THESIS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY

Two-scale Convergence and Homogenization of SomePartial Differential Equations

Hermann Douanla

Department of Mathematical SciencesDivision of Mathematics

Chalmers University of Technology and University of GothenburgGothenburg, Sweden 2013

Two-scale Convergence and Homogenization of Some PartialDifferential EquationsHermann DouanlaISBN 978-91-7385-850-2

c©Hermann Yonta Douanla, April 2013.

Doktorsavhandlingar vid Chalmers Tekniska HogskolaNy serie Nr 3531ISSN 0346-718X

Department of Mathematical SciencesChalmers University of Technology| University of GothenburgSE – 412 96 Gothenburg, SwedenPhone: + 46 (0)31–772 1000Fax: + 46 (0)31–161973

douanla@chalmers.sehdouanla@gmail.com

Printed in Gothenburg, Sweden 2013

Two-scale Convergence and Homogenization of Some PartialDifferential Equations

Hermann Douanla

Department of Mathematical SciencesChalmers University of Technology and University of Gothenburg

AbstractThis thesis consists of four papers devoted to the homogenization of some partial differentialequations by means of the two-scale convergence method and theΣ-convergence method.In the first two papers, I investigate the asymptotic behavior of second order self-adjointelliptic eigenvalue problems with periodic rapidly oscillating coefficients and with indefinite(sign-changing) density function in periodically perforated domains.

In the first paper, we have a Steklov condition on the boundaries of the holes. I provethat the spectrum of the Steklov problem under consideration is discrete and consists of twosequences, one tending to−∞ and another to+∞, then I study the limiting behavior ofpositive and negative eigencouples which crucially depends on the sign of the average ofthe weight over the surface of the reference hole.

In the second paper, we have a homogeneous Neumann conditionon the boundary ofthe holes. The spectrum is also discrete and consists of two sequences, one tending to−∞and another to+∞. I study the asymptotic behavior of positive and negative eigencoupleswhich critically depends on the sign of the average of the weight over the solid part of thereference cell. The third and fourth papers deal with deterministic homogenization.

In the third paper, we study the existence and almost periodic homogenization of somemodel of generalized Navier-Stokes equations. We establish an existence result for nonsta-tionary Ladyzhenskaya equations with a given nonconstant density and an external forcedepending nonlinearly on the velocity. In the case of a nonconstant density of the fluid, westudy the asymptotic behavior of the velocity field.

In the fourth paper, we first introduce a framework to study homogenization problemsin general deterministic fissured media composed of blocks of an ordinary porous mediumwith fissures between them. Next, we study homogenization for nonstationary Navier-Stokes systems in a fissured medium of general deterministictype. Assuming that the blocksof the porous medium consist of deterministically distributed inclusions and the elasticitytensors satisfy general deterministic hypotheses, we prove that the macroscopic problem isa Navier-Stokes type equation for Newtonian fluid in a fixed domain. Our setting includesthe classical periodic framework, the weakly almost periodic one and some others.

Keywords: Homogenization. Two-scale convergence. Perforated domains. Eigenvalueproblem. Ergodic algebra. Algebra with mean value. Gelfandtransform.Σ-convergence.Navier-Stokes equation. Deterministic fissured medium.

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iv

List of Papers

This thesis consists of an introduction and the following papers.

Paper A: H. Douanla,Homogenization of Steklov spectral problems with indefinitedensity function in perforated domains, Acta Applicandae Mathematicae,123(2013)261-284.

Paper B: H. Douanla,Two-scale convergence of periodic elliptic spectral problemswith indefinite density function in perforated domains, Asymptotic Analysis,81(2013)251-272.

Paper C: H. Douanla and J.L. Woukeng,Almost periodic homogenization of a gener-alized Ladyzhenskaya model for incompressible viscous flow, Journal of Mathemati-cal Sciences (N. Y.),189(2013) 431-458.

Paper D: H. Douanla, G. Nguetseng and J.L. Woukeng,Incompressible viscous New-tonian flow in a fissured medium of general deterministic type, Journal of Mathemat-ical Sciences (N. Y.),191(2013) 214-242.

In addition to the above, there are three other papers by the author, which arenot included in this thesis.

• H. Douanla and N. Svanstedt,Reiterated homogenization of linear eigenvalue prob-lems in multiscale perforated domains beyond the periodic setting, Commun. Math.Anal., 11 (2011) 61-93.

• H. Douanla,Two-Scale convergence of Stekloff eigenvalue problems in perforateddomains, Boundary Value Problems(2010), Article ID 853717, 15 pages.

• H. Douanla and Nils Svanstedt,Homogenization of a nonlinear elliptic problem withlarge nonlinear potential, Applicable Analysis,91 (2012) 1205-1218.

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vi

Acknowledgements

First of all, I would like to thank my advisor Grigori Rozenblioum for his guidance, forsharing his mathematical style and philosophy with me, for being a good proofreader, andfor the new breath he gave me during my Ph.D. studies. I owe a big thanks to my co-advisorJean Louis Woukeng for his support, for suggesting some of the problems discussed in thisthesis, and also, for being a good (gossip) friend. Nils Svanstedt was my advisor for theLicentiate Exam, I appreciate the discussions we had together.

I hereby express my gratitude to Mathematical Sciences at Chalmers University of Tech-nology and University of Gothenburg for giving me the opportunity to do a Ph.D. in Math-ematics. I also express my gratitude to my fellow Ph.D. students, other colleagues andthe staff at Mathematical Sciences for the great time we had during my doctoral studies.A special thanks to the board of graduate studies and the staff, namely: B. Johansson, B.Wennberg, J.A. Svensson, P. Hegarty, T. Gustafsson, J. Madjarova and L. Turowska for theirunconditional support during hard times.

I am also thankful to the Departments of Mathematics of the University of Yaounde Iand the University of Dschang, both in Cameroon, for the background they provided mewith during my Master studies and Bachelor studies, respectively. I am deeply grateful toGabriel Nguetseng for his constant support and priceless encouragements.

Last but not least, a great thanks goes to my family and friends all over the world forbelieving in me, and for their friendship and constant encouragements. A huge thanks tomy wife, Mirene Douanla, for all her invaluable support, love and understanding.

Hermann DouanlaGothenburg, April 2013

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I dedicate this thesis to my wife

Mirene Douanla

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x

Contents

Page

Abstract iii

List of Papers v

Acknowledgements vii

Dedication ix

Table of contents xi

Introduction 13

1 Brief history 15

2 Homogenization techniques 172.1 TheΓ-convergence method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192.2 TheH-convergence and theG-convergence methods . . . . . . . . . . . . . . . . . 202.3 The asymptotic expansion method . . . . . . . . . . . . . . . . . . . .. . . . . . . 232.4 The two-scale convergence method . . . . . . . . . . . . . . . . . . .. . . . . . . . 262.5 The periodic unfolding method . . . . . . . . . . . . . . . . . . . . . .. . . . . . . 302.6 TheΣ-convergence method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

3 Overview of Papers 403.1 Homogenization of Steklov spectral problems with indefinite density function in

perforated domains,Acta Appl. Math.,123 (2013) 261-284.. . . . . . . . . . . . . . 403.2 Two-scale convergence of elliptic spectral problems with indefinite density function

in perforated domains,Asymptot. Anal.,81 (2013) 251-272.. . . . . . . . . . . . . . 433.3 Almost periodic homogenization of a generalized Ladyzhenskaya model for incom-

pressible viscous flow,J. Math. Sci. (N. Y.),189 (2013) 431-458.. . . . . . . . . . . 443.4 Incompressible viscous Newtonian flow in a fissured medium of general determinis-

tic type,J. Math. Sci. (N. Y.),191 (2013) 214-242.. . . . . . . . . . . . . . . . . . . 46

Bibliography 50

Paper A 59

Paper B 87

Paper C 113

Paper D 147

xi

xii

Introduction

The homogenization theory aims at describing the effective(macroscopic) properties ofcomposite materials by means of the characteristics of their micro-structures. Compositematerials are widely used in engineering construction because of their properties that, ingeneral, are better that those of their individual constituents. Well known examples of com-posites are the superconducting multifilamentary composites which are used in the com-position of optical fibers. Under external loads, the behavior of a composite material onits microscale is often quite different from that of a uniform material with the same effec-tive properties. Cracking, for instance, starts and develops in these materials in differentways [15]. The homogenization theory cover a wide range of applications ranging from thestudy of the characteristics of composites to optimal design. The homogenization theorymade it possible to predict properties of a composite material even before it is engineered,and consequently, to consider the problem of developing a composite with predeterminedor optimum characteristics. In mathematical terms, the homogenization theory study theasymptotic behavior of partial differential equations with varying coefficients depending ona small parameterε > 0 (representing the size of the heterogeneities), as this parametertends to zero.

The following classical example (see e.g., [15]) clarifies how studying the limiting be-havior of partial differential equations with varying coefficients helps to describe the effec-tive properties of composites.

Figure 1: Anε-periodic composite.

We consider stationary heat conduction in a material with a periodic structure of periodε > 0 in all directions (see figure 1). This could, for example, be acut across the sectionof a unidirectionally reinforced fibrous composite which isa system of unidirectional fibersof one compound and the matrix of another compound filling thespace between the fibers.We assume that the thickness of fibers and that of the fibers spacing are of the same orderε > 0. Thus, if the temperature of the fibrous composite is constant along the orientation ofthe fibers, the two-dimensional thermal field in the composite is described by the Poisson

14 H. Douanla

equation −∇ · (Aε(x)∇uε) = f(x) in Ω

uε = 0 on ∂Ω,(1)

everywhere outside the interface of the two materials. Herewe have assumed that thetemperature on the boundary of the domainΩ is kept at 0, andAε(x) is the conductivity atthe pointx. We have

Aε(x) =

Af if x is on a fiber,Am if x is on the matrix,

whereAf andAm are the conductivity of the fibers and matrix, respectively.The functionf(x) is the density of the heat sources in the composite anduε(x) is the temperature at thepoint x. Continuity conditions are satisfied on the interface for the temperature[uε] = 0and for the heat flux density

[Aε

∂uε

∂n

]= 0, where[h] stands for the jump of the function

h at the matrix-fibers interface. The conductivity coefficient Aε(x) changes by|Af − Am|whenx changes by a value of orderε << 1. Thus the conductivityAε oscillates rapidly,making the numerical solvability of (1) (say by a finite difference or finite element method)very expensive or even impossible because of the discontinuities in the functionAε at thefibers-matrix interface. In order to get reasonable resultsusing a numerical scheme, thediscretization step∆(x) must satisfies∆(x) << ε. Otherwise, the microstructure is nottaken into account and the computations are not realistic. Choosing for exampleε = 10−4

and∆(x) = 10−1ε, and assuming that theε-periodic bidimensional domain has orderof magnitude 1, the order of degrees of freedom is(10 · 104)2 = 1010 ! This is veryexpensive. Moreover, the variations of the temperature at that tiny microscale is of nointerest, and composites are often heterogeneous on several scales, making the numericalapproach tricky.

In the asymptotic process (asε→ 0), we look for an equation−∇ · (A(x)∇u) = f(x) in Ω

u = 0 on ∂Ω,(2)

whose coefficients are not oscillating, and whose solution is ‘close’ to that of theε-problem(1). The new equation (2) is the averaged (or homogenized) equation and its coefficientsdescribe the effective properties of the composite under consideration.

Let l andL denote the characteristic lengthes of the micro-cell and the whole domain,respectively (l << L). In the early work of Babuska on the homogenization approach inengineering [8], he pointed out that: ‘l is a given parameter, with physical meaning whichcannot be changed, e.g., cannot be made “sufficiently” small’. Moreover, he mentionedin [7, 10] thatε in (1) is the ratio of the microscale to the macroscale:ε = l

L . Thusε→ 0 isequivalent toL→∞, for this reason, the homogenization procedure is sometimes referredto as “upscaling” or “zooming out”. For an extensive description and further queries aboutthe homogenization theory, we refer to leading books in thisresearch area like, e.g., [15, 21,29, 30, 46, 51, 83, 106, 128, 131, 143, 164]. We particularly recommend the introduction

Introduction 15

to homogenization by Cioranescu and Donato [46], which is the best start (at least to myopinion) to study the homogenization theory of partial differential equations.

1 Brief history

Following [15, 21, 46, 83], we give a short (possibly incomplete) historical developmentof homogenization methods. The problem of replacement of a heterogenous material byan “equivalent” homogeneous one dates back to at least the 19th century. This was raisedin works by Poisson [135], Maxwell [105] and Rayleigh [138].In 1881, Maxwell [105]studied the effective conductivity of media with small concentrations of randomly arrangedinclusions, and Rayleigh [138] studied the same problem with periodically distributed in-clusions in 1892. In 1906, Einstein [71] investigated the effective viscosity of suspensionswith hard spherical particles in incompressible viscous fluids. A good survey of results onthis question until 1926 can be found in [97].

Striking contributions were made in the 1930s. Voight [157]calculated effective pa-rameters of polycrystal, such as, the stiffness tensor, by averaging the appropriate valuesover volume and orientation, while Reuss [139] used averaging of the component of the re-verse tensor (compliance) for the same problem. Later on, Hill [80, 81] and Il’iushina [86]rigorously proved that Voight’s and Reuss’ methods give theupper bound and the lowerbound, respectively, of those effective parameters. For results in the direction of the so-called Reuss-Voight inequality (Hill’s fork), such as the Hashin-Shtrikman bounds, we referto [164, Section 1.6 and Chapter 6] and [153, Chapter 25], andreferences therein. It shouldbe noted that iterated homogenization type problems were considered for the first time byBruggeman in 1935 [32].

The first asymptotically exact scheme for calculating effective parameters of laminatedmedia was proposed in 1946 by Lifshits and Rozentsveig [98, 99]. Complex analysis basedmethods were used by Van Fo Fy [154], Grigoliuk and Filshtinskii [76] for exact solutionsof two-dimensional problems of elasticity for composite materials and perforated plates andshells.

In 1964, Marchenko and Khruslov [103] introduced a general approach based on asymp-totic tools which could handle numerous physical problems,including for example (for thefirst time), boundary value problems with fine-grained boundary [103, 104]. From the early1970s, further development of the mathematical study of phenomena in heterogeneous me-dia is done by averaging differential equations with rapidly oscillating coefficients, and thefirst results (according to e.g., [4, 15]) are in [8, 12, 14, 13, 20, 22, 54, 126, 142].

The name “homogenization” was introduced in 1974 by Babuska[9]. Several homog-enization methods were developed in the 1970s, and homogenization became a subject inMathematics. The methods introduced include:

• the multiple scales asymptotic expansion set out by engineers and mechanical sci-entists (see e.g., [26]), and systematically formalized tohandle homogenization ofboundary value problems with periodic rapidly oscillatingcoefficients by Bensous-

16 H. Douanla

san, Lions and Papanicolaou [21], see also Keller and Larsen[92, 93], and Sanchez-Palencia [143];

• the G-convergence of Spagnolo [146, 147] well adapted to problem involving Greenkernel;

• theΓ-convergence of De Giorgi [55, 56] suitable for homogenization of optimizationproblem;

• the H-convergence of Tartar and Murat [107, 108, 109, 150, 151] (a generalization ofthe G-convergence method to non symmetric problems).

In 1989, Nguetseng [116, 117] introduced the two-scale convergence method to study ho-mogenization of boundary value problems with periodic rapidly oscillating coefficients.The name “two-scale convergence” was later on introduced in1992 by Allaire [1] who pro-posed an elegant proof of Nguetseng’s compactness theorem and further studied propertiesof the two-scale convergence method. Nguetseng’s breakthrough was “groundshaking andinspiring to periodic homogenization as was the fall of the Berlin wall in 1989 to worldhistory”, quoting Stelzig [149, Page 38]. The two-scale convergence method was extendedto periodic surfaces in 1992 by Neuss-Radu [114], this extension has been further studiedin [5, 115]. In 1994, Bourgeat, Mikelic, and Right [28] introduced the “stochastic two-scale convergence” to study random homogenization. The two-scale convergence has beendeveloped in many other papers, among them, [1, 2, 3, 28, 100,156, 162, 163].

Recently in 2002, Damlamian, Griso and Cioranescu [43, 44] introduced the “unfoldingmethod” to study the homogenization of periodic heterogeneous composites while Nguet-seng [119] extended the two-scale convergence method, under the label “Σ-convergence”, totackle deterministic homogenization beyond the periodic setting. In 2009, Wellander [159]introduced the two-scale Fourier transform (for periodic homogenization in Fourier spaces)which connects some existing techniques for periodic homogenization, namely: the two-scale convergence method, the periodic unfolding method and the Floquet-Bloch expansionapproach to homogenization.

The homogenization methods utilized in this thesis are the two-scale convergence methodand theΣ-convergence method. In the following section, we briefly present the asymptoticexpansion method, the two-scale convergence method and theΣ-convergence method. Butwe will also recall the definition and main compactness theorems of the H-convergencemethod and theΓ-convergence method. As for the periodic unfolding method,its main in-gredient, the unfolding operator, will be presented. For the sake of simplicity, we assumethroughout (except otherwise specified) that all vector spaces are real vector spaces, andscalar functions are assumed to take real values.

Introduction 17

2 Homogenization techniques

We start this section by giving a flavour of homogenization. We consider the followingtwo-point boundary value problem (see e.g., [21, 46]):

−d

dx

(a(x

ε)duεdx

)= f in (0, 1)

uε(0) = uε(1) = 0,

(3)

wheref ∈ L2(0, 1), and where the coefficienta ∈ L∞(0, 1) is 1-periodic iny and satisfies

0 < α ≤ a(y) ≤ β <∞ a.e. in(0, 1) (4)

for some constantsα, β ∈ R. Owing to (4), we have

‖uε‖H1

0(0, 1) ≤

1

α‖f‖L2(0, 1). (5)

Then, there exists a subsequence still denoted byε such that

uε → u0 weakly inH10 (0, 1) (6)

asε→ 0. We set

pε(x) = a(x

ε)duεdx

(x) a.e. inx ∈ (0, 1).

We have

−dpε

dx= f in (0, 1),

which together with (5) imply

∥∥∥∥dpε

dx

∥∥∥∥L2(0, 1)

+ ‖pε‖L2(0, 1) ≤β

α‖f‖L2(0, 1) + ‖f‖L2(0, 1).

Therefore, up to a subsequence still denoted byε we have by the Rellich-Kondrachov com-pactness theorem,

pε → p0 strongly inL2(0, 1)

asε→ 0. Hence, passing to the limit, asε→ 0, in the weak formulation of (3)

∫ 1

0a(x

ε)duεdx

dv

dxdx =

∫ 1

0fvdx ∀v ∈ H1

0 (0, 1) (7)

yields

−dp0

dx= f in (0, 1). (8)

18 H. Douanla

We recall that, as the functiona is 1-periodic and bounded from above and away from zero,the following convergence result hold

1

a( ·ε)→

∫ 1

0

1

a(x)dx ≡M(

1

a) weakly∗ in L∞(0, 1),

so that we can pass to the limit in the product

duεdx

=1

a( ·ε)pε

using the well-known weak-strong convergence result. We obtain

duεdx→M(

1

a)p0 weakly inL2(0, 1),

asε→ 0. Consequently, from (6) we have

du0dx

=M(1

a)p0,

which together with (8) yield

−d

dx

(1

M( 1a)

du0dx

)= f in (0, 1).

Hence, the limitu0 is fixed by this equation and does not depend on any particularsubse-quence extracted in the process. Therefore, we have prove the following homogenizationresult.

Theorem 1. Letuε ∈ H10 (0, 1) be the solution to (3) with hypothesis (4). Then, asε→ 0,

we haveuε → u0 weakly inH1

0 (0, 1),

whereu0 is the unique solution inH10 (0, 1) to the problem

−

d

dx

(1

M( 1a)

du0dx

)= f in (0, 1)

u0(0) = u0(1) = 0.

Following the homogenization result in Theorem 1, we are tempted to claim that in theN -dimensional case(N > 1) the homogenized coefficients can be obtained in terms of themean value of the inverse matrixA−1 of A. This is not true as we will see in the followingsubsections where we briefly present popular homogenization techniques. In the sequel, theletterE stands for a sequence of strictly positive real numbers(εn)n∈N with εn → 0 asn→∞.

Introduction 19

2.1 TheΓ-convergence method

Introduced by De Giorgi [55, 56] in the early 1970s, theΓ-convergence is an abstract no-tion of functional convergence aiming at describing the asymptotic behavior of families ofminimum problems usually depending on some parameters whose nature may be geomet-ric or constitutive, deriving from a discretization argument, an approximation procedure,etc [29]. TheΓ-convergence has several applications in many research areas including forexample the calculus of variation and the homogenization ofpartial differential equations.A good introduction to theΓ-convergence is [29] and several applications can be found in[30, 33, 51]. It should be noted that theepiconvergenceintroduced by Attouch [6] in 1984is a functional convergence notion close to theΓ-convergence. We now give the defini-tion of theΓ-convergence, its fundamental theorem and hint how it is utilized to handle thehomogenization of partial differential equations.

Definition 1. Let W be a metric space endowed with a distanced. Let (Fε)ε∈E be asequence of real functions defined onW . The sequence(Fε)ε∈E is said toΓ-converges to alimit functionF0 if for anyx ∈W , the following hold:

1. (lim inf inequality) any sequence(xε)ε∈E converging tox in W asε→ 0 satisfies

F0(x) ≤ lim infε→0

Fε(xε),

2. (existence of a recovering sequence) there exists a sequence(xε)ε∈E converging toxasε→ 0 and such that

F0(x) ≥ limε→0

Fε(xε).

Indeed, theΓ-convergence and theΓ-limit depend on the choice of the distance. AΓ-limit F0 is a lower semi continuous function onW , viz, F0(x) ≤ lim infε→0 F0(xε) forany sequencexε → x as ε → 0. TheΓ-limit of the constant sequence F is the lowersemicontinuous envelope (relaxation) ofF .

To formulate the main (as regards the homogenization theory) theorems of theΓ-convergence,we recall that a sequence(Fε)ε∈E of real functions defined onW is said to beequi-mildlycoercive onW if there exists a compact setK (independent ofε) such that

infx∈W

Fε(x) = infx∈K

Fε(x).

Theorem 2. Let (Fε)ε∈E be an equi-mildly coercive sequence onW whichΓ-converges toa limit F0. Then,

1. the minima ofFε converges to that ofF0, viz,

minx∈W

F0(x) = limε→0

(infx∈W

Fε(x)

),

20 H. Douanla

2. the minimizers ofFε converge to those ofF0, viz, ifxε → x inW andlimε→0 Fε(xε) =limε→0(infx∈W Fε(x)), then,x is a minimizer ofF0.

Theorem 3. Assume that the metric spaceW endowed with the distanced is separable.For any sequence of functions(Fε)ε∈E defined onW there exists a subsequenceE′ of Eand aΓ-limit F0 such that(Fε)ε∈E′ Γ-converges toF0 asE′ 3 ε→ 0.

The proofs of the above theorems can be found in [51]. Looselyspeaking, to utilize theΓ-convergence in homogenization, one usually transforms the partial differential equationinto a minimization problem. To illustrate this, we consider the model problem

−∇ ·

(A(x

ε)∇uε

)= f in Ω

uε = 0 on ∂Ω,(9)

whereΩ is a bounded domain inRN , and the matrixA is coercive, bounded and symmetric.Forε ∈ E, we set

Fε(u) =1

2

∫

ΩA(x

ε)∇u · ∇u dx−

∫

Ωfu dx u ∈ H1

0 (Ω). (10)

It is well known that, when the matrixA is symmetric, the problem (9) is equivalent to thefollowing minimisation problem:

Finduε ∈ H10 (Ω) such that

Fε(uε) ≤ Fε(v)

for all v ∈ H10 (Ω).

(11)

Hence, theΓ-convergence of the sequences of functionals(Fε)ε∈E (defined by (10)) inL2(Ω)-strong is equivalent to the homogenization of the partial differential equation (9).TheΓ-convergence method applied to homogenization theory is neither restricted to linearequation nor to periodic structure. Albeit theΓ-convergence method is one of the mostutilized homogenization technique, it is sometimes blamedfor being of limited interest toreal-life problems of continuum physics. For example, Tartar [152, 153] believes that theenergy minimization approaches to homogenization problems (or to continuum mechanicsin general) is fake mechanics [153, Page 31], since from the first principle of thermodynam-ics, nature conserves energy rather than minimizing it. However, Benamou and Brenier [19]were able to formulate evolution problems of continuum physics as minimization problemby means of the optimal transportation theory.

2.2 TheH-convergence and theG-convergence methods

As said earlier, theH-convergence of Tartar and Murat [107, 108, 109, 150, 151] isageneralization to non symmetric problems of theG-convergence of Spagnolo [146, 147].

Introduction 21

The lettersG andH stand for ‘Green’ and ‘Homogenization’, respectively. These con-vergence methods are equivalent to the convergence of the associated Green kernel. Webriefly present theH-convergence method for specific simple example of operators. Foran advanced description and several applications of these types of convergence, we referto [128, 153].

Let Ω be a bounded open set inRN , and let0 < α ≤ β be two positive constants. WedefineM(α, β,Ω) to be the set of allN×N -matrices defined onΩ with uniform coercivityconstantα andL∞(Ω)-boundβ.

M(α, β,Ω) = A ∈ L∞(Ω;RN×N ) : α|ξ|2 ≤ A(x)ξ·ξ ≤ β|ξ|2 ∀ξ ∈ RN and a.ex ∈ Ω.

We consider a sequence(Aε)ε∈E ⊂M(α, β,Ω) without any periodicity assumption or anysymmetric hypothesis either. Givenf ∈ L2(Ω), there exists, by the Lax-Milgram lemma, aunique solutionuε ∈ H1

0 (Ω) to

−∇ · (Aε∇uε) = f in Ω

uε = 0 on ∂Ω.(12)

Definition 2. The sequence(Aε)ε∈E is said toH-convergence to a limitA∗ asE 3 ε→ 0,if, for any right-hand sidef ∈ L2(Ω) in (12), we have

uε → u0 weakly inH10 (Ω)

Aε∇uε → A∗∇u0 weakly inL2(Ω)N ,

asE 3 ε→ 0, whereu0 is the solution to thehomogenizedequation associated with A:

−∇ · (A∗∇u0) = f in Ω

u0 = 0 on ∂Ω.(13)

Among useful properties of theH-convergence, we have the following:

1. if (Aε)ε∈E ⊂M(α, β,Ω) H-converges, itsH-limit is unique,

2. let (Aε)ε∈E and(Bε)ε∈E be two sequences inM(α, β,Ω) whichH-converge toA∗

andB∗, respectively. IfAε = Bε in ω Ω, thenA∗ = B∗ in ω,

3. theH-limit does neither depend on the source term nor the boundary condition on∂Ω,

4. if (Aε)ε∈E ⊂ M(α, β,Ω) H-converges toA∗, then the associated density of energyalso converges, viz,Aε∇uε ·∇uε converges toA∗∇u·∇u in the sense of distributionsin Ω.

22 H. Douanla

Indeed, theH-convergence is a local property. Note that the definition oftheH-convergencediffers from that of theG-convergence by requiring the convergence of the flux(Aεuε)ε∈Ein addition to that of the sequence(uε)ε∈E . This additional requirement is essential when re-moving the symmetry hypothesis (say, when passing fromG-convergence toH-convergence)since it ensures the uniqueness of theH-limit (see e.g., [83, Page 232]).

Without the following compactness result, the concept ofH-convergence would be use-less.

Theorem 4. For any sequence(Aε)ε∈E ⊂M(α, β,Ω), there exists a subsequenceE′ ofE

and a homogenized limitA∗, belonging toM(α, β2

α ,Ω), such thatAε H-converges toA∗

asE′ 3 ε→ 0.

TheG-convergence version of Theorem 4 was proved for the first time by Spagnolo [146]by means of the convergence of the Green functions associated with (12). It can also beproved using theΓ-convergence. Tartar [107, 151] proposed a simpler proof inthe generalframework ofH-convergence. We present the idea (in the periodic setting)of tartar’s proofwhich has been later called theenergy method, and sometimes, theoscillating test functionmethod. We assume that the matrixAε in (12) is given byAε(x) = A(xε ), whereA is1-periodic in all coordinate directions. The variational formulation of (12) reads

∫

ΩA(x

ε)∇uε · ∇ϕdx =

∫

Ωf(x)ϕ(x) dx for all ϕ ∈ H1

0 (Ω). (14)

The coercivity of the matrixA implies the boundedness of the sequence(uε)ε∈E inH10 (Ω).

Hence, it weakly converges (up to a subsequence) to someu0 ∈ H10 (Ω). We recall that,

with the hypothesis onA, we have

Aε →

∫

ΩA(y) dy weakly inL2(Ω),

so that the left-hand side of (14) involves a product of two weakly converging sequencesin L2(Ω): (A( ·ε))ε∈E and(∇uε)ε∈E . Therefore we cannot pass to the limit by means ofclassical arguments. Tartar’s idea is to use the oscillating test function defined by

ϕε(x) = ϕ(x) + ε

N∑

i=1

∂ϕ

∂xi(x)ω∗

i (x

ε) (ε ∈ E),

whereϕ ∈ D(Ω), and whereωi (i = 1, · · · , N) solve the so-called dual cell problem−∇y · (A

t(y)(ei +∇yωi(y))) = 0 in (0, 1)N

y → ωi(y) (0, 1)N -periodic,

with ei = (δij)Nj=1. Utilizing this test function, Tartar was able to eliminatethe ‘bad terms’

in (14), and to pass to the limit and get a macroscopic equation. However, constructing

Introduction 23

the oscillating test functions is not always as easy as in this simple periodic problem. Tobe convinced, see e.g., [109, 151, 153, 155] where this method is utilized to handle dif-ferent non trivial homogenization problems. We conclude this subsection by mentioningthat Donato et al. [31, 42, 53] have extended theH-convergence method (under the labelH0-convergence) to the case of perforated domains with a Neumann condition on the holes.

2.3 The asymptotic expansion method

The asymptotic expansion method in homogenization is aformal procedureto derive thelimit problem. One usually makes use of further devices to justify the homogenizationresult. To sketch the method, we reconsider the model problem:

−∇ ·

(A(x

ε)∇uε

)= f in Ω

uε = 0 on ∂Ω,(15)

whereΩ is an open bounded domain inRN with smooth boundary∂Ω, and where weassume that the matrixA is symmetric, coercive, bounded, andY -periodic, withY =(0, 1)N (i.e.,A is 1-periodic in all coordinate directions). To derive the limit problem for(15), we assume that the unknown functionuε ∈ H1

0 (Ω) has a two-scale expansion (withrespect toε) of the form

uε(x) = u0(x,x

ε) + εu1(x,

x

ε) + ε2u2(x,

x

ε) + · · · (16)

where the coefficients functionsui(x, y) areY -periodic with respect to the local variabley = x

ε . Next we plug (16) into (15) to get the formula

ε−2∇y · (A(y)∇yu0(x, y))

+ ε−1(∇y · (A(y)∇yu1(x, y)) +∇y · (A(y)∇xu0(x, y))

+A(y)∇x · ∇yu0(x, y)) (17)

+ ε0(∇y · (A(y)∇yu2(x, y) +A(y)∇xu1(x, y)) +A(y)∇x · ∇yu1(x, y)

+A(y)∇x · ∇xu0(x, y))

+ ε1(· · · ) + · · · = −f(x)|y=xε.

The next step is to compare the coefficients of different powers of ε in the above equation.The terms withε−2 gives

∇y · (A(y)∇yu0(x, y)) = 0 in ∈ Y,

which admits only constant functions asY -periodic solutions. Thus,u0(x, y) = u0(x) (theleading termu0 does not depend on the local variable). Therefore, the equation from theterms withε−1 reduces to

−∇y · (A(y)∇yu1(x, y)) = ∇y · (A(y)∇xu0(x)) in Y

u1 Y -periodic iny.(18)

24 H. Douanla

Since ∫

Y∇y · (A(y)∇xu0(x)) dy = 0,

the equation (18) admits a unique (up to an additive constant) solution by the Fredholmalternative. Now, we rewrite the equation in (18) as follows

−∇y · (A(y)∇yu1(x, y)) =

N∑

j=1

∂

∂yjA(y)

∂u0∂xj

(x) for y ∈ Y.

This suggests we introduce forj = 1, · · · , N the so-calledcell problems−∇y · (A(y)∇yωj(y)) = ∇y · (A(y)ej) in Y

ωj Y -periodic iny,(19)

so that comparing (18) and (19), and using the notationω = (ω1, · · · , ωN ), we obtain

u1(x, y) = ω(y) · ∇xu0(x) + u(x), (20)

whereu(x) is independent ofy. As for the terms withε0, we integrate the resulting equation(from (17)) overY ,

∫

YA(y)∇x · ∇yu1(x, y) dy +

∫

YA(y)∇x · ∇xu0(x) dy = −f(x) in Ω. (21)

Note that some integrals have vanished after integrating bypart therein and using the peri-odicity hypothesis of functions in the integrands. From (20), we have

∇x · ∇yu1(x, y) =

N∑

i,j=1

∂ωj

∂yi(y)

∂2u0∂xixj

,

so that (21) writes

N∑

i,j=1

∫

YA(y)

∂ωj

∂yi(y)

∂2u0∂xi∂xj

dy +

∫

YA(y)∆xu0 dy = −f(x) in Ω.

We introduce the following notation

aij =

∫

YA(y)(δij +

∂ωj

∂yi(y))dy (1 ≤ i, j ≤ N),

with which, it appears thatu0 solves the homogenized equation

−N∑

i,j=1

aij∂2u0∂xi∂xj

= f(x) in Ω

u0 = 0 on ∂Ω.

(22)

Introduction 25

The constant matrixA = (aij)1≤i,j≤N is the homogenized tensor, and is symmetric andpositive definite (this follows from the hypothesis onA). The asymptotic expansion methodappears here to be simple but it is not that simple in practicesince it is based onestablishingtheansatz(16). However, it is usefulto guessthe homogenized equation. As said earlier,a further step is needed to prove the convergence of the sequence(uε)ε∈E to u0. This stepcan be done by means of several methods, such as, theΓ orG-convergence, the oscillatingtest function methods, maximum principle, etc.

We present a justification of the formal expansion (16) by means of the maximum prin-ciple [21]. We further assume that the matrixA and the source termf , henceuε (ε ∈ E)are smooth. To start, we takeu1 as in (20) withu = 0 and put

vε = uε − (u0 + εu1 + ε2u2).

With this notation, we have

−∇ · (A(x

ε)∇vε)

= ε∇y · (A(

x

ε)∇xu2) +∇x · (A(

x

ε)∇yu2) +∇x · (A(

x

ε)∇x(u1 + εu2))

≡ εvε.

Sinceu1 andu2 are smooth we have

|vε| ≤ C in Ω.

On the boundary∂Ω we havevε = −(εu1 + ε2u2). Hence

|vε| ≤ Cε on ∂Ω

The maximum principle yields|vε| ≤ Cε in Ω.

We have proved that|uε − u0| ≤ Cε in Ω.

Therefore, the sequence(uε)ε∈E uniformly converges tou0 in Ω. Generally, it holds that

uε → u0 weakly in H10 (Ω)

asE 3 ε → 0. As extensively discussed in [21], the previous convergence does not holdstrongly inH1

0 (Ω), but with the correctionεu1 it holds (asE 3 ε→ 0) that

[uε(·)− u0(·)− εu1(

·

ε)]→ 0 strongly in H1

0 (Ω).

Unlike theG andΓ-convergence methods, the asymptotic expansion method canonly han-dle linear homogenization problems. Let now discuss the two-scale convergence method.

26 H. Douanla

2.4 The two-scale convergence method

In 1989, Nguetseng [116, 117] introduced a type of convergence which was later (1992)named “two-scale convergence” by Allaire [1]. Nguetseng’stwo-scale convergence methodblends the formal asymptotic expansion and the test function method and give the homog-enization result in one step: it constructs the homogenizedproblem and proves the con-vergence of solutions ofε-problems to the solution of the limit problem, simultaneously.Flagged in its early years as restricted to periodic homogenization, it was extended to thestochastic setting of homogenization in 1994 by Bourgeat, Mikelic, and Wright [28]. Re-cently in 2003, Nguetseng proposed adirectextension of the two-scale convergence method(under the labelΣ-convergence), to a more general setting of deterministic homogenization,which can tackle almost periodic homogenization, weakly almost periodic homogenization,homogenization in Fourier-Stieltjes algebra, etc,just likethe two-scale convergence methodhandles periodic homogenization problems. Unlike the asymptotic expansion method, thetwo-scale convergence method can handle nonlinear homogenization problems. We de-scribe the two-scale convergence method below and present theΣ-convergence method ina further subsection. For a full exposition of the the two-scale convergence method, werecommend [1, 2, 100, 116, 162, 163].

We first introduce some notations.Ω is an open bounded set inRN , andY = (0, 1)N .We write C∞per(Y ) and Cper(Y ) for the space of functions respectively inC∞(RN ) andC(RN ) that areY -periodic. As for Lebesgue spaces, for a given Lebesgue function spaceF (RN ), we denote byFper(Y ) the space of functions inFloc(R

N ) (when it makes sense)that areY -periodic, and byFper(Y )/R the space of those functionsu ∈ Fper(Y ) with∫Y u(y)dy = 0. We will sometimes omit the subscript ‘per’, and write for exampleL2(Y )

in place ofL2per(Y ). Now let1 < p <∞ andp′ = p

p−1 .

Definition 3. A sequence(uε)ε∈E ⊂ Lp(Ω) is said to:

• two-scale converge inLp(Ω) to someu0 ∈ Lp(Ω;Lpper(Y )) (≡ Lp(Ω× Y )) if

limE3ε→0

∫

Ωuε(x)ϕ(x,

x

ε) dx =

∫∫

Ω×Yu0(x, y)ϕ(x, y) dxdy, (23)

for all ϕ ∈ Lp′(Ω; Cper(Y )). This is expressed in the sequel byuε2s−→ u0 in Lp(Ω).

• strongly two-scale converge inLp(Ω) to someu0 ∈ Lp(Ω × Y ) if it two-scale con-verges tou0 in Lp(Ω) and

limE3ε→0

‖uε‖Lp(Ω) = ‖u0‖Lp(Ω×Y ).

Before formulating the compactness theorems that give sense to Definition 3, we givesome examples of two scale convergent sequences.

1. Letu0 ∈ Lp(Ω; Cper(Y )) and defineuε(x) = u0(x,xε ) for ε ∈ E. Then, the sequence

of traces(uε)ε∈E is well defined and two-scale converges tou0 in Lp(Ω). However,

Introduction 27

the sequence(vε)ε∈E defined byvε(x) = u0(x,xε2) two-scale converges inLp(Ω) to

its weakLp(Ω) limit:∫Y u0(·, y) dy. The oscillations are lost in the limit since they

are not in resonance with those of the test function.

2. Assume thatuε : Ω→ R (ε ∈ E) admits the following asymptotic expansion

uε(x) = u0(x,x

ε) + εu1(x,

x

ε) + ε2u2(x,

x

ε) + · · · ,

where theui belong toLp(Ω; Cper(Y )) . Then,

uε2s−→ u0 in Lp(Ω)

anduε(·)− u0(·,

·ε)

ε

2s−→ u1 in Lp(Ω).

3. If the sequence(uε)ε∈E strongly converges tou0 in Lp(Ω), thenuε2s−→ u0 in Lp(Ω).

The concept of two-scale convergence would be useless if thefollowing compactness resultswere not available.

Theorem 5. Let 1 < p < +∞. Any sequence(uε)ε∈E , bounded inLp(Ω), admits asubsequence which two-scale converges inLp(Ω).

Theorem 6. Let 1 < p < +∞ and assume that the sequence(uε)ε∈E is bounded inW 1,p(Ω). Then, a subsequenceE′ can be extracted fromE such that, asE′ 3 ε→ 0,

uε → u0 weakly inW 1,p(Ω), (24)

uε2s−→ u0 in Lp(Ω), (25)

∂uε∂xi

2s−→

∂u0∂xi

+∂u1∂yi

in Lp(Ω) (1 ≤ i ≤ N), (26)

whereu0 ∈W 1,p(Ω) andu1 ∈ Lp(Ω;W 1,pper(Y )/R).

The original proof of Theorem 5 is by Nguetseng [116] and usesadvanced distribu-tion theory and some density arguments. A simpler proof (similar to that of the existenceof Young measures by Ball [16]) was proposed by Allaire [1] in1992. Allaire [1] furtherdeveloped the two-scale convergence method and applied it to various periodic homoge-nization problems. The following proposition (see e.g., [122]) is of capital interest whenutilizing the two-scale convergence method.

Proposition 1. If the sequence(uε)ε∈E two-scale converges to someu0 ∈ Lp(Ω×Y ), then(23) holds for anyϕ ∈ C(Ω;L∞

per(Y )).

28 H. Douanla

We apply the two-scale convergence method, by way of illustration, to the followinglocally periodic model problem

−∇ ·

(A(x,

x

ε)∇uε

)= f in Ω

uε = 0 on ∂Ω.(27)

The matrixA ∈ C(Ω;L∞per(Y )N×N ) satisfies for almost everyy ∈ RN and for allx ∈ Ω

α|ξ|2 ≤ (A(x, y)ξ, ξ) ≤ β|ξ|2 for all ξ ∈ RN .

With the boundedness of the sequence(uε)ε∈E in H10 (Ω), Theorem 6 applies, and there is

a subsequenceE′ of E such that

uε → u0 weakly inH10 (Ω), and ∇xuε

2s−→ ∇xu0 +∇yu1 in L2(Ω× Y )N , (28)

asE′ 3 ε→ 0, for some(u0, u1) ∈ H10 (Ω)× L

2(Ω;H1per(Y )/R). Let (ϕ,ψ1) ∈ D(Ω)×

D(Ω)⊗ C∞per(Y )/R, and define

ψε(x) = ϕ(x) + εψ1(x,x

ε) (ε > 0, x ∈ Ω).

We have∫

Ω

A(x,x

ε)∇xuε(x) ·

(∇xϕ(x) +∇yψ1(x,

x

ε) + ε∇xψ1(x,

x

ε))dx =

∫

Ω

f(x)ψε(x)dx.

A limit passage asE′ 3 ε→ 0 (use Proposition 1 in the left-hand side) yields

∫∫

Ω×YA(x, y) [∇xu0 +∇yu1] · [∇xϕ+∇yψ1] dxdy =

∫

Ωf(x)ϕ(x)dx,

which, by density, still holds for(ϕ,ψ1) in H10 (Ω) × L

2(Ω;H1per(Y )/R), a Hilbert space

when endowed with the norm

‖(ϕ,ψ1)‖ =(‖∇xϕ‖

2L2(Ω) + ‖∇yψ1‖

2L2(Ω×Y )

) 1

2

.

SinceA is coercive, we claim by the Lax-Milgram lemma that(u0, u1) is the unique solu-tion to

(u0, u1) ∈ H10 (Ω)× L

2(Ω;H1per(Y )/R)∫∫

Ω×YA(x, y) [∇xu0 +∇yu1] · [∇xϕ+∇yψ1] dxdy =

∫

Ωf(x)ϕ(x) dx

for all (ϕ,ψ1) ∈ H10 (Ω)× L

2(Ω;H1per(Y )/R).

(29)

Introduction 29

Hence, (28) holds for the whole sequenceE. We now derive the macroscopic problem.Taking(ϕ,ψ1) in (29) such thatϕ = 0, yields

−∇y · (A(x, y)∇yu1(x, y)) = ∇y · (A(x, y)∇xu0(x)) in Ω× Y

y → u1(x, y) Y -periodic.

Therefore, we can representu1 in terms ofu0 as follows,

u1(x, y) =N∑

i=1

∂u0∂xi

(x)ωi(x, y) a.e. inΩ× Y, (30)

whereωi (i = 1, · · · , N) is defined, at each pointx, as the unique solution inH1per(Y )/R

to the cell problem−∇y · (A(x, y)∇yωi(x, y)) = ∇y · (A(x, y)ei) in Y

y → ωi(x, y) Y -periodic.

Now, we take take(ϕ,ψ1) in (29), such thatψ1 = 0 and use (30) to get, after easy calcula-tions, the so-called macroscopic problem

−∇ · (A∗(x)∇u0(x)) = f(x) in Ω

u0 = 0 on∂Ω,

where the homogenized diffusion tensor is given by

A∗i,j(x) =

∫

yA(x, y)[ei+∇yωi(x, y)]·[ej+∇yωj(x, y)] dy (x ∈ Ω, 1 ≤ i, j ≤ N).

Like in Subsection 2.3, corrector type results can be obtained by means of the two-scaleconvergence method, namely, asE 3 ε→ 0, we have

[uε(·)− u0(·)− εu1(·,

·

ε)]→ 0 strongly inH1

0 (Ω).

It should be noted that, unlike all the homogenization methods presented in previoussubsections, the two-scale convergence exhibits the weak limit of the gradients,∇uε, thatis exactly the local behavior ofuε, which is interesting from the physical point of view.However, with the two-scale convergence method, the homogenization procedure is usuallystraightforward that one cannot obtain convergence rates type results, which measure themagnitude of the error we commit when replacing the solutionto theε-problem by that ofthe homogenized equation. Other extensions of the two-scale convergence method can befound in [3, 37, 75, 77, 82, 129, 134, 156]

In 1990, Arbogast, Douglas, and Hornung [5] developed an idea similar to the two-scaleconvergence. They defined a so-calleddilation operator to study homogenization problemsin porous media. That dilation operator has been also used for example in [27, 36, 94, 95,

30 H. Douanla

113]. Later on, Cioranescu, Damlamian, and Griso [43, 44] expanded and formalized thatidea, and introduced theperiodic unfolding methodin homogenization, a dual formulationof the two-scale convergence method. Before we present Nguetseng’s approach to generaldeterministic homogenization (theΣ-convergenve method), we briefly present the unfold-ing operator and formulate the dual definition of the two-scale convergence, which is thecornerstone of the periodic unfolding method.

2.5 The periodic unfolding method

For an extensive presentation and some applications of the unfolding method in periodichomogenization, we refer to e.g., [41, 43, 44, 47, 48, 52, 73,74, 113]. Loosely speaking,the main ingredient of the unfolding method in periodic homogenization is the unfoldingoperator which transforms a function of one variable into a function of two variables, mak-ing it possible to reduce the concept of two scale-convergence to usual notions of weak andstrong convergence inLp(Ω× Y ).

LetΩ be an open set inRN and letY = [0, 1)N . More generallyY can be replaced byanN -dimensional parallelepiped

Y = λ1b1 + · · ·+ λN bN : 0 ≤ λi < 1, i = 1, · · · , N,

whereb1, · · · , bN ∈ RN is anN -tuple of independent vectors. The cellY has thepavingproperty, i.e., the collectionYξ = Y + ξ : ξ ∈ Ξ of cellsY shifted by a vectorξ from aset of shifts

Ξ = ξ = k1b1 + · · ·+ kN bN : k1, · · · , kN ∈ Z

is a partition ofRN : the shifted cellsYξ are disjoint and coverRN . For z ∈ RN , let [z]Ydenotes the unique integer combination

∑Nj=1 kjbj such thatz− [z]Y belongs toY , and set

zY = z − [z]Y . In the case whenY = [0, 1)N , we haveΞ = ZN and the decompositionz = [z]Y + zY is the usual decomposition into the integer and fractional parts. Then, forx ∈ RN andε > 0, we have

x = ε([xε

]Y+xε

Y

).

We introduce the following notations

Ξε = ξ ∈ ZN : ε(Y +ξ) ⊂ Ω, Ωε = Interior

∪ξ∈Ξε

ε(Y + ξ), Λε = Ω\Ωε. (31)

With theses notations we can now define the unfolding operator.

Definition 4. For u Lebesgue measurable onΩ, the unfolding operator is defined a follows,

Tεu(x, y) =

u(ε[xε

]Y+ εy

)a.e. for(x, y) ∈ Ωε × Y,

u(x) a.e. for(x, y) ∈ Λε × Y,(32)

Introduction 31

Remark 1. The unfolding operatorTε defined by (32) for eachε > 0 has the followingproperties:

1. Tε(f) is Lebesgue measurable onΩ× Y ,

Tε(αf + βg) = αTε(f) + βTε(g),

Tε(fg) = Tε(f)Tε(g), α, β ∈ R, f, g ∈ L1(Ω);

2. integral conservation

1

|Y |

∫∫

Ω×Y(Tεu)(x, y) dxdy =

∫

Ωf(x) dx, f ∈ L1(Ω). (33)

The primary definition of the unfolding operator has been revisited by Francu [73,74]. The primary definition by Cioranescu, Damlamian and Griso [43, 44] requires thatTε(u)(x, y) = 0 almost everywhere for(x, y) ∈ Λε × Y , which prevents theintegral con-servationproperty (33) to holds, making some proofs in the unfolding method in homog-enization needlessly technical. By means of the unfolding operator we can now formulatethe dual definition of the two scale convergence.

Definition 5. LetE be as above.

1. The sequence(uε)ε∈E is said to two-scale converge tou0 in Lp(Ω), if the sequence(Tεuε)ε∈E converges tou0 weakly inLp(Ω× Y ).

2. The sequence(uε)ε∈E is said to strongly two-scale converge tou0 in Lp(Ω), if thesequence(Tεuε)ε∈E converges tou0 strongly inLp(Ω× Y ).

Indeed, it is an easy exercise to check that this definition agrees with Definition 3, a keyobservation is that

Tε[ϕ(·,·

ε)](x, y) = ϕ

(ε[xε

]Y+ εy, y

)for ϕ ∈ Lp(Ω; Cper(Y )) (1 < p <∞).

2.6 TheΣ-convergence method

Introduced in 2003 by Nguetseng [119], theΣ-convergence method is the generalization ofthe two scale convergence method which can handle general deterministic (beyond the pe-riodic setting) homogenization problems in the same way thetwo-scale convergence tackleperiodic homogenization problems. The main ingredient in this extension is the Gelfandrepresentation theory of bounded uniformly continuous functions. We first give a shortintroduction to the concept of algebra with mean value whichwas introduced in 1983 byZhikov and Krivenko [165] to study the problem of averaging of partial differential equa-tions with almost periodic coefficients. Nguetseng [118, 119, 120, 121] has formalized anddeveloped this concept of algebra with mean value which turns out to be the right candidate

32 H. Douanla

for replacement ofCper(Y ) when moving from periodic homogenization to general de-terministic homogenization. However, Nguetseng sometimes utilizes the slightly differentconcept ofH-algebra. It is worth mentioning that in their paper [165], Zhikov and Krivenkoalready addressed the question of embedding the theory of deterministic homogenizationinto the theory of stochastic homogenization. We quote themwithout further comments:“ . . . if the coefficientsaαβ(x) are realizations of a homogenous stochastic field, then theyare not necessarily almost-periodic in the sense of Besicovitch and have only rather simpleand general properties of the type of existence of mean values”.

Definition 6. A closed subalgebraA of theC*-algebra of bounded uniformly continuousfunctionsBuc(RN ) is analgebra with mean valueonRN if it satisfies the following proper-ties:

• A contains the constants,

• A is translation invariant,u(·+ a) ∈ A for anyu ∈ A and eacha ∈ RN ,

• for anyu ∈ A, the sequence(uε)ε>0 (uε(x) = u(xε ), x ∈ RN ) weakly∗-converges

in L∞(RN ) to some constant real numberM(u) asε→ 0 (the real numberM(u) iscalled the mean value ofu ∈ A).

Endowed with the sup norm topology,A is a commutativeC*-algebra with identity (thisis a classical result of the theory of commutative Banach algebras). We therefore denote by∆(A) its spectrum and byG the Gelfand transform onA. We recall that∆(A) (whichis a subset of the topological dualA′ of A) is the set of all nonzero multiplicative linearfunctionals onA, andG is the mapping ofA into C(∆(A)) such thatG(u)(s) = 〈s, u〉(s ∈ ∆(A)), where〈 , 〉 denotes the duality pairing betweenA′ andA. Equipped withthe relative weak∗ topology onA′, ∆(A) is a compact topological space, and the GelfandtransformG is an isometric∗-isomorphism identifying theC*-algebrasA andC(∆(A)). Inaddition, the mean valueM defined onA is a nonnegative continuous linear functional, andthere exists a Radon measureβ of total mass1 such that the following integral representationholds [119]:

M(u) =

∫

∆(A)G(u)dβ (u ∈ A).

We define Besicovitch spaces and some Sobolev type spaces. Let A be an algebra withmean value and letm ∈ N. We define the following subspaces ofA,

Am = ψ ∈ Cm(RN ) : Dαy ψ ∈ A, ∀α = (α1, ..., αN ) ∈ NN with |α| ≤ m,

whereDαyψ = ∂|α|ψ/∂yα1

1 · · · ∂yαN

N . Endowed with the norm

‖|u|‖m = sup|α|≤m

∥∥Dαyψ∥∥∞

(u ∈ Am),

the spaceAm is a Banach space. We also define

A∞ = ψ ∈ C∞(RN ) : Dαyψ ∈ A ∀α = (α1, ..., αN ) ∈ NN,

Introduction 33

a Frechet space when endowed with the locally convex topology defined by the family ofnorms‖|·|‖m ,m ∈ N. We consider the Besicovitch spaceB2

A(RN ) defined as the closure

of A with respect to the (Besicovitch) seminorm

‖u‖2 =

(lim supr→+∞

1

|Br|

∫

Br

|u(y)|2 dy

) 1

2

,

whereBr is the open ball ofRN of radiusr centered at the origin. It is well-known thatB2

A(RN ) is a complete seminormed vector space which, in general, is not separated since

u 6= 0 does not entail‖u‖2 6= 0. However, the following properties hold [119, 123, 144].

(1) The Gelfand transformG : A → C(∆(A)) extends by continuity to a unique contin-uous linear mapping (still denoted byG) of B2

A(RN ) into L2(∆(A)), which induces

an isometric isomorphismG1 of B2A(R

N )/N = B2A(RN ) onto L2(∆(A)) (where

N = u ∈ B2A(R

N ) : G(u) = 0). Moreover, ifu ∈ B2A(R

N ) ∩ L∞(RN ) thenG(u) ∈ L∞(∆(A)) and‖G(u)‖L∞(∆(A)) ≤ ‖u‖L∞(RN ).

(2) The mean valueM defined onA, extends by continuity to a positive continuouslinear form (still denoted byM ) onB2

A(RN ) satisfyingM(u) =

∫∆(A) G(u)dβ (u ∈

B2A(R

N )), andM(τau) = M(u) for eachu ∈ B2A(R

N ) and alla ∈ RN , whereτau(y) = u(y + a) for almost ally ∈ RN . Furthermore, foru ∈ B2

A(RN ) we have

‖u‖2 =[M(|u|2)

]1/2, and foru + N ∈ B2A(R

N ), we define its mean value (which

we still denoted byM ) as expected:M(u+N ) =M(u).

Let us justify the following important equality:

‖u‖2 =[M(|u|2)

]1/2for u ∈ B2

A(RN ).

Let r be a positive real number. Setε = 1/r. Then,ε→ 0 asr→ +∞. Sinceuε →M(u)in L∞(RN )-weak∗, we have

∫u1/rχB1

dx → M(u) |B1| asr → +∞, whereB1 denotesthe unit open ball inRN andχB1

the characteristic function ofB1. But∫u1/rχB1

dx =∫B1u(rx)dx, and a change of variabley = rx yields

1

|B1|

∫

B1

u(rx)dx =1

rN |B1|

∫

Br

u(y)dy =1

|Br|

∫

Br

u(y)dy,

hence our claim is justified. We define Sobolev type spaces associated to the algebraA. Todo this, we first consider theN -parameter group of isometriesT (y) : y ∈ RN defined by

T (y) : B2A(RN ) → B2A(R

N )

u+N 7→ T (y)(u+N ) = τyu+N (u ∈ B2A(R

N )).

34 H. Douanla

Since the elements ofA are uniformly continuous,T (y) : y ∈ RN is a strongly contin-uous group inL(B2A(R

N ),B2A(RN )) (the Banach space of continuous linear functionals of

B2A(RN ) into B2A(R

N )), viz:

T (y)(u+N )→ u+N in B2A(RN ) as |y| → 0.

Likewise, theN -parameter groupT (y) : y ∈ RN defined by

T (y) : L2(∆(A)) → L2(∆(A))

G1(u+N ) 7→ T (y)G1(u+N ) = G1(T (y)(u+N )) (u ∈ B2A(R

N )),

is also strongly continuous. The infinitesimal generator ofT (y) (resp.T (y)) along thei-thcoordinate direction, denoted by∂/∂yi (resp.∂i), is defined as follows

∂u

∂yi= lim

t→0

(T (tei)u− u

t

)in B2A(R

N )

(resp. ∂iv = lim

t→0

(T (tei)v − v

t

)in L2(∆(A))

),

where, for the sake of simplicity, the equivalence classu+N of an elementu ∈ B2A(R

N )is still denoted byu, andei = (δij)1≤j≤N (δij is the Kroneckerδ). Indeed, this is justifiedby

∂(u+N )

∂yi= lim

t→0

(T (tei)(u+N )− (u+N )

t

)= lim

t→0

((τteiu+N )− (u+N )

t

)

= limt→0

%

(τteiu− u

t

),

where% denotes the canonical mapping ofB2A(R

N ) ontoB2A(RN ), that is,%(u) = u+ N

for u ∈ B2A(R

N ). The domain of∂/∂yi (resp.∂i) in B2A(RN ) (resp.L2(∆(A))) is denoted

byDi (resp.Wi). The following result is just a consequence of the semigroup theory; seee.g., [68, Chap. VIII, Section 1].

Proposition 2. Di (resp.Wi) is a vector subspace ofB2A(RN ) (resp.L2(∆(A))), ∂/∂yi :

Di → B2A(R

N ) (resp. ∂i : Wi → L2(∆(A))) is a linear operator,Di (resp.Wi) is densein B2A(R

N ) (resp. L2(∆(A))), and the graph of∂/∂yi (resp. ∂i) is closed inB2A(RN ) ×

B2A(RN ) (resp.L2(∆(A)) × L2(∆(A))).

For1 ≤ i ≤ N andu ∈ A1, we have%(u) ∈ Di and

∂

∂yi(%(u)) = %

(∂u

∂yi

), (34)

which shows that∂/∂yi % = % ∂/∂yi. We can also define higher order derivatives by

settingDαy = ∂

α1

∂yα1

1

· · · ∂αN

∂yαNN

(resp.∂α = ∂α1

1 · · · ∂αN

N ) for α = (α1, ..., αN ) ∈ NN

Introduction 35

with ∂αi

∂yαii

= ∂∂yi · · · ∂

∂yi, αi-times. The spaces of smooth functions onA and on∆(A)

are respectively defined by

DA(RN ) = %(A∞) andD(∆(A)) = G1(DA(R

N )).

SinceG1 % = G, it holds thatD(∆(A)) = G(A∞). Likewise, foru ∈ A∞ we haveD

αy %(u) = %(Dα

y u) for anyα ∈ NN (this is the generalization of (34)), hence∂αG(u) =

G(Dαy u). Due to the density ofA∞ in B2

A(RN ) (A∞ is dense inA since the elements of

A are uniformly continuous andA is translation invariant), the spaceDA(RN ) is dense in

B2A(RN ) . For anyu ∈ Di we haveG1(u) ∈ Wi and

G1(∂u/∂yi) = ∂iG1(u),

which, together with (34) show thatDA(RN ) ⊂ Di (henceD(∆(A)) ⊂ Wi) for any1 ≤

i ≤ N . From now on, we writeu either forG1(u) if u ∈ B2A(RN ) or for G(u) if u ∈

B2A(R

N ). The following properties hold [160, Proposition 4]:

(i)∫∆(A) ∂

αudβ = 0 for all u ∈ DA(RN ) andα ∈ NN ,

(ii)∫∆(A) ∂iudβ = 0 for all u ∈ Di and1 ≤ i ≤ N ,

(iii) ∂∂yi

(uφ) = u ∂φ∂yi

+ φ ∂u∂yi

for all (φ, u) ∈ DA(RN )×Di and1 ≤ i ≤ N .

Formulae (i) and (iii), above imply∫

∆(A)φ∂iudβ = −

∫

∆(A)u∂iφdβ for all (u, φ) ∈ Di ×DA(R

N ),

which is a hint for the adaptation of the concepts of distribution and weak derivative to ourfunctional setting. We endowDA(R

N ) = %(A∞) with its natural topology defined by thefamily of normsNm(u) = sup|α|≤m supy∈RN

∣∣Dαyu(y)

∣∣, m ∈ N, which makesDA(RN )

a Frechet space. The topological dual ofDA(RN ) is denoted in the sequel byD′

A(RN )

and is endowed with the strong dual topology. The elements ofD′A(R

N ) are calledthedistributions onA. Forf ∈ D′

A(RN ), we define the weak derivativeDαf (α ∈ NN ) of f

as expected, viz:

〈Dαf, φ〉 = (−1)|α|⟨f,D

αyφ⟩

for all φ ∈ DA(RN ).

SinceDA(RN ) is dense inB2A(R

N ), it is immediate thatB2A(RN ) ⊂ D′

A(RN ) with contin-

uous embedding so that we can define the weak derivative off ∈ B2A(RN ). It satisfies the

following functional equation:

〈Dαf, φ〉 = (−1)|α|∫

∆(A)f∂αφdβ for all φ ∈ DA(R

N ).

36 H. Douanla

In the case wheref ∈ Di, we have

−

∫

∆(A)f∂iφdβ =

∫

∆(A)φ∂if dβ for all φ ∈ DA(R

N ),

and∂f/∂yi is identified withDαif , αi = (δij)1≤j≤N . Conversely, iff ∈ B2A(RN ) is such

that there existsfi ∈ B2A(RN ) with 〈Dαif, φ〉 = −

∫∆(A) fiφdβ for all φ ∈ DA(R

N ), then

f ∈ Di and∂f/∂yi = fi. This being done, we define the following Sobolev type space,

B1,2A (RN ) = ∩Ni=1Di ≡

u ∈ B2A(R

N ) :∂u

∂yi∈ B2A(R

N ) for all 1 ≤ i ≤ N

,

a Hilbert space under the norm

‖u‖B1,2A

(RN ) =

(‖u‖22 +

N∑

i=1

∥∥∥∥∂u

∂yi

∥∥∥∥2

2

)1

2

(u ∈ B1,2A (RN )).

Likewise, we defineW 1,2(∆(A)) = ∩Ni=1Wi as expected and endow it with its naturalnorm. The spaceDA(R

N ) (resp. D(∆(A))) is obviously a dense subspace ofB1,2A (RN )(resp.W 1,2(∆(A))).

With the Sobolev spaces defined, we proceed and recall some facts aboutergodicalge-bras. A functionf ∈ B2A(R

N ) is said to beinvariant if for any y ∈ RN , T (y)f = f .

Definition 7. An algebra with mean valueA is said to beergodicif every invariant functionf ∈ B2A(R

N ) is constant inB2A(RN ).

The set of invariant functionsf ∈ B2A(RN ) is denoted in the sequel byI2A, and is a

closed vector subspace ofB2A(RN ) satisfying the following property (see e.g., [28]):

f ∈ I2A if and only if∂f

∂yi= 0 for all 1 ≤ i ≤ N . (35)

Let u ∈ B2A(RN ) (resp.v = (v1, ..., vN ) ∈ (B2A(R

N ))N ). We define the gradient operator∇yu and the divergence operatordiv yv by

∇yu =

(∂u

∂y1, ...,

∂u

∂yN

)and div yv =

N∑

i=1

∂vi∂yi

.

The Laplace operator is defined by∆yu = div y(∇yu), and we have∆y%(u) = %(∆yu) forall u ∈ A∞, where∆y denotes the usual Laplace operator onRN

y . The following propertiesare satisfied.

Introduction 37

• The divergence operatordiv y sends the space(B2A(RN ))N continuously and linearly

into the space(B1,2A (RN ))′, and

⟨div yu, v

⟩= −

⟨u,∇yv

⟩for v ∈ B1,2A (RN ) and u = (ui) ∈ (B2A(R

N ))N , (36)

where⟨u,∇yv

⟩=

N∑

i=1

∫

∆(A)ui∂ivdβ.

• Taking in (36)u = ∇yw with w ∈ B1,2A (RN ), we obtain

⟨∆yw, v

⟩=⟨div y(∇yw), v

⟩= −

⟨∇yw,∇yv

⟩=⟨w,∆yv

⟩

for all v,w ∈ B1,2A (RN ).

The gradient operator∂ = (∂1, ..., ∂N ), the divergence operatordiv , and the Laplaceoperator∆ on L2(∆(A)) are defined similarly by replacingB2A by L2(∆(A)), ∇y by∂ = (∂1, ..., ∂N ), div y by div and∆y by ∆ in the above definitions. Moreover, similarproperties to the above ones hold, and we have the following relations between the justdefined operators (when they make sense):

∂ = G1 ∇y, div = G1 div y and ∆ = G1 ∆y.

Now, we recall the definition and main compactness results oftheΣ-convergence method.We eventually give some examples of ergodic algebras with mean value.

Definition 8. A sequence(uε)ε∈E ⊂ L2(Ω) is said to :

• Σ-converge inL2(Ω) to someu0 ∈ L2(Ω;B2A), if asE 3 ε→ 0 we have

∫

Ωuε(x)f

(x,x

ε

)dx→

∫∫

Ω×∆(A)u0(x, s)f(x, s) dxdβ (37)

for everyf ∈ L2(Ω;A), and wheref = G1 (% f) = G f . We express this bywriting uε → u0 in L2(Ω)-weakΣ;

• stronglyΣ-converge inL2(Ω) to someu0 ∈ L2(Ω;B2A), if it Σ-converges tou0 inL2(Ω) and

limE3ε→0

‖uε‖L2(Ω) = ‖u0‖L2(Ω×∆(A)),

whereu0 = G1 u0. We express this by writinguε → u0 in L2(Ω)-strongΣ

38 H. Douanla

Remark 2. Due to the equalityG1(B2A) = L2(∆(A)), the right-hand side of (37) is equalto ∫

ΩM(u0(x, ·)f(x, ·)) dx.

Therefore, as expected,uε → u0 in L2(Ω)-weakΣ impliesuε → M(u0(x, ·)) in L2(Ω)-weak. In particular, whenA = Cper(Y ), we have (up to a group isomorphism)∆(A) = TN ,theN -dimensional torus. Thus, (37) reduces to

∫

Ωuε(x)f

(x,x

ε

)dx→

∫∫

Ω×Yu0(x, y)f(x, y) dxdy

whereu0 ∈ L2(Ω× Y ), which is the definition of the two-scale convergence.

The following compactness results are of great interest in practice.

Theorem 7. Any bounded sequence(uε)ε∈E in L2(Ω) admits a subsequence whichΣ-converges inL2(Ω).

Theorem 8. LetΩ be an open subset inRN . LetA be an ergodic algebra with mean valueonRN , and let(uε)ε∈E be a bounded sequence inH1(Ω). There exist a subsequenceE′ ofE and a pair(u0, u1) ∈ H1(Ω)× L2(Ω;B1,2A (RN )) such that

uε → u0 in H1(Ω)-weak, (38)

∂uε∂xj→

∂u0∂xj

+∂u1∂yj

in L2(Ω)-weakΣ (1 ≤ j ≤ N), (39)

asE′ 3 ε→ 0.

We end this subsection with some examples of algebras with mean value.

The periodic algebra with mean value. Let A = Cper(Y ) be the algebra ofY -periodiccontinuous functions onRN . It is classical thatA is an ergodic algebra with meean value.The mean of a functionu ∈ Cper(Y )) is given by

M(u) =

∫

Yu(y) dy.

The almost periodic algebra with mean value. Let AP (RN ) be the algebra of Bohrcontinuous almost periodic functionsRN . We recall that a functionu ∈ B(RN ) is inAP (RN ) if the set of translatesτau : a ∈ RN is relatively compact inB(RN ) (theC∗-algebra of bounded continuous functions onRN ). Equivalently [24],u ∈ AP (RN ) ifand only ifu may be uniformly approximated by finite linear combinationsof functions inthe setγk : k ∈ RN, whereγk(y) = exp(2iπk · y) (y ∈ RN ). It is also a classical

Introduction 39

result thatA is an ergodic algebra with mean value ([165]). The mean valueof a functionu ∈ AP (RN ) is given by

M(u) =

∫

KG(u)dβ,

whereK is the spectrum ofAP (RN ), which more precisely, is the Bohr compactificationof RN andβ is the Haar measure on the compact topological abelian groupK.

LetR be any subgroup ofRN . We denote byAPR(RN ) the space of those functions

in AP (RN ) that can be uniformly approximated by finite linear combinations in the setγk : k ∈ R. ThenA = APR(R

N ) is an ergodic algebra with mean value [144].

The algebra with mean value of convergence at infinity.LetB∞(RN ) denote the space ofall continuous functions onRN that converge finitely at infinity, that is the space of allu ∈B(RN) such thatlim|y|→∞ u(y) ∈ R. The spaceB∞(RN ) is an ergodic algebra with meanvalue [70]. The mean value of a functionu ∈ B∞(RN ) is given byM(u) = lim|y|→∞ u(y).

The weakly almost periodic algebra with mean value.The concept of weakly almostperiodic function is due to Eberlein [70]. A continuous function u onRN is weakly almostperiodic if the set of translatesτau : a ∈ RN is relatively weakly compact inC(RN ).Endowed with the sup norm topology,WAP (RN ) is a Banach algebra with the usual mul-tiplication. As examples of Eberlein’s functions we have the Bohr continuous almost peri-odic functions, the continuous functions vanishing at infinity, the positive definite functions(hence Fourier-Stieltjes transforms).WAP (RN ) is a translation invariantC∗-subalgebra ofC(RN ) whose elements are uniformly continuous, bounded and possess a mean value

M(u) = limR→+∞

1

|BR|

∫

BR

u(y + a)dy (u ∈WAP (RN )),

the convergence being uniform ina ∈ RN . Moreover, everyu ∈ WAP (RN ) admitsthe unique decompositionu = v + w, v being a Bohr almost periodic function andw acontinuous function with zero quadratic mean value:M(|w|2) = 0. Hence denoting byW0(R

N ) the complete vector subspace ofWAP (RN ) consisting of functions with zeroquadratic mean value, we have

WAP (RN ) = AP (RN )⊕W0(RN ).

With this in mind, letR be a subgroup ofRN and set

WAPR(RN ) = APR(R

N )⊕W0(RN ) (40)

(bear in mind thatWAPR(RN ) = WAP (RN ) whenR = RN ). ThenWAPR(R

N ) is anergodic algebra with mean value [144].

Remark 3. • In practice, it is possible to sum (perturbed) algebras withmean value [119]to get the algebra with mean value that exactly meets the structural hypothesis of the ho-mogenization problem under consideration.

40 H. Douanla

• A generalisation of Theorem 8 without the ergodicity hypothesis has been proved [66,Theorem 3.9]. In this case, the “macroscopic” limit function u0 belongs toH1(Ω; I2A) anddepend on the “local” variable. As already pointed out by Zhikov andKrivenko [165],the problem of averaging in nonergodic algebra is tricky andstill open.

3 Overview of Papers

During the first part of my doctoral studies, I investigated the asymptotic behavior of lin-ear elliptic eigenvalue problems in perforated domains by means of the two-scale con-vergence method and theΣ-convergence method. The results obtained are published in[59, 60, 61, 62, 64]. It should be noted that many nonlinear problems lead, after lineariza-tion, to elliptic eigenvalue problems (see e.g., the surveypaper by de Figueiredo [72] andthe work of Hess and Kato [78, 79]). A vast literature in engineering, physics and appliedmathematics deals with such problems arising, for instance, in the study of transport the-ory, reaction-diffusion equations and fluid dynamics. Generally speaking, spectral asymp-totics is a two folded research area. On the one hand, it dealswith asymptotic formulas(estimates) and asymptotic distribution of the eigenvalues, see e.g. [25, 140, 141] and thereferences therein. On the other hand, it is concerned with homogenization of eigenvaluesof operators with oscillating coefficients on possibly varying domains such as perforatedones. My study falls within the second framework: the homogenization theory. The asymp-totics of eigenvalue is a very important problem and has beenwidely explored (see e.g,[11, 23, 38, 87, 89, 90, 110, 111, 112, 127, 130, 155] and the references therein). In a fixeddomain, the homogenization of spectral problems with point-wise positive density functiongoes back to Kesavan [89, 90]. In perforated domains, the eigenvalue asymptotics was firstconsidered by Rauch and Taylor [136, 137] but the first homogenization result in that di-rection pertains to Vanninathan [155]. Before presenting my results, it is worth pointingout that I did not only address new problems in spectral asymptotics but I also utilized thetwo-scale convergence method (which dramatically simplified my proofs) for the first timefor such problems, albeit I missed results of convergence ofrate type.

3.1 Homogenization of Steklov spectral problems with indefinite density func-tion in perforated domains, Acta Applicandae Mathematicae, 123 (2013)261-284.

In this paper, I studied the asymptotic behavior of second order self-adjoint elliptic Stekloveigenvalue problems with periodic rapidly oscillating coefficients and with indefinite (sign-changing) density function in periodically perforated domains. To be more precise, letΩbe a bounded domain inRN , with integerN ≥ 2 and letT ⊂ Y = (0, 1)N be a compactsubset ofY in RN . We assume thatΩ andT haveC1 boundaries∂Ω and∂T , respectively.Forε > 0, we define

tε = k ∈ ZN : ε(k + T ) ⊂ Ω, T ε =⋃

k∈tε

ε(k + T ) and Ωε = Ω \ T ε.

Introduction 41

We denoteY \T by Y ∗ andn = (ni)Ni=1 stands for the outer unit normal vector to∂T with

respect toY ∗. Theε-problem reads

−N∑

i,j=1

∂

∂xi

(aij(

x

ε)∂uε∂xj

)= 0 in Ωε

N∑

i,j=1

aij(x

ε)∂uε∂xj

ni(x

ε) = ρ(

x

ε)λεuε on ∂T ε

uε = 0 on ∂Ω,

(41)

wherea = (aij) ∈ L∞(RN )N×N is symmetric, elliptic andY -periodic. The densityfunction ρ ∈ Cper(Y ) changes sign on∂T , that is, both the setsy ∈ ∂T, ρ(y) < 0 andy ∈ ∂T, ρ(y) > 0 are of positive surface measure. I proved that the spectrum of problem(4) is discrete and consists of two infinite sequences, one tending to−∞ and another to+∞, viz,

−∞← λn,−ε ≤ · · · ≤ λ2,−ε ≤ λ1,−ε < 0 < λ1,+ε ≤ λ2,+ε ≤ · · · ≤ λn,+ε → +∞.

The limiting behavior of positive and negative eigencouples crucially depends on whetherthe average of the weight over the surface of the reference hole

M∂T (ρ) =

∫

∂Tρ(y)dσ(y),

is positive, negative or equal to zero. To handle the caseM∂T (ρ) = 0, I proved the followingconvergence result [59, Lemma 2.9].

Lemma 1. Let (uε)ε∈E ⊂ H1(Ω) be such that, asE 3 ε→ 0,

uε2s−→ u0 in L2(Ω),

∂uε∂xj

2s−→

∂u0∂xj

+∂u1∂yj

in L2(Ω) (1 ≤ j ≤ N),

for someu0 ∈ H1(Ω) andu1 ∈ L2(Ω;H1per(Y )). Then

limε→0

∫

∂T ε

uε(x)ϕ(x)θ(x

ε)dσε(x) =

∫∫

Ω×∂Tu1(x, y)ϕ(x)θ(y)dxdσ(y)

for all ϕ ∈ D(Ω) andθ ∈ Cper(Y ) with∫∂T θ(y)dσ(y) = 0.

The results obtained in this paper generalize previous results by Pastukhova [130] andmyself [61] where the same problem was considered (withC∞ andL∞ coefficients, respec-tively) but with a constant positive density, hence dealingonly with one positive sequenceof eigenvalues tending to+∞. In short,

42 H. Douanla

i) if MS(ρ) > 0, then the positive eigencouples behave like in the case of point-wisepositive density function, i.e., fork ≥ 1, λk,+ε is of orderε and 1

ελk,+ε converges

asε→ 0 to thekth eigenvalue of the limit Dirichlet spectral problem, correspondingextended eigenfunctions converge along subsequences.

As for the “negative” eigencouples,λk,−ε converges to−∞ at the rate1ε and the

corresponding eigenfunctions oscillate rapidly. I utilized a factorization technique([45, 155]) to prove that

λk,−ε =1

ελ−1 + ξk,−ε + o(1), k = 1, 2, 3, . . .

where(λ−1 , θ−1 ) is the first negative eigencouple to the following local Steklov spec-

tral problem

−div(a(y)Dyθ) = 0 in Y ∗

a(y)Dyθ · n = λρ(y)θ on ∂T

θ Y -periodic,

(42)

andξk,±ε ∞k=1 are eigenvalues of a Steklov eigenvalue problem similar to (41). With

this trick, I proved that thatλk,−ε

ε −λ−

1

ε2 converges to thekth eigenvalue of a limitDirichlet spectral problem which is different from that obtained for positive eigenval-

ues. As for eigenfunctions, extensions of uk,−ε

(θ−1)εε∈E - where(θ−1 )

ε(x) = θ−1 (xε ) -

converge along subsequences to thekth eigenfunctions of the limit problem.

ii) If M∂T (ρ) = 0, then the limit spectral problem is a quadratic operator pencil problem,andλk,±ε converges to the(k,±)th eigenvalue of the limit operator, extended eigen-functions converge along subsequences as well. This case requires a new convergenceresult as regards the two-scale convergence theory, Lemma 1.

iii) The casesM∂T (ρ) < 0 andM∂T (ρ) > 0 are equivalent, just replaceρ with −ρ.

Remark 4. Problems similar to the one study in this paper have been independently (andat the same time) investigated in [35, 40].

In [35], Cao et al. studied the asymtotics of steklov eigenvalue for second order linearperiodic operators with a zero lower order term, but, in a fixed domain, and with a constantdensityρ = 1. The Steklov boundary condition is set on a fixed nonnegligeable part of theboundary∂Ω. Their work deals with one sequence of positive eigenvalue and has somenumerical results.

In [40], Piatnitski et al. studied the problem considered inthis paper, but for the Laplaceoperator (aij = δij). They combined the formal asymtotic expansion method witha justifi-cation procedure.

Introduction 43

3.2 Two-scale convergence of elliptic spectral problems with indefinite den-sity function in perforated domains, Asymptotic Analysis, 81 (2013) 251-272.

In this paper, I studied the spectral asymptotics of linear periodic elliptic operators with asign-changing density function in perforated domains. More precisely, letΩ be a boundeddomain inRN (N ≥ 2) and letT ⊂ Y = (0, 1)N be a compact subset ofY in RN . Weassume thatΩ andT haveC1 boundaries∂Ω and∂T , respectively. Forε > 0, we define

tε = k ∈ ZN : ε(k + T ) ⊂ Ω, T ε =⋃

k∈tε

ε(k + T ) and Ωε = Ω \ T ε.

We denoteY \T by Y ∗ andn = (ni)Ni=1 stands for the outer unit normal vector to∂T with

respect toY ∗. Theε-problem reads

−N∑

i,j=1

∂

∂xj

(aij(

x

ε)∂uε∂xi

)= ρ(

x

ε)λεuε in Ωε

N∑

i,j=1

aij(x

ε)∂uε∂xj

ni(x

ε) = 0 on ∂T ε

uε = 0 on ∂Ω,

(43)

wherea = (aij) ∈ L∞(RN )N×N is symmetric, elliptic andY -periodic. The densityfunction ρ ∈ L∞(RN ) is Y -periodic and changes sign onY ∗, that is, both the setsy ∈Y ∗, ρ(y) < 0 andy ∈ Y ∗, ρ(y) > 0 are of positive Lebesgue measure. This hypothesismakes the problem nonstandard. The spectrum of (43) is discrete and consists of two infinitesequences

−∞← λn,−ε ≤ · · · ≤ λ2,−ε ≤ λ1,−ε < 0 < λ1,+ε ≤ λ2,+ε ≤ · · · ≤ λn,+ε → +∞.

The limiting behavior of positive and negative eigencouples crucially depends on whetherthe average of the weight over the solid part

MY ∗(ρ) =

∫

Y ∗

ρ(y)dy,

is positive, negative or equal to zero. I proved concise homogenization results in all threecases.

• In the case whenMY ∗(ρ) = 0, problem (43) generates in the limit (asε → 0) aquadratic operator pencil problem.

• In the case whenMY ∗(ρ) > 0 both parts of the spectrum (positive and negative)generate Dirichlet eigenvalue problems in the fixed domainΩ, though the negative

44 H. Douanla

part of the spectrum requires a non-obvious scaling trick involving the following localeigenvalue problem with sign-changing density function

−div(a(y)Dyθ) = λρ(y)θ in Y ∗

a(y)Dyθ · n = 0 on ∂T

θ Y -periodic.

(44)

More precisely,λk,−ε − 1ε2λ−1 converges to thekth eigenvalue of the limit problem,

a Dirichlet eigenvalue problem, where(λ−1 , θ−1 ) is the first negative eigencouple to

(44).

• The caseMY ∗(ρ) < 0 reduces to the caseMY ∗(ρ) > 0 by replacingρ with −ρ.

My results in this paper simplify and generalize those in [112] where similar results whereobtained in a fixed domain (no perforation !) via a combination of formal asymptotic ex-pansion and a heavy justification procedure.

After working on spectral asymptotics for a while, I startedbroadening my interest inthe homogenization theory of partial differential equations. This gave birth to [63, 65, 66].

3.3 Almost periodic homogenization of a generalized Ladyzhenskaya modelfor incompressible viscous flow,Journal of Mathematical Sciences (N. Y.),189 (2013) 431-458. Translated from: Problemy Matematicheskogo Anal-iza, 68 (2013) 89-112.

The Navier-Stokes equations model the motion of Newtonian fluids. In order to understandthe phenomenon of turbulence related to the motion of a fluid,several mathematical mod-els have been developed and studied over the years. We refer to [18, 39, 69, 91], just tocite a few. In this paper, we proved an existence result for a nonstationary generalized La-dyzhenskaya equations with a given nonconstant density andan external force dependingnonlinearly on the velocity, and then study the limiting behavior of the velocity field in analmost periodic settingby combining some compactness arguments with theΣ-convergencemethod. As for my contribution, I studied the whole problem under the supervision of myco-author who suggested the problem. Grigori Rozenblioum improved the presentation ofthe paper.

We considerN -dimensional problem,N = 2, 3 and assume that all the function spacesare real-valued spaces and scalar functions assume real values. Letε > 0 be a small param-eter representing the scale of the heterogeneities, and let1 + 2N

N+2 ≤ p < ∞ andT > 0

be real numbers. PutQT = Q × (0, T ) whereQ is a bounded smooth domain inRN andconsider the following well-known spaces:V = ϕ ∈ C∞0 (Q)N : divϕ = 0; V = closureof V in W 1,p(Q)N ; H = closure ofV in L2(Q)N . In view of the smoothness ofQ, it isknown thatV = u ∈ W 1,p

0 (Q)N : divu = 0 andH = u ∈ L2(Q)N : divu = 0 andu|∂Q · n = 0, whereu|∂Q denotes the trace ofu on∂Q andn is the outward unit vector

Introduction 45

normal to∂Q. The spaceV is endowed with the gradient norm andH with theL2-norm.With all this in mind, we introduce the functionsa, b, ρ andf constrained as follows.

(A1) The matrixa = (aij)1≤i,j≤N ∈ L∞(RN+1)N×N satisfiesaij = aji and

(a(y, τ)λ) · λ ≥ ν0 |λ|2 for all λ ∈ RN and a.e.(y, τ) ∈ RN+1

for some positiveν0.

(A2) The functionb ∈ L∞(RN+1) is such thatν1 ≤ b ≤ ν2 for some positiveν1 andν2.

(A3) The density functionρ belongs to∈ L∞(RN ) and satisfiesΛ−1 ≤ ρ(y) ≤ Λ foralmost everyy ∈ RN , for some positiveΛ.

(A4) The mappingf : R × RN → RN , (t, r) 7→ f(t, r) is continuous with the followingproperties:

(i) f maps continuouslyR×H intoH (f(τ, u) ∈ H for anyu ∈ H)

(ii) There is a positive constantk such that

|f(τ, 0)| ≤ k for all τ ∈ R|f(τ, r1)− f(τ, r2)| ≤ k |r1 − r2| for all r1, r2 ∈ RN andτ ∈ R.

(A5) Almost periodicity . We assume that the functionsaij andb lie in B2AP (R

N+1y,τ ) ∩

L∞(RN+1y,τ ) for all 1 ≤ i, j ≤ N . We also assume that the functionρ belongs to

B2AP (R

N ) ∩ L∞(RN ) with My(ρ) > 0. Finally the functionτ 7→ f(τ, r) lies in(AP (Rτ ))

N for anyr ∈ RN .

Givenu0 ∈ H, we are interested in the asymptotic behavior of the sequence of velocity field(uε)ε>0 of the following generalized Ladyzhenskaya equation

ρε∂uε

∂t−div

(aε∇uε+b

ε |∇uε|p−2∇uε

)+(uε · ∇)uε+∇qε = ρεf ε(·,uε) in QT

divuε = 0 in QT

uε = 0 on∂Q× (0, T )

uε(x, 0) = u0(x) in Q,

(45)

where the scaled functionsρε ∈ L∞(Q), aε = (aεij)1≤i,j≤N ∈ L∞(QT )

N×N , f ε(·, r) ∈C(0, T ) (for anyr ∈ RN ) andbε ∈ L∞(QT ) are defined as follows:

ρε(x) = ρ(xε

), aεij(x, t) = aij

(x

ε,t

ε2

), f ε(·, r)(t) = f

(t

ε2, r

)and

bε(x, t) = b

(x

ε,t

ε2

)for (x, t) ∈ QT .

46 H. Douanla

The equation (45) models various types of motion of non-Newtonian fluids. We citea few examples. Ifa = (ν0δij)1≤i,j≤N (δij is the Kronecker delta) andb = ν1 whereν0andν1 are positive constants, then we get the Ladyzhenskaya equations. In that case, theanalysis conducted in [69] reveals that one may either letν1 → 0 (and get the usual Navier-Stokes equations) or letν0 → 0, and get the power-law fluids equations. In particular whenp = 3, we get the Smagorinski’s model of turbulence [145] withν0 being the molecularviscosity andν1 the turbulent viscosity. Another model included in (45) is the equation ofincompressible bipolar fluids [18]. It should be noted that if in equation (45) we replace thegradient∇u by its symmetric part

1

2(∇u+∇Tu),

then thanks to the Korn’s inequality, the mathematical analysis does not change, althoughthe model becomes in this case, physical. The following existence theorem is the first resultof this paper.

Theorem 9. Let 1 + 2NN+2 ≤ p < ∞. Supposeu0 ∈ H. Under assumptions (A1)-(A4),

there exists (for each fixedε > 0) a couple(uε, qε) ∈ Lp(0, T ;V) ∩ L∞(0, T ;H) ×W−1,∞(0, T ;Lp′(Q)) solution to(45). The functionuε also belongs toC([0, T ];H) andqε is unique up to a constant function ofx:

∫

Qqεdx = 0.

The second result is the homogenization result.

Theorem 10. Assume that(A1)-(A5) hold. Let1 + 2NN+2 ≤ p < ∞. For eachε > 0

let (uε, qε) be a solution of(45). Then there exists a subsequence of(uε, qε)ε>0 whichconverges strongly inL2(QT )

N (with respect to the first componentuε) and weakly inLp′(QT ) (with respect toqε) to the solution to

My(ρ)∂u0

∂t − div(mDu0) +M(Du0)) +B(u0) +∇q0 = M(ρf(·,u0))divu0 = 0 in QT

u0 = 0 on∂Q× (0, T )u0(x, 0) = u0(x) in Q,

(46)

where we refer to the appended version of the paper for the definition of the limit operators.Moreover, any limit point inL2(QT )

N × Lp′(QT ) (in the above sense) of(uε, qε)ε>0 is asolution to(46).

3.4 Incompressible viscous Newtonian flow in a fissured medium of generaldeterministic type, Journal of Mathematical Sciences (N. Y.), 191 (2013)214-242. Translated from: Problemy Matematicheskogo Analiza, 70 (2013)105-128.

Jointly with Gabriel Nguetseng and Jean Louis Woukeng, I generalized the usual approachfor upscaling flow in periodic porous media and proposed a framework which can handle

Introduction 47

general deterministically fissured media. However, I should stress that my contribution inthis paper does not include the new developments in the theory of Σ-convergence herein,but is the a priori estimates, the compactness results and the homogenization procedure.Grigori Rozenblioum improved the presentation of the paper.

A fissured porous medium is a structure made up of blocks of an usual porous mediumwith fissures between them. The blocks form theporous matrixwhile the fissures are char-acterized by substantially higher flow rates and lower relative volume. In order to study theflow of a fluid in a fractured porous medium, several mathematical models have been sug-gested and upscaled by homogenization. The classical and most studieddouble diffusionmodelfor fissured porous rock domain was introduced by Barenblatt, Zheltov and Kochinain 1960 [17], and further developed by Warren and Root in 1963[158], Coats and Smith in1964 [50], and some other researchers, including [5, 85, 88,102, 101, 125, 132, 133, 148,161].

All the previous models are characterized by the common factthat they have been in-vestigated under the classicalequidistribution(periodicity) assumption on the geometry ofthe medium under consideration. In this paper we propose an approach to handle the gen-eral case where the medium is fissured in a deterministic manner including the special caseof equidistribution commonly known in the literature as theperiodic fissured medium. Wethen apply the approach to a double diffusion type problem.

To be precise, let us describe the geometry of the problem. Let N = 2 or 3 and letY =(0, 1)N be the reference cell. LetY1 andY2 be two open disjoint subsets ofY representingthe local structure of the porous matrix and the local structure of fissures, respectively. Weassume thatY 1 ⊂ Y , Y = Y 1 ∪ Y2, Y2 is connected and that the boundary∂Y1 of Y1 isLipschitz continuous. Moreover, we assume thatY1 andY2 have positive Lebesgue measure.Let S ⊂ ZN be an infinite subset ofZN and put

Gj = ∪k∈S(k + Yj) (j = 1, 2).

ThenGj is an open subset ofRN and moreover,G2 is connected. Since the cells(k+Y )k∈Sare pairwise disjoint, the characteristic functionχj of the setGj in RN satisfies

χj =∑

k∈S

χk+Yj,

or more preciselyχj =

∑

k∈ZN

θ(k)χk+Yj,

whereθ is the characteristic function ofS in ZN . The functionθ is thedistribution functionof the fissured cells. The special caseθ(k) = 1 for all k ∈ ZN (that is whenS = ZN )leads to the equidistribution of fissured cells, commonly known as theperiodic fissuredmedium. The functionθ may also be assumed to be periodic, that is, there exists a discretesubgroup ofRN of rankN , R (such anR is called a network inRN ), with R ⊂ ZN andθ(k + r) = θ(k) for all k ∈ ZN and allr ∈ R. We get in that case a periodic distribution

48 H. Douanla

of fissured cells. One can consider the case whenθ is almost periodic (which correspondsto the almost periodic distribution of fissured cells). Someother assumptions may also beconsidered.

Bearing this in mind, letΩ be an open bounded Lipschitz domain inRN and letε > 0 bea small parameter. We define ourε-fissured medium as follows:Ωε

j = Ω ∩ εGj (j = 1, 2).The setΩε

1 (resp. Ωε2) is the portion ofΩ occupied by the matrix (resp. fissures) and we

haveΩ = Ωε1 ∪ Γε

12 ∪ Ωε2 (disjoint union) whereΓε

12 = ∂Ωε1 ∩ ∂Ω

ε2 ∩ Ω is the part of the

interface ofΩε1 with Ωε

2 lying in Ω. Let νj denote the outward unit normal vector to∂Ωεj . It

is clear thatν1 = −ν2 onΓε12. With the geometry of the medium described, let

Q = (aij)1≤i,j≤N and B = (bij)1≤i,j≤N

be two real symmetric matrices with entries inL∞(RN ), satisfying

Q(y)ξ · ξ ≥ α |ξ|2 , B(y)ξ · ξ ≥ α |ξ|2 (47)

for almost everyy ∈ RN , and for allξ ∈ RN , whereα is a given positive constant indepen-dent ofy andξ. Forj = 1, 2, letρj : RN → R be a bounded continuous function satisfying

Λ−1 ≤ ρj(y) ≤ Λ for all y ∈ RN , (48)

whereΛ is a positive constant independent ofy. It is well known that the functionx 7→Q(x/ε) (resp. x 7→ B(x/ε), x 7→ ρj(x/ε)) defined fromΩ into RN×N (resp. RN×N ,R) is well-defined as an element ofL∞(Ω)N

2

(resp. L∞(Ω)N2

, C(Ω)) denoted byQε

(resp. Bε, ρεj). Next, letT > 0 be freely fixed and putQ = Ω × (0, T ). Assume thatf, g ∈ L∞(0, T ;L2(Ω)N ) andu0,v0 ∈ L2(Ω)N with divu0 = div v0 = 0 in Ω.

A nonstationary flow of an incompressible viscous Newtonianfluid is governed by theNavier-Stokes system. We study the evolution state of the medium Ω which is saturatedwith viscous Newtonian fluids satisfying a slip condition onthe rough part of the boundaryas well as on the fluid-solid interface. The Navier-Stokes equations inΩε

j (j = 1, 2) and thecontinuity equations of the normal stress and velocity at the solid-fluid interfaceΓε

12 are:

ρε1∂uε

∂t− div (Qε∇uε) + (uε · ∇)uε +∇pε = ρε1f in Ωε

1 × (0, T ) (49)

divuε = 0 in Ωε1 × (0, T ) (50)

ρε2∂vε∂t− div (Bε∇vε) + (vε · ∇)vε +∇qε = ρε2g in Ωε

2 × (0, T ) (51)

div vε = 0 in Ωε2 × (0, T ) (52)

uε = vε on Γε12 × (0, T ) (53)

Introduction 49

(Qε∇uε − pεI) · ν1 = (Bε∇vε − qεI) · ν1 on Γε12 × (0, T ) (54)

(uε ⊗ uε) · ν1 = (vε ⊗ vε) · ν1 on Γε12 × (0, T ) (55)

uε = 0 on (∂Ωε1 ∩ ∂Ω)× (0, T ), vε = 0 on (∂Ωε

2 ∩ ∂Ω)× (0, T ) (56)

uε(x, 0) = u0(x) for x ∈ Ωε1, vε(x, 0) = v0(x) for x ∈ Ωε

2 (57)

where on the matrix (or solid part) the fluid has the stress tensorσε1 = Qε∇uε−pεI, density

ρε1, velocityuε, pressurepε, and the fluid partΩε2 has the stress tensorσε2 = Bε∇vε − qεI,

velocityvε and pressureqε; I is the identity tensor,ρε1f (or ρε2g) is the external body forceper volume. The above system consists of a nonstationary incompressible Navier-Stokesequations inΩε

1 coupled across the interfaceΓε12 to another nonstationary incompressible

Navier-Stokes system inΩε2. The structure of our fissured medium produces very high fre-

quency spatial variations of pressures in the matrixΩε1, hence leads to corresponding varia-

tions of velocity fields. The positive definite symmetric elasticity tensorsQ = (aij)1≤i,j≤N

andB = (bij)1≤i,j≤N provide a model for general anisotropic materials. Conditions (53)-(55) are the interface conditions stating the continuity ofthe fluid flow at the interfaceΓε

12

while condition (56) is the homogeneous Dirichlet boundaryconditions on the outer bound-ary, and (57) is the initial condition.

The limit passageε → 0 in the above system is done under suitable hypotheses onthe elasticity tensors and on the density of the fluids when the medium is subjected to ageneral deterministic hypothesis. Each of these hypotheses covers a great set of patternsof deterministic behavior such as the periodicity, the almost periodicity, the convergenceat infinity, the weak almost periodicity, and many more. It should be noted that such amathematical analysis has never been done previously for our model, even in the so-calledperiodic setting. By means of theΣ-convergence method we prove the following theorem.

Theorem 11. LetA be an ergodic algebra with mean value and assume that

χj ∈ B2A(R

N ) with M(χj) > 0 , j = 1, 2,

Q, B ∈ (B2A(R

N ))N2

and ρj ∈ A for all j = 1, 2.

For eachε > 0 let (uε, pε) and (vε, qε) be the sequence of solutions to(49)-(57). Letuε

andpε denote the global velocity and global pressure, respectively defined by

uε = χε1uε + χε

2vε and pε = χε1pε + χε

2qε.

There exists a subsequence of(uε, pε) still denoted here by(uε, pε) such that, asε → 0,we have

uε → u0 in L2(Q)N -strong

50 H. Douanla

pε → p in L2(Q)-weak

where(u0, p) is a solution to the following Navier-Stokes system

ρ∂u0

∂t− div(C∇u0) + (u0 · ∇)u0 +∇p = F in Q

divu0 = 0 in Q

u0 = 0 on∂Ω × (0, T )

u0(x, 0) = χ1u0(x) + χ2v

0(x), x ∈ Ω.

Moreover, any limit point of(uε, pε) in L2(Q)N ×L2(Q) (in the above sense) is a solutionto the above system. The matrixC = (cij)1≤i,j≤N is the effective homogenized elasticitytensor, and has constant entries.

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