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
NORTHHOLLAND PUBLISHING COMPANYAMSTERDAM
NEW YORK ’ OXFORD
ASYMPTOTIC ANALYSIS FOR PERIODIC STRUCTURES
ALAIN BENSOUSSAN Universite' de Paris IX and IRIA
JACQUESLOUIS LIONS CollPge de France and IRIA
GEORGE PAPANICOLAOU Courant Institute of Mathematical Sciences New York University
1978 NORTHHOLLAND PUBLISHING COMPANYAMSTERDAM . NEW YORK . OXFORD
0 NorthHolland 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
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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 problemsNumerical solutions. 2. Differential equations, PartialNumerical 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’ 35 785 5 98 ISBN
[email protected] 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 illconditioned 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 clearcut 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. 352361, for a general expansion
procedure using multiple space scales. ttJ. Math. Phys.,
16
(1975), pp. 14211427.
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 socalled 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 BN821, 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. 331351.
tt~oll.U.M.I., 8 (1973), pp. 391411, and in Numerical Methods
of Partial Differential Equations, 111, B. Hubbard, editor, Academic Press, New York, 1976, pp. 469498. tttAnnali Scuola Norm. Sup. Pisa, IV (19751, p. 617638. ttttBoll. U.M.I.
,2
(1977).
INTRODUCTION
xiii
Some results in Chapter 4 are well known in solidstate 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, selfcontained 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 onedimensional 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 nonuniformly 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 ................................. Nonlocal homogenized operator ......................... Homogenization theorem .................................
Nonlocal limits
16.3
.................... Formal general formulas ................................ Compact perturbations .................................. Noncompact perturbations ..............................
16.4
Nonlinearities in the higher derivatives
16
16.1
16.2
Introduction to nonlinear problems
..............
188 188 189 191 194 194 196 199 200 200
202 20 3 206
LIST OF CONTENTS
.
xix
............... ............................................
Homogenization of multivalued 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 ............................................. Nonlinear 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 KleinGordon 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 ChapmanEnskog expansion
645
Lattice waves
......................... Spatially localized data and stationary phase .......... Behavior of probability amplitudes ..................... The acoustic equations ................................. Dual homogenization formulas ........................... Maxwell's equations .................................... A onedimensional 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 (pseudodifferential 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 Yperiodic 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 Yperiodic
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 Yperiodic
,
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 socalled 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
i1
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 LaxMilgram'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 onedimensional 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,, +
/
[email protected],
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
Yperiodic in y, and we associate to $(x,y) the function @(x,x/E). We shall look for uE
=
u,(x)
in the form of the asymptotic
expansion
where the functions u. (x,y) are Yperiodic in y, V x 3
E 0
.
Remark 2.1. It is technically complicated to keep track of boundary conditions when seeking uE in the form (2.1) and this is actually the source of serious technical difficulties in justifying the method.
The method will nevertheless give the "right answer" because
it will turn out that, in this sort of problems, the boundary conditions are somewhat irrelevant. The idea of the method is (simply) to insert (2.1) in equation (1.5) and to identify powers of E. In order to present these computations in a simple form, it is useful to consider first x and y as independent variables and to replace next y by X/E. Applied to a function +(x,x/E), the operator a becomes ax,
.
J
With this in mind, one can write AE = E  2 ~ 1
where
+
E  ~ A+~ E 0A~
,
13
ASYMPTOTIC EXPANSIONS
a
a 5 1
A 1 = ayi  [ a i j (Y)
(2.3)
A2 =
A
3
=
I
 q a[ a i j (Y) ..) a  5I.i a 1  [aij a axi
(y) 2 1+ a. ax j
y] a
.
t h e e q u a t i o n ( 1 . 5 ) becomes
Using (2.1)
(2.21,
(2.4)
A u = 0 , 1 0
(2.5)
A u
+
A2u0 = 0
(2.6)
A1u2
+
A2u1 + A3u0 = f ;
11
j (Y)
,
one c a n o f c o u r s e ( f o r m a l l y ) proceed A u
(2.7)
1 3
+
A2u2
+
,
A3u1 = 0
etC.
W e are now g o i n g to see t h a t t h e homogenized o p e r a t o r
be c o n s t r u c t e d from (2.4)
,
(2.5)
,
(2.6).
L e t u s remark t h a t t h e e q u a t i o n A @ = F
1
in
Y ,
(2.8) @
periodic i n
Y
,
,
admits a u n i q u e s o l u t i o n (up t o a n a F(y)dy = 0
(2.9)
a
ve constant) ,
.
Y
Indeed l e t u s i n t r o d u c e (2.10)
W(Y) =
1 {@
[email protected] E H (Y),@
"periodic"}
i.e. 0 t a k e s e q u a l v a l u e s on o p p o s i t e faces of Y. F o r @,
J, E W ( Y )
, let
us set
ff
a
can
14
ELLIPTIC OPERATORS
(2.11) and let us also set
Then (2.8) is equivalent to:
@ E W(Y) (2.13)
r
al($,$) = ( 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
Yperiodic
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 Yperiodic, 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)
~
€
= 2Er~
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 ktimes 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 Yperiodic 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 Yperiodic
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 Hl(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 sufficeand 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 Hl 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 Xgrad 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
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 socalled
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 Yperiodic. 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 = a1v
0
which can be solved in u1 iff rot grad u = rot (a1v,) Y X Y i.e. (5.36)
rot (alv0) = 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)
a1 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 Yperiodic
.
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
,
@
Yperiodic 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 thatby
We
the standard projection theoremthere
exists $, such that $ is Yperiodic and div $ = 0 and Y (rotY $,rotY
(roty G o , $ )
=
J,)
(5.48)
.
V II, such that J, is Yperiodic, 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 a1 (rot $ + Y Y (5.52)
div
Y
$ = 0
,
p) = 0
$
,
Yperiodic
.
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 al rot jiP(y) = rot(alep) (5.53)
div
'2
= 0
,
,
jip is Yperiodic ,
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 (al(I 
rot Y
2 ) ) ~
3tl (al(I  rot 2 ) ) 1grad u Y
.
ELLIPTIC OPERATORS
64 Then (5.59) gives (5.60)
(al(I  rot ?))'grad . Y
div
u
+
Xu = f
.
Remark 5.8. Since we necessarily obtain the same homogenized operator, it followsif (5.61)
(5.60) is justifiedthat
~
(a1 (I

rot Y
1l =
[
41
.
But, according to usual formulae
[ a ]= (5.62)
grad Y
x
Ih(a(1 =
a1(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 (al(I  rot Y Y
(5.65)
div (al(I  rot Y Y
2))
=
rot (I  grady Y
='
div a(I Y

x) [ a 1 1 ,
grad Y
x) [ a 1l ;
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 ( a1 ( 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
+ EldivY 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 .
LionsMagenes
W e w i l l need
(5.78)
IU1
I IH1/2 ( r )
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 Yperiodic 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 Yperiodic 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 stepfunctions 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 "approximate"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)
Yperiodic
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
(663)
( 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 Yperiodic 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 Yperiodic 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,...,Xnl~ ,
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
,
n1
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 n1) can be different from the periods
Notations. We introduce the space (7.14)
1 1 n1 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 = Rn1
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 (n1) Let us denote by
ff = homogenized operator (in Rn ) of AE (7.21)
Let and
B
.
n
=
,
homogenized operator (in Rnl) 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 n1 w = u(x',O) in H ( R ) weakly
.
au 1/2 n1 t We define 2 E H (R ) by av
A"
(7.27)
(Rnl)
I$ E
9
+
H1/2
,
@ E
HI(&)
,
@(x',o) = 9 #
being a continuous linear mapping from
@
n1
(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. LionsMagenes [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 Rnl.
We can apply the theory of Sections 1 to 3 . (7.31)
av
+
in H;ica1(Rnl)
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 Yperiodic 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 (N1
,...,Z
= X/E
Z ( l )
(8.5)
a
ij
E
d i f f e r e n t periodsin 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 2periodic 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 wlvp(&)
,
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 WltP(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)
YZ
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
Yperiodic
,
(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
Yperiodic
.
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 cutoff 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 AgmonDouglis 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, Yperiodic, 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. DenyLions 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)
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
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
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
HI( 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)
,
km = 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 lkml2e am(x) :
satisfies (12.11) in Rn, x playing the role of a parameter:
x
is
aperiodic in y. We introduce the meanvalue (in the sense of almostpesiodic 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+ 2cl
Y
AYx
+
E
0
Ax)
where Ayx =
a2
ay.ax. * 3 3
Therefore (12.64)
A
+
p
+
E
2 W
=
E  ~ A + ~E3A3
+
hlA4
+
+
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 Yperiodic ,
r
by
(12.86)
A1(X) + W = O
x
I
Yperiodic
,
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 Yperiodic,
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
Yperiodic
,
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 Yperiodic.
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 CauchySchwarz. One has
and one assumes that
Other sufficient conditions,
but only bi E L " ( Y ) ,
are obtained by
184
ELLIPTIC OPERATORS
BE = Elgl (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 Yperiodic
,
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 + Yperiodic 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 Yperiodic
,
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 Yperiodic
= 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.
Nonlocal
+
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 Yperiodic f u n c t i o n s i n L m ( R n ) , such t h a t
NONLOCAL
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 Hl(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 integropartial differential equation,
i.e., a nonlocal 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 selfadjoint).
Nonlocal homogenized operator. We use Fourier transform in A ;
if g E L2 (RhrV)we introduce
+m
3 g(ll) =
B(')
e2niX'g(X)dX
= m
,
6
E
L2(R ,V) P
.
NONLOCAL
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
Yperiodic
,
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
NONLOCAL 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 nonlinear problems. Formal general formulas. Let us consider a nonlinear elliptic equation
(16.2)
A'
is given as in Section 1
,
B is a nonlinear 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 NONLINEAR 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 NONLINEAR 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
Noncompact 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 NONLINEAR 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~ , ~ , Dum,(~ 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
Nonlinearities 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 
P2 au
5 1
and
u1 being Yperiodic 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 MULTIVALUED OPERATORS Homogenization of multivalued operators.
17.
17.1
Orientation. What we have now in mind is the homogenization of the socalled
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 (nonempty)
.
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 (LionsStampacchia [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 MULTIVALUED
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
,
Yperiodic 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 MULTIVaUED 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 Yperiodic 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 Yperiodic,
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 MULTIVALUED 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 equivlent 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 MULTIVALUED 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
+
IuErl 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
+ r1 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 MULTIVALUED 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 quasivariational inequalities has been considered in BensoussanLions [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 othersand
even in a much
more general setting of calculus of variationsunder of Gconvergence.
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 herewith
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 "multiscale" methodswell 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 MuratTartar
;
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 MarcelliniSbordone [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 Yperiodic 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 Yperiodic solution (defined up to
an additive constant) of
Then
where
(3)
Another extension of the results of Sections 1 to 11 is
relative to quasielliptic 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, Yperiodic, 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 nonhomogeneous 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 Hl 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 Yperiodic
.
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
1ul
+
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 + A3ul)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 selfadjoint 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], CioranescuSaint 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 pseudodifferential operators, a question not considered here. In Sections 16 and 17 we give some indications on the homogenization of nonlinear 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 foundat
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 casesby
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] BourgatDervieux
[ 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], Sobolevspaces. 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],TConvergenza e G convergenza... ~011.U.M.I. ( 5 ) i 3  ~( 1 9 7 6 1 , p. 111. 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 Gconvergence. 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 .
...
[ 3 ] C.R.A.S.
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 quasivariationnelles. C.R.A.S. Paris, 1 9 7 7 . 228
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L. Boccardo and I. Capuzzo Dolcetta [l] Gconvergenza e problema di Dirichlet unilaterale. Boll. U.M.I. ( 4 ) , 12, (1975), p. 115123. 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. 137159. 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. BourgatA. 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] rConvergence 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. 119132. 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), 947950. 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. 391411. S . Kesavan [ l ] Homog6n6isation et valeurs propres. C.R.A.S. Paris 285 (19771, p. 229232.
...
...
...
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. 4414134. S.M. Kozlov [l] Averaging of differential operators with almost periodic, fast oscillating coefficients. Doklady Akad. Nauk. 236 (1977), NO 5, p . 10681071. 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. 1103. [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. 39101. J.L. Lions and G. Stampacchia [l] Variational inequalities. C.P.A.M. 20, 3 (1967), p. 493519. 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. 157182. [2] Sur quelques questions de Gconvergence et d'homogingisation non lin6aire. C.R.A.S. Paris, t. 284, 1977, p. 535537. [ 3 ] Homogenization of nonuniformly elliptic operators. Applicable Analysis, 1977. [el Dualita e perburbazione di funzionali integrali. Istituto Caccioppoli. June 1977. N.G. Meyers [l] An LPestimate for the gradient of solutions of second order elliptic divergence equations. Ann. Sc. Norm. Sup. Pisa, 17 (1963), p. 189206. 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. 11221126. 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
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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. 221236. 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. 711731. J. Planchard [l] Etudes et Recherches de l'Electricit6 de France Internal Report, 1977. To appear. E. SanchezPalencia [l] Solutions pgriodiques par rapport aux variables d'espace et applications. C.R.A.S. Paris (2711, (1970), p. 11291132. [2] Equations aux d6rivGes partielles dans un type de milieux hgtgrogsnes. C.R.A.S. Paris (272) (1970), p. 14101611. C. Sbordone [l] Su alcune applicazioni di un tipo di convergenza variazionale. Annali Scuola Norm. Sup. Pisa, IV (1975), p. 617638. 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. 571597. [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, FrenchJapan 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.
This Page Intentionally Left Blank
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 Yr0 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 ) ,
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;W1rp(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 Yr0 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 Yro 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 Yperiodic
,
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
+ E3kvl +
...
,
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  ~ ,
E3kvl +
E'
...
gives
,
(2.38)
AluO = 0
(2.39)
Alul + A2u0 = 0
(2.40)
A1U2
+
€2kv
A2U1
+
,
A3u0
au, + aw, = +at ar
Identifying next the coefficients of
E
f
.
k
,
E
1k
I
E
k2
, E k1,
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
...
+
... +
E3kvl
+
... +
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+lpkvo
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)
+
E3kA(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 Yperiodic and B and
T
X/E
being
I
YTo
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
(ayi) = 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 Yperiodic
(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
iYperiodic ,
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 Dirichst, 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;Wl'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;W1rP(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 = P1F
then (2.91) is equivalent to (2.93)
u
+
(Pl(A 1
+
A ) + PlA2)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 IPll 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 RieszThorin'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 constructusing
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 Yr0 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 Yperiodic 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 cutoff 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
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
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 "nonlocal"
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 Yperiodic 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], LionsMagenes [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 Yperiodic (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 Yperiodic 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 isthe 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 Yr0 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 Hweakly,
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 fashionthe
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
+
= E2Q1
+
ElQ2
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 ar
+ A 1@
@
which is Yro 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 Yr0 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 Yro 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 smoothwe
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 Yto
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 Yr0 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 Yperiodic
,
j
y . ) = 0 3
and we use cutoff 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
DuvautLions [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 likelybut
it is not provedthat
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) 
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:
= E2~11
A :
=
E2
+
E1~12
+
,
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 CarrollShowalter 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 nonlocal 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 Yr0 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. 317322. [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. FrenchJapan 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. 657673. V . G . Markov and O.A. Qleinik [l] On the heat diffusion in a one dimensional dispersive medium. P.M.M. 39 (19751, p. 10731081.
...
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. SanchezPalencia 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 Gconvergenza 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 , 657699. [ 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),
11791183.
This Page Intentionally Left Blank
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, ItoMacKean
[
11, Priouret
GikhmanSkorokhod [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 StroockVaradhan [ l
I.
Other basic references are WatanabeYamada [I], YamadaWatanabe [l], 345
346
PROBABILISTIC PROBLEMS AND METHODS
Meyer [ 1 1 , DoleansDadeMeyer
[
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 ualgebras 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
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 nonnegative 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 oalgebra 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
a1 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 al 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 oalgebras 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 pointwiseuniquenessof 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~ 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)
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 BensoussanLions
[
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 selfadjoint).
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 ualgebra
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)
,pX)
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 CauchySchwarz 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(: =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 :11
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 ,
11;
+
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 Eaij(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 oalgebra R
sets {y(tj), 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 GikhmanSkorokhod [ 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 StroockVaradhan
[
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',
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 :11
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 selfadjoint 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 checksthat 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
(811)
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. 1EIx *
= o(x(A) , A
5
s)
.
This is the submartingale formulation of the reflected diffusion process (StroockVaradhan
[ 2 ])
.
We want to study the limit of
11;
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. 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
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
2k
x1
+
E
42k
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. 3d~ 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) 2p1 < 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 42kx2(y,r), and define
E
2 kxl(yIT)
By Ito's formula we have (taking into account the equations of
+ j [I%€ S
t
53k t Let now @
E
&:(Rn
x
2k axl aY
€42k
%IudW€ aY
% dl ar
[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
L2l
4
W
$
Id
c
N
X
v
tr
h
u
u
I
N
ffl10
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 11
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 PP
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 selfadjoint 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+1k =
>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)
+
A2XN1 +
A 1 ~ N + l+ A2xN A1X2N1
+
... + Anx1 =  cON
+
A2X2N2 +
+
AN+lxl =
...
+
'
 = 0,N1 '
A2N1X1 =

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 DonskerVaradhan
[l] o r a s i n AzencottRuget
[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 HamiltonJacobi 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, DelebecqueQuadrat 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. 154. 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 FrancoJapanese 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. DoleansDade, 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. 147 (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. 121125. 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, SpringerVerlag, 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, NewYork, 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. 155. S. Watanabe [l] On stochastic Differential Equations for Multi Dimensional Diffusion Processes with Boundary Condition, J. Math. Kyoto Univ. 11 (1971) p . 169180 : 11, p. 545551. 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 2.112.14 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 socalled 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 KleinGordon 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
(complexvalued) 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 complexvalued.
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 ndimensional torus (end points of intervals
identified).
The 2atorus 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 realvalued and Cm and f(x,y), g(x,y) are complexvalued, Cm(Rn
x
2aY) (i.e. they are
2nperiodic 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 KleinGordon 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 realvalued, f and g complexvalued 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
)
, + ic2VSV 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 HamiltonJacobi) 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:
+ = kx W(x) = constant, w = w(k) and S(x,t)
w(k)t.
g(x) = kx,
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 2point 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 + VFE = 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)
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 complexvalued
where S(x',t) is real and
Cm
compact support in Rnl
[O,m).
x
Cm
and of
One can also consider spacetime
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 spacetime 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 spacetime 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
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 Nvector 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

'CourantHilbert, 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 Nreal solutions w1 (krx),w2(k,x),
...,aN(k,x), counting multiplicities.
Moreover
there exist for each x and k Nlinearly 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) 1kIl 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 spacetime, 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
‘CourantHilbert, 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. CourantHilbert, Vol. 11).
The situation for
initialboundary 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), scalarvalued, 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 initialboundary 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
V11 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), vectorvalued, 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
'LandauLifschitz [ 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 Nvector 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 initialboundary value problems.
The infinite dimen
sional case is not really difficult given enough information on B. The initialboundary value problems are, however, harder. It
~~
'P.
D. Lax
;
cf. also CourantHilbert, [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 Nvector f
c
components and any t > 0,
Remark 2.1. Note that the result does not hold uniformly in [O,T], T
,
p . ~P. ( x ),
JES,
J
w = 1,2
pyfj(x) ,
=
,...,r
,
=
I
J
fj = jth 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 rdimensional nullspace 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!Jl 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 nullspace 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 Boltzmannlike systems. We now consider briefly (2.102) when f, (x) = f (x) and B(x) is skewsymmetric 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) skewsymmetric, for any Nvector 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 Nvector functions
 Pu = u.
We shall give
two proofs
with
of this theorem in order to illustrate
some points that play an important role in periodicstructure 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 quasiperiodic 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 eBs >: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
0 for x tics of the vector field
E
it, 0 < 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 semidefinite. 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 initialboundary value
problem for (2.224) over Rnl
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 WeinbergBurridge 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 skewsymmetric.
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) ,
Rnl and y E Rnl 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 selfadjoint 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
Rnl 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 possibilitiesmetallic 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 realvalued 2nperiodic smooth function.
W(y)
Let
We shall find
the spectral resolution of the closure of this operator in L2(Rn) (complexvalued). The spectral resolution is given in terms of the Bloch waves associated with A as we now describe. Let Y be the unit ndimensional 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)
.
(complexvalued)
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 )
(complexvalued).

?! m (k) = 1.i.m.
(3.9)
Then
dy f (y)eik'Y $,,(y;k)
N+m
.
Moreover, Parseval's identity holds m
Proof
(
Gelfand [l], OdehKeller 111)
.
In view of (3.10) it is enough to show that (3.8)(3.10) with f
E
&?(Rn), the space of Cm rapidly decreasing functions on Rn.
For such an f, the function (3.11)
hold
?(y;k) =
belongs to 2aY.
yE2az"
f(y
+
y)eik. (y+y)
I
We expand it in terms of {$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 $m does not depend on k. (3.17)
2 wm(k) = Ik
+
ml
2
,
m e zn,
I
The eigenvalues are m
E
Zn
,
k
E y
.
=
6Pq
HIGH FREQUENCY WAVE PROPAGATION
618
The theorem above is now the PlancherelParseval theorem for the n Fourier transform in R and it is deduced from the corresponding result for Fourier series. We note that (3.1) and (3.2) are connected by E
eikyA(k)
r
which explains the terminology "shifted cell problem". (3.18)
Aeik"$
m
(y;k) = eikyum(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 21rperiodic,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 2nperiodic 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
[pl(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 ZITperiodicvector 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 PerronFrobenius
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 2nperiodic coefficients. Using multiindex 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 KreinRutman [l] , T. Harris [11.
SPECTRAL THEORY OF DIFFERENTIAL OPERATORS (3.35)
623
eik.Y A = A(k)eik'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)eik.Y U(Y) = eik.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 2nperiodic
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
eik. (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
,(AX)'
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 IuumI
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 (nkpwm(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
ptq1
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 $m(y;k) are the Bloch eigenfunctions of Section 3.1.
We
assume that for some N 2 N o (4.10)
ko is such that w (k ) : m=O
We have
e
?m(k)$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 5.15.8. 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 mth 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 kdifferentiations 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 5.15.8.
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 mth 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) ZITperiodicand
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
SIWLE 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 ChapmanEnskog expansion. Theorem 4.1 of the previous section is an example of a Hilbert
expansion while Theorem 4.2 an example of a ChapmanEnskog 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 LeontovichFocktt 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 againup first time caustics formbut
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 ChapmanEnskog 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 ) willnot 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 finitemode 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 6function 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)
$luEI2dxdt
Rn tends asymptotically as
E +
0 to
R"), T
O ,
x E R
3
.
(i.e.,
SIMPLE APPLICATIONS OF THE SPECTRAL EXPANSION
651
We shall assume that
(4.69)
pE(x,O) = eik'X/E
2 m'=O
fm, (x)$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) + iwmp1p1(m) + Vxuo 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 ql 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 [P1(6ij
.
+ vxj)i)]
It is easily verified that ql (or q) is positive definite (we use (4.67)). Define the operator q
1
0
a
(4.100) 57 = 0
R"V11
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
=
(vu + 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 vectorvalued 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 vectorvalued 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(11) 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
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