Die Grundlehren der mathematischen Wissenschaften

in Einzeldarstellungen mit besonderer Berücksichtigung

der Anwendungsgebiete

Band 181

Herausgegeben von

J. 1. Doab . 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

Geschäftsführende Herausgeber

B. Eckmann und B. 1. van der Waerden

J. L. Lions . E. Magenes

Non-Homogeneous Boundary Value Problems

and Applications

Translated from the French py P.Kenneth

Volume I

t· . .

"

Springer-Verlag Berlin Heidelberg New York 1972

Primary

]. L. Lions E. Magenes University of Paris University of Pavia

Tille of the Freneh Original Edition: Problemes aux limites non homog~nes et applications (tarne Ij

Publisher: S. A. Dunod, Paris 1968

Translator:

P. Kenneth Paris

Geschäftsführende Herausgeber:

B.Eckmann Eidgenössische Technische Hochschule Zürich

B. L. van der Waerden Mathematisehes Institut der Universität Zürich

AMS Subject Classifications (1970)

35J20, 35J25, 35J30, 35J35, 35J40, 35K20, 35K35, 35L20

Secondary 46E35

ISBN -13 :978-3-642-65163-2 e-ISBN -13:978-3-642-65161-8

DOI: 10.1007/978-3-642-65161-8

This work is subject to copyright. All rights are rcserved, whether the whole or part of the material is concerned,specifically those of translation, reprinting, rc-tise 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. Softeover reprint of the hardeover 1st edition 1972

Library of Congrcss Catalog Card Number 71-151407

Preface

1. We describe, at first in a very formaI manner, our essential aim. Let m be an op en subset of Rn, with boundary am. In m and on am we introduce, respectively, linear differential operators

P and Qj' 0 ~ i ~ 'V.

By "non-homogeneous boundary value problem" we mean a problem of the following type: let f and gj' 0 ~ i ~ 'v, be given in function space s F and Gj , F being a space" on m" and the G/ s spaces" on am" ; we seek u in a function space u/t "on m" satisfying

(1) Pu = f in m, (2) Qju = gj on am, 0 ~ i ~ 'v«])).

Qj may be identically zero on part of am, so that the number of boundary conditions may depend on the part of am considered 2.

We take as "working hypothesis" that, for fEF and gjEGj ,

the problem (1), (2) admits a unique solution u E U/t, which depends continuously on the data3 .

But for alllinear probIems, there is a large number of choiees for the space s u/t and {F; Gj } (naturally linke d together).

Generally speaking, our aim is to determine families of spaces 'ft and {F; Gj }, associated in a "natural" way with problem (1), (2) and con­venient for applications, and also all possible choiees for u/t and {F; Gj }

in these families. Let us make this explicit by means of two examples, chosen as the

simplest possible ones, but which already demonstrate th" utility of non-homogeneous probIems.

«1» The Q/s will be called "boundary operators". Such problems are called non-homogeneous because if we consider the setting of unbounded operators, then Pu=t, uED(P) (=domain ofP) implies null boundary conditions; hence a certain difference bctwcen t and the boundary data gJ.

2 This will obviously be the case for most problems of evolutian. 3 At least in general; for elliptie problems uniqueness conditions will not be

satisficd, but in any case wc shall deal with operators with indiees and therefore stin have uniqueness on passing to the quotient by finite-dimensional subspaces. We shall verify that this "working hypothesis" is satisfied in each particular situatian.

VI Preface

2. Examples

2.1 For rP, we take an open subset D of Rft , with boundary r, and for P the operator fl., fl. = Laplacian; we take 11 = 0 and Qo = identity. Then, the problem corresponding to (1), (2) is the classical Dirichlet problem lor L1:

(3) fl.u = I in D,

(4) u = gu = g on r. We then ask in what spaces I and g may be chosen so that (3), (4) admits

a unique solution (in an appropriate sense). Classical answers are furnished by potential theory and the" Dirichlet

principle" : for example, we may choose I, g and certain of their derivatives in D and r respectively to be square integrabIe and obtain u, the solution of (3), (4), as well as certain of its derivatives to be square integrable in D.

Therefore, a natural famiIy for problem (3), (4) must be (if we Iimit ourselyes to "P theory", that is Hilbert theory) the Hilbert lamily 01 the Sobolev spaces H' (D) and H' (r), where H' (D) (resp. H' (r)) is, if s is an integer ~ 0, the space of u's such that u and its deriva­tives (in the sense of distributions) up to order s are square integrable in D (resp. r) (this definition is generalized to all real s by introducing the derivative of order s by Fourier transform; see Chapter 1).

Here is one of the resuIts we shall prove (see Chapter 2) :

Let s be any real, non-negative number; il I E H' (D) and g E HH 3/2 (r) , then there exists a unique u E Hs+2 (D) which is a solution 01 (3), (4) (having given an appropriate sense to (3) and (4) separately, by a natural generatization of the classical definitions). M ore precisely: the operator u -+ {fl. u, u Ir} is an isomorphism 01

Hs+2 (D) onto H' (D) x Hs+3/2 (r) .

It must be pointed out that the derivatives of non-integer order necessarily enter the problem if we want the optimal result for each s, since s + 2 and s + ~ cannot be integers simultaneously!

Furthermore, it is equally natural to study the case "negative s", since many problem s deal with (3), (4), where, for example, with 1= 0, g is very irregular: such as g square integrable on r (this is the case for optimal control theory), or g = the Dirac-mass at the point Xu E r (then the solution u yields the Poisson kernel of the problem) or more generally g = an arbitrary distribution on r.

It is still possible to solve problem (3), (4) when s is a negative real number, the space s tfI and G remaining of the same type (Le. tfI = H'+2 (D) , G = H'+3/2 (r), but with negative s) and the space F being an appropriate

Preface VII

subspace Ss (Q) of H' (Q) consisting of elements whioh do not grow too rapidly "in the neighborhood of r" (see Chapter 2, Seetions 6 and 7).

It follows that (3), (4) is solvable with g an arbitrary distribution on r, since then g necessarily belongs to a space HS (F), for an ap­propriate s.

In faet, in volume 3 of this book, we shall see that g may beIong to the space of analytic functionals on r (and this space is the most general for which, at least for f = 0, it is possible to give meaning to problem (3), (4)).

2.2 As a second example, we consider the heat operator

in

a P=--tl. at x

(9 = Q X JO, T [ e Rft+I;

the part of the boundary a (9 on which boundary conditions are given splits up into

il and 1: = r x JO, T[.

Then, a problem corresponding to (1), (2) is

(5)

(6) •

(7)

au - - !l.xu = f in (9, at u (x, 0) = Uo (x) m Q,

u = g on 1:.

One of our aims is to obtain the largest possible famiHes of spaces for f, Uo and g such that (5), (6), (7) admits a unique solution, in an appropriate sense. We shall see, in Chapter 4 of Volume 2, that a .. nat­ural" Hilbert family of space s <ft for the solution is the family H2s, s ((9) , with s any real number, where H2S.S((9) is, if s is a non-negative in­teger, the space of u's such that u and its derivatives up to order s in t and up to order 2s in x are square integrable in (9. The family H2S, s ((9) plays, for problem (5), (6), (7), an analogous role to the family HS (Q) for problem (3), (4). Also, remarks analogous to those we made for problem (3), (4) are valid for problem (5), (6), (7).

3. We now specify which are the principal systems {P, Qj} studied in this book.

3.1 We consider the case where P is an elliptie operator (denoted by A) and the Q/s are normal boundary operators (denoted BJ), where A and Bj verify suitable ellipticity conditions (see Chapter 2).

VIn Preface

3.2 We consider the case where

a P=-+ A, at

a parabolic operator with suitable boundary eonditions (Chapters 3 and 4).

3.3 We also consider the eases

a2

P=-+A at2

and

P = ~ + iA at ' where A is a self-adjoint elliptie operator, and still with suitable boundary conditions (Chapters 3 and 5).

4. For all these probIems, we proceed systematicaUy as lollows (exeept for possibly different techniques):

(i) we study the regularity of problem (1), (2), i.e.: assuming the data I and gJ to be regular (in a sense to be speeified), we study the corresponding regularity 01 u;

(ii) by transposition of (i) (for the "adjoint problem") we deduce therefrom (with a suitable technique and in partieular the obtainment of "trace theorems") the solution of problem (1), (2) for data belonging to spaces of distributions;

(iii) by interpolation between (i) and (ii), we obtain "intermediate" results.

Of course, the systematic setting-up of such a program is an enormous task and many possibilities had to be put aside (we have formulated them in lists of problem s in the last sections of each chapter).

In general, we consider for (i):

in volumes 1 and 2: data which are finitely often dillerentiable in the sense of P (spaces such as H· (Q), H2.,. (@), H· (r), ... )

in volume 3: analytic data or data belonging to suitable Gevrey elasses.

5. As we have seen, the present volume depends on regularity theorems in "dillerentiable in the sense 01 P" spaces (Sobolev spaces).

Therefore, the basic tools are:

- Sobolev spaces constructed on L2, - the theory of interpolation of corresponding spaces.

This is the subject of Chapter 1, where we study interpolation only lor the Hilbert cases; the introduction of interpolation between (non-

Preface IX

"hilbertizable") Banach space s and its applications to Sobolev space s constructed on LP, p =!= 2, would have complicated this work con­siderably.

Once in possession of these tools we need to prove regularity theorems (stage (i)) and then to implement stages (ii) and (iii).

This is done for the situations described in Section 3, aboveo Let us be more precise.

The elliptic case is the subject of Chapter 2. Stage (i) is studied completely by the method of J. Peetre [2J, under the hypotheses that A is properly eUiptic and that the E/s cover A in the sense of Lopatinskii­Shapiro and Agmon-Douglis-Nirenberg.

Stages (ii) and (iii) follow our previous papers on these subjects: see Lions-Magenes [lJ, [2J and [3J (where we also study the LP case, for I < p < oc, which we disregard here for p =!= 2).

"Variational" evolution operators and their applications are studied in Chapter 3.

Partial differential equations of evolution are studied in more detail in Chapters 4 and 5 of Volume 2 and applications to optimal control theory in Chapter 6 of Volume 2. Other applications will be given in Volume 3.

6. We have made an effort to make the book readable in "local" fashion; indications about the logical relations between the different subjects are given at the beginning of each chapter.

7. Each chapter ends with a section of comments and a section of probiems.

The comments give bibliographical indications, which, although numerous, by no means cover the subjeet. This is especially the case for research work cited in the comments but not studied in thi" book.

The rather large number of problems to which we call attention are very unequal in difficulty. For cases where results 01 type (i) are already available, the execution of stages (ii) and (iii) may offer great technical difficulties if one looks for optimal results, but is certainly much more accessible if one is satisfied with results in the "neigh­borhood" of the optimal results. Situations for which the results of type (i) are lacking (and we indicate a number of such problem s) may, of course, be much more difficult.

We wish to thank C. Baiocchi, M. S. Baouendi and G. Geymonat for reading various parts of the manuscript and giving us their com­ments.

ParisJPavia, luly 1967 J. L. LIONS E. MAGENES

Preface to the English Translatjon

The present translation follows the French edition without change, except for some corrections which were suggested to us by the remarks of M. S. Baouendi, G. Geymonat, C. Goulaouic and P. Schapira, to all of whom we express our sincerest thanks. We have added a complement­ary bibliography. We also wislr to thank P. Kenneth for his excellent work of translation.

Paris/Pavia, October 1971 J. L. LWNS E. MAGENES

Contents

Chapter 1

Hilbert Theory of Trace and Interpolation Spaces

1. Some Function Space s . . .

1.1 Sobolev Space s .... 1.2 The Case of the Entire Space 1.3 The Half-Space Case . 1.4 Orientation. . . . . .

2. Intcrmediate Derivatives Theorem

2.1 Intermediate Spaces 2.2 Density and Extension Theorems. 2.3 Intermediate Derivatives Theorem 2.4 A Simple Example . . . 2.5 Interpolation lnequaJity .

3. Trace Theorem .

3.1 Continuity Properties of the Elements of W (a, b) 3.2 Traec Theorem. .. . .....

4. Traee Spaees and Non-Intcger Order Derivatives.

4.1 4.2

Orientation. Definitions . "Intermediate Derivatives" and Trace Theorems

5. Intcrpolation Theorem. . . . .

5.1 Main Theorem ..... . 5.2 Interpolation of a Family of Operators .

6. Reiteration Properties and Duality of the Spaees [X, Y]O

6.1 Reiteration 6.2 Duality . . . .

7. The Spaees HS(Rn ) and HS(r)

7.1 HS (Rn)-Spaces . . . . . 7.2 Traees on the Boundary of a Half-Space 7.3 HS(F)-Spaees ....

8. Trace Theorem in H m (D)

8.1 Extension and Density Theorems. 8.2 Traee Theorem. . . . . .

9. The Spaees HS (.Q), Real s ;;;; 0 .

9.1 Definition by Interpolation 9.2 Trace Theorem in H' (D)

1

1 4 6 8

9

9 10 14 18 19

19 19 21

23

23 24

27 27 27

28

28 29

30

30 33 34

38 38 39

40 40 41

XII Contents

9.3 Interpolation of HS (.Q)-Spaees . . 9.4 Regularity Properties of HS (.Q)-Funetions .

10. Some Further Properties of the Spaees [X, Y]o

10.1 Domains of Semi-Groups . 10.2 Applieation to HS (R") . . . . . 10.3 Applieation to HS (0, oo) . . . .

11. Subspaees of HS (!J). The Spaees H~ (!J) .

11.1 H~ (.Q)-Spaees . . . . . . . . . 11.2 A Property of HS(.Q), 0;;:;; s <! 11.3 The Extension by 0 outside .0 . 11.4 Charaeterization of H~ (.Q)-Spaees 11.5 Interpolation of H~ (.Q)-Spaees

12. The Space s H-S (.0), s > o. . . . . . 12.1 Definition. First Properties. . . 12.2 Interpolation between the Spaees H-s (.0), s > 0 12.3 Interpolation between H~' (r) and H-s, (.0), sl > 0 12.4 Interpolation between HS' (.0) and H-s, (.0), sl > 0 12.5 Interpolation between HS'(.Q) and (HSI(.Q))' 12.6 Interpolation between H~'(.Q) and (HSI(.Q))' 12.7 A Lemma ............... . 12.8 Differential Operators on HS (.0). . . . . . 12.9 Invarianee by Diffeomorphism of H' (.Q)-Spaees .

13. Interseetion Interpolation . . . .

13.1 A General Result . . . . . 13.2 Example of Applieation (I) . 13.3 Example of Applieation (II) 13.4 Interpolation of Quotient Space s

14. Holomorphic Interpolation . . .

14.1 General Result ..... 14.2 Interpolation of Spaees of Continuous Funetions with

Range ................... . 14.3 AResult Pertaining to Interpolation of Subspaees

15. Another Jntrinsic Definition of the Space s [X, Y] •.

16. Compactness Properties

17. Comments

18. ProbIems.

Chapter 2

Elliptic Operators. Hilbert Theory

43 45

47

47 51 54

54

54 57 60 62 64

70

70 71 72 73 ,76 77 79 85 85

86

86 87 ~7 90

91

91 Hilbert

94 96

98

99

103

106

1. Elliptic Operators and Regular Boundary Value Problems 109

1.1 Elliptic Operators. . . . . . . . . . . . . . . . . 109 1.2 Properly and Strongly Elliptic Operators . . . . . . 110 1.3 Regularity Hypotheses on the Open Set .0 and the Coefficients of

the Operator A 111 1.4 The Boundary Operators . . . . . . . . . . . . . . . . . . 112

Contents XIII

2. Green's Formula and Adjoint Boundary Value Problem s . . . .. 114

2.1 The Adjoint of A in the Sense of Distributions or Formai Adjoint 114 2.2 The Theorem on Green's Formula 114 2.3 Pro of of the Theorem. . . . . . . . . . . . . . . . .. 115 2.4 A Variant of Green's Formula. . . . . . . . . . . . .. 120 2.5 FormaI Adjoint Prubkms with Respcet to Green's Formula 121

3. The Regularity of Solutions of Elliptic Equations in the lnterior of D 121

121 123

4.

Two Lcmmas . . . . . A priori Estimates in R"

3.1 3.2 3.3 The Regularity in the lnterior of

Elliptic Operators. . . . . . . D and the Hypoellipticity 01

A priori Estimates in the Half-Space .

4.1 A new Formulation of the Covering Condition 4.2 A Lemma on Ordinary Differential Equations. 4.3 First Application: Pruof of Theorem 2.2 4.4 A priori Estimates in the Half-Space for the Case

Coefficients. . . . . . . . . . . . . . . . . . . of Constant

125

127

127 130 133

136 4.5 A priori Estimates in the Half-Space for the Case of Variablc

Coefficients. . . . . . . . . . . . . . . . . . . . . . . . . 142

5. A priori Estimates in the Op en Set D and the Existcnce of Solutions in HS (D)-Spaces, with Real s ;;;; 2m . . . . . . . . . . . . . . .. 148

5.1 A priori Estimates in the Open Set il ............ 148 5.2 Existence of Solutions in HS (D)-Spaees, with Integer s ;;;; 2m. . 152 5.3 Precise Statcment of the Compatibility Conditions for Existence 155 5.4 Existence of Solutions in HS(D)-Spaces, with Real s;;;; 2m . .. 165

6. Application of Transpm;ition: Existcnce of Solutions in HS (D)-Spaces, with Real s ~ O. . . . . . . . . . . . . . 166

6.1 The Transposition Method; Generalities. 166 6.2 Choice of the Form L. . . . 167 6.3 The Spaees E (D) and DA (D) . . . . . 170 6.4 Density Theorem. . . . . . . . . . . 173 6.5 Trace Theorem, and Green's Formula for the Space DA (D), s ~ 0 175 6.6 Existence of Solutions in DA (D)-Spaces, with Real s ~ O. . . . 177

7. Application of Interpolation: Existencc of Solutions in HS (D)-Spaces, with Real s, 0 < s < 2 m. . . . . . . . . . 180

7.1 New Properties of ES(D)-Spaces . . 180 7.2 Use of Interpolation; First Results 185 7.3 The Final Results 187

8. Complements and Generalizations . . . . . . . . . . . . 191

8.1 Continuity of 'Iraees on Surfaees Neighbouring r . . 191 8.2 A Generalization; Application to Dirichlet's Problem. 194 8.3 Remarks on the Hypotheses on A and B J 195 8.4 The Realization of A in L2 (D) 196 8.5 Some Remarks on the Index of fl!. . 198 8.6 Uniqueness and Surjectivity Theorems 199

XIV Contents

9. Variational Theory of Boundary Value Probiems.

9.1 Variational Problems 9.2 The Problem. . . . . . . . . . . . . . . 9.3 A Counter-Example. . . . . . . . . . . . 8.4 Variational Formulation and Green's Formula . 9.5 "Concrete" Variational Problems 9.6 Coereive Forms and Problems 9.7 Regularity of Solutions 9.8 Generalizations (I) 9.9 Generalizations (II)

10. Coniments

11. ProbIems.

Chapter 3

VariatioDal EvolutiOD EquatiODS

1. An Isomorphism Theorem .

1.1 Notation ...... . 1.2 Isomorphism Theorem. 1.3 The Adjoint A* .. 1.4 Proof of Theorem 1.1

2. Transposition. . . . . .

2.1 Generalities. . . . . 2.2 Adjoint Isomorphism Theorem . 2.3 Transposition

3. Interpolation . . .

3.1 General Application . 3.2 Charaeterization of Interpolation Space s 3.3 The Case "0 = t". . . .. . ....

200

200 203 203 204 207 209 212 212 214

216

225

227

227 230 230 230

231

231 232 232

233

233 233 234

4. Example: Abstract Parabolic Equations, Initial Condition Problem (1) 234

4.1 Notation. . . . 234 4.2 The Operator M . . . . . . . . . . . . 235 4.3 The Operator A . . . . . . . . . . . . 237 4.4 Applieation of the I8omorphism Theorems 238 4.5 Choice of L in (4.20) . . . . 239 4.6 Interpretation of the Problem . . . . . . 241 4.7 Examples . . . . . . . . . . . . . . . 243

5. Examplc: Abstract Parabolie Equation8, lnitial Condition Problem (Il) 255 5.1 Some Interpolation Results . . . . . . . . . . . . 5.2 Interpretation of the Spaces c;tJ l /2, c;tJ}.'2 • ••••••

6. Example: Abstrad Parabolie Equations, Periodie Solutions

6.1 Notation. The Operator A . ...... . 6.2 Application of the 18omorphism Theorems 6.3 Choice of L . . . . . . . . . . . . . 6.4 lnterpretation of the Problem . . . . . 6.5 The Isomorphism of cp l /2 onto its Dual

255 257

258

258 259 259 260 261

Contents XV

7. Elliptic Regularization. . 261

7.1 The Elliptic Problem 261 7.2 Passage to the Limit 262

8. Equations of the Second Order in t. 265

8.1 Notation. . . . . . . . . . . 265 8.2 Existence and Uniqueness Theorem. 265 8.3 Remarks on the Application of the General Theory of Seetion 1 270 8.4 AdditionaI Regularity Results . . . . . . . . . . . . . 275 8.5 Parabolie Regularization; Direet Method and Application 280

9. Equations of the Second Order in t; Transposition. 283

9.1 Adjoint Isomorphism 283 9.2 Transposition. . 283 9.3 Choice of L . . . . 284 9.4 Traee Theorem. . . 285 9.5 Variant; Direet Method 287 9.6 Examples . . . . . . 292

10. Schroedinger Type Equations. 299

10.1 Notation . . . . . . . 299 10.2 Existence and Uniqueness Theorem 299

11. Sehroedinger Type Equations; Transposition . 302

11.1 Adjoint Isomorphism. 302 11.2 Transposition of (11.5) 303 11.3 Choice of L . 303

12. Comments

13. Problenis.

Bibliography. .

304

307

309

Contents of Volume II

Chapter 4 Parabolic Evolution Operators. Hilbert Theory

Chapter 5 Hyperbolic Evolution Operators, of Petrowski and of Schroedinger. Hilbert Theory

Chapter 6 Applications to Optirnal Control Problerns

Appendix Boundary Value Problerns and Operator Extensions

Contents of Volume III

Chapter 7 Scalar and Vector Ultra-Distributions

Chapter 8 Elliptic Boundary Value Problerns in Spaces of Distributions and Ultra-Distributions .

Chapter 9 Evolution Equations in Spaces of Distributions and Ultra-Distri­butions

Chapter 10 Parabolic Boundary Value Problernsin Space s of Ultra-Distributions

Chapter 11 Evolution Equations of the Second Order in t and Schroedingcr Type Equations

Appendix Calculus of Variations in Gevrey Type Spaces