Table Of ContentCompact Textbooks in Mathematics
Leonid Berlyand
Volodymyr Rybalko
Getting
Acquainted with
Homogenization
and Multiscale
CompactTextbooksinMathematics
Compact Textbooks in Mathematics
This textbook series presents concise introductions to current topics
inmathematicsandmainlyaddressesadvancedundergraduatesand
masterstudents.Theconceptistooffersmallbookscoveringsubject
matterequivalentto2-or3-hourlecturesorseminarswhicharealso
suitableforself-study.Thebooksprovidestudentsandteacherswith
new perspectives and novel approaches. They feature examples and
exercisestoillustratekeyconceptsandapplicationsofthetheoretical
contents. The series also includes textbooks specifically speaking to
theneedsofstudentsfromotherdisciplinessuchasphysics,computer
science,engineering,lifesciences,finance.
• compact:smallbookspresentingtherelevantknowledge
• learningmadeeasy:examplesandexercisesillustratethe
applicationofthecontents
• usefulforlecturers:eachtitlecanserveasbasisandguidelinefor
asemestercourse/lecture/seminarof2–3hoursperweek.
Moreinformationaboutthisseriesathttp://www.springer.com/series/11225
LeonidBerlyand•VolodymyrRybalko
Getting Acquainted
with Homogenization
and Multiscale
LeonidBerlyand VolodymyrRybalko
MathematicsandMaterialsResearchInstitute MathematicalDivision
PennsylvaniaStateUniversity BVerkinInstituteforLow
UniversityPark,PA,USA TemperaturePhysicsand
EngineeringofNationalAcademy
ofSciencesofUkraine
Kharkiv,Ukraine
ISSN2296-4568 ISSN2296-455X (electronic)
CompactTextbooksinMathematics
ISBN978-3-030-01776-7 ISBN978-3-030-01777-4 (eBook)
https://doi.org/10.1007/978-3-030-01777-4
LibraryofCongressControlNumber:2018957824
MathematicsSubjectClassification:35-01,35B27,35B40,35Q35,35Q74,74Q05,74Q15,76M50
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Toourlatemothers
vii
Preface
TheideaofthisbookcameaboutwhenL.Berlyandtaughtseveralgraduate
courses on Multiscale and Homogenization at Penn State University. The
classes were typically a mixture of first year mathematics and engineering
graduate students as well as some advancedundergraduates. Most of these
studentswerenotnecessarilythinkingofwritingtheirthesisonmultiscaleor
homogenization,butratherwereinterestedinlearningaboutthesesubjects.
ThiswasnosurpriseasGooglesearchonmultiscaleshowsseveralmillion
hitsandcontinuestogroweveryyear.Althoughtheconceptofscaleplays
a fundamental role in the natural sciences, engineering, and computational
sciences,itisnottypicallyintroducedinmathematicaltextbooks.Homoge-
nizationtheorywas“bornoutofthecomputationalbattlewithfinescales”in
themodelingofheterogeneousmediainengineering,physicalandbiological
sciences. Typical examples include composite materials, porous media,
fracturedmaterials,andsuspensions/colloids.Alloftheseproblemspossess
fine scale structure whose resolution results in computational challenges.
MathematicalmodelsofsuchproblemsleadtoPDEandvariationalproblems
with small parameters (corresponding to fine scales), and analysis of such
problemsrequiresthedevelopmentofnovelasymptotictechniques.
Itturnedoutthatwhiletherearemanyadvancedmathematicaltextbooks
on homogenization and multiscale (and thousands of research papers), the
students in those classes wanted an introduction “from scratch" that did
not require any significant experience in PDEs, calculus of variations, or
functional analysis. One of the first textbooks on homogenization [93]
was written for such an audience but since then several areas such as Γ-
convergence for nonlinear problems, stochastic homogenization as well as
continuumlimitsfordiscreteproblemshaveseenrapiddevelopments.Atthis
pointwerecognizedthechallengeofdesigningaone-semestercoursefora
quickintroductiontothesubject.Thatputusbetweenarockandahardplace
of making the presentation accessible for beginners and providing an idea
aboutthebroadscopeofresearchwithanoverviewofbasichomogenization
techniques.
Our goal is to help a novice graduate student navigate through a quite
mature field. We attempt to present mathematical ideas and techniques
well known to experts while avoiding technicalities as much as possi-
ble. This is done in two ways. First, several basic techniques (two-scale
asymptotic expansions, two-scale convergence, compensated compactness
forbothperiodicandstochasticproblems)areintroducedusingacasestudy
conductivityproblem.ThisprovidesthesimplestPDEandvariationalsetting
from which one can move on to more technically involved problems, e.g.,
viii Preface
arisinginfluidmechanicsandelasticitytheory.Second,weuse1Dsettingsto
introducethereadertoacomplexworldofnonlinearhomogenizationviaΓ-
convergence for problems in both continuum and discrete settings. Further
these ideas and approaches are illustrated by examples that are of interest
for material scientists and engineers, e.g., double porosity and plasticity
problems.Moreover,theproofsareintendedtobeeasytofollow:nostepsare
skipped,andrigorousargumentsaresupplementedbyintuitiveexplanations
andfigures.Also,wechoosetosacrificegeneralityforthesakeofsimplicity
ofpresentation.Insteadofthemostgeneralformulationsofthemainresults,
we present particular cases that still convey the main ideas but are simpler
tofollow,e.g.,weconsiderscalarratherthanvectorialsettings.Inshort,we
presentknownresultsinanewwaysuitableforaone-semestercourseforthe
specificaudienceofbeginninggraduatestudentsinappliedmathematicsand
engineering.Thepresentationisself-containedsothattheonlyprerequisites
are undergraduate courses in PDEs and analysis, and all other necessary
mathematics is contained in the Preliminaries section. The book contains
many exercises (often with hints), which are designed to help the reader
master the techniques and better understand the presentation. While our
primaryfocusisonintroducingmathematicaltoolsandtechniques,modeling
is also emphasized. We present several fundamental multiscale models
arisinginapplications,e.g.,doubleporosityandplasticitymodels,aswellas
basicstochasticmodelsofheterogeneousmedia,e.g.,randomcheckerboard
andPoissoncloud.
We use material from well-known books on homogenization theory
such as [15–17, 33, 37, 40, 62, 69, 82, 93, 97]. We focus our attention on
several basic asymptotic approaches in homogenization, so that a number
of important developments in homogenization theory are not presented in
our book. We refer the reader to the books and monographs [4, 20, 34,
44, 50, 58, 72, 75, 87, 88, 98] for further reading on variational bounds,
networkapproximation,optimization,meso-scaleapproximation,andmulti-
scaleapproachesforrandomanddeterministicmedia.Wealsomentionthe
fieldofnumericalhomogenizationthatbuildsonanalyticalhomogenization.
Whileitishardlypossibletoreviewtheliteratureinthisfastgrowingfield,
we mention only a few subfields and selected references (a comprehensive
listmaybeabookbyitself!):multiscalefiniteelementmethods(MsFEM)
[45, 59, 60, 73], heterogeneous multiscale methods (HMM) [9, 44, 48],
adaptive multiscale methods [81], PDEs with rough coefficients [21, 84–
86],localization[11,68],andBayesiannumericalhomogenization[83].The
researchers entering the field of numerical homogenization would benefit
fromanalyticalperspectiveinthisbook.
FinallywementionthatthiscoursewastaughtbyL.Berlyandatsummer
schools in Kyoto and Haifa as well as in Beijing (Renmin University) and
Shanghai(SJTU).
ix
Preface
Wenowoutlinethecontentsofthe11chaptersofthebook.
InChapter1,Preliminaries,werecallnecessarynotionsandresultsfrom
functional analysis, PDEs, calculus of variations, and probability theory.
Special attention is paid to the notion of weak convergence which plays
a key role in homogenization theory. We also review several results from
elementary analysis which are typically not part of standard undergraduate
courses(e.g.,convexanalysis,semicontinuity).
InChapter2,WhatIsHomogenizationandMultiscale?FirstExamples,
we introduce and discuss the concept of scales and provide numerous
examples. We also present the two-scale conductivity problem which is
usedasacasestudythroughoutthetextbookandstatethehomogenization
theoreminarbitrarydimension.Weintroducethereadertorigorousanalysis
by proving the homogenization theorem for a 1D conductivity “baby”
problem and provide numerical examples. We conclude this chapter with
an interesting example of an inverse homogenization problem for thermo-
elasticityina1Dsetting.
Chapter3,BriefHistoryandSurprisingExamplesinHomogenization,is
devotedto the colorful history of homogenizationthat dates backto giants
such as Poisson, Faraday, Maxwell, Rayleigh, and Einstein. Here we also
mention the origins of the modern homogenization theory that started in
the 1960s and is actively developing nowadays. Finally we present several
examplesofsurprisingpredictionsbyhomogenizationtheory.
InChapter4,FormalTwo-ScaleAsymptoticExpansionsandtheCorrec-
tor Problem, the fundamental idea of separation of fast and slow variables
is used in the analysis of the multidimensional case study conductivity
problem. We state a stronger version of the homogenization theorem that
provides the rate of convergence of the homogenization approximation.
Thisisdonebyintroductionofacorrectingtermintheso-calledcorrector
problem.
InChapter5,CompensatedCompactnessandOscillatingTestFunctions,
we describe a general concept of proving convergence in PDE problems
basedontheso-calledDiv-CurlLemma.Intheframeworkofhomogeniza-
tion theory, this approach led to what is known as the energy method. It
providesmuchshorterargumentsdealingwiththeweakformulationofthe
problemwhencomparedtodirectaprioriestimatesfortwo-scaleexpansions
originallydevelopedforhomogenizationproblems.
InChapter6,Two-ScaleConvergence,wepresentyetanotherapproach
toestablishingthehomogenizationlimitbysynergybetweenthedirecttwo-
scaleexpansionandtheenergymethod.Althoughitisrestrictedtoperiodic
problems, it provides a simple framework for many practical problems
x Preface
where direct asymptotic expansions become prohibitively cumbersome. In
particularwedescribeaninterestingexampleofhomogenizationinadouble
porositymodel.
In Chapter 7, 2D explicit example of effective conductivity via convex
duality, we present an elegant example of explicit calculation of effective
conductivityfora2D,two-phaseproblem.Weusethisexampletointroduce
thereadertothebeautifulideaofconvexduality,whichappliestomanyareas
ofmathematics.
In Chapter 8, Introduction to Stochastic Homogenization, we attempt
to introduce the reader to a field which is currently very active. While
there are a number of mathematical books on periodic homogenization,
this is not the case for stochastic homogenization which, in part, is due to
complexityofthesubjectthatrequiresdualexpertiseinbothPDEs/calculus
ofvariationsandprobability.Inthischapterweexplainthedistinctfeatures
of stochastic vs periodic homogenization and prove a classical theorem on
existenceofthehomogenizedlimitforproblemswithstationaryandergodic
random coefficients. Before proving the theorem we introduce the reader
to stochastic models of heterogeneous media. This is done by providing
detailed descriptions of basic examples of the random checkerboard and
thePoissoncloud.Theseexamplesareeasyontheintuitivelevel,buttheir
rigorousmathematicalunderstandingrequiressignificanteffortforsomeone
with no experience in stochastic modeling. We conclude the chapter by a
briefdiscussionofrecentdevelopmentsandprovideanumberofreferences
forfurtherreading.
In Chapter 9, Γ-Convergence in Nonlinear Homogenization Problems,
we introduce a very powerful tool in the study of nonlinear variational
problems. This is done in an abstract framework because Γ-convergence
appliesfarbeyondhomogenization,e.g.,inthetheoryofphasetransitions.
NextwedemonstratehowΓ-convergenceworksinnonlinearhomogeniza-
tionproblems.Hereweusea1Dsettingtoexplainideaswhileminimizing
thetechnicalities.InthesubsequentChapter10,AnExampleofaNonlinear
Problem:HomogenizationofPlasticityandLimitLoads,wepresentrigorous
analysis of a practical problem in dimensions n ≥ 2. Here we are not
using directly the classical Γ-convergence approach but rather introduce
techniques inspired by Γ-convergence. We also demonstrate the power
of convex duality techniques that are further extended from the linear
conductivityproblempresentedinChapter7.
In Chapter 11, Continuum Limits for Discrete Problems with Fine
Scales,weuseΓ-convergencetechniquestoprovethediscretetocontinuum
limit in a 1D setting. We emphasize the “range of interaction parameter”
which provides a key difference with the Γ-limit in continuum problems
