Table Of Contentv.
V. Jikov S. M. Kozlov O. A. Oleinik
Homogenization
of Differential Operators
and Integral Functionals
Translated from the Russian
by G.A. Yosifian
With 13 Figures
Springer-Verlag
Berlin Heidelberg New York
London Paris Tokyo
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Budapest
v.
V. Jikov (= V. V. Zhikov)
Pedagogical Institute of Vladimir
Department of Mathematics
600024 Vladimir, Russia
S. M. Kozlov
Universite Aix-Provence I
3 Place Victor Hugo
F-13331 Marseille, France
O. A. Oleinik
Moscow State University
Department of Mathematics and Mechanics
119899 Moscow, Russia
G. A. Yosifian
Institute of Problems in Mechanics
Russian Academy of Sciences
Prospekt Vemadskogo 101
117526 Moscow, Russia
Mathematics Subject Classification (1991): 35B27, 35115, 35120, 73B27
ISBN-13:978-3-642-84661-8 e-ISBN-13 :978-3-642-84659-5
DOl: 10.1 007/978-3-642-84659-5
Library of Congress Cataloging-in-Publication Data
Zhikov, Vasilir Vasil'evich. Homogenization of differential operators and integral functionalsl
V. V. Jikov, O. A. OIeinik, S. M. Kozlov; translated from the Russian by G. A. Yosifian.
p. cm. Includes bibliographical references and index.
ISBN-13:978-3-642-84661-8
I. Homogenization (Differential equations) 2. Differential equations, Elliptic. 3. Continuum mechanics.
I. OIeinik, O. A. II. Kozlov, S. M. (Sergei M.) III. Title.
QA377.Z45 1994 531' .0!'515353-dc20 93-19395
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Copyright Law.
© Springer-Verlag Berlin Heidelberg 1994
Softcover reprint of the hardcover! st edition 1994
SPIN: 10051388 4113140 -543210 -Printed on acid-free paper
Preface
It was mainly during the last two decades that the theory of homogenization
or averaging of partial differential equations took shape as a distinct mathe
matical discipline. This theory has a lot of important applications in mechanics
of composite and perforated materials, filtration, disperse media, and in many
other branches of physics, mechanics and modern technology. There is a vast
literature on the subject.
The term averaging has been usually associated with the methods of non
linear mechanics and ordinary differential equations developed in the works of
Poincare, Van Der Pol, Krylov, Bogoliubov, etc.
For a long time, after the works of Maxwell and Rayleigh, homogeniza
tion problems for· partial differential equations were being mostly considered
by specialists in physics and mechanics, and were staying beyond the scope of
mathematicians. A great deal of attention was given to the so called disperse
media, which, in the simplest case, are two-phase media formed by the main
homogeneous material containing small foreign particles (grains, inclusions).
Such two-phase bodies, whose size is considerably larger than that of each sep
arate inclusion, have been discovered to possess stable physical properties (such
as heat transfer, electric conductivity, etc.) which differ from those of the con
stituent phases. For this reason, the word homogenized, or effective, is used in
relation to these characteristics. An enormous number of results, approximation
formulas, and estimates have been obtained in connection with such problems
as electromagnetic wave scattering on small particles, effective heat transfer in
two-phase media, etc.
During the last decades, some disciplines in mechanics and physics have
stimulated the appearance of a concept of microscopically non-homogeneous
media of general type, and encouraged intensive development of the methods
of the effective medium, averaged field, etc., which are in agreement with the
modern homogenization theory.
In mathematical models of microscopically non-homogeneous media, various
local characteristics are usually described by functions of the form a(C1x),
where E > 0 is a small parameter. The function a(x) can be periodic, almost
periodic, or even a realization of a stationary random field; it can also belong
to some other specific class. To compute the characteristics of a micro non
homogeneous medium is an extremely difficult task, since the coefficients of the
corresponding differential equations are rapidly oscillating functions. Therefore,
vi Preface
it is necessary to apply asymptotic analysis to the problems of micro non
homogeneous media, which immediately leads to the concept of homogenization.
At present, numerous publications can be found on the mathematical as
pects of the homogenization theory for strongly non-homogeneous media in
the multi-dimensional case. Several monographs dedicated to the mathematical
problems of homogenization have appeared recently, in particular: Bakhvalov
& Panasenko [1], Bensoussan, Lions & Papanicolaou [1], Lions [2], Marchenko
& Khruslov [1], Oleinik, Shamaev & Yosifian [7], Sanchez-Palencia [lJ. These
books also contain an extensive bibliography on the theory of homogenization,
G-convergence and related topics. Most of the material included in the present
book cannot be found in other monographs written on the subject either by
mathematicians, or specialists in mechanics.
This book embraces a fairly wide scope of problems connected with the
theory of homogenization, and it also reflects results obtained by its authors.
Of special care were the choice of the material and the method of exposition,
so as to make the book accessible to a possibly wider audience. The subject
matter of the book pertains mainly to, the stationary problems involving lin
ear second order elliptic equations and variational functionals associated with
second order nonlinear operators. Among the nonstationary problems, only dif
fusion processes in periodic and random media and homogenization of Maxwell
equations have been chosen.
The decision to write this book dates back to the beginning of the 80's;
however, its realization extended over many years. For the most part, the ac
tual implementation of the project is due to O.A. Oleinik and V.V. Zhikov;
S.M. Kozlov contributed to the manuscript some new topics. A large portion of
the monograph bears the influence of prolonged and numerous joint discussions.
The authors express their gratitude to 0.0. Barabanov, V.V. Demidov,
O.A. Nazarova, A.L. Pyatnitsky, M.M. Sirazhudinov, and G.A. Yosifian for
their great and varied assistance during the preparation of the manuscript.
The system of notation used throughout the book is a standard one. Each
chapter is provided with its own double numeration of formulas and proposi
tions, the first number being that of the chapter. When enumerating the propo
sitions, we do not distinguish between theorems, lemmas, remarks, etc.
V. V. Zhikov S. M. Kozlov O. A. Oleinik
Publisher's Note: The spelling JIKOV is used on the cover and the title pages
of this book, whereas the alternate spelling ZHIKOV is used throughout the
book as in numerous other publications.
Table of Contents
Chapter 1
Homogenization of Second Order Elliptic Operators
with Periodic Coefficients . . . . . . . . . . . 1
1.1 Preliminaries ................. 1
1.2 Setting of the Homogenization Problem. . . 12
1.3 Problems of Justification; Further Examples 19
1.4 The Method of Asymptotic Expansions . . . 24
1.5 Explicit Formulas for the Homogenized Matrix in the
Two-Dimensional Case . . . . . . . . . . . . . . . . . 35
1.6 Estimates and Approximations for the Homogenized Matrix 39
1. 7 The Rayleigh-Maxwell Formulas . 45
Comments . . . . . . . . . . . . . 53
Chapter 2
An Introduction to the Problems of Diffusion 55
2.1 Homogenization of Parabolic Operators . . . 55
2.2 Homogenization and the Central Limit Theorem . 61
2.3 Stabilization of Solutions of Parabolic Equations . 64
2.4 Diffusion in a Solenoidal Flow . . . . . . . . . . . 68
2.5 Diffusion in an Arbitrary Periodic Flow. . . . . . 73
2.6 Spectral Approach to the Asymptotic Problems of Diffusion 76
2.7 Diffusion with Absorption 83
Comments . . . . . . . . . 84
Chapter 3
Elementary Soft and Stiff Problems 86
3.1 Homogenization of Soft Inclusions . 86
3.2 Homogenization of Stiff Inclusions. 96
3.3 Virtual Mass ........... . 106
3.4 The Method of Asymptotic Expansions . 109
3.5 On a Dense Cubic Packing of Balls . . . 112
3.6 The Dirichlet Problem in a Perforated Domain. 119
Comments .................... . 131
viii Table of Contents
Chapter 4
Homogenization of Maxwell Equations 133
4.1 Preliminary Results . . . . . . . . . 133
4.2 A Lemma on Compensated Compactness 138
4.3 Homogenization ............ . 141
4.4 The Problem of an Artificial Dielectric 145
Comments. 148
Chapter 5
G-Convergence of Differential Operators 149
5.1 Basic Properties of G-Convergence . 149
5.2 A Sufficient Condition of G-Convergence 156
5.3 G-Convergence of Abstract Operators .. 160
5.4 Compactness Theorem and Its Implications 163
5.5 G-Convergence and Duality ........ . 166
5.6 Stratified Media ............... . 168
5.7 G-Convergence of Divergent Elliptic Operators
of Higher Order 179
Comments .................... . 186
Chapter 6
Estimates for the Homogenized Matrix 187
6.1 The Hashin-Shtrikman Bounds ... 187
6.2 Attainability of Bounds. The Hashin Structure. 195
6.3 Extremum Principles . . . . . . . . . . . . . . . 199
6.4 The Variational Method . . . . . . . . . . . . . 202
6.5 G-Limit Media; Attainment of the Bounds on Stratified
Composites . . . . . . . . . . . . 205
6.6 The Method of Quasi-Convexity. . . . 210
6.7 The Method of Null Lagrangians ... 212
6.8 The Method of Integral Representation 217
Comments . . . . . . . . . . . . . . . . 220
Chapter 7
Homogenization of Elliptic Operators with Random
Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . 222
7.1 Probabilistic Description of Non-Homogeneous Media 223
7.2 Homogenization .................. . 227
7.3 Explicit Formulas in Two-Dimensional Problems .. . 233
7.4 Homogenization of Almost-Periodic Operators ... . 238
7.5 The General Theorem of Individual Homogenization. 242
Comments ....................... . 248
Table of Contents IX
Chapter 8
Homogenization in Perforated Random Domains 250
8.1 Homogenization.................. 252
8.2 Remarks on Positive Definiteness of the Homogenized Matrix. 264
8.3 Central Limit Theorem. . . . . . . . . . . . . . . 268
8.4 Disperse Media . . . . . . . . . . . . . . . . . . . 279
8.5 Criterion of Pointwise Stabilization; A Refinement
of the Central Limit Theorem . . . . . . . . . . . 281
8.6 Stiff Problem for a Random Spherical Structure . 284
8.7 Random Spherical Structure with Small Concentration 287
Comments ........................ . 296
Chapter 9
Homogenization and Percolation 298
9.1 Existence of the Effective Conductivity 299
9.2 Random Structure of Chess-Board Type 304
9.3 The Method of Percolation Channels . . 308
9.4 Conductivity Threshold for a Random Cubic Structure in IR3 313
9.5 Resistance Threshold for a Random Cubic Structure in IR3 318
9.6 Central Limit Theorem for Random Motion in an Infinite
Two-Dimensional Cluster. 319
Comments 321
Chapter 10
Some Asymptotic Problems for a Non-Divergent Parabolic
Equation with Random Stationary Coefficients 323
10.1 Preliminary Remarks . . . . . . . . . . . . . . . . . 323
10.2 Auxiliary Equation A'p = 0 on a Probability Space 325
10.3 Homogenization and the Central Limit Theorem. 330
10.4 Criterion of Pointwise Stabilization 332
Comments 336
Chapter 11
Spectral Problems in Homogenization Theory 338
11.1 Spectral Properties of Abstract Operators Forming a Sequence 338
11.2 On the Spectrum of G-Convergent Operators. 345
11.3 The Sturm-Liouville Problem . . . . . . . . . . 349
11.4 Spectral Properties of Stratified Media . . . . . 355
11.5 Density of States for Random Elliptic Operators 357
11.6 Asymptotics of the Density of States 359
Comments . . . . . . . . . . . . . . . . . . . . . 365
x Table of Contents
Chapter 12
Homogenization in Linear Elasticity 367
12.1 Some General Facts from the Theory of Elasticity 367
12.2 G-Convergence of Elasticity Tensors ....... . 372
12.3 Homogenization of Periodic and Random Tensors 375
12.4 Fourth Order Operators ............. . 381
12.5 Linear Problems of Incompressible Elasticity .. . 382
12.6 Explicit Formulas for Two-Dimensional Incompressible
Composites ..................... . 386
12.7 Some Questions of Analysis on a Probability Space 389
Chapter 13
Estimates for the Homogenized Elasticity Tensor 391
13.1 Basic Estimates . . . . . . . . . . . . . . . . . . 392
13.2 The Variational Method ............ . 395
13.3 Two-Phase Media; Attainability of Bounds on Stratified
Composites ....................... . 399
13.4 On the Hashin Structure . . . . . . . . . . . . . . . . . 403
13.5 Disperse Media with Inclusions of Small Concentration 406
13.6 Fourth Order Operators; Systems of Stokes Type 409
Comments ..................... . 414
Chapter 14
Elements of the Duality Theory 415
14.1 Convex Functions ..... . 415
14.2 Integral Functionals .... . 420
14.3 On Two Types of Boundary Value Problems 423
14.4 Dual Boundary Value Problems . 427
14.5 Extremal Relations ....... . 431
14.6 Examples of Regular Lagrangians 433
Comments ............ . 437
Chapter 15
Homogenization of Nonlinear Variational Problems 438
15.1 Random Lagrangians . . . 438
15.2 Two Principal Lemmas .. 444
15.3 Homogenization Theorems 449
15.4 Applications to Boundary Value Problems
in Perforated Domains . . . . . . . . . 452
15.5 Chess Lagrangians; Dychne's Formula. 455
Comments ............... . 458
Table of Contents xi
Chapter 16
Passing to the Limit in Nonlinear Variational Problems 460
16.1 Definition of T-Convergence of Lagrangians; Formulation
of the Compactness Theorems . . . . . . 460
16.2 Convergence of Energies and Minimizers ........ . 465
16.3 Proof of the Compactness Theorems .......... . 468
16.4 Two Examples: Ulam's Problem; Homogenization Problem 476
16.5 Compactness of Lagrangians in Plasticity Problems;
Application to LI-Closedness .... 480
16.6 Remarks on Non-Convex Functionals 487
Comments. 490
Chapter 17
Basic Properties of Abstract T-Convergence 492
17.1 T -Convergence of Functions on a Metric Space . . . . . 492
17.2 T-Convergence of Functions Defined in a Banach Space 495
17.3 T-Convergence of Integral Functionals 498
Comm~~ ............... . 501
Chapter 18
Limit Load 502
18.1 The Notion of Limit Load . . 502
18.2 Dual Definition of Limit Load 507
18.3 Equivalence Principle . . . . . 509
18.4 Convergence of Limit Loads in Homogenization Problems . 512
18.5 Surface Loads . . . . . . . . . . . . . . . . . 518
18.6 Representation of the Functional F on BVo . 521
18.7 T -Convergence in BVo 529
Comments . . . . . . . . . . . . . . . . . . 534
Appendix A. Proof of the Nash-Aronson Estimate 536
Appendix B. Weak Convergence in £1 and Weak Convergence
of Measures . . . . . . . . . . . . . . . . . . . . . . . 540
Appendix C. A Property of Bounded Lipschitz Domains 542
References 544
Subject Index 569
