9.25)
(9.26)
dp
OV
dp
dv
d I dp
z dt \dv
P2{-Ay\z-yl2),
= 0, for t = 0 and t = T
p{xtT) =0.
dp
Equations (9.24) and (9.25) define p \s and —-
with (9.23) and (9.26) defines p in Q. □
dv
, which together
10. Non-Parabolic Cases. Statement of the Problems.
Generalities
Here, we introduce the framework of the theory in a somewhat
different axiomatic fashion, in order to insist on the numerous
possible variants of control problems, when the controls appear in
the boundary conditions, and because of the inherent difficulties of
non-homogeneous, non-parabolic problems noted in Chapter 5.
10.1 Notation
For a control
= {«i, ^2* Us> u4.} eJ^ = Yl ^f
Jf ( a Hilbert space on R, we consider a system for which the state y
is given by
(10.1) Ay + y"^E±ul9
(10.2) By =E2u2i
(10.3) y{x,0) =E3u3,
(10.4) y'(x,0) =E^u^

10.2 Cost Function 179
where:
the operators A, {Bj} = B satisfy the conditions of Chapter 5,
Section 1, and the operators Et satisfy
(10.5) EteSP{Jti$SPt)9
where:
J27! = J?i(Q) is a space of functions or distributions or "
functional' on Q,
J£2 = j£?2 (27) is a space of functions or distributions on 27,
jg?3 = jg?3 (Q) t j^4 = j^4 (Q) are spaces of functions or
distributions on Q.
We assume the spaces j£?f to be chosen so that (10.1), . . ., (10.4)
admits a unique solution, either the "usual" solution, or the solution
defined by transposition, i.e. (see Chapter 5):
(10.6) (y, A* O + 0")Q = (Ex ult 0)Q + (E2 u2, T 0)£ +
+ (£4«4,*(0)x,- {E3u3,0{O))Q
for every 0 such that
[ A* 0 + 0 = y>,
(10.7) j S0 = 0,
[ 0(T) =0, 0'(T) =0,
where ip describes the dual of the space described by y. □
Remark 10.1. Even in practical examples, we have some freedom
in the choice of the J^/s. □
The solution of (10.1), . . ., (10.4) (possibly in the sense of (10.6))
is denoted by
(10.8) y = y(u) = g(x,t;u).
10.2 Cost Function
4
When {Ex ux, E2u2, E3u3, EA u±} e\\^iy the solution y(u) be-
i = l
longs to a space ty (the complete characterization of which may be
delicate; see Chapter 5). □
Formally, the control problem is the following: as u describes a set
4
°tt a Y\ <^i> minimize the "distance" from y(u) to yd, with yd given in <W.
i = l
We shall always assume that the J^f/s are Hilbert spaces, and
therefore so is ?!/. Thus we want to minimize \\y(u) — y&\<&. □

180 10. Non-Parabolic Cases. Statement of the Problems. Generalities
(10.11)
As for the parabolic case (see Section 1), it is useful to consider, not
only y on Q, but also 5 y (on E) and y (T; u) and y' (T',u).
We introduce the following spaces (compare with Sections 2 and 8):
(10.9) K1(Q) z> W (with the possibility of equality),
(10.10) K2(Z) such that S e&(<8f;K2(Z))t
K3 (Q), X4 (Q) such that y (u) -* {y (T; u), y' (T; u)}
is a continuous mapping of <& -* i£3 (i2) x i£4 (i2).
We now introduce the cos£ function'.
J(u) = ^ || y («) - y,1 ||it + 0a \\Sy(u) - y] ||£a +
+ /33 \\y(T; u) - y\ ||*, + /?4 \\y'(T; u) - y\
0t>O, I/8,>0,
i = l
4
where ^ = {y\,y\9 y\, y%) is given in \\ Kt. U
i = l
The control problem now is:
(10.12)
4 ii 2
K4>
(10.13)
let % be a closed convex set in Y] J^t\ find
Inf /(«).
% satisfies
If u° e
(10.14) J(u°) ^J{u), "iue®,
then w° is called an optimal control. □
Remark 10.2. The analogue to Remark 1.2, Section 1 holds. □
10.3 Optimality Condition (I)
The remarksvJeading to Proposition 4.1 are evidently valid here and
yield:
Proposition 10.1 A necessary and sufficient condition for u° to be
an optimal control is that:
\ ^(y(u°) - yly(u) - y(u°))Kl +
+ Pi(5y(«°) -ylSy(u)-Sy(«<>))*, +
+ p3(y{T; u°) - yly(T; u) - y(T; u°))Ki +
+ ^(y'(T;u°) - y$,y'(T;u) - y'(T; u°))K4^ 0 V« 6 <*.
(10.15)

10.4 Adjoint Problem
181
10.4 Adjoint Problem
As in Definition 4.1 of Section 4.2, we define At to be the canonical
isomorphism of Kt onto K't.
Next, we define p(x, t ;u)=p(u) to be the solution of (the adjoint
problem):
A*p(u) + p"(u) = ^ Ax{y(u) - yi),
(10.16)
(10.17)
(10.18)
(10.19)
Cp(u)=p2A2($y(u) -yl),
p(x,T;u)=p3A3(y(x,T;u) -yl),
p'(x, T; u) = -pt A4(?(x, T; u) - y$).
Let us specify in what sense p(u) is a solution of (10.16), .
Consider co, solution of
(10.19).
(10.20)
(10.21)
A co + co" = %, % given in Se 1 (Q),
Sco = 0,
co(*,0) = 0, co'(*,0) = 0.
Then (by definition) p(u) satisfies (10.16), . . ., (10.19) if
(p(u),A co + co") = /8i</li(yM - yi), co> +
+ j82<5y(«)-y2,5«>>-
-P3<y(T;u)-yla>'(T)y-
-Pt<y(T{u)-y},a>(T)>.
More precisely:
Proposition 10.2. There exists a unique p (u) e <£[ (Q), satisfying
(10.21) Vco given by (10.20) (for arbitrary xe^i(Q))-
Proof. For # e 3?^ (Q), problem (10.20) admits a unique solution in ty
(by definition of ^; see Section 10.2). Then (see (10.9), . . ., (10.11))
4
co -* {co, 5 co, co(T), co' (T)} is a continuous mapping of <& -► Y[ Kt>
and therefore the right-hand side of (10.21) is a linear form, say co -► M (co),
which is continuous on <&.
If <%/0 (c <%/) is the space described by co as % describes ^'lJ the
d2
operator A H is an isomorphism of <%/0 onto ££x and therefore,
dt2
by transposition (see Chapter 5), there exists a unique p(u) eJ^KQ)
satisfying (10.21). □

182 10. Non-Parabolic Cases. Statement of the Problems. Generalities
10.5 Green's Formula
We now make the hypothesis:
for {l2, h > U} e <&?2 x °^3 x °^4 > there exists %eKx (Q)
such that
Ax + t'e&i.
B% = h, x(0)=l3, z'(0) = /4.
5% = 0, X(T) = 0, X'(T)=0,
the mapping {Z2, l3, l4} -> % being linear and continuous.
Then, if p e «S?i with A* p + p" eK[, we can define
Tp ese'2>
(10.22)
(10.23)
*(0)eJS?;.
£'(0)e^.
with the help of
<P,Ax + x"y~(A*p + p",xy
= -<Tp,i2y-<p(o),uy + <pf(o),i3y.
(10.24)
Remark 10.3. In fact, for each particular case, we shall need to
demonstrate (as we have done in Chapter 4 and 5) that the mappings
p -> T p, p (0), p' (0) may-be defined by extension by continuity.
Furthermore, the property of belonging to j£?2 x <^3 x ^4 may contain com-
patibility relations (see Chapter 4 and 5), so that we cannot "separate''
the "components" Tp, p(0), f (0) in (10.23): {? p,p(0), f {0)}
belongs to a distribution space quotient. U
Under hypothesis (10.22) (and within the restrictions of Remark 10.3)
we have (done what was necessary to have) Green s formula:
Proposition 10.3. For u and u° given in £F:
' {A y(u) + y"(u), p(«°)> -<y{u),A*p(u°) + p"(u°)y
= ($y(u),Cp(u°)y-<By(u),Tp(u°)y +
+ (y'(T;u),p(T',u°)y - (y'{Q\u),p{Q',uQ)y -
- <y(T; u),p'(T; u°)} + <y(0; u),p'(0; «°)>,
(10.25)
where on the left-hand [resp. right-hand) side of the equation the brackets
denote, in order, the duality between^x and££\, Kx and K\ (resp. K2,K'2\
~?2> -*2\ Kit K*'* °£±> ~£'±\ Ks> K^', o^f3, 0^3). D

11.2 Choice of K1 183
10.6 Optimality Condition (H)
Theorem 10.1 Hypotheses- of Section 10.1 and (10.22). A necessary
and sufficient condition for u° to be an optimal control is that u° e %
satisfies
I <£(«°),£i(«i ~ «?)> + <Tp(u<>),E2(u2 - u°2)} +
(10.26) j +<-p'(0',u°),E3(u3-uo3)> +
[ + <p{0; u°),Et(u4 - u$)} ^ 0 \/u e «f,
where p(u°) is the solution of (10.16), . . ., (10.19) (in the sense of (10.21))
(with u = u°).
Proof. In (10.25), substitute u — u° for u; the equality of the left-
hand sides of (10.15) and (10.26) follows, whence the theorem. D
10.7 Consequences
Results analogous to Sections 5 and 6 follow from Theorem 10.1. D
11. Applications. Examples
11.1 Control in the Boundary Conditions
From now on, we shall assume that the control appears only in the
boundary conditions.
We take
U, = o, £3 = o, £4 = o,
j AT2 = {L2{E))m, £e2 = (L2{S))m, E2 = identity.
Thus, setting u2 = u, the state y of the system is defined by
A y + y" = 0,
By =ue(L2(Z))m,
y(0) = 0,
1/(0) =0.
We take the cost function:
(11.3) /(«) = /yy(«) - yJh + P2 \\Sy(u) - yj\\2K2
and we now have to specify how to choose Kx and K2.
11.2 Choice of Kt
We apply Theorem 6.1, Chapter 5 (for the case of Dirichlet boundary
conditions, we can also apply Theorem 6.2, Chapter 5; see also
Remark 5.3, Chapter 5).
(11.2)

184 11. Applications. Examples
Thus, let
(11.4) 0 = 1 minnij.
6m 3m
Then
(11.5) yeH-e(0, T\ H2m^^{Q)) n H2-*e(0, T\ H°(Q)) = ^,
except for the two following particular cases:
m = 1, m0 = 1; then
y e #oo1/2(0, T\ H1 (Q)) n tf 1/2(0, T\ H° (Q)) = «r
(11.6)
and
m— 1
w-odd, minwj =
(11.7) J ' J 2
y e #"5/6(0, T; Hm<*(Q)) n ^oO1/2(0, T; H°(Q)) =
It is required to choose K± = K1(Q), a "simple" space such that
®J c: Kx.
We distinguish two cases, according to whether 2 — 30 ^ 0 or <0.
First case:
2 - 30 = 0, 0 defined by (11.4), is equivalent to
minwj _■ w — -J-,
that is
(11.8) minw,. = m
(note that this also corresponds to (11.6)).
Then in any case (i.e. (11.5) and (11.6)) we have: <& c L2(Q) and
therefore:
Proposition 11.1. If (11.8) holds, we may take
(11.9) K^L'iQ). Q
Second case.
Assume that 2 - 30 < 0. We still have
2 - 30 > -1
and in any case (i.e. (11.5) and (11.7)) we have
<2J a H-1(0>T\H°{Q)).

11.3 Choice of K2 185
Then :
Proposition 11.2. // (11.8) does not hold, we may take
(n.io) k± =^-1(o,r;^°(fi)). □
Remark 11.1. For the case (11.10) we have:
K[ = H1o(0,T;H°(Q));
if we provide K[ with the scalar product
T
J [(^^)flo(fi) + (u',v')H0(Q)]dt,
0
the operator A1 of the general theory is given in the following way:
let / be given in tf-^O, T\ H°(Q))] let co be the solution of
-co" + co = /
co(0) = 0, co(T) = 0.
Then
co = A1f. □
11.3 Choke of K2
Now, the problem is to "characterize" the space described by 5 y
as y describes W and to choose K2 = K2(Z), a "simple" space such
that
$<&czK2. U
The result (11.5) does not apply to all cases; indeed for the present
case (A having symmetric principal part) we have:
order Sj = 2m — ms — 1
and we cannot define 5j y by using only (11.5), unless
2m(I — 6) — \ > 2m — md — 1, 0 given by (11.4),
i.e. unless
ntj ^ 2m — | minw,. + \,
which is not always realized for all / (and may not be realized for any j). □
We shall therefore reconsider the question directly by interpolation,
using the results of Chapter 5.
First of all, Theorem 3.1, Chapter 5 yields:
if UjeH^O, T; H2m-mJ-^2(D) n H$m-iM>+ll2»iM(0, T\ H°(r)),
(11.11)

186
11. Applications. Examples
to which we must add the global relations —j^ufjeL2(0, T;H°(r))
in the exceptional case: V^
m-odd and mj =
3m — 1
(11.12)
j = o
then y e H2m>2 (Q) and therefore
m- 1
5 v G TT f72m~(2m~mJ~1)~1/2»(2m~(2m~mJ~1)~1/2)/m/,r'\
j = o
from which it follows that:
u -> 5 y is a continuous linear mapping of
m-l
m-l
-> J-[^^+l/2;(o^+l/2)/m(r)> Q
Let us now apply Theorem 4.1, Chapter 5. We see that if
m-l
Wen(^*+1/2:omj+1/2)/mK)' then yetf0'"1^).
In the present case, we have: A y + y" = 0, therefore in particular
4y + y"eL2(<2).
For the moment, we denote by F the space:
F = {y | y e H»>-* (Q) ,A y + y" e L2 (Q)}
and (see Chapter 5, Section 11 for a similar situation)
F = closure of 3{Q) in F.
(we do not know whether F is really distinct from F). Then y eF and
we shall verify that
for y eF, we have
m-l
5 y e n (#5(0, T; H2"-"'-1'2 (T)) n
(11.13)
J=o
n ^"—'-^'/"(OTT; H°(r)))',
the mapping y -♦ 5 y being continuous for the natural
topologies.
Indeed, we use Green's formula:
(A y + y", co)-(y,A*<o + co") = <5y, C^y^ - (By,Tri>E,

11.3 Choice of K2
187
where we must take (so that the scalar products on the left-hand side
of the equation are well-defined):
(11.14)
a)eL2(0, T; H2m{Q))t co' e H2m>2 (Q),
a)(x, 0) = cd'(x, 0) = cd"(x, 0) = cd(x, T)
= co'{x, T) = co"{xt T) = 0.
Then, if ip is given in the space
m- 1
(11.15) sfi = n^o(o,r;^2m-m^1/2(r))n^3m-mj-1/2)/m(o,r,^0(r))
j = o
(with possibly the global relations which we already noted), we can
find (see Chapter 5, Proposition 3.3) co which satisfies (11.14) and
C co = ip and T co = 0, and which depends continuously on ip. (11.13)
follows. D
Thus, it follows from (11.12) and (11.13) that
i -> 5 y is a continuous linear mapping of
(11.16)
m-1
^ = YlHmj+ll2](0mj+ll2)lm(Z)
J = 0
and of
v2 -> *!•
By interpolation we obtain:
Proposition 11.3. Let ^ and &2 be defined as in (11.5) and (11.16).
Then, for 0 < 6 < 1,
u-*$y, y solution of (11.2),
is a continuous linear mapping of
(11.17) [^i,^W[^2,^]e. □
Remark 11.2. We recall that (duality theorem. Chapter 1, Section6.2)
(n.18) [»2.ari]. = [^i.^]'i-.. □
Now we choose 6 such that
(11.19) L2(20"«=l?i,Sry..
But
TT rj3m-mj-l/2,(3m-mj-l/2)/m/yi\

188 11. Applications. Examples
(with the global relations already noted) and therefore
(11.20)
m-l
\?i,9i*\e => 11 [H3m-mi-ll2'£m-m*-ll2)lm{Z)t (Hm^ll2'p+ll2)lm(Z))^et
j = o
a space which, by Chapter 1, Section 12, contains L2{Z)m if
3w(l - 0) - mj - \ ^ 0, V/,
i.e. if
1 1
6 = 1 minmj;
6w 3w
we recover the 6 given by (11.4).
Consequently: v
f if ueL2 {E)m, then 5 y e [&2, ^i],,
(11.21)
[ 0 given by (11.4).
But, by duality, it follows from (11.20) that
(11.22)
m-l
^/-i ^ rr^-3ffl(l-fl) + mJ+l/2,(-3m(l-fl) + mj+l/2)/m/r.v
J = 0
or in the dual of the space containing the eventual
global relations,
from which, together with (11.18), we obtain:
Proposition 11.4. If ueL2{Z)m, the trace Sy belongs to
m-l
T-jr Tjmj + minnij-3m+ l((mj + minmj—3m + l)fm /y\
j = 0
or to the dual of the space containing the eventual global relations.
Note that we always have
ntj + minwj — 3m + 1 < 0,
with equality in the only case: m = 1, m0 = 1 (which is not an
exceptional case). Therefore:
Proposition 11.5. If m = 1, m0 = 1, we, may take
(11.23) K2 = L2(Z).
For the other cases, we may take
m-l
(11.24) K2 = [] H-i3m-mi-minmt-l\Z) <(1». D
i = o
((1)) For positive s, we set H~S(Z) = (HS£(Z))'.

11.4 Examples
11.4 Examples
Example 11.1
The state y is given by
' -Ay + y" = 0.
189
(11.25)
dy
dv
= u,
y(x,0) = 0, y'(x,0) = 0.
We have: m = 1, m0 = 1; according to Propositions 11.1 and 11.5
we may take
K^L^Q), K2 = L2{Z).
The adjoint problem is:
-Afi + p- =Pily-y\),
dp
(11.26)
dv
Pifyk-yi).
p(x,T)=0, p'{x,T)=0. D
Example 11.2
The state y is given by
f -Ay + y" = 0,
(11.27) \y\z =«.
[y(x,0) = 0, y'(*,0) = 0.
Then m = 1, w0 = 0. According to (11.10), we take
and according to (11.24):
K2 = H~2(Z).
Let Ar be the Laplace-Beltrami operator on J". Then
yl2 = (canonical isomorphism of i?2 onto K'2)
d2 \-2
(11.28)
= -Ar -
dt2
with Dirichlet boundary conditions,

190 11. Applications. Examples
i.e. if / e K2, A2f = a) is the solution of
d2 ,2
(11.29)
-Ar-
dt2
(O = f,
co{x, 0) = (o'(x, 0) = 0, co(x, T) = (o'{x, T) = 0.
Then, the adjoint problem is
-Ap + p" = p! A,(y - y'd), A^ defined by (11.11),
(11.30)
P lz = j8 A2 (-^- - y2), yl2 defined by (11.29),
P(x,T)=0, p'(x,T)=0.
Example 11.3
The state y is given by
' A2y + y" = 0,
(11.31) {
Then:
dy
= m2, w = {u1;u2},
y(x,0)=0, y'(*,0)=0.
m = 2, m0 = 0, »»! = 1.
We still take K1 according to (11.10). By (11.24):
K2 = H-s(Z)xH-+{F),
and the adjoint problem is
A2p + p" ^^A^y-yl),
(11.32)
V \s
dt2
fih=Pz
dAy
dv
yl
P,...,
d+p
dt2
zero at 0 and T,
-Ar~ ,
82 \* dp
dW) ~Jv~
= p2(-Ay\z-yl),
dp
83 ldt \ +n at
zero at 0 and 1, ■
dv '"" dt3 \dv
P(x,T)=0, P'(x,T)=0

12. Comments
191
Example 11.4
The state y is given by
" A2y + y" = 0,
(11.33)
Ay\
dAy
dv
y(x,0) = 0, y'{x,0) = 0.
Then m = 2, m0 = 2,m1 = 3. We still take Kt according to (11.10)
and, according to (11.24), we take
And the adjoint problem is given by
' A2p + p" =p1A1{y-y%, A± defined by (11.11),
(11.34)
-yl
d2 \ I dy
Ap zero at 0 and T,
d A „ , . dAp
— Ap\E = p2{y\E-y22), —!- zero at 0 and T,
dv dv
P(x,T) =0, p'{x,T) =0.
12. Comments
The bibliography for control problems for systems governed by
ordinary differential equations is enormous; the basic reference is
Pontryagin-Boltyanskii-Gamkrelidze-Mischenko [ 1 ].
Most control problems considered in the literature pertaining to
systems governed by ordinary differential equations lead to analogous
problem statements for systems governed by partial differential evolution
equations. The subject is therefore immense, and its analysis is only in
its beginnings.
Thus, the present chapter necessarily has a very incomplete aspect.
We study only linear systems, with fixed terminal time and quadratic
cost function. For optimal time and controllability problems, consult
Balakrishnan [1], Conti [1], Ju. V. Egorov [1, 2], Falb [1], Fattorini
[1-3].
In this chapter, we present the problems resulting from the study of
control problems when, in particular, the controls appear in the boundary
conditions (boundary control) (the necessity of using the theory of non-
homogeneous boundary value problems being evident in this case).

192 13. Problems
More general results are given in Lions [26]; we have restricted the
present discussion to certain points which we believe to be rather
significant : the essential result is that, after (rather considerable and
seemingly unavoidable) technical difficulties bound to the justification of
Green's formula used in each case, we are led to new types of boundary
value problems (see, for example, Theorem 6.2), which replace the
"two-point boundary value problems" of the usual theory for systems
governed by ordinary differential equations — see Lions [26, 28, 30].
For problems of existence of optimal controls for systems governed
by partial differential equations, consult Cecconi [1], Cesari [1, 2],
Lions [24, 25], Schmaedeke [1], Vallee [1], Zolezzi (1]. For systems
described by elliptic equations considered from the point of view of
this chapter, see Lions [26], note (II) and [30].
13. Problems
We only indicate some of the very numerous, still unsolved problems
in the direction taken in this chapter.
13.1 Extension of the situation of Section 11 to the case of
nonzero Elt E3, E4. and where J(u) contains norms in K3 and KA.
13.2 The preceding considerations extend, without difficulty, to
the Schroedinger equations
i A y + y' = E1 u1,
with the initial and boundary conditions corresponding to Chapter 5,
Section 10 (in this case, consider complex Hilbert spaces and replace
(3.5) with
Re7i(u°, u - u°) ^ ReM{u - u°), Vwe^).
However, the best choice of spaces seems rather delicate.
13.3 In general, we shall obtain an analogous theory to the one in
this chapter, every time a new class of non-homogeneous evolution
problems will be solved.
Consequently, we shall be able to study optimal control problems
in connection with problems pertaining to coupled equations, equations
with transmission boundary conditions, etc., as stated in Chapter 4,
Section 17 and in Chapter 5, Section 14.
It will also be of interest to study control problems (especially
when the control appears in the boundary conditions) in connection
with first-order hyperbolic systems, general parabolic equations, coupled
systems (first-order hyperbolic, parabolic). For first-order hyperbolic
systems, see Russell [1], Schmaedeke [1] and partial results in Lions [30].

Appendix
Boundary Value Problems and Operator Extensions
1. Statement of the Problem. Well-Posed Spaces
1.1 Notation
Let: E be a Banach space on C, ^ be a locally convex, topological
vector space such that E c ^ and P be given with:
(1.1) Pe^(E]^).
Set
(1.2) EP = {u\ueE,PueE}.
Provide Ep with the norm
(1-3) \\u\\ + \\Pu\\ = \\u\\Epl
where
|| . || = norm in E\
this makes EP a Banach space (indeed, it is easy to verify that the
space EP is complete in the norm ||«/ ||£j,).
Roughly speaking, the problem to be studied here is the following:
find all closed vector subspaces U of EP such that
(1.4) P is an isomorphism of U onto E.
Such U's will be called well-posed spaces of EP\ if (1.4) holds, the
problem (the boundary value problem for the applications we have in
mind):
(1.5) Pu = f, f given in E, u e U,
admits a unique solution which depends continuously on /.
Evidently, we cannot expect results different from the definition
unless we assume:
there exists a closed subspace U1 of EP such that P is an
isomorphism of U1 onto E.
(1.6)

194 Boundary Value Problems and Operator Extensions
1.2 First Condition for U to be Well-Posed
We introduce
(1.7) Z = {u\ueE,Pu = 0}.
Then:
Proposition 1.1. Assume that (1.6) holds.
A necessary and sufficient condition for U to be a well-posed space
is that there exists a k with
(4.8) ke&iU^Z),
such that
(1.9) U=(I + k)U,
(i. e. U = image of Ui under I + k, I = identity).
Proof. 1) The condition is necessary.
Let u1 e U1. Assume U to be well-posed. Then there exists a unique
u e U, solution of
(1.10) Pu = Pux.
Therefore
f = u — u1 e Z
and
u1 ->C
is a continuous linear mapping of U1 -> Z\ let k be this mapping. Then
u = u1 + f = (/ + k) u1,
and 4he result follows (since, as u1 describes Ult P u1 describes Et
therefore Pu describes E and finally u describes U).
2) The condition is sufficient.
Now let the space U be defined by (1.9). We verify that U is closed
in EP. Indeed, let une U with
un -» u in E, P un -> / in is..
By definition
**„ = ^n + £*V ^etfi,
and therefore
Pun = Pvn->f.
Since P is an isomorphism of U1 onto E, we see that
vn-* v in £7j

1. Statement of the Problem. Well-Posed Spaces 195
and then
u = v + kv e U and / = P u\ therefore U is closed.
It remains to be shown that P is an isomorphism of U onto E. Let eeE.
We seek u e U with
(1.11) Pu = e.
But, by hypothesis, there exists a v e U1 with u — v + k v. Then
P u = P v, therefore P v = e, therefore a exists and is unique,
therefore (1.11) admits a unique solution in U. D
1.3 The Space D0
We consider a closed subspace D0 of C7X:
(1.12) D0 c ^
and we seek the well-posed spaces U such that in addition
(1.13) D0a U.
Remark 1.1. In the examples, D0 will be the closure in EP of the
space of infinitely differentiable functions with compact support in the
domain in which the boundary value problems are studied. D
Remark 1.2. Let P0 be the restriction of P to D0. The search for
well-posed U's such that (1.13) holds is equivalent to the search for
all extensions of P0 which are isomorphisms. D
We have:
Proposition 1.2. Assume that (1.6) and (1.12) are satisfied. A necessary
and sufficient condition for U to be a well-posed space containing D0 is
that U, be of the form (1.9), with k satisfying (1.8) and furthermore
(1.14) & = 0 on Z>0.
Proof. 1) Let U be of the form (1.9) with (1.14). Then, if d0 eD0>
d0 + k d0 = d0 e U, therefore D0 c U".
2) Conversely, let D0 c U. Let us again consider the construction
of k given in 1) of the proof of Proposition 1.1. Let d0 eD0; we consider
the solution in U of
(1.15) Pu = Pd0
and then
kd0 = u — d0.

196 Boundary Value Problems and Operator Extensions
Since D0 c U, the solution of (1.15) in 17 is u = d0, so that
kd0 = 0. D
Following result (1.14) it is, of course, advisable to pass to the quotient
by D0. Thus, we introduce
(1.16) U\ = t/i/A,
and the quotient mapping of k:
(1.17) hreSe{JU\\Z).
Then
Corollary 1.1. Assume that (1.6) and (1.12) are satisfied. A necessary
and sufficient condition for U to be a well-posed space containing D0 is
that there exists a km eJ?(U\',Z) such that
(1.18) U =U,+ k*U\. D
2. Abstract Boundary Conditions
2.1 Boundary Spaces and Operators
We are given Banach spaces 0, W and operators /?, n, such that
(2.1) fi e ^(C^; 0), p surjective with kernel D0>
(2.2) n e 3? (EP; W), n surjective with kernel U1.
The spaces 0, W are the boundary spaces and the operators ft, n the
boundary operators.
Let /?* be the quotient operator of $ by D0:
(2.3) (5* e&{U\\ 0), (}\ isomorphism of U\ onto 0.
Let 7tz be the restriction of n to Z. Since we have the decomposition
into direct sum
(2.4)
EP = U1 + Z,
0 = ^x + f, P^x = P e, u1 eU1,
we see that
(2.5) ttz is aw isomorphism of Z onto W.
2.2 Characterization of Well-Posed Spaces
We are now in a position to prove the main "abstract" theorem:
Theorem 2.1. Assume that (1.6) and (1.12) are satisfied. Assume that
we are given the boundary spaces and operators with (2.1), (2.2). A necessary

2. Abstract Boundary Conditions 197
and sufficient condition for U to be a well-posed space containing D0 is
that there exists a K with
(2.6) Ke^(0>lF)>
in such a way that U is the space of u's e EP such that
(2.7) n u = K (I (u - Ttz 1 n u).
Remark 2.1. Condition (2.7) has meaning, since
(2.8) u — Ttz 1 n u e U1
by virtue of the fact that
n(u — Ttz 1 n u) = 0. D
Proof of Theorem 2.1. 1) Let U be well-posed and contain D0. Then,
there exist
ke&iUuZ), k-e&(U\\Z), with (1.18).
Set
(2.9) K = 7tz^(n~1'>
K belongs to &{0; W).
Then every u e U may be written :
(2.10) u = u1 + f, f = k* u\, u1eU1.
Since u1eU1, we have: nu1 = 0, therefore
(2.11) nu = ttC
and
(2.12) f = Ttz 1 n u.
But, according to (2.9), k = n^1 K (}', therefore
n C = n k* u\ = K j8# w; = K ^ u1 = Kp{u - £),
which, together with (2.11), (2.12), yields (2.7). D
2) Conversely, let U be the (closed) space of elements u e EP
satisfying (2.7). It follows immediately that D0 c= U.
Let us this time define k* by
*• ^Ttz1 Kp", kme&{U\\Z)\

198 Boundary Value Problems and Operator Extensions
and for u given in U, let us define:
C = Ttz1 7t u e Z,
u± = u — Ttz1 7t u = u — £ e U! (see Remark 2.1).
Then
(2.13) u - (% + k* u\) = C - *# u\ e Z.
We verify that
(2.14) 7t[u - («x + k* u[)] = 0.
Indeed
(2.15) n\_u — (% + k* u\)] = Tiu — 7i km u\ .
Now, by (2.7):
n u = i£ /? (u — ttz * 7T u) = K ft u± = K /?# u\ — 7tzk* u[ = n k* u[,
therefore (2.15) implies (2.14).
But, according to (2.5), it follows from (2.13), (2.14) that
u = u1 + k* u[.
This completes the proof of the theorem. □
3. Example 1. Elliptic Operators
3.1 Notation
We adopt the notation of Chapter 2.
Let Q be a bounded open set in R", with infinitely differentiate
boundary J7, and let A be a properly elliptic operator in D, with regular
coefficients.
In the notation of Sections 1 and 2, we choose:
E = L2(fi), & = 9'{Q),
P = A.
We take
(3.2) U± =H2m(Q)nHZ(Q)
and we assume that
(3.3) A is an isomorphism of U1 onto L2{Q).
(3.1)

3. Example 1. Elliptic Operators
199
Remark 3.1. Thus we assume the Dirichlet problem to be well-posed
for the operator A. We know (see Chapter 2, Section 8), for example
if A is strongly elliptic, that A + A is an isomorphism of U1 onto
L2(Q) for sufficiently large real X. We can therefore apply the present
theory to A + X. D
The space D0 is chosen as follows:
DQ = closure of @(Q) in EP = {u \ u, A u eL2(Q)}.
We have (see Chapter 2, Section 5):
(3.4) D0 = H20m{Q).
3.2 The Boundary Operators and Spaces
For u e Ulf we set
, dJu
(3.5) fiu =
m <L j ^ 2m —If,
dvJ
2m- 1
(3.6) <Z>= [] H2m-J-^2(D
j = m
(see Chapter 1, Section 7 for these spaces).
Condition (2.1) is satisfied by Chapter 1, Section 8.2.
Next, for u e EP, let
dju
(3.7) 7i u =
dvJ
0 ^ / ^ m — 11,
m-l
(3.8) w = n#~j~i/2cn-
Condition (2.2) is satisfied according to Chapter 2, Section 6. Note that,
changing / to 2m — j — 1,
m-l
J = 0
so that
3.3 Consequences
Let us write out (2.7) for the present situation.
Let u be given in Z,2(Q), with iwel2(i2). Let
' K= \\Kji\\
(3.9) {
1 XJleJ2?(^2,,,-|-1/2(r);^--'-1/2(r))

2001 Boundary Value Problems and Operator Extensions
Sjra let f be the solution of
dJ£ dJu
(3.10)
dvJ d vj
0 ^ / ^ m — 1.
Then u — f e #2m (i2) and C7 « defined as the space of elements
u e L2 (Q) such that A u e L2(Q), which satisfy
dJu 2^x dl
^ i=m 3r
By varying if in (3.9), we obtain all well-posed spaces.
3.4 Various Remarks
Remark 3.2. In (3.1) we started from hypothesis (3.3).
We could assume that another problem, different from the Dirichlet
problem, is well-posed. For example (in the notation of Chapter 2),
assume that the problem
(3.12)
Au = /, / given in L2 (Q), ue H2m{Q),
BjU = 0, 0 g / g w - 1,
is well-posed.
In that case, we take
(3.13) DY= {u\ueH2m(Q),BjU = 0,0 g / g w - 1},
(3.14) j8«={Si/«|0g/gw-l} (notation of Chapter 2, Section 2).
If the order of Sj = //,-, we have
m-l
(3.15) 0 = n#2m~"'~1/2cn-
J = o
Next, for u e EP, we take
(3.16) nu = {B,.«|0 g / g w - 1},
m-l -
(3.i7) y=n^~m'~1/2(r)«
J = 0
^4Z/ well-posed spaces are given by the elements u e L2 (Q) such that
A u e L2 (Q) and »
m-l
(3.18) B,« = X^i5i(« - 0' 0^/^ w - 1,
1=0

3. Example 1. Elliptic Operators 201
where
(3.19) Kn e^(H2m~^-ll2(r);H-m^ll2(r))t
(3.20) C given by A £ = 0, Bj£ = BjU, Og/gw-1. □
Remark 3.3. If we assume that problem (3.12) is well-posed for two
distinct families Bj1} and Bj2) of boundary operators, we can define all
well-posed spaces by two families of relations of the type (3.18), with
two families Kjf* and Kjf} of operators. Since the well-posed spaces
are evidently the same starting from the two families, there is a one-
to-one correspondence between K(1) and Ki2) yielding the same space. □
Remark 3.4. We started from the choice E = L2 (Q). We could also
start from the (Hilbert) choice:
(3.21) E = Hs{Q)t 5>0.
In that case, we consider problem (3.12) with / e HS{Q). We assume
this problem to be well-posed. Then
(3.22) U1 = {u | ueHs+2m(Q),Bj w = 0,0^'^-l}
and this time (compare with (3.15))
m-l
(3.23) 0 = 0s = [] H2m+s-v-lj2(r)
and /? is given by (3.14). Next, with n given by (3.16),
m-l
(3.24) W = W* = Yl Hs-m*~1/2 (r).
j = o
All well-posed spaces are given by (3.18), where f is still given by (3.20),
but instead of (3.19), we have
(3.25) Kn eJ^(^2m+s-^-1/2(r);^s-m^1/2(r)). D
Remark 3.5. If we compare Remarks 3.2 and 3.4, then we see that
the well-posed spaces depend on the initial choice of E. For example,
for Kji, we can take a multiplier of
jjlm-m- 1/2/jH\ _^ #-»*,—1/2/jTT)
which is not a multiplier of
jjlm + s-iH- 1/2 /jn\ _^ jjs-mj- 1/2 /jn\ ^ g

202 Boundary Value Problems and Operator Extensions
Remark 3.6. The boundary conditions (3.11) or (3.18) are not in
general of a local nature. D
Remark 3.7 (This remark uses notions which are not fully developed
in this book).
We can also choose
E = LP(Q), 1 < p < oo,~ p*2,
and assume that (3.12) is well-posed in LP(Q). Then
ueW2m>p(Q) = {v\D«veLp{Q),\<x\ g 2m}
and
U1 = {u\ueW2m>p(Q);BjU = 0,0 <L / ^ m - 1}.
The formulas are /&£ same as in Remark 3.2, but this time with:
m-l
0 = [7 W2m-w-1/iM'(jr»)
i = o
m-l
y = [] w-m'-1/M(r)
i = o
(for the spaces Ws,p(r), see for example Lions-Magenes [1]).
The well-posed boundary value problems depend on p. D
4. Example 2. Parabolic Operators
4.1 Notation
Now, consider the setting of Chapter 4.
Let Q be the cylinder
(4.1) 6=flx]0J[
and P the operator
/ d\ d d d
(4.2) P = 4 *,*,— +—= 4 +
dx J dt dt
with the hypotheses of Chapter 4, Section 1.
We choose (notation of Sections 1 and 2)
(4.3) E=L2(Q), & = 9'(Q).
Then
(4.4) EP = \u | u eL2((2), A u +^eL2(Q)\.

4. Example 2. Parabolic Operators 203
We take
(4.5) U1 = \u\ueH2m>l(Q),u(x,0) = 0,-^- =
I dvJ
= 0 on 27, 0 <; / g w -1
(Recall that 27 is the lateral boundary of Q:
z = rx]o,r[).
The operator P being assumed parabolic, we know (Chapter 4,
Section 1) that
(4.6) P is an isomorphism of U1 onto L2 (Q).
The space D0 is chosen as
D0 = closure of @(Q) in Ep.
We have:
Lemma 4.1. The space D0 coincides with the closure of @(Q) in
H2"1'1^), that is
dJv
dvJ
(4-7)
Do = H^'MQ) = \v\veH2^(Q)>—T = 0 on E,
for 0 <; / g 2m - 1, v(*, 0) = 0, v(*, T) =0
Proo/. Indeed (see Chapter 4, Section 1), we have
\\P<p\\Li«»*c\\<p\\B3m.liQ} V<pe®(Q). D
4.2 The Boundary Operators and Spaces
For u e Ulf we set:
(4.8) flu = on E, m g / ^ 2m - l]u{x,T)\-
[ dvJ J
And, we let
!2m-l \
JJ H2m-J-l/2.l2m-J-ll2)l2mp) x H%(Q)\.
Then condition (2.1) is satisfied.

204 Boundary Value Problems and Operator Extensions
For u e EP, we define (formally at first) n u by
\dJu I
(4.10) nu = -on27, 0 <; / ^ m - 1;«(*,0)|-
We know (Chapter 4, Section 10) that for ueEP, we have
eH-U+l/2),-U+l/2)/2m^)> 0 ^ / ^ m - 1,
(4.11)
dJu
dvj
u(x,0) eH-m{Q).
But we can state this more precisely (see Chapter 4, Section 12.3).
Define:
(4.12)
W1 = space described by Tj v, 0 ^ j ^ m — 1, and
v(x,0) as v describes H2m'1(Q), with
djv
r = 0 on 27, 0 < / < m — 1,
v(x,T) =0,
where 7^ corresponds to B* = in Green's formula (see Chapter 2,
dvJ
Section 2 and Chapter 4, Section 1).
Therefore
(4.13) W1 = {y>j,(p\y>jeW+1s2-u+1'2v2m(Z),<peH'Z(Q)}.
Then we can define
(4.14)
dJu
7tu= —-, u{x,0)\eW[
dv
= 11 ^-•/-1/2--(-/+1/2)/2m(27) x H~m{Q)
i = o
with the help of Green's formula and of a continuous linear right-inverse of
f dJv )
W1 -> \v \veH2m^(Q)f—- = Oon27,0 ^ / ^ m - 1, v(*, T) = 0 •
I or )
Now, we can take
(4.15) ^=^,
<zmZ condition (2.2) « satisfied.

5. Example 3. Evolution Operators of the Second Order in t 205
4.3 Consequences
Let u be given in EP. We define £ by
(4.16)
^l-
djC dJu
dvJ dvJ
, 0 ^ / ^ w — 1,
(not formally, but in the sense of Remark 12.3 of Chapter 4).
Then, if K e J?(0', W), the well-posed space U corresponding to K
is defined by
(4.17) nu = K0{u - C),
which decomposes into
(4.18)
dJu 2m-1 8l(u - t)
J = I gji a... - + *,(«(*• T) - t(x, T)),
dv
dvl
2m-1 flUu ___ £\
o^i^-i, «(*,o) = E ^i a , +
+ K(u(x,T) -f(*,71).
In this way, we obtain a// well-posed spaces. D
5. Example 3. Evolution Operators of the Second Order in t
5.1 Notation
We now consider the setting of Chapter 5. The cylinder Q is defined
as in Section 4 and
/ d \ d2
with the hypotheses of Chapter 5.
The interpretation of the abstract results are delicate, because of
the difficulties with the regularity and the trace theorems noted in
Chapter 5.

206 Boundary Value Problems and Operator Extensions
5.2 Formal Results
Formally, we take
(5.2)
U, =
dJu
u\ueHm>1(Q),PueL2(Q), - = 0, 0 ^ / ^ m - 1
dvJ
du
u{x,0) = 0, —(*,0) = 0 ,
, dJu du ]
(5.3) pu = \——onZ,m^ j ^ 2m - \,u{%, T), (*, T) ,
dt
(5.4) ttw = —— onl, 0 |||w- 1,«(*,0), (*,0)
3W
dt
The space 0 is the " trace'' space described by /? u as u describes U1,
and the space W is the "trace" space described by n u as u describes
EP = {u\ueL2(Q),PueL2{Q)}.
For w given in EP, the function f is an (appropriately interpreted;
see Chapter 5) solution of
d2C
(5.5)
^C +
a;2
0,
&Z dJu
r = -, 0^j^m—l, on 27,
f(*,0) = «(*,0),
— (*,0) = —(*,0).
The well-posed spaces are defined by a relation of the form (4.17), where
Remark 5.1. Similar (also formal) results can be developed for
Schroedinger-type operators (see Chapter 5.) D
6. Comments and Problems
"All" extensions leading to well-posed boundary value problems
have been specified, for second-order elliptic operator by Vishik [2].
For elliptic operators of order 2m, m > 1, the corresponding problem
has been solved by Peetre and Lions (see Peetre [1]).

6. Comments and Problems
207
For points of view different from this Appendix, see Bade-Free-
man [1], Beals [1], Birman [1], Browder [7], Calkin [1], Fishel [1],
Freeman [2], Friedrichs [4], M. G. Krein [1], Schechter [1]. We also
recall the non-local elliptic problems which come up in probability theory
and have been noted in the Comments to Chapter 2.
We have given a very general (and very simple) abstract presentation
which also applies to evolution operators; the results of Section 4 are
apparently new. The same is true for Section 5, but the spaces appearing
there should be specified] this should be possible at least for certain
particular cases, but we have not attempted to do so. Similarly, it
would be interesting to apply the preceding considerations to quasi-
elliptic operators.
For elliptic operators, a much more complete study is done in
Grubb [1]; it would be of interest to extend the results of this author
to the parabolic case.
Along the same lines, we call attention to the work of Phillips [2]
on maximal extensions of contraction operators, and to the work of
Carroll [1] which constitutes a kind of "abstract analogue" to Section 4.

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Additional Bibliography
Chapter 4
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1. 2., 3., already quoted in this additional bibliography, and
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Chapter 5
There has been a large amount of effort concerned with the solution of general
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240 Bibliography
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di Schroedinger. Boll. U.M.I. 4 1, 591—609 (1968).
2. Valutazioni in B% della soluzione del problema di Cauchy per I'equazione
delle onde. Boll. U.M.I. 4 1, 780-790 (1968).
For scattering theory cf.:
Kato, T.
• 1. Scattering theory and perturbation of continuous spectra. Proc. Int. Congress
of Math., Nice, 1970.
Phillips, R.
1. Scattering theory for the wave equation in the exterior of an obstacle.
Proc. Int. Congress Math., Nice, 1970.
For the theory of lacunas for hyperbolic operators cf.:
Atiyah, M. F., Bott, R., Garding, L.
1. Lacunas for hyperbolic differential operators with constant coefficients, I.
Acta Math. 124, 109-189 (1970).
Garding, L.
1. Lacunas for hyperbolic differential operators. Proc. Intern. Congress of
Math., Nice, 1970.
Finally we must quote the recent generalization of the theory of
pseudo-differential operators developed by Hormander (cf. also Maslov, Eskin, Egorov,
Nirenberg-Treves) which also contains hyperbolic operators and which seems to
be also useful for boundary value problems for hyperbolic operators:
Egorov, Yu. V.
1. On canonical transformations of pseudo-differential operators. Uspehi Mat.
Nauk 25, 235-236 (1969).
2. On non degenerate hypoelliptic pseudo-differential operators. Dokl. Akad.
Nauk 186, 1006-1007 (1969).
Eskin, G. I.
1. The Cauchy problem for hyperbolic systems in convolution. Mat. Sbornik
74 (116), 262-297 (1967).
2. Pseudo differential systems of equations with simple real characteristics.
Mat. Sbornik 77, 174-200 (1968).
Hormander, L.
1. On the singularities of solutions of partial differential equations. Conf. on
Funct. Anal, and related Topics, Tokyo 1969, 31—40.
2. The Calculus of Fourier Integral Operators. Acta Math. 127, 79—183 (1971).
Maslov, V. P.
1. Perturbation theory and asymptotic methods, Moscow 1965.

242
Bibliography
Nirenberg, L., Treves, F.
1. On local solvability of linear partial differential equations. Part I:
Necessary conditions, Part II: Sufficient conditions. Comm. pure appl. math. 23,
1-38 and 459—510 (1970).
Treves, F.
1. Hamiltonian fields, bicharacteristic strips in relation with existence and
regularity of partial differential equations, Proc. Intern. Congress of Math.,
Nice, 1970.
Chapter 6.
For the optimal control of systems governed by stochastic partial differential
equations, cf.:
Bensoussan, A.
Filtrage optimal des systemes lineaires, Paris: Dunod 1971.
and the bibliography of this book.
For systems with delays, let us mention:
Delfour, M. C, Mitter, S. K.
Hereditary differential systems with constant delays, I: General Case, Univ.
de Montreal, 1970, and the bibliography of this work.
For systems with several "controllers" and relations to game theory cf.
Lemaire, B.
Problemes min-max et applications au controle optimal de systemes gouvernes
par des equations aux derivees partielles, Thesis, Paris 1970.
cf. also the work of Bensoussan, Yvon and Lions, to appear, a short account of it
being given in a survey paper by Lions for the IFAC Symposium, Banff, June
1971.
For optimal control theory in Banach spaces cf.
Friedman, A.
1. Optimal control in Banach spaces. J. Math. Anal, and Appl. 19, 35 — 55
(1967). /
2. Optimal control in Banach spaces with fixed end-points. J. Math. Anal,
and Appl. 24, 161 — 181 (1968).
And for existence theorems cf.:
Fleming, W. H.
1. Optimal continuous-parameter stochastic control. S.I.A.M. Review 11,
470-509 (1969).
Zolezzi, T.
1. Teoremi di esistenza per problemi di controllo ottimo retti da equazioni
ellittiche o paraboliche. Rend. Sem. Mat. Univ. Padova 44, 155—175 (1970).
721/3/72 - III/18/203

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