Asymptotic Analysis for Periodic Structures

Asymptotic Analysis for Periodic Structures STUDIES IN MATHEMATICS AND ITS APPLICATIONS VOLUME 5 Editors: J. L. LION...

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Asymptotic Analysis for Periodic Structures

STUDIES IN MATHEMATICS AND ITS APPLICATIONS VOLUME 5

Editors: J. L. LIONS, Paris G. PAPANICOLAOU, New York R. T . ROCKAFELLAR, Seattle

NORTH-HOLLAND PUBLISHING COMPANY-AMSTERDAM

NEW YORK ’ OXFORD

ASYMPTOTIC ANALYSIS FOR PERIODIC STRUCTURES

ALAIN BENSOUSSAN Universite' de Paris IX and IRIA

JACQUES-LOUIS LIONS CollPge de France and IRIA

GEORGE PAPANICOLAOU Courant Institute of Mathematical Sciences New York University

1978 NORTH-HOLLAND PUBLISHING COMPANY-AMSTERDAM . NEW YORK . OXFORD

0 North-Holland Publishing Company, 1978 AN rights resewed. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior permission of the copyright owner.

ISBN 0 444 851 72 0

Pu blishers:

NORTH-HOLLAND PUBLISHING COMPANY AMSTERDAM *NEW YORK- OXFORD Sole distributors for the U.S.A. and Canada: ELSEVIER NORTH-HOLLAND, INC. 5 2 VANDERBILT AVENUE, NEW YORK, N.Y. 10017

L i b r a r y of Congress Cataloging i n P u b l i r a t i o n D a t a

Bensoussan, Alain. Asymptotic analysis f o r periodic structures. (Studies in mathematics and i t s applications ; v. 5) Bibliography: p. 1. Boundary value problems-Numerical solutions. 2. Differential equations, Partial-Numerical solutions 3. Asymptotic expansions. 4. F’robabilities. I. Lions, Jacques Louis, joint author. 11. Papanicolaou, George, joint author. 111. l”.tle. IV. Series. w79.B45 515’ -35 78-5 5 98 ISBN 0-44.4-@5172-0

PRINTED IN THE NETHERLANDS

Introduction.

1.

In Mechanics, Physics, Chemistry and Engineering, in the

study of composite materials, macroscopic properties of crystalline or polymer structures, nuclear reactor design, etc., one is led to the study of boundary value problems in media with periodic structure. If the period of the structure is small compared to the size of the region in which the system is to be studied, then an asymptotic analysis is called for:

to obtain an asymptotic expansion of the

solution in terms of a small parameter

E

which is the ratio of the

period of the structure to a typical length in the region.

In other

words, to obtain by systematic expansion procedures the passage from a microscopic description to a macroscopic description of the behavior of the system. 2.

In mathematical terms, the above'problems can be formulated,

typically, as follows. A',

A family of partial differential operators

depending on the small parameter E, is given.

The partial

differential operators may be time independent or time debendent, steady or of evolution type, linear or nonlinear, etc.

These

operators have coefficients which are periodic (or sometimes almost periodic) functions in all or in some variables with periods proportional to

E.

Since E is assumed to be small, we have a family

of operators with rapidly oscillationg coefficients. In a domain R , we have a boundary value problem

,

(1)

A

~ =U f ~in

(2)

u

subject to appropriate boundary conditions V

,

vi

INTRODUCTION

which we assume is well set when

E

> 0 is fixed.

The problem is now

to obtain, if possible, an expansiont of uE u

(3)

E

= uo

+

€U1

+

...

,

which would be asymptotic in general, or at least obtain the first term of this expansion along with a convergence theorem as 3.

E

* 0.

The type of results that one obtains in many cases is that

with a suitable definition of convergence (necessarily of a

weak type

as we shall see i.e., convergence of suitable averages), uE converges as

E:

* 0 to uo where uo is the solution of Quo = f in SI

(4)

,

uo subject to appropriate boundary conditions

(5) (4)

.

tt

a is, in generalIttt a partial differential operator

with simpletttt coefficients; it is called the homogenized operator of the family AE because, in a well defined sense, we approximate uE by uo which satisfies an equation with simple coefficients. coefficients of

The

a are called, by definition, the effective

coefficients or effective parameters that describe the macroscopic properties of the underlying medium.

tThe form of the expansion may be more complicated than ( 3 ) . ttWhich, of course, depend on the conditions imposed on uE. tttWe shall see examples where the AE are partial differential operators but a is an integrodifferential operator. may depend on

E,

so we write sometimes a'.

ttttIn many cases they are constants.

Also a itself

INTRODUCTION

vi i

The most important aspect of the passage from (l), ( 2 ) to ( 4 ) ,

(5) is the explicit analytical construction of

a (i.e., its

coefficients) and not merely the assertion that such an operator exists.

Throughout this book we give analytical formulas for the

construction of the coefficients of

a.

This construction requires,

typically, the solution of a boundary value problem within a single period cell.

We call this the cell problem.

Thus, the solution of (11, ( 2 ) when

E

is small is replaced by

the solution of a cell problem and then of ( 4 ) , surprising that both the cell problem and ( 4 ) ,

(5).

It is hardly

(5) must be treated

numerically since, except in trivial cases, exact solutions in closed form are not available.

However, whereas the direct numerical

solution of (l), ( 2 ) , when

E

is small, is an ill-conditioned and

complicated computational problem, the solution of the cell problem plus ( 4 ) ,

4.

( 5 ) is, usually, a standard problem in numerical analysis.

In order to obtain U from A€, in this book we use four

methods which we now describe.

The distinctions between these

methods are not, however, sharp or clear-cut and the description that follows should be considered as a rough one. The first method is based on the construction of asymptotic expansions using multiple scales.

The use of multiple scales is well

known in many specialized contexts (but may not be cleatly articulated) as well as in modern perturbation theory.

For many

problems in ordinary differential equations, multiple scale methods give the same results as the well known method of averaging.t

In the

present context there are at least two natural spatial length scales.

'N.

N. Bogoliubov and Yu. Mitropolsky, Asymptotic methods in non-

linear mechanics, Gordon and Breach, New York, 1961.

vili

INTRODUCTION

One measuring variations within one period cell (the fast scale) and the one measuring variations within the region of interest (the slow scale). The use of multiple space scales (along with multiple time scales) to treat systematically boundary value problems with rapidly varying periodic structure was introduced and exploited by us. effectiveness in this context was anticipated by J. B. Keller.t

Its The

method was also used by E. Larsentt in transport theory (we do not treat transport problems here). The second method is based on energy estimates.

Since the

coefficients of A E are rapidly oscillating, derivatives of the coefficients are multiplied by powers of E-’. to obtain estimates independent of

E.

This makes it difficult

One must then pass to the

limit in a weak sense and for this one uses integration by parts and suitable test functions.

The prototype of this argument is due to

L. Tartar. Very often we use the two methods together, especially in Chapters 1 and 2. operator c7

The multiple scales method is used to obtain the

and expansions under liberal regularity conditions on the

data and the coefficients.

It is also used to construct special test

functions in order to pass to the limit by the energy methods yielding convergence results (without expansions) under minimal regularity hypotheses.

tPrivate communication; see also, SIAM J. Appl. Math.,

S.

Kogelman and J. B. Keller,

(1973), pp. 352-361, for a general expansion

procedure using multiple space scales. ttJ. Math. Phys.,

16

(1975), pp. 1421-1427.

INTRODUCTION

ix

The third method is based on probabilistic arguments and works whenever

the problem admits a probabilistic formulation or has a

probabilistic origin.

Again, one can construct expansions using

multiple scales when a lot of regularity is assumed.

With the

probabilistically natural notions of weak solution and weak convergence one can also obtain convergence results, without expansions, using test functions constructed by multiple scales under minimal regularity conditions. The fourth method is based on the spectral decomposition of operators with periodic coefficients, the so-called expansion in Bloch waves.

This method is not intended as an alternative to the

above methods since its applicability

s

more restricted. It is,

however, indispensable in the study of high frequency wave propagation in rapidly varying periodic media

By high frequency we mean

here that another length scale becomes relevant in the problem, namely the typical wavelength, which is now assumed to be small and comparable to the period of the structure of the medium. 5.

The material in this book is organized in three units that

have been written so that they could be read independently by readers with more specialized interests. Chapters 1 and 2 form the first unit which deals with elliptic, parabolic and hyperbolic (but not high frequency) problkms with emphasison the energy methods (methods 1 and 2 mostly). 2 and 3

Sections 1,

of Chapter 1 are basic to the whole book, however, and should

be at least looked at by readers more interested in Chapters 3 or 4 . Chapter 3 is the second unit which treats problems probabilistically i.e., by method 3 .

A separate introduction to the contents

is provided at the beginning of this chapter. Chapter 4 is the third unit which deals with high frequency problems, i.e., method 4 mostly.

Here again a separate introduction

to the contents is provided at the beginning of the chapter.

INTRODUCTION

X

In Chapter 1 we consider, among other things, elliptic systems, operators with coefficients that have multiple periodic structure (with coefficients that depend on several variables and are periodic 2 N with periods E , E ,..., E , in each variable, respectively), some nonlinear operators and variational inequalities.

Chapter 2 follows

essentially the lines of Chapter 1, first for parabolic equations and next for hyperbolic operators. operators with period of order E~

in the time variable.

In the parabolic case we study E

in the space variables and of order

For second order parabolic operators one

has essentially the 3 cases k < 2, k = 2, k > 2, at least as far as the first term in the expansion ( 3 ) is concerned.

In both Chapters

1 and 2 we give examples of partial differential operators for which the corresponding homogenized operator

a is an integrodifferential

operator. 6.

A number of questions which can be analyzed by methods

similar to the ones in this book are not studied here.

In particular,

for the analysis of transport problems we refer to a paper by ust and the references cited therein.

For problems that deal with periodic

distribution of holes, we refer to D. Cioranescutt and to our own papers. ttt A systematic treatment of the numerical and computational aspects

of the problems considered here would go beyond the scope of this book.

Some references are cited in the bibliography of Chapter 1.

'J.

Publ. RIMS, Kyoto Univ., Japan, 1978.

ttThesis, Paris, October 1977.

tt t ~ o appear.

xi

INTRODUCTION 7.

history.

The problems considered in this book have a rather long The first attempt to construct "effective parameters" for

complicated media seems to go back to Poisson.

We refer to

I. Babugkat where a brief account of the historical development is given (along with several references).

In Babuska's paper one sees

clearly that the notion of "effective parameters" or "effective coefficients" depends very much on how one chooses to model a physical problem. This means that a given physical problem may be modelled by imbedding it into a family of problems (parametrized by form (l), (2) in many different ways.

E)

of the

For example, modelling a

problem by multiple periodic structure (cf. Chapter 1, Section 8 time variations proportional to ck with k < 2, k = 2 or k > 2, static or high frequency excitation, etc., constitutes a specific choice.

The homogenized problems, and hence the effective parameters,

are different in each case. The modelling question is not, in this form,tt a mathematical one and it is important to keep in mind that the definition of effective parameters is a relative one.

The formulas change by

changing the scaling of a problem, i.e., by adopting another family of problems (l), ( 2 ) in which to imbed a given physical problem. t

'I.

Babugka, Technical Note BN-821, July 1975, Institute of Fluid

Dynamics and Appl. Math., Univ. of Maryland, College Park, Maryland 20742.

"The

inverse problem:

given a particular homogenization

algorithm, find a family of problems (scaling) in some class, whose asymptotic limit is the given homogenized problem, is frequently interesting, difficult and need not have a "solution" in general.

xii

INTRODUCTION For interesting partly heuristic argument leading to effective

t

parameters, we refer to E. Sanchez Palencia.

There are connections between the asymptotic problems considered here and the very general viewpoint of E. de Giorgi which is called and G convergence. S. Spagnolo,

r

We refer to the work of E. de Giorgi and

S. Spagnolo,tt C. Sbordone,ttt

cited in a recent papertttt

and to the references

of de Giorgi.

Some questions studied in Chapters 1 and 2 have also been studied by I. Babuska and N. S. Bakbalov (cf. the Bibliography at the end of Chapters 1 and 2 ) . A particular case of the probabilistic problems analyzed in Chapter 3 has been studied previously by M. I. Freidlin (cf. the bibliography of Chapter 3).

The method of multiple scales that is

followed frequently here was not used by Freidlin.

His motivation

seems to have been the generalization of the averaging method to stochastic equations with spatially rapidly oscillating periodic coefficients. Some of the material of Chapters 1, 2 and 3 has been announced in notes at the C.R.A.S.,

Paris, and has been presented in various

lectures by the authors, since 1975.

tint.

J. Engng. Sci., 1_2 (1974), pp. 331-351.

tt~oll.U.M.I., 8 (1973), pp. 391-411, and in Numerical Methods

of Partial Differential Equations, 111, B. Hubbard, editor, Academic Press, New York, 1976, pp. 469-498. tttAnnali Scuola Norm. Sup. Pisa, IV (19751, p. 617-638. ttttBoll. U.M.I.

,2

(1977).

INTRODUCTION

xiii

Some results in Chapter 4 are well known in solid-state physics (for example, the notion of effective mass).

We give a systematic

t

treatment here using the W.K.B. or geometrical optics methods combined with multiple scale methods.

A

preliminary version of this

chapter circulated among colleagues as a preprint dilring 1977. 8.

A large number of questions remain open.

Some of them are

indicated in the text and follow the present lines of development. Of particular importance is the analysis of the behavior of solutions near boundaries and, possibly, any associated boundary layers. Relatively little seems to be known"

about this problem.

We wish to thank I. Babuska, J.B. Keller, F. Murat, E. Sanchez Palencia, and L. Tartar for several interesting discussions

and useful advice while this work was being carried out. We also would like to thank Dr. Breton, from Soci6t6 Nationale des Industries Abrospatiales (SNIAS), for showing to us several very interesting examples arising in Industry.

tA brief, self-contained introduction to these methods, along with references to the voluminous literature, is given in Chapter 4 , Section 2.

W.K.B. stands for Wentzel, Kramers and Brillouin who used I

these methods in the 1920's for the solution of some quantum mechanical problems.

The methods were widely used much earlier by

Liouville, Rayleigh and others but the terminology WKB persists.

tt

A. Bensoussan, J.L. Lions, G. Papanicolaou, Boundary Layer

Analysis of the Dirichlet problem for elliptic equations with rapidly varying coefficients, Proceedings of the Kyoto Conference on Stochastic differential equations, Kyoto July 1976, to be published.

This Page Intentionally Left Blank

TABLE OF CONTENTS

.................................................... Contents ..............................................

Introduction Table of

Chapter 1

:

............................................. 1. Setting of the "model" problem .......................... 1.1 Setting of the problem (I) .............................. 1.2 Setting of the problem (1I):boundary conditions ......... 1.3 An example: a one-dimensional problem ..................

.

2.1 2.2 2.3 2.4

2.5 2.6

.

2 4

8

Justification of the asymptotic expansion for Dirichlet's boundary conditions

19

................... Higher order terms in the expansion ..................... Extensions ..............................................

3.4

Comparison results

3.2

2

16

3.3

3.1

1

................................... Orientation ............................................. Asymptotic expansions using multiple scales ............. Remarks on the homogenized operator ..................... Asymptotic expansions

.............. Orientation: statement of the main result .............. Proof of the convergence theorem ........................ A remark on the use of the "adjoint expansion" ..........

3

xv

Elliptic Operators

Orientation

2

v

Energy proof of the homogenization formula

...................................... xv

11 11 12

21 22

23 23 24 28

31

xvi

LIST OF CONTENTS

4.4

............................................ Estimates for the Dirichlet problem ..................... Reduction of the equation ............................... Proof of Theorem 4.2 .................................... Local estimates .........................................

4.5

Extensions

5.

Correctors

4. 4.1 4.2 4.3

5.1 5.2 5.3 5.4 5.5 6.

LP estimates

..............................................

.............................................. Orientation ............................................. Structure of the first corrector: Statement of theorem

7.2 7.3

44

46 49 49

71

Complements on boundary conditions

7.1

40

Second order elliptic operators with non-uniformly oscillating coefficients

7.

6.3

38

65

6.4

6.2

35

.................................. Proof of Theorem 5.1 .................................... First order system and asymptotic expansion ............. Correctors: Error estimates for Dirichlet's problem ....

.............................. Setting of the problem and general formulas ............. Homogenization of transmission problems ................. Proof of Theorem 6.1 .................................... Another approach to Theorem 6.1 .........................

6.1

35

...................... A remark on the nonhomogeneous Neumann's problem ........ Higher order boundary conditions ........................ Proof of ( 7 . 6 ) , (7.7) ...................................

49 53 59

71 77 82 84 87 87 89 94

LIST OF CONTENTS

.

xvii

.............................. the problem: Statement of main result ........

Reiterated homogenization

96

8.1

Setting of

96

8.2

Approximation by smooth coefficients

103

8.3

Asymptotic expansion

108

8.4

Proof of the reiteration formula for smooth coefficients ..............................

112

.............................................

116

.....................

117

.................................

117

8

8.5 9

.

Correctors

................... ...................................

Homogenization of elliptic systems

9.1

Setting of the problem

9.2

Statement of the homogenization procedure

120

9.3

Proof of the homogenization theorem

123

9.4

Correctors

.

.............. ....................

............................................. .................. ............................................

126

Homogenization of the Stokes equation

129

10.1

Orientation

129

10.2

Statement of the problem and the homogenization theorem

10.3

Proof of the homogenization theorem

132

10.4

Asymptotic expansion

136

10

.

.......................

.................... ...................................

11.4

.......... Setting of the problem ................................. Asymptotic expansions .................................. Another asymptotic expansion ........................... Compensated compactness ................................

11.5

Homogenization theorem

11

11.1 11.2 11.3

11.6 11.7

Homogenization of equations of Maxwell's type

I

................................. Zero order term ........................................ Remark on a regularization method ......................

129

138 138 140 144 147 150 155 157

LIST OF CONTENTS

xviii 12

.

Homogenization with rapidly oscillating potentials

.....

............................................

12.1

Orientation

12.2

Asymptotic expansion

12.3

.......... Correctors ............................................. Almost periodic potentials ............................. Neumann's problem ...................................... Higher order equations ................................. Oscillating potential and oscillatory coefficients ..... A phenomenon of uncoupling .............................

12.4

12.5 12.6 12.7

12.8 12.9

13

.

...................................

Estimates for the spectrum and homogenization

............................. ............................................

158 158 159 161 167 168 170

173 176 179

Study of lower order terms

181

13.1

Orientation

181

13.2

Asymptotic expansion

183

13.3

Energy estimates

185

.

14

14.1 14.2 14.3

15

.

15.1 15.2

15.3

.

................................... .......................................

.............. Orientation ............................................ Asymptotic expansion ................................... 2 Homogenization with respect to A ...................... Singular perturbations and homogenization

....................................... Setting of the problem ................................. Non-local homogenized operator ......................... Homogenization theorem .................................

Non-local limits

16.3

.................... Formal general formulas ................................ Compact perturbations .................................. Non-compact perturbations ..............................

16.4

Non-linearities in the higher derivatives

16

16.1

16.2

Introduction to non-linear problems

..............

188 188 189 191 194 194 196 199 200 200

202 20 3 206

LIST OF CONTENTS

.

xix

............... ............................................

Homogenization of multi-valued operators

207

17.1

Orientation

207

17.2

A formal procedure for the homogenization of problems of the calculus of variations

17

............ 17.3 Unilateral variational inequalities .................... 18. Comments and problems .................................. Bibliography Chapter 2

:

...........................................

1.3

............................................ Parabolic operators: Asymptotic expansions ............ Notations and orientation .............................. Variational formulation ................................ Asymptotic expansions: Preliminary formulas ...........

1.4

Asymptotic expansions: The case k

1.5

Asymptotic expansions:

1.6

Asymptotic expansions: The case k

1.7

Other form of

1.8

The role of k

.

1.1

1.2

2

.

2.1 2.2 2.3 2.4 2.5 2.6

214 218 228

Evolution Operators

Orientation

1

209

=

1

The case k = 2

................. .................

................. homogenization formulas .................. .......................................... = 3

Convergence of the homogenization of parabolic equations

............................... Statement of the homogenization result ................. Proof of the homogenization when k = 2 ................. Reduction to the smooth case ........................... Proof of the homogenization when 0 < k < 2 ............. Proof of the homogenization when k > 2 ................. Proof of the homogenization formulas E Lp(Rn x RT) using Lp estimates when a ij Y

2.1

............ The Lp estimates .......................................

2.8

The adjoint expansion

..................................

233 234 234 235 243 245 246 248 250 252

253 253 253 257 259 266

269 271 276

xx

LIST OF CONTENTS 2.9

2.10 2.11 2.12 2.13

.

........................... Higher order equations and systems ..................... Correctors ............................................. Non-linear problems .................................... Remarks on averaging ................................... Use of the maximum principle

277 278 282 287 293

Evolution operators of hyperbolic. Petrowsky or Schrodinger type

299

3.1

Orientation

299

3.2

Linear operators with coefficients which are regular in t

300

3.3

Linear operators with coefficients which are irregular in t

304

3

3.4 3.5 3.6 3.7 3.8 3.9

4

.

.................................. ............................................ ...............................

............................. Asymptotic expansions (I) .............................. Asymptotic expansions (11) ............................. Remarks on correctors .................................. Remarks on nonlinear problems .......................... Remarks on Schr6dinger type equations .................. Nonlocal operators ..................................... Comments and problems .................................. Bibliography

Chapter 3:

.

308 312 314 317 318 325 342

Probabilistic Problems and Methods

Orientation 1

...........................................

306

..................................................

Stochastic differential equations and connections with partial differential equations

345

..

................................... ..........................................

348

1.1

Stochastic integrals

348

1.2

It8's formula

350

1.3

Strong formulation of stochastic differential equations

1.4

............................................ Connections with partial differential equations ........

351 353

LIST OF CONTENTS

xxi

2.

Martingale formulation of stochastic differential equations

2.1

Martingale problem

2.2

Weak formulation of stochastic differential equations

2.3

Connections with P. D. E.

3.

...............................

357

.....................................

357

..

359

..............................

361

Some results from ergodic theory

364

3.1

General results

....................... ........................................

364

3.2

Ergodic properties of diffusions on the torus

3.3

Invariant measure and the Fredholm alternative

4.

Homogenization with a constant coefficients limit operator

..........

370

.........

377

....................................... Diffusion without drift ................................ Diffusion with unbounded drift .........................

384

4.3

Convergence of functionals and probabilistic proof of homogenization

405

5.

Analytic approach to the problem (4.76)

414

5.1

The method of

414

5.2

The method of

6.

Operators with locally periodic coefficients

429

6.1

Setting of the problem

429

6.2

Probabilistic approach

6.3

Remarks on the martingale approach and the adjoint expansion method

4.1 4.2

................ asymptotic expansions .................... energy ...................................

........... ................................. .................................

6.4

Analytic approach to problem 6.5

7.

Reiterated homogenization

7.1 7.2

............

383

399

419

431

.....................

442

.......................

444

.............................. Setting of the problem ................................. Proof of Theorem 7.1 ...................................

455 455 463

xxii 8

.

8.1 8.2 9

.

LIST OF CONTENTS

............................... A variant of Theorem 6.3 ............................... A general problem with potentials ...................... Problems with potentials

Homogenization of reflected diffusion processes

........

................................. ...................................

467 467 470 475

9.1

Setting of the problem

475

9.2

Proof of convergence

478

9.3

Applications to partial differential equations

10

.

10.1 10.2 10.3 10.4 10.5 10.6

..................................... Notation and setting of problems ....................... Fredholm alternative for an evolution operator ......... Case k 2 ............................................. Case k = 2 ............................................. Case k > 2 ............................................. Applications to parabolic equations .................... Evolution problems

.............................................. 11.1 Setting of the problem ................................. 11.2 Proof of Theorem 11.1 .................................. 11.3 Remarks on generalized averaging ....................... Comments and problems .................................. 12 . Bibliography ........................................... 11

.

.........

Averaginq

Chapter 4

:

.

1.1 1.2

488 488 489 496 505 509 515 516 516 518 525 529 534

High Frequency wave Propagation in Periodic Structures

............................................ Formulation of the problems ............................ High frequency wave propagation ........................ Propagation in periodic structures ..................... Orientation

1

483

.

537 538 538 545

LIST OF CONTENtTS 2

.

xxiii

. K . B . or geometrical optics method Expansion for the Klein-Gordon equation ................ Eiconal equation and rays .............................. The W

2.1 2.2

.................................... Connections with the static problem .................... Propagation of energy .................................. Spatially localized data ............................... Expansion for the fundamental solution ................. Expansion near smooth caustics ......................... Transport equations

2.3 2.4 2.5 2.6 2.1 2.8

......................................... Symmetric hyperbolic systems ........................... Impact problem

2.9

2.10

541 550 552 556 557 559 563 565 566 568

Expansions for symmetric hyperbolic systems (low frequency)

574

2.12

Expansion for symmetric hyperbolic systems (probabilistic)

581

2.13

Expansion for symmetric hyperbolic systems (high frequency)

587

2.14

W

2.11

2.15 3

.

3.1 3.2 3.3

3.4 3.5 3.6

......................................

......................................

.....................................

. K . B . for dissipative symmetric hyperbolic systems .. Operator form of the W . K . B ...........................

Spectral theory for differential operators with periodic coefficients

.............................

The shifted cell problem for a second order elliptic operator

.................................... The Bloch expansion theorem ............................ Bloch expansion for the acoustic equations ............. Bloch expansion for Maxwell's equations ................ The dynamo problem ..................................... Some nonselfadjoint problems ...........................

600 609

614 614 616 618 619 620 621

xxiv 4

.

4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9

4.10

4.11 4.12

5

.

5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 6

.

LIST OF CONTENTS

Simple applications of the spectral expansion

.......................................... Schrodinger's equation ................................. Nature of the expansion ................................ Connections with the static theory ..................... Validity of the expansion ..............................

640

Relation between the Hilbert and Chapman-Enskog expansion

645

Lattice waves

......................... Spatially localized data and stationary phase .......... Behavior of probability amplitudes ..................... The acoustic equations ................................. Dual homogenization formulas ........................... Maxwell's equations .................................... A one-dimensional example ..............................

627 629 632 637

646 649 650 654 661 667

The general geometrical optics expansion

................... Eiconal equations and rays ............................. Transport equations .................................... Connections with the static theory ..................... Spatially localized data ............................... Behavior of probability amplitudes ..................... Expansion for the wave equation ........................ Expansion for the heat equation ........................ Comments and problems .................................. Bibliography ........................................... Expansion for Schrodinger's equation

671 677 678 682 684 685 685 687 694 696

Chapter 1

:

Elliptic Operators

Orientation. Sections 1, 2 , 3 are basic for the reading of all other parts

of the book. Section 1 presents the simplest "model" problem : the multiple scale method is applied to this problem in Section 2 and an "energy" proof of convergence is given in Section 3. The reader mainly interested in Probalistic Methods (Chapter 3) should read Section 5 (after Sections 1, 2 , 3), some parts of Chapter 2 (indicated in the Orientation of Chapter 2 ) and then proceed with Chapter 3. The same can be said for those mostly interested in high frequency wave propagation, before reading Chapter 4 . Section 4 gives Lp estimates, p > 2 (not too large) : these estimates are not indispensable tion 8, but

-

;

we use them in some parts of Sec-

as we show in that Section

-

one can avoid these esti-

mates by making some stronger regularity assumptions on the coefficients. Section 5 gives "correctors" which are improving the approximation. These correctors

(

or variant of them) are indispensable in nuI

merical computations (not reported in this book). Sections 6 , 7 , 9, 10, 11 give extensions and variants to elliptic equations of higher order or to some elliptic systems of interest in the applications. Section 8 treats the case when there are more than 2 different scales

;

this Section is technically difficult but the final result

is quite simple. Some even more general situations (but for second order operators only) are considered in Chapter 3 . Sections 12, 13, 14 give variants : Section 14 shows (among other things) that one has to be quite careful when working with pro1

2

E L L I P T I C OPERATORS

blems which involve at the same time "homogenization" and "singular perturbations". Section 15 gives examples where the homogenized ope-

=

rator or Partial Different a1 Operators (local operators) is a

local (pseudo-differential operator. Sections 16 and 17 study the homogenization of some non linear problems, in particular of some Variational Inequalities.

1. Setting of the "model" problem. 1.1

Setting of the problem ( I ) . Let 0 be a bounded open set of Rn; p is assumed to be bounded

to simplify the exposition but this hypothesis is by no means indispensable; we shall return to this point on several occasions. In B we are going to consider various boundary value problems associated to operators AE which are uniformly elliptic when the coefficients of AE

E

are rapidly oscillating (with "period"

-t

0 and

E).

More precisely, we define n y = v]O,y;[ j=O a function

f:

Rn

+

C

(1.1)

;

Rm is said to be Y-periodic if it admits period

j = 1,...,n: yo in the direction y I j' i,j = l,...,n, such that

aij (y) E R

Rn

,

we consider functions aijl

aij is Y-periodic

aij(y)SiSj 2 aSiSi

,

a > 0

,

,

aij

E

a.e. in y

We do not assi.lme, for the time being, that aij =

L~(R")

,

.t aji*

~

'We

adopt here and in what follows the summation convention.

3

SETTING OF THE "MODEL? PROBLEM

We also consider a . a . (1.2)

E

such that

Lw(Rn)

,

1. uo

> 0

ao(y)

To the functions aij and a .

where

E

. a

is Y-periodic

,

a.e. we associate the family of operators

is a "small" positive parameter.

Remark 1.1. As

it was said in the Introduction, operators (1.3) mode1 the

simplest possible situations of composite materials, when the period of the structure can be chosen to be E. Remark 1.2. It is immediately seen from (1.1) and (1.2) that the family AE consists of second order elliptic operators, which are uniformly elliptic in

E.

Remark 1.3. In many cases, it will be possible to take

(one could even consider negative functions ao, provided they are not too large). The first problem we want to study in this book is, roughly speaking, as follows : We consider the equation

ELLIPTIC OPERATORS

4

A%,

= f

in

5

,

(1.5) uE subject to boundary conditions on and we want to study the behavior of uE

as E

r

= a 0

t

,

* 0.

A typical "result" we shall obtain (in a more precise form!)

is:

one can construct (with constructive formulas) a second order elliptic operator 4 such that uE * u (in an appropriate topology) where u is the solution of

u

is subject to boundary conditions (which of course will depend on those boundary conditions imposed on uE)

.

The operator0 is the so-called homogenized operator of the family A€. We now proceed with a more precise formulation of the boundary conditions in (1.5).

1.2

Setting of the problem (1I):boundary conditions. We use a variational formulation. The presentation is partially

imposed by the structure of (1.3) since, even assuming the aij's regular (which will not be the case in many results obtained below), 1 the functions aij(g) would have derivatives of order therefore the

z;

a priori estimates on u '--khichare independent of

on the regularity (if any) of the coefficients.

E

will not be based

Therefore a

variational formulation (of a weak type) is indispensable here. Sobolev spaces.

(cf. S. L. Sobolev 111, J. Nezas [l],

J. L. Lions and E. Magenes [l], R. Adams [l]).

TThe boundary conditions are made precise in Section 1.2 below.

5

SETTING OF THE "MODEL", PROBLEM

1 We s h a l l d e n o t e by H ( 8 ) t h e s p a c e 1

(1.7)

a

1 =

H (0

provided w i t h t h e norm g i v e n by

IvI2 =

I

v 2 dx

t

1

H ( 09 i s a H i l b e r t s p a c e

.

We define

(1.9)

1 H~ ( 8

(1.10)

C O () ~= C

= c l o s u r e of

m

c:,

(0

1 in H (B )

,

m

f u n c t i o n s w i t h compact s u p p o r t i n

.

One h a s ( c f . B i b l i o g r a p h y ) (1.11)

1

~

~1 = ( { v l0 v E H'(o),~ =

o

on

i-1

I

1 and it i s known ( P o i n c a r g ' s i n e q u a l i t y ) t h a t on Ho(B

) I

t h e norms

W e s h a l l introduce

V = closed

1 subspace of H ( 0 )

(1.12) Hi((P) C V 5 H 1 ( 0 )

such t h a t

.

B i l i n e a r form a s s o c i a t e d to AE. For u , v E H 1( 8 . )

'For

valued.

I

we define

t h e t i m e b e i n g , a l l f u n c t i o n s are supposed t o be

6

ELLIPTIC OPERATORS

(1.13) where we have set (1.14)

E (x) = aij(x/E) aij

,

agE(x) = ao(x/E)

.

We observe that, by virtue of (1.11, (1.2)

Remark 1.4. Let

r0

C

I' be a subset of

(actually one could take

ro

r

of positive surface measure

of positive capacity), and let us assume

that (1.16)

v

= { V ~ VE H 1 ( 8 j,v =

0 on

ro3

.

The precise meaning of (1.16) is clear if To is "smooth"; if not, V consists of the closure in H 1 ( 0 ,) of functions which are zero in a (variable) neighborhood of

ro.

Then if a :

= 0, i.e.

if

(1.17) one has (1.18) where c does not -

depend on

E.

Boundary value problem. The function uE is defined as the solution oft

iWe have existence and uniqueness of a solution by the Lax-Milgram's Lemma, since one has (1.15) (or (1.18)).

7

SETTING OF THE "MODEL" PROBLEM U E E V '

(1.19)

E

a (uElv) = (f,v)

I

b v E V

,

where (1.20)

Examples. Example 1 1

1

Then (1.19) is t h e c l a s s i c a l D i r i c h l e t ' s problem.

V = Ho(b).

Example 1 . 2 .

1 V = H ( 0 ) . Then (1.20) i s t h e Neumann's problem; assuming t h e boundary

r

and t h e c o e f f i c i e n t s a A'U~ =

ij

f

' s smooth enough, t h e problem i s

in 0

I

(1.21) F8% = O

o n r ,

AE

av av

where

av axi

E

= aij

C O S ( ~ ~ Xv ~= ) o~u t e r

normal t o

r.

AE

Example 1.3. V i s g i v e n by ( 1 . 1 6 ) . A

~ =U

UE

= 0

(1.22)

Then (1.19) means:

f~

in on

0 ,

r0

'

a v = ~o n r l = r - r o . AE

'We

s h a l l c o n s i d e r more g e n e r a l r i g h t hand s i d e s l a t e r on.

a

ELLIPTIC OPERATORS

Example 1.4. Let us suppose that

v

(1.23)

= {V]V

E

1

H (a),v = constant on

rl

(where the value of the constant depends of course on v). Then (1.19) means A

u

(1.24)

~

E

=U f

= c

~

E

in 0 on

,

r ,

c

= constant (not given)

The first problem we want to study is now in its precise form derive the behavior of uE ' solution of (1.19),

as E

+

,

to

0.

Before entering the study of the general case, let us consider a very simple particular case. 1.3

An

example:

a one-dimensional problem.

We consider the case n = 1 in the above problem, for the Dirichlet's boundary condition (to fix ideas). 0 = Jxo,xl[,

Therefore if

we have

(1.25) uE (xo) = UE(X1) = 0

(1.26)

I

where a(y) is Y periodic (i.e. admits a period yo) and a(y) 2 a > 0 a.e. The variational formulation is

(1.27) U

E

E

1

Ho(0)

.

9

SETTING OF THE "MODEL", PROBLEM

By t a k i n g v = uE i n (1.27) one immediately sees t h a t (1.28) Therefore one c a n e x t r a c t a subsequence, s t i l l denoted by u E , such t h a t u

(1.29)

-+

E

i n H i ( 0 ) weakly

u

.

W e a l s o n o t i c e ( t h i s is a simple e x e r c i s e ) t h a t

aE + n(a) =

(1.30)

From ( 1 . 2 9 ) ,

-

a(y)dy

i n L m ( O ) weak s t a r

. tt

( 1 . 3 0 ) , (1.25) it i s tempting t o b e l i e v e t h a t i n

t h e l i m i t one h a s (1.31)

u s a t i s f y i n g t h e boundary c o n d i t i o n s analogous t o ( 1 . 2 6 1 , u ( x o ) = u(x,)

(1.32)

= 0

But t h i s i s u n t r u e ( i n g e n e r a l ) .

i.e.

.

The c o r r e c t answer i s a s

we introduce

follows: (1.33) s i n c e a'

remains i n a bounded set of Ls( 0 ' ) and s i n c e on& h a s (1.281,

i s bounded i n L 2 ( 6 ) and by (1.25) w e have (1.34)

tWhere h e r e and i n what f o l l o w s , t h e c ' s d e n o t e c o n s t a n t s which do n o t depend on E .

1

"In

g,@dx 0

g e n e r a l , i f g,, +

/og@dx,

4

g E Lm( 0 )

E L1( 0 )

.

, g,

+

g i n LQ(O ) weak s t a r means

10

E L L I P T I C OPERATORS

so that

1 H ( 0)

cE

+

is bounded in H 1 ( 8 ) . Since the identity mapping from 2

L ( 8) is compact,'

cE

(1.35)

+

it follows that one can assume that in L 2 ( 8 ) strongly

5

,

so that

15'

(1.36) (Since 1 a

aE

+

1 in m (z)

+

1 m(,)c

in L2 ( 0 ) weakly

Lm( 0 ) weak star).

But

1cE a

= d U E so that

(1.36) and (1.29) imply

.

du - rn($ azOn the other hand (1.36) gives

-

2

= f,

so that

(1.37)

The homogenized operator associated to AE is given by (1.38)

Since Q is uniquely defined, uE

+

u in Ht(6

)

weakly, without

extracting a subsequence. Remark 1.5.

We notice that (1.39)

(a)

1 2' 7 , with R (g)

strict inequality in general

.

'This is true if 0 is bounded. But in the unbounded case the local compactness is s u f f i c i ' n e . the compactness of the injection mapping H 1 ( 0 ) -+ L 2 ( o ' ) , 0 ' bounded, F 1C 0 ) .

11

ASYMPTOTIC EXPANSIQNS Remark 1.6.

The p e r i o d i c i t y of a ( y ) d i d n o t p l a y a fundamental r o l e i n t h e

W e can more g e n e r a l l y assume t h a t t

above r e s u l t .

a E remains i n a bounded set o f LOD(0 ) (1.40)

Then

aE(x)

1 also aE

2

a >

o

a.e.

remains i n a bounded set of LOD( 0 )

-

e x t r a c t a subsequence

,

.

Therefore we can

such t h a t

aE'

$

(1.41)

Then uE,

+

i n LOD(0 ) weak s t a r + p

u i n Hi ( d

(1.42)

)

.

weakly, where u i s t h e s o l u t i o n of % ] = f .

But t h i s t i m e , c o n t r a r y t o t h e preceding case, t h e l i m i t u i s n o t unique.

2.

2.1

Asymptotic expansions. Orientation. W e i n t r o d u c e i n t h i s s e c t i o n a method based on asymptotic ex-

pansions u s i n g m u l t i p l e scales (i.e. "slow" and " f a s t " v a r i a b l e s ) . As

w e s h a l l see a l l o v e r t h i s book, t h e method we a r e going t o

develop i s t h e most convenient and t h e most u s e f u l t o o b t a i n t h e r i g h t answers.

'We

The j u s t i f i c a t i o n of t h e formulas

do n o t assume t h a t a E ( x ) = a ( x / E ) .

obtained by t h i s

ELLIPTIC OPERATORS

12

method can sometimes be made directly, but in general other tools will be needed; these tools will be introduced later on in this book.

2.2

Asymptotic expansions using multiple scales. we introduce functions @(x,y) , x

€0

,y

E

Rn, which are

Y-periodic in y, and we associate to $(x,y) the function @(x,x/E). We shall look for uE

=

u,(x)

in the form of the asymptotic

expansion

where the functions u. (x,y) are Y-periodic in y, V x 3

E 0

.

Remark 2.1. It is technically complicated to keep track of boundary conditions when seeking uE in the form (2.1) and this is actually the source of serious technical difficulties in justifying the method.

The method will nevertheless give the "right answer" because

it will turn out that, in this sort of problems, the boundary conditions are somewhat irrelevant. The idea of the method is (simply) to insert (2.1) in equation (1.5) and to identify powers of E. In order to present these computations in a simple form, it is useful to consider first x and y as independent variables and to replace next y by X/E. Applied to a function +(x,x/E), the operator a becomes ax,

.

J

With this in mind, one can write AE = E - 2 ~ 1

where

+

E - ~ A+~ E 0A~

,

13

ASYMPTOTIC EXPANSIONS

a

a 5 1

A 1 = -ayi - [ a i j (Y)

(2.3)

A2 =

A

3

=

I

- q a[ a i j (Y) ..-) a - 5I.i a 1 - -[aij a axi

(y) 2 1+ a. ax j

y] a

.

t h e e q u a t i o n ( 1 . 5 ) becomes

Using (2.1)

(2.21,

(2.4)

A u = 0 , 1 0

(2.5)

A u

+

A2u0 = 0

(2.6)

A1u2

+

A2u1 + A3u0 = f ;

11

j (Y)

,

one c a n o f c o u r s e ( f o r m a l l y ) proceed A u

(2.7)

1 3

+

A2u2

+

,

A3u1 = 0

etC.

W e are now g o i n g to see t h a t t h e homogenized o p e r a t o r

be c o n s t r u c t e d from (2.4)

,

(2.5)

,

(2.6).

L e t u s remark t h a t t h e e q u a t i o n A @ = F

1

in

Y ,

(2.8) @

periodic i n

Y

,

,

admits a u n i q u e s o l u t i o n (up t o a n a F(y)dy = 0

(2.9)

a

ve constant) ,

.

Y

Indeed l e t u s i n t r o d u c e (2.10)

W(Y) =

1 {@I@ E H (Y),@

"periodic"}

i.e. 0 t a k e s e q u a l v a l u e s on o p p o s i t e faces of Y. F o r @,

J, E W ( Y )

, let

us set

ff

a

can

14

ELLIPTIC OPERATORS

(2.11) and let us also set

Then (2.8) is equivalent to:

@ E W(Y) (2.13)

r

al($,$) = ( F r $ I y

r

v$

E

W(Y)

-

Condition (2.9) is clearly necessary since periodic. (2.14)

If (2.9) holds, we can consider W'(Y)

I

(Al$)dy = 0 if 0 is

Y

= W(Y)/R

(F,$+c)~, V c E R, becomes a continuous linear form 2 on w * ( Y ) (assuming F E L (Y)). Since a,($,$) 2 cI 141 . , 1 2 (y), c > o r and $

+ (F,$)y =

it follows

that

(2.13) admits a unique solution in W'(Y)

(i.e. $ is

defined up to an additive constant). We apply now this remark to the solution of (2.41, (2.51, (2.6). Solution of (2.4). By virtue of (2.8), (2.9)

, the

only periodic solution of (2.6)

is uo = constant where x is a parameter, i.e.

Remark 2.2. One could directly justify the fact that, in this case, the first term of the expansion (2.1) should not depend on y but, as we shall see later on, this need not be always the case.

15

ASYMPTOTIC EXPANSLONS Solution of (2.5). Using (2.15), (2.5) reduces to (2.16)

Due to the separation of variables in the right hand side of (2.16), we can represent u1 in a simple form. We define XI = XI(,)

as the solution (up to an additive

constant) of A1x3 = AIYj = (2.17)

xj

- a

Y-periodic

aij(Y) '

.

I

(A y.)dy = 0, x3 exists by virtue of (2.8) and (2.9). 1 3 Y the general solution of (2.16) is

Since

(2.18)

Then

U,(X,Y)

Solution of (2.6). We now consider (2.6) where we think of u2 as the unknown,

x being a parameter. (2.19)

1

Y

(A2U1

By virtue of (2.8), (2.9), u2 exists iff

+ Ajuo)dy

and using (2.18)

fdy = lYlf

,

I

Condition (2.19) is the homogenized

equation we are looking for.

Y

I

Y

where IYI = measure of Y.

A2uldy =

=

-

We observe that

I

Y

aik(Y)

aul dy ayk

16 SO

ELLIPTIC OPERATORS that (2.19) becomes

Conclusion. The formal rule (which will be justified below) to compute the homogenized operator

is as follows:

(1) Solve (2.17) on the unit cell Y, (2) U

for j

= l,...,n;

is given by (2.20).

A few remarks are now in order.

2.3

Remarks on the homogenized operator.

Remark 2.3. In the one dimensional case one has (xj = x)

hence a(y) iff

I

Y

g] -

aj;[a(y) d

= a(y)

=

Y

+ c and this equation admits a periodic solution

(1 + z)dy = 0, i.e.

Y

(2.21)

l+cm(1 a)=o,

Then the coefficient of

I

IYI y

and we find (1.38).

(a

-

a

2

dx2 -

g=1+:. Y in 0 is, according to (2.20):

c)dy = -c =

1 1

m (5)

17

ASYMPTOTIC EXPANSTONS Remark 2.4. I f w e set

t h e s t r u c t u r e of q i j i s i n t e r e s t i n g :

(2.23)

As

w e have a l r e a d y s e e n i n t h e one dimensional c a s e ,

a p p e a r s as a " c o r r e c t o r " .

T h i s remark w i l l be made

p r e c i s e l a t e r on i n t h i s c h a p t e r .

Remark 2.5. I n (2.20) t h e t e r m ? ? l ( a o ) u is t r i v i a l , s i n c e uE

+

u i n L2($)

s t r o n g l y (cf. S e c t i o n 3 ) . Remark 2.6. A t t h i s s t a g e , t h e e l l i p t i c i t y of

i s n o t e n t i r e l y obvious.

I n o r d e r t o v e r i f y t h a t , i n d e e d , c7 i s an e l l i p t i c o p e r a t o r , l e t us f i r s t prove t h a t (2.24)

qij

-

2al(xj -

yj,x

i

-

Yi)

.

IYI

Indeed (2.17) i s e q u i v a l e n t t o (2.25)

al(Xj

-

yj,+) =

+we s e t i n g e n e r a l

o

($1 =

v

lul

J,

I

y

E W(Y)

4(Y)dY.

.

ELLIPTIC OPERATORS

18

Taking JI =

x1

in (2.25) we see that

which proves (2.24). The ellipticity of

But al(w,w)

2

follows. Indeed

[ % ] ' d y

a

so that qijSiEj

2

0 and

Y g.

-5.6

11 1

j

= 0 if€

aw

ayk

= 0 V k, i.e. w = c.

is Y-periodic, i.e.

ti

= 0

But then Siyi =

Sixi

-

c

V i so that qijSiCj is positive

definite. Remark 2.7. In case one has

(i.e. A E is symmetric) then al($,$) = a,($,$)

Y$,$

1

E H (Y) so that

(2.24) implies

i.e. 0 is also symnetric. Remark 2.8.

If

AE

is "diagonal" (i.e. aij = 0 if i # j), this is not

necessarily the case f o r a ; indeed for i # j, one has

19

ASYMPTOTIC EXPANSIONS

which is not necessarily equal to zero. Remark 2.9. The operator 0 does not depend on 8 .

2.4

Justification of the asymptotic expansion for Dirichlet's boundary conditions. We consider uE,solution of the Dirichlet's problem

(2.29)

AEuE = f

uE = O

,

onr,

and we are going to show that, assuming all data and all functions to be smooth enough,'

uE converges in the uniform topology, towards

u, solution of (2.30)

O u = f ,

U = O

onr.

We take u,(x,y)

as defined by (2.18) with

(2.31)

= UE

ZE

-

(u

+

EU1

+ E 2w) ,

c,

= 0, and we set:

w = w(x,y)

,

where we choose w so that AEzE is "as small as possible'.v

A

natural

candidate is (2.32)

w = u z ;

then (2.33)

~

€

= 2--Er~

E

'

'We do not make efforts to minimize here the regularity hypothesis, ;or reasons explained at the end of this section.

20

ELLIPTIC OPERATORS

with r

(2.34)

= A2w

E

+

+ E A ~ W.

A3u1

We can define uniquely w if we add the condition (2.35)

w(x,y)dy = 0

.

Y If f is smooth, say f E C k ( T ) , + then x

-

+

w(x,y) will be k times

continuously differentiable in 0 with values in C1(?).

For this

choice of w, (2.34) gives (2.36) On the other hand, 0” z

r,

= -(Ell1

+

E

2 w)lr

I

hence

I lZEl ILm(r) -< C E

(2.37)

.

It follows from (2.33), (2.37) and from the maximum principle that

and therefore (2.38) This estimate proves the desired result and moreover, it gives an error estimate between u

E

and the solution u of the homogenized

equation.

-

‘Ck( 0 ) 5 space of k-times continuously differentiable functions in 8

.

21

ASYMPTOTIC EXPANSIONS Remark 2.10.

The difficulty in extending the above proof to other boundary conditions is obvious; if uE (respectively u) is the solution of the Neumann problem associated to ' A satisfy different conditions on

(respectively 0

r,

)

I

then uE and u

and zE, defined as in (2.311,

does not satisfy simple boundary conditions. The way out of this difficulty is to work with weak (variational) solutions. Remark 2.11. The above proof is slightly misleading, since it uses the maximum principle and since we will see below that ue converges to the solution of a homogenized equation also for higher order elliptic operators (where the maximum principle no longer applies), but uE will converge in a different topology (in general).

2.5

Higher order terms in the expansion. proceed with the computation of in the expansion (2.1). aij (Y) - aik(y)

If we set

+- a j yk

a (akixj yk

then we see from (2.6) that (2.40) But

(since

m [q a (akixj)]

= 0)

so that ( 2 . 4 0 ) becomes

22

ELLIPTIC OPERATORS

(2.41)

A1u2 =

xi’

If we introduce

a s a Y-periodic solution (defined up to an

additive constant) of

then (if a .

= 0):

(2.43) We can now compute iil in such a manner that the computation can proceed. We consider (2.7).

I

(2.44)

We can solve for u2 iff

(A2u2 + A3ul)dy = 0

.

Y

But by an explicit computation, this condition is equivalent to

aii,

(2.45)

3

=

m[ake

-

a u aijXk] ax. x.axk la

I

-

If the boundary conditions are of Dirichlet’s type, a natural choice for iil on

r

would be

but it gives a 6, which depends on E and

2.6

has “bad” derivatives.

Extensions. It is clear by now that the method presented above can be

extended in many ways and in several directions such as for the solution of higher order partial differential operators, to systems, to nonlinear problems, etc. (and also to evolution equations). shall therefore return many times to this method.

We

ENERGY PROOF OF THE HOMOGENIZATION FORMULA

3.

Energy p r o o f of t h e homogenization formula.

3.1

Orientation:

S t a t e m e n t o f t h e main r e s u l t .

We now r e t u r n t o ( 1 . 1 9 ) . (3.1)

Q

( u ~ v )= I , g i j

q i j d e f i n e d by ( 2 . 2 4 )

I

xi

1 For u I v E H ( 0 ) w e set au av axj

I

dx

b e i n g d e f i n e d by ( 2 . 1 7 ) .

L e t u be t h e s o l u t i o n o f t h e homogenized problem: U E V ,

(3.2)

Q(U,V) = ( f , v )

'd v E

v

.

W e s h a l l prove:

Theorem 3.1. W e assume t h a t a i j

a. s a t i s f y (1.1)I ( 1 . 2 ) .

( r e s p e c t i v e l y u ) be t h e s o l u t i o n of Then as

E +

(1.19)

Let

uE

(respectively (3.2)).

0 one h a s u

(3.3)

E

+

u

&

V weakly

.

Remark 3.1. S i n c e t h e c o e f f i c i e n t s a i j ' s belong t o Lm(Y) t h e y can be d i s c o n t i n u o u s on e a c h c e l l Y each c e l l E ( Y

i5 )

i . e . w e have a t r a n s m i s s i o n problem on

5 E 2".

Remark 3.2. au The example of 1 . 3 shows t h a t i n g e n e r a l s t r o n g l y towards

au T.

does n o t converge xi

T h e r e f o r e (3.3) c a n n o t be improved w i t h o u t

adding e x t r a terms (of t h e " c o r r e c t o r " t y p e ) .

23

24

ELLIPTIC OPERATORS

Remark 3.3. Let us consider mare generally a family f, E V' (dual of V); let uE be the solution of a E (uE,v) = (fErv)

(3.4)

vv

E

V

uE E V

;

let us assume that .

(3.5)

f,

+

f

in

V' strongly

and let u be the solution of

Then one still h a s (3.3). 3.2

Proof of the convergence theorem. We consider the more general case of Remark 3.3.

Taking

v = u E in (3.4) and using (1.18) we have

I.,

(3.7)

II II

where

2c

denotes the norm in V (or H'(0

))

.

We set

it follows from (3.7) that (3.9) (where

I I

denotes the norm in L2 (0)).Therefore we can extract a

subsequence, still denoted by u f l 6; such that

ENERGY PROOF O F THE HOMOGENI~ZATION FORMULA

(3.10)

,

uE

-+

u

ui

+

Si in L2 ( 6 ) weakly

in

V

weakly

25

.

Equation (3.4) can be written t

which gives in the limit

(here the weak convergence in V' of fE to f would suffice). We now compute ti using adjoint equations.

Let P (y) be a

homogeneous polynomial of degree 1 and let A; be the adjoint of A1:

We define w as "the" solution of

*

(3.13)

A1w = 0

such that w

-P

is Y periodic.

If

then (3.15)

(i is

*n

*

A1X = Alp

I

is Y-periodic

defined up to an additive constant).

We define next

'We

do not write the term (aiuEIv)whose limit is ?!I (a,) (u,v).

ELLIPTIC OPERATORS

26

W e have (we can always assume t h a t a. = 0 ) : (A')*W~ = 0-

(3.17)

in

0

.

W e now t a k e i n (3.4) , v = $wE, $ E C8z( 0 ) (so t h a t QwE E V s i n c e 1 $wE 6 €I (o 0 ) V) and w e t a k e t h e s c a l a r p r o d u c t o f (3.17) w i t h

WE.

Taking t h e d i f f e r e n c e , w e o b t a i n

a (uE,$wE) - a ( O U , , ~ , ) E

(3.18)

E

= (fE,$wE)

.

The l e f t hand s i d e of (3.18) e q u a l s (3.19)

but

[Ci, ( e w E ]

a i j 7=

e]

a i j :$ $I u,dx

-

E

[aij

where

;

g E ( x ) = g ( x / E ) so t h a t t h e

e x p r e s s i o n i n (3.19) c o n v e r g e s , as

E +

0, to

(3.20)

indeed wE + P i n L 2 ( 0 ) s t r o n g l y , u + u i n H 1( 0 ) weakly, t h e r e f o r e E, 2 i n L ( 0 ' ) s t r o n g l y f o r e v e r y 0 ' C 0 ' C 0 , 0' bounded and

k] E

[aij

converges t o

m [a i j ay awi]

i n Lm(O ; weak s t a r .

On t h e o t h e r hand

OwE

(3.21) i n d e e d 6wE

and

+

+

$P

in

$P i n L2 ( 0

$[$-IE

+

Q 7;1

H1 o(O )

weakly ;

and

[g] = 0

in

L 2 ( 0 ) weakly

.

ENERGY PROOF OF THE HOMOGENI!3ATION FORMULA

T h e r e f o r e (3.21) and (3.5) imply (fErOWE) + ( f r 4 P )

By u s i n g (3.111,

(f,@P) =

[Ei,v] so t h a t

Therefore

i.e. (3.22)

ci

ap

m

=

If w e t a k e P ( y ) = yir

w e have w = yi (3.24)

si

aw

au

[aij

and i f we d e f i n e

- iii

and

=

[akj

a ayk

(Yi

.

-

Consequently, u s i n g (3.24) i n t o ( 3 . 1 1 I f w e see t h a t u s a t i s f i e s

I t remains o n l y t o v e r i f y t h a t (3.25) c o i n c i d e s w i t h (3.6),

i.e. t h a t (3.26)

[ai k QJ ayk

=

m[akj

But t a k i n g t h e scalar p r o d u c t of

. (3.23) w i t h

x3

gives

27

28

ELLIPTIC OPERATORS

and taking the scalar product of (2.17) with

ii

gives

Therefore (3.26) is equivalent to saying that

which is true. completed

The proof of Theorem 3.1 (and of Remark 3.3) is

.

Remark 3.4. It easily follows from the equivalence of (3.15) and (3.6) that "the homogenized of the adjoint equals the adjoint of the homogenized operator." 3.3

A remark on the use of the "adjoint expansion". We consider again (1.19) with 1( 0) V = Ho

(3.28)

(Dirichlet's problemt)

.

In this case, the subspace (3.29)

H:omp(

0 ) = (vlv E HI( 0 ) ,V with compact support in 0

1

is dense in V and (1.19) is equivalent with

We can of course in (3.30) take a variable family vE of test functions:

'Cf.

Remark 3.5 below for the case of other boundary conditions.

29

ENERGY PROOF OF THE HOMOGENILATION FORMULA We now choose vE in a particular manner, using the "adjoint expansion" t

.

We shall verify below that, given v

E

C,"(O )

, one

can

construct (by using the method of asymptotic expansions) functions vE such that 1 vE E Hcom( 6 )

,

vE

* v in L 2 ( 0 )weakly (for instance) ,

(3.32) A € * V ~+ o*v in H - ~ (6 ) strongly

.

Using this choice of vE in (3.31) givesin the limit, since uE

+

u in H t ( 0 ) weakly, (u, 4*V) = (f,v)

vv

E C;( 8 )

i.e. u satisfies (3.2) for V given by (3.18). Construction of vE. We use the notations of Section 2. (3.33)

VE =

v

+

EV1

I

v EC; ( 8 )

,

We look for vE in the form v1 = v(x,y)

,

We choose v1 such that

*

*

A 1v 1 + A 2 v = 0 and we can take, using (3.231,

'We assume the coefficients aij to be smooth enough. The weakest hypotheses are given by the method of Section 3.2.

30

ELLIPTIC OPERATORS

-$

v1 =

(3.35)

This choice gives

*

AE*vE = A2Vl

(3.36)

=

%.

VE-€

+

*

A3V

HiOrnp(8)

+

V

E.

Then (3.34) reduces to

*

€A3v1

f *

One easily checks that one has AE*vE

+

(3.37) in L~ ( 8 ) -

weakly

i

1 ( 8) + L 2 ( 8 since the injection of Ho that AE*vE

+

)

is compact, (3.37) implies

c7*v in H-l(8 ) strongly and the result follows.

Remark 3.5. If V # H1o ( B ) , a similar method leads to difficulties in dealing with boundary conditions of Neumann's type to impose (on some part of the boundary) on vE.

The method of Section 3.2 is

then simpler. Remark 3.6. In situations where the identity mapping V + H is not compact (we shall meet such a situation in Section 11) then the weak it is then necessary to convergence in L 2 ( 8 )does not suffice-and take higher order terms in the "expansion" (3.33)

.

ENERGY PROOF OF THE HOMOGENfZATrON FORMULA

31

Remark 3.7. The idea of taking variable test functions constructed in a special manner is useful in several c o n t e x t s .

3.4

Comparison results. We are going to show:

Theorem 3.2. If we assume (1.1) and if qij are the homogenized coefficients then one has (3.38) with the same a

&

(3.33) and in (1.1).

This inequality can be directly verified, but we are going to show it is a particular case of a general "comparison theorem", due to L. Tartar. Theorem 3.3. Let us consider a second family of operators

(3.39)

, [bij (y)I ,..., the n x (y), , we suppose that

We denote by [aij(y)I coefficients ai (y), bi

'We

...

do not assume that [aij]* = [aij].

n matrices with

32

ELLIPTIC OPERATORS [bijl 5 [aij] t

(3. 41)

.

Then if [ a], [ B ] denote the homogenized matrices, one has (3.42)

[nl

5 [a1

.

Remark 3.8. If we take [b. . I = a (identity matrix) 11 and if the aij's satisfy (1.1) , we are in the situation of Theorem 3.3, and since of course [ B ] = a (identity), (3.38) follows. Remark 3.9. We are going to give a proof of Theorem 3.3 which is somewhat longer than it could be but which can be extended to the "generalized" convergence in the sense of de Giorgi and Spagnolo [l], cf. L. Tartar [ 11.

It can also be applied to situations met in the

subsequent sections. Proof of Theorem 3.3. Given X E Rn, one can find a sequence of functions uE such that uE

(3.43)

* u in H 1 (

,

0 ) weakly

grad uE * X in (L2 ( 8))" weakly [aijlgrad uE

-t

,

[ U l x in ( L * ( o ) ) weakly ~

div[azjlgrad uE converges in H-l strongly 'I.e.,

X

E

Rn.

the matrix M = [aij]

-

[bij] is

, ,

0, i.e. M X * X

0,

33

ENERGY PROOF OF THE HOMOGENIZATION FORMULA and such that (3.44)

i.e.

,

(3.44b)

E [aij]grad uE

I

[aEj]grad uE

grad uE

*

+

[alA*A"vaguely"

grad uE @ dx

I

+

,

[alA*A$dx

v

$ E GCS

1

.

This will be verified below. We can also find a sequence vE with similar properties but with aij (respectively

a) replaced

.

by bij (respectively 8 )

We have

also (this is verified below) (3.45)

[b!.lgrad

vE

13

grad uE

+

[nlA*A "vaguely" .

Assuming these properties to hold true, we observe that

But XE = [b: .]grad vE 13

-

-

grad vE

[bf . I grad uE 13

-

-

[b! .]grad vE =I

grad vE

+

+

[bf .]grad uE 13

grad vE

-

2[ b ! .

grad uE

11

11

13

grad uE

[bf . lgrad uE

By virtue of the symmetry of [b..], one can write XE = [bf .I grad vE

*

13

lgrad vE

.

I

grad uE

grad uE

and by virtue of (3.411, we have (3.46)

0 5 XE 5 [bijlgrad E vE * grad vE

+

[aij]grad uE

grad uE

.

-

Z[b!.Igrad 13

vE

grad uE

34

ELLIPTIC OPERATORS Since positivity is conserved in the vague topology, it follows

from (3.44), (3.45) and (3.46) that 0

5

- [ B]X*X

+

[Q]X*X

t/

X

,

i.e.

(3.37) ,

It remains to verify (3.43) , (3.44) , (3.45). Given X E Rn, we define = EU(X/E)

u,(x) (3.47)

u(y) = Xj(yj

-

r

,

xj)

x j defined as in Section 2

.

We observe that AEuE = 0 and (3.44) follows easily. Let us prove more than (3.44). such that w E

/

+.

w in HI( 8 ) weakly.

([aijlgrad uE

-

Let wE be given in H

1

(

Then

grad wE)4dx

0

and by virtue of (3.43) , this integral converges to

-

L$

([

( [ alX*grad w)@dx

QIX)iwdx =

so that (3.48)

[ a i j grad u E

Taking wE = u E

grad w E

gives (3.44).

+

[ U ]Amgrad w “vaguely”

.

8)

35

Lp ESTIMATES In the same manner [b! .]grad vE 13

grad wE

and taking wE = uE gives (3.44).

4.

*

I X-grad w "vaguely"

[B

The proof is completed.

LP estimates.

4.1

Estimates for the Dirichlet problem. We are going to show in this section that one can slightly

improve the estimate (3.7) for the Dirichlet problem (the case of other boundary conditions is considered in Section 4.4 below).

More

precisely we shall prove the following. Theorem 4.1.

u

E

We assume that 8 is bounded with a smooth boundary. 1 E Ho ( 0 ) be the solution of

(4.1)

E

a (uEfv)= (ffv) I

V v

E

under the hypotheses of Theorem 3.1. p > 2 (and independent of (4.2)

E)

1

Ho(0)

& e J

I

Then there exists a number

such that if

a fi

f=fo+-

The number p can be (roughly) estimated in terms of the aij's and the dimension (cf. Section 4.3).

In (4.3) the constant c does not depend of

E.

36

ELLIPTIC OPERATORS

Remark 4.1. The estimate ( 4 . 3 ) - i s indeed an improvement w i t h r e s p e c t t o ( 3 . 7 ) ; more i m p o r t a n t l y t h i s estimate w i l l be used a s a

tool t o

prove

other results. A c t u a l l y Theorem 4 . 1 i s a p a r t i c u l a r c a s e of a more g e n e r a l r e s u l t due t o Meyers [ l 1, which i s r e l a t i v e t o g e n e r a l second o r d e r e l l i p t i c o p e r a t o r s w i t h L~ c o e f f i c i e n t s . I n o r d e r t o p r e s e n t t h i s r e s u l t , it i s n e c e s s a r y t o i n t r o d u c e c l a s s i c a l Sobolev s p a c e s W l r P ( 8 1 by t a k i n g some c a r e i n t h e p r e c i s e d e f i n i t i o n of t h e norm of t h e s e s p a c e s . Spaces w ” P ( o ) ,

WlrP(aJ (4.5)

[

w ~ ’ P 0(1

i s a Banach s p a c e w i t h t h e norm n

IIvIIpLp(a)

+

W e introduce next

The norm w e p u t on W:Ip(

0 ) (which i s e q u i v a l e n t t o ( 4 . 5 ) ) i s

as f o l l o w s ; w e set

V$ =

(4.7)

IIfl

r

I f ( x ) 1% Y

rp

where

n If ( x ) I = Euclidean norm of f ( x ) i n R and we d e f i n e

,

37

Lp ESTIMATES

By virtue of Poincarg's inequality, (4.8) is equivalent to (4.5) on Wi'P( 0 ) .

We take 1 < p <

m

and we define p' by l/p

il / p '

=

1.

We

define W - l r P ( O ) = (WirP'(0))'= dual space of Wit"' where (4.9)

L2 ( 8 ) is identified with its dual.

We observe that the mapping (4.10) is

onto.

@

-+

div

@

from

(L'(L)))~

+

W-lnp(a)

We provide then W-lrP( 0 ) with the "quotient" norm asso-

ciated to (4.10), namely

I I f 1 Iw-lfp (4.11)

=

( 8 )

infllgll

(LP(8

div g = f, which is of course one way of defining the norm on W-lIp( 8 ) . The operator A. We consider now a family of functions a such that ij

and we define

,

ELLIPTIC OPERATORS

38

W e c o n s i d e r t h e D i r i c h l e t ' s problem

(4.14)

,

Au = f

1

u E Ho(0)

,

f E H-I(0)

.

W e s h a l l prove

Theorem 4.2. W e assume t h a t (

,12

that 0 holds t r u e an -

i s bound 1 w i t h a

There e x i s t s a number p > 2 (which d e p e n d s on a ,

smooth boundary.

on t h e Lm norm o f t h e a . . I s and on 0 and on t h e d i m e n s i o n ) s u c h 17 ___ that, i f f E W - l r p ( O ) , then the solution u ( 4 . 1 4 ) belongs t o

Wifp (d

of

) and s a t i s f i e s

(where C depends on t h e s a m e q u a n t i t i e s t h a n p d o e s ) .

Remark 4 . 2 . Theorem 4.2 c o n t a i n s Theorem 4 . 1 as a p a r t i c u l a r c a s e . The proof o f Theorem 4.2 i s p r e s e n t e d i n two s t e p s :

i n the

f i r s t s t e p ( S e c t i o n 4 . 2 ) w e make an " a l g e b r a i c " r e d u c t i o n of

(4.14)

t o a s p e c i a l form and t h e theorem i t s e l f i s proved i n S e c t i o n 4 . 3 .

4.2

Reduction of t h e e q u a t i o n . L e t u s i n t r o d u c e t h e f o l l o w i n g n o t a t i o n s ; w e d e n o t e by [ a i j ( x ) ]

t h e m a t r i x w i t h e n t r i e s t h e a i j ' s and w e set (4.16)

M(x) =

W e d e n o t e by

I

IM(X)

I

t h e norm d e f i n e d by M(x)S(

I

39

LP ESTIMATES and w e s e t (4.17)

1

We d e f i n e M1(x) =

[M(x) + M * ( x ) ] , M2(x) = 1 [M(x)

-

M*(x)l and w e

write t h e i d e n t i t y M = M 1 + C I + M 2 - C I ,

(4.18)

c t o be chosen l a t e r , I = i d e n t i t y m a t r i x

W e replace

(4.19)

1 8+c AU

.

( 4 . 1 4 ) by =

1 f B+C

.

W e set

(4.20)

1 B+c (M1 p1 = -

+

cI)

,

P2 =

1 (M2 8+C

cI)

and w e d e n o t e by A1 and A2 t h e o p e r a t o r s c o r r e s p o n d i n g t o P1 and P 2 i

if 2 P2 ( x ) = [ a i j

(XI 1 ,

then

W e are g o i n g t o v e r i f y t h a t w e c a n choose c i n s u c h a f a s h i o n t h a t

(4.25)

1

-

l~

+

w < 1

Indeed P 1 ( x ) * = P,(x)

and

( i . e . w < 11)

.

40

ELLIPTIC OPERATORS

(4.26)

a+c B+c

p =

c,o.

'

We next have

v

2 - B' -

+ c '

(B +

Cl2

-

It remains to choose c such that

(4.27)

c > B2 2c-1 a2 ~

Summarizing: (4.28)

2

(B + d2

( B + c)

-

the equation (4.14) is equivalent to

(A1 + A2)u =

f ,

f=-1 f, B+c

where c is chosen to satisfy (4.27) and where A1, A2 satisfy (4.23), (4.24), (4.25).

4.3

Proof of Theorem 4.2. We replace (4.28) by

(4.29)

-AU

+ (A + A1 + A2)U

=

and we use the fact that (cf. Agmon-Douglis-Nirenberg [l], J. Peetre [I], J. L. Lions-E. Magenes [21) v p such that 1 < p <

We set (4.31)

(-A)-' = G

.

,

-A

is an isomorphism from

41

Lp ESTIMATES

?

Then (assuming t h a t (4.32)

u

+

+

G(A

E W-l

A1

+

p(0 ) )

(4.29)

i s equivalent t o

A2 u = G z .

W e a r e g o i n g t o show t h a t t h e r e e x i s t s p > 2 such t h a t (4.33)

I IG(A

+

II

+ A ~ )

For s u c h a p , I

k(p)

5

G(A

+

A1

+

A2)

)

is invertible in W i " ( 0 )

IIGII

W e are g o i n g t o v e r i f y t h a t (4.35)

I I (A

II

+ Al)

<

l - p

and

8 1 ;w-l'P(s ) 1

< v . -

I f w e a d m i t t h e s e e s t i m a t e s f o r a moment, and i f w e s e t (4.37)

g(p) =

then ( 4 . 3 4 ) , (4.38) It

I

(4.35

k(p)

5 g ( p ) (1 -

p

+ v)

.

is straightforward to verify t h a t

(4.39)

g(2) = 1

.

.

;w;'p( 0 ) )

T h e r e f o r e e v e r y t h i n g rests on ( 4 . 3 3 ) .

the r e s u l t follows. (4.34)

+

= k(p) < 1

,L(w;'p(s

and

W e have

ELLIPTIC OPERATORS

42

-

+ we can prove thatt Since 1

w < 1 (cf. ( 4 . 2 5 ) ) , we shall have proved ( 4 . 3 3 ) if

there exists a function Q(p), continuous of p L 2 , (4.40)

such that g(p)

5 G(p) and

G ( 2 ) = g(2) = 1

.

Proof of (4.40). We take po > 2 arbitrarily large and we take p such that 2

p 5 po.

(4.41)

For h

(LP(6))n, we define u E W i r P ( 0 ) by

E

-Au = div h

and we consider the mapping (4.42)

h

-t

ah = Vu

from (Lp(O ))" into itself.

Let ;(p) be the norm of this mapping.

By virtue of the Riesz-Thorin interpolation theorem, we have

(4.43)

1 = 1 P

-

e(p)

PO

+

e;p)

,

(since W ( 2 ) = 1). For f E W-l'p(O Gf = ah

)

, we write f

=

div h; then

so that

hence

'It seems likely-but it does not appear to be proven somewhere in the literature-that the function p * g(p) itself is continuous. But ( 4 . 4 0 ) is sufficient for our purpose.

Lp ESTIMATES

43

so t h a t w e c a n t a k e i n ( 4 . 4 0 ) (4.44)

G(p) = w ( p , ) l -

(For p = 2 ,

Proof o f

8 ( p ) = 1 and i j ( 2 ) = 1 ) .

(4.35).

L e t u be i n

(A (4.45)

e(p)

+

Wirp( 8

)

, we

A1)u = d i v g

,

au gi = (6: . - a i1j (x)) F(x) j

Therefore

hence ( 4 . 3 5 )

Proof of

have

follows.

(4.36).

W e have, f o r u E

W;"(o),

9 = ISi}

,

I

6: = Kronecker i n d e x

.

ELLIPTIC OPERATORS

44

,

A u = -div g

2

(4.47)

so t h a t

I g ( x ) I 5 u I Vu ( x ) I .

hence ( 4 . 3 6 ) f o l l o w s s i n c e

4.4

Local e s t i m a t e s . W e consider

a.

(4.48)

and w e s e t , (4.49)

E

v

,

Lm(O)

u,v

E

a(u,v) =

H

I

2

ao(x)

0

a.

1

( B ) ,

au av dx a x . axi

aij(x)

+

3

I

aOuvdx

8

8

For f E V', w e d e n o t e by u t h e s o l u t i o n o f (1.12)

.

( n o t a t i o n s of

:

(4.50)

a(u,v) = (f,v)

,Vv

E V

,

u

E

V

.

m

W e o b s e r v e t h a t i f @ E &, (

( @ f , v )= ( f , @ v )

,

V

3 ) we define @f E H - l ( 8 v

E

H:(O)

) by

.

W e have

Theorem 4.3. W e assume t h a t t h e h y p o t h e s i s of Theorem 4 . 2 h o l d t r u e ' ( 4 . 4 8 ) t a k e s place.

'But

Then t h e r e e x i s t s p such t h a t

n o smoothness o n

r

= 3 8

i s needed h e r e .

and t h a t

Lp ESTIMATES 2 < p < & - n - 2 '

(4.51)

so t h a t , i f Q EC:

45

t

( 6 )i s g i v e n and Qf E W - l r p ( G ) ,

where c d e p e n d s o n l y on 8

,

then

Q, and on t h e a i j ' s .

Remark 4.3. I f w e t a k e @ = 1 on a s e t

-

o',&'

C

8

,

(4.52) gives a l o c a l

estimate.

Proof o f Theorem 4.3. By a s i m p l e c o m p u t a t i o n , w e have

we observe t h a t a i j a i j U ag ax E L q ( 8 )

,

j

v i r t u e of

5 axi

E L 2 ( B ) C W-1'p(8)tt

1 - 1n.

= 2

and t h a t

( S o b o l e v ' s t h e o r e m ) , s o t h a t , by

(4.51),

axi a

(aiju

ax) a@ E

.

w - l ~ p ~ s )

j

The r e s u l t t h e n f o l l o w s from Theorem 4 . 2 .

t T h i s u p p e r bound becomes i r r e l e v a n t i f n = 2 . ttThis

i n c l u s i o n f o l l o w s from t h e i n c l u s i o n W i r p ' ( 8 ) C L 2 ( ( 0)

which, a c c o r d i n g t o t h e S o b o l e v ' s theorem i s t r u e i f which i s e q u i v a l e n t t o t h e u p p e r bound i n ( 4 . 5 1 ) .

1 p

- 1-n -<2 -1'

46

4.5

ELLIPTIC OPERATORS

Extensions. The above r e s u l t s can be e x t e n d e d t o h i g h e r o r d e r e l l i p t i c

e q u a t i o n s o r systems.

L e t u s g i v e one example.

We consider

where la1 = I f 3 1 = m, and where

W e set

l e t N = number of i n d i c e s a s u c h t h a t la1 = m.

We define

which i s a Banach s p a c e f o r t h e norm

W e then introduce

and w e p u t on

GIp 6 (

)

$" ( 6 )

t h e norm ( e q u i v a l e n t t o ( 4 . 5 8 ) )

where

m * , d e f i n e d by

The o p e r a t o r ( D )

by

47

Lp ESTIMATES (4.62)

(Dm)*f = ( - l ) m D a f a

maps ( ~ ~ ( o6n t)o ) ~ - ~~ ' ~ ( c ?where ) , (4.63)

w - ~ ' ~ (=od)u a l

space of

t h e r e f o r e w e can provide W-mrP

(

$Ip'

( 8)

;

8 ) w i t h t h e norm

( LP

( A l l t h e remarks a r e e n t i r e l y p a r a l e 1 t o t h o s e p r e s e n t e d i n

Section 4 . 1 . ) W e c o n s i d e r now t h e D i r i c h e t s problem: (4.65)

AU = f

,

f E H-m(O

t ,

U E

H:(O.)

.

W e have

Theorem 4 . 4 . W e assume t h a t ( 4 . 5 5 ) h o l d s t r u e and t h a t

smooth boundary. f E W-m'P(0),

8 i s bounded w i t h a

Then t h e r e e x i s t s a number p > 2 s u c h t h a t i f

then t h e s o l u t i o n u of

(4.65) belongs t o $ " ( O )

and s a t i s f i e s

, The p r o o f i s a l o n g t h e l i n e s o f t h e p r o o f of Theorem 4 . 2 . r e p l a c e s f i r s t ( 4 . 6 5 ) by (4.67)

(A1

+

A2)u =

,

One

48

ELLIPTIC OPERATORS

where

We introduce next (4.71)

E = (-l)mDaDa ,

la1 = m

,

and we use the fact (cf. Agmon, Douglis, Nirenberg [l]; the regularity hypothesis on from $ ' P ( o )

r

onto W-~'P(O

is used here) that E is an isomorphism ),

v

p finite.

We replace (4.67) by (4.72)

(E

+ (A1 -

E)

+ A2)U

I

=

f

or, equivalently, if (4.73)

E-1 = G

(4.74)

[I

+

,

G(A1

-

E

+ A2)]U

=

-

Gf

.

We then show, by an argument similar to that of Section 4.3, that one can find p > 2 such that

49

CORRECTORS

and t h e Theorem f o l l o w s .

5.

Correctors.

5.1

Orientation. W e now r e t u r n t o t h e n o t a t i o n s o f S e c t i o n 3 . As w e a l r e a d y s a i d , u E

-

u d o e s n o t converge t o 0 i n V s t r o n g l y

( b u t o n l y w e a k l y ) ; t h e r e s u l t s j u s t o b t a i n e d show t h a t

a (uE axi

u)

-+

0 i n L p ( 8 ) ( o r i n Lyoc ( 8 ) t ) weakly f o r s o m e p > 2 ,

b u t t h i s d o e s n o t imply s t r o n g convergence i n V of u

is actually false).

-

u (and t h i s

But w e c a n i n t r o d u c e c o r r e c t o r s , s a y 0

E'

such -

that (5.1)

u

-

u

-

OE

-f

0

&v

strongly

.

Of c o u r s e a remark l i k e ( 5 . 1 ) c a n be e n t i r e l y t r i v i a l , i f one t a k e s BE = u

-

u!!

But w e are g o i n g t o show t h a t one c a n a c h i e v e

(5.1) w i t h a c o r r e c t o r O E which i s of o r d e r

5.2

E

i n the L ' ( 8 )

S t r u c t u r e of t h e f i r s t c o r r e c t o r - S t a t e m e n t

norm.

of theorem.

The f i r s t i d e a which comes t o mind i s t o t a k e

eE

= E U ~,

u1 =

(5.2) u1 d e f i n e d i n ( 2 . 1 8 ) , (ul =

tLyoccg)

,

Ul(X,X/E)

-

-

x j au F) j

= s p a c e of f u n c t i o n s which are l o c a l l y Lp i n 8 ' .

ELLIPTIC OPERATORS

50

Assuming that x j E Wl'"(Y)

and that u E H

2

O E belongs to

( 8 )

1

H ( 0 ) but in general does not belong to V.

Let us first remark that one can always-by transformation-take

a corrector in V.

a small

We introduce to this effect

cut-off functions mE; they have the following properties mE E

(5.3)

b(o), t

mE(x) = 0

if d(x,r) = (distance from x to

mc(x) = 1

if d(x,r) 2 2~

E

IDYmE (x)I

5 cY, V

y

,

r) 5

functions exist, provided

r

,

where c depends on Y but Y does not depend on

Such

6

E

.

is smooth enough.

We then define (5.4)

B E = EmEul

(=

-

EmE

xj

au (-)-IxE ax

.

1

We shall prove Theorem 5.1. We assume that the hypothesis of Theorem 3.1 hold true.

More-

over we assume that

(5.6)

x j defined by (2.17) belongs to Wlfrn(Y)

and the boundary

r 02 8

is smooth enough.

j

I

Then if B E is defined by

(5.4) we have

+Actually it would be sufficient here to take m E E C1( 8 ) and to take the y's with IyI = 1.

51

CORRECTORS z

(5.7)

E

The f u n c t i o n

= u

-

E

€IE i s

u

-

eE

+

0

3v

strongly.

c a l l e d a f i r s t o r d e r corrector.

W e s h a l l p r o v e t h i s r e s u l t i n S e c t i o n 5 . 3 below.

Remark 5 . 1 . I t f o l l o w s from ( 5 . 5 ) and from t h e smoothness o f

r

that the

solution u s a t i s f i e s

T h e r e f o r e B E E V and so d o e s z E .

Remark 5 . 2 . I f w e do n o t i n t r o d u c e t h e cut-off

f u n c t i o n s m E , and i f w e

consider 5 , = u

(5.9)

- u - E U 1

t h e n we have (5.10)

5,

1 i n H ( 6 )s t r o n g l y

0

+

.

1 [ T h e r e f o r e if V = H ( 8 )it i s c l e a r t h a t w e s h o u l d t a k e mE = 1, I

also

although we can

i n t r o d u c e mE e v e n i n t h a t case.]

Indeed z

-

5 , = ~ ( - 1m ) u l

w e have t o show t h a t

But t h i s e q u a l s

,

ELLIPTIC OPERATORS

52

a (u,)

amE

- E -axi u

1

+

-

E(l

+

mE)

axi

( l - m ) -aul --+~(l-m)-. E aYi E axi

By virtue of the hypothesis (5.6), (5.3) it follows from the Lebesque theorem that strongly and since u

and (1

E

E

-

aul 2 mE) - + 0 in L ( 8 ) aYi

H

= O(E).

L2(8) Remark 5.3. The corrector B e (which is of course not unique) is a smoothing operator: of uE

-

it corrects for the rapid oscillations in the derivatives

u, and the boundary does not play any role; B E is completely

different from a boundary layer corrector.

Remark 5.4. We can "change the scaling" in m

(there are really

parameters, that we took equal in Theorem 5.1). number.

(5.11)

We can choose m

small

Let X be a positive

such that

X

mE(x) = 0

if d(x,r) 5

E

mE(x) = 1

if d(x,r)

2E

without changing Theorem 5.1. in other situations.

two

,

x

A convenient choice of X can be useful

CORRECTORS

53

Remark 5.5.

As we said in Section 5.1, one has

loE/

L ( 8 )

< CE.

Remark 5.6. This remark completes Remark 2.6.

3 ax

Section 3 , 6 ; = a:.

11

Theorem 5.1,

-+

j

9.. au in L 1 3 ax.

With the notations of 2 ( 6) weakly.

1

will have the same limit that a;j

By virtue of

& (u +

OE),

and

j

this converges to qij au j

ax .

5.3

Proof of Theorem 5.1. We consider two cases; we begin with the symmetric case

(5.12)

,

a17 . . = aji

V

i,j

where the proof is slightly simpler, and we consider next the nonsymmetric case. We write a'(@)

for a'($,$).

By virtue of (4.15). it will be

sufficient to prove that (5.13)

aE(zE)

+.

0

where we do not consider zero order terms. But, using the symmetry (5.12), we can write: aE(z

and since zE, u

=

aE(uE,zE) - aE(u + eE,uE) + aE(u

=

aE(uE,zE) - aE(uE,u

)

+

OE E V, we have

+ eE)

+ eE) + aE(u + e,)

ELLIPTIC OPERATORS

54

aE(zE) = (f,uE - 2(u

(5.14)

+ e E ) ) + aE(u +

€IE)

.

Let us admit for a moment that

Then (5.14) gives:

aE(zE)

-+

-(f,u) + Q(u,u) = 0, and (5.13)

is proven. Proof of (5.15). We have a'(u

+ eE)

=

I

a(u + e E ) a(u + eE) akll dx axk aXR

.

8

But

Using the properties of mE, u, x i and Lebesgue's theorem, one sees that rEk * 0 in L2 ( 8 ) strongly, so that aE (u + has the same limit as

But

55

CORRECTORS

0 Therefore

where w e have s e t i n g e n e r a l J I E ( x ) = JI ( x / E ) .

By v i r t u e of t h e d e f i n i t i o n o f X I , w e have

and ( 5 . 1 6 ) g i v e s t h e d e s i r e d r e s u l t .

Remark 5 . 7 . L e t u s emphasize t h a t t h e symmetry w a s n o t used i n ( 5 . 1 5 ) . W e now p a s s t o t h e nonsymmetric case.

According t o Remark 5 . 2 , W e therefore introduce

it amounts t o t h e same t h i n g t o work w i t h 5 , . a'(<

= aE(u )

+

aE(u

-

a E ( u + Eul,uE)

+

E U ~ ):

-

a E ( u E , u+ Eul)

w e have a E ( u E ) = ( f , u E ) s o t h a t w e a l r e a d y know t h a t aE(u )

+

a E ( u + tul)

+

(f,u)

+

Q(U,U)

.

= ~C~IU,U)

T h e r e f o r e w e s h a l l have t h e d e s i r e d r e s u l t i f w e show t h a t

56

ELLIPTIC OPERATORS

Let

be given in

@

m

c,

(

8)

, with

measure [El which tends to zero. aE(u,v) =

0

j a. E

ij

8

@ =

1 except on a set E of

We set

au av ax. ax l i

'

and we use the analogous notation for ff

@

.

(u,v)

We are going to

verify that aE (u (5.19)

+ €ul,uE) - aE (u + @

uniformly in

Let us notice that

E ul,uE)

-f

o

as

IEI

o ,

+

E.

there exists q finite such that

and that we have a similar estimate for aE(uE,u + E U ~ )

-

ai(uE,u +

EU~).

The difference in (5.19) equals to

8

The second term in (5.20) is bounded by

CE.

Since

xk

E

the first term in (5.20) is bounded by

since u

E

follows.

H 2 ( 0 ) we have

a xi

E

Lp( 0 ) for p > 2 and (5.19b)

W1'"(Y),

57

CORRECTORS

T h e r e f o r e , i n order t o p r o v e ( 5 . 1 7 ) , prove t h e s i m i l a r r e s u l t s for

aE

P‘

(5.181, it s u f f i c e s t o

a:@,w i t h

a fixed @

B ( 8 ), i . e .

E

that

+

(5.21)

aE(u

(5.22)

ai(uE,u

@

+

U (u,u)

,

E U ~ )+

U0 ( u , u )

.

cul,u

+

)

@

W e have

(5.23)

a @E ( u

+

€ul,uE)

a

(u = i a iEj. ax j “t

+

EU

)

1

The s e c o n d t e r m i n ( 5 . 2 3 ) c o n v e r g e s t o

AE(u

The f i r s t t e r m i n ( 5 . 2 3 ) e q u a l s

+

EU

)

1

= A u

3

+

A2u1

E U ~ () u E @ ) d x .

we have

By v i r t u e of t h e c o n s t r u c t i o n o f u l , AE(u

+

+

2 a u i n L ( 8 )w e a k l y

so t h a t , w i t h ( 5 . 2 4 ) , w e o b t a i n , aE(u

@

+

€ul,uE)

+

( Uu,u@)

t

-

u P-

which p r o v e s ( 5 . 2 1 ) . For proving (5.22) we observe t h a t

?k a xi d x

=

4 (u,u) $

58

ELLIPTIC OPERATORS

Corollary 5.1. We suppose that the hypothesis of Theorem 5.1 hold true. F be any measurable set

5

8

Let

. Then

Proof. According to Theorem 5.1, the expression in (5.25) has the same limit as

and one completes the proof as in the proof of (5.15). Orientation. A natural goal is now to obtain error estimates for z or for results for 5, in H1 ( 8 ) . We shall give in Section 5.5 below the Dirichlet's boundary conditions (and we shall return to this question in Section 18).

We found it convenient for getting these

error estimates to write the equation AEuE = f as a first order system; this is made in Section 5 . 4 , as we shall see it leads to two results of independent interest:

(i) the so-called

dual

CORRECTORS

59

formulas (cf. Remarks 5.8 and 5.9 below); (ii) a strong approximation 2 in (L aE grad uE (cf. Theorem 5.2, Section 5.5).

for

5.4

First order system and asymptotic expansion. We consider equation (1.5), where for simplicity (the zero

order terms do not causehere any difficulty) we assume that (5.26)

ao(y) = X > 0

.

We suppose that we are in R 3 , but this is just for the convenience of writing.

We write (1.5) as a first order system with

the idea of getting a better approximation of the first derivativest Since (1.5) is equivalent tott (5.27)

-div(aE grad uE) +

XUE = f

we can write the first order system a" grad uE (5.28)

-div v

-

vE = 0

+ Xue

= f

,

.

We look for an expansion of the form (as in Section 2) =

uo

+

EU1

+

...

V" =

vo

+

"V1

+

...

u (5.29)

,

where u .(x,y), v . (x,y) are Y-periodic. 7 I By identification, we obtain

'We

write here a" instead of [a"] as in Section 3 . 4 .

ttIt is this idea which leads to the mixed approximation techniques in Finite Elements.

.

ELLIPTIC OPERATORS

60 (5.30)

a grad uo = 0 Y

(5.31)

-div vo = 0 Y

(5.32)

a grad u1 Y

(5.33)

-divY v1

-

+

,

, a gradx uo

divX vo

+

-

vo = 0 ,

Xuo = f

.

Since a is invertible, (5.30) is equivalent to grad uo = 0, Y i.e. (as it should!) (5.34)

uo = u(x)

.

Using (5.34) in (5.32) gives: (5.35)

grad

Y

u1 + gradx u = a-1v

0

which can be solved in u1 iff rot grad u = rot (a-1v,) Y X Y i.e. (5.36)

rot (a-lv0) = 0 Y

.

We have now two ways of proceeding, either by exploiting (5.36) first, or by exploiting (5.31) first. Let us start with (5.36). (5.37)

a-1 vo = wo

We set

,

and we define g (wo) by taking the average of each component. rot (wo - hi(w0)) = 0 Y (5.38)

?a

(wo -

h (w,))

= 0

,

,

Then

61

CORRECTORS hence it follows that there exists p such that (5.39)

w0

-

,

q(wo) = grady p

p(x,y) being Y-periodic

.

We now use (5.37) and (5.39) in (5.31) to get (5.40)

-div a grad p = -div a ~ ( w , ) Y Y

.

We can solve (5.32) forul iff (5.41)

grad u = hl(a-’v0)

=

Iml(Wo)

-

Therefore (5.40), (5.41) give, with the usual notations (5.42)

p = -xJ

&

(defined up to the addition

7

.

of a function of x )

We can solve (5.33) forvl iff (5.43)

-div 3;i(vo) + Xu = f

.

Using vo = awo and (5.39), (5.41), (5.42), (5.43) gives the usual equation U u + Xu = f. Let us start now with (5.31). (5.44)

‘m

(5.45)

Go = vo

(vo) = P

-

We set

t

.

p

Then (5.46)

div Go = 0 Y

,

?7l ( G o ) = 0

.

We claim that (5.46) is equivalent to (5.47)

Go = rot @

Y

,

div @ = 0 Y

,

@

Y-periodic in y

.

ELLIPTIC OPERATORS

62

It is obvious that (5.47) implies (5.46). Conversely, if Go satisfies (5.46), we can find $ such that (5.47) holds true. begin by observing that-by

We

the standard projection theorem-there

exists $, such that $ is Y-periodic and div $ = 0 and Y (rotY $,rotY

(roty G o , $ )

=

J,)

(5.48)

.

V II, such that J, is Y-periodic, divY J, = 0, rotY J, E (L*( Y ) ) ~

Then

- Go)

rot (rot $ Y Y

grad Y n ’

div grad n = An = 0 so that n = constant in Y

n being Y periodic, and

y.

=

Therefore

(5.49)

-

rot (rot $ Y Y

Go) = 0

.

By virtue of (5.46) (5.50)

div (rot $ Y Y

-

Go) = 0

and (5.51)

(rot $ Y

-

Go) = 0

.

But (5.49), (5.50), (5.51) are equivalent to rot $ Y and (5.47) follows.

-

Go = 0

We now use (5.47) in (5.36): rot a-1 (rot $ + Y Y (5.52)

div

Y

$ = 0

,

p) = 0

$

,

Y-periodic

.

We use now (5.52) to express $ in terms of p . e

We define

1 = {l,O,O}, e2 = {O,l,O}, e3 = {O,O,l} and we define gp(y) as the solution of

63

CORRECTORS rot a-l rot jiP(y) = rot(a-lep) (5.53)

div

'2

= 0

,

,

jip is Y-periodic ,

3;1 ( 2 ' )

=

.

0

Then

Therefore (5.55)

vo = p

-

rot 2P(y)pp(x) Y

.

If we define the matrix rot 2(y) by Y (5.56)

rot i(y)-s = roty iP(y)C P Y

we can write (5.55) in the equivalent form (5.57)

vo = (1

-

rot :(y))p(x) Y

.

We take the y averages in (5.32) and (5.33) after multiplying by -1 a ; we obtain (5.41) which can now be written (5.58)

grad u =

-1 (a (I

-

rot ?)p(x)) Y

Also (5.43) with use of (5.44) becomes (5.59)

-div p

+

Xu = f

.

If we define the average of a matrix by

(5.58) gives

grad u = i.e., p =

a (a-l(I -

rot Y

2 ) ) ~

3tl (a-l(I - rot 2 ) ) -1grad u Y

.

ELLIPTIC OPERATORS

64 Then (5.59) gives (5.60)

(a-l(I - rot ?))-'grad . Y

-div

u

+

Xu = f

.

Remark 5.8. Since we necessarily obtain the same homogenized operator, it follows--if (5.61)

(5.60) is justified-that

~

(a-1 (I

-

rot Y

1-l =

[

41

.

But, according to usual formulae

[ a ]= (5.62)

grad Y

x

Ih(a(1 =

a-1(I

-

grad x)) Y

matrix(grad XI) Y

This is indeed true. (5.63)

-

.

We canobtaina little bit more, namely that

rot Y

f)

=

(1 - grad Y

x) [ a I -1

which gives (5.61) after taking averages. Since both sides of (5.63) consist of periodic matrices, it is enough to verify that (5.64)

rot (a-l(I - rot Y Y

(5.65)

div (a-l(I - rot Y Y

2))

=

rot (I - grady Y

='

div a(I Y

-

x) [ a 1 -1 ,

grad Y

x) [ a 1-l ;

but using div rot = 0, rot grad = 0 and the definitions of x and Y Y Y Y ?, one sees that all expressions which appear in (5.64) and in (5.65) are zero, hence the result follows. Remark 5.9. We have therefore two formulas for computing these dual formulas.

[ a ] ; we shall call

65

CORRECTORS Remark 5.10. We can choose for a solution v1 of (5.33) (5.66)

v1 = -rotx(%P)

.

Indeed, we note that div rotx = -divx rot Y Y defined by (5.66) then

SO

that if v1 is

-div v1 = divx rot ( g p ) = (according to (5.55)) div Y Y - divx v

p

0

-

(according to (5.59))hu

=

f

-

divx v0

i.e., v1 satisfies (5.33).

5.5

Correctors:

Error estimates for the Dirichlet's problem.

We now return to the question of correctors, in the sense of Section 5.2.

We confine ourselves to the Dirichlet's problem, but

we do not assume symmetry.

The notations are those of Theorem 5.1.

Theorem 5.2. The hypothesis are those of Theorem 5.1. 1 We consider the Dirichlet's problem (V = H o ( B ) ) .

,Then, mE

being chosen as in (5.3), one has (5.67)

z E = uE

(5.68)

qE =

a '

-

u

EmEul

grad uE

where (5.69)

-

u1 = -xJ ax

j

-

+

vo

-

0

1 & Ho( 8

E V ~ -+

0

) strongly

&

,

(L2 (8))3 strongly

,

66

ELLIPTIC OPERATORS vo = ( I

(5.70)

p =

v1

-

rot

m ( a-1 ( I -

rot

-rotx(ip)

.

=

, .

g)p

Y

Y

g))-'grad

u

,

Remark 5.11. For g e n e r a l boundary c o n d i t i o n s , w e have a s i m i l a r r e s u l t ( w i t h o u t m E ) on e v e r y 8

'

-

s u c h t h a t 8"

C

0

.

Remark 5.12. The r e s u l t ( 5 . 6 8 ) d o e s n o t f o l l o w from ( 5 . 6 7 ) . s u p p r e s s t h e t e r m cul

W e may a l s o

i n (5.68).

Remark 5.13. W e c a n o b t a i n e r r o r estimates a s f o l l o w s .

2 S i n c e w e know t h a t u E H ( O ) , i t f o l l o w s t h a t

q f i n i t e i f n = 2.

T h e r e f o r e w e c a n assume t h a t t h e r e e x i s t s

p > 2 such t h a t (5.71)

E Lp(8)

.

Then

T h e r e f o r e t h e b e s t p o s s i b l e r e s u l t (by t h e t e c h n i q u e s used h e r e )

i s o b t a i n e d when w e have ( 5 . 7 1 ) w i t h p =

estimate i s

cE1I2.

+m,

i n which case t h e error

61

CORRECTORS Proof of Theorem 5.2. We set

and we shall show below that (5.74)

I blEI I (L2

3 -

(5.75)

I lgZEl I

- CE

2 L

CE

U2P'

.

Let us assume these estimates for a moment.

We multiply the

first (respectively, second) equation (5.73) by grad zE (respectively, zE).

+ (div nE,zE)

=

Since z

1

E HO( 0 )

we have (n,grad z E )

0, so that

hence it follows that

hence (5.72) follows (the estimate in nE follows from qE =

aE grad z E

-

glE).

Proof of (5.751. We have gZE = -div(aE grad u E )

+

(div, vo

+

XuE

+ E-ldivY vo

+ divY v1 - Xu) + c(divX v1 - Am €u 1)

.

68

ELLIPTIC OPERATORS

But - d i v ( a '

grad u E )

+

Au

g 2 € = c ( d i v x v1

= f and u s i n g

--

(5.311,

(5.33) w e o b t a i n

,

AmEul)

hence ( 5 . 7 5 ) f o l l o w s .

Proof of

(5.74).

W e have glE = a'

-

g r a d uE

vE

-

EaE grad(mEul) grad u

= -a'

+

-

+

Vo

-

a E grad u

+

+

vo

€aE grad u

EV

1

+

1 Ea'

grad(1

-

m )u E

l

.

EV1

Using ( 5 . 3 2 ) , w e o b t a i n glE

= bE

-

cE

+

dE

,

b

= E ( V ~

c

= -€a' ( g r a d mE)ul

dE = a'(1

E

+

g r a d mE = 0 i f m

-

mEaE g r a d x u l )

5

,

,

mE)grad y u1

-

= 1, t h e e s t i m a t i o n o f c E and of d E r e d u c e s

(assuming ~1 E W 1 ' " ( Y ) )

Using ( 5 . 7 1 ) , YE

where

t o t h e e s t i m a t i o n of

C E ~ " ~ ' .

W e c o n s i d e r now t h e s i t u a t i o n o f Theorem 5.2 b u t w i t h o u t

introducinq mE.

W e have

CORRECTORS

69

Theorem 5 . 3 . The h y p o t h e s i s a r e t h o s e o f Theorem 5.2. (5.76)

and n,

5, = u

-

as i n ( 5 . 6 8 ) ,

u

-

ELI

W e define

1 '

( 5 . 7 0 1 , u1 b e i n g d e f i n e d as i n ( 5 . 6 9 ) .

W e have

Remark 5.14. S i n c e w e d i d n o t i n t r o d u c e m E , 5,

9

1

V = H O ( s 8 ) ; b u t on t h e

1

o t h e r hand w e o b t a i n a b e t t e r H ( 8 ) a p p r o x i m a t i o n .

Proof. for s

W e use here spaces H S ( r )

=

.+ 1

we r e f e r , f o r instance to

[l] for t h e main p r o p e r t i e s of t h e s e s p a c e s .

Lions-Magenes

W e w i l l need

(5.78)

IU1

I IH1/2 ( r )

<

CE

-1/2

(5.79) I

W e a s s u m e t h e s e e s t i m a t e s f o r a moment and w e show t h a t t h e y imply ( 5 . 7 7 ) . a'

W e observe t h a t g r a d 5,

- nE

- hls

(5.80)

- d i v TI, + with

= h2€

,

,

ELLIPTIC OPERATORS OPERATORS ELLIPTIC

70 70

hl, = = EE (( VV ~~- ' gradx uu,,)) hl, aa E gradx

-

= ~ ( d i v ~ vXul) ~~ h2E = ~ ( d i v ~ vXul) h2€

Therefore IhlEl Ihl,l Therefore

++

,

.

IhZE] << CE C E and and ((5.80) gives IhZE] 5 . 8 0 ) gives

by CC EE]]IICCEE I] 1 . with aa right right hand hand side side bounded bounded by with But But r

so that

By virtue of (5.79), and since 5 , = (5.82)

I (n,,grad

6,)

+ (div

Using (5.82) into (5.81) gives

hence the the result result follows. follows hence

nEl1;,)

-

u1 on

I 5

r,

we have

CEI lull

IHll2

(r)

< CE 1/2

.

71

SECOND ORDER ELLIPTIC QPERATORS

Proof of (5.78). It follows from (5.69) (where ~1 = ~1 ( x / E )

)

that

Proof of (5.79). Since

I Ivo +

E

V

I~

But aE grad uE = v 2

is bounded in L

(

~

< C, we have only to verify that

LZ(r) -

is bounded in (L2 ( 8 ))3 and div vE = AuE

8). It is known (cf. Lions-Magenes, loc

-

cit.)

that

hence (5.83) follows. Remark 5.15. We shall obtain in Section 18.2 below a better estimate in symmetric case.

6.

6.1

Second order elliptic operators with non-uniformly oscillating coefficients. Setting of the problem and general formulae. We consider now a family of operators of the type

the

f

ELLIPTIC OPERATORS

72

with the following hypotheses on the coefficients aij, ao. We denote 0 m n by C ( 8; L (R 1 ) the space of functions I$ (x,y) defined in 8 x Rn, P real valued, such that $(x,.) E L ~ ( R ~ ) ,x E 5 , x + $(x,.) is m n continuous from 8 + L (R ) , and I$ is Y-periodic in y,V x (we recall

-

-

this fact by the index "p" in Lm(Rn)). P the norm

0

We provide C (z';Lm(Rn))with

P

we now assume that

x E B

,

a.e. in y ,

and that a .

E

C

0

(6.4)

(In what follows, we can take a. = 0 if V consists of functions in H 1 ( 8 )which are zero on a part r0 of r , measure r o > 0.) For u,v E H 1 ( 8 ) ,we set (6.5)

aij(x,F) au av dx ax. axi 3

and we denote by uE the solution of

+

ao(x,:)uvdx

,

73

SECOND ORDER ELLIPTIC OPERATORS where f is given in V'. We want to study the behavior of uE

when

E

+

0.

Remark 6.1. In case the a . . (and ao) are constant in x, the problem reduces 13

to the one solved in Sections 1 to 3 . We shall prove the existence of an homogenized operator constructed as follows:

4.. (x) is constructed as follows; we define 13

and we define x3(x,y) as the unique solution (up to an additive constant) of

XJ depends continuously on x with values in W(Y) For $,$ (6.11)

E

.t

H 1 (Y) we set

aij(xty)

al(x;$,$) = Y

a ~a@ j dy aYi

,

tThis will be the case if we define for instance xj(x,y) by

ELLIPTIC OPERATORS

14

then q . . (x) is given by 13

Remark 6.2. The functions x

-

+

q..(x) are continuous in 6 17

.

The polynomial

q..(x)C.S.being positive definite for every x (cf. Remark 2.6) it 17

1

1

follows that there exists y > 0 such that

We set, v u,v E (6.14)

a (u,v) =

~'(0)

I

qij(x)

au axj

av

dx

+

I

qo(x)uvdx

.

8

8

By virtue of (6.13) and since (6.4) implies qo(x) 2

ao,

it

follows that there exists a unique u such that

The main result of this Section is as follows: Theorem -6.1. Under the hypotheses (6.3), (6.4), the solution uE fo converges in V weakly towards u, solution of (6.151, where

.. .,(6.12).

d l are constructed by (6.7),

called the homogenized form of

AE.

The operator

(6.6) ( a &

a is still

SECOND ORDER ELLIPTIC OPERATORS

75

Remark 6.3. With t h e n o t a t i o n s o f S e c t i o n 2 , one c a n c o n s t r u c t a n a s y m p t o t i c expansion.

We have AE = €-*A1[x)

A (X) =

1

+

E

-1

A2

a - aYi

I f w e look f o r uE i n t h e form

(6.17)

uE = u o ( x , y ) +

EU~(X,Y)

+

E

2

u2(xry)

+

...

,

we f i n d ( a s i n S e c t i o n 2 ) t h a t (6.18)

etc.

A1(x)u0(x,y) = 0

Equation (6.18) g i v e s

so t h a t ( 6 . 2 0 ) r e d u c e s t o

S i n c e x p l a y s h e r e t h e r o l e of a p a r a m e t e r , it f o l l o w s t h a t (6.21)

u,(x,Y) = - x ' ( x , Y )

au ax (x) j

+

ii,(X)

.

76

ELLIPTIC OPERATORS Equation (6.20) admits a Y-periodic solution in u2, iff.

which leads to a u = f . Remark 6.4. Assuming all data to be smooth, one can prove, as in Section 2.4 that, for the Dirichlet's problem

Remark 6.5:

(Correctors).

Let us assume that AE is symmetric (aij = aji, V i,j) and let us assume that

We use the same notations (6.23)

BE = - E X

j

as

in Section 5 and we introduce

x au (x,~) (x)mE(x) j

ax

.

Then we have Theorem 6.2. We assume that the hypotheses of Theorem 6.1 hold true and that 2 V i,j and that we have (6.22). Then if f E V' n Lloc(B)t

=

(6.24)

zE =

uE

-

(u + B E )

+

0

in V strongly.

SECOND ORDER ELLIPTIC OPERATORS Proof:

77

(Assuming Theorem 6.1 proven).

We argue as in Section 5.

We have

We observe that by virtue of (6.22) one can check that

that the solution u of (6.15) satisfies (if suitable V ) (6.27) u E H2(c+)

so

r

is smooth and for

.

It remains, as in Section 5, to prove that X

+

But

Q(U).

where rEk contains the same terms as in the proof of (5.15) and the term

-

E

a j -x-

a Xk

(x,x/E)

&j (x)mE(x), which tends to 0 in L2

(

8)

.

Hence

the result follows. Before proceeding to the proof of Theorem 6.1, we prove a t

variant of this theorem, which is of independent interest. 6.2

Homogenization of transmission problems. Let us consider a partition of 8

t

'Equality (6.29) below does not take into account the interfaces between the sets 8 k

.

ELLIPTIC OPERATORS .

N

o= u

(6.29)

o'k,

k=l

and let us give a family of functions alj(y) such that ak ij (y) E Lm(Rn)

k aij(y) is Y-periodic V i,j,k

,

(6.30)

aij(Y)Citj 2 aCiCi

,

a > 0

,

a.e. in y

, k

=

...,N

1,

and also a family of functions ak ( y ) : 0 :a (6.31)

E Lm(Rn)

k ao(y)

2 a.

k a . is Y periodic

,

> 0

.

a.e. in y

,

bk

,

For u,v E H 1 ( 8 ) , we define (6.32)

aE(u,v) =

& N

a:j(~) x au av dx + 1 ax. axi I

a

Ok

We take V as in preceding sections.

By virtue of (6.30),

(6.31), there exists a unique uE E V such that (6.33)

aE(uE,v) = (f,v)

, v v

E

V

I

f E

V'

.

We want to "homogenize" this problem, which is indeed possible. Remark 6.6. If we define (6.34)

k aij(x,y) = aij(y) in 8 k

(6.35)

k aO(x,y) = ao(y) in

,

...,N

k = 1,

,

, b k,

problem (6.33) enters into the framework of (6.6), but with functiors aij which are not in C0 ( -.0 ;Lm (Rn ) ) , but step-functions with values in P

SECOND ORDER ELLIPTIC OPERATORS

79

Actually Theorem 6.3 below will show that (as one could

in L;(Rn).

easily surmise) the "regularity" condition a. E 11

not necessary in order that Theorem 6.1 be true. m

(s

;Lm(Rn)) is P (The condition

Co

m

aij E L ( 0 ; L (R")) is too general, since then aij (x,x/E) need not

P

even be measurable in x). The homogenization of (6.33) proceeds a s follows. Q k = -

-?!-

qij ax.ax. 1

=

We define

homogenized operator associated to

7

(6.36)

and we define N

dx

(6.37)

+

Ik

qiuvdx]

,

8

k k where qo = f i (ao). Let u be the solution of (6.38)

u E V

,

a ( . , . )

=

(f,v)

,

V

v E V

.

One has Theorem 6.3. We assume that (6.29), (6.301, (6.31) hold true.

Let uE

(respectively, u) be the solution of (6.33) (respectively, (6.38)). Then one has u

+

u

in V weakly

.

Remark 6.7. Problem (6.33) is a transmission problem; there are transmission boundary conditions on the interfaces between adjacent sets 0'.

Problem (6.38) is the homogenized problem.

ok and

80

ELLIPTIC OPERATORS

Remark 6.8. Applying (formally) the rule of Theorem 6.1 to the present situation leads to (6.38). Remark 6.9. One can also introduce (at least in the symmetric case, i.e., k k V i,j, V ' k ) correctors. One has then to introduce a . . = aji 13

(6.39)

mk (x)

=

"truncation" function in 8

We define k k a .(y) -)ayj a (a. A 1 = - -aYi 11

Then we introduce the corrector N

and we can prove that (6.41)

uE

-

(u

+

BE)

Proof of Theorem 6.3. We have (6.42)

I Iu, Iv -< c .

If we set (6.43)

C iE t k

+

0

in V strongly.

(cf. (5.3))

.

SECOND ORDER ELLIPTIC OPERATORS

81

w e c a n e x t r a c t a s u b s e q u e n c e , s t i l l d e n o t e d by u E , s u c h t h a t u (6.44)

u

+

Sifk

-f

i n V weakly

5:

in L

2

I

( sk )

weakly

,

k = 1,2

,...,N

,

E q u a t i o n (6.33) g i v e s i n t h e l i m i t

and it s u f f i c e s t o p r o v e t h a t

But t h i s i s p r o v e n e x a c t l y l i k e w e p r o v e d (3.24).

We introduce w

such t h a t k *

(A1)

w = 0

,

w

-

P(y) is Y periodic

;

W e define next

w (x) =

where A E r k = i n (6.33).

EW(X/E)

a - axi

.

( ak. . ( x / E )

=I

W e multiply

a ax).

W e then take v =

$ E

8(5 k ) ,

(6.47) by @mE a n d w e i n t e g r a t e o v e r 8 k.,

a f t e r s u b t r a c t i n g , we can p a s s t o t h e l i m i t i n

as i n Section 3 .

WE,

j

6

a n d w e o b t a i n (6.46)

82 6.3

ELLIPTIC OPERATORS

Proof of Theorem 6.1.' We are going to "app-roximate"the a. ' s by step functions and

11

use Theorem 6.3.

We introduce a "small" parameter htt and a

"triangulation" of 3 :

co (6.48)

=

N (h) u k= 1

0 ; nCO;'

6 ; (equality up to interfaces) ,

(diameter of Of course N(h)

+

-

as h

,

if k # k'

L3 ,

=

.k eh) 5 Ch . +

0.

For every h we define step functions aijh(x,y) by k k aijh(x,y) = a . . (y) in 8, 1lh

I

k

=

.k measure of B h

,

,

and we define (6.50)

au aijh(x,-) E ax axi dx j

-

ai(u,v) =

4

+

I

X

aOh(x,E)uvdx

0

where aOh is defined in a similar manner. ttt Let uEh be the solution of

tAnother approach is presented in Section 6 . 4 below. ttWhich plays the role of the parameter h in finite element methods. tttFor the symmetry of the procedure. straightforward anyway.

The zero order term is

SECOND ORDER

(6.51)

ai(uah,v) = (f,v)

,

ELLIPTIC

V v E

a3

OPERATORS

v ,

u ah E V .

Then s i n c e t h e a

i j

' s a r e u n i f o r m l y c o n t i n u o u s from 0

+

L;(Rn),

one h a s (6.52)

uE

-

uEh

+

in V as h

0

+

0

,

uniformly i n

E.

For h > 0 , f i x e d , w e c a n homogenize ( 6 . 5 1 ) , u s i n g Theorem 6.3.

We

find that (6.53)

uEh

+

uh

i n V weakly

,

where uh i s t h e s o l u t i o n of

and where 'Oh i s computed a s f o l l o w s .

W e define

and w e i n t r o d u c e x r ( y ) a s t h e s o l u t i o n ( u p t o a n a d d i t i v e c o n s t a n t )

of (6.56)

Alh(y) k (x?(y)

Then i f w e set

then

-

y.) = 0 3

,

X?

(y)

Y-periodic

ELLIPTIC OPERATORS

a4

One t h e n v e r i f i e s t h a t uh

(6.59)

-

u

+

i n V as h

0

+

0

,

and t h e theorem f o l l o w s .

6.4

A n o t h e r a p p r o a c h t o Theorem 6 . 1 . W e c a n g i v e a p r o o f of Theorem 6 . 1 a l o n g t h e l i n e s of S e c t i o n 3 .

The s o l u t i o n u E o f

(6.6) s a t i s f i e s

If we set

w e have

1C;l

(6.61)

5

( L 2 ( 8 ) norm)

C

and we can e x t r a c t a s u b s e q u e n c e , s t i l l d e n o t e d b y u E , s u c h t h a t uE

-+

c:

+

u

i n V weakly

(6.62)

,

2 i n L ( 8 ) weakly

ci

.

The e q u a t i o n ( 6 . 6 ) , w h i c h c a n be w r i t t e n v E

v

(6-63)

( s i av ,5) E

= (f,v)

I

gives in the l i m i t av

(Sit=)

=

(frv)

1

A s i n Section 3.2,

functions.

,

V

v

E

V

-

t h e main p o i n t i s now t o u s e " a d j o i n t "

I n S e c t i o n 3.2 w e i n t r o d u c e d w

'Assuming

t h a t a.

such t h a t

t

= 0 , which d o e s n o t restrict t h e g e n e r a l i t y .

85

SECOND ORDER E L L I P T I C OPERATORS

(6.64)

(AE)*wE = 0 ,

and wE has a known behavior as

E

+

0.

Due to the fact that the coefficients depend on x and on y, the construction of wE as in Section 3.2 is no longer possible. with a known behavior as

are going to construct w (6.65

(AE

I_

+

But we

0 and satisfying

* WE = g E

where gE has a known asymptotic behavior.+

It suffices to look for wE

in the form (6.66)

WE =

E(Yp

-

B)

p = given index

t

.

Using the "adjoint identity" of (6.16), an identification computation leads to

and then (6.68)

gE = A;(yp

-

*

B ) + €A3(yp

-

B)

.

If we define (compare to (6.10)), ip(XrY) by A~(X)*(XP - yp) = 0 (6.69)

and

ip

,

;p Y-periodic in y

satisfies for instance

I

, I

ip(x,y)dy = 0

,

Y we can take 8 =

ip.

We have:

'We will use several times in the following of this book this important method.

86

ELLIPTIC OPERATORS

in

*

where aij

=

LA(&) weakly

,

aji and where in (6.71) we assume all functions to be

smooth enought s o that (6.72)

cAi(yp

-

6)

=

We now take v = @w

-

* A

A ~ X P+

o

in ~ ' ( 6 )

.

in (6.6), we multiply (6. 5) by @uc and we

integrate over d : after substracting, we obtain

(6.73)

Passing to the limit, we obtain

'We

have

The last term in g,

ip is C2

-

+

0, in, say, L2 ( 8 ) (this can be improved) if

in x E 8 with values in W(Y).

use the fact that if

@ E

In obtaining (6.71) we also

C0(~;Lm(Rn)), then

P

where

The regularity hypotheses needed here on the aij's can then be weakened by a density argument as in Section 8 below.

COMPLEMENTS ON BOUNDARY ,CONDITIONS

But using (6.73), (6.84) reduces to

10

and therefore

It is now a simple exercise (similar to Section 3.2) to verify that we obtain the same formulas

7.

in (6.12) , (6.14).

as

Complements on boundary conditions.

7.1

A remark on the nonhomogeneous Neumann's problem.

Let us consider problem (3.4), Remark 3.3, Section 3.1, with V = HI( 6 ) and with

(7.2)

fi is Y-periodic in y

,

and where we assume that the boundary

r is

smooth enough.

a7

ELLIPTIC OPERATORS

88

The interpretation of (3.4) is formallyt X

(7.3)

A

~ =U fo(x,E) ~

(7.4)

-aauE v-

6,

in

X on - fl(x,-)

.

r

AE

if

Then one can apply Theorem 3.1

(3.5) takes place.

As we already noticed X

(7.5)

fo(x,;)

(where

+.

ny(f0(x,y)) in L m ( d ) weak star 1

Ry(fo(x,y)) = -

.

fo(x,y)dy

IYI y

The limit of the surface integral in (7.1) is now completed. We have to distinguish a "generic" case from an "exceptional" one: (7.6)

(Generic behavior):

We assume that

r

does not contain

flat pieces or that it contains finitely many flat pieces with conormal not proportional to any m E Zn; then fl(x,r) X +'Ry(fl(x,y)) in Lm ( (7.7)

(Exceptional behavior): flat piece, say part of

r,

ro,

E',E'',

(respectively, E " , X

We assume that

say rl, being generic.

subsequences

+

weak star

with conormal mo

the generic behavior on

fl(x,E)

r )

..., . .. )

rl;

on

ro,

E

r

.tt

contains onettt

Zn, the remaining

Then one has of course one can extract

such that if

E =

E'

,

f' (x) (respectively, f" (x),

. ..

in L m ( r o )

weak star (where the f',f", ..., are different) tThe interpretation is formal under the hypothesis a ttHere fl(x,:)

denotes in fact its restriction to

ij

E

.

La.

r.

tttExtension to the case of a finite number of pieces is obvious.

COMPLEMENTS ON BOUNDARY CONDITIONS

The proof of (7.6), (7.7) is given in Section 7.3 below.

89

The

application to the problem (7.3), (7.4) is straightforward. (7.8)

In the generic case (7.6), one has uE

+

u in H1(6)

weakly where u is the solution of

au

=

m Y (fo(x,y))

au= (7.9)

in 8 ,

y(fl(x,~)) on

r

;

In the exceptional case (7.7), one can extract subsequences uc,,uE,,, ., which converge weakly fn H 1 ( 8 ) to

..

u' ,u",..

7.2

., where

Higher order boundary conditions. We consider a boundary value problem in the half space

(7.10)

8

=

1 x 1 >~ 0~1

,

r

=

{x~x, = 0 1

We shall set x' = {X1,...,Xn-l~ ,

y' = X'/E

we consider functions (7.11)

aij,ao satisfying (l.l), (1.2)

,

;

.

ELLIPTIC OPERATORS

90

bij(y') (7.12)

E

,

Lm(Rn-')

bij is Y'-periodic

,

n-1

Y' =

TT 10,Y;OI j=l

,

b0 is Y' periodic ,

bo

E

Lm(Rn-')

,

Remark 7.1. The period y:' 0 Yj

3

-

(1

5

j

5 n-1) can be different from the periods

Notations. We introduce the space (7.14)

1 1 n-1 V = {vlv E H ( 6 ) ,v(x',O) E H (R

which is a Hilbert space for the norm given by

For u,v

E

V, we Set

and we consider the boundary value problem

I

91

COMPLEMENTS ON BOUNDARY CONDITIONS

By virtue of (7.11), (7.12)

(7.13)

problem (7.18) admits a unique

solution. The interpretation of (7.18) is as follows: ~ =U f ~ in 0 ,

A

(7.19)

% . + B ~ u ~ = oon r = Rn-1

av

,

AE

where BEuE =

a - a b . . (-)X I ax u E (x',O) + axi 11 E

bo(t)uE(x',O) X'

j

(7.20)

where the summation in i,j extends here from 1 to (n-1) Let us denote by

ff = homogenized operator (in Rn ) of AE (7.21)

Let and

B

.

n

=

,

homogenized operator (in Rn-l) of BE

a (u,v),

.

B(u,v) be the bilinear forms associated to c?

Then the homogenized problem is:

(7.22) U E V ,

and (7.23)

u

+

u

in V weakly when

The proof is as follows.

E

-t

0

.

One has first

I luEl Iv 5

C; we can

therefore extract a subsequence, still denoted by u E r such that

.

ELLIPTIC OPERATORS

92

u

-+

u

in V weakly

,

if we set (7.25)

wE (x') = uE ( X ' ,o)

we have (7.26)

w

-+

1 n-1 w = u(x',O) in H ( R ) weakly

.

au -1/2 n-1 t We define 2 E H (R ) by av

A"

(7.27)

(Rn-l)

I$ E

9

-+

H1/2

,

@ E

HI(&)

,

@(x',o) = 9 #

being a continuous linear mapping from

@

n-1

(R

)

+

H1(ct)

.

This gives a precise meaning to the boundary condition in (7.19), and it implies that

where

8

= Fourier transform of I$.(cf. Lions-Magenes [I].)

COMPLEMENTS ON BOUNDARY CONDITIONS

93

where 5 is given by

The boundary condition in ( 7 . 1 9 ) (7.30)

BEwE =

-

is written

au av AE

and we think of ( 7 . 3 0 )

as an equation in Rn-l.

We can apply the theory of Sections 1 to 3 . (7.31)

av

-+

in H;ica1(Rn-l)

-5

Indeed by virtue of

Strongly

AE

which is sufficient to prove the desired results (in Section 3 ) , we have w (7.32)

+

Bw

w in H ~ ( R " - ~ )weakly, where w is the solution of

=

-5

.

On the other hand the reasoning of Section 3 gives (7.33)

si =

using ( 7 . 3 2 )

axj

q ij au

and ( 7 . 3 3 )

,

q . . = coefficient of 13

(-

a

in ( 7 . 2 9 ) gives

V rP E V, i.e., since w = u(x',O) , the equation ( 7 . 2 2 ) .

)

;

94 7.3

ELLIPTIC OPERATORS Proof of ( 7 . 6 ) ,

(7.7).

L e t u s p r o v e f i r s t (7..6)

(generic behavior).

Since fl(x,x/E)

b e l o n g s (under assumption ( 7 . 2 ) ) t o a bounded set o f L m ( r ) , w e have o n l y t o show t h a t

v smooth and w i t h compact s u p p o r t on

V

k Since C ( 6 )

.

r

n (with p e r i o d i c functions i n y) i s dense Y 0 .i n C ( 8 x Rn) ( w i t h p e r i o d i c f u n c t i o n s i n y ) , and s i n c e f o r m u l a s i n Y 0 (7.34) a r e c o n t i n u o u s f o r t h e C ( 8 x Rn) t o p o l o g y , it s u f f i c e s t o Y prove (7.34) f o r

@

k

C (R )

f l ( x , y ) = f i n i t e sum of f u n c t i o n s a ( x ) B ( y )

(7.35)

,

a and B b e i n g Ck ;

h e r e w e can f a k e k a s l a r g e as w e want and even k =

m.

Replacing av

by v , w e are l e d t o p r o v i n g t h a t

where B i s C

m

and Y-periodic w i t h a l l i t s d e r i v a t i v e s .

We d o n o t r e s t r i c t t h e g e n e r a l i t y by assuming Y = 1 0 , 1 [ " and w e

can u s e F o u r i e r series:

I n (7.37) t h e series i s a b s o l u t e l y and u n i f o r m l y c o n v e r g e n t ( t o g e t h e r w i t h a l l i t s d e r i v a t i v e s ) and one i s reduced t o p r o v i n g ( 7 . 3 6 ) f o r B(y) = f i n i t e sum of

(7.38)

i.e.,

that

P

e x p ( 2 ~ i p y ),

95

COMPLEMENTS ON BOUNDARY CONDITIONS (7.39)

exp(2aip f)v(x)dr(x)

-+

0

,

ifpfO.

r This is a classical result (stationary phase approximation; cf. Bibliography of Chapter 4) under the hypothesis made on

r

in ( 7 . 6 ) .

Proof of (7.7). Let us consider a single particular case, where (7.40)

fl(x,y) = exp(2vipy)

and let us assume that (7.41)

m x = y 0

If p # kmo, k

(7.42)

E

r

mo E Z"

,

.

I " r

fl(x,:)v(x)dr(x)

r

p fixed in Zn

is contained in the hyperplane

,

Z, then

,

f,(~,~)v(x)dr(x)+ 0.

=

exp(F)

1

But if p = kmo, then

v(x)dr(x)

,

r

which admits indeed infinitely many different limits depending on the way

E

tends to 0.

96

ELLIPTIC OPERATORS

8

Reiterated homogenization.

8.1

Setting of the problem:

Statement of main result.

We define n

Y =

77lo,+ j=l

z

-fjIO,z?[ j=1

=

,

3

,

and we consider functions aij(y,z) such that ai j

(8.1)

E

,

L ~ ( R " x R")

a . . is Y - Z periodic 13

,

We also consider ao(y,z) such that

a

0

E

L ~ ( R "x R")

,

and we define nowtt A' by

'As usual this hypothesis is not necessary if, for instance, V consists of functions which are zero on a set r0 of positive measure on r .

ttSee Remark 8.1 below.

97

REITERATED HOMOGENIZATION W e want t o s t u d y t h e b e h a v i o r o f t h e s o l u t i o n uE of boundary

v a l u e problems a s s o c i a t e d t o A E a s

E

+

0.

Remark 8.1. x x Under t h e m e r e h y p o t h e s e s (8.1), (8.2), t h e f u n c t i o n s a i j ( F , T ) E

a (5,x) need n o t even be measurable. 0 E €2 the hypothesis t h a t

W e s h a l l make from now on

Remark 8 . 2 . I f functions aij,

a.

a r e c o n s t a n t i n z , t h e problem r e d u c e s t o

t h e one c o n s i d e r e d i n S e c t i o n s 1 t o 3 .

Remark 8 . 3 . A l l what w e are g o i n g t o s a y r e a d i l y e x t e n d s t o t h e c a s e of

operators with coefficients ai

X x x (-,T,. .. ,+ E

E

E

which a r e p e r i o d i c - w i t h 2 (N-1

,...,Z

= X/E

Z ( l )

(8.5)

a

ij

E

d i f f e r e n t periods-in y = X/E, N = X/E W e would assume t h a t

.

cO(R" z

Remark 8 . 4 . W e can a l s o e x t e n d t h e r e s u l t s t o f o l l o w t o t h e c a s e of

x x c o e f f i c i e n t s a i j ( x , - , ~ ) , assuming t h a t L

E

ELLIPTIC OPERATORS

98 aij(x,y,z)

(8.6)

E

c0(Zx

Statement of main result:

x

R::L~(R~))

.

Homogenization with parameter.

Let us consider first a family of functions aij(A,y),

X

=

parameter: X

E

A = topological set.

A,

We assume that

and we also consider ao(X,y)

Y periodic ,

Then the operator X

(8.9)

is homogenized by the standard formulas, X being a parameter.

This

(entirely obvious) operation is cal ed "homogenization with parameter"

.

We homogenize A'-as

8.3)-in

given by

We consider first the operator

"reiteration of the homogenization" (8.10)

a a (x,X) - axi ij E ax

+ a j

two steps, by

( A , -X ) O

E

which is homogenized with X as a parameter.

More precisely we

introduce (8.11)

A 1,x =

-

a azi aij

a

X,z) a z + a0 (X,z)

i

and we define x j ( z ) as the 2-periodic solution of

REITERATED HOMOGENXZATION AlrA(xi(z) (8.12)

2. 71

such that

\

= 0

xJ(z)dz = Ot

;

z for

E

H 1(Z), we set

then the homogenized operator of (8.10) is given by (8.14)

qA)

a

-

- - axi

a +

ql.(A)

ax

11

ao(A)

I

1

where

and

We consider then the operator (8.17) which is homogenized by (8.18

a- a a = - -axi qij

j

+

90

where qij is obtained as follows. We introduce (8.19)

a

a l = - - ayi qij(Y)

+This implies that

!x

a 8yj

+

1

.

~ O ( Y )8

depends continuously on A.

99

ELLIPTIC OPERATORS

100

and we define XJ as a solution of (8.21)

-

U,(X’

y.) = 0 7

x7

,

is Y periodic

.

Then (8.22)

qij - - Ql(XJ IYI

-

Yj‘X

i

-

Yi)

and

It can be verified that C7 is elliptic. (8.24)

a“(u,v) =

(8.25)

Q.(u,v) =

au

1

av dx +

For u,v E HI( 8 ) we set

qOuvdx

.

The boundary value problems we consider are: (8.26)

u

(8.27)

u

E

E

,

V

V

aE(uc,v) = (f,v)

a

,

,V

v

,

t/

(u,v) = (f,v)

E

V E

,

f

E

V’

,

v ;

the problem (8.27) is called the homogenized problem of (8.26). According to Section 4, there exists p > 2 such that if V = H 1 ( 6 ), 0

and if f E w-lvp(&)

,

10 1

REITERATED HOMOGENIZATION

The v a l u e " p " ( j u s t a l i t t l e l a r g e r t h a n 2 ) is u s e d i n Theorem 8 . 1 below.

Theorem 8 . 1 . W e assume t h a t ( 8 . 1 ) , ( 8 . 2 1 ,

,

f E W-ltP(8)

(8.30)

if

(8.4) hold t r u e .

v

=

W e assume t h a t

~1 ~ ( 6 )

or t h a t

Let

c7 be c o n s t r u c t e d by ( 8 . 1 0 )

( r e s p e c t i v e l y , u ) be t h e s o l u t i o n of

,..., ( 8 . 2 5 ) (8.26)

a n d l e t uE

(respectively,

(8.27)).

Then (8.32)

uE

+

u

&

V

weakly

;

moreover (8.33)

auE au ax + ax i i

(8.34)

@ axi -P @

in

au ax. in

Lp(

8 ) weakly,

L p ( 6 ) weakly,

i n case of

Cz(S)

@ E

(8.4).

,

1

i n case o f The h y p o t h e s i s ( 8 . 3 0 )

(8.30) ;

(8.31).

( o r ( 8 . 3 1 ) ) c a n be s u p p r e s s e d i f o n e r e i n f o r c e s

W e have

+Here p i s r e s t r i c t e d t o p

5

%, b u t o n e

t a k i n g p a s close t o 2 as p o s s i b l e .

is interested i n

102

ELLIPTIC OPERATORS

Theorem 8.2. We assume that (8.1),,(8.2) hold true, together with (8.35)

aij

E

O n C (RY

X

n RZ)

.

Remark 8.5. The practical meaning of the construction (8.10),

..., (8.15) of

4 can be summarized in saying that one homogenizes first with respect to z, considering y as a parameter and that one homogenizes next with respect to y.

This is the "reiterated homogenization."

Remark 8.6. Theorem 8.2 is much simpler to state than Theorem 8.1 and its proof will not use (contrary to Theorem 8.1) the results of Section 4.

Its defect is that it does not contain the "model" problem

aij WE) with aij E L ~ ( R ~ ) . Remark 8.7. One obtains results similar to those of Theorem 8.1 and 8.2 when considering coefficients of the form, say: (8.36)

x x aij(r,F)

.

But one would have then to modify the first corrector. The proof of Theorems 8.1 and 8.2 is given in two steps.

We

first show (Section 8.2) that one can reduce the problem to the case when the coefficients aij are

Cm

in y and in z, and we prove the

Theorems in Section 8 . 4 under this assumption, after using in Section 8 . 3 asymptotic expansions.

103

REITERATED HOMOGENI&ATION 8.2

Approximation by smooth coefficients.

1st Case:

Case of Theorem 8.2.

By virtue of (8.35) we can find a sequence of functions a‘’) E c ~ ( R x ~ ij (8.37)

Y-Z

periodic V i,j

,

such that

We can do the same thing with ao, approximated by smooth functions a(’), 0

although this is by no means indispensable.

We then define (8.39)

I

aE(B)(u,v) =

0 f

I

a : ’ )

(:,$)uvdx

,

3 and we introduce u : ~ ) as the solution of (8.40)

u:~)E V

,

Then it is standard to verify that (8.41)

I IU:’)

-

uEl I v

m

Let us assume Theorem 8.2 proven when the coefficients are C Then we know that

.

104

ELLIPTIC OPERATORS

(8.42)

u:’)

+

u‘’)

i n V weakly

,

as

E +

0

,

B fixed

,

is t h e s o l u t i o n of

w h e r e u(’)

where 0 (’)

i s o b t a i n e d by t h e r e i t e r a t e d h o m o g e n i z a t i o n p r o c e d u r e .

I t i s e a s i l y v e r i f i e d t h a t i f q i( jB )., q : ’ )

d e n o t e t h e c o e f f i c i e n t s of

a!”,then (B qij ,qiB)

(8.44)

qij,qo

as B

+

ulIv+ 0

if 8

+

+

m

.

Therefore

I lu

(8.45)

’)

-

- ,

and w e s h a l l have p r o v e d ( 8 . 3 2 ) .

Case of Theorem 8 . 1 .

2nd C a s e :

Smooth f u n c t i o n s b e i n g n o t d e n s e i n L m , ( 8 . 3 7 ) , longer

(8.38)

no

t h e case w h e r e w e assume ( 8 . 4 ) a n d w e h a v e t h e n t o

apply to

r e l y on t h e r e s u l t s of S e c t i o n 4 . L e t u s c o n s i d e r f i r s t t h e case V = H 1 O( 8 ) under h y p o t h e s i s

(8.30). (8.46)

W e u s e ( 8 . 2 8 ) and w e i n t r o d u c e q s u c h t h a t

1 + 1 = 1 q

P

2 .

W e c a n f i n d a s e q u e n c e a!’)

s a t i s f y i n g ( 8 . 3 7 ) and s u c h t h a t

lj

(8.47)

I la:;)

-

aijl

I

z

L q (Y;CO ( ) )

C o n d i t i o n ( 8 . 4 7 ) means t h a t

+ O

a s ’ + m .

REITERATED HOMOGENI!AATION

W e c o n s t r u c t ( f o r t h e symmetry o f t h e p r o c e d u r e ) a s e q u e n c e

aAB) w i t h s i m i l a r p r o p e r t i e s . W e d e f i n e n e x t a‘‘’)

( u , v ) and u : ~ ) by ( 8 . 3 9 ) and ( 8 . 4 0 ) .

W e have t h e n

(8.49)

I n d e e d w e have t h e i d e n t i t y

L e t u s e s t i m a t e t h e g e n e r a l t e r m o f t h e r i g h t hand s i d e o f

(8.50) containing f i r s t o r d e r d e r i v a t i v e s .

I t i s bounded by

it remains t o e s t i m a t e

105

ELLIPTIC OPERATORS

106

8

where we have set

But

< C

I

$(’I (y)dy ,

Y

and

g”)

(y)dy is the expression (8.481, hence (8.49) follows.

Y

Assuming Theorem 8.1 to be proven for aij smooth, we can therefore “homogenize” problem (8.40) and we shall have (8.421, (8.43). It remains to verify (8.44) under (8.48). to (8.11))

and define ~ 1 ‘ (z) ~ )by Y

We set

We define (compare

REITERATED HOMOGENIZATION

107

If we can show that (8.56)

qtjB) (y)

+

1

qij (y)

in Lq(Y)

then, by an argument similar to the one leading to

8.49)

we shall

have again ( 8 . 4 5 ) and the desired result. In order to verify ( 8 . 5 6 ) , and by virtue of (8 4 8 ) , it remains to show that

where Cij(y) =

a xj aik(y,z) 2 dz Z azk

I

But

We have

so that (8.58) gives

and therefore

.

108

ELLIPTXC OPERATORS L e t u s c o n s i d e r now t h e case when w e h a v e ( 8 . 3 2 ) .

We still

a p p r o x i m a t e a . . by a!!) as i n (8.47). The r e s u l t ( 8 . 5 6 ) ( w h i c h d o e s 17 13 W e replace n o t depend on boundary c o n d i t i o n s ) s t i l l h o l d s t r u e . ( 8 . 4 9 ) and ( 8 . 4 5 ) by l o c a l r e s u l t s :

8.3

Asymptotic expansion. W e u s e now t h e m e t h o d s of S e c t i o n 2 , w i t h t h e v a r i a b l e s x , y , z .

One h a s (8.59)

A E = E - ~ A+ ~ E -

3A2

+

E - ~ A+ ~ € - l A q

where

(8.60)

W e a r e l o o k i n g for u

(8.61)

where

u

= uo

+

+

E U ~

i n t h e form 2

E U2

+

...

,

+

E 0 A5

,

REITERATED HOMOGENIZATION (8.62)

u

1

= u.(x,y,z) is

1

109

Y periodic in y and Z periodic in z

An identification in powers

of

E

.

in the equation AEuE = f

leads to

,

(8.63)

AluO = 0

(8.64)

Alul + A2u0 = 0

(8.65)

A1u2

+

A2u1 + A3u0 = 0

(8.66)

A1u3

+

A2u2 + A3u1

+

A4u0

= 0

(8.67)

A1u4

+

A2u3

+

+

A4u1

+

,

A3u2

, ,

A5u0 = f

,

etc.

In the equation (8.63), x and y are parameters, and the general solution of (8.63) is (8.68)

uo = G0(x,y)

.

Using (8.68) into (8.64) gives (8.69)

aa. . a; A u = 13 1 1 azi aYj

.

We introduce (notations are consistent with (8.12)) ~ ' ( z as ) the Y Z periodic solution in z of

Then (8.69) gives

We can solve (8.65) for u2 iff (8.72)

2 121

I

(A2u1

+

A3uO)dz = 0

.

ELLIPTIC OPERATORS

110

Using (8.71) i n t o (8.72) l e a d s t o t h e i n t r o d u c t i o n o f

which c o i n c i d e s w i t h (8.19) generality)

( i f a.

= 0 which d o e s n o t

restrict t h e

.

With t h i s n o t a t i o n (8.72) i s e q u i v a l e n t t o

a p0= i.e.

0

I

I

(8.74)

Go

= u(x)

.

Then (8.71) r e d u c e s to (8.75)

u1 = ii,(x,y)

and (8.65) r e d u c e s t o

hence

+ A3u1 + A4uO)dz = 0 ,

W e c a n s o l v e (8.66) for u3, i f f ,

i.e.

Z

,

i.e.

,

aYi

I Z

aij(y,z)dz

ax

= 0

j

REITERATED HOMOGENIZATION

111

We introduce XJ = xJ(y) by t

-

al(xJ

(8.79)

yj) = 0

,

XJ

Y-periodic

,

(xj is defined up to an additive constant), and we have

A necessary condition for being able to proceed with the computations in (8.67) is: Zlf

Since

II

A1u4dydz

YXZ

=

II

A2u3dydz = 0, (8.81) reduces to

YXZ

We use (8.80) into (8.82) and we obtain in this manner (8.83)

-qij A

2

axiax =

f ,

j

with

9. +

aik

aZk

d]dydz ayL

But (8.22) gives

tThis coincides with (8.21)

.

.

.

112

ELLIPTIC OPERATORS

and since

121

qf . (y) =

should!) qij =

13

Gij.

1

(aij

-

axj aik xazk ) d z , we see that (as it

Z

Remark 8 . 8 . For the Dirichlet boundary condition one can prove (as in Section 2.4) that

8.4

Proof of the reiteration formula for smooth coefficients. We assume now that a . . 13'

a.

satisfy (8.1), (8.2) and moreover

that

We use a technique similar to that of Section 6.4. given index in [1,2,...,n]. (8.87)

Let X be a

We look for a function w E ' solution of

(AE)*wE =

with (8.88)

a, 6, y,

w

=

ECL

+ E 2(zA + B) +

6 to be defined.

identification leads to

E

3

y

+

E

4 6

,

Using the adjoint expansion of ( 8 . 5 9 ) ,

an

113

REITERATED HOMOGENLZATION

,

(8.89)

Ala = 0

(8.90)

AI(B

(8.91)

A 1y

+

A2(B

(8.92)

A;6

+

Aly

*

+

z,)

*

+

Ala = 0

+

2,)

+

A3a = 0

A;(B

+

2,)

+

,

+

,

Aaa = 0

,

and then we have

Equation (8.89) is satisfied iff a = a ( y ) .

Equation (8.90) in

6 means

*

A ~ B= (where we write a

a

*

* aa 13 aYj

a

aiX + - a . .

azi

*

= aji); if we introduce (compare to (8.70))

ij

XiCz) by Af(?i(z)

-

z.) = 0 3

,

;j(z)

Y

is Z periodic and

(8.94)

z then

We can compute y solution of (8.91) iff

which is equivalent to

ELLIPTIC OPERATORS

114

W e r e p l a c e i n (8.96) B by i t s v a l u e from (8.95); i t becomes

But u s i n g (8.73) one sees t h a t (8.97) i s e q u i v a l e n t t o

* U,a

*

=

- UlY,

*

W e d e f i n e j x ( y ) by

x A

Y-periodic

.

Then w e can t a k e

so t h a t (8.95) becomes

I t i s t h e n p o s s i b l e t o s o l v e (8.91)for y .

We can next solve

(8.92) f o r y i f f

and t h i s i s an e q u a t i o n i n

E.

W e can then solve f o r

b

iff

REITERATED HOMOGENIZATION

115

a condition which is satisfied, so that the calculation is possible. Moreover by virtue of ( 8 . 8 6 ) we can assume that a, 5, y,

(8.101) gE

6 are Cm functions in y and in

is also

Cm

in y and z

.

We can now pass to the limit.

We know that

set, as usual,

:C

=

x x aij (TI--) 2

ax j

and we can assume that (8.102)

uE

+

u in V weakly

,

c:

+

ti

2

in L ( 0 ) weakly

,

and we have (assuming a . = 0):

We take $ E S : ( 8

and we choose v = $w

)

multiply ( 8 . 8 7 ) by $uE.

We obtain

But

aw

azx + a5 + E = ag + axi

P,(y,z)

ayi

azi

E Cm

azi

in y and z

Ep E

(y,z)

,

and wE * xx in L 2 ( 6 )strongly

.

,

in ( 8 . 2 6 ) and we

ELLIPTIC OPERATORS

116

Therefore (8.104) g i v e s

t

Using ( 8 . 1 0 5 ) i n t o ( 8 . 1 0 3 ) g i v e s t h e d e s i r e d r e s u l t p r o v i d e d w e v e r i f y t h a t ( u s i n g ( 8 . 9 9 ) and ( 8 . 1 0 0 ) ) :

This i d e n t i t y is a simple computational e x e r c i s e .

8.5

Correctors. W e introduce the f i r s t order c o r r e c t o r (using notations of

S e c t i o n 5)

eE

(8.107)

=

+

( E U ~

) m (x)

E 2u

2

E

i.e., (8.108)

W ' e

=

OE =

-EX

j x (-)

E

au ax

(x)mE(x)

j

x x

use t h e f a c t t h a t , f o r i n s t a n c e , a i j ( E , z )

1l y l 1

I Z I yxz

a i j ( y , z ) d y d z i n Lm( P

) weak s t a r .

-t

h

Yr=

(aij)

117

HOMOGENIZATION OF ELLIPTIC SYSTEMS

Then, one can verify, by the methods of Section 5, that (8.109)

uE

-

u

-

t?

+

0 in V strongly as

E

-+

0

.

As observed in Section 5 , the cut-off functions are not at all

indispensable.

9.

Homogenization of elliptic systems.

9.1

Setting of the problem. We now proceed to the homogenization of higher order elliptic

operators and, more generally, of some elliptic systems.t Notations. We consider functions

, (9.1)

ail (y) E L-(R~) , aB

real valued

,

Y -periodic

,

where in (9.1) we assume that

(9.2)

it] = 1,.

..,N

la1 = m

,

ml,..

'We

(we shall consider anN

161 = m

.,% given

,

integers

x

N system)

a = Ial,...,anj

1. 1

, 8

= {B1,

...,'n'

.

do not consider here general elliptic systems in the sense

of Agmon-Douglis -Nirenberg [ 2 1 .

ELLIPTIC OPERATORS

118

With the functions (9.3)

E

we associate

ij (x) = aij (x/E) aa% a%

,

and introduce vector functions: u = {U1,...,UN}

...,VN I

v = {Vl,

,

ui, vi defined in 0

,

.

We consider the system (9.4)

m (-1) jD8(a:i($)DauiE)

= f

j

in 8

,

j = l,...,N

,

with uE = {uiE} subject to appropriate boundary conditions and with a suitable ellipticity hypothesis. We introduce a Hilbert space V such that N

(9.5)

N

'llH:i(o) i=l

C_ V C_

i=l

Hmi(8)

N For u,v

(9.6)

E

i=l

aE(u,v) =

Hmi(0), we set

I

Ea:i(x)DauiDBv.dx 3

We shall assume

'Provided

with the norm

.

119

HOMOGENIZATION OF ELLIPTIC SYSTEMS

(9.7)

aE(v,v)

aI IvI

I v2

, a

> 0

a independent of

, E

v € V ,

.

Remark 9.1. Condition (9.7) will be satisfied if the following conditions hold true : N

v

E

v

implies DYV. = 3

o

on a part

rj

of

r

=

ao

with positive measure (or capacity) on IYI

2 mj - 1

r ,

-

Remark 9.2. If one has (9.8) but not (9.9) one would add in (9.6) zero order

terms : (9.10)

where!:a (9.11)

I

EaO0(x)u.v.dx ij

,

1 3

(y) = aij ( y ) is Lm, Y-periodic, and if aij (y)cicj 1. aocici

,

a

0

> 0

,

a.e. in y

.

Let f be given in V', dual of V, and let uE be the solution of

'One can relax this condition. as an example.

We just present this remark

120

ELLIPTIC OPERATORS

, V v

aE(uE,v) = (f,v)

(9.12)

We want to study the behavior of uE

9.2

.

E V

when

E

* 0.

Statement of the homogenization procedure. We introduce some more notations.

] $I$

W(Y) =

N

E

i=l

Hmi(Y)

,

$i periodic in Y

y

E

i=l

such that I y I

5 mi - 1 1 ;

m. H l(Y), we set

and we introduce (9.15)

+-j B' B = { O ,...,0 , 5 , 0 P.(y) 3

,...,01 .

We then define

xy

'

E W(Y)

,

is defined mod

such that

p

where

6 = set of polynomials IP1,

(9.17)

, i.e.

DY$i takes equal values on opposite faces of Y for every

(9.13)

for $ , q

n'

We define

Pi of degree

5 mi

-1.

...,PNl

,

HOMOGENIZATION OF ELLIPTIC SYSTEMS

Indeed we introduce Wo (Y) = W(Y)/ 6 a1 ( 4 , $1 becomes coercive on Wo (Y).

121

, and we observe that

We then uniquely define

and ijDauiD6v.dx

(9.19)

3

Let U

.

be the system associated to

U is the homogenized system of

a (u,v).

We will show that

(associated to aE (u,v)).

AE

Remark 9.3. The above formulas extend those previously given for the second order scalar operators.

We could as well consider systems with

...,-M'

coefficients functions of x ,X ~ ,

X

we shall prove in Section 9.3 below the Theorem 9.1.

Let u (9.20)

be the solution of

u E

v ,

a (u,v) = (f,v) ,

V v E

v .

Under hypothesis (9.71, one has (9.21)

u

.+

u

&V

weakly, as

E

.+

0

.

Remark 9.6. One can verify that U.(u,v) as defined by (9.19) is coercive

v.

ELLIPTIC OPERATORS

122

Remark 9.5. It is useful to notice that coefficients qti can be given an "adjoint" form. We define m. (9.22)

a;($,$)

and we define (9.23)

=

xB* j

a*(XB* 1 1

(mod 6

-

,

a,($,$)

V

$,$

E

by

)

8 P.,$) = 0

,

3

V J,

E

W(Y)

.

We have then the formula

Indeed let us denote by qiJ* the right hand side of (9.24). aB

We

have YlqiJ = -a aB

1

(x? 1

* a * YlqiJ* = -al(Xi aB

,

P ! , P : ) 3

-

a

..

and in order to prove that 4 : ;

B 3

P ~ , P . ), =

qij* it remains only to verify that aB

* a* B a (xB,pq) = al(Xi ,P.) 1 J 3

But if we take $ =

xB1

in (9.23) (for

xy*

instead of xB*) we see j

that the left hand side of (9.25) equals (9.26)

* a* ,x!) al(xi

if we take 8 = (9.25) equals

xs*

in (9.16), we find that the right hand side of

HOMOGENIZATION OF ELLIPTIC SYSTEMS

123

and since the quantities in (9.26) and in (9.27) are equal, (9.25) is proven. One encounters in a natural fashion the formula (9.18) (respectively, (9.24)) when using the method of asymptotic expansions (respectively, of energy)

9.3

.

Proof of the homogenization theorem. We have

(9.28)

IIUEIIV

5

C

-

If we set

we see that €5; is bounded in L2 ( 8 ), therefore we can extract a subsequence, still denoted by uE, such that uE * u in V weakly (9.30)

,

€5: * 5; in L2 ( 8 )weakly

.

Equation (9.12) can be rewritten (9.31)

('5;,D

Bv.) = (f,v) 3

,

(where parentheses denote either the scalar product in L2 ( 0' ) or the duality between V' and V).

Passing to the limit in (9.31), we obtain

We compute now 5; using "adjoint functions". We introduce:

124

ELLIPTIC OPERATORS

(9.33)

P = IP.(y)1 3

,

P . ( y ) = homogeneous polynomial 3 of degree m j '

and we define w such that

*

A w = 0 in Y

1

(9.34)

w

-

P

E

W(Y)

I

.

If we set (9.35)

w

-

P =

-x

then (9.34) is equivalent to

We then introduce

We observe that (9.38)

AE*wE = 0

,

and that (9.39)

WE(X) = P(X)

For 4 E

(9.40)

-

m. {E

X 3 E

'X.(-)j

.

cE( 8 ' ) we set

$u = {$u ll...,$uNl ;

we choose

v

= $wE in (9.12)

and we multiply (9.38) by $uE.

We obtain:

125

HOMOGENIZATION OF ELLIPTIC SYSTEMS

aE(uE,$wE)

(9.41)

-

aE($uE,wE) =

(ft$WE)

or, more explicitly :

1

(9.42)

‘C;(DB($wEj)

8

-

j

E

-

$D Bw .)ax €3

qcrB ijDBw E j (Da($uEi)

-

$DaUEi)dX = (ft$wE)

-

8

But one verifies that (using (9.39)) DB(~W,~)-

(9.43)

$

~

-,D’($P.) ~ w - $D ~ B pj~ , 3

IBI

=

mj

I

in L 2 ( 8 ) strongly 8 ) into H ” , A ( 8

and that (compactness of the injection of H’( (9.44)

D”($uEi)

-

$D uEi+ Da(@ui)

in L 2 ( 8 strongly

-

$Daui

I

=

))

mi ‘

.

On the other hand,

But

Da($ui)dx = 0 and the right hand side of (9.46) equals (using J8

(9.32)) (CJ,DB($P.)); therefore (9.46) reduces to B 3

ELLIPTIC OPERATORS

126

-

1

ciDBPj$dx +

(aiiD:wj)

8

1

$Dauidx = 0

,

8

i.e.

We now take P = PB j .

- xjB * and

Then w = Pj

ci SO

=

R (aikD(P!~

( 9 . 4 7 ) gives

-

= xjk))~"ui B* l Y1l a1* (xj

-

p8,-pq)~"ui j

that (using ( 9 . 2 4 ) (gJ,DBvj)= O.(u,v) B

and ( 9 . 3 2 ) proves the theorem.

9.4

Correctors. We introduce

(9.48)

where !x (5.3).

=

B {xkjl

is defined in ( 9 . 1 6 ) and where mE is defined in

We have

Theorem 9 . 2 . We suppose that the hypotheses of Theorem 9 . 1 hold true and that al(@,$) is symmetric.

We also assume that

127

HOMOGENIZATION OF ELLIPTIC SYSTEMS B xjk

(9.50)

mk'm W (Y)

E

,

and that the boundary of 0 is smooth enough. Then if (9.51)

zE

is defined by (9.48), one has

OE

-

-

uE

-

u

BE + 0

5

V strongly ;

8, is called the first corrector.

Remark 9.6

(similar to Remark 5.2).

It follows from (9.49) that u satisfies (9.52)

u

j

E

H

2m

J ( 0 ),

V

j

so that B E E V and therefore (9.53)

2,

E

v

.

Remark 9.7 (similar to Remark 5.3). m If V = n H

(Neumann's problem) one can kake m

=

1 in

(9.48). Proof of Theorem 9.2. I

The method is entirely analogous to the one in Section 5.3. We have (9.54)

aE (zE = (f,uc - 2(u

(9.55)

xE

= aE(u

+

8,)

+ e E ) ) + xE ,

,

and all we have to prove is that (9.56)

XE

+

Q(u,u)

.

ELLIPTIC OPERATORS

128 But (9.57)

XE =

j

E a aiBjD

a (ui.+ BEi)D 6 ( u . I

+

8

E j

)dx

,

0 and (9.58)

Da(ui

+

-

= Daui

Oei)

,

-

x 6 . ) D6uk Y kl

mE(D

where

m. (9.59)

=

lD a (rn,xkiD 6

E

-

B u,)

a6

mE(Dyxki)D

B

uk

.

By v i r t u e of t h e c o n s t r u c t i o n o f m E a n d p r o p e r t i e s o f

we have (9.60)

r:,

+

2 0 i n L (6)

.

Therefore, i f we set (9.61)

YE =

1

Eai j

(Daui

6

-

(D u j

-

mE(Dyxki)DYuk) a y

mE(D

x 6 . ) D6 u Q ) d x ,

Y Ql

we have x E - Y E + o .

But we can p a s s t o t h e l i m i t i n ( 9 . 6 1 ) ; w e o b t a i n (9.62)

where

i.e.,

l i m YE =

,

i j D a u . D 6u . d x

pa6

x k6,

HOMOGENIZATION OF THE STOKES EQUATION B

-

XB) = a 3 1

(x'1 -

129

P~,-P?)

so that (9.62) proves that

lim YE = ,g(u,u and the theorem is proven Remark 9 . 8 . The results of this section apply in particular to the equations of elasticity in composite materials.

10. 10.1

Homogenization of the Stokes equation. Orientation. We give in this section the homogenization of an elliptic system

which does not fit into the framework of Section 9 (where, as we said, we did

not consider the most general elliptic systems). This

system is related to the Stokes equations and can be used in the modelling of flows in porous media. 10.2

Statement of the problem and of the homogenization theorem. In the bounded open set

(10.1)

lf=

{$I$

0

of Rn we consider

E ( C p )",div $ = 0 1

,

H = closure of T (10.2)

v

= closure of " ;in

( H( ~ o )"

.

One can check that (10.3)

V =

{ V ~ V=

{vi],vi

E

H1 (R),div v = 0 1 0

.

ELLIPTIC OPERATORS

130

We consider the scalar operator ' A

defined by

with the hypothesis (1.1).t A Given u in (H1 ( 0 ))n, we define '

acting on u in a "diagonal"

way, i.e., (AEu). = AEui

(10.5)

,

i = 1

,...,n .

For u,v E (H1(0))n, we set aE(u,v) =

(10.6)

0

x auk avk aij (F) ax axi dx

.

j

If f E V', we denote by (f,v) the scalar product between V' and 2 n V, and also the scalar product in (L ( 0 ) )

.

Given f, there exists a unique u, E V such that (10.7)

aE(uE,v) = (f,v)

, v

v E V

The interpretation of (10.7) is:

. there exists a distribution

p, in 0 (the pressure) such that AEuE = f (10.8)

-

grad p, in 0

diu u, = 0 in 0 uE = 0 on

r

I

,

.

The homogenization of this problem is given as follows. We define

'We

could

X

.

X

also consider here coefficients aij ( x , ~ , . . ,TI. E

HOMOGENIZATION OF THE STOKES EQUATION (10.9)

{ @ I @E

Wdiv(Y) =

W(Y),div

@ =

131

0 in Yl

(where W(Y) denotes here the subspace of H1 (Y))” which consists of periodic functions).

For @,$

E

(Hl(Y))”, we set

and we introduce (with the notations of Section 9) +j-

(10.11)

pj(y) = { O

xBj

We define

,...,O,yB,O,...,01

,

j,B = 1

,...,n .

(up the addition of a vector with constant

components) by a (XB 1 1 (10.12)

xj

E

-

B

P.,JI) =

3

WdiV(Y)

o ,

v

$ E

wdiv(y)

,

.

Then we set

and

Let u (10.15)

E V

be the solution of

lii(u,v)= ( f , v )

v € V ;

I

(10.15) is the homogenized problem of (10.8).

If c7 denotes the

operator associated to U , (10.15) can be interpreted as a.u= f (10.16)

-

div u = 0

grad P

,

u=OonI’.

I

132

ELLIPTIC OPERATORS

Remark 10.1. Formula (10.13) formally coincides with the one of Section 9. But the proof of Section 9 does not apply directly in the present context. We shall prove in Section 10.3 below the Theorem 10.1. Under the hypothesis (1.1),

uE (respectively, u) is the

solution of (10.7) (respectively, (10.15)), one has (10.17)

uE

-+

u

&

V weakly

.

We shall give in Section 10.4 (after the "energy" proof of Section 10.3) some indications on the asymptotic expansion of uE.

10.3

Proof of the homogenization theorem. It immediately follows from (10.7) and hypothesis (1.1) that

Then AEu, remains bounded in ( H - l ( S ) In, and therefore by (10.8) one has:

(10.19)

grad p, bounded in (H-'(S))"

.

Therefore (cf. Deny-Lions 111 for instance) one can choose p (which is defined up to an additive constant) such that

We introduce

133

HOMOGENIZATION OF THE STOKES EQUATION

and by v i r t u e o f (10.22)

( 1 0 1 8 ) one h a s C C .

15;

In -

Equation

can be w r i t t e n

W e can e x t r a c t a subsequence,

s t i l l d e n o t e d by u,,

p,,

such

that u (10.24)

<: p,

+

+

+

u i n V weakly

,

2 n i n (L ( 0 ) )w e a k l y

ci

,

2 p i n L ( 8 )w e a k l y ;

i n t h e l i m i t (10.23) g i v e s

and ( 1 0 . 8 ) g i v e s

(10.26)

aci

- a xi -

f

-

grad p

The p r o b l e m i s now t o compute

. ci.

W e i n t r o d u c e a ; ( @ , $ ) = a l ( $ , $ ) and P ( y ) s u c h t h a t

P

j

=

homogeneous p o l y n o m i a l of f i r s t d e g r e e

Given P w e d e f i n e

x

.

as t h e s o l u t i o n (up t o an a d d i t i v e v e c t o r

w i t h c o n s t a n t c o m p o n e n t s ) of

134

ELLIPTIC OPERAXORS

v

$ ) = O ,

(10.28) Y)

'div

*

(10.29)

div

x

*

AIP

= 0

Wdl"(Y)

I

. is:

The interpretation of (10.28 A1x =

J, E

-

2 there exists n (E L (Y)) such that

grad n ,

,

.

X and n periodic We now set

w = P

(10.30)

-x

and (10.31)

w (x) = EW(X/E)

,

,

x,

Given $ E

a:(

n E ( x ) = I(X/E)

= EX(X/E)

.

We have (A€ ) * wE = grad n E

(10.32)

.

We now use (10.8) and (10.32).

8 ) ,wB set

$v = {$v.); we multiply (10.8) (respectively, (10.32)) by $wE 3

(respectively, $uE); after integrating over 8 and subtracting, we obtain

Using the fact that div side of (10.33) equals

x,

= 0

and div u

= 0,

the right hand

135

HOMOGENIZATION OF THE STqKES EQUATION Since w n

+

2

P in (L ( 0 ) )" strongly, and since

m

+

and using (10.24 (10.35)

(n) in Lm( 0 ) weak star

, one

,

has for the limit of (10.34)

(f,@P) + (p,?

Pi + @ div P) + ( R (n),div(@u))

axl

But

(

.

m ( v ) ,div(@u)) = 0 and (10.35) reduces to

(10.36)

(f,$P)

+ (p,div(@P))

.

Using (10.26), we see that (10.36) equals

The left hand side of (10.33) equals (this is the usual calculation)

(where g" (x) = g(x/E)), and the limit of this term is given by

-

1

1 (aij %)*$ aw

0

j

udx

.

37) and of (10.38) gives

ax)=

(Sit$ ap i

-

I s m

(aij

%)*$

0

i.e., (10.39)

ap 61 . ayk ax, axl = m(a ki *)-k

.

We take now P = P* 1 '

j

udx

I

t/

@

c3:(0)

136

ELLIPTIC OPERATORS

x

Then

=

x?*3

and (10.39) gives (if S B = { S B j } ) :

which (after a short computation) proves the desired result.

10.4

Asymptotic expansion. We look for uE, p, in the form u, = uo +

(10.41)

EU1

+

p, = Po + EP1 +

E

2u2 +

...

...

,

,

where uk = uk(x,y), pk = pk(x,y) are vectors or functions Y-periodic. We write (10.8) in the form (€-'Al (10.42)

+ ,-lA2 +

(E-ldiv Y

E 0A3)uE

+ divX)uE

-

= f

(€-'grad

Y

+ gradx)p,

.

= 0

We identify the coefficients of the powers E-',

E-',

gives (10.43)

(10.44)

AluO = 0 Alul

+ A2u0

=

div

uo = 0

,

A

(10.45)

,

Y

~ +U A2u1 ~

-grady po

+ A3u0

= f

div u1 + divx uo = 0 Y

-

.

It follows from (10.43) that

,

grad

X

Po - grady P1 ,

EO.

This

HOMOGENIZATION OF THE STOKES EQUATION (10.46)

137

.

uo = u ( x )

Then ( 1 0 . 4 4 ) r e d u c e s t o ALul

+

A2u0 = - g r a d y p o

.

The s e c o n d c o n d i t i o n ( 1 0 . 4 5 ) i m p l i e s

i

+

( d i v y u1

divx uo)dy = 0

Y

and s i n c e t h i s i n t e g r a l e q u a l s l Y l d i v u o n e h a s (10.47)

.

div u = 0

Then t h e s e c o n d c o n d i t i o n ( 1 0 . 4 5 ) i s e q u i v a l e n t t o d i v i.e.,

finally for u

u1 = 0 ,

1

Alul

+

A2u0

=

div

u1 = 0

.

(10.48)

Y

Y

-grad

Y

p

0 '

More e x p l i c i t l y AIUl

au aaij o ayi ax -

=

j

(10.49) div

Y

u

1

= 0

grady Po

. (10.49) is:

With p r e v i o u s n o t a t i o n s , t h e g e n e r a l s o l u t i o n o f

(10.50)

u1 = -x!(y)

au

3 (x) +

fii,(x)

.

axf3

The f i r s t e q u a t i o n i n ( 1 0 . 4 5 ) c a n b e s o l v e d i n u2 i f (10.51)

1

(A2u1 + A3uO)dy = f l Y l

Y

Using ( 1 0 . 5 0 ) , with (10.47)).

-

lYlgradx

pOdy

.

Y

( 1 0 . 5 1 ) g i v e s t h e homogenized s y s t e m ( t o g e t h e r

ELLIPTIC OPERATORS

138

Remark 10.2. Expression (10.50) gives the first term in the asymptotic expansion.

We have found a technical difficulty in the precise form

of the first corrector, since the multiplication by a function mE does not leave invariant the space of vectors with zero divergence.

Remark 10.3. One can also use the "adjoint expansion", as in Section 3 . 3 , since we have the Dirichlet's boundary conditions.

Remark 10.4. One can also consider other boundary value problems, by taking other spaces V.

v

(10.52) As

For instance one can take: 1

= IV~V E (H (B))",div v = 0, v.v =

o

on

ri

.

usual, the homogenized operator does not depend on V.

11. 11.1

Homogenization of equations of Maxwell's type.

.

Setting of the problem In 8

C

R3 we consider two matrices

(11.1)

aE(x) = [a:j(x)~

(11.2)

ai(x) = [aiij(x)l

, ,

aE (x) = aij(x/€) ij

,

aoij(x) E = aOij(x/E)

,

where the functions aij(y), aoij(y) are Y-periodic and satisfy the analogs of (1.1). Given a vector function u, we set (11.3)

rot u = V

x

u

HOMOGENIZATION OF EQUATIONS OF MAXWELL'S TYPE

139

and w e c o n s i d e r boundary v a l u e problems a s s o c i a t e d w i t h t h e e q u a t i o n

r o t uE)

rot(a'

(11.4)

+

aiuE = f

W e want t o s t u d y t h e b e h a v i o r of uE a s

Variational formulation.

8

in E

-+

.

0.

Functional spaces.

We introduce 2

3

H = (L ( 8 ) )

(11.5)

where t h e s c a l a r p r o d u c t i s d e n o t e d by ( f , g ) .

W e introduce

.I ( 8 , r o t ) = { v l v E H , r o t v E H I

(11.6)

provided w i t h t h e norm g i v e n by

I IvI 1'

(11.7)

=

I

IVI

I:,

+

I

lrot

3 ( 8, r o t We introduce next

H O(O,rot)

(11.8)

=

One can v e r i f y ( c f . f o r i n s t a n c e Duvaut - L i o n s 3 o ( 8 ,rot) =

(11.9)

where i n ( 1 1 . 9 ) n = u n i t normal on of

r

d i r e c t e d toward t h e e x t e r i o r

8 t o f i x ideas. W e introduce V such t h a t

For u , v (11.11)

[ l ], Chapter 7 ) t h a t

E

3 ( 8 , r o t ) we set aE(u,v) =

a'rot 8

u - r o t v dx

+ 8

ELLIPTIC OPERATORS

140

The operator associated to (11.11) is (11.12)

=

.

rot(aE rot) + a :

The boundary value problem we study is: aE(uE,v) = (f,v)

(11.13)

where u

+

,

v E V

V

,

u

E V ,

(f,v) is a continuous linear form on V (f E V').

In order to study the asymptotic behavior of uE as

E

+

0, we

begin with the method of asymptotic expansions. In order to simplify the exposition, we shall consider first the case (11.14)

a: = A1

,

A > O ,

and we shall return later on to the more general case (11.2).

11.2

Asymptotic expansions. We have = E-2~1

+

+

,

E0

A1 = rot a rot Y Y' (11.15)

A2 = rot a rotx + rotx a rot Y ' Y A3 =

rotx a rotx + A

.

We look for uE in the usual form: u . = u.(x,y). 3 7

We obtain:

AluO = 0 (11.16)

, ,

Alul

t A2uO = 0

A1u2

+ A2u1 + A3u0

=

f

.

uE = uo +

E U ~ +

...,

HOMOGENIZATION O F EQUATIONS OF MAXWELL'S But ( A 1 ~ O , ~ O ) y te q u a l s ( a r o t

Y

141

TYPE

u o , r o t y uoly so t h a t t h e f i r s t

equation (11.16) is equivalent t o rot

Y

uo = 0

.

Then t h e second e q u a t i o n ( 1 1 . 1 6 ) r e d u c e s t o r o t (a r o t

Y

Applying d i v

Y

Y

u1 + a r o t x u,)

.

= 0

t o b o t h s i d e s of t h e t h i r d e q u a t i o n ( 1 1 . 1 6 ) w e

obtain

+

div rotx(a rot u Y Y l

a rotxuo)

+

X div

Y

uo = 0

,

and s i n c e d i v r o t x = - d i v x r o t we obtain Y Y' div

Y

uo = 0

.

Hence uo = u ( x ) and

(11.17)

rot

Y

a(rot u Y 1

+

rotx u) = 0

.

W e set (11.18)

w = a(rot

u1

+ rotx u)

and (11.19)

n

( w ) = i? = G ( x )

I t f o l l o w s from ( 1 1 . 1 7 )

'We

. and from t h e d e f i n i t i o n of w , t h a t

2 3 t a k e t h e s c a l a r p r o d u c t i n (L ( Y ) )

.

142

ELLIPTIC OPERATORS

,

(11.20)

rot w = 0 Y

(11.21)

div (a-lw) = 0 Y

. m

Then rot (w - G ) = 0, (w - G ) = 0, so that there exists a Y Y-’periodic function J, (x,y) such that (11.22)

w

-

G

=

-grad J, Y

.

Using (11.22) into (11.21) gives (11.23)

-div a-1 grady J, = -div a-lG Y Y

.

We introduce, in order to solve (11.23), the functions 8 1 needed for finding the homogenized operator associated with a-l: (11.24)

-div (a-’ grad 8j) = -div (a-le.) , Y Y Y 3

If we define the matrix 8 = {81,02,83}, and if we define grad 8 by grad 8.c Y Y (11.25)

=

grad ( e g ) , we obtain

Y

J, = 8.3

and (11.22) gives (11.26)

w = (I

Applying

m

-

grad 0 ) 3 Y

.

to (11.18) after multiplying by a-l gives:

If we define 1 (a) to be the homogenized matrix associated with aij, we obtain (11.27)

rot u = 1 (a-’)G

.

143

HOMOGENIZATION O F EQUATIONS QF MAXWELL’S TYPE Applying

*m

(11.28

rot

+

(w)

rot hence, using

t o (11.11) g i v e s xu = f

1 1 . 1 9 ) and ( 1 1 . 2 7 ) :

-’ r o t u + xu

8 (a-’

.

= f

The ho ogenizl 9 o p e r t o r ( o b t a i n e d i n a forma

+

j u s t i f i e d below) a s s o c i a t e d t o r o t a E r o t r o t H (a

-1 -1

+

rot

)

manner, whic

A 1

wi

be

&

.

A

Asymptotic e x p a n s i o n s f o r t h e f i r s t o r d e r s y s t e m . L e t u s w r i t e now t h e e q u a t i o n AEuE = f a s t h e f i r s t o r d e r

s y s tern : a E r o t uE

(11.29)

+

rot v

-

vE

=

.

AuE = f

+

W e l o o k f o r uE = u o

o ,

+

E U ~

..., vc

= vo

+

+

E V ~

and w e o b t a i n (11.30)

a rot

Y

uo = 0

,

roty vo = 0

a r o t y u1 + a r o t (11.31)

r o t y v1 + r o t x v o

X

uo - vo = 0

+

Xuo = f

,

,

,

By t h e s a m e methods t h a n a b o v e , o n e f i n d s

v0 = ( I

( i n f a c t vo = w ) .

-

grad

Y

8)Co

,

vo = Go

I

..., a s

usual

144

ELLIPTIC OPERATORS

11.3

Another asymptotic expansion. Let us prove a simple.Lemma:

Lemma 11.1. We assume that a: = XI, X > 0,

that r

= 38

is smooth enough

and that (11.32)

div f

E

2

L (8)

.

Then we do not restrict the generality in (11.13) by assuming that (11.33)

div f = 0

.

Proof. We introduce the solution z of XAz = div f

(11.34)

,

z = O o n T ,

which satisfies (11.35)

1

z E H2(8)

n HO(8)

.

Therefore

6

(11.36)

=

uE

-

Vz belongs to V

indeed n / \ V z = 0 on vz

!t o ( 8,rot

E

V.

r

,

2 1 if z E H ( 8 )0 H o ( 8 ) so that

Then (11.13) is equiva ent to

dE(iiE,V) = (?,v) where

2

=

f

-

XVz satisfies div

f

= 0

.

We now assume that (11.33) holds true and therefore div(AEu

E

) =

div f = 0 so that

HOMOGENIZATION OF EQUATIONS OF MAXWELL'S TYPE (11.37)

div u

.

= 0 =

W e look f o r u

145

uo

+

E

+~

U

...,

(11.37).

W e obtain, with the notation

(11.38)

AIUO

= 0

(11.39)

Alul

+

(11.40)

A1U2

+ A2U1 + A3UO =

div

r

A2u0 = 0

u

Y

,

O

div

f

Y

(11.15) :

,

0

=

of

s o l u t i o n o f AEuE = f

U1

+ divx

divY

t

u2

Uo

+

= 0

r

divx u

1

= 0

.

Now ( 1 1 . 3 8 ) i s e q u i v a l e n t t o uo = u ( x ) , and 0 =

R(div u Y 1 rot

Y

+

d i v u) = d i v u.

a r o t u1 = - r o t

(11.41) div

Y

,

u1 = 0

Therefore (11.39) reduces t o

( a r o t u)

Y

,

u1 i s Y - p e r i o d i c

.

W e now u s e t h e v e c t o r s G p ( y ) (compare t o ( 5 . 5 3 ) ) :

rot

Y

(11.42)

a rot

Y

TeP = r o t ( a e ) Y P

0

,

GP

div Y

Gp

u1

-6(y)rot u

=

,

i s Y-periodic

q (6')

,

= 0

.

Then (11.43)

=

+ ;,(x)

.

Applying 'I? t o(be t h e f i r s t e q u a t i o n ( 1 1 . 4 0 ) , w e o b t a i n (11.44)

r o t R (a

(11.45)

div u = 0

-

a rot

Y

-€!)rot

u

+

Xu = f

,

.

Remark 11.1. F o r m u l a s ( l l . 2 8 ) and ( 1 1 . 4 4 ) (11.46)

ti

=

m

(a

-

are c o m p a t i b l e s i n c e

a rot

Y

:

ELLIPTIC OPERATORS

146

t h i s i s f o r m u l a ( 5 . 6 1 ) w i t h a r e p l a c e d by a - l . Remark 1 1 . 2 . I n t h e c a s e of

(11.33), we can formulate

functional setting. (11.47)

(11.13) i n a d i f f e r e n t

W e introduce

.

W = { v l v E V,div v = 0 1

Then (11.48)

aE(uE,v) = (f,v)

,

There i s a n i m p o r t a n t

V v E W

,

uE

E

d i f f e r e n c e between

W

.

( 1 1 . 1 3 ) and (11.481.

W e have

t ,

W

+

H i s l o c a l l y compact

V

+

H i s n o t l o c a l l y compact

(11.49)

.

The l o c a l c o m p a c t n e s s i s enough t o j u s t i f y i n t h e p r e s e n t s i t u a t i o n t h e expansion obtained i n t h i s s e c t i o n , a s i n Section 9.

Therefore

w e have p r o v e n ( u s i n g Remark 11.1):

Theorem 11.1. I f w e assume a:

= AI,

r

smooth enough and ( 1 1 . 3 2 )

then

u

-f

i n V w e a k l y , where u i s t h e s o l u t i o n of (11.50)

'By

(

(a-')-'rot

t h i s w e mean:

2 3 (L ( 8 ' ) ) s t r o n g l y f o r

u , r o t v)

i f vE

-

8 '

C

+

+

h(u,v) = (f,v)

,V

v i n W weakly, t h e n vE

8

.

W e can t a k e 8

'

=

v E V

+

.

v in

8 with

s u i t a b l e boundary c o n d i t i o n s b u t t h i s i s n o t u s e f u l h e r e .

u

HOMOGENIZATION OF EQUATIONS OF MAXWELL'S TYPE

147

Orientation. In what follows we give a more direct proof of (11.50); which

r

does not assume

smooth nor (11.32); it is based on a result of

"compensated compactness" which is of independent interest.

11.4

Compensated compactness. We introduce the space 2 1: (8,div) = {vlv E H,div v E L ( 0 ) ) ,

(11.51)

with the norm given by

Theorem 11.2.

Let ua and

va be two sequences of vector functions which satisfy

(11.53)

ua

+

u

&

(11.54)

va

+

v

& a

uava

+

uv

1: ( 8,rot) weakly

( 8,div) weakly

,

.

Then (11.55)

.

in

.B' ( 6 )

Remark 11.3. In fact we have a little more than (11.55). is continuous with compact support in 8 (11.56)

1 3

(u"v")$dx

-+

i

0

(uv)$dx

.

, we have

For every $ which

ELLIPTIC OPERATORS

148

Remark 11.4. If we suppose that uci+ u in (H1 (8)13 weakly 2

va

v in (L

+

(

,

8)13weakly ,

then (11.55) is immediate because ua + u in (L2 ( 8) ) strongly (compactness of the injection H 1 ( 8 )* L2( 8 ) or, if 3 0 is irregular, in Llocal ( 8 ) ) .In Theorem 11.2 above we obtain (11.55) by using some information on the derivatives of ua and some other information on the derivatives of vci,hence the terminology for Theorem 11.2:

we call it a result of "compensated" compactness.

Proof of Theorem 11.2. m

Let $ ECO ( 8 ) . We set $v = {$vl,$v2,$v3}.

The mapping v

+

$v

is linear and continuous from kl ( 8,rot) (respectively, kl ( 8,div)

into itself.

.

We have to show (11.56) the support of $.

But since v

+

We take J,

m

E

0, ( 8 ) such that JI

$v (where we extend $v by 0 outside 8 ) is a continuous

3 8 ( R ,rot) (respectively,

the following.

+

(0,div)) into

3

(R ,div)), we are lead to showing

We assume that

a c i

u ,v ua

1 on

Then (11.56) amounts to showing that

linear mapping from 8 (8,rot) (respectively,

(11.58)

=

have their support in a fixed compact set K of R

u (respectively, va * v) in kl (R3,rot) weakly

(respectively in

a

(R3,div) weakly)

,

,

3

,

HOMOGENIZATION OF EQUATIONS OF MAXWELL'S TYPE

149

then (11.59)

1,

u"v"dx

uvdx

+

.

R3

R

We can also consider complex valued functions.

G

If

- - -

= ~ v l l v 2 , v 35 ~ l= complex conjugate of $, we are going to show that

(11.60)

j3 uaGadx

I

-t

uGdx

.

R3

R

We now use the Fourier transform. We define:

where in general

By virtue of Plancherel'sformula, (11.60) is equivalent to (11.62)

ra?dc

Ia =

+

R3

j3 rsdc . R

The information (11.58) is now "transformed" by Fourier ,transform. Since the ua, 'v (11.63)

r"(c)

have their support in K, we have +

, ~ " ( 6 )+

r(c)

in R3 uniformly

s(c)

for 6 E bounded set

.

Moreover, cjrE (11.64)

-

ckr;

~ . s -t a cjsj

3 3

+

cjrk

-

ckrj

in L2 (R3 ) weakly

in L 2 ( R3 ) weakly

.

,

150

ELLIPTIC OPERATORS We observe that, V k:

(11.65) Hence

Ekr"sa = (ckr?

-

cjrE)sg

+

r;CjS;

.

t follows, using (11.64), that

(11.66

We can write (11.67

This inequality shows that (11.62) will be proven if we verify that

which follows from (11.63).

11.5

Homogenization theorem.

Theorem 11.3. Let uE that f

~

E

V'

be the solution of (11.13), where a :

and r

is not necessarily smooth.

= AI.

Then uE

We assume -f

u

&

V

weakly, where u is the solution of (11.50). Proof. We have

I

Iu,I

Iv 5

C, so that we can extract a subsequence,

still denoted by uE, such that

HOMOGENIZATION OF EQUATIONS OF MAXWELL'S TYPE (11.68)

uE

-+

u

,

i n V weakly

a" r o t uE

5 in

-+

2 ( L ( 8 ))

151

weakly

and (11.69)

f = r o t ( a " r o t u E ) t XuE

rot

+

5

+

Xu = f

.

W e introduce

(11.70)

bE

=

;

= (a"*)-'

8 ) w e d e f i n e v E as t h e s o l u t i o n o f

given g i n (11.71)

[ a ? I-1 li

,

- d i v b" g r a d v E = g

.

r

v E = 0 on

W e know t h a t

v

-+

1 i n H o 8 ) weakly

v

,

(11.72) bE g r a d v E

-f

; (b)grad v

i n (L2 ( 8 ) )

weakly

.

W e s t a r t from t h e i d e n t i t y

(11.73)

a E rot u *b" grad vE = r o t u E * g r a d vE

By v i r t u e of

( 1 1 . 6 8 ) , ( 1 1 . 6 9 ) , a " r o t uE

-+

. 5 in 8 (8 ,rot)

weakly and by v i r t u e of

( 1 1 . 7 2 ) , ( 1 1 . 7 1 1 , b" g r a d v E

i n . H ( 8 , d i v ) weakly.

Using Theorem 1 1 . 2 ,

(11.74)

a" grad u E - b E g r a dvE

-+

5.

-+

it f o l l o w s $ h a t

8 (blgrad v

i n B ' ( 8)

S i n c e d i v r o t uE = 0 , r o t g r a d vE = 0 , w e have

r o t uE

rot u

-+

U (0,div)

in

weakly

and g r a d vE

-+

grad v

so t h a t Theorem 1 1 . 2 g i v e s

in

3 (8,rot)

2 (blgrad v

weakly

,

.

,

ELLIPTIC OPERATORS

152

(11.75)

rot uE*grad vE

+

rot u-grad v

in

.

B' ( 8 )

It follows from (11.73), (11.74), (11.75) that Y (b)grad v = rot u-grad v

t;.

i.e., (b)*t;-gradv = rot u-grad v

(11.76)

,

1 where v is the solution of -div Y (b)grad v = g , v E HO( 8 ) ;since -1 1 g is arbitrary in H ( 8) , v spans Ho ( 0 ) and (11.76) implies

*5

Y (b)

= rot u

,

Remark 11.5.

a

ij

We have the analogous result (with the same proof) when N (x) = a . . (X,X/E,X/EZ,. .,X/E ) . 13

Remark 11.6.

.

Use of the adjoint expansion.

We consider problem (11.13) with

Then we can use the method of Section 3.3; u

is characterized

by (11.78)

(uE,AE*vE) = (f,vE)

'Or vE

E

,

V with compact support.

3

t

HOMOGENIZATION O F EQUATIONS O F MAXWELL'S TYPE

153

and by t a k i n g enough t e r m s i n t h e a s y m p t o t i c e x p a n s i o n ( t h i s i s made more p r e c i s e below) w e c a n a l w a y s c o n s t r u c t ( a t l e a s t i f c o e f f i c i e n t s a r e smooth enough) a s e q u e n c e o f f u n c t i o n s v E which b e l o n g t o V , have

8 and s a t i s f y

compact s u p p o r t s i n vE

+

2 i n (L ( S ) ) 3 strongly

v

,

v

E

2 i n (L ( 8 )) 3 s t r o n g l y

.

( s ~ ( o3) )

(11.79)

A ~ * V + ~ u *v

W e c a n p a s s t o t h e l i m i t i n ( 1 1 . 7 8 ) and w e o b t a i n

which i s t h e d e s i r e d r e s u l t . I n o r d e r t o o b t a i n ( 1 1 . 7 9 ) w e look f o r v E i n t h e form (11.80)

VE

=

V

+

EV

-k E L V

1

,

v

1

=

v . (x,y) 1

i

two t e r m s are needed h e r e ( c f . Remark 3 . 6 ) . I f a ( y 1 i s smooth enough, t h e i d e n t i f i c a t i o n i s p o s s i b l e : w e impose

*

+

Alvl

0

=

*

*

(11.81)

*

A2v

+ A2v1 +

A1v2

, *

*

A 3V = U v

,

so t h a t (11.82)

AE*v

E

=

Q*v

+ E(A*v 2 2 + A*v 3 1) +

E

2 " A3v2

,

and one can t h e n v e r i f y ( 1 1 . 7 9 ) .

Remark 1 1 . 7 . The r e s u l t of compensated compactness p e r m i t s t o g i v e a new p r e s e n t a t i o n of t h e s o - c a l l e d 5.8 and 5 . 9 ) .

d u a l f o r m u l a ( c f . Remarks

154

ELLIPTIC OPERATORS We consider simultaneously the problems:

(11.83)

-div aE grad u rot bE rot v

+ XuE

+ AvE

=

f

= q

,

, (div q = 0)

,

(11.84)

div v

,

= 0

with appropriate boundary conditions. (11.85)

bE*a" = I

We choose bE such that

.

Then (11.86)

aE grad u *bE rot vE = grad u -rot v

.

Let us call, as usual, X (a) the homogenized matrix of aE'and let us introduce the notation (11.87)

X (b) = homogenized matrix for bE in problem (11.84)

.

We have aE grad u b E rot vE

+

+

H (a)qrad u

in H

(

8 ,div) weakly

X (b)rot v

in kJ

(

8 ,rot) weakly

,

so that by virtue of Theorem 11.2: aE grad uE-bErot vE

+

H (a)qrad

U.

x (b)rot v in A ' ((9

Since grad u

+

grad u

in

Y

(

8,rot) weakly

and rot vE

+

rot v

Theorem 11.2 shows that

in

n

( 6 ,div) ,

)

.

155

HOMOGENIZATION OF EQUATIONS OF MAXWELL'S TYPE

grad uE-rot v

+.

grad umrot v

in B ' ( 0 )

,

and therefore (11.86) gives in the limit

.

3 (a)grad u - X (b)rot v = grad usrot v

Hence it follows that (11.88)

*

X (b) U (a) = I

.

But as we saw in Section 11.3, the natural formula for

m

X (b) =

*-1

(a

(I

-

rot Y

x

(b) is

2))

(we use (11.44) with a replaced by a*-'

so that

e

becomes

as

defined in (5.53)+), so that (11.88) gives

This gives a new proof of (5.61).

11.6

Zero order term.

Orientation. We now return to (11.13) without the hypothesis (11.14). We have

I

u

+.

aE

+.

u

in

3 (0,rot) weakly

,

and 0

'In

m

(ao) in (Lm( 0 ) ) weak star

.

fact it should be 2 * obtained by replacing a by a* in

(5.53); it is then

which appears in (11.89).

156

ELLIPTIC OPERATORS Since t h e i n j e c t i o n of

li

(

2

8 , r o t ) i n t o ( L ( 8)

compact, i t d o e s n o t f o l l o w t h a t a i u E weakly.

is n o t

)

2

-+

-

3

711 ( a o ) u , i n ( L ( 8 ) )

W e show now how t o s o l v e t h i s q u e s t i o n .

Theorem 1 1 . 4 . Let u -

be t h e s o l u t i o n o f

Then u

(11.13).

-+

u

&

V weakly,

where u i s t h e s o l u t i o n o f (11.90)

( 3 ( a-1) -1r o t u , r o t v )

+

( Y (aO)u,v) = (f,v)

,

V v

E

V

.

Proof. By a p p r o x i m a t i n g i n V' t h e f u n c t i o n f by a s e q u e n c e f J o f smooth f u n c t i o n s , w e d o n o t r e s t r i c t t h e g e n e r a l i t y by assuming t h a t (11.91)

d i v f E L2( 8 ) ( 0 ),

Given g i n

(11.92)

. we d e f i n e vE a s t h e s o l u t i o n of

,

-div(ai)*grad vE = g

v

= 0 on

r ,

and w e c o n s i d e r t h e i d e n t i t y (11.93)

aiuE.grad v

Since a i u E

+

rl i n

.

= uE-ai*grad v

2

( L ( 0 ))

div(aEu ) = div f O E

E

weakly and s i n c e

L

2

( 8 )

,

w e have aEu O E

-+ 0

i n 3 ( 8 , d i v ) weakly

Since r o t grad vE = 0 , grad v

-+

.

grad v i n

3 ( 0 , r o t ) weakly and

a c c o r d i n g t o Theorem 1 1 . 2

1

157

HOMOGENIZATION OF EQUATIONS 0F:MAXWELL'S TYPE

(11.94)

aiuE.grad vE

We have u

+

ai*grad vE

+

+.

qsgrad v

in

J ' ( (9)

.

u in H (8,rot) weakly and according to (11.92), 3 (a:)grad

v in W

( 0

,div) weakly, so that according

to Theorem 11.2 (11.95)

E* uE*a0 grad v

+.

US

W (ai)grad v

in

&'(B)

.

*

q-grad v = us W(ao)grad v

* *

hence we deduce that 11 = H (a,) u = 3 (ao)u and (11.90) follows. Remark 11.8. A reduction to the case "div f = 0 " analogous to the one in

Lemma 11.1 is possible only if we assume ao(y) smooth enough (say aOij (y) E W""(Y)), 11.4.

11.7

an hypothesis which is unnecessary in Theorem

Remark on a regularization method. Let us consider (11.3) with

(11.96)

a :

= A1

,

V = 3 ( 8,rot)

,

div f

E

L2( 8 )

.

In that case we saw that we can always reduce the problem to the case when div f = 0, div u

= 0.

But there is another possibility of

reducing the problem to a situation where we can apply the method of Section 9.

We introduce the regularized problem defined as follows.

We introduce (11.97)

=

f

{vlv E V,div v

and for u > 0 we consider

E

L2 ( 8 ) ),

158

ELLIPTIC OPERATORS

(11.98)

wE = w

E EtU

?

which i s t h e s o l u t i o n o f t h e ( r e g u l a r i z e d p r o b l e m ) (11.99)

aE(wE,v) + u(div w ,div v) = (f,v)

,

V .v

.

E

W e w i l l v e r i f y below t h a t

(11.100)

I

/WE

-

WEl

I

<

a

cv5.

(9,rot)

(

I t f o l l o w s t h a t w e c a n homogenize ( 1 1 . 9 9 ) f i r s t ,

fixed).

S i n c e t h e i n j e c t i o n of

a p p l y S e c t i o n 9 , and n e x t w e l e t u

V e r i f i c a t i o n of

+

H i s l o c a l l y compact, w e c a n

+

0.

(11.100).

Given v E V , w e h a v e , u s i n g ( 1 1 . 3 ) (11.101) a E ( u E - w E , v )

Since d i v u (11.191).

(with u > 0

=

+

x

(-div

nd ( 1 1 . 9 9 ) :

x(uE - w E , v ) =

f,

.

(div w ,div)

-

E ~ ~ ( 3w e1 c a , n take v = u

w

Therefore a E ( u E - wE,uE

-

w E ) + A(uE

-

wE)2

=

uldiv wEl

2

= u ( w , d i v we)

Therefore

12.

12.1

in

ldiv wEl <

div f

and (11.100) f o l l o w s e a s i l y .

Homogenization w i t h r a p i d l y o s c i l l a t i n g p o t e n t i a l s .

Orientation. W e consider f i r s t i n t h i s section the equation

.

HOMOGENIZATION WITH RAPIDLY OSCILLATING POTENTIALS A U ~ +1 - EW U + v u E = f (12.1)

u

in

0

159

I

= O o n r ,

where (12.2) (12.3)

W is a Y-periodic Lm function such that (12.4)

1

W(y)dy = 0

,

Y

and

u is a constant. We want to study the behavior of uE as

E +

0.

We shall consider next similar questions with other operators A and other boundary conditions. If we multiply (12.1) by u E l we find

This expression does not give any simple a priori estimate, and it is even unclear at this stage if we can solve (12.1) for all with a fixed U. of

E l

12.2

E +

0

In order to obtain a priori estimates independant

,

we shall introduce first the asymptotic expansion of u

Asymptotic expansion. We use the notation

of Section 2.

We have

.

ELLIPTIC OPERATORS

160

I f w e look f o r u E i n t h e f o r m (12.6)

= uo

u

+

+ ELu2 +

E U ~

...

,

u

j

= u . (x,y)

i

,

we o b t a i n (12.7)

,

AluO = 0

Equation (12.7) is e q u i v a l e n t t o uo = u ( x ) SO

,

t h a t (12.8) reduces t o

(12.10)

Alul

+ Wu

By v i r t u e o f

= 0

.

(12.4),

we can f i n d

x (y)

such t h a t

A1x+W=O, (12.11)

x

which d e f i n e s (12.12)

u

,

i s Y-periodic

x

up t o a n a d d i t i v e c o n s t a n t .

1 = xu

+

ij1(X)

.

W e can s o l v e (12.9) i n u2, i f f .

i.e.,

It follows t h a t

HOMOGENIZATION WITH RAPIDLY OSCILLATING POTENTIALS

1G1

where as usual

Equation (12.13) is the homogenized equation of (12.11, when

p

~ ( W X ) > X1 = first eigenvalue of -A

+

(12.14)

in 0

for Dirichlet's boundary condition

.

This will be justified in Section 12.3 below. Remark 12.1. It follows from (12.11) that

Remark 12.2. We justify below (12.14) using energy estimates.

One could

also justify this result directly from the asymptotic expansion, by methods

similar to those of Section 2.4.

Remark 12.3. 1 The oscillatory potential WE has a "penalty" factor F.

Therefore

we can also think of the results of this section as results of the singular perturbations type, for the equation (12.16)

12.3

€(A

+

p)uE

+

WEuE = ~f

,

uE = 0 on

r

Estimates of the spectrum and homogenization. We shall prove

.

162

ELLIPTIC OPERATORS

Theorem 1 2 . 1 . W e assume f o r

(12.4)

(12.14) hold t r u e .

equation (12.1) admits a unique s o l u t i o n f o r uE

(12.17)

-+

u i n H1 o ( ( P ) weakly

1 where u i s t h e s o l u t i o n i n H o (

(9 )

E

Then, as

E

+

0,

s m a l l enough, and

,

of

(12.13)

.

W e s h a l l d e r i v e Theorem 1 2 . 1 from t h e f o l l o w i n g

Theorem 1 2 . 2 . A

(E)

1

b e t h e f i r s t e i g e n v a l u e o f t h e o p e r a t o r -A

t h e D i r i c h l e t boundary c o n d i t i o n .

Then

Proof of Theorem 1 2 . 2 . L e t u s set

(12.19)

nE(v) =

1

[lVvI2

.

+ F 1 WE v 2 l d x

0 Given v w e d e f i n e $,

by

(which i n d e e d u n i q u e l y d e f i n e s $, W e have

and

so t h a t

for

E

small enough).

+

$ WE

for

HOMOGENIZATION WITH RAPIDLY OSCILLATING POTENTIALS

+

/

wEXE(i + ExE)$,dX 2

163

8

8

which gives

Let us introduce J,

(J,

=

$(y) solution of

exists, and it is defined up to an additive constant).

(12.23)

+ wExE -

-E~AJ,~

711

Therefore

It follows from (12.24) that

(w) =

0

.

Therefore

164

E L L I P T I C OPERATORS

v l z , so t h a t

i.e.,

To o b t a i n a n u p p e r e s t i m a t e f o r X 1 ( ~ ) ,

w e remark t h a t (12.24)

imp 1i e s

(12.29)

rE(v)

5 (1 +

[j

C ~ E )

IV$E12dx + 74 (Wx)

9

= (1 +

EX^)$.

$:dx]

-

c7E

0

I n (12.29) w e now t h i n k of 0, g i v e n by v = v

j

Then

given f i r s t

($E

= $ ) and of v

2

.

HOMOGENIZATION WITH RAPIDLY OSCILLATING POTENTIALS t h i s i n e q u a l i t y being s a t i s f i e d ,

V vE such t h a t

I(1+

165 = 1.

Therefore

n (v)

5

(1 +

C8€

[ ( I+

C6E)

hence t h e " r e v e r s e " i n e q u a i t y o f

( A 1 + ~ ( W X ) -) C ~ E I r

(12.28

f o l l o w s , and ( 1 2 . 1 8 ) i s

proven.

Proof o f Theorem 1 2 . 1 . I t f o l l o w s from Theorem 1 2 . 2 t h a t , u n d e r c o n d i t i o n ( 1 2 . 1 4 ) ,

for

E

s m a l l enough. This proves t h a t e q u a t i o n (12.1) admits a unique s o l u t i o n f o r

s m a l l enough and t h a t

W e now o b s e r v e t h a t

t

I n d e e d it f o l l o w s from ( 1 2 . 1 1 ) t h a t

-€AX€

'It

+ 1 WE

=

o ,

f o l l o w s from ( 1 2 . 3 2 ) t h a t -A

1 ' from HO( 19

+

+

p

+ 1 WE

i s a n isomorphism

H-I( 19') f o r p l a r g e enough, b u t ( 1 2 . 3 2 ) d o e s n o t g i v e

t h e b e s t p o s s i b l e e s t i m a t e f o r p ( a s g i v e n ' i n Theorem 1 2 . 2 ) .

E

166

ELLIPTIC OPERATORS

so that after multiplying by v2:

c so that

hence (12.32) follows.

It follows from (12.31), (12.32) that, if

we set (12.33)

1

=

W EuE

then

We can extract a subsequence, still denoted by uE, such that u

-+

u in 'H

Q,

-+

Q

(12.35)

0 9 ) weakly

,

in L2 5 ) weakly

,

and

-Au +

QU

+

UU = f

In order to compute

Given $

E

11,

. we observe that, if we introduce

C " ( G ) , we multiply (12.1) by $ O s 0

After integrating and subtracting, we obtain

and (12.37) by $uE.

HOMOGENIZATION WITH RAPIDLY OSCILLATING POTENTIALS

167

and we obtain

i.e., (12.13).

It follows that

Remark 12.4. One can avoid the introduction of

in the above proof but nE

q,

will be needed in Theorem 12.3 below.

12.4

Correctors. We have now a result proving the first order corrector.

Theorem 12.3. Under the conditions of Theorem 12.1, one has (12.40)

z E = uE

-

(u +

where x E ( x ) = x(x/E),

x

EX

E

u)

+

0 in H1o ( B ) strongly

,

,

defined by (12.11).

Proof. According to Theorem 12.1, we have only to show that (12.41) But

((A

+

1 WE

+

p)zE,ZE) = XE

+

0

.

168

ELLIPTIC OPERATORS

Theref o r e

of t h e p r o o f of Theorem 1 2 . 2 , w e have

But w i t h t h e n o t a t i o n

Hence, u s i n g ( 1 2 . 2 1 )

so t h a t

XE

+

-(f,u)

+

Y = 0

,

and (12.41) is proven.

12.5

Almost periodic potentials.

Theorems 1 2 . 1 and 12.2 c a n b e e x t e n d e d t o t h e f o l l o w i n g situation;

( 1 2 . 1 ) i s r e p l a c e d by

HOMOGENIZATION WITH RAPIDLY OSCILLATING POTENTIALS where W(x,y) is defined in

169

Rn, and satisfies the following

8'x

properties: (12.45)

W(x,y) =

>' m€ Z

ikmY

am(x)

,

k-m = -k

lk,l?S

e

=

a-,,(x)

ifm#O

> 0 ,

W(x,y) is almost periodic in y.

,

,

a.

= 0

,

;

We define (compare to (12.11))

X(X,Y) by (12.47)

x

x(x,y) =

-

m

ikmY lkml-2e am(x) :

satisfies (12.11) in Rn, x playing the role of a parameter:

x

is

a-periodic in y. We introduce the mean-value (in the sense of almost-pesiodic functions)

:

one has

(it is now a function of x). We have the extension of Theorem 12.2: Theorem 12.4. 1 ( E ) be the first eigenvalue of the operator A + - W(X,X/E) 1 E for the Dirichlet boundary conditions. We assume that (12.45), Let -

170

ELLIPTIC OPERATORS

(12.46) hold true. (12.50)

IX1(€)

-

Then (A, + ~.ap(Wxl) I 5 CE

.

The proof is analogous to the one of Theorem 12.2, with some more technical difficulties. papanicolaou

[

1,

[

We refer to Bensoussan, Lions and

1.

Corollary 12.1. Under the hypothesis of Theorem 12.4,

of

uE denotes the solution

(12.44), one has

(12.51)

uE

+

1 u in Ho ( 8 ) weakly

,

1 where u is the solution in Ho( 8 ) of (12.52)

AU

+

m aP (Wx)u +

uu = f

.

Remark 12.5. One has also a result analogous to the one of Theorem 12.3 for the first order corrector.

12.6

Neumann's problem. We come back to the periodic caset but with the Neumann's

boundary condition:

(12.53)

_ _ -- 0 on r ,

'tResultsof this section extend to the a.p. caseof Section 12.5.

HOMOGENIZATION WITH RAPIDLY OSCILLATING POTENTIALS

a where a v = normal derivative to 8 .t

r,

171

directed towards the exterior of

We have Theorem 12.5. We assume that (12.4) holds true. not contain any piece o_f (n

-

We also assume that

r does

1) dimensional hyperplane with

rational conormalt+. Let XYeum(c) be the first eigenvalue of the operator A

+

$ wE for the Neumman's boundary

condition.

Then

Remark 12.6. The first eigenvalue of A = - A in 8 condition is Xyeum

= 0;

for the Neumann's boundary

the analogy between (12.54) and (12.18) is

therefore complete. Remark 12.7. The difference between Theorems 12.2 and 12.5 lies in the fact that in Theorem 12.5 we Theorem 12.5.

have to assume on r the special condition in

If this condition does not hold, the estiqates depend

on the manner in which

E +.

0.

Proof of Theorem 12.5. We consider

tWe assume here that the boundary of 8 is smooth enough. tti.e., with all cosinus directors rational.

ELLIPTIC OPERATORS

172

2 + 1F € ~ ~2 ] d x ,v E D ( A ) ,

of

A

for Neumman's boundary condition

.

With the notation (12.20), we find (compare to (12.21)) that (12.55)

-

n (v) =

+

1r 1 1

,":E

(1 + EXE)2$EdT

(1 + EXE)21V$E12dx

8

+

WExE(1

+ EXE)$2dX

.

8

But if v E

D(A)

,

= 0

and therefore

a a and therefore (12.55) becomes: But av = vi(x) ax.

Under the conditions of the Theorem, (12.58)

(k)' 0 in aYi

+

Lm(r) weak star

.

1 If v E bounded set of H (e),I$€ belongs also to a bounded set 1 2 of H ( 0 ) , so that E compact set of L ( r ) and therefore

1 uniformly for v E bounded set of H ( 8 )

;

173

HOMOGENIZATION WITH RAPIDLY OSCILLATING POTENTIALS (12.59) shows that we can neglect in (12.57) the surface integral and the proof is completed as in Theorem 12.2. Corollary 12.2. Under the hypothesis of Theorem 12.5, and if (12.60)

p

+

~ ( W X >) 0

the solution uE

of

(12.53) converges in H 1 ( 8) weakly towards a

solution of

(12.61)

-a u - o on av

r

.

Remark 12.8. One can also introduce

EXU

as first order corrector.

Remark 12.9. One can prove similar results for boundary conditions of "mixed" type, i.e., Dirichlet's boundary condition on Neumann's on l'

12.7

ro

and

- ro.

Higher order equations. The preceding remarks can be extended in many directions.

of them is obtained by

replacing the second order elliptic

operator A by a higher order elliptic operator. We confine ourselves to the following example. We consider (12.62)

A = A2

One

174

ELLIPTIC OPERATORS

with Dirichlet boundary conditions and the problem (12.63)

+ uuE +

A U ~

4 wEuE

= f

,

u

au

=

=

o

on

r ,

E

wE(x) = W(X/E)

, where W satisfies (12.4)

,

Let us consider the method of asymptotic expansions. A = ( E - ~ A+ 2c-l

Y

AYx

+

E

0

Ax)

where Ayx =

a2

ay.ax. * 3 3

Therefore (12.64)

A

+

p

+

E

-2 W

=

E - ~ A + ~E-3A3

+

h-lA4

+

+

E 0 (A5

E -2 (A3

+ w)

+ u)

with (12.65)

A1 = A2 Y'

A2 = 4A A Y YX '

A4 = 4A A yx x

'

A5 = :A

A3 = 2A A Y X

.

If we look for uE in the form (12.66)

u

=

u0 +

E U ~ + E

we obtain (12.67)

AIUO = 0

,

2

u2 +

...

,

+ 482

YX '

We have

HOMOGENIZATION WITH RAPIDLY OSCILLATING POTENTIALS A u

(12.70)

1 3

+ A2u2 + (A3 + W)ul + A4u0

= 0

175

,

It follows from (12.67), (12.68), that

Then (12.69) gives

+

A1u2 We define

x

(12.72)

A2

.

= 0

0

by

x

which defines (12.73)

Wu

+ W

x

= 0

x

,

periodic

up to an additive constant.

Then

.

u2 = xu + G 2 ( x )

We can also compute u3 from (12.70) but this is not indispensable. Equation (12.71) admits a solution u4, iff

1

[AZu3 + (A3 + W)u2 + A4u1 + (A5 + v)uoldy = lYlf

Y

which reduces to wuZdy + IYI(A2 + v)u = lYlf Y and using (12.73) it follows that [A2

(12.74)

+ v + IYI

[We have

I

Y

x. 1

wxdy =

-

i

[i

Wxdy]]u

=

f

.

y

2

(AX) dy which does not depend on the choice of

Y

This procedure can be justified.

One can prove:

if X1(c)

176

ELLIPTIC OPERATORS

denotes the first eigenvalue of the operator, A

+

2 wEl and if X1

denotes the first eigenvalue of A ( f o r the Dirichlet's boundary conditions), then

(12.76)

p

+

'i?i(Wx) > X1

then (12.77)

uE

-+

u in H,,2 ( 8 ) weakly

where u is the solution in Ho2 ( 8 ) of (12.78)

12.8

Au

+ [ u + 1 (Wx)lu = f

.

Oscillatory potential and oscillatory coefficients. We consider now the problem AEuE

+

1

E

W uE = f

,

(12.79) uE=Oonr, where AE is now given by (1.3), and W E is given by (12.2) and (12.3). We also assume that (12.80)

aij = aji

,

V

i,j

.

The asymptotic expansion proceeds as follows. notations of Section 2.

and the equations,

We have

We use the

HOMOGENIZATION WITH RAPIDLY OSCJLLATING POTENTIALS (12.82)

AluO = 0

(12.83)

Alul

177

,

+ (A2 + W)u0

= 0

,

It follows from (12.82) that uo = u(x).

If we introduce XJ as

in Section 2: (12.85) and

x

A,(X'

-

yj) = 0

XJ Y-periodic ,

r

by

(12.86)

A1(X) + W = O

x

I

Y-periodic

,

then (12.87)

u1 = -xj

* ax.

+ XU + iil(X)

.

3

The compatibility condition in (12.84) gives: (A2 + W)uldy

+

i

A3udy = lYlf

Y

Y

hence

But one easily checks, using (12.85), (12.86) and the symmetry (12.80) of the coefficients, that

SO

that (12.88) implies

(12.89)

L?

ur+

711 (Wx)u = f

.

178

ELLIPTIC OPERATORS

One h a s t h e f o l l o w i n g r e s u l t :

Theorem 1 2 . 6 . (1.1) h o l d s t r u e a n d t h a t a.

W e assume t h a t

denotes the f i r s t eigenvalue of

G

0" 8

b o u n d a r y c o n d i t i o n , a n d i f 111 ( W x ) > X1, solution for

E

s m a l l e n o u g h a n d uE

1

the solution i n H o ( 8 )

of

+

Then i f X 1

= 0.

for the Dirichlet

then

u

(12.79) admits a unique 1 H ( 8 )w e a k l y , w h e r e u 0

(12.89).

L e t u s only sketch t h e proof.

One v e r i f i e s f i r s t t h a t

b y i n t r o d u c i n g u E = (1 + € x E ) G E a n d a r g u i n g a s b e f o r e .

Using (12.321,

it f o l l o w s t h a t

T h e r e f o r e w e c a n e x t r a c t a s u b s e q u e n c e , s t i l l d e n o t e d by u E l 1 s u c h t h a t uE + u i n H O ( 8 ) w e a k l y a n d (12.92)

$ WEuE

+

q

2 i n L ( 0 ) weakly

.

T h e r e f ore (12.93)

AEuE = f

-

a n d u s i n g Remark 3 . 3 , (12.94)

W uE

+

cr

= a:j

-

q

i n L2 ( 8 )w e a k l y

,

it f o l l o w s t h a t

u = f - q .

W e proceed as i n (12.36).

if

f

ax, w e auE

j

W e i n t r o d u c e BE by ( 1 2 . 3 6 ) ; we have

o b t a i n as u s u a l l y

a, ( 0 ) ) : u

(@ E

'

HOMOGENIZATION WITH RAPIDLY OSCTLLATING POTENTIALS

12.9

179

A phenomenon of uncoupling.

We give in this section a simple example where a (rather unsuspected) phenomenon of "uncoupling" arises when passing to the limit in a system of two equations with oscillatory potentials. m

We denote by Wi(y), i = 1,2, two functions L

and Y-periodic,

such that (12.96)

J

Wi(Y)dY = 0

i = 1,2

r

.

Y We denote by {uE,vE} the solution of

(12.98)

u

E

=

v

E

Using (12.32) for

=

o

on

wir we

r

.

see that (12.97), (12.98) admit a unique

solution for X1, A2 large enough. We introduce (12.99)

-Axi

xi

by

4 Wi = 0

,

xi

Y-periodic

,

i = 1,2

.

180

ELLIPTIC OPERATORS

Then, a s (12.100)

E

+

uE

0 , one h a s

+

u

,

vE

+-

v

Hk(B

in

) weakly

,

where f

,

A2v = g

.

-Au

+ (7'l (W2x1)u + Alu

-Av

+ ! R ( W1 x 2 ) V +

(12.101)

=

Remark 1 2 . 1 0 . The s y s t e m (12.101) i s _uncoupled and c o n s i s t s of t w o s e p a r a t e equations. W e i n t r o d u c e iiE,

W e c a n p r o v e t h e above a s s e r t i o n as f o l l o w s .

GE by u

E

=

6E +

E X p E

,

v

=

CE +

EEX2UE

,

and w e p r o v e t h a t

W e c a n t h e r e f o r e assume t h a t , a f t e r e x t r a c t i n g a s u b s e q u e n c e ,

one h a s ( 1 2 . 1 0 0 ) . 00

L e t @ and J, be g i v e n i n Co ( 8 ) .

W e multiply the f i r s t

( r e s p e c t i v e l y , t h e s e c o n d ) e q u a t i o n (12.97) by 41 J,

+

EX'@)

1

and w e i n t e g r a t e ; a d d i n g up, w e o b t a i n

+

EX;$ ( r e s p e c t i v e l y ,

STUDY OF LOWER ORDER TERMS

The terms in in Lm( 0

)

1 -

181

drop out and we obtain in the limit (since

weak star)

.

hence (12.101)

13. 13.1

Study of lower order terms. Orientation. In this section we consider elliptic operators with rapidly

varying coefficients of all orders.

We confine our study to

second order operators, but the methods are general. Let us consider the equation (13.1)

AEuE

+ BEuE + CEuE + a; uE

AE =

- axi a

=

f in 8

,

where (13.2)

[aij

+],

a:j(x)

=

aij(x/E)

the aij's satisfy to (1.1) ,

q a ,

(13.3)

BE = b :

(13.4)

C E = ,axi ( ' i )

a

b:(x)

'

=

bi(x/E)

,

CE(X) = Ci(X/E)

,

,

182

ELLIPTIC OPERATORS a k ( x ) = ao(x/E)

(13.5)

, c i , a.

where t h e c o e f f i c i e n t s bi,

are a l l Y-periodic.

I n ( 1 3 . 1 ) uE

i s s u b j e c t t o a p p r o p r i a t e boundary c o n d i t i o n s . From t h e v i e w p o i n t o f homogenization, w e do n o t r e s t r i c t t h e g e n e r a l i t y by assuming t h a t (13.6)

CE = 0

.

Indeed, i f w e o b t a i n t h e a p r i o r i e s t i m a t e (13.7

IIUEII

(11 I(

C'

= norm i n H 1(

t h e n , a f t e r e x t r a c t i n g a subsequence, u in L

2

8 ) s t r o n g l y , and c ? u 1

(13.8)

CEuE

+

E

+

+.

u i n H 1(

8)) )

weakly, hence

??l ( c i ) u so t h a t

72 ( c i ) aaxi u i n L2 ( 8 ) weakly

.

Therefore, we s h a l l only consider t h e equation

The boundary c o n d i t i o n s are as f o l l o w s . Then w e c o n s i d e r t h e problem

S e c t i o n 1. u

E

E V ,

W e now assume

and (13.12)

v E V

+

v = 0 i f bini

5

0

;

W e i n t r o d u c e V as i n

183

STUDY OF LOWER ORDER TERMS

since in (13.12) V should be independent of

E,

a simple particular

case where (13.12) holds true is: (13.13)

V = H 1o ( d )

Under the conditions (13.14)

13.11), (13.12)t one has

aE(v,v) + (BEv,v)

min(al,ao)I IvI I

since

Consequently, under conditions (13.11), (13.12), problem (13.10) admits a unique solution uE, which satisfies (13.7). We want now to study the behavior of u

13.2

as

E +

0.

Asymptotic expansion. Using notations of Section 2, we have

'We

emphasize the fact that these conditions are sufficient but

not necessary in order to obtain (13.7). which do not assume bi E W1'"(Y) using Cauchy-Schwarz. One has

and one assumes that

Other sufficient conditions,

but only bi E L " ( Y ) ,

are obtained by

184

ELLIPTIC OPERATORS

BE = E-lgl (13.15) B1 = bi

+

E

a ayi ,

0

B~

,

-.

a ' B 2 = b1. ax.

If we look for uE = uo +

EU

1

+ ...,

u. = u.(x,y), (13.9) becomes: 3

3

,

(13.16)

AluO = 0

(13.17)

AIUl + (A2 + B1)uO = 0

(13.18)

A1U2

,

+ (A2 + B1)ul + (A3 + B2)u0

=

f

.

Equation (13.16) gives uo = u(x), and (13.17) gives the same result than in Section 2 for ul; namely, u1 = -xJ &ax j

(13.19)

+ Cl(x)

.

Equation (13.18) admits a solution in u2, iff.

1

(A2 + B1)uldy +

Y

I

(A3 + B2)udy = lYlf

.

Y

We obtain: (13.20) where

a

Q u

+

n

u

+

74 (ao)u = f

,

is the homogenized operator of A€, and where

9 is given

!2Y

This result will be justified in Section 13.3 below. 1-

'One can give a direct justification of this computation for Dirichlet's boundary conditions.

185

STUDY O F LOWER ORDEV TERMS Remark 1 3 . 1 .

8

The o p e r a t o r

depends o f c o u r s e on BE b u t a l s o on A E .

homogenization of B E i s " r e l a t i v e t o A'".

The

Another example of t h i s

t y p e of p r o p e r t y w i l l b e m e t a g a i n i n S e c t i o n 1 4 below.

Energy estimates.

13.3

W e p r o v e now

Theorem 1 3 . 1 .

and

W e assume t h a t ( 1 3 . 1 1 )

(13.12) t a k e p l a c e , o r t h e c o n d i t i o n s

of b e f o r e ( 1 3 . 1 4 ) . Then t h e s o l u t i o n ue (13.22)

uE

of

u i n V weakly

+

(3.10) s a t i s f i e s

,

where u i s t h e s o l u t i o n of (13.23)

u E V

,

rg(u,v)

+

(Ru,v) = (f,v)

, v

v E

v

.

Proof. W e set

(13.24)

rlE = b?

1

By v i r t u e o f

au

2 axi

.

( 1 3 . 7 ) , w e c a n e x t r a c t a subsequence u

such t h a t

one h a s ( 1 3 . 2 2 ) and s u c h t h a t (13.25)

'I, -+ Q i n L ' ( 0 )

Then A'uE that

= f

-

aEuE

.

weakly

-

rlE

-+

f

- % (ao)u -

rl i n L

2

(

0 ' ) weakly s o

ELLIPTIC OPERATORS

186

We now define B(y) as the solution (defined up to an additive constant) of

*

(13.27)

ab.

B Y-periodic

,

i.e.,

We define next

We cbserve that (we implicitly assume that a . = 0 to simplify slightly) (13.30)

(A')*y,

=

abE 2 . j

We multiply (13.9) by @ y E , $ E by $uE.

C31(0),and we multiply

After subtracting, we obtain

Passing to the limit we obtain

(13.30)

STUDY OF LOWER ORDER TERMS

187

Therefore

It remains to verify that (13.32) coincides with (13.21). IYI m ( b k

8) =

But

(using (13.28)) = ai(B,xJ)

= al(xj,B) = (by definition of XI) =

a1 (Y1 , B ) =

IYI

6 R (aij -)aaYi

,

hence the result follows.

Remark 13.2. One can also understand the homogenization (13.21) of BE by using correctors, at least in the case when AE is symmetric. If au converges strongly to u in H1 ( 8 ) , then uE + EX]. (xF ) loc ax j j au j au ax j E EX B (ax.)so that, locally, BEuE = BE(uE + E X bk ayk ax 7 j 7

-

B

~ converges U ~ to

which is formula (13.21).

au

188

ELLIPTIC OPERATORS S i n g u l a r p e r t u r b a t i o n s and homogenization.

14.

Orientation.

14.1

W e c o n s i d e r now boundary v a l u e problems a s s o c i a t e d t o t h e

equation (14.1)

c2A2uE

+

AEuc = f

in

8

,

i s g i v e n a s i n S e c t i o n 1.

where A

Remark 1 4 . 1 . W e c o n s i d e r ( 1 4 . 1 ) a s a n example.

Very many v a r i a n t s are

p o s s i b l e , by t h e same methods t h a n t h o s e g i v e n below. L e t u s f i r s t make p r e c i s e what a r e t h e boundary c o n d i t i o n s we

consider. (14.2)

W e introduce H1( 8 ; A

.

= Iv v E H1(8),Av E L L ( 0 ) } ,

which i s a H i l b e r t s p a c e f o r t h e norm g i v e n by

(1 I

= L

1 t h e c l o s u r e of Ci ( 8 ) i n H ( 8; A )

2 coincides with H o ( 8 ) 1 c o n s i d e r Vo, c l o s e d s u b s p a c e of H ( ; A ) s u c h t h a t

(14.3)

5

H i ( O . 1 EV,,

1 For u , v E H ( 8 ; A ) (14.4)

nE(u,v) =

E

2

H1(O;A)

(where

nE(V,Vf

II II

( 0 )norm)

, and

we

.

w e i n t r o d u c e t h e b i l i n e a r form E

(Au,Av) + a ( u , v )

.

Under h y p o t h e s i s (1.1) and, when n e c e s s a r y , (14.5)

2

E21AVI2

+

CIIVlI

d e n o t e s H 1 ( 8 )norm).

2

( 1 . 2 1 , w e have

SINGULAR PERTURBATIONS AND ,HOMOGENIZATION

189

It follows that given f in, say, Vo, there exists a unique ug such that

, v v

VE(UE,V) = (f,v) (14.6)

.

vo

UE E

We want to study uE as (weak) limit u

vo ,

E

E +

0.

of uE satisfies an

We are going to show that the

equation

2

,gA u

(14.7)

= f

where the homogenized operator a A2 depends of course on AE but also

on AL.

(Compare to Remark 13.1.)

As usual, the best way to obtain the formulas is to use the asymptotic expansion.

This is what we do in Section 14.2 below.

14.2 Asymptotic expansion. Using notations of Section 2, we have

A

=

E - ~ A + ~ E - ~ A+ Y

YX

E

0

Ax ,

where -2

Then (14.8)

E~A'

+

A'

=

E

-2

2

(Ay + A1) +

E

-1

+

EO(ZA A + 4A2 Y X YX

+

E(ZA A Y X

If we look f o r uE = u0 + Y-periodic in'y, we obtain:

+

4A2 ) YX

E U ~ + E

+ A2)

(4A A Y YX

+ A3)

-

+ :A'€

2 u2 +

...,

uj = u.(x,y), 3

190

ELLIPTIC OPERATORS

,

+ A1)uO

= 0

(Ay

+ A1)ul

+ (4AyAyx

+

A2)U0

2 (Ay

+

+

+

A2)u1 + ( 2 8 A

(14.9)

(A;

(14.10)

(14.11)

2

A1)u2

(48 A

Y

YX

= 0

, +

Y X

..........

Equation (14.9) i s e q u i v a l e n t t o uo = u ( x ) .

48'

YX

+ A3)u0

= f

Then ( 1 4 . 1 0 )

reduces t o (14.12)

(A

2

Y

aa..

+

A1)ul

=

au .

3 a y j ax

j

W e i n t r o d u c e p J by

(14.13)

(A2 Y

+

A1)pl

= Alyj

,

p j Y-periodic

,

which d e f i n e s pJ up t o an a d d i t i v e c o n s t a n t .

Remark 1 4 . 2 . The f u n c t i o n s p j ( y ) " r e p l a c e " i n t h e p r e s e n t s i t u a t i o n t h e f u n c t i o n s x j ( y ) i n t r o d u c e d i n S e c t i o n 2. Using t h e f u n c t i o n s pJ,

14.12) gives:

E q u a t i o n ( 1 4 . 1 1 ) c a n be s o l v e d f o r u E i f f

1

(4AyAyx

Y

+ A2)uldy +

I

(2AyAx

Y

which r e d u c e s t o

Using ( 1 4 . 1 4 ) ,

(14.15) g i v e s (14.7) w i t h

+

4A2 yx

+

A 3 ) u0 d y =

I

Y

fdy

,

SINGULAR PERTURBATIONS AND HOMOGENIZATION (14.16)

a A2

=

-

1 IYI

1

[aij

-

..1$

-+

aik

y

j

1%

(ao)

191

.

Remark 14.3. We obtain formulas analogous to those of Section 2 , with replaced by p j . Remark 14.4. One can verify, as in Section 2, the ellipticity of U A

2

.

We

are going to justify the above procedure.

14.3

Homogenization with respect to A 2

.

Theorem 14.1. We assume that (1.1), (1.2) hold true. of -

(14.16).

(14.17)

us

Then, as -+

u

&

E +

0, one has

1. H ( 6 )weakly

where u is the solution of

where

U A2(u,v) is given by

,

Let uE be the solution

192

ELLIPTIC OPERATORS

Proof. I t follows f r o m (14.5) t h a t

(14.20)

I (uEl

W e s e t , a s u s u a l , t;:

=

a i j ax. . auE

W e can e x t r a c t a subsequence

3

u

such t h a t u

' U

(14.21)

si and s i n c e

E

si

+

2

i n H 1( ( 9 ) w e a k l y

,

2 i n L ( 8 ) weakly

,

(AuE,Av) -+ 0 ( b y v i r t u e of

and t h e r e f o r e

v

(14.20)), we obtain I

vo.

v E

W e also o b s e r v e t h a t u E

vo,

s o t h a t i t o n l y r e m a i n s t o show

that

W e introduce w(y) such t h a t

(A (14.24)

w

2 Y

+

* A1)w

-

P i s Y-periodic

= 0

,

,

P ( y ) = homogeneous p o l y n o m i a l o f f i r s t degree.

If w e s e t w (14.25)

(A'

Y

-

P = -p,

+ A;)p

W e now i n t r o d u c e

= A;P

then

,

p is Y periodic

.

193

SINGULAR PERTURBATIONS AND HOMOGENIZATION W e ver f y t h a t

(14.27

( E 2A 2

+

( A E ) * ) w E= 0

.

T h e r e f o r e , m u l t i p l y i n g (14.1) by $wE, $

,

and (14.27) by

$uc, we o b t a i n E2

(AuE,A($wE)) - E 2 (AwE,A($uE

a$

8

which g i v e s i n t h e l i m i t

hence

Taking P ( y ) = yi g i v e s w = yi

If

Gij

-

pi

so t h a t (14.28) g i v e s

ax

d e n o t e s t h e c o e f f i c i e n t o f au

i n (14.29), w e have I

j

and one v e r i f i e s t h a t t h i s f o r m u l a c o i n c i d e s w i t h (14.19). theorem f o l l o w s .

Remark 14.5.

Correctors.

L e t us introduce, with t h e notations of Section 5 ,

Hence t h e

194

ELLIPTIC OPERATORS

2 i s symmetric and i f u E Hloc(B

Then, i f A (14.31)

z E = uE

-

(u +

eE)

+

),

w e havz

1 0 i n H ( 8 ) strongly

.

For t h e p r o o f , w e compute

TE(ZE)

=

VE(UE,ZE

+

TE(U

-

(f,u)

+

so t h a t

=E ( z E

+

and ( 1 4 . 3 1 ) fol ows

15.

Non-local

+

QA2(U,U) = 0

,

.

limits.

S e t t i n g of t h e problem.

15.1

W e c o n s i d e r o p e r a t o r s A E , BE d e f i n e d by

(15.1)

a i j , b i j being Y-periodic f u n c t i o n s i n L m ( R n ) , such t h a t

NON-LOCAL

195

LIMIT6 O x R X the operator

We consider in the cylinder

and the equation

In order to define precisely the boundary conditions, we use a We introduce V such that

variational formulation.

and we assume that aE(v,v)

c~ lv112

, v v

E

v ,

b'(v,v)

cI Iv1I2

,

E

V

c >

o ,

(15.6) V

v

,

II 11

, 1 ' [We add > 0 zero order terms if V = H ( 8 ) . ]

b" (u,v) =

hij 8

au

av

axi dx axj -

-

We introduce next

which is a Hilbert space for the norm given by

For u,v

E

1 H (Rl,V),

we set

=

norm in V

.

In (15.6), we have set

ELLIPTIC OPERATORS

196

The problem we consider is now this (15.11) u

:

given f such that

(dual of H 1 (Rh,V)) ,

f E H-l(R,,V1)

is the solution of

(15.12)

CE(uE,v) =

+r

(f,v)dh

, v

H1 (RXrV),

v

-m

u

E

1

H (Rh,V)

.

This equation admits indeed a unique solution, since by virtue of (15.6) we have

We want to study the behavior of u limit u

of

as

E +

0.

We shall show that the

uE satisfies an integro-partial differential equation,

i.e., a non-local equation. Remark 15.1. The operators AE, BE could be replaced by other operators such as those introduced before, and the operator A could also be replaced by other operators (it suffices then to replace the Fourier transform in h by spectral diagonalization if A

15.2

is self-adjoint).

Non-local homogenized operator. We use Fourier transform in A ;

if g E L2 (RhrV)we introduce

+m

3 g(ll) =

B(')

e-2niX'g(X)dX

= -m

,

6

E

L2(R ,V) P

.

NON-LOCAL

197

LIMITS

Then (15.4) is equivalent to (15.14)

(AE

+

.

2 2 6 4n p B ) G E ( p ) = ? ( p )

by Fourier transform in A,

The variational form is as follows: H1 (RA,V) becomes

For u,v E

3 HL(Rp,V), we define

The Fourier transform

and

(p)

2

of f satisfies

is the solution of +m

(15.18)

y(;,,v)

(E,v)dp

=

,

V

v E H

1 ( R v p v ) -

-m

For each j, we define G J ( y , p ) (up to an additive constant chosen, for instance, by adding the condition

I'

;'(y,p)dy

= 0)

Y as the solution oft (15.19)

2 2 (A1 + 4n p B1) (G1(y,u) - y . ) = 0 3

Y-periodic

,

We set next (this is the homogenization of A'

+ 4a2p2BE, i.e.,

homogenization with parameter 11; cf. Section 8.1)

tA

1

is defined as in Section 2 and B

1

=

-

-[b. a

.

.

11

.I;"

aY

198

ELLIPTIC OPERATORS

4w2p 2bl(G3.

+

-

-

yi)l

,

is bounded by c(l

+

yj,ii

and we define

The function p that

I I i h I Iw(y)

(15.22)

6.1 3. ( p )

-+

2

p )

(one verifies

< c) so that we can define (cf. L. Schwartz [l])

9.. = 5 13

A

j)

(a distribution on R A )

,

and (writing distributions as functions) (15.23)

0

(A)

= -9.. (A) 11

2

a axiax j

Then, as we shall show below, the homogenized equation of (15.4) is:

or

(15.24 bis)

-

'j -m

9 . .(A 13

- A')

aZu ax.ax. (x,A')dA' = 1

The variational formulation is that

7

f(x,A)

.

199

NON-LOCAL LIMITS where we have set +m

Remark 15.2. A

The functions qij(p) are not, in general, polynomials, so that 0 (A) is generally not a partial differential operator.

15.3

Homogenization theorem.

Theorem 15.1. We assume that (15.2), (15.6) hold true. Let uE (respectively, 1 u) be the solution in H (RA,V) of (15.12) (respectively, (15.25) via i

Fourier transform). (15.27)

uE

-+

u

Then H1 (RA,V)weakly

.

Proof. By virtue of (15.13) we know that

(15.28)

IIluEIII

5 c

so that, by Fourier transform, (15.29) GE(p)

remains bounded in 5

;,(U)

1 H (Rp,V) ;

is characterized by (15.18); hence a.e. in

(15.30)

,.

2 2 (u),v) + 4a p b (u,(p) ,v) =

a'.(;,

(

(11)

u ,v)

,' V v

E V

It follows from (15.29) that, after extracting a subsequence, (15.31)

G,

+ ;,

in

1

5 H (R ,V) weakly !J

,

.

200

ELLIPTIC OPERATORS

and by the homogenization with parameter, for a.e. p , G E ( p ) converges of in V weakly towards j ( l ~solution ) (15.32)

L7 (v;G(l~),v)=

But necessarilyt

=

,V)

i , hence

V

I

v

-

the theorem follows.

Remark 15.3. We can find the above formulas using asymptotic expansions.

Remark 15.4. We can also introduce correctors of first order.

16. 16.1

Introduction to non-linear problems. Formal general formulas. Let us consider a non-linear elliptic equation

(16.2)

A'

is given as in Section 1

,

B is a non-linear perturbation , (16.3)

Du = Vu = grad u

.

is bounded in, say, L 2 (R ,v) and lJ 2 2 L (RIJ,V),then @, + @ in L (R ,V) weakly.

tWe can use the fact that if @,(l~)

+

@ ( u ) , a.e.,

$ E

@,

lJ

201

INTRODUCTION TO NON-LINEAR PROBLEMS Remark 16.1. All what follows can be extended to higher order operators A', and accordingly higher order perturbations. Remark 16.2. We shall assume that the function (16.4)

y

+

B(y,u,p) is Y periodic

.

(More precise hypothesis will be given below.)

One can easily extend

the remarks given below to perturbations of the form

Remark 16.3. Boundary conditions are, as it is usual, expressed in variational form: u

E V ,

(16.6) aE(uE,v)

+

X

(B(~,u~,DU~), = V (f,v) )

,v

v E

v

.

If we look for uE in the form u

=

uo +

EU1

+

...

,

u. = u. (x,y) 7

1

,

we obtain (with the notations of Section 2, and implicitly assuming that B(y,u,p) is "smooth" in u and in p) : (16.7)

AluO = 0

,

i.e., u

0

=

u(x1

,

202

ELLIPTIC OPERATORS Equation (16.8) gives, as in the linear case

One can solve equation (16.9) in u2, iff.

i.e.,

where

The variational form of the homogenized equation is: U E V , (16.13) 0 (u,v) +

(

n

(u,Du),v) = (f,v)

,

‘d

v

E

V

.

We now justify this formula on two examples.

16.2

Compact perturbations. Let us consider AEu

(16.15)

uE = 0 on

E

au

+ ucaxl 2=

(16.14)

r

f

,

f

E

L2(d)

.

The homogenization is straightforward.

v

= 0

on

r,

,

av Since (v q , v ) = 0 if

one easily shows the existence of (weak) solutions of

(16.14), (16.15), such that

203

INTRODUCTION TO NON-LINEAR PROBLEMS

(One does not necessarily have uniqueness of the solution uE.) 1 1 1 (or for every q Then uE remains bounded in Lq ( 8 ) , = z r finite if n = 2) and u auE/axl remains in a bounded set of L ( 8 ) ,

4

-

Then we can extract a subsequence, still denoted by u

r=i7+5-

,

such that (16.17)

uE

+

AEuE

u +

in

in Lr ( G ) weakly

g

Necessarily g = ff u. au uE E ax, (since u (16.18)

+

1

,

Ho ( 8 ) weakly

.

On the other hand

a 2 ax, * -

=lL

in

B ' ( 8 )(for instance)

2

u strongly in L (8) ) so that u is one solution of

0u

+

a

u au = f x1

.

We can apply the general formula (16.12). We obtain (16.19)

and since

16.3

B (u,Du) =

I d.

Y ay1

1 -

(x)

IYI y

-

j

aY1

(y)

&j (x)]dx

(y)dy = 0, (16.19) reduces to

Non-compact perturbations. We consider now problem (16.1) or, more precisely, (16.6), with

the following hypothesis: (16.20)

uE exists and IluEII

(16.21)

AE is ,symmetric ,

5 c ,

(1 I 11

= V's

norm)

I

,

Then -u

-f

u i n V w e a k l y , w h e r e u is a s o l u t i o n of

Proof. I f we set

w e can e x t r a c t a subsequence u E ,

such t h a t

qL,

,

u

-+

u

i n V weakly

qE

-+

‘1

2 i n L ( 8 )w e a k l y

(16.26)

.

We introduce (cf. Section 5) (16.27)

eE

= --EmE

XI(:)

ax j

a n d a s l i g h t v a r i a n t of S e c t i o n 5 shows t h a t (16.28)

z

L

= u

-

E

(u + O E )

-f

0 i n V strongly.

W e have

(16.29)

Qu

+

rl = f

a n d , i n v a r i a t i o n a l form, (16.30)

O.(U,V)

+ ( n , ~ )=

I t o n l y r e m a i n s t o show that

(f,v)

,

t/

v

E

V

.

(6.13).

205

INTRODUCTION TO NON-LINEAR PROBLEMS (16.31)

rl = B

(u,Du) (given in (16.12))

.

By virtue of (16.24), we have

-

(16.32)

I B X( ~ , u ~ , D U ~ B) C ~ ,+ U e,,D(u < c(lzE(x)I + IDzE(x)1

eE))I

+

) ,

e same L ' and since we have (16.28), it follows that rlE as the

1 nit limit

as X

(16.33)

cE(x) = B ( ~ ; u+

e ,D(U

+ BE))

.

But using again (16.24) we see that I BX ( ~+, eE,D ~ u

-

+ e,)) - B (X~ , ~ , D-um,(~ x')(~) . x ax au (x))I Y

j

CElvU

so that that 5, 5, has has the the same same L2 L2 limit limit as as so (16.34)

p,(x)

=

X

B(z,u,Du

-

au ax

m D XI(:) E Y

j

(x))

.

It suffices suffices to to show show that that It

(16.35) (16.35)

and for for and

8' ' cc 8 EE

SS ''

8 ,, cc 8

small enough, enough, mE mE == 11 on on 8 8 small

11

II

so that that finally finally it it remains remains to to so

show that, that, if if we we set set show

then then (16.37) (16.37) If we we set set If

2

oE + 88 (u,Du) (u,Du)in in LL'(8( 8 ' ')) weakly weakly oE +

..

so that

1

F(x,x/€) * - F(x,y)dy in L2 ( 0 ' ) weakly IYI y and (16.37) follows.

16.4

Non-linearities in the higher derivatives. Let us consider the equation

with u

= 0

in

r,

and p # 2.

A

formal computation leads to the

homogenized equationt (16.41)

au

-a xi a [ r/l(aij) IaxjI -

P-2 au

5 1

and

u1 being Y-periodic in y

.

For results along these lines, we refer to L. Tartar [l].

'It

is actually a system in u and u1'

207

HOMOGENIZATION OF MULTI-VALUED OPERATORS Homogenization of multi-valued operators.

17.

17.1

Orientation. What we have now in mind is the homogenization of the so-called

variational inequalities (V.1). Let AE be given as in Section 1 and let K be a subset of V such that (17.1)

K is a closed convex subset of V (non-empty)

.

By a V.I. related to AE we mean the problem of finding u

such

that u

E K ,

(17.2) aE(uE,v

-

uE) 2 (f,v

-

uE)

,

H

v E K

.

It is known (Lions-Stampacchia [l]) that, under hypothesis

(l.l), (1.2), (17.2) admits a unique solution uE. We want to study u

as

E

+.

0.

Remark 17.1. If

AE

is symmetric, (17.2) is equivalent to

the minimum is attained in a unique point uE' Remark 17.2. One can think of (17.2) as an equation for a multivalued operator, introducing the subdifferential of the characteristic function of K.

208

ELLIPTIC OPERATORS

Remark 17.3. The problem o f t h e b e h a v i o r o f u (17.4)

K = compact s e t i n V

becomes t r i v i a l i f

.

au au 2 Indeed one h a s t h e n 2 -+ - i n L ( 8 ) s t r o n g l y , so t h a t ax ax j

-

a' ( u E v

u )

-+

I

8

j

au a ( v - U ) dx R ( a 11 ..) ax. ax 1 i

.

I f we s e t (17.5)

(

R ' a

w e have (17.6)

u

,

u in V

-+

where u i s t h e s o l u t i o n of (17.7)

(

n

a ) (u,v

-

u)

2 (f,v

-

u)

V

I

v E K

u

K

-

I n t h i s c a s e t h e homogenized o p e r a t o r i s simply

a - -axi

n

a

(aij) ax j

I

which i s ( i n g e n e r a l ) d i f f e r e n t from

a.

Remark 1 7 . 4 . I f K = V, (17.8)

uE

E

(17.2) r e d u c e s t o t h e e q u a t i o n

V

,

aE(uc,v) = (f,v)

,

V v E V

.

T h e r e f o r e i f w e c o n j e c t u r e t h e e x i s t e n c e o f an homogenized problem, i t w i l l depend on K :

we indeed conjecture t h e e x i s t e n c e K of a c o n t i n u o u s c o e r c i v e b i l i n e a r form a ( u , v ) 0" Vi depending on

HOMOGENIZATION O F MULTI-VALUED

such t h a t one h a s (17.6) where u i s t h e s o l u t i o n of

A E and on K ,

u

(17.9)

209

OPERATORS

E

K

,

~7

K (U,V

-

U)

2

(f,V

-

U)

V

I

K

.

W e g i v e i n t h e n e x t s e c t i o n s some e v i d e n c e t o s u p p o r t t h i s

conjecture.

Remark 17.5. One could a l s o c o n s i d e r i n ( 1 7 . 2 ) a f a m i l y o f convex s e t s KE depending on

17.2

E,

i.e.,

A f o r m a l p r o c e d u r e f o r t h e homogenization o f problems

of t h e c a l c u l u s of v a r i a t i o n s .

Notations. W e d e n o t e by K E t h e s e t of f u n c t i o n s v = v ( x ) on 8

(17.11)

g,(x)

5 v ( x ) 5 g,(x) ,

where gi a r e g i v e n on

such t h a t 8

and where q1 ( r e s p e c t i v e l y , g 2 ) c a n t a k e t h e value

o

(17.12)

v =

(17.13)

IDv(x) I on

-m

r0

on

( r e s p e c t i v e l y , +-)

c

r

=

a 0

,

5 m(x,x/E) ,

-8

x Rn

,

Y

,

where m ( x , y ) i s c o n t i n u o u s

-

periodic i n y

.

L e t t h e r e be g i v e n a f u n c t i o n @ ( x , y , A , p ) which s a t i s f i e s :

ELLIPTIC OPERATORS

210 (17.14)

@

5

is continuous+ on

x

R"

x

R

Rn

x

,

Y-periodic in y ,

We also restrict the class K E to functions which are such that (17.15)

$(x,x/~,v,Dv)E L1 ( ' 0 ) ,

v v

E

KE

,

and we consider the problem (17.16)

i

inf

$(x,x/~,v,Dv)dx ,

vEKE.

8

Remark 17.6. If we take 1

@ (XtypVtD~)= -

aij = aji KE =

K = closed convex subset of

Hi(@) C V

C

H'(0)

v ,

,

problem (17.16) coincides with (17.2) (cf. (17.3)).

Remark 17.7. The remarks to follow extend to functionals of the form (17.17)

inf

1

@(x,:,?,v.Dv)dx

8 where @(x,y,z,v,Dv) is Y periodic in y and Z periodic in z.

'This

hypothesis can be weakened.

211

HOMOGENIZATION OF MULTI-VaUED OPERATORS We suppose t h a t problem ( 1 7 . 1 6 ) a d m i t s a s o l u t i o n uE and w e want t o d e f i n e dn homogenized problem a s s o c i a t e d t o it. W e c o n s i d e r t e s t f u n c t i o n s uE of t h e form

(17.18)

+

‘vE = v ( x )

EV~(X,X/E)

where v i s s u c h t h a t (17.19)

g1

5

v

5 g 2 on 8 ,

v = O

on

ro ‘

and where v1 ( x , y ) s a t i s f i e s

i s Y-periodic i n y

v,(x,y)

,

v1 i s bounded t o g e t h e r w i t h Dxvl and

(17.20)

IDV(X)

+

D v

Y l

(x,y) I

5 m(x,y) ,

x

E

8

,

Then vE d o e s n o t b e l o n g t o KE b u t “ b e l o n g s t o K E up t o g1

-

EM

5

vE

5

g2

+

EM

,

y E R”

E”.

(17.21)

inf

I

0,

E -+

Indeed

i f M i s an upper bound f o r Iv

It i s therefore reasonable t o think t h a t t h e probleq

t h e same 1 m i t , when

.

17.16) has

as

~ ( X , ~ ~ V ~ D V+ ( DX v) ( x ,Xr ) ) d x

,

Y l

0 where v and v1 s a t i s f y t o ( 1 7 . 1 9 ) ,

(17.20).

We now d e f i n e

W e have D J, = D v Y l y ,

+

I ,

1

Dv so t h a t ( 1 7 . 2 1 ) c a n be r e f o r m u l a t e d :

212

ELLIPTIC OPERATORS

(17.23)

inf

$(x,:,v,D

Y

$)ax

,

0 where v s a t i s f i e s (17.19) and where j , s a t i s f i e s j,

(17.24)

-

,

y Dv i s Y - p e r i o d i c

$,D,J,,D

Y

iii a r e bounded

IDy$(x,y)

I 5

m(x,y)

Using t h e f a c t t h a t F ( x , f )

+

I

. -

1

F(x,y)dy i f f o r instance

IYI y

0 n F E C ( 8 x R 1, F being Y-periodic,

it i s a g a i n r e a s o n a b l e t o t h i n k

t h a t p r o b l e m ( 1 7 . 2 3 ) h a s t h e same l i m i t t h a n

v and J, b e i n g s u b j e c t t o ( 1 7 . 1 9 ) a n d ( 1 7 . 2 4 ) . This leads, i n conclusion, t o t h e following general r u l e .

F i r s t step. G i v e n X E R a n d l~ E Rn, w e c o n s i d e r (17.26)

where i n ( 1 7 . 2 6 ) , 8 ( y ) is s u b j e c t t o

J, c a n t a k e t h e v a l u e

+m

if

( 1 7 . 2 7 ) d e f i n e s t h e empty s e t .

213

HOMOGENIZATION O F MULTI-VALUED OPERATORS

Second s t e p . W e consider (17.28)

I

inf

$(x,v,Dv)dx

,

0 where v i s s u b j e c t t o (17.29)

q1

5

v

5

,

g 2 on 0

The p r o b l e m ( 1 7 . 2 8 ) ,

v = 0 on

r0

'

( 1 7 . 2 9 ) i s t h e homogenized p r o b l e m a s s o c i a t e d

to (17.17).

Remark 1 7 . 8 . I n a l l t h e p a r t i c u l a r c a s e s ~ w h e r et h e a b o v e c o n j e c t u r e i s p r o v e d , i t g i v e s t h e same r e s u l t s t h a n t h o s e o b t a i n e d b e f o r e .

Remark 1 7 . 9 . L e t us s u p p o s e t h a t

1

KE = K = i v l v E H o (

(17.30)

=

r,

v

5

0 on 8

1

9 i s g i v e n a s i n Remark 1 7 . 6 [ i . e . , g1

and t h a t

ro

B ) ,

m

=

+-I.

A

~

u

= O o n r ,

=

g 2 = 0,

-m,

Then t h e p r o b l e m i s

-U

f

~

5 o ,

uE

5 o ,

-

( A ~ U ~ f)

uE =

o

in

o ,

(17.31)

a n d t h e homogenized p r o b l e m a c c o r d i n g t o t h e f o r m a l r u l e o b t a i n e d above i s :

214

ELLIPTIC OPERATORS We are going to prove this fact, even with

17.3

AE

not symmetric.

Unilateral variational inequalities. We consider problem (17.31) with

AE

given as in Section 1 ;

(17.31) is equiv-lent to (17.2) when K is given by (17.30). Theorem 17.1. We assume that (l.l), (1.2) hold true.

Let U be the homo-

genized operator associated to A E and let uE (respectively, u) be the solution of the unilateral V.I. Then, as

E

(17.33)

-t

uE

(17.31), (respectively, (17.32)).

0,

1 ( 0 ) weakly u in Ho

-f

.

Several proofs can be given of this result (cf. the Bibliography given in the last section).

We give here a proof based on the penalty

method because of the independent interest of the following error estimate.

Theorem 17.2.

For

AEUEIl

(17.34)

uEn be the solution of t

> 0

u

En

+ L0 u +E n = f

= o

on

r

in

0 ,

;

then

t $ + = sup($,O). term

n

u+

En

Equation (17.34) admits a unique solution.

is called a penalty term.

The

215

HOMOGENIZATION OF MULTI-VALUED OPERATORS

I luEn -

(17.35)

5 cfi

UEII

w h e r e t h e c o n s t a n t C does n o t d e p e n d o n

E.

Proof. W e t a k e t h e scalar p r o d u c t o f E

+

a (uEr-)+

(17.36)

+ 2 n1 tuE,,

(17.34) w i t h u

+

= (f,uE,,)

+ Er-

.

We obtain

.

Hence

+

IuEr-l 5 I f l r l

(17.37)

and

+ I IuErI I2 5

+ 1,

CIuE,,

+

I I u E r - lI

(17.38)

5

hence

C f i

.

We write u

E

- u

and by v i r t u e o f (17.39)

IIu,

€0

= u

E

by

UE

Er-

- u

+ €17

.

+ u i r -I I ' C 6

i n (17.2) (v i s

Erl

+ UEr-.

-

( 1 7 . 3 8 ) , it r e m a i n s o n l y t o p r o v e t h a t

W e c h o o s e v = -u-

-

+ u

5

0 ) and we m u l e i p l y (17.34)

W e o b t a i n a f t e r a d d i n g up

a E ( u E r -- u E I u E +

LIE,,)

+

6 1 ( u+ Er-'uE +

uir-)

2

0

i.e., (17.40)

S i n c e uE

a E ( u E + ui,,)

5

0 , we have

+ r-1 1 17

(uE,,,-uE +

+ (uEr-,-uE)

2

implies

Remark 17.10. We have the same estimate (17.35) for a family of operators AE with coefficients aFj

E

Lm( 8 ) and a:

E

Lm( 8 ) such that

The periodicity of a..(y), ao(y) does not play any role. 13

We can now proceed to the Proof of Theorem 17.1. Let u

Q

be the solution of the penalized problem associated to

(17.32): u

rl

+ A u + = f

n

o

i n o ,

(17.42) u

n

= O

onr.

We have of course

Therefore, in order to prove the Theorem, it suffices to prove that

for n

(17.44)

> fixed, one has

uEn

+

uQ

1 in H o ( o ) weakly

,

as

E

+

0

.

But (17.44) is a simple example of compact perturbations as in Section 16.2, and (17.44) follows.

217

HOMOGENIZATION OF MULTI-VALUED OPERATORS Remark 17.11. Let us consider the convex set (17.45)

1 K = {V(V E Ho(8),v 5 g}

,

the only hypothesis made on g being that K is not empty.

Then one

can prove (cf. F. Murat [2]) that if uE is the solution of the V.I. corresponding to K given by

17.45) one has (17.33) where u is

the solution of (17.46)

u E K

,

Q(u,v

-

U)

2 (f,v -

U)

, v

V E

K

.

[If g is smooth, this result is immediate by translation.]

Remark 17.12. The homogenization of quasi-variational inequalities has been considered in Bensoussan-Lions [l], and Biroli 111.

218 18.

ELLIPTIC OPERATORS Comments and problems. The problem of homogenization is actually a particular case of

the general study of partial differential operators with coefficients in a bounded set of Loo,and such that they are (in the framework of the present chapter) uniformly elliptic. In the case of symmetric operators, this approach is the one followed by E. de Giorgi, Spagnolo and others-and

even in a much

more general setting of calculus of variations-under of G-convergence.

We refer to E. de Giorgi and

S.

the terminology

Spagnolo [l],

S. Spagnolo [ l l , P. Marcellini and C. Sbordone [l], [21, C. Sbordone

[ll, and to the Bibliography therein. The special structure studied here-with

coefficients which are

periodic or connected to periodic or to almost periodic functionsallows to obtain precise results on the structure of coefficients of the homogenized problem.

Some quantitative results have been

obtained in the general case by Murat and Tartar [l]. The method of asymptotic expansions presented in Section 2 is of the type of the "multi-scale" methods-well perturbations.

known in the theory of

The special way of using this general framework in

the present context has been introduced, and used, by the authors in a series of notes and reports; these papers of the authors are quoted in the Bibliography.

Its use in the present context was anticipated

by J. B. Keller as we mentioned in the Introduction. Similar problems have been studied by I. Babuska, in a series of reports at the University of Maryland and by Bahbalov (cf. Bibliography)

.

The idea of the proof of Section 3.2 is due to L. Tartar [ll, [2], together with the method of Section 3 . 4 . The Lp estimates given in Section 4 are due to N. G. Meyers [11 for second order operators.

The extension to higher .order operators

given here, although very simple, seems to be new.

COMMENTS AND PROBLEMS

219

Particular cases of correctors (as presented in Section 5) have been given in the notes of the authors. The results of Section 7 are given here for the first time. The results of Section 8 were announced in one of the notes of the authors [l].

Although the hypothesis made here on the regularity

of aij(x,y,z) improve those of this note, the question of the most general hypothesis under which the formulas of Section 8 are valid is unclear. Formulas ofsections 9, 10, 11 are given here for the first time. Those of Sections 9 , 10 were presented in lectures of the authors and those of Section 11 are due to Murat and Tartar.

The results of

Section 11.4 about compensated compactness follow Murat-Tartar

;

cf-

Murat [l]. Particular cases of Section 9 may be useful in the theory of elasticity.

The theory of Sections 1 to 11 can be extended in several other directions: (1) One can consider elliptic operators which have singularities or degeneracies (for instance near the boundary). First results in this direction are given in Marcellini-Sbordone [2], [31 ;

(2) One can consider elliptic operators with coefficients which have a "stratified periodic structure". More precisely,' let us consider the operator (18.1)

A '

=

- %[aiaj[*]%]

where (18.2)

p E

C2(

5

,Rm)

,

(m 5 n)

,

a. . E L - C R ~ ),

a is Y-periodic in R" ij

aij(y)ninj 2 aninj

,

13

(18.3)

a > 0

,

a.e. in y

,

.

ELLIPTIC OPERATORS

220

Let us assume that

If p(x) = x, we have the situation of Section 1.

If p(x) = 1x1,

(m = l), we have a periodic structure in spheres, and (18.4) is

-

valid if 0 f 0 . The homogenized operator c7 of

AE

is constructed as follows.

We set (18.5)

A1 =

and we define

-1

- a

ayk

x7

=

x3(x,y) as the Y-periodic solution (defined up to

an additive constant) of

Then

where

(3)

Another extension of the results of Sections 1 to 11 is

relative to quasi-elliptic operators. (18.9)

A'

=

a2 b(E) x 7 a2 , - a a(5) -?+ 7 ax, E ax, ax2

m

where a(y), b(y) are L b(y)

a > 0.

For instance let us consider

ax2

functions, Y-periodic, a(y) 2 a > 0,

Boundary conditions associated to

variational formulation.

We consider

AE

are defined by a

COMMENTS AND PROBLEMS

221

p r o v i d e d w i t h it's n a t u r a l H i l b e r t s p a c e s t r u c t u r e . W

d e n o t e s t h e c l o s u r e of C0 ( 8

If H i r 2 ( 8 )

i n H l f 2 ( 8 1, w e t a k e

)

Hir2(IO) C - V C - Hlt2(0)

(18.11)

and w e s e t (18

Then ( w e add a z e r o o r d e r t e r m i f , f o r i n s t a n c e , V = H I p 2 ( 8 ) ) uE i s d e f i n e d by (18.13) When

a E ( u E r v )= ( f , v ) E

+

0 , one h a s u

, +

V

v

E

V

,

u

E V .

u i n V weakly, where u i s t h e s o l u t i o n

o f t h e homogenized e q u a t i o n

i n (18.141, one h a s (18.15)

( u , v ) = q1

1

au av ax, ax,

8

with

and s i m i l a r n o t a t i o n f o r ??l y2

.

2 2 a u a v

222

ELLIPTIC OPERATORS One can also consider non-homogeneous boundary value

(4)

problems.

For instance, with the notations of Section 1, we can

consider the problem

(18.17) $E

=

(18.18)

g

on

( r ).

where g E H 1 ' 2 +

,

in 8

A ~ $ ,= o

r ,

Then 1 in H ( 8 )weakly

@

,

where @ is the solution of (18.19)

f f $ = O

in 8

Indeed, if we introduce G u

= $,

-

(18.20)

,

$=gon

E

r .

1 H ( 8 )with G = g on

r,

and if we set

G, then AEuE = -AEG

,

U

1( 8 Ho

E

But AEG remains in a bounded set of H-l 8 ) , so that UE remains in 1 < C. Then we can a bounded set of H o ( 8 ) , so that I I H1 (.O ) extract such that (18.18) holds true, and we have (18.18). In connection with (18.20), one can more generally consider the equation (18.21)

AEuE =

E

-1 fE

+

9,

where (18.22)

X

fE(x) = f(X,,)

,

X

g,(x) = g(x,,)

,

and (18.23)

f,g E Co(s ;L~(R;))

,

f,g being Y-periodic

.

COMMENTS AND PROBLEMS

223

The method of asymptotic expansion (which can be justified) gives the following. (18.24)

u

=

E

We look for uE in the form

-1u-l

+

uo +

Identifying the powers in

E,

EU1

+

...

u. = 3

u.(x,y) 3

.

we obtain

,

(18.25)

A 1 ~ - l= 0

(18.26)

AluO

(18.28)

AlU2 + A2U1 + A3UO = S(XrY)

+

,

A 2 ~ - 1= 0

,

................. Hence (18.26) gives

It follows from (18.25) that u - ~ = u-,(x).

and (18.27) can be solved in ul, iff.

1

(A2u0 + A3u-l)dy =

Y

I

f(x,y)dy

Y

hence

One can then proceed with (18.28).

If we compute explicitly

AEG in (18.201, we find (18.21) with

(18.31) gE =

-

a2G aij (y) axiaxj

y =

X/E

.

ELLIPTIC OPERATORS

224

Therefore

f(x,y)dy = 0, so that u - ~ = 0. Y

The behavior of eigenvalues in the homogenization procedure (for self-adjoint operators) is studied in Kesavan [l] , Planchard 111. The homogenization of problems with oblique derivatives in the boundary conditions is not entirely solved and not presented here. Some aspects of that problem are developed in Chapter 111. One can also study homogenization problems for boundary value problems in media with "small" holes having a periodical structure. We refer to Cioranescu [l], Cioranescu-Saint Jean Paulin [l]. Asymptotic expansions giving higher order estimates have been introduced int, and are currently used in several directions. The results of Section 12, and others, are given in the notes of the authors. Results not proved here and complements will be given elsewhere. Problems similar to those of Section 12 but with a nonsymmetric operator A'

do not seem to be solved.

Results of Section 13 are taken from Bensoussan, Lions, Papanicolaou [ 2 I

.

In Section 14 we did not make any attempt to study the boundary layers arising in these questions. The results of Section 15 were announced in one of the notes of the authors.

They indicate that it could be interesting to consider

the general homogenization problem for pseudo-differential operators, a question not considered here. In Sections 16 and 17 we give some indications on the homogenization of non-linear problems. The general conjecture presented in Section 17.2 is somewhat similar to the "averaging principle" of Whitham [l].

An approach

bearing some resemblance with this conjecture is followed in Berdichevski [ 11

.

t J.L. Lions, Introduction remarks to asymptotic analysis of periodic structures, Symp. on Trends in App. of Math. to Mech., Kozubnik (Poland), Sept. 7 7 .

COMMENTS AND PROBLEMS A

225

proof of the conjecture of Section 17.2 has been announced by

L. Carbone [l] for the case when g1 =

-03,

g2 =

+a

(i.e. conditions

(17.11) are absent) and when in (17.13) m(x,y) = 1. There is a "non symmetric" analogue of the conjecture, which can be explained as follows. Let us consider the variational problem

We use the "ansatz" u

E

=

u +EU 0

+...,u j

1

=

t

u. (x,y) and we take 1

v = v(x,y) in (18.32). We obtain

We identify the various powers of

6.

We obtain first

hence it follows (if we assume that uo E V so that we can take v=u0) U

0

=

u(x). The coefficient of

in (18.33) gives now

:

:

(18.34) r)

But assuming that v = @(x)$(y), (18.34) is "approximately" indentical to

so that -1 We emphasize that aij is not necessarily equal to aji.

226

ELLIPTIC OPERATORS au av aij (y)(2+ -) ayj axj

?!k.-

ayi

(y)dy = 0 V @

, JI periodic.

This gives (with the notations of the text)

One takes next the coefficient of

(18.36)

aij

au2 (-

ayj

+

aul av + 3 ayi ax.

-)-

6'

in (18.33) and one obtains au, (-+

ayj

aul av

a x ,7 axi

-)-

dx

=

if~ dx-

The compatibility condition for solving (18.36) in u2 leads to the usual homogenized equation. The interest of this formal procedure (but we conjecture it can be justified) is that it can be applied to variational inequalities for non symmetric operators. Another type of homogenization has been considered(for reinforced plates) by energy methods by Artola and Duvaut 111. We shall return to the problem of boundary layer terms in separate papers. In case L9 = Rn the asymptotic expansion constructed in the text gives an approximation of arbitrary high order in

E

(in

Sobolev's norms). The homogenization fcrmula given in the text for periodic coefficients has been extended by S.M. Kozlov [l] to almost periodic coefficients (using the method of asymptotic expansion, the average over the torus being replaced by the average for almost periodic functions and using results of E. Muhamadiev [l] extending the theorems of Favard on the existence of almost periodic solutions of P.D.E.). A

survey of homogenization problems (both for the stationary

and for the evolution case) will be presented in the Ouspechi Mat. Nauk. by O.A. Oleinik [ 2 ] .

COMMENTS AND PROBLEMS

227

The r e s u l t s o f S e c t i o n 1 7 . 3 were announced i n t h e f i r s t of t h e n o t e s of t h e a u t h o r s .

O t h e r r e s u l t s a l o n g t h e s e l i n e s are g i v e n i n

Murat [21, H . Attouch and Y. K o n i s h i 1 1 1 , L. Boccardo and I . Capuzzo D o l c e t t a

[ l ] ,L. Boccardo and P . M a r c e l l i n i [ l ] . With a

convex K of t h e t y p e IDv(x) I

5

1 in 8

I

t h e problem h a s been s o l v e d

o n l y i n dimension 1; c f . Carbone [ l ] , Desgraupes and Salmona [ l ] . Homogenization f o r m u l a s c a n be found-at

l e a s t f o r second o r d e r

o p e r a t o r s , b u t a l s o , i n p r i n c i p l e , f o r o t h e r cases-by

u s i n g "Bloch

waves"; s i n c e t h i s t h e o r y i s n a t u r a l l y c o n n e c t e d w i t h wave propagat i o n , remarks a l o n g t h e s e l i n e s w i l l o n l y be g i v e n i n C h a p t e r 4 . Numerical c o m p u t a t i o n s and n u m e r i c a l problems c o n n e c t e d w i t h homogenization of e l l i p t i c problems a r e n o t s t u d i e d h e r e .

We refer

t o Bourgat [l], Bourgat and Dervieux [ l ] ,Kesavan [ l ] ,P l a n c h a r d [ l ] , Vaninnathan [ l ] and t o t h e r e p o r t s o f I . Babuska. One i s l e d , i n p r a c -

t i c a l c a s e s , t o problems w i t h s e v e r a l s m a l l p a r a m e t e r s ; c f . L i o n s [2] Bourgat-Dervieux

[ l ] . Another a p p r o a c h t o t h e s e q u e s t i o n s i s p r e s e n -

.

t e d i n Bamberger [ l ]

BIBLIOGRAPHY OF CHAPTER 1

-

R. Adams [l], Sobolev-spaces. Acad. Press, 1 9 7 5 . S. Agmon, A. Douglis and L. Nirenberg, [l] Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. I. C.P.A.M. 1 2 ( 1 9 5 9 ) , p. 6 2 3 - 7 2 7 . [ 2 ] 11. C.P.A.M. 1 7 ( 1 9 6 4 1 , p. 3 5 - 9 2 . A. Ambrosetti and C . Sbordone [l],T-Convergenza e G convergenza... ~011.U.M.I. ( 5 ) i 3 - ~( 1 9 7 6 1 , p. 1-11. Y . Artola and G . Duvaut [l], Homogdn6isation d'une plaque renforcge - C.R.A.S. Paris, 2 3 4 ( 1 9 7 7 ) , p. 7 0 7 - 7 1 0 . H. Attouch [l] Convergence de fonctionnelles convexes. Journges Analyse non lindaire, Besanson, June 1 9 7 7 . H. Attouch and Y. Konishi [l] Convergence d'op6rateurs maximaux monotones et ingquations variationelles. C.R.A.S. Paris, 2 8 2 , p. j 6 7 - 4 6 9 . I. Babuska [l] Homogenization approach in Engineering, in Computing Methods in Applied Sciences and Engineering. Lecture Notes in Economics and Math. Systems, 1 3 4 , 1 9 7 6 , p. 1 3 7 - 1 5 3 . [ 2 ] Articles of SIAM, 1 9 7 7 . N.S. Bakhvalov [l] Averaged characteristics of bodies with periodic structure. Sov. Phys. Dokl. 1 9 (101, 1 9 7 5 , p. 6 5 0 - 6 5 1 . [ 2 ] Averaging of non linear partial differential Doklady Akad. Nauk. 2 2 5 ( 1 9 7 5 1 , p. 2 4 9 - 2 5 2 . equations A. Bamberger [l] Approximation des coefficients d'op6rateurs elliptiques, stable pour la G-convergence. To appear. A. Bensoussan and J.L. Lions [l] InGquations quasi variationnelles dependant d'un paramstre. Annali Scuola Norm. Sup. Pisa, 1 9 7 7 , p. 2 3 1 - 2 5 5 . A. Bensoussan, J.L. Lions and C,. Papanicolaou [l] C.R.A.S. Paris, 2 8 1 , ( 9 7 5 ) , p. 8 9 - 9 3 , p. 3 1 7 - 3 2 2 , 2 8 2 , ( 1 9 7 6 1 , p. 1 2 7 7 - 1 2 8 2 . [Z] Homogenization and ergodic theory, Stefan Banach Symp. on probabilities, Warsaw, 1 9 7 6 .

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Paris, February 7 , 1 9 7 7 . V.L. Berdichevski [l] Periodical structures. Doklady Akad. Nauk. 2 2 2 (No 3 ) , 1 9 7 5 , p. 5 6 5 - 5 6 7 . M. Biroli [l] Sur l'homog6n6isation des ingquations quasi-variationnelles. C.R.A.S. Paris, 1 9 7 7 . 228

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L. Boccardo and I. Capuzzo Dolcetta [l] G-convergenza e problema di Dirichlet unilaterale. Boll. U.M.I. ( 4 ) , 12, (1975), p. 115-123. L. Boccardo and I. Capuzzo Dolcetta [2] Stabilita'delle soluzioni... Univ. dell'Aquila. Report 10/77. L. Boccardo and P. Marcellini [l] Sulla convergenza delle soluzioni di disequazione variazionali. Anna11 di Mat. Pura ed Appl. (IV), CX, 1976, p. 137-159. J.F. Bourgat [l] Numerical experiments of the homogenization method for operators with periodic coefficients. Proceedings of third International Colloquium on Computing Methods in Applied Sciences and Engineering, Versailles, December 1977, SprinqerVerlag, Berlin. J. F. Bourgat-A. Dervieu:: [l] M6thode d'homog6n6isation des opgrateurs h coefficients p6riodiques. Etude des correcteurs provenant du d6veloppement asymptotique. Laboria Report. To appear. J.F. Bourgat and H. Lanchon [l] Application of the homoqenization method to composite materials with periodic structure. Laboria Report No 208, (1976) J.M.P.A., 1977. C. Carbone [l] r-Convergence d'integrales [2] Paper to appear, Rend. Lincei., 1978. L. Carbone and C. Sbordone [l] Un teorema di compatezza Rend. Acc. Naz. Lincei., 1978. D. Cioranescu, H. Lanchon and J. SaintJean Paulin [I] Torsion To appear. elastoplastique d'arbres cylindriques non homog&es. D. Cioranescu and J. Saint Jean Paulin [l] To appear. D. Cioranescu [l] Thesis, Paris, 1977. , Desgraupes and Salmona [l] To appear. G. Duvaut [l] Analyse fonctionnelle en m6canique des milieux continus... Theoretical and Applied Mech, W.T. Koiter, ed. North Holland Pub. Company (1976), p. 119-132. G. Duvaut and J.L. Lions [l] Les ingquations en Mgcanique et en Physique. Paris, Dunod, 1972. English translation, Springer, 1975. G. Duvaut and A.M. Metellus [l] Homog6n6isation d'une plaque C.R.A.S., Paris, 283 (1976), 947-950. mince E. de Giorqi and S . Spagnolo [l] Sulla converqenza degli integrali dell'enerqia per operatori ellittici del secundo ordine. Boll. U.M.I. 8 (1973), p. 391-411. S . Kesavan [ l ] Homog6n6isation et valeurs propres. C.R.A.S. Paris 285 (19771, p. 229-232.

...

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230

BIBLIOGRAPHY [2] Laboria Report, 1978. Kesavan and M. Vaninathan 111 L'homog6n6isation d'un problbme de contr6le optimal. C.R.A.S. Paris 285 (1977), p. 441-4134. S.M. Kozlov [l] Averaging of differential operators with almost periodic, fast oscillating coefficients. Doklady Akad. Nauk. 236 (1977), NO 5, p . 1068-1071. J.L. Lions [l] Remarks on the theory of optimal control of distributed systems. In Control theory of systems governed by S.

partial differential equations. Acad. Press. 1977, p. 1-103. [2] Remarks on some numerical aspects of the homogenization method in composite materials. Novosibirsk 1976. J.L. Lions and E. Magenes [l] Problbmes aux limites non homoghes et applications. Vol. I and 11, Paris, Dunod, 1968 ; English translation, Springer, 1970. 121 Problgmes aux limites non homog&nes (111) Ann. Scuola Norm. Sup. Pisa, 15 (1961), p. 39-101. J.L. Lions and G. Stampacchia [l] Variational inequalities. C.P.A.M. 20, 3 (1967), p. 493-519. P. Marcellini and C. Sbordone 111 An approach to the asymptotic behaviour of elliptic parabolic operators. J. de Math. Pures et Appliquges, 56, 1977, p. 157-182. [2] Sur quelques questions de G-convergence et d'homogingisation non lin6aire. C.R.A.S. Paris, t. 284, 1977, p. 535-537. [ 3 ] Homogenization of non-uniformly elliptic operators. Applicable Analysis, 1977. [el Dualita e perburbazione di funzionali integrali. Istituto Caccioppoli. June 1977. N.G. Meyers [l] An LP-estimate for the gradient of solutions of second order elliptic divergence equations. Ann. Sc. Norm. Sup. Pisa, 17 (1963), p. 189-206. E. Muhamadiev [l] On the invertibility of elliptic partial differential operators. Doklady Akad. Nauk. SSR 205 (1972), No 6 Soviet Math. 13 (19721, No E - , p. 1122-1126. F. Murat [I] Compacitg par compensation. Ann. Scuola Norm. Sup. Pisa, 1977. [21 Sur l'homog6n6isation d'ingquations elliptiques du 2Gme ordre... To appear. F. Murat and L. Tartar [l] To appear. J. Negas [ll Les mgthodes directes dans la thiorie des gquations b

BIBLIOGRAPHY

231

elliptiques. Ed. de 1'Acad. Tech. des Sciences, Prague, 1967. O.A. Oleinik [l] In 'Problems of Mechanics and Mathematical Physics', Book dedicated to Professor Petrowsky, Moscow, 1976, p. 221-236. O.A. Oleinik [ 2 ] Survey on homogenization. Ouspechi Mat. Nauk. to appear. J. Peetre [l] Another approach to elliptic boundary problems. C.P.A.M. 14 (1961), p. 711-731. J. Planchard [l] Etudes et Recherches de l'Electricit6 de France Internal Report, 1977. To appear. E. Sanchez-Palencia [l] Solutions pgriodiques par rapport aux variables d'espace et applications. C.R.A.S. Paris (2711, (1970), p. 1129-1132. [2] Equations aux d6rivGes partielles dans un type de milieux hgtgrogsnes. C.R.A.S. Paris (272) (1970), p. 1410-1611. C. Sbordone [l] Su alcune applicazioni di un tipo di convergenza variazionale. Annali Scuola Norm. Sup. Pisa, IV (1975), p. 617-638. L. Schwartz [l] Thgorie des distributions. t. 1 et 2. Paris, Hermann, 1950, 1951. S . L . Sobolev [l] Application de 1'Analyse Fonctionnelle aux Gquations de la Physique Mathgmatique. Leningrad. 1950. S. Spagnolo [ l Sulla convergenza di soluzioni di equazioni paraboliche ed el ittiche. Ann. Sc. Norm. Sup. Pisa, I11 22 (19651, p. 571-597. [2 Convergence in energy for elliptic operators, in Num. Sol. of Partial Diff. Equations. 111, Synspade 1975, Acad. Press 1976. I L. Tartar [l] Cours Peccot, CollGge de France, February 1977. [2] Quelques remarques sur l'homog&e'isation, French-Japan Symposium. Tokyo, September 1976. Y.Vanninathan [l] Numerical problems in homogenization, Laboria Report, to appear. Xa Tien Ngoan [l] Vestnik Moskorskogo Univ., Math. and Mech. Series, 5, 1977.

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Chapter 2

:

Evolution Operators

Orientation. The reader mostly interested in Probabilistic Methods (Chapter 3) should read Section 1 and Sections 2.1 to 2.5, and then proceed with Chapter 3. The reader mostly interested in High Frequency wave propagation should read Sections 3.1 to 3.5 and Section 3 . 8 ,

and then proceed

with Chapter 4 . Sections 1 and 2 treat the case of parabolic operators with coefficients rapidly varying in the space and in the time variables, with different scales. The method of asymptotic expansion shows that there are three cases to be considered. Proofs of convergence are given in Section 2; Sections 2 . 6

and 2 . 7 give the proof under the

weakest possibly hypothesis on the coefficients and are somewhat complicated. Section 3 studies the operators of hyperbolic type, or of Petrowsky or of Schrodinger type. Some non linear problems are considered in Sections 2.12 and 3.7. Section 3.9 gives an example (somewhat connected with Chapter 1, Section 15)

of a situation where

local operators,of evolution admit for homogenized operator a non local evolution operator. 233

234

EVOLUTION OPERATORS Many other situations can be studied by using ideas and

techniques of Chapters 1 and 2. Some of them are briefly indicated in Section 4 (comments). Another situation will be studied by these techniques in Chapter 3, Section 5 (since the motivation for such a study appears in the Probabilistic context).

1.

Parabolic operators.

1.1

Asymptotic expansions.

Notations and orientation. Let 8 be a bounded (to fix ideas!) open set of Rn, with boundary

r

(which will be assumed smooth or

not, depending on the situation).

Let T > 0 be a given finite number.

We set

We will consider first, equations of the following type:

with initial condition (1.3)

UE(X,0)

=

u 0 (x)

,

x E 8

,

and with suitable boundary conditions, defined in Section 1.2 below. On the functions aij we shall always assume that . ! a

17

remains in a bounded set of Lm( S 1

,

(1.4)

aTj(x,t)cicj 2 a

~

in , ~ a.e. c ~

s ,

a >

o ,

si E

R

.

In order to obtain constructive formulas, we shall assume much more on the structure of functions aE namely, ij;

PARABOLIC OPERATORS: (1.5)

ASYMPTOTIC EXPANSIONS

aE (x,t) = aij(x/E.t/E k ) ij

235

,

where k > 0 is for the moment unspecified, and where: aij(y,T) is Y-r0 periodic

aij(y,T)ciEj

clcici

,

, t

a.e. in y and

We want to study the behavior of uE

E

-+

T

.

0.

We begin by giving a precise definition of uE (Section 1.2) and we study next the asymptotic expansion of uE.

As we shall see the

formulas obtained depend on k (which appears in (1.5)).

Remark 1.1. We shall explain in the comments on Section 4 the results one of the form

obtains when considering functions a:j (1.7)

aij (x,t) = aij (x,x/E,~,~/E k)

and more generally aij(x,t) = aij(x,-,x x

(1.8

E

1.2

,...,t,-t

t EkF1--*) *

E2

Variational formulation. We need some notation.

We shall use, as in Chapter 1, spaces

V such that

'That

is aij(y,r) is Y periodic as a function of y and aij(y,r

+

T

0

)

=

a. .(y,r) , 17

.

a.e. in y , ~

236

EVOLUTION OPERATORS

Given a H i l b e r t s p a c e L ~ ( O , T ; 8 ) t h e s p a c e of (0,T)

-t

( o r a Banach space!) w e d e n o t e by

(classes o f ) functions t

-t

v ( t ) from

which a r e measurable and which s a t i s f y

8

= ess.supl l u ( t )

II

8

Provided w i t h t h e norm g i v e n i n ( l . l O ) ,

<

m

,

Lp(0,T;8

if p = )

m

.

is a

Banach s p a c e (and it i s a H i l b e r t s p a c e i f p = 2 and i f M i s a Hilbert space). W e s h a l l set

(1.11)

2 H = L ( 0 )

,

V C H C V '

and w e o b s e r v e t h a t

1 For u , v E H ( C ? ) , w e d e f i n e :

(1.12)

a'(u,v)

=

au av dx (x,t) ax. 3 axi

.

W e have t o be c a r e f u l s i n c e a E( u , v ) depends on t and s i n c e , even

i f t h e f u n c t i o n s a . 's a r e smooth ( f o r e v e r y E ) t h e t - d e r i v a t i v e s of lj

t o t h e s t r u c t u r e of a . given i n (1.5) 1j t h a t , i n what f o l l o w s , t h e a p r i o r i e s t i m a t e s s h o u l d n e v e r u s e t h e a i j ' s are unbounded-due

d a E( u , v ) . dt For u

E

V w e d e f i n e A'u

E

V'

by

PARABOLIC OPERATORS:

I n f a c t A'

ASYMPTOTIC EXPANSIONS

231

depends on t :

(1.14)

AE = A E ( t )

.

T h e " a b s t r a c t " f o r m u l a t i o n of t h e p r o b l e m w e w a n t t o s t u d y i s

l e t f a n d u o be g i v e n , s a t i s f y i n g

now: (1.15)

f E L2(0,T;V')

(1.16)

uo

Then

E

H

,

.

u E i s t h e s o l u t i o n of 2 L (0,T:V)

(1.17)

u

(1.19)

UE(0) = u

E

0

,

.

Remark 1.2. 2 I f v E L (0,T;V)

a bounded s e t of L m ( S

then AEv )

E

2 L (0,T;V')

and s i n c e a .

11

remains in

,

w h e r e here and i n w h a t f o l l o w s , t h e c ' s d e n o t e v a r i o u s c o n s t a n t s w h i c h do n o t depend on

E.

T h e r e f o r e (1.18) gives:

238

EVOLUTION OPERATORS

I c[ I

(1.22)

IUE

II

+ 11

.

L2 (0,T;V) But it is known (cf. for instance Lions-Magenes [l]) that: (1.23)

if v

6

2 L (0,T;V) and

2

L (O,T;V'), then v

is a.e. equal to a function, still denoted by v, such that t

+

v(t) is continuous from [O,Tl

+

H.

Therefore we see that (1.17), (1.21) imply that (1.19) makes sense.

Remark 1.3. It is proven in Lions [l] that the problem (1.171, (1.18), (1.19) admits a unique solution. One has the estimate

The verification of ( 1 . 2 4 ) is easy. (1.18) by uE.

We multiply both sides of

We denote by (f,g) the scalar product in H and we set

But we observe thatt

'This @ E

2

is obvious if

L (0,T;V) and

2

E

@

is smooth and it can be justified if

L'(0,T;V').

PARABOLIC OPERATORS:

ASYMPTOTIC EXPANSIONS

so t h a t (1.26) g i v e s ( w e w r i t e a'($)

2 39

for

Hence t

b But d u e t o (1.4), w e c a n f i n d A t (1.28)

aE

V)

2 a1

W e d e n o t e by

1\71

l2 -

I 1 I I*

A v/

2

such t h a t

v

,

t h e norm i n V'

.

E

V

.

W e o b t a i n from ( .27) 2

+;

/

IluEl

2do

'

0

1 +One c a n t a k e X = 0 i f V = H O ( 0 ) or i f V = s p a c e w h i c h d o e s not contain the constants.

240

EVOLUTION OPERATORS

Using G r o n w a l l ' s lemma,

(1.24)

follows.

Remark 1 . 4 . One c a n r e p l a c e ( 1 . 1 8 ) by t h e e q u i v a l e n t f o r m u l a :

1

V = Ho(B').

Example 1.1:

Then ( 1 . 1 7 ) i m p l i e s (1.31)

u

= 0 on C

.

I t i s t h e D i r i c h l e t ' s boundary v a l u e problem.

Example 1 . 2 :

V

1 = H (8).

Then one deduces from (1.30) t h a t , i n a "weak" s e n s e , au

(1.32)

2 = 0 on C

where

a

-=

av

a

a?.(x,t)v. 11 7 axi

,

AE

(1.33)

v = { v l , . . . , ~ n } = normal t o

r

d i r e c t e d toward t h e e x t e r i o r

PARABOLIC OPERATORS:

ASYMPTOTIC EXPANSIONS

241

I t i s t h e Neumann's b o u n d a r y v a l u e p r o b l e m .

Remark 1 . 5 . ( 1 . 2 4 ) w e see t h a t w e c a n e x t r a c t a s u b s e q u e n c e ,

By v i r t u e o f

s t i l l d e n o t e d by u E , s u c h t h a t (1.34)

u

2 u i n L (0,T;V) w e a k l y

-+

au

$ is

(and ( 1 . 2 4 ) ) ,

Using ( 1 . 2 2

.

2 bounded i n L ( 0 , T ; V ' ) and t h e r e f o r e

w e c a n assume t h a t +

I t i s known t h a t ,

.

i n L2 ( 0 , T ; V ' ) weakly

when

V

-+

H i s compact, it f o l l o w s from ( 1 . 3 4 ) ,

(1.35) t h a t (1.36)

uE

+

.

2 u i n L (0,T;H) s t r o n g l y

I f we set

w e c a n a l s o assume t h a t (1.38)

6;

+

ci

2 i n L (0,T;H) w e a k l y

.

One v e r i f i e s t h a t , i f a ? . i s g i v e n by ( 1 . 5 ) , w i t h ( 1 . 6 ) , t h e n 13 (1.39)

a!.

13

+

where i n g e n e r a l

74 ( a i j )

&

L m ( S ) weak s t a r

,

242

EVOLUTION OPERATORS

topology).

k).

ci

m

( a1. 3. ) &.a x . ( i n any 3 W e s h a l l o b t a i n below t h e l i m i t of ST (which depends on

But i t d o e s n o t f o l l o w t h a t

-+

The o n l y t h i n g w e c a n s a y f o r t h e t i m e b e i n g i s t h a t

(1.41)

at -

Remark 1 . 6 .

ac. axi

.

= f

The case of c o e f f i c i e n t s a:. i n d e p e n d e n t of t . 13

L e t u s assume t h a t

(1.42)

,

a ? . = a . . (x/E) 13 =I

where t h e a i j ' s

a r e Y-periodic

( i . e . w i t h t h e hypotheses of

Chapter 1 ) . Then t h e answer i s immediate.

Let

g e n i z e d o p e r a t o r of A E , c o n s t r u c t e d i n Chapter 1.

(1.44)

% at +

Q

be t h e homo-

=

,

4 u = f

d e f i n e s t h e l i m i t of u E . m

I n d e e d , l e t $ be g i v e n in?,,

(10,T[). For any f u n c t i o n

v E L ~ ( O , T ; li ) , w e d e f i n e

W e remark t h a t au

(1.47)

a t (0) =

-u

E

d0 (-)

dt

s o t h a t w e deduce from ( 1 . 1 8 ) t h a t

PARABOLIC OPERATORS:

But since

AE

ASYMPTOTIC EXPANSIONS

does not depend on t, (A'u,)

(@) =

AE (uE( $ ) ) and

therefore

(we use (1.36)).

We also know that

(1.50)

u($) in V weakly

uE($)

+

.

hence (1.44) and the result follow.

1.3

Asymptotic expansions.

Preliminary formulas.

We shall consider the three most interesting cases for k appearing in (1.7). P E r k=

In order to avoid confusion, we shall set:

a +

A

(x)

E t E

(1.51) E

and we shall consider the values (1.52)

k = 1,2,3

We shall s e t

'I

. = t/Ek and

243

EVOLUTION OPERATORS

244

.5

all these operators depend on (1.54)

PErl =

Q1+E

E - ~

It follows that

parametrically.

T

-1 0 Q2+EQ3r

where (1.55)

Q1 = A1

(1.56)

PCr2 =

,

Q2

=

-

R1 ~ +

E

E

a + A2

-lR2 +

E

a+ A 3 Q 3 = -at

,

a?

0

;

R3 ,

where

a +

A1

,

,

(1.57)

R1 =

R2 = A2

(1.58)

P E r 3 = E - ~ +S E~-2 s2 + €-Is3

R3 =

+

E

0

a +

A3 ;

s4 ,

where

a ,

(1.59)

S1 = aT

S2 =

A1

,

S3 =

,

A2

S4 =

a +

.

A3

We use now these formulas to find an asymptotic expansion of uE.

We distinguish the three cases k = 1,2,3. ' We look for uE in

the form (1.60)

u

€

=

u

0

+

EU1

+

E2U2

+

. ..

,

U. =

u . being Y periodic in y and 3

'Another

3

T~

U.(X,Y,t,T)

,

periodic in

T

3

.

ansatz will be necessary when k is not an integer.

PARABOLIC OPERATORS: 1.4

Asymptotic expansions:

ASYMPTOTIC EXPANSIONS

245

The case k = 1.

By identifying powers of

in the expansion of PEV1u = f , we

E

obtain:

,

(1.61)

QluO = 0

(1.62)

QlUl + Q2UO = 0

(1.63)

Q1u2

t

+ Q2u1 + Q3uo

=

f ,

... .

In (1.61), x , ~ , Tare parameters, so that (1.61) is equivalent to (1.64)

uo does not depend on y

We denote by tively,

T ) so

.

m

(respectively, Y that ??I = h T O

the average in y (respec-

T)

Y

=

.

By taking

n

of

(1.62), we obtain

i.e., since we have (1.641, (1.65)

uo = u(x,t)

auO

=

0 and therefore

.

Then (1.62) reduces to: (1.66)

Alul = -A2u =

au a , .( y , ~ )+ aYi a11 ax j

(x,t)

.

We can write the general solution of (1.66). We introduce

XJ

=

XI ( y , ~ )as the Y-periodic solution, defined up to an additive

constant, of

and we choose the constants such that (1.68)

xJ(Y,T)

+ T~

periodic

;

246

EVOLUTION OPERATORS

(for instance we can take ?!4

Y

( ~ J ( Y , T )=) 0, v

T ) .

Then (1.66) gives (1.69)

au u1 = - ~ J ( Y , T ) (x,t) + Gl(x,t,~) j

.

ax

We can solve (1.63) for u2 iff,

?4 y(Q2u1 + Q 3 u )

= f

,

i.e., (1.70)

aGl

ar +

?4 Y Q 2 (-XI &) 3

74 y(Q3u)

+

We can solve (1.70) forG1 (with a

T~

=

f

.

periodic function of

T)

i.e.,

We observe that

so that we obtain

(1.71)

au

-

9 [aij - aik

$1

2

axiax. a u = f . 3

This is the homogenized operator (as we shall justify below).

1.5

Asymptotic expansions:

The case k = 2.

We use now (1.56), (1.57).

,

(1.72)

RluO = 0

(1.73)

Rlul + R2uO = 0

,

We obtain

iff

PARABOLIC OPERATORS: (1.74)

R1u2

+ R2ul + R3u0

=

f

ASYMPTOTIC EXPANSIONS

247

.

We multiply (1.72) by uo and we integrate over Y.

Let us set

We obtain

Since uo is

au, aYi

= 0,

(1.77)

T~

i.

periodic,

'0

au [g,u0],

d-r = 0, so that (1.76) implies

0 auO Then AluO = 0 and a?

- + AIUO

:yo

= - - 0,

so that

.

uo = u(x,t)

Then (1.73) reduces to

We introduce 8j

( y , ~ ),

Y - T ~periodic

, solution of

(1.78)

(0'

is defined up to an additive constant).

Then

We can solve (1.74) for u2 ifft

tR1@ = F admits a Y-T,, periodic solution iff m ( F ) = 0

(observe that RZji = 0 admits the constants as only Y - T ~per odic solution).

248

EVOLUTION OPERATORS (R2u1

+ R3uO)

,

= f

i.e., au (1.80)

-

n [ a1.3. -

a 2u

aej] a ik

ayk

_ .

~

-

ax.ax. 1

3

This is the homogenized operator of below).

in

(1.71)

P E P 2 (as

for P E r l ,

but coefficients are different.

The case k = 3 .

We now use (1.58), (1.59).

We obtain:

,

(1.81)

S u = 0 1 0

(1.82)

Slul + S2uo = 0 ,

(1.83)

S1U2 + S2U1 + S3Uo

(1.84)

S1u3 + S2U2 + S3U1 + S4Uo

=

0

t

= f

,

... .

It follows from (1.81) that uo does not depend on a

T

0

we shall justify

It has of course the same structure as the one obtained

Asymptotic expansions:

1.6

.

-

T:

(1.82) admits

-periodic solution in u1 iff

I

Ts2uo

=

0

,

i.e.,

We define (1.86)

6,

=

a

It is of course.an elliptic operator in Y, so that (1.85) is au equivalent to 2 = 0, V j and therefore uo = u(x,t)'. Then (1.82) aYj

PARABOLIC OPERATORS:

ASYMPTOTIC EXPANSIONS

reduces to __ a-r = 0, i.e. (1.87)

u1 does not depend on

T

.

We can solve (1.83) for u2 iff RT(S2U1 + S3Uo)

=

0

$ 1 , ~ )is Y-periodic

,

,

i.e.

If we introduce @ J ( y )by:

(1.88) J

,

- y . )

= o ,

(which defines tJj up to an additive constant), we see that (1.89)

u1

'

= -@'(y)

au ax. +

u,(x,~)

.

3

Then (1.83) can be solved for u2; if 7

we have (1.91)

0 u2 = u2 + u2(x,y,t)

.

Equation (1.84) can be solved for U3 iff (1.92)

R T ( s2u 2 + s3u1 + s4u 0)

and (1.92) can be solved for (1.93)

R

1.r r(S2U2

+

G2

s3u1

+

= f

iff S4U0) = f

.

249

EVOLUTION OPERATORS

250

But

m'Y

r

(S2u2) =

RT m

Y (Szu2) = 0, so that (1.93) reduces to

We have again the same structure for the homogenized operator, with again different formulas.

1.7

Other form of homogenization formulas. Let us denote by q k. . = homogenized coefficient of 17

k = 1,2,3.

Let us check the following formulas:

where al is defined in (1.75) and where

For (1.961, we will actually prove that

This is sufficient since

a2 u

(-----ax.ax.

1

7

I

for

PARABOLIC OPERATORS:

251

ASYMPTOTIC EXPANSIONS

taking into account that

= o . Therefore, by comparing the expressions (1.71), (1.80), (1.94) with (1.95), (1.96)I , (1.97), we have to verify that

(1.101)

;+$’ - Yj&)

= 0

This is straightforward.

al(xj

-

.

Indeed (1.67) implies that

yj,x i) =

o

for every

T

,

hence (1.99) follows. If we multiply (1.78) by 0

i

and integrate in y and in T, we

obtain (1.100) and (1.101) follows from (1.88). Formulas (1.95), (1.96), (1.97) imply the ellipticity I Of k a’ f f k = -qij axiax j

(1.1021

~

-

Indeed, if we set:

+ we have

=

cj(xj

-

yj)

,

J, =

cj(ej

-

yj)

,

P =

c j @j

252

EVOLUTION OPERATORS

Hence the result follows.

1.8

The role of k. Let us consider now k > 0 not necessarily an integer. We set

(1.106)

ui = solution of (1.171, (1.181, (1.19) relative to AE =

A(t/E

k)

,

and we denote by uk , k = 1,2,3, the solution of (1.107)

*+

(1.108)

Uk E

(1.109)

uk (0) = u0

k at

,

qkuk = f

L2(0,T;V)

,

.

Then we shall see that the limit of uz when k < 2 (respectively, k > 2 ) does not depend on k, i.e. that :u

-f

u1 (in the appropriate

topology) when k < 2 (respectively, -+ u* when k

> 2).

But the

asymptotic expansion depends on k, and actually new "ansatz" will be necessary for the cases when k is not an integer.

CONVERGENCE OF THE HOMOGENIZATION OF PARABOLIC EQUATIONS 2. 2.1

253

Convergence of the homogenization of parabolic equations. Statement of the homogenization result. We shall prove in several steps the

Theorem 2.1. We denote by :u k solution of (1.17), (1.18), (1.19) relative to AE = A(t/E ) We assume that (1.5), (1.6) hold true.

the solution of (1.107), (1.108), (1.1091, k = 1,2,3, where (respectively, 0 ’, respectively,

a

3

)

the k u

a1

is given by (1.71)

(respectively, (1.80), respectively, (1.94)). Also assume that there exists p > 2 such that f E Lp(0,T;W-1rp(C3) ) and that uo or at least that f E

-+

and

E

uo belong locally to these spaces.+

WitP(b 1 , Then, as

0, one has

(2.1)

:u

(2.2)

u :

(2.3)

:u

+

-+

-+

u1

in L2 (0,T;V) weakly if k < 2

,

u2

2 in L (0,T;V) weakly if k = 2

,

u3

2 in L (0,T;V) weakly if k > 2

.

We begin by the case (2.2) which is the simplest; we shall next proceed with (2.1) and (2.3). 2.2

Proof of the homogenization when k = 2. Given P(y) = homogeneous polynomial in y of first degree, we

define w(y,.r) as the solution (defined up to an additive constant) of

t1.e. if ~3

1

c

5

I

c

0 , f E LP(O,T;W-~’P(O I ) ) ,

This hypothesis is not necessary if k = 2 or if k # 1 (cf. (2.22) below).

uo E W’~P(S aij E C

0

1 ) .

EVOLUTION OPERATORS

254

- -:y +

(2.4)

,

AIw = 0

-

w

P i s Y-r0 p e r i o d i c

.

We write:

(2.5)

AE(t/E

2

) = AE

,

u2 = u E

E

.

I f w e set

*

(2.6)

w - P = - ~

then

* * * - -aare + A1(B -

(2.7)

,

P) = 0

b e i n g Y-ro p e r i o d i c

O*

.

W e now d e f i n e wE by

w (x,t) =

(2.8)

2

EW(X/E,~/E

.

)

W e observe t h a t

- -aa wt E + AE*wE =

(2.9)

For any (0 E C ? z ( S ) , and ( 2 . 9 ) by $ u E .

s

.

0

=

8

x

] O , T [ , w e m u l t i p l y (1.18) by $wE

W e i n t e g r a t e over

s

;

w e o b t a i n (compare w i t h

C h a p t e r 1, S e c t i o n 2 ) :

+

[[>]@w,

(2.10)

s +

Is

[c:

[?]$u,]dxdt

wE

-

aij

2$

uE]dxdt =

j

f$wEdxdt

.

0

The f i r s t term i n ( 2 . 1 0 ) e q u a l s (2.11)

Since w

+.

P in L

2

( s ) s t r o n g l y and

t o t h e 1 m i t i n a l l terms of

s i n c e w e have (1.36

(2.10), ( 2 . 1 1 ) .

W e obtain

I

w e c a n pass

CONVERGENCE OF THE HOMOGENIZATION OF PARABOLIC EQUATIONS

(2.12)

I

-

UP

a+

dxdt

s

i

f$Pdxdt

K )* axj u l d x d t

( a i j aw

Q =

‘

+

.

n

Q

Using (1.41) w e see t h a t

J s

dxdt

+

s

sJ

5,

so t h a t (2.12) r e d u c e s t o

Therefore

(2.13)

k)au.

6. ap = 1

311 ( a k j ayk

ax,

ax j

L e t us take

W e define 8

(2.14)

i* by

- -aei* ar

+

A;(ei*

ei* b e i n g Y - r o Then w = -(Bi*

or

-

-

yi) = 0

periodic

,

.

yi) and (2.13) g i v e s :

a ax. l

($P)dxdt

8

255

EVOLUTION OPERATORS

256

(2.15)

The variational formulation of the problem is:

[>,v]

+

[ti&]

= (f,v)

,

v v

E

v ,

,

\J

v

E

v

and passing to the limit we find (2.16)

au (=,V)

av + (5.,-) 1 axi

= (f,v)

This proves the Theorem (for k

=

.

2) if we verify that we obtain the

same formula than in (1.80), i.e. that

This is equivalent to proving that

or that

We multiply (2.14) by 8 1 ,

(1.78) by Bi*.

This shows that the

left hand side of (2.19) equals

and that the right hand side of (2.19) equals

and the two expressions ( 2 . 2 0 ) ,

(2.21)

are indeed equal.

CONVERGENCE OF THE HOMOGENIZATION OF PARABOLIC EQUATIONS

257

Orientation. We consider now the cases when k # 2.

We begin with the study

of this question under the extra hypothesis (2.22)

2.3

,

a . E Co(F x [ O , - r 0 1 ) ij

.

periodic

Reduction to the smooth case. We are going to show

Lemma 2.1. Under the supplementary hypothesis (2.22), it suffices to prove Theorem 2.1 under the hypothesis (2.23)

aij

E

C"(Y

,

x [O,.rO1)

periodic together with all its derivatives

.

Proof.

of (1.30), k fixed arbitrarily. [ O , T ~ ] ) by j!a E C"(Y x [ O , T ~ ~ ) aij ,B

We denote by uE the solution !u

We approximate aij in C"(Y x being periodic together with all its derivatives and satisfying (2.24)

agjcitj 5 a l ~ i t iI

al > 0

,

y,T

b!

.

B k We denote by uE (not to be confused with uE) the solution in

L'(o,T;v) (2.25) where

of aUE

IK,v]

+ aEB(u!,v)

=

(f,v)

,

tl

v

E

v

250

EVOLUTION OPERATORS

We shall check below that

B

(2.27)

HUE

-

UEII

L2 (O',T;V)

< -

c

B supllaij i,j

-

a

iJ

11

co(iX

Let us consider now the case k = 1 to fix ideas.

[o,Tol)

.

We define

x j r B by A1(~JfB y.) = 0 7

where A!

=

- a aYi

(aBj(y,T)

"1,

,

aYj

xJtB

Y - T ~

One .,bows easily that (normalizing x

n Y (XI)

= 0,

nY

,

and we define

-

for instance

periodic

=

o

yi)d.r

.

J , ~ 1 in ' the ~ same manner,

CONVERGENCE OF THE HOMOGENIZATION OF PARABOLIC EQUATIONS

259

It follows from (2.27), (2.28), (2.29) that it is enough to 2

prove that u l P B + u l t B in L (0,T;V) weakly when E

E

+

0, 6

fixed.

The same remarks apply to all values of k, so that it remains only to prove (2.27), which is standard. m

E

=u'-u E

Indeed if we set

E

we have

where

so that

and therefore

I IfE Thus (2.27) fol ows from (1.24).

2.4

Proof of the homogenization when 0 < k < 2 . We now prove (2.1) under the assumption (2.22), by Lemma 2.1.

We can therefore assume that ( 2 . 2 3 ) holds true.

260

EVOLUTION OPERATORS

2.4.1

The case 0 < k

5 1.

Let P(y) be given as in Section 2.2.

Let w be the solution

(defined up to an additive constant) of (2.30)

*

A1w = 0

,

w

-

P(y)

is Y - T ~periodic

k We use notation (2.5) and we set uE

-

w

=

uE.

.

If we set

p = -x*

then (2.31)

*

A1(X*

-

P)

=

0

,

x* being Y - T ~periodic

.

We now define w E by x -1t w (x,t) = w(-

.

E 'Ek

We observe that AE*wE = 0

.

The same procedure as for (2.10) leads to

f@wEdxdt

=

.

n

l4

The first term in (2.32) equals (2.33)

-

I

s

uEwE

aw

2 dxdt s

Since we have assumed the coefficients aij to be smooth, it follows from (2.30) that

CONVERGENCE OF THE HOMOGENIZATION OF PARABOLIC EQUATIONS

261

so that we have in particular

11

at awE

dxdtl 5 CE'-~

awE If k = 1, =

[ElEand aw

uE@

+

0

,

if k < 1

.

s

aT

aw

+

(=)=

0 in Lm(

s)

weak star

so that

I

uE@ at dxdt awE

+

0

.

s

In all cases, the limit of (2.33) is

-

UP

2 dxdt ,

s SO

that (2.41) gives in the limit

-

I

UP

2 dxdt + I

[ci

P

-

s

s

f@Pdxdt

=

.

s This is the same formula as (2.12), with a different w. i* by obtain (2.13) and if we define x (2.341 then

A:(xi*

-

y.)

=

0

,

xi* being Y-periodic

,

We

262

EVOLUTION OPERATORS

(2.35)

ci

=

7/1 [aij

-

a

The Theorem is proven, provided we verify that (2.35) furnishes the same formula as in (1.71), i.e. that

The verification is made along similar lines as for (2.17).

2.4.2

The case 1 < k < 3/2. We prove now the result for 1 < k < 3/2 and we shall indicate

next how to proceed when k

+

2.

We begin by general considerations on the asymptotic expansions. With the notation (1.51) we have (2.36)

P E r k = €-'A1

+

E - ~ A+ ~E O ( A ~+ =) a +

E -k

a

We are looking for an asymptotic expansion of uE where PErkuE= f. The usual expansion using terms

E ~ U (x . ,y ,t,T)

7

is impossible

with-

out additional terms, since there is otherwise no term to "compensate" - ~ would appear.t the powers ~ j which Therefore it is natural to look for u UE =

uo + EU1

+

E2u2

+

... +

where all €unctions depend on x, y, t, T ~ .

in the form:

P v 0 T

+ E3-kvl +

...

,

and are periodic in y and in

But if we use the above "ansatz" for uE, we do not have T deriva-

tives coming in the identification for uo, ul, u2 and one easily

tExcept if functions u fication is impossible.

j

do not depend on

T,

but then the identi-

CONVERGENCE OF THE HOMOGENIZATION OF PARABOLIC EQUATIONS

263

checks that the computations are impossible. Therefore one should add k to the above "ansatz",terms of the type E wo + ~ ~ + + so ~ that w

...,

finally we look for uE in the form UE

=

(2.37)

+

uo

+

Ell1

€kWO

+

+ E 2u2 +

... +

+

... .

Ek+1Wl

Identifying the coefficients of

0

+

E-',

E - ~ ,

E3-kvl +

E'

...

gives

,

(2.38)

AluO = 0

(2.39)

Alul + A2u0 = 0

(2.40)

A1U2

+

€2-kv

A2U1

+

,

A3u0

au, + aw, = +at a-r

Identifying next the coefficients of

E

f

.

-k

,

E

1-k

I

E

-k-2

, E -k-1,

we obtain

(2.43)

AlwO = 0

(2.44)

Alwl + A2w0 = 0

I

.

It follows from (2.38) that uo does not depend on y; then, by integrating (2.41) in y, we obtain

au,/aT

= 0 and therefore

Then (2.41) reduces to AlvO = 0 and v 0 does not depend on y. If we use the functions x j ( y , ~ )satisfying (1.67) , (1.681, we have from (2.39)

~

EVOLUTION OPERATORS

264

I n o r d e r t o be a b l e t o s o l v e ( 2 . 4 2 ) f o r v1 w e need

[?+

A2vO]dy = 0

.

Y But s i n c e v o d o e s n o t depend on y ,

J

AZvOdy = 0 , so t h a t t h e o n l y way

Y t o s a t i s f y t h i s c o n d i t i o n i s t o choose x j s u c h t h a t

I

XJ(y,T)dy

f o r every T

0

=

,

Y

and G1 i n d e p e n d e n t o f T. Then we can compute v1 no m a t t e r how w e choose v o .

Therefore w e

choose vo = 0 and v1 i s d e f i n e d (up t o t h e a d d i t i o n of a f u n c t i o n of x , t , T) by

AIVl

-

a

j

aT

au ax = 0 . j

W e see from (2.43) t h a t wo i s i n d e p e n d e n t o f y and t h e n w1

can

be computed by ( 2 . 4 4 ) . I t only remains t o s a t i s f y t o (2.40)

(observe t h a t t h i s i d e n t i t y

i s i m p o s s i b l e t o s a t i s f y i f t h e r e i s no w,,!).

By i n t e g r a t i o n

and i n T w e f i n d t h a t u s a t i s f i e s t o ( 1 . 7 1 ) . "

Summing u p :

&

y

with t h e

a n sa t z " u

(2.45)

= U + E U

+

Ek+1Wl

+ +

EZU2

...

+

... +

E3-kvl

+

... +

E kwo

,

where t h e c o e f f i c i e n t s are computed as above, one h a s

CONVERGENCE OF THE HOMOGENIZATION OF PARABOLIC EQUATIONS

PErkuE= f

+

terms of order 2 k

-

1 in

E

265

.

Remark 2.1. 1) We have introduced the above ansatz on a rather "algebraic" basis: another motivation will be obtained in Chapter 3 (probabilistic methods). 2)

One sees from the above computation that the limit of uE does

not depend on k (when 1 < k < 3/2)+ but that the expansion itself depends on k. 3)

For k satisfying

the "ansatz" for the asymptotic expansion will be: = u + E U 1 + E 2u 2 + E 2p+l-pkvo

u

+

... +

EkWO +

... .

We can now proceed with the proof of the Theorem. We construct a function w E such that (2.46)

(-

a

at

+

AE)*w

=

E

k- 1 gE '

2 where g E is bounded in L ( Q ) as 2

L (Y

x

( 0 , ~ ~ )as )

E

-+

E

-+

0 and where wE

+ .

yi in

0.

One looks for wE in the formtt (2.47)

wC = E(CX(Y,T)- yi

+

EZB(Y,T)

+

E3-kA(y,T)

.

tOf course once the above (formal) proof is justified. ttWe do not need in this expansion terms in E~ or in

E

k

.

266

EVOLUTION OPERATORS

One finds: (2.48)

A;(a

-

- -a~ + ar

One defines xi* by (2.34) take

ci =

x i*

*

,

yi) = 0

*

AIB + A2(" A;A

= 0

-

yi) = 0

,

.

j Xi*(y,T)dy

V T.

= 0,

Then one can

Y

and one can compute X from the third equation (2.48).

One computes 6 using the second equation (2.48) and one obtains the result.

2.5

The proof is then completed along the usual lines.

Proof of the homogenization when k > 2.

(Preliminary) Remark 2.2. 1) We are going to prove the theorem only for k = 3.

The proof

for the general case proceeds along the lines of Section 2.4.2. 2) We assume again that (2.22) holds true.

See Section 2.6 for

the general case. According to Lemma 2.1 we assume (2.23). We are going to construct a function wE satisfying

0

such that as

E -+

(2.50)

g,

is bounded in L2 ( S

(2.51)

wE

+

)

I

2 yi = P(y) in L ( S

We use asymptotic expansions

Y = c1

being Y-periodic and B and

T

X/E

being

I

Y-To

periodic

.

CONVERGENCE OF THE HOMOGENIZATION OF PARABOLIC EQUATIONS Using the notations ( 1 . 5 8 1 ,

Since a does not depend on

+

S;(CY

-

(2.54)

S*B

(2.55)

s y + s 2 5 + s

*

Yi) = 0

*

*

1

Then we find ( 2 . 4 9 )

(2.56)

the coefficient of

E - ~

is zero.

3

I

(a-yi) = o .

(assuming we can solve ( 2 . 5 4 ) ,

+

g, = h

T,

we have

and of EO are zero iff

The coefficients of 1

(1.59)

267

EL

*

+

with

,

E2m

*

(2.55))

*

h = S2 y + S3 B + S 4 ( c Y - Y ~ )

*

* L

=

s*y 3

+

S4B

m = S y . 4

I

*

We can solve ( 2 . 5 4 ) for B iff (S1 =

a : - E)

(2.57)

*

*

But S2 = A1 and since a does not depend on 74 A

(a

-

-*

A1(a

We denote by

-

is equivalent to

y.) = 0

, this means

.

(compare to ( 1 . 8 8 ) ) the solution which is Y-periodic

(defined up to an additive constant) of ( 2 . 5 8 ) . (2.59)

(2.57)

yi) = 0

Using the notation ( 1 . 8 6 ) (2.58)

T,

i* a = @

.

Now we can solve ( 2 . 5 4 ) ; if we set

Then

268

EVOLUTION OPERATORS

then

.

~

(2.61)

B

=

Bo + B(x,y,t)

We can solve (2.55) for y iff

i.e., (2.62)

i *1g +

mT(S;Bo

+

Sl(a

-

yi)) = 0

,

V

iY-periodic ,

which defines (up to an additive constant) the function All the functions introduced are Cm Then we can solve (2.55) for y.

& I

or in ?

Functions

X

i(y). [o,To]. m

a,B,y

are C

so that

in particular

and

We now use (2.49).

We take $

E

C?z(s )

and, as before, we

multiply the equation for uE by $wE and (2.49) by @uE.

=

\

s

f$wEdxdt

-

E

i s

gE$uEdxdt

We can pass to the limit and we find

.

We find

CONVERGENCE OF THE HOMOGENIZATION OF PARABOLIC EQUATIONS

-

I

dxdt +

uw

I

Ci

269

Pdxdt

s

s

and finally (2.64)

Si

=

[

74 aij

-

akj

a$i*

au

T]

It remains only to verify that this formula coincides with what we have obtained in (1.94), which is left to the reader

2.6

Proof of the homogenization formulas when a E Lm ij using LP estimates. for

We show now that the assumption (2.22) Theorem 2.1 to be true.

The hypothesis aij E Lm(Rn x R ) is Y T sufficient. We assume f o r a moment the following estimate: (2.65)

for the Dirichlet's boundary conditions there exists p > 2 ,

independent of

E

,

such that

The proof of this result is given in Section 2.7 below. We introduce q such that

We can find a sequence of functions aB such that ij

2 70

EVOLUTION OPERATORS

(2.67)

B a . .E C " ( Y

x

13

[O,rO1)

,

B is periodic together with aij

all its derivatives , sup l r 7

B Ilaij

-

aij

I I Lq(y

= p(B)

*

0 as B

and

* "

.

[o,rol)

With the notations of Section 2.3 we show that

Then Lemma 2.1 is still valid and the Theorem is proven. With the notations of the end of Section 2 . 3 we observe that

I (fE,V)I

5

CP(B

so that (2

By virtue of ( 2 65) it follows that (2.70)

I Ifc

so that in part cular

and (2.68) follows. For other boundary conditions than Dirich-st, the analogous of (2.65) is valid locally, i.e. for

S'

-

C

8 ' x lO,T[, 0 ' C

and the analogous of (2.68) is valid locally, i.e. in

8

,

CONVERGENCE OF THE HOMOGENIZATION OF PARABOLIC EQUATIONS 2.7

271

The Lp estimates. We now prove the Lp estimates used in Section 2 . 6 above.

are of independent interest.

They

The method of proof is analogous to the

one used in Chapter 1, Section 4, for elliptic problems.

The

notations are those of this Section 4, Chapter 1; in particular, we use norms ( 4 . 5 ) ,

(4.7),

(4.81,

of that section. We define

(4.11)

provided with the norm

We introduce

and we observe that the mapping v Lp(O,T;W-l'p(O

(2.74)

)).

Ilfl l y

p

divx v maps (Lp( (j

-+

We then provide Y

=

inf divx g = f

P

) )

onto

with the norm

I 191 I (LP((j ))n

We consider a family of functions a ij such that

We denote by [a. (x,t)] the n 11 and we set

x

n matrix with entrices aij(x,t)

272

EVOLUTION OPERATORS

We set

Theorem 2 . 2 .

Let f and u o be given such that f uo

H = L

E

2

( 0 ) .

u

(2.80)

u(0) = u0

Then assuming only on a , %

L2 (O,T;H-'(S))

and

Let u be the solution of

(2.79)

E

E

L 2 (0,T;H0(8)) 1

I

.

r to be smooth enough there exists 8 , such that if

p > 2,

depending

(2.82)

then

u

when f

E

Lp(O,T;WkrP(O )

and remains in a bounded set of this space

)

& uo remain bounded in LP(0,T;W-1rP(8)) and in WirP(O).

Remark 2 . 3 . We can reduce the problem to the case when uo = 0. can find I$ E Lp(O,T;W;'p(O

)),

$

E

Lp(0,T;Wi'p(8

)),

@

Indeed we depending

continously on uo in the corresponding topology, so that @(O) = u 0 and then we consider u

'This

-

@

instead of u.

hypothesis can be very much improved.

CONVERGENCE O F THE HOMOGENIZATION OF PARABOLIC EQUATIONS

273

P r o o f of T h e o r e m 2 . 2 .

of C h a p t e r 1, w e c a n r e d u c e t h e p r o b l e m t o

A s i n S e c t i o n 4.2

where

&i [ a l j ( x , t )

-

(2.84)

Ak =

(2.87)

v < p .

,

k = 1,2

,

By c h a n g i n g t h e scale o f t i m e , w e see t h a t w e a r e r e d u c e d t o proving t h e theorem f o r

a t + A 1u +

A u = f

2

,

u(0) = 0

,

f s a t i s f y i n g (2.81)

(2.88) u = 0 on C and w i t h (2.841,.

r

=

x

]O,T[

,

. .,( 2 . 8 7 ) .

W e set

(2.89)

a

P = - - A .

at

I t i s known t h a t , g i v e n F

E

Yp,

( w h e r e p i s a r b i t r a r y f o r the

t i m e being), t h e r e e x i s t s a unique u such t h a t P u = F i n

q ,

(2.90) U E X

P '

u(0) = 0

.

,

274

EVOLUTION OPERATORS Equation (2.88) is equivalent to

(2.91)

Pu

f

(A1 + A ) u - + A2u = f

,

and if the solution u of (2.90) is denoted by (2.92)

u = P-1F

then (2.91) is equivalent to (2.93)

u

+

(P-l(A 1

+

A ) + P-lA2)u = P-’f

.

We introduce

and the Theorem will be proven if we verify that (2.95)

one can find p > 2 such that k ( p ) < 1

.

We have

But exactly as in Chapter 1, Section 4,3 (t plays the role of a parameter) we have

so that

(2.97)

CONVERGENCE OF THE HOMOGENIZATION OF PARABOLIC EQUATIONS It is straightforward to verify that that, since 1 (2.98)

-

p

+

1 IP-ll 1

v < 1, we shall have ( 2 . 9 5 )

2 ( Y 2 ;x2)

275

= 1 so

if we can prove that

there exists a function p ( p ) which is continuous

and p ( 2 ) = 1

.

Indeed we have then k(p) < p ( p ) and we can find p0 > 2 such that k(po) < 1.

Proof of ( 2 . 9 8 ) . We take F = div g, g

E

(Lp(

in ( 2 . 9 0 ) and we consider the

mapping TT

(2.99)

g

+

gradx u

,

Using Riesz-Thorin's theorem it follows that

Therefore, according to ( 2 . 7 2 ) and to ( 2 . 7 4 ) , we have

276

EVOLUTION OPERATORS

so that we can take p(p) = w:-'(~)

and the result follows.

Remark 2.4. We have, for arbitrary boundary conditions, a local result (as there exists p > 2 (and not too large:

in Theorem 4.3, Chapter 1): p

5

%)

such that if f and uo satisfy locally (2.81) and (2.82),

then (2.103)

$u

E

L p ( O , T ; W ~ ' p () ~)

.

Remark 2.5. .It follows of course from these estimates that, if f and uo satisfy conditions of the type (2.81), (2.82), then (2.104)

au

E -f

1

2.8

au ax i

in Lp(

s)

weakly or in Lp(O,T;Lp( 8

I ) )

weakly

,

The adjoint expansion. Let us remark that convergence can be proven for Dirichlet's

boundary conditions and when coefficients a

are smooth enough ij (cf. Section 2.3) using the method of the adjoint expansion as in

Chapter 1, Section 3.3.

We present this briefly, since the idea is

exactly analogous to the one in the elliptic case.

We can always

assume that uo = 0, and we write the equation in the equivalent form

where here (f,v) =

/

J

s

fvdxdt.

277

CONVERGENCE OF THE HOMOGENIZATION 6 F PARABOLIC EQUATIONS

Given v E

J

(S

),

asymptotic e x p a n s i o n s - a

we construct-using

sequence of f u n c t i o n s v

such t h a t vE

+

v i n L L ( )~ w e a k l y ( s a y ) , a n d

such t h a t

[- ata +

(2.106)

ff

where

*

[- 2 + at

+

a * ] v i n L2 (

ff

= a d j o i n t of t h e operator

0 = Ol, ff

i.e.

AE*]vE

S)

weakly

,

c o n s t r u c t e d i n S e c t i o n 1,

o r f f 3 a c c o r d i n g t o t h e case k < 1, k = 2 , k > 3.

Then

and t h e r e s u l t f o l l o w s

U s e o f t h e maximum p r i n c i p l e .

2.9

Assuming a l l d a t a t o

be v e r y s m o o t h a n d f o r t h e D i r i c h l e t ' s

boundary c o n d i t i o n s , we have

Indeed let u s w r i t e (2.108) valid

PE =

a a t + AE

method a n d a d d a n

E

€-'Pa

+

,

E 0 P3

for k = 1 and 2.

(with d i f f e r e n t Pi's) 3

+

= €-'Pl

(For k = 3 use t h e same

u3 t e r m i n t h e proof b e l o w . )

We introduce (with

n o t a t i o n s of S e c t i o n 1 , (2.109)

ZE

-

= u

(u

+

EU1

+

E

2

u,)

and we o b s e r v e t h a t

(2.110)

PEZE = -E(P2U2

I f all data m

the L (

S

)

norm.

+

P u )

3 1

-

E

2

P3U2

.

are Cm, t h e r i g h t hand s i d e of But

(2.110) i s O ( E )

in

EVOLUTION OPERATORS

and

so that, by using the maximum principle

hence (2.107) follows.

2.10

Higher order equations and systems. Except in Section 2.9 above, we never used the maximum

principle.

All results obtained sofar extend to higher order

equations and to systems, i.e. to the "parabolic analogue" of the problems considered in Chapter 1, Sections 9 and 10. Section 6, Chapter 1 also extend easily.

The results of

Let us consider an equation

of the type (2.111)

auE a at - axi

[a..(x,-,t,-) Ek

11

aUE1

-

=

axj

f

,

where we suppose that

m

where Lrn(Rnx RT) = space of L P Y periodic. We also assume that (2.113)

a . .(x,y,t,r)C.C.> aEiCi 1 7 -

13

Then we denote by (2.114)

functions of y and

k

,

7

a > O ,

a k (x,t) the homogenized operator

(x,t) =

a k a - axi sij(x,t) ax

j

which are Y

SiER.

x T~

CONVERGENCE OF THE HOMOGENIZATION W PARABOLIC EQUATIONS k where the coefficients qij(x,t) are computed as the qij's in Section 1, with x and t playing the role of parameters.

If u

k

denotes the solution of uk

L~(o,T;v) ,

E

k

k a u + a (x,t)uk = at

(2.115)

k

u (0)

=

u

0

f

,

,

then (2.116)

u

+

2

uk in L (0,T;V) weakly as

E

+

0

.

One can also extend, always with the same type of method, the results of Chapter 1, Section 8. Let us consider now briefly the "parabolic analogue" of the situation of Section 11, Chapter 1. We assume that the space dimension equals 3 , and we denote by aE(x,t) a 3

x

We

3 matrix.

assume that

a . .( y , ~ )having the same properties than in (1.6) 13

.

I

We consider, with the notations of Chapter 1, Section 11:

and we observe that there exists a unique function uE such that

(2.119)

u

E

L~(o,T;v) ,

2 where f E L ( O , T ; V ' ) ,

uo

E

uE(0) = u 0 3

H = (L2(0))

.

,

279

EVOLUTION OPERATORS

280

Asymptotic expansions. We give only the results of the asymptotic expansion; one has to consider three cases.

Case k = 1. We define jiP(y,r) as a Y-r0 periodic solution of

-

roty a(y,r) (rot jiP Y

(2.120)

div jip = 0 Y

e

P

)

=

o ,

.

Then

U 1 (u,v) =

(2.121)

(

-

R(a

a rot ?)rot u,rot v) Y

and the homogenized equation is (%v) at

(2.122)

Case k

+ Q 1 (u,v) = (f,v)

,

tJ v E

v

-

= 2.

We define BP(y,r) as a Y - r 0

aa.tirp + (2.123)

periodic solution of

rot a(y,r) (rot GP Y Y

div BP(y,r) = 0 Y

-

e ) P

=

o ,

.

Then (2.124)

U

2 (u,v) = ( 7;1 (a

-

-

a rot 8)rot u,rot v) Y

and the homogenized equation is the analogous of ( 2 . 1 2 2 ) instead of

with 0 2

CONVERGENCE OF THE HOMOGENIZATION QF PARABOLIC EQUATIONS Case k = 3 . We define

and we define ip(y) as a Y-periodic solution of rot Z(rot Y Y

5' -

div '$I

.

e

P

=

)

o ,

(2.126)

Y

= 0

Then (2.127)

u

3 (u,v) =

( n (a - a 17

rot $)rot u,rot v) Y

and the homogenized equation is the analogous of (2.122) with U 3 instead of

a'.

The justification of these formulas is simple if we assume: (2.128)

,

divx f E L 2 ( S )

div uo

E

L2( 8 )

.

Indeed, we define z o as the solution of (2.129)

Azo = div u 0

,

and assuming the boundary (2.130)

zo E ~

z

r

0

E

1

HO(8)

smooth enough, we have

.

~ ( n 8H ~) ( D )

We then define z as the solution of (2.131)

a Az

=

divx f

,

(This is an elliptic problem:

Z ( O ) = zo

Az =

1 0

div

X

,

z =

f(x,u)do

o

on

c

.

+ Az 0 ) ; we have

281

EVOLUTION OPERATORS

282

ii

= UE

-

vz

we see that

aiiE

+ (a rot iiE,rot v)

[K,v] where ? = f =

div uo

-

a - at P,z0

= (f

-

Vz, so that divx f = 0.

=

a - at vz,v)

=

( ~ , v ),

Since div fiE ( 0 )

0, we see that div GE = 0.

Therefore we can assume (if (2.118) holds true) that divx f = 0 ,

divx uo = 0

,

divx u

= 0

.

Then one can prove the above formulas by arguments similar to those of Sections 2.2, 2.4, 2.5.

Remark 2.6. proof along the lines of Chapter 1, Sections 11.4, 11.5, does

A

not seem to work in the present situation.

The homogenization

problem seems to be open if we do not assume (2.128).

2.11

Correctors.

Orientation. We now return to the situation of (1.17), (1.18), (1.19). want to introduce first order correctors '8 (2.132)

u

-

(u +

OE)

2

-+

1

0 in L (0,T;H ( 8)

We

such that )

strongly

.

Remark 2.7. By using cut-off functions m E ( x ) , as in Chapter 1, Section 5.1, we can have correctors giving an approximation in LL(O,T;V) strongly.

CONVERGENCE OF THE HOMOGENIZATION OF PARABOLIC EQUATIONS Remark 2.8. We did not make any attempt to find the analogous results to those of Chapter 1, Section 18.

Notation. We shall denote by u1 the function: 1

(2.133)

&L

=

ax

ax

u1 = - e l

if k = 1

,

if k = 2

,

j

j

We shall also denote by the ~ a m enotation u2,u3,..., the next terms of the asymptotic expansion, for k = 1,2,3. We shall use (2.134)

GE

=

u +

=

u

+

if k = 1,2

EU 1

+ 2u2

E U ~ E

,

if k = 3

Theorem 2.3. We assume the hypothesis of Theorem 2.1 to hold true and we assume all data (2.135)

uE

-

smooth enough.' (u

+ EU~) -f

0

&

Then L2 ( 0 , T ; V )

strongly

Proof. We introduce

'This

is made more precise in the proof below.

.

283

284

EVOLUTION OPERATORS

(2.136)

- 6E

uE

<,=

and we consider (2.137)

XE =

(T

-

t)[(
.

0

We observe that m

Since 1<,(0)l2

= 6 2 lu1(0)

we prove that X E replaced by T

-

0 as

+

or c 2 lul(0)

E -+

+ eu2(0] ', we see that, if

0 , then we shall have (2.135) with T

but since T does not play any role, this

q > 0,

q,

m

gives the general result. We can write XE =

1

(T

-

+ aE(uE)ldt

t) [ (u:,uE)

0

+

(T

-

+ aE(GE)]dt

-

+ aE(uE,GE)ldt

,

t) [(G;,GE)

YE

-

ZE

,

0

T

)

The first term in X

1

(T

-

.

equals

t) f,uE)dt

+

T

(T

0

0

+ aE(GE,uE)ldt

=

-

1

t ) (f,u)dt

(T

-

t) [ (u' ui

+ a ( ~ , ~ ) i d .tt

0

tWe set

=

a k , k = 1,2,3 and c7Xu,v

= (

a u,v).

CONVERGENCE OF THE HOMOGENIZATION QF PARABOLIC EQUATIONS

The second term in X

equals

(5.15), that

One verifies, exactly as in Chapter 1, Section 5, T

285

T (T

-

t)aE(GE)dt

I

-f

n

0

tends to

-

5

1u0l2

+

(T

T 0

-

(T

-

t) Q(u,u)dt, so that the second term

t) Q(u,u)dt

=

T 0

(T - t)[(u',u) +Q(u,u)ldt.

Therefore, if we show that (2.138)

YE

(2.139)

ZE

-t

-f

T T 0

0

(T - t) [(u',U) + Q(utu)ldt

(T

-

it will follow that X E

t) [(u',u)

+

+

Q ( ~ , ~ ) l d ,t

0, hence the Theorem follows.

Let $ be a given function in & g ( S ) , (near

r)

r

$ =

1 except on a set E

of measure /El. We define (compare to Chapter 1, (5.21),

(5.22)) T

T

We see exactly as in Chapter 1, Section 5,3, that (2.138), (2.139) will be proven if we verify that, for fixed $, Y

+

E$

(2.140)

1

(T

-

t) [(u',u$) + 0 $(u,u)ldt

(T

-

t) [(u',u$) + Q $ (u,u)ldt

0

zE$ 0

,

EVOLUTION OPERATORS

286

But

s

If we denote by q .

Ij

the coefficients of ftk, k = 1,2,3, we have

hence (2.140) follows for Y

EtJ.

We observe next that

But from the definition of GE it follows that +

au

2

+ C7u in L ( Q ) weakly,

hence the result

aii

$+

A Gc

so that (2.141) gives

follows.

Remark 2 . 9 . We can also write the equation as a first order system and develop arguments along the lines of Chapter 1, Sections 5.5 and 5.6.

CONVERGENCE OF THE HOMOGENIZATION OF PARABOLIC EQUATIONS

287

For Dirichlet's boundary conditions we obtain in this manner error estimates which do not rely on the maximum principle.

2.12

Non-linear problems.

Orientation. We confine ourselves to

one aspect of

the homogenization of

non-linear parabolic equations (or inequalities), namely to Variational Inequalities (V.1.) of evolution.

Many other cases can

be treated by using the methods introduced below.

Setting of the problem. We consider operators AE as in Section 1.1; A E = A

X/E,t/E k 1 .

Formally we want to consider V.I. of the following type 3at+ A E u E

-

,

f 0 '

UE(0

I

(2.143) E

u

+

AeuE

-

fluE = o

Q ,

in

being subject to usual boundary and initial conditions.

To

simplify a little, we shall take: (2.144)

uE(x,O) = 0

.

The variational form of (2.143) is as follows: 2 in L (O,T;H), then we are looking for uE such that: u (2.145)

and

E L 2 (0,T;V)

auE E at

let f be given

,

2 L (0,T;V')

,

uE 5 0

,

a.e. in Q

,

EVOLUTION OPERATORS

200 m

2 V V E L (0,T;V)

,

in S ,

v 2 0

and such that ( 2 . 1 4 4 ) holds true. We shall show below that this problem admits a unique solution. The main result we want to prove is Theorem 2 . 4 . We assume that the hypothesis of Theorem 2 . 1 hold true.

We

denote by U the homogenized operator found in Theorem 2 . 1 (i.e.

U

=

Uk, k

on V. -

and by

= 1,2,3)

U(u,v) the corresponding bilinear form

Let u be the solution of the V.I. of evolution

u

u(x,O) = 0

.

0

(2.149)

Then when E

I

-

[I%,.

(2.148)

-f

u

+

0,

+

(2.150) -auE at + -

u in L2 (0,T;V) weakly

,

au . L2 (0,T;V') weakly at in

.

The proof is divided in several steps. We first introduce the penalized equation associated to the V.I. (2.146).

equation

For 11 > 0, we denote by uEn the solution of the nonlinear

CONVERGENCE OF THE HOMOGENIZATION OF PARABOLIC EQUATIONS

289

(2.151)

u

En

(X,O)

= 0

,

u

En

E

L2 (0,T;V

(and as usual, the boundary conditions which correspond to the variational formulation).

The equation (2.1 1) is called the It admits a unique solu-

penalized equation associated to ( 2 . 1 4 6 ) .

tion, for which we now obtain a priori estimates. If we multiply ( 2 . 1 5 1 ) =

E

+

+

a (v ,v

=

+

E

+

by uEn, we obtain (since a (v,v

)

+ aE(v ) ) :

hence

We deduce from ( 2 . 1 5 2 ) (2.153)

1 u+ rl En

that

is bounded in L 2 ( Q

It follows from ( 2 . 1 5 3 )

)

,

V

8

and V n

I

that

(and uE satisfies initial and boundary conditions) so that the usual a priori estimates are valid:

If we let n such that

-+

0,

we can extract a subsequence, still denoted by uEn,

290

EVOLUTION OPERATORS

(2.156)

u

Erl

+

u

E

2 in L (0,T;V) weakly

au

,

au

2 3 + 2 in L (0,T;V') weakly , at at (2.157)

-+

0 in L

2

(S

)

.

2 It follows from (2.156) that uEll+ uE in L ( 0 ) strongly, so that u+

El1

-f

+ in . L2(S

u

E

)

and by comparison with (2.157) we obtain

u+ = 0, i.e. uE -< 0. Therefore uE satisfies (2.146) and (2.144). 2 Let v be in L (O,T;V), v 5 0. We multiply (2.151) by v - u we

-

El1'

obtain :

hence

or

Since

CONVERGENCE OF THE HOMOGENIZATION OF PARABOLIC EQUATIONS it follows from (2.158) that uE satisfies (2.145).

291

Therefore we have

shown the existence of uE, satisfying (2.145), (2.1461, (2.144) and such that

The uniqueness of uE is straightforward. We now establish an error estimate of independent interest:

Proof.

We have

UE

-

UE, = UE

+ u-E, - u+E n and by virtue of (2.154) we

have only to show that (2.161)

I

We take v = -uiq in (2.146) and we multiply (2.151) by uE

-

+ uE,.

Adding up, we obtain

+ PUE

+

uEq

J

+

+ u,,]

-

aE(uE + ui,,)

i.e.

1 + and integrating over (O,t), we obtain (since (uE n ,-uE) 1. 0 ) r7

292

EVOLUTION OPERATORS

Using (2.154) and (2.159), the right hand side is bounded by C h i , hence (2.161) follows. We have also proved that

It follows from (2.160) that it suffices to study the limit of u

€17

as

E

+

0

for Q

fixed.

But this is immediate.

By virtue of

(2.155), we can suppose, by extracting a subsequence, still denoted by uEn, that U

(2.164)

-f

En

u

in L2 (0,T;V) weakly ,

rl

au nE:' 9 in L2 (0,T;V') weakly . +

Then uErl+ u

n

in L2 ( F

)

strongly, so that f

-

II

u+

f

-f

En

- n u+n

in

L2( Q ) and then

au ].,$I where

+

a (u17 ,v)

=

(f

-

1 + - u / rl

, v v

E

v ,

Q(u,v) is defined as in the statement of the Theorem.

proof is completed.

The

CONVERGENCE OF THE HOMOGENIZATION OF PARABOLIC EQUATIONS

293

Remark 2.10. Theorem 2.4 gives, in particular, the homogenization of the Stefan's free boundary problem for a "granular" medium-i.e. coefficients are functions of

when

t aij(x/E).

X/E:

Remark 2.11. The result of Theorem 2.4 extends to the case of coefficients of x t the form aij (x,-,t,x). E

E

Remark 2.12. Theorem 2.4 can be extended to obstacles of the type b1(x,t) when $,

5

v

5 b,(x,t)

and $2 are smooth enough.

, The case of the "most general"

obstacles yi under which the analogous of Theorem 2.4 still holds does not seem to be known.

Remark 2.13. The homogenization of V.1 of evolution for "arbitrary constraints"

v

E

2.13

K(t) is largely open.

Remarks on Averaging.

Setting of the problem. We consider now the case T =

'This

+w:

is because the Stefan's free boundary problem can be

transformed into a V.I. of evolution.

294

EVOLUTION OPERATORS

s

(2.165)

=

8

x

lo,+-[

and we consider the equation

(2.166)

av

For u,v (2.167)

E

H1(S), we set now

aE(u,v) = 8

8

The precise problem we consider is then: u

(2.168)

E

L~(o,T;v),

v

T finite ,

[auc w , v ] + aE(uE,v) = (f,v) UE(0) =

u0

,

V

v E V

,

t > 0

,

.

We assume the following: (2.169)

aij

and of course

Example 2.1. We take :

,

aa! axk

,

b;

remain in a bounded set of I,-( Q

)

295

CONVERGENCE OF THE HOMOGENIZATION OF PARABOLIC EQUATIONS a . .( x , ~ ), (2.172)

a aij(x,T)

b .( x , ~ ),

13

axk

3 m

belong to L ( 8 x R T ) and are almost periodic in

T

.

We extract a subsequence such that

(2.173) in L m ( 6 ) weak star

.

We want to study the behavior of uE

as

E

-t

0.

In the case of Example 2.1 we have (2.1741

iij =

nT(aij)

,

j =,R',(bj) ,

where we have set here (2.175)

m T ( $ )=

x

\

lim X++m

$(x,a)do

.

0

Theorem 2.5.

t We assume that (2.169), (2.170) hold true and that (2.176)

1 V = Ho(8)

1 V = H ( 8 )

.

We also assume that f , g

(2.177)

E

i

L2( 8

x

lO,T[)

,

V

T finite,

UOEV.

hypothesis is by no means indispensable. I f V consists of 1 the functions of H ( 8 )which are zero on a subset ro of r , the 'This

result is still valid with a slightly more complicated proof.

296

EVOLUTION OPERATORS

Then a s (2.178)

E

-+

u

0 , one h a s +

,

u i n L2(0,-T;V) w e a k l y f o r e v e r y T f i n i t e

where u i s t h e s o l u t i o n o f

(2.179)

I&[

+ i(u,v)

u(0) = u where (2.180)

i(u,v) =

0

= (f,v)

, v

v E

v ,

,

I

aij

ax a u ax av

8

j

i

dx

+

I Gj

vdx

.

j

0

Remark 2 . 1 4 . I n f a c t w e s h a l l p r o v e more: n a m e l y , (2.181)

uE

+

2 u i n L 2 ( 0 , T ; H ( 0 ) )w e a k l y

, v

T <

m

,

Remark 2 . 1 5 . I n case o f Example 2 . 1

(formulas (2.171)) t h e theorem h o l d s

a

( 2 . 1 8 2 ) ) w i t h o u t c o n d i t i o n s on axk a i j ' u s e s t h e s e m e t h o d s a l o n g t h e l i n e s o f t h o s e of S e c t i o n 2.2. t r u e (without (2.181),

P r o o f o f Theorem 2 . 5 . Thewhole problem i s reduced t o p r o v i n g t h a t , VT f i n i t e ,

One

CONVERGENCE OF THE HOMOGENIZATION OF PARABOLIC EQUATIONS

297

Indeed, if we admit (2.183), then one can extract a subsequence, still denoted by uE, such that one has (2.178), (2.181) and (2.182). It follows that, V T finite auc

+

1

au ax

in L2 ( 8 x ]O,T[) strongly

j

and therefore

in L2 ( 0

x

]O,T[) weakly

and the result follows. Estimates (2.183) are obtained by using first local mappings, to reduce the problem to the same question when (2.184)

0

=

Ixlx > 0 1 n

.

Then one uses the method of translations parallel to the boundary. t We suppress the index h

=

"E"

to simplify the writing.

Given

{hl,...,hn-l,Oj, we define

We have

, g + ~ u = f i nS ,

tThis is a classical method, introduced by L. Nirenberg 111 for the study of the regularity of elliptic equations.

When V does not

satisfy (2.176) but consists of functions which are zero on

ro

C

r,

then there is a difficulty because V is not "translation invariant". One can then use a compensation method, due to Aronszajn and Smith, and reported in'lions 121.

298

EVOLUTION OPERATORS

and with obvious notations

3 at + AhUh

fh in

=

q

.

Therefore (2.185)

a(uh at

U)

+ Ah(uh - U) + (Ah - A)u

=

f h

-

f

.

It follows from (2. 8 5 ) and from the hypothesis that

1 IUh so that by letting (2.186)

hl

+

0, we see that when

2

a remains axjax

E

-+

0

in a bounded set of L2 ( 8

x

]O,T[)

,

j

where x‘ i = x i ,

i L n - 1 .

We now return to the equation, which we write (2.187)

au

-

ann

2 2

=

a xn

g *

where q remains in a bounded set of L2 ( 8 x lO,T[) (we use ( 2 . 1 8 6 ) ) ; au we obtain (since a > y > 0) multiplying (2.187) by 1 at nn -

au = 0 on (since u = 0 or axn (2.188)

r)

hence it follows that

au remains in a bounded set of L2 ( 8 x lO,T[)

But then ( 2 . 1 8 7 ) ,

which can be written:

.

OPERATORS OF HYPERBOLIC, PETROWSKY OR SCHRODINGER TYPE

299

imp1ies 2 9

E

bounded set of L2( 8

lO,T[)

x

a xn

and (2.183) is proven.

Remark 2.16. In the case of almost periodic coefficients, Theorem 2.5 is related to the classical theory of averaging for ordinary differential equations with almost periodic coefficients.

Cf. Bogoliubov

and Mitropolsky 111 and the Bibliography therein.

Remark 2.17. We confine ourselves to remarks made in the comments of the last section of this chapter for the case of coefficients aij(x/E,t/E k ) , where the aij's are almost periodic in

3.

T

(and periodic in y).

Evolution operators of hyperbolic, Petrowsky or Schrodinger type. I

3.1

Orientation. We present now some very simple remarks connected with the

homogenization problem for hyperbolic operators.

Further results,

using some different ideas and techniques, are given in Chapter 4 of this book.

We also give in this section some results for operators of

Petrowsky or of Schrodinger type, the hyperbolicity not being used in an essential manner.

We give next some examples of "non-local"

operators which appearin the homogenization process.

300

EVOLUTION OPERATORS

3.2

Linear operators with coefficients which are regular in t. Let us consider the operator

where the a . . ' s satisfy: 13

,

a. , (y) E Lm(R;) 17

(3.3)

a. . is Y-periodic 13

aij(y)SiEj 2 clSiSi

a > 0

I

1 We consider V as usual (Ho(Q) _c V

,

,

a.e. in y

5 H 1 ( 6 ) )and

.

with the usual

notations we consider the following problem:

0

where f, u

, u1 are given such that

(3.7)

f

E

L~(o,T:H)

,

uo

E

v ,

u1 E H

.

It is known (cf. for instance Lions [l], Lions-Magenes [11) that problem ( 3 . 4 ) , (3.5), ( 3 . 6 ) admits a unique solution.

A

priori estimates. We shall write $',$I"

for -, at

a2e at2'

Taking v = u~ in (3.5) gives

OPERATORS OF HYPERBOLIC, PETROWSKY'OR SCHRODINGER TYPE

30 1

We emphasize that in order to obtain (3.8) we used in an essential manner the fact that aE(u,v) is symmetric and does not depend on t.

It follows immediately from (3.8) that

The behavior of uE

E

-t

0 is very simple to study.

It follows

from (3.9) that we can extract a subsequence, still denoted by uE, such that uE

+

u

in L~(o,T;v)weak star

,

u'

+.

u'

in Lm(O,T;H) weak star

.

(3.10)

We use next the method of Remark 1.6 which is entirely general. With the notation (1.46) one has, V

@ E

C,"(]O,T[):

in H strongly, one has: uE ( @ ) + u ( @ ) in

V weakly

, where

where Q(u,v) = homogenized form associated to aE(u,v). u is the solution of 2

(3.12)

IF,.]

(3.13)

u

(3.14)

u(0) = u0

E

v v

6

L-(o,T;H)

,

+ Q(U,V) = (f,v)

L~(o,T;v) ,

,

u'

E

,

u' ( 0 ) = u1

Let us consider now the case when

.

v ,

Therefore

EVOLUTION OPERATORS

302 (3.15) We assume that (3.16)

aij(x,y,t) is Y-periodic (3.17)

a1.7.E Co( 5

We set, t/ u,v (3.18)

E

X

[O,T];Lm(Rn)) Y

x,t

,

,

H1 ( 8 ) :

aE(t;u,v) =

J

8 We can

, v

au av aij(x,x,t) E ax axi dx j

.

efine

(3.19)

8 We consider again (3.4), (3.5),

(3.6) (with aE(u,v) replaced by

aE (t;u,v)), and we have existence and uniqueness of the solution. The a priori estimates are now the following.

Equation (3.8)

is replaced by

and by virtue of the last condition in (3.17) we obtain the same a priori estimates (3.9). By the same technique

as

in Chapter 1, Section 6, one proves

that uE satisfies (3.10) where u is the solution of (3.13), (3.14) and

OPERATORS OF HYPERBOLIC, PETROWSKY OR SCHRODINGER TYPE (3.22)

( a (t)u,v) if

u(t;u,v) =

u,v E

303

cr(8) ,

where G(t) is the homogenized operator of AE(t) for fixed t.

Remark 3.1.

a2 + AE is a second order hyperbolic operator. 7

The operator

at

But the results indicated so far really do not depend at all on the hyperbolicity.

All what we sa d remains unchanged (except obvious

modifications) i f one considers (3.23)

aE(t;u,v) =

I

aaB(x,E X

t)DauDBvdx ,

la1 =

161 = m

,

8 a

aB

a

aB

(3.24)

aaB

= a

'

(x,y,t) being Y-periodic in y Co( 5

E

aa at

Ba

E

Lm( 8

[O,T];Lm(Ry))

x

x

(0,T)

x

Rn) Y

,

,

.

The homogenized operator is computed for every fixed t according to the results of Chapter 1, Section 9. We remark that the corresponding operator is not oft hyperbolic type when m > 1; it is an operator of Petrowskytype.

Remark 3.2. One can also add to the operator AE lower order terms. return to that in Section 3.3 which follows.

We

EVOLUTION OPERATORS

304 3.3

Linear operators with coefficients which are irregular in t. A natural problem is-the following:

Instead of the situation

( 3 . 1 ) we consider functions a..(y,T) which satisfy ( 3 . 2 ) and 11

(3.25)

aij(y,r)

E

Lm(Rn Y

x

RT)

,

and the usual ellipticity condition.

Y - r 0 periodic

We then consider the operator

AE defined by

If we assume that (3.27)

a

- aij

E

L-(R;

x

R ~ ),

then problem ( 3 . 4 ) , ( 3 . 5 ) , ( 3 . 6 ) admits a unique solution. are nof a priori estimates independent of

But there

Indeed if we define

E.

(3.28)

then ( 3 . 2 0 ) is replaced by

which does not give estimates independent of

as E

-t

E.

0 does not seem to be known in general.

The behavior of uE We shall give in

Section 3 . 4 below some formal computations (which can be justified in

cases). It is possible to consider lower order terms which have

coefficients irregular in t.

Let us consider

+At least there do not seem to be known a priori estimates strong enough to pass to the limit in

E.

OPERATORS OF HYPERBOLIC, PETROWSKY.OR SCHRODINGER TYPE A'

(3.30)

given by (3.1)

305

,

a x t BE = b. (-,-) 7 E E k ax. 3

where (3.31)

b.(y,r) is Y--r0 periodic 3

,

b. E L m ( R y 3

x

RT)

.

there exists a function uE and only one which

We have then:

satisfies (3.4), (3.6) (3.32)

(u2.v) + aE(uE,v) + (B'U',~)

=

(f,v)

The a priori estimates are as follows.

, v v

E

v

Taking v = u

I

. in (3.32),

we obtain

But

so that (3.33) gives:

t

and we obtain (3.9). Therefore one can extract a subsequence, still denoted by uE, such that one has (3.10) and (3.36)

BEuE

m

+

rl

Then (3.32) gives, V

in L (0,T;H) weak star C$

EC;(]O,T[)

.

306

EVOLUTION OPERATORS

and ( B E u E )( @ )

+

Q(@)

i n H-weakly,

follows f r o m (3.37) t h a t ,

t h e r e f o r e i n V' s t r o n g l y a n d it

a ( u , v ) b e i n g t h e homogenized f o r m

a s s o c i a t e d t o aE

Therefore

and t h e p r o b l e m i s now t o compute

We s t u d y t h i s q u e s t i o n i n t h e

Q.

following s e c t i o n s .

3.4

.

A s y m p t o t i c e x p a n s i o n s (I)

a purely formal fashion-the

L e t us consider f i r s t - i n

(3.4),

(3.5),

( 3 . 6 ) f o r A E g i v e n by ( 3 . 2 6 ) .

W e have w i t h t h e u s u a l

notations:

(3.39)

a2 + A E

+

= E-2Q1

+

E-lQ2

E 0Q 3

,

at2

a

Q1 =

a2 2 + aT

Q3 =

at2+ A 3

a a 11 . . (y,T) A 1 = - -ayi aYj

A 1 '

I

a2

'

1

Then, i f w e look f o r uE = uo

0

(3.40)

QIUO

=

(3.41)

QlUl

+ Q2UO = 0

r

r

+

E

U

+ ~

problem

..., w e

f i n d as u s u a l :

OPERATORS OF HYPERBOLIC, PETROWSKY OR SCHRODINGER TYPE (3.42)

Q1u2 +

+ Q3u0 = f

Q2Ul

307

.

We are thus lead to the following question:

given a function F

such that (3.43)

F

E

L2(Y

( 0 , ~ ~ ) )

x

is it possible to find (3.44)

9 a-r

+ A 1@

@

which is Y--ro periodic and satisfies

,

= F

i.e., @ takes equal values on opposite faces of Y and @(Y,O) = @(YtTo) , A necessary condition for

(3.45)

7?( (F) = 0

@

a@

(Y,O) =

a@

(y,-ro)?

to exist is

.

Let usassume that this condition is sufficient (so that (3.44) defines @ up to an additive constant). "generically" the case.

It can be proven that this is

Then (3.40) is equivalent to uo = u(x,tl and

(3.41) reduces to Q u + A 2 u = 0 . 1 1

If we introduce XJ as some Y--r0 periodic solution of

then

and (3.42) can be solved for u2 iff

+ ,Q ~ u =~ f) ~ ( Q ~ u

,

i.e.

,

308

EVOLUTION OPERATORS

Of course we add the appropriate initial and boundary conditions.

In

which sense u is an “approximation“ of uE is an open question.

3.5

Asymptotic expansions (11). We now consider the situation (3.30) and we take

(3.49)

k = l .

Then

We find

,

(3.51)

RluO = 0

(3.52)

Rlul + R2u0 = 0

(3.53)

R1u2 + R2u1 + R3u0 = f

,

-

We are lead again to (3.44) but this time with

of T.

A1

independent

It follows from de Simon [l] that for almost every T ~ condi,

tion (3.45) is necessary and sufficient for (3.44) to admit a Y-ro periodic solution.

We therefore assume

OPERATORS OF HYPERBOLIC, PETROWSKY OR SCHRODINGER TYPE (3.54)

309

condition (3.45) is necessary and sufficient for the existence of $ which is a Y - T ~ periodic solution of (3.44)

.

Then (3.52) gives u1 by formula (3.47), (3.46); but since A1 does not depend on

T,

being defined by

we have

and

Then (3.53) admits a solution for u2 iff

7/1 (R2u1 + R3UO)

= f

,

i.e. 2

*+

(3.56)

a u +

at2

nC

This can actually be proven: Theorem 3.1. I

We assume that AE is given by (3.1), (3.2), (3.3) and that BE is given by (3.30)

that

bj

E

C1(?

x

with

k = 1.

[O,T~].

solution of (3.38)

with

0

Then

We assume that (3.54) holds true and uE satisfies (3.10) where u is the

given by

Sketch of Proof. We introduce BE* = adjoint of B i.e., BE*v = such that

a (b.(x,t)v). € - fax J j

in the sense of distributions,

We construct a family of functions wE

EVOLUTION OPERATORS

310

+

a2

+ BE*]wE

AE

=

€4,

,

,

gE bounded in L 2 ( Q )

(3.58) w

+ 1

o

-+

in L ' ( Q )

as

E

-+

o

.

For the construction of w, we need all coefficients aij and b smooth-we

j

to be

can always reduce the problem to this case by

approximation.

We look for wE in the form:

(3.59)

-1

W

=

4-

+ E 2@ ( y , T ) ,

ECL( Y,T

a and B being Y--to

periodic

.

*

We find, using (3.50) and since R1 = R1:

*

(3.60)

R a 1

+ R2(-l)

(3.61)

RIB

+

= 0

,

*

*

R2a + R3(-l) = 0

and g,

=

*

.

R2B

We define a as a Y--r0 periodic solution of (3.60), i.e. (3.62)

R1a = -

4 ab

(YtT)

*

Then (3.61) gives (using the fact that R3(-l) = 0):

*

R1B = - R2a which admits a Y - T ~ periodic solution by virtue of (3.54).

Then we

have (3.58). We now use $

E&,;(

Q )

and we take v = Ow, in (3.32).

multiply (3.58) by $uE. We obtain, by subtracting

We

OPERATORS OF HYPERBOLIC, PETROWSKY,OR SCHRODINGER TYPE

The first term in ( 3 . 6 3 ) equals

In ( 3 . 6 4 ) , the first term converges to 0, since u

aw = aa strongly and 2 at aT

+

E

aB + aT

Therefore ( 3 . 6 4 ) converges to If we set E:

=

aij

aa

-

(=)

?

I

2 = 0 in L (

-+

u in L

(u,@")dt.

0

au

$, the

second term in ( 3 . 6 3 ) equals

The third term in ( 3 . 6 3 ) equals b;uEwE

s

$ dxdt 1

(s)

s ) weakly.

j

-

2

+

(bj)

1

s

u'$

j

dxdt

.

311

EVOLUTION OPERATORS

312

The right hand side in (3.63) converges to

(3.38)) =

-

(u",@)dt 0

(citq) lo!u,@), =

f 0

Q(u,@)dt =

T 0

-

(n,@)dt.

T 0

(f,$)dt = (using

Since

it follows that

It remains to show the identity of (3.65) with (3.57), i.e. that

We multiply (3.62) by

xi.

It follows that

hence the result follows.

3.6

Remarks on correctors. We present here only a very preliminary result on the question

of correctors. (3.67)

1

=

We consider the case (3.1).

2~ ax ,

We have

xJ(y) being solution of A1(xJ

-

and being Y-periodic

,

j

y . ) = 0 3

and we use cut-off functions mE(x) as in Chapter 1, Section 5.2. set (3.68)

-

u

= U + E ~ U E

l

'

More generally we suppose that ue is the solution of

We

OPERATORS OF HYPERBOLIC, PETROWSKY OR SCHRODINGER TYPE (3.69)

313

(uE,v) + aE(uE,v) = (fE,v)

where (3.70)

fE,f E L2 (0,T;H) ,

fE

f in L1 (0,T;H) strongly

+

We have again (same proof), (3.10) and (3.12).

.

If we assume

that (3.71)

u E L ~ ( o , T ; H ~ ( o ),)

at

E

L~(o,T;v),

then -

-

= uE - uE

z

(3.72)

I

zE

+

-t

2

0 in L (0,T;V strongly

,

0 in L 2 0,T;H) strongly

The proof of this result relies on the following identity. 1 (T - t)2 $(t) = 2 We have

We set

.

uE,$)]dt

To prove this identity, we can assume that f

.

and f are smooth and witt

values in H (by approximation of these functions).

It is then

obtained by integration by parts. Since T does not play any role here, the result will be proved if we verify that X E

+

0.

314

One verifies that

T

EVOLUTION OPERATORS

0

@'aE(iiE)dt +

T 0

@ ' a(u)dt and one can pass

to the limit directly in all other terms.

3.7

The result follows.

Remarks on nonlinear problems. Let us give a simple example.

We consider the situation of

Section 3 . 6 and we set (3.73)

2

@(v) =

lgrad vI dx

.

8

Let us consider the nonlinear equation (ui,v) + aE(uE,v) + @(uE)(udfv)= (f,v) (3.74)

uE E L-(o,T;v) UE(0) = 0

,

,

u:

E

UE(0) = 0

,

L ~ ( o ~ T ; H,)

.

We assume that (3.75)

af

f,E

E

2

L (0,T;H)

.

Then there exists a unique solution of ( 3 . 7 4 ) which satisfies (3.76)

u :

E

L~(o,T;v),

u" E L-(o,T;H) E

.

The proof of this fact is standard. One obtains the a priori estimates (3.77)

I bElI L- ( 0, T;v)

+

1l~:Il

L~(o,T;v)

OPERATORS OF HYPERBOLIC,

PETROWSKP OR SCHRODINGER TYPE

315

Therefore we can extract a subsequence, still denoted by uE, and such that u

+

u and u'

+

u

+

u' in L m ( O , T ; V )

,

weak star

(3.78) 1

uE

I,

m

in L ( 0 , T ; H )

weak star

.

Since grad uE is bounded in Lm(O,T;(L2 ( 8 ) )"), $(uE) is bounded in L m ( O , T ) and we can assume that (3.79)

since u

$(uE) + 5 in Lm(O,T) weak star I

* u' in L2 ( S

)

;

strongly, we have $(uE

weakly and if we write (3.80)

(u:,v)

+ aE(uE,v) = (f

-

$(uE)u;,v)

gu' in L 2 ( S ) weakly.

we have fE * f

But one can improve this

result; indeed d

u

;iT@

= 2

)

i

I

grad u .grad u dx

8 m

remains in a bounded set of L (0,T) so that we can assume that $(uE) * g in Lm(O,T) strongly and therefore (3.81)

fE

+

f

-

CU' in L 2 ( S

)

strongly

.

Therefore, using (3.72), we see that $(UE)

-

1 $(fiE )+ 0 in L (0,T) for instance

.

But

a 8

(u

+

Em u E

1

)

a (u + Ern u )dx axi E l

,

EVOLUTION OPERATORS

316

so that

in L 1 ( 0 , ~ )

.

Therefore u is the solution of

where $(u) is given in ( 3 . 8 2 ) .

Remark 3 . 3 .

Homogenization of variational inequalities.

The general question of the homogenization in variational inequalities of "hyperbolic type" is largely open. simple result along the following lines. (3.84)

j(v) =

lvldx

g

I

We give only a

We set

g > 0

8

and, always with the conditions of Section 3 . 6 , we consider the solution uE of the V.I.

OPERATORS OF HYPERBOLIC, PETROWSKF OR SCHRODINGER TYPE We assume that f satisfies (3.75).

317

It is known (cf. for instance

Duvaut-Lions [l]) that (3.85) admits a unique solution.

Then one

can prove that (3.78) holds true, with u being the solution of the homogenized V.I. ( u " , -~ u') + U(U,V (3.86)

-

u')

+ j(v) - j(u') L (f,v - u')

u(0) = 0 ,

V V E V ,

u'(0)

=

0

.

Remark 3.4. If in the V.I. of the preceding Remark, we replace j(v) given by (3.84) by (3.87)

j,(v)

=

g

I

lgrad vldx

9

it is likely-but

it is not proved-that

one obtains for the limit a

V.I. of type (3.86) with j replaced by

3.8

Remarks on Schrodinger type equations. We can consider, now with complex valued functions, the equation

(of the Schrodinger type)t

'Of course one has now: (f,v) =

I

8

f Gdx

,

aE(u,v) = 8

318

EVOLUTION OPERATORS

where aE is given as in (3.1)

(3.2), (3.3).

[We can also consider

other cases considered in the preceding sections for aE . I

If we

assume (3.75), we can obtain the following a priori estimate: v = u

I

taking

in (3.89) and taking the real part

1 2 dt

aE(uE) = Re(f,ui)

Hence it follows that (one obtains a preliminary estimate by taking v = u (3.90)

in (3.89))

I bElI L~ ( o ,T ;v) -<

c .

Then we can extract a subsequence, still denoted by uEf such that (3.91)

uE

co

-t

u in L (0,T;V) weak star

and (3.92)

3.9

Nonlocal operators.

Setting of the problem. We consider: (3.93)

A E given with (3.1), (3.3)t

tThe symmetry is now necessary.

OPERATORS OF HYPERBOLICf PETROWSKY OR SCHRODINGER TYPE

bij = bji

(3.94)

m

,

b1.3. E L (R") Y

,

bij Y-periodic

319

,

B > 0

bij(y)S.S. 1 3 2 @Sici and we taket V = H 1o ( b )

(3.95)

.

We set bE (u,v) =

(3.96)

b : j

(x)

au av ax.

dx

3

8

and we consider the equation

(3.97)

UE(0) = 0

.

Remark 3.5. The ellipticity of A E is not necessary in what follows. The problem (3.97) admits a unique solution uE (3.98)

f E L'(O~T;V~)

E L

2 (O,T;V), if

,

and (3.99)

'We

I lUEl I

L~ ( o T;V)

1 can take HO(Q

)

< c .

5V 5

1

H ( 6 )if 1 9 V.

1 If V = H ( a ) , what

we are going to say is valid if one replaces BE by BE + bo (x/E) bo(y) 2 B 0 > O.,

EVOLUTION OPERATORS

320

Indeed taking v = uE in (3.97) we have

hence (3.99) follows.

Therefore one can extract a subsequence, still

denoted by uE, such that uE

(3.100)

+

u in Lm(O,T;V) weak star

.

The problem is now to see what is the "homogenized" equation which characterizes u.

We are going to show it is not a partial

differential equation but an equation with pseudo-differential operator. We take the Laplace transform in t of (3.97).

I

Let us set

W

(3.101)

uE(p) =

e-PtuE(t)dt

,

R e p > O .

0

We do not restrict the generality by assuming that (3.97) is taken for every t > 0 and with f E LL(O,m;H) with compact support. also consider (3.97) as an equation over Rt, with u t < 0.

We can

and f zero for

We then set

and we write (3.97)

This leads to "homogenization with parameter" as in Chapter 1, Sections 8.1 and 15.

We define xJ(p) as a Y-periodic solution of

T

HYPERBOLIC, PETROWSKYiOR SCHRODINGER TYPE

normalized by, say,

I’

x’(p)dy = 0 ; in (3.103) we use the standard

I .

%].

1

notation for A1 and B1 = - -[bij a (3.104)

321

Then we define

qij(p) =

and

Q (p) =

(3.105)

We can take the inverse Laplacetransform of q..(p) and set 17

(3.106)

Q

=

2 - l Q (PI

e - l = inverse Laplace transform

I

Then the homogenized equation is u

E

L~(R;v),

(3.107) Q(t)u = f

u =

o

for t <

o ,

.

The proof proceeds by methods similar to those of Chapter 1, Section 15.

Remark 3.6. The operator Q(;)

is not, in general, a partial differential

operator.

Remark 3.7. Let us consider materials which have a composite periodical structure and with a long memory (cf. for this last point, for instance, Duvaut-Lions [l], Chapter 3 , Section 7). simple model.

We consider a

These general cases will be treated with similar

.

322

EVOLUTION OPERATORS

methods.

We are given functions aij(y) satisfying (3.1), (3.2),

(3.3). We are also given functions bij(y,t) such that

bij - bji

, v

(No ellipticity is assumed). (3.109)

BE(t)(;)v =

.

i,j

Given v

- axi a j

E

x bij(;,t

Lioc (Rt;V) , we set

-

S)

av ax. (x,s)ds 3

0

and we consider the problem:

u

= O f o r t < O ,

where (3.111)

2 f E Lloc(Rt;H)

,

f = 0 for t < 0

This problem admits a unique solution.

.

One has the following

a priori estimates: (3.112)

I luElI

L-(o,T;V)

L- (0,T;H)

< c .

Indeed, if we set b E t;u,v) =

1 8

';1

t;u,v) =

x

bij(F,t)

au av ax 5 dx

(E,t) ax 8

I

j

j

$ dx =

,

OPERATORS OF HYPERBOLICf PETROWSKY OR SCHRODINGER TYPE

323

we obtain

and

so that (3.113) gives:

and therefore, for t 5 T finite (arbitrary)

I t

1 lu,'(t)

l2 +

1

aE (uE(t)) +

bE(t-s;uE(s)fuE(t))ds

0

hence *

hence (3.112) follows. Therefore we can extract a subsequence, still denoted by uE, which satisfies (3.10).

We show now that u is characterized by the

solution of a pseudo differential operator equation.

EVOLUTION OPERATORS

324 We define, for Re p

0

m

2. ,(y;p) = 13 (3.1

J

,

e Ptb. .(y,t)dt

0

-

13

Then (3.110) can be written (3.115)

(AE

+

gE(p)

+ p 2 I)GE(p) = g(p)

.

(We can always assume that f admits a Laplace transform in t.) We use now "homogenization with parameter".

I

the Y-periodic solution (normalized by

We define

Xl(p)dy = 0) of

Y

we have set

We observe that (3.118)

A1

+ g,(p)

Therefore (3.116

gij (y;p)I

5

& so that the operator

is elliptic for Re p large enough

.

defines XJ (y;p)

(3.119)

and

4 (p) and

Q as in (3.105), (3.106).

We define

.

x3 (p) as

COMMENTS AND PROBLEMS

325

Then the homogenized equation is

(3.120) 2

u E Lloc(Rt;V)

,

u = 0 for t < 0

.

Comments and problems.

4.

Most of the results presented in this chapter are given for the first time.

Preliminary announcements were made in notes by the

authors (cf. Bibliography of Chapter 1). The Lp estimates of Section 2.7 were obtained, in the symmetric case, by Pulvirenti [l], [2].

The method given here extends to

parabolic equations of higher order or to systems. Given a family of elliptic operators, say A E , one can say that A E "G converges" to

c 7 if

V f, (AE)-lf -+

a-lf in a weak Sobolev

space, where the inverse are taken for a given set of homogeneous boundary conditions.

If one considers next a family of elliptic

operators depending on t, say AE (t), one can say that A' converges" to U -1 + A E ) f -t

(where "P" stands for "parabolic") if V f, -1 + a f in a weak Sobolev space (where again the

[A

[&

"P-G

]

inverse are taken for a given set of homogeneous boundary conditions). Then one can compare the "P-G convergence" with the "G-convergence V t " (in general these are different notions!) S.

Spagnolo [l].

Cf. F. Colombini and

Cf. also for this general approach, Spagnolo 111,

121, Sbordone [l]. In case the coefficients a..(y,-r)are Y-periodic in y and almost 13

periodic in

T,

one obtains formulas similar to those of the text,

with a difficulty for k = 2.

In the case k = 1, one considers

xJ(y,-r) as the solution of (1.67) which satisfies, say

EVOLUTION OPERATORS

326

I'

Y

XI (y,T)dy = 0. Then XI is almost periodic in

T

(T

plays the role

of a parameter) and one obtains

where

In case k = 3 , one introduces (compare to (1.86))

and

U

is given again as in (1.94) where @ =

lim A+=

-

1

@(y,o)dody

.

lYlA y 0

In case k = 2 , one has to consider the T-almost periodic solution of

which is periodic in y, and many problems remain open in this direction.

327

COMMENTS AND PROBLEMS One can of course in the .r-almost periodic case consider correctors, as in the case of periodic coefficients.

Let us give now some indications on possible extensions of the resu ts given in the text. b is Y--r0

(4.5

Let b(y,r) be given satisfying

periodic

b E Lm(Rn Y

,

RT)

x

b(y,r) '> B > 0

,

,

et us consider the operator

and (4.6

a

bE&--

axi(arj

a

k b" = b(x/Elt/E 1

x) , j

aFj = aij (x/E,t/Ek

,

.

(If one considers the case bE(x,t) = b(x/E) when b does not depend on

T,

the homogenization is immediate and is given by

711

Y

(b)

a + ak ,

k = 1,2,3

)

.

The general case (4.6) leads to a number of interesting questions. In case k = 1, the asymptotic expansion, with usual notations, leads to

,

(4.7)

AluO = 0

(4.8)

A U + b -auO +Au = O , 1 1 aT 2 0

(4.9)

A1u2

+

(b

a + A2)u1 + aT

(b

a +

A3)u0 = f

.

It follows from (4.7) that uo does not depend on y . Taking I Y au (b) 2 = 0, so that uo = u(x,t) and with the of (4.8) gives:

Y

aT

notation (1.67), one will have:

328

EVOLUTION OPERATORS We use (4.10) into (4.9).

m

Taking

Y

of the result, we obtain:

1 (b) and we take the average in 2 y

We then multiply by

~

T:

(4.11)

a

1

my(aij

-

aik

el]% 7

One sees the appearance of first order derivatives in x in the homogenization process.

One also sees that one needs neither

regularity on the a. ' s of the type: lj

&

a

to make sense nor regularity of b:=ab

can integrate by parts in

'I

a. . E Lm(Rn 13 Y E

Lm(Ry

x

x

RT)

, in order

R T ) (since one

au in the coefficients of F ).

The

j

justification can then be made along the lines of Section 2 .

If

k = 2, one runs into the difficulty of finding Y - T ~periodic solutions of

a question whose general study does not seem to have been made (the same is a fortiori the case for the almost periodic situation). In the case k = 3 , one defines:

and the functions

d

by (1.88) (with this new

El).

Then the

COMMENTS AND PROBLEMS

329

homogenized equation is

Singular perturbations and homogenization. Let us consider first the problem

u

being subject, say, to Dirichlet's boundary conditions (with usual

boundary conditions).

Then the homogenized equation is obtained by

a

formulas analogous to those for homogenizing ; i i + AE but where A1 replaced by A1 + A 2 . For instance if k = 2, one considers the Y solution x 3 ( y , ~ ) , which is Y - T ~periodic, of

and

a2= a

2 p A 2 is now given by

(4.17) 2td2 =

qij

[aij

-

d].

aik ayk

One can also consider the problem: (4.18)

au

E

k 2 at + AE(t/E ) u E = f ,

with usual boundary conditions and with uE(0) = 0. If k = 2 , one has (4.19)

uE

+

u in L2 (0,T;V) weakly

&

EVOLUTION OPERATORS

330

where u is the solution of (4.20)

C7 u = f = f (x,t)

,

(t plays the role of a parameter)

(More precisely, (4.21)

u E

If k = 3

v , a 1 (u,v)=

or

(f,v)

,v

v

E

v

)

.

k = 4, one has again (4.19), with u given by

(4.22)

C12(u,v) = (f,v)

,V

E

V

,

if k = 3

,

(4.23)

0 3 (u,v) = (f,v)

, V'v E

V

,

if k = 4

.

v

Actually the proof of (4.22) has been obtained only for the Dirichlet's boundary condition by the method of the adjoint expansion, and assuming all data

to be smooth.

Proofs of (4.21), (4.23)

proceed along the lines of Section 2. There is one more a priori estimate one can obtain when

[One will obtain the same estimates for the equation

Indeed if we multiply (4.18), for k = 1, by AE(t/E)UE, we obtain (we use the symmetry): (4.26) Hence (4.27)

caE [2%~ , u € +] IAE(t/E)u,12 = (f,AE(t/E)uE)

.

.

331

COMMENTS AND PROBLEMS

=

. E (a) (uE,uE). It follows from (4.27) that

2

(4.28)

AE(t/c)uE remains in a bounded set of L ( 0 , T ; H ) .

Using (4.18) and (4.28) we see that (4.29)

E

au 2 in a bounded set of L ( 0 , T ; H ) 2 at remains

.

The behavior of the solution of (4.18) for k = 1 is unsettled.

Reiteration. Let us consider for instance the equation

with the usual boundary conditions and initial conditions, where the aij's satisfy the following conditions: aij(ylz,Tl,T2,T3)is Y-Z periodic in y,z and admits (4.31)

the period T O in the variable j

'lj,

j = 1,2,3 ;

and the usual hypothesis

-

'One

could weaken this hypo~esis but the weakest hypothesis is

not known.

332

EVOLUTION OPERATORS In (4.30) we assume that

(4.33)

0 < kl < k2 < k3

. 2

we have then the "usual" result that uE + u in L (0,T;V) weakly, where u is the solution of

au at + au

=

f subject to appropriate

boundary conditions and where 2 is computed as follows. One considers the operator with coefficients (4.34)

X t a. .(l,I,,v,v,x)

13

E

where k = k3/2.

One homogenizes the corresponding operator according

to the rules of Section 1; therefore we have three cases to consider: k < 2, k = 2 or k > 2,

(4.35)

k3 < 4

,

i.e.

k3 = 4

or

k3 > 4

.

Let us denote by b..(A,iJ,v) the coefficients of the homogenized 17

operator obtained in this way.

We then consider the operator with

coefficients

and we homogenize the corresponding operator: therefore there are again three cases to consider: (4.37)

k2 < 2

,

k2=2

Let us denote by c .

11

k 2 > 2 .

the coefficients of the homogenized operator

(11)

obtained in this way.

or

The coefficients of the operator we are looking

for are then given by 0

I

71 (4.38) 71

o

~

~

~

( = ~dij~

.)

d

7

~

COMMENTS AND PROBLEMS

333

The proof is technically long but the ideas are those given in Chapters 1 and 2 . In case of coefficients of the form

one will apply the same rule to aij(x0,y,z,t0,~l,~2,T3) and one will obtain coefficients d . . (xo,to). Then the homogenized operator is 11

d . .(x,t). 11

Homogenization with rapidly oscillating potentials. One can extend the considerations of Chapter 1, Section 12, to the evolution problems, and this for parabolic and hyperbolic equations, and also forschrodinger equations.

For instance let uE

be the solution of

-at-

AuE + -1 W Eu

(4.40)

uE = 0 on

C

=

f

O

in

E

E

,

uE(xIO) = 0

Q

= Ox

lO,T[

,

.

2 We assume that WE = W(x/E,t/E ) (one can also consider 2) ) is almost W(X/E,~/Ek ) , k # 2 , and a function W (x,~,x/E,~/E periodic. y and in

Then one considers the almost periodic solution T)

and we set

of

=

average in y and A

Then

T

of Wx A

x

(a.p. in

334

EVOLUTION OPERATORS

(4.42

uE

.+

u in L2(0,T;Hi(8)) weakly

,

where u is the solution of

cf. Bensoussan, Lions, Papanicolaou [21 (where one will find analogous formulas also for the hyperbolic case).

Homogenization an8 penalty. Let us consider

(4.44)

where the aijk(y,r) satisfy the usual hypotheses.

We consider the

system : aUEl

at

+

1 (u A h 1 + 2 E

l

-

UE2) =

1'

(4.45)

with boundary conditions, say the Dirichlet's boundary conditions but this is without importance, (4.46)

uEl = uE2 = 0 on E

and (4.47)

UEl(X,0) = Uc2(X,O) = 0

.

This problem admits a unique solution.

Multiplying the first

(respectively, second) equation (4.45) by uEl (respectively, uc2) orLe obtains:

335

COMMENTS AND PROBLEMS

Hence it follows that

and 1 (uEl

(4.49)

-

uE2) is bounded in L2 (0,T;H)

Estimate (4.49) comes from the penalty terms 5 (4.35).

.

1 7

(uEl

-

uE2) in

It follows from (4.38), (4.39) that one can extract a

subsequence still denoted by uEi such that uEi

(4.50)

+

L2 (0,T;V) weakly (where we have the same

u

limit for uEl and for uE2)

.

The limit u is the solution of the following homogenized We give the formulas only for the case k = 2.

problem.

Alj, A2j by

and

{;x

x

;I

A:

= E-2~11

A :

=

E-2

+

E-1~12

+

,

E0

A21 + E - ~ +A E~ 0A23 ~

is the solution which is Y - T ~periodic of

Then the equation for u is (4.52)

au

2 - at -

qij

a 2u ZQT j

= fl + f2

#

One defines

EVOLUTION OPERATORS

336

where (4.53)

qij =

nm

ax;

1

a i k l - K + aij2

-

aik2

$1

.

The proof proceeds along the lines of Section 2. In (4.52) the boundary condition is (4.54)

u = O o n C .

If in (4.45) one takes boundary conditions which are the same for u

El and for uc2, that is if we take a variational formulation with a space (4.55)

V = W

x

W

,

Hi(0)

5 W 5 H1(S) ,

(with the same W), the result extends:

u will be characterized by

u(t) E W and

In case we take V = V1

x

Vz with different spaces V

and V 12

one obtains (4.57)

u(t)

E

v1

n

v2 ,

and

Problems where one has rapi ly oscillating coefficients in t an a penalty term are studied by Simonenko [l], [21. One can also study the evolution problems with degenerate elliptic part and also with domains with "periodic holes".

Homogenization and regularization. It is known that the solution of a parabolic equation can be obtained as the limit of the solution of elliptic equations, via the

COMMENTS AND PROBLEMS

337

elliptic regularization. Applying this idea to operators of the type

a + A' at

leads to the behavior as

(4.58)

-Em

aLuc

au

at2

at

-- k E + A E u

Ue(O) = 0

E +

0 of equations of the type

= f ,

au EaT( T ) = 0

,

.

The behavior of uE will depend on m and on k, if ' A k cients a. . ( x / ~t/E , )

is with coeffi-

.

13

For instance if we consider

(4.59)

-€

2 3% at2

au au +at > - -axia aij ( E L2 ) E ax ' f , E'

j

then the limit value u of uE is solution of a new parabolic problem obtained in the following manner.

One defines $J(y,.r)as the Y - T ~

periodic solution of

Then (4.61)

& + B u = f at

where B u =

-

,

77i

The results presented for the homogenization of V.1 are sketchy. Other results are announced in the first note of the authors in the Bibliography of this chapter. The problem studied in Section 3.9 corresponds to operators which have been studied (without homogenization) by Showalter and Ting [l]; cf. also other equations of this type, or which present somewhat analogous properties, in Carroll-Showalter 111. results for the homogenization of V . I .

Some

connected with these operators

EVOLUTION OPERATORS

338

are given in the first note of the authors in the Bibliography.

Many

questions arise for these equations. It would be interesting to understand "intrinsically" when the homogenization of partial differential operators leads to non-local operators in the limit (this question could also have a physical interest).

For instance let us

consider the equation

BE given as in Section 3.9 and the aijvs not necessarily elliptic. We set:

B1 =

a - ayi

[bij (y) "aYj -1

and let us assume that there exists

a function ~ j ( ~ , rwhich ) is Y-r0 periodic and which satisfies

"

(i.e., 0

aa.. ay,

(y,-c)dT= 0). Then the homogenized equation (found by

asymptotic expansion) is formally

For the homogenization of first order hyperbolic systems, we refer to Chapter 4.

One can also study (cf. Bensoussan, Lions and Papanicolaou

[ 3 1 ) a case when one has simultaneously "homogenization" and

"penalty":

COMMENTS AND PROBLEMS

339

a u ~ 2 aE a u ~ 2 bE E ax 2 (UEl - UE2) = f2 at (4.64)

(0,l)

x E

,

UEl(O,t) = 0

u (x,O) = 0 Ei

,

I

UE2(l,t) = 0

,

,

aE,bE E Lm(O,l)

,

aE,bE 1. a > 0

1 m - -+ p in L (0,l) weak star aE

,

v in L (0,l) weak star

.

,

(4.65) -+

aC

afi at E L2((0,1) x Then, assuming that fi, __ m

UEi

-+

(O,T)), one has

u in L (0,T;L2(0,1)) weak star

,

(same limit u for i = 1,2) and where u is the solution of the parabolic equation

(4.66)

In ( 4 . 6 4 ) we have only ( 4 . 6 5 ) ,

without periodic, or almost

periodic, structure. For problems of this type for parabolic equations, we refer to Markov and Oleinik [l]. Very many questions arise in connection with the homogenization of second order hyperbolic operators. the text.

Some of them are indicated in

We shall return to this question in Chapter 4 .

For a

340

EVOLUTION OPERATORS

recent work on the regularity of the solution of hyperbolic equations with irregular coefficients we refer to Colombini 1 1 1 . One can also study the homogenization of coupled hyperbolic parabolic systems: cf. Bensoussan, Lions, Papanicolaou [l]. Another problem is the following. Consider the equation

where k E i (4.68)

E

Lm( a ) , k :

-+

Gl

in Lm( 8 ) weak star

.

We assume that (4.69)

kT,B>O

and that (4.70)

kz 1. 0 (but kh can be zero)

.

In (4.67) A is a fixed second order elliptic operator.

Then one

can add to (4.67) standard boundary conditions on uE and for the initial conditions

(4.72)

k9

at

auE where uo and v1 are given: (4.72) gives a condition on ar (X,O) only on the set where ki # 0. This is an equation of "hyperbolic-

parabolic" type.

Cf. V. N. Vragov [l] for the case of fixed

coefficients kl, k2.

One can show the following (cf. Bensoussan,

Lions, Papanicolaou [4]): we do notrestrict the generality in assuming that

COMMENTS AND PROBLEMS

@* x

( 4 . 73)

in Lm(B ) weak star

341

.

Then us * u in Lm(O,T;V) weak star, ui * u' in L2 (0,T;H) weak star, where u is the solution of

+

1u'

(4.74)

i2u"

(4.75)

u(0) = u0 ,

(4.76)

u'(0) =

+

;1L Vl

Au = f

,

.

k2 We remark that in ( 4 . 7 6 ) , domain 0

.

u' (x,O) = u' ( 0 ) is given over the whole

Comparing with ( 4 . 7 2 )

we see that there is, here, some

sort of "increase" in the initial data. Other evolution equations are studied from the new point of homogenization in separated papers by the authors.

For the transport

equations, cf. already the authors' [51 and the Bibliography therein. For other physical aspects of these questions, cf. Sanchez-

.

Palencia [l]

We also refer to the survey of O.A. Oleinik [I], to appear in the Ouspechi Mat. Nauk. We also mention applications of the asymptotic expansion method to problems in turbulence?

.

-t P. Perrier and 0. Pironneau, C.R. Acad. Sc. Paris, 1978.

BIBLIOGRAPHY OF CHAPTER 2 A. Bensoussan, J.L; Lions and G. Papanicolaou [l] Sur quelques problbmes asymptotiques d'dvolution. C.R.A.S. Paris, t. 281 (1975), p. 317-322. [2] Sur la convergence d'opgrateurs diffgrentiels avec potentiel oscillant, C.R.A.S., Paris, February 7, 1977. 131 Remarques sur le comportement asymptotique de systsmes d'6volution. French-Japan Symposium, Tokyo, September 1976. [ E l Perturbations et "augmentation" des conditions initiales. Lyon, December 1976. 151 Boundary layers and homogenization of transport processes. R.I.M.S. Kyoto, 1978. N.N. Bogoliubov and Y.A. Mitropolsky [l] Asymptotic methods in the theory of non linear oscillations. (Translated from the Russian) Hindustan Pub. Corp. Delhi 1961. R.W. Carroll and R.E. Showalter [l] Singular and degenerate Cauchy problems. Math. in Sc. and Eng., Vol. 127, Acad. Press 1976. F. Colombini [l] On the regularity of solutions of hyperbolic equations with discontinuous coefficients variable in time. Scuola Normale Superiore Report, 1976. F. Colombini and S. Spagnolo 111 Sur la convergence de solutions d'gquations paraboliques. To appear. G. Duvaut and J.L. Lions [l] Les Idquations en Mdcanique et en Physique, Paris, Dunod 1972. English translation, Springer, 1975 J.L. Lions [l] Equations differentielles op6rationnelles et problsmes aux limites. Springer, 1961. [2] Lectures on elliptic differential equations. Tata Institute, Bombay, 1957. J.L. Lions and E. Magenes 111 ProblSmes aux limites non homogenes et applications. Vol. 1, 2, Paris, Dunod, 1968. English translation, Springer 1970. P. Marcellini and C. Sbordone [l] An approach to the asymptotic behaviour of elliptic parabolic operators. To appear A. Marino and S. Spagnolo [l] Un tipo di approssimazione dell' operatore Ann. S.N. Sup. Pisa, XXIII (1969), p. 657-673. V . G . Markov and O.A. Qleinik [l] On the heat diffusion in a one dimensional dispersive medium. P.M.M. 39 (19751, p. 1073-1081.

...

342

BIBLIOGRAPHY

343

L. Nirenberq 111 Remarks on strongly elliptic partial differential equations. C.P.A.M. 8 ( 1 9 5 5 ) , p. 6 4 8 - 6 7 6 . O.A. Oleinik [l] Survey on homogenization. Ouspechi Mat. Nauk. to appear. G. Pulvirenti [ l l Sulla sommabilits Lp.. Le Matematiche. 2 2 ( 1 9 7 1 ) , p. 2 5 0 - 2 6 5 . [ 2 1 Ancora sulla sommabilits Lp.. Le Matematiche, 2 3 ( 1 9 6 8 ) , p. 1 6 0 - 1 6 5 . E. Sanchez-Palencia 111 Comportement local et macroscopique d'un type de milieux physiques h6t6rogSne.s. Int. J. Enqrg. Sciences, 1 2 ( 1 9 7 h ) , p. 3 3 1 - 3 5 1 . C. Sbordone [ l l Sulla G-convergenza di equazioni ellittiche e paraboliche. Ric. di Mat. 2 4 ( 1 9 7 5 ) , p. 7 6 - 1 3 6 . R.E.Showalter and T.W. Ting [l] Pseudo parabolic partial dif-

.

.

ferential equations. SIAM J. Math. Anal. 1 ( 1 9 7 0 ) , p. 1 - 2 6 . L. de Simon [l] Sull'equazione delle onde con termine noto periodico. Ren. 1st. di Mat. Univ. Trieste, Vol. 1, fasc. 11, ( 1 9 6 9 ) , p. 1 5 0 - 1 6 2 . I.B. Simonenko [ l ] A justification of the averaging method for abstract parabolic equations. Mat. Sbornik. ( 1 2 3 ) ( 1 9 7 0 ) , p. 5 3 - 6 1 . [ 2 ] A justi€ication of the averaging method ..., Mat. Sbornik, ( 1 2 9 ) ( 1 9 7 2 ) , p. 2 4 5 - 2 6 3 . S. Spaqnolo [l] Sul limite delle soluzioni di problemi di Cauchy relativi all'equazione del calore. Ann. Sc. Norm. Sup. Pisa, 2 1 ( 1 9 6 7 1 , 657-699. [ 2 ] Sulla convergenza di soluzioni di equazioni pakaboliche ed ellittiche. Ann. Sc. Norm. Sup. Pisa, 2 2 ( 1 9 6 8 ) , 5 7 1 - 5 9 7 . V.N. Vragov [ l ] On a mixed problem for a class of hyperbolicparabolic equations. Doklady, 2 2 2 ( 1 9 7 4 ) , Soviet Math. Doklady, 16 (1975),

1179-1183.

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Chapter 3:

Probabilistic Problems and Methods

Orientation. In this chapter we will develop the probabilistic approach to homogenization.

This approach is based upon the probabilistic inter-

pretation of the solution of elliptic and parabolic P. D. E., as averages of functionals of the trajectory of a diffusion process. Therefore, the problem is reduced to the study of the behavior of diffusion processes with rapidly varying drift and diffusion terms.

In

paragraphs 1 and 2, we give a brief survey of the theory of stochastic differential equations and their connections with P. D. E. the survey as brief as possible, no proofs are given.

To keep

This survey

should not be considered as complete, even for the reading of the chapter, where some other notions will be used and briefly recalled when necessary.

But for the reader with no probabilistic background, I

the survey will give a quick overview of the basic results and Paragraph 1 gives the elements of stochastic differential

concepts.

and integral calculus, and the theory of strong solutions of stochastic differential equations. This material is quite standard and the reader may use for example the following references: Friedman

[

11, Ito-MacKean

[

11, Priouret

Gikhman-Skorokhod [l 1, [

1

I.

Paragraph 2 presents

the weak formulation of stochastic differential equations and the martingale formulation, which turns out to be instrumental in the sequel.

The martingale formulation is due to Stroock-Varadhan [ l

I.

Other basic references are Watanabe-Yamada [I], Yamada-Watanabe [l], 345

346

PROBABILISTIC PROBLEMS AND METHODS

Meyer [ 1 1 , Doleans-Dade-Meyer

[

1 I , Girsanov [ l I . The study

of the behavior of stochastic differential equations (S. D. E.) with rapidly varying coefficients is very much connected to ergodic theory.

We consider it in Section 3 and we give all the results

necessary for the sequel (with proofs). Doob

[

1 1 , Neveu

[ 1I

The reader may consult

for other results and technical details.

In the applications we have in mind, we are mainly concerned with the ergodic properties of diffusions on the torus.

An important

analytic counterpart is the Fredholm alternative for differential operators on the torus, which is also a basic tool in the study of homogenization.

In Section 4 , we consider the theory of homogeniza-

tion where the limit operator has constant coefficients. pcssible to use directly ergodic theory.

It is then

Also since the limit

operator has constant coefficients, the limit process is a Gaussian diffusion, and therefore direct identification of the limit finite dimensional distributions is possible.

Here we do not use the

martingale formulation of diffusions. The probabilistic approach naturally leads to the study of operators which are not written in divergence form.

The analytic

approach is therefore not included in Chapters 1 and 2. reason why we give

This is the

in Section 5 the analytic approach for the case

of operators which are not written in divergence form.

A serious

drawback of all these theories is the need of quite strong regularity assumptions on the coefficients.

This is also the reason

why the results of Chapters 1 and 2 (where minimal regularity assumptions were made) are not a particular case of what is considered in this chapter.

On the other hand, if regularity of the

coefficients is introduced, then the results of Chapters 1 and 2 become a particular case of those of the present chapter, where we give more general homogenization formulas.

ORIENTATION

347

In Section 6 , we consider the situation of operators with nonuniformly oscillating coefficients.

Some of the methods developed in

the previous paragraphs for uniformly oscillating coefficients cannot be applied anymore.

The methods we use here require in general more

regularity on the coefficients.

Since the limit process is a general

diffusion, the use of the martingale approach is very important to identify the limit process,

It turns out that the martingale

approach is related to the ideas of the "adjoint expansion" considered in Chapter 1. We also give the analytic approach to the problem of homogenization with nonuniformly oscillating coefficients. It requires some nontrivial generalizations of the "uniformly oscillating" case. In Section 7, we consider the problem of reiterated homogenization for operators not written in divergence form.

Only the

probabilistic approach is given completely, to avoid too lengthy developments.

Tedious calculations are necessary, but the general

structure of the homogenization formulas is interesting, and it cannot be easily guessed from what has been done in Chapter 1 (divergence form operators).

Paragraph 8 is devoted to the study of

problems with potentials. Up to Section 8 , only the Dirichlet problem has been considered. In Section 9, we give some results for the Neumann problem. This involves the study of reflected diffusion processes with rapidly varying coefficients. However we always assume here that the direction of reflection is not rapidly varying. Therefore we cannot recover the general results of Chapter 1, even if we assume regularity of the coefficients. In Section 10, we turn to evolution problems.

Our purpose is to

give the probabilistic counterpart of Chapter 2 (Sections 1 and 2). We recover the three cases, depending on how the coefficients depend on time.

However we do not consider necessarily the case of

348

PROBABILISTIC PROBLEMS AND METHODS

operators in divergence form.

Paragraph 11 is devoted to the problem

of averaging, where some specific methods can be developed.

Also we

give a general framework, that we call generalized averaging which includes most of the problems considered before. paid by the complication of the arguments.

So

The generality is

we do not give all

the developments, but the framework is useful as a general reference.

1.

Stochastic differential equations and connections with partial differential equations.

1.1

Stochastic integrals. Let

(Cl,

4 ,P) be a probability space, and

family of sub u-algebras of

,

( 5 t1 C

5

5

be an increasing

, if tl 5 t,).

A

stochastic process w(t) with values in Rn, which is continuous and satisfies (1.1)

w(0) = 0

where I is the identity matrix, is called a Wiener process with respect to 5 . '

Here

* stands for transpose.

Such a process is a Gaussian process, i.e., V tl,t2, random vector w(tl),

...,w(tk)

is Gaussian.

expresses the fact that w(t) is an

at

...,tkl the

The property (1.2)

martingale.

Wiener processes will be the basic process in the following. We now introduce the spaces

STOCHASTIC DIFFERENTIAL EQUATIONS (1.4)

349

$(t;w)I$ is measurable,V t, $(t) = $(t,*)

3

is Z t measurable, a.s.

T

I$(t)lPdt <

m

0

(1.5)

Mp (0,T;R") = 3

J

$ E

Lp3 (0,T;R") IE

7 0

I$(t) lPdt <

t.

The space M? is isomorphic to a Banach subspace of 5 Lp( (0,T) x n,dt @ dP;Rn). In particular M 23 is a Hilbert space. If $ E M 23 is a step process, i.e. if there exists a deterministic splitting of (0,T) to = 0 < tl < t2

... < t N = T

such that $(t) = $n in [tnltn+l[ ,

n = 0, ...,N-I

,

then one defines the stochastic integral as follows

and the following properties hold true

The relation (1.8) permits to extend the stochastic integral to the whole space M 23 , and the mapping I is an isometry from M 2 3 to L2 (a, C7 ,P;Rn) The notation

.

350

PROBABILISTIC PROBLEMS AND METHODS

is standard. Remark 1.1. The stochastic integral extends to integrands $ which are random matrices. Remark 1.2. The stochastic integral extends to integrands $ which belong only to L;

.

It is clear that one can define for any t E [ O , T ] (1.9)

I(t) =

$(s)*dw(s)

.

0

One can show that the process I(t) is continuoust an 5

1.2

.

Moreover I(t) is

martingale.

Ito's formula. The stochastic integral leads to a stochastic differential

calculus, which differs from the usual differential calculus.

Let us

consider 1 a E L~ (o,T;R") (1.10)

,

b

E

52

(o,T;,L(R~;R"))

c ( 0 ) random variable with values in Rn, 5

We define

'tUp to a stochastic equivalence.

measurable

.

351

STOCHASTIC DIFFERENTIAL EQUATIONS

which is a continuous, adapted process. a function on Rn

Let now $(x,t) be

[O,T], belonging to C2'l(Rn

x

[O,T]); then the

x

following formula holds: t (1.12)

q(c(t),t) =

~(c(o),o)

+

j [g 2ax- a +

0

+

tr

(S)

t

2 9 bb*(s)] (c(s),s)ds + 2 (S(s),s)*b(s)dw(s) ax

I

0

H

t

E

[o,ti

,

a.s.

.

The formula (1.12) is called Its's formula. A standard notation for (1.11) is: (1.13)

d€,(t)= a(t)dt

+ b(t)dw(t)

and one says that the right hand side of (1.13) is the stochastic differential of the process c(t).

Then (1.12) expresses the fact

that the process $(€,(t)lt)has also a stochastic differential whose integral form is (1.12).

1.3

Strong formulation of stochastic differential equatibns. Let (n,Q , P I at,w(t)) be as above, and let g(x,t), u(x,t) be two

functions such that g:

Rn

X

[O,T]

+

Rn

,

(5:

Rn

X

[O,T]

+

&(Rm;Rn)

352

PROBABILISTIC PROBLEMS AND METHODS

We define the following problem: find a process y(s) = yxt(s), s

E

for x,t given t

y(s) is adapted and continuous

(1.16)

y(s)

x +

I

t

[O,T], to

[t,T] such that

(1.15)

=

E

g(y(A),A)dA +

T

t

,

u(y(A),A)dw(X)

s

E

[t,T]

,

, a.s.

.

Equation (1.16) is called a stochastic differential equation. Under the assumptions (1.14) and (1.14)' there exists one and only one process satisfying (1.15), (1.16). stood in the following sense:

The uniqueness is under-

if yl(s), y2 ( s ) are two processes

satisfying (1.16) then one has y,(s) = Y2(S)

,

s E

[t,T]

,

a.s.

.

Remark 1.3. The formulation (1.16) is a natural extension of the formulation of ordinary differential equations.

We will usually write (1.16) in

the more intuitive way

A stopping time

T

with respect to 5

variable such that

+la12 = tr uu*.

is a nonnegative random

STOCHASTIC DIFFERENTIAL EQUATIONS

353

To a stopping time T we associate the cr-algebra

which is interpreted as the a-algebra of events which take place before the random time

T.

Let T ~ T~ , be two stopping times such that 0

5

5

T~

T~

5

a.s.

T

one can define the stochastic integral in the interval

[ T ~ , T ~ ] as,

where X,(t)

=

1

if

t < T ,

0

if

t

2

~

.

It is then possible to solve a stochastic differential equation like (1.17) with stochastic initial conditions, i.e., the pair (x,t) in (1.17) can be changed into a pair time with respect to 5 ,'

1.4

(S,T), where

T

is a stopping

and 5 is 5 ' measurable.

Connections with partial differential equations.

,

Let 8 be an open bounded subset of Rn, whose boundary class C

2

.

In

5

r

is of

functions aij(x), gi, a0 (x) are given such that

(1.18)

aij = aji '

aijSjci 2 a

>1 51 i

, v a>O

We denote by A the second order differential operator

PROBABILISTIC PROBLEMS AND METHODS

354

Let f be such that

We consider the Dirichlet problem AU = f in 8

,

o ,

uIr =

(1.21) u

6

W2'P(8) n

c l m ,

which has one and only one solution.

One can give an explicit

formula for u(x), in terms of a functional of the solution of a stochastic differential equation. One defines a

n by n matrix u(x) which is symmetric, positive

definite and Lipschitz such that a = o L / 2 .

An explicit formula for

u(x) is given by m

(1.22)

u(x) =

A-'"

(2a(x) + X)-l2adA

.

0

Given a system ( a , Q ,P, 5 t,w(t)), one can solve the stochastic differential equation

y(0) = x Let now (1.24)

T~

T~ =

which is an 5

.

be the exit time from 8 ,i.e., infItlyx(t) 9 8 1 stopping time.

, Then the following formula holds

STOCHASTIC DIFFERENTIAL EQUATIONS

355

Remark 1.4. In (1.23) , g, a must be defined on Rn. extending g, a , defined on

,

This can be done by

so that they remain globally bounded

and Lipschitz. Let us now turn to the evolution case.

Z

=

r

x

(1.26)

We set Q = 0

x

lO,T[,

]O,T[ and we consider the family A(t) of operators

A(t) = -aij(x,t)

a2 j

-

a .

gi(x,t) axi

We will assume that a . . is symmetric, strongly elliptic, and regular 17

enough.

At this level, we do not make precise the regularity

assumptions, since the weakest assumptions will be given later on. Let f be defined on

5,

regular enough and such that

Then one can find one and only one u (1.27)

-

at

+ A(t)u

One can also give

=

f in Q

,

E

C2”(Q), uIc = 0

u E C

,

an explicit formula for u(x,t).

0

;Ir

= 0.

(5) such

that

u(x,T) = If a,(x,t) is

again the square root of a(x,t), then one can solve the equation

(1.28)

y(t) = x

.

.

356

PROBABILISTIC PROBLEMS AND METHODS

We set

Then we have

Remark 1.5. One can consider a zero order term in A(t), and modify the relation (1.30) along the lines of (1.25). From the explicit formulas (1.25), (1.30) and standard estimates in P. D. E., one obtains the following probabilistic estimates,

'

0

Txt* (1.32)

IE

[

JI(Yxt s),s)ds

where C does not depend on JI.

5C

The estimates (1.31), (1.32) prove

that the mathematical expectations on the left hand side of (1.31), (1.32) make sense for functions @ not necessarily continuous, but in Lp, with p large enough.

This permits to give an integrated form of

Ito's formula, which holds true for functions $(x,t) which are not necessarily in C2''(8),

with Q = Rn

x

]

O,T[.

Indeed let JI be such

that

Let 0 5 8

5

8'

5

T, be two stopping tines and let

from a regular bounded domain 0 ; then one has

T

be the exit time

357

MARTINGALE FORMULATION

The result (1.34) is very useful in giving a probabilistic interpretation of the solution of elliptic or parabolic equations, with the weakest possible assumptions.

We will do this in the next

paragraph.

2. 2.1

Martingale formulation of stochastic differential equations. Martingale problem. We suppose that two functions

are given.

We assume that

(2.2)

, a are Borel measurable and bounded

g

;

a is symmetric non-negative definite

.

We now set (2.3)

R = C([O,m[;Rn) ;

it is a Frechet space for the topology of uniform convergence on compact subsets of [ 0 , m [

(2.4)

5

S

=

,

u algebra generated by w ( A ) , A

E

[t,sl

, 5

m

=zt .

One can identify 3; with the Borel o-algebra on n. The family (a, z t , 5;)

will be called the canonical space.

canonical process is then defined as follows

The

358

PROBABILISTIC PROBLEMS AND METHODS

We will say that a probability measure P E Pxt on ( Q ,5 t) is solution of the martingale problem, starting from (x,t) if

In ( 2 . 7 ) we have set

where a = (a.. ) . 13

Remark 2 . 1 . Many other formulations of the martingale problem are possible. The one we have chosen is the most convenient for our purpose. The most important result which we will use about the martingale If besides ( 2 . 2 ) we assume that

problem is the following. a is continuous

(2.9)

I

a-1 exists and it is continuous

and bounded

,

then there exists one and only one solution of the martingale problem.

Remark 2 . 2 . The existence of a-l is instrumental for obtaining the uniqueness of the solution of the martingale problem. (2.2)

(2.10)

and a continuous

I

g continuous

If we have only

359

MARTINGALE FORMULATION then there exists a solution of the martingale problem, but this solution is not necessarily unique.

The invertibility of a is also

useful in removing the assumption of continuity of g.

2.2

Weak formulation of stochastic differential equations. Let us assume now that there exists a factorization of the

matrix a such that

u E

&(R";R"

)

,

We consider a set 8 = ( R o t set, tj t, TI

s

measurable and bounded

s

.

s E,w(s),y(s), v ) , where R o s t = s :2 if s1 I S2' s t

tl

are o-algebras on R o .

is a

s1

m

=s,t

is a probability measure on (no,qt), w(s) is a Wiener process with

respect to

s :,

adapted to Q (2.12)

with values in Rn, and y(s) is a continuous process,

:,

such that

y(s) = x

,

Y(S) = x +

I

t

g(y(X)iA)dX +

I

t

o(y(X)tX)dw(A)

*

A set 6 satisfying all the above conditions is called a weak solution of the stochastic differential equation (2.121, wit; initial

.

conditions (x,t)

Remark 2.3. If one can choose arbitrarily the subset ( R o , Q then 8 reduces to y(s), in which case we recover the concept of strong solution. As far as the uniqueness of weak solutions is concerned, two types of uniqueness are considered.

PROBABILISTIC PROBLEMS AND METHODS

360

Pointwise uniqueness. There is pointwise-uniquenessof the weak solutions of (2.12) if for any two weak solutions 8,,

8

such that

then one has

Uniqueness in law. There is uniqueness in law, if for any weak solutions

then y1 and y2 have the same probability law (i.e., define the same image probability on C([O,-[;Rn)). The relation between the martingale formulation and the concept

of weak solutions is given by the following statement. If (2.2) & (2.11) hold true, then the martingale problem has

one and only one solution, if and only if there is existence and uniqueness in law of the weak solutions of (2.12). to know that under (2.2)

-

and

(2.11),

It is important

then

pointwise uniqueness implies uniqueness in law

;

existence of a weak solution and pointwise uniqueness imply existence of a strong solution

.

361

MARTINGALE FORMULATION 2.3

Connections with P. D. E. We can now give the probabilistic interpretations of weak solu-

tions of P. D. E. in terms of functionals of weak solutions of stochastic differential equations or solutions of the martingale problem. Let 0 be an open bounded subset of Rn, whose boundary class C

(2.13)

2

.

r

is of

Let g and u be such that g:

Rn * Rn continuous and bounded

u:

R"

+.

L(R";R")

I

,

u continuously differentiable and bounded

Let also a . be given satisfying (2.14) For f

E

(2.15)

,

ao(x) 2 0

continuous and bounded

.

Lp( 8), p > 1, there exists one and only one solution of 2 Laij(x) axiax j u

E

W2'P(0)

-

g. 1 axi

+

a0u = f

I

ulr = 0

,

.

Now under the assumption (2.13), we know that there fxists one and only one solution of the martingale problem ( 2 . 6 ) , P

(2.71,

Pxo (starting from 0)I or one and only one (in law) weak solution

of the stochastic differential equation

y(0) = x

.

c

tActually only the values on 8 will play a role.

362

PROBABILISTIC PROBLEMS AND METHODS

is continuous and is given explicitly by Then for p > 11, 2 u(x)

A similar result holds true for parabolic equations.

(2.18)

g:

Rn

x

]O,T[

U:

Rn

x

]O,T[

+

-f

Rn

,

is continuous bounded

d(Rn;Rn)

u is continuous on R"

a = - uu* -->BI, 2 -

We assume that

x

,

, [O,T] and bounded

,

B > O .

Let also a. be given satisfying (2.19) For f

E

a (x,t) continuous and bounded 0

Lp(Q), Q = 0

X

.

lO,T[, p 1. 1, and 6

E

W2"(0)

n Wi"(0)

there exists one and only one solution of (2.20)

au - at uIc = 0

2 a u aij(x,t) - gi(x,t) au axiax j

3+

I

u(x,T) =

u

a0 (x,t)u = f

,

,

Under the assumption (2.18) we also know that there exists one and only one solution of the martingale problem (2.61, (2.71, P E pxt (starting from x at time t), or one and only one (in law)

weak solution of the stochastic differential equation

363

MARTINGALE FORMULATION

Then, for p >

5 + 1, u(x,t) is continuous and is given explicitly by AT

(2.22)

f(y(s),s)

u(x,t) =

ds

1.

+ E(Y(T))X~<~

Remark 2.4.

We will be interested particularly in P. D. E. written in variational form (of Chapters I and II), i.e. instead of (2.15), or (2.20) we will have

(2.24)

-

au

a q au aij(x,t) -= ax j

au a.(x,t) axi + a0u = f

.

We will then assume that besides (2.131, or (2.18) that

In that case the forms (2.23), (2.24) are equivalent to (2.15), (2.20) provided that we set gi = ai

+

a ax aji j

*

PROBABILISTIC PROBLEMS AND METHODS

364

3.

Some results from ergodic theory.

3.1

General results. Let

x

E

(S,C)

be a probability space, on which a family P(x,E),

S, E E C , of probability measures is given.

(3.1)

x

-+

P(x,E) is measurable

,

t/

E E

C

We assume that

.

An invariant (probability) measure n(E) is a probability on

(S,C),

such that

To P(x,E) corresponds a linear bounded operator on B(S) (set of bounded measurable functions on sup norm.

S),

which is a Banach space for the

This operator, denoted by P, is defined by

Clearly (3.4)

IIP 1 '

*

Of course, we have PXE(x) = P(x,E)

,

where XE is the characteristic function of E. One of the main objectives of ergodic theory is to study the limit of the sequence Pn as n * +*.

We will give only the result

which we are going to use in homogenization. (3.5)

S is a metric compact space

and

C

is the Bore1

We assume that

,

o-algebra

,

SOME RESULTS FROM ERGODIC THEORY

(3.6)

there exists a probability measure

(3.7)

p(x,y) :

(3.8)

there exists a ball Uo such that and

s

x

s

(S,C)

such that

R+ is continuous

+

p(x,y) > 0

on

365

,

x E

s ,

u(u0) y

H

E

> 0

uo

.

Since S is compact we can replace (3.8) without loss of generality by there exists a ball U o such that u ( U o ) > 0

(3.8)'

and a positive number 6 > 0 (depending on U o ) such that P(X,Y) 2 6 ,

x

E

s

'Y

r

E

uo

-

The next result is basic for the following. found in Doob 111.

We give

Its proof can be

it for the convenience of the reader.

Theorem 3.1.

where

0, K > 0 are independent of $.

p >

Proof of Theorem 3.1. If there exists a probability n for which (3.9) holds true, then n is necessarily invariant.

Indeed taking in (3.9) $

state that n(E) = lim P%x,(x) n+m

,

'd

x

.

= x E rwe

can

PROBABILISTIC PROBLEMS AND METHODS

366

Obviously also a ( ~ = ) lim P"PX,(X)

n+m

.

From ( 3 . 9 ) again with @ = PXE, we have

=

I

a(dx)P(x,E)

.

Hence

which proves that n is invariant. Let us prove that the invariant measure is unique. %(ax) is an invariant probability measure, then (3.10)

I

%(dx)P$(x) =

I

%(dx)

1

P(x,dz)$(z)

We can use Fubini theorem, since ?(dx)P(x,dz) = p(x,z)%(dx)p(dz) Therefore

Also

.

.

Indeed, if

SOME RESULTS FROM ERGODIC THEORY Since Pn$(x)

+.

i

367

$a(dx) (which is a constant) , and is bounded in the

sup norm, it follows from Lebesgue's theorem that

hence

TI =

?.

Let us prove now (3.9).

(3.11)

For any E

Mn(E) = sup PnxE(x) X

E C,

we set

.

For x, y fixed, we set

Now A is a measure on finite VE. So

(S,C)

(not necessarily positive), and A(E) is

Therefore by the Hahn decomposition theorem; there exists

such that A+(E) = A(ESO)

(where S o

(=

S

Moreover

-

S 1. 0

,

A-(E) = -A(ES;))

,

368 368

PROBABILISTIC PROBLEMS PROBLEMS AND AND METHODS METHODS PROBABILISTIC

Now we we have have Now X(S0) = Pxs (XI -- P% 0

0

1

-

Px

< 1

-

Pxs,u (XI

=

s;,

(x)

-

0 0

(Y) P%

-

0

(y)

Pxs 0 0

Therefore, collecting results we obtain (3.14)

X(So)

5 1 -

6p(Uo)

.

(Y)

.

369

SOME RESULTS FROM ERGODIC THEORY

T h ee rreeffoorree w e h ave have ( 3 . 1 55))

Mn(E)

-

mn(E)

5

(1 - d v ( U o ) ) n - l

,

n

1. 1

.

From From ( 3 . 1 22)) and ( 3 . 11 55)) i t f o l l o w s t h a t

Therefore we obtain IPnxE(x)

(3.17)

Clearly

TI

-

a(E)I

5

Mn(E)

<

Cl(l

-

-

mn(E)

c,)

n

c2

I

i s a p r o b a b i l i t y measure on ( S , Z ) .

-< where where P" P"

-

TI, TI,

IIP"

-

E

10,1[

.

F o r @ E B ( S ) , w e have,

~lIlI@lI

hheerr ee ddee nnoott ee ss tt hh ee m e a s u r e PnxE measure PnxE

-

aa (( E E))II EE EE CC ..

But But

370

PROBABILISTIC PROBLEMS AND METHODS

Hence from (3.18) and the above estimates, we obtain (3.91, with K=2c

3.2

2

=

1

-

1 Sll(U0)

-

Ergodic properties of diffusions on the torus. Let us consider functions aij (x), bi(x) , i,j = 1,.

. .,n on Rn

such that (3.19)

E a ij

bi (3.20)

E

C

1 (Rn) with holderian derivatives

1 n C (R )

,

with holderian derivatives

aij = aji '

aijSiSj B ' O ,

(3.21)

aij

,

bi

,

8151

2

,

,

5ERn,

are periodic in all variables with period 1t

.

We define u(x) such that

. (3.22)

a = Lf2 .

By virtue of the regularity assumptions, u and b are lipschitz functions, and bounded.

Therefore we car. solve on an arbitrary

probability space and for an arbitrarlr Wiener process, the stochastic differential equation

'We

will consider homogenization only in the case when the

period is 1. This is only for simplicity, the general case is obtained by an easy extension.

With the notation of preceding

chapters, Y = 10,1[", (cf. Section 1.1).

SOME RESULTS FROM ERGODIC THEORY (3.23)

dy = b(y)dt

+ a(y)dw(t) ,

y(0) = x

371

.

We recall that Y = 10,1[", and we denote by ? the torus obtained by identifying the opposite faces of Y.

?

=

Rn/Zn

We can also write

.

? is a compact metric space. The Borel a-algebra on

Then

can be

identified with the sub a-algebra of periodic Borel sets of Rn (i.e., whose characteristic function is periodic), by the formula E =

kl.

U

..kn€Z

6 +

kiei

is a Borel subset of ?, and E is a periodic Borel subset of

where

Rn (elr...,e n are the unit coordinate vectors). (3.24)

A =

-

a a' ij X Q T j

-

bi

Let us set

a

axi .

We will use the fact (cf., A Friedman [ 2 1 for instance) that there exists a unique Green's function p(x,t,y) from Rn x

Rn

+

X

]O,m[

R such that p(xrtry) > 0

,

p is continuous on

(3.25)

{t > 0,x E Rn,y

pisc'inx,

c1

E

Rn}

,

int,

as a function of x,t

,

p satisfies the relations

g + A p = O

(3.26)

v

x

r

1

p(x,t,y)f(y)dy

continuous and boundedt

'The

* f(x) as t *

0

.

periodicity does not play any role here.

,

V f

PROBABILISTIC PROBLEMS AND METHODS

372

Moreover we have the representation formula Ef(yx(t)) =

(3.27)

1

t

f(Y)P(x,ttY)dY

I

Rn f measurable and bounded

.

Suppose now that f(y) is periodic (i.e., f is a bounded Bore1 function on the torus); then we have

A point in the torus will be represented by

representative of the equivalence class

4, where

x E Rn is a

4.

Since b, o are periodic

hence yx(t) can be identified with a process on the torus denoted by ix(t); moreover the function p(x,t,y) is periodic in x, hence depends only on x.

We can thus rewrite ( 3 . 2 7 ) as follows (for f periodic)

where (3.29)

po(x,t,y) =

tt kl.. .knEZ

an& d$ is the Lebesgue measure on the torus (which is a probability on the torus).

tThe function u(x,t) = E + A u = 0, U(X,O) = f(x).

I

f(y)p(x,t,y)dy is solution of

'+Note that the right hand side of ( 3 . 2 9 ) is a function of x, y convergent by virtue of the exponential decay of p(x,t,y) in y as

IYI

-+

373

SOME RESULTS FROM ERGODIC THEORY

As a consequence of the general results of the preceding paragraph, we can state the Theorem 3.2. Under the assumptions (3.19) , (3.20) , (3.21), there exists one and only one invariant probability measure on the torus ?, denoted by T,

such that for any f periodic bounded Borel function on Rn

f is Borel bounded on (3.30)

?),

suplEf(yx(t)) X

(&,

one has

-

I

fa(d$) I

5

KI

If1

I

Y

where K, p are positive constants depending only on t3 and the bounds

of a . . 11 '

bi.

Proof. Let us define on Y

..

P(x,E) =

I-

the family of probability measures

pO(4,1,$)X.(y)dy E

Y

,

i.e. (3.31)

P(;,d$)

=

po(G,l,$)d$

;

then (3.5), (3.6), (3.7) are satisfied with

..

P(X,Y) = Po(Ll,$) p(d$) = d$

I

.

Moreover ( 3 . 8 ) is also sati Ei 3 , since for ball Uo of

6 > 0,

x

?, whose closure is contained in E Y,

and y E U , , ,

stance for any

Y, then p(x,l,y)

and from (3.29) I

p(&,l;$) 2 6 > 0

j

,

E Y

,

$

E

uo

.

2 6

> 0,

314

PROBABILISTIC PROBLEMS AND METHODS

The constant 6 depends only on B and on the bounds of aijf bi.

Now

we have

which follows easily from the Chapman-Kolmogorov property of p(x,t,y) , i.e.,

I

p(x,t+s,y) =

p(x,t,z)p(z,s,y)dz

.

Therefore applying Theorem 3.1, we can state that

and there exists one and only one invariant measure on the torus, such that (3.32) holds true. Now for t

n, we have

Ef(yx(t)) =

I

p(x,t-n,z)Ef(y,(n))dz.

therefore

Applying (3.33) with t and n = [tl and

possibly

modifying the

constant K, we complete the proof of the desired result.

Remark 3.1. The measure

TI (a;)

is invariant in the sense of (3.2) , i.e.

SOME RESULTS FROM ERGODIC THEORY

375

Also

As a consequence of the Chapman Kolmogorov property, stated in the preceding theorem, we have

hence it easily follows from ( 3 . 3 5 ) that

at least for t

1.

Actually ( 3 . 3 6 ) is true for any t > 0, since one

can take in the proof of Theorem 3 . 2 ,

and necessarily obtain the same invariant measure.

Theorem 3 . 3 . The assumptions are those of Theorem 3 . 2 .

Let $(x) be a Bore1

periodic bounded function such that (3.37)

J

$a(d$) =

o

.

I

Y

Then there exists one and only one (up to a constant) solution of

376

PROBABILISTIC PROBLEKS AND METHODS

(3.38)

,

.

p <

m

,

Without assuming (3.371, but only that $

E

Lm(Rn), one can solve

Z

E

p 2 1

W2"'p(Rn)t'l

z periodic

Proof.

for a > 0, the equation

(3.39) z

E

.

W2'pr'(Rn)

There exists one and only one solution of (3.39).

This can be proved

by using variational techniques and the iterative scheme

as in Bensoussan-Lions

[

11.

Details are omitted.

Furthermore za(x)

has a probabilistic representation, namely t))dt

.

Now if (3.37) holds true, then from (3.30) we have IE $(Y,(t))

'Sobolev

I

5

KI

161 I

.-Pt

space with weights; z E W2'p'p

az

a 2z

L

~

I

~

377

SOME RESULTS FROM ERGODIC THEORY which along with (3.41) easily leads to Iza(x) I S C , as a

-t

0.

This estimate and the equation (3.39) insures that za remains bounded in W2rprv(Rn), p 2 1, p < solution of (3.38).

41

= 0,

m.

One can then let a

-t

0 and obtain a

The uniqueness is obtained as follows.

Take

then by Ito's formula (see (1.34)) we have

Hence the limit z(x) is also

Now z (x) is periodic, for any a. periodic.

Applying Theorem 3.2 again, it follows that as t

-t

m,

Y

i.e., z is a constant.

3.3

This completes the proof of the theorem.

Invariant measure and the Fredholm alternative. We can

interpret

the results of the preceding paragraph

(especially Theorem 3.3) in terms of the Fredholm alternative. will also allow us to describe explicitly the unique invariant measure. We rewrite the operator A (given by (3.24)) as follows

The formal adjoint operator is (since aij are symmetric) (3.44)

A* =

a --

a

a

We recall (cf. Chapter I, (2.10), (2.14)), that

This

378

PROBABILISTIC PROBLEMS AND METHODS

Theorem 3.4. The assumptions are those of Theorem 3.2.

We consider the

homogeneous equations (3.45)

Az = 0 ,

(3.46)

A*m = 0

z

,

E

W2,ptp(Rn)

p 2 2

m E w~~P~'(R~)p

,

p > 0

2 2 ,

,

p > 0

z

periodic

, m periodic.

There exists one and only one solution (up to a multiplicative constant) of (3.45), (3.46) (actually z = 1). Let

@,

J, be Bore1

periodic bounded functions such that (3.47)

@(y)m(y)dy = 0

,

Y

then there exists one and only one solution of the inhomogeneous equations (3.49)

Azo = @

,

z

0

E

W2rp'p(Rn) zo

,

p 2 2

periodic

,

,

J

p > 0

,

zOdy = 0

I

Y (3.50)

A*mo = J,

,

m 0 E W2""(Rn)

,

p 2 2

mo periodic

,

,

p > 0

I

mOdy = 1

Y

,

,

.

SOME RESULTS FROM ERGODIC THEORY

379

Remark 3.2. Obviously 1 is solution of (3.45).

The normalization constant

of the solution of (3.46) is taken such that

Remark 3.3. Theorem 3.4 expresses the Fredholm alternative for the operator A, with periodic conditions.

Remark 3.4. Let f be Bore1 periodic bounded; we can consider the Cauchy problem (3.51)

at

+

Au = 0

,

u(x,O) = f(x)

,

which has one and only one solution which is bounded periodic, and 1 It is given for instance which is C2 in x, C in t in Rn x ] 0 , m [ . explicitly by

According to Theorem 3.2 we have (3.52)

u(x,t) * C (a constant) as t *

m

,

V x

.

Now let m be the solution of (3.46) satisfying the condition of Remark 3.2; we have from (3.51) and (3.46)

& hence

u(x,t)m(x)dx = 0

,

380

PROBABILISTIC PROBLEMS AND METHODS

Letting t

.+

+m,

it f o l l o w s from L e b e s g u e ' s theorem t h a t

However w e know a l r e a d y t h a t

Therefore

I n p a r t i c u l a r m(y)

2

I n f a c t m(y) > 0 ,

0.

v

y.

Indeed if f ( x ) i s

p o s i t i v e and # 0 on a s e t of p o s i t i v e Lebesgue measure, t h e n u ( x , t ) > 0, b' x ,

V t > 0.

Since f o r t p o s i t i v e u ( x , t ) i s continuous

and p e r i o d i c i n x , t h e n w e c a n s a y f o r i n s t a n c e t h a t u ( x , l )

v

For t

x.

L

2 6

> 0,

1 w e have by t h e maximum p r i n c i p l e

Therefore l e t t i n g t

.+

w e can assert t h a t C > 0.

+m,

Since t h i s is

t r u e f o r any f a s above, it f o l l o w s t h a t m(y) > 0 a . e .

Actually

m(y) c a n n o t be 0 i n s i d e Y , s i n c e 0 would be t h e minimum, c o n t r a d i c t i n g t h e maximum p r i n c i p l e .

I t c a n n o t be 0 on t h e boundary because o f t h e

p e r i o d i c i t y , hence m(y) > 0 , V y.

Proof of Theorem 3 . 4 . L e t A be l a r g e enough.

(3.53)

+

A Z ~

aZA

= g

,

Then w e c a n s o l v e t h e problem zA E W(Y)

.

SOME RESULTS FROM ERGODIC THEORY

381

For any g E L2 (Y), there exists one and only one solution of ( 3 . 5 3 ) . Thus we define an operator GA: L2(Y)

+.

,

L2(Y)

by setting

G g = z

x

A '

Since the injection of W(Y) into L2 (Y) is compact, the operator

GA

is compact. Let (3.54)

@ E

L2 (Y); the problem

,

Az = @

z E W(Y)

is equivalent to the problem (3.55)

(I - hGx)Z = Gx@

Similarly let J, (3.56)

E

A*m = J,

,

Z E

L2 (Y)

.

L2 (Y); the problem

,

m

W(Y)

E

,

is equivalent to the problem (3.57)

(I

-

AG;)m

=

GlJ, ,

m E W(Y)

,

.

where GI is the adjoint of Gx (in L2 (Y))

Since Gx is compact, the Fredholm alternative applies.

We must

find the number of linearly independent solutions of (I

-

AGA)Z = 0

,

i.e., (3.58)

Az = 0

,

z

E

W(Y)

.

Let us consider a solution of ( 3 . 5 8 ) . and (from Green's formula),

5

aij

By regularity, z

E

H2(Y)

is periodic in xi' This insures 2 that if one extends z by periodicity then Az E Lloc(Rn) and Az=O,

a.e.

in R"

.

PROBABILISTIC PROBLEMS AND METHODS

382

Furthermore z E W1f2f’(Rn),

0.

?.I>

But then we can consider

that z satisfies the equation Az

(3.59)

For A.

+

X 0z = X 0 z

,

E

w1t211J(Rn)

.

large enough, and for a right hand side which is given in

L2” (Rn), the equation (3.59) has one and only one solution in

W1f28’(Rn). E

This solution is actually in WZr2”(Rn).

w212,’ (R”)

.

Hence

Suppose for instance that n > 4, then z

E

Lqr’ (R”),

I, = 1. - 2, i.e., q

The solution of (3.59) is then in > 2. q 2 n W21q1p(Rn). Step by step, by a bootstrap argument which is standard, with

we see that actually z E W2rprp(Rn)lV p 2 2,

> 0.

)I.

Thus (3.58) is

equivalent to (3.60)

AZ = 0

,

z E W2’Pr’(Rn)

p

V

Suppose that z

!j

2

2

constant.

,

I

u

> 0

,

z

periodic

.

By the strong maximum principle, z

cannot reach its maximum or its minimum inside Y.

Suppose that it

reaches its maximum on the facet xi = 0 and in a point inside the facet. x

j

Then consider the function z on the cube

E 10,1[

-$

xi < I,

for j # 1; we see that it reaches its maximum inside the

cube (by the periodicity) contradicting the strong maximum principle. By similar reasonings we see that necessarily the solution of (3.60) is a constant. The Fredholm alternative implies that there exists one and only one solution of A*m = 0

,

m

E

W(Y)

,

I

mdy = 1

.

Y

By the same bootstrap argument as above, we complete the proof of the first part of the theorem (i.e., proving that (3.45), (3.46) have one and only one solution up to a multiplicative constant).

HOMOGENIZATION WITH A CONSTANT COEFF>ICIENTSLIMIT OPERATOR

383

The second part is a direct consequence of the Fredholm alternative, and the regularity argument.

Remark 3 . 5 . The fact that ( 3 . 4 5 ) has one solution and only one (up to multiplicative constant) has been proved also in Theorem 3 . 3 using a probabilistic argument (see 3 . 4 2 ) ) .

Since the solution of ( 3 . 4 5 )

is also a constant, the fact that the constant is multiplicative or additive does not matter.

Remark 3 . 6 . Suppose that (3.61)

aa bi -ij ax

.

j

Then A = A* (the operator is self-adjoint).

In that case m = 1, and

the invariant measure is the Lebesgue measure.

4.

Homogenization with a constant coefficients limit operator.

Orientation. In this paragraph we start the study of homogenization using probabilistic methods.

Throughout the paragraph the limit operator

will be an operator with constant coefficients. will be considered in the next paragraph.

The general case

384 4.1

PROBABILISTIC PROBLEMS AND METHODS Diffusion without drift.

We consider functions .a . ( x ) satisfying the conditions (3.191, i~ (3.20), (3.21), and defining u by (3.22), we introduce the stochastic differential equation (depending on a small parameter

E)

The solution of (4.1) is constructed on an arbitrary system (

n , u ,P,5 t,w(t))

.

The solution yE(t) is a continuous process with values in Rn. If one defines

then J, is a separable Frechet space for the topology of uniform convergence on compact sets of [ O , - [

(cf. (2.3)).

The Borel u-algebra

on J,, denoted by B is generated by the sets {y(t . ) 1 3

E

B

1'

where y(.)

denotes an element of P I and tj E R, Bj Borel subset of Rn. denote by p:

We

the probability measure on J, defined by the stochastic

.

process : y (t)

We want to study the behavior of p:,

as

E +

0.

We have

S

by standard properties of stochastic integrals, and since u is bounded.

From Prokhorov's compactness theorem (cf. Prokhorov [l])

it follows that (4.3)

V x

1

remains in a weakly compact set of

I

77( + ( $ I

=

set of probability measures on J,

We introduce the differential operator A defined by

.

HOMOGENIZATION WITH A CONSTANT COEFFICIENTS LIMIT OPERATOR

a2 A = -aij (x) ___ axiax j

(4.4)

385

-

According to Theorem 3.4, there exists one and only solution of (4.5)

A*m = 0

,

p 2 2

I

,

m E W2""(Rn)

,

LI > 0

,

m periodic

,

,

m(y)dy = 1

Y

and we know from Remark 3.4 that

We define

Clearly qij is symmetric positive definite matrix. the Gaussian measure on

)I

We denote by p X

defined by the process

Theorem 4.1. If the aij satisfy the conditions (3.19), (3.20), (S.21),

then

we have (4.9)

v

x

,

:p

-+

ux

1 weakly in R + ( @ )

.

Remark 4.1. In Section 4.3 we will see that the result (4.9) (and similar results later on) implies homogenization results for elliptic P.D.E. along the lines of the results of Chapter 1.

PROBABILISTIC PROBLEMS AND METHODS

386

The proof of Theorem 4.1 will rely on the following.

Lemma 4.1.

Let (4.10)

be Bore1 periodic bounded, such that

@

1

.

@(y)m(y)dy = 0

Y

Then for any s 5 t, we have

We are going to give two proofs of Lemma 4.1, using different aspects of ergodic theory. more regular $.

One of the proofs will be valid only for

t

First proof of Lemma 4.1. For this proof we require the additional assumption (4.12)

4 differentiable and

$

E

Lyoc(Rn)

,

p >

.

Since (4.10) is satisfied, we can assert, using Theorem 3.4, that there exists one and only one function such that (4.13)

Az = @

,

Z

E

WZrp"(Rn)

,

By virtue of (4.12), z is actually more regular, namely z E W:AE(Rn), p >

5.

In particular z E C2(Rn).

Therefore we can apply Ito's

formula, which yields

'In

general this will be enough for the applications.

HOMOGENIZATION WITH A CONSTANT COEFFICIENTS LIMIT OPERATOR

hence

Therefore

hence (4.11) follows.

Second proof of Lemma 4.1. We define the process (4.14)

1 qE(t) = F

hence from (4.1),

But, if we set

g

E

(E

qE (t)

2 t) is solution of the Ito equation

387

PROBABILISTIC PROBLEMS AND METHODS

300

then w'(t)

is also a Wiener process (although it is not a martingale

with respect to 5 t). (4.16)

dq'(t)

=

We can write

,

o(q'(t))dw'(t)

qE(0) =

X

.

It is easy to check that the conditions of application of Theorem 3.2 are realized, and therefore we canassert that (by virtue of (4.10)) suplE$(qt(t)) I

(4.17)

5

5

KI

$1

where qE(t) stands for the solution of (4.16) with initial condition

5

q;(O)

=

5.

The constants K, p do not depend on

E.

From (4.17) it follows in particular that

where yz(t) = y'(t)

(defined by (4.1)).

We now consider

If we set

then it follows from the Markov property of yz(t) that

[ [ 1I

E $

3.

=

HE(y'(X)

,p-X)

for p 2 X

.

HOMOGENIZATION WITH A CONSTANT COEFFICIENTS LIMIT OPERATOR

389

Using (4.18), we can state

hence again (4.11).

Remark 4.2. The second proof of Lemma 4.1 is due to Freidlin [ 11. We can now use the results of Lemma 4.1, to prove the following.

Lemma 4.2. The finite dimensional distributions of the process : y (t) (solution of 4.1) converge towards the finite dimensional distributions of the process yx(t) (solution of (4.8)).

Proof. We have to show that v k, V tl

,...,tk,

V A1

,...,Ak

(Ai

E

Rn)

then

+

exp i(A1

+

... +

xk)*x exp

[- 1

Since

showing (4.20) is equivalent to showing that

I

qAi*A min(ti,t.) j

3

.

PROBABILISTIC PROBLEMS AND METHODS

390

We can assume without loss of generality that

5 tl 5 t2

0

... 5 tk .

We can prove (4.21) by

induction on k.

Let us take k = 1, then we

have to prove that t.

More generally, we will prove the following result.

Let us assume

that h is given such that (4.23)

h is a Bore1 periodic bounded function from Rn

+

Rn

then we have (4.25)

iT

E exp i

If we prove (4.25 t = tl, h = o*hl.

6

=

1

,

then we can apply this result with s = 0,

Then

ua*hl*hl m(y)dy =

Y

I

2a(y)h1*X1 m(y)dy

Y

= 29X1'h1

hence (4.22) follows.

,

.

HOMOGENIZATION WITH A CONSTANT COEFFICIENTS LIMIT OPERATOR Let us prove (4.25).

391

We start with the identity

1

t lh/2($)dX13

s

1

= 1

.

Let us set

c

= exp 1 h(t -

-

S)

=

.

exp

x,

I

From Lemma 4.1, we can assert that

But

where

x,

(4.27),

(4.29)

is bounded, (0

5

x, 5

constant, since h is bounded).

From

(4.28) and Cauchy-Schwarz inequality, it follows that E[(YE

-

C)2/5 '1

0

+

,

a.s.

We can now complete the proof of (4.25), which in our notation, amounts to proving that (4.30) But

Hence

E[XE

-

61 3 '1

+

0

I

a.s.

PROBABILISTIC PROBLEMS AND METHODS

392

IEIXE

-

1

I 5

'1

s)11/2[E((-Y, + C ) ' 1 3

1

1 - -< c-[E(-YE

+

C)'1

3s]1/2 -+ 0 t

from ( 4 . 2 9 ) . W e t h e r e f o r e have proved

induction.

(4.22),

and w e c a n proceed w i t h t h e

W e have

I

0

* E tk = EZERE

with

rt.

+ ...

+

J

(A,

1

+ ~~+~)*o(<)dw(s)

0

RE = E exp i

xk+l'o (--Y E dwl

stk]

.

tk

IX

E

I

i s t h e modulus of t h e complex number X,

..

HOMOGENIZATION WITH A CONSTANT COEFFICIENTS LIMIT OPERATOR

393

Applying again (4.25), we can assert that

However from the induction hypothesis, we have k-1 EZE

i,j=l

Hence the result follows.

Proof of Theorem 4.1. E is This follows immediately from (4.3) and Lemma 4.2, since p X

the measure of

YE(.),

px is the measure of yx(.), p;

is weakly

compact, and the finite dimensional distributions of :y

(.)

converge

towards the finite dimensional distributions of yx(.) (cf. GikhmanSkorokhod

[

1I ) .

We can now give an extension of Theorem 4.1. We are going to consider the equation (4.31)

dy‘ = b(<)dt

+ u(c)dw(t) ,

~‘(0) = x

.

In (4.31), the functions b and u (derived from a) satisfy the conditions (3.191, (3.20), (3.21). We denote again by p z the probability measure of yi(.). b is bounded, we again have (4.3).

Since

39 4

P R O B A B I L I S T I C PROBLEMS AND METHODS

We still consider m as defined by

4.5), qij defined by

4.7)

and

6

(4.32)

b(y)m(y)dy

=

.

Y

We next introduce the process y(t) = yX t) solution of dy = 6dt

(4.33)

and let

ux

6 dw(t) ,

+

y 0) = x ,

be the corresponding Gaussian measure on $.

the following

Then we have

:

Theorem 4.2. Under the assumptions (3.191, (3.20), (3.21) , then we have (4.34)

v x ,

E

px

+

p x weakly in

R +1 ( @ )

.

Proof. We first extend Lemma 4.1, whose statement remains unchanged. Let us see what are the modifications in the two proofs which have been given.

In the first proof, which carries over easily, we still

consider the function defined by (4.13).

Applying Ito's formula, we

obtain c

S L

S

from which the desired result easily follows.

HOMOGENIZATION WITH A CONSTANT COEFFICIENTS LIMIT OPERATOR

395

In the second proof, we still define qE(t) by (4.14) and now nE(t) is solution of

We then introduce the operator (4.36)

'a AE = -aij ax.ax. - Ebi ~

1

(4.37)

(AE)* = A*

+

a

3

a (b.) axi 1

E

.

Let mE be defined analogously to m, i.e.

(4.38)

1

mE periodic mE(y)dy = 1

,

.

Y

Denoting by qE(t) the process solution of (4.35), with initial

5

data q E ( 0 ) = 6 , then we can replace

5

(4.39)

supIE$(n;(t))

-

I

4.17) by

$mE(y)dy

Y

Let us prove that (4.40)

mE

-t

m

in W2'pr'(Rn)

,

We first obtain a priori estimates. solution of

p

2

,

p > 0

,

weakly

We can consider mE as the

.

396 (4.41)

PROBABILISTIC PROBLEMS AND METHODS A*mE =

a (bimE) axi

-E

,

mE i n W(Y)

.

W e make t h e change o f f u n c t i o n s

.

(note t h a t m > 0 )

mE = mr?iE

Then w e have s i n c e A*m = 0 , A*mE =

-

axi

[a. .m 13

$1

- a axj

j

ahiE ( a . .m) 13 axi

Theref o r e (A*mE,hiE

=

=

I

aijm

I

From ( 4 . 4 2 )

aijm

ax. aiii' axi akE dx 3

=

-

E

a ax

( a . .m) 13

.@ 6'dx axi

J

+ z

J

Hence w e o b t a i n t h e i d e n t i t y

I

Y

ax a i i E axi afiE dx

Y

(4.42)

-

afiE afiE dx aijm a x . axi Y 3

.

2&

(aijm) (hiE)'dx

Y

I

(bimriiE)fiEdx

Y

it f o l l o w s t h a t

hence

S i n c e m i s p o s i t i v e c o n t i n u o u s and p e r i o d i c , i t i s bounded below by a p o s i t i v e c o n s t a n t , hence ( 4 . 4 3 ) i m p l i e s

HOMOGENIZATION WITH A CONSTANT COEFEICIENTS LIMIT OPERATOR

397

But

Hence

and since 6 ' 1. 0,

I

&iEdx = 1 and m is bounded below, it follows that

Therefore from ( 4 . 4 4 ) , we see that

hence

It follows from ( 4 . 4 5 ) that (at least for a subsequence) 1

15~ +. fi in H (Y) weakly (4.46)

fi

=

constant

,

2

in L (Y) strongly

.

Since

we see that fif (4.47)

mE

+.

+.

1 1 in H (Y) weakly.

m in H 1 (Y) weakly

.

It follows that

,

398

PROBABILISTIC PROBLEMS AND METHODS

The statement (4.40) follows from a bound in W2"' deduced from (4.7) and the equation (4.41).

which can be

Actually (4.47) is

sufficient to assert chat (4.48)

I

$mE(y)dy

+

Y

I

$m(y)dy

.

Y

Therefore from (4.39), and (4.48) we can deduce that

0 as E 0, and does not depend on t. We can then where h ( @ ) @ easily transform the end of the argument of the second proof of -+

Lemma 4.1.

-+

We have instead of (4.18), using (4.10) L

The end of the argument can be carried over without any difficulty. We next prove Lemma 4.2, where of course yE(t) now denotes the solution of (4.311, and yx(t) the solution of (4.33). We have to show that (analogue of (4.21))

-+

If we set

1

exp i(Al*Etl +

... +

x,-ljtk)

-

>: =r3

qxi.xjmin(ti,tj)

1.

HOMOGENIZATION WITH A CONSTANT COEFFICIENTS LIMIT OPERATOR

399

then since Lemma 4 . 1 holds, the proof of Lemma 4 . 2 shows immediately that

xE €YE

exp

-+

+

exp

i,j

...

+

i(i,*t;t,

+ ik*6tk) in L2 ( Q , Q ,P) ,

qXi*X min(ti,tj) j

.

Since the limits are constants and since lX,l (4.51).

= 1,

we can conclude

From the weak compactness and the convergence of the finite

dimensional distributions, we obtain ( 4 . 3 4 ) . Remark 4 . 3 . Theorems 4 . 2 (and of course Theorem 4 . 1 which is a particular case) are due to Freidlin

4.2

[

11.

Diffusion with unbounded drift. We now consider, instead of ( 4 . 3 1 ) , the equation

Again we assume that ( 3 . 1 9 1 , ( 3 . 2 0 ) , (3.21) hold true. We denote by contrary :1-1

to

the probability measure of YE(.).

This time,

the cases considered above, we are not guaranteed that

remains in a weakly compact subset of

1 m+($).

In general, this will

not be true unless we make an additional assumption on b, which we will state below. We introduce the differential operator A given by

(4.53)

A =

-

aL aij(x) ___ axiax. 7

-

bi(x)

a 5

and m will again be the unique solution of A*m = 0

,

m

E

w2,P,lJ(~"),

(4.54)

m periodic

,

I

P'2

m(y)dy = 1

.

8

l J ' 0 ,

400

PROBABILISTIC PROBLEMS AND METHODS

Before we state a theorem of convergence concerning p E 1 let us note that Lemma 4.1 is still valid mutatis mutandis.

We repeat it for

convenience (we will only emphasize the differences with the proof already given).

Lemma 4.1'. Let 4 be Bore1 periodic bounded, such that

1

(4.55)

$(y)m(y)dy = 0

.

Y

Then for any s - t, we have (4.56)

Proof. First proof. We assume (4.12) and apply Ito's formula to the function z solution of (4.57)

Az = C$

z periodic

I

I

zdy = 0

I

Y

and we know that there exists a solution which is C 2

.

We apply Ito's formula and we follow exactly the same calculation as in Lemma 4.1 (we have changed the definition of A in the adequate manner)

.

Hence (4.56) follows.

tThe property (4.56) does not depend on the fact that the measure p z remains or not in a weakly compact set of 1

n+($).

HOMOGENIZATION WITH A CONSTANT COEFFICIENTS LIMIT OPERATOR

401

Second proof. We again define q'(t)

by (4.14).

This time we get that q'(t)

is

solution of

We see that m has been defined in the right way to make the calculations of the second proof of Lemma 4.1 valid, without any change. We now make the assumption

1

(4.59)

b(y)m(y)dy = 0

.

Y

Appling Theorem 3.4, we can then assert that there exist functions xQ (x) such that (4.60)

AX'

= -b

x II

'X

Q'

periodic

E

W2'p'p(Rn)

,

,

p 2 2

xQ(y)dy = 0

,

U ' O ,

.

Y

We denote by

x

1 the vector (x ,...,Xn).

We define now the matrix q by (compare with (4.7))

Let then px be the Gaussian measure on $ defined by the process (4.62)

dy =

6 dw(t) ,

y(0) = x

.

Then we have the Theorem 4.3. Under the assumptions (3.19), (3.20), (3.21) (4.63)

V

x ,

1-1;

+

px weakly in

m +1( J I ) .

(4.59), we have

402

PROBABILISTIC PROBLEMS AND METHODS

Proof. We first prove that :1

nt($).

remains in a weakly compact subset of

For this we note that

x a (y) are

C'functions,

hence we can

apply Ito's formula, which gives using ( 4 . 6 0 )

Combining ( 4 . 6 4 ) and ( 4 . 5 2 ) to eliminate the b term, we obtain

To show that p:

remains in a weakly compact subset of 77( 1 +,we can

apply the following criterion (cf. Parthasarathy ill):

The second condition is trivially realized since y:(O) next

SUP t -t 1 2

+

Hence

= x.

We have

HOMOGENIZATION WITH A CONSTANT COEFFICIENTS LIMIT OPERATOR

403

But

and (4.66) follows immediately. We turn now to the convergence of the finite dimensional distributions.

We have to prove (4.20), where yE(t) is the process given

by (4.65) and where q is defined by (4.61). is clear that it is enough to prove that t.

Since

x

is bounded, it

PROBABILISTIC PROBLEMS AND METHODS

404

But from (4.1)', from the proof of Lemma 4.2, and by virtue of the choice of q, it follows that (4.68) holds true, which completes the proof of the theorem. We can give an extension of Theorem 4.3 similar to the one of Theorem 4.2 with respect to Theorem 4.1.

We only state it, the

proof being left to the reader. We consider the equation

We assume that (3.19), (3.201, (3.21) are satisfied and that c satisfies the same conditions as b. (4.54).

We again assume (4.59).

We denote by m the solution of

Define the

functions by (4.60),

q by (4.61) and

We consider the Gaussian process yx(t) solution of (4.71)

dy = cdt +

and we denote by :p

a 6 dw(t) and

vx

,

y(0) = x

the measures of (:y

-)

, and yx(*). We have

Theorem 4.4. Under the assumptionst (3.19), (3.20), (3.21) , c satisfying the same assumptions as b,

t

In particular aij

then

(4.59)

6

w2tm( R")

.

HOMOGENIZATION WITH A CONSTANT COEFFICIENTS LIMIT OPERATOR

405

Remark 4.4. In all the above Theorems 4.1 through 4.4, one cannot hope for a stronger convergence such as (4.73)

yt

,

y:(t)

-f

y,(t)

in

L*(Q,Q ,P;R")

.

Indeed, let us consider for example yE defined by (4.1).

If (4.73)

was correct then we would have

But

which is different from 0, since$// ady # Y

4.3

I

&dy.

Convergence of functionals and probabilistic proof of homogenization. I

be an open bounded subset of Rn, whose boundary is

Let 0 denoted by

r

;

8 will be supposed regular in the following sense:

Defining aij, bi, ci as in Theorem 4 . 4 ,

and

PROBABILISTIC PROBLEMS AND METHODS

406 (4.75)

a . continuous bounded

periodic, a.

a.

> 0

.

We may consider the solution uE(x) of the Dirichlet problem

(4.76)

a2u E-aij(E) axiax j

x

au axi

1. b. (X) 2 E

1 E

c. (-)x 1

E

a + x i

X a (-)uE = f O E

,

where (4.77)

.

f E CO(%

We know (cf. Section 1.4) t st t ere ex ats one and only one solution of (4.76) in W2"(8)r The homogenized (4.78)

a =-

qij

v P 2 2. operator is defined by the formula

aL axiax

-i a +ao axi

where

We also consider the solution u of the homogenized equation (4.80)

a u = f

I

ulr=o.

We can state the following Theorem 4.5. Under the assumptions of Theorem 4.4, (4.77), we have

and

(4.74), (4.75),

HOMOGENIZATION WITH

A

CONSTANT COEFFICIENTS LIMIT OPERATOR

407

Remark 4.5. We can relate the result of Theorem 4 . 5 , with the homogenization results obtained in Chapter I.

Let us take

Then (4.76) can be rewritten as

which is the problem considered in Chapter I (of course here we require much more regularity on the data than in Chapter I).

If we

turn to the operator A defined by (4.53), we see that

a aij(x) A = - -axi

a = ax

A*

j

so that m ( x )

The equations for the

1.

'x

functions (see (4.60))

reduce to the ones obtained in Chapter I (cf. (2.17)), and we recover the usual homogenization formulas.

One advantage of the probabilistic

method is to obtain a pointwise convergence result.

A

very serious

drawback is the need of stringent regularity assumptions.

Remark 4.6. is not really a loss of The assumption of symmetry a . = a ij ji generality, since (4.76) can always be rewritten with symmetric coefficients.

This is not true for the divergence form.

Proof of Theorem 4 . 5 . We consider the process yE(t), defined by (4.69) and the process

1 I

(4.83)

EE(t) =

0

ao[*]ds

.

408

PROBABILISTIC PROBLEMS AND METHODS

From Lemma (4.1)' we can assert that

We denote by

the exit time of yE(t) from 8

T:

.

Then we can write

an explicit formula for us (x) (see (1.25)), namely ,,.E

It will be convenient to give another formulation of the right hand side of (4.85).

We introduce

C([O,m[;Rn)

=

An element of ?, namely

We provide

x

.

C([O,m[;R)

5, will

be a pair

-.

with the Borel o-algebra R

sets {y(tj),<(t.)3 7

process YE(.), SE

It is generated by the

Borel sets of Rn+l. The j' tj R, j induces on ? a probability measure denoted by

B

E

( a )

ii;. Similarly, we denote by Jx the measure induced by yx(*), S ( . ) where yx(t) is defined by (4.71) and (4.86)

of

.

c(t) = sot

The family IJ; (for fixed x) remains in a weakly compact subset 1 To prove it, it is enough to modify slightly the proof of

n+(+).

Theorem 4.3.

One has then to prove that

c

sup lim sup P 6+0 E + O Oztl
for any

> 0,

T <

m.

-

Since

(Iy:(t2)

-

y:(tl)I

+ 1SE(t2)

-

cE(tl)I)

HOMOGENIZATION WITH A CONSTANT COEFFICIENTS LIMIT OPERATOR

409

this can be done easily by extending the argument of Theorem 4.3. The next thing to prove is that the finite dimensional distributions of (y:

( )

,SE (

-

) )

converge towards the corresponding finite dimen-

sional distributions of (yx(.),5(*)). This is also easy to obtain as in Theorem 4.3, taking into account the property (4.84). We now define on (4.87)

F(;)

=

1

(

y

,5

a functional F ( $ ) by

)

(exp -S(t))f(y(t))dt

, if S(t) 1. a0t ,

V t

,

0

+m

otherwise ,

where we have set if $ = (y(*),S(-)) (4.88)

~(y= )

T =

inf{tlt 2 0, y(t) 9 8 1

.

Then we can assert the following properties (4.89)

F is

measurable

,

a.s. with respect to (4.90)

it is bounded

cz

-

and Cx '

F is continuous a.s. with respect to C z and

The property (4.89) is clear, since a

2

a.

> 0.

Ex

.

The property (4.90)

is a standard result for diffusions, and it is where the specific assumptions on the domain (4.74) are used (cf. Lemma 4.3 below). can then rewrite uE(x) as follows

Y

and s milarly (4.92

u(x) =

i

?

F

.

We

PROBABILISTIC PROBLEMS AND METHODS

410

We can now rely on a theorem of Gikhman-Skorokhod [ 1 1 , to assert that since

ci

converges weakly towards

c,

and since F is

Ex

a.s.

continous and bounded, then

i.e., (5.81)

Lemma 4 . 3 . The functional F

%

;,a.s.

continuous.

Proof. Let

be the complement of 8 and

ir

0

C

be the interior of C.

Let

us set

(Y) = infIt 2 r(y)

(4.93)

IY

t)

E

.

;I

We shall prove that A(y) = r(y)

(4.94)

,

.-.

!-I,

If ( 4 . 9 4 ) holds true, then y (4.95)

yn

-+

r(y) =

r(y) is continuous.

-+

y in C([Olm[;Rn)

To prove ( 4 . 9 5 ) , a)

a.s.

,

then ~(y,)

-f

Indeed if

.

~(y)

let us consider three cases.

+m.

Then y(t)

8

I

V t

Let T be an arbitrary number; the set

2 0.

KT = Iy(t),t E [ O , T I } is compact and contained i n 8

if 5 E KT and

-

.

5 - 51 < dT, then

Hence there ex sts aT > 0 such that E 8

.

For n 2 NT, one has

HOMOGENIZATION WITH A CONSTANT COEFFICIENTS LIMIT OPERATOR lyn(t)

-

y(t)

1

< dTf

v t

E [O,Tl, hence yn(t) E 8

,

411

V t E [O,T],

which implies n 2 NT

-f

.

~(y,) 2 T

Since T is arbitrary, we have proved that r(yn)

By ( 4 . 9 4 ) we also have A(y) = 0. that 0 < t6

0

+

yn(t6)

E

t

+-.

Therefore V 6 > 0, 3t6 such

But then 3N6 such that

5 6 and y(t6) E X. 2 Ng

+

+

r(yn) 5 A(yn)

5

6

.

Therefore ~(y,) * 0.

Let 0 < 6 < ~ ( y and ) K 6 = {y(t),t E [O,r(y)-6]} which is a compact subset of 8. Let d6 be such that

There exists N6 such that for n 2 N6, then Iyn(t)

-

5 dg

v

t

E

Hence for t E [O,r(y)-6] one has yn(t)

E

(4.96)

n 2 Ng

+

y(t)I

I

T(Yn) 2 T(Y)

-

6

[o,T(Y)l

8

*

, which implies

-

On the other hand since

0

[r(y),b+~(y)] such that y(t6) E X. 0 1 1 Also, there exists N6 such that n 2 N6 -+ yn(t6) E Z, hence

there exists t6

E

PROBABILISTIC PROBLEMS AND METHODS

412

and (4.96), (4.97) prove that T(yn) It remains to prove (4.94).

iix a.s. ,

y(t) = x

T(y).

This will follow from the fact that

+ JT fi w(t) Here the special assumptions on

and properties of Brownian motion. 0

-+

will be explicitly used. We will follow ideas of Stroock-Varadhan

[

2I.

Actually, we

shall prove that

Let 6 o < 6 (where 6 is defined in (4.74)).

which makes sense since (since Y(T) E 3 0 (4.100)

T'

= T

)

. T

A

Let us set

2 (y(t)) is continuous and

1 1 %ax

(Y(T))

II 2

6

Let us set

,

I \ ' = ~ A T A& T ,

0

which are stopping times with respect to 5 t+o.

Let us set for X > 0

where

XS(t) =

If

B #

0, then

T'

t

l i f t c s , 0 otherwise

.

< A ' and

= T,

implies that y(t) E

5 ,

T'

hence [ T ' , A ' ]

,

t E [T',A']

C

and therefore IB(t) I

w(t) is a 3 t+o martingale, we have (4.101)

E !.xe

B(t)*w(t)

-

] 0

[T,A],

lB(t)12d.]

=

1

.

which

5

C.

Since

HOMOGENIZATION WITH A CONSTANT COEFFICIENTS LIMIT OPERATOR

413

From Ito's formula, one has

If A ' > (4.103)

-

one has since y(t) E 8 , for t E

TI,

X

j'

5 XM(A' -

6 $$ql/*dw(t)

[T',A']

T')

TI

taking into account that Again if A ' >

T',

T'

= T,

and

$(Y(T'))

one has

and therefore

Therefore we can assert that

Clearly (4.104) is also true if A ' =

T'.

It follows from (4.102) and (4.104) that (4.105)

E exp(XM

-

2M160) ( A '

-

T')

1

.

= 0,

and $(y(A')) 5 0.

414

PROBABILISTIC PROBLEMS AND METHODS

If Q o = { w l h ' >

-

p(R

Letting h

-+

+m,

P(R

then (4.105) can be rewritten as

TI)},

+

0,)

f

- exp(A'

-

T')

(AM - A2M16,)dP(u)

2 1

.

we obtain

- no)

=

1

-

which proves that T' = A ' , p x a.s. But

> T

T~

0

2

T,,

T, hence also

i.e. (4.98) and the proof of the Lemma is complete.

5.

Analytic approach to the problem (4.76). The problem (4.76) has not been considered in Chapter I (aside

the case when bi = aa. ./ax.). 13

7

We will show in this paragraph that

the method of asymptotic expansions as well as the energy method can be carried over to this case, with some adequate changes.

5.1

The method of asymptotic expansions. We write (4.76) as AEuE = f

,

UElr

= 0

.

We set (as in (2.2), (2.3) of Chapter 1) (5.1) where

A€ = E - 2 ~ 1

+ 8 - l ~+ ~E 0

,

415

ANALYTIC APPROACH TO THE PROBLEM

(5.2)

a2 j

ci(Y)

+

(XI--)+

A3 = -aij (y) axiax

a +

a0 .

We define (5.3)

3 (x) = u (X,X) O

E

EU

X

1

E

E

2

X

U2(X'F)

.

The functions uol ulI u2 are chosen to satisfy if possible the conditions (5.4)

AluO = 0

(5.5)

A 1u 1

+ A2U0

= 0

(5.6)

A1u2

+

+ A3U0

I

A2ul

From Theorem 3.4 (5.7)

uo(x,y)

5

I

= f

.

t we can deduce that

u(x)

.

Then (5.5) becomes AIUl

(5.8)

-

au bi(Y) axi = 0

.

Since b. (y) satisfies the solvability condition (

I

bi(y)m(y)dy = 0) ,

Y and recalling the definition of the functions

(cf. (4.60)) I we can

assert that

'A1

is the operator denoted by A in Theorem 3.4.

PROBABILISTIC PROBLEMS AND METHODS

416

We now turn to (5.6), which yields

-

c.

:zi

-+

a0u = f

.

Before we write the solvability condition for (5.101, we notice that we can rewrite (5.10) as follows

au

Writing the solvability condition (3.47) we obtain

where we have set

(5.14)

ci

=

\

m(y) [ci

Y

-

c

d]dy aYk

.

Clearly (5.14) and (4.70) are identical.

We need to show that (5.13)

and (4.61) are identical, as it should be (although it is not completely obvious). We use the equation for ,j which is

ANALYTIC APPROACH TO THE PROBLEM 2 J

- a x kg aYkaYR Multiplying by

,xi

417

j

b

&=-b k aYk

j.

and integrating over Y, we obtain

=IxJ-

aYkaYQ 'a

Y

(akgxim)dy -

I

Y

XI

a ak'

(bkXim)dY

and by virtue of the equation satisfied by m (5.15)

I

Y

bjXimdy =

I

Y

-

2 i

xJakam

I xJ

Y

$-&

dy

+

2

I

Y i ?Lbkmdy ak'

XJ

i

& (akRm)dy yk

.

But

By addition of (5.15) and (5.16) we obtain

Using the expression in (5.13) we obtain

and it is easy then to check that this last expression is the same as (4.70).

418

PROBABILISTIC PROBLEMS AND METHODS W e t u r n back t o t h e s o l u t i o n of

(5.11).

We take

Gl

= 0 , and w e

can t a k e

where

xi j ,

$i,

xo

a r e d e f i n e d by

xij periodic

,

&] + ci aYk

(5.20)

Alqi

= -[ci

-

(5.21)

Alx0

= -(ao

- a,)

c

x0

,

q1 p e r i o d i c ,

,

periodic

.

With t h e p r e c e d i n g c h o i c e , w e c a n check t h a t (5.22)

AEGE = f = f

+

EA2u2

+ EgE

+

+

EA3u1

E

2

A3U2

,

where (5.23)

-

g E - -a

-

+

i j

3 a u ax.axkaxll 3

a kt ayi X kt

biX

a3u axiaxkaxll

a RsaxiaxR a2 u

a i j ay

-

+

j

0

-

aij

-

a3u a x i a x . axR 3 kR

EaijX

j X

aij

a

x

ki

a 3u

ki

axiaXkaxll

a2u

+

a+‘

a

i j ayi

aXkaXR

a2u ax

2 R a u biJ’

9. u a ~ i +a ci~ X ayi ~ aaxR

0

- au xo bi 5 5L ayj

j

aij x

-

ci

au 3 . L - a au ZT ayi ij

a +

a

2

au

+

4

a u axiax. aXkaXR 3

- “ix

-

kR

a 3u

axiaxkaxa

0

C,U 1

R

?x-

aYi

4 19

ANALYTIC APPROACH TO THE PROBLEM

I f f i s i n W3'p(

8 ), p > n , t h e n t h e d e r i v a t i v e s o f u ( s o l u t i o n

of ( 6 . 1 2 ) w i t h u l r = 0 ) up t o t h e o r d e r 4 w i l l be bounded, and t h u s w e may a s s e r t t h a t

Therefore i f we set 2

E

= u

E

-ii

E

w e see t h a t

W e can then state t h e

Theorem 5.1.

t

Under t h e a s s u m p t i o n s of Theorem 4.5, and i f f E W3"(

0 ),

p > n , w e have

5.2

The method o f e n e r g y . I t i s b a s e d upon o b t a i n i n g a p r i o r i e s t i m a t e s and " a d j o i n t "

e x p a n s i o n ( c f . C h a p t e r 1, S e c t i o n 3 . 3 ) .

W e n o t e t h a t s i n c e w e have

420

PROBABILISTIC PROBLEMS AND METHODS

assumed that a. 2 a.

> 0, the maximum principle gives an estimate,

namely that

However we are not going to use here this estimate, since we assume that (5.26)

,

a. 2 0

a . bounded and periodic

.

Also we will weaken the assumptions on f to (5.27)

f E ~'(8)

.

On the coefficients aij, bi, ci we will keep the same assumptions as in Theorem 4.4.

This guarantees the existence of functions m'(y)

m(y) solutions of

mE E w ~ * P ~ ~ , ( R ~p)2 2 ' m

periodic

,

mE(y)dy = 1 Y

Moreover we know that

,

p >

.

o ,

,

ANALYTIC APPROACH TO THE PROBLEM

T h i s f o l l o w s from Theorem 3.4,

Remark 3.4,

421

and t h e proof o f Theorem

I n p a r t i c u l a r w e can assert t h a t

4.2.

M > mE(y)

(5.31)

2

y > 0

I

v

y

. by m E ( y ) , and w e s e t

W e then multiply t h e equation (4.76)'

'2

-

(5.32)

ii = a 0mE

gi

,

i j - aijmE

EE

,

= bimE

= fmE

-E

I

ci = c . m E

,

.

We obtain

W e now set

so t h a t uE is. s o l u t i o n o f

W e n o t e t h a t from (5.28) w e have

;;=o. Thus m u l t i p l y i n g v i r t u e of

5.34) by uE and i n t e g r a t i n g o v e r 8

, we

o b t a n l by

(5.35)

From ( 5 . 3 1 ) it f o l l o w s t h a t

'The

s o l u t i o n of

2

1

(4.76) i s i n H ( 8 )n H o ( 8 ) i f & i s r e g u l a r .

422

PROBABILISTIC PROBLEMS AND METHODS IuEll,O 5 c

and since uE E H1( 8 ) ,it follows from Poincare's inequality, that

I luEl I

(5.36)

Hi( 8 )

< c .

We now set instead of (5.32)

sij (5.37)

so

= a. .m 13

b. = b.m 1 1

Ei

,

fE = fm(x)

aom

=

-.

,

=

c.m

,

.

We next set (5.38)

Eii(y) = Gi(y)

a sij(y) . - "yj

With these definitions, andby d o i n g a calculation similar to the one for (5.341, we obtain that uE is solution of (5.39)

au

-a H axi i j (E5 ) E ax j

UE&

= 0

1

E

.

6, (5) 1 E

au

x E axi - E 1. (-)E axi + 5 o (X)U E E

= fE

.

The variational form of (5.39) is then

+

I

2iO(Z)uEvdx =

f'vdx

,

V v E Ho(8) 1

-

We introduce next the function x(y) solution of

x J Y

Xdy = O

.

periodic

,

#

423

ANALYTIC APPROACH TO THE PROBLEM The problem (5.41) has one and only one solution, by virtue of Theorem 3 . 4 , applied to the operator

and noting that the corresponding m = 1, since

a pi .+

by virtue of the fact that For

E

aYi

0.

small enough, the problem (5.40) is equivalent to the one

with v changed into v(l

+

EX);(

Next writing (5.41) at point

X

)

.

$ and

Hence we have

recalling that

a

=

1 a, we E aYi

have

We multiply by uEv and integrate over 8

.

We obtain, using

integration by parts

X/E

.

tWe recall that in (5.42) the argument of Feriodic functions is

PROBABILISTIC PROBLEMS AND METHODS

424

I

(5.43)

aEi

uEv - dx aYi

+ 1

0

1

uEv

gi % ayl dx

1 %2

+

Zij

vdx

8

0

8 Combining (5.42) and (5.431, we obtain

0

fE(l

+ EX)V d x ,

0

0

a'

Let now 0

E

(5.45)

v E

C, ( 8 )and let us take in (5.44)

VE

E

@(x) + E$l(x'x) E

.

The function @,(x,y) will be chosen below.

Let us denote by

where

*

a

A 1 = - - ayi (5.46)

*

A2

=

%. , ( y ) 11

a

ayj +

-

a

Bi(Y) aYi

,

a a a a a -axi a11 . .(y) aYj - ayi 1. 1 3.(y) ax + B 1. (y) axi j

a

-

+ BiX

a aYi

I

425

ANALYTIC APPROACH TO TBE PROBLEM

*

-

A3 = -aij ( y )

+

-'ix

-

a

.

The f u n c t i o n $1 w i l l be chosen t o s a t i s f y

i.e.

I

where t h e

iR f u n c t i o n s -*-i A1x =

(5.48)

iR E

a r e d e f i n e d by

-a B ayj

W2rp"(Rn)

(y)

+

gII(y)

P L 2

I

,

i ap e r i o d i c

I

p > o .

Such f u n c t i o n s e x i s t , s i n c e by t h e a s s u m p t i o n s on bi, t h e r i g h t hand s i d e of

(5.49)

- *

AE v E = +

( 5 . 4 8 ) o v e r Y is 0 .

-

-1

&+

axiaxII a i j ay

j

2

ax.ax, 3

I

t h e i n t e g r a l of

W e have

a Z . .iaayi 11

axiaxII

BiFk

W e a l s o n o t e t h a t ( 5 . 4 4 ) c a n be r e w r i t t e n a s

avE Xdx ax axi

j

I luEl I

+

1

uE

3 avE

(E~i)Xdx

8

< C , w e can e x t r a c t a subsequence c o n v e r g i n g weakly Ii; towards u s a t i s f y i n g

Since

-

426

PROBABILISTIC PROBLEMS AND METHODS

We set

which is in (5.51) the coefficient (in symmetric form) of Let us identify the expression (5.52) with the

dx.

expression obtained in (5.13).

First of all, we remark that (5.52)

simplifies as follows (5.53)

qij =

I

m(y) [aij (y) + T1 x - j bi + 1

.

2 ibj]dy

Y

W e note that

a - aYiaYj

(5.54) whereas

2% is solution of (cf. (5.48))

~

xk

-R a R (a. .mX ) + ) = 13 aYi (b.mg 1

is solution of

-

a

2 ayj (a. J R m)

+

bR m

,

ANALYTIC APPROACH TO THE PROBLEM

427

From (5.54) and (5.55), it easily follows that

-

1

xkFllrn-

bkmZady =

Y

2

Y

.

a (ajllmg d y ayj

Hence qij =

1

m(y) [aij(y)

Y

-

1

x ibj -

i a

j

+ Xm- aYk (akJ.m) + X-

xjbi

aYk (akim;ldy

and integrating by parts we recover the expression (5.13). We next set (5.56)

E,

=

1 Fix - % iiEj

Y

+ cddy

Y

which is the coefficient of We must identify the expression (5.56) with (5.14).

We first

notice that

"Bx - 't.j hence (5.57)

c,

=

k = i ; x - L ( H ayj

1 1 [a, + 1

(EL

ayj

II

Y

1

(aijx) dy

Y

+ i;,x

-

Ei d ] d y

Y

2'

a

-

+ b,X)dy

Y

=

X), aj

Yi

[bi $ - 2 as aYi

But from (5.41) we deduce that

?xaYi -

Z ij aYiaY, dy

.

.

428

PROBABILISTIC PROBLEMS AND METHODS

hence from ( 5 . 5 7 ) (5.58)

C,

=

1

we see that ( E , + bex)dy

.

Y

However an easy calculation shows that

Since

x9. is

solution of

we see by integration by parts that

hence finally =

1

(E,

-

9.

? aYiLEi]dy

,

Y

which is nothing other than (5.14), since

a,

= mcR.

Therefore we can

write (5.51) as follows

m

and since $ is arbitrary in

C,(

81, we see that u is solution of the

homogenized equation. We can summarize the results which have been obtained in the following

OPERATORS .WITH LOCALLY PERIODIC COEFFICIENTS

429

Theorem 5.2. We assume (5.26), (5.27) that 8 is regular and that the coefficients a..(y), bi(y)r ci(y) satisfy the assumptions of Theorem 4.4. 11 Then the solution u fo (4.76) converges in H 1o ( B ) weakly towards the solution of the homogenized equation ( 4 . 8 0 ) .

Remark 5.1. The method of energy requires no regularity assumptions on f, and does not require that a .

be bounded below.

Like in the self

adjoint case, it has therefore a considerable advantage with respect to the probabilistic method or with respect to the method of asymptotic expansions.

However the type of convergence is different

in the three methods.

Remark 5.2. For operators considered here, none of the three methods seem to extend to boundary conditions other than Dirichlet.

This shows the

importance of having the problem in the framework of Chapters 1 and 2 (divergence form of the operator)

6. 6.1

. t

Operators with locally periodic coefficients. Setting of the problem. We consider the analogue of problem (4.76 with coefficients

depending on x, ci(x,y)

.:

We will consider functions aij (x,Y)r bi(xrY) r

ao(xry). We assume that

'Also referred to as nonuniformly oscillating coefficients.

430

PROBABILISTIC PROBLEMS AND METHODS

(6.1)

for x fixed, aij, bi, ci satisfy as functions of y the assumptions (3.19), (3.20), (3.21) (ci satisfying the same conditions as bi). strictly elliptic

(6.2)

The aij are uniformly

;

aij, bi, ci are continuous, bounded in x, y and for y fixed are twice continuously differentiable in x. All the partial derivatives with respect to x or y or mixed are globally bounded and continuous

(6.3)

a .

;

is continuous in x, y, bounded and periodic in y,

a

> a > O . 0 - 0

Let 0 be a domain of Rn and f a function on 0 (6.4)

o satisfies (4.74)

;

f E ~'(5)

We are going to study the behavior as

E +

of the Dirichlet problem (6.5)

We denote by A the differential operator (in y) (6.6) and by A* its formal adjoint

,

such that

. 0, of the solution uE

431

OPERATORS WITH LOCALLY PERIODIC COEFFICIENTS Applying Theorem 3 . 4 ,

there exists for any x fixed,one and only one

function m(x,y) such that

I

m(y)dy = 1

,

y

+

m(x,y) periodic

,

m(x,y) > 0

.

Y

The function m(x,y) is globally continuous in x, y.

the values for x E 8

will matter.

Actually only

Since 8 is bounded, we can

without loss of generality assume that

As in the preceding paragraphs the vector b will satisfy

6.2

Probabilistic approach. We define o(x,y) such that

(6.11)

1 7 a(x,y)a*(x,y)

=

a(x,y)

.

We construct on an arbitrary probability system the Ito process y'(t)

(

Cl

, a ,P, 5 t,w(t))

= yE(t) solution of

We denote by p E the probability measure induced by YE(.) on Y =

CIO,m;Rn].

432

PROBABILISTIC PROBLEMS AND METHODS

Lemma 6.1. The family :11

E

varies, remains in a weakly compact subset of

qw). Proof. We consider the functions

xR (x,y) solutions

of

By the regularity on the coefficients we can assert that there exists one and only one solution of (6.13) which is C2 (Rn x Rn) and bounded. Applying Ito's formula, we obtain, (noting

x

= (x',...,~ n I

But

hence

I [g

t

+

c + ?X ax b] (y",$)ds

0

),

OPERATORS WITH LOCALLY PERIODIC COEFFICIENTS

1

t

+

[tr a

2

433

2

$ + tr a &] (yE,g)ds

0

1 1% t

+

E

c

2

+ tr a %](y',<)ds ax .

0

we eliminate the term

r

$ b between

(6.14) and (6.12).

One can then

0

use a reasoning similar to the one in Theorem 4 . 3 , proof.

to complete the

Actually one has

hence the compactness property follows.

Lemma 6.2.

Let @(x,y) be a function which is C2 2 x, C 1

y, with all

partial derivatives continuous and bounded globally in x , y. also that 4 is periodic in y for fixed x, and satisfies (6.16)

I

Y

Then we have

@(x,y)m(x,y)dy = 0

.

Suppose

434

PROBABILISTIC PROBLEMS AND METHODS

Proof. We can solve the equation (6.18)

A$ = $

,

periodic

J,

.

From the regularity assumptions on the coefficients and $, we can assert that J, is C2(Rn

Rn) and J, bounded as well as its partial

x

derivatives. From Ito's formula we have

S

From (6.18) it follows that 2

3 - b + tr a

= -$

aY Multiplying (6.19) by

E

~

.

and , reasoning as in the proof of Lemma 4.1

(first proof), one easily obtains the desired result.

Remark 6.1. The second proof of Lemma 4.1 does not seem to extend easily to the present context. the function $

.

This explains why we assume more regularity on

OPERATORS WITH LOCALLY PERIODIC COEFFICIENTS

We

435

consider the Ito process defined by dy = r(y)dt + fl

(6.22)

Jg(y)dw(t) ,

y(0) = x

,

and let px be the probability measure on C(IO,-[;Rn) defined by y,(.).

Then we have the

Theorem 6.1. Under assumutions (6.11, (6.21, (6.10) we have (6.23)

v xrpE +

px in

R1(Y) weakly +

.

Remark 6.2. The situation is more complex than in the situation of Section 4 , since the limit p x is the measure of a diffusion process which is no longer Gaussian. distributions.

Hence we cannot express easily its finite dimensional

Here is where the martingale formulation of diffusions

is quite useful (cf. Section 2).

Proof of Theorem 6.1. We know that p: Let us set

remains in a weakly compact subset of ! R1+ ( Y ) .

PROBABILISTIC PROBLEMS AND METHODS

436 (6.24)

zE(t) = yE(t)

-

E

.

~ ~ [ y ~ ( t ) , q ]

It follows from (6.14) and (6.12) that we have

We are going now to use the martingale formulation of diffusions. For this we consider the space 4' = CIO,m;Rn] and the canonical process x

ti$)=

o-algebra Q

$(t).

We set p

= o

(x(s),O

5

5 t). The

s

= B = o-algebra of Bore1 sets on 1 .

From the theory of weak solutions of stochastic differential equations (see Section 2), we can construct a process wE (t) adapted to g

',

such that when 1 , B is equipped with the probability

wE (t) is a Wiener process with respect to Q ,' (6.26)

1 b(xfx)dt dx(t) = F E + o(x,:)dwE(t)

and it is clear that <'(t) ( 6 . 2 5 ) 1,

,

! J : ,

then

and E

( ~ ( 0 )= x) = 1

will satisfy the relation (analog to

.

OPERATORS WITH LOCALLY PERIODIC COEFFICIENTS

I

t (6.28)

sE(t) = x

-

+

EX(X,:)

-

[(I

- 3 ax b

$c

-

437

2 tr a %i

0

-

Let @

E

CZ(Rn).

-

&- 2 c -

2

2

tr a

E

tr a ~axl ] ( x ( xs ( )s ), ~ ) d s

E

By Ito's formula w e have

2 tr a $-tr

a

2

LX-

axay

-

E

2c -

e tr a

$1

(x(A),%)dl

ax

We define (6.30)

G ( x , y ) = (I

-

g)a(I

(6.31)

r(x,y) = (I

-

a)c aY

We then obtain from (6.29)

-

-

%)*

b

-

-

q(x)

,

2 tr a !ayax -x-

-

2

tr a

$&- r ( x ) .

PROBABILISTIC PROBLEMS AND METHODS

438

5 CE, a.s., C being a deterministic constant. We notice

where

that

'EZ

(respectively, Ex) denotes the mathematical expectation with

respect to :p

(respectively, px).

OPERATORS WITH LOCALLY PERIODIC COEFFICIENTS

439

from Lemma 6.2. Similarly we have

Using (6.331, (6.34) and (6.27), it easily follows from (6.32) that

Since at least for a subsequence

uz

+

ux

weakly in

1 (6.34) n +,

implies

Hence lJx

solves the martingale problem relatively to the pair q(x),

r(x) and by the uniqueness theorem of Stroock-Varadhan Section 2), it follows that the whole sequence

v:

[

11, (see

converges towards

px, which is the probability measure on Y induced by yx(*) defined

by (6.22).

This completes the proof of Theorem 6.1.

440

PROBABILISTIC PROBLEMS AND METHODS We can now obtain the limit of uE(x), solution of (6.5).

introduce the homogenized equation (6.37)

2

-qij (x)

-

=

I

au + r. (x) 1 axi

ao (x)u =

f(x)

,

where

We then have the following

Theorem 6.2. Under the assumptions of Theorem 6.1 and (6.31, (6.4),we have (6.39)

uE(x)

+

,

u(x)

Vx E 8

Proof. We have the explicit formula

Let us first prove that

-

exp

-

1a. 0

(yx(s)

a s ~ + O .

We

441

OPERATORS WITH LOCALLY PERIODIC COEFFICIENTS We note that

j

m

(6.42)

XE

5 CE

YE(t)dt

,

0

where

But

-

EYE(t) 5 Ell a o [ y E ( s ) , ~ ] d s

from Lemma 6.2. EYE(t)

1

ao(yE(s))ds

+

0

Moreover,

5

2 exp

-

clot

.

From Lebesgue ' s theorem, we obtain (6.41). It is thus enough to study the limit of E

(6.43)

vE(x)

But we can rewrite vE(x) as vE:(x)=

I

F(+)duz

Y where

As in the proof of Theorem 4.5, we can then assert that F is px a.s. continuous and bounded. imply (6.39).

This and the result of Theorem 6.1,

PROBABILISTIC PROBLEMS AND METHODS

442 6.3

Remarks on the martingale approach and the adjoint expansion method. The method of "adjoint expansion" used in the proof of Theorem

5.2, can also be used within the context of the martingale approach.

Indeed the martingale formulation of (6.121, leads to

, which we rewrite as t (6.45)

EEk(x t)) +

j AE@(x S

and (6.45) is valid for any @

E

&z(Rn

.

Clearly by approximation, (6.45) is valid for any We are going to take then @ =

where @o E C;(Rn),

=

@,(XI

@ E

C2(Rn).+ b

as follows X

+ E@l(X,$

,

and @,(x,y) periodic in y determined from an

asymptotic expansion, namely (6.46)

-aij(x,y)

-aYiaYj

Therefore using the 'x

a+O bi(XiY) a@1 - b ayi i axi

.

functions defined in (6.13), we choose

'Functions which are twice continuously differentiable , bounded as well as its derivatives up to the order 2 .

OPERATORS WITH LOCALLY PERIODIC COEFFICIENTS

443

Hence we have 2

- a200 axiax

i axi ~

+

aij

a@, aXiaXR

ayj

2 a @ o 211 a x + a ago + aij aXiaY ij axsax, ayi 3 j

ax,

+

with

I 5

CE

aij

'$0

a2xR

ET ax.ayi 1

+

bi

a200

axiaxll

X

R +

bi

where C is a constant depending on 9,.

a@o 5 axi

Hence

(6.45), we obtain

where C1 is a deterministic constant. Using Lemma 6.1, and 6.2 as in the proof of Theorem 6.1, it follows that if for a subsquence

u:

-+

ux, then

where we have set

444

PROBABILISTIC PROBLEMS AND METHODS

This formula for q . is the same as (6.20), as it has been shown in lj (5.13). Here x is merely a parameter which plays no role. The above reasoning thus leads to an alternative proof of Theorem 6.1 (but Lemmas 6.1 and 6.2 are used in both proofs).

6.4

Analytic approach to problem 6.5. We do not repeat here the method of asymptotic expansions, which

is an easy extension of what has been done in Section 5.1.

We will

concentrate on the energy method for which some differences with Section 5.2 will be encountered. We will assume that (6.50)

ao(x,y) 2 0

(6.51)

f

E

L2( 8 )

, ,

a . bounded

,

periodic in y

8 regular and bounded

,

.

By virtue of the regularity assumptions on the coefficients and (6.50), (6.51) there exists one and only one solution uE of

(6.52)

X + ao(x,,)uE

= f

,

OPERATORS WITH LOCALLY PERIODIC COEFFICIENTS

445

Similarlythere exists one and only one solution u of the homogenized equation -qij (x)

a 2u - ri(x) au axiax axi + aO(x)u = f

,

j

(6.53)

We are going to prove the following.

Theorem 6.3. Under the assumptions (6.1), (6.2), (6.10) then uE -t u & H 10 ( 8 ) weakly. -

(6.50), (6.51),

Proof. Our first objective is to obtain a priori estimates in H1o ( B ) . Let mE:(x,y)be a function which will be made precise below and such that (6.54)

M 2 mE(x,y) 2 mo > 0

.

We set

i4j = (6.55)

aijmE

-

aE = a mE 0

0

,

,

6:

1

=

bimE

= fmE

13; = c.mE

,

,

.

We multiply (6.52) by mE and rewrite the result in variational form. We obtain (6.56 where we have set

446 (6.57)

PROBABILISTIC PROBLEMS AND METHODS

g:(x,y)

=

g: +

EE'

i

-

a Szj ayj

-

E

a a- E. . . axj 11

We are going to choose mE in order that

:;a ai? - + -2 = € 2 ayi E axi

O(E)

.

Expliciting, we obtain the desired relationship (6.58)

a2 i a E 1 - E 2 ayiay, (aijmE) + - - (him ) + € 2 aYi ~

3

a (c.m ) +axi 1

~

aL .mE)= axiax (a. 13

l a E - (c.m ) E ayi 1

.

O(E)

We are going to choose mE as follows (6.59)

mE(x,y) = mO(x,y) + Eml(xfy) +

E 2rn,(x,y)

,

where

a2

a

(6.60)

-

(6.61)

-

(6.62)

'a - ___ aYiaYj (a..m 1 + 13 2

(bim o ) = 0

aYiaYj (aijmo) + ayi

~

a2

ayiayj

'a - -axiax

(a m ij 1

j

,

mo periodic in y

,

a (b m ) + + (c m ) ayi i 1 ayi i 0

a aYi

(aijmo) = O

a (bim2) + ayi (cim 1)

I

a2 - ax.ay (aijml)

m2 periodic in y

1

.

1

OPERATORS WITH LOCALLY PERIODIC COEFFICIENTS

From the fact that

I

bimdy = 0

447

V x, it follows that the solvability

condition for (6.61) i s satisfied, hence (6.61) has a solution which we write as m,(x,y)

-

ax liij j

+

~ii

n(x,y) are solutions of

where G’(x,Y), (6.63

=

-- a

aYiaYj

(aijfiR

.-R m periodic

ii periodic in y

,

fi(x,y)dy = 1

.

Y We turn next to (6.62), which is rewritten as follows

+ a

axi

(ciXm)

-

axiaxj a2 (aijXm) =

0

,

m periodic in y

We obtain the following solvability condition

.

PROBABILISTIC PROBLEMS AND METHODS

448

ah

(6.66)

1

a + axi

.

fi’dy

Y

+

Xbiiidy

Y

&j

ciXmdy

-

1

axiax a2

Y

aijhmdy.= 0

.

J Y

Comparing (6.63) with ( 5 . 4 8 ) , we see that EL and miL are identical when the coefficients do not depend on x.

From the

identification made to obtain (5.53), we can assert since x does not play any specific role in such an identification, that (6.67)

a. 1 3.mdy + 1 2

qij(x) = Y

I

+

bifijdy

Y

We next consider the quantity si(x) =

1

biiidy

Y

1

+

b.Eidy 3

.

Y

cimdy

+ a ax

Y

Recall ng the equations for the

xR

1

.

biIjdy

I Y

functions (6.13), the equa-

tion for ii, (6.64), it follows from an integration by parts calculation that (6.68)

Similarly, we can check that (6.69)

1

bifiJdy =

-

Y

1 1

Xi[bjm

-

a

2

ayk

Y =

-

Y

Xibjmdy

-

2

1

(a m) jk i

.% ajQmdy

Y ay!2

We easily deduce from (6.68), (6.69) that

.

OPERATORS WITH LOCALLY PERIODIC COEFFICIENTS si(x) =

1

Y

[ci

i

- ?x-

ayk

ck]mdy

-

2

1& 2 i

449

akkmdy

Y

i Y

Comparing with (6.21), one checks that s . (x) = ri(x)

.

Taking

this and (6.67) into account, we can rewrite (6.66) as follows

Our problem is to find a function A(x) such that (6.59) is guaranteed.

We notice that (6.70) is an equation without boundary

conditions.

But of course the values of x which matter are those in

-

8

.

Therefore we must find a function A(x) which satisfies (6.70)

-

in 8 , bounded below and above by positive constants in 8 . that we consider a regular domain

4

E

Lm(z )

= 0

on 8

I

4 1. 0, 4 #

.

s"

such that

5 C s"

a function

0 in a set of positive Lebesgue measure and

We then solve the problem

The existence and uniqueness of X follows by duality frow the existence and uniqueness of

'Note

To do

that qij (x), ri (x) have been defined in Rn.

PROBABILISTIC PROBLEMS AND METHODS

450

h

where

E

Lm(g 1 .

routine way.

The Fredholm alternative must be applied in a

Moreover we have the relation

1

Bcdx =

gxdx

.

0

Since for

ii ->

2 0, it follows that if

0, then

hidx 2 0, for any

h2

0, hence i[

We can always assume that

2 0 if 4 2 0.

is sufficiently smooth (since 8

regular), and therefore we may assume that strong maximum principle it follows that

A

X I -8 '

=

(6.73)

2 0, then

x

i>

E

C

is

4 -

0 in

( 5 ) . From the

5

.

Taking

we see that we can construct a function A such that

A

E C4(5)

,

A > 0 in

5,

and A satisfies (6.70) in 0 . We can then exhibit a choice for m2 as usual in the method of asymptotic expansions. involved in (6.58). IO(E)

I 5

CE.

We can also define what is the function O ( E )

By virtue of the regularity of h(x) we have

We thus have completely defined the function mE(x,y).

We can now turn back to (6.56).

Multiplying by uE and in-

tegrating we obtain, by virtue of the choice of mE, auE auE

(6.74)

dx

8

E

that

1 0

=

For

+

1

fEuEdx

O ( E ) ( U € )+~

1

FiE(uE)2dx

0

.

small enough, we deduce from (6.74) and Poincarg's inequality

451

OPERATORS WITH LOCALLY PERIODIC COEFFICIENTS We next proceed as in the proof of Theorem 5.2.

-

-

bi = bim

aij = aijm

-

(6.76)

= a m

a .

-

ci = c.m

,

fE = f(x)m(x,t)

0

We define

,

.

Thus uE is solution of

Let v

E

au

X E1 . (x,X) E 2 axi + ao(x,F)uE

1 Ho (

H2 ( 0 )

) n

.

-E uE(A

)

* vdx

=

1

fE

,

uElr = o

If AE denotes the operator in ( 6 . 7 7 ) ,

then we have, since u E l v vanish on (6.78)

=

r,

f'vdx

where

(AE)*

=

E

-2-* A1

+

E

- li,*

+

0-* E

A3

with

a

-*

a2 (aij') + - (b..) A 1 = - -aYiaYj aYi 1

-*

a2

-*

'a

a a2 - aYiaXj (gij-)+ axi (b.

A 2 = - - axiaYj ('ij"

a

)

- .

axi (Ei*) + a. A 3 = - - axiax (Sij*)+ j

The unique solution of

is J, = 1 (up to

;t

multiplicative constant) (by virtue of the unique-

ness of the solution of ( 6 . 8 ) )

.

The equation

452

PROBABILISTIC PROBLEMS AND METHODS Ale = 0

has also

@ =

constant).

where @(x)

@

periodic

,

1 as a solution (thus unique up to a multiplicative

We

m

E

,

C, (

look for a function v in (6.78) of the form

8 ) and @l(x,y) are defined such that

-*

*

A1@1 + A2@ = 0

r

@

1

periodic in y

We have

+ % bai

where

We have next

and

- +

@ - ba i +axi

aa.

@ >=

aYi

0 ,

.

OPERATORS WITH LOCALLY PERIODIC COEFFICIENTS

-*

+

The mean value of A3@ (6.79)

*

A3@

+

*

-*

A2@l in Y, is equal to

a2

-

axiax

+

a axi

A2@1 =

453

j

(@biX)

.

We notice that (cf. (5.53)) -

(6.80)

qij = 1 ij

-+ 1 - i + 1. 6.zj b.i 2

1

2

1

.

Let us set (6.81)

s.

-

(x) = ci +

We recall that (cf. 6

Combining this with the

Hence

x

equation and integrating by parts, yields

PROBABILISTIC PROBLEMS AND METHODS

454

Using now the equations of

xi

and

2 R we obtain I

hence

Using this in (6.821, we obtain

si =

I [ai

Y

-

E, $]dy

-

I

b,dy

Y

+

2

I

2 i

Y ax, Yj

Zkjdy

and thus si(x) = ri(x) (cf. (6.21)). Collecting results we can assert that

Therefore from (6.78) and for a convergent subsequence uE

+

u, we

have

From the uniqueness of the limit we conclude the proof of the theorem.

REITERATED HOMOGENIZATION

455

Remark 6.2. We have used in the proof of Theorem 6.3 a method slightly different from the one used in Theorem 5.2.

The functions

have been used in both proofs, but in a different manner.

i a and x But both

proofs can be used in each theorem.

7.

Reiterated homogenization. We consider in this paragraph, the probabilistic counterpart of

Section 8 , Chapter 2.

7.1

Setting of the problem. Let us consider functions b'(x,y,z),

Rn

-+

Rn, and u (x,y,z): Rn

-+

bL(x,y,z), c(x,y,z) from

d (Rn;Rn), satisfying the following

conditions (7.1)

1 br

2 b ,

C,

0

are "sufficiently" smooth

in x, y , z with bounded partial derivatives (7.2)

, ' b

b2,

c,

u

are periodic in y

, and periodic

in z with period 1 in each component (7.3)

1 uo* a = 2

-

CII,

CI

> 0

,

,

constant

, tt

.

We construct on an arbitrary probability space ( S l ,

a ,P, 3 t,w(t)),

the Ito process yE(t) = yi(t) solution of

tThis of course can be made more precise, .consideringwhat will be needed in the following. ttAgain the choice of 1 as the period is a matter of convenience.

PROBABILISTIC PROBLEMS AND METHODS

456

We denote by 1;

the probability measure induced by YE(.) on

Y = C ( [O,-;Rn]).

We want to study the behavior of :1-1

for fixed x, as

E

goes to 0.

As we have seen in the previous paragraphs, the weak convergence of :11

towards some px in 111 t ( Y ) ,

is related to the convergence of the

solution of Dirichlet problems, with rapidly varying coefficients. We introduce the differential operator connected with ( 7 . 4 ) namely A '

=

-[$+ b2 + c1.V

-

div a0

x x we write with standard and operating on functions of x, E'

notation

with

€2'

,

457

REITERATED HOMOGENIZATION

a -

A5 = -ci

aij

a2

axiaxj

'

We first define m(x,y,z) such that

*

A m = 0 1

(7.6)

,

m periodic in z

I

,

m(x,y,z)dz = 1

.t

z

From Theorem 3.4, we know that there exists one and only one function m of z (x, y being merely parameters), which is "sufficiently" smooth and m > 0.

The smoothness of m is easily determined by the

smoothness of the coefficients. We first assume that

I

(7.7)

b'(x,y,z)m(x,y,z)dz

= 0

.

Z

The assumption ( 7 . 7 ) guarantees the existence of functions such that (7.8)

A1x R = -bi

,

xR

periodic in z

,

I

X'(x,y,z)dz

x R (x,y,z)

= 0

.

z

To proceed in a reasonably intuitive manner, we will recall the result obtained in the self-adjoint case (cf. Section 8, Chapter 1). The homogenized coefficients were obtained in a two stage procedure: first homogenization in z , letting y be a parameter, next homogenization in y of the operator obtained after the first stage. Within the present context, we can rely on the results of the previous paragraphs to assert that the homogenized

coefficients

after the first stage are given by (cf. (6.49) and (6.21))

+Z =

n

Y = 10,1[

.

PROBABILISTIC PROBLEMS AND METHODS

458

Formulas (7.9), (7.10) correspond to the homogenization problem where X

y is merely a parameter, the diffusion term is a ( x , ~ , ~ )the , drift 1 1 2 X term is 7 b (x,y,:) + b (x,y,;). We next define

We write with standard notation (7.12)

E

g1 =

l

1

3 Q1 + E

Q

E

2 + a 3 '

where

1-

a 3 = -qij

aL

axiax

a

j

We then define the function m 1 (x,y) satisfying the analogue of (7.6), i.e. (7.14)

Iml = 0

,

m1 periodic in y ,

J m1(x,y)dy Y

We have then to assume that

=

1

.

REITERATED HOMOGENIBATION (7.15)

I

1 rl(xfy)m (x,y)dy = 0

459

.

Y

9,

Then we can define the functions K (x,y) solutions Of (7.16)

alKR = -r

R f

KR periodic in y

I

,

KR(x,y)dy = 0

.

Y

The final homogenization diffusion term will be defined as follows (7.

To make precise the final homogenization drift term, we need some other functions.

We introduce the functions 5 % solutions of

g R periodic in z

,

J

gRdy = o

.

2

The functions g R are well defined since the mean value in' z of the right hand side of (7.18), in z using the weight m(x,y,z) ,is nothing other 1 which is 0 according to the choice of the functions than aLKR + rR, KR

. With this definition we set

PROBABILISTIC PROBLEMS AND METHODS

460

With the coefficients qijl ril one can define the Ito process y(t) = yx(t) given by (7.20) Let

dy = r(y)dt

ux

+

a

Jg(y)dw(t)

y(0) = x

.

be the corresponding probability measure on '4.

One can

state the following Theorem 7.1. Under the assumptions we have (7.21)

u:

-+

ux

in

%'(+1

Y ) weakly

.

Remark 7.1. To obtain the

uij

one uses the reiterated homogeni-

X We first consider the coefficients aij(x,y,E), 2 where y is a parameter, then we define the b:(x,y,,) X + bi(x,y,E) 1 1 corresponding homogenization coefficients qij(x,y), ri(x,y).

zation rule.

REITERATED HOMOGENIZATION

461

We next consider the homogenization problem for the diffusion 1. (x, x~ ) , rt(x,:). We apply the standard homowith coefficients q.

2

13

genization rule to obtain qij(x).

To obtain rl(x) a more complicated

procedure is required.

Remark 7.2. Let us consider the case when

This is the situation when

We clearly have m = 1, and (7.7) is satisfied. We have

since

Next we have 1 a 1 ri(x,y) = - qij ayj It follows that

a,

=

satisfied. We then have

.

and m1 = 1.

The assumption (7.15) is

PROBABILISTIC PROBLEMS AND METHODS

462 i.e., (7.22)

dy

q = Y

1

Z

aK dz(I-- -)(I aY

-

.

%)a

Let us compute the drift term ri(x), using (7.19).

aa.k ar;i+ 3 aa.k aci + 2ajk =3ayk azj azk ayj

and the mean

il

We first have

a2ci k

j

dydz vanishes.

YXZ

Then we have

a2 i - a jk aYkaYj

a and the corresponding mean is again 0.

azk

-

From (7.19)

ax

a2Ki 2ajk axjaYk

i aajk aK ayk axj

aa. i ]k aK axk ayj

+

ax

REITERATED HOMOGENIZATION

463

Thus the homogenized operator is

where qij is given by the reiterated homogenization procedure.

We

recover the results obtained in Chapter 1 , Section 8.

7.2

Proof of Theorem 7.1.

We write K(x,y) for the vector (K1,. n 1 . We set for (x1 ,...,x (7.23)

L(x,y,z) = + (I

aK - -)x aY

..rKE,.. .,Kn) and

.

Then by construction L is solution of (7.24)

A 1L

+

A 2K = -b1

.

We next define M(x,y,z) as being the solution of (7.25)

A 1M

+

A L 2

+ A3K

= -b2

M periodic in z

To show that M is well defined, we note that

.

x(x,y,z)

PROBABILISTIC PROBLEMS AND METHODS

464

Taking the mean in z (witfi the weight m), we obtain

f

-

m(bi

A2Le

-

A3K R )dz = r,(x,y) 1

+

QIKR = 0

,

2

and thus the solvability condition for (7.25) is satisfied. Let us define next the vector function

We have from the expression of AE, and (7.24), (7.25)

-

AEeE =

a- E2

E

where 10E(l)I 5 C. We then apply Ito's formula to compute OE[yE(t) and we obtain

Therefore (7.27)

yE(t)

L

= x

-

x x eE(x,-,,) E

E

465

REITERATED HOMOGENIZATION Since

it follows from (7.27), reasoning as after (4.65), that 1 a compact subset of ?(+(Y). To prove that in Section 6.3.

uE remains in

has a limit and to identify it, we proceed as

From the martingale formulation we have

m

where $ o ECo(Rn), and +1, $ 2 , +3, + 4 are determined by an asymptotic expansion.

We write the conditions

(7.29)

Ale2 + A2$1

(7.30)

A1$3 + A2$2 + A3$, + A4$ = 0

(7.31)

A104

+ A ~ $=

0

,

+ A2$3 + A3$2 + A4@1 + A 5$

Easy but tedious calculations show that $,(Xty) =

-

* axk

k

K (x,Y)

,

= function of x only

.

PROBABILISTIC PROBLEMS AND METHODS

466

Writing the solvability condition for (7.71)’ we get an equation

for e2(x,y),

for which a solvability condition is again needed.

Expressing it, one checks-that one must choose in (7.31) the function of x only,to be equal to a@,, where of course

with the definitions (7.19) and (7.17).

Taking a subsequence of

u;,

Therefore (7.28) yields

1 converging weakly in7;1+(Y) towards l.~X’

we obtain

and from the uniqueness of the solution of the martingale problem, we obtain that pX is the diffusion connected with (7.20), which completes the proof of the theorem.

Remark 7.3. We leave to the reader to express the convergence result for the corresponding Dirichlet problem.

Remark 7.4. We can consider in the same manner drift terms of the form -1b

€ 3 1

+ , b 1 E

2

+ -1 b + c , € 3

with adequate assumptions.

PROBLEMS WITH POTENTIALS 8.

467

Problems with potentials. We consider here the situation of potentials a . which are not

always positive and we consider also some generalizations to include bounded potentials.

8.1

A

variant of Theorem 6.3.

We will first notice that Theorem 6.3 holds true with the assumption (6.50) replaced by (8.1)

ao(x,y) measurable and bounded

,

periodic in y

twice continously differentiable in x

,

,

once differentiable

in y, with all the derivatives being globally bounded,

We can state the Theorem 8.1. Under the assumptions of Theorem 6.3 with (6.50) replaced by (8.1), (8.2), then the same conclusions hold true.

Remark 8.1. Theorem 8.1 is not exactly a generalization of Theorem 6.3, since some smoothness is required for ao, unlike Theorem 6.3. as far as the positivity of a .

However,

is concerned, it states that only the

positivity in the average is important.

468

PROBABILISTIC PROBLEMS AND METHODS

Proof of Theorem 8.1. We have denoted by a,(x) the quantity (cf. (6.38))

We consider the equation (6.52). existence of u

Since a. is no more positive, the

is not a consequence of standard results.

However

we will first prove that at least for

E

one and only one solution of (6.52).

This will follow from a change

of unknown functions.

small enough, there exists

First of all we introduce the function O(x,y)

solution of

--

-a a2e ij ayiay. 3

bi

ae

=

-a 0

+ o f

(8.3) 8 periodic in y

.

Ody = 0 Y

BY virtue of the regularity assumptions on ao, we can assert that (8 are bounded globally in x,y

.

We look for uE in the following form

For

E

of Cc.

small enough the knowledge of uE is equivalent to the knowledge But from (6.52) we easily check that

(taking (8.3) into account)

must be solution of

469

PROBLEMS WITH POTENTIALS

-a

(8.6)

Since

a0 ->

ij

(1 +

E

2

8)

a2aE

axiax j

- E -

aaixiE iEaij

+

u

O E

+

6

+

EiiEEzaij

E

E bi

aiiE

aiiE

c i axi

-

aaxe a i j j

a2e axiay

+ E

bi

aij

+

EBci

a2e - bi a e ax.ax

7

1

0, it follows t h a t a t l e a s t f o r

E

1

j

s m a l l enough t h e problem

(8.6) with t h e conditions

i s w e l l posed. unique way. H;,

T h e r e f o r e fiE (and a l s o u ) a r e w e l l d e f i n e d , i n a

I t i s a l s o c l e a r t h a t uE and CE have t h e same l i m i t i n

since

1 p r o v i d e d t h a t w e p r o v e t h a t GiE r e m a i n s i n a bounded s u b s e t of H o ( 8 ) .

T h e r e f o r e it i s enough t o s t u d y t h e l i m i t of ill i n H o ( B )

.

be done by a n e a s y v a r i a n t of t h e proof o f Theorem 6 . 3 .

We w i l l just

1

b r i e f l y s k e t c h t h e main s i m p l e changes. Using t h e n o t a t i o n -€

-E

ci, f

zijl

w i t h t h e same meaning a s i n t h e proof of Theorem 6.3, w e

e a s i l y check t h a t iiE i s s o l u t i o n of (8.8)

This can

- -axia

(Hij

where w e have set

+

aii

b- €. . ) 3 11 ax j

aii

- 1. fjE 2 E

i axi

WE

'Yi

aiiE axi

PROBABILISTIC PROBLEMS AND METHODS

470

(8-11)

bO(x,y) = -2aij

a2e

j

- c ae i ayi

--

a2e -

a ij axiaxj

bi

ae

c *+Eae. 0 i axi

Multiplying (8.8) by GE and integrating, we get by virtue of the properties of

Since

6 i j ,?:

@Tr that

bi

are bounded, we deduce from (8.12) and from

Poincard's inequality, that at least for

Therefore also

I luEl I

< C.

E

small enough, we have

To identify the limit of u E r we

HO( 8 )

proceed with the equation of uE (and not with the equation of G E ) r as in the proof of Theorem 6.3.

The proof is actually identical, hence

the result follows.

8.2

A general problem with potentials. We consider here the following problem

PROBLEMS WITH POTENTIALS (8.14)

2 X a uE -a..(X,-) ___ 11 E ax.ax 1

-

[1 X F bi(x,F)

471

+

1

We will show mainly that (under some adequate assumptions) one can reduce problem (8.14) to the problem considered in Theorem 8.1, although equation (6.52) seems to be only a very particular case of (8.14), (when wl, w2 are equal to 0).

We assume that

(8.16)

there exists a solution +(x,y) of

periodic in y, $(x,y) 1. 6 > 0, and $ is sufficiently regular in x, y, so that the quantities aij$

,

2a + $bi ij ay j

,

2aij aJI ax j

+

$Ci

,

satisfy the same regularity assumptions than aij, bi, cil respectively : (8.17)

a a2+ ~ i abi~

ci aJI ayi + w2+ is j periodic in y, is twice continuously differentiable

the quantity 2aij

+

+

in x, once in y, all derivatives bcing bounded W

We denote by Awl (or All) the differential operator

(8.18)

a2 AW1 = -a ij aYiaYj ~

-

bi

a

ayi

+ w1

,

.

472

PROBABILISTIC PROBLEMS AND METHODS

and by A = A1 the differential operator

When w1 = 0, we have

+

=

1 and Ao = A = All the usual operator

which has been considered up to now.

Since aij

, 2aii. a+ + $bi' aYj

satisfy the same regularity assumptions than a . . bi respectively and 13'

since $ 1. 6 > 0, the Fredholm alternative applies to the operator A, and therefore there exists one and only one function m such that (8.20)

-

___ a (aijJlm)+

aYiaYj

m(x,y)dy = 1

,

a ayi

[ (2a..* + $bi)m] ayj 11

= 0

,

I

m periodic in y

,

m > O ,

Y

with the standard regularity assumptions on m. A simple calculation shows that m is solution of (8.21)

-

~

a2

aYiaYj

(a..m) + 11

a ayi

(b m) i

+

W m = 0 1

.

Since conversely a solution of (8.21) satisfies (8 20), it follows from the uniqueness of the solution of (8.20), that (8.21) has a unique solution satisfying the additional conditions of (8.20). Applying again the Fredholm alternative to (8.21) which is the formal adjoint of Awl, we can assert that the function 9 satisfying the conditions (8.16) is necessarily unique (up to a multiplicative constant in y). We then assume that

(8.23)

1

+ W2$ dy = 0 Y

3

.

PROBLEMS WITH POTENTIALS

473

From the assumption ( 8 . 2 3 ) it follows that there exists one and only one solution (8.24)

x 0 (x,y) of

Aw1x 0 = 2aij

x0

the problem

aa 2J,~ ji +abi~ a$

I

,

periodic in y

Xody = 0

.

Y

We will finally assume that

We also define the functions (8.26)

Aw1x 9. = -2a .

9.1

x 9.

*ayj

periodic in y

,

xR

solutions of

,

b,J,

I

X'dy

= 0

.

Y

We define the homogenized coefficients as follows (8.27)

(8.28)

(8.29)

PROBABILISTIC PROBLEMS AND METHODS

474

Let u be the solution of

Then we can state the following

Theorem 8.2. Under the assumptions (8.15), (8.16), (8.171, (8.22), (8.231, (8.25) then we have (8.31)

UE (x) X * J, (X,T)

1

in HO( 8 ) -

u(x)

weakly

.

Proof. We will reduce the proof of Theorem 8.2 to the one of Theorem 8.1, by successive transformations. We look for uE in the following form

Again for

E

small enough, (8.32) defines iiE in a unique way.

into account the equations for J, and

0

x ,

Taking

it follows from an easy

calculation that GE satisfies (8.33)

-a..( J , 13

+

2

EX

)

a " - - - (2a. ?!/!ax.ax E axi ij ayj i

aii E

+ jbi]

j

+ xOci] + cE(ao$ - ci

-

0

ci

HOMOGENIZATION OF REFLECTED DIFFUSION PROCESSES

475

Therefore GE satisfies a problem similar to the one considered in Theorem 8.1.

The assumptions made, especially (8.22) and (8.25)

guarantee that GE * u in Hi ( 8 ) weakly. From (8.32) we see that

U

%remains in a bounded subset of

9 (x,;) 2 1 Ho ( 8) ant has the same limit as GE in L ( 8 ) strongly.

Hence the

desired result follows.

9.

Homogenization of reflected diffusion processes.

9.1

Setting of the problem. Let 8 be an open bounded subset of Rn, satisfying the conditions

of ( 4 . 7 4 ) .

We consider functions aij (x,y), bi(x,y) satisfying the

assumptions (6.1), (6.2). Let next y(x) be a function such that y:

a 8

* Rn ,

Y(X)*V@(X)

,

continuous bounded

(9.1) 6 > 0

1

X E a 8

and such that

,t

Defining o(x,y) such that 1 2

(9.2)

aa* = a

,

we can construct on an arbitrary system ( S l , f f

,PI Zt,w(t)) the diffu-

sion process reflected at the boundary of 8. More precisely, for

E

> 0 fixed, there exists a unique pair

yE(t), cE(t) of processes satisfying the conditions

-= v4J

lv4Jl

n = unit outward normal.

PROBABILISTIC PROBLEMS AND METHODS

476

y'(t)

,

cE(t) are adapted continuous processes

with values in Rn and R respectively; cE(t) is an increasing process y'(t)

E

s

;

, v t ,

xo(yE(t))dSE(t) = 0

. t2

The last condition (9.3) means that

x8

(yE(t))dgE(t)= 0,+ a.s.,

v tl, t2 with tl 5 t2. For details see Watanabe [ l 1 . We will denote by p: Y'(*)

on cO[O,~;SI.

the probability measure defined by

-

Setting Y = C 0 [ O , m ; O l ,

and J,

5

x(*) E Y, x(t) = x(t;J,), one can

characterize p z as the unique probability measure on Y, such that for any @(x,t) E C2'l(Rn

'This

x

[O,m[)

with compact support, satisfying

is a Stieltjes integral.

HOMOGENIZATION OF REFLECTED DIFFUSION PROCESSES

,

x(0) = x We recall that

C

477

a.s. 1-EIx *

= o(x(A) , A

5

s)

.

This is the submartingale formulation of the reflected diffusion process (Stroock-Varadhan

[ 2 ])

.

We want to study the limit of

1-1;

as

E

goes to 0.

We introduce

the operator (9.6)

A =

-aij(x,y)

=

~

a2

aYiaYj

and let m(x,y) be the unique solution of (9.7)

*

Alm = 0

,

m periodic in y

I

m(x,y)dy = 1

,

Y (cf. (6.8)).

Let us consider the homogenized coefficients defined as

follows

(9.8)

ri(x) =

I

bi(x,y)m(x,y)dy

.

Y

Let us consider now the reflected diffusion process corresponding to the coefficients qij(x), ri(x)' i.e., the pair of processes y(t),

5 (t) satisfying the conditions

y(t)

, C(t) are adapted continous process with values

in Rn and R, respectively; E(t) is an increasing process

-

Y(t)

E 0

;

, v

t

,

PROBABILISTIC PROBLEMS AND METHODS

478

Then we can

Let ux be the probability measure defined by y(t) on Y .

.

state the following Theorem 9.1.

aij, bi satisfy (6.1),

We assume that 0 satisfies ( 4 . 7 4 ) (6.2).

We also assume (9.1).

(9.10) +

9.2

px

in

4 1( +

Y )

Then we have weakly

.

Proof of convergence.

Lemma 9.1. The family :p

remains in a weakly compact subset of

n+1 ( Y ) .

Proof. Let us apply Ito's formula to the functions $ and $ 2

t2

+j

t2 +

and since

$Ir

= 0

t r q ax

2 (YE

.

We have

HOMOGENIZATION OF REFLECTED DIFFUSION PROCESSES

I

479

t2 (9.12)

@ 2 (YE (t,))

- 02 (YE (t,))

=

2@ g.(b(yE.<)ds

tl E

+ o(yE,%)dw(s)!

+

2

I

[a

2-2+ @tr a qax] d s

We note the identity

(@(YE(t2))

-

@(YE(tl)))

.

Since -@(yE(tl)) 2 0 and y * g l r 1. 0, it follows from (9.11), (9.12) and (9.13) that

From (9.14) it follows that, using standard estimates on stochastic integrals, w e have (9.15)

E(@ (yE(t2

Going back to (9.11 we deduce that

.

PROBABILISTIC PROBLEMS AND METHODS

480

(9.16)

E

I n (9.15) and (9.16) C d e n o t e s a c o n s t a n t i n d e p e n d e n t of v a l u e need n o t b e made p r e c i s e .

E,

whose

But

hence w e can a s s e r t t h a t

I f w e go back t o t h e e q u a t i o n ( 9 . 3 ) , w e deduce from ( 9 . 1 7 )

l6

5

which i n s u r e s t h a t t h e f a m i l y

uE

(9.18)

EIy'(t2)

-

yE(tl)

C(t2

-

t l )3/2

that

,

remains i n a weakly compact s u b s e t

o f 4 +1( S ) . t W e can now s t a t e t h e a n a l o g u e o f Lemma 6 . 2 f o r t h e r e f l e c t e d

diffusion process.

Lemma 9.2.

L e t @(.x,y) be a f u n c t i o n which i s C

2

&

x, C

1

&

y, with a l l

p a r t i a l d e r i v a t i v e s c o n t i n u o u s and bounded g l o b a l l y i n x , y.

@ @

p e r i o d i c i n y and s a t i s f i e s @ ( x , y ) m ( x , y ) d y= 0

(9.19)

r

Y

'We

have used i n t h e proof some i d e a s of Stroock-Varadhan

i n a d i f f e r e n t context.

[2

Ir

HOMOGENIZATION OF REFLECTED DIFFUSION PROCESSES

then we have

Proof. We proceed as in the proof of Lemma 6.2. equation A $ = $

r

and use Ito's formula.

JI periodic , We obtain

By construction we have

hence it follows from (9.21) that

We first solve the

481

482

PROBABILISTIC PROBLEMS AND METHODS t

(9.22)

1

$[yE(A),e]dA

=

e2$[yE(s)

,*] - EZ$[yE(t) ,*]

S

L

But from (9.17) it follows that

Using this estimate in (9.22), we easily obtain that

hence (9.20) follows.

Proof of Theorem 9.1. We can now proceed with the proof of Theorem 9.1.

We will rely

on the submartingale formulation of the d ffusion with reflection. Let $(x,t) E

&:(Rn

x

[O,-[)

satisfying (9.4).

From (9.5) we have

HOMOGENIZATION OF REFLECTED DIFFUSION PROCESSES

483

where we have set

From Lemma 9.2, it follows that

This and (9.23) imply at least for a converging subsequence of 1; that

I

Ex $(x(t),t) t

-

-

$(x(s),s)

-

2 (x(A);A).r(x(A))dA

s

and from the uniqueness of the solution of the submartingale problem we complete the proof of the theorem.

9.3

Applications to partial differential equations.

9.3.1

Homogenization of a Neumann problem.

With the notation of the previous paragraphs, we take

PROBABILISTIC PROBLEMS AND METHODS

484

,

a(x,y) = a h )

(9.24)

Let us consider (9.26)

f E

a .

0 -

C ( 0)

, ao(x,y) 2

> 0 large enough

a.

,

continuous and bounded

,

a. periodic in y

.

We consider the following Neumann problem

aij(x)ni

auE

=

o ,

j r

The structure of this operator is very special and actually the analytic treatment is easy (see Remark 9.1). We can rewrite (9.27) in divergence form, namely au

a aij (XI J - -axi ax

j

au

+ 2 axi

(% ax -

X X bi(x,z)] + a0 (X,T)U€ = f

3

Setting (9.28)

ai(x,E) =

aa. & - bi(x,T) j X

,

we can give the variational formulation of (9.271, namely (9.29)

I

8

auE av a11 . .(x) ax axi dx j

=

I

fvdx

,

+

I

x - vdx a. (x,-) 1 E axi

8

Vv E H’(8)

+ 8

.

0

For a.

large enough, the bilinear form on the left hand side of

(9.29) is coercive and therefore uE is defined in a unique way in

.

HOMOGENIZATION OF REFLECTED DIFFUSION PROCESSES HI( 8 )

.

485

The regularity follows from standard results (Agmon-

Douglis-Nirenberg [ 11 1. Considering the process yE (t) solution of (9.3) , the function uc(x) is given explicitly by the following formula

Clearly m = 1 is the unique solution of (a..(x)m) = 0 17

,

m periodic

, Y

We set (9.32)

ri(x) =

I

bi(x,y)dy

,

ao(x) =

i

ao(x,y)dy

.

Y

Y

We consider the homogenized Neumann problem (9.33)

a2u -aij(x) -axiaxj

-

au rI.(x) axi + ao (x)u = f

,

= o , P 2 2

-

We then have the following

Theorem 9 . 2 . Under the assumptions of Theorem 9.1 and (9.241, (9.25) we have (9.34)

UE(X)

+

u(x)

,

X G 8 .

Proof. The proof is identical to the proof of Theorem 6 . 2 , as a consequence of Theorem 9.1.

PROBABILISTIC PROBLEMS AND METHODS

486

Remark 9.1. It is not surprising that we can give a convergence result which is valid pointwise up to the boundary of 8 .

Indeed if we look at

(9.27) from the point of view of a priori estimates, we can see that uE remains in a bounded subset of W2”

(

8)

,

p 2 2.

Therefore we

can pass to the limit in (9.29) without any difficulty, obtaining the convergence of u

towards u in W2lp( 3

)

weakly for any p

2.

The Neumann problem in a half space.

9.3.2

We take here 8 = {xn > 0).

We note that (4.74) is satisfied if

for instance we set

The fact that 8 was supposed bounded in the previous paragraph was actually not used. y = (0,

...,0,-l),

Only (4.74) has played a role.

We take

and consider the Neumann problem

We assume that

ao(x,y) 2

a.

> 0 large enough, continous and bounded

Then (9.35) has one and only one solution in (9.37)

u

E

E

W2%Cs)

,

v p

2

.

.

HOMOGENIZATION OF REFLECTED DIFFUSION PROCESSES

407

Indeed the problem (9.35) admits a variational formulation as follows

+

aOuEvdx =

1

fvdx

, Vv

H'(0)

E

.

Indeed the conormal derivative corresponding to the matrix a. ij and the domain 8 is ann a and therefore applying Green's formula,

ax., 3

we see that the boundary condition corresponding to (9.38) is

= 0.

and since ann > 0, this is equivalent to follows from standard arguments.

The regularity

We again have the explicit formula

(9.30). The function m solution of (9.7) is no more identical to 1. We set (9.39)

ao(x) =

1

ao(x,y)m(x,y)dy

,

Y

and consider the homogenized Neumann problem

= 0 if j # n). nj We can state the following.

(note that q

Then again u

E

W2rP(0),

p 1. 2 .

Theorem 9.3. Under the assumptions of Theorem 9.1 and (9.36), we have

PROBABILISTIC PROBLEMS AND METHODS

4aa

10.

Evolution problems.

Orientation. We will consider in this paragraph

some of the problems studied

in Chapter 2 , from the probabilistic viewpoint.

We will not give a

treatment as detailed as in the stationary case, and we shall mainly emphasize the differences between the stationary and the evolution case.

10.1

Notation and setting of problems. We consider here functions aij(y,T), bi(ylr), ci(Y,T) Satisfying

the following conditions (10.1)

aij, bi, ci are continously differentiable in y,

T

aij are twice continuously differentiable in y, aij, bi are twice continously differentiable in

T

.

(10.2)

2 a..c.c 1 1 1 j -> a l c I

(10.3)

aij, bi, ci are periodic in y with period 1 in all

,

5

E

R"

,

components, and periodic in

aij

'I

-

aji

,

with period

T

0'

Setting a s usual (10.4)

uu* a = 2 '

we can construct on an arbitrary probability space ( Q , Q , P , 3:,w(s)) the solution of the stochastic differential equation

EVOLUTION PROBLWS The process yxt(s)

489

defines a probability measures t:u

! y'(s)

on

0

the space I = C [+Z,~;R~].We will denote as usual by x(*) the the rs-algebra generated by x(A) ,

canonical element of I, and by S t 5 A 5 s. &,

as

E

We are mainly concerned in studying the weak limit of

*

As in the previous paragraphs, this will imply

0.

convergence results for P.D.E.

(here parabolic P.D.E.).

Fredholm alternative for an evolution operator.

10.2

We will be concerned in this paragraph with the Fredholm alternative for the operator (10.6)

R = R

1

=

a -a-r

bi

a ayi

a - 'a ij aYiaY

t

j

with periodic boundary conditions in 'I, y

We denote by R* the

formal adjoint aij-)

.

Theorem 10.1. We assume that a. .

17'

bi satisfy (10.1), (10.2).

We consider the homogeneous equations (10.8)

Rz = 0

(10.9)

R*p = 0

,

z E CZt1

,

,

p E C2"

z

,

periodic in y,

T

m periodic in y,

, T

.

There exists one and only one solution (up to a multiplicative constant) of (10.8), (10.9) ~~

.+'

Let

$,

J, be continously

~-

tThe notation R1 has been chosen in connection with Chapter 2 (1.57). ttz is obviously a constant.

490

PROBABILISTIC PROBLEMS AND METHODS

differentiable periodic functions such that

!o 1 io 1

(10.10)

(10.11)

O

Y

O

Y

$(y.T)p(yiT)dydT = 0

I

,

J,(y,T)dydT = 0

then there exists one and only one solution of the inhomogeneous equations

,

(10.12)

R5 = @

(10.13)

R*n = J,

5 E C2'l

II E

C2"

,

5 periodic

,

n periodic

,

,

r

= l .

Proof. Let X be large enough; we consider the problem (10.14)

for g

E

RzX

+

AzX = g

,

L2 (O,r0;L2 (Y)). To give a meaning to the equation (10.141,

we first rewrite R under divergence form, namely

with periodic conditions z x ( 0 ) = zA ('I 0 ) . then

If g = 0 then zx = 0 since

EVOLUTION PROBLEMS :O

o = J

O

+ A

49 1

TO az. azA Ja.zAdydr + aij 1 aYi Y O Y

j

io

O

dydT

aYj

z:dydr Y

and for A large enough, this implies z A = 0. T o prove the existence, we notice that the problem

- 2 + aT

(10.15)

2

E

A(T)Z

+ Az

L2(0,m;W(Y))

=

h

E

L2(0,m;L2 (Y))

,

,

2 2 has one and only one solution.' However g E L ( O , T ~ ; L(Y)) implies 2 2 ge-YT E L (Olm;L (Y)) assuming that g has been extended by periodicity.

Taking h = ge-yr and setting z = Ce-Y' we obtain the existence of 5 such that Ce-yT E L2 (O,m;W(Y)) and

Changing A into A

-

y

(y

is > 0 arbitrary), we see that there exists

one and only one solution of

+

The function z is clearly periodic (since z(r solution).

T

0

)

is also a

Hence we can define a mapping

TA:

2 2 L (OITO;L (Y))

-+

2 2 L ( O , T ~ ; L(Y)) by setting

T g = z A A '

+Use a truncation argument; consider T

z ( T ) = 0 and extend z

solution of (10.15)

.

T

(T)

by 0 f o r

T

-

T

az

+

A(-r)zT

> T ; then as T

-+

+ AzT

+-,

zT

= h, +

z

PROBABILISTIC PROBLEMS AND METHODS

492

The operator Tx is compact.

2 2 For $,$ E L ( O , T ~ ; L (Y)), the problems

Rz=@, t (10.16) R*P = $

r

are equivalent to the problems (I

-

XTx)z = Tx I$

The Fredholm alternative applies, hence we must find the number of linearly independent solutions of (10.18)

Rz = 0

z

z'

E

E

L 2 (O,rO;W(Y))

2

L (O,r0;W'(Y))

,

z(0) =

Z(T0)

.

Since the solution of (10.18) satisfies

and a..=a..and the coefficients are differentiable in T , we see that 17 7 1 2 E L ( O , T ~ ; L ~ ( Y ) ) .Since the coefficients are differentiable in y, aT 2 2 z E L ( O , T ~ ; H(Y)). Using the Sobolev spaces with weights we see

az

that we can consider that

- a? +

z E

A ( T ) ~+ ~z = ~z

a.e.

in R"

'Solutions are of course in the space of functions 2 az 2 L ( O , T ~ ; W ( Y ) )with E L ( 0 , ~ ;W'(Y)) and ' ( 0 ) = z ( T ~ ) . aT 0

EVOLUTION PROBLEMS 2f1f2”(Rn

lb

and z E

493

By a bootstrap argument like in the

X

proof of Theorem 3 . 4 , we obtain that z

b 2’1fp1p, p

E

1. 2,

p > 0.

Since the a..‘s are twice continuously differentiable in y, the bi’s 17

are continously differentiable in y, and both coefficients are twice continously differentiable in aTt

tt

E

ayi

2r”p”.

Hence z

we can also show that

T, E

Czv1(Rn

principle implies that z is a constant.

x

(0,m))-

The strong maximum

The remainder of the results

is a consequence of the Fredholm alternative and regularity arguments. We shall need in some of the results to follow (Theorem 10.3) the assumption, (10.19)

the solution p of (10.9) is such that

Let us notice that if (10.19) is satisfied then from the strong maximum principle it follows that (10.20)

p > 0

Let p = 1.

US

.

notice that if a i

=

aa.. 0 (bi = 2) then R = R*, hence aY

It is possible to give a sufficient condition for (10.19) to

hold true, which is less restrictive than ai = 0.

Indeed we have

494

PROBABILISTIC PROBLEMS AND METHODS

Proposition 10.1. If C laiI2 i

5 k2, gr&

.

1 3 1< k where ayi -

k is small enough then

(10.19) holds true.

Proof. We proceed using elliptic regularization.

We consider pE solu-

tion of

From the elliptic theory, we know that pE

0.

But multiplying by

pE and integrating we obtain

hence (10.21)

CrlVpEl

L (0.ro;L2(Y))

Multiplying theequation by

Integrating in

T

yields

<

klP"I 2 2 L ( O , T ~ ; L(Y))

$ and integrating in y only we obtain

-EVOLUTION PROBLEMS

Hence

From (10.21), (10.23) it follows that

But from PoincarG's inequality

and therefore if k is small enough (1 > PC"k 2 ) , we obtain

We can then extract a subsequence, which clearly converges weakly towards p.

Hence p

2 0.

495

PROBABILISTIC PROBLEMS AND METHODS

496

10.3

The case k < 2. We begin now the study of (10.5), assuming here that k < 2 .

We

set

We consider

a parameter.

T as

From the elliptic theory there exists

m(y,.r) solution of (10.24)

A*,m = 0

,

m periodic in y

,

I

mdy = 1

.

Y

We have m(y,r) > 0, V y,

T,

and m is also periodic in

T.

We make

the assumption (10.25)

b(y,T)m(y,T)dy = 0

.

Y

We can then define the functions x3(y,.r) as the solutions of (10.26)

A1x7 = -b j'

1

XI periodic in y ,

mxjdy = 0

.

Y

From the regularity of the coefficients, we can insure that ~1

E

C2'l, and is periodic in y,

T.

Lemma 10.1. The family of measures pit remains as subset of 74

('4)

E

-t

0, in a weakly compact

.

Proof. We first assume that k

5 1. From Ito's formula, we have

EVOLUTION PROBLEMS

Using (10.26) and setting

x

=

497

(x1,...,x n ) , it follows from (10.27)

that

1& S

+

1$ S

+

bdh

t

.

3.odw

t

From (10.5) and (10.28) it follows, eliminating the b term, that

1

+

(I - 2 ) o d w

.

t Since 1

-

k 2 0, it easily follows from (10.29) that pzt remains in a

1 compact subset of 4 +(Y). 3 We next consider the case when 1 5 k < 2.

define

x1 (y,T )

(10.30)

We set r = 2

-

k, and

by

Alxl =

2,

x1

periodic

1

r

Xlmdy = 0

.

Y

Since we have chosen

x

in order that

I

Xmdy = 0, then the solvability

Y

condition is satisfied for (10.30), hence apply Ito's formula to

x+

x1

to obtain

is well defined.

We

PROBABILISTIC PROBLEMS AND METHODS

498

t

t

t

1 b, and From (10.5) and (10.31), we can again eliminate the term in F obtain the compactness. From here on, we need to assume one more degree of differenti5 ability in T than in (10.1). We consider the case 3 5 k < 3. We define

x2

(10.32)

by axl Alx2 = aT

Y

and we apply ItGIs formula to Reasoning as before 5 consider the case when 3 enough differentiability

x

+

E

2-k

x1

+

E

4-2k

x2.

we obtain again the compactness. We next 7 < k < 7 and so on. Provided that there is in 'I, we can consider any k < 2. Hence the

desired result follows.

Lemma 10.2.

Let $(x,y,~,s) be periodic in y, (10.33)

T

a function which is C1

and such that

j $(y,'I)m(y,T)dy

Y

then we have

= 0 ;

& I y,

T, s,

C

2

& I x,

499

EVOLUTION PROBLEMS

Proof. We define $ such that (10.35)

A1$

= @

,

J, periodic

.

From the regularity on @ and on the coefficients it follows that J, E CZf1. From ItGIs formula, we have (for convenience we just

consider the case when

@

is independent of x , s )

+ S. 1

I S

s2

L[”*b E 2 ay

+

tr a q ] d X

aY

+

2

1E. 3-d~ aY

S.

and from (10.35) it follows that

Taking the square and the conditional expectation we easily obtain the property (10.34).

500

PROBABILISTIC PROBLEMS AND METHODS We now define the homogenized coefficients

then we can define the measure pxt on I corresponding to the diffusion with constant coefficients q, r, namely dy =

(10.38)

6 dw(s) ,

r ds +

y(t) = x

.

We can now state the

Theorem 10.2. We assume (lO.l), (10.2), (10.3), (10.25) 2p-1 < k < P -

9 (p 2

2)

, then we assume that aij, bi

continously differentiable in (10.39)

V

x, t

&

,

k < 2.

+

T.

uxt

rf

are p+l

times

Then we have in

1 m + ( Y ) weakly

.

Proof. We use the martingale formulation of diffusions as in the proof

of Theorem 6.1. Q

= u(x(X) ,t

We consider the canonical process x(s;$) = $ ( s ) and

5

h

5

s)

.

From the theory of weak solutions of

stochastic differential equations, we can construct a process w which is a Wiener process when I is equipped with the probability measure &,

and such that 1 b(z,L)ds dx = E E Ek

+ o(z'S)dw'(s) E

(10.40)

lJEt(x(t) = x) = 1

.

€k

x s + c(-,-)ds E

€k

,

EVOLUTION PROBLEMS

z5 k

To fix the ideas we assume that 3

501

5 (but the reasoning is < 5

completely general). We consider the function x(y,T) + + E 4-2kx2(y,-r), and define

E

2 -kxl(yIT)

By Ito's formula we have (taking into account the equations of

+ j [I-%-€ S

t

5-3k t Let now @

E

&:(Rn

x

2-k axl aY

€4-2k

%IudW€ aY

% dl a-r

[O,m[).

From Ito's formula we have using

(10.42)

+

tr

s1

2 9 (I - %)a(I ax

-

%)*

+

O(E)/ds

50 2

PROBABILISTIC PROBLEMS AND METHODS

W e set

Y

W e o b t a i n using ( 1 0 . 4 1 ) and (10.43) r

where

F r o m Lemma 1 0 . 2 , w e have

hence

m 0

In

N

a ffl

ffl

X

Y

Y

L!

u

ffl + f f l I

rl

ffl rl (0 Y

X 8

Y

I

N

ffl N

ffl Y

X Y

u

wx

u

W

w x W

rl

0

4

ffl

a

3: +ffl

N

rn I

rl

c

ffl 4

-4

3

c

rl

rd

C

0

u

-4

C W

c

a a 7

0

a ffl

7

0 C

u

4

G

10

ffl

a

"IXW Y

L!

c,

0

4 X -

f f l l

_h*

I

a

X

u a 4

u

3

w x W

a,

0

c

a,

7

m ffl

a

2 6" .4

e

a

3: I

rl

l u l l

N

ffl + f f l

rl ffl rl ffl

X

Y

Y

I

8

w x W

'u

2

Y

X

3

-

Y

ffl

N

(0

N

L!

L!

u

3

z

0

u

rl

a,

m

ffl

W

0

m

G

>

.4

L!

I

v

ffl \ffl

X

-

0

ffl I X W rl

0

N

I

m m

X

u W

v Y

X

II 3 Y

X W H

rl

10 rl

ffl

X

v Y

I

8

N

ffl

-

N

ffl

X

Y

Y

X

u

_N

L2-l

4

W

$

Id

c

N

X

v

tr

h

u

u

-I

N

ffl-10

L!

tr

u

II

rl

m c

.4

u u

X

:

a,

L! a,

a,

w

>

w x

m .s

0

rl

PROBABILISTIC PROBLEMS AND METHODS

504

we have

and

pE(s)

is bounded, hence

p‘(s)

-+

p(s)

in L2 (sl,s2;d (Rn;Rn)). But

2 in L (sl,s2;=f(Rn;Rn)) weakly

q(5)q +

E

.

Therefore we can assert that

-+

s2 EXt[i tr q 1

9 ax

I

(x(s),s)ds H

,

and a similar result holds true for the term containing r in (10.46). Gathering results, we obtain

s1

This implies

EVOLUTION PROBLfMS

505

Therefore pXt is the diffusion corresponding to (10.38).

From the

uniqueness, we obtain (10.39), which completes the proof.

The case k = 2.

10.4

Here we use the results of Section 10.2.

We assume that instead

of (10.25), we have -

Therefore we can define the solutions €J’(y,r) of (10.51)

- 2 ej + aT

Alej =

-

bj

,

6’

periodic in Y,T

.

From the regularity of the coefficients we can insure that 0’ E CZt1.

Lemma 10.3. The family of measures t:p 1 subset of 111 +(Y).

remains as

E

+

0, in a weakly compact

Proof. We use Ito’s formula on the 01, hence ( 0 =

and from (10.51), it follows that

(e1,...,8

n)

)

PROBABILISTIC PROBLEMS AND METHODS

506

Combining with (10.5), we obtain

S

(I

+

- za e) o d w

t and this relation implies by standard arguments that t:p

remains in

a compact subset of 74 f ( Y ) .

Lemma 10.4.

Let @(x,y,~,s)be a function which is C1 & y, periodic in y,

(10.55)

lo O

T

and such that

@(x,y,~,s)p(y,~)dydr = 0

i

Y

then we have

Proof.

We define $I such that (10.57)

-

*+ ar

A1$I =

@

,

(I

periodic in y,

T

.

T, s ,

C

2

x,

EVOLUTION PROBLEMS Then J, is in C2 in x, y, C1 in

T,

s.

507

Using Its's formula in a routine

way, the result follows as in Lemma 10.2. We define the homogenized coefficients as follows

(10.59

j0I

r =

Too

ae - -)cpdydT aY

(I

Y

.

on Y corresponding to the diffusion with xt constant coefficients q, r, i.e., (10.38) and we can state the

We def ne the measure 1-1

Theorem 10.3. We assume (10.1), (10.21, (10.3), (10.19)

(10.50).

Then we

have (10.60)

v

x, t

,

uEt

+

pxt

in

1

m + ( Y ) weakly

.

Proof. The argument is now standard and we will only sketch it briefly. We use the canonical process, hence

We define CE(S)

hence

=

x(s)

-

I$,*[

€0

508

PROBABILISTIC PROBLEMS AND METHODS

[",&-I

(10.62)

c E ( s ) = X-€6

E

1 S

E

+

1 S

[I

-

g]cdA +

t

[I

-

g]adw

.

t

m

Let '$ E a o ( R n

X

[O,m[).

Applying again Ito's formula we have

ae

-)

aY

+ tr

9 ax

(I

-

g)a(I

C

-

hence

Using Lemma 10.4 and the compactness of ,'p

we easily obtain (10.60).

EVOLUTION PROBLEMS 10.5

509

The case k > 2. A situation similar to the case when k < 2, will be encountered.

We set now r = k

-

2,

and consider an expansion of the form (vector

expansion)

Applying It81s formula, we obtain (10.63)

E

@

[

E~ r Ek b ]

E

t Our objective is to adjust the expansion (10.63) in order that the only term which has a positive power of denominator be equal to b. (10.64)

?aT !?! =

(10.66)

a@l

-

0

,

A1@ = b

.

E

equal to 1 in the

Hence we try to satisfy the conditions

510

PROBABILISTIC PROBLEMS AND METHODS

At this level a discussion is necessary. 2r

+

1

3 > k

-

2

-

k = r 5

L

3 , then r

0, and the objective is satisfied.

2 1,

If

then we need to add the condition a@2

(10.67)

-

A1@l = 0

.

i, we need the additional condition

5 If 2 > k 2

(10.68)

1

If k

a@3

-

A1Q2

= 0

s,

2 +1 > k and so on, if -P-P

2 i3

we need to go to

We must insure that conditions (10.64), (10.66 realized.

We have

@ =

, (10.69) can be

@(y) and to satisfy (10 66) we must have

1

TO @dr =

-

0

bdT

,

0

or (10.70)

where

We introduce the function a(y) solution of (10.71)

-*

A1n = 0

,

II

periodic

,

I

ady=l,

Y

and the solvability condition for (10.70) is

a > O ,

EVOLUTION PROBLEMS

511

If we need to satisfy (10.67) then we must guarantee that

or

To solve (10.74) we need the following solvability condition

We then have

1 T

(10.76)

$2(y,~)=

AIQlds + $,(y)

,

0

and if we need to solve (10.68) we must have the analogue for (10.75), i.e. 0.

2 +3 In general if 2p+l > k > P - p+l we will need the condition

The conditions (10.72), (10.75) and (10.78) are conditions on b. We can now state the convergence result. coefficients are defined by

The homogenized

PROBABILISTIC PROBLEMS AND METHODS

512

(10.79)

TO

Theorem 10.4. We assume (10.1), (10.2), (10.3), (10.72). we assume in addition that a.

ij

differentiable in y, bi y

are 3p+2 -

vx,t

,

P:~

+1 > k 2 2p+3 P p+l'

times continously

are 3p+l times continously differentiable in

(10.75), (10.78) hold true.

(10.80)

If 2

+

Then we have

R +1( Y ) weakly

uxt

.

Proof. The family

vEt

remains in a compact subset of

construction explained above. ' C

m +1( Y ) ,

by the 1 Now if $(x,y,~,s)is C in y, T, s ,

in x and satisfies

then we can state the analogue of Lemma 10.1 or 10.2, i.e.,

Indeed let +(y) be independent of

T

such that

EVOLUTION PROBLEMS

513

and let $l(y,~)be defined by

then $l is periodic in y,

T

and

- 3a? + A 1 $ - $ = O . Therefore, from Ito's formula we have (omitting x,

s

to simplify)

and (10.82) follows by standard arguments. We next proceed as in the proof of Theorem 10.2.

We define

cE and by Ito's formula

From Ito's formula again we obtain

PROBABILISTIC PROBLEMS AND METHODS

514

W E ( S 2 ) ’ S 2 ) = +(SE(Sl),S1) +

Hence

Using the property ( 1 0 . 8 2 ) desired result.

we easily complete the proof of the

EVOLUTION PROBLEMS 10.6

515

Applications to parabolic equations. We can apply the preceding results to the homogenization of the

parabolic equation

where

C =

r

x

]o,T[, Q =

8

x

lO,T[.

The homogenized problem will be (10.84 We can state the

Theorem 10.5. We assume that 8 satisfies (4.74), f E assumptions of Theorem 10.2

(if k

< 2), 10.3

Co(6),

(if k

and that the =

2), 10.4

(if

k > 2) are satisfied, then we have

The proof of Theorem 10.5 is now standard and is left toethe reader.

Remark 10.1. The formulas for q , r depend on k in the following way.

There

is one formula for k < 2, (independent of k), one formula for k = 2, one formula for k > 2 (independent of k).

516

PROBABILISTIC PROBLEMS AND METHODS

Remark 10.2. Let us consider the self-adjoint case, i.e., c = 0, bi = Then we have m = 1, p = 1, n = 1.

Then (10.25) is satisfied, also

(10.50) and (10.72) are satisfied.

Let us check that (10.75) and all

the conditions (10.78) for any p are also satisfied. dy function $.

aa.

A. aYj

=

Indeed

0 for any period

C

sufficiently regular, the

Therefore when the

results of Theorem 10.5 apply for the problem

= o . We recover the results of Chapter 2.

Remark 10.3. We can also give some results in the case of Neumann boundary conditions by probabilistic methods.

This can be done as in the

elliptic case (cf. Section 9) with extensions to the parabolic case similar to those of the present paragraph.

11. 11.1

Averaging. Setting of the problem. We consider the following problem

uElc = 0

,

We will assume that 8

uE(x,T) = 0

.

satisfies (4.74) and

AVE RAG1NG

517

,

(11.2)

a i j , bi

(11.3)

i f h ( x , t ) d e n o t e s one of t h e f u n c t i o n s a i j ( x , t ) ,

a.

,

a r e m e a s u r a b l e and bounded

b i ( x , t ) , a ( x , t ) t h e n w e assume t h a t 0

(11.4)

co

x,x'

E

where p ( 6 )

+

0 as 6

+

0

,

h ( x , t ) d e n o t i n g t h e same f u n c t i o n s a s i n (11.31, then

re

(h(x,s)

t

-

a i j , bi,

-

f

E

CO(0)

-f

+m

,

-

a i j , bi a r e c o n t i n o u s i n x , t ,

and bounded

(11.6)

B

,

f o r x, t fixed

(11.5)

K ( x ) ) d s i s bounded a s

,

.

S i n c e t h e problem i s p a r a b o l i c w e can w i t h o u t l o s s of g e n e r a l i t y , assume t h a t (11.7)

a

> 0 0 -

.

The a v e r a g e d problem is d e f i n e d a s f o l l o w s

.

518

PROBABILISTIC PROBLEMS AND METHODS

The functions uE and u belong to ln "lrP(Q), V p 1. 2, hence in particular are continuous in

6.

We can state the following Theorem 11.1. Under the assumptions (11.2), (11.3), (11.4), (11.5)

(11.6)

we have

11.2

Proof of Theorem 11.1. We consider o(x,t) such that

(11.10)

o(J* 2

a

,

and we extend the functions a . . bi, a. outside 17'

5 ,

in such a way

that the conditions (11.2), (11.3), (11.4), (11.5) are satisfied on Rn x [O,Tl. We set Y = C 0 [t,T;Rn ] and consider the canonical process

6

=

x(*), x(s) = J l ( s ) = x(s,$) and

S

t

=

(x(X),t 5 A

exists one and only one probability measure on

Y,

5

s).

There

m

S t, denoted by

vlt, such that there exists a standard Wiener process with respect to

S

t , denoted by wE ( s ) , for which one has llEt(x(t) = x) = 1

.

Similarly, we denote by pxt the probability measure associated with

519

AVERAGING (11.12)

dx(s) = L(x(s))ds

+

pxt(x(t) = x) = 1

.

6

,

t

V x,t

.

(x(s))dw(s)

We are going to prove that (11.13)

pEt

+

pxt

in

Z +1( Y ) weakly

,

It is clear that pxt remains in a weakly compact subset of Let

a,

(I

ECo(Rn).

n +1 ( Y ) .

We have

Hence

'Since

the coefficients are not Lipschitz, only the weak sense

of stochastic differential equations (11.111, (11.12) can be considered.

520

PROBABILISTIC PROBLEMS AND METHODS

where we have set

We are going to prove that

from which, and the weak compactness,the desired result will follow. It is of course enough to prove

[i'

?!(x(X)).b(x(X),F)dh ?i

E L s1 ax

l2

+

0

,

This will follow from the

Lemma 11.1.

Let +(x,T) be a measurable bounded function such that

(11.20)

rT

+(x,s)ds is bounded as

0

Then we have

T

+

+m

,

for x,t

fixed

.

521

AVERAGING

Remark 11.1. Lemma 11.1 is the analogue of Lemma 4.1, and again two proofs will be given.

One easy proof, requiring more regularity on

@

and a

more technical one using just the assumptions of the Lemma.

First proof of Lemma 11.1. We assume here that @(x,T) is C2 in x, and that

and

a$

a%

5' axiaxj

are bounded in x,

T.

From ItGIs formula, we have S,

I

I

Multiplying by

E,

taking the square and the mathematical expectation,

we obtain (11.21) (and also an estimate of the rate of convergence).

PROBABILISTIC PROBLEMS AND METHODS

522

Second proof of Lemma 11.1. For simplicity of notation we take s1 = 0, s2 = T, and we denote by 1;

=

The measures

u:

uE0.

We rewrite (11.21) as

areweakly compact and therefore, there exists, for

given 6 > 0 a compact set K6

C P'

such that

(x is fixed, so we do not indicate the dependence on x). we have if (11.23)

I$] 5

XE

Therefore,

M

5 (MT)26 +

dpE($) Kg

Since K6 is a compact subset of a complete metric space, we can write for any given K 6 =

rl

m U

j=l

K

j '

where the K. are disjoint and K j is contained in a sphere of center 3

$.

I

and diameter

q.

Therefore

AVERAGING AVERAGING

We consider consider aa splitting splitting of of (O,T), (O,T), But x(s,Qj) X(S,$.) E C(0,T;R"). C(O,T;Rn). We But 3

...,Nk Nk == T, T, and and ...,

O,k,2k, O,k,2k,

[I

with 5L.(k) as kk with . (k) ++ 00 as 33

(11.25)

But

+ +

0. Therefore Therefore we we have have 0.

l2

@(x(s,$~) , ~ ) d s

523 523

524

PROBABILISTIC PROBLEMS AND METHODS

hence (n+l)k

Collecting results we may write

+ 2 E n=O 2 p Letting first

E

.*I2

.

go to 0, then k, then

Q,

then 6, we obtain X E

+.

0,

which completes the proof of the Lemma. We thus have completely proven (11.13).

To complete the proof

of the theorem, we write UE(X,t

and we know that a0 -> 0.

We first assert that

E'O.

But

using again Lemma 11.1.

The remainder of the proof is an immediate

consequence of (11.13).

The proof of Theorem 11.1 is complete.

525

AVERAGING

11.3

Remarks on generalized averaging. We can put homogenization and averaging in a common framework,

which could also include other examples, provided that one can check some assumptions which will be stated below. We will remain a little bit formal ir. the presentation, since we are mre interested in giving ideas that precise statatm-~ts(there will actually be no new results). We consider a pair of diffusion processes as follows

(11.27)

I

dxE =

[k bl(xE,yE) + cl(xE,yE) dt + ul(xE,yE

dyE =

[$b2(xEryE) + 1 c2(x ,y u2 E + 7 (X r~

(11.28)

E)

E

~'(0) = x

,

E

+

d2(xE ,yE

)dw2

~'(0) = yO

Y

+ 2 E

.

The process xE will be called the slow varying process, and y E the fast varying process.

The Wiener processes wlr w2 will in

general be correlated, as follows (11.29)

*

Ewl(t)w2(s) = Q12 min(t,s)

,

Let us give some examples. First we take b2=bl=b, c2=c1=c, d 2 = O r (11.30)

u2

=

u1

=

u

w1 = w2 = w

+

Q12 = I

I

then dxE = E

-

b(x E , ) a tXE

x (0) = x

+ c(xE ,)atXE

XE + u(x E ,y)dw

,

YE(t) = XE(t) 7, and we recover the homogenization problem.

PROBABILISTIC PROBLEMS AND METHODS

526

A second example is the following bl=O

b2=.1

c2=0,

d 2 = 0 ,

u l = u r

wl=w,

Yo=Y1=O

(11.31) I

then we have yE(t) = t and E

+

dxE = c(x",%)dt

u(xE,T)dw t

E

XE(O) = x

,

E

,

and we recover the averaging example. We can even consider a more general model than (11.27) which will include for instance reiterated homogenization.

Indeed we

. ..,yi(t) solution of

consider diffusion processes yi (t),YE (t),

the

system dy?(t) = 3

+

k

k= 0

I

Ej

(11.32) yE(o) = yo 3

+ 7 Y1 +

j = 0,

+

..*

.

...,N

If we take N = 1, and

u o = u1

,

u1

xE(t) = Y p )

,

then we reobtain (11.27).

= u2

,

w 0 = w1 '

yE(t) =

w1 = w2

,

If we take N = 2 and

I

527

AVERAGING

dxE =

1 b dt + 1 b2dt 2 1 E

y p ) = x (t)

,

icdt 4

yf(t) =

odw

,

7 E

then we recover the reiterated homogenization. Therefore the model (11.32) is the more general one in the sense We

that it contains all the problems considered in this chapter.'+

still call yi(t) the slow process, and the problem of interest is to study the limit of of y:(-),

).I:

as

E

-+ 0,

where u E is the probability measure

defined on Y = C 0 [O,m;Rn].

The first objective is to check under what condition 1 remain in a weakly compact subset ofV(+(Y).

can

).I:

To do this, we introduce

a vector function xE(yo,yl,...,yN) and apply Ito's formula, to yield

0

k=O

tThe precise values of c

E

jk

are easily defined.

ttExcept however the evolution problems with k # 2.

PROBABILISTIC PROBLEMS AND METHODS

528

and

xE

= O(E).

xE

We look for

as follows

...,YN) . Let us set N

(11.35)

-A2N+1-k =

>1 c. j = o lk

aYj

+

+ >:

i+j=k i,j N

aYiaYj 'a ai~ijo;

,

for N+1 5 k 5 2N

.

tr

We try to realize the following conditions AIXN

(11.36)

+

A2XN-1 +

A 1 ~ N + l+ A2xN A1X2N-1

+

... + Anx1 = - cON

+

A2X2N-2 +

+

AN+lxl =

...

+

'

- = 0,N-1 '

A2N-1X1 =

-

C

0,l

-

If we define ~ ~ r . . . , x in ~ ~order - ~ that (11.36) be satisfied, then one can easily check that the family u z remains in a weakly 1 compact subset of ?l+(Y). This will also be sufficient to obtain the

COMMENTS AND PROBLEMS

limit.

529

We omit the calculations which are quite tedious.

It is easy

to check that (11.36) correspond to the conditions obtained in homogenization, reiterated homogenization or averaging.

12.

Comments and problems. As has been noted in the text of the Chapter, Freidlin was the

first to consider the problem 2

(12.1)

-

aij ( F )

~

a uE axiaY j

au b. ( 5 ) 2 + a (-)X l € a x i O E

u

E

= f

and the limit of uE was obtained by probabilistic arguments. As stated in the general introduction, problem (12.1) is to some extent a natural extension of the averaging problem (where coefficients are rapidly varying in time in a periodic manner). In a completely independent way,analysts and numerical analysts have been working on "homogenization" with motivation coming from physics and mechanics as explained in the general introduction. In that context operators in divergence form appear more naturally than (12.1). The authors were the first to study the general framework (which includes (12.1) and divergence form operators)

To obtain a limit for u E l we need an assumption on the vector field b(y) I namely that

I

b(y)m(dy) = 0, where m(dy) is a suitable probabi-

lity measure on the torus. The study of the problem when this assumption is not satisfied is largely open.

PROBABILISTIC PROBLEMS AND METHODS

530

If w e use t h e method of a s y m p t o t i c e x p a n s i o n s , i t i s e a s y t o

check t h a t t h e a n s a t z should be o f t h e form

(12.3)

U

?I

E

U,(X)

-k

2

E U2

x (--,X)+

...

where u1 i s s o l u t i o n of

If a o ( x ) i s bounded below by a p o s i t i v e c o n s t a n t , t h e e q u a t i o n (12.4)

makes s e n s e , w i t h some a p p r o p r i a t e boundary c o n d i t i o n s . The i n t e r e s t U

i n g problem i s t h e n t h e s t u d y o f t h e convergence o f

,

expecially

t h e boundary l a y e r a n a l y s i s . Another q u i t e i n t e r e s t i n g problem conc e r n s t h e c a s e when t h e s u p p o r t o f f d o e s n o t c o i n c i d e w i t h 0 , and when t h e c h a r a c t e r i s t i c s o f

(12.4) do

not necessarily m e e t the

s u p p o r t of f . These problems a r e r e l a t e d t o t h e s t u d y of e l l i p t i c problems w i t h s m a l l d i f f u s i o n c o e f f i c i e n t s , a s c o n s i d e r e d by V e n t s e l and F r e i d l i n [l]. R e l a t e d c o n s i d e r a t i o n s f o r t h e Cauchy problem c o r responding t o (12.2) w i l l be g i v e n i n C h a p t e r 4

.

The p r o b a b i l i s t i c methods d e a l w i t h t h e s t u d y o f t h e convergence o f a f a m i l y of measures uE on C ( [O,-) ; R n ) convergence r e s u l t s . They a r e s u f f i c i e n t

. We

have o b t a i n e d o n l y weak

t o s t u d y t h e convergence

o f t h e e x p e c t e d v a l u e of f u n c t i o n a l s which a r e c o n t i n u o u s on C ( [0,-)

vior

;Rn)

.

But n o t h i n g more can be s a i d on t h e p r o b a b i l i s t i c beha-

o f t h e f u n c t i o n a l i t s e l f . I t would be v e r y i n t e r e s t i n g t o

approach t h e s t u d y o f t h i s b e h a v i o r from t h e p o i n t of view o f l a r g e d e v i a t i o n s , as i n Donsker-Varadhan

[l] o r a s i n Azencott-Ruget

[I].

Along t h e s e l i n e s , w e can mention t h e problem o f t h e b e h a v i o r of t h e p r i n c i p a l e i g e n v a l u e o f t h e o p e r a t o r (12.2) (non s e l f a d j o i n t c a s e ) . Does t h e p r i n c i p a l e i g e n v a l u e converge towards t h e p r i n c i p a l e i g e n v a l u e o f t h e homogenized o p e r a t o r ? T h i s seems v e r y l i k e l y , b u t it h a s n o t y e t been proved.

COMMENTS AND PROBLEMS

531

The same problem can be posed for the transport operator. It is a very important problem for the applications, and one should look for methods which can be carried over from diffusion operators to transport operators. Our methods (probabilistic or asymptotic expansions) can be used to recover classical results on the asymptotic behavior of random evolutions (see the survey by M. Pinsky [I]).

A

more general

problem can be studied within the framework of Transport Theory (see A.

Bensoussan, J.L. Lions, G.C. Papanicolaou 111 for details). The study of the behavior of reflected diffusions with rapidly

varying coefficients is also largely open. In the present chapter, we have considered only the averaging problem, i.e., we have not considered the case of reflected processes with unbounded drifts. We have also not considered the problem of reflection with rapidly varying directions. This prevents us from using probabilistic techniques to study the problem

x auE aij(F) axj ni[, =

o

.

To introduce a reflected process to study (12.5) is probhbly not a a good approach since we know that the boundary conditions do not play a significant role in the convergence of operators given in divergence form. It would be interesting to see if this comment can be interpreted in probabilistic terms. In any case we know nothing about the behavior of uE(x) near the boundary. It is likely that the behavior of the boundary process plays a fundamental role. This involves homogenization for Levy processes, which is an interesting problem by itself, probably not out of reach. It is not easy to study nonlinear problems by probabilistic

PROBABILISTIC PROBLEMS AND METHODS

532

techniques because this involves the study of nonlinear functionals of probability measures u E , which converge only in the weak sense. For instance, one can consider stochastic control problems or stopping time problems for diffusions with rapidly varying coefficients. What happens to the optimal control or the optimal cost as E

tends to 0

?

It is not true in general that the optimal cost, for

instance, will converge towards the optimal cost of the same stochastic control problem for the homogenized diffusion. This can be seen by studying the behavior of the Hamilton Jacobi equation, by analytic techniques, using the methods of Chapter I, I 16. The limit problem may not even be an Hamilton-Jacobi equation. In some cases, we can assert by analytic techniques that the limit problem is indeed the optimal cost of the same stochastic control problem for the homogenized operator. This is the case for instance, for optimal stopping time problems, which are related to variational inequalities (see

A.

Bensoussan

-

J.L. Lions 111). One

expects that it should be possible to use probabilistic techniques to obtain the desired convergence results. These questions are also related to the use of singular perturbation techniques in stochastic control problems. We refer to Kokotovic 111, O'Malley 111, Delebecque-Quadrat 1 1 1 for a precise description of the problems and results. Some nonlinear equations can be handled by probabilistic methods without using stochastic control techniques. These are the equations which can be interpreted in terms of branching processes.

As

we have seen in this chapter, probabilistic techniques require

a great deal of regularity on the coefficients of the diffusion. It would be interesting to relax these assumptions as m u c h as possible. We have seen in

§

11 that this is possible for averaging. The tech-

nical proof of Lemma 11.1,

is of course much more complicated than

COMMENTS AND PROBLEMS the proof which is available in the case of regular coefficients. It does not seem to extend easily to homogenization problems.

533

BISLIOGR9PHY OF CHAPTER 3

P. Azencott and G. Ruget, Mdlange d'dquations diffdrentielles et grands Bcarts a la loi des grands nombres, Wahrscheinlichkeits theorie, Band 3 8 Heft 1, 1977, p. 1-54. A. Bensoussan, J.L. Lions [l] ContrBle impulsionnel et indquations quasi variationnelles, Dunod, Paris, to be published. A. Bensoussan, J.L. Lions, G.C. Papanicoalou [l] Remarques sur le comportement asymptotique de systgmes d'bvolution, Proceedings of Franco-Japanese Colloquium, Tokyo, Sept. 1976. J.H. Chow, P.V. Kokotovic [l] Decomposition of near optimal state regulator for systems with slow and fast modes, IEEE Automatic Control, 1976. F. Delebecque, J.P. Quadrat, [l] Contribution of stochastic control, singular perturbation, averaging and team th ries to an example of Large Scale System : Management of Hydropower production, Special issue on large scale systems, IEEE on Automatic Control, April 1978. C. Doleans-Dade, P.A. Meyer [l] Intdgrales stochastiques par rapport aux martingales locales, Sdminaire de probabilitds IV, Lecture Notes in Mathematics, Springer Verlag, 1970. M.D. Donsker, S.R.S. Varadhan [l] Asymptotic Evaluation of Certain Markov Process Expectations for Large Time, Comm. on pure and Applied Math., vol XXVIII, p. 1-47 (1975). J.L. Doob [l], Stochastic Processes, Wiley, New York, 1967. M. I. Freidlin [l] Dirichlet's problem for an equation with periodic coefficients depending on a small parameter, Theory of Prob.and its Applications, 1963, p. 121-125. A. Friedman [l], Stochastic Differential Equations and Applications, vol.1 and 2, Academic Press, New York, 1976. [2] Partial Differential Equations of Parabolic Type, Prentice 534

BIBLIOGRAPHY

535

Hall, Englewood Cliffs, New Jersey, 1 9 6 4 . K. Ito, H.P. Mc Kean [l] Diffusion Processes and Their Sample Paths, Springer-Verlag, Berlin, 1 9 6 5 . 1.1. Gikhman, A.V. Skorokhod [l] Stochastic Differential Equations, Springer Verlag, NewYork, 1 9 7 2 . [ 2 ] Introduction to the Theory of Random Processes, Saunders, 1969.

I.V. Girsanov 1 1 1 On Ito's stochastic integral equations, Soviet Math. Dokl., 1 9 6 1 , p. 5 0 6 - 5 0 9 . P.A. Meyer [l] Probabilitbs et Potentiel, Paris, Hermann, 1 9 6 6 ; English Translation, Blaisdell, 1 9 6 6 . J. Neveu [l] Bases Mathgmatiques du Calcul des Probabilitgs, GauthierVillars, Paris, 1 9 6 8 . R.E. O'Malley [l] The singular perturbated linear state regulator problem, SIAM J. Control, vol. 1 3 , no 2, Feb. 1 9 7 5 . K.R. Parthasarathy [l] Probability Measures on Metric Spaces, Academic Press, New York, 1 9 6 7 . M. Pinsky, Multiplicative operator functionals and their asymptotic properties, Advances in probability and related topics, Vol. 3 , Peter Ney and Sidney Port ed., Marcel Dekker, New-York, 1 9 7 4 . R. Priouret [l] Problkmes de Martingales, Lecture Notes in Math., no 390, Springer Verlag, 1 9 7 4 . Y.V. Prokhorov [l] Convergence of Random Processes and Limit Theorems in Probability theory. Theory of Probabili y and its Applications, Vol. I, no 2, 1 9 5 6 . D. Stroock, S.R.S. Varadhan [l] Diffusion Processes with Continuous Coefficients, Comm. Pure Appl. Math. , Part I, 2 2 ( 1 9 6 9 ) , p. 3 1 5 - 4 0 0 ; Part 11, ibid., p. 4 7 9 - 5 3 0 . [ 2 1 Diffusion Processes with Boundary Conditions, Comm. Pure Applied Math., vol. XXIV, p. 1 4 7 - 2 2 5 ( 1 9 7 1 ) .

536

BIBLIOGRAPHY A.D. Ventsel, M.I. Freidlin, [l] On small random perturbations of dynamical systems, Russian Math. Survey 1970, vol. 25, no 1, p. 1-55. S. Watanabe [l] On stochastic Differential Equations for Multi Dimensional Diffusion Processes with Boundary Condition, J. Math. Kyoto Univ. 11 (1971) p . 169-180 : 11, p. 545-551. S. Watanabe, T. Yamada [l] On the uniqueness of solutions of stochastic differential equations 11, Journal of Mathematics of Kyoto University, vol. 11, n03, 1971. T. Yamada, S. Watanabe [I] On the uniqueness of solutions of stochastic differential equations, Journal of Mathematics of Kyoto University, ~01.11, nol, 1971.

Chapter 4 :

High Frequency Wave Propagation in Periodic Structures

Orientation. In this chapter we study asymptotic problems in periodic structures involving wave propagation when the typical wavelength of the motion is comparable to the period of the structure and both are small.

We also study several other related problems, including some

long wavelength (or low frequency) cases that were analyzed in previous chapters.

They are now seen in a somewhat different context

which should provide more insight into their structure. We begin in section one with a review of some notions and conventions regarding scaling.

It is impossible to be absolutely

precise about such things; we simply attempt to convey a point of view that we have found useful. Section 2 is intended as a general introduction to the WKB or geometrical optics methods.

It is hoped that by selecting for brief

presentation only the basic ideas, or the ones that we have found particularly useful in periodic structures, it will be of assistance to the reader.

It should be pointed out that there are ramifications,

both regarding the mathematical techniques as well as the physical applications, that are not dealt with at all here (or dealt with in very simplified forms).

For example we examine diffraction effects

only in a very indirect way.

The reason is simply that such problems

are very difficult already in homogeneous media; in periodic media they seem to be impenetrable.

In Sections 2.11-2.14 we Present Some 537

HIGH FREQUENCY WAVE PROPAGATION

538

results that deal with the asymptotic simplification (order reduction) of hyperbolic systems. In Section 3 we analyze the spectral properties of differential

operators with periodic coefficients and give the so-called Bloch expansion theorem. The Bloch expansion is used directly in Section 4 to solve several problems and analyze their asymptotic behavior.

We make

frequent contact with results of previous chapters by specializing to low frequency.

For the reader's convenience we provide the necessary

information here in order to avoid tedious cross references. Section 5 contains the general expansion process where the full force of the WKB method enters.

It is hoped that the close connec-

tions between the results in this section and those of Section 2 have been made clear.

There is a good deal of methodological and

conceptual unity linking all these problems closely. The phenomena, mathematical or physical, contemplated in this chapter come in much greater variety than they did in previous chapters. Consequently our results are much less complete or refined compared to the ones in Chapters 1, 2 or 3 . There is an enormous number of unsolved problems, as far as we know, in connection with practically every topic discussed here.

1. 1.1

Formulation of the problems. High frequency wave propagation. Consider the following form of the Klein-Gordon equation n

539

FORMULATION OF THE PROBLEMS w i t h f ( x ) and g ( x ) smootht f u n c t i o n s of compact s u p p o r t and c L ( x ) > 0 , W(x) 2 0 smooth.

W e know t h a t (1.1) h a s a u n i q u e smooth

s o l u t i o n i n e a c h f i n i t e t i m e i n t e r v a l . tt W e are i n t e r e s t e d i n t h e b e h a v i o r o f t h e s o l u t i o n of

(1.1) when

t h e c o e f f i c i e n t f u n c t i o n s c 2 ( x ) , W(x) and t h e d a t a have s p e c i a l forms.

A c o n v e n i e n t way t o d e s c r i b e t h e forms of i n t e r e s t i s by

i n t r o d u c i n g a s m a l l parameter

> 0.

E

W e now c o n s i d e r t h e f o l l o w i n g

problems. Suppose t h a t c2 and W are s l o w l y v a r y i n g f u n c t i o n s o f x s o w e 2 r e p l a c e c 2 , W i n (1.1) by c EX)^ W ( E X ) w i t h E > 0 and E s m a l l . Suppose a l s o t h a t t h e d a t a depend on

6

i n t e r e s t e d i n t h e b e h a v i o r o f u E , t h e s o l u t i o n of changes when

E

-+

0.

W e are

so w e w r i t e f E and g E .

( l . l ) ,under t h e s e

However, t h i s would be a t r i v i a l problem if w e

w e r e i n t e r e s t e d i n t h e b e h a v i o r of uE a s

E -c

0 w i t h x and t f i x e d .

The i n t e r e s t i n g and much more d i f f i c u l t problem i s t h e a n a l y s i s of uE as

E -c

0 when t and x a l s o change and become l a r g e a s

6 -c

0.

A

r e s c a l i n g o f s p a c e and t i m e v a r i a b l e s i s c a l l e d f o r . F o r t h e r e s c a l i n g w e i n t r o d u c e t ' and x' by t' =

(1.2)

E t

I

x' =

EX

,

and rewrite (1.1) i n t h e new v a r i a b l e s .

Dropping t h e p r i m e s , t h i s

y i e l d s t h e problem

'This

means Cm.

A ' g e n e r a l r e s u l t a b o u t r e g u l a r i t y of s o l u t i o n s o f h y p e r b o l i c s y s t e m s , c o v e r i n g a l l t h e n e e d s o f t h i s c h a p t e r , i s g i v e n by Rauch and ~~~~~y [ l ] . S i m p l e r theorems which s u f f i c e most of t h e t i m e can be found i n C o u r a n t - H i l b e r t r [ l ] .

540

HIGH FREQUENCY WAVE PROPAGATION

(1.3)

We emphasize that, despite the appearance of a large term in (1.31, when

E

is small, problem (1.3) is a rescaled version of one with

slowly varying coefficients. Of course the discussion is incomplete without specification of the dependence of fE and gE on

E.

Said another way, the terms slowly

varying, etc., are relative to scales of variation of the data since there are no other length scales in (1.1) (we have no boundaries). We now classify the data as follows. Class A. f (x) and gE(x) have asymptotic power series expansions in (1.4)

E

fE(X)

-

fo(x)

+ Efl(X) +

...

,

The terms in the expansions are Ci functions. Problem (1.3) with data (1.4), class A data by definition, is the static or low frequency problem.

This problem is

discussed further in Section 2.4 where it is shown that the terminology is not fully consistent (but merely convenient)

.

The expansions in (1.4) do not have to start with

E

0

.

Since the

problem is linear this makes no difference; the solution is simply multiplied by a power of

E.

FORMULATION OF THE PROBLEMS

541

I n t h i s c h a p t e r w e a d o p t f r e q u e n t l y ( b u t n o t a l w a y s ) t h e convent i o n t h a t t h e d a t a a r e t o be m u l t i p l i e d by a s u i t a b l e power of

E

( p o s i t i v e o r n e g a t i v e ) i n o r d e r t o make a l l terms i n t h e a p p r o p r i a t e e n e r g y i d e n t i t y of e q u a l s t r e n g t h i n E

-+

E,

i f p o s s i b l e , and f i n i t e as

0. L e t (,)

d e n o t e i n n e r p r o d u c t o v e r Rn

(complex-valued) and

t h e norm b a s e d on t h i s i n n e r p r o d u c t , i . e . L‘

I I

The e n e r g y

norm.

i d e n t i t y f o r ( 1 . 3 ) i s o b t a i n e d by m u l t i p l y i n g by uEt i n t e g r a t i n g o v e r Rn,

e t c . , which y i e l d s

+

(1.5)

2

(c

VUE,VUE)

1 + 7

(WUEIUE)

E

IUEtl

= 19,1*

2

+ (c VfEIVfE) +

W e see t h a t on t h e r i g h t s i d e of ( 1 . 5 ) t h e t e r m due t o t h e p o t e n t i a l

w

dominates ( w e assume W > 0 ) .

m u l t i p l i e d by a power of

E,

To make t h e e n e r g y f i n i t e as C l a s s B.

With d a t a of t h e form (1.4), o r ( 1 . 4 )

o n e c a n n o t change t h i s l a c k of b a l a n c e . E

-f

0 w e must t a k e f o ( x )

f E ( x ) and g E ( x ) have t h e form ( i =

0 i n (1.4).

a)

(1.6)

r e a l - v a l u e d and Cm,

ZE and GE a r e

complex v a l u e d and C i and have a s y m p t o t i c power

series e x p a n s i o n s l i k e ( 1 . 4 ) . ( 1 . 6 ) are complex-valued.

Note t h a t t h e d a t a

S i n c e (1.3) i s l i n e a r ,

b o t h t h e r e a l and i m a g i n a r y p a r t s o f u E are s o l u t i o n s b u t t h e complex n o t a t i o n i s c o n v e n i e n t and w i d e l y used. c a l l e d c l a s s B.

D a t a of t h e form ( 1 . 6 )

are

542

HIGH FREQUENCY WAVE PROPAGATION To understand the physical significance of (1.6) suppose

(1.7)

I

S(x) = k * x

,

.

where k is a constant vector and dot stands for the inner product of vectors in Rn.

The function g(x) in (1.6) is called the phase

function. The particular phase function (1.7) gives data (1.6) that are spatially modulated plane waves with rapidly varying phase. term rapidly varying corresponds to having .? (x)/E rather than in (1.6).

The

s (x)

In general the surfaces of constant phase .?(x) = constant

are not planes, as with (1.7), and the initial data are spatially modulated waves with rapidly varying phase. Note that all three terms on the right side of (1.5) are now of -2 order E To make the energy of order one as E 0 we must either

.

take T o

-f

4,

I

0 or replace uE by E U ~ .

Initial data of the form (1.6) may seem at first too special.

As we noted, (1.3) with class B data does correspond to high frequency wave propagation in a slowly varying medium.

However, we

shall see that by linear superposition of asymptotic solutions of such problems, solutions to more general problems can be obtained. Class C.

Consider the case where fE and gE have the form fE(X) =

E1 -n'2f

,

(x,;)

Here f and g are Cm functions of compact support in x and y

(5

X/E) ,(x,y)

E

R2n.

We could have taken f and g depending onE with an expansion like (1.4) but (1.8) will suffice.

The scaling in (1.8) is chosen so that

the terms on the right side of (1.5) are of order one as

E +

0.

This

543

FORMULATION OF THE PROBLEMS

is easily seen by noting that for any function h(x) E

Cm

with compact

support,

where the convergence is in the distribution sense. Problem (1.3) with data of the form (1.8), class C data, corresponds to wave propagation in a slowly varying medium with spatially localized data. of the coefficients c '

Said another way, the scale of variation

and W and the typical diameter of the support

of the data are comparable and both are small. Definet

which are

Cm

functions of compact support in x.

We may write (1.8)

in the form

(1.11)

By linearity, the solution of (1.3) with data (1.11) is the integral over all

'When Rn

.

K

of the solutions of (1.3) with data

the domain of integration is not shown it is assumed to be

HIGH FREQUENCY WAVE PROPAGATION

544

(1.12)

This is just a family of problems, parametrized by of class B (i.e. (1.6)). We conclude that:

K E

Rn, with data

the analysis of (1.3)

with class C data reduces effectively to the analysis of (1.3) with class B data. So far we have restricted ourselves to the pure initial value

problem (1.3).

First, it is clear that the above scaling notions

carry over to

u (X,O) =

f E M,

which is the forced problem corresponding to (1.3).

The forcing

function may be rapidly varying in both space and time.

A typical

example is

-

m

where w > 0 is fixed, h(x,y), y = X/E, is C

and has compact support

and a is suitably chosen to make uE behave in some prescribed way as E

-t

0 (for example, with energy O(1)1 .

Second, one can consider (1.3) in the presence of boundaries where homogeneous or inhomogeneous boundary conditions are satisfied. In the inhomogeneous case, the data may be rapidly varying in space or time.

In the simplest cases the phenomena of interest are reflec-

tion and transmission.

More complicated phenomena involve diffraction.

FORMULATION OF THE PROBLEMS

545

We shall not discuss such problems in this chapter, as was mentioned in the introduction, except for some simple examples.

1.2

Propagation in periodic structures. We consider directly the scaled problem (1.3) and ask the

If c2 and W are not only functions of x, i.e. slowly varying functions,but also functions of X/E, c2 = c2 (x,;x ), X 2 w = W(x,r), such that c (x,y), W(x,y) are periodic in y of period 2a following question.

in all directions,'

how does the corresponding solution uE behave?

(so that W = W(x,y) evaluated at y = X/E) as above are called locally periodic. If c2 = c2 (y),

Coefficients c2 and W depending on

W = W(y) with y =

X/E

E

then they are called globally periodic.

We point out that the new problem

has three natural length scales associated with it: of variation of the coefficients, the

fast scale of

the

slow

scale

variation of the

coefficients whichis proportional to 2 7 ~ and ~ the scale of variation of the data which has not yet been specified. As in Section 1.1 we distinguish three classes of data

'It

is convenient in this chapter to depart from the convention

of previous chapters where the period was one.

Y

=

We denote by

(lO,l[)", the n-dimensional torus (end points of intervals

identified).

The 2a-torus is denoted by 2aY.

546

HIGH FREQUENCY WAVE PROPAGATION

Class A.

Data of the form (1.4).

Problem (1.15) with such.

class A data, is the static or low frequency homogenization problem.

The scale of variation of

the data is comparable to the slow scale (cf. Section 2 . 4 for additional remarks regarding class A). Class B.

Data of the form

(1.16)

where s(x) is real-valued and Cm and f(x,y), g(x,y) are complex-valued, Cm(Rn

x

2aY) (i.e. they are

2n-periodic in y) and of compact support in x. As in Section (1.1) this class of data, class B, is called the spatially modulated waves with rapidly varying phase.

Note that the

modulation is not necessarily slowly varying, i.e. we have dependence.

X/E

However, the rapid variation in the modulation is

periodic and as such it does not induce propagation of energy (these matters are taken up in detail in the following sections). convention, we do not call data of the form (1.16) with

2

By 0 “high

fequency” data; we call them rapidly varying periodic data. Class C.

Data of the form (1.8) with precisely the hypotheses introduced there.

This class, class C, is called

the class of spatially localized data. Of course, the above distinctions are to some extent arbitrary. For example, why not data of the form, say,

THE W . K . B .

OR GEOMETRICAL OPTICS METHOD

with the hypotheses of (1.16)?

541

The reason is that if k < 1 then we

are essentially dealing with rapidly varying periodic data. Similarly if k > 1 then the coefficients are relatively slowly varying relative to the phase so the rapidly varying periodic coefficients play a minor role, i.e. we are reduced to problems of the form discussed in Section 1.1.

The "interesting" case therefore

is precisely the one we have singled out; namely, the one where the induced waves interact genuinely with the periodic structure. A

final remark about class A data (low frequency data).

For

many equations, including (1.3) and (1.15) when W Z 0, the fact that s(x)

0 does

a later time.

not imply that rapid phase variations do not appear at This again points out that class B data is a basic

class among the three (cf. Section 2.4 for additional discussion).

2.

2.1

The W . K . B .

or geometrical optics method.

Expansion for the Klein-Gordon equation. To illustrate the methods in the title we shall work consistently

with one example (until Section (2.10)) which is at first the following:

To analyze as

E +

0 the behavior of uE, solution of the

equation

x E R n 1

548

HIGH FREQUENCY WAVE PROPAGATION

with s(x), Cm and real-valued, f and g complex-valued Cm with compact support. Also cL(x) 2 co > 0 and W(x) > 0. We begin by attempting to construct an expansion for uE of the form

Inserting (2.4) into (2.1) yields the following equation for vE (2.6)

(E-'A~

+

~ = ) o

E - ~ A+~~

, v

~

where (2.7) (2.8) (2.9)

A1 = -c2 (VS)2 + (Stl2 - W =

+

-2istat

A3 = V*(C2V

)

iV. (c2VS

- at2

)

, + ic2VS-V -istt '

.

Inserting (2.5) into (2.6) and equating coefficients of equal powers of

E

yields the following sequence of problems

,

(2.10)

AlvO = 0

(2.11)

A 1v1 + A2v0 = 0

(2.12)

Alv2 + A2v1

+

,

A3v0

,

etc.

.

From (2.10) we conclude that if vo is not to be identically zero we must have (2.13)

St +[C~(X)(OS)~+ WI1I2 = 0

.

THE W.K.B.

OR GEOMETRICAL OPTICS METHOD

549

This is the eikonal equation for the phase function (actually two of them,one with +,one with - ) .

Its solution is discussed further in

the next section. Note that this choice of S makes the operator A1 identically zero. Hence (2.11) is directly an equation for vo(x,t), i.e. (2.14)

A2V0 = 0

.

This equation is called the transport equation and is discussed Note the coefficients of A2 depend on S.

further in Section 2.3.

When the plus sign is used in (2.13) we write

S

+

(and similarly S-)

.

The corresponding vo is denoted by v6. Equation (2.12) and the succeeding ones of higher order are called the higher order transport equations. S-

Corresponding to

S+

or

-I-

we have v1 or vy, etc. The solution now is represented in the form

+ t)

E 2v,(x,t) +

+

+

E 2 v,(x,t) -

...I +

...1

with uEt(x,t) asymptotic to the formal derivative of the terms on the right side of (2.15).

Comparison with the initial conditions in (2.1)

yields initial conditions for ' S

+

and vk, k 1. 0.

The validity of (2.15) as an asymptotic expansion, and the corresponding one for uEtr follows immediately after it has been shown that such an expansion, with smooth ' S

and vi, k 2 0, exists.

The energy identity (1.5) shows in fact that

N where uE consists of the sum of terms including the one proportional

to

E~

in (2.15).

HIGH FREQUENCY WAVE PROPAGATION

550

2.2

Eiconal equation and rays. We now consider (2.13) in some detail.

+

We pick the one with the

sign and set

(2.17)

St

+

w(VS,x) = 0

,

S(0,x) = Z(X)

,

X E

Rn

,

where w(k,x) is the Hamiltonian (or radian frequency) (2.18)

w(k,x) = [c2 (x)k2 + W(x)l 'I2,

k E R n ,

xERn.

The eiconal (or Hamilton-Jacobi) equation (2.17) is a first order

t

nonlinear PDE that controls the evolution of the phase function.

To solve (2.17) we introduce Hamilton's system of ODE'S for the x(t) and momenta (or wavenumbers) k(t).

x (0) = x

P

P'

(2.19)

p = 1,2 Recall that W is assumed positive and function of k and x.

Cm

,...,n .

which makes w(k,x) a

Cm

System (2.19) has a unique solution for any

finite time interval which depends smoothly on the initial value of x and k.

'The

Suppose (2.17) has a smooth solution.

S+

is called the forward and

S-

Then

the backward moving phase

by extending the terminology of the plane wave case:

+ = k-x W(x) = constant, w = w(k) and S-(x,t)

w(k)t.

g(x) = k-x,

THE W.K.B.

OR GEOMETRICAL OPTICS METHOD

551

n

=

-w(k(t),x(t))

+

>:

aw(k(;Lrx(t)) kp(t) P

p= 1

,

where we identify (2.21)

k(t) = VS(x(t),t)

.

If we also introduce the Lagrangian

where k is considered as a function of x and

G

obtained fromt

x = V kw(k,x), then we have the representation L

(2.23)

S(t,x(t)) = g(x) +

i

0

L(x(s),G(s))ds

.

From (2.18) we find the following formulas:

(2.25)

c2

(2.26)

k =

-

x2 = c2W(c2k2

+

W)-'

> 0

,

-C

Consider the system (2.19) with the 2-point boundary conditions (2.28)

'This

x (t) = x P

,

k (0) = a'(x) , ax P

p = l,Z,...,n

P

is possible because w(k,x) is convex in k.

,

552

HIGH FREQUENCY WAVE PROPAGATION

and suppose it has a unique solution for 0 The mapping 2 (2.29)

-t

5 t 5 to for some to

x(t,%), x(O,%) = % is one to one for 0 2 t

S(t,x) = ?(XI

+

1 0

,

L(x(s),;(s))ds

is the unique smooth solution of (2.17).

0

5 t 5 to

> 0.

5 to and 8

In (2.29) the path x(s) is

the one satisfying (2.19) with boundary conditions (2.28). A point (x,t) at which the mapping 2

-t

x(t,G), ~(0,;) = 2 be-

comes singular (its Jacobian vanishes) is a conjugate or focal point of the Hamiltonian system. solutions.

At such points (2.17) fails to have

In fact the whole expansion scheme (2.4), (2.5) is

inappropriate there and different types of expansions are needed.

We

restrict discussion to the local expansion at present, i.e. up to a time t

0

2.3

> 0 without focal points.

Transport equations. We consider now equation (2.11) which in view of (2.10) and

(2.8) becomes (2.30)

-2StvOt + V*(C 2VSv0)

+ c2VS.Vv0 - S ttV 0

+

This equation is satisfied by both vo and vi. 0

= 0

.

Multiplying (2.30) by

t

and adding to the conjugate of (2.30) multiplied by vo, we obtain

(2.31)

2 2 2 (wlvo )t + V*(c klvOl 1 = 0

,

where we write (2.32)

'Bar

k = k x,t) = VS(x,t)

,

w = w(x,t) = w(k(x,t),x)

denotes complex conjugate.

.

THE W.K.B.

OR GEOMETRICAL OPTICS METHOD

553

This abuses notation to some extent but the meaning will be clear Note in particular that k(t) = k(x(t),t) with

from the context.

k(t), x(t) as in ( 2 . 1 9 ) . From ( 2 . 2 4 )

it follows that ( 2 . 3 1 )

(WlV0l2 )t + V.(&

(2.33)

ak

W[V

0

I 2)

= 0

has the form

.

2 The quantity WIv 0 I will be identified in Section 2 . 5 with the first approximation to the wave action. Thus ( 2 . 3 3 ) is a conservation

equation for the first approximation of the wave action.

The

velocity field is identified as the group velocity

From ( 2 . 3 2 )

and ( 2 . 1 7 ) we obtain, in addition to ( 2 . 3 3 1 ,

the following

kinematic equations for the wavenumber and radian frequency fields k(x,t), w(x,t):

n

(2.36)

ak $ +q=1c & s+ g =p P 0 ,

9

Using (2.3'7),

= 1,2

,...,n .

9

we obtain from ( 2 . 3 3 )

an equation for w 2 IvOl2 which

will be identified later as the first approximation of the wave energy (2.38)

(W

2

2

aW

2

[ v o [)t + V * ( s w lv0l

2

) =

0

.

Consider an infinitesimal tube of rays between t = tl and t = t2, o c t

< t 2 5 to (to > 0 as in Section 2 . 2 ) .

Integrating ( 2 . 3 8 )

over

this tube and using the divergence theorem we obtain the usual law of

554

HIGH FREQUENCY WAVE PROPAGATION

energy propagation of geometrical optics: The energy density at a point x is inversely proportional to the density of rays at the point. If 2

+

x(t,%), x(O,%) = 2 is the ray transformation, with t

2

to,

then the above statement translates to

where J(t,%) is the Jacobian determinant of the transformation. Note that the w2 cancels in (2.39) because w = w(k,x) does not depend on t and is hence a constant of the motion (2.19). From (2.39) it follows that vO(t,x), determined from (2.33), (2.38) or (2.39) , becomes singular at focal points of (2.19). is why such points are also called caustic points:

This

the wave

intensity at these points is higher on account of the focusing of the rays.

At such points the expansion (2.4), (2.5) fails.

The higher order transport equation (2.12) may be treated in much the same way to obtain the second term vl(xft). It satisfies the inhomogeneous equation (2.40)

2wvlt

+

=

wtvl

+ V.(ao wv ) + ak 1

itV*(c2Vvo

-

VOttl

w **VV

ak

1

-

Multiplying by vl, rearrang ng and integrating over an infinitesimal tube or rays we obtaint (2.41)

'J(t)

2 wvl(x(t) ,t)J(t) = wv 2 1(x,O) + i

= J(t,%) is the Jacobian of x

x(t) satisfies (2.19).

r

vl(x(s) , s ) ov0(x(s) ,s)ds

0

+

x(t,x), x(0,x) = x where

THE W.K.B.

555

OR GEOMETRICAL OPTICS METHOD

where (2.42)

0

=

v.

(c2V . )

-

.

a:

Equation (2.41) is equivalent to

as can be verified by direct calculation. We see therefore that as long as J(t) does not vanish (no caustics), the expansion (2.4) and (2.5) can be constructed by solving ordinary differential equations only. We turn now to the determination of the initial conditions, i.e.

.

of vi(x,O), v;(x,O),...

From (2.15) and (2.21, (2.3) we obtain,

evaluating functions at t = 0,

+ E(VO

+ vo) - +

- -iw

(EVA

+ iw

(EVO

E 2 (vl +

+ v;,

+

... =

Ef

,

and

(2.45)

f

v1

=

+ E2V; +

-+

E 2v1 -

+

...) ...)

+ E 2Vlt

+

(wit +

E 2Vlt -

+

+

i + +- 2w (vOt + vOt) ,

+ + The specification of vOI vi,...,

+ (EV;t +

etc.

.

is now complete.

...) ...)

= 9

.

556 2.4

HIGH FREQUENCY WAVE PROPAGATION Connections with the static problem. Suppose first that the potential W is strictly positive as we

have been assuming and that g(x)

0 in (2.2) and ( 2 . 3 ) , i.e., the

data has no rapid phase variation. With w as in (2.18) we see that for t > 0 S(t,x) f 0 even when i(x)

E

That is:

0.

the system itself creates rapid phase variations

whether or not the data have them. This explains why we remarked that the classification of data in Section 1.1 is not entirely appropriate. On the other hand: (2.46)

If the Hamiltonian is a homogeneous functiont of k,then slowly varying data (s(x) : 0) produce slowly varying solutions (S(t,x)

0, t > 0).

This is immediate from (2.17) since by homogeneity (2.47)

w(0,x) : 0

.

Moreover

and hence the transport equation (2.30) is trivially satisfied. Indeed the problem (2.1) is now trivial since W : 0 and

s

: 0 leave

us with a problem without a small parameter, i.e. the wave equation which is to be solved exactly.

'Of

degree one, two, etc., this means W(x) E 0 and 2

2

w(k,x) = c (x)k in (2.18).

THE W.K.B. OR GEOMETRICAL OPTICS METHOD

557

The example (2.1) at hand is too simple to illustrate some interesting phenomena that take place when the Hamiltonians

are

homogeneous but at the same time the static problem is nontrivial. The problems with periodic structure have this feature.

We return to

this point in Section 2.13.

2.5

Propagation of energy. Consider the energy identity for (2.1) obtained by multiplying

(2.1) by

uE and

integrating over a region '0

conjugate of (2.1) multiplied by u

=

where

jao

.

,

and by adding to it the

This yields

1

k2(uEtviiE + iiEtVuE) dS(x)

& is the unit outward normal at the boundary

, ab

.

Define the

energy density EE(x,t) and the flux densi y F € ( x , ~ )(a vector) , respectively, as follows

I

IVUE(X,t) 2

(2.51)

F (x,t) =

-

Et

The differential form of (2.49) is

(2.52)

a E E + V-FE = 0 at

.

558

HIGH FREQUENCY WAVE PROPAGATION We now use the expansion (2.15) in (2.50) and (2.51).

We find

thatt

2

+ 2 +

= w lvol

(2.54)

F,

aw+ -- ak

2

+

O(E)

+ rapidly oscillating terms ,

+

Ivo

+ rapidly oscillating terms

.

Comparing (2.52), (2.53), (2.54) and (2.38), we see that (i) Each of the two modes

(+ =

forward, backward)

satisfies its own energy equation to highest order so that (2.52) holds to highest order; (ii) The rapidly oscillating terms drop out because (2.52) is always interpreted in the integral form (2.49). We also note that the terminology used in Section (2.3) is indeed appropriate if we recall that action is, by definition, energy divided by (radian) frequency.

THE W.K.B. 2.6

OR GEOMETRICAL OPTICS METHOD

559

Spatially localized data. Consider now (2.1) with data f

(2.55)

fE(x) = E ~ - f~(X'$/ ~

(2.56)

gE(x) = E-"/'g(x,:)

and gE given by

, ,

with f(x,y) and g(x,y), Cm functions on Rn support.

x

Rn and with compact

This corresponds to class C data (cf. (1.8)).

Let ?(x,K) and G(x,K) be defined by (1.10) and similarly let fE(X,K)I gE(x,K) be defined by (1.12).

With

K E

Rn fixed let

uE(X,t;K) be the solution of (2.1) with initial data equal to fE(XIK), gE(x,K). (2.57)

-

Clearly in this case

S(x) : K - x

,

The asymptotic expansion of uE takes the form

' [EVi(X,t;K)

with

K

a parameter.

+

E 2V1(Xnt;K) -t

Note that the factor

E-~"

...I

8

has been pulled out-

side on the right in (2.59) so that vo, vl, etc. do not depend on

+

+

E.

The initial values of vOl vi, etc. are determined by (2.441, (2.45) I etc. with f

+

?(x,K), g

+

G(x,K).

560

HIGH FREQUENCY WAVE PROPAGATION The solution of (2.1) with data (2.55), (2.56) is then, by

linearity,

K

+ eis- (x,t;K It remains therefore to simplify the integral in (2.60 convergent since

? and +

inherited by the v-

0'

decay r pidly as

IKI

+

-

and

, which is his is

etc.

Consider one integral and let us drop the superscripts (2.61)

I (x,t) =

J

E - ~ / ~

(x,t;K ) / E Vo(X,t;K)dK

-

We analyze this integral asymptotically as

.

0 by the method of

E +

stationary phase.

The region of integration in (2.61) may be taken n as any compact subset of R which is sufficiently large, in view of

the rapid decay of vo as

IKI

+ m.

We fix nowt (x,t), t > 0, and consider the system of equations p = 1,2 If there does not exist a

K

,...,n .

satisfying (2.62), then I E at this point

(x,t) decays to zero faster than any power of

E

which means that at

such points the integral IE is interpreted as zero. I f there is a unique point

K*

=

K*(x,t) such that (2.62) is

satisfied and such that the matrix

~

tRecall that 0

5 t 5 to and to

> 0 is sufficiently small so that

the phase function S ( X , ~ ; K )exists in 0 5 t 5 to and is uniquely defined

.

THE W.K.B.

561

OR GEOMETRICAL OPTICS METHOD

t

is nonsingular, then we have (2.64)

IE(x,t)

(2a)-n/2 ldet Q(Xrt)I 1/2

-

'VO(X,t;K*)

iS(x,t;~*)/e+ iua/4

.

Here u = o(x,t) is the signature of the matrix Q(x,t), i.e. the number of positive minus the number of negative eigenvalues.

If Q is

singular at (x,t) then (2.64) is not valid. Let us examine in detail the nature of equation (2.62) and its solutions K*. (2.65)

Define

U (x,~;K= )

P

as (X,t;K) aK

I

p = 1,2

P

,...,n .

tt From (2.17) it follows that (2.66)

au

+

au >:Eq 2 q=1

= 0

I

9

p = 1,2,

U (X,O;K) = X

P

P I

..., n .

To solve (2.66) we use x(t;x,~)which is the solution of the ray equations (2.19) with x(0) = x and k ( 0 ) =

K.

Then U (x(t;x,~)~t;K) P

is independent of t and hence (2.67)

U (x(t;x,~),t;K)= x P' P

tR. M. Lewis

[

1]

p = l,2,...ln

.

V. P. Maslov, J.J. Duistermaat [I].

ttThe group velocity

c

is defined by (2.34).

562

HIGH FREQUENCY WAVE PROPAGATION

Thus (2.68)

U (x(t;O,K),t;K) = 0 P

.

Given (x,t) fixed with t > 0, if there exists

K*

then this K* also solves (2.65) and conversely. immediately from (2.68).

= K * ( x , ~ )such that

This follows

Note also that p = l,21...ln

.

Consider now the pencil of rays x(t;O,K) issuing out of the origin and such that k(O;O,K) = (2.19)).

K

(i.e. family of solutions to

This family defines a conoid in (x,t) space t > 0 and since

W > 0 and 0 < c(x) <

m,

the local speed along the rays never exceeds the local phase velocity c(x)

.

This conoid is called the ray conoid (or light cone or forward

light cone). exists no

K*

If (x,t) is not in the interior of the light cone there and hence the solution us is zero.

If (x,t) is in the

interior of the light cone and t is sufficiently small then there will be only one ray that connects it to the origin and so uE behaves asymptotically like (2.64) along that ray (with that the

S+

and

fact S-(Xlt;K) =

S-

K*

constant).

phases cannot be stationary at the same

+ -S (x,tl-~).Said

another way:

since in

If for a given

corresponding to a fixed (x,t) and

S+

forward cone, the ray with this

corresponding to

K*

K

Note

the ray with this S-

K*

K*

is in the

is in the

backward cone and hence does not contribute. In conclusion, inside the forward light cone the solution ue has the form

THE W.K.B.

where

K*

OR GEOMETRICAL OPTICS METHOD

= K * ( x , ~ )is the (assumed unique) solution of (2.65) for -I

this (x,t) (and with S = S ) . 0 < t

The time ranges over some interval

5 to before caustics appear. Also, (2.70) is not valid at

t = 0 since the right hand side is singular. u

563

is zero.

Outside the light cone

On the boundary of the light cone (2.70) is not valid

because the integral in (2.61) is taken over a compact set of

K'S.

The expansion (2.60) is valid up to some time to > 0 and an estimate like (2.16) is easily obtained (with T = to).

The subse-

quent approximation of uc by the stationary phase method is not uniformly valid as we have just explained.

We note finally that the

result (2.70) could have been obtained directly by constructing special solutions to (2.17) and then matching the singularityt in (2.70) with the singularity of the exact solution of (2.1) with constant coefficients c2 = c2(0),

2.7

w

=

~(0).

Expansion for the fundamental solution. Consider the problem (2.1) with initial data

(2.71)

'This

uE(x,O) = 0

,

is frequently called the method of canonical problems,

cf. R. M. Lewis [ 11, J. B. Keller [ 11.

HIGH FREQUENCY WAVE PROPAGATION

564

(2.72)

at (X,O) = ,-lS(x)

.

Here S(x) is the delta function centered at the origin and the factor -1 E is inserted to make u of order one as E .+ 0. We note that this problem has distribution-valued initial data and consequently the solution itself will be a distribution.

The question is how does

this distribution solution behave as

O?

E

+

We clarify first what is meant by "behave". Suppose f(x) and m

g(x) are Co functions. Let G, be the distribution solution of (2.1), (2.71) and (2.72). (2.73)

Thent

u, = GEt*f

+ G,*g ,

is the solution of (2.1) with slowly varying data u,(x,O) = f(x), -1 uEt(x,O) = E g(x). For each f and g E Ct the expansion of u,(x,t) (for t sufficiently small) was obtained in previous sections. However the estimate of the error depends each time on f and g and their derivatives. Here we want to construct a distribution GN such N that G, G is smooth and becomes smoother as N increases. Moreover N this is to hold uniformly as E. -+ 0. As a consequence, G, contains

-

the singular part of G

for each

E

i.e. we have precise information

about the character of the singularity of G

.

There are a number of ways GN can be constructed.+'

Since we

already know in sufficient detail the behavior of smooth solutions and since the nature of the singularities of GE in media with periodic structure appears to be very complicated, we shall not discuss the analysis of G, further.

tHere

*

denotes convolution.

ttP. D. Lax [l], Zauderer [l], J.J. Duistermaat [l].

565

THE W.K.B. OR GEOMETRICAL OPTICS METHOD 2.8

Expansion near smooth caustics. The construction of asymptotic expansions of uE, solution of

(2.1), (2.2) and (2.31, valid globally in t, and not merely until the first time to that caustics appear, can be obtained by the general methods of Keller, Maslov and Hormander.’t This expansion reduces at each non-caustic point (x,t) to a sum of terms of the form (2.15) with their phases appropriately adjusted by factors of s/4.

In the

immediate vicinity of a smooth caustic the field does not behave like (2.15) but rather in a manner first described by Ludwigtt and Kravtsov. Instead of (2.4) and (2.5) one uses the ansatz (2.74)

where v

uE = eis(x t)/E

[V( E-2’3p

(x,t )

,t)

) V E (x

and wE have asymptotic power series expansions as in (2.5)

and p(x,t) = 0 is assumed to be the equation defining the smooth caustic surface. argument and V “

+

The function V(r;) is the Airy function of minus its r;V = 0.

The Airy function models locally the

behavior of the field across the caustic from the illuminated side p > 0 to the shadow p < 0.

The expression (2.74) is inserted into (2.1) snd codfficients of equal powers of E of V and V ’ are set to zero separately.

This leads

to eiconal equations, rays, transport equations, etc. analogous to the ones of the usual WKB with some important differences.

We refer

to the paper of Ludwig for the complete analysis.

‘cf. V. P. Maslov 111, J.J. Duistermaat [l], J. Leray [1,2] and references herein. See also, J.B. Keller [2]. ++D. Ludwig [ 11 , Yu. A. Xravtsov 111.

566

H I G H FREQUENCY WAVE PROPAGATION

In media with periodic structure the ansatz of the form (2.74), suitably generalized, works just as well as it does for (2.1).

2.9

Impact problem. This is an initial-boundary value problem consisting of equation

,...,xn)),

(2.1) for t > 0 and x E Rn such that (with x = (x1,x2 xn > 0, the initial conditions (2.75)

uE(x,O) = 0

,

uEt (X,O) = 0

,

and the boundary condition (2.76)

uE(x',O,t) = @(x',t)

,

x = (x',xn) ,

t > 0

with @(x',t) a given smooth function such that @(x',O)

Z ,

,

0 or

@(x',O) E 0 but some t derivative of @ at t = 0 is not zero. Notice that the impact function $I does not depend upon

E.

For n = 1, one space dimension, this problem was analyzed in detail by Reiss.+

These results have not been generalized to more

dimensions.

A simpler version of the impact problem consists of (Z.l), (2.75) and

and f (x',t) is complex-valued

where S(x',t) is real and

Cm

compact support in Rn-l

[O,m).

x

Cm

and of

One can also consider space-time

localized data on the hyperplane xn = 0 by Fourier representation and

'E.

L. Reiss

[

1 1, D. Ahluwalia, S. Stone and E. L. Reiss 111.

THE W.K.B. OR GEOMETRICAL OPTICS METHOD stationary phase analysis as in Section (2.6).

567

It follows that the

impact problem with space-time data (2.77) is a basic problem and we now consider it briefly. First we note that (2.1), (2.75) and (2.77) has a smooth solution for each t

This follows from the fact that the data (2.77)

vanish identically in the neighborhood of the space-time corner xn = 0, t = 0 (cf. Rauch and Massey [l])

.

Next we note that if we obtain an expansion which satisfies (2.77) identically, then the energy identity (1.5) can be applied to the difference of the exact minus the approximate solution.

Now the

inner product is over x = (x',xn) with xn > 0 and the boundary terms vanish.

The approximate solution (i.e.f the expansion) must be valid

throughout xn > 0 for this argument to work.

This puts a serious

restriction that one expects however should be removable. Consider again the ansatz ( 2 . 4 ) , Section

2.1

at first.

(2.5).

Everything goes as in

It will be clear however that in (2.15) only

the terms with superscript

+

should appear.

Consider (2.17).

We

prescribe initial conditions g(x) : 0 and boundary condition (2.78)

S(x',t) = S(x',t) on xn = 0

,

t > 0

.

We assume that the eiconal equation (2.17) with (2.78) has a smooth solution globally for xn > 0, 0 5 t

5T

<

-.

This rules out the

formation of caustics for rays issuing out of the hyperplane xn = 0 (t > 0 ) .

With this assumption the construction of the expansion

proceeds as before.

+

- ...,

The transport equations for vof vo, particular, initial values for f and for zero.

+ vlf...

zero.

+ vo

etc., are as before.

In

are given on xn = 0 by the function are identically The quantities vif v

;,...

This is the only way that (2.75) will be satisfied.

HIGH FREQUENCY WAVE PROPAGATION

568

The above generalize to periodic media under the assumption that no caustics form (for each mode under consideration) and a few others as we discuss in detail later (such as modal nondegeneracy).

Symmetric hyperbolic systems.

2.10

Many equations of mathematical physics can be written as symmetric hyperbolic systems which admit an elegant theory.'

First

we describe the general framework and then give some examples.

..., An (x) be hermitian N

Let A1 (x), A2 (x), of x

E

x

N matrix functions

0

Rn and let A (x) be hermitian and positive definite and B(x)

an arbitrary N functions.

x

N matrix.

The entries are assumed to be smooth

Consider the system of partial differential equations n

where u(x,t) and f(x) are N-vector functions and matrix multiplication is implied in (2.79). Suppose (2.79) has a differentiable solution. We show that this solution is unique and define the domain of dependence and range of influence of a point. Let k

E

Rn be a fixed vector and consider the characteristic

equation associated with (2.79) (2.80)

n Q(w,k) = det[Aow + c A P k p ] = 0 p= 1

-

'Courant-Hilbert, Vol. I1 [ 1 I and the many references cited therein.

THE W.K.B.

OR GEOMETRICAL OPTICS METHOD

569

This is an eigenvalue equation for each k fixed with the eigenvalue parameter being w . depend on x.

Note that w depends on k and x since the matrices

From the assumption that the matrices Ap are hermitian

and Ao is positive definite it follows that (2.80) has N-real solutions w1 (krx),w2(k,x),

...,aN(k,x), counting multiplicities.

Moreover

there exist for each x and k N-linearly independent eigenvectors for the matrix on the right side of (2.80), unique up to a constant factor (which may depend on x and k). Since (2.80) is an algebraic equation depending analytically on k, the roots wm(k,x)are differentiable functions of k and x at all points where they are distinct.

Similarly the eiqenvectors can be

chosen to be differentiable functions of x and k locally in the neighborhood of a point where the corresponding w is isolated.

It is

difficult to say, in general, much more about the eigenvalues

and

Wm

the corresponding eigenvectors as regards their global behavior with respect to k and x (except in the case n = 1).

We shall assume that

for all k and x the wm(k,x) 1kI-l are uniformly bounded in absolute value. are solutions of the

The mth bicharacteristic curves or Hamiltonian equations

(2.81) dk(m) (t) = dt

awm (k(m)(t) ax

(t)

,

k(m)(0) = k

.

The system of rays issuing out of a point x forms a multiply sheeted conoid.

The convex hull of this conoid is called the (forward)

conoid (or light cone).

Similar definitions hold for the backward

ray conoid. Let G be a bounded region of space-time, t 2 0. identity associated with ( 2 . 7 9 )

The energy

is obtained by taking the inner

570

HIGH FREQUENCY WAVE PROPAGATION

product of (2.79) by

u,

integrating over G and integrating by parts,

If we use for G the region between t = 0, t = tl

etc.

m

and the

surface of the backward ray conoid issuing from a point (x,t2), t2 > t, we deducet that the domain

of dependence of the solution at

a point is contained in the intersection of the backward ray conoid with t = 0.

As a consequence data with compact support produce

solutions with compact support and the way the support expands with time, i.e. the first arrival of a signal, is controlled by the ray conoid.

The range of influence of a region

the union

D C Rn at time t > 0 is

of the forward ray conoids issuing from each point of D

and extended up to time t. The energyidentity for u of (2.79) over a fixed spatial region 8 takes the form

+

where

(,)

2Re(i(x,t),[B(x) - T1 n -]u(x,t))dx P=l P 8 denotes Euclidean inner product of vectors and

components of the unit outward normal to a 0 at x.

(x) are the P On the basis of

this identity and the domain of dependence considerations just discussed we conclude that the norm

of any solution u of (2.79) satisfies

‘Courant-Hilbert, vol. I1

[

11.

THE W.K.B. OR GEOMETRICAL OPTICS METHOD

where c1 and c2 are constants independent of f.

571

Similarly L2 norms

0 (with respect to A ) of derivatives of u satisfy inequalities such as

(2.84) involving the corresponding derivatives of f.

Thus, for

symmetric hyperbolic systems we have simple a priori estimates which give uniqueness at once.

Existence also follows by using these

identities (cf. Courant-Hilbert, Vol. 11).

The situation for

initial-boundary value problems is more complicated and will not be discussed here. We consider now two examples:

the equations of acoustics and

Maxwell's equation. Let u(x,t), with values in R3 , denote the small variations in the velocity field and p(x,t), scalar-valued, the excess negative pressure over a fixed reference state.

The linear acoustic equations

for u and p are put = VP (2.85)

pt=uV*ur t > O , u(x,O) = u0 (x)

(2.86) Here

p =

I

,

X E R3

, 0

p(x,O) = P (x)

.

p(x) and p = p(x) are the material density and the density

times the square of the sound speed, respectively. that p and

A

y

We shall assume

are given smooth functions and

typical initial-boundary value problem for (2.85), (2.86) in the

interior of a smooth, bounded domain 8 boundary conditions

C

R

3

is provided by the

572

HIGH FREQUENCY WAVE PROPAGATION

(2.88)

;(x)*u(x,t) = 0

,

x

,

3 0

E

t > 0

h

n = unit outward normal

,

.

Equations (2.85) constitute a symmetric hyperbolic system with (u,p) the solution vector (N = 4 ) and in the notation (2.79)

(2.89)

Ao(x) =

PI

0

0

V-11 0

0

11

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

1

0

0

0

O

lo

1

0

0,

0

0

0'

0

0

0

0

0

1 '

0

1

0,

1 A (x) =

(2.90)

AL(x) =

A3(x) =

I = 3

I

x

,

3 identity

'

f

We write in short

"iI"] O

P

I

t > 0

I

The energy density is defined by (2.92)

1 2 E(X,t) = 2 P(X) lU(Xrt) I +

2 11 ~ 0Ip(Xrt) l 2

r

THE W.K.B. OR GEOMETRICAL OPTICS METHOD

573

while the energy flux density is defined by (2.93)

F(x,t) = p(x,t)u(x,t)

.

The energy identity (2.82) becomes (2.94)

& 1 E(x,t)dx = 8

G(x).F(x,t)dS

.

ao

Maxwell's equations for the electric and magnetic fields E(x,t) t and H(x,t), vector-valued, have the following form

(2.96)

Here

Oh

E(x,O) = E0 (x)

,

H(x,O) = H0 (x)

,

denotes the rot or curl operator,E, p and

0

are, respec-

tively, the dielectric tensor, magnetic permeability tensor and the conductivity tensor (3

x

3 matrices).

Note that (2.97) imposes a

restriction on the data which is preserved by the equation for all t > 0.

Noting also that

a

VAE =

ax3

0

a

'Landau-Lifschitz [ 11. We assume that there are no sources of charges or currents in (2.95) and that E , p and u are symmetric and c f P positive definite.

HIGH FREQUENCY WAVE PROPAGATION

574

it follows immediately that (2.95) is a symmetric hyperbolic system. The usual boundary conditions across interfaces are: (2.98)

tangential components of E are continuous : tangential components of H differ by a surface current density : normal components of EE differ by a surface charge density : normal components of pH are continuous

.

We shall deal mostly with unbounded media in the sequel (except in Section 4.11)

.

The energy identity takes the following form.

Let 62 and P be

the field energy density and energy flux respectively

Then we have for any region 8 (2.101)

-& 1 fi(x,t)dx + 3

2.11

C

R3

Iao *

n(x)-P(x,t)dS =

-

I

o(x)E(x,t).E(x,t)dx

8

Expansions for symmetric hyperbolic systems (low frequency). As was discussed in

Section 1.1, equations with slowly varyin'

coefficients lead, upon rescaling, to equations with large terms. For symmetric hyperbolic systems with smooth coefficient matrices the scaled problem takes the form

t > O , u (X,O) =

f E M,

xERn.

THE W.K.B. OR GEOMETRICAL OPTICS METHOD

575

The data f (x) is N-vector valued and may have one of three forms, according to the discussion in Section 1.1: (2.103)

(2.104)

A.

f (x) = f(x) with the components of f(x),

ern0 functions B.

;

f (x) = e iS(X)/Ef (x) where g(x) is real valued and

Cm

and the components of f(x) are complex-

valued and in (2.105)

C.

fE(x) = E-"/'f f(x,y) are

Note that when B(x)

W

Co ;

(x,:) W

Co

i

where the components of

functions

.

0 the equation does not depend on E at all

and in case A one must solve the problem exactly; no asymptotics is involved.

In case B we recover the problem of oscillatory initial

datat while in case

C

we are concerned with the approximation of the

fundamental solution (scaled so that the energy is of order one as E

-f

0).

Consider now class A data with B(x)

0.

We shall illustrate

the asymptotic analysis with two theorems regarding model Boltzmannlike equations.

Most of the content of theorems one and two below

carries over to (i) infinite dimensional hyperbolic systems (N = ") and (ii) mixed initial-boundary value problems.

The infinite dimen-

sional case is not really difficult given enough information on B. The initial-boundary value problems are, however, harder. It

~~

'P.

D. Lax

;

cf. also Courant-Hilbert, [l], Vol. 11.

'+A. Bensoussan, J. L. Lions

, G.

C. Papanicolaou [ 11.

The term

infinite dimensional hyperbolic systems really means transport equations here.

576

HIGH FREQUENCY WAVE PROPAGATION With no loss in generality we may take A.

= I.

We assume that

B(x) is symmetric and negative semidefinite: (2.106)

(v,Bv) 5 0

, v v

..

.

Let $1 (x),$2(XI,. ,@N (x), 1 5 N o < N be a set of vectors spanning 0

the null space of B(x) with No independent of x and with $..(x) smooth J

in x .

Moreover, we assume that

(2.107)

($i(x),$.(x)) = 6 ij

i,j = 1,2

3

Define the symmetric N o

*

No matrices xp(x) by

x

j,k = 1,2,

,...,No

...,N O

,

p = l,2,...fn

.

Define also (2.109)

jk

j,k = 1,2 ,..., N o

(x) = P

...,N 0

be the solution vector of the N 0 dimensional symmetric hyperbolic system Let (i.(xft)), j = 1,2, 3

t > O , ii.(x,O) = f.(x) 3

7

,

with (2.111)

fj(X) = (f(x),$.(x)) 3

.

.

577

THE W.K.B. OR GEOMETRICAL OPTICS METHOD Theorem 2.1. L e t u(x,t) be defined by

Under the above conditions,'

for any N-vector f

c

components and any t > 0,

Remark 2.1. Note that the result does not hold uniformly in [O,T], T <

m,

because there is an initial layer involved.

Remark 2.2. The point of Theorem 1 is the ord r-reduc ion which can be rather dramatic.

For example, (2.102) could be m-dimensional (N =

m)

(which

we have not set up here but is not difficult to do) whereas (2.110) could be finite dimensional.

The most famous example of this is the

hydrodynamical limit of the linearized Botzmann equation':

where

(2.110) corresponds to the inviscid, linearized Euler equations and No = 5.

+Recall Ap (x), p = 1,. bounded entries. "H.

Grad [l].

. .,n and

B (x) are matrices with Cm and

HIGH FREQUENCY WAVE PROPAGATION

578

Proof of Theorem 1. Let P(x) be the N

x

N matrix projecting into the null space of

B(x) i.e. NO

(2.114)

P(x)v =

2' j=1

.

(vf$.(x))$.(x) 3

3

BY a simple argument involving an initial layer+ analysis we may assume that the data f satisfy Pf = f which means that

with f.(x) defined in (2.111). 1

Now we construct an expansion of uE(x,t) of the form (2.116)

u (x,t) = u (Xtt) + EU1(Xtt) + 0

E

2 U2(xrt) +

-.. -

From (2.102) and (2.116) the following sequence of problems obtains for u0,u1,u2,

...,

(2.117)

Bug = 0

(2.118)

Bul

,

P (2.119)

Bu2

+

A'

P

'This Proof 1).

au 2 - at aul -

0

etc.

P

is discussed more fully in Section 2.13 (Theorem 2.4,

THE W.K.B.

OR GEOMETRICAL OPTICS METHOD

519

From (2.117) we conclude that uo(x,t) belongs to the null space of B, i.e., Pug = uo and we write

(2.120) with the

u0(x1t) =

i:ii. j=1

u.1 (x,t) to be

3

(x,t)9.(x) 1

,

determined.

From (2.118) we see that for u1 to exist the component of the inhomogeneous term in the null space of B must vanish.

This provides

restrictions on uo which are exactly equations (2.110) for u.(x,t). 1

Thus ul(x,t) is well determined up to a vector function in the null space of B, which we take identically zero, and satisfies (2.118). Note that ;(x,t)

= uo(x,t) now.

Define

It is easily verified that (2.122)

aw ~

=

P

Ap

2 + 1. P

E

BW E

+

Egl(x,t)

,

where gl(xlt) and g2(x) are smooth and bounded functions of compact support.

By the hyperbolicity of (2.102) and (2.110)l wE has compact

support.

Hence the energy identity over Rn gives

580

HIGH FREQUENCY WAVE PROPAGATION

Since B is negative semidefinite it follows that

This implies that

and hence (2.113) is proved.

Remark 2 . 4 . Note that our estimates are sharper than the regularity hypotheses on coefficients and data required. Note also that we have for the norm I IuE uI I . If we had taken the error estimate O(&)

-

care of the initial layer terms, we could have easily obtained

where ;IL (x,t) denotes the initial layer corrections.

We shall not

carry out the details for this.

Remark 2.5. Sometimes it is possible to approximate u

of (2.102) by the

of a lower dimensional parabolic system where the coeffisolution iE cients of the second spatial derivatives are proportional to

E.

Under

favorable circumstances, far more special than in Theorem 1 above, the advantage gained by this is that the approximation

cE is valid

for a

THE W.K.B. OR GEOMETRICAL OBTICS METHOD much longer time interval of order E-l.

581

In the context of the

linearized Boltzmann equation this is the Chapman-Enskog expansiont (up to second order) whereas Theorem 1 is the Hilbert expansion.

We

return to this in Section 4.6.

.

7.12

Expansions for symmetric hyperbolic systems (probabilistic)

Consider again uE(x,t), solution of (2.102) with fE(x) = f ( x ) an N-vector function with Ci components. We shall prove a theorem similar to Theorem 2.1 of the previous section under different conditions on B.

Note that we are still in the "low frequency" case, i.e.

class A data.

systems

Most of what follows goes through for --dimensional

.

An

important restriction in the probabilistic treatment is that

the matrices Ap (x), p = 0,1,2,. 0

A (x)

..,IT are diagonal.

We may take

I, the identity, and

(2.124)

(AP (XI) = (AP(x)S ) Ik 7 jk

,

p = 1,2

,...,n

,

j , k = 1,2

,...,N .

We assume that (2.125)

the matrix B has non-negative off-diagonal elements with row sums normalized to equal one.

The normalization of the row sums to one is not indispensable and is introduced for convenience.

Instead of taking B in (2.102) as inde-

pendent of E, in this section we shall assume that B(x) depends on

E

also and that ~

'In

~~

-

the optics and acoustic literature this corresponds to the

Leontovich-Fock or parabolic approximation (cf.Tappert 111).

582

HIGH FREQUENCY WAVE PROPAGATION

(2.126)

BE(x) = B0 (x) + EB1 (x)

.

The matrices Bo and B1 are-specified below where it is also explained why we introduce (2.126). A matrix B satisfying (2.125) is the infinitesimal generator of

As a consequence, and

a continuous time Markov chain with N states.t

because of (2.124), uE(x,t), solution of (2.1021, has the following properties: if each component of f(x) is nonnegative then each component of ug(x,t), t > 0, is nonnegative , (2.127) supluE(x,t) I X

where in this section

5 suplf(x) I ,

t 2 0

X

I I

stands for the maximum absolute value of a

row vector (and not the Euclidean norm).

We say now that the hyper-

bolic system satisfies the maximum principle. (2.127) is independent of

,

E;

Note that estimate

it will allow us to obtain an asymptotic

expansion much as in Section 2.11. A matrix B satisfying (2.125) has the vector 1 (all coefficients equal to one) as a right null-vector.

If the off-diagonal elements

of B are strictly positive, B has a one-dimensional null-space and the rest of the eigenvalues have negative real parts.tt

The

components of the, unique up to normalization, left null-vector of B pi, i = 1,2,...,N, are strictly positive and we take them so that

tThe hyperbolic system (2.102) under (2.1241, (2.125) is also called a random evolution (cf. R. Hersh [ll, M. Pinsky 111 and T.G. Kurtz 111). ttThis is a special case of the Perron-Frobenius theorem (cf. Gantmacher [l]).

583

THE W.K.B. OR GEOMETRICAL OPTICS METHOD N (2.128)

1=1

pi

= 1

.

If some off-diagonal elements of B vanish then the set of states {1,2,

...,Nl

can be decomposed into classes of states such that by Each

permutations of rows and columns B has block diagonal form.

block along the diagonal may or may not have positive off-diagonal entries (within the block): however, the null-space of each block is one-dimensional without excluding pure imaginary eigenvalues (periodically moving states)

.'

We shall assume that BE (x) of (2.126)

satisfies the following: (2.129)

(i) When

E

...,N}

is positive the set of states {1,2,

is irreducible with no periodically moving states. (ii) When

E

= 0

(i.e., regarding B 0 ) the states

{lI2,...,N1 decompose into r classes S1,S 21...,Sr, 1 5 r < N, of irreducible and aperiodic states

with N1,N 2,...,N r components, respectively (N1 + N2 + + Nr = N).

...

Note that as a result of (2.129) B has no imaginary eigenvalues for E E

2

0 and it has a null-space which is one-dimensional when

is r-dimensional when

E

pl,

.

Let vectors

5'

E

> 0 and

= 0.

i = 1,2,. .,N, v = 1,2,.

..,r denote the components of

spanning the null-space of Bo.

These vectors can be

chosen so that

'W. Feller

I

Vol. I,[l]. The states composing each block are said

to form an irreducible set of states.

584

HIGH FREQUENCY WAVE PROPAGATION

where '1 We say that the

has components '1 = 1, j E j

p"

1" = 0, j 9

S,

j

have disjoint support with union

S,

.

...,N}.

S =

{1,2,

P

1,2,.-.rn

Define (2.131)

(2.132)

q(X) =

z,(x)

>-,

p . ~P. ( x ),

JES,

J

w = 1,2

pyfj(x) ,

=

,...,r

,

=

I

J

fj = j-th component of f

,

3

(2.133)

We have now the following theorem which is an asymptotic orderreduction result by lumping of components.

Theorem 2.2. Under the above hypotheses, each component u

j E

(x,t),

j = 1,2,...,N, of the solution of (2.102) with fE(x) = f(x) converges for each t > 0 the index

!.I

E

-+

0

;v(j)

of the unique class

(x,t), uniformly in x, where v ( j ) S

?J

to which j belongs.

The functions

iv(x,t), v = 1,2,...,r are solutions of the symmetric hyperbolic system

THE W.K.B. OR GEOMETRICAL OPTICS METHOD

585

We shall not give the proof of Theorem 2.2 in detail since it does not differ in any essential way from the one of Theorem 2.1, Section 2.11.

We point out a couple of things that are important in the

proof.

The first is smoothness of all coefficients and data which

gives smoothness of solutions. The second is that BE:has at most an r-dimensional null-space with the rest of the eigenvalues having negative real parts independently of

This allows us to construct

E.

a simple initial layer expansion which gives the lumping of the data and the restriction t > 0 in Theorem 2.2 above. In the event (2.135)

=

x:(x)

v = 1,2,...,r ,

0

p = 1,2

,...,n

Theorem 2.2 does not have much content, evidently.

It turns out that

(2.135) occurs in some important applications in transport theory and, in any case, it is of interest to know if an improvement to Theorem 2.2 can be obtained. It is convenient to rescale (2.102) so that the final result appears in neat form. It is necessary to rescale time (t 2 1 Then (2.102) becomes BE (BE + Bo + E B )

+

t/E) and

.

(2.136)

3

=

u

jE

$

n p=l

Ay(x)

(X,O) = fj(X)

au p

+ 1

N

(B;k(~)

7 k=l

+

E

2Bjk(x))ukE 1

,

...,N .

j = lf2,

f

The matrix Bo can be written as (2.137)

Bo =

r

,,gl

o

Bv

,

i.e. as the direct sum of matrices Bt for the transitions within S v V

only.

On vectors v such that (p ,v) = 0 the matrix (B!J-l is well

defined up to a constant which we take to be zero (Fredholm alternative)

.

We write Cv = -(Bv)

-'for short.

HIGH FREQUENCY WAVE PROPAGATION

586

For each g(x), a Ci function, we define diffusion operators

a,v

as follows n

(2.138)

QVg(x) =

El

(pv,Aq

P19'

&q (CVAPlvw]) , P v = 1,2,...,r

This is indeed a we1 Note that pv, Ap and

.

defined diffusion operatort in view of (2.135). C,

all depend smoothly on x.

Theorem 2.3. Under the hypotheses of Theorem 2.2 and (2.135) the solution u

(x,t) of (2.136) for each t > 0 converges uniformly in x as

E -+

0

to the solution i v(x,t) ( j )(cf. Theorem 2 . 2 ) where iv(x,t) is the solution of the parabolic system

5 at (2.139)

r =

a G + v

E E V U G V, ?J=1

iv(X,O) = fv(x)

,

t > O ,

v = 1,2,...fr

.

Proof of Theorem 2.3. First we observe that since ?V (x) E C i and the coefficients in (2.139) are

the solution

Cm,

to zero as 1x1

-t

m

G

(x,t) is

Cm

for each t > 0 and tends

faster than any power of 1x1.

This is easily seen

because the system (2.139) is coupled only through the undifferentiated terms, i.e. it has particularly simple structure.

tWe leave it to the reader to verify that not necessarily strictly elliptic.

aV

is elliptic, but

THE W.K.B.

OR GEOMETRICAL, OPTICS METHOD

587

As in Theorems 1 and 2, we may assume that f(x) has the form

r (2.140)

f (x) =

,

lvFv(x)

V=l

since this can be achieved by a simple initial layer expansion (which explains why we must have t > 0). The proof now goes much the same way as most proofs in the previous chapters as well as the one of Theorem 2.1.

We construct an

expansion UE.=

uo

+

Ell1 iE 2u2

by the usual process. uo(x,t) =

+

...

I

Our hypotheses guarantee that

2

lVGv(x,t) I

v=1

where

iV

satisfy (2.139), and that ul(x,t) and u2(x,t) can be properly

constructed and are smooth. UE

-

uo

-

EU1

-

E

We find finally that

2u2 -

WE

satisfies the inhomogeneous version of (2.136) with the inhomogeneous term of order

E

and with initial data of order E.

The theorem then

follows from the a priori estimate (2.127).

2.13

Expansion for symmetric hyperbolic systems (high frequency). The systems of Sections 2.11 and 2.12 had one common feature

regarding the matrix B:

it had a null-space but we did not allow

purely imaginary eigenvalues.

It turns out that in the asymptotic

analysis of systems with periodic structure, as we shall see in Sections 4 and 5, the B that is involved has pure imaginary eigenvalues (both in static and high frequency cases).

In genera1,systems

588

HIGH FREQUENCY WAVE PROPAGATION

that deal with waves also have such B's.

It is appropriate therefore

to call problems with B as in the previous sections dissipative or Boltzmann-like systems. We now consider briefly (2.102) when f, (x) = f (x) and B(x) is skew-symmetric and has an N0 -dimensional null space as described below (2.106), including (2.107).

Theorem 4. Under the hypotheses of Theorem 2.1 (Section 2.11) but with B(x) skew-symmetric, for any N-vector test function h(x,t) that vanishes identically for t

and

E

N (Cm(Rnx]O,m[))

1x1 large and Ph(x,t) = h(x,t)

(cf. (2.114)), we have m

Here ;(x,t)

is the solution of the symmetric hyperbolic system (2.110)

as in Theorem 2.1.

Remark 2.6. The difference between Theorems 2.1 and 2.4 is the mode of convergence. In Theorem 2.1 we have strong convergence in Lm (0,T; (L2(R") ) N, In Theorem 2.4 we have weak star convergence in Lm(O,T; (L2(R")),), N 2 n N where (L (R ) I p are square integrable N-vector functions

- Pu = u.

We shall give

two proofs

with

of this theorem in order to illustrate

some points that play an important role in periodic-structure problems later.

.

THE W.K.B. OR GEOMETRICAL OPTICS METHOD

589

Proof I. System (2.102) has a smooth solution for

E

> 0; as before coef-

ficients and data are Cm and the data have compact support. We construct an asymptotic expansion for uE as follows. t t 2 t uE(x,t) = U ~ ( X , ~ , E + ) E U ~ ( X , ~ , -+) E u 2 (x,t,-)

(2.142)

+

Let

...

,

and put (2.143)

T =

t/E

.

We do not assume that f ( x ) has the form (2.115) since there will be no initial layer now; in fact this is the basic difference here as we shall see.

From (2.102) and (2.142) the following sequence of

problems obtains for u o,u1'u2 I . . (2.144)

Bug

.

- --

From (2.144) we get (cf. (2.112)),

Here u.(x,t), j = 1,2,..., No are to be determined so that 3

(2.147)

c 3. (x,O) =

f .(x) = (f(x), $ . ( x ) ) 3

3

and vo(x,t) is a vector function satisfying

H I G H FREQUENCY WAVE PROPAGATION

590

and is also to be determined. We write u1 in the form

We consider next (2.145). (2.150)

ul(x,t,T) = eBTv,(x,t)

a - at (6 + We must impose conditions on the integral term in (2.150) so that it be uniformly bounded in

'1.

Since B is skew symmetric the components

of eBT are quasi-periodic functions of

T.

Hence the integrand on the

right side of (2.150) is quasiperiodic in s and for u1 to be uniformly bounded in

T

(pointwise in x and t) it is necessary and sufficient

that (for each x and t fixed)

We rewrite this as conditions on

I '

n e-Bs >:AP 0 p= 1 T

(2.153)

lim Tfm

1

a

axp

and onvO separately as follows

at (eBsvo)ds = -

.

But (2.152) is nothing more than system (2.110); hence (2.152) is satisfied if

ii

is as in the statement of the theorem.

The left

hand side of (2.153) is a differential operator acting on vo(x,t) and

THE W.K.B.

OR GEOMETRICAL OPTICS METHOD

591

the system (2.153) with initial condition (2.149)satisfies

(2.148).

Therefore vo(x,t) is well defined and thus uo(x,t,~)in (2.146) is completely determined. 2.11.

One can similarly determine u1 as in Theorem 2.1, Section Exactly as it was proved in that theorem, we obtain that (2.154) sup T't'O for any T <

m.

have lim EJ.0

1

luE(x,t)

-

t 2 u o ( x l t , ~I )dx

+

0

as

E

+

0

R" The proof of (2.141) from (2.154) is as follows. We

1I

(h(x,t),uE(x,t)- u(x,t))dxdt

Rn t t,F))dxdt

dxdt

.

The first term on the right goes to zero because of (2.154).

The

second goes to zero by the Riemann-Lebesgue lemma since, by (2.1481, all terms in the integral have oscillatory factors.

Proof #I is

complete. Although Proof I is direct, constructive and shows that we can compute even the rapidly oscillating parts of the solution and get strong convergence (2.154), it has the disadvantage that it uses too much information if all one wants is (2.141) and nothing more. give next an indirect proof.

We

592

HIGH FREQUENCY WAVE PROPAGATION

Proof 11. From the energy identity for (2.102) we conclude that

where C depends on f but not on 0 <

E

This means that the family uE,

E.

5 1 say, is weak star compact in Lm(O,T; (L2(Rn))N). Let uE

denote a convergent subsequence. We shall identify the limit uniquely (so that then any sequence converges) as the solution

u of

(2.110).

For the identification we give a weak formulation of (2.102) and use adjoint expansion. theorem.

Let h(x,t) be as in the statement of the

Taking the inner product of (2.102) integrating over space

and time and integrating by parts we obtain

I1

m

(2.156)

(ht(x,t) ,uE(x,t))dxdt

+

0 Ril

I

(h(x,O),f (x))dx

R" n _.

m

=iI

P=l

Rn

& (Aph),uE]dxdt + P

II

m

(BhfuE)dxdt

.

Rn

Fix h(t,x) as in the statement of the theorem and let h(x,O) = ho(x). (2.157)

' A

=

Now let Ph(x,t) = h(x,t).

aat

x&

Let

n

p=l

p

(AP.)

-

B

,

which is the negative of the adjoint operator associated with (2.102). Let

a be defined by

'But

without Ph = h, yet.

THE W.K.B. OR GEOMETRICAL OPTICS METHOD

a

(2.158)

a -

=

n

.

-

ax (lip.) P

P=l

593

This is meant to be a coordinate free representation of the adjoint of the operator (2.110) acting on 6 .

Acting on vector functions v

with P v = v , i.e. in the null space, it is exactly the formal adjoint of (2.110), by definition.

Acting on v such that Pv = 0, i.e. off

the null space, U v = 0, by definition. By perturbation expansion we construct a function h (x,t) such 1 that if we set

then (2.160)

U h +

AEhE =

O(E)

,

hE(x,O) = ho + O ( E )

.

Note that PhE # hE here but Ph = h by hypothesis. We leave it to the reader to verify that hl can be so constructed (a computation analogous to the one for u

1 and (2.159)-(2.160) give us

I II

in (2.116)).

m

(2.161)

0 =

(AEhE,uE)dxdt+

Rn

I

Now (2.156) (with h

-+

hE)

(hE(x,O),f(x))dx

Rn

m

=

(c7h,uE.)dxdt+

Rn

(ho(x),f(x))dx

+

O(E)

,

Rn

where the constant in 0 ( E ) depends only on f and h and not on uE, in view of (2.155).

II

Passing to the limit as

m

(2.162)

Rn

( ~ h,;)dxdt 7

+

I

E +

0 in (2.161) we obtain

(ho(x),f(x))dx = 0

.

R"

This is just the weak form of (2.110) and since h(x,t) is in a

594

HIGH FREQUENCY WAVE PROPAGATION

determining class of test functions and (2.110) has a unique solution, the result (2.141) follows.

Remark 2.7. The second proof uses very few of our regularity hypotheses and only the weak existence and uniqueness for (2.102) and (2.110). This is a major advantage that it has over the first proof. One objective of Theorem 2.4 is the clarification of what it really means to have static or high-frequency data, i.e. that in the wave propagation context nothing damps out and we must either construct the full oscillatory form of the expansion (as in (2.142))

weaken

the notion of convergence sufficiently to kill the oscillations. Notice that in Theorem 4, fE(x) = f (x), i.e. we have still class A data. We now pass to class B data (2.104) and consider (2.102) anew. We assume that B(x) is skew symmetric but we do not specify its spectrum further for the moment. 0

We also assume for simplicity that

Ao = A (x) but the Ap are constant matrices for v = 1.2... ..N

.

Inserting this into (2.102) yields the following problem for v (2.163)

Let (2.164)

Set

THE W.K.B. OR GEOMETRICAL OPTICS METHOD and consider the N

595

N matrix eigenvalue problem ( w is the eigenvalue

x

parameter) (2.165)

[- 2

l o

APkp + iB v = wA v

p=l

.

The variable x E Rn is suppressed here; it is merely a parameter. n Similarly, we regard k = (k ) E R as a parameter. P Recall that,'A p = 0,1,2,. ,N are real and symmetric, A 0 is

..

positive definite and B is skew-symmetric so that iB is hermitian. Thus for each k and x there exist N real eigenvalues

and N linearly independent eigenvectors

which can be chosen orthonormal relative to A 0

The inner product here is the hermitian inner product N

(2.167)

(u,v) =

j=l

u

j j

.

As k and x vary so do the eigenvalues and the eigenfunctions. It is possible to show that the eigenvalues are continous and the eigenfunctions can be constructed' of x and k.

so as to be measurable functions

In the neighborhood of a point (k,x) where an eigenvalue

w(k,x) is isolated, it and its corresponding eigenvector @(k,x) (as constructed by Wilcox) are differentiable functions.

'cf.

C. Wilcox

[

11 and T. Kato 111.

596

H I G H FREQUENCY WAVE PROPAGATION

Now consider the eiconal equations (2.168)

Sy

+ wm(VSm,x)

=-0

,

t > 0

,

m

1,2,. ..,N

.

=

Sm(x,O) = s(x)

,

We shall assume that the following holds regarding (2.168): (2.169)

The gradient of the initial data Vs(x) varies smoothly within a neighborhood of a fixed vector k and in which the wm(k,x) are distinct.

Moreover there is a to > 0

such that (2.168) has a unique smooth solution in m 0 5 t < to (no caustics in 0 5 t < to) and VS (x,t) stays near k. If we fix m and lett S satisfy (2.168) then (2.163) becomes

Letting vE = vo +

(2.171)

[-

E V ~+ E

2v2

+

..., we

+

iB

-

uAO].l

obtain, as usual,

n P=l

P

Ap

-

ilo

$ - >: Ap av

p= 1

etc.

tWe frequently drop the superscript m, to simplify notation, when discussing one mode at a time, i.e. when m is fixed.

THE W.K.B.

597

OR GEOMETRICAL OPTICS METHOD

From (2.170) we conclude that

where u,(x,t)

is a scalar function to be determined and $(x,k) is

the eigenvector corresponding to the eigenvalue w (subscript m omitted). Next we use (2.172) in (2.171).

Since

w

is an isolated eigen-

value, the solvability condition yields one equation for uo(x,t), the usual transport equation:

Using the fact that (2.174)

awO a kP

=

- ( $ (k,x),Ap$(k,x)) ,

p = 1,2,. ..,N

,

obtained by differentiating (2.165), it follows that uo(x,t) satisfies

where b(x,t) is defined by

n

598

HIGH FREQUENCY WAVE PROPAGATION

It is easily verified that b(x,t) is purely imaginary. Let Cm(k,x) be defined by (2.177)

Cm(k,x) =

[-

I'

n p=l

k Ap p

+

iB(x)

-

acting on vectors orthogonal to $,,,(k,x).

wm(k,x)A (x) O

We can write vl, in view

of (2.173), in the form

where vlO(xrt) is undetermined but we have

The function ul(x,t) is determined by the solvability condition for v2 and it satisfies an inhomogeneous transport equation of the form (2.1751, namely (2.180)

aul(xrt) at +

n

>7 P=l

P

(VS(xrt)rx)

aul(x,t)

ax

P

where vo is given by (2.172) with uo solution of (2.175). Now we consider the assignment of initial data.

u

of (2.102) is asymptotically of the form

The solution

THE W.K.B.

OR GEOMETRICAL OPTICS METHOD

with entries as just described.

599

To satisfy (2.104) we must have

Hence

We make two remarks regarding the asymptotic solution (2.181) of (2.102), (2.104).

First it is considerably more involved than the

low frequency case as we must expect in view of the analysis of the Klein-Gordon equation in previous sections.

The proof of its validity

is routine under (2.169) which says that everything in sight is smooth

so the energy estimate applies immediately. Second, we would like to point out the significance of the inhomogeneous term in (2.180) and to carrying out the expansion correctly through order

E.

Because of (2.1691, each wave mode (i.e.

term in (2.181) labelled by m) propagates inc!ependently of the other modes as far as the zero order term in coupling to zero order.

The order

E

E

goes.

We have no mode

term includes mode coupling

effects and more specifically the way the energy is transferred from one mode to another; it is the lowest order term in which this occurs. Hence its significance.

In Section

4.6

we shall see that sometimes

HIGH FREQUENCY WAVE PROPAGATION

600

one can elevate the mode coupling effect to order one in

E.

The

static case is, of course such an instance but there are also others.

2.14

WKB for dissipative symmetric hyperbolic systems. Consider the system of Section (2.12), i.e.

(2.185)

3

=

n p=l

AP(x)

au

+

P

3

$

N R= 1

BjR(x)ukE

,

t > O ,

with initial data u

jE

(X,O) = f .(x) 3

Here f .(x), AP(x) and B 3

3

,

j = 1,2

(x) are Cm.

support and 0 5 f.(x) 5 1. 7

,...,N .

Moreover the f .(x) have compact 3

We define

which is open with 6 compact.

The matrix B . R ( ~ ) is assumed to 3

havet positive off diagonal elements and unit row sums.

Note also

that the matrices Ap in (2.185) are assumed to be diagonal. Theorem 2.2 of Section

...,N,

each j = 1,2,

ujE(x,t)

2.12 +

implies that for each t > 0 and

;(x,t) uniformly in x where

and N (2.188)

;(x,O) =

3=1

pjfj(x)

.

tHence a dissipative system (cf. beginning of Section 2.13).

THE W.K.B. Here

5j

> 0

(Cpj

OR GEOMETRICAL OPTICS METHOD

601

= 1) are the components of the left null-vector of

B (null-space has dimension one) and N

(2.189)

AP(x) =

J=l

GjAfi)(x)

.

Since ;(x,O) has compact support G, the support Gt of c(x,t) will be precisely the set of points x that are translates of points of G along the orbits of the vector field A(x) = (xp(x)). Note that G is the union of the supports of the f.(x), j = 1,2,...,N, i.e. as defined 3

by (2.186). On the other hand the union of the supports of u. (x,t) at t > 0,

-

bigger than Gt (it is independent of u

E j

(x,t) -t 0 as

zero in

Et -

E

-t

0 for x E

Gt

-

We have just seen that

E).

Gt.

...,N

(x,t) tend to

How does u

This is the question we examine in this section.t

Gt?

The answer to this question is this. 2,

3E

for j = 1,2,. ..,N, call it Gt, is in general much

> 0

E

For each x

E

Gt, j = 1,

and t > 0

(2.190)

lim

E

ESO

log u

-

j E

(x,t) = -S(x,t)

,

where S(x,t) is the solution of a calculus of variations problem depending only on the support G of the initial data (and of course the coefficients in (2.185)). in Gt; hence u

j E

Moreover, S(x,t) > 0 in

-.

Et -

decays to zero exponentially in Gt

-

Gt and

Gt.

This -

result is valid for all t > 0 fixed, no matter what the support of the f.'s looks like.

-

3

'This [

Here we shall prove it only for aG smooth and

problem is treated in great generality by M. I. Freidlin

11 by probabilistic methods.

0

602

HIGH FREQUENCY WAVE PROPAGATION

t small.

Our method fails in general on account of the formation

of caustics as we have seen several times already.

By employing

certain probabilistic toolst a general proof can be given. Recall that for problem (2.185) we have a maximum principle which will be used repeatedly below.

The first step is to

majorize and minorize the solution of (2.185) with the solution of two other problems; this may be called a penalty argument. The second step is to analyze these two problems by a WKB asymptotic expansion.

In the end we remove the penalty and recover (2.190).

Let h(x) be a

function such that

Cm

and such that IVh(x) [ L 1 on

aG

.

Such a function exists because G is smooth by hypothesis. Fix a sufficiently large integer y and for each a > 0 put

and (2.192)

sA1’(x) = ga(h(x))

.

This is a function that is differentiable as often as desired (by taking y large)

‘M.

. Moreover

1.Freidlin

M.D.

[

1 1 , S. R. S.Varadhan [I], A.D. Ventsel 111,

Donsker and S . R . S .

Varadhan [l].

THE W.K.B.

OR GEOMETRICAL OPTICS METHOD

603

Similarly let

which is also differentiable and has the property that if

and

with x E G

-

(G(B) U

aG(’))

fixed and B > 0 fixed

.

Consider next the eigenvalue problem t n kpAp(x)

+

B(x) $(k,x) = w(k,x)$(k,x)

for each k E Rn and x E Rn fixed.

,

By the Perron-Frobenius theoremtt

we know that there exists an isolated maximal eigenvalue, which we call w(k,x), which is smooth in k and x and (2.198)

w(0.x)

E

0

.

Moreover, w(k,x) is a convex function of k for each x, being a maximal eigenvalue.

There exists also a right eigenvector @(k,x)

whose components $.(k,x) are positive and smooth functions of k, x. I

tHere A‘(%)

= (AP(x)S. . ) are diagonal matrices. 7 13

ttGantmacher [l], Vol. 11.

HIGH FREQUENCY WAVE PROPAGATION

604

A left eigenvector c(k,x) also exists with positive and smooth components.

We have also that

where 1 is the vector with all components equal to one and p(x) =

( 57. (x)) is

the left null vector of B(x)

.

We have the

normalization N

Let S(x,t) denote the solution of the Hamilton-Jacobi equation

st + w ( v s f ~ )= o , t

(2.201)

> n

(x) or i(2) (x) and with superscripts and subn,O scripts suppressed temporarily. This equation has J smooth solution

with S(x,O) equal to

up to some time

fo

when caustics form. We discuss P(x,t) now in so-

me detail. Since w(k,x) is convex we may define the Lagrangian function

which is the conjugate convex function corresponding to w . Because of (2.198) and the convexity of w it follows that

(2.203)

L(x,x) 2

0

and

L(x,x)

= 0

only when

From (2.197) we find by differentiation that (cf.(2.189))

THE W.K.B. (2.204)

~

'

ao (k,x) = akp Ik=O

605

OR GEOMETRICAL OPTICS METHOD

-

iiP(x)

, p=1,2 ,...,n.

Thus : the Lagrangian L(x,G) is strictly positive along all trajectories x(t) except along the characteristics of (2.187) (with reversed time in view of the minus sign in (2.204)) on which it vanishes. The Lagrangian L(x,G) =

fm

for directions 2 that are inaccessible and this

follows automatically from (2.202). Let (x(t),k(t)) be the solution of the Hamiltonian system

(2.206

S(x(t) ,t) = g(x)

+

:j

L(x(s) ,x(s))ds

satisf es the Hamilton-Jacobi equation (2.201). The function

dsl

,

with the infinimum taken over all differentiable pa hs 5 that

ermina-

te at x at time t, is the unique smooth solution of (2.201) up to the time to > O when caustics form.

where to > O is the first time uniqueness breaks term for this variatio-

606

H I G H FREQUENCY WAVE PROPAGATION

-

n a l problem. I f Gt

(2.185) a t t i m e t , as d e f i -

is t h e support set of

-

I

ned b e f o r e , t h e n E ( x , t )

< m

S ( x , t ) >II f o r x c G t - G t ,

z ( x , t ) = 0 for x cGt.

-

for x cGt,

S(x,t) =

f o r x{Gt

+m

and

We also note that,

u s i n g s u p e r s c r i p t and s u b s c r i p t n o t a t i o n now,

for t
W e r e t u r n now t o t h e p e n a l t y a r g u m e n t

2t

t
ixed an

l e t a s u f f i c i e n t l y l a r g e be f i x e d a l s o . W e w r i t e t h e s o l u t i o n u . ( x , t ) 7E

of

( 2 . 1 8 5 ) i n t h e form

I t f o l l o w s t h a t va

( x , t ) s a t i s f i e s t h e problem

From ( 2 . 2 1 1 ) w e o b t a i n

(x,t)

(2.213)

+

E

+

E

log va ( x , t ) j E

log

.

( v s y ( x , t ) ,x) +

$l.

1

THE W.K.B. (2.197)

Because o f

OR GEOMETRICAL OPTICS METHOD

t h e system ( 2 . 2 1 2 )

607

obeys t h e maximum p r i n c i p l e

and t h e r e f o r e

lim

(2.214)

E

EJ.0

Letting a

+

-

I E

(x,t) s - S y x , t ) .

we o b t a i n

lim

(2.215)

log u .

E

log u .

E $0

IE

(x,t)

5

-S(x,t)

,

0 < t< t o

which i s one h a l f of t h e r e s u l t ( 2 . 1 9 0 ) .

To o b t a i n t h e lower bound on F i x 0 < t< t o, x

cGt

and

E

l o g ujE w e p r o c e e d a s f o l l o w s .

> 0 . The p a r a m e t e r a i s chosen l a r g e

enough below and i s t h e n f i x e d a l s o . I f w e r e p l a c e SA1) by S L l b i n ( 2 . 2 1 1 ) we o b t a i n

Here v a r B ( x , t ) i s t h e s o l u t i o n of

I€

( 2 . 2 1 2 ) w i t h S(I) r e p l a c e d by S a( 2f B) .

a

Now choose a s o l a r g e t h a t i f x ( s ) d e n o t e s t h e o p t i m a l p a t h i n (2.207)

(with

,

0

with x ( t ) = X ,

5 s S t ,

s r e p l a c e d by s ( 2 ) ) ,t h e n a,B

x ( o ) E G . T h i s i s p o s s i b l e b e c a u s e B i s p o s i t i v e and ( 2 . 2 0 9 ) With a and B s o f i x e d now w e l o o k a t s y s t e m ( 2 . 2 1 2 ) . i n t h e form of theorem 2 . 2 ,

Section 2.12,

It is precisely

so w e can a p p l y t o it t h i s

theorem ( a n e x t r a u n d i f f e r e n t i a t e d t e r m of o r d e r one as r i g h t s i d e ( 2 . 2 1 2 ) is harmless). W e f i n d t h a t v?"(x,t) 3E

0 < t< t oa s

E

+

holds.

E

+

-t

0 on t h e

v a f B( x , t )

0 where G a f B ( x , t ) s a t i s f i e s a s y s t e m of t h e form

( 2 . 1 8 7 ) . NOW, however, t h e v e c t o r f i e l d x p ( x ) i s t i m e dependent and i n s t e a d of

( 2 . 2 0 4 ) w e have ( t h i s i s e a s i l y v e r i f i e d from ( 2 . 1 9 7 ) )

f

608

HIGH FREQUENCY WAVE PROPAGATION

These observations and the selection of a large enough imply that G a r (x,t) > 0 for x tics of the vector field

E

it, 0 < t
fip in (2.217) are precisely the optimal paths

of (2.207). Therefore from (2.216) we obtain

(2.218)

.

log u . (x,t) z-S(2)(x,t) I€ ar8

=E

€40

~

In view of (2.210) this means that for x

E

Gt and 0 < t
and hence, by (2.215)

(2.220)

lim

E

log ujE(x,t) = -S(x,t)

, o < t
.

E+O

We summarize the results as follows.

Theorem 2.5. The solution u

that G

j E

(x,t) of (2.185) under the hypothesis

of (2.1861 is smooth and t
, satisfies

(2.190) where the

decay rate S(x,t) is the cost function of the variational problem (2.208).

The Lagrangian L(x,x) is defined bv ( 2 . 2 0 2 )

and the

Hamiltonian w(k,x) is the maximal eiqenvalue of (2.197). Recall that Theorem 2 of Section 2.12 dealt with the case where B(x) = BE(x) and the null space became r > 1-dimensional when

E

Theorem 2.5 can be extended to that more complicated situation.

= 0.

THE W.K.B. OR GEOMETRICAL OPTICS METHOD 2.15

609

Operator form of the WKB. Consider the symmetric hyperbolic system

with some initial conditions.

In Section 1.1 we saw how slowly

varying coefficients lead to scaled systems like (2.221).

The notion

of slowly varying coefficients can take many other forms however.

We

shall now examine two of them. The first corresponds to horizontally stratified media with slow variation of parameters in the horizontal direction.

Let xn be the

coordinate perpendicular to the stratification and write x = (x',xn with x'

E

Rn-l the coordinates within a horizontal plane of

stratification.

(2.222)

Suppose that

A 0 = A 0 (6x',xn)

,

B = B(Ex',x~)

.

Define (2.223)

y =

and rescale t

EX'

+.

,

t/E.

y

E

Rn-l

,

Then (2.221) becomes

t > O , with some initial conditions and/or boundary conditions.

Here as

usual the matrices Ap are symmetric, A 0 is positive definite and B negative semi-definite. Assume that

HIGH FREQUENCY WAVE PROPAGATION

610 (2.225)

An has r positive eigenvalues and N eigenvalue (no zero eigenvalues)

-

r negative

.

We may then, with no loss in generality, take An to be diagonal already with +1 in the first r diagonal positions and -1 in the last N

-

r diagonal positions. A well posed mixed initial-boundary value

problem for (2.224) over Rn-l

x

[a,bl, a, b finite, is the following:

One easily verifies that the following energy identity holds (when fE(y,xn) has compact support in y and satisfies (2.227))

b

N

Using this identity, uniqueness follows immediately for (2.2241, (2.226) and (2.227) as well as an

E

independent bound.

Existence of f

a smooth solution also follows but is more complicated.

'K.

0. Friedrichs 111, H.O. Kreiss 111.

THE W.K.B. OR GEOMETRICAL OPTICS METHOD

611

A problem of the form (2.224), (2.226) and (2.227) arises in underwater acoustics and a detailed asymptotic analysis (including extensive numerical work) has been given by Weinberg-Burridge 111. Other problems of the same form have been analyzed by Shen and J. B. Keller [ll. As we shall see the analysis of problems with periodic structure leads to an equation of the form (2.224).

This explains

why we consider this problem here. Assume that B is skew-symmetric.

Then the asymptotic analysis

of (2.224) follows exactly the pattern of analysis in Section 2.13: both as regards the static problem described in Theorem 4 and the high frequency expansion (2.181).

A major role is now played by the

eigenvalue problem

=

W

0

(kly)A (y,xn)@ (xn;kly) ,

Rn-l and y E Rn-l are parameters. The formal operator on N the left is over L2 (a,b;R ) with boundary conditions as in (2.227)

where k

E

and has a unique self-adjoint extension.

For each k, y fixed,the

spectrum is discrete and the eigenvalues depend smoothly on k, y near points where they are distinct.

Eigenvectors can also be selected to

t

be smoothly dependent on k, y I locally.

Before going to the second example we note that it is appropriate to refer to (2.224) I (2.226)

(2.227) as horizontal propagation

in a horizontally stratified medium. It is also of interest to study

'The

work of Wilcox

[

2 ] applies directly to (2.229 ) even though

it is aimed at the analogous problem for Bloch waves.

612

HIGH FREQUENCY WAVE PROPAGATION

vertical propagation in a horizontally stratified medium.

This is a

scattering problem whereby Ao and B are assumed to be independent of xn for lxnl large and at t'= 0 a wave with support in the lower homogeneous region is prescribed travelling in the +xn direction thereby creating at time t later a reflected waveform and a transmitted waveform

(if t is large enough) in the upper homogeneous

region. The second problem corresponds to a waveguide structure with slow variation of parameters in the direction of its axis.

Let xn

now be the coordinate along the waveguide structure and let x' E Rn-' be the transverse coordinates. Ao = A 0 (X',EXn)

(2.230)

I

Suppose that B = B(x',Ex~)

.

Define y = Exn,

(2.231)

and rescale t

-f

t/E.

YER1, Then (2.221) becomes

t > O , with some initial and boundary conditions. where 0

C

Rn-l is a bounded open set.

conditions on 3 8 so that the operator

Suppose that x'

E

We introduce boundary

8

THE W.K.B. OR GEOMETRICAL OPTICS METHOD

613

with k E Rl, y E R1 parameters, has a unique selfadjoint extension 2 N on L (8;R ) . Physically, there are many possibilities-metallic waveguides, dielectric waveguides, etc., and (2.233) may be considered on all of R"-l with interface conditions on a 8

.

When the spectrum

of (2.233) (relative to A 0)is discrete for each k, y,the asymptotic analysis goes exactly as in Section 2.13.

Otherwise new phenomena

arise which we shall not discuss here (since that has no immediate analog in periodic structures). An example of a waveguide problem with periodic structure is given in Section 4.12.

HIGH FREQUENCY WAVE PROPAGATION

614

3.

Spectral theory of differential operators with periodic coefficients.

3.1

The shifted cell problems for a second order elliptic operator. Let A be the formal differential operator

m

defined on Co(Rn) with (a (y)) symmetric, smooth and uniformly P9 positive definite matrix of periodic functions of period 2n in each variable (i.e. functions on 2aY in the notation of Section 1.2). 0 be a real-valued 2n-periodic smooth function.

W(y)

Let

We shall find

the spectral resolution of the closure of this operator in L2(Rn) (complex-valued). The spectral resolution is given in terms of the Bloch waves associated with A as we now describe. Let Y be the unit n-dimensional torus and let 2nZn be the 2nlattice in Rn centered at the origin.

Let H1(2aY) be the Hilbert

space of functions on 2aY that have square integrable derivatives. For each k E Y define

Consider the eigenvalue problem, the shifted cell problem, (3.3)

A(k)@ = w 2@

,

@ E

H1(2sY)

for each k E Y.

,

Since W 2 0 the operator A(k) is nonnegative and hence u2 in (3.3) is used consistently. For any f E H1 (2nY) 'and with (,)

the usual inner product over 2rY we have

SPECTRAL THEORY OF DIFFERENTIAL OPERATORS , (3.4)

615

n

(A(k)f,f) = 2 aY

for all k E Y with al and

a2

positive constants that do not depend on

k. From (3.4) it follows that the essentially selfadjoint operator A(k) has for each k E Y a compact resolvent ( A + A ( k ) ) -1I Re A # 0. As a consequence, for each k the eigenvalue problem (3.3) has a discrete sequence of eigenvalues

with corresponding eigenfunctions, the Bloch waves,

which we take to be orthonormalized, i.e.

The eigenfunctions are smooth functions y and they are complete in 2 L (2nY)

.

(complex-valued)

2

The dependence of wm(k) and $m(y;k) I m 1. 0,on k requires special attention; it has been analyzed by Wilcox [ 2 1 .

He finds that the

are real analytic functions of k everywhere except on Y subsets m of measure zero where their multiplicity changes;they are continuous w ‘s

for all k.

The eigenfunctions $m(y;k) can be constructed so as to be

measurable functions of k, in general; in the component of null sets

HIGH FREQUENCY WAVE PROPAGATION

616

in Y they are analytic functions of k with values in C(2aY), the continuous functions on 2aY.

3.2

The Bloch expansion theorem. All considerations regarding asymptotics in this chapter are

local in character.

It is always assumed that the relevant k's are

restricted so that the urn's and 0,'s

are smooth.

The following

theorem, however, requires only the measurability of the 4 1 ~ ' s . Theorem 3.1 (Bloch expansion).

Let f

2

n

E L (R )

(complex-valued).

-

?! m (k) = 1.i.m.

(3.9)

Then

dy f (y)e-ik'Y $,,(y;k)

N+m

.

Moreover, Parseval's identity holds m

Proof

(

Gelfand [l], Odeh-Keller 111)

.

In view of (3.10) it is enough to show that (3.8)-(3.10) with f

E

&?(Rn), the space of Cm rapidly decreasing functions on Rn.

For such an f, the function (3.11)

hold

?(y;k) =

belongs to 2aY.

yE2az"

f(y

+

y)e-ik. (y+y)

I

We expand it in terms of {$m(y;k)l

k E Y

617

SPECTRAL THEORY OF DIFFERENTIAL OPERATORS

where

From (3.11) and (3.12) we also have (3.14)

f(y) =

J

dk eik'Yf(y;k)

Y =

1

m

dk

im(k)eik'Y$m(y;k)

m=O

.

This proves (3.8) and (3.9) for f E&(Rn). To prove (3.10) we note that from (3.12) we have m

Using (3.11) on the left in (3.15), integrating over k

E

P and

rearranging yields (3.10). A s a simple example of this theorem consider the case a

Pq

and W = 0 so that A = - A . (3.16)

$m(y;k)

=

Evidently -n/2 immy e

(27~)

and $m does not depend on k. (3.17)

2 wm(k) = Ik

+

ml

2

,

m e zn,

I

The eigenvalues are m

E

Zn

,

k

E y

.

=

6Pq

HIGH FREQUENCY WAVE PROPAGATION

618

The theorem above is now the Plancherel-Parseval theorem for the n Fourier transform in R and it is deduced from the corresponding result for Fourier series-. We note that (3.1) and (3.2) are connected by E

eik-yA(k)

r

which explains the terminology "shifted cell problem". (3.18)

Aeik"$

m

(y;k) = eik-yum(k)41m(y;k) 2

Thus,

.

The above theoremt then and (3.18) give the spectral resolution of A is) which for f E ~ ( R ~ m

(3.19)

3.3

Bloch expansion for the acoustic equation. For the acoustic system (2.85) with p and ~.r 21r-periodic,the

previous results apply with slight modifications.

The operator at

hand has the form

acting on &vector

functions (u,p) on R

3

.

If we replace (u,p) by

eikoy(urp) the shifted cell problem analoqus to (3.3) becomes

(V

+

ik)@ = -iup$

,

I$

E H1(2nY)

In fact for 4I we have the eigenvalue problem

t See also Section 3 . 6 for further remarks regarding this theorem.

SPECTRAL THEORY OF DIFFERENTIAL OPERATORS which is in a slightly more general form than ( 3 . 3 ) p-’

on the right.

619

on account of the

However, upon defining a weighted inner product

with weight p - l , all goes as before and we obtain the Bloch eigenfunctionsand eigenvalues for ( 3 . 2 1 ) 2

wo(k)

2

5 wl(k) 5

$o(Y;k) From ( 3 . 2 0 )

..

$1(Y;k)

, ,

... .

we also find that

This definition of Jlm holds for all m = Oll12,...,

and all k E Y

2 except for Jl (y;O) which is not defined since here wo(0) = 0, and

0

(3.23)

$o(y;O) =

But from ( 3 . 2 0 ) (3.24)

[

-1/2

p-’(y)dy]

.

2 Trry

we see that when k = 0,

ii)

= 0

$o(y;o) = vA,J)(y)

where $ (y) is an arbitrary differentiable vector function. This abnormality at k = 0 for the m = 0 mode makes the static problem quite interesting as we shall see in Section 4.10.’

3.4

Bloch expansion for Maxwell’s equation. The shifted cell problem for Maxwell’s system (2.95)

u

0) analogous to ( 3 . 3 )

(with

has the following f o m (analogous to ( 3 . 2 0 ) )

620

HIGH FREQUENCY WAVE PROPAGATION

1 with (E,H) E (H ( ~ I T Y )and ) ~ ~ ( y ) ,~ ( y )smooth symmetric positive definite 3

x

3 matrices of 2n-periodic functions.

Eliminating H in (3.25) we obtain the following eigenvalue problem for E (k E Y is fixed)

(V

(3.26)

+

ik)

and

A

[p-l(V

+

2 ik) AE] = w EE

+ ik)*(EE)

(V

= 0

,

E E (H1(21Ty))3,

.

Once E has been determined then (3.27)

H =

-1 1 W

(V

provided that w # 0. For k

+

ik) A E

,

This can happen only when k = 0.

*

Y fixed,the eiqenvalue problem (3.26), relative to the inner product over (L2 ( ~ I T Y )weighted )~ by (the positive definite

matrix)

E,

E

has discretse eiqenvalues and Corresponding eigenfunctions

as did (3.3).

The static case k = 0, w = 0 is singular as it was for

(3.20) and this leads to some interesting phenomena as we discuss in Section 4.11.

3.5

The dynamo problem. This problem is treated in detail by Childress [l] and G.O. Roberts

[I] and is as follows (kinematic dynamo). Given a ZIT-periodicvector function u(y) in ( C " ( ~ I T Y )the ) ~ qaqnetic field B(y,t) satisfies the initial value problem aBt-_ a (3.28)

V ~ ( u ~ B ) + A A B t, > O ,

B(y,O) = BO(y)

,

V*Bo = 0

A > O ,

.

The shifted cell problem corresponding to (3.28) is now the eiqenvalue problem

SPECTRAL THEORY OF DIFFERENTIAL OPERATORS (3.29)

X(V

+

with

2

ik) U

+ (V + ik) A (u

U E (H'(2aY))

3

A

,

U) = wU

k

E

621

,

Y fixed

.

The operator on the right of (3.29) has compact resolvent, hence discrete spectrum but is not selfadjoint.

The dynamc problem consists of showing that (3.28) has solutions whose L2 norm over Rn grows with t.

While one can develop spectral results as is done by Roberts[l],

the conclusion that one seeks is an essentially static phenomenon (in our

terminology).

It can be therefore treated directly by an expansion

processes which are much more flexible than spectral analysis and work, in particular, even in the presence of boundaries.

We shall

not, however, treat this problem in detail here.

3.6

Some nonselfadjoint problems. Consider the general elliptic operator

(3.30)

A =

2

n apq(y)

Prq=l (y)) , (b (y)) and c(y) smooth functions on 2nY and (a ) P P9 positive definite and c(y) 5 0. The shifted cell problem for this and (a

P9

operator is not a selfadjoint problem and hence an expansion for it is not easily constructed.

In Section 5.8 we shall be concerned

however with the shifted cell problem for imaginary k, i.e.

acting on Cm(2nY) with k E Rn fixed.

The operator (3.31) has a

closure in C(~ITY)which generates a continuous semigroup, but not of

622

HIGH FREQUENCY WAVE PROPAGATION

contractions,on C(2nY) which is however positivity preserving (strong maximum principle). From the Perron-Frobenius

theoryt we conclude that A(k) has an

isolated maximal eigenvalue w(k) (which is a convex function of k) and strictly positive right and left eigenfunctions corresponding to it. We shall use this result in Section 5.8. In the remainder of this section we shall give a general representation theorem for the solution of equations with periodic coefficients that does not depend on spectral theory. Let A be a differential operator of order m with 2n-periodic coefficients. Using multi-index notation we write

where

Let f

E

$(Rn)

(3.34)

I

hence not a periodic -mction, and consider the problem

Au = f.

Assume for the moment that (3.34) has a solution u 6 A ( R n ) so that the calculations that follow make sense. With k E Y fixed we define A(k), as in Section 3.2, by

'cf.

for example Krein-Rutman [l] , T. Harris [11.

SPECTRAL THEORY OF DIFFERENTIAL OPERATORS (3.35)

623

e-ik.Y A = A(k)e-ik'Y

so that

with (D+ik)"

=

a aY1 +

(

ik1 )"1

...

a n aYm + ikn ) U

(

.

For each z cRn let TZ be the translation by z operator

Since a cCm(2vY), i.e. are periodic and smooth, it follows that

which means that A and A(k) commute with translations by any lattice n

vector (vector y in 2vZ

).

With k E Y fixed we multiply ( 3 . 3 4 ) by eViksY and use ( 3 . 3 5 ) . This yields

(3.39)

A(k)e-ik.Y U(Y) = e-ik.y f(y)

Now we apply T

.

to both sides of ( 3 . 3 9 ) , we use ( 3 . 3 8 ) and then sum Y over all y c2vZn. We obtain the following equation

Here

HIGH FREQUENCY WAVE PROPAGATION

624

(3.41)

and

(3.42)

By h y p o t h e s i s t h e sums are c o n v e r g e n t a n d h e n c e b o t h 2n-periodic

6 and ? are

functions.

L e t u s assume t h a t p r o b l e m ( 3 . 4 0 ) a s a p r o b l e m o n t h e t o r u s

2 n Y h a s a s o l u t i o n w h i c h w e w r i t e as

(3.43)

6(y;k) = (A-'(k)f(-;k)) ( y )

.

M u l t i p l y i n g ( 3 . 4 3 ) by eikmY, i n t e g r a t i n g w i t h r e s p e c t t o k o v e r Y and u s i n g ( 3 . 4 1 ) w e o b t a i n :

1

=

eikeY A-'(k)[

1

e-ik. (y+')

f (y+y)]dk

y E2.irZ"

Y

.

W e summarize a n d make t h e a b o v e p r e c i s e i n t h e f o l l o w i n g t h e o r e m w h i c h i s t y p i c a l o f w h a t o n e may e x p e c t i n o t h e r s i m i l a r s i t u a t i o n .

Theorem 3 . 2 . L e t A b e d e f i n e d by

_ .

2

n L (R ) . For e a c h k

E

( 3 . 3 2 ) w i t h Cm(2.irY) c o e f f i c i e n t s , Y

let A ( k )

b e d e f i n e d by

a c t i n g on

2 ( 3 . 3 6 ) a c t i n g o n L (2nY).

Assume t h a t t h e r e i s a c o n s t a n t C i n d e p e n d e n t of k E Y s u c h t h a t

625

SPECTRAL THEORY OF DIFFERENTIAL OPERATORS Assume a l s o t h a t f o r some complex number 2

X

,(A-X)-'

n

e x i s t s and i s a

bounded operator o n L ( R 1 . Assume f i n a l l y t h a t A - l ( k ) measurable a s a function of k

E

is strongly

Y.

2 n Then e q u a t i o n ( 3 . 3 4 ) h a s a u n i q u e s o l u t i o n u E L ( R ) f o r e a c h f

E

L2 (R")

*

. Moreover

t h i s s o l u t i o n i s g i v e n by t h e r e p r e s e n t a t i o n f o r -

(3.44).

Proof

:

(3.44) is w e l l defined as an element of

We note f i r s t t h a t u of 2

n

Moreover t h e o p e r a t o r A o f

L (R ) because of o u r hypotheses.

(3.32)

i s c l o s e d s i n c e i t s r e s o l v e n t set i s non empty. L e t & ( A ) d e n o t e t h e domain o f A i n L 2 ( R n ) . With f E L2 ( Rn ) g i v e n , l e t f m

E

d ( R n ) be a s e q u e n c e s u c h t h a t

2 n f m * f i n L ( R ) a s m ' m ,

(3.46)

D e f i n e u m ( y ) by ( 3 . 4 4 ) w i t h f r e p l a c e d by f m ( y ) . I t i s e a s i l y seen, s i n c e A i s c l o s e d so w e c a n t a k e i t i n s i d e t h e i n t e g r a l , t h a t

(3.47)

Thus um

Au

E

m

= f

m '

&(A) and Aum * f i n L2 ( Rn ) a s m

Now

=

c

a n d h e n c e Iu-umI

jRn l f - f m l +

L2 ( R n )

0 as m

2

*

dY

m.

+

m.

HIGH FREQUENCY WAVE PROPAGATION

626

S i n c e A i s c l o s e d w e c o n c l u d e t h a t Au = f and u i s i n d e e d t h e s o l u t i o n o f o u r problem ( 3 . 3 4 ) . The p r o o f i s c o m p l e t e . The main p o i n t of t h i s e l e m e n t a r y r e s u l t i s t h i s : problem ( 3 . 3 4 ) i s e f f e c t i v e l y r e d u c e d t o t h e s o l u t i o n of a f a m i l y of s h i f t e d

c e l l problems ( 3 . 4 0 ) , p a r a m e t r i z e d by k which b e l o n g s t o t h e compact

s e t Y ; t h e f i n a l r e s u l t i s o b t a i n e d by i n t e g r a t i o n o v e r k , i . e . by (3.44).

SIMPLE APPLICATIONS OF THE SPECTRAL EXPANSION 4. 4.1

627

Simple applications of the spectral expansion. Lattice waves. To motivate better the physical significance of the results in

later sections we analyze here a very simple example for which all computations can be carried out explicitly by elementary means.

t

Let u (t), p E 2,be the displacements from equilibrium of P oscillators located at p, with mass M and coupled to each other by P linear springs. The equations of motion are (in suitable units)

uP (0) =

up ,

du (0)

+-

=

vP '

uP 'vP

o

for IpI large

.

We shall assume that there are two masses Mo and M1 periodically distributed:

Mo at the even locations, M1 at the odd locations.

The

system has periodic structure with period 2, i.e. M = M P+2 P' First we construct the spectral resolution for the operator in (4.1). We put

with k E [0,1] and @(p+2;k) = @(p;k) in (4.1).

This leads to the

"shifted cell" eigenvalue problem

tThis is done in practically all solid state physics books, e.g. Ziman [l], as well as in Brillouin 111.

628

HIGH FREQUENCY WAVE PROPAGATION

which is now a 2

2 matrix eigenvalue problem for each k E [0,1].

X

We rewrite it in the form

The two eigenvalues w 2 (k) 5 w2 (k) are usually called the acoustical 0 1 and optical modes respectively given by ( - with 0,+ with 1)

(4.4)

2 M0 + M1 T d(Mo+M1) ~ ~ , ~ =( k ) 2MOM1

-

4MOM1sin2 .rrk

The eigenfunctions Q (p;k), Q (p;k) are chosen orthonormal relative 0 1 to Mo, M1, i.e. (4.5)

-

p=o,1

,

MpQm(P;k)Qm,(p;k)= 6m,

m,m' = 0,l

.

Given u with u 0 for IpI large we can construct its expanP P sion in discrete Bloch waves as in Section 4.2. We obtain

Now we can use (4.6) to solve (4.1).

This is elementary.

obtain

(4.7)

u (t) =

P

1 -

dk

c

2'

m=0,1

Am (k)e

i (akp+wm(k)t)

+ Bm(k)e

1

i (nkp-wm(k)t)

We

SIMPLE APPLICATIONS OF THE SPECTRAL EXPANSION

629

with

, (4.8)

Here Um(k) and Vm(k) are given by (4.6) with u replaced by U and V P P P respectively. Formula (4.7) is of course a representation of the exact solution to (4.1) but the wm(k) (cf. 4.4)) are quite complicated. However the integral can be evaluated asymptotically by stationary phase and this is the primary utility of (4.7). More general and multidimensional problems on a lattice can be treated the same way.

4.2

Schrodinger equation. We shall now introduce a problem which we shall investigate in

some detail until Section 4.8.

We have chosen this problem for

computational convenience but the results, although simple, should be useful at least in understanding the various expansion processes that we have used so far. Consider the Schrodinger equation with modulated plane wave initial data:

(4.9)

i E

n auE

at

ptq-1

6 30

HIGH FREQUENCY WAVE PROPAGATION

Here (a (y) and W(y) 2 0 are smooth functions on 2aY, fm(x) are in Pq CO(Rn) and the (a ) matrix is positive definite. Equation (4.9) has P9 a unique smooth solution with t and E independent L2(Rn) norm. The functions $m(y;k) are the Bloch eigenfunctions of Section 3.1.

We

assume that for some N 2 N o (4.10)

ko is such that w (k ) <_ 0 0

Since the solution u

.. . 5 wN (kO )

are distinct.

belongs to L2 for each t it can be

E

expanded in eigenfunctions using the theorem of Section 3.2. (4.11)

uE (x,t) =

I

i(k*x-wi(k)t)/E

m

dk

>: m=O

We have

e

?m(k)$m(z:k)

,

The sum in (4.11) is understood convergent in the L2 sense. We are interested in the behavior of the exact solution (4.11) as E

-+

0.

Although writing the exact solution down and then

expanding it is contrary to our general methodology,+ we carry it out here to illustrate the more general results of Sections 5.1-5.8. We shall first carry out the expansion of uE in (4.11) formally. In Section 4.5 we show how one proves its validity. By simple changes of variables we rewrite (4.11) and (4.12) as follows:

‘We want methods that give the asymptotic behavior directly, especially when we do not have formulas for the exact solution.

SIMPLE APPLICATIONS OF THE SPECTRAL EXPANSION

We now expand formally in powers of

E.

631

The exponential inside the

integral in (4.13) becomes

where (4.15)

(4.16)

c(m)(ko) = P

2 awm(ko)

7 I

ig)

(ko) =

P

2 a 2wm(ko) ak ak P 9

,

p,q = 1,2

,...,n .

It is assumed that these derivatives exist although hypothesis (4.10) covers only the case m

5 N. We shall clarify this in Section 4.5.

We expand next the terms outside the exponential in the integral in (4.13)

HIGH FREQUENCY WAVE PROPAGATION

632

Here we use the notation

Pm, (k) is @,,,

for (complex) inner product over 2 a Y and

the Fourier transform over Rn of fm, (x), the coefficient of

in the data in ( 4 . 9 ) .

smooth on Rn

4.3

(,)

x

We also use the fact that if g(x,y) is

2aY and has compact support in x then

Nature of the expansion. We collect the results just obtained and note that uE of ( 4 . 1 3 )

has the form m

(4.18)

and

00

uE(x,t) = x e m= 0

i(ko*x

-

2

wm(ko)t)/€

satisfies the Schrodinger equation

SIMPLE APPLICATIONS OF THE SPECTRAL EXPANSION

(4.20)

633

Fi

Dependence of vAmtE)on ko is not shown explicitly. The terms v Y t E ) in (4.18) can be written, after some simple rearrangements, in the following form:

Here xP( ~ ) (y;k0) are functions on ZITY given by

The functions

x ( ~(y;ko) ) are

them in more detail.

P

important so we shall now examine

We begin with the shifted cell eigenvalue

problem

where A(k) is given by (3.2).

The vector k

E

Y is to be confined to

a neighborhood of ko so that (4.10) holds and we can differentiate (4.23) with respect to k.

Differentiating once we obtain

H I G H FREQUENCY WAVE PROPAGATION

634

where A

IP

(k) is an operator defined by

Taking inner products with I$m f in ( 4 . 2 4 ) we obtain

and

Now from the definition ( 4 . 2 2 ) of

P

and ( 4 . 2 4 ) we deduce that

X F ) is the unique solution of

on 2vY

$m(y;ko) = 0

,

subject to the normalization (4.29)

(x~).I$,) = 0

,

p = lr2,...,n

.

The eigenvalue problem ( 4 . 2 3 ) i B of course the shifted cell eigenvalue problem.

We shall now s h W the following important facts :

SIMPLE APPLICATIONS OF THE SPECTRAL EXPANSION (4.30)

In order to obtain the first two terms in (4.18) one has to solve the eigenvalue problem (4.23) only at ko and it is

not necessary to differentiate

2

either wm or $m at ko.

The derivatives can be

obtained by integration and by solving other cell problems.

Of course one must also solve (4.20)

which is not a cell problem but has constant coefficients.

Equation (4.20) is the effective

Schrodinger equation for the m-th mode (or band) at ko. Specifically:

does not involve k differentiation. (ii)

(m) is given by solving (4.28) subject to

xP

(4.29).

No k differentiation is involved. 2 2

pq (k (iii) Si(m)

) =

-

a wm(k0)

akpakq by (4.16).

We now

show how this matrix can be obtained without k differentation (cf. (4.33)). First we differentiate (4.24) with respect to k q'

This gives

635

636

HIGH FREQUENCY WAVE PROPAGATION

Taking inner products with $m we obtain

Using (4.22) and (4.20) we obtain

This is the desired formula. There are two important reasons why we do not want to differentiate with respect to k.

First, the eigenfunction $m in (4.23) is

determined only up to a phase factor that could depend on k.

We have

just shown that v F t E ) in (4.18) are determined independently of how one chooses the $m's (still under hypothesis (4.10)).

For numerical

computations avoiding k-differentiations is a very important objective. Second, the above results can be easily modified to deal with the case that at ko degeneracy at ko.

the w

i are not distinct, i.e. when we have

The form (4.18) changes then and instead of

decoupled effective Schrodinger equations for each mode (or band) m as in (4.20) we get coupled effective SchrGdinger equations (their number being equal to the multiplicity of the particular degenerate

w2 m at ko). Unfortunately we have not been able to generalize this observation to the inhomogeneous (locally periodic) caset of Sections 5.1-5.8.

Therefore we shall not discuss the degenerate case further.

t Where exact solutions are not available as they are here.

637

SIMPLE APPLICATIONS OF THE SPECTRAL EXPANSION When it exists, the tensor

is called the effective mass tensort of the m-th band. This tensor along with the group velocity vector c (m) (cf. (4.15)) are basic

P

quantities that one can use to study many complex phenomena associated with transport properties of solids.

4.4

Connection with the static theory. Let us specialize the results of Section

4.3

,

i.e. the expan-

sion (4.18) for problem (4.9), to the following case:

-n/2 UE(X,0) = (2a) fo(X) This corresponds to setting W with k

0

= 0.

.

0 in (4.9) and fm,(x)

3

0 for m'

Note also that we have rescaled the time:

t

+

2 1

t/E,

because this is the scaling used throughout Chapters 2 and 3.

Note

also that in this casett (4.35)

2 w0(O)

= 0

,

-42

l $ o ( y ; o ) E (2a)

which explains the factor ( 2 ~ 2)'n-

,

in uE(x,0) in (4.34).

The results of Section 4.3 specialize as follows.

The expansion

(4.18) has now the form

tcf. Ziman [11 or M. Lax [I]. ttRecall that we have normalization in (4.35).

= 6m,.

This explains the

638

HIGH FREQUENCY WAVE PROPAGATION

(4.36)

(xrt) +

uE(xrt) = )::v

EVjo)

Note the effect of rescaling of t xio)(y;O) = $(y)

(4.37)

AXp(Y)

+

X

(Xrztt)

t/E in (4.36). Note also that

satisfies, from (4.28) and (4.35

-

2 q=l

aa

pq

(Y)

ayq

= o ,

p = 1,2

I

...,n

,

xP (y) ZIT-periodicand

Here A = A(O), clearly (cf. (3.1), (3.2) with W : 0 ) . In (4.36) )::v

satisfies the Schrodinger equation (cf. (4.20))

The coefficients (2")

w

( 0 ) ) are given by (4.33) as follows:

SIMPLE APPLICATIONS OF THE SPECTRAL EXPANSION

639

The reader can verify easily that the above results agree precisely with the ones obtained by direct expansion in previous Of course we do not have boundary conditions: the spectral

chapters.

expansions do not work in that case, at least not in the form given here.

The expansion processes of previous chapters work just as well,

however, even in the presence of boundaries.

Remark 4.1. In previous chapters we constructed expansions of the form

but we did not have infinite sums with oscillatory time factors like in (4.36

(note that we have no

with O ( E

terms).

satisfy

via) term in

(4.36) since we stopped

The reason is that expansions like (4.41) cannot

nitial conditions beyond the E O level.

Since the problem

does not have initial layers the oscillatory terms like in (4.36) are necessary to satisfy initial conditions.

As we saw in Section 2.13,

if we do not look for expansions or results valid pointwise in t the oscillatory terms drop outt and we recover the ansatz (4.41).

Remark 4.2. The time rescaling in the analysis of the static problem (i.e., ko = 0, which is all that counts) is a consequence of the fact that

'In

particular if we consider weak convergence results only.

HIGH FREQUENCY WAVE PROPAGATION

640

2

in the case of distinct wm since they are even about k

=

0 and

differentiable. When the group velocity c ( ~ )vanishes we can have P oscillatory solutions but they are stationary; they do not propagate. The dispersive effects that come from S i ( m ) (the reciprocal of the P9 effective mass tensor) are one order of magnitude smaller than the propagation effects except when k = 0 (or near zero).

4.5

Validity of the expansion. Consider problem ( 4 . 9 ) with condition (4.10).

The solutions

(")(y;k0), p = 0,1,..., n, m = 0,1,..., N of (4.28) are smooth bounded

XP

functions. This smoothness and the smoothness of the data fm(x) n (E Ci(R ) ) suffice for the following theorems which summarize results of previous sections.

Theorem 4.1.

Let )::v

Let -

(x,t) be the solution of

(y;ko) be defined by ( 4 . 2 8 ) , P solution of

(4.29)

and let v$)

(x,t) be the

t > O ,

SIMPLE APPLICATIONS OF THE SPECTRAL EXPANSION

641

Define (4.44)

(4.45) and (4.46)

Then if uE (x,t) is the solution of (4.9), under the hypothesis stated there including (4.10), we have

Here CN(f) , C ( f ) 0

are constants that depend on the data fm(x),

5 m 5 No but do not depend on

E.

Moreover, the constant EN(f)

can

be made as small as desired, for given data fm, 0 ( m 5 No, by taking N sufficiently large.

Proof. The proof is routine under our assumptions and the elementary L estimate (energy estimate) for problem (4.9).

It is possible to

construct a smooth function v p ) (x,y,t) such that

and if we define

2

642

HIGH FREQUENCY WAVE PROPAGATION

then

E The functions g1 and gg are such that

From these facts (4.47) follows immediately and the proof is complete. The explicit construction of vim) (as well as be given in a more general context in Section 5.1.

"Am)

and v ( ~ ) )will 1

It is, in any

case, simply the "next term" in the expansion (4.18) (slightly modified). We should explain why we have chosen to state our results in the form of Theorem 4.1; in particular we mu6t explain the role of N. This is because hypothesis (4.10) is a restrictive one and it is unlikely that it would be valid for all N. with cnly finitely many modes.

This necessitates working

Theorem 4.1 can be generalized to deal

SI-WLE APPLICATIONS OF THE SPECTRAL EXPANSION

643

with degenerate modes, i.e. without (4.10), as we mentioned below (4.33). The second feature of Theorem 4.1 that departs from (4.18) is the fact that we do not use )"!:v it further.

(x,t) of (4.20) but rather expand

Thus, in Theorem 4.1, (4.42) and (4.43) are simple first

order linear (homogeneous and inhomogeneous, respectively) partial differential equations which can be solved explicitly trivially.

For

example

where c ( ~ )= (cm(ko)) is the group velocity vector.

P

The reason for

doing this is that in the more general context of Section 5.1 an analog for Theorem 4.2 below does not exist in general. Therefore Theorem 4.1 is the prototype of what we can get in general.

Theorem 4.2.

)~'!:v

for 0 0

5 t 5 T/E, T

5m 5

here. -

(x,t) be the solution of (4.20) and define

c

a,

with C a constant depending on fm(x),

No, but not on E.

Hypothesis (4.10) is, of course assumed

644

HIGH FREQUENCY WAVE PROPAGATION

Remark 4.3. Note that (4.50) is valid on a much longer time interval than (4.47).

On the other hand the error in (4.47) is actually better

than O ( E ) since CN(f) can be controlled with N.

is a better theorem.

Anyway, Theorem 4.2

It also generalizes to degenerate modes (i.e.,

without (4.10)).

Proof. The proof is again routine as in Theorem 4.1; here also because )"::v

(x,t) of (4.20) has a smooth solution with L2 norm and L2 norm

of derivatives independent of

E

((4.20) has constant coefficients).

As is shown in Section 5.1 we can construct functions

vpte)(x,y,t) and

(x,y,t) such that

and such that if

m= 0

then (4.48) holds again.

and

The new functions gi and gi satisfy

SIMPLE APPLICATIONS OF THE SPECTRAL EXPANSION

645

From these facts (4.50) follows immediately and the proof is complete.

4.6

Relation between the Hilbert and the Chapman-Enskog expansion. Theorem 4.1 of the previous section is an example of a Hilbert

expansion while Theorem 4.2 an example of a Chapman-Enskog expansion. This terminology is not standard and readers are cautioned not to look for very clear analogies with problems in other areas.'

In acoustics

and optics Theorem 4.1 is the usual geometrical optics (WKB) expansion while Theorem 4.2 is the Leontovich-Focktt or parabolic approximation. In the globally periodic case of problem (4.9) one can have either theorem; in fact one can even write down the exact solution (4.11). In the locally periodic case when

as described in Section 5.1, Theorem 4.1 is valid again-up first time caustics form-but

to the

Theorem 4.2 is more delicate and in

general is false. It is of interest to know when Theorem 4.2 can be obtained more generally.

One reason for this is that Theorem 4.2 is valid

uniformly through the static regime (cf. Section 4.4) and.includes diffraction and mode coupling effects.

'For

the Boltzmann equation see Chapran and CoWlincJ [l], (Chapter 7).

What is called Chapman-Enskog here is the second order case there (p. 116).

.

ttF. Tappert [l]

HIGH FREQUENCY WAVE PROPAGATION

646

A

look at the proof of Theorem 4.2 shows that in the locally

periodic case two things break down:

(i) caustics may form and

(ii) v;itm) of ( 4 . 2 0 ) will-not have derivatives with €-independent norms over 0 5 t 5 T/E.

These two thing are not unrelated.

The rays, solutions of

are straight lines in the globally periodic case. are curved.

In general they

If they curve very slowly and if they have no envelopes

in the relevant time intervals then Theorem 4.2 will be valid in general.

Because of the way we have set up our problems here this

statement cannot be made completely precise since either the coefficients in ( 4 . 2 0 ) will be constant or variable: no in between situation can arise.

To have ( 4 . 2 0 ) with slowly varying coefficients

requires a more elaborate scaling of the problems.

We shall not do

this here. The general idea, however, that Theorem 4 . 2 should hold when the rays are nearly straight lines is very useful.

4.7

Spatially localized data and stationary phase. We shall now consider the following problem:

where f(y) E C;

(not periodic and complex valued).

We shall assume

specifically, in order to employ (4.10), that f (y) has a finite-mode Bloch expansion

SIMPLE APPLICATIONS OF THE SPECTRAL EXPANSION

647

Nn

The scaling in (4.51) is chosen so that the initial probability amplitude

I

E - ~f (x/E)

I

is approximately a 6-function for E small and

No large (cf. Section 1.2). Using the Bloch expansion theorem we can write the exact solution of (4.51) as follows:

The analysis we shall give for (4.53) is entirely analogous to the one given in Section 2.6. The solution (4.53) is a finite sum of integrals of the form

where

For each t > 0 and x fixed we consider the system of equations

If there is no

K

E Y satisfying (4.57) then the corresponding ) : 1

goes to zero faster than any power of

*

*

E

as

E

-t

0.

If there is a

unique point K~ = ~ ~ ( x , tin) Y satisfying (4.57) and such that

HIGH FREQUENCY WAVE PROPAGATION

648

is nonsingular, then we have

Here am = om(x,t) is the signature of the matrix Q ( m ) i.e., the number of positive minus the number of negative eigenvalues of Q (m)

.

I f there are a finite number of isolated solutions to (4.57) then

(4.59) is replaced by a finite sum of terms of the same form.

If

however Q ( m ) is singular then (4.59) is not valid and other types of expansions are necessary (using the Airy function in many cases). In the particular case where S ( m ) is given by (4.56), equations (4.57) take the form 2

"

aUm ( K m )

(4.60)

___ -

Since the

w,(K),

aK

2

0

t '

5 m 5 N, are bounded and smooth, when (x,t) is

inside the forward light cone, (4.60) will have a solution. We conclude that:

the asymptotically localized disturbance at

the origin creates No modes (because of the special form (4.52)) and each mode has waves (possibly several depending on the shape of ui(k)) which propagate out of the origin with their group velocities as described by (4.59).

The solution (4.53) has then the same

asymptotic form as (4.18). (as it was in Section 2.6).

This is the main point of this section

SIMPLE APPLICATIONS OF THE SPECTRAL EXPANSION 4.8

649

Behavior of probability amplitudes. Consider the problem n

t > O ,

xER",

NO

>:

u (x,O) = eiko'X'E

as in Section

(4.62)

m= 0

4.2

m'=O

,

fm, ( x ) $ ~(' :;ko)

, including (4.10). Assume that

2 Ifm(x) I dx = 1

.

R"

Then I

tends asymptotically to one as

E

* 0 at t

= 0.

independent of t, the same holds for any t.

Since (4.63) is

Note that luE(x,t) l 2

is

the probability density that the quantum particle will be at x at time t.

We shall use Theorem

4.1

of Section 4.5 and (4.10) to obtain

the following result.

Theorem 4.3.

Let $(x,t)

I1

be in C;((O,T)

X

T (4.64)

$luEI2dxdt

Rn tends asymptotically as

E +

0 to

R"), T <

a.

Then

650

HIGH FREQUENCY WAVE PROPAGATION

and )!;v

Here

are a s i n T h e o r e m 4 . 1 ,

S e c t i o n 4.5.

Proof. L e t z E r N be a s i n ( 4 . 4 6 ) .

(4.66)

7J

Us ng ( 4 . 4 7 ) it f o l l o w s e a s i l y t h a t

d x d t @ l u E / 2- IzEfN 2 1 - 0 ,

a s € - 0 .

Rn

I t s u f f i c a s therefore t o show t h a t

behaves l i k e ( 4 . 6 5 ) .

T h i s follows e a s i l y b y u s i n g ( 4 . 1 7 ) ,

o r t h o g o n a l i t y of t h e Gm's and t h e n o r m a l i z a t i o n of , ( m ) P (4.29)).

4.9

the

(cf. (4.28),

The acoustic e q u a t i o n s . L e t P ( y ) a n d u ( y ) be smooth 2 n - p e r i o d i c

f u n c t i o n s o n R3

o n 2aY) s u c h t h a t

W e s h a l l a n a l y z e t h e a c o u s t i c e q u a t i o n s ( c f . S e c t i o n 2.10) (4.68)

X

P(,)uEt

'Et

= VpE X

= U(F)V'UE

,

' t > O ,

x E R

3

.

(i.e.,

SIMPLE APPLICATIONS OF THE SPECTRAL EXPANSION

651

We shall assume that

(4.69)

pE(x,O) = eik'X/E

2 m'=O

fm, (x)$Jrn, (t;k)

.

It will turn out that when k # 0 asymptotic solutions of (4.68) satisfying (4.69) must necessarily have u,(x,O) given in terms of p , ; one cannot give uE(x,O) arbitrarily. 2 In (4.69), $m(y;k) are the eigenfunctions corresponding to urn in the shifted cell problem

1 2 in H (271Y), subject to (4.10) and k # 0 so that wm(k) # 0.

The

normalization here is

We also have

We look for solutions of (4.68) of the following form where each term in the sum is a solution ( N 2 No):

Define operators Al(k) and A2 by

HIGH FREQUENCY WAVE PROPAGATION

652

Then ( 4 . 6 8 )

and ( 4 . 7 3 )

yield the following sequence of problems (we

fix attention on one term in the sum)

These equations ( 4 . 7 6 ) (4.78)

(4.79)

and ( 4 . 7 7 )

are more explicitly as follows:

(VY + ik)pAm) + iwm puAm) = 0

,

(V Y

+

ik) *uAm) + iw F(-’ (m) = 0 m PO

(V Y

+

ik)pp) + iwmpul(m)

+

,

Vxpo(m)

-

pub:'

(m) (Vy + ik)*up) + iwmp-1p1(m) + Vx-uo From ( 4 . 7 8 ) it follows that (4.80)

pAm) = O,(y;k)pA:)

(x,t)

,

-

lJ

= 0

,

-1 (m) = Pot

.

SIMPLE APPLICATIONS OF THE SPECTRAL EXPANSION We see therefore that when k # 0 and p,(x,O)

653

has the form (4.691,

expansions of the form (4.73) leave no freedom for uAm); it is completely determined by p t : ) .

The solvability condition for (4.79)

yields a transport equation for pi:)

(x,t).

2 Assume that (4.10) holds, i.e. that the wm(k) are distinct for 0

5 m 5 N. Multiplying the first equation in (4.79) by

(iup)-’(V

Y

+

ik)$Jm, the second by

Tm,

integrating over y E 2aY and

subtracting we obtain

+

I$,,

(V Y

+

ikl (7 1 $ ) ]. V IP m

p(m) = 0 x 00

,

which is the same thing as (cf. (4.26))

To get the next terms we define (4.83)

(v + Y

ik)*[$

P

(Vy + ik)~(~)]+ P

(y;k), p = 1,2,3 by 2 -1 ( m )

WmV

xP

where e p = 1,2,3 are the column vectors (l,O,O), (0,1,0) and P’ (O,O,l), respectively. This is the same as (4.28), (4.29). In terms of ) : ,y

we can write the solutions p p ) and u ( ~ )of (4.79) as follows 1

654

HIGH FREQUENCY WAVE PROPAGATION

(4.85)

py) =

(4.86)

u?)

=

w

lump

[p~(~) Ot

- V

p(m) x o

- (V +

ik)pj.m)]

Y

.

Note again that u p ) is completely determined by pjm) and p r ) . functions p$)

The

(x,t) satisfy inhomogeneous transport equations obtained

by the solvability condition for pam), uam), etc.

In conclusion: expansions for (4.68) of the form (4.73) can be constructed as usual when k # 0.

The proof of their asymptotic

validity in the form of Theorem 4.1 is essentially the same as the proof of that theorem. at all.

The system form of the problem plays no role

In particular the velocities u, are completely determined by

the pressure p, and initial values cannot be prescribed for u

.

This

is a special feature of asymptotic solutions of the form (4.73).

4.10

Dual homogenization formulas. The case k = 0 requires special treatment.

studied in detail in Chapters 1 and 2.

It is the static case

For comparison purposes with

the previous section we give the results here in somewhat different form. We consider the problem

(4.88)

pE(x,O) = f(x)

,

uE(x,O) = g(x)

.

Here p and y satisfy (4.67) and are smooth and f 3 3 g E (Ci(R 1 )

.

If we look for an expansion of the form

E

a ( R3 ) Co

,

SIMPLE APPLICATIONS OF THE SPECTRAL EXPANSION p,(x,t)

= P(O) (x,:,t)

(4.89)

655

+ ep(l) (x,F,t) + c2p(2) (X,+

+

...

,

+

+

...

,

+

(x,:,t)

E?d2)(X,;,t)

we find that we cannot satisfy the initial conditions uE(x,O) = g(x) even to principal order in

E.

Another ansatz will not work here; it

is necessary to either consider expansions for special classes of data (not (4.88)) as in the previous section

or

vergence results with general data like (4.88).

to seek week conWe shall obtain here

weak convergence results. We begin with some constructions and then state and prove the main theorem.

The results go through even for (4.87) with boundary conditions as follows. Let 0 C R3 be a bounded open set with a8 its smooth boundary. (4.90)

We suppose that

equations (4.87) hold in 0 and p E (x,t) = 0, x

E

a8

with f

E

C;(S).

The weak form of (4.87), (4.88), (4.90) is as follows. v(x,t), n(x,t) be in (Cm(R3

x

[O,-))I3 and Cm(R3

x

[O,m)),

Let

respectively,

such that they vanish identically for t large and a(x,t) vanishes outside a compact subset of 8 . Denote by V the class of all such functions. Multiplying by v and n , integrating and integrating by parts we obtain

If we let

H I G H FREQUENCY WAVE PROPAGATION

656

(4.92)

AE =

t h e n w e c a n w r i t e (4.91) i n t h e form (p,

-x j ( y )

= P

X

(F)I

PE = !J(,))

X

W e i n t r o d u c e two sets of c e l l p r o b l e m s f o r v e c t o r f u n c t i o n s

and scalar f u n c t i o n s x . ( y ) , j = 1 , 2 , 3 d e f i n e d on 2aY a s 3 t follows

T h e s e f u n c t i o n s are u n i q u e l y d e f i n e d by ( 4 . 9 4 ) and ( 4 . 9 5 ) . Define further" (4.96)

=

q;;

711

[

t h e c o n s t a n t m a t r i x q-l w i t h e n t r i e s

p(Gij

+ (V

A

xj'i .

1

By d i r e c t c o m p u t a t i o n w e f i n d t h a t t h e f o l l o w i n g i d e n t i t i e s h o l d ( p o i n t w i s e o n 2nY)

tHere e

j

are column v e c t o r s el =

( ~ , o , o ) , e2

e3 = ( 0 , 0 , 1 ) a n d , as i n p r e v i o u s c h a p t e r s ,

[fl =

= (o,I,o),

1 7 2aY

tt(V

A

xjIi

i s t h e i - t h component o f V A

ij '

f(y)dy.

657

SIMPLE APPLICATIONS OF THE SPECTRAL EXPANSION .)

From these identities it follows that q is also given by the dual formula to (4.96): (4.99)

qij = ?4 [P-1(6ij

.

+ vxj)i)]

It is easily verified that q-l (or q) is positive definite (we use (4.67)). Define the operator q

-1

0

a

(4.100) 57 = 0

R"V-11

at

I

This is the homogenized operator corresponding to (4.92).

I:[

Let (u,p)

be the solutiont of the problem

(4.101)

= 0

,

t > 0

,

x

E

d ,

Theorem 4.4. For each (v,n) E V fixed the solution

of (4.87), (4.88)

and (4.90) satisfies

tThis problem has a unique smooth solution

(ad

is smooth)

.

HIGH FREQUENCY WAVE PROPAGATION

658

11

m

m

(4.102)

lim €40

jj

(v.uE

+ npE)dxdt

=

(v-u + ap)dxdt

,

0 6

0 0

where (u,p) is the solution of (4.101).

Remark 4.4. The convergence is weak star convergence.

Note also that very

little regularity is needed here but we shall not attempt to give the optimal conditions.

Proof. From the energy identity for (4.87) and (4.67) we conclude there is a subsequence of (uE,pE), also denoted by (uE,pE),which converges weak star in Lm ([O,T]; ( L 2 ( 8) ) 4 ) for each T < It

-.

remains, as usual, to identify the limit as the unique solution (u,p) of (4.101) which completes the proof.

This is done by adjoint

expansion as follows. Fix (v,a) in V.

We define v ( ' )

(x,y,t) and

follows:

i = 1,2,3 where Mi(y) are periodic functions such that

,

7")

(x,y,t) as

SIMPLE APPLICATIONS OF THE SPECTRAL EXPANSION

6 59

Define further

(4.107)

GE(x,t) = n(x,t)

+

this is not true for GE so we must

Even though n vanishes near cut it off as follows.

.

E*(l)(x,:,t)

Since 3 0

is smooth there exists a function

m E ( x ) such that

(4.108)

mE(x) = 0

,

when dist(x,ao

)

5 fi ,

mE (x) = 1

,

when dist(x,aO

)

2

2 f i

,

Now put

It can be verified by direct computation that the above constructions produce the following result (which was the objective of the constructions all along):

where

AE

and c7 are defined by (4.92) and (4.100) and

(4.111)

hE =

(4.112)

hi =

E ( p

v(l) E t

-

m

E ( V -1mE.rrt (1)

v E X

.rr(l))

-

(mE

-

1 ) r(l) ~ Y

-

EVmEa (1)

,

660

H I G H FREQUENCY WAVE PROPAGATION

Note that (4.113)

sup O l t

and (4.114)

sup

o5t

NOW w e g o

I I

I h i ( x , t ) L2dx

+

0

as

E

.+

0

,

Ih;(x,t)I

+

0

as

E -+

0

.

2dx

t o ( 4 . 9 3 ) and u s e it w i t h

T h i s y i e l d s , t a l o n g w i t h (4.1101,

passing t o the l i m i t

E

-+

(V,IT)

r e p l a c e d by (v , n E )

(4.106) and ( 4 . 1 0 9 ) ,

0 and u s i n g ( 4 . 1 1 3 ) ,

(4.114)

(4.96),

we obtain

-

t W e u s e summation c o n v e n t i o n i n ( 4 . 1 1 5 ) and t h e n o t a t i o n

.

SIMPLE APPLICATIONS OF THE SPECTRAL EXPANSION

661

This is precisely the weak form of (4.101) so the proof of Theorem 4.4 is complete.

4.11

Maxwell's equations. We shall now consider Maxwell's equations in a periodic medium,

i.e., when the dielectric "constant" and the magnetic permeability are rapidly varying periodic functions of space. To avoid confusion with the small parameter

E

we depart the

traditional notation and (4.117)

denote by p(y) the dielectric "constant" and by u(y) the magnetic permeability; we assume that (4.67) holds.

Maxwell's equations are then (cf. (2.95)-(2.97))

(4.118)

where p,(x)

[:' ,", =

a -

EE

EE],

at

HE

~ ( x / E )times the 3

t > O ,

HE

x 3

identity and similarly for p E .

We shall deal with the static case only here; the analogs of the results of Section 4.9 carry over without much change. case we can also handle boundary conditions.

In the static

We now describe the

initial and boundary conditions for (4.118) that. we shall adopt. Let Let

8 C R 3 be a bounded open set with smooth boundary 2 0 .

k denote the unit outward normal to a 8

hold in 8

.

We take (4.118) to

subject to the boundary condition (conducting walls)

662

HIGH FREQUENCY WAVE PROPAGATION

(4.119)

;IAE

forxEa8

= O

.

For initial conditions we take

with f E ( C i ( S ) ) 3 and g (4.121)

E

(Cm(8) ) 3 and satisfying

V.f(x) = V.g(x) = 0

.

Conditions (4.121) are imposed so that

which are the usual complementing conditions on data and are preserved by (4.118) at later times. Let V denote the class of all functions (u(x,t),v(x,t)) where u and v are vector-valued and smooth, i.e. in (Cm([O,m) x 8 ) ) 3 , such that u and v are identically zero for t large and identically near 3 0

GAU vanishes

.

With (u,v) E V we introduce the weak form of (4.118), (4.119), (4.120) (this problem has a unique smooth solution as it stands when E

> 0 is fixed).

Multiplying by (u,v) integrating and integrating by

parts as usual we obtain

(4.12 3 )

I

[p,ut.E,

O O

+

+

pEvt'HE

- VA

v'E&

+ VA

u*HE]dxdt

1

E(O,x)-f(x) + v(O,x).g(x) dx = 0 8

This is the weak form of (4.118), (4.1191, (4.120).

,

(u,v) E V

If we define

.

663

SIMPLE APPLICATIONS OF THE SPECTRAL EXPANSION

we can write it in operator form as follows:

Now we continue with the necessary construction of cell functions which will enter in the proof of the theorem that follows, i.e. in the adjoint expansion (the constructions are, of course, dictated by the usual perturbation expansions). We define two sets of cells functions; one associated with p and one with p: (4.126)

xj (Yip

, xj(y;u) , jij(y;p) , ;j(y;p) ,

The x's are scalar and the

2

are vector-valued functions on 2aY.

They are defined uniquely by (cf. (4.94)

(4.127)

(4.128)

(4.129)

qij ( P ) =

" P

6ij

j = 1,2,3

ax. (yip) +

aYi

(4.95))

.

HIGH FREQUENCY WAVE PROPAGATION

664

I1

(4.130)

i,j = 1,2,3

.

'

It can be verified easily that q(p) and q(p) are positive definite matrices. We also have flux identities (cf. (4.97), (4.98)),

and the dual homogenization formulas (4.133)

-1

'6ij +

(VA

%.(y:P))i) 3

-1 -1 qij ( p ) = Q [u (6ij +

(vA

~ ~ ( y : p ) ) ~,)

Given (u,v) E V, define u(')

1;

i,j = 1,2,3

/

(x,y,t) and v(l) (x,y,t) as follows

.

SIMPLE APPLICATIONS OF THE SPECTRAL EXPANSION

665

Let mE(x) be the cutoff function defined by (4.108) and define

(4.136)

UiC(X,t) = Ui(X,t) +

(4.137)

VjE(X,t) = Vi(X,t)

[$

axk(Y;P) aYi

+

I

Uk(X,t)

v,(x,t)

y=x/E

1

y=X/E

+ "(I) (x,E,t) ,

i = 1,2,3

Define also the homogenized operator CI

Note that q ( p ) and q(1-1) are 3

x 3

.

by

matrices now given by (4.129),(4.130)

The above constructions lead to the following facts which can be verified by direct computation. For (u,v) E V given, (i)

(uElvE)belongs also to V

,

and

E where hi, i = 1,2 are such that

(4.140)

lim sup E+O t2o

Ihi(x,t)I 2dx = 0

,

i = 1,2

.

Let (E,H) be the solution of the homogenized problem corresponding to (4.118), (4.119) , (4.120):

HIGH FREQUENCY WAVE PROPAGATION

666

?.

(4.141)

~ A E 0= ,

x ~ a s,

Note that problem (4.141) satisfies the constraints

as should be in view of (4.122) and (4.121).

Theorem 4.5. The solution (EE(x,t),HE(x,t)) of (4.118), (4.119), (4.120) converges weak star in Lm ([O,T];L6 ( O ) ) , T <

-

and arbitrary, to

(E(x,t),H(x,t)) solution of (4.141) i.e., for each (u,v) E V m

as E -

-c

m

0.

Proof. The proof is a verbatim repetition of the one of Theorem 4.4 of the previous section.

The construction of the cell functions

x

and

that lead to (4.139) is the basic element of the proof and, as was pointed out already, this is straight perturbation theory.

SIMPLE APPLICATIONS OF THE SPECTRAL EXPANSION 4.12

667

A one dimensional example. It is too difficult to deal in the high frequency context with

boundary value problems even in one dimension.

This is due to the

complexity of the phenomena that take place near boundaries.

If the

variations of the coefficients from a uniform background (i.e., constant values) are small then a good deal can be said; even for higher dimensions and random media.'

It is very likely that good

progress will be made in such cases but we confine the discussion here to the following simple example. In a one dimensional medium let the inhomogeneities occupy the interval [ O , L l .

With the time factor e-iWt/E omitted the wave field

uE(x) = u (x;k) satisfies

(4.144)

2 d'~ k 2 2 x T + (r) n (E)uE = dx uE =

u

.-ikx/~ E

o ,

o

< x < L ,

x < o ,

I

du and 2 dx continuous at x = 0

Here n(y) is the index of refraction.

,

x = L

.

We assume it is a seal-valued

almost periodic function such that (4.145)

2 n (Y) = 1 + EP(Y)

'See Kohler-Papanicolaou wave problems.

,

[ 1 ]

and references therein for random

Formally one can treat ever? multidimensional random

wave problems a s in Burridge-Papanicolaou [ 11 but techniques for proofs have not been developed yet.

668

HIGH FREQUENCY WAVE PROPAGATION

with l u ( y ) I 5 1, u ( y ) real-valued and almost periodic. To solve (4.144) we write uE and duE/dx in terms of new functions A E , B

as follows:

uE =

AE

ikx/E -ikx/E + BE(x)e (x)e

It follows then that

(4.147) dx dBE

-

';" LI(:)

[AE (x)e2ikx/E + BE(x)] ,

0 < x < L ,

with boundary conditions

We also have that RE and T E , the reflection and transmission coeffic ents respectively (RE valued) are given by t

=

RE(L;k) and TE = TE(L;k) are complex

(4.149) One can easily analyze directly the asymptotic behavior of the solution of the two-point boundary value problem (4.147) as

E

-L

0.

We are primarily interested in RE(L;k), the reflection coefficient. This can be handled directly as follows.

+Note that because p is real valued and

5

1, for

E

2 have I R E ] + lTEI2 = 1; there is no loss or gain in energy.

< 1 we

SIMPLE APPLICATIONS OF THE SPECTRAL EXPANSION From (4.147) we find that UE(x)

669

AE(x)/BE(x) satisfies the

following equation (4.150)

d UE(x) = dx

$ p(E)[uE(x)e ikx/E

+

.-ikx/e

1’

, O < X L L ,

and

Therefore to study RE(L;k) as

E

* 0 it is enough to analyze the

asymptotic behavior of the nonlinear initial value problem (4.150). The asymptotic analysis of (4.150) amounts to a direct and simple application of the method of averaging.+

It is quite

reasonable to assume that T

(4.152)

while we define (4.153)

1 lim T4m

1

T

p(y)eZikYdy

E

S(2k)

,

which is the Fourier coefficient of p at frequency 2k.

Since l~ is

real (4.154)

S(-Zk) = S(2k)

.

We also have that N.N. Bogoliubov and Ju. A. Mitropolski [l]. For (4.150), the limit equation (4.157) is obtained by fixing right side of (4.159) with respect to Y=X/E.

UE

and averaging the

670

HIGH FREQUENCY WAVE PROPAGATION

(4.155)

s(2k) = 0 except when 2k equals one of the discrete values that form the spectrum of the almost periodic function p .

Applying the averaging method to (4 150) we find that for any L < (4.156)

lim ECO

sup IuE(x) OXLL

-

m

U(X) = o ,

where U(x) satisfies (4.157)

dx

*

= 2

U(0) = 0

[S(2k)U2(x)

+

,

S(Zk)]

O < X L L ,

,

and hence

From (4.157) and (4.155) we conclude immediately that (4.159)

RE(L;k) is asymptotically zero whenever 2k is not a point in the spectrum of

u.

Physically,

this says that the medium appears asymptotically to be transparnt except at resonant frequencies (frequencies that match, i.e. resonate with, the inhomogeneities in the above way). At the resonant frequencies when S(2k) # 0 we do get reflection. fact by solving (4.157) we find that RE(L;k) + R(L;k) where (4.160

a s ~ + O ,

In

6 71

THE GENERAL GEOMETRICAL OPTICS EXPANSION

This is the main result of this section. The power reflection coefficient in the limit (4.161)

IR(L;k)l 2

tanh2(la(k)IL)

=

E

has the form

= 0

I

which shows that for L large compared to la(k) 1-l the amount of power reflected is quite substantial.

5. 5.1

The general geometrical optics expansion. Expansion for Schrodinger's equation. We shall construct an asymptotic expansion for the solution

u (x,t) of the problem

X

+ 7W ( X , ~ ) U=~ 0

,

x

,

Rn

,

t > 0

,

g (x) real-valued ,

E

-

(5.2)

u (x,O) = eiS(X)'Ef

(x,:)

Here W(x,y) 2 0 and a (x,y) are smooth functionst on Rn Pq is positive definite (5.4)

'2?rY

2

p,q=l

a 5 5 > alcI PqPq-

,

x

a a positive constant

2aY, a

Pq

,

the n-dimensional torus with basic set the parallelepiped

all of whose sides are 2n.

672

HIGH FREQUENCY WAVE PROPAGATION

and the functions fm(x), 0

5 m 5 No are in Ci(Rn), complex-valued, and

g(x) E Cm(Rn) real-valued. The functions $m (y;k,x) are the eigenfunctions of the shifted cell problem (cf. Section 3.1) (5.5)

A(k,x)$Jm= ~,(krx)$~ 2 t

where n

(5.6)

A(k,x) =

-

c1[”

P19’

ayP + ikp] [apq(x,y)

where k E Y and x E Rn are parameters.

a

[T+

ikq

It will be assumed throughout

what follows that: (5.7)

for 0 < m < N with some N 2 No,the eigenvalues 2

wm(k,x) are distinct for all k x E Rn.

E

K

C

Y and all

Moreover, the eigenfunctions $m(y;k,x)

are differentiable functions of y, k, X, y

E

ZrY, k E K and x E Rn.

t and makes the expansion process This assumption is quite restrictive rather simple.

It would be interesting to find appropriate expansion

forms in general when the multiplicity is variable. We shall construct an asymptotic expansion for uE in the form

where each term in the sum is to satisfy (5.1). the phases S ( m )(x,t), 0

First we construct

5 m 5 N. They are to satisfy the

t Recall that it is not necessary in the globally periodic case , Section

4.

THE GENERAL GEOMETRICAL OPTICS EXPANSION

673

Hamilton-Jacobi equations

I

We shall assume that S(x) is such that (5.10)

there exists a time to > O for which ( 5 . 9 ) have a unique smooth solution and VS(m) (x,t) E K for all x

E

R ~ o, 5 t < to,

o

z m LN.

Thus all our results are local in time and in wave-number space.

The

closer Vg(x) is to being a constant (i.e., the more "plane" the initial wavefunction (5.2) is) the longer the interval of existence [O,to) for (5.9) will be if Vs(x)

E

K, x E Rn.

discuss (5.9) a bit further. With the S(m) so determined we find that

and hence for 0 5 m 5 N,

In Section 5.2 we

674

HIGH FREQUENCY WAVE PROPAGATION

We assume that the v ( ~ )are 21r-periodic as functions of y and that they have asymptotic power series expansions (5.15)

~(~)(x,y,t= ; ~vo )( m ) (x,y,t) + ~v:~)(x,y,t)

+

E 2 v p (x,y,t) +

... .

Inserting (5.15) into (5.11) and,equating coefficients of equal powers of

E

(5.16)

AP)vAm) = 0

(5.18)

A:m)

yields the following sequence of problems

Vim)

+

,

A2( m ) v1(m

Equation (5.16) implies that (5.19)

v(m) 0 = $,(y;~s(~) (x,t),x)v$)

(x,t)

o5

,

where the function vit) (x,t) is to be determined.

m

5

N

,

From (5.3) we see,

however, that v(m) (x,O) = f,(X) 00

,

O l m l N o

r

(5.20) "(m) 00

(X,O) = 0

,

No

+

We consider next equation (5.17). equation for

vim). (

Since A?)

1 5m 5 N

.

It is an inhomogeneous

has a one-dimensional null-space the

inhomogeneous term must satisfy a solvability condition. and (5.7) we conclude that this condition is

From (5.5)

675

THE GENERAL GEOMETRICAL OPTICS EXPANSION

where the inner product is over 2nY with (x,t) parameters. (5.21) for

):Av

Equation

(x,t) along with (5.20) constitutes the homogenized

transport equation for the slowly varying amplitudes )::v

in the

expansion (5.81, (5.15), (5.19), i.e.,

Note that v;i)(x,t)

(5.23)

E

0 for N 0 + 1

in view of (5.21) and (5.20).

a

Am)

is given in Section 5.3.

5m 5N ,

A more explicit form for the operators They are homogeneous, first order

linear partial differential operators. We return to (5.17) where now the solvability condition is satisfied.

We have that

where the functions ) : i v

(x,t), 0 5 m 5 N, are to be determined again.

A more explicit form for the first term on the right side of (5.241, in terms of cell problems, is given in Section 5.3. We continue with (5.18) and apply the solvability condition to its inhomogeneous term. transport equation for v i : )

Here

a

Am)

This yields an inhomogeneous (x,t) of the form

is defined as in (5.21) and

HIGH FREQUENCY WAVE PROPAGATION

676

Initial conditions for v l : ) as follows.

From (5.8

(x,t), solution of (5.25), are obtained

, (5.15) and

(5.3) it follows that

(5.2),

= o .

Using (5.24) this condition takes the form

We must therefore set

This provides initial conditions for )::v completely from (5.25).

(5.27).

which are now determined

However, these conditions do not suffice for

This is a natural limitation since we are working with only

N modes while infinitely many modes are required to satisfy (5.27). We are precisely in the situation of Theorem 4.1, Section 4.5. Let

THE GENERAL GEOMETRICAL OPTICS EXPANSION

677

Theorem 5.1. The statement of Theorem 4.1, Section 4.5, is valid here again provided t

[O,tO) and (5.7), (5.10) hold.

E

where CN(f)

and

( c c o (R”)),

0

Explicitly,

C(f) are constants that depend on the data fm(x)

5 m 5 No but do not depend on

E.

Moreover the constant

CN(f) can be made as small as desired by taking N sufficiently large. The function v y ’ required in the proof of Theorem 4.1 is, of course, the next term vim) in (5.15) and is obtained by carrying the above expansion process one step further.

5.2

Eiconal equations and rays. The analysis of the eiconal or Hamilton-Jacobi equations (5.9)

parallels almost word for word the one given in Section 2 . 2 , except 2

that the Hamiltonians are now wm(k,x), 0 5 m 5 N, the eigenvalues of the shifted cell problems (5.5). Let ( x ( ~(t) )

(t)) be solutions of the Hamiltonian system

,...,n .

p = 1,2

HIGH FREQUENCY WAVE PROPAGATION

678

Locally, the solution of (5.9) can be written by using the rays x ( m ) (t;x)

2

Here L(m) (x,x) is the Lagrangian associated with wm (k,x) as in (2.22).

Absence of caustics means that for 0 x

+.

5 m 5 N the mappings

(where x ( ~ (0;x) ) = x) are one to one on Rn and hence x ( ~ (t;x) )

S ( m ) can be found at any point if it is known along rays by (5.32).

5.3

Transport equations. we now give more explicit forms for the operators

a

p) defined by

QAm)

and

(5.21) and (5.26), respectively.

We observe that Ajm) has the form

where VS(m) = vS(m)(x,t) and A(k,x) is given by (5.6). local group velocity of the m-th mode (or band) by

Direct computation yields now

Define the

THE GENERAL GEOMETRICAL OPTICS EXPANSION

679

In (5.35) the functions b(m) (x,t) are real-valued and are given by

where

and

Equation (5.35) has the usual form of a transport equation of geometrical optics. follows :

It can be written as a conservation law as

HIGH FREQUENCY WAVE PROPAGATION

680

W e continue with t h e operator $)

a?)

d e f i n e d by ( 5 . 2 6 ) .

Let

( y ; k , x ) be t h e s o l u t i o n s o f t h e ( p a r a m e t r i c a l l y x-dependent) c e l l

problems ( c f . ( 4 . 2 8 ) ) G w 2 (k,x)

(5.40)

m

+

1

( m ) ,$Jm) (XP Here A

IP

1

+ A ( k , x )) : x

rP

(k,x)

= 0

-

(y;k,x) aw;(k,x)

akP

I

p = 1,2

1

$ J m ( y ; k , x )= 0

,..., n

,

I

O c m c N .

( k , x ) i s d e f i n e d by

The c e l l f u n c t i o n s XK(y;k,x) a r e w e l l d e f i n e d by ( 5 . 4 0 ) . Define a l s o t h e c e l l f u n c t i o n s X m ( y ; x l t ) by

where B ( m ) ( x , t ) are g i v e n by

681

THE GENERAL GEOMETRICAL OPTICS EXPANSION

We note that

x ( ~ of )

(5.43) is well defined because (B(m),@m) = 0.

Now we can write vim) of (5.24) using the cell functions ,(m)

P

and

x ( ~ as )

follows:

From this it is easily seen that we can express

ffp) of

(5.26) as

follows

Using the identity (4.33, with additional parametric x-dependence which changes nothing, we can rewrite (5.46) in the following more revealing form

682

H I G H FREQUENCY WAVE PROPAGATION

( m ) and 2. (m) are g i v e n by c e r t a i n e a s i l y o b t a i n e d P b u t l o n g e x p r e s s i o n s which w e do n o t w r i t e s i n c e t h e y d o n o t seem t o

The c o e f f i c i e n t s

be p a r t i c u l a r l y i n t e r e s t i n g . W e c l o s e t h i s s e c t i o n by o b s e r v i n g t h a t a l l p r e s e n t r e s u l t s

s p e c i a l i z e , a s t h e y s h o u l d , t o t h o s e of S e c t i o n s 4.1-4.5

i n the

g l o b a l l y p e r i o d i c case (apq = a p q ( y ) , W = W(y) i n 5 . 1 ) .

5.4

Connections w i t h t h e s t a t i c theory. W e now s p e c i a l i z e problem ( 5 . 1 ) - ( 5 . 3 )

(5.50)

uE(x,O) = ( 2 ~ ) - " ' ~ f , ( x ) E C i ( R n )

t o t h e following

.

W e assume a g a i n t h a t ( a p q ( x , y ) ) i s smooth on Rn x 2nY and u n i f o r m l y

e l l i p t i c (cf. (5.4)). Note t h a t ( 5 . 4 9 ) ,

( 5 . 5 0 ) i s o b t a i n e d from ( 5 . 1 ) - ( 5 . 3 )

by s e t t i n g

THE GENERAL GEOMETRICAL OPTICS EXPANSION

and by rescaling time t

+

explained in Section 4.4. Chapter 2. (5.52)

Since W 2 wo(O,x)

E

t/E.

683

The rescaling of the time was

The new time scaling is the one used in

0, it follows that 0

,

-n/2 $O(y;k,x) : (2n)

-n/2 This explains the factor ( 2 ~ ) in (5.50). By going through the steps in the expansion in Section (5.1)

we see that: (5.53)

the operator -ia/at is now part of A3 (5.14) and not of (5.13), because of the change t

+

t/E

, ,

(5.54)

the phase function S(O)(x,t) E 0

(5.55)

the transport equation (5.21) is automatically satisfied for m = 0

(5.56)

,

equation (5.25) with m = 0 is the determining equation for ) :v

(x,t)

.

The determining equationt has explicitly, in view of (5.47), the following form

'It

is easily seen that i ( O ) and Z ( O 1 in ( 5 . 4 7 ) are now zero.

HIGH FREQUENCY WAVE PROPAGATION

684

This is the homogenized Schrodinger equation and the coefficients 5(O)(x) are given by ( 5 . 4 8 ) . Going back to the form ( 4 . 3 3 ) of the P9 2 2 second derivatives of wm (k,x) and specializing to w o at k = 0 (and using ( 5 . 5 2 ) ) we find that

p,q

=

1,2 ,...,n

.

This is precisely the result obtained in Chapters 1 and 2. functions

xio) (y;O,x) satisfy

The cell

( 5 . 4 0 ) with m = 0, k = 0 (and we make

use of ( 5 . 5 2 ) ) .

5.5

Spatially localized data. consider equation (5.1) but now assume that

where fm(K,x) are smooth functions on Y

x.

x

Rn with compact support in

In view of the Bloch expansion theorem in Section 3 . 2 the initial

wavefunction ( 5 . 5 8 ) corresponds to data of class C (Section 1.2) but we have specialized them further to simplify the analysis.

'With

m = 0 and k = VS'O) = 0.

Problem

THE GENERAL GEOMETRICAL OPTICS EXPANSION

685

(5.1), (5.58) is the locally periodic analog of (4.51), (4.52) in Section 4.7. Essentially all the analysis of Section 4.7 carries over with minor modifications.

These minor modifications introduced by the

locally periodic nature of the problem are actually spelled out in Section 2.6.

They now have to be applied to each term in the sum N

which forms an asymptotic solution for (5.1), (5.58) when the integrands in (5.59) are solutions for each

5.6

and m.

K

Behavior of probability amplitudes. Given the result (5.30), Theorem 4.3, Section 4.8, extends without

any change, including the proof, to the locally periodic problem (5.1)-(5.3).

5.7

Expansion f o r the wave equation. We consider now the initial value problem

(5.60)

a

2 uE

at2 -

n

Ela

Pr9-

axp[aps(X,;)

ax

under the same hypotheses as in Section 5.1. take

W(x,-)u X E

= 0

;

t > 0

E

For initial data we

,

686

HIGH FREQUENCY WAVE PROPAGATION

.

We note that fE is proportional to

with fm and gm in Ci (Rn)

E

so

that the initial energy

is of order one as

E +

0.

The energy at time t

is bounded independently of t 2 0 and

E

(recall that W 2 0).

gives the basic a priori estimate f o r the asymptotic analysis. The expansion corresponding to (5.8) takes now the form

+ e

is-(m (x,t)/E v-(m) x (X,r,t;E)

1.

The phases S ' ( m ) satisfy the Hamilton-Jacobi equations

as' (m) at

+

-

wm(vs'(m),x)

(5.65)

Sf(m)(x,O) =

S(X)

,

= 0

,

x E Rn

t > 0

.

,

This

THE GENERAL GEOMETRICAL OPTICS EXPANSION The functions v'(~)

687

have expansions like (5.15) and each term in the

sum (5.64) is to satisfy (5.60) (the separately solutions).

+

and

-

terms are to be

Nothing special, not encountered in Section

5.1, arises for the present problem.

5.8

Expansion for the heat equation. Consider the heat equation

where the coefficients apq(x,y) bp(xry), c (x,y) are smooth functions

P

2nY, (a ) is uniformly positive definite and f E Ci(Rn). We Pq point out the factor E - ~in front of auE/at in (5.66); this makes it on Rn

X

a problem different from the ones studied in Chapter 3 .

The actual

differences which necessitate time rescaling are explained in detail below. Consider the operator

acting on smooth functions on 2rrY (i.e., 2~-periodicin each direction) with x E Rn a parameter.

This operator is the generator

of a contraction semigroup on C(2nY), the continuous functions on 2rY. By the strong maximum principle (implied by the uniform ellipticity) this semigroup maps nonnegative functions that do not vanish identically to strictly positive functions. We saw in Chapter 3 that

688

HIGH FREQUENCY WAVE PROPAGATION

the strong maximum principle and the smoothness of solutions of av/at = A1v, vl(y,O) = g(y), (x a parameter) for t > 0, independently of the smoothness of g, imply that the only nonzero solutions of (5.68)

A1$ = 0

are the y-independent functions, say @(y;x)

Z

1, and the only

solution of the adjoint equation (5.69)

R-

A1@ = 0

is a strictly positive function

5 (y;x), parametrically dependent

on

x, and rendered unique by the normalization

It was shown in Chapter 3 in several different ways and in much greater generality that under the present conditions (5.71)

uE(x,t)

uniformly in x

E

+

u(x,t)

a s ~ + O ,

Itn, 0 5 t 5 T, T <

m

but arbitrary, where u(t,x) is

the solution of n (5.72)

au( x,t)= at

b p=l

p

au(x,t) , (x) -

a xP

and (5.73)

-

bP (x) =

bp(x,y)G(y;x)dy

,

p = 1,2

,...,n .

2nY Most of the analysis of Chapter 3 dealt with the case (5.74)

bP (x) E

0

.

THE GENERAL GEOMETRICAL OPTICS EXPANSION

689

Indeed, if the right side of (5.66) is a formally self-adjoint operator then i(y;x) E 1 and (5.74) is automatically satisfied since

In this section we shall study (5.66) when

bP (x) $

0 s o that

(5.71)-(5.73) holds; more specifically we shall concentrate on the following problem which is analogous to the one in Section 2.14. Let (5.76)

G = {x

E

Rnlf(x) > 0)

.

We know that G is an open set with compact closure and smooth boundary.

We shall restrict t
condition is not necessary for the validity of the theorem that follows; it results from a weakness in the method of analysis that we use.

Let Gt denote the support of u(x,t).

It is precisely the

set of points x that are translates of points of G along the orbits of the vector field g(x) =

( gP (x)) for a time of length t. Gt is

also an open set with compact closure. Now when

E

is positive uE(x,t) is strictly positive for any x

and any t > 0 by the strong maximum principle. answer here is this:

how does uE(x,t) go to zero as

(x,t) is fixed, t > 0 and x in Section

2.14

The question we shall

E

Rn

-

Gt?

E

-+

0 when

References and remarks made

are relevant here also.

Define the operator A(k,x) by

t

For the "smoothed" Hamilton-Jacobi equation as in Section 2.14.

HIGH FREQUENCY WAVE PROPAGATION

690

acting on smooth functions on 2nY with k E Rn and x E Rn parameters. This operator, like A1 of ( 5 . 6 7 ) ,

generates a semigroup on C(2nY),

which is not a contraction in general but is positivity preserving. From this fact one can deduce rather easily that A(k,x) has an isolated maximal eigenvalue

,

w(k,x) with strictly positive eigenfunction $(y;k,x)

,

The eigenfunction is unique up to a multiplicative factor and can be chosen to be smooth in (k,x) E R2n because the maximal eigenvalue is isolated.

Similarly there is a strictly positive function

6 (y;k,x)

such that

where A*(k,x) is the formal adjoint of A(k,x). (defined on 2aY

x

Rn

x

The function i(y;k,x)

Rn) is unique up to a multiplicative factor.

Because of the uniform ellipticity assumption we have that

THE GENERAL GEOMETRICAL OPTfCS EXPANSION w(kIx) is strictly convex in k

(5.80)

69 1

.

Let L(x,A) be its conjugate convex function L(x,G)

(5.81)

=

sup(k*x - w(k,x)) k

.

Since w(0,x) r 0 we have that L(x,G) 2 0 and L(x,x) = 0 only when =

aw(k,x)/ak at k = 0 .

(5.82)

=

But from ( 5 . 7 7 )

-

k=O

1; (x) P

{I

,

and ( 5 . 7 8 )

we find that

p = l,2,...,n

.

For t
(5.83)

sup x(t)=x x(O)EG

L(x(s) ,&(s)fds

From the observations just made we conclude that S(x,t)

(5.84)

0

and S(x,t) = 0

only when x

E

Gt (defined below ( 5 . 7 6 ) )

.

It is also clear that we may replace G by aG in ( 5 . 8 3 ) when x k Gt.

Theorem 5.2. The solution uE(x,t) tends to zero as x E R" (5.85)

-

E

-+

0 with t > 0 , t < t o I

Gt fixed, and we have lim

E

log uE(x,t) = -S(x,t)

EJ.0

where S(x,t) is defined by ( 5 . 8 3 ) .

,

for each x,t

t
I

HIGH FREQUENCY WAVE PROPAGATION

692 Proof.

The proof of this theo'rem is entirely analogous to the one of Theorem 2.5, Section 2.14, so it will be omitted.

Remark 5.1. The physical significance of this theorem is the following. We may take (5.66) in the form

and suppose uE(x,t) represents the temperature of a composite medium1. which can sustain heat transfer by conduction and convection (like a porous medium).

Suppose that

E

measures the relative size of the

period of the structure and is small and also the conductivity is small i.e., of order

To principal order in €,heat is transferred

E.

by convection along the average flow field

b ( x ) (cf. (5.71)-(5.73)).

The theorem above gives the precise rate at which heat is transferred by conduction into regions that are inaccessible to transfer of heat by convection.

The amount of heat so transferred is, of course,

exponentially small in

'With

E.

periodic structure.

THE GENERAL GEOMETRICAL OPTICS EXPANSION

693

Remark 5 . 2 . The f o r m a l r e s u l t i n Theorem 5 . 2 i s e a s i l y o b t a i n e d by t h e WKB

as i n S e c t i o n 2 . 1 4 .

S p e c i f i c a l l y , t h e s o l u t i o n uE of

( 5 . 6 6 ) is

expanded ( f o r m a l l y ) i n t h e form

To o b t a i n a proof one m u s t u s e t h i s e x p a n s i o n a f t e r one h a s c a r r i e d o u t a "smoothing" of t h e r e g i o n G a s i s done i n t h e proof of Theorem 2 . 5 . The r e s t r i c t i o n t < t i s a v e r y s e r i o u s d i s a d v a n t a g e of 0

t h i s , r a t h e r s i m p l e , p r o o f . The r e s u l t i s v a l i d f o r any f i n i t e t .

Remark 5 . 3 . I n t h e g l o b a l l y p e r i o d i c c a s e , where a

and b a r e f u n c t i o n s P9 P of Y E ~ I T oYn l y , t h e Hamiltonian w = w(k) d o e s n o t depend on x . I f t h e s u p p o r t o f f = G i s convex t h e n t h e above theorem i s v a l i d g l o b a l l y i n t s i n c e t h e r e w i l l be no c a u s t i c o r c o n j u g a t e p o i n t s f o r ( 5 . 8 3 ) .

694 6.

HIGH FREQUENCY WAVE PROPAGATION Comments and P r o b l e m s . I n r e v i e w i n g t h e m a t e r i a l o f t h i s c h a p t e r w e would l i k e t o

p o i n t o u t a g a i n t h e p r i n c i p a l d e f e c t s i n t h e m e t h o d s p r e s e n t e d and d i s c u s s them b r i e f l y . The f i r s t d e f e c t i s t h e a p p e a r a n c e of c a u s t i c s and t h e n e c e s s i t y

t o r e s t r i c t d i s c u s s i o n up t o t h e t i m e t h a t t h e y f i r s t a p p e a r . One knows how t o overcome t h i s d i f f i c u l t y a t p r e s e n t ( c f . r e f e r e n c e s i n S e c t i o n 2.8)

i n many cases o f i n t e r e s t . P e r i o d i c m e d i a p r o b l e m s a r e n o t d i f -

f e r e n t from o t h e r p r o p a g a t i o n p r o b l e m s a s f a r as t h i s p o i n t i s concerned. Therefore a t least i n p r i n c i p l e d i f f i c u l t i e s with c a u s t i c s c a n b e overcome. F o r p r o b l e m s a d m i t t i n g a p r o b a b i l i s t i c f o r m u l a t i o n ,

a s i n S e c t i o n s 2 . 1 4 and 5 . 8 ,

t h e p r o b a b i l i s t i c methods ( c f . r e f e r e n -

ces i n S e c t i o n 2 . 1 4 ) a r e , i n o u r v i e w , t r u l y s u p e r i o r t o a n y t h i n g else. The s e c o n d d e f e c t , t h e m o s t s e r i o u s o n e i n o u r v i e w , c o n c e r n s t h e n e c e s s i t y t o impose mode n o n d e g e n e r a c y i n t h e l o c a l l y p e r i o d i c

case ( S e c t i o n 5 . 1 , a s s u m p t i o n ( 5 . 7 ) ) . R e c a l l t h a t i n t h e g l o b a l l y p e r i o d i c case, S e c t i o n s 4 . 2 - 4 . 8 ,

t h i s is frequently

necessary.

T h i s i s n o t s u r p r i s i n g s i n c e i n t h e g l o b a l l y p e r i o d i c case o n e c a n i n f a c t w r i t e down t h e e x a c t s o l u t i o n i n t e r m s o f Bloch waves. The p r o b l e m o f mode d e g e n e r a c y and mode c o u p l i n g i n t h e g e o m e t r i c a l o p t i c s

l i m i t i s e s s e n t i a l l y a n open p r o b l e m i n a n y c o n t e x t e v e n i n r e l a t i v e l y s i m p l e e x a m p l e s (compared t o media w i t h p e r i o d i c s t r u c t u r e ) . W e r e f e r t o t h e p a p e r o f Ludwig and G r a n o f f

[ l ] f o r some f i r s t s t e p s i n t h e

s o l u t i o n o f t h i s problem. There i s c u r r e n t l y a g r e a t d e a l o f a c t i v i t y

i n t h i s area. The t h i r d d e f e c t c o n c e r n s t h e a b s e n c e o f r e s u l t s i n bounded r e g i o n s ( e x c e p t i n one dimension) w i t h g e n e r a l b o u n d a r i e s . W e p o in te d o u t repeatedly i n previous Chapters t h a t behavior near boundaries is v e r y much a m y s t e r y e v e n i n t h e l o w f r e q u e n c y o r s t a t i c case. The

COMMENTS AND PROBLEMS

69 5

only general problem that one expects may be solvable is the periodic half-space problem i.e. the determination of reflection and transmission operators for periodic half-spaces. Unfortunately even if this was solved it would not be of very much help in general. It is important here to introduce new simplifying assumptions, like small periodic perturbations from uniform media, as we pointed out in Section 4.12.

The fourth and last major defect concerns the constant use of smoothness assumptions on coefficients and to a lesser degree on data. In previous Chapters a good deal of effort was spent in removing regularity hypotheses and as a result it was possible to obtain a number of theorems with optimal conditions in this respect. Experience with geometrical optics methods in the usual setting (cf. for example R. Lewis and J.B. Keller [ l ] ) shows that essentially new phenomena

arise in the presence of discontinuities (diffraction effects) that require a great deal of insight into special, canonical, problems for their solution (cf. J.B. Keller [l]). It appears therefore at present that geometrical optics results in periodic media with discontinuous (non smooth) structure are difficult to obtain. If one restricts attention to the propagation of energy only, which is different from a geometrical optics analysis, one expects that discontinuity effects would then be blurred out leading to tractable analysis. We have not discussed this viewpoint here and as far as we know it constitutes an open problem in media with periodic structure.

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