Homogenization of Partial Differential Equations

406 Pages·2006·6.76 MB·English

by  Vladimir A. Marchenko, Evgueni Ya. Khruslov (auth.)

Progress in Mathematical Physics
Volume 46
Editors-in-Chief
Anne Boutet de Monvel, Universite Paris VII Denis Diderot
Gerald Kaiser, Centerfor Signals and Waves, Austin, TX
Editorial Board
C. Berenstein, University ofMaryland, College Park
SirM. Berry, University ofBristol
p. Blanchard, UniversitiitBielefeld
M. Eastwood, University ofAdelaide
A.S. Fokas, University ofCambridge
D. Sternheimer, Universite de Bourgogne, Dijon
C. Tracy, University ofCalifornia, Davis
Vladimir A. Marchenko
Evgueni Ya. Khruslov
Homogenization of
Partial Differential Equations
Translatedfrom the original Russian
by M. Goncharenko and D. Shepelsky
Birkhauser
Boston • Basel • Berlin
VladimirA. Marchenko Translatedby: MariyaGoncharenko
Evgueni Ya. Khruslov DmitryShepelsky
B.VerkinInstituteforLowTemperature B.VerkinInstituteforLowTemperature
PhysicsandEngineering PhysicsandEngineering
MathematicalDivision MathematicalDivision
61103 Kharkov 61103 Kharkov
Ukraine Ukraine
MathematicsSubjectClassicifications(2000): 35B27, 35Bxx, 35125, 35140, 35K20, 35P15, 36Pxx,
74Q05,74QIO,74Q15,74Qxx
LibraryofCongressControlNumber:2005935041
ISBN-IO0-8176-4351-6 eISBN0-8176-4468-7 Printedonacid-freepaper.
ISBN-13978-0-8176-4351-5
$
©2006BirkhauserBoston Birkhauser ®
BasedontheoriginalRussianedition,ycpe,n:HeHHbleMo,n:eJIHMHKpOHeo,n:Hopo,n:HbIXcpe,n:. KHeB:
HayK. lIyMKa, 2005
Allrights reserved. This workmaynotbe translatedorcopied in wholeorin part withoutthe writ
tenpermissionofthepublisher(BirkhauserBoston,cloSpringerScience+BusinessMediaInc.,233
Spring Street,New York, NY 10013,USA), exceptforbriefexcerptsinconnectionwithreviewsor
scholarlyanalysis. Useinconnectionwith anyform ofinformation storageandretrieval,electronic
adaptation,computersoftware,orbysimilarordissimilarmethodology nowknownorhereafterde
velopedisforbidden.
Theuseinthispublicationoftradenames,trademarks,servicemarksandsimilarterms,evenifthey
arenotidentifiedassuch,isnottobetakenasanexpressionofopinionastowhetherornottheyare
subjecttoproprietaryrights.
PrintedintheUnitedStatesofAmerica. (TXQIEB)
987654321
www.birkhauser.com
Preface
This book is devoted to homogenization problems for partial differential equations
describing various physical phenomena in microinhomogeneous media. This direc
tion in the theoryofpartialdifferentialequationshas been intensivelydevelopedfor
the lastforty years; it finds numerous applications in radiophysics, filtration theory,
rheology, elasticity theory, and many other areas ofphysics, mechanics, and engi
neeringsciences.
Amediumiscalledmicroinhomogeneousifitslocalparameterscanbedescribed
by functions rapidly varying with respect to the space variables. We will always
assume that the length scale ofoscillations is much less than the linear sizes ofthe
domainin whicha physical process isconsideredbutmuch greaterthan the sizes of
molecules, so that the process can be described using the differential equations of
the mechanics ofsolids. These differential equations either have rapidly oscillating
coefficients (with respect to the space variables) orare considered in domains with
complex microstructure, such as domains withfine-grained boundary [112] (called
later by the better-known term stronglyperforateddomains). The microstructure is
understood as the local structure ofadomain or the coefficients ofequations in the
scaleofmicroinhomogeneities.
Obviously, itis practically impossibleto solvethecorresponding boundary (ini
tial boundary) value problems by eitheranalytical or numerical methods. However,
ifthemicroscaleis much lessthanthecharacteristicscaleoftheprocessunderinves
tigation (e.g., the wavelength), then itis possible to give a macroscopic description
ofthe process. Ifit is the case, the medium usually has stable characteristics (heat
conductivity,dielectricpermeability,etc.),which,ingeneral,maydiffersubstantially
from the localcharacteristics. Such stablecharacteristics are referred to as homoge
nized, oreffective, characteristics, because they are usually determined by methods
of the homogenization theory for differential equations or the relevant mean field
methods,effectivemedium methods,etc.
The term homogenization is associated, first of all, with methods of nonlin
ear mechanics and ordinary differential equations developed by Poincare, Krylov,
Bogolyubov,andMitropolskii(see,e.g.,[21, 123]).Forpartialdifferentialequations,
homogenization problems have been studied by physicists from Maxwell's times,
vi Preface
butthey remainedfor along timeoutsidethe interests ofmathematicians. However,
since the mid I960s, homogenization theory for partial differential equations began
to be intensively developed by mathematicians as well, which was motivated not
only by numerous applications (first ofall, in the theory ofcomposite media [142])
but also by the emergence of new deep ideas and concepts important for mathe
matics itself. Currently, there is a great number of publications devoted to mathe
matical aspects of homogenization such as asymptotic analysis, two-scale conver
gence, G-convergence, and r-convergence. Making no claimto cite all ofthe avail
able monographs onthe subject, we would like to mention thebooks by Allaire [3],
Bakhvalov and Panasenko [9], Bensoussan, Lions, and Papanicolaou [13], Braides
and Defranceschi [26], Cioranescu and Donato [42], Cioranescu and Saint Jean
Paulin [45],Dal-Maso[46],MarchenkoandKhruslov [113],Oleinik, Iossifiyan, and
Shamaev [131], Pankov [133], Sanches-Palencia [148], Skrypnik [161],
Zhikov, Kozlov, andOleinik[181].
In the mathematical description ofa physical phenomenon in microinhomoge
neous media, the local characteristics depends on a small parameter£, which is the
characteristic scaleofthe microstructure. Itis the asymptotic analysis, as £ ---+ 0, of
the problemthatleads tothehomogenizedmodeloftheprocess. Itturns outthatthe
limits ofsolutionsofthe originalproblemcan bedescribed bycertain new differen
tialequationswithcoefficientssmoothlyvaryinginsimpledomains.Theseequations
constitute a mathematical model ofthe physical process in a microinhomogeneous
medium, their coefficients being effective characteristics ofthe medium. Forexam
ple,in the simplestcase, the localcharacteristicsofa microinhomogeneous medium
are described by periodic functions ofthe form aU), X E JRn. The corresponding
effectivecharacteristics appearto be independent ofx; moreover, the homogenized
equations have the same structure as the original ones. Therefore, in this case, the
main problem ofmathematical modeling is to determine the coefficients ofthe ho
mogenizedequations; these coefficientscanthen beviewedas theeffectiveparame
ters ofthe medium. This situationis typicalfor various microinhomogeneous media
encounteredin nature.
However, there exist media with more complicated microstructure, the macro
scopic description ofwhich cannot be reduced to the determination ofthe effective
characteristics only, sincehomogenization leads to equations substantially different
from the original ones. Such a situation usually occurs when the microstructure is
characterized by several small parameters, ofdifferent order ofsmallness; artificial
composite materials as well as some natural media provide the relevant examples.
The correspondinghomogenized models differsubstantially from the original, "mi
croscopic," ones; depending onthe microstructure, they appeartobeeithernonlocal
models or multicomponent models or models with memory. This book is basically
devotedtothestudyofstructureofmicroinhomogeneous medialeadingto"nonstan
dard" models; therefore, it has almost no intersections with the monographs cited
above,except [113].Webeganto writethisbook(whichwas initiallythoughtofas a
revised edition of[113]) in the late 1980s; butsince then, new results havebeenob
tained, which now constitutethe maincontents ofthe book, the needed results from
rI13] being presentedin moreconvenientfashion.
Preface Vll
In the book, we restrict ourselves mainly to physical phenomena described by
theDirichletandNeumannboundaryvalueproblemsinstronglyperforateddomains
andbylinearellipticandparabolicdifferentialequationswithrapidlyoscillatingco
efficients;butthedevelopedmethodscanbeappliedaswellinthestudyofboundary
value problems ofelasticity theory, electrodynamics, Fourierboundary value prob
lems, nonlinearproblems, etc.
Acknowledgments
We are grateful to MariaGoncharenko and Dmitry Shepelsky for preparing the text
forpublicationandtranslating thebookintoEnglish.
Kharkov, VladimirMarchenko
March 2004 Evgenii Khruslov
Contents
Preface......................................................... v
1 Introduction. ................................................ 1
1.1 The SimplestHomogenizedModel. ........................... 1
1.2 NonlocalHomogenizedModel. .. ............................ 5
1.3 Two-ComponentHomogenizedModel. ........................ 7
1.4 HomogenizedModel withMemory 10
1.5 HomogenizedModel withMemory: TheCaseofViolatedUniform
Boundedness ofbe(x) ....................................... 12
1.6 Homogenization ofBoundary Value Problems in Strongly
PerforatedDomains. ........................................ 13
1.7 Strongly ConnectedandWeaklyConnectedDomains: Definitions
and QuantitativeCharacteristics 18
1.7.1 StronglyConnectedandWeaklyConnectedDomains. ..... 18
1.7.2 LocalMesoscopic Characteristics ofStrongly Connected
Domains........................................... 20
2 The Dirichlet Boundary Value Problem in Strongly Perforated
Domainswith Fine-GrainedBoundary 31
2.1 MethodofOrthogonal Projections andan AbstractScheme for the
DirichletProblemin StronglyDerforatedDomains 32
2.1.1 MethodofOrthogonalProjections. ..................... 32
2.1.2 An AbstractScheme. ................................. 34
2.2 Asymptotic BehaviorofSolutions ofthe DirichletProblem in
Domains withFine-GrainedBoundary. ........................ 42
2.2.1 ProblemFormulationandMainResult 42
2.2.2 Auxiliary Statements ................................. 44
2.2.3 Validityofthe Assumptions ofTheorem2.3 .............. 49
2.2.4 ProofofTheorem2.3 ................................. 55
2.3 The DirichletProblemin Domains with Random Fine-Grained
Boundary ... .............................................. 56
x Contents
2.3.1 ProblemFormulation andMainResult . 56
2.3.2 AssumptionsofTheorem 2.3 "inProbability" . 58
2.3.3 Convergence in Probability ofSolutions ofProblem
(2.23)-(2.24) . 63
3 The Dirichlet Boundary Value Problem in Strongly Perforated
DomainswithComplexBoundary 67
3.1 Necessary andSufficientConditions forConvergenceofSolutions
oftheDirichletProblem. .................................... 67
3.1.1 ProblemFormulation andMainResult 67
3.1.2 SufficiencyofConditions 1and 2. ...................... 68
3.1.3 NecessityofConditions 1and 2 ........................ 77
3.1.4 Higher-OrderEquations. .............................. 80
3.2 Asymptotic BehaviorofSolutions ofVariational Problems for
NonquadraticFunctionalsinDomainswithComplexBoundary. ... 82
3.2.1 The Sobolev-OrliczSpaces: Preliminaries ........ 82
3.2.2 Problem StatementandMainResult 84
3.2.3 TheProofofTheorem3.7 ............................. 86
3.3 Asymptotic Behaviorofthe Potential ofthe Electrostatic Field
in aWeakly NonlinearMedium withThin Perfectly Conducting
Filaments ................................................. 96
4 Strongly ConnectedDomains 105
4.1 PreliminaryConsiderations 106
4.1.1 OnOnePropertyofLattices withColoredNodes 106
4.1.2 SomeProperties ofDifferentiableFunctions. ............. 107
4.2 StronglyConnectedDomains 114
4.2.1 ConvergenceandCompactness ofSequences ofFunctions
GiveninVaryingDomains 114
4.2.2 DomainsAdmittingExtensionofFunctions. ............. 116
4.2.3 Domains Admitting Extension ofFunctions with Small
Distortion 121
4.3 StronglyConnectedDomains ofDecreasingVolume ............. 125
4.3.1 ConvergenceandCompactness ofSequences ofFunctions
DefinedinDomains ofDecreasingVolume. .............. 125
4.3.2 ExamplesofDomains ofDecreasingVolume ThatSatisfy
the StrongConnectivityCondition A.................... 128
5 The NeumannBoundary Value Problems in Strongly Perforated
Domains " 137
5.1 Asymptotic Behaviorofthe Neumann Boundary Value Problems
in StronglyConnectedDomains. ............................. 137
5.1.1 TheConductivityTensor: MainTheorem 137
5.1.2 ProofofTheorem5.1 139
5.1.3 ConvergenceofEnergies andFlows 145
Contents
Xl
5.1.4 NecessityofConditions I and2ofTheorem5.1 147
5.2 Calculation ofthe Conductivity Tensorfor Structures Close to
Periodic 151
5.3 AsymptoticBehaviorofthe Neumann BoundaryValue Problems
in WeaklyConnectedDomains 158
5.3.1 Weakly ConnectedDomains 158
5.3.2 Quantitative Characteristics ofWeakly Connected
Domains: MainTheorem 160
5.3.3 AuxiliaryConstructionsandStatements. ................ 163
5.3.4 ProofofTheorem5.7................................. 173
5.3.5 ConvergenceofEnergiesand Flows. .................... 178
5.4 AsymptoticBehavioroftheNeumann Boundary Value Problems
inDomains with Accumulators (Traps) 180
5.4.1 Weakly ConnectedDomains withAccumulators and Their
QuantitativeCharacteristics: MainTheorem. ............. 180
5.4.2 Auxiliary Constructions and Statements 183
5.4.3 ProofofTheorem5.13................................ 189
5.4.4 AGeneralizationofTheorem5.13 196
5.5 Asymptotic Behaviorofthe Neumann Boundary Value Problems
in StronglyConnectedDomainsofDecreasingVolume. .......... 198
5.5.1 QuantitativeCharacteristics ofDomains and Main Theorem 198
5.5.2 Examples........................................... 202
6 NonstationaryProblemsandSpectralProblems 211
6.1 Asymptotic BehaviorofSolutions ofNonstationary Problems in
TubeDomains 211
6.1.1 ConvergenceofSpectral Projections .................... 211
6.1.2 TheDirichletInitial Boundary ValueProblem ............ 213
6.1.3 TheNeumannInitialBoundaryValueProblem 216
6.1.4 TheNeumannInitialBoundaryValueProbleminDomains
with Accumulators. .................................. 218
6.2 Asymptotic Behavior ofSolutions ofDirichlet Problems in
Varying Strongly PerforatedDomains 220
6.2.1 Problem Formulationand MainResult 220
6.2.2 Estimatesfor Derivativesofu(s)(x, t) .. ................. 222
6.2.3 ProofofTheorem6.7 '" 226
6.3 Asymptotic BehaviorofEigenvalues ofBoundary Value Problems
in Strongly Perforated Domains. .............................. 228
6.3.1 Strongly ConnectedDomains 228
6.3.2 WeaklyConnectedDomains 233
6.3.3 Domainswith Accumulators. .......................... 234
7 DifferentialEquations with RapidlyOscillatingCoefficients 237
7.1 Asymptotic BehaviorofSolutions ofDifferentialEquations with
CoefficientsThatAreNotUniformlyElliptic 239