G-Convergence and HomogenizationGof some Monotone Operators

Marianne Olsson

DEPARTMENT OF ENGINEERING, PHYSICS AND MATHEMATICS

Mid Sweden University Doctoral Thesis 45

i

Thesis for the degree of Doctor of Philosophy Östersund 2008

G-CONVERGENCE AND HOMOGENIZATION OF SOME MONOTONE OPERATORS

Marianne Olsson

Supervisors: Associate Professor Anders Holmbom, Mid Sweden University

Professor Nils Svanstedt, Göteborg University Professor Mårten Gulliksson, Mid Sweden University

Department of Engineering, Physics and Mathematics Mid Sweden University, SE‐831 25 Östersund, Sweden

ISSN 1652‐893X, Mid Sweden University Doctoral Thesis 45

ISBN 978‐91‐85317‐85‐1

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Akademisk avhandling som med tillstånd av Mittuniversitetet framläggs till offentlig granskning för avläggande av filosofie doktorsexamen torsdagen den 28 februari 2008, klockan 13.00 i sal Q221, Mittuniversitetet, Östersund. Seminariet kommer att hållas på engelska. G-CONVERGENCE AND HOMOGENIZATION OF SOME MONOTONE OPERATORS Marianne Olsson © Marianne Olsson, 2008 Department of Engineering, Physics and Mathematics Mid Sweden University, SE‐831 25 Östersund Sweden Telephone: +46 (0)771‐97 50 00 Printed by Kopieringen Mittuniversitetet, Sundsvall, Sweden, 2008

G-convergence and Homogenizationof some Monotone Operators

Marianne OlssonDepartment of Engineering, Physics and MathematicsMid Sweden University, SE-831 25 Östersund, Sweden

AbstractIn this thesis we investigate some partial differential equations

with respect to G-convergence and homogenization. We study a fewmonotone parabolic equations that contain periodic oscillations onseveral scales, and also some linear elliptic and parabolic problemswhere there are no periodicity assumptions. To begin with, we exam-ine parabolic equations with multiple scales regarding the existenceand uniqueness of the solution, in view of the properties of somemonotone operators. We then consider G-convergence for elliptic andparabolic operators and recall some results that guarantee the ex-istence of a well-posed limit problem. Then we proceed with someclassical homogenization techniques that allow an explicit characteri-zation of the limit operator in periodic cases. In this context, we proveG-convergence and homogenization results for a monotone parabolicproblem with oscillations on two scales in the space variable. Then weconsider two-scale convergence and the homogenization method basedon this notion, and also its generalization to multiple scales. This isfurther extended to the case that allows oscillations in space as well asin time. We prove homogenization results for a monotone parabolicproblem with oscillations on two spatial scales and one temporal scale,and for a linear parabolic problem where oscillations occur on one scalein space and two scales in time. Finally, we study some linear ellipticand parabolic problems where no periodicity assumptions are madeand where the coefficients are created by certain integral operators.Here we prove results concerning when the G-limit may be obtainedimmediately and is equal to a certain weak limit of the sequence ofcoefficients.

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Acknowledgements

First, I would like to thank my main supervisor Anders Holmbom for hisguidance and advice, for finding something positive to say even when it meantfinding a needle in a haystack, for bearing with me when I was not quite aspositive, and for being there as a sounding board. I would also like to expressmy gratitude to my supervisor Nils Svanstedt for his encouragement and forcontributing with helpful suggestions and inspiring discussions. I am alsograteful to my supervisor Mårten Gulliksson for interesting discussions andfor his support of my research.

In addition, I want to thank the Graduate School in Mathematics andComputing for their financial support.

I would also like to thank my colleagues and friends at the departmentfor making it a pleasure to work here during my time as a Ph.D. student. Iwould especially like to mention my fellow graduate students Lotta Flodén,with whom I have cooperated a lot, Marie Lund and David Bergström. Myheart and grateful mind also go to Jeanette Silfver.

Finally, I want to thank my family for always encouraging me, withoutpushing me.

Östersund, January 2008Marianne Olsson

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Notations

For the convenience of the reader, here are some symbols and sets used inwhat follows. For some fundamental definitions and results from functionalanalysis, the reader is referred to the appendix.

X : Any linear space.

kukX : The norm of u ∈ X, when X is a normed space.

X 0 : The dual space of X.

H : Any Hilbert space.

V : Any Banach space such that the embedding V ⊆ H is continuous.©uhª: A sequence of functions uh.

uh → u :©uhªconverges strongly to u.

uh u :©uhªconverges weakly to u.

uh∗u :

©uhªconverges weakly* to u.

ε : A sequence ε (h) such that ε = ε (h)→ 0 as h →∞.O : Any open bounded subset of RM with smooth (at least Lipschitz)

boundary.

∂O : The boundary of O.O : The closure of O.Ω : Any open bounded subset of RN with smooth (at least Lipschitz)

boundary.

ΩT : The set Ω× (0, T ).ΩT : The set Ω× [0, T ].Y∗ : Unit cell in RM .

Y, Y1, Y2, ...Yn : Unit cells in RN .

Y n : The set Y1 × ...× Yn.

Yn,m : The set Y n × (0, 1)m.a · b : The scalar product of two vectors a and b in RN .

(u, v)H : The inner product of u and v in a Hilbert space H.

h·, ·iX0,X : The duality pairing between X 0 and X.

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Below is a list of function spaces and their norms. All functions u are assumedto be measurable.

F (O) : Any space of functions u : O→ R.

Floc(RM) : All functions u : RM → R such that their restriction to anyopen bounded subset O of RM belongs to F (O).

F (Y∗) : All functions in Floc(RM) which are the periodic repetitionof some function in F (Y∗).

F (O) /R : All functions u in F (O) such thatRO u (x) dx = 0.

Lp (O) : All functions u : O→ R such thatkukLp(O) =

¡RO |u (x)|

p dx¢1/p

<∞.

L∞ (O) : All functions u : O→ R such thatkukL∞(O) = ess supx∈O |u (x)| <∞.

W 1,p (O) : All functions u in Lp (O) such that their first-order distribu-tional derivatives belong to Lp (O).kukW 1,p(O) = kukLp(O) + k∇ukLp(O)M

W 1,p0 (O) : All functions u in W 1,p (O) such that u = 0 on ∂O.

kukW 1,p0 (O) = k∇ukLp(O)M

W−1,q (O) : The dual space of W 1,p0 (O), 1

p+ 1

q= 1.

H1 (O) : The function space W 1,p (O) for p = 2.

H10 (O) : The function space W 1,p

0 (O) for p = 2.

H−1 (O) : The space W−1,q (O) for p = q = 2.

C(O) : All continuous functions u : O→ R.kukC(O) = supx∈O |u (x)|

C0(O) : All continuous functions u : O→ R with compact support in O.

C∞ (O) : All infinitely differentiable functions u : O→ R.

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D (O) : All infinitely differentiable functions u : O→ R with compactsupport in O.

D0 (O) : All distributions in O.

C (Y∗) : All continuous and Y∗-periodic functions u : RM → R.kukC (Y∗)

= supy∈Y∗ |u (y)|

C∞ (Y∗) : All infinitely differentiable and Y∗-periodic functionsu : RM → R.

H1 (Y∗) : All Y∗-periodic functions in H1loc(RM).

H1 (Y∗) /R : All functions u in H1 (Y∗) such thatRY∗u (y) dy = 0.

kukH1(Y∗)/R = k∇ukL2(Y∗)M

L∞ (Y∗) : All Y∗-periodic functions in L∞(RM).

L2 (O;X) : All functions u : O→ X such thatkukL2(O;X) = (

RO ku (x, ·)k

2X dx)1/2 <∞.

L2 (0, 1;X) : All (0, 1)-periodic functions u : R→ X such thatkukL2(0,1;X) = (

R 10ku (x, ·)k2X dx)1/2 <∞.

L∞ (O;X) : All functions u : O→ X such thatkukL∞(O;X) = ess supx∈O ku (x)kX <∞.

C(O;X) : All continuous functions u : O → X.kukC(O;X) = supx∈O ku (x, ·)kX

C0(O,X) : All continuous functions u : O→ X with compact supportin O.

D (O;X) : All infinitely differentiable functions u : O→ X withcompact support in O.

H1 (0, T ;V, V 0) : All functions u in L2 (0, T ;V ) such that ∂tu belongsto L2 (0, T ;V 0).kukH1(0,T ;V,V 0) = kukL2(0,T ;V ) + k∂tukL2(0,T ;V 0)

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Contents

1 Introduction 11.1 The idea of homogenization . . . . . . . . . . . . . . . . . . . 21.2 The snag . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.3 G-convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.4 Periodic homogenization . . . . . . . . . . . . . . . . . . . . . 71.5 Outline of the thesis . . . . . . . . . . . . . . . . . . . . . . . 9

2 Monotone operators 122.1 The concept of monotone operators . . . . . . . . . . . . . . . 122.2 Monotone operators and evolution problems . . . . . . . . . . 15

2.2.1 General evolution problems . . . . . . . . . . . . . . . 162.2.2 Parabolic PDEs with multiple scales . . . . . . . . . . 18

3 G-convergence 243.1 G-convergence for elliptic operators . . . . . . . . . . . . . . . 25

3.1.1 Linear elliptic operators . . . . . . . . . . . . . . . . . 263.1.2 Monotone elliptic operators . . . . . . . . . . . . . . . 31

3.2 G-convergence for parabolic operators . . . . . . . . . . . . . . 323.2.1 Linear parabolic operators . . . . . . . . . . . . . . . . 333.2.2 Monotone parabolic operators . . . . . . . . . . . . . . 34

4 Classical homogenization techniques 374.1 The method of asymptotic expansions . . . . . . . . . . . . . . 384.2 Tartar’s method of oscillating test functions . . . . . . . . . . 414.3 Reiterated homogenization . . . . . . . . . . . . . . . . . . . . 49

5 Homogenization by multiscale convergence techniques 565.1 Homogenization with two scales . . . . . . . . . . . . . . . . . 56

5.1.1 Two-scale convergence . . . . . . . . . . . . . . . . . . 575.1.2 Homogenization using two-scale convergence . . . . . . 665.1.3 Two-scale convergence and asymptotic expansions . . . 68

5.2 Homogenization with multiple scales . . . . . . . . . . . . . . 745.2.1 Multiscale convergence . . . . . . . . . . . . . . . . . . 765.2.2 Reiterated homogenization using multiscale convergence 77

5.3 Homogenization with multiple scales in space and time . . . . 805.3.1 Evolution multiscale convergence . . . . . . . . . . . . 81

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5.3.2 Homogenization of some monotone parabolic operatorswith three spatial and two temporal scales . . . . . . . 92

5.3.3 Homogenization of some linear parabolic operators withtwo spatial and three temporal scales . . . . . . . . . . 110

6 Characterization of the G-limit for some non-periodic linearoperators 1176.1 Linear elliptic operators . . . . . . . . . . . . . . . . . . . . . 117

6.1.1 Some numerical experiments . . . . . . . . . . . . . . . 1196.1.2 A theoretical result . . . . . . . . . . . . . . . . . . . . 122

6.2 Linear parabolic operators . . . . . . . . . . . . . . . . . . . . 1256.2.1 A theoretical result . . . . . . . . . . . . . . . . . . . . 1256.2.2 Some further results and open questions . . . . . . . . 127

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1 Introduction

In fields such as physics and engineering, there is often cause to study phys-ical phenomena in highly heterogeneous media. Typical problems are heatconduction, elasticity or torsion in composites and flow in porous media, justto mention a few. Composites consist of two or more different materials thatare mixed together in e.g. a fibered, layered or crystalline structure andare widely used because of their properties which may combine beneficialfeatures of the individual components. Some well-known composites are re-inforced concrete, plastic reinforced by glass or carbon fiber and engineeredwood such as plywood. Heterogeneous materials with a fine microstructurealso occur naturally, such as for example in wood, porous rocks and bone.

Figure 1. A material with a fine microstructure.

The microscopic structure makes problems of the type mentioned abovedifficult to treat directly. Typically, they can be modeled by some partialdifferential equation (PDE), where the presence of the microscopic variationsis reflected in oscillations in the coefficient of the equation (or in a perforateddomain in the case of porous media). Solving such a problem using somenumerical method would require a very fine mesh to capture the changes inthe structure. Thus, the solution may be costly to find or be out of reach.

When the microstructure is very fine compared to the sample of material,as in Figure 1, the material may at first glance seem to be homogeneous, andwe can also expect it to behave as if homogeneous at the macroscopic level.This means that another approach to these problems might be to search forthis global behavior; that is, to look for the macroscopic or so-called effectiveproperties of the material. Thus, using the information about the microstruc-ture, such as the properties of the constituents and the arrangement of these,the aim is to "homogenize" the material in the sense of finding the proper-ties of a homogeneous material that gives the same overall response as theheterogeneous one.

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1.1 The idea of homogenization

The main source of inspiration for the mathematical theory of homogeniza-tion is to find the effective properties in problems such as those mentionedabove. A prototype problem is stationary heat diffusion in a composite ma-terial with a periodic microstructure. We briefly describe some features ofhomogenization illustrated by this problem.

We let the set Ω be occupied by our composite material. If we assumethat the heterogeneities are evenly distributed, we can model the material asperiodic, as in Figure 2.

Figure 2. The material sample and the representative cell.

As illustrated in the figure, this means that we can think of the materialas being built up of small identical cubes, the side length of which we callε. We let a (y) be the periodic repetition of the function that describes howthe ability to conduct heat varies over the representative cell Y ; see Figure2. For simplicity, we choose Y to be the unit cube. Hence, substituting x

ε

for y, we obtain a function a(xε) that oscillates periodically with period ε

as the variable x passes through Ω, describing the oscillations of the heatconductivity in the composite.

If the material sample is located in an environment with temperaturezero and is subject to a heat source given by a function f , after a while thetemperature distribution in the material will stabilize to the solution uε tothe problem

−∇ ·³a³xε

´∇uε (x)

´= f (x) in Ω, (1)

uε (x) = 0 on ∂Ω.

This is the usual stationary heat equation, governed by a¡xε

¢, the heat con-

ductivity of our periodic material.

2

For small values of ε, equation (1) will be very difficult to treat numeri-cally, which means that we might not be able to compute uε. On the otherhand, when ε is small we can expect that the material would behave likea homogenous one and hence, from a macroscopic point of view, could bedescribed by an equation of the form

−∇ · (b∇u (x)) = f (x) in Ω, (2)

u (x) = 0 on ∂Ω,

with b constant. Since this problem does not contain any rapid oscillations,it is much easier to solve than the original one. Thus, if b was known, wecould find u, which would give us a good approximation of the temperaturedistribution.

So how could we find the effective property represented by b? For theperiodic case a first, and not unreasonable, guess might be that b is given bythe arithmetic mean value of a over a period, i.e. over Y . We try this for aone-dimensional example with Ω = (0, 1), f(x) = x2 and

a (y) =1

2 + sin 2πy(3)

and solve

− d

dx

µZY

a (y) dyd

dxu (x)

¶= f (x) in Ω, (4)

u (x) = 0 on ∂Ω.

As we can see from Figure 3 below, the solution u does not give a goodapproximation of the true temperature distribution uε, which means thatour guess was wrong.

Figure 3. The solutions uε and u to (1) and (4), respectively.

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To find b, the thermal conductivity of the fictitious homogeneous material,we imagine that the structure becomes increasingly finer, as in Figure 4.

Figure 4. Homogenizing the mixture.

This corresponds to the study of a sequence of equations (1) for successivelysmaller values of ε. If the sequence uε of solutions close in on some functionu that does not have any rapid oscillations depending on ε, this functionshould give the desired approximation of the temperature distribution. Thisis illustrated in Figure 5 for example (3).

Figure 5. uε for some values of ε and the limit u.

Thus, if by letting ε approach zero in (1), we could find a problem of theform (2), which provides us with the limit function u as its unique solution,we would obtain the proper macroscopic description of the original problem.In the following sections we will study some mathematical aspects of thishomogenization problem in more detail; that is, what type of convergencewe can expect for uε, and how we can pass to the limit and find the so-called homogenized problem.

1.2 The snag

To find the limit equation, we want to let ε pass to zero in (1). To do this,we will apply certain types of weak convergence, and the proper form of theequation is then the so-called weak form. If we let f ∈ L2(Ω), this meansthat we are searching for uε ∈ H1

0 (Ω) such thatZΩ

a³xε

´∇uε (x) ·∇v (x) dx =

ZΩ

f (x) v (x) dx (5)

for all v ∈ H10(Ω).

4

Under standard assumptions on boundedness and coercivity of a, seeSection 3.1.1, this problem has a unique solution uε, also called the weaksolution to (1). From the properties of a we also get that uε satisfies aso-called a priori estimate; that is, boundedness in a certain function space,in this case H1

0 (Ω). Since H10 (Ω) is a reflexive Banach space, this guarantees

that there is a function u such that, at least for a subsequence, it holds that

uε (x) u (x) in H10 (Ω). (6)

Next, we want to pass to the limit in (5) to find the problem for whichthis limit u is the unique solution. As it turns out, this is a rather intricatetask. Through the assumptions made and the separability of L1(Ω) we have

a³xε

´∗ZY

a (y) dy in L∞(Ω)N×N ,

which implies weak convergence in L2(Ω)N×N to the same limit. Furthermore,the a priori estimate gives that ∇uε (x) is bounded in L2(Ω)N and henceit holds, up to a subsequence, that

∇uε (x) ∇u (x) in L2(Ω)N ,

where u is the limit in (6). These convergence properties mean that if onlyone of the sequences

©a¡xε

¢ªand ∇uε (x) was present in (5), we could

immediately pass to the limit, at least for a subsequence, just by replacingthe sequence with its weak limit. However, when taking the product, as in(5), it is not that easy. Generally, the product of two only weakly convergentsequences does not converge to the product of the limits. Even though thesequence

©a¡xε

¢∇uε (x)

ªof products is bounded inL2(Ω)N , see Section 3.1.1,

in general we have

a³xε

´∇uε (x) 6

ZY

a (y) dy ∇u (x) in L2(Ω)N ,

which means that usually

b 6=ZY

a (y) dy.

This is the explanation as to why our initial guess was not very good afterall. We remark that in one dimension b turns out to be the harmonic meanof a, but in higher dimensions this does generally not hold true.

5

Thus, the homogenized coefficient b is usually not any conventional typeof mean of a. Nor is it in general any usual kind of weak limit of

©a(x

ε)ª,

and obviously there is no trivial way of passing to the limit in (5) and hencefinding the homogenized problem. Motivated by the identification of thismajor snag in the homogenization process, we now turn our interest to someof the tools developed for proving the existence of, and finding, the limit b.

1.3 G-convergence

The matter of the existence and the character of the limit coefficient can beembedded in the general theory of G-convergence, which is a type of con-vergence of operators, first introduced by Spagnolo in [Spa1] and [Spa2] forsecond-order elliptic and parabolic operators. It can be applied for exampleto problems of the form

−∇ ·¡ah (x)∇uh (x)

¢= f (x) in Ω, (7)

uh (x) = 0 on ∂Ω,

where the coefficient matrices ah do not have to meet any periodicity require-ments. In its simplest form, G-convergence of ah means that the sequenceuh of solutions to (7) satisfies

uh (x) u (x) in H10 (Ω)

where u solves an equation of the type

−∇ · (b (x)∇u (x)) = f (x) in Ω, (8)

u (x) = 0 on ∂Ω,

and b, which we call the G-limit of ah, has appropriate properties so that(8) has a unique solution. This type of convergence will be investigated inChapter 3.

If ah is a sequence of symmetric matrix functions that fulfill certain con-ditions of uniform boundedness and coercivity, we will have G-convergenceup to a subsequence, and there is also a similar result for the case withoutsymmetry assumption. This means that at least for a subsequence, we areguaranteed the existence of a limit problem with suitable properties. This isthe case for the model problem, for example. However, we do not know whatb looks like and we have seen that it is not trivial to determine b even in theperiodic case.

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1.4 Periodic homogenization

Much effort has been made to develop methods for finding the limit matrix b,especially for periodic problems. We will mention some of the most importantones here, but we first take a look at their common result for the prototypeproblem (1).

It turns out that the information needed to characterize b in (2) can beattained from a so-called local problem, which is an equation defined on onerepresentative cell Y , in our case the unit cube. In the homogenized problemthe coefficient has the entries

bij =

ZY

aij (y) +NXk=1

aik (y) ∂ykzj(y) dy. (9)

Each zj can, under certain assumptions of periodicity, be determined from acorresponding local problem

−∇y · (a (y) (ej +∇yzj (y))) = 0 in Y ,

where ejNj=1 is the usual orthonormal basis inRN , and hence we can identifythe limit coefficient b.

In (9) there is an additional term compared to the kind of mean that weused in (4). The fact that b does not coincide with this mean is also obviousfrom Figure 6, where we show these numbers together with a

¡xε

¢for example

(3).

Figure 6. The coefficients a¡xε

¢and b, and the arithmetic mean of a.

Once b has been identified, we can easily compute the solution u to thehomogenized problem (2). Now u agrees well with the solution uε, as illus-

7

trated in Figure 7 for ε = 0.05.

Figure 7. The solutions uε and u.

Since the limit problem is created in such a way that it is solved by the weakH10(Ω)-limit, and hence the strong L

2(Ω)-limit, of uε, the correspondencebetween the functions will improve for smaller values of ε. Obviously, thesecond term in (9) has captured the effect of the oscillations in a

¡xε

¢.

Thus, homogenization provides us with the G-limit, representing the ef-fective property discussed in Section 1.1. Also, the numerical computation isconsiderably simpler than solving the original equation (1) for ε close to zero,in the sense that there are no rapid oscillations in the equations we have tosolve, i.e. the local and the homogenized problem.

A classical method for derivation of the local and the homogenized prob-lem, widely used in mechanics and physics, is the asymptotic expansionmethod; see e.g. [BLP], [SaPa] and [CiDo]. With this approach, one as-sumes that the solution uε admits a two-scale asymptotic expansion of theform

uε (x) = u0

³x,

x

ε

´+ εu1

³x,

x

ε

´+ ε2u2

³x,

x

ε

´+ ..., (10)

where each uk is Y -periodic in the second argument. Plugging (10) intoequation (1) and equating equal powers of ε, we formally obtain both thelocal problem and the homogenized problem. The main features of thismethod are given in Chapter 4.

To achieve this result rigorously, one passes to the limit in (5)–makingappropriate choices of test functions v. One method from classical homog-enization theory, which uses this approach, is the method of oscillating testfunctions, originally introduced by Tartar; see e.g. [Tar1] and [CiDo]. Thistechnique is also sketched in Chapter 4.

8

Another method, a generalization of which will be used to prove some ofthe main results in this thesis, is the method based on so-called two-scaleconvergence. This type of convergence was first presented by Nguetseng in[Ngu1]. If uε is bounded in L2(Ω), it holds, up to a subsequence, that forsome u0 in L2 (Ω× Y )Z

Ω

uε (x) v³x,

x

ε

´dx→

ZΩ

ZY

u0 (x, y) v (x, y) dydx

for all v in L2 (Ω× Y ) that are sufficiently smooth and periodic in the secondvariable. The most significant property of this type of convergence is thatthe limit u0 contains both the global and the local variable. This makesthe method well suited for problems with periodically oscillating coefficients.We will return to this homogenization technique and its generalizations inChapter 5.

1.5 Outline of the thesis

In this thesis, we will study a few different equations with respect to G-convergence and homogenization. Some of the main contributions concernperiodic parabolic problems with several micro-scales, i.e., problems of theform

∂tuε (x, t)−∇ · aε (x, t,∇uε) = f (x, t) in ΩT ,

uε (x, 0) = u0 (x) in Ω, (11)

uε (x, t) = 0 on ∂Ω× (0, T ) ,

for some choices of aε with periodic oscillations that depend on ε in vari-ous ways. We also investigate some elliptic and parabolic problems withoutperiodicity assumptions.

The thesis is organized as follows. In Chapter 2 we discuss the propertiesof some monotone operators and parabolic PDEs. In particular, we will seeunder which assumptions we will have a unique solution to equation (11) forour choices of aε.

In Chapter 3 we define G-convergence for elliptic and parabolic operatorsand recall some results on the existence of the G-limit. In connection withthis, we also comment on some properties of these limits. Finally, we discussmethods to obtain the G-limit for parabolic problems from the correspondingresults for elliptic equations.

9

The most well-studied case of G-convergence is periodic homogenization.In Chapter 4, we become acquainted with a couple of classical homoge-nization techniques, i.e. the method of asymptotic expansions and Tartar’smethod of oscillating test functions. These allow an explicit characterizationof the G-limit in periodic problems and we illustrate how they can be ap-plied to homogenize the model problem (1). We then proceed with problemswith periodic oscillations on several microscopic scales. Adopting the termi-nology of [BLP], the homogenization of problems containing multiple scaleswill be referred to as reiterated homogenization. After recalling a result formonotone elliptic problems with two microscopic spatial scales, we study thecorresponding parabolic case, that is, problem (11) with

aε (x, t,∇uε) = a³t,x

ε,x

ε2,∇uε

´. (12)

We prove the existence of a G-limit for this problem and also the corre-sponding homogenization result, using the result for elliptic problems and acomparison theorem for G-limits.

Chapter 5 is devoted to the more recent homogenization method basedon the concept of two-scale convergence and its generalizations. First, wegive a brief description of two-scale convergence and demonstrate how it canbe applied for the homogenization of the model problem. We also investigatehow a mode of convergence of two-scale type can contribute to the interpre-tation of the asymptotic expansion. We then study the extension of two-scaleconvergence to the multiscale case and illustrate reiterated homogenizationfor an elliptic problem with two spatial microscopic scales. Furthermore, weintroduce evolution multiscale convergence, which allows oscillations also intime, and study the homogenization of two parabolic problems where suchoscillations occur. First, we make the preparations necessary to perform thehomogenization of these problems. In particular, we prove a compactnesstheorem for sequences of gradients under certain boundedness assumptions.We then consider a problem where there, in addition to the two spatial micro-scales, is one fast temporal scale; that is, we have chosen

aε (x, t,∇uε) = a

µx

ε,x

ε2,t

εr,∇uε

¶. (13)

We carry out the homogenization procedure for this problem, where we iden-tify local problems and characterize the limit function b for 0 < r < 3. Here

10

we distinguish three different cases: 0 < r < 2 , r = 2 and 2 < r < 3,for which we have different local problems. We also study a linear parabolicproblem which contains one spatial scale at the microscopic level and wherewe have introduced a second fast temporal scale, i.e. (11) with

aε (x, t,∇uε) = a

µx

ε,t

ε,t

εr

¶∇uε(x, t). (14)

We end this chapter by homogenizing this problem for r > 0, r 6= 1, wherewe consider the cases 0 < r < 2 with r 6= 1, r = 2 and r > 2.

Finally, in Chapter 6 we study G-convergence for certain problems wherethere are no periodicity assumptions. We investigate some linear elliptic andparabolic problems with coefficients created by means of operators of thetype

ahij(x) =

ZA

whij(x, y)Kij(x, y) dy

andahij(x, t) =

ZA

whij(x, t, y)Kij(x, t, y) dy,

respectively, where A is an open bounded set in RM and wh convergesweakly in suitable Lebesgue spaces. Here, we will see under which circum-stances we can immediately conclude that the G-limit is equal to the weaklimit of ah in L2(Ω) and L2(ΩT ), respectively.

Most results in this thesis can also be found in papers [FHOSv], [FlOl2],[FlOl3] or [HOS], which have been published in international journals, or inthe contribution [FHOSi1] to the proceedings of an international conference.The investigation in Chapter 2 of existence and uniqueness of solutions toparabolic PDEs with multiple scales is a slightly more general version of aresult found in [FlOl2]. The result in Chapter 4 on G-convergence for case(12) is proven along the lines of the proof of a theorem in [FlOl2], whereasthe homogenization result for the same problem is found in [FHOSv]. Mostresults in Chapter 5 can be found in [FlOl2] and [FlOl3]. Case (13) is treatedin [FlOl2], where both the existence of the G-limit and the characterizationof this limit by means of homogenization are investigated, and in [FlOl3] thehomogenization result for case (14) is proven. Chapter 6 is based on [HOS]and [FHOSi1] and extensions of the results found there.

11

2 Monotone operators

Some of the PDEs we will investigate in this thesis contain operators whichmay depend on∇u in a nonlinear way. In this chapter we review some theoryof monotone operators which will allow us to study such problems, especiallyregarding what we need to require to be assured of the existence of a uniquesolution.

In Section 2.1 we start out with an elementary case of a real equationcontaining a monotone function, where we know the conditions sufficientfor a unique solution to exist. We also consider the generalization of thisresult to stationary monotone operator equations, which can be applied e.g.to elliptic PDEs. Then we proceed with the case in which the monotoneoperator appears in an evolution problem in Section 2.2. Finally, we applythe result on the existence of a unique solution for general evolution problemsto study the parabolic PDE (11) for the monotone cases (12) and (13).

2.1 The concept of monotone operators

Let us start with an example from elementary calculus. Consider

A(u) = f,

where u and f are inR, and we assume that A fulfills the following conditions:

(I∗) A : R→ R is strictly monotone.

(II∗) A is continuous.

(III∗) A(u)→ ±∞ as u→ ±∞.

From (II∗) and (III∗) it follows, by the intermediate value theorem, thatthis equation has a solution for any f in R. Moreover, since A is strictlymonotone the solution is unique.

This case, which is quite easily understood, can be generalized to problemsof the type

Au = f, (15)

where u is in X, which may now be any Banach space, and f is in the dualspaceX 0. To be precise, since this equation is an equality between functionals

12

in X 0, acting on elements in X, (15) means that

hAu, viX0,X = hf, viX0,X (16)

for all v ∈ X. In other words, for u ∈ X, the corresponding operator Au ∈ X 0

should have the same effect on all v ∈ X as the functional f .

Monotonicity of the operator A : X → X 0 means that

hAu−Av, u− viX0,X ≥ 0 (17)

for all u, v ∈ X. This is of course consistent with the real case. Indeed, forX = X 0 = R, (17) simply expresses that A(u)−A(v) and u− v always havethe same sign, and hence A is increasing.

As in the real case, we need the stronger assumption of strict monotonicityto obtain uniqueness of the solution to (15). This is given in (I) below foroperators on general Banach spaces. We also give conditions (II∗) and (III∗)in generalized form.

(I) The operator A : X → X 0 is strictly monotone on X, i.e.,

hAu−Av, u− viX0,X > 0

for all u, v ∈ X.

(II) A is hemicontinuous, i.e., the function

g(δ) = hA(u+ δv), wiX0,X

is continuous on [0, 1] for all fixed u, v, w ∈ X.

(III) A is coercive, i.e.,

limkukX→∞

hAu, uiX0,X

kukX=∞. (18)

These conditions will guarantee the existence and uniqueness of the solu-tion to (15) as stated in the next theorem.

Theorem 1 If A : X → X 0 satisfies the conditions (I)-(III) and X is areal reflexive Banach space, then (15) possesses a unique solution u ∈ X.

Proof. See, e.g., Theorem 26.A in [ZeiIIB], or [Min] and [Bro].

13

Let us also mention the concept of maximal monotone operators, whichwe will refer to in Section 4.2. An operator A : X → X 0 is called maximalmonotone if for any (u, f) ∈ X ×X 0 such that

hf −Av, u− viX0,X ≥ 0

for every v in X, it follows that f = Au; see e.g. A.3.2. in [Def]. This meansthat if A is maximal monotone, we have an alternative way of expressing theequation Au = f .

The following proposition tells us that one way of knowing that an opera-tor is maximal monotone is to show that it is monotone and hemicontinuous.A proof can be found, e.g., in [Lio].

Proposition 2 Let X be a Banach space and A : X → X 0 a monotone andhemicontinuous operator. Then A is maximal monotone.

The theory of monotone operators can be applied, e.g., to elliptic PDEsof the type

−∇ · a(x,∇u) = f (x) in Ω, (19)

u (x) = 0 on ∂Ω.

More precisely, we study the weak form of (19) which states that forf ∈ H−1(Ω), we search for u in H1

0 (Ω) such thatZΩ

a (x,∇u) ·∇v (x) dx = hf, viH−1(Ω),H10 (Ω)

(20)

holds for all v ∈ H10(Ω). This means that the PDE can be interpreted as an

equality between functionals. Let us consider the operator

A : H10 (Ω)→ H−1(Ω)

given by

hA (·) , ··iH−1(Ω),H10 (Ω)

=

ZΩ

a (x,∇ (·)) ·∇ (··) dx.

Following the pattern above for the general case, we obtain for eachu ∈ H1

0 (Ω) a functional Au ∈ H−1(Ω) defined by

hAu, ·iH−1(Ω),H10 (Ω)

=

ZΩ

a (x,∇u) ·∇ (·) dx

and hence equation (20) is of the type (16).

14

If we impose certain requirements on a the operator A fulfills the sufficientconditions (I)-(III) given above, and hence there is a unique u ∈ H1

0 (Ω) suchthat (20) holds for all v ∈ H1

0 (Ω). The following conditions on the Lebesguemeasurable function

a : Ω×RN → RN ,

where α, β > 0, 0 < k1 ≤ 1 and 2 ≤ k2 < ∞ are enough to ensure this; seeLemma 1 in [LNW].

(i) a(x, 0) = 0 for a.e. x ∈ Ω.

(ii) |a (x, ξ)− a (x, ξ0)| ≤ β (1 + |ξ|+ |ξ0|)1−k1 |ξ − ξ0|k1for a.e. x ∈ Ω and for all ξ, ξ0 ∈ RN .

(iii) (a (x, ξ)− a (x, ξ0)) · (ξ − ξ0) ≥ α (1 + |ξ|+ |ξ0|)2−k2 |ξ − ξ0|k2for a.e. x ∈ Ω and for all ξ, ξ0 ∈ RN .

Remark 3 A solution u ∈ H10 (Ω) to (20) is called a weak solution to (19).

In the following, all solutions to PDEs are solutions in this sense.

Remark 4 In the discussion above we have considered the Hilbert space set-ting of elliptic PDEs; that is, for right-hand sides in H−1(Ω) and solutionsin H1

0(Ω). The corresponding can be done for equations where we considersource terms in W−1,q(Ω) and the solution is searched for in W 1,p

0 (Ω) for1 < p <∞; see, e.g., [LNW] and [ZeiIIB]. Let us also remark that the con-ditions (i)-(iii) above are sometimes restricted to special cases, which simpli-fies the form of the conditions. This will be done in Chapter 3, where we fixk2 = 2 and define G-convergence for the corresponding functions.

2.2 Monotone operators and evolution problems

In order to study equation (11), which is a time-dependent problem, we wantto extend the results in the previous section to evolution problems. We startwith a general evolution problem. The theory for the general case is thencarried over to equation (11). In particular, we will state conditions that areenough to ensure the existence of a unique solution.

In later chapters, we will see that these conditions are also sufficient toobtain certain a priori estimates and, at least for a subsequence, to haveG-convergence of the operators corresponding to (11). Except for being aprerequisite for G-convergence, the a priori estimates are necessary to per-form the homogenization procedures in Section 5.3.

15

2.2.1 General evolution problems

Let us consider the problem

∂tu (t) +A (t)u (t) = f(t) for a.e. t ∈ (0, T ) , (21)

u (0) = u0,

for f ∈ L2 (0, T ;V 0) and u0 ∈ H, where H is a Hilbert space containing theBanach space V and

A(t) : V → V 0

creates operators A(t)u ∈ V 0 for each t ∈ (0, T ) and any u ∈ V . Our aim isto identify conditions on A that provide (21) with a unique solution

u ∈ H1 (0, T ;V, V 0) =©u ∈ L2(0, T ;V )|∂tu ∈ L2(0, T ;V 0)

ª.

The two spaces V and H that appear in this evolution problem are as-sumed to fulfill the conditions of a so-called evolution triple, which can bedescribed as follows. Consider a real, separable and reflexive Banach space Vand a real, separable Hilbert space H such that V is continuously embeddedin H; that is, V ⊆ H and

kukH ≤ C kukVfor all u ∈ V . Furthermore, let V be dense in H. We then say that

V ⊆ H ⊆ V 0

forms an evolution triple.

Note also that the derivative ∂tu should be understood as a generalizedderivative; see Section 23.5 in [ZeiIIA]. This will also be the interpretationthroughout the thesis.

Again, the problem (21) should be interpreted as that we are searchingfor u ∈ H1 (0, T ;V, V 0) such that for almost every t ∈ (0, T ), the equation

h∂tu, viV 0,V + hA(t)u, viV 0,V = hf(t), viV 0,V , (22)

u (0) = u0

holds for all v ∈ V .

We assume that the operator A has the following properties, where thefirst three essentially correspond to the conditions listed for the stationarycase, and the two additional ones concern the time-dependence.

16

(I) The operator A(t) : V → V 0 is monotone, i.e.

hA(t)u−A(t)v, u− viV 0,V ≥ 0 (23)

for all u, v ∈ V and any t ∈ (0, T ).

(II) A(t) is hemicontinuous; that is, the function

g1(δ) = hA(t)(u+ δv), wiV 0,V

is continuous on [0, 1] for all fixed u, v, w ∈ V and any t ∈ (0, T ).

(III) A(t) is coercive; that is, it satisfies

hA(t)u, uiV 0,V ≥ C1 kuk2V (24)

for all u ∈ V , some C1 > 0 and any t ∈ (0, T ).

(IV ) A(t) satisfies the growth condition that there exists a non-negativefunction g2 ∈ L2(0, T ) and a constant C2 > 0 such that

kA(t)ukV 0 ≤ g2(t) + C2 kukV

for all u ∈ V and any t ∈ (0, T ).

(V ) The functiong3(t) = hA(t)u, viV 0,V

is measurable on (0, T ) for all fixed u, v ∈ V .

Theorem 5 Consider the initial boundary value problem (21) whereV ⊆ H ⊆ V 0 forms an evolution triple. If A satisfies the conditions (I)−(V ),this problem possesses a unique solution u ∈ H1 (0, T ;V, V 0).

Proof. See Theorem 30.A. in [ZeiIIB].

Remark 6 Condition (III) above is somewhat stronger than in Theorem30.A. in [ZeiIIB], to be in better accordance with our investigations in theremainder of the thesis. Let us also mention that the coercivity condition ofthe form (18) used in the stationary case follows from (24) if we divide bythe norm of u in the latter form. Note also that this evolution setting doesnot require strict monotonicity; see Section 30.3a. in [ZeiIIB].

17

2.2.2 Parabolic PDEs with multiple scales

The theory for general evolution problems from the preceding section can beapplied to the study of equations of the character

∂tu (x, t)−∇ · a (x, t,∇u) = f (x, t) in ΩT ,

u (x, 0) = u0 (x) in Ω, (25)

u (x, t) = 0 on ∂Ω× (0, T ) ,i.e. parabolic PDEs, which is a special case of the equation given in (21). Tostudy a problem of this type, we may choose V to be H1

0 (Ω), which is a real,separable and reflexive Banach space. Also, we can choose H to be equal toL2(Ω) since this space fulfills the properties required. It is a real, separableHilbert space and, moreover, it holds that H1

0(Ω) is dense in L2(Ω) and, bythe Poincaré inequality,

kukL2(Ω) ≤ C kukH10 (Ω)

for all u ∈ H10 (Ω). Hence, we can form the evolution triple

H10(Ω) ⊆ L2(Ω) ⊆ H−1(Ω).

As in the stationary case, it is possible to specify conditions for a such that(25) has a unique solution for any f ∈ L2 (0, T ;H−1(Ω)) and any u0 ∈ L2(Ω).

In this thesis we will investigate a couple of different equations of thetype (25), which are governed by operators containing rapid oscillations inspace or in both time and space, i.e. problem (11) for the cases (12) and(13). These problems are covered by the equation

∂tuε (x, t)−∇ · a

µx, t,

x

ε,x

ε2,t

εr,∇uε

¶= f (x, t) in ΩT ,

uε (x, 0) = u0 (x) in Ω, (26)

uε (x, t) = 0 on ∂Ω× (0, T )and the purpose of this section is to list sufficient criteria for this equationto possess a unique solution for any f ∈ L2 (0, T ;H−1(Ω)) and u0 ∈ L2(Ω).

Rewriting (26) in the weak form, we have thatZΩT

−uε (x, t) v (x) ∂tc (t) + a

µx, t,

x

ε,x

ε2,t

εr,∇uε

¶·∇v (x) c (t) dxdt = (27)Z T

0

hf (t) , viH−1(Ω),H10 (Ω)

c (t) dt

18

for all v ∈ H10 (Ω) and c ∈ D (0, T ), where the solution is searched for in

H1 (0, T ;H10 (Ω),H

−1(Ω)) and uε (x, 0) = u0(x). Thus, the correspondenceto A(t) in (22) is

Aε(t) : H10(Ω)→ H−1(Ω)

defined by

hAε(t) (·) , ··iH−1(Ω),H10 (Ω)

=

ZΩ

a

µx, t,

x

ε,x

ε2,t

εr,∇ (·)

¶·∇ (··) dx,

which for any u ∈ H10 (Ω) creates operators A

ε(t)u in H−1(Ω).

Remark 7 Equation (27) is equivalent to a corresponding equality betweenoperators in L2(0, T ;H−1(Ω)) acting on test functions v ∈ L2(0, T ;H1

0(Ω)).The corresponding holds true for general operator equations of the type inSection 2.2.1 acting on some suitable space V of the kind introduced thereinstead of H1

0(Ω); see Theorem 30.A. (c) in [ZeiIIB].

We will see that the operators Aε satisfy the conditions (I)-(V ) for everyfixed ε > 0, if the function a fulfills the structure conditions given below. Thismeans that there is a unique uε ∈ H1 (0, T ;H1

0(Ω), H−1(Ω)) for every ε > 0

such that uε (x, 0) = u0(x) and (27) holds for all v ∈ H10 (Ω), c ∈ D (0, T ),

i.e. (26) has a unique weak solution.

We assume that

a : RN × R+ ×R2N ×R×RN → RN

agrees with the following conditions, where 0 < k ≤ 1 and α and β arepositive constants:

(i) a (x, t, y1, y2, s, 0) = 0 for all (x, t, y1, y2, s) ∈ RN ×R+ ×R2N ×R.

(ii) a (·, ·, ·, ·, ·, ξ) is Y2,1-periodic in (y1, y2, s) and continuous for all ξ ∈ RN .

(iii) a (x, t, y1, y2, s, ·) is continuous for all (x, t, y1, y2, s) ∈ RN×R+×R2N×R.

(iv) |a (x, t, y1, y2, s, ξ)− a (x, t, y1, y2, s, ξ0)| ≤ β (1 + |ξ|+ |ξ0|)1−k |ξ − ξ0|k

for all (x, t, y1, y2, s) ∈ RN × R+ ×R2N ×R and all ξ, ξ0 ∈ RN .

(v) (a (x, t, y1, y2, s, ξ)− a (x, t, y1, y2, s, ξ0)) · (ξ − ξ0) ≥ α |ξ − ξ0|2

for all (x, t, y1, y2, s) ∈ RN × R+ ×R2N ×R and all ξ, ξ0 ∈ RN .

19

Theorem 8 If a satisfies (i)-(v), then the problem (26) possesses a uniquesolution uε ∈ H1 (0, T ;H1

0 (Ω), H−1(Ω)) for every fixed ε > 0.

Proof. We will show that Aε satisfies the conditions (I)-(V ) in Section 2.2.1for every ε > 0 and hence the problem (26) will have a unique solution byTheorem 5.

To show that condition (I) is satisfied, we use the knowledge that a fulfillscondition (v), which means thatZΩ

µa

µx, t,

x

ε,x

ε2,t

εr,∇u

¶−aµx, t,

x

ε,x

ε2,t

εr,∇v

¶¶·∇ (u (x)− v (x)) dx≥ (28)

α

ZΩ

|∇u (x)−∇v (x)|2 dx = α ku− vk2H10 (Ω)≥ 0

and hencehAε(t)u−Aε(t)v, u− viH−1(Ω),H1

0 (Ω)≥ 0

for all u, v ∈ H10(Ω).

From (iv) it follows that¯hAε(t)(u+ δv), wiH−1(Ω),H1

0 (Ω)− hAε(t)(u+ δ0v), wiH−1(Ω),H1

0 (Ω)

¯=¯Z

Ω

µa

µx, t,

x

ε,x

ε2,t

εr,∇u+ δ∇v

¶−aµx, t,

x

ε,x

ε2,t

εr,∇u+ δ0∇v

¶¶·∇w(x)dx

¯≤Z

Ω

¯a

µx, t,

x

ε,x

ε2,t

εr,∇u+ δ∇v

¶−aµx, t,

x

ε,x

ε2,t

εr,∇u+ δ0∇v

¶¯|∇w(x)| dx≤Z

Ω

β (1+|∇u(x)+ δ∇v(x)|+|∇u(x)+ δ0∇v(x)|)1−k|(δ− δ0)∇v(x)|k |∇w(x)|dx

which tends to zero as δ − δ0 does, and thus condition (II) is satisfied.

To prove that condition (III) is fulfilled, we make use of (28). Choosingv = 0 and using (i), we get

hAε(t)u−Aε(t)0, u− 0iH−1(Ω),H10 (Ω)

=ZΩ

µa

µx, t,

x

ε,x

ε2,t

εr,∇u

¶− a

µx, t,

x

ε,x

ε2,t

εr, 0

¶¶·∇u (x) dx =Z

Ω

a

µx, t,

x

ε,x

ε2,t

εr,∇u

¶·∇u (x) dx ≥ α kuk2H1

0 (Ω),

20

that ishAε(t)u, uiH−1(Ω),H1

0 (Ω)≥ α kuk2H1

0 (Ω)

and hence the operator is coercive.

Aε also satisfies condition (IV ). First, we observe that

kAε(t)ukH−1(Ω) = supkvk

H10(Ω)≤1

¯hAε(t)u, viH−1(Ω),H1

0 (Ω)

¯=

supkvk

H10(Ω)≤1

¯ZΩ

a

µx, t,

x

ε,x

ε2,t

εr,∇u

¶·∇v (x) dx

¯≤

supkvk

H10(Ω)≤1

ZΩ

¯a

µx, t,

x

ε,x

ε2,t

εr,∇u

¶¯|∇v (x)| dx.

From (iv) it follows, by choosing ξ0 = 0 and using (i), that

|a (x, t, y1, y2, s, ξ)| ≤ β (1 + |ξ|)1−k |ξ|k < β (1 + |ξ|)1−k (1 + |ξ|)k

and hence|a (x, t, y1, y2, s, ξ)| ≤ β (1 + |ξ|) . (29)

Thus, by (29) and the Hölder inequality we get

supkvk

H10(Ω)≤1

ZΩ

¯a

µx, t,

x

ε,x

ε2,t

εr,∇u

¶¯|∇v (x)| dx ≤

supkvk

H10(Ω)≤1

ZΩ

β (1 + |∇u (x)|) |∇v (x)| dx ≤

supkvk

H10(Ω)≤1

β k1 + |∇u|kL2(Ω) k∇vkL2(Ω)N = β k1 + |∇u|kL2(Ω) .

Hence, by the triangle inequality,

kAε(t)ukH−1(Ω) < β k1 + |∇u|kL2(Ω) ≤

β³k1kL2(Ω) + k∇ukL2(Ω)N

´= β1 + β kukH1

0 (Ω)

and since β and β1 are non-negative constants, condition (IV ) is met.

21

From the continuity assumptions on a and the growth condition (29), itfollows that a

¡x, t, x

ε, xε2, tεr,∇u

¢belongs to L2(ΩT )

N . Thus, for v ∈ H10 (Ω)

g3(t) =

ZΩ

a

µx, t,

x

ε,x

ε2,t

εr,∇u

¶·∇v(x) dx

is in L1(0, T ) and hence obviously measurable, which means that condition(V ) is fulfilled.

Remark 9 We have established the existence of a unique solution to (11) forthe possibly nonlinear monotone cases (12) and (13). Case (14) can easilybe included in the theorem and proof above, just by adding a third temporalscale. However, for this case linear theory is enough. Consider the linearparabolic system

∂tu (x, t)−∇ · (a (x, t)∇u(x, t)) = f (x, t) in ΩT ,

u (x, 0) = u0 (x) in Ω, (30)

u (x, t) = 0 on ∂Ω× (0, T ) ,

where f ∈ L2(0, T ;H−1(Ω)) and u0 ∈ L2(Ω). Here, the conditions are madedirectly on the coefficient and are the following. We require thata ∈ L∞(ΩT )

N×N and satisfies the coercivity condition

a (x, t) ξ · ξ ≥ α |ξ|2

for some α > 0, a.e. (x, t) ∈ ΩT and all ξ ∈ RN . Under these assumptions,(30) has a unique solution u ∈ H1 (0, T ;H1

0 (Ω),H−1(Ω)); see Proposition

23.30 in [ZeiIIA].

Studying problem (11) with aε given by (14), we have a problem of theform (30) with the coefficient a

¡xε, tε, tεr

¢. We assume that the function

a : RN × R2 → RN×N

belongs to L∞ (Y1,2)N×N and fulfills

a (y, s1, s2) ξ · ξ ≥ α |ξ|2

for some α > 0, all (y, s1, s2) ∈ RN×R2 and all ξ ∈ RN . From this it is easilyseen that the conditions above are satisfied for our coefficient a

¡xε, tε, tεr

¢with

oscillations depending on ε, for every ε > 0. Hence, we have existence anduniqueness of a solution uε ∈ H1 (0, T ;H1

0(Ω), H−1(Ω)) to (11) also for case

(14).

22

Remark 10 Monotone elliptic and parabolic problems of the kind consideredin this chapter have been carefully studied by Zeidler in [ZeiIIA] and [ZeiIIB]for both linear and nonlinear cases.

23

3 G-convergence

The mathematical discipline of homogenization dates back to the late 1960swhen Spagnolo introduced a type of convergence of operators called G-convergence. As noted in Chapter 1, homogenization is a special case ofG-convergence. Before we discuss this concept of convergence in more math-ematical terms, let us consider an illustration, which highlights the sense inwhich G-convergence is more general than periodic homogenization.

The sequence of equations studied in Section 1.1 could be regarded asdescribing a phenomenon in a series of materials with a successively finerperiodic microstructure. In a more general context we can imagine that wehave a sequence of materials that are not periodically built up, but where asimilar kind of stabilization takes place. This could for example look like thecase illustrated in Figure 8.

Figure 8. Stabilization towards a limit structure.

Thus, except for omitting the periodicity assumption, we can allow sequencessuch that the limit structure is not homogeneous.

Let us now make a little more precise what we mean by convergence ofoperators. To this end, we consider a sequence of equations of the form

Ahuh = f

with unique solutions uh belonging to some function space X. For a sequenceof operators Ah, we obtain a corresponding sequence of solutions uh to theseequations. If we look at this from a slightly different angle, this means thatwe have a sequence of operators Ah that determine a sequence of functionsuh such that the same image f is obtained. If the sequence uh stabilizes toa limit u in X, i.e. if

uh → u

in some reasonable sense, one might wonder if there is an operator B thatproduces the same image f of this limit function u, i.e. if for some B

Bu = f. (31)

24

If this is the case with the same B for any f that may be in question, Bcould be seen as the limit of Ah regarding the capability of transformingfunctions in X, i.e. in the sense of operators.

When searching for the limit equation solved by the limit function u, wehave to deal with two issues. The first one is to find assumptions underwhich there exists a limit B, with properties such that (31) has a uniquesolution. The second difficulty is the one of determining B, and has beenstudied mainly for periodic problems. In this chapter, we deal primarily withthe first issue.

Although it has been defined for abstract operators, G-convergence ismostly considered for partial differential operators, for which it was originallyintroduced. This will also be the interest in the remainder of this chapter.In Section 3.1, we define the notion of G-convergence for elliptic operatorsand see some results on G-convergence compactness, which assure us of theexistence of a well-posed limit problem for certain sequences of elliptic PDEs.In Section 3.2, we do the corresponding for parabolic operators.

Remark 11 For elliptic and parabolic PDEs, we may consider G-convergenceas convergence of sequences of operators of the types

Ah(·) = −∇ · ah (x,∇(·))and

Ah(·) = ∂t(·)−∇ · ah (x, t,∇(·)) ,respectively, in the sense described in the reasoning above. For such equationsthis is, however, often expressed as that the sequence of functions ah G-converges, or the sequence of coefficients in ah if ah is linear. The operatornotion is perhaps the most consistent one since it is not restricted to PDEsof this type. However, since the differential operators above are determinedby ah, there should be no confusion about which operators are in question ifwe speak in terms of ah, as long as we know whether we are in the ellipticor the parabolic setting. Also, since this is more convenient, we will use thisway of expressing ourselves in what follows.

3.1 G-convergence for elliptic operators

In this section we start by discussing linear elliptic problems regarding theprospect of finding a limit in the sense described above. We recall the defin-ition of G-convergence for such problems and some results concerning the

25

assumptions under which there exists a G-limit. We also discuss some prop-erties of this limit. Then we consider G-convergence for possibly nonlinearmonotone elliptic problems, where similar conditions can also be identified.

3.1.1 Linear elliptic operators

Let us consider a sequence of equations of the type

−∇ ·¡ah (x)∇uh (x)

¢= f (x) in Ω, (32)

uh (x) = 0 on ∂Ω,

where we let f ∈ L2(Ω), and ah is assumed to fulfill conditions that guaranteethe existence of a unique solution and also an appropriate a priori estimatefor the sequence of solutions. To be precise, we assume that the coefficientsare functions

a : Ω→ RN×N

satisfying the following conditions on boundedness and coercivity, where αand β are constants such that 0 < α ≤ β <∞:(M1) a ∈ L∞(Ω)N×N .

(M2) |a (x) ξ| ≤ β |ξ| for a.e. x ∈ Ω and for all ξ ∈ RN .

(M3) a (x) ξ · ξ ≥ α |ξ|2 for a.e. x ∈ Ω and for all ξ ∈ RN .

We name the set of such functionsM(α, β,Ω).

From (M3) it follows that the sequence of solutions uh is bounded inH10(Ω), that is, °°uh°°

H10 (Ω)≤ C

for some positive constant C; see Theorem 4.16 in [CiDo]. Hence, there is afunction u such that, for a subsequence,

uh (x) u (x) in H10(Ω).

If there is a function b such that u uniquely solves the problem

−∇ · (b (x)∇u (x)) = f (x) in Ω,

u (x) = 0 on ∂Ω,

we could regard b as the limit of this subsequence of ah in the capacityof representing an operator, following the more abstract reasoning in theintroduction to this chapter.

26

We recall that we are actually studying the weak form of (32), i.e.ZΩ

ah (x)∇uh (x) ·∇v (x) dx =

ZΩ

f (x) v (x) dx (33)

for all v ∈ H10(Ω). Thus, we are looking for a limit problem of the formZ

Ω

b (x)∇u (x) ·∇v (x) dx =

ZΩ

f (x) v (x) dx. (34)

Remark 12 There would be nothing to hinder us from passing to the limitin (33) if

ah (x)→ a (x) in L∞(Ω)N×N .

We can then easily deduce that

ah (x)∇uh (x) a (x)∇u (x) in L2(Ω)N .

Hence, u solves the problemZΩ

a (x)∇u (x) ·∇v (x) dx =

ZΩ

f (x) v (x) dx

for all v ∈ H10 (Ω). Since a belongs to M(α, β,Ω) (see Remark 6.1 in [Def]

and Remark 5.1 in [CiDo]), u is the unique solution. Thus, in this case wesimply have that b = a.

From the properties of ah, we can only establish weak* convergence inL∞(Ω)N×N . Thus, in general we face the same problem as in Section 1.2,that we have a product of two only weakly convergent sequences, and hencewe cannot proceed in as straightforward a manner as in Remark 12.

What we can say is that the sequence of products has a weak L2(Ω)N -limit. From the a priori estimate of uh and the condition (M2), it followsthat

©ah (x)∇uh (x)

ªis bounded in L2(Ω)N (see Section 5.1 in [CiDo]) and

thus there is some σ ∈ L2(Ω)N such that, up to a subsequence,

ah (x)∇uh (x) σ (x) in L2(Ω)N .

Hence, for such a subsequence we can pass to the limit in (33) obtainingZΩ

σ (x) ·∇v (x) dx =

ZΩ

f (x) v (x) dx

for all v ∈ H10(Ω).

27

What remains to be investigated is whether we can express σ in terms ofu in an appropriate way, i.e., if for some b

σ (x) = b (x)∇u (x) ,

where b is of the same character as ah, so that there is in fact a well-posedlimit problem of the desired form (34).

The first important results on this topic were given by Spagnolo in [Spa1]and [Spa2], where G-convergence was introduced for linear elliptic and par-abolic operators with symmetric coefficients. We will reproduce a result forthe linear elliptic case which shows that, for such operators, the assump-tion that ah belongs toM(α, β,Ω) is actually enough to secure the existenceof a well-posed limit problem, at least for a subsequence. Thus, followingSpagnolo, we first consider problems where the coefficients ah are symmetric,i.e.

ahij (x) = ahji (x)

for all x ∈ Ω. Let us also state the exact definition of G-convergence for suchmatrices.

Definition 13 A sequence ah of symmetric matrices belonging toM(α,β,Ω) is said to G-converge to the symmetric matrix b ∈M(α0, β0,Ω)(called the G-limit) if, for every f ∈ H−1(Ω), the sequence uh of solutionsto the equations

−∇ ·¡ah (x)∇uh (x)

¢= f (x) in Ω,

uh (x) = 0 on ∂Ω,

satisfiesuh (x) u (x) in H1

0 (Ω), (35)

where u is the unique solution to

−∇ · (b (x)∇u (x)) = f (x) in Ω,

u (x) = 0 on ∂Ω.

The following result on the compactness of the set of symmetric matricesinM(α, β,Ω) with respect to G-convergence is proven in [Spa2].

Theorem 14 Any sequence ah of symmetric matrices inM(α, β,Ω) pos-sesses a subsequence G-converging to some limit b in the same set.

28

Under the name of H-convergence, a generalization of G-convergence tothe non-symmetric case has been made by Murat and Tartar. The definitionof H-convergence requires–besides (35), i.e. the weak convergence of uhin H1

0 (Ω)–that

ah (x)∇uh (x) b (x)∇u (x) in L2(Ω)N . (36)

The reason is that if we omit the symmetry assumption in Definition 13, itmay happen that the limit b is not unique. An illustrative example of this canbe found in [Def]. For a G-convergent sequence of symmetric matrices theproperty (36) always holds true, and for such matrices G- andH-convergenceare equivalent.

Remark 15 Since H-convergence is a natural generalization of G-conver-gence rather than a different kind of convergence, in the remainder of thisthesis we will keep to the term G-convergence for both symmetric and non-symmetric cases.

The setM(α, β,Ω), including non-symmetric matrices, is compact withrespect to G-convergence in the sense given in the following theorem, provenin [MuTa].

Theorem 16 Any sequence ah in M(α, β,Ω) has a subsequence that G-converges to some limit b inM(α, β

2

α,Ω).

Note that in this case, though preserving the properties of boundednessand coercivity, the limit belongs to a larger set, unless α = β.

For problems with sequences of coefficients belonging toM(α, β,Ω), thecompactness results mentioned guarantee the existence of a well-posed limitproblem, at least for a subsequence. To draw the conclusion that the entiresequence G-converges, and not only a subsequence, we need the followingtheorem; see e.g. [Spa2].

Theorem 17 A sequence in M(α, β,Ω) G-converges if and only if all G-convergent subsequences have the same limit.

The results above can be applied, for example, to periodic problems likethe model problem (1). This problem is treated in Sections 4.1, 4.2 and 5.1.2,where we also find an explicit expression for the G-limit. In Section 5.2.2,we give a similar result for a periodic problem with two microscopic scales.Another problem with coefficients in M(α, β,Ω), and without periodicityassumptions, is investigated in Section 6.1.

29

In Remark 12, we saw a situation where we could immediately determinethe G-limit. A similar result can also be obtained under some more modestassumptions, as stated in the proposition below.

Proposition 18 Let ah be a sequence in M(α, β,Ω). If ah convergesto a either in L1(Ω)N×N or almost everywhere in Ω, then ah G-convergesto a.

Proof. See the proof of Lemma 1.2.22 in [All3].

We end this section by listing some important results on properties pos-sessed by the G-limit. These qualities are required from all types of G-convergence in order to obtain a meaningful concept of convergence.

• The limit of a G-convergent sequence is unique.

• By definition, the G-limit does not depend on the source term f .

• The G-limit does not depend on the boundary conditions.

• Assume that ah and ah are two sequences in M(α, β,Ω), whichG-converge to b and b, respectively. If, for some open subset ω in Ω,

ah (x) = ah (x) in ω,

thenb (x) = b (x) in ω.

• If a sequence ah G-converges to a limit b in Ω, then ah G-convergesto the same limit in any subset of Ω (referring now to the restrictionsto the subset).

Remark 19 The properties listed above are of interest in the case of comput-ing effective properties. The limit coefficient is unique and the determinationcan be done once and for all. This is, of course, a desirable quality. We donot want to derive effective properties over and over if we would like to studya problem for different source terms. Nor do we want the derived coefficientto depend on the sample size or on the boundary conditions, following thesame reasoning.

30

3.1.2 Monotone elliptic operators

G-convergence can be formulated analogously for possibly nonlinear monotoneelliptic problems under certain assumptions on the governing function a,such as strong monotonicity and a certain continuity condition. We defineN (α, β, k,Ω) to be the set of functions

a : Ω×RN → RN

such that, for constants α and β satisfying 0 < α ≤ β < ∞ and 0 < k ≤ 1,it holds that:

(N1) a(x, 0) = 0 for a.e. x ∈ Ω.

(N2) a (·, ξ) is Lebesgue measurable for every ξ ∈ RN .

(N3) |a (x, ξ)− a (x, ξ0)| ≤ β (1 + |ξ|+ |ξ0|)1−k |ξ − ξ0|kfor a.e. x ∈ Ω and for all ξ, ξ0 ∈ RN .

(N4) (a (x, ξ)− a (x, ξ0)) · (ξ − ξ0) ≥ α |ξ − ξ0|2for a.e. x ∈ Ω and for all ξ, ξ0 ∈ RN .

We are now prepared for the following definition.

Definition 20 A sequence ah in N (α, β, k,Ω) is said to G-converge tob ∈ N (α0, β0, k0,Ω) if, for every f ∈ H−1(Ω), the sequence uh of solutionsto the equations

−∇ · ah¡x,∇uh

¢= f (x) in Ω,

uh (x) = 0 on ∂Ω,

satisfies

uh (x) u (x) in H10(Ω),

ah¡x,∇uh

¢b (x,∇u) in L2(Ω)N ,

where u is the unique solution to

−∇ · b (x,∇u) = f (x) in Ω,

u (x) = 0 on ∂Ω.

31

For k = 1, the condition (N3) becomes more transparent and takes theform

|a (x, ξ)− a (x, ξ0)| ≤ β |ξ − ξ0|and we denote the corresponding set of functions by N (α, β,Ω). This is theclass that was studied by Tartar in the original result for possibly nonlinearmonotone problems. The theorem below is proven in [Tar1].

Theorem 21 Every sequence ah belonging to N (α, β,Ω) G-converges, upto a subsequence, to some b in N (α, β2

α,Ω).

A similar compactness result for the set N (α, β, k,Ω) can be concludedfrom Lemma 2.3.2 in [Pan].

Theorem 22 Any sequence ah in N (α, β, k,Ω) has a subsequence that G-converges to some b ∈ N (α, β, k,Ω), where k = k/(2 − k) and α and β arepositive constants depending only on α and β.

In Section 4.2, we study the monotone correspondence to the model prob-lem with ah in N (α, β,Ω), where the G-limit is determined by classical ho-mogenization techniques. In Section 4.3, we give an account of a resultwhere the G-limit is characterized for another periodic problem with ah inN (α, β, k,Ω) and where oscillations occur on two microscopic scale levels.Remark 23 G-convergence of linear elliptic operators is investigated in e.g.[Spa2] and [Spa3] by Spagnolo, in [GiSp] by De Giorgi and Spagnolo, andby Jikov et al. in [JKO]. The nonlinear case was first studied by Tartarin [Tar1]. A reproduction of the proof of the compactness result can also befound in [Del]. G-convergence of nonlinear monotone, possibly multivaluedoperators is treated by Chiadò Piat et al. in [CDD] and [ChDe]. For somefundamental results onH-convergence and its theoretical background, we referto [Mur1] and [Mur2] by Murat, [Tar1] and [Tar2] by Tartar, and [MuTa] byMurat and Tartar. See also the book [Eva] by Evans.

3.2 G-convergence for parabolic operators

The notion of G-convergence as defined in Section 3.1 can also be extended toparabolic problems. As we did for elliptic operators, we first consider linearproblems and then we proceed with possibly nonlinear monotone problems.Finally, we consider a result which, under certain assumptions, allows us toobtain the G-limit in the sense of parabolic operators if we know the G-limitin the elliptic sense.

32

3.2.1 Linear parabolic operators

As in the linear elliptic case, we first define the set of coefficients we willconsider. LetM(α, β,ΩT ) be the set of functions

a : ΩT → RN×N

satisfying the following conditions, where α and β are constants such that0 < α ≤ β <∞:(M1) a ∈ L∞(ΩT )

N×N .

(M2) |a (x, t) ξ| ≤ β |ξ| for a.e. (x, t) ∈ ΩT and for all ξ ∈ RN .

(M3) a (x, t) ξ · ξ ≥ α |ξ|2 for a.e. (x, t) ∈ ΩT and for all ξ ∈ RN .

G-convergence for linear parabolic operators is defined as follows.

Definition 24 A sequence ah in M(α, β,ΩT ) is said to G-converge tob ∈ M(α0, β0,ΩT ) if, for every f ∈ L2 (0, T ;H−1(Ω)) and u0 ∈ L2(Ω), thesequence uh of solutions to the equations

∂tuh (x, t)−∇ ·

¡ah (x, t)∇uh (x, t)

¢= f (x, t) in ΩT ,

uh (x, 0) = u0 (x) in Ω,

uh (x, t) = 0 on ∂Ω× (0, T ) ,satisfies

uh (x, t) u (x, t) in L2¡0, T ;H1

0 (Ω)¢,

ah (x, t)∇uh (x, t) b (x, t)∇u (x, t) in L2 (ΩT )N ,

where u is the unique solution to

∂tu (x, t)−∇ · (b (x, t)∇u (x, t)) = f (x, t) in ΩT ,

u (x, 0) = u0 (x) in Ω,

u (x, t) = 0 on ∂Ω× (0, T ) .The definition is motivated by the following compactness result for the

setM(α, β,ΩT ); see [Spa4] and [Sva1].

Theorem 25 A sequence ah belonging toM(α, β,ΩT ) possesses a subse-quence that G-converges to some b inM(α, β

2

α,ΩT ).

This will be applied to a periodic problem with two scales in space andthree scales in time in Section 5.3.3, and to a problem without periodicityassumptions in Section 6.2.

33

3.2.2 Monotone parabolic operators

For our purposes, we also need to define G-convergence of parabolic opera-tors for the monotone possibly nonlinear case, which we formulate followingSvanstedt; see [Sva1]. We define N (α, β, k,ΩT ) as the set of functions

a : ΩT × RN → RN

satisfying the following conditions, where 0 < k ≤ 1 and α and β are positiveconstants:

(N1) a (x, t, 0) = 0 for a.e. (x, t) ∈ ΩT .

(N2) a (·, ·, ξ) is Lebesgue measurable for every ξ ∈ RN .

(N3) |a (x, t, ξ)− a (x, t, ξ0)| ≤ β (1 + |ξ|+ |ξ0|)1−k |ξ − ξ0|kfor a.e. (x, t) ∈ ΩT and for all ξ, ξ

0 ∈ RN .

(N4) (a (x, t, ξ)− a (x, t, ξ0)) · (ξ − ξ0) ≥ α |ξ − ξ0|2for a.e. (x, t) ∈ ΩT and for all ξ, ξ

0 ∈ RN .

The definition of G-convergence for the monotone parabolic case is givenbelow.

Definition 26 A sequence ah in N (α, β, k,ΩT ) is said to G-converge tob ∈ N (α0, β0, k0,ΩT ) if, for every f ∈ L2 (0, T ;H−1(Ω)) and u0 ∈ L2(Ω), thesequence uh of solutions to the equations

∂tuh (x, t)−∇ · ah

¡x, t,∇uh

¢= f (x, t) in ΩT ,

uh (x, 0) = u0 (x) in Ω,

uh (x, t) = 0 on ∂Ω× (0, T ) ,

satisfies

uh (x, t) u (x, t) in L2¡0, T ;H1

0(Ω)¢,

ah¡x, t,∇uh

¢b (x, t,∇u) in L2 (ΩT )

N ,

where u is the unique solution to

∂tu (x, t)−∇ · b (x, t,∇u) = f (x, t) in ΩT ,

u (x, 0) = u0 (x) in Ω,

u (x, t) = 0 on ∂Ω× (0, T ) .

34

As proven in [Sva1], the set N (α, β, k,ΩT ) is sequentially compact in thesense given in the following theorem.

Theorem 27 A sequence ah in N (α, β, k,ΩT ) possesses a subsequencethat G-converges to a limit b ∈ N (α, β, k,ΩT ), where k = k/(2 − k) andβ is a positive constant depending on α, β and k only.

Proof. See the proof of Theorem 5.2 in [Sva1].

This will be used for example in Section 4.3, where we proveG-convergencefor a monotone parabolic problem with multiple scales in space. A similarproblem with several scales in both time and space is investigated in Section5.3.2.

The G-limits for parabolic operators will have properties essentially cor-responding to those of the limit in the linear elliptic case, listed in Section3.1. For the parabolic case, we can also add the independence of the timeinterval; see [Sva1] and [Sva2].

So far, we have considered some fundamental results on existence andproperties of the G-limit for sequences of elliptic and parabolic operators.There exist also results on comparisons between the G-limits in these twocases. These make it possible to obtain the G-limit for a parabolic case oncethe corresponding elliptic result has been established, provided that we makea certain continuity assumption for the t-dependence. We have the followingresult for monotone problems.

Theorem 28 Let B : R+ → R+ be an increasing continuous function suchthat B(t)→ 0 as t→ 0+. Assume that¯

ah(x, t, ξ)− ah(x, t0, ξ)¯≤ B(t− t0)(1 + |ξ|) (37)

for all ξ ∈ RN , a.e. x ∈ Ω and all t and t0 such that 0 < t0 < t < T . Assumealso that ah G-converges to b in the sense of parabolic operators, and thatah(t) G-converges to b0(t) in the sense of elliptic operators for every fixedt ∈ (0, T ). Then b = b0.

Proof. See Theorem 6.6 in [Sva1].

We will return to this in Section 4.3, where we use this result to provehomogenization of the parabolic problem with several spatial scales men-tioned above. In this connection, we also need the following theorem onG-convergence for parameter-dependent elliptic problems, that is, ellipticproblems where t appears in ah in the role of a parameter.

35

Theorem 29 Let B be as in Theorem 28. Assume that ah(t) belongs toN (α, β, k,Ω) for every fixed t ∈ (0, T ) and that¯

ah(x, t, ξ)− ah(x, t0, ξ)¯≤ B(t− t0)(1 + |ξ|)

for all ξ ∈ RN , a.e. x ∈ Ω and all t and t0 such that 0 < t0 < t < T . Then,there exists a subsequence such that ah(t) G-converges to a function b (t),with the same subsequence for all t ∈ (0, T ).Proof. See Theorem 4.2 in [Sva1].

Remark 30 Results of comparisons for the linear case, similar to the onein Theorem 28, can be found in [CoSp]. Similar results are also contained in[Par]. A linear correspondence to Theorem 29 is given in [CoSp].

Remark 31 Parabolic G-convergence for linear problems is studied by Spag-nolo in [Spa1], [Spa2] and [Spa4], and by Colombini and Spagnolo in [CoSp].See also [ZKO] by Zhikov et al. Results on G-convergence for the possiblynonlinear monotone parabolic case are proven by Svanstedt in [Sva1] and[Sva2], and in [Pan] by Pankov.

36

4 Classical homogenization techniques

For problems with ah in the sets defined in the preceding chapter, we areguaranteed G-convergence up to a subsequence, i.e. the existence of a limitproblem for this subsequence, where the G-limit b has the desired properties.There are, however, not many clues as to how to compute this limit. In theremainder of the thesis we will proceed with this issue and develop techniquesfor the characterization of the limit operator. In this chapter we will see acouple of classical methods to find b explicitly in periodic problems. In otherwords, we now turn our interest to periodic homogenization.

Let us again consider the model problem

−∇ ·³a³xε

´∇uε (x)

´= f (x) in Ω, (38)

uε (x) = 0 on ∂Ω,

where we assume that a ∈M(α, β,RN) is Y -periodic and f ∈ L2(Ω). Underthese assumptions a(x

ε) belongs to M(α, β,Ω). From Theorem 16 we then

know that, at least for a subsequence, we have G-convergence and hence alimit problem of the form

−∇ · (b(x)∇u (x)) = f (x) in Ω, (39)

u (x) = 0 on ∂Ω,

with b ∈M(α, β2

α,Ω). The well-known asymptotic expansion method allows

us to proceed one step further, in the sense that we can obtain formally boththe local and the homogenized problem and hence the limit matrix b. Asketch of this method is given in Section 4.1. This is, however, still only aguess and is not proven in a mathematically rigorous way. In Section 4.2we show how the proof is done by a classical method put forward by Tartar,known as the oscillating test function method.

These methods can also be applied to equations with periodic oscillationson several microscopic scales, that is, reiterated homogenization problems.Such problems are considered in Section 4.3. Here we recall a homogenizationresult for an elliptic problem with two micro-scales. We then study thecorresponding parabolic case, i.e. problem (11) with aε given by (12). Weprove the existence of a G-limit for this problem, and we also characterizethe limit operator by using the homogenization result for elliptic operatorsand a comparison theorem for G-limits.

37

4.1 The method of asymptotic expansions

The method of asymptotic expansions is based on the assumption that thesolution uε to (38) can be expanded in a power series in ε of the form

uε (x) = u0³x,

x

ε

´+ εu1

³x,

x

ε

´+ ε2u2

³x,

x

ε

´+ ..., (40)

where each uk is Y -periodic in the second argument. The idea is to substitutethe asymptotic expansion into the problem (38) and equate the terms thatcontain the same powers of ε. This will lead us to the local and the homog-enized problem. The ansatz (40) is reasonable in the sense that it takes intoaccount the two different length scales in the problem, i.e. the global and thelocal. However, we cannot be sure that uε does in fact admit an expansionof this kind, which means that the validity of the result has to be provenafterwards.

First, we observe that by using the chain rule the operator

Aε(·) = −∇ ·³a³xε

´∇(·)

´can be written as (see Section 2.2 in [PPSV])

Aε = ε−2A0 + ε−1A1 +A2,

where

A0(·) = −∇y · (a (y)∇y(·)) ,A1(·) = −∇y · (a (y)∇(·))−∇ · (a (y)∇y(·)) ,A2(·) = −∇ · (a (y)∇(·)) .

Now, inserting the expansion (40) into (38), we have

ε−2A0u0 + ε−1 (A0u1 +A1u0) + (A0u2 +A1u1 +A2u0)+

ε (A1u2 +A2u1) + ... = f.

Equating equal powers of ε, we get a series of equations, one for each power.The first three become

A0u0 = 0, (41)

A0u1 = −A1u0, (42)

A0u2 = −A1u1 −A2u0 + f. (43)

38

These problems all have Y -periodicity in the second variable as boundarycondition. Problems of this kind possess a unique solution, up to an additiveconstant, if and only if the integral mean value over Y of the right-hand sideequals zero; see e.g. Lemma 3.2. in [Def]. Thus equation (41) has a uniquesolution up to the addition of a function depending only on x. Since onesolution of (41) is u0 = 0, it follows that

u0 (x, y) = u (x) ,

i.e. that the first term in the expansion actually depends only on x. Takingthis into account and rearranging equation (42), we get the problem

−∇y · (a (y) (∇u (x) +∇yu1 (x, y))) = 0 in Ω× Y.

Using separation of variables, we have

u1 (x, y) = ∇u (x) · z (y) , (44)

where z is Y -periodic and satisfies

−∇y · (a (y) (ej +∇yzj (y))) = 0 in Y . (45)

This is the so-called local problem. Finally, using the requirement that theright-hand side should have integral mean value zero over Y in equation (43)and inserting (44), we get

−∇ · (b∇u (x)) = f (x) in Ω,

where b is given by

bij =

ZY

aij (y) +NXk=1

aik (y) ∂ykzj(y) dy, (46)

which means that we have found the homogenized problem.

This suggests that we are able to compute the limit b by solving PDEson one representative unit Y . Let us exemplify this with a two-dimensionalproblem, where we choose

a(y) =

µ2 + 1.95 sin 2π(y1 + y2) 0

0 2 + 1.95 cos 2π(y1 + y2)

¶,

39

f(x) = x21 and Ω = (0, 1)2. Rewriting (45), we obtain the elliptic problems

−∇y · (a(y)∇yzj(y)) = ∇y · (a(y)ej) in Y . (47)

For our choice of a, we get the right-hand side

∇y · (a(y)e1) = 2π · 1.95 cos 2π(y1 + y2)

for j = 1 and

∇y · (a(y)e2) = −2π · 1.95 sin 2π(y1 + y2)

for j = 2. Inserting the corresponding solutions z1 and z2 to (47) in (46), weobtain

b =

µ1.4485 00 1.4485

¶. (48)

If the result above gives the proper homogenized equation, the solutions uε

to (38) for

a³xε

´=

Ã2 + 1.95 sin 2π(x1+x2)

ε0

0 2 + 1.95 cos 2π(x1+x2)ε

!should be well approximated by the solution u to (39) with b given by (48),if ε is small.

Figure 9. The entries a11(xε ) for ε = 0.1 and b11.

A comparison between uε for ε = 0.1 and u (see Figure 10) confirms this, andfor any reasonable right-hand side f in (38) and (39) the same phenomenonappears.

40

Figure 10. uε for ε = 0.1 and u.

4.2 Tartar’s method of oscillating test functions

The result from the previous section is heuristically obtained. Let us nowconsider the following theorem, which will be proven in a mathematicallyrigorous way.

Theorem 32 Let uε be a sequence of solutions in H10(Ω) to the sequence

of problems

−∇ ·³a³xε

´∇uε (x)

´= f (x) in Ω, (49)

uε (x) = 0 on ∂Ω,

where a ∈M(α, β,RN) is Y -periodic and f ∈ L2(Ω). Then it holds that

uε (x) u (x) in H10 (Ω).

The function u ∈ H10 (Ω) is the unique solution to

−∇ · (b∇u (x)) = f (x) in Ω, (50)

u (x) = 0 on ∂Ω,

with

bij =

ZY

aij (y) +NXk=1

aik (y) ∂ykzj(y) dy, (51)

where, for j = 1, ..., N , zj ∈ H1 (Y ) /R uniquely solves the local problem

−∇y · (a (y) (ej +∇yzj (y))) = 0 in Y. (52)

41

In this section we show how the proof is done for the symmetric caseusing Tartar’s method of oscillating test functions. We will also reproduce adifferent proof of this theorem in Section 5.1.2 in order to illustrate a morerecent homogenization method based on so-called two-scale convergence.

Let us first sum up what we already know. As we noted in the discussionin Section 3.1.1, there are u and σ such that

uε (x) u (x) in H10 (Ω) (53)

anda³xε

´∇uε (x) σ (x) in L2(Ω)N (54)

up to a subsequence. Hence, for such a subsequence, there exists a limitproblem of the formZ

Ω

σ (x) ·∇v (x) dx =

ZΩ

f (x) v (x) dx, (55)

holding for all v ∈ H10 (Ω). By the G-convergence result, we know that σ can

be expressed asσ (x) = b (x)∇u(x)

for some b with the desired properties. This means that what remains to beproven is that b is in fact the particular one given in (51). If this can be doneindependently of the subsequence, we have found the homogenized problem.

To prove the result in Theorem 32 rigorously, we want to pass to the limitin the weak form of (49), i.e.Z

Ω

a³xε

´∇uε (x) ·∇v (x) dx =

ZΩ

f (x) v (x) dx (56)

for any v ∈ H10 (Ω). The essential difficulty, which we saw already in the intro-

duction, is that we have the product of two only weakly convergent sequences.The trick is to construct certain test functions that will allow us to study thelimit process of (56). In Tartar’s method, the approach for creating thesetest functions is to make use of an equation that is generated from the localproblem (52) and defined on all of Ω. We then make a crosswise choice oftest functions in this problem and (56), in the sense that we let the solutionsto the respective problems (multiplied by a smooth function φ) act as testfunctions in the other. Then, subtracting one problem from the other makesthe terms that contain products of two only weakly convergent sequencescancel out.

42

The following two lemmas on extensions of periodic functions, see e.g.the appendix in [Def], will be useful in the remainder of this section.

Lemma 33 A function f ∈ H1 (Y ) /R can be extended by periodicity to afunction in H1

loc(RN).

Lemma 34 Let g ∈ L2(Y )N and assume thatRYg · ∇v dy = 0 for every

v ∈ H1 (Y ) /R. Then g can be extended by periodicity to a function inL2loc(RN)N , still denoted by g, such that −∇ · g = 0 in D0(RN).

First, we note that the local problem has a unique solution in H1 (Y ) /R.This is realized if we rewrite it in the form

−∇y · (a(y)∇yzj(y)) = ∇y · (a(y)ej) in Y .

The right-hand side belongs to (H1(Y )/R)0 and an application of the Lax-Milgram theorem then gives that there exists a unique solution in H1 (Y ) /R;see Theorem 4.26 and Section 7.1 in [CiDo].

Let us now see how to use the local problem to construct appropriate testfunctions. The weak form of the local problem states thatZ

Y

a (y) (ej +∇yzj (y)) ·∇yv (y) dy = 0

for all v ∈ H1 (Y ) /R. Lemma 33 allows us to make a Y -periodic extension ofz to a function in H1

loc(RN)N . Similarly, by Lemma 34 there is a Y -periodicextension of a (y) (ej +∇yzj (y)) in L2loc(RN)N such thatZ

RNa³xε

´³ej +∇yzj

³xε

´´·∇v (x) dx = 0

for all v ∈ D(RN). Thus, defining

wεj (x) = xj + εzj

³xε

´for j = 1, 2, ..., N we haveZ

RNa³xε

´∇wε

j (x) ·∇v (x) dx = 0

for all v ∈ D(RN), and henceZΩ

a³xε

´∇wε

j (x) ·∇v (x) dx = 0 (57)

for any v ∈ H10 (Ω).

43

For wεj we have

wεj (x)→ xj in L2(Ω) (58)

and∇wε

j (x) ej in L2(Ω)N .

Now, for φ ∈ D(Ω), we choose test functions v = φwεj in (56) and v = φuε in

(57), and obtainZΩ

a³xε

´∇uε (x) ·∇φ (x)wε

j (x) + a³xε

´∇uε (x) ·∇wε

j (x)φ (x) dx = (59)ZΩ

f (x)φ (x)wεj (x) dx

andZΩ

a³xε

´∇wε

j (x)·∇φ (x)uε (x)+a³xε

´∇wε

j (x)·∇uε (x)φ (x) dx = 0, (60)

respectively. Subtracting (60) from (59) and using the fact that a is sym-metric, we arrive atZ

Ω

a³xε

´∇uε (x) ·∇φ (x)wε

j (x)− a³xε

´∇wε

j (x) ·∇φ (x)uε (x) dx = (61)ZΩ

f (x)φ (x)wεj (x) dx.

Note that through this procedure we get rid of the problematic terms andwe are left only with terms that consist of one weakly and one stronglyconvergent sequence. Indeed, by (54) and (58), we note that the first termin (61) is under control. Furthermore, we have³a³xε

´∇wε

j (x)´k=

NXi=1

aki

³xε

´∂xiw

εj (x) =

NXi=1

aki

³xε

´³δij + ∂yizj

³xε

´´and thus³

a³xε

´∇wε

j (x)´k

ZY

akj(y) +NXi=1

aki (y) ∂yizj (y) dy = bkj in L2(Ω)

with b as in (51). Also, by (53) together with the Rellich embedding theorem,we get

∇φ (x)uε (x)→∇φ (x)u (x) in L2(Ω)N ,

and hence we can handle the second term also. This means that we can let

44

ε tend to zero in (61) and obtainZΩ

NXk=1

σk (x) ∂xkφ (x)xj dx−ZΩ

NXk=1

bkj∂xkφ (x)u (x) dx =

ZΩ

f (x)φ (x)xj dx.

Taking (55) with v = φxj into account in the right-hand side, we haveZΩ

NXk=1

(σk (x)xj − bkju (x)) ∂xkφ (x) dx =

ZΩ

NXk=1

σk (x) ∂xk (φ (x)xj) dx

for all φ ∈ D(Ω). Integration by parts in both sides givesZΩ

NXk=1

(−∂xkσk (x)xj − σk (x) δjk + bkj∂xku (x))φ (x) dx =

ZΩ

NXk=1

−∂xkσk (x)φ (x)xj dx

and hence, for every j = 1, ..., N ,ZΩ

Ãσj (x)−

NXk=1

bkj∂xku (x)

!φ (x) dx = 0.

Since this holds for all φ ∈ D(Ω) and b is symmetric, we conclude that

σ (x) = b∇u (x) .Now, since b has the necessary properties (see Proposition 4.2 i [Def] orProposition 6.12 in [CiDo]) to provide us with a unique solution to (50), theentire sequence uε converges and hence we are done.Remark 35 The reason that the troublesome terms canceled out in (61)is that the operator is symmetric. In the non-symmetric case the adjointoperator of the local problem is used to obtain the corresponding cancellation.For details, see e.g. [CiDo].

This method may also be used for possibly nonlinear monotone problems.However, in this case the procedure of subtracting the two problems with thecrosswise choice of test functions, in order to be able to pass to the limit,is not applicable. This means that one needs an alternative way to handlethe limit process in the more general setting. The key to this is so-calledcompensated compactness, which was first introduced by Murat and Tartar;see e.g. [Mur2] and [Tar2].

45

Compensated compactness makes it possible to, in a certain sense, obtainconvergence of the products of two sequences uh and vh to the product ofthe respective limits, without any of the sequences being strongly convergent.Instead, suitable convergence properties can be imposed on the derivativesin both sequences. A prototype example of compensated compactness is thefollowing result, which is known as the div-curl lemma.

Lemma 36 Let uh and vh be two sequences in L2(Ω)N such that

uh(x) u(x) in L2(Ω)N ,

vh(x) v(x) in L2(Ω)N .

Define the curl operator by ∇× v =³∂vi∂xj− ∂vj

∂xi

´1≤i,j≤N

and assume that

∇ · uh(x) → ∇ · u(x) in H−1(Ω),

∇× vh(x) ∇× v(x) in L2(Ω)N×N .

Then ZΩ

uh (x) · vh (x)φ (x) dx→ZΩ

u (x) · v (x)φ (x) dx

for every φ ∈ D(Ω).

Wewill briefly look at the monotone correspondence to the model problemto see how this concept can be applied. More precisely, we will use the lemmabelow, which is a certain type of compensated compactness; see [MuTa].

Lemma 37 Let gh and uh be two sequences in L2(Ω)N and H1(Ω), re-spectively, such that

gh(x) g(x) in L2(Ω)N ,

uh(x) u(x) in H1(Ω)

and∇ · gh(x)→∇ · g(x) in H−1(Ω).

Then it holds thatZΩ

gh (x) ·∇uh (x)φ (x) dx→ZΩ

g (x) ·∇u (x)φ (x) dx

for every φ ∈ D(Ω).

46

Let us now consider the problem

−∇ · a³xε,∇uε

´= f (x) in Ω,

uε (x) = 0 on ∂Ω,

where a ∈ N (α, β,RN) is Y -periodic in the first argument and f ∈ H−1(Ω).By Theorem 21, G-convergence up to a subsequence is at hand. Furthermore,we are led by asymptotic expansion to a homogenized operator such that

b(∇u) =ZY

a (y,∇u+∇yu1) dy (62)

and a local problem of the form

−∇y · a (y,∇u+∇yu1) = 0 in Ω× Y . (63)

This is carefully carried out in [ChSm]. Using arguments similar to thoseused in the linear case one can show that there are functions u and σ suchthat, up to a subsequence,

uε (x) u (x) in H10 (Ω) (64)

anda³xε,∇uε

´σ (x) in L2(Ω)N . (65)

Hence, we need to prove that

σ (x) = b (∇u(x))

with b given by (62).

As in the linear case, let us first note that the local problem has a uniquesolution, see e.g. [Def], Remarks 5.2 and 5.4. Now let u1,η ∈ H1(Y )/R bethe solution to (63) for a fixed η ∈ RN in the place of ∇u. If we let u1,η stilldenote the periodic extension of this function to RN , then by Lemma 33 wehave that u1,η ∈ H1

loc(RN). Using Lemma 34, as in the linear case, we haveZΩ

a³xε, η +∇yu1,η

³xε

´´·∇v (x) dx = 0 (66)

for all v ∈ H10 (Ω). Let us now define test functions

wεη (x) = η · x+ εu1,η

³xε

´

47

in a way similar to the above. Obviously,

wεη (x) η · x in H1(Ω) (67)

and from the periodicity of u1,η and of a, we get

∇wεη (x) η in L2(Ω)N (68)

anda³xε,∇wε

η

´b(η) in L2(Ω)N . (69)

The monotonicity of a (see (N4) in Section 3.1.2) givesZΩ

³a³xε,∇uε

´− a

³xε,∇wε

η

´´·¡∇uε (x)−∇wε

η (x)¢φ (x) dx ≥ 0 (70)

for all φ ∈ D(Ω), φ ≥ 0. We recall that

−∇ · a³xε,∇uε

´= f(x) in Ω

and (see (66))

−∇ · a³xε,∇wε

η

´= 0 in Ω.

Hence −∇ ·a¡xε,∇uε

¢ and −∇ ·a

¡xε,∇wε

η

¢ converge strongly in H−1(Ω)

to f and zero, respectively. Since we also have the convergence properties(64)-(69) above, we can now apply the compensated compactness Lemma 37to (70), which results inZ

Ω

(σ(x)− b(η)) · (∇u (x)− η)φ (x) dx ≥ 0

for all φ ∈ D(Ω), φ ≥ 0. This means that

(σ(x)− b(η)) · (∇u (x)− η) ≥ 0

for every η ∈ RN and a.e. x ∈ Ω. From the properties of a, one can show (seeRemark 5.4 in [Def]) that b : RN → RN is monotone and hemicontinuous,and hence, by Proposition 2, maximal monotone. Thus, we have

σ(x) = b(∇u (x)).

Also, the function b is strictly monotone (see [Def], Proposition 5.5), whichmeans that the limit problem has a unique solution. Since this result doesnot depend on the choice of convergent subsequence we have accomplishedwhat we set out to do.

48

The homogenization methods described in this and the previous section,i.e. the method of asymptotic expansions and Tartar’s method of oscillatingtest functions, can also be used to study parabolic homogenization problems.However, once the result for the elliptic case is available, there is sometimesa shortcut to the parabolic case via the comparison result in Theorem 28 inSection 3.2.2. In [Sva2], this theorem is combined with the homogenizationresult above to obtain a corresponding result for a monotone parabolic prob-lem with oscillations on one microscopic scale. This approach will also beused in the next section for a parabolic problem with two different micro-scales.

Remark 38 In this section we have considered Tartar’s method from the pe-riodic homogenization point of view. Let us point out that this method is quitea general one and can be applied to problems with no periodicity assumptions.It can be used, for example, to prove G-convergence compactness for the setM(α, β,Ω); see [MuTa]. Although there is of course no local problem thatcan be employed in the general case, the main idea remains the same, i.e. tochoose test functions with certain convergence properties in order to be ableto pass to the limit. However, the construction of the test functions is donein a more abstract way in the sense that the operators T ε : H1

0 (Ω)→ H−1(Ω)used to define them are not explicitly given, but proven to exist and to havecertain key properties. This also means that in the non-periodic case, thismethod does not provide any way to actually determine the limit operator.

4.3 Reiterated homogenization

So far, we have studied homogenization of problems that are characterizedby two different length scales, i.e. the global scale and one microscopic scale.However, these procedures can be extended to cases in which there are morethan two scale levels.

Also, when dealing with problems from the real world there is nothing tosuggest that there should be a restriction to merely two scales. The hetero-geneities in a material can of course occur on several different microscopiclevels. This is, for example, the case if in a fiber composite there is a scale ofeven thinner fibers inside the fibers, or if a composite is built up of a matrixmaterial with inclusions of different sizes and different periodicities. Figure11 illustrates the build-up of the material in these two cases.

49

Figure 11. Materials which are heterogeneous on more than one level.

Homogenization problems of this kind have become known as reiteratedhomogenization problems, and was first studied by means of asymptotic ex-pansions by Bensoussan et al. in the case of two microscopic scales. Here, theasymptotic expansion contains two local variables, see [BLP], Section 8.3.

A homogenization result similar to that in Section 4.2, which is alsoproven using the same methods, is given in [LLPW] for a monotone ellip-tic case with two micro-scales. Here, we reproduce a version of this result,which we will use to study a similar parabolic case.

Theorem 39 Consider the problem

−∇ · a³xε,x

ε2,∇uε

´= f (x) in Ω, (71)

uε (x) = 0 on ∂Ω,

where we assume that f ∈ H−1(Ω) and that a ∈ N (α, β, k,R2N) is Y1-periodicin the first variable and Y2-periodic in the second variable. Furthermore, letω : R → R be an increasing continuous function such that ω(0) = 0 andassume that

|a(y1, y2, ξ)− a(y01, y2, ξ)|2 ≤ ω(|y1 − y01|)(1 + |ξ|

2)

for all y1, y01 ∈ Y1, a.e. y2 ∈ RN and all ξ ∈ RN . Then, for a sequence uεof solutions in H1

0(Ω) to (71), it holds that

uε (x) u (x) in H10(Ω)

where u ∈ H10 (Ω) is the unique solution to the problem

−∇ · b(∇u) = f (x) in Ω,

u (x) = 0 on ∂Ω.

50

Here, b is defined by

b(∇u) =ZY 2

a (y1, y2,∇u+∇y1u1 +∇y2u2) dy2dy1,

where u1 ∈ L2¡Ω;H1 (Y1) /R

¢and u2 ∈ L2

¡Ω× Y1;H

1 (Y2) /R¢uniquely

solve the local problems

−∇y2 · a (y1, y2,∇u+∇y1u1 +∇y2u2) = 0,

−∇y1 ·ZY2

a (y1, y2,∇u+∇y1u1 +∇y2u2) dy2 = 0.

Proof. See Theorem 3.1 in [LLPW].

Remark 40 A generalization of this result to operators of the form−∇ · a(x, x

ε, xε2,∇(·)) is given in Theorem 4.1 in [LLPW], where multiscale

convergence techniques are used.

Let us now investigate a parabolic equation that also has two scales on themicroscopic level and where a is time-dependent. We consider the problem

∂tuε (x, t)−∇ · a

³t,x

ε,x

ε2,∇uε

´= f (x, t) in ΩT ,

uε (x, 0) = u0 (x) in Ω, (72)

uε (x, t) = 0 on ∂Ω× (0, T ) ,

where we let u0 ∈ L2(Ω) and f ∈ L2 (0, T ;H−1(Ω)). We will prove a homoge-nization result for this equation, benefiting from the result above concerningthe elliptic problem.

We assume thata : R+ ×R3N → RN

satisfies the following conditions, where 0 < k ≤ 1 and α and β are positiveconstants:

(i) a (t, y1, y2, 0) = 0 for all (t, y1, y2) ∈ R+ × R2N .

(ii) a (·, ·, ·, ξ) is Y 2-periodic in (y1, y2) and continuous for all ξ ∈ RN .

(iii) a (t, y1, y2, ·) is continuous for all (t, y1, y2) ∈ R+ ×R2N .

51

(iv) |a (t, y1, y2, ξ)− a (t, y1, y2, ξ0)| ≤ β (1 + |ξ|+ |ξ0|)1−k |ξ − ξ0|k

for all (t, y1, y2) ∈ R+ ×R2N and all ξ, ξ0 ∈ RN .

(v) (a (t, y1, y2, ξ)− a (t, y1, y2, ξ0)) · (ξ − ξ0) ≥ α |ξ − ξ0|2

for all (t, y1, y2) ∈ R+ ×R2N and all ξ, ξ0 ∈ RN .

(vi) |a(t, y1, y2, ξ)− a(t, y1, y02, ξ)| ≤ B(t− t0)(1 + |ξ|)

for all (y1, y2) ∈ R2N , all ξ ∈ RN and all t, t0 such that 0 < t0 < t < T ,where B : R+ → R+ is an increasing continuous function such thatB(t)→ 0 as t→ 0+.

(vii) |a(t, y1, y2, ξ)− a(t, y01, y2, ξ)|2 ≤ ω(|y1 − y01|)(1 + |ξ|

2)for all y1, y01 ∈ Y1, a.e. y2 ∈ RN , all ξ ∈ RN and all t ∈ (0, T ), whereω : R→ R is an increasing continuous function such that ω(0) = 0.

This means, for example, that a belongs to N (α, β, k,R+ × R2N). Wewill begin by making sure that the sequence of operators G-converges. Thefollowing theorem concerns also uniqueness of the solution and a priori esti-mates.

Theorem 41 Consider the problem (72). It holds that

(a) the problem has a unique solution uε ∈ H1 (0, T ;H10(Ω), H

−1(Ω)) forevery fixed ε > 0.

(b) the solutions uε are uniformly bounded in L∞ (0, T ;L2(Ω)) andH1 (0, T ;H1

0 (Ω), H−1(Ω)), i.e., for some positive constant C it holds

thatkuεkL∞(0,T ;L2(Ω)) ≤ C

andkuεkH1(0,T ;H1

0 (Ω),H−1(Ω)) ≤ C.

(c) the sequence of functions

ah(x, t, ξ) = a³t,x

ε,x

ε2, ξ´

where ε = ε (h) → 0 as h → ∞ and (x, t, ξ) ∈ ΩT × RN belongs toN (α, β, k,ΩT ). Hence ah G-converges up to a subsequence, to a limitb ∈ N (α, β, k,ΩT ), where k = k/(2 − k) and β is a positive constantdepending on α, β and k only.

52

Proof. For (a) we refer to Theorem 8 in Section 2.2.2, and for the first apriori estimate in (b) to Lemma 30.3 in [ZeiIIB]. Let us now define

ah(x, t, ξ) = a³t,x

ε,x

ε2, ξ´

(73)

where (x, t, ξ) ∈ ΩT × RN and with ε > 0, where ε = ε (h) → 0 as h → ∞.We will prove that this function belongs to N (α, β, k,ΩT ), i.e., that it fulfillsthe requirements (N1)− (N4) in Section 3.2.2.From assumption (i) we immediately get

a³t,x

ε,x

ε2, 0´= 0,

which means that condition (N1) is satisfied.

We have that a¡·, ·

ε, ·ε2, ξ¢is measurable for every fixed ξ ∈ RN because

a(·, ·, ·, ξ) is continuous for every such ξ by (ii). Hence, condition (N2) isfulfilled.

Furthermore, from the fact that a satisfies (iv) it follows that¯a³t,x

ε,x

ε2, ξ´− a

³t,x

ε,x

ε2, ξ0´¯≤ β(1 + |ξ|+ |ξ0|1−k) |ξ − ξ0|k

holds for all (x, t) ∈ ΩT , all ξ, ξ0 ∈ RN and 0 < k ≤ 1. This means that aagrees with condition (N3).

Finally, condition (N4) is satisfied because³a³t,x

ε,x

ε2, ξ´− a

³t,x

ε,x

ε2, ξ0´´· (ξ − ξ0) ≥ α |ξ − ξ0|2

for all (x, t) ∈ ΩT and all ξ, ξ0 ∈ RN since a fulfills (v).Hence, ah given by (73)

belongs to N (α, β, k,ΩT ). By Proposition 5.3 and Corollary 5.1 in [Sva1], wecan then deduce that the solutions uε and the derivatives ∂tuε are boundedin L2(0, T ;H1

0(Ω)) and L2(0, T ;H−1(Ω)), respectively, and hence the seconda priori estimate follows. Also, Theorem 27, on parabolic G-convergence upto a subsequence, applies and the proof is complete.

Remark 42 Note that since we have proven that the conditions (N1)-(N4)in Section 3.2.2 are satisfied for all t ∈ (0, T ), we know, apart from the factthat ah is in N (α, β, k,ΩT ), that it belongs to N (α, β, k,Ω) for every fixed t;see the corresponding conditions in Section 3.1.2.

53

Thus, we know that we have a limit problem of the desired form; thatis, a problem of the same type as (72) with the governing function b hav-ing the properties necessary to imply a unique solution, for example. Nowwe will characterize the limit function b further by proving the homogeniza-tion result below. The proof will be done by utilizing the conclusion aboutG-convergence from Theorem 41 and the homogenization result for ellipticreiterated problems in Theorem 39, together with the comparison result forG-limits in Theorem 28.

Theorem 43 Let uε be a sequence of solutions inH1 (0, T ;H10 (Ω),H

−1(Ω))to (72). Then it holds that

uε (x, t) u (x, t) in L2¡0, T ;H1

0(Ω)¢

anduε (x, t)→ u (x, t) in L2 (ΩT ) (74)

where u ∈ H1 (0, T ;H10 (Ω),H

−1(Ω)) is the unique solution to the homogenizedproblem

∂tu (x, t)−∇ · b (t,∇u) = f (x, t) in ΩT ,

u (x, 0) = u0 (x) in Ω,

u (x, t) = 0 on ∂Ω× (0, T ) .

Here, b is given by

b(t,∇u) =ZY 2

a (t, y1, y2,∇u+∇y1u1 +∇y2u2) dy2dy1,

where u1(t) ∈ L2¡Ω;H1 (Y1) /R

¢and u2(t) ∈ L2

¡Ω× Y1;H

1 (Y2) /R¢uniquely

solve the local problems

−∇y2 · a (t, y1, y2,∇u+∇y1u1 +∇y2u2) = 0,

−∇y1 ·ZY2

a (t, y1, y2,∇u+∇y1u1 +∇y2u2) dy2 = 0.

Proof. So far, we know–by Theorem 41–that we have G-convergence, atleast for a subsequence, of the operators corresponding to (72), i.e. for thesequence ah given by

ah(x, t, ξ) = a³t,x

ε,x

ε2, ξ´,

54

where ε = ε(h) → 0 as h → ∞, we have G-convergence in the sense ofparabolic operators to some b, up to a subsequence. To apply the comparisonresult in Theorem 28, we must also prove that the condition¯

ah(x, t, ξ)− ah(x, t0, ξ)¯≤ B(t− t0)(1 + |ξ|), (75)

where B : R+ → R+ is an increasing continuous function such that B(t)→ 0as t → 0+, is satisfied for all ξ ∈ RN , a.e. x ∈ Ω and all t and t0 such that0 < t0 < t < T . Since a satisfies (vi) it holds that¯

a³t,x

ε,x

ε2, ξ´− a

³t0,

x

ε,x

ε2, ξ´¯≤ B(t− t0)(1 + |ξ|)

for all x ∈ Ω, all ξ ∈ RN and t, t0 such that 0 < t0 < t < T . Hence (75) isfulfilled.

Finally, we need to know that we have G-convergence in the sense ofelliptic operators for t fixed. This is provided by Theorem 29 in Section 3.2.We have already proven that the conditions stated for ah in Theorem 29are satisfied; that is, (75) is fulfilled and ah(t) belongs to N (α, β, k,Ω) forevery fixed t; see Remark 42. Thus, we have G-convergence of ah(t) in thesense of elliptic operators to a function b0(t), at least for a subsequence, andwith the same subsequence for all t ∈ (0, T ).Now we know that the requirements needed to apply Theorem 28 are all

fulfilled. Hence b = b0 and since all assumptions made in Theorem 39 con-cerning the elliptic case are satisfied, we can employ this to determine b0(t)for any fixed t. Thus the desired result follows, at least for a subsequence, ex-cept for the convergence (74). This convergence property is a consequence ofthe second a priori estimate in (b) in Theorem 41; see e.g. Problem 23.13b in[ZeiIIA]. Since the same result will follow for any G-convergent subsequence,the entire sequence G-converges and the proof is complete.

Remark 44 Parabolic homogenization problems similar to the one treatedin this section will be studied in Section 5.3. In these problems, there areoscillations also in the time variable and the homogenization results will beproven by multiscale convergence techniques.

55

5 Homogenization by multiscale convergencetechniques

In this chapter we will investigate an alternative method for periodic homog-enization. A technique based on so-called two-scale convergence has provento be efficient for problems with one microscopic scale, such as the modelproblem studied in Chapters 1 and 4. As in Tartar’s method, the choiceof test functions plays an essential role, but in contrast to that approach,this choice does not rely on the local problem. This means that we are notdependent on the method of asymptotic expansions to find the local problembeforehand. Instead, the local and the homogenized problem are immediatelyobtained in the coupled form, after passing to the limit for certain choices oftest functions.

In Section 5.1 we recall some fundamentals about two-scale convergenceand show how it can be used to perform the homogenization of the modelproblem. We also investigate the connection between a mode of convergenceof two-scale type and the asymptotic expansion. Two-scale convergence andthe homogenization method based on this have been generalized to multiplescales and used to solve corresponding reiterated homogenization problems.In Section 5.2, we illustrate this for an elliptic problem with two micro-scales.We then extend this approach in Section 5.3 to fit problems that may haveperiodic oscillations in time as well. In particular we prove a compactnessresult for gradients adapted to a problem with two microscopic scales in spaceand one fast scale in time, such as the parabolic problem (11) with aε givenby (13). We then proceed with the homogenization of this problem. Finally,we perform the homogenization of (11) for the choice of aε given in (14),where we have one microscopic spatial scale and two fast temporal scales.

5.1 Homogenization with two scales

In 1989, Nguetseng introduced what has come to be known as two-scale con-vergence. A significant property of this type of convergence is that whilethe functions in the sequence depend on one variable, x ∈ RN only, thelimit function contains two variables. As we will see, this means that it cap-tures oscillations on a microscopic scale, which are averaged away in usualweak limits. This explains the name, and is also what makes two-scale con-vergence particularly suitable for periodic homogenization. After studying

56

two-scale convergence with respect to some properties of the test functionsand its relationship to other modes of convergence in Section 5.1.1 we applyit to homogenization in Section 5.1.2. Finally, we discuss the relationshipbetween two-scale convergence and asymptotic expansions in Section 5.1.3.

5.1.1 Two-scale convergence

In [Ngu1], Nguetseng presented a result on compactness with respect to a con-vergence of two-scale type. He proved that for every bounded sequence uεin L2(Ω), there is a function u0 in L2(Ω×Y ) such that, up to a subsequence,

limε→0

ZΩ

uε (x) v³x,

x

ε

´dx =

ZΩ

ZY

u0 (x, y) v (x, y) dydx

for all v ∈ C0(Ω, C (Y )) and all v = v1v2, v1 ∈ C0(Ω), v2 ∈ L2(Y ) whenΩ is bounded, and for all v ∈ C0(Ω, C (Y )) and all v = v1v2, v1 ∈ C0(Ω),v2 ∈ L2(Y ) when Ω is equal to RN .

Allaire extended this result to hold for some less restricted classes of testfunctions, and the proof is also significantly different; see [All1]. One of thesespaces of test functions is L2(Ω;C (Y )), and this class has been chosen inthe exact contemporary definition of two-scale convergence.

Definition 45 A sequence uε in L2(Ω) is said to two-scale converge to alimit u0 ∈ L2 (Ω× Y ), called the two-scale limit, if

limε→0

ZΩ

uε (x) v³x,

x

ε

´dx =

ZΩ

ZY

u0 (x, y) v (x, y) dydx (76)

for all v ∈ L2 (Ω;C (Y )). This is written

uε (x) u0 (x, y) .

We have the following theorem on compactness with respect to two-scaleconvergence.

Theorem 46 A sequence uε bounded in L2(Ω) has a two-scale convergentsubsequence.

Proof. See the proof of Theorem 1.2 in [All1].

To see what a two-scale limit might look like, we study the sequence uεgiven by

uε(x) = ln

µx+ sin

2πx

ε

¶. (77)

57

This sequence is weakly convergent in L2(Ω) e.g. for Ω = (2, 3). We obtainthe following appearance of uε for some different values of ε.

Figure 12. uε for some values of ε.

To gain a better understanding of the two-scale limit and its connection tothe weak L2(Ω)-limit, we first recall that weak convergence of uε to u inL2(Ω) means that Z

Ω

uε(x)v(x) dx→ZΩ

u(x)v(x) dx

for all v ∈ L2(Ω). For v = 1, this simply means that the sequence of meanvalues converges–or the sequence of areas under the graph for our choice ofuε. Since this must also hold, for example, for functions v that are equal toone on any small interval of Ω and zero elsewhere, the convergence of themean values of uε will also take place locally and hence the weak limit ufollows the trend of the functions in the sequence, as in Figure 13.

Figure 13. The weak L2(Ω)-limit of uε.

Thus, this limit of uε could be understood as a way to represent the limit ofthe mean value locally, which implies that it does not contain any informationabout what the rapid oscillations look like.

58

For two-scale convergence, we require thatZΩ

uε (x) v³x,

x

ε

´dx→

ZΩ

ZY

u0 (x, y) v (x, y) dydx (78)

for all v ∈ L2 (Ω;C (Y )). Again, choosing v = 1, this obviously means thatthe areas under the graphs of the functions in uε approach the volumeunder the graph of the function u0, or, equivalently, that the sequence ofmean values of uε converges to the mean value of u0. Thus, our first estimateof the two-scale limit might be a plane surface with height equal to the limitof the mean values of uε.

Figure 14. A first estimate of the two-scale limit.

In a way similar to that described above, we conclude that this convergencewill take place locally in Ω; i.e. in any small interval, the area under the graphof uε approaches the volume of the corresponding slice under the graph ofu0. This means that u0 will be different at different locations in Ω.

Figure 15. Trend in the x-direction.

59

A naive guess at the two-scale limit for uε in (77) might then be the functionabove. This suggestion has the essential feature that the mean value of u0over Y coincides with the weak L2(Ω)-limit u; see (84). It remains for us tofind out how the two-scale limit is actually shaped in the y-direction.

Our choice of uε contains oscillations of the same frequency as v(x, xε).

This is reflected in the fact that u0 varies in the y variable for x fixed. Thisis perhaps most easily realized if we study Y -periodic test functions thatare equal to one on some small fraction of the period and zero elsewhere.To capture the local variations together with the global tendency observedabove, we choose v as the product between such a function and a cutofffunction, whose support is contained in a small interval in Ω.

Figure 16. v(x, xε) for two choices of test functions v.

For this kind of test functions, (78) means that the area under the graph of uε

on the small intervals where the function v(x, xε) is equal to one approaches

the volume under the graph of u0 on the small rectangle in Ω× Y where thetest function v has support.

Figure 17. The product uε(x)v(x, xε) for v(x, x

ε) as above.

Thus, the mean height of u0 on such a rectangle is determined by the height ofthe bars under the graph of uε, see Figure 17. In Figure 18, the two leftmostbars correspond to the two choices of test functions above. Obviously, theheight differs depending on where in the period v equals one.

60

Figure 18. Variation in the x- and y- directions.

We also get different heights for different choices of small intervals in Ωdepending on the global tendency of uε, and duly we recognize the trend inthe x-direction from Figure 15 in the picture to the right in Figure 18. Weend up with the two-scale limit u0 displayed below.

Figure 19. The two-scale limit of uε.The trend in the x-direction seems to be in accordance with the weak limit,but in the two-scale limit we also see that the local fluctuations are reflected inoscillations in the y-direction, which is typical for sequences uε of oscillatingfunctions of the kind studied above. However, two-scale convergence is notrestricted to sequences of this type. Any function u0 ∈ L2(Ω × Y ) can beobtained as a two-scale limit to some bounded sequence uε in L2(Ω) (see[All1], Lemma 1.13), and we already know that any such sequence has atwo-scale convergent subsequence. The character of the y-dependence in u0depends on whether uε contains oscillations that the oscillations in v(x, x

ε)

can capture.

61

The choice of test functions is crucial in order to obtain a useful conceptof convergence. Firstly, we need to require that v is periodic in the y variable.We must also be able to substitute x

εfor y and still get a measurable function.

This means that the test functions v must satisfy the following:

v(·, y) is measurable for all y ∈ Y ,

v(x, ·) is continuous for a.e. x ∈ Ω.

These conditions are known as Carathéodory conditions and allow us toreplace y with a measurable function of x and still obtain a measurablefunction; see p. 1013 in the appendix in [ZeiIIB].

Within the scope of these assumptions, it is possible to define differ-ent classes of test functions for which compactness results of the kind givenin Theorem 46 hold true. Examples of such spaces of test functions areL2(Ω;C (Y )), L2(Y ;C(Ω)) andC(Ω;C (Y )) forΩ bounded, andL2(Ω;C (Y ))when Ω may be unbounded, e.g. when Ω is equal to RN ; see [All1] and [All2].

The proof is based on some properties possessed by these spaces of so-called admissible test functions. If we denote any of these classes by B(Ω, Y ),B(Ω, Y ) is a separable Banach space that is dense in L2(Ω × Y ). We alsohave for all v ∈ B(Ω, Y )°°°v ³x, x

ε

´°°°L2(Ω)

≤ kv(x, y)kB(Ω,Y ) (79)

and °°°v ³x, xε

´°°°L2(Ω)

→ kv(x, y)kL2(Ω×Y ) . (80)

Let us summarize the proof given in [All1]. Consider the integral in theleft-hand side of (76) as a linear functional F ε acting on v ∈ B(Ω, Y ). ByHölder’s inequality and (79), F ε is bounded with respect to theB(Ω, Y )-normand thus belongs to (B(Ω, Y ))0. For uε bounded in L2(Ω), the sequence offunctionals

hF ε, vi(B(Ω,Y ))0,B(Ω,Y ) =ZΩ

uε (x) v³x,

x

ε

´dx

is bounded in (B(Ω, Y ))0, and since B(Ω, Y ) is separable F ε has a weak*limit F ∈ (B(Ω, Y ))0, up to a subsequence. Finally, from (80) it followsthat F is bounded also with respect to the L2(Ω × Y )-norm, and thus bydensity we can extend F to a functional G ∈ (L2(Ω× Y ))0. Hence, by Riesz

62

representation theorem, we deduce that there is a unique u0 ∈ L2(Ω × Y )such that

hG, vi(L2(Ω×Y ))0,L2(Ω×Y ) =ZΩ

ZY

u0 (x, y) v (x, y) dydx.

We will now investigate the class L2 (Ω;C (Y )) of admissible test func-tions used in the definition in more detail. Some important properties ofthese functions, such as (80), are based on the fact that for any functionf ∈ L1(Ω;C (Y )) it holds that

limε→0

ZΩ

f³x,

x

ε

´dx =

ZΩ

ZY

f (x, y) dydx; (81)

see e.g. Theorem 2 in [LNW]. This implies that for v ∈ L2 (Ω;C (Y )), andhence v2 ∈ L1 (Ω;C (Y )), we have

limε→0

ZΩ

v2³x,

x

ε

´dx =

ZΩ

ZY

v2 (x, y) dydx

and obviously (80) follows. Also, if we let w ∈ L2(Ω), we have–still forv ∈ L2 (Ω;C (Y ))–that vw ∈ L1(Ω;C (Y )) and thus we obtain from (81)that

limε→0

ZΩ

v³x,

x

ε

´w (x) dx =

ZΩ

ZY

v (x, y)w (x) dydx

for any w ∈ L2(Ω), which means that

v³x,

x

ε

´ ZY

v (x, y) dy in L2(Ω). (82)

We will see that this allows a link between the two-scale limit and the stronglimit.

Let us now consider a couple of results relating two-scale convergence tostrong and weak convergence in L2(Ω), respectively. If

uε (x)→ u (x) in L2(Ω),

the power of possible oscillations can be forced to be arbitrarily small bychoosing ε small enough; see Figure 5 for example. This means that for such

63

a sequence, there should not be any substantial local oscillations for the two-scale limit to capture, i.e., we could expect that it is constant in y. Thus, itis not surprising that for a sequence strongly convergent in L2(Ω) we have

uε (x) u (x) ; (83)

that is, a strongly convergent sequence uε two-scale converges to its stronglimit u and hence, for such a sequence, the second variable vanishes in thetwo-scale limit. To see this, we simply use (82) and get for v ∈ L2 (Ω;C (Y ))

limε→0

ZΩ

uε (x) v³x,

x

ε

´dx =

ZΩ

u(x)

ZY

v (x, y) dy dx =ZΩ

ZY

u(x)v (x, y) dydx,

where the passage to the limit is easily done since we have a product of onestrongly and one weakly convergent sequence in L2(Ω).

If uε does not converge strongly, we can still get a connection to theweak limit. If

uε (x) u0 (x, y) ,

thenuε (x)

ZY

u0 (x, y) dy in L2(Ω); (84)

that is, the weak limit of the sequence is obtained by integrating the two-scale limit over Y . This follows immediately from the definition of two-scaleconvergence if we choose test functions independent of y, i.e. in L2(Ω). Obvi-ously, when deriving the weak limit from the two-scale limit the informationabout the local oscillations that we saw in Figure 19 is lost.

Two-scale convergence has sometimes been defined as the requirementthat (76) should hold for all functions v in the set D(Ω;C∞ (Y )). This is,however, too restricted a choice of test functions in the sense that we lose theconnection to the weak limit. Indeed, a limit in this sense can be obtainedalso for unbounded sequences in L2(Ω), and since an unbounded sequencedoes not possess a weak limit there is no such limit to relate to as in (84).However, for a sequence bounded in L2(Ω), the fact that (76) holds for everyv in D(Ω;C∞ (Y )) guarantees that the entire sequence two-scale convergesto u0, and not just a subsequence. See also comments on this in [LNW].

64

When using two-scale convergence for the purpose of homogenization,we will need information about the two-scale limit of the gradient of uε.For a sequence uε bounded in H1(Ω), we have the characterization in thefollowing theorem.

Theorem 47 Let uε be a sequence bounded in H1(Ω). Then there existsu ∈ H1(Ω) and u1 ∈ L2(Ω;H1(Y )/R) such that, for a subsequence, it holdsthat

uε (x) u (x) in H1(Ω)

and∇uε (x) ∇u (x) +∇yu1 (x, y) .

Proof. See Theorem 3 in [Ngu1] and Proposition 1.14 in [All1].

Before we turn our interest to the homogenization let us point out thattwo-scale convergence can be defined in a more general setting, where noperiodicity assumptions are made. If we consider the left-hand side in (76)to be a special case of functionals of the type

F h(v) =

ZΩ

uh (x) (τhv)(x) dx

where τh is a sequence of maps acting on functions v ∈ L2(Ω;C (Y )), i.e.

τh(v(x, y)) = (τhv)(x) = v³x,

x

ε

´,

where ε = ε (h) → 0 as h → ∞, we obtain a generalization of periodictwo-scale convergence in a natural way.

Imposing certain conditions on the maps τh making up the key propertiesnecessary for two-scale compactness to hold, i.e. properties corresponding to(79) and (80), it is possible to obtain a compactness result of the same kindas for periodic two-scale convergence. We assume that uh is a boundedsequence in L2(Ω), thatX is a separable Banach space contained inL2(Ω×A),where A is an open bounded subset of RM , and τh : X → L2(Ω) linear mapssuch that the conditions °°τhv°°

L2(Ω)≤ C kvkX ,

limh→∞

°°τhv°°L2(Ω)

≤ C kvkL2(Ω×A)

65

are met. Then there is u0 ∈ L2 (Ω×A) such that, up to a subsequence,

limh→∞

ZΩ

uh(x)(τhv)(x) dx =

ZΩ

ZA

u0(x, y)v(x, y) dydx

for all v ∈ X; see e.g. [Hol1], [HSSW], [Sil1] and [HOS].

Let us point out that in order to achieve a connection to the usual weaklimit, like (84), additional assumptions are needed, see [Sil1] and [Sil2].

Finally we remark that other generalizations of two-scale convergenceallowing non-periodic functions have been made, such as scale convergence(see [MaTo]) and Σ-convergence (see [Ngu2]).

5.1.2 Homogenization using two-scale convergence

Now let us see how two-scale convergence can be applied to carry out thehomogenization for the model problem

−∇ ·³a³xε

´∇uε (x)

´= f (x) in Ω, (85)

uε (x) = 0 on ∂Ω,

where f ∈ L2(Ω) and a ∈M(α, β,RN) is Y -periodic. The point of departureis the weak form of the equation, namely that the equalityZ

Ω

a³xε

´∇uε (x) ·∇v (x) dx =

ZΩ

f (x) v (x) dx

should hold for all v ∈ H10(Ω). Using two different classes of test functions

we will get the homogenized problem and the local problem as two coupledPDEs.

Recall that uε is bounded in H10 (Ω) and hence in H1(Ω). This means

thatuε (x) u (x) in H1

0 (Ω)

up to a subsequence, and furthermore that the compactness result for thegradients in Theorem 47 is applicable. First, we choose test functions v withno microscopic oscillations. All products of the components in a and ∇vcreate admissible test functions (see Theorem 16 in [LNW]) and hence weget, after passing to the limit, the global problemZ

Ω

ZY

a (y) (∇u (x) +∇yu1 (x, y)) ·∇v (x) dydx =

ZΩ

f (x) v (x) dx (86)

for all v ∈ H10(Ω).

66

If we choose test functions oscillating in time with a¡xε

¢, more precisely

v (x) = εv1 (x) v2³xε

´,

where v1 ∈ D(Ω) and v2 ∈ C∞ (Y ) /R, we get after differentiation of vZΩ

a³xε

´∇uε(x) ·

³ε∇v1(x)v2

³xε

´+ v1(x)∇yv2

³xε

´´dx =Z

Ω

f(x)εv1(x)v2

³xε

´dx.

When passing to the limit, the right-hand side vanishes as well as one termin the left-hand side. Using Theorem 47 for the remaining term, we haveZ

Ω

ZY

a (y) (∇u (x) +∇yu1 (x, y)) v1(x)∇yv2 (y) dydx = 0.

Applying the variational lemma to eliminate the integral over Ω, we end upwith the local problemZ

Y

a (y) (∇u (x) +∇yu1 (x, y)) ·∇yv2 (y) dy = 0

holding for all v2 ∈ C∞ (Y ) /R and hence, by density, for all v2 ∈ H1 (Y ) /R.We now have a system of two PDEs and two unknowns, u and u1.

In some lucky cases the problems can be decoupled using separation ofvariables, enabling us to rewrite the local problem without the appearanceof x. Using the ansatz

u1 (x, y) = ∇u (x) · z (y) ,where z ∈ (H1 (Y ) /R)N in (86) we obtain the homogenized problem whichmeans that Z

Ω

b∇u (x) ·∇v (x) dx =

ZΩ

f (x) v (x) dx (87)

for all v ∈ H10(Ω), where

bij =

ZY

aij (y) +NXk=1

aik (y) ∂ykzj(y) dy.

The variable separated version of the local problem becomesZY

a (y) (ej +∇yzj (y)) ·∇yv2 (y) dy = 0,

still for all v2 ∈ H1 (Y ) /R.

67

Thus, we can determine z from the local problem and use it to identifythe deviation of b from the arithmetic mean of a. Hence, z tells us how u1represents the impact of the rapid oscillations in a(x

ε) on the limit coefficient.

Since the characterization of b is not dependent on the subsequence and thelimit equation (87) has a unique solution (see e.g. Section 6.3 in [CiDo]) wehave convergence of the entire sequence.

Summing up, we have, writing in classical form, that for a sequence uεof solutions to (85) it holds that

uε (x) u (x) in H10 (Ω),

where u uniquely solves the homogenized problem

−∇ · (b∇u (x)) = f (x) in Ω,

u (x) = 0 on ∂Ω,

with

bij =

ZY

aij (y) +NXk=1

aik (y) ∂ykzj(y) dy.

Here, zj ∈ H1 (Y ) /R is the unique solution to

−∇y · (a (y) (ej +∇yzj (y))) = 0 in Y .

This means that we have now proven Theorem 32 by two-scale convergencetechniques.

5.1.3 Two-scale convergence and asymptotic expansions

In this section we investigate the connection between the asymptotic expan-sion discussed in Section 4.1 and two-scale convergence. In particular, we willsee how a convergence of two-scale type can contribute to the interpretationof the second term in the expansion. Generalizations of these observationswill turn out to be useful in the homogenization procedures in Section 5.3.

We have seen that the two-scale convergence method and the method ofasymptotic expansions both provide us with the homogenization result forthe model problem. To enhance the understanding of the relation betweenthese two methods, we consider this result with the homogenized and thelocal problem in the coupled form, where the function u1 is visible.

68

Let us again recall the problem

−∇ ·³a³xε

´∇uε (x)

´= f (x) in Ω, (88)

uε (x) = 0 on ∂Ω.

We have obtained the homogenized problem

−∇ · (b∇u (x)) = f (x) in Ω, (89)

u (x) = 0 on ∂Ω,

where, by (86),

b∇u(x) =ZY

a (y) (∇u (x) +∇yu1 (x, y)) dy

and the local problem

−∇y · (a (y) (∇u (x) +∇yu1 (x, y))) = 0 in Ω× Y, (90)

where u is the weak H10 (Ω)-limit of uε and u1 ∈ L2(Ω;H1(Y )/R) in two

different ways.

In the two-scale convergence method we used the result from Theorem47, that for a sequence uε bounded in H1(Ω) it holds, up to a subsequence,that

∇uε (x) ∇u (x) +∇yu1 (x, y) , (91)

where u ∈ H1(Ω) and u1 ∈ L2(Ω;H1(Y )/R). As we saw in the previoussection, this allows us to pass to the limit in the weak form of (88) for certainchoices of test functions and thereby obtain the homogenized problem (89)and the local problem (90).

In the earlier method of asymptotic expansion one assumes (see Section4.1) that the solution uε can be expanded in a power series in ε of the form

uε (x) = u (x) + εu1

³x,

x

ε

´+ ε2u2

³x,

x

ε

´+ ..., (92)

where uk is Y -periodic in the second argument. Inserting (92) into the modelproblem (88) and equating equal powers of ε gives both the homogenizedproblem and the local problem, solved by u and u1, respectively. Actually, it

69

is in this process that u turns out to only depend on x. Furthermore, u andu1 are identical to the functions with the same names that appear in (91).

Let us now proceed by studying the separate terms in (92). Concerningthe first term in the expansion, we know that it is the limit of the sequenceuε of solutions to (88) in the sense that

uε (x) u (x) in H10 (Ω),

but it remains to be explored whether u1 also has significance in the sense ofsome suitable kind of limit. The truncated form

uε (x) ≈ u (x) + εu1

³x,

x

ε

´of (92) can be written

uε (x)− u (x)

ε≈ u1

³x,

x

ε

´, (93)

which indicates that©uε−uε

ªapproaches u1 in a sense similar to two-scale

convergence. However, we cannot expect ordinary two-scale convergence,since

©uε−uε

ªis not bounded in L2(Ω).

According to Proposition 3.2 in [AlBr], we can identify a class of contin-uous functions v : Ω× Y → R such that

F ε (·) = 1

ε

ZΩ

v³x,

x

ε

´(·) dx

is bounded in H−1(Ω). The key properties of these functions are that theyare Y -periodic for any fixed x ∈ Ω and that they have integral mean valuezero over Y in their second argument. Hence, for any bounded sequence αεin H1

0 (Ω) and

βε =1

ε

ZΩ

v³x,

x

ε

´αε (x) dx,

βε converges up to a subsequence. This means that, still up to a subse-quence, the limit

limε→0

ZΩ

uε (x)− u (x)

εv³x,

x

ε

´dx

exists for suitable test functions v.

70

It turns out that the choice of test functions is crucial. We will investigatethis using the example from Section 1.1, i.e. (88) with Ω = (0, 1), f(x) = x2

anda (y) =

1

2 + sin 2πy.

Figure 20 shows the left-hand side together with the right-hand side of (93),where uε and u are the solutions to (88) and (89), respectively.

Figure 20. uε(x)−u(x)ε

together with u1(x,xε).

They apparently have similar patterns of oscillations but differ in the globaltendency. The difference between these two functions, i.e.

gε(x) =uε (x)− u (x)

ε− u1

³x,

x

ε

´,

is shown in Figure 21.

Figure 21. The function gε(x).

71

For the test function

v (x, y) = x2 cos (2πy)

the graph of gε (x) v¡x, x

ε

¢is shown in Figure 22.

Figure 22. The product gε (x) v¡x, x

ε

¢,RYv (x, y) dy = 0.

Obviously, the integral of gε (x) v¡x, x

ε

¢over Ω is close to zero for small values

of ε. The high-frequency variations of gε are of vanishing amplitude; hence,the effect of these on the integral tends to zero. Furthermore, the function vhas mean value zero in its second variable; thus, for small ε the slower globaltendency of gε is filtered away by the rapid oscillations in v

¡x, x

ε

¢.

For this choice of v and for ε = 0.05 we obtainZΩ

uε (x)− u (x)

εv³x,

x

ε

´dx ≈ 0.00224 ≈Z

Ω

ZY

u1(x, y)v (x, y) dydx ≈ 0.00221,

and hence a limit of two-scale type consistent with the approach of asymptoticexpansion seems to be at hand for this kind of test functions.

If we omit the requirement that v should have integral mean value zeroover Y and choose, e.g.

v (x, y) = x2 (3 + cos (2πy)) ,

we obtain the function illustrated in Figure 23, whose integral over Ω obvi-ously does not vanish.

72

Figure 23. The product gε (x) v¡x, x

ε

¢,RYv (x, y) dy 6= 0.

In this case, ZΩ

uε (x)− u (x)

εv³x,

x

ε

´dx ≈ −0.02441 6=Z

Ω

ZY

u1(x, y)v (x, y) dydx ≈ 0.00221

for ε = 0.05. This shows that the class of test functions must be morerestricted than for usual two-scale convergence. Our investigation abovereveals that

©uε−uε

ªis approaching u1 only in a certain weak sense, namely

limε→0

ZΩ

uε (x)− u (x)

εv³x,

x

ε

´dx =

ZΩ

ZY

u1(x, y)v (x, y) dydx,

where the delicate question is to identify the appropriate class of test func-tions.

In [HSW], it is proven that for a bounded sequence uε in H1(Ω) andwith u and u1 defined as in Theorem 47, it holds that, up to a subsequence,

limε→0

ZΩ

uε (x)− u (x)

εv1 (x) v2

³xε

´dx =

ZΩ

ZY

u1(x, y)v1 (x) v2 (y) dydx (94)

for all v1 ∈ D(Ω) and v2 ∈ C∞ (Y ) /R. Thus, u1 appears as a limit oftwo-scale type to

©uε−uε

ª, which contributes to the interpretation of the as-

ymptotic expansion (92).

More general versions of this result (see Theorems 63 and 68) will be usedto prove the homogenization results in Sections 5.3.2 and 5.3.3. For problemswith more than one micro-scale, higher order terms in a multiscale expansioncan be identified as limits in a way similar to (94); see [HSW].

73

Finally, we note that the limit u1 is reminiscent of an ordinary two-scalelimit in the sense that it captures the local oscillations in uε−u

ε; see Figure

20, but, as observed earlier, the global tendency in x is not caught.

Figure 24. The function u1(x, y).

Remark 48 In the investigations in this section we have used test functionsthat are not zero on the boundary. Thus, they do not belong to D(Ω) as inthe convergence result above. However, if one assumes that u and uε havethe same boundary conditions, the factor uε−u

εwill be zero on the boundary,

and under these conditions the convergence result will also hold true. This isthe case, for example, if uε belongs to H1

0(Ω).

Remark 49 The considerations in this section are made in the context of thehomogenization of the model problem, where uε converges weakly in H1

0 (Ω),and hence strongly in L2(Ω). A relation between two-scale convergence andasymptotic expansions, which is not revealed in this connection, is that anyfunction that admits a two-scale asymptotic expansion of the type (40)–withuk(x, y) smooth and also Y -periodic in y–two-scale converges to the firstterm in the expansion, i.e. to u0(x, y); see [All1].

5.2 Homogenization with multiple scales

We have already encountered homogenization problems with several micro-scopic scales in Section 4.3, where we considered such problems in the contextof classical homogenization methods and G-convergence techniques. As wesaw in the discussion in Section 5.1.2, the method of two-scale convergenceis an efficient tool for homogenizing PDEs with periodic oscillations on onemicro-scale. In this section, we will see how this method can be extended toproblems with several micro-scales.

74

Adjusting our model problem to the case of multiple scales, we arrive at

−∇ ·µa

µx

ε1, ...,

x

εn

¶∇uε (x)

¶= f (x) in Ω, (95)

uε (x) = 0 on ∂Ω,

where f ∈ L2(Ω), and where a (y1, ..., yn) belongs to M(α, β,RnN) and isYk-periodic in the argument yk, k = 1, 2, ..., n.

Here, all εk are supposed to be functions depending on the common vari-able ε. The way in which the scales are related to each other is of significance,for example, for the proofs of the compactness results for gradients in The-orem 51 (Section 5.2.1) and Theorem 62 (Section 5.3.1). Apart from theassumption that the scales are microscopic, i.e. that

limε→0

εk = 0

for k = 1, 2, ..., n, we assume from now on that the scales εk are separated,i.e. that

limε→0

εk+1εk

= 0

for k = 1, 2, ..., n− 1. This means that even though they are all microscopic,they can be distinguished from one another, making it meaningful to refer todifferent scales. We will also use the notion of well-separated scales, whichmeans that there exists an integer m > 0 such that

limε→0

1

εk

µεk+1εk

¶m

= 0

for k = 1, 2, ..., n− 1.One can show that, under the assumption that a ∈ M(α, β,RnN), the

sequence uε of solutions to (95) satisfies suitable a priori estimates, i.e.boundedness in H1

0 (Ω) and hence in H1(Ω), and also that for a sequenceof equations (95), we have G-convergence up to a subsequence. However,it is not obvious how the homogenization procedure should be performedor what the local problems will look like. After introducing the appropri-ate generalization of two-scale convergence in Section 5.2.1, we perform thehomogenization of a problem of this form in Section 5.2.2.

75

5.2.1 Multiscale convergence

For reiterated homogenization problems, when we are dealing with multiplescales of periodic oscillations, we need a generalization of two-scale conver-gence. The so-called multiscale convergence was first introduced by Allaireand Briane in [AlBr] for an arbitrary number of microscopic scales.

Definition 50 A sequence uε in L2(Ω) is said to (n+ 1)-scale convergeto u0 ∈ L2 (Ω× Y n) if

limε→0

ZΩ

uε (x) v

µx,

x

ε1, ...,

x

εn

¶dx = (96)Z

Ω

ZY n

u0 (x, y1, ..., yn) v (x, y1, ..., yn) dyn...dy1dx

for all v ∈ L2 (Ω;C (Y n)). We write

uε (x)n+1

u0 (x, y1, ..., yn) .

A compactness result similar to that in the two-scale case holds true,namely that a bounded sequence uε in L2(Ω) possesses a subsequence that(n+ 1)-scale converges to some limit u0 ∈ L2 (Ω× Y n); see Theorem 2.4 in[AlBr].

We also have the corresponding result on the multiscale convergence ofsequences of gradients, proven in [AlBr].

Theorem 51 Let uε be a bounded sequence in H1(Ω). Then there exist nfunctions uk ∈ L2

¡Ω× Y k−1;H1(Yk)

¢such that, up to a subsequence,

uε (x)n+1

u (x)

and

∇uε (x) n+1 ∇u (x) +nX

k=1

∇ykuk (x, y1, . . . , yk) ,

where u is the weak H1(Ω)-limit of uε.Proof. See the proof of Theorem 2.6 in [AlBr].

76

Remark 52 If uε is a bounded sequence in L2(Ω) and (96) holds for everyv ∈ D

¡Ω;C∞ (Y n)

¢, then uε (n+ 1)-scale converges. The proof is in line

with the proof of Proposition 1 in [LNW].

Remark 53 The proof of Theorem 51 is done in different ways for the casesof separated and well-separated scales, respectively, but the result is the same;see the proofs of Theorem 2.6 in [AlBr], for these two cases.

5.2.2 Reiterated homogenization using multiscale convergence

Multiscale convergence allows us to pass to the limit in the homogenizationprocedure of problem (95). By analogy with the two-scale case, this is doneusing different classes of test functions, each class being responsible for acertain scale. In this case, we will end up with a system of local problems, thesolutions of which are used to characterize the coefficient in the homogenizedproblem.

To illustrate this method, there follows a short review of the homoge-nization of a problem of the type (95) following [AlBr]. A case in point, yetsimple, is when we have two microscopic spatial scales with periodicities εand ε2, respectively, which are obviously separated, i.e. the problem

−∇ ·³a³xε,x

ε2

´∇uε (x)

´= f (x) in Ω,

uε (x) = 0 on ∂Ω.

In this case the coefficients can, for two different values of ε, have a patternof oscillations such as in Figure 25.

Figure 25. Pattern of oscillations of a(xε, xε2).

77

For this problem, we will need two different local problems to identifythe limit coefficient. We start out by giving the weak form of the problem,namely to find uε ∈ H1

0 (Ω) such thatZΩ

a³xε,x

ε2

´∇uε (x) ·∇v (x) dx =

ZΩ

f (x) v (x) dx (97)

holds for all v ∈ H10 (Ω). The sequence uε of solutions satisfies the appro-

priate a priori estimate, and thus we know that, at least for a subsequence,we have weak convergence in H1

0(Ω) of the solutions uε to some limit u.

To find the homogenized problem solved by u, we first choose test func-tions with no microscopic oscillations. By Theorem 51, we can pass to thelimit in (97). Hence, returning to the classical form we achieve

−∇ · (b∇u (x)) = f (x) in Ω,

u (x) = 0 on ∂Ω,

(see Remark 54) where

b∇u(x) =ZY 2

a (y1, y2) (∇u(x) +∇y1u1 (x, y1) +∇y2u2 (x, y1, y2)) dy2dy1.

Next, we choose test functions that have oscillations in time with themicro-scales, more precisely

v (x) = ε2v1 (x) v2³xε

´v3³ xε2

´,

where v1 ∈ D(Ω), v2 ∈ C∞ (Y1) and v3 ∈ C∞ (Y2) /R. Thus, we haveZΩ

a³xε,x

ε2

´∇uε (x) ·∇

³ε2v1 (x) v2

³xε

´v3

³ xε2

´´dx =Z

Ω

f (x) ε2v1 (x) v2³xε

´v3³ xε2

´dx.

We can now proceed as we did in Section 5.1.2 for the case with one micro-scale. Carrying out the differentiation of v, applying Theorem 51 and usingthe variational lemma, we obtain as our limit the weak form of the first localproblem

−∇y2 · (a (y1, y2) (∇u (x) +∇y1u1 (x, y1) +∇y2u2 (x, y1, y2))) = 0,

78

which we recognize as an elliptic equation for u2 in Y2. In a similar way tothat observed in Section 5.1.2–that the solution to the local problem repre-sented the influence of the oscillations on the G-limit b–this means that wehave characterized the contribution to b from the oscillations in the ε2-scale.

To capture the oscillations in the ε-scale, let us now choose test functionswhere we leave out oscillations corresponding to the ε2-scale. We choose

v (x) = εv1 (x) v2

³xε

´,

where v1 ∈ D(Ω) and v2 ∈ C∞ (Y1) /R, which means that we haveZΩ

a³xε,x

ε2

´∇uε (x) ·∇

³εv1 (x) v2

³xε

´´dx =

ZΩ

f (x) εv1 (x) v2³xε

´dx.

Again, using Theorem 51, we obtain

−∇y1 ·ZY2

a (y1, y2) (∇u (x) +∇y1u1 (x, y1) +∇y2u2 (x, y1, y2)) dy2 = 0.

This is the other local problem, which we observe is also an elliptic equation,but for the function u1 in Y1.

Hence, up to a subsequence we know that the weak H10 (Ω)-limit u of uε

solves the problem

−∇ · (b∇u (x)) = f (x) in Ω,

u (x) = 0 on ∂Ω,

with

b∇u(x) =ZY 2

a (y1, y2) (∇u(x) +∇y1u1 (x, y1) +∇y2u2 (x, y1, y2)) dy2dy1,

where u1 ∈ L2(Ω;H1(Y1)/R) and u2 ∈ L2(Ω×Y1;H1(Y2)/R) are the solutionsto the local problems

−∇y2 · (a (y1, y2) (∇u (x) +∇y1u1 (x, y1) +∇y2u2 (x, y1, y2))) = 0,

−∇y1 ·ZY2

a (y1, y2) (∇u (x) +∇y1u1 (x, y1) +∇y2u2 (x, y1, y2)) dy2 = 0.

Arguing as we did in Section 5.1.2, we have convergence of the entire se-quence, which means that the homogenization is completed. See Theorem2.11 in [AlBr] for the corresponding result for an arbitrary finite number ofscales.

79

Remark 54 From our observations on the G-convergence of©a(x

ε, xε2)ª, it

follows that the homogenized problem can be written as

−∇ · (b (x)∇u (x)) = f (x) in Ω,

u (x) = 0 on ∂Ω.

This also means that we can expect that it is possible to extract b from thehomogenized problem as we found it. This is done by successively using thelocal problems, starting with the one corresponding to the smallest scale. Itturns out that b is constant. For details, see the inductive homogenizationformula in Corollary 2.12 in [AlBr].

Remark 55 In the examples studied in this section and in Section 5.1.2, wefind the local and the homogenized problems in a straightforward way, choos-ing the appropriate test functions. In a nonlinear case, the correspondingprocedure does not provide us with the complete characterization of the limitoperator. For this purpose, one uses so-called perturbed test functions; seee.g. [All1]. This is done in the proof of Theorem 72 in Section 5.3.2.

5.3 Homogenization with multiple scales in space andtime

In Chapter 4, we homogenized a monotone parabolic problem with oscilla-tions in space on two scales. In this section we will investigate some problemswhere oscillations appear also in the time variable, and for such problems theapproach we used in Chapter 4 is not applicable. Instead, we use multiscaleconvergence techniques. In Section 5.3.1 we extend the multiscale conver-gence to include temporal oscillations, in order to fit problems of this kind.We also make the necessary preparations for the homogenization procedures.In Section 5.3.2, we consider a possibly nonlinear case where oscillations intwo spatial scales and one temporal scale appear, i.e. of the type

∂tuε (x, t)−∇ · a

µx

ε,x

ε2,t

εr,∇uε

¶= f (x, t) in ΩT ,

uε (x, 0) = u0 (x) in Ω, (98)

uε (x, t) = 0 on ∂Ω× (0, T ) .

We perform the homogenization of this problem for 0 < r < 3, where westudy the three cases 0 < r < 2, r = 2 and 2 < r < 3, which require different

80

sets of local problems in order to identify the limit operator. Restrictingourselves to the linear case, we proceed by introducing a second fast temporalscale. A problem with rapid oscillations in one spatial scale and two temporalscales, that is, an equation of the form

∂tuε (x, t)−∇ ·

µa

µx

ε,t

ε,t

εr

¶∇uε(x, t)

¶= f (x, t) in ΩT ,

uε (x, 0) = u0 (x) in Ω, (99)

uε (x, t) = 0 on ∂Ω× (0, T ) ,

is studied in Section 5.3.3, where we carry out the homogenization procedurefor r > 0, r 6= 1. Also for this problem, we distinguish three different cases:0 < r < 2, where r 6= 1, r = 2 and r > 2.

An example of the pattern of oscillations is shown in Figure 26 for thecase with oscillations in two spatial scales and one temporal scale.

Figure 26. Oscillations in space and time.

If we permute the x and t axes, this could also illustrate what the coefficientin (99) might look like.

5.3.1 Evolution multiscale convergence

In the problems above, we have oscillations in space as well as in time. Thissuggests that we will need a further generalization of the notion of multiscaleconvergence, where there may also be temporal oscillations. We give thefollowing definition of evolution multiscale convergence, which includes n+1spatial scales and m+ 1 temporal scales, i.e. n and m micro-scales in spaceand time, respectively.

81

Definition 56 A sequence uε in L2 (ΩT ) is said to (n + 1,m + 1)-scaleconverge to u0 ∈ L2 (ΩT ×Yn,m) if

limε→0

ZΩT

uε (x, t) v

µx, t,

x

ε1, ...,

x

εn,t

ε01, ...,

t

ε0m

¶dxdt =Z

ΩT

ZYn,m

u0 (x, t, y1, ..., yn, s1, ..., sm) ·

v (x, t, y1, ..., yn, s1, ..., sm) dyn...dy1dsm...ds1dxdt

for all v ∈ L2 (ΩT ;C (Yn,m)) . This is denoted by

uε (x, t)n+1,m+1

u0 (x, t, y1, ..., yn, s1, ..., sm) .

3,2-scale convergence

Let us first prepare for the homogenization of problem (98) where we havetwo microscopic scales in space and one fast temporal scale. Thus, adjust-ing for this problem we choose n = 2 and m = 1, which means that theappropriate form of evolution multiscale convergence in this case is 3,2-scaleconvergence. Obviously, for a sequence uε in L2 (ΩT ), 3,2-scale convergenceto u0 ∈ L2 (ΩT ×Y2,1) means that

limε→0

ZΩT

uε (x, t) v

µx, t,

x

ε1,x

ε2,t

ε01

¶dxdt =Z

ΩT

ZY2,1

u0 (x, t, y1, y2, s) v (x, t, y1, y2, s) dy2dy1dsdxdt

for all v ∈ L2 (ΩT ;C (Y2,1)), and the notation of this is

uε (x, t)3,2

u0 (x, t, y1, y2, s) .

The following theorem on a special case of 3,2-scale convergence, wherethe scales are chosen as in (98), will be useful in the sequel.

Theorem 57 Let uε be a bounded sequence in L2 (ΩT ) and assume that wehave ε1 = ε, ε2 = ε2 and ε01 = εr, where r > 0. Then there exists a function

u0 ∈ L2 (ΩT ×Y2,1) such that, for a subsequence, it holds that uε3,2

u0.

Proof. This follows directly from Theorem 2.4 in [AlBr].

82

As we saw in Section 5.2.2, the result on the multiscale limit of the gra-dients was crucial to be able to pass to the limit in the homogenizationprocedure. In the next section, where we perform the homogenization ofthe parabolic problem (98), we need a similar result, adapted to this time-dependent case. This is stated in Theorem 62 below.

To prove this theorem, we need a few preliminaries. We will, for example,make use of a property similar to (82), i.e. that v(x, x

ε) converges weakly

in L2 (Ω) to the mean of v in the local variable over Y if v is an admissibletest function.

If we assume that v (x, y1, ..., yn) is Yk-periodic in yk for all k = 1, 2, ..., n,and sufficiently smooth and the scales are separated, then according to [AlBr],

v

µx,

x

ε1, ...,

x

εn

¶→ZY n

v (x, y1, ..., yn) dyn...dy1 in D0(Ω).

Moreover, if v(x, xε1, ..., x

εn) is bounded in L2(Ω), we obtain

v

µx,

x

ε1, ...,

x

εn

¶ ZY n

v (x, y1, ..., yn) dyn...dy1 in L2(Ω).

A transition to our evolution setting is given in the following proposition,where the function v has oscillations in time as well as in space.

Proposition 58 For every v ∈ C(ΩT ;C (Y2,1)) and r > 0, it holds that

limε→0

ZΩT

v

µx, t,

x

ε,x

ε2,t

εr

¶φ (x, t) dxdt =Z

ΩT

ZY2,1

v (x, t, y1, y2, s)φ (x, t) dy2dy1dsdxdt

for all φ ∈ L2 (ΩT ).

Proof. See Proposition 5 and Remark 6 in [FlOl1]. See also Corollario 3.5in [Don] and Lemma 4.2.2 in [Pan].

To prove the result on the convergence of gradients, we also need thecorrespondence to (83), i.e. that a strongly convergent sequence two-scaleconverges to its strong limit. We have the following proposition concerning3,2-scale convergence.

83

Proposition 59 Assume that uε converges strongly to u in L2(ΩT ) andthat ε1 = ε, ε2 = ε2 and ε01 = εr, where r > 0. Then uε 3,2-scale convergesto u.

Proof. According to Proposition 58, it holds that

v

µx, t,

x

ε,x

ε2,t

εr

¶ ZY2,1

v (x, t, y1, y2, s) dy2dy1ds in L2 (ΩT )

for all v ∈ C(ΩT ;C (Y2,1)). By assumption,

uε (x, t)→ u (x, t) in L2 (ΩT )

and thus we have

limε→0

ZΩT

uε (x, t) v

µx, t,

x

ε,x

ε2,t

εr

¶dxdt =Z

ΩT

ZY2,1

u (x, t) v (x, t, y1, y2, s) dy2dy1dsdxdt

for all v ∈ C(ΩT ;C (Y2,1)), and hence (see Remark 52) for allv ∈ L2 (ΩT ;C (Y2,1)) .

Finally, we will need the following two lemmas.

Lemma 60 Let H be the space of generalized divergence-free functions inL2(Ω;L2 (Y 2))N defined by

H =

½v ∈ L2

¡Ω;L2(Y 2)

¢N ¯∇y2 · v = 0,ZY2

∇y1 · v (x, y1, y2) dy2 = 0

¾.

H has the following properties:

(i) D(Ω;C∞ (Y 2))N ∩H is dense in H.

(ii) The orthogonal of H is

H⊥ =©∇y1u1 +∇y2u2|u1 ∈ L2(Ω;H1(Y1)), u2 ∈ L2(Ω× Y1;H

1(Y2))ª.

Proof. See Lemma 3.7 in [AlBr].

84

Lemma 61 Let φ ∈ D(Ω;C∞ (Y 2)) be a function such thatZY2

φ (x, y1, y2) dy2 = 0

and assume that the scales ε1, ε2 are well-separated. Then

1

ε2φ

µx,

x

ε1,x

ε2

¶is bounded in

¡H1(Ω)

¢0.

Proof. See Theorem 3.3 in [AlBr].

We are now prepared to give the following theorem on 3,2-scale conver-gence of gradients, which will be used while proving the homogenizationresults for equation (98), i.e. Theorems 72, 73 and 74 in Section 5.3.2.

Theorem 62 Let uε be a sequence bounded in H1 (0, T ;H10(Ω),H

−1(Ω))and assume that we have ε1 = ε, ε2 = ε2 and ε01 = εr, r > 0. Then it holds,up to a subsequence, that

uε (x, t) → u (x, t) in L2 (ΩT ) ,

uε (x, t) u (x, t) in L2¡0, T ;H1

0(Ω)¢

and

∇uε (x, t) 3,2 ∇u (x, t) +∇y1u1 (x, t, y1, s) +∇y2u2 (x, t, y1, y2, s) ,

where u ∈ L2 (0, T ;H10 (Ω)) , u1 ∈ L2(ΩT × (0, 1) ;H1 (Y1) /R) and

u2 ∈ L2(ΩT ×Y1,1;H1 (Y2) /R).Proof. Since uε is bounded in H1 (0, T ;H1

0 (Ω), H−1(Ω)), it is obviously

bounded in L2 (0, T ;H10(Ω)). Hence, we know by Theorem 57 that uε and

∇uε both possess a 3,2-scale limit, up to a subsequence. This means thatthere exist u0 ∈ L2 (ΩT ×Y2,1) and w0 ∈ L2 (ΩT ×Y2,1)N such that, at leastfor a subsequence,

uε (x, t)3,2

u0 (x, t, y1, y2, s)

and∇uε (x, t) 3,2 w0 (x, t, y1, y2, s) .

From the assumption of boundedness in H1 (0, T ;H10 (Ω),H

−1(Ω)), it alsofollows that ∂tuε is bounded in L2 (0, T ;H−1(Ω)). For uε bounded in

85

L2 (0, T ;H10(Ω)) with ∂tuε bounded in L2 (0, T ;H−1(Ω)), it is proven in

Lemmas 8.2 and 8.4 in [CoFo] that for a suitable subsequence

uε (x, t)→ u (x, t) in L2(ΩT ). (100)

See also Problem 23.13b in [ZeiIIA]. Furthermore, the boundedness inL2 (0, T ;H1

0(Ω)) gives that, up to a subsequence,

uε (x, t) u (x, t) in L2(0, T ;H10 (Ω)).

By (100), we can apply Proposition 59 which yieldsZΩT

uε (x, t) v³x,

x

ε,x

ε2

´c1 (t) c2

µt

εr

¶dxdt→Z

ΩT

u (x, t)

ÃZY2,1

v (x, y1, y2) c1 (t) c2 (s) dy2dy1ds

!dxdt =Z

ΩT

ZY2,1

u (x, t) v (x, y1, y2) c1 (t) c2 (s) dy2dy1dsdxdt

for all c1 ∈ D (0, T ), c2 ∈ C∞ (0, 1) and v ∈ D(Ω;C∞ (Y 2))N .

Our next aim is to characterize the limit w0. For this purpose, we choosefunctions v belonging to the set D(Ω;C∞ (Y 2))N ∩H introduced in Lemma60. We know that, up to a subsequence,Z

ΩT

∇uε (x, t) · v³x,

x

ε,x

ε2

´c1 (t) c2

µt

εr

¶dxdt→Z

ΩT

ZY2,1

w0 (x, t, y1, y2, s) · v (x, y1, y2) c1 (t) c2 (s) dy2dy1dsdxdt

and we carry out the characterization by studying the limit process forZΩT

∇uε (x, t) · v³x,

x

ε,x

ε2

´c1 (t) c2

µt

εr

¶dxdt.

For t fixed, integration by parts over Ω gives

−ZΩ

uε (x, t)¡∇+ ε−1∇y1 + ε−2∇y2

¢· v³x,

x

ε,x

ε2

´c1 (t) c2

µt

εr

¶dx =

−ZΩ

uε (x, t)¡∇+ ε−1∇y1

¢· v³x,

x

ε,x

ε2

´c1 (t) c2

µt

εr

¶dx,

86

due to the definition of H. We will now see that the contribution from theterm Z

Ω

uε (x, t) ε−1∇y1 · v³x,

x

ε,x

ε2

´c1 (t) c2

µt

εr

¶dx

tends to zero. If φ ∈ D¡Ω;C∞ (Y 2)

¢andZ

Y2

φ (x, y1, y2) dy2 = 0,

then ε−2φ(x, xε, xε2) is bounded in (H1(Ω))0, by Lemma 61, i.e.¯ZΩ

ε−2φ³x,

x

ε,x

ε2

´ρ (x) dx

¯≤ C kρkH1(Ω)

for all ρ ∈ H1(Ω). Since ∇y1 · v fulfills the conditions on φ above, we get¯ZΩ

ε−2∇y1 · v³x,

x

ε,x

ε2

´ρ (x) dx

¯≤ C kρkH1(Ω) ,

that is ¯ZΩ

ε−1∇y1 · v³x,

x

ε,x

ε2

´ρ (x) dx

¯≤ Cε kρkH1(Ω) .

For t fixed and ρ = uε, we have¯ZΩ

ε−1∇y1 · v³x,

x

ε,x

ε2

´uε (x, t) dx

¯≤ Cε kuεkH1(Ω)

and it follows that¯ZΩT

ε−1∇y1 · v³x,

x

ε,x

ε2

´uε (x, t) dxdt

¯2≤

C1

Z T

0

¯ZΩ

ε−1∇y1 · v³x,

x

ε,x

ε2

´uε (x, t) dx

¯2dt ≤

C2ε2

Z T

0

kuε (·, t)k2H1(Ω) dt = C2ε2 kuεk2L2(0,T ;H1(Ω)) ≤ C3ε

2 → 0

as ε→ 0. We have proven that

−ZΩT

uε (x, t)¡∇+ ε−1∇y1 + ε−2∇y2

¢· v³x,

x

ε,x

ε2

´c1 (t) c2

µt

εr

¶dxdt→

−ZΩT

ZY2,1

u (x, t)∇ · v (x, y1, y2) c1 (t) c2 (s) dy2dy1dsdxdt =

87

ZΩT

ZY2,1∇u (x, t) · v (x, y1, y2) c1 (t) c2 (s) dy2dy1dsdxdt

for all v ∈ D(Ω;C∞(Y 2))N ∩H, all c1 ∈ D(0, T ) and all c2 ∈ C∞(0, 1). Thus,we haveZ

ΩT

ZY2,1

w0 (x, t, y1, y2, s) · v (x, y1, y2) c1 (t) c2 (s) dy2dy1dsdxdt =

ZΩT

ZY2,1∇u (x, t) · v (x, y1, y2) c1 (t) c2 (s) dy2dy1dsdxdt

for every such v, c1 and c2. By the variational lemma, we get for almost all sand t Z

Ω

ZY 2

w0 (x, t, y1, y2, s) · v (x, y1, y2) dy2dy1dx =ZΩ

ZY 2∇u (x, t) · v (x, y1, y2) dy2dy1dx

for all v ∈ D(Ω;C∞(Y 2))N ∩H and by density (see (i) in Lemma 60), thisholds for all v ∈ H. Hence, w0 and ∇u can only differ by elements in H⊥,that is, we have

w0 (x, t, y1, y2, s)−∇u (x, t) = ∇y1u1 (x, t, y1, s) +∇y2u2 (x, t, y1, y2, s)

for almost all s and t; that is

w0 (x, t, y1, y2, s) = ∇u (x, t) +∇y1u1 (x, t, y1, s) +∇y2u2 (x, t, y1, y2, s) .

It remains for us to localize u1 and u2 into suitable function spaces. If wechoose u1 and u2 with average values of zero over Y1 and Y2, respectively,this means that we should prove that u1 ∈ L2(ΩT × (0, 1);H1(Y1)/R) andu2 ∈ L2(ΩT ×Y1,1;H1(Y2)/R). We have, for all v ∈ D(ΩT , C

∞ (Y1,1))N ,ZΩT

ZY2,1(w0 (x, t, y1, y2, s)−∇u (x, t)) · v (x, t, y1, s) dy2dy1dsdxdt =Z

ΩT

ZY2,1(∇y1u1 (x, t, y1, s)+∇y2u2 (x, t, y1, y2, s)) · v (x, t, y1, s) dy2dy1dsdxdt=Z

ΩT

ZY1,1∇y1u1 (x, t, y1, s) · v (x, t, y1, s) dy1dsdxdt,

88

where after integration over Y2 the second term has vanished due to theY2-periodicity of u2. Again applying the variational lemma, we have foralmost every (x, t, y1, s) ∈ ΩT ×Y1,1

∇y1u1 (x, t, y1, s) =

ZY2

w0 (x, t, y1, y2, s) dy2 −∇u (x, t)

and hence ∇y1u1 ∈ L2 (ΩT ×Y1,1)N , that is, u1 ∈ L2(ΩT × (0, 1);H1(Y1)/R).It follows that

∇y2u2 (x, t, y1, y2, s) = w0 (x, t, y1, y2, s)−∇u (x, t)−∇y1u1 (x, t, y1, s) ,

i.e. ∇y2u2 ∈ L2(ΩT × Y2,1)N , and hence u2 ∈ L2(ΩT × Y1,1;H1(Y2)/R).The functions u1 and u2 are bounded in L2 (ΩT ×Y1,1) and L2(ΩT × Y2,1),respectively, by the Poincaré-Wirtinger inequality.

For the homogenization procedure in the next section we also need thefollowing theorem, which generalizes our observations in Section 5.1.3 to aparticular evolution case. A similar result, also including non-periodic cases,has been proven recently by Nguetseng and Woukeng in [NgWo2].

Theorem 63 Let uε be a bounded sequence in H1 (0, T ;H10 (Ω),H

−1(Ω)),and let u and u1 be defined as in Theorem 62. Then, up to a subsequence,

limε→0

ZΩT

uε (x, t)− u (x, t)

εv1 (x) v2

³xε

´c1 (t) c2

µt

εr

¶dxdt =Z

ΩT

ZY1,1

u1 (x, t, y1, s) v1 (x) v2 (y1) c1 (t) c2 (s) dy1dsdxdt

for all v1 ∈ D(Ω), v2 ∈ L2 (Y1) /R, c1 ∈ D (0, T ) and c2 ∈ L2 (0, 1) .

Proof. See Corollary 3.3 in [Hol2] and Theorem 3 in [HSW].

Our last preparation concerning problem (98) is the following proposition,proven in [NgWo2]. This will be used for the case where r = 2.

Proposition 64 Let u and v belong to L2(0, 1;H1(Y )/R) and assume that∂su and ∂sv are in L2(0, 1; (H1(Y )/R)0). ThenZ 1

0

h∂su(s), v(s)i(H1(Y )/R)0,H1(Y )/R ds+

Z 1

0

h∂sv(s), u(s)i(H1(Y )/R)0,H1(Y )/R ds = 0.

Proof. See Corollary 4.1. in [NgWo2].

89

2,3-scale convergence

The results we have established so far in this section have been with a viewto homogenizing (98). We also need similar results to prepare for the homog-enization of (99). Adapting to this problem, we obtain 2,3-scale convergencethat includes two spatial scales and three temporal scales. Following thegeneral definition of evolution multiscale convergence, 2,3-scale convergenceof a sequence uε in L2 (ΩT ) to a function u0 ∈ L2 (ΩT ×Y1,2) means that

limε→0

ZΩT

uε (x, t) v

µx, t,

x

ε1,t

ε01,t

ε02

¶dxdt =Z

ΩT

ZY1,2

u0 (x, t, y, s1, s2) v (x, t, y, s1, s2) dyds2ds1dxdt

for all v ∈ L2 (ΩT ;C (Y1,2)). We denote this by

uε (x, t)2,3

u0 (x, t, y, s1, s2) .

The following compactness result holds true.

Theorem 65 Let uε be a bounded sequence in L2 (ΩT ) and assume thatε1 = ε, ε01 = ε and ε02 = εr, where r > 0 and r 6= 1. Then there exists afunction u0 ∈ L2 (ΩT ×Y1,2) such that, up to a subsequence, uε 2,3-scaleconverges to u0.

Proof. This is an immediate consequence of Theorem 2.4 in [AlBr].

Remark 66 We omit the case r = 1 to obtain separation between the scales,see [AlBr] for details.

We also have the theorem below on 2, 3-scale convergence of gradients.

Theorem 67 Let uε be a sequence bounded in H1 (0, T ;H10(Ω),H

−1(Ω))and assume that ε1 = ε, ε01 = ε and ε02 = εr, where r > 0 and r 6= 1. Then itholds, up to a subsequence, that

uε (x, t) → u (x, t) in L2 (ΩT ) ,

uε (x, t) u (x, t) in L2¡0, T ;H1

0(Ω)¢

and∇uε (x, t) 2,3 ∇u (x, t) +∇yu1 (x, t, y, s1, s2) ,

where u ∈ L2 (0, T ;H10 (Ω)) and u1 ∈ L2(ΩT × (0, 1)2 ;H1 (Y ) /R).

90

Proof. The proof follows directly along the lines of the proof of Theorem3.1 in [Hol2].

Finally, we have the following correspondence to Theorem 63.

Theorem 68 Let uε be a bounded sequence in H1 (0, T ;H10 (Ω),H

−1(Ω)),and let u and u1 be defined as in Theorem 67. Then, up to a subsequence,

limε→0

ZΩT

uε (x, t)− u (x, t)

εv1 (x) v2

³xε

´c1 (t) c2

µt

ε

¶c3

µt

εr

¶dxdt =Z

ΩT

ZY1,2

u1(x, t, y, s1, s2)v1 (x) v2 (y) c1 (t) c2 (s1) c3(s2) dyds2ds1dxdt

for all v1 ∈ D(Ω), v2 ∈ L2 (Y ) /R, c1 ∈ D (0, T ) and c2, c3 ∈ L2 (0, 1) ifr > 0, r 6= 1.Proof. The proof is performed in exactly the same way as the proof ofCorollary 3.3 in [Hol2].

Theorems 65, 67 and 68 above will be used while proving the homoge-nization result in Theorem 76 in Section 5.3.3.

Remark 69 The first results on two-scale convergence were presented byNguetseng in [Ngu1], where the efficiency of this homogenization techniqueis illustrated on linear elliptic problems. In [All1] Allaire gives a differentproof for two-scale compactness, where some new sets of test functions areintroduced. The method is applied to several non-trivial problems such as thehomogenization of linear elliptic problems in perforated domains (see alsoe.g. [AlBr], [Ngu3] and [FHOSi2]) and homogenization of nonlinear ellipticproblems under certain monotonicity assumptions. In [AlBr], Allaire andBriane introduce a generalization to the multiple scale case and use this forhomogenization of linear elliptic problems with multiple scales; see Section5.2. The nonlinear monotone case is studied by Lions et al. in [LLPW].In [CaGa], the extension to the almost periodic case is treated by Casado-Diaz and Gayte, and in [BMW] Bourgeat et al. develop a type of stochastictwo-scale convergence. A careful investigation of the properties of periodictwo-scale convergence is made by Lukkassen et al. in [LNW]. A general-ization to certain non-periodic cases is made in [Ngu2] under the name ofΣ-convergence and a corresponding extension of the results in Theorems 63and 68 can be found in [NgWo2]. In [CDG], the closely related approach ofso-called unfolding is introduced by Cioranescu et al. This is also studied byNechvatal in [Nec].

91

5.3.2 Homogenization of some monotone parabolic operators withthree spatial and two temporal scales

We are now prepared to further investigate the parabolic equation

∂tuε (x, t)−∇ · a

µx

ε,x

ε2,t

εr,∇uε

¶= f (x, t) in ΩT ,

uε (x, 0) = u0 (x) in Ω, (101)

uε (x, t) = 0 on ∂Ω× (0, T )

where u0 ∈ L2(Ω) and f ∈ L2 (ΩT ). The function

a : R2N ×R×RN → RN

is assumed to satisfy the following conditions, where 0 < k ≤ 1 and α and βare positive constants:

(i) a (y1, y2, s, 0) = 0 for all (y1, y2, s) ∈ R2N × R.

(ii) a (·, ·, ·, ξ) is continuous and Y2,1-periodic for all ξ ∈ RN .

(iii) a (y1, y2, s, ·) is continuous for all (y1, y2, s) ∈ R2N × R.

(iv) |a (y1, y2, s, ξ)− a (y1, y2, s, ξ0)| ≤ β (1 + |ξ|+ |ξ0|)1−k |ξ − ξ0|k

for all (y1, y2, s) ∈ R2N ×R and all ξ, ξ0 ∈ RN .

(v) (a (y1, y2, s, ξ)− a (y1, y2, s, ξ0)) · (ξ − ξ0) ≥ α |ξ − ξ0|2

for all (y1, y2, s) ∈ R2N ×R and all ξ, ξ0 ∈ RN .

In a way similar to that in Section 4.3, we observe that this means that abelongs to N (α, β, k,R2N ×R).

Remark 70 Note that when we studied this problem with respect to ex-istence and uniqueness of the solution in Chapter 2 we considered f inL2 (0, T ;H−1(Ω)), whereas we now have chosen right-hand sides in L2(ΩT )to fit the multiscale convergence approach. It can be shown (see Example 23.4in [ZeiIIA]) that these can be identified with functionals in L2 (0, T ;H−1(Ω)).

We will establish the existence of a G-limit for the sequence of opera-tors corresponding to (101) and also characterize this limit by performingthe homogenization procedure for this problem. This will be done for dif-ferent values of the constant r, that is for different frequencies of the timeoscillations. We consider the three cases 0 < r < 2, r = 2 and 2 < r < 3.

92

Let us start by giving the following theorem.

Theorem 71 For problem (101), it holds that

(a) the problem has a unique solution uε ∈ H1 (0, T ;H10(Ω), H

−1(Ω)) forevery fixed ε > 0.

(b) the solutions uε are uniformly bounded in L∞ (0, T ;L2(Ω)) andH1 (0, T ;H1

0 (Ω), H−1(Ω)), i.e. for some positive constant C

kuεkL∞(0,T ;L2(Ω)) ≤ C

andkuεkH1(0,T ;H1

0 (Ω),H−1(Ω)) ≤ C.

(c) the sequence of functions

ah(x, t, ξ) = a

µx

ε,x

ε2,t

εr, ξ

¶where (x, t, ξ) ∈ ΩT × RN and ε = ε (h) → 0 as h → ∞ belongsto N (α, β, k,ΩT ). Hence, ah G-converges up to a subsequence, toa limit b ∈ N (α, β, k,ΩT ), where k = k/(2 − k) and β is a positiveconstant depending on α, β and k only.

Proof. The proof of the result (a) on the existence of a unique solution iscarried out in Section 2.2.2, Theorem 8. The proof of (b) and (c) is done inan analogous way to the proof of Theorem 41. See also [FlOl2].

The result in (a) means that we have a unique weak solution to (101), i.e.that there is a unique uε ∈ H1 (0, T ;H1

0 (Ω), H−1(Ω)) for every ε > 0 such

that uε (x, 0) = u0(x) andZΩT

−uε (x, t) v (x) ∂tc (t) + a

µx

ε,x

ε2,t

εr,∇uε

¶·∇v (x) c (t) dxdt = (102)Z

ΩT

f (x, t) v (x) c (t) dxdt

for all v ∈ H10(Ω) and c ∈ D (0, T ).

93

We also know from the G-convergence result in (c) that the sequence uεof solutions converges weakly in L2(0, T ;H1

0 (Ω)), up to a subsequence, to alimit u, which solves a well-posed parabolic problem. What remains to bedone is to determine the G-limit b. We will do this by applying a generaliza-tion of the multiscale convergence method described in Section 5.2.2, and wewill see that it can be done by means of local problems reminiscent of thosefound there. We first consider the case when 0 < r < 2.

Theorem 72 Let uε be a sequence of solutions inH1 (0, T ;H10 (Ω),H

−1(Ω))to (101) for 0 < r < 2. Then it holds that

uε (x, t) → u (x, t) in L2 (ΩT ) ,

uε (x, t) u (x, t) in L2¡0, T ;H1

0(Ω)¢

and

∇uε (x, t) 3,2 ∇u (x, t) +∇y1u1 (x, t, y1, s) +∇y2u2 (x, t, y1, y2, s)

where u ∈ H1 (0, T ;H10 (Ω),H

−1(Ω)) is the unique solution to the homogenizedproblem

∂tu (x, t)−∇ · b (x, t,∇u) = f (x, t) in ΩT ,

u (x, 0) = u0 (x) in Ω,

u (x, t) = 0 on ∂Ω× (0, T ) ,

where in turn

b (x, t,∇u) =ZY2,1

a (y1, y2, s,∇u+∇y1u1 +∇y2u2) dy2dy1ds.

Here, u1 ∈ L2(ΩT × (0, 1) ;H1 (Y1) /R) and u2 ∈ L2(ΩT × Y1,1;H1 (Y2) /R)are the unique solutions to the system of local problems

−∇y2 · a (y1, y2, s,∇u+∇y1u1 +∇y2u2) = 0, (103)

−∇y1 ·ZY2

a (y1, y2, s,∇u+∇y1u1 +∇y2u2) dy2 = 0. (104)

Proof. By the properties of a, we can successively apply (b) in Theorem 71and Theorem 62 to conclude that, up to a subsequence,

uε (x, t) → u (x, t) in L2 (ΩT ) ,

uε (x, t) u (x, t) in L2¡0, T ;H1

0 (Ω)¢

94

and

∇uε (x, t) 3,2 ∇u (x, t) +∇y1u1 (x, t, y1, s) +∇y2u2 (x, t, y1,y2, s) ,

where u ∈ L2(0, T ;H10(Ω)), u1 ∈ L2(ΩT × (0, 1) ;H1 (Y1) /R) and

u2 ∈ L2(ΩT × Y1,1;H1 (Y2) /R). Moreover, the growth condition (29) andthe boundedness of uε in L2 (0, T ;H1

0 (Ω)) imply that a(xε ,xε2, tεr,∇uε) is

bounded in L2 (ΩT )N . Thus, by Theorem 57, there exists a function

a0 ∈ L2 (ΩT ×Y2,1)N such that

a

µx

ε,x

ε2,t

εr,∇uε

¶3,2

a0 (x, t, y1,y2, s) (105)

up to a subsequence. In (102) we choose v ∈ H10 (Ω) and c ∈ D (0, T ) to be

independent of ε. When passing to the limit, we getZΩT

−u (x, t) v (x) ∂tc (t)+ÃZY2,1

a0 (x, t, y1, y2, s) dy2dy1ds

!·∇v (x) c (t) dxdt = (106)Z

ΩT

f (x, t) v (x) c (t) dxdt,

which means that

b (x, t,∇u) =ZY2,1

a0 (x, t, y1, y2, s) dy2dy1ds.

The next step is to find the local problems (103) and (104), the solutionsu1 and u2 of which we will use to characterize a0. We look for one of the localproblems, choosing test functions in (102) according to

v (x) = ε2v1 (x) v2

³xε

´v3

³ xε2

´, c (t) = c1 (t) c2

µt

εr

¶,

where v1 ∈ D(Ω), v2 ∈ C∞ (Y1), v3 ∈ C∞ (Y2) /R, c1 ∈ D (0, T ) and

95

c2 ∈ C∞ (0, 1), and getZΩT

−uε (x, t) ε2v1 (x) v2³xε

´v3

³ xε2

´∂t

µc1 (t) c2

µt

εr

¶¶+

a

µx

ε,x

ε2,t

εr,∇uε

¶·∇³ε2v1 (x) v2

³xε

´v3³ xε2

´´c1 (t) c2

µt

εr

¶dxdt =Z

ΩT

f (x, t) ε2v1 (x) v2

³xε

´v3

³ xε2

´c1 (t) c2

µt

εr

¶dxdt.

Hence, ZΩT

−uε (x, t) v1 (x) v2³xε

´v3³ xε2

´µε2∂tc1 (t) c2

µt

εr

¶+

ε2−rc1 (t) ∂sc2

µt

εr

¶¶+ a

µx

ε,x

ε2,t

εr,∇uε

¶·³ε2∇v1 (x) v2

³xε

´v3

³ xε2

´+

εv1 (x)∇y1v2³xε

´v3³ xε2

´+ v1 (x) v2

³xε

´∇y2v3

³ xε2

´´c1 (t) c2

µt

εr

¶dxdt =Z

ΩT

f (x, t) ε2v1 (x) v2

³xε

´v3

³ xε2

´c1 (t) c2

µt

εr

¶dxdt

and when ε tends to zero, we obtainZΩT

ZY2,1

a0 (x, t, y1, y2, s)·v1 (x) v2 (y1)∇y2v3 (y2) c1 (t) c2 (s) dy2dy1dsdxdt = 0.

Finally, applying the variational lemma, we getZY2

a0 (x, t, y1, y2, s) ·∇y2v3 (y2) dy2 = 0 (107)

almost everywhere in ΩT ×Y1,1 for all v3 ∈ C∞ (Y2) /R, and, by density, forall v3 ∈ H1 (Y2) /R. In what follows, the corresponding conclusion concerningdensity will be understood implicitly while deriving local problems. Equation(107) will turn out to be the weak form of (103).

To find the second local problem we now study the difference between(102) and (106) for test functions

v (x) = εv1 (x) v2³xε

´, c (t) = c1 (t) c2

µt

εr

¶,

96

where v1 ∈ D(Ω), v2 ∈ C∞ (Y1) /R, c1 ∈ D (0, T ) and c2 ∈ C∞ (0, 1), and getZΩT

(uε (x, t)− u (x, t)) εv1 (x) v2³xε

´∂t

µc1 (t) c2

µt

εr

¶¶dxdt+

ZΩT

ÃZY2,1

a0 (x, t, y1, y2, s) dy2dy1ds− a

µx

ε,x

ε2,t

εr,∇uε

¶!·

∇³εv1 (x) v2

³xε

´´c1 (t) c2

µt

εr

¶dxdt =Z

ΩT

(f (x, t)− f (x, t)) εv1 (x) v2³xε

´c1 (t) c2

µt

εr

¶dxdt = 0.

Carrying out the differentiations and after some rewriting, we haveZΩT

1

ε(uε (x, t)− u (x, t)) v1 (x) v2

³xε

´·µ

ε2∂tc1 (t) c2

µt

εr

¶+ ε2−rc1 (t) ∂sc2

µt

εr

¶¶+ÃZ

Y2,1a0 (x, t, y1, y2, s) dy2dy1ds− a

µx

ε,x

ε2,t

εr,∇uε

¶!·

³ε∇v1 (x) v2

³xε

´+ v1 (x)∇y1v2

³xε

´´c1 (t) c2

µt

εr

¶dxdt = 0.

For 0 < r < 2 we obtain by Theorem 63, or alternatively by Lemma 61, thatthat the first term vanishes, and by (105) we getZ

ΩT

ZY2,1

ÃZY2,1

a0 (x, t, y1, y2, s) dy2dy1ds− a0 (x, t, y1, y2, s)

!·

v1 (x)∇y1v2 (y1) c1 (t) c2 (s) dy2dy1dsdxdt = 0

as ε tends to zero. Due to the Y1-periodicity of v2 and the variational lemma,we obtain Z

Y1

µZY2

a0 (x, t, y1, y2, s) dy2

¶·∇y1v2 (y1) dy1 = 0 (108)

for all v2 ∈ H1 (Y1) /R, i.e. we have the weak form of (104) if we can provethat

a0 (x, t, y1, y2, s) = a (y1, y2, s,∇u+∇y1u1 +∇y2u2) .

97

For the characterization of the limit a0, we use perturbed test functions

pk (x, t, y1, y2, s) =

pk,0 (x, t) + pk,1 (x, t, y1, s) + pk,2 (x, t, y1, y2, s) + δc (x, t, y1, y2, s) ,

where pk,0 ∈ D (ΩT )N , pk,1 ∈ D(ΩT ;C

∞ (Y1,1))N , pk,2, c ∈ D(ΩT ;C∞ (Y2,1))N

and δ > 0. These sequences are chosen such that

pk,0 (x, t)→∇u (x, t) in L2 (ΩT )N ,

pk,1 (x, t, y1, s)→∇y1u1 (x, t, y1, s) in L2 (ΩT ×Y1,1)N

and

pk,2 (x, t, y1, y2, s)→∇y2u2 (x, t, y1, y2, s) in L2 (ΩT ×Y2,1)N ,

and such that they converge almost everywhere to the same limits as k →∞.We define

pkε (x, t) = pkµx, t,

x

ε,x

ε2,t

εr

¶.

From the monotonicity property (v) we getµa

µx

ε,x

ε2,t

εr,∇uε

¶− a

µx

ε,x

ε2,t

εr, pkε

¶¶·¡∇uε (x, t)− pkε (x, t)

¢≥ 0,

which after integration and expansion takes the formZΩT

a

µx

ε,x

ε2,t

εr,∇uε

¶·∇uε (x, t)− a

µx

ε,x

ε2,t

εr,∇uε

¶· pkε (x, t)−

a

µx

ε,x

ε2,t

εr, pkε

¶·∇uε (x, t) + a

µx

ε,x

ε2,t

εr, pkε

¶· pkε (x, t) dxdt ≥ 0.

Replacing vc with uε in (102), we get an alternative way of expressing thefirst term and the above inequality can be written asZ

ΩT

f (x, t)uε (x, t)− a

µx

ε,x

ε2,t

εr,∇uε

¶· pkε (x, t)−

a

µx

ε,x

ε2,t

εr, pkε

¶·∇uε (x, t) + a

µx

ε,x

ε2,t

εr, pkε

¶· pkε (x, t) dxdt−Z T

0

h∂tuε (x, t) , uε (x, t)iH−1(Ω),H10 (Ω)

dxdt ≥ 0.

98

We recall that a(y1, y2, s, pk), pk and their product are admissible test func-tions and get, up to a subsequence, thatZ

ΩT

ZY2,1

f (x, t)u (x, t)− a0 (x, t, y1, y2, s) · pk (x, t, y1, y2, s)−

a¡y1, y2, s, p

k¢· (∇u (x, t) +∇y1u1 (x, t, y1, s) +∇y2u2 (x, t, y1, y2, s))+

a¡y1, y2, s, p

k¢· pk (x, t, y1, y2, s) dy2dy1dsdxdt− (110)Z T

0

h∂tu (x, t) , u (x, t)iH−1(Ω),H10 (Ω)

dxdt ≥ 0

when ε tends to zero. Our next step is to let k tend to infinity. We have

pk (x, t, y1, y2, s)→∇u (x, t) +∇y1u1 (x, t, y1, s) +∇y2u2 (x, t, y1, y2, s) + δc (x, t, y1, y2, s)

in L2 (ΩT ×Y2,1)N . Moreover,a¡y1, y2, s, p

k¢→ a (y1, y2, s,∇u+∇y1u1 +∇y2u2 + δc)

and

a¡y1, y2, s, p

k¢· pk (x, t, y1, y2, s)→ a (y1, y2, s,∇u+∇y1u1 +∇y2u2 + δc) ·

(∇u (x, t) +∇y1u1 (x, t, y1, s) +∇y2u1 (x, t, y1, y2, s) + δc (x, t, y1, y2, s))

almost everywhere in ΩT ×Y2,1. The condition (iv) yields that¯a¡y1, y2, s, p

k¢¯≤ β

¡1 +

¯pk (x, t, y1, y2, s)

¯¢,

which, together with the Cauchy-Schwarz inequality, gives us¯a¡y1, y2, s, p

k¢· pk (x, t, y1, y2, s)

¯≤¯a¡y1, y2, s, p

k¢¯ ¯

pk (x, t, y1, y2, s)¯≤

β¡1 +

¯pk (x, t, y1, y2, s)

¯¢ ¯pk (x, t, y1, y2, s)

¯=

β³¯pk (x, t, y1, y2, s)

¯+¯pk (x, t, y1, y2, s)

¯2´.

For k →∞, it holds thatZΩT

ZY2,1

¯pk (x, t, y1, y2, s)

¯+¯pk (x, t, y1, y2, s)

¯2dy2dy1dsdxdt→Z

ΩT

ZY2,1

|∇u (x, t) +∇y1u1 (x, t, y1, s)+

∇y2u2 (x, t, y1, y2, s) + δc (x, t, y1, y2, s)|+

99

|∇u (x, t) +∇y1u1 (x, t, y1, s) +∇y2u2 (x, t, y1, y2, s)+

δc (x, t, y1, y2, s)|2 dy2dy1dsdxdt.

Lebesgue’s generalized majorized convergence theorem now gives thatZΩT

ZY2,1

a¡y1, y2, s, p

k¢· pk (x, t, y1, y2, s) dy2dy1dsdxdt→Z

ΩT

ZY2,1

a (y1, y2, s,∇u+∇y1u1 +∇y2u2 + δc) ·

(∇u (x, t) +∇y1u1 (x, t, y1, s) +∇y2u2 (x, t, y1, y2, s)+

δc (x, t, y1, y2, s)) dy2dy1dsdxdt.

We conclude that when k tends to infinity (110) turns intoZΩT

ZY2,1

f (x, t)u (x, t)− a0 (x, t, y1, y2, s) ·

(∇u (x, t) +∇y1u1 (x, t, y1, s) +∇y2u2 (x, t, y1, y2, s)+

δc (x, t, y1, y2, s))− a (y1, y2, s,∇u+∇y1u1 +∇y2u2 + δc) ·(∇u (x, t) +∇y1u1 (x, t, y1, s) +∇y2u2 (x, t, y1, y2, s))+

a (y1, y2, s,∇u+∇y1u1 +∇y2u2 + δc) ·(∇u (x, t) +∇y1u1 (x, t, y1, s) +∇y2u2 (x, t, y1, y2, s)+

δc (x, t, y1, y2, s)) dy2dy1dsdxdt−Z T

0

h∂tu (x, t) , u (x, t)iH−1(Ω),H10 (Ω)

dxdt ≥ 0.

Some of the terms cancel out and, using (106) with vc replaced by u, we getZΩT

ZY2,1−a0 (x, t, y1, y2, s) ·∇y1u1 (x, t, y1, s)−

a0 (x, t, y1, y2, s) ·∇y2u2 (x, t, y1, y2, s)− (111)

a0 (x, t, y1, y2, s) · δc (x, t, y1, y2, s)+a (y1, y2, s,∇u+∇y1u1 +∇y2u2 + δc) ·δc (x, t, y1, y2, s) dy2dy1dsdxdt ≥ 0.

Here, the first and second term vanish due to (108) and (107), respectively.We have left:Z

ΩT

ZY2,1

¡−a0 (x, t, y1, y2, s) + a (y1, y2, s,∇u+∇y1u1 +∇y2u2 + δc)

¢·

δc (x, t, y1, y2, s) dy2dy1dsdxdt ≥ 0.

100

Dividing by δ and letting δ → 0, we get

a0 (x, t, y1, y2, s) = a (y1, y2, s,∇u+∇y1u1 +∇y2u2) .

By the uniqueness of u the whole sequence converges and the proof is com-plete.

Now we proceed with the case when r = 2. One might wonder whetherthere is any reason to believe that this case would be different from theprevious one; that is, if we can expect to get other local problems. In thecase with one micro-scale in space and time, respectively, we get a localproblem which is parabolic when r = 2; see e.g. [DaMu], [Hol1], [Hol2],[Sva2] and [Wel]. Hence, it is not unreasonable to assume that the situationwould be different for this choice of r compared to the case in Theorem 72.

Theorem 73 Let uε be a sequence of solutions inH1 (0, T ;H10 (Ω),H

−1(Ω))to (101) for r = 2. Then

uε (x, t) → u (x, t) in L2 (ΩT ) ,

uε (x, t) u (x, t) in L2¡0, T ;H1

0(Ω)¢

and

∇uε (x, t) 3,2 ∇u (x, t) +∇y1u1 (x, t, y1, s) +∇y2u2 (x, t, y1, y2, s) ,

where u ∈ H1 (0, T ;H10 (Ω),H

−1(Ω)), u1 ∈ L2(ΩT × (0, 1) ;H1 (Y1) /R) andu2 ∈ L2(ΩT × Y1,1;H1 (Y2) /R), and where u is the unique solution to thehomogenized problem

∂tu (x, t)−∇ · b (x, t,∇u) = f (x, t) in ΩT ,

u (x, 0) = u0 (x) in Ω, (112)

u (x, t) = 0 on ∂Ω× (0, T ) .Here,

b (x, t,∇u) =ZY2,1

a (y1, y2, s,∇u+∇y1u1 +∇y2u2) dy2dy1ds,

where u1 and u2 are the unique solutions to the system of local problems

−∇y2 · a (y1, y2, s,∇u+∇y1u1 +∇y2u2) = 0, (113)

∂su1 (x, t, y1, s)−∇y1 ·ZY2

a (y1, y2, s,∇u+∇y1u1 +∇y2u2) dy2 = 0. (114)

101

Proof. Following the same line of reasoning as in the proof of Theorem 72,we have, up to a subsequence, that

uε (x, t) → u (x, t) in L2 (ΩT ) ,

uε (x, t) u (x, t) in L2¡0, T ;H1

0 (Ω)¢

and

∇uε (x, t) 3,2 ∇u (x, t) +∇y1u1 (x, t, y1, s) +∇y2u2 (x, t, y1, y2, s) ,

where u ∈ L2(0, T ;H10(Ω)), u1 ∈ L2(ΩT × (0, 1) ;H1 (Y1) /R) and

u2 ∈ L2(ΩT×Y1,1;H1 (Y2) /R). In addition we get that, up to a subsequence,

a

µx

ε,x

ε2,t

εr,∇uε

¶3,2

a0 (x, t, y1, y2, s)

for some a0 ∈ L2 (ΩT ×Y2,1)N . Furthermore, we know that if we choosev ∈ H1

0 (Ω) and c ∈ D (0, T ) in (102), we end up with the limit (106); that is,

b (x, t,∇u) =ZY2,1

a0 (x, t, y1, y2, s) dy2dy1ds.

To find the functions u1 and u2 we need local problems. In order to findthe first local problem, we choose the test functions

v (x) = ε2v1 (x) v2

³xε

´v3

³ xε2

´, c (t) = c1 (t) c2

µt

ε2

¶,

in (102), where v1 ∈ D(Ω), v2 ∈ C∞ (Y1), v3 ∈ C∞ (Y2) /R, c1 ∈ D (0, T ) andc2 ∈ C∞ (0, 1). Carrying out the differentiations yieldsZΩT

−uε (x, t) v1 (x) v2³xε

´v3³ xε2

´µε2∂tc1 (t) c2

µt

ε2

¶+ c1 (t) ∂sc2

µt

ε2

¶¶+

a

µx

ε,x

ε2,t

ε2,∇uε

¶·³ε2∇v1 (x) v2

³xε

´v3

³ xε2

´+

εv1 (x)∇y1v2³xε

´v3³ xε2

´+ v1 (x) v2

³xε

´∇y2v3

³ xε2

´´c1 (t) c2

µt

ε2

¶dxdt =Z

ΩT

f (x, t) ε2v1 (x) v2

³xε

´v3

³ xε2

´c1 (t) c2

µt

ε2

¶dxdt

102

and letting ε pass to zero we getZΩT

ZY2,1−u (x, t) v1 (x) v2 (y1) v3 (y2) c1 (t) ∂sc2 (s)+

a0 (x, t, y1, y2, s) · v1 (x) v2 (y1)∇y2v3 (y2) c1 (t) c2 (s) dy2dy1dsdxdt = 0.

The first term vanishes since c2 is periodic. Applying the variational lemmato the remaining part, we obtainZ

Y2

a0 (x, t, y1, y2, s) ·∇y2v3 (y2) dy2 = 0 (115)

for all v3 ∈ H1 (Y2) /R, which will turn out to be the weak form of (113).

To find the next local problem, we choose the test functions

v (x) = εv1 (x) v2

³xε

´, c (t) = c1 (t) c2

µt

ε2

¶(116)

in (102) as well as in (106), where v1 ∈ D(Ω), v2 ∈ C∞ (Y1) /R, c1 ∈ D (0, T )and c2 ∈ C∞ (0, 1). For the difference between these two equations, we getZ

ΩT

1

ε(uε (x, t)− u (x, t)) v1 (x) v2

³xε

´·µ

ε2∂tc1 (t) c2

µt

ε2

¶+ c1 (t) ∂sc2

µt

ε2

¶¶+ÃZ

Y2,1a0 (x, t, y1, y2, s) dy2dy1ds− a

µx

ε,x

ε2,t

ε2,∇uε

¶!·

³ε∇v1 (x) v2

³xε

´+ v1 (x)∇y1v2

³xε

´´c1 (t) c2

µt

ε2

¶dxdt = 0.

Applying Theorem 63 and 3,2-scale convergence yieldsZΩT

ZY1,1

u1(x, t, y1, s)v1 (x) v2 (y1) c1 (t) ∂sc2 (s) dy1dsdxdt+ZΩT

ZY2,1

ÃZY2,1

a0 (x, t, y1, y2, s) dy2dy1ds

!·

v1 (x)∇y1v2 (y1) c1 (t) c2 (s) dy2dy1dsdxdt−ZΩT

ZY2,1

a0 (x, t, y1, y2, s) · v1 (x)∇y1v2 (y1) c1 (t) c2 (s) dy2dy1dsdxdt = 0

103

as ε tends to zero. Due to the periodicity of v2, the middle term vanishesand we getZ

ΩT

ÃZY1,1

u1(x, t, y1, s)v2 (y1) ∂sc2 (s)−µZ

Y2

a0 (x, t, y1, y2, s) dy2

¶·

∇y1v2 (y1) c2 (s) dy1ds) v1 (x) c1 (t) dxdt = 0.

By the variational lemmaZY1,1

u1(x, t, y1, s)v2 (y1) ∂sc2 (s)− (117)µZY2

a0 (x, t, y1, y2, s) dy2

¶·∇y1v2 (y1) c2 (s) dy1ds = 0

for all v2 ∈ H1 (Y1) /R, c2 ∈ C∞ (0, 1) and almost everywhere in ΩT , i.e. wehave the weak form of (114) if we can prove that

a0 (x, t, y1, y2, s) = a (y1, y2, s,∇u+∇y1u1 +∇y2u2) .

For the characterization of the limit a0, we use the method with perturbedtest functions carried out in detail for the case when 0 < r < 2. As in thatcase we arrive at (111), i.e.Z

ΩT

ZY2,1−a0 (x, t, y1, y2, s) ·∇y1u1 (x, t, y1, s)−

a0 (x, t, y1, y2, s) ·∇y2u2 (x, t, y1, y2, s)−a0 (x, t, y1, y2, s) · δc (x, t, y1, y2, s)+ (118)

a (y1, y2, s,∇u+∇y1u1 +∇y2u2 + δc) ·δc (x, t, y1, y2, s) dy2dy1dsdxdt ≥ 0.

From (117) it follows that we may replace a0 · ∇y1 in (118) with ∂su1 andsince ∂su1(x, t, ·, ·) ∈ L2(0, 1; (H1(Y1)/R)0) (c.f. Definition 3.1, Lemma 3.4and Section 4.2 in [NgWo2]) apply Proposition 64. Hence, the first term in(118) vanishes. By (115), so does the second term. Dividing by δ and lettingδ → 0, we have

a0 (x, t, y1, y2, s) = a (y1, y2, s,∇u+∇y1u1 +∇y2u2) .

This means that (115) and (117) are the weak formulations of the localproblems (113) and (114), respectively, which completes the proof.

104

Our next concern is to study the homogenization procedure for (101) when2 < r < 3. Again, looking at the case with one spatial and one temporalmicro-scale, we observe that for r > 2 the local time variable is averagedaway in one of the local problems. It turns out that this is the case also forour problem, as we can see in the next theorem.

Theorem 74 Let uε be a sequence of solutions inH1 (0, T ;H10 (Ω),H

−1(Ω))to (101) for 2 < r < 3. Then it holds that

uε (x, t) → u (x, t) in L2 (ΩT ) ,

uε (x, t) u (x, t) in L2¡0, T ;H1

0(Ω)¢

and

∇uε (x, t) 3,2 ∇u (x, t) +∇y1u1 (x, t, y1) +∇y2u2 (x, t, y1, y2, s) ,

where u ∈ H1 (0, T ;H10 (Ω),H

−1(Ω)) is the unique solution to the homogenizedproblem

∂tu (x, t)−∇ · b (x, t,∇u) = f (x, t) in ΩT ,

u (x, 0) = u0 (x) in Ω,

u (x, t) = 0 on ∂Ω× (0, T ) ,

where in turn

b (x, t,∇u) =ZY2,1

a (y1, y2, s,∇u+∇y1u1 +∇y2u2) dy2dy1ds.

Here, u1 ∈ L2(ΩT ;H1 (Y1) /R) and u2 ∈ L2(ΩT × Y1,1;H1 (Y2) /R) are the

unique solutions to the system of local problems

−∇y2 · a (y1, y2, s,∇u+∇y1u1 +∇y2u2) = 0, (119)

∂su1 (x, t, y1, s) = 0, (120)

−∇y1 ·Z 1

0

ZY2

a (y1, y2, s,∇u+∇y1u1 +∇y2u2) dy2ds = 0. (121)

Proof. As in the proof of Theorem 72, we know that, up to a subsequence,

uε (x, t) → u (x, t) in L2 (ΩT ) ,

uε (x, t) u (x, t) in L2¡0, T ;H1

0 (Ω)¢

105

and

∇uε (x, t) 3,2 ∇u (x, t) +∇y1u1 (x, t, y1, s) +∇y2u2 (x, t, y1, y2, s) ,

where u ∈ L2(0, T ;H10(Ω)), u1 ∈ L2(ΩT × (0, 1) ;H1 (Y1) /R) and

u2 ∈ L2(ΩT ×Y1,1;H1 (Y2) /R). Moreover, up to a subsequence

a

µx

ε,x

ε2,t

εr,∇uε

¶3,2

a0 (x, t, y1, y2, s)

for some a0 ∈ L2 (ΩT ×Y2,1)N . Also, we know that if we choose v ∈ H10(Ω)

and c ∈ D (0, T ) in (102), we end up with the limit (106) and hence

b (x, t,∇u) =ZY2,1

a0 (x, t, y1, y2, s) dy2dy1ds.

To find the first local problem, we study the difference between (102) and(106) for the test functions

v (x) = ε2v1 (x) v2³xε

´v3³ xε2

´, c (t) = c1 (t) c2

µt

εr

¶,

where v1 ∈ D(Ω), v2 ∈ C∞(Y1), v3 ∈ C∞(Y2)/R, c1 ∈ D(0, T ) andc2 ∈ C∞(0, 1). We getZ

ΩT

(uε (x, t)− u (x, t)) ε2v1 (x) v2³xε

´v3³ xε2

´·µ

∂tc1 (t) c2

µt

εr

¶+ ε−rc1 (t) ∂sc2

µt

εr

¶¶+ÃZ

Y2,1a0 (x, t, y1, y2, s) dy2dy1ds− a

µx

ε,x

ε2,t

εr,∇uε

¶!·³

ε2∇v1 (x) v2³xε

´v3³ xε2

´+ εv1 (x)∇y1v2

³xε

´v3³ xε2

´+

v1 (x) v2

³xε

´∇y2v3

³ xε2

´´c1 (t) c2

µt

εr

¶dxdt = 0,

106

and hence ZΩT

µ1

ε(uε (x, t)− u (x, t)) v1 (x) v2

³xε

´v3³ xε2

´¶·µ

ε3∂tc1 (t) c2

µt

εr

¶+ ε3−rc1 (t) ∂sc2

µt

εr

¶¶+ÃZ

Y2,1a0 (x, t, y1, y2, s) dy2dy1ds− a

µx

ε,x

ε2,t

εr,∇uε

¶!·³

ε2∇v1 (x) v2³xε

´v3

³ xε2

´+ εv1 (x)∇y1v2

³xε

´v3

³ xε2

´+

v1 (x) v2³xε

´∇y2v3

³ xε2

´´c1 (t) c2

µt

εr

¶dxdt = 0.

When ε tends to zero, we obtainZΩT

ZY2,1

ÃZY2,1

a0 (x, t, y1, y2, s) dy2dy1ds− a0 (x, t, y1, y2, s)

!·

v1 (x) v2(y1)∇y2v3(y2)c1 (t) c2(s) dy2dy1dsdxdt = 0

and since v3 is Y2-periodic, the first term vanishes and we obtainZΩT

ZY2,1

a0 (x, t, y1, y2, s) ·v1 (x) v2(y1)∇y2v3(y2)c1 (t) c2(s) dy2dy1dsdxdt = 0.

Due to the variational lemma,ZY2

a0 (x, t, y1, y2, s) ·∇y2v3(y2) dy2 = 0 (122)

for all v3 ∈ H1 (Y2) /R.

Our next concern is the second local problem.We now study the differencebetween (102) and (106) for the test functions

v (x) = εr−1v1 (x) v2³xε

´, c (t) = c1 (t) c2

µt

εr

¶,

where v1 ∈ D(Ω), v2 ∈ C∞ (Y1) /R, c1 ∈ D (0, T ) and c2 ∈ C∞ (0, 1), and

107

arrive atZΩT

(uε (x, t)− u (x, t)) εr−1v1 (x) v2³xε

´∂t

µc1 (t) c2

µt

εr

¶¶+ÃZ

Y2,1a0 (x, t, y1, y2, s) dy2dy1ds− a

µx

ε,x

ε2,t

εr,∇uε

¶!·

∇³εr−1v1 (x) v2

³xε

´´c1 (t) c2

µt

εr

¶dxdt = 0.

Differentiating and rewriting, we haveZΩT

1

ε(uε (x, t)− u (x, t)) v1 (x) v2

³xε

´·µ

εr∂tc1 (t) c2

µt

εr

¶+ c1 (t) ∂sc2

µt

εr

¶¶+ÃZ

Y2,1a0 (x, t, y1, y2, s) dy2dy1ds− a

µx

ε,x

ε2,t

εr,∇uε

¶!·

³εr−1∇v1 (x) v2

³xε

´+ εr−2v1 (x)∇y1v2

³xε

´´c1 (t) c2

µt

εr

¶dxdt = 0

and for 2 < r < 3, by Theorem 63, as ε tends to zero, we obtain thatZΩT

ZY1,1

u1 (x, t, y1, s) v1 (x) v2 (y1) c1 (t) ∂sc2 (s) dy1dsdxdt = 0.

Finally, applying the variational lemma we getZY1,1

u1 (x, t, y1, s) v2 (y1) ∂sc2 (s) dy1ds = 0 (123)

for all v2 ∈ H1 (Y1) /R and c2 ∈ C∞ (0, 1), i.e. we have the weak form of(120). We deduce that u1 does not depend on the local time variable s.

To find the third local problem, we now choose the test functions

v (x) = εv1 (x) v2³xε

´, c (t) = c1 (t)

108

in (102), where v1 ∈ D(Ω), v2 ∈ C∞ (Y1) /R and c1 ∈ D (0, T ). Afterdifferentiations, we getZ

ΩT

−uε (x, t) εv1 (x) v2³xε

´∂tc1 (t) + a

µx

ε,x

ε2,t

εr,∇uε

¶·³

ε∇v1 (x) v2³xε

´+ v1 (x)∇y1v2

³xε

´´c1 (t) dxdt =Z

ΩT

f (x, t) εv1 (x) v2³xε

´c1 (t) dxdt.

When ε tends to zero, we obtainZΩT

ZY2,1

a0 (x, t, y1, y2, s) · v1 (x)∇y1v2 (y1) c1 (t) dy2dy1dsdxdt = 0.

By applying the variational lemma, we getZY1

µZ 1

0

ZY2

a0 (x, t, y1, y2, s) dy2ds

¶·∇y1v2 (y1) dy1 = 0 (124)

for all v2 ∈ H1 (Y1) /R.

For characterization of the limit a0, we again use the method with per-turbed test functions carried out in detail for the case when 0 < r < 2, anddeduce that

a0 (x, t, y1, y2, s) = a (y1, y2, s,∇u+∇y1u1 +∇y2u2) .

This means that (122) and (124) are the weak formulations of the localproblems (119) and (121), respectively.

Remark 75 Note that with the multiscale convergence method used in thischapter, the existence of the solution to the local problems is immediate sincethe functions u, u1 and u2 appear as a multiscale limit and constituents ofsuch limits of uε, respectively. Methods to prove existence and uniquenessfor solutions to local problems of the kind found above can be concluded fromthe considerations in Sections 4.2.1 and 4.2.3 in [Pan].

109

5.3.3 Homogenization of some linear parabolic operators with twospatial and three temporal scales

In this section, we study the homogenization of the parabolic equation

∂tuε (x, t)−∇ ·

µa

µx

ε,t

ε,t

εr

¶∇uε (x, t)

¶= f (x, t) in Ω× (0, T ) ,

uε (x, 0) = u0 (x) in Ω, (125)

uε (x, t) = 0 on ∂Ω× (0, T ) ,

where f ∈ L2 (ΩT ), u0 ∈ L2(Ω) and

a : RN × R2 → RN×N

is assumed to satisfy the following conditions, where 0 < α ≤ β:

(i) a ∈ L∞ (Y1,2)N×N .

(ii) |a (y, s1, s2) ξ| ≤ β |ξ| for all (y, s1, s2) ∈ RN ×R2 and all ξ ∈ RN .

(iii) a (y, s1, s2) ξ · ξ ≥ α |ξ|2 for all (y, s1, s2) ∈ RN ×R2 and all ξ ∈ RN .

As we noted in Remark 9, this problem allows, for any fixed ε > 0,a unique solution uε ∈ H1 (0, T ;H1

0(Ω), H−1(Ω)) if (i) and (iii) are fulfilled.

This means that there exists a unique function uε ∈ H1 (0, T ;H10(Ω),H

−1(Ω))such that uε (x, 0) = u0 (x) andZ

ΩT

−uε (x, t) v (x) ∂tc (t) + a

µx

ε,t

ε,t

εr

¶∇uε (x, t) ·∇v (x) c (t) dxdt =Z

ΩT

f (x, t) v (x) c (t) dxdt (126)

for all v ∈ H10 (Ω) and c ∈ D (0, T ). Under the assumptions above, we also

havekuεkL∞(0,T ;L2(Ω)) ≤ C

andkuεkH1(0,T ;H1

0 (Ω),H−1(Ω)) ≤ C (127)

for some positive constant C; see Propositions 23.23 and 23.30 in [ZeiIIA] orTheorems 11.2 and 11.4 in [CiDo].

110

Furthermore, a belongs toM(α, β,RN ×R2). From this, it can be seen thata(x

ε, tε, tεr) satisfies the conditions for G-convergence for linear parabolic

problems, i.e. it belongs to M(α, β,ΩT ), which we defined in Section 3.2.This means that, at least for a suitable subsequence, there exists a well-posedlimit problem of the same type as (125) governed by a coefficient matrix b;see Theorem 25. As in the previous section, our aim is to characterize theG-limit b further by means of homogenization procedures.

Theorem 76 Let uε be a sequence of solutions inH1 (0, T ;H10 (Ω),H

−1(Ω))to (125). Then

uε (x, t) → u (x, t) in L2 (ΩT ) ,

uε (x, t) u (x, t) in L2¡0, T ;H1

0(Ω)¢

and∇uε (x, t) 2,3 ∇u (x, t) +∇yu1 (x, t, y, s1, s2) ,

where u ∈ H1 (0, T ;H10 (Ω), H

−1(Ω)) and u1 ∈ L2(ΩT × (0, 1)2 ;H1 (Y ) /R).Furthermore, u is the unique solution to the homogenized problem

∂tu (x, t)−∇ · (b∇u (x, t)) = f (x, t) in ΩT ,

u (x, 0) = u0 (x) in Ω,

u (x, t) = 0 on ∂Ω× (0, T ) ,

where

b∇u (x, t) =ZY1,2

a (y, s1, s2) (∇u (x, t) +∇yu1 (x, t, y, s1, s2)) dyds2ds1.

For 0 < r < 2, r 6= 1, the function u1 is determined by the local problem

−∇y · (a (y, s1, s2) (∇u (x, t) +∇yu1 (x, t, y, s1, s2))) = 0, (128)

for r = 2 by

∂s2u1(x,t,y,s1,s2)−∇y ·(a(y,s1,s2)(∇u(x, t)+∇yu1(x,t,y,s1,s2)))=0, (129)

and for r > 2 by the system of local problems

∂s2u1(x, t, y, s1, s2) = 0, (130)

−∇y ·µµZ 1

0

a (y, s1, s2) ds2

¶(∇u (x, t) +∇yu1 (x, t, y, s1))

¶= 0. (131)

111

Proof. The a priori estimate (127) allows us to apply Theorem 67. Hence,up to a subsequence,

uε (x, t) → u (x, t) in L2 (ΩT ) ,

uε (x, t) u (x, t) in L2¡0, T ;H1

0 (Ω)¢

and∇uε (x, t) 2,3 ∇u (x, t) +∇yu1 (x, t, y, s1, s2) ,

where u ∈ L2 (0, T ;H10 (Ω)) and u1 ∈ L2(ΩT × (0, 1)2 ;H1 (Y ) /R). Choosing

v ∈ D(Ω) and c ∈ D (0, T ) to be independent of ε in (126) and letting ε tendto zero, by Theorem 67 we getZ

ΩT

−u (x, t) v (x) ∂tc (t)+ÃZY1,2

a (y, s1, s2) (∇u (x, t) +∇yu1 (x, t, y, s1, s2)) dyds2ds1

!· (132)

∇v (x) c (t) dxdt =

ZΩT

f (x, t) v (x) c (t) dxdt.

In order to find the local problems, we study the difference between (126)and (132), i.e. Z

ΩT

(uε (x, t)− u (x, t)) v (x) ∂tc (t)+ÃZY1,2

a (y, s1, s2) (∇u (x, t) +∇yu1 (x, t, y, s1, s2)) dyds2ds1 − (133)

a

µx

ε,t

ε,t

εr

¶∇uε (x, t)

¶·∇v (x) c (t) dxdt = 0.

We choose the test functions

v (x) = εv1 (x) v2³xε

´, c (t) = c1 (t) c2

µt

ε

¶c3

µt

εr

¶,

where v1 ∈ D(Ω), v2 ∈ C∞ (Y ) /R, c1 ∈ D (0, T ) and c2, c3 ∈ C∞ (0, 1) in

112

(133) and getZΩT

1

ε(uε (x, t)− u (x, t)) v1 (x) v2

³xε

´µε2∂tc1 (t) c2

µt

ε

¶c3

µt

εr

¶+

εc1 (t) ∂s1c2

µt

ε

¶c3

µt

εr

¶+ ε2−rc1 (t) c2

µt

ε

¶∂s2c3

µt

εr

¶¶+ÃZ

Y1,2a (y, s1, s2) (∇u (x, t) +∇yu1 (x, t, y, s1, s2)) dyds2ds1 − (134)

a

µx

ε,t

ε,t

εr

¶∇uε (x, t)

¶·³

ε∇v1 (x) v2³xε

´+ v1 (x)∇yv2

³xε

´´c1 (t) c2

µt

ε

¶c3

µt

εr

¶dxdt = 0.

Next, we let ε pass to zero.

For the case when 0 < r < 2 we have, using Theorems 68 and 67 for thefirst and second term above, respectively,Z

ΩT

ZY1,2

ÃZY1,2

a (y, s1, s2) (∇u (x, t) +∇yu1 (x, t, y, s1, s2)) dyds2ds1 −

a (y, s1, s2) (∇u (x, t) +∇yu1 (x, t, y, s1, s2))) ·v1 (x)∇yv2 (y) c1 (t) c2 (s1) c3 (s2) dyds2ds1dxdt = 0

and due to the periodicity of v2, we obtainZΩT

ZY1,2−a (y, s1, s2) (∇u (x, t) +∇yu1 (x, t, y, s1, s2)) ·

v1 (x)∇yv2 (y) c1 (t) c2 (s1) c3 (s2) dyds2ds1dxdt = 0.

By the variational lemma,ZY

a (y, s1, s2) (∇u (x, t) +∇yu1 (x, t, y, s1, s2)) ·∇yv2 (y) dy = 0

for all v2 ∈ C∞ (Y ) /R, and hence, by density, for all v2 ∈ H1 (Y ) /R, a.e. inΩT × (0, 1)2. This is the weak form of (128).

113

For r = 2, according to Theorems 67 and 68, (134) approachesZΩT

ZY1,2

u1(x, t, y, s1, s2)v1 (x) v2 (y) c1 (t) c2 (s1) ∂s2c3 (s2) dyds2ds1dxdt+ZΩT

ZY1,2

ÃZY1,2

a (y, s1, s2) (∇u (x, t) +∇yu1 (x, t, y, s1, s2)) dyds2ds1

!·

v1 (x)∇yv2 (y) c1 (t) c2 (s1) c3 (s2)− a (y, s1, s2) (∇u (x, t)+∇yu1 (x, t, y, s1, s2)) · v1 (x)∇yv2 (y) c1 (t) c2 (s1) c3 (s2) dyds2ds1dxdt = 0,

when ε tends to zero. Since v2 is periodic, the middle term vanishes and wehave Z

ΩT

ZY1,2

u1(x, t, y, s1, s2)v1 (x) v2 (y) c1 (t) c2 (s1) ∂s2c3 (s2)−

a (y, s1, s2) (∇u (x, t) +∇yu1 (x, t, y, s1, s2))·v1 (x)∇yv2 (y) c1 (t) c2 (s1) c3 (s2) dyds2ds1dxdt = 0.

Applying the variational lemma, we arrive atZY1,1

u1(x, t, y, s1, s2)v2 (y) ∂s2c3 (s2) −

a (y, s1, s2) (∇u (x, t) +∇yu1 (x, t, y, s1, s2)) ·∇yv2 (y) c3 (s2) dyds2 = 0,

for all v2 ∈ H1 (Y ) /R and all c3 ∈ C∞ (0, 1), a.e. in ΩT × (0, 1). We havefound the weak form of (129).

For the case when r > 2, we choose the test functions

v (x) = εr−1v1 (x) v2

³xε

´, c (t) = c1 (t) c2

µt

ε

¶c3

µt

εr

¶,

in the difference (133), where v1 ∈ D(Ω), v2 ∈ C∞ (Y ) /R, c1 ∈ D (0, T ) andc2, c3 ∈ C∞ (0, 1) and obtainZ

ΩT

1

ε(uε (x, t)− u (x, t)) v1 (x) v2

³xε

´µεr∂tc1 (t) c2

µt

ε

¶c3

µt

εr

¶+

εr−1c1 (t) ∂s1c2

µt

ε

¶c3

µt

εr

¶+ c1 (t) c2

µt

ε

¶∂s2c3

µt

εr

¶¶+

114

ÃZY1,2

a (y, s1, s2) (∇u (x, t) +∇yu1 (x, t, y, s1, s2)) dyds2ds1 −

a

µx

ε,t

ε,t

εr

¶∇uε (x, t)

¶·³εr−1∇v1 (x) v2

³xε

´+

εr−2v1 (x)∇yv2

³xε

´´c1 (t) c2

µt

ε

¶c3

µt

εr

¶dxdt = 0.

When ε passes to zero, we get according to Theorem 68ZΩT

ZY1,2

u1(x, t, y, s1, s2)v1 (x) v2 (y) c1 (t) c2 (s1) ∂s2c3 (s2) dyds2ds1dxdt = 0

and hence Z 1

0

u1(x, t, y, s1, s2)∂s2c3 (s2) ds2 = 0

for all c3 ∈ C∞ (0, 1), a.e. in ΩT × Y × (0, 1). This is the weak form of (130)and means that u1 is independent of s2.

Next, we study (126) for the test functions

v (x) = εv1 (x) v2

³xε

´, c (t) = c1 (t) c2

µt

ε

¶where v1 ∈ D(Ω), v2 ∈ C∞ (Y ) /R, c1 ∈ D (0, T ) and c2 ∈ C∞ (0, 1). WeobtainZ

ΩT

−uε (x, t) v1 (x) v2³xε

´µε∂tc1 (t) c2

µt

ε

¶+ c1 (t) ∂s1c2

µt

ε

¶¶+

a

µx

ε,t

ε,t

ε2

¶∇uε (x, t) ·

³ε∇v1 (x) v2

³xε

´+ v1 (x)∇yv2

³xε

´´·

c1 (t) c2

µt

ε

¶dxdt =

ZΩT

f (x, t) εv1 (x) v2

³xε

´c1 (t) c2

µt

ε

¶dxdt

and when ε goes to zero, we getZΩT

ZY1,2−u (x, t) v1 (x) v2 (y) c1 (t) ∂s1c2 (s1)+

a (y, s1, s2) (∇u (x, t) +∇yu1 (x, t, y, s1))·v1 (x)∇yv2 (y) c1 (t) c2 (s1) dyds2ds1dxdt = 0.

115

The periodicity of c2 and the variational lemma imply thatZY

µZ 1

0

a (y, s1, s2) ds2

¶(∇u (x, t) +∇yu1 (x, t, y, s1)) ·∇yv2 (y) dy = 0

for all v2 ∈ H1 (Y ) /R a.e. in ΩT × (0, 1). This is the weak form of (131).

Remark 77 Homogenization of linear parabolic equations with oscillationsin both space and time was first investigated by means of asymptotic ex-pansions by Bensoussan et al. in [BLP]. This is also studied in [PrTe] byProfeti and Terreni, where a different proof is provided. The homogeniza-tion of monotone nonlinear problems, with two scales in space and time,respectively, is proven by means of G-convergence by Svanstedt in [Sva1] and[Sva2]. Similar results can also be found in [Pan]. For corrector results forlinear parabolic equations with oscillation in space only, we refer to [BFM]by Brahim-Otsmane et al. Such results for problems with oscillations also intime have been proven by means of two-scale convergence methods by Holm-bom in [Hol2], and with H-convergence techniques by Dall’Aglio and Muratin [DaMu]. The corresponding result for nonlinear problems is presented bySvanstedt in [Sva3]. Equations of this kind are also considered by Wellanderin [Wel]. In [HSW] Holmbom, Svanstedt and Wellander proved a reiter-ated homogenization result for linear parabolic operators with two scales ofoscillation in space, and different frequencies of oscillation in time, whichis the linear correspondence to the result in Section 5.3.2. Some results onnonlinear parabolic problems with one spatial scale and one temporal scalecan be found in [NaRa] by Nandakumaran and Rajesh, and in [NgWo1] and[NgWo2] by Nguetseng and Woukeng concerning such problems without pe-riodicity assumptions.

116

6 Characterization of the G-limit for somenon-periodic linear operators

So far, we have seen a few different examples where it has turned out to bepossible to determine the G-limit. In Chapter 3, we came across some caseswhere the convergence of the sequence ah of coefficients to some limit a wasstrong enough to ensure that a was also the G-limit. Parts of Chapters 4 and5 concerned some linear periodic homogenization problems where ah wasin general only weakly convergent in some suitable Lebesgue space. For suchproblems the determination of the G-limit required non-trivial techniques,and it turned out that the G-limit was not equal to the weak limit of ah.Thus, it might be of interest to see if we can find cases in which it is

possible to relax the strength of the convergence of ah compared to, forexample, the situations in Remark 12 and Proposition 18, and where theG-limit is still some conventional kind of limit of ah. Another interestingarea to investigate is non-periodic problems. In this chapter, we will studysome linear, not necessarily periodic, problems where the coefficients ah arecreated by means of some special integral operators. In Section 6.1, we studyelliptic equations with such coefficients and give a criterion for when the G-limit is equal to the weak L2(Ω)N×N -limit of ah, and in Section 6.2 we dothe corresponding for parabolic problems and discuss some open questions.

6.1 Linear elliptic operators

The interest in this section is to study G-convergence of the operators asso-ciated with the equations

−∇ ·¡ah (x)∇uh (x)

¢= f (x) in Ω, (135)

uh (x) = 0 on ∂Ω,

where ah is generated by certain integral operators. We start out with a classof coefficients with entries

ahij(x) =

ZA

whij(y)Kij(x, y) dy,

where A is an open bounded set in RM with smooth boundary, K belongsto L2(Ω×A)N×N and for some w in L2(A)N×N

wh(y) w(y) in L2(A)N×N .

117

This can be interpreted as a so-called Hilbert-Schmidt integral operatordefined by K and acting on wh; see Remark 78. By the compactness of suchoperators, we have strong convergence of ah in L2(Ω)N×N and hence

ahij(x)→ aij(x) =

ZA

wij(y)Kij(x, y) dy in L2(Ω).

If ah belongs toM(α, β,Ω), Proposition 18 now gives that a is the G-limit.

If the convergence of wh is disturbed in the sense that we let wh dependon x, the result becomes more difficult to predict. In the next two subsections,we will investigate G-convergence for sequences ah with the entries

ahij(x) =

ZA

whij(x, y)Kij(x, y) dy (136)

for some different choices of wh.

Remark 78 A compact operator T : X → Z from one Banach space toanother has the property that, if X is reflexive, it creates a strongly convergentsequence from a weakly convergent one when acting on it; that is, if

uh u in X,

thenTuh → Tu in Z.

Hilbert-Schmidt integral operators provide an example of a class of compactoperators (see [Alt] 8.9). An operator T : Lp(A) → Lq(Ω), 1 < p < ∞,1p+ 1

q= 1, of this kind is defined by

(Tw)(x) =

ZA

w(y)K(x, y) dy

where K ∈ Lq(Ω × A). Since Lp(A) is reflexive and T is compact, we havethat if

wh(y) w(y) in Lp(A)

then(Twh)(x)→ (Tw)(x) in Lq(Ω),

that is, ZA

wh(y)K(x, y) dy →ZA

w(y)K(x, y) dy in Lq(Ω). (137)

118

6.1.1 Some numerical experiments

The questions concerning ways in which we can relax the strength of theconvergence of ah until there appear substantial difficulties in decidingthe G-limit can be illuminated by suitable choices of wh and K in (136).We study some two-dimensional examples, where we choose Ω = (1, 3)2,A = Y = (0, 1)2 and f(x) = 100x1.

Example 1First, we consider coefficients of the form

ahij(x) =

ZY

whij(y)Kij(x, y) dy,

which can be associated with Hilbert-Schmidt integral operators. We let

wh(y) =

µ2 + sin(2πh(y1 + y2))

32+ sin(2πh(y1 + y2))

32+ sin(2πh(y1 + y2)) 2 + sin(2πh(y1 + y2))

¶and

Kij(x, y) = 2 + sin(x1 + x2 + y1 + y2). (138)

Obviously,

wh(y) w =

µ2 3

2322

¶in L2(Y )2×2, (139)

and hence

ahij(x)→ aij(x) =

ZY

wij(y)Kij(x, y) dy in L2(Ω); (140)

see (137). By Proposition 18, this means that we have found the G-limit.In Figure 27 below, we see a12 and the solution u obtained for the limitcoefficient a.

Figure 27. a12 and the solution u.

119

Example 2Next, we study an example where we allow wh to depend on x as well. For

ahij(x) =

ZY

whij(x, y)Kij(x, y) dy, (141)

withwh(x, y) =µ

2 + sin(2πh(x1 + x2) + y1 + y2)32+ sin(2πh(x1 + x2) + y1 + y2)

32+ sin(2πh(x1 + x2) + y1 + y2) 2 + sin(2πh(x1 + x2) + y1 + y2)

¶(142)

we obtain

wh(x, y) w =

µ2 3

2322

¶in L2(Ω× Y )2×2. (143)

This means that we still have weak convergence of wh to the same matrixas in (139), now in the space L2(Ω×Y )2×2, but since we can no longer inter-pret the coefficient as the result of the action of a Hilbert-Schmidt integraloperator, we are not guaranteed strong convergence of ah. However, ahstill converges weakly, that is, with K defined by (138), we obtain

ahij(x)

ZY

wij(x, y)Kij(x, y) dy in L2(Ω), (144)

where the limit coincides with aij in (140) for our choice of wh in (142). Nowthe question is: does this mean that we obtain the same G-limit, that is, isthe G-limit in this case the weak L2(Ω)N×N -limit of ah obtained in (144)?In Figure 28, to the left, we have the entry ah12 of a

h where we can see theoscillations in the x1 and x2 variables.

Figure 28. ah12 and the solution uh for h = 10.

120

The other entries have essentially the same pattern of oscillations. To theright we see the solution uh to (135) for ah given by (141). It apparently doesnot coincide with u in Figure 27, the solution obtained for the coefficient a,and the same phenomenon will remain for larger h. Thus, it seems that theG-limit differs from a in this case, and hence from the weak limit. Obviously,the oscillations in x1 and x2 have affected the G-limit.

Example 3

Finally, we consider a second example where wh is dependent on x, namely

wh(x, y) =µ2 + sin(2πh(x1 + x2) + y1 + y2)

32− sin(2πh(x1 + x2) + y1 + y2)

32− sin(2πh(x1 + x2) + y1 + y2) 2 + sin(2πh(x1 + x2) + y1 + y2)

¶(145)

and with the sameK as before. We obtain the same weak limit of wh givenin (143), and hence the same weak limit of ah, which is identical to the lim-its in (140) and (144). In Figure 29, we see ah12 together with the solution u

h.

Figure 29. ah12 and the solution uh for h = 10.

The oscillations in the coefficient have the same magnitude as in the lastexample. However, in this case they do not appear to have any effect onthe solution. We get a solution very much like u, which indicates that theG-limit is again the function a obtained already in (140); that is, for thisexample the G-limit appears to be the weak L2(Ω)N×N -limit of ah. Thus,the two problems where we have dependence of x in wh do not seem to lead tothe same G-limit, even though the weak limit for the respective sequences ofcoefficients coincides. In Example 3, the G-limit appears to be the functiona in (140) whereas in Example 2 we have a deviation from a. In the nextsection, we will explain this phenomenon in theoretical terms.

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6.1.2 A theoretical result

Through a couple of examples, we have shown that the G-limit might beaffected by introducing the dependence of x in wh even when the weak limitof wh remains the same. Below, we provide a result that distinguishessome cases where the determination of the G-limit is straightforward.

Theorem 79 Let ah be a sequence inM(α, β,Ω) such that

ahij(x) =

ZA

whij(x, y)Kij(x, y) dy, (146)

where wh, K ∈ C1(Ω× A)N×N and

wh(x, y) w(x, y) in L2(Ω×A)N×N (147)

for some w ∈ H1(Ω×A)N×N . Assume also that for i = 1, 2, ..., N

NXj=1

ZA

∂xjwhij(x, y)Kij(x, y) dy 0 in L2(Ω) (148)

and that b ∈M(α, β2

α,Ω) has the entries

bij(x) =

ZA

wij(x, y)Kij(x, y) dy.

Then ah G-converges to b.Proof. Since ah belongs toM(α, β,Ω), we know that there exists at leasta subsequence that G-converges. To identify the G-limit, we want to studythe limit process for the problem (135), i.e.

NXi,j=1

ZΩ

ahij(x)∂xjuh(x)∂xiv(x) dx =

ZΩ

f(x)v(x) dx

for all v ∈ D(Ω), for such a sequence. Integrating by parts, the left-handside reads

−NX

i,j=1

ZΩ

uh(x)∂xjahij(x)∂xiv(x) + uh(x)ahij(x)∂

2xixj

v(x) dx.

122

Inserting (146), we get

−NX

i,j=1

ZΩ

ZA

uh(x)¡∂xjw

hij(x, y)Kij(x, y) + wh

ij(x, y)∂xjKij(x, y)¢∂xiv(x)+

uh(x)whij(x, y)Kij(x, y)∂

2xixj

v(x) dydx,

where we use (148) and the convergence properties of wh and uh toobtain, after letting h→∞,

−NX

i,j=1

ZΩ

ZA

u(x)wij(x, y)∂xjKij(x, y)∂xiv(x)+

u(x)wij(x, y)Kij(x, y)∂2xixj

v(x) dydx.

Integrating by parts in the second term, we have

NXi,j=1

ZΩ

ZA

−u(x)wij(x, y)∂xjKij(x, y)∂xiv(x) +¡∂xju(x)wij(x, y)Kij(x, y) +

u(x)∂xjwij(x, y)Kij(x, y) + u(x)wij(x, y)∂xjKij(x, y)¢∂xiv(x) dydx,

where the first and the last term cancel out and we are left withNX

i,j=1

ZΩ

ZA

(wij(x, y)Kij(x, y)∂xju(x)+u(x)∂xjwij(x, y)Kij(x, y))∂xiv(x) dydx.

By rearrangement, this can be written

NXi,j=1

µZΩ

bij(x)∂xju(x)∂xiv(x)dx+

ZΩ

ZA

u(x)∂xjwij(x, y)Kij(x, y)∂xiv(x)dydx

¶.

This means that it remains to be proven that

NXi,j=1

ZΩ

ZA

u(x)∂xjwij(x, y)Kij(x, y)∂xiv(x) dydx = 0. (149)

By assumption, we have

NXj=1

ZA

∂xjwhij(x, y)Kij(x, y) dy 0 in L2(Ω)

123

and hence

NXi,j=1

ZΩ

ZA

∂xjwhij(x, y)Kij(x, y)u(x)∂xiv(x) dydx→ 0

for all v ∈ D(Ω). Integration by parts gives us

NXi,j=1

ZΩ

ZA

whij(x, y)∂xjKij(x, y)u(x)∂xiv(x)+ wh

ij(x, y)Kij(x, y)∂xju(x)∂xiv(x)+

whij(x, y)Kij(x, y)u(x)∂

2xixj

v(x) dydx→ 0.

Using (147) in the left-hand side, we have

NXi,j=1

ZΩ

ZA

wij(x, y)∂xjKij(x, y)u(x)∂xiv(x)+ wij(x, y)Kij(x, y)∂xju(x)∂xiv(x)+

wij(x, y)Kij(x, y)u(x)∂2xixj

v(x) dydx = 0.

A final integration by parts results in

NXi,j=1

ZΩ

ZA

∂xjwij(x, y)Kij(x, y)u(x)∂xiv(x) dydx = 0

and (149) is proven. Hence b is the G-limit for the chosen subsequence, andsince the same result will follow for any convergent subsequence the entiresequence G-converges to b.

It is now clear how we could get results that indicated different G-limitsin the two examples with wh dependent on x in the previous section. InExample 2 the criterion (148) is not fulfilled, whereas in Example 3 it isobviously satisfied since the derivatives ∂xjw

hij cancel out when summing

over j for the choice of wh given in (145). Thus, the G-limit is truly equal tothe weak limit of ah in the Example 3.

124

6.2 Linear parabolic operators

In this section we will investigate sequences of equations of the form

∂tuh (x, t)−∇ ·

¡ah (x, t)∇uh(x, t)

¢= f (x, t) in ΩT ,

uh (x, 0) = u0 (x) in Ω, (150)

uh (x, t) = 0 on ∂Ω× (0, T ) ,

with respect to G-convergence, where the coefficients are created in a waysimilar to that in the preceding section. A slight modification of the operatorsstudied there, to fit the evolution setting, yields

ahij(x, t) =

ZA

whij(x, t, y)Kij(x, t, y) dy. (151)

If wh and K are chosen such that ah belongs toM(α, β,ΩT ), the exis-tence of a G-convergent subsequence is evident by Theorem 25. Moreover, ifwh is independent of x and t and

whij(y) wij(y) in L2(A),

then, for any K ∈ L2(Ω×A),

ahij(x, t)→ZA

wij(y)Kij(x, t, y) dy in L2(ΩT )

and hence the entire sequence G-converges to the unique limit

b(x, t) =

ZA

wij(y)Kij(x, t, y) dy.

In the following sections, we will investigate the limit behavior of equationsof the type in (150) when wh may depend also on x and t.

6.2.1 A theoretical result

Below, we prove a mathematical criterion for the parabolic case, similar tothe one given in Section 6.1.2, for when the G-limit is trivial to determine,namely for when it coincides with the weak L2(ΩT )

N×N -limit of ah. Wenote that the comparison theorem given in Section 4.3 provides a shortcut,if we make a certain continuity assumption.

125

Theorem 80 Assume that ah belongs to M(α, β,Ω) for every fixedt ∈ (0, T ) and that¯¡

ah(x, t)− ah(x, t0)¢ξ¯≤ B(t− t0)(1 + |ξ|) (152)

for all ξ ∈ RN and all t0, t such that 0 < t0 < t < T , where B : R+ → R+ is anincreasing continuous function such that B(t)→ 0 as t→ 0+. Furthermore,let ah be a sequence such that

ahij(x, t) =

ZA

whij(x, t, y)Kij(x, t, y) dy,

where wh, K ∈ C1(ΩT × A)N×N and for some w(t) ∈ H1(Ω×A)N×N

wh(x, t, y) w(x, t, y) in L2(Ω×A)N×N (153)

for every fixed t ∈ (0, T ). Assume also that

NXj=1

ZA

∂xjwhij(x, t, y)Kij(x, t, y) dy 0 in L2(Ω) (154)

for all t ∈ (0, T ). Then ah G-converges in the sense of parabolic operatorsto

bij(x, t) =

ZA

wij(x, t, y)Kij(x, t, y) dy. (155)

Proof. From the assumption that ah is in M(α, β,Ω) for every fixedt ∈ (0, T ), it follows that it belongs to M(α, β,ΩT ). This means that, upto a subsequence, we have G-convergence in the parabolic sense to some baccording to Theorem 25. By assumption, ah fulfills (152) and hence weknow from Theorem 29 that it G-converges in the elliptic sense, up to asubsequence, for every fixed t to some limit b0(t). Thus, by Theorem 28, bcoincides with b0. Moreover, since ah agrees with (152), (153) and (154) itfollows, in view of Theorem 79, that b is given by (155).

This result covers possible heavy oscillations in x and parameter depen-dence of t. Thus, we can handle a large class of parabolic problems as well.However, oscillations in t require some more investigations. This will beexplored in the next section.

126

6.2.2 Some further results and open questions

Let us now introduce temporal oscillations in wh from Example 2 in Section6.1.1. We choose wh as the matrix given in (142) with the second term ineach component multiplied by sin(2πht), i.e. we have for instance

wh11(x, t, y) = 2 + sin(2πh(x1 + x2) + y1 + y2) sin(2πht). (156)

We still let Ω = (1, 3)2, A = Y = (0, 1)2 and f(x, t) = 100x1. If we chooseu0(x) = 0 and solve problem (150) for this choice of ah, we obtain the solutionin Figure 30 below for t = 1. In contrast to the corresponding elliptic case,the G-limit appears to be close or even identical to the weak limit a obtainedin (144).

Figure 30. The solution uh for h = 20.For this example, the criterion (154) is not fulfilled for any fixed t such that2t 6∈ Z. Actually, since we have introduced temporal oscillations, condition(152) is not satisfied and thus Proposition 80 is no longer applicable.

The temporal oscillations obviously have an effect in this example, andit seems that they, so to speak, have neutralized or at least significantlyreduced the effect of the oscillations in space. Thus, it might be of interestto investigate whether we could find some alternative to the criterion (154).We try multiplying the sum in (154) with test functions v, depending alsoon t and integrating over ΩT . For different choices of v in C(ΩT ), we obtain

NXj=1

ZΩT

ZY

∂xjwhij(x, t, y)Kij(x, t, y) dy v(x, t) dxdt→ 0.

Thus, it seems that we might get something new out of integrating also overthe time interval.

127

A result that is in line with this is given below. It shows that there arecriteria for time-dependent problems, corresponding to (147) and (148), butwhere the convergences take place in L2(ΩT × A)N×N and L2(ΩT ), respec-tively. This result also means that we can omit the condition (152).

Theorem 81 Let ah be a sequence inM(α, β,ΩT ) such that

ahij(x, t) =

ZA

whij(x, t, y)Kij(x, t, y) dy,

where wh, K ∈ C1(ΩT × A)N×N and

wh(x, t, y) w(x, t, y) in L2(ΩT ×A)N×N

for some w ∈ L2(0, T ;H1(Ω×A))N×N . Assume also that

NXj=1

ZA

∂xjwhij(x, t, y)Kij(x, t, y) dy 0 in L2(ΩT ) (157)

and that b ∈M(α, β2

α,ΩT ) has the entries

bij(x, t) =

ZA

wij(x, t, y)Kij(x, t, y) dy. (158)

Then ah G-converges to b.Proof. The proof can be carried out by combining the methods used in[FHOSi1] and [Sil2], which treat the case when w = 1, and the technique ofthe proof for the elliptic case in Proposition 79.

Actually, for our example, we do not obtain the weak convergence inL2(ΩT ) given in (157). This is realized if in place of the test function, we tryfor example with the sequence vh of functions

vh(x, t) = x21 +1

hsin(2πh(x1 + x2 + t)), (159)

which is strongly convergent in L2(ΩT ) towards x21. Indeed, we have

limh→∞

NXj=1

ZΩT

ZY

∂xjwhij(x, t, y)Kij(x, t, y) dy vh(x, t) dxdt ≈ 0.655,

and evidently (157) is not fulfilled.

128

So what can we say about this particular example? By Proposition 3.2 in[AlBr], for this choice of wh and K we obtain that°°°°°

NXj=1

ZA

∂xjwhij(x, t, y)Kij(x, t, y) dy

°°°°°L2(0,T ;H−1(Ω))

≤ C,

and hence we have at least weak* convergence in L2(0, T ;H−1(Ω)), up to asubsequence. Different choices of test functions v ∈ L2(0, T ;H1

0 (Ω)) give

NXj=1

ZΩT

ZA

∂xjwhij(x, t, y)Kij(x, t, y) dy v(x, t) dxdt→ 0, (160)

which indicates that the limit is 0, that is

NXj=1

ZA

∂xjwhij(x, t, y)Kij(x, t, y) dy

∗0 in L2(0, T ;H−1(Ω)). (161)

This should mean that for any fixed test function in L2(0, T ;H10 (Ω)) we

obtain (160), but if we multiply by a sequence that is weakly convergent inL2(0, T ;H1

0(Ω)) we have a product of two only weakly convergent sequencesand thus there is no obvious way to predict the limit.

If we perform the proof of Proposition 81, we see that what we reallyneed is that

NXj=1

ZΩT

ZA

∂xjwhij(x, t, y)Kij(x, t, y) dy ∂xjv(x)c(t)u

h(x, t) dxdt→ 0, (162)

and thus the convergence (161) is generally not enough since the other se-quence in (162) is weakly convergent in L2(0, T ;H1

0 (Ω)). On the other hand itis now clear that if we had strong convergence in L2(0, T ;H−1(Ω)) everythingwould work–that is, if

NXj=1

ZA

∂xjwhij(x, t, y)Kij(x, t, y) dy → 0 in L2(0, T ;H−1(Ω)). (163)

Thus, we have an alternative to the key criterion (157). This is not fulfilledin our example, however. To see this, we can employ some sequence of test

129

functions vh, which is weakly convergent in L2(0, T ;H10(Ω)). It remains an

open question whether conditions (157) and (163) could be replaced by someweaker alternative. One key to this may be contained in the properties ofthe specific sequence of solutions uh and the relation between wh and uh,which follows from the fact that uh solves a PDE where wh generates theoscillations in the coefficient matrix.

Remark 82 The examples we have studied in this section can also be re-garded as quasiperiodic homogenization problems. If we replace ht in (156)with hrt, r > 2, it is possible to prove that b is actually given by (158).

Remark 83 The numerical experiments have been carried out on powerfulcomputers. Still, due to the strong oscillations the results are not completelystable and hence should be regarded as a source of inspiration for an exper-iment of thought rather than really reliable computations. This also empha-sizes the importance of further theoretical investigations.

Remark 84 The considerations in this chapter open up new approaches inthe study of convergence for operators without periodicity assumptions, whichcould also be regarded as non-periodic homogenization. What would happenif the criterion (148) is not fulfilled and we for instance have

NXj=1

ZA

∂xjwhij(x, y)Kij(x, y) dy gi(x) in L2(Ω)?

Could the corresponding limit term then be integrated in the coefficient in anelliptic equation and hence be a part of the G-limit? Or might it be possibleto obtain a different limit equation with a so-called strange term? In thatcase, the limit problem would not give the G-limit, but if it was well-posedwe could still obtain a solution that approximates uh for h large. There areclearly many challenges ahead in continuation of research along these lines,the roots of which can be found in [Sil1], [Sil2] and [HOS].

130

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Appendix

Below is a list of some fundamental definitions and theorems that are usedin the thesis. If nothing else is stated, Ω is open and bounded with at leastLipschitz boundary.

Definition 85 (Strong convergence) A sequence uh in a normed spaceX is said to converge strongly, or in norm, to u if°°uh − u

°°X→ 0.

This is denoted byuh → u in X.

Definition 86 (Weak convergence) A sequence uh in a normed spaceX is said to converge weakly to u if

F (uh)→ F (u) for all F ∈ X 0.

We denote this byuh u in X.

Definition 87 (Weak* convergence) A sequence F h in the dual X 0 ofa normed space X is said to converge weakly* to F if

F h(u)→ F (u) for all u ∈ X.

We use the notationF h ∗

F in X 0.

Theorem 88 Let X be a reflexive Banach space. Then every bounded se-quence uh in X has a weakly convergent subsequence.

Proof. See [Alt], 5.7. Satz.

Theorem 89 Let X be a separable normed space. Then every bounded se-quence F h in X 0 has a weakly* convergent subsequence.

Proof. See [ZeiIIA], Theorem 21.E.

138

Theorem 90 Lp(Ω) is a Banach space for 1 ≤ p ≤ ∞. Furthermore, it isreflexive for 1 < p <∞ and separable for 1 ≤ p <∞.Proof. See Theorems 2.7.1, 2.8.1 and 2.10.1 in [Kuf].

Theorem 91 For 1 ≤ p ≤ ∞, W 1,p(Ω) is a Banach space. It is reflexive for1 < p <∞ and separable for 1 ≤ p <∞. Furthermore, the space W 1,p

0 (Ω) isreflexive for 1 < p <∞, and for 1 ≤ p ≤ ∞ it is separable.

Proof. See Theorems 5.4.2, 5.2.2, 5.2.4 and 5.4.4 in [Kuf].

Theorem 92 For 1 ≤ p < ∞, the dual of Lp(Ω) can be identified withLq(Ω), where 1

p+ 1

q= 1. Moreover, (L1(Ω))0 can be identified with L∞(Ω).

Proof. See Theorems 2.9.1 and 2.11.8 in [Kuf].

Theorem 93 (Rellich embedding theorem) Let 1 ≤ p <∞. If

uh u in W 1,p(Ω),

thenuh → u in Lp(Ω).

Proof. See [Alt], A 5.4.

Theorem 94 (Cauchy-Schwarz inequality) For any u, v in a Hilbertspace H it holds that

|(u, v)H | ≤ kukH kvkH .Proof. See Lemma 3.2-1 in [Kre].

Theorem 95 (Hölder’s inequality) Let u ∈ Lp(Ω) and v ∈ Lq(Ω), where1 < p <∞, 1

p+ 1

q= 1 and Ω is a non-empty, measurable set in RN . Then¯Z

Ω

u(x)v(x) dx

¯≤ZΩ

|u(x)v(x)| dx ≤ kukLp(Ω) kvkLq(Ω) . (164)

Moreover, if u ∈ L1(Ω) and v ∈ L∞(Ω) (164) holds true for p = 1 andq =∞.Proof. See [ZeiIIA], Proposition 18.13.

139

Theorem 96 (Poincaré’s inequality) For any u ∈ H10 (Ω), where Ω is an

open bounded set in RN , there is a positive constant C such thatZΩ

u(x)2 dx ≤ C

ZΩ

|∇u(x)|2 dx

where C depends only on Ω.

Proof. See 4.10 in [Alt].

Theorem 97 (Poincaré-Wirtinger inequality) Let 1 ≤ p < ∞. Thenthere exists a positive constant C such that

ku−MΩ(u)kLp(Ω) ≤ C k∇ukLp(Ω)N

for every u ∈ W 1,p(Ω), where MΩ(u) denotes the integral mean value of uover Ω.

Proof. See p. 194 in [McO].

Theorem 98 (Riesz representation theorem) Let F be a bounded lin-ear functional on the Hilbert space H (i.e. let F ∈ H 0). Then there is aunique element u ∈ H such that

F (v) = (u, v)H for every v ∈ H

withkFkH0 = kukH .

Proof. See [Mad], Theorem 13.

Theorem 99 (Lax-Milgram theorem) Let H be a Hilbert space and leta be a continuous bilinear form on H ×H satisfying

|a(u, v)| ≤ β kukH kvkHand

a(u, u) ≥ α kuk2Hfor all u, v ∈ H and some α > 0, and let F ∈ H 0. Then there is a uniqueu ∈ H such that

a(u, v) = hF, viH0,H

for all v ∈ H.

Proof. See [CiDo], Theorem 4.6.

140

Theorem 100 (Variational lemma) Let Ω be a non-empty open set inRN , let u ∈ L2(Ω) and assume thatZ

Ω

u(x)v(x) dx = 0 for every v ∈ C∞0 (Ω).

Thenu(x) = 0 for almost every x ∈ Ω.

Proof. See [ZeiIIA], Proposition 18.2.

Theorem 101 (Lebesgue’s generalized majorized convergence theorem)Let Ω ⊂ RN be a measurable set and fh : Ω → R be measurable for all hand assume that fh converges to f a.e. in Ω. Assume also that there areintegrable functions gh : Ω→ R such that¯

fh(x)¯≤ gh(x) for a.e. x ∈ Ω,

gh converges to g a.e. in Ω andZΩ

gh(x) dx→ZΩ

g(x) dx.

Then

limh→∞

ZΩ

fh(x) dx =

ZΩ

limh→∞

fh(x) dx.

Proof. See [ZeiIIB], (19a) in the appendix.

141

Doctoral Thesis 45

Marianne O

lsson |G-C

onvergence and Hom

ogenization of some M

onotone Operators

G| 2008

Mid Sweden University Doctoral Thesis 45, 2008ISSN 1652-893X, ISBN 978-91-85317-85-1

www.miun.se