# Homogenization in Sobolev Spaces with Nonstandard Growth: Brief Review of Methods and Applications

Authors: Brahim Amaziane; Leonid Pankratov
Journal: International Journal of Differential Equations
 (2013)

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## Abstract

We review recent results on the homogenization in Sobolev spaces with variable exponents. In particular, we are dealing with the Γ-convergence of variational functionals with rapidly oscillating coefficients, the homogenization of the Dirichlet and Neumann variational problems in strongly perforated domains, as well as double porosity type problems. The growth functions also depend on the small parameter
characterizing the scale of the microstructure. The homogenization results are obtained by the method of local energy characteristics. We also consider a parabolic double porosity type problem, which is studied by combining the variational homogenization approach and the two-scale convergence method. Results are illustrated with periodic examples, and the problem of stability in homogenization is discussed.


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## Body

## 1. Introduction

In recent years, there has been an increasing interest in the study of the functionals with variable exponents or nonstandardp(x)-growth and the corresponding Sobolev spaces, see for instance [1–8] and the references therein. In particular, the conditions under which C0∞ functions are dense in W1,p(·) have been found. Also, Meyers estimates, which are used in the homogenization process, have been obtained in [6]. Let us mention that such partial differential equations arise in many engineering disciplines, such as electrorheological fluids, non-Newtonian fluids with thermoconvective effects, and nonlinear Darcy flow of compressible fluids in heterogeneous porous media, see for instance [1].This paper discusses problems of homogenization in Sobolev spaces with variable exponents. Attention is focussed on the homogenization and minimization problems for variational functionals in the framework of Sobolev spaces with nonstandard growth. The material is essentially a review with some new results.Γ-convergence and minimization problems for functionals with periodic and locally periodic rapidly oscillating Lagrangians of p-growth with a constant p are well studied now, see for instance [9, 10] and the bibliography therein.The works [11–15] (see also [16]) focus on the variational functionals with nonstandard growth conditions. In particular, the homogenization and Γ-convergence problems for Lagrangians with variable rapidly oscillating exponents p(x) were considered in [13, 14]. It was shown that the energy minimums and the homogenized Lagrangians in the spaces W1,r might depend on the value of r (the so-called Lavrentiev phenomenon). For example, such a behavior can be observed for the Lagrangian |∇u|p(x/ε) with a periodic “chess board” exponent p(y) and a small parameter ε>0.Another interesting example of Lagrangian with rapidly oscillating exponent was considered in [11]. Namely, for the functional
(1)𝒥ε[u]=∫‍|∇u|p(x/ε)dx
with a smooth periodic p(y) such that p(y)>1, it was shown that the limit functional is bounded on Sobolev-Orlicz space of functions with gradient in an Lαlog space, where α is the fiber percolation level of p(x).Variational functionals with nonstandard growth conditions have also been considered in [9]. Chapter 21 of this book focuses on the Γ-convergence of such functionals in Lp spaces.In this paper, we are dealing mainly with theΓ-convergence of variational functionals with periodic rapidly oscillating coefficients, the homogenization of the variational problems in strongly perforated domains (Dirichlet and Neumann problems), and nonlinear double porosity type problems, that is, with the problems where the coefficient of the differential operator asymptotically degenerates on a some specially defined subset (e.g., the set of periodically distributed inclusions) of the domain under consideration. The paper is based on the results obtained in papers [17–26].The paper is organized as follows. In Section2, for the sake of completeness, we recall the definition and main results on the Lebesgue and Sobolev spaces with variable exponents which will be used in the sequel. Then in auxiliary Section 3, we give some definitions which will be used in the paper. In Section 4, we study the question of the Γ-convergence and homogenization of functionals with rapidly oscillating periodic coefficients. In Section 5, we are dealing with the homogenization of the Dirichlet variational problem in strongly perforated domains. The main result of the section (see Theorem 8) is then applied to the study of nonlocal effects in the homogenization (see Theorem 9). In Section 5.3, we give a periodic example, when all the conditions of Theorems 8 and 9 are satisfied and all the coefficients of the homogenized problem are calculated explicitly. Moreover, the question of stability in homogenization is also discussed here. Theorems 8 and 9 are proved by using the so-called method of local energy characteristics proposed earlier by Marchenko and Khruslov for linear homogenization problems (see [27] and the references herein). This method is close to the Γ-convergence method. Briefly, it is based on the derivation of the lim-inf and lim-sup estimates for the variational functional under consideration along with the assumptions on the behavior of the local energy characteristics. In Section 6, the homogenization of the Neumann problem in strongly perforated domains is considered. In this section, the closeness of the method of local energy characteristics is shown directly. In Section 7, we are dealing with a variational problem with high contrast coefficients (nonlinear double porosity type model). The main result of the section (Theorem 23) is also obtained by the method of local energy characteristics. As an application of this result, we consider the periodic case, where we focus our attention on the question of stability in homogenization. Finally, in Section 8, we are dealing with the homogenization of a class of quasilinear parabolic equations with nonstandard growth. The main results of the section are obtained by combining the two-scale convergence method and the variational homogenization approach.


## 2. Sobolev Spaces with Variable Exponents

In this section, we introduce the function spaces used throughout the paper and describe their basic properties. We refer here to [1, 4, 5, 7, 8].LetΩ be a bounded Lipschitz domain in ℝn (n≥2). We introduce the function p=p(x) and assume that this function is bounded such that
(2)1<p-=infΩp(x)≤p(x)≤supΩp(x)=p+<+∞.
We also assume that the function p(x) satisfies the log-Hölder continuity property. Namely, for all x,y∈Ω,
(3)|p(x)-p(y)|≤ω(|x-y|)withlim¯τ→0ω(τ)ln(1τ)≤C.
Notice that this property was introduced by Zhikov to avoid the so-called Lavrentiev phenomenon (see, e.g., [15]).(1)
ByLp(·)(Ω) we denote the space of measurable functions f in Ω such that
(4)Ap(·),Ω(f)=∫Ω|f(x)|p(x)dx<+∞.
The space Lp(·)(Ω) equipped with the norm
(5)∥f∥Lp(·)(Ω)=inf{λ>0:Ap(·),Ω(fλ)≤1}
becomes a Banach space.(2)
The following inequalities hold:(6)min(∥f∥Lp(·)(Ω)p-,∥f∥Lp(·)(Ω)p+)≤Ap(·),Ω(f)≤max(∥f∥Lp(·)(Ω)p-,∥f∥Lp(·)(Ω)p+),min(Ap(·),Ω1/p-,Ap(·),Ω1/p+)≤∥f∥Lp(·)(Ω)≤max(Ap(·),Ω1/p-,Ap(·),Ω1/p+).(3)
Letf∈Lp(·)(Ω), g∈Lq(·)(Ω) with
(7)1p(x)+1q(x)=1,1<p-≤p(x)≤p+<∞,1p(x)+1q(x)==1<q-≤q(x)≤q+<+∞.
Then the Hölder’s inequality holds
(8)∫Ω|fg|dx≤2∥f∥Lp(·)(Ω)∥g∥Lq(·)(Ω).(4)
According to (8), for every 1≤q=const<p-≤p(x)<+∞(9)∥f∥Lq(Ω)≤C∥f∥Lp(·)(Ω)withtheconstantC=2∥1∥p(·)/(Lp(·)-q(Ω)).
It is straightforward to check that for domains Ω such that measΩ<+∞,
(10)∥1∥Lp(·)(Ω)≤2max{[measΩ]2/p-,[measΩ]1/2p+}.(5)
The spaceW1,p(·)(Ω), p(·)∈[p-,p+]⊂]1,+∞[, is defined by
(11)W1,p(·)(Ω)={f∈Lp(·)(Ω):|∇f|∈Lp(·)(Ω)}.
If condition (3) is satisfied, W01,p(·)(Ω) is the closure of the set C0∞(Ω) with respect to the norm of W1,p(·)(Ω). If the boundary of Ω is Lipschitz continuous and p(x) satisfies (3), then C0∞(Ω) is dense in W01,p(·)(Ω). The norm in the space W01,p(·) is defined by
(12)∥u∥W01,p(·)=∑i‍∥Diu∥Lp(·)(Ω)+∥u∥Lp(·)(Ω).
If the boundary of Ω is Lipschitz and p∈C0(Ω), then the norm ∥·∥W01,p(·)(Ω) is equivalent to the norm
(13)∥u∥~W01,p(x)(Ω)=∑i‍∥Diu∥Lp(·)(Ω).(6)
Ifp∈C0(Ω¯), then W1,p(·)(Ω) is separable and reflexive.(7)
Ifp,q∈C0(Ω¯),
(14)p*(x)={p(x)nn-p(x)ifp(x)<n,and1<q(x)≤supΩq(x)and1<infΩp*(x),+∞ifp(x)>n,
then the embedding W01,p(·)(Ω)↪Lq(·)(Ω) is continuous and compact.(8)
Friedrich’s inequality is valid in the following form: ifp(x) satisfies conditions (2) and (3), then there exists a constant C>0 such that for every f∈W01,p(·)(Ω)(15)∥f∥Lp(·)(Ω)≤C∥∇f∥Lp(·)(Ω).


## 3. Definitions

In this auxiliary section, we introduce the necessary definitions that will be used in the paper. We start by introducing the class of the variable exponentspε, where ε is a small positive parameter characterizing the microscopic length scale.Definition 1 (class𝔏p0(·)ε).
A sequence of functions{pε}(ε>0) is said to belong to the class 𝔏p0(·)ε if this sequence possesses the following properties:(i)
for anyε>0, pε is bounded in the following sense:
(16)1<p-≤pε-≡minx∈Ω¯pε(x)≤pε(x)≤maxx∈Ω¯pε(x)≡pε+≤p+<+∞inΩ¯;(ii)
for anyε>0, pε satisfies the log-Hölder continuity property;(iii)
the functionpε converges uniformly in Ω to a function p0, where the limit function p0 is assumed to satisfy the log-Hölder continuity property.We also recall the definition of theΓ-convergence (see, e.g., [9, 10, 28] and the bibliography therein). In our case this definition takes the following form.Definition 2 (Γp(·)-convergence).
The functionalIε:Lpε(·)(Ω)→ℝ∪{∞} is said to Γp0(·)-converge to a functional I:Lp0(·)(Ω)→ℝ∪{∞} if(a)
(“liminf”-inequality) for any u∈Lp0(·)(Ω) and any sequence {uε}ε>0⊂Lp0(·)(Ω) which converges to the function u strongly in the space Lp0(·)(Ω) we have
(17)lim_ε→0Iε[uε]≥I[u],(b)
(“limsup”-inequality) for any u∈Lp0(·)(Ω), there exists a sequence {wε}ε>0⊂Lp0(·)(Ω) such that {wε}ε>0 converges to the function u(·) strongly in the space Lp0(·)(Ω), and
(18)lim¯ε→0Iε[wε]≤I[u].We define the strong convergence inLp0(·)(Ωε) in the following way.Definition 3 (strong convergence inLp0(·)(Ωε)).
The sequence{ωε}ε>0∈Lp0(·)(Ωε) is said to converge strongly in the space Lp0(·)(Ωε) to a function ω∈Lp0(·)(Ω) if
(19)limε→0∥ωε-ω∥Lp0(·)(Ωε)=0.Finally, we recall the definition of the two-scale convergence (see, e.g., [29]).Definition 4.
LetY=]0,1[n be a basic cell and ΩT=Ω×]0,T[. A function, φ∈L2(ΩT;C#∞(Y)), which is Y-periodic in y and which satisfies
(20)limε→0∫ΩT|φ(x,xε,t)|2dxdt=∫ΩT×Y|φ(x,y,t)|2dydxdt
is called an admissible test function.Definition 5 (two-scale convergence).
A sequence of functionsvε∈L2(ΩT) two-scale converges to v∈L2(ΩT×Y) if for any admissible test function φ(x,y,t),
(21)limε→0∫ΩTvε(x,t)φ(x,xε,t)dxdt=∫ΩT×Yv(x,y,t)φ(x,y,t)dydxdt.
This convergence is denoted by vε(x,t)⇀2sv(x,y,t).The method of the local energy characteristics, generally speaking, deals with nonperiodic structures. We often make use of the following definition.Definition 6 (distribution in an asymptotically regular way).
The setℱε is said to be distributed in an asymptotically regular way in Ω, if for any ball V(y,r) of radius r centered at y∈Ω and ε>0 small enough (ε≤ε0(r)), V(y,r)∩ℱε≠∅ and V(y,r)∩(Ω∖ℱε)≠∅.


## 4.Γ-Convergence and Homogenization of Functionals with Rapidly Oscillating Coefficients in Sobolev Spaces with Variable Exponents

LetΩ be a bounded domain in ℝn (n≥2) with a sufficiently smooth boundary and denote that Y=]0,1[n. We assume that a family of continuous functions {pε}(ε>0) belongs to the class, 𝔏p0(·)ε, and we also suppose that (H.1)
a=a(y) and b=b(y) are Y-periodic measurable functions such that
(22)0<a0≤a(y)≤a1,0<b0≤b(y)≤b1.(H.2)
f∈C(Ω).For the notational convenience, we set(23)apεε(x)=1pε(x)a(xε),bpεε(x)=1pε(x)b(xε).In the spaceLpε(·)(Ω), we define the functional Jε:Lpε(·)(Ω)→ℝ:
(24)Jε[u]≝{∫Ω{{12}apεε(x)|∇u|pε(x)+bpεε(x)|u|pε(x)-f(x)u{12}}dx,ifu∈W1,pε(·)(Ω),+∞,otherwise.We study the asymptotic behavior ofJε and its minimizers as ε→0. Our analysis relies on the Γ-convergence approach in Sobolev spaces with variable exponents. The main result of the section is the following.Theorem 7.
Let assumptions (H.1) and (H.2) be fulfilled. Then (i)
The functionalJε, Γp0(·)-converges to the functional Jhom:Lp0(·)(Ω)→ℝ∪{+∞} given by
(25)Jhom[u]={∫Ω{T(x,∇u)+βp(x)|u|p0(x)-f(x)u{T(x,∇u)+βp(x)|u|p0(x)}}dx,ifu∈W1,p0(Ω),+∞,otherwise,
where
(26)T(x,b)=inf{1p0(x)∫Ya(y)|∇v(y)=inf+b{∇v(y)}|p0(x)dy:v∈W#1,p0(x)(Y){1p0(x)∫Ya(y)|∇v(y)}},βp(x)=1p0(x)∫Yb(y)dy.(ii)
The minimizeruε of the functional Jε converges to the minimizer u of the functional Jhom strongly in the space Lp0(·)(Ω).The Scheme of the Proof of Theorem7 (See [18] for More Details) Is as Follows.We start our analysis by proving the “liminf”-inequality. The proof of this inequality is done in two main steps. First we introduce an auxiliary functional J~ε:Lπε(·)(Ω)→ℝ:
(27)J~ε[u]={∫Ω{apεε(x)|∇u|πε(x)+bpεε(x)|u|pε(x)-f(x)u{apεε(x)|∇u|πε(x)+bpεε(x)|u|pε(x)}}dx,ifu∈W1,πε(·)(Ω);+∞,otherwise,whereπε(x)=min{pε(x),p0(x)} and prove the “liminf”-inequality for this functional. Then, at the second step, we show that the “liminf”-inequality for the auxiliary functional J~ε implies the “liminf”-inequality for Jε. Then, using a special test function and the Meyers estimate (see [6]), we obtain the “limsup”-inequality. Finally, we prove the convergence of the minimizers. This completes the proof of Theorem 7.


## 5. Homogenization of the Dirichlet Problem and Related Questions

### 5.1. Homogenization of the Dirichlet Problem

LetΩ be a bounded domain in ℝn(n≥2) with sufficiently smooth boundary. Let ℱε be an open subset in Ω. Here ε is a small parameter characterizing the scale of the microstructure. We assume that ℱε is distributed in an asymptotically regular way in Ω and we set
(28)Ωε=Ω∖ℱ¯ε.Letpε=pε(x) be a continuous function defined in the domain Ω¯. We assume that, for any ε>0, it satisfies the following conditions:(A.1.1)
the functionpε(x) is bounded in the following sense:
(29)1<p-≤pε-≡minx∈Ω¯pε(x)≤pε(x)≤maxx∈Ω¯pε(x)≡pε+≤p+≤ninΩ¯;(A.1.2)
the functionpε satisfies the log-Hölder continuity property;(A.1.3)
the functionpε converges uniformly in Ω to a function p0, where the limit function p0 is assumed to satisfy the log-Hölder continuity property;(A.1.4)
the functionpε is such that pε(x)≥p0(x) in Ω.We consider the following variational problem:(30)Jε[u]≝∫ΩεFε(x,u,∇u)dx⟶min,u∈W01,pε(·)(Ωε),
where
(31)Fε(x,u,∇u)=1pε(x)|∇u|pε(x)+1pε(x)|u|pε(x)-f(x)u,
and f∈C1(Ω). It is known from [1–3, 6] that for each ε>0, there exists a unique solution (minimizer) uε∈W1,pε(·)(Ωε) of problem (30). Let us extend uε in ℱε by zero (keeping for it the same notation). Then we obtain the family {uε}⊂W1,pε(·)(Ω). We study the asymptotic behavior of uε as ε→0.Instead of the classical periodicity assumption on the microstructure of the perforated domainΩε, we impose certain conditions on the local energy characteristic of the set ℱε. To this end we introduce Khz an open cube centered at z∈Ω with length equal to h (0<ε≪h<1), and we set
(32)cε,h(z,b)=infvε∫Khz{1pε(x)|∇vε|pε(x)+h-p+-γ|vε-b|pε(x)}dx,
where γ>0, and the infimum is taken over vε∈W1,pε(·)(Ω) that equal zero in ℱε. We assume that(C.1.1)
there exists a continuous functionc(x,b) such that for any x∈Ω, any b∈ℝ, and a certain γ=γ0>0,
(33)limh→0lim¯ε→0h-ncε,h(z,b)=limh→0lim_ε→0h-ncε,h(z,b)=c(x,b);(C.1.2)
there exists a constantA independent of ε such that for any x∈Ω,
(34)limh→0lim¯ε→0h-ncε,h(z,b)≤A(1+|b|p0(x)).The first main result of Section5 is the following.Theorem 8.
Let conditions (A.1.1)–(A.1.4) on the functionpε and conditions (C.1.1) and (C.1.2) on the local characteristic be satisfied. Then uε the solution (minimizer) of the variational problem (30) (extended by zero in ℱε) converges weakly in W1,p0(·)(Ω) to u the solution (minimizer) of
(35)Jhom[u]≝∫Ω{F0(x,u,∇u)+c(x,u)-f(x)u}dx⟶min,Jhom[u]≝∫Ω{F0(x,u,∇u)(x,u)-f(x)u}u∈W01,p0(·)(Ω),
where
(36)F0(x,w,∇w)≝1p0(x)|∇w|p0(x)+1p0(x)|w|p0(x)-g(x)w.The Scheme of the Proof of Theorem8 (See [19] for More Details) Is as Follows. First, it follows from (6), (30), and the regularity properties of the functions f,pε that ∥uε∥W1,pε(·)(Ωε)≤C. We extend uε by zero to the set ℱε and consider {uε} as a sequence in W1,pε(·)(Ω). Then ∥uε∥W1,pε(·)(Ω)≤C. Condition (A.1.4) immediately implies that
(37)∥uε∥W1,p0(·)(Ω)≤C.
Hence, one can extract a subsequence {uε,ε=εk→0} that converges weakly to a function u∈W1,p0(·)(Ω). We will show that u is the solution of the variational problem (35). The proof is done in two mains steps. On the first step, we prove the “limsup”-inequality (the upper bound for the functional Jε). To this end, we cover the domain Ω by the cubes Khα of length h>0 centered at the points xα, where {xα} be a periodic grid in Ω with a period h′=h-h1+γ/p+ (ε≪h≪1, 0<γ<p+). We associate with this covering a partition of unity {φα}: 0≤φα(x)≤1; φα(x)=0 for x∉Khα; φα(x)=1 for x∈Khα∖∪β≠αKhβ; ∑α‍φα(x)=1 for x∈Ω; |∇φα(x)|≤Ch-1-γ/p+. Then we construct the function whε using the minimizers vαε=vαε(x) of (32) for z=xα. We show that, for any w∈W01,p0(·)(Ω),
(38)limh→0lim¯ε→0Jε[whε]≤Jhom[w],
and, therefore,
(39)lim¯ε→0Jε[uε]≤Jhom[w].
On the second step, we prove the “liminf”-inequality (the lower bound for the functional Jε) using the definition of the local energy characteristic and the convexity of our functional:
(40)lim_ε=εk→0Jε[uε]≥Jhom[u].
Finally it follows from (39) and (40) that
(41)Jhom[u]≤Jhom[w],
for any w∈W1,p0(·)(Ω) such that w=0 on ∂Ω. This means that any weak limit of the solution of problem (30) extended to the set ℱε by zero is the solution of the homogenized problem (35). This completes the proof of Theorem 8. The generalization of Theorem 8 is given in [17].


### 5.2. Nonlocal Effects in Homogenization ofpε(x)-Laplacian in Perforated Domains

LetΩ be a domain in ℝn (n≥3) defined in the previous section. Let ℱε be an open connected subset in Ω like a net. We assume that the set ℱε satisfies the following conditions: (F.1.1)
ℱε⋐Ωd, where Ωd={x∈Ω:dist(x,∂Ω)≥d} with d>0 independent of ε;(F.1.2)
ℱε is distributed in an asymptotically regular way in Ωd;(F.1.3)
the subdomainΩε=defΩ∖ℱε¯ is a connected set.We assume that for anyε>0, the function pε=pε(x) satisfies the conditions (A.1.1)–(A.1.4) from Section 5.1. In the space W1,pε(·)(Ωε), we define the functional Jε:W1,pε(·)(Ωε)→ℝ,
(42)Jε[u]={∫ΩεFε(x,u,∇u)dxifu∈W1,pε(·)(Ωε),+∞otherwise,
where Fε(x,u,∇u) is defined in (31) and we consider the following variational problem:
(43)Jε[uε]⟶inf,uε∈W1,pε(·)(Ωε);uε=Aεon∂ℱε,uε=0on∂Ω,
where Aε is an unknown constant. It is known from [1, 2] that for each ε>0, there exists a unique solution uε∈W1,pε(·)(Ωε) of the variational problem (43). We extend uε by the equality uε=Aε in ℱε and keep for it the same notation. Thus, we obtain the family {uε}(ε>0)⊂W1,pε(·)(Ω). We study the asymptotic behavior of the family {uε}(ε>0) as ε→0.Let us introduce the functionalJhom:W1,p0(·)(Ω)→ℝ,
(44)Jhom[u,A]={∫Ω{F0(x,u,∇u)+1d(x)c(x,u-A)dx}ifu∈W1,p0(·)(Ω),+∞otherwise,
where 1d=1d(x) is the characteristic function of the domain Ωd and F0(x,u,∇u) is defined in (36).The main result of the section is the following.Theorem 9.
Letuε be a solution of (43) extended by the equality uε(x)=Aε in ℱε. Assume that the conditions (F.1.1)–(F.1.3), (A.1.1)–(A.1.4), and(C.1.1)-(C.1.2) on the set ℱε, the functions pε, and the local characteristic cε,h(x,b) are satisfied. Then there is a subsequence {uε,ε=εk→0} that converges weakly in W1,p0(·)(Ω) to a function u such that the pair {u,A} is a solution of
(45)Jhom[u,A]⟶inf,u∈W01,p0(·)(Ω)withA=limε→0Aε.Remark 10.
It is important to notice that the constantA in (45) remains unknown. Suppose, in addition, that the function c(x,b) is differentiable with respect to the argument b. Then it is easy to see that Euler’s equation for the homogenized problem (45) reads
(46)-div(|∇u|p0(x)-2∇u)+|u|p0(x)-2u+1d(x)cu′(x,u-A)=f(x)inΩ;u=0on∂Ω,∫Ωdcu′(x,u-A)dx=0,
where cu′ denotes the partial derivative of the function c with respect to u. This means that the homogenized problem (45) is nonlocal.Theorem9 is proved by arguments similar to those from the proof of Theorem 8.


### 5.3. A Periodic Example

Theorems8 and 9 provide sufficient conditions for the existence of the homogenized problem (45). The goal of this section is to prove that, for appropriate examples, all the conditions of Theorems 8 and 9 are satisfied and to compute the function c(x,u) and the constant A in the homogenized problem (45) explicitly.LetΩ be a bounded Lipschitz domain in ℝ3 and let Ωd be the subdomain of Ω defined in condition (F.1.1). Let 𝔉ε be an ε-periodic coordinate lattice in ℝ3 formed by the intersecting circular cylinders of radius rε,
(47)rε=e-1/ε2.
We set ℱε=𝔉ε∩Ωd and Ωε=Ω∖ℱε¯.Let{pε}(ε>0) be a class of continuous functions defined in the domain Ω¯. We assume that, for any ε>0, pε satisfies the log-Hölder continuity condition and the following condition: (B.1.2)
the functionspε are given by
(48)pε(x)={2+ℓ(x)ε2in𝒩(ℱε,ε2),2+ℓε(x)elsewhere,where 𝒩(𝒪,δ) denotes the cylindrical δ-neighborhood of the set 𝒪 and where ℓ and ℓε are smooth strictly positive functions in Ω¯, moreover, maxx∈Ω¯ℓε(x)=o(1) as ε→0.Consider the boundary value problem(49)-div(|∇uε|pε(x)-2∇uε)+|uε|pε(x)-2uε=f(x)inΩε,uε=Aεon∂ℱε;uε=0on∂Ω,∫∂ℱε|∇uε|pε(x)-2∂uε∂ν→ds=0,
where Aε is an unknown constant. It is known from [1, 2] that, for each ε>0, there exists a unique solution uε∈W1,pε(·)(Ωε) of problem (49). We extend uε by the equality uε=Aε in ℱε and keep for it the same notation. The asymptotic behavior of the solutions of problem (49) is given now by the following theorem.Theorem 11.
Letuε be a solution of (49) extended by the equality uε(x)=Aε in ℱε. Assume that the condition (B.1.2) on the function pε is satisfied. Then the sequence {uε}(ε>0) converges weakly in H1(Ω) to a function u solution of
(50)-Δu+u+μ(x)(u-Aℓ(u))=g(x)inΩ,u=0on∂Ω,
where
(51)μ(x)=6π1d(x)eℓ(x)-1ℓ(x),Aℓ(u)=(∫Ωdμ(x)dx)-1∫Ωd‍μ(x)u(x)dx.Theorem11 is proved in [23] and is based essentially on the calculation of the local energy characteristic for this periodic structure.Remark 12.
It is easy to see from the proof of Theorem11 that if in (48) we replace ℓ(x)ε2 by a function ℓ~ε with maxx∈Ω¯ℓ~ε=o(ε2), then μ(x)=6π1d(x) as in the linear case.Remark 13.
It is shown in [27] Paragraph 3.3 that for the integrands of growth |∇uε|2ln|∇uε|, the 3D lattice becomes extremely thin. Moreover, for the integrands of growth |∇uε|2+δ, where δ>0 is a parameter independent of ε, there is no 3D lattice which admits nontrivial homogenization result, because the capacity of the lattice goes to infinity as ε→0. Theorem 11 shows the maximal possible polynomial growth of the integrand (in a small neighborhood of the lattice) which admits a nontrivial homogenization result.


## 5.1. Homogenization of the Dirichlet Problem

LetΩ be a bounded domain in ℝn(n≥2) with sufficiently smooth boundary. Let ℱε be an open subset in Ω. Here ε is a small parameter characterizing the scale of the microstructure. We assume that ℱε is distributed in an asymptotically regular way in Ω and we set
(28)Ωε=Ω∖ℱ¯ε.Letpε=pε(x) be a continuous function defined in the domain Ω¯. We assume that, for any ε>0, it satisfies the following conditions:(A.1.1)
the functionpε(x) is bounded in the following sense:
(29)1<p-≤pε-≡minx∈Ω¯pε(x)≤pε(x)≤maxx∈Ω¯pε(x)≡pε+≤p+≤ninΩ¯;(A.1.2)
the functionpε satisfies the log-Hölder continuity property;(A.1.3)
the functionpε converges uniformly in Ω to a function p0, where the limit function p0 is assumed to satisfy the log-Hölder continuity property;(A.1.4)
the functionpε is such that pε(x)≥p0(x) in Ω.We consider the following variational problem:(30)Jε[u]≝∫ΩεFε(x,u,∇u)dx⟶min,u∈W01,pε(·)(Ωε),
where
(31)Fε(x,u,∇u)=1pε(x)|∇u|pε(x)+1pε(x)|u|pε(x)-f(x)u,
and f∈C1(Ω). It is known from [1–3, 6] that for each ε>0, there exists a unique solution (minimizer) uε∈W1,pε(·)(Ωε) of problem (30). Let us extend uε in ℱε by zero (keeping for it the same notation). Then we obtain the family {uε}⊂W1,pε(·)(Ω). We study the asymptotic behavior of uε as ε→0.Instead of the classical periodicity assumption on the microstructure of the perforated domainΩε, we impose certain conditions on the local energy characteristic of the set ℱε. To this end we introduce Khz an open cube centered at z∈Ω with length equal to h (0<ε≪h<1), and we set
(32)cε,h(z,b)=infvε∫Khz{1pε(x)|∇vε|pε(x)+h-p+-γ|vε-b|pε(x)}dx,
where γ>0, and the infimum is taken over vε∈W1,pε(·)(Ω) that equal zero in ℱε. We assume that(C.1.1)
there exists a continuous functionc(x,b) such that for any x∈Ω, any b∈ℝ, and a certain γ=γ0>0,
(33)limh→0lim¯ε→0h-ncε,h(z,b)=limh→0lim_ε→0h-ncε,h(z,b)=c(x,b);(C.1.2)
there exists a constantA independent of ε such that for any x∈Ω,
(34)limh→0lim¯ε→0h-ncε,h(z,b)≤A(1+|b|p0(x)).The first main result of Section5 is the following.Theorem 8.
Let conditions (A.1.1)–(A.1.4) on the functionpε and conditions (C.1.1) and (C.1.2) on the local characteristic be satisfied. Then uε the solution (minimizer) of the variational problem (30) (extended by zero in ℱε) converges weakly in W1,p0(·)(Ω) to u the solution (minimizer) of
(35)Jhom[u]≝∫Ω{F0(x,u,∇u)+c(x,u)-f(x)u}dx⟶min,Jhom[u]≝∫Ω{F0(x,u,∇u)(x,u)-f(x)u}u∈W01,p0(·)(Ω),
where
(36)F0(x,w,∇w)≝1p0(x)|∇w|p0(x)+1p0(x)|w|p0(x)-g(x)w.The Scheme of the Proof of Theorem8 (See [19] for More Details) Is as Follows. First, it follows from (6), (30), and the regularity properties of the functions f,pε that ∥uε∥W1,pε(·)(Ωε)≤C. We extend uε by zero to the set ℱε and consider {uε} as a sequence in W1,pε(·)(Ω). Then ∥uε∥W1,pε(·)(Ω)≤C. Condition (A.1.4) immediately implies that
(37)∥uε∥W1,p0(·)(Ω)≤C.
Hence, one can extract a subsequence {uε,ε=εk→0} that converges weakly to a function u∈W1,p0(·)(Ω). We will show that u is the solution of the variational problem (35). The proof is done in two mains steps. On the first step, we prove the “limsup”-inequality (the upper bound for the functional Jε). To this end, we cover the domain Ω by the cubes Khα of length h>0 centered at the points xα, where {xα} be a periodic grid in Ω with a period h′=h-h1+γ/p+ (ε≪h≪1, 0<γ<p+). We associate with this covering a partition of unity {φα}: 0≤φα(x)≤1; φα(x)=0 for x∉Khα; φα(x)=1 for x∈Khα∖∪β≠αKhβ; ∑α‍φα(x)=1 for x∈Ω; |∇φα(x)|≤Ch-1-γ/p+. Then we construct the function whε using the minimizers vαε=vαε(x) of (32) for z=xα. We show that, for any w∈W01,p0(·)(Ω),
(38)limh→0lim¯ε→0Jε[whε]≤Jhom[w],
and, therefore,
(39)lim¯ε→0Jε[uε]≤Jhom[w].
On the second step, we prove the “liminf”-inequality (the lower bound for the functional Jε) using the definition of the local energy characteristic and the convexity of our functional:
(40)lim_ε=εk→0Jε[uε]≥Jhom[u].
Finally it follows from (39) and (40) that
(41)Jhom[u]≤Jhom[w],
for any w∈W1,p0(·)(Ω) such that w=0 on ∂Ω. This means that any weak limit of the solution of problem (30) extended to the set ℱε by zero is the solution of the homogenized problem (35). This completes the proof of Theorem 8. The generalization of Theorem 8 is given in [17].


## 5.2. Nonlocal Effects in Homogenization ofpε(x)-Laplacian in Perforated Domains

LetΩ be a domain in ℝn (n≥3) defined in the previous section. Let ℱε be an open connected subset in Ω like a net. We assume that the set ℱε satisfies the following conditions: (F.1.1)
ℱε⋐Ωd, where Ωd={x∈Ω:dist(x,∂Ω)≥d} with d>0 independent of ε;(F.1.2)
ℱε is distributed in an asymptotically regular way in Ωd;(F.1.3)
the subdomainΩε=defΩ∖ℱε¯ is a connected set.We assume that for anyε>0, the function pε=pε(x) satisfies the conditions (A.1.1)–(A.1.4) from Section 5.1. In the space W1,pε(·)(Ωε), we define the functional Jε:W1,pε(·)(Ωε)→ℝ,
(42)Jε[u]={∫ΩεFε(x,u,∇u)dxifu∈W1,pε(·)(Ωε),+∞otherwise,
where Fε(x,u,∇u) is defined in (31) and we consider the following variational problem:
(43)Jε[uε]⟶inf,uε∈W1,pε(·)(Ωε);uε=Aεon∂ℱε,uε=0on∂Ω,
where Aε is an unknown constant. It is known from [1, 2] that for each ε>0, there exists a unique solution uε∈W1,pε(·)(Ωε) of the variational problem (43). We extend uε by the equality uε=Aε in ℱε and keep for it the same notation. Thus, we obtain the family {uε}(ε>0)⊂W1,pε(·)(Ω). We study the asymptotic behavior of the family {uε}(ε>0) as ε→0.Let us introduce the functionalJhom:W1,p0(·)(Ω)→ℝ,
(44)Jhom[u,A]={∫Ω{F0(x,u,∇u)+1d(x)c(x,u-A)dx}ifu∈W1,p0(·)(Ω),+∞otherwise,
where 1d=1d(x) is the characteristic function of the domain Ωd and F0(x,u,∇u) is defined in (36).The main result of the section is the following.Theorem 9.
Letuε be a solution of (43) extended by the equality uε(x)=Aε in ℱε. Assume that the conditions (F.1.1)–(F.1.3), (A.1.1)–(A.1.4), and(C.1.1)-(C.1.2) on the set ℱε, the functions pε, and the local characteristic cε,h(x,b) are satisfied. Then there is a subsequence {uε,ε=εk→0} that converges weakly in W1,p0(·)(Ω) to a function u such that the pair {u,A} is a solution of
(45)Jhom[u,A]⟶inf,u∈W01,p0(·)(Ω)withA=limε→0Aε.Remark 10.
It is important to notice that the constantA in (45) remains unknown. Suppose, in addition, that the function c(x,b) is differentiable with respect to the argument b. Then it is easy to see that Euler’s equation for the homogenized problem (45) reads
(46)-div(|∇u|p0(x)-2∇u)+|u|p0(x)-2u+1d(x)cu′(x,u-A)=f(x)inΩ;u=0on∂Ω,∫Ωdcu′(x,u-A)dx=0,
where cu′ denotes the partial derivative of the function c with respect to u. This means that the homogenized problem (45) is nonlocal.Theorem9 is proved by arguments similar to those from the proof of Theorem 8.


## 5.3. A Periodic Example

Theorems8 and 9 provide sufficient conditions for the existence of the homogenized problem (45). The goal of this section is to prove that, for appropriate examples, all the conditions of Theorems 8 and 9 are satisfied and to compute the function c(x,u) and the constant A in the homogenized problem (45) explicitly.LetΩ be a bounded Lipschitz domain in ℝ3 and let Ωd be the subdomain of Ω defined in condition (F.1.1). Let 𝔉ε be an ε-periodic coordinate lattice in ℝ3 formed by the intersecting circular cylinders of radius rε,
(47)rε=e-1/ε2.
We set ℱε=𝔉ε∩Ωd and Ωε=Ω∖ℱε¯.Let{pε}(ε>0) be a class of continuous functions defined in the domain Ω¯. We assume that, for any ε>0, pε satisfies the log-Hölder continuity condition and the following condition: (B.1.2)
the functionspε are given by
(48)pε(x)={2+ℓ(x)ε2in𝒩(ℱε,ε2),2+ℓε(x)elsewhere,where 𝒩(𝒪,δ) denotes the cylindrical δ-neighborhood of the set 𝒪 and where ℓ and ℓε are smooth strictly positive functions in Ω¯, moreover, maxx∈Ω¯ℓε(x)=o(1) as ε→0.Consider the boundary value problem(49)-div(|∇uε|pε(x)-2∇uε)+|uε|pε(x)-2uε=f(x)inΩε,uε=Aεon∂ℱε;uε=0on∂Ω,∫∂ℱε|∇uε|pε(x)-2∂uε∂ν→ds=0,
where Aε is an unknown constant. It is known from [1, 2] that, for each ε>0, there exists a unique solution uε∈W1,pε(·)(Ωε) of problem (49). We extend uε by the equality uε=Aε in ℱε and keep for it the same notation. The asymptotic behavior of the solutions of problem (49) is given now by the following theorem.Theorem 11.
Letuε be a solution of (49) extended by the equality uε(x)=Aε in ℱε. Assume that the condition (B.1.2) on the function pε is satisfied. Then the sequence {uε}(ε>0) converges weakly in H1(Ω) to a function u solution of
(50)-Δu+u+μ(x)(u-Aℓ(u))=g(x)inΩ,u=0on∂Ω,
where
(51)μ(x)=6π1d(x)eℓ(x)-1ℓ(x),Aℓ(u)=(∫Ωdμ(x)dx)-1∫Ωd‍μ(x)u(x)dx.Theorem11 is proved in [23] and is based essentially on the calculation of the local energy characteristic for this periodic structure.Remark 12.
It is easy to see from the proof of Theorem11 that if in (48) we replace ℓ(x)ε2 by a function ℓ~ε with maxx∈Ω¯ℓ~ε=o(ε2), then μ(x)=6π1d(x) as in the linear case.Remark 13.
It is shown in [27] Paragraph 3.3 that for the integrands of growth |∇uε|2ln|∇uε|, the 3D lattice becomes extremely thin. Moreover, for the integrands of growth |∇uε|2+δ, where δ>0 is a parameter independent of ε, there is no 3D lattice which admits nontrivial homogenization result, because the capacity of the lattice goes to infinity as ε→0. Theorem 11 shows the maximal possible polynomial growth of the integrand (in a small neighborhood of the lattice) which admits a nontrivial homogenization result.


## 6. Homogenization of the Neumann Problem

### 6.1. Statement of the Problem and Main Results

LetΩ be a bounded Lipschitz domain in ℝn (n≥2). Let {ℱε}(ε>0) be a family of open subsets in Ω; in the sequel ε is a small positive parameter characterizing the microscopic length scale. We assume that the set ℱε consists of Nε (Nε→+∞ as ε→0) small isolated components such that their diameters go to zero as ε→0 and ℱε is distributed in an asymptotically regular way in Ω. We set
(52)Ωε=Ω∖ℱε¯.We assume that a family of continuous functions{pε}(ε>0) belongs to the class 𝔏p0(·)ε. On the space Lpε(·)(Ωε), we define the functional Jε:Lpε(·)(Ωε)→ℝ∪{+∞}:
(53)Jε[u]={∫ΩεFε(x,u,∇u)dx,ifu∈W1,pε(·)(Ωε),+∞,otherwise,
where Fε(x,u,∇u) is defined in (31) and f∈C(Ω¯).We study the asymptotic behavior ofJε and their minimizers as ε→0. The classical periodicity assumption is here substituted by an abstract one covering a variety of concrete behaviors, such as the periodicity, the almost periodicity, and many more besides. For this, we assume that Ωε⊂Ω is a disperse medium; that is, the following assumptions hold: (C.2.1)
the local concentration of the setΩε has a positive continuous limit, that is, the indicator of Ωε converges weakly in L2(Ω) to a continuous positive limit. This implies that there exists a continuous positive function ρ=ρ(x) such that
(54)limh→0limε→0h-nmeas(Khx∩Ωε)=ρ(x),for any open cubeKhx centered at x∈Ω with lengths equal to h>0;(C.2.2)
for anyq∈[p-,p+], there exists a family of extension operators Pqε:W1,q(Ωε)→W1,q(Ω) such that.global: for any uε∈W1,q(Ωε),
(55)∥uε∥Lq(Ω)≤C∥uε∥Lq(Ωε),∥∇uε∥Lq(Ω)≤C∥∇uε∥Lq(Ωε),uniformly inε>0, where uε=Pεuε and uε=uε in Ωε;local: for any h>0 there is ε0(h)>0 such that for all ε<ε0(h), z∈Ω and any function u∈W1,q((z+[-2h,2h]n)∩Ωε), the estimates hold
(56)∥uε∥Lq((z+[-h,h]n)∩Ω)≤C∥uε∥Lq((z+[-2h,2h]n)∩Ωε),∥∇uε∥Lq((z+[-h,h]n)∩Ω)≤C∥∇uε∥Lq((z+[-2h,2h]n)∩Ωε).Remark 14.
Notice that in condition (C.2.2), we require the existence of extension operators only in usual Sobolev spacesW1,q with constant q. In this case, the extension condition is well studied in the mathematical literature (see, e.g., [27, 30–32]). For instance, it holds for a wide class of disperse media (see, for instance, [27]).One more condition is imposed on the local characteristic of the setℱε associated to the functional (53). In order to formulate this condition, we denote by Khz an open cube centered at z∈Ω with edge length h (0<ε≪h≪1) and introduce the functional
(57)cpε(·)ε,h(z,b)=infvε∫Khz∩Ωε{1pε(x)|∇vε|pε(x)∫x{1pε(x)+h-pε(x)-γ|vε-(x-z,b)|pε(x){1pε(x)}}dx,
where γ>0, b∈ℝn, and the infimum is taken over vε∈W1,pε(·)(Khz∩Ωε). Here (·,·) stands for the scalar product in ℝn. We assume that(C.2.3)
there is a continuous, with respect tox∈Ω¯, function T(x,b) and γ=γ0 (0<γ0<p-) such that for any {pε}(ε>0)⊂𝔏p0(·)ε, any x∈Ω and any b∈ℝn,
(58)limh→0lim¯ε→0h-ncpε(·)ε,h(x,b)=limh→0lim_ε→0h-ncpε(·)ε,h(x,b)=T(x,b).Remark 15.
Condition (C.2.3) is always fulfilled for periodic and locally periodic structures.Remark 16.
It is crucial in condition (C.2.3) that the limit functionT(x,b) does not depend on the particular choice of the sequence pε→p0. Notice that this is always the case for periodic and locally periodic perforated media. These media will be considered in detail in the last section of the paper.Now we are in position to formulate the first convergence result of the section.Theorem 17.
Assume that{pε}(ε>0)⊂𝔏p0(·)ε, and let conditions (C.2.1)–(C.2.3) be satisfied. Then the functionals Jε defined in (53), Γp0(·)-converge to the functional Jhom:Lp0(·)(Ω)→ℝ∪{+∞} given by
(59)Jhom[u]={∫Ω{T(x,∇u)+ρ(x)p0(x)|u|p0(x)-ρ(x)f(x)u{ρ(x)p0(x)|u|p0(x)}}dx,ifu∈W1,p0(·)(Ω),+∞,otherwise.Now let us formulate the convergence result for the minimizers of the functionalsJε. Consider the variational problem
(60)Jε[uε]⟶min,uε∈W1,pε(·)(Ωε).
According to [1–3, 6], for each ε>0, problem (60) has a unique solution uε∈W1,pε(·)(Ωε).The following convergence result holds.Theorem 18.
Under the assumptions of Theorem17, the solution uε of the variational problem (60) converges strongly in Lp0(·)(Ωε) to a solution of the problem
(61)Jhom[u]⟶min,u∈W1,p0(·)(Ω).The Scheme of the Proof of Theorems17 and 18 (See [22] for More Details) Is as Follows. The “liminf”-inequality is proved in two steps as in Theorem 7 by introducing an auxiliary functional. The “limsup”-inequality is proved by the arguments similar to those from the proof of Theorem 8.


### 6.2. A Periodic Example

Theorems17 and 18 of Section 6.1 provide sufficient conditions for the existence of the Γ-limit functional (59) and for the convergence of minimizers of the variational problem (60) to the minimizer of the homogenized variational problem (61). It is important to show that the class of functions which satisfy the conditions of these theorems is not empty. The goal of this section is to prove that for periodic and locally periodic media all conditions of the above-mentioned theorems are satisfied and to compute the coefficients of the homogenized functional (59) in terms of solutions of auxiliary cell problems. In fact, conditions (C.2.1) and  (C.2.3) are always satisfied in the periodic case if the boundary of inclusions is regular enough, and that the extension condition (C.2.2) can also be replaced with the assumption on the regularity of the inclusions geometry.LetΩ be a bounded domain in ℝn (n≥2) with sufficiently smooth boundary. We assume that in the periodic cell Y=]0,1[n, there is an obstacle F⋐Y being an open set with a sufficiently smooth boundary ∂F. We assume that this geometry is repeated periodically in the whole ℝn. The geometric structure within the domain Ω is then obtained by intersecting the ε-multiple of this geometry with Ω. Let {xk,ε} be an ε-periodic grid in ℝn: xk,ε=εk, k∈ℤn. Then we define ℱε as the union of sets ℱkε⊂Kεk obtained from εF by translations with vectors εk, k∈ℤn, that is,
(62)ℱkε=εk+εF,ℱε=⋃k‍(ℱkε∩Ω),
and Kεk=εk+εY. Notice that the geometry of the inclusions having a nontrivial intersection with the domain boundary might be rather complicated. In particular, the extension condition (C.2.2) might be violated for these inclusions. To avoid these technical difficulties, we assume that the domain Ω is not perforated in a small neighborhood of its boundary ∂Ω. We set
(63)Ωε≝Ω∖ℱ¯ε.Let a family of continuous functions{pε}(ε>0) belongs to the class 𝔏p0(·)ε. On the space Lpε(·)(Ωε), we define the functional Jε:Lpε(·)(Ωε)→ℝ∪{+∞}:
(64)Jε[u]={∫Ωε{1pε(x)(|∇u|pε(x)+|u|pε(x))-f(x)u{1pε(x)}}dx,ifu∈W1,pε(·)(Ωε),+∞,otherwise,
where f∈C(Ω¯).We study the asymptotic behavior of the functionalJε and its minimizer as ε→0. To formulate the main result of this section we will introduce some notation. We denote by Ub=Ub(p,y) a minimizer of the following variational problem:
(65)∫Y⋆1p|∇yUb-b|pdy⟶min,u∈Wper1,p(Y⋆),
where Y⋆=Y∖F¯, and b=(b1,b2,…,bn) is a vector in ℝn, and p>1 is a parameter.Ifp≥2, then the solution Ub coincides with a unique solution in Wper1,p(Y⋆) of the following cell problem:
(66)divy(|∇yUb|p-2∇yUb)=0inY⋆,(|∇yUb|p-2∇yUb-b,ν→)=0on∂F,y⟶Ub(y)Y-periodic,
here ν→ is the outward normal to ∂F.The following result holds.Theorem 19.
The sequence of functionals{Jε}(ε>0) defined in (64) Γp0(·)-converges to the functional Jhom:Lp0(·)(Ω)→ℝ∪{+∞} given by
(67)Jhom[u]={∫Ω{T(x,∇u)+ρp0(x)|u|p0(x)-ρf(x)u{ρp0(x)}}dx,ifu∈W1,p0(·)(Ω),+∞,otherwise,
where
(68)ρ=measY⋆,T(x,b)=∫Y⋆1p0(x)|∇yUb(p0(x),y)-b|p0(x)dy.
Moreover, a minimizer uε of the functional (64) converges strongly in the space Lp0(·)(Ωε) to u the minimizer of the homogenized functional (67).Notice that Theorem19 (see [22]) can be proved in two different ways. One of them is to check that under the assumptions of Theorem 19 conditions (C.2.1)–(C.2.3) are satisfied and that the characteristics introduced in conditions (C.2.1) and (C.2.3) coincide with those defined in (68). On the other hand, in the periodic case, the direct Γ-convergence techniques can be applied. This allows us to obtain formula (68) by means of Γ-convergence approach used in periodic homogenization.


## 6.1. Statement of the Problem and Main Results

LetΩ be a bounded Lipschitz domain in ℝn (n≥2). Let {ℱε}(ε>0) be a family of open subsets in Ω; in the sequel ε is a small positive parameter characterizing the microscopic length scale. We assume that the set ℱε consists of Nε (Nε→+∞ as ε→0) small isolated components such that their diameters go to zero as ε→0 and ℱε is distributed in an asymptotically regular way in Ω. We set
(52)Ωε=Ω∖ℱε¯.We assume that a family of continuous functions{pε}(ε>0) belongs to the class 𝔏p0(·)ε. On the space Lpε(·)(Ωε), we define the functional Jε:Lpε(·)(Ωε)→ℝ∪{+∞}:
(53)Jε[u]={∫ΩεFε(x,u,∇u)dx,ifu∈W1,pε(·)(Ωε),+∞,otherwise,
where Fε(x,u,∇u) is defined in (31) and f∈C(Ω¯).We study the asymptotic behavior ofJε and their minimizers as ε→0. The classical periodicity assumption is here substituted by an abstract one covering a variety of concrete behaviors, such as the periodicity, the almost periodicity, and many more besides. For this, we assume that Ωε⊂Ω is a disperse medium; that is, the following assumptions hold: (C.2.1)
the local concentration of the setΩε has a positive continuous limit, that is, the indicator of Ωε converges weakly in L2(Ω) to a continuous positive limit. This implies that there exists a continuous positive function ρ=ρ(x) such that
(54)limh→0limε→0h-nmeas(Khx∩Ωε)=ρ(x),for any open cubeKhx centered at x∈Ω with lengths equal to h>0;(C.2.2)
for anyq∈[p-,p+], there exists a family of extension operators Pqε:W1,q(Ωε)→W1,q(Ω) such that.global: for any uε∈W1,q(Ωε),
(55)∥uε∥Lq(Ω)≤C∥uε∥Lq(Ωε),∥∇uε∥Lq(Ω)≤C∥∇uε∥Lq(Ωε),uniformly inε>0, where uε=Pεuε and uε=uε in Ωε;local: for any h>0 there is ε0(h)>0 such that for all ε<ε0(h), z∈Ω and any function u∈W1,q((z+[-2h,2h]n)∩Ωε), the estimates hold
(56)∥uε∥Lq((z+[-h,h]n)∩Ω)≤C∥uε∥Lq((z+[-2h,2h]n)∩Ωε),∥∇uε∥Lq((z+[-h,h]n)∩Ω)≤C∥∇uε∥Lq((z+[-2h,2h]n)∩Ωε).Remark 14.
Notice that in condition (C.2.2), we require the existence of extension operators only in usual Sobolev spacesW1,q with constant q. In this case, the extension condition is well studied in the mathematical literature (see, e.g., [27, 30–32]). For instance, it holds for a wide class of disperse media (see, for instance, [27]).One more condition is imposed on the local characteristic of the setℱε associated to the functional (53). In order to formulate this condition, we denote by Khz an open cube centered at z∈Ω with edge length h (0<ε≪h≪1) and introduce the functional
(57)cpε(·)ε,h(z,b)=infvε∫Khz∩Ωε{1pε(x)|∇vε|pε(x)∫x{1pε(x)+h-pε(x)-γ|vε-(x-z,b)|pε(x){1pε(x)}}dx,
where γ>0, b∈ℝn, and the infimum is taken over vε∈W1,pε(·)(Khz∩Ωε). Here (·,·) stands for the scalar product in ℝn. We assume that(C.2.3)
there is a continuous, with respect tox∈Ω¯, function T(x,b) and γ=γ0 (0<γ0<p-) such that for any {pε}(ε>0)⊂𝔏p0(·)ε, any x∈Ω and any b∈ℝn,
(58)limh→0lim¯ε→0h-ncpε(·)ε,h(x,b)=limh→0lim_ε→0h-ncpε(·)ε,h(x,b)=T(x,b).Remark 15.
Condition (C.2.3) is always fulfilled for periodic and locally periodic structures.Remark 16.
It is crucial in condition (C.2.3) that the limit functionT(x,b) does not depend on the particular choice of the sequence pε→p0. Notice that this is always the case for periodic and locally periodic perforated media. These media will be considered in detail in the last section of the paper.Now we are in position to formulate the first convergence result of the section.Theorem 17.
Assume that{pε}(ε>0)⊂𝔏p0(·)ε, and let conditions (C.2.1)–(C.2.3) be satisfied. Then the functionals Jε defined in (53), Γp0(·)-converge to the functional Jhom:Lp0(·)(Ω)→ℝ∪{+∞} given by
(59)Jhom[u]={∫Ω{T(x,∇u)+ρ(x)p0(x)|u|p0(x)-ρ(x)f(x)u{ρ(x)p0(x)|u|p0(x)}}dx,ifu∈W1,p0(·)(Ω),+∞,otherwise.Now let us formulate the convergence result for the minimizers of the functionalsJε. Consider the variational problem
(60)Jε[uε]⟶min,uε∈W1,pε(·)(Ωε).
According to [1–3, 6], for each ε>0, problem (60) has a unique solution uε∈W1,pε(·)(Ωε).The following convergence result holds.Theorem 18.
Under the assumptions of Theorem17, the solution uε of the variational problem (60) converges strongly in Lp0(·)(Ωε) to a solution of the problem
(61)Jhom[u]⟶min,u∈W1,p0(·)(Ω).The Scheme of the Proof of Theorems17 and 18 (See [22] for More Details) Is as Follows. The “liminf”-inequality is proved in two steps as in Theorem 7 by introducing an auxiliary functional. The “limsup”-inequality is proved by the arguments similar to those from the proof of Theorem 8.


## 6.2. A Periodic Example

Theorems17 and 18 of Section 6.1 provide sufficient conditions for the existence of the Γ-limit functional (59) and for the convergence of minimizers of the variational problem (60) to the minimizer of the homogenized variational problem (61). It is important to show that the class of functions which satisfy the conditions of these theorems is not empty. The goal of this section is to prove that for periodic and locally periodic media all conditions of the above-mentioned theorems are satisfied and to compute the coefficients of the homogenized functional (59) in terms of solutions of auxiliary cell problems. In fact, conditions (C.2.1) and  (C.2.3) are always satisfied in the periodic case if the boundary of inclusions is regular enough, and that the extension condition (C.2.2) can also be replaced with the assumption on the regularity of the inclusions geometry.LetΩ be a bounded domain in ℝn (n≥2) with sufficiently smooth boundary. We assume that in the periodic cell Y=]0,1[n, there is an obstacle F⋐Y being an open set with a sufficiently smooth boundary ∂F. We assume that this geometry is repeated periodically in the whole ℝn. The geometric structure within the domain Ω is then obtained by intersecting the ε-multiple of this geometry with Ω. Let {xk,ε} be an ε-periodic grid in ℝn: xk,ε=εk, k∈ℤn. Then we define ℱε as the union of sets ℱkε⊂Kεk obtained from εF by translations with vectors εk, k∈ℤn, that is,
(62)ℱkε=εk+εF,ℱε=⋃k‍(ℱkε∩Ω),
and Kεk=εk+εY. Notice that the geometry of the inclusions having a nontrivial intersection with the domain boundary might be rather complicated. In particular, the extension condition (C.2.2) might be violated for these inclusions. To avoid these technical difficulties, we assume that the domain Ω is not perforated in a small neighborhood of its boundary ∂Ω. We set
(63)Ωε≝Ω∖ℱ¯ε.Let a family of continuous functions{pε}(ε>0) belongs to the class 𝔏p0(·)ε. On the space Lpε(·)(Ωε), we define the functional Jε:Lpε(·)(Ωε)→ℝ∪{+∞}:
(64)Jε[u]={∫Ωε{1pε(x)(|∇u|pε(x)+|u|pε(x))-f(x)u{1pε(x)}}dx,ifu∈W1,pε(·)(Ωε),+∞,otherwise,
where f∈C(Ω¯).We study the asymptotic behavior of the functionalJε and its minimizer as ε→0. To formulate the main result of this section we will introduce some notation. We denote by Ub=Ub(p,y) a minimizer of the following variational problem:
(65)∫Y⋆1p|∇yUb-b|pdy⟶min,u∈Wper1,p(Y⋆),
where Y⋆=Y∖F¯, and b=(b1,b2,…,bn) is a vector in ℝn, and p>1 is a parameter.Ifp≥2, then the solution Ub coincides with a unique solution in Wper1,p(Y⋆) of the following cell problem:
(66)divy(|∇yUb|p-2∇yUb)=0inY⋆,(|∇yUb|p-2∇yUb-b,ν→)=0on∂F,y⟶Ub(y)Y-periodic,
here ν→ is the outward normal to ∂F.The following result holds.Theorem 19.
The sequence of functionals{Jε}(ε>0) defined in (64) Γp0(·)-converges to the functional Jhom:Lp0(·)(Ω)→ℝ∪{+∞} given by
(67)Jhom[u]={∫Ω{T(x,∇u)+ρp0(x)|u|p0(x)-ρf(x)u{ρp0(x)}}dx,ifu∈W1,p0(·)(Ω),+∞,otherwise,
where
(68)ρ=measY⋆,T(x,b)=∫Y⋆1p0(x)|∇yUb(p0(x),y)-b|p0(x)dy.
Moreover, a minimizer uε of the functional (64) converges strongly in the space Lp0(·)(Ωε) to u the minimizer of the homogenized functional (67).Notice that Theorem19 (see [22]) can be proved in two different ways. One of them is to check that under the assumptions of Theorem 19 conditions (C.2.1)–(C.2.3) are satisfied and that the characteristics introduced in conditions (C.2.1) and (C.2.3) coincide with those defined in (68). On the other hand, in the periodic case, the direct Γ-convergence techniques can be applied. This allows us to obtain formula (68) by means of Γ-convergence approach used in periodic homogenization.


## 7. Homogenization of Quasilinear Elliptic Equations with Nonstandard Growth in High-Contrast Media

### 7.1. Statement of the Problem and the Main Result

LetΩ=Ωfε∪Ωmε¯ be a bounded domain of ℝn (n≥2) with Lipschitz boundary ∂Ω. Here {Ωmε}(ε>0) is a family of open subsets in Ω. We assume that the set Ωmε is distributed in an asymptotically regular way in Ω; moreover, for the sake of simplicity, we suppose that Ωmε∩∂Ω=∅.Remark 20.
In the framework of the method of local energy characteristics presented in the section, we do not specify the geometrical structure of the setΩmε. Generally speaking, it may consist of Nε (Nε→+∞ as ε→0) small isolated components such that their diameters go to zero as ε→0 or it may be defined as fibres becoming more and more dense as ε→0, such that the diameters of the fibers go to zero as ε→0.We assume that the family of functions{pε}(ε>0) belongs to the class 𝔏p0(·)ε. Let σ∈C(Ω¯) be such that(A.3.4)
there exist two real numbersσ- and σ+ such that the function σ is bounded in the following sense:
(69)0<σ-≡minx∈Ω¯σ(x)≤σ(x)≤maxx∈Ω¯σ(x)≡σ+<minx∈Ω¯p0(x)nn-p0(x)inΩ¯,(A.3.5)
the functionσ satisfies the log-Hölder continuity property.Let us now define the variational problem under consideration. To this end, we consider the functionalJε:W1,pε(·)(Ω)→ℝ∪{+∞},
(70)Jε[u]≝{∫ΩGε(x,u,∇u)dxifu∈W1,pε(·)(Ω),+∞otherwise,
where
(71)Gε(x,u,∇u)≝ϰε(x)|∇u|pε(x)+1σ(x)|u|σ(x)-gε(x)uwithϰε(x)≝Kε(x)pε(x).
Here the function gε is defined by gε(x)=def1fε(x)g(x), g∈C(Ω). We denote by 1kε the characteristic function of the set Ωkε, k=f,m. Kε is a measurable function in Ω such that(K.1)
there exists a real numberk0 such that 0<k0≤Kε(x)≤k0-1 in Ωfε;(K.2)
for anyε>0, there exists a real number kε such that supx∈ΩmεKε(x)=kε>0 and kε→0 as ε→0.We consider the following variational problem:(72)Jε[uε]⟶min,uε∈W01,pε(·)(Ω).
It is known from [1] that for each ε>0, there exists a unique solution uε∈W01,pε(·)(Ω) of the variational problem (72). We aim to study the asymptotic behavior of the family {uε} as ε→0, bearing in mind that the geometry of Ω=Ωfε∪Ω¯mε depends on ε. So, we have to specify this geometry. Most of the papers dealing with homogenization assume that Ω is a periodic repetition of a standard cell. This classical periodicity assumption is here substituted by an abstract one covering a variety of concrete behaviors including periodicity and almost periodicity. We thus make the following assumptions: (C.3.1)
the local concentration of the setΩfε has a positive continuous limit; that is, the indicator of Ωfε converges weakly in L2(Ω) to a continuous positive limit. This implies that there exists a continuous positive function ρ=ρ(x) such that
(73)limh→0limε→0h-nmeas(Khx∩Ωfε)=ρ(x)for any open cubeKhx centered at x∈Ω with lengths equal to h>0;(C.3.2)
for any{pε}(ε>0)⊂𝔏p0(·)ε, there is a constant Cpε≥0 such that if the function pε⋆ is defined by pε⋆(x)=pε(x)-Cpε in Ω, then:(i)
the sequence{pε⋆}(ε>0) belongs to 𝔏p0(·)ε, that is, limε→0Cpε=0;(ii)
there exists a family of extension operatorsPε:W1,pε⋆(·)(Ωfε)→W1,pε⋆(·)(Ω) such that, for any vε∈W1,pε(·)(Ωfε),
(74)Pεvε=vεinΩfε,∥Pεvε∥W1,pε⋆(·)(Ω)≤Φ(∥vε∥W1,pε(·)(Ωfε)),whereΦ=Φ(t) is a strictly monotone continuous function in ℝ+ such that Φ(0)=0 and Φ(t)→+∞ as t→+∞.We also impose several conditions on the local characteristic of the setΩfε and Ωmε associated to the functional (70). Let Khz be an open cube centered at z∈Ω with lengths equal to h (0<ε≪h≪1). We introduce the following functionals:(i)
the functionalcpε(·)ε,h associated to the energy in Ωfε is defined in Ω×ℝn by
(75)cpε(·)ε,h(z;a)≝infvε∫Khz∩Ωfε{ϰε(x)|∇vε(x)|pε(x)+h-pε(x)-γ|vε(x)-(x-z,a)|pε(x)}dx,
for z∈Ω, a∈ℝn, where γ>0 and the infimum is taken over vε∈W1,pε(·)(Khz∩Ωfε).(ii)
The functionalbpε(·)ε,h associated to the energy exchange between Ωfε and Ωmε is defined in Ω×ℝn by
(76)bpε(·)ε,h(z;β)≝infwε∫Khz{ϰε(x)|∇wε|pε(x)+1mε(x)σ(x)|wε|σ(x)+h-pε+h-pε(x)-γ1fε(x)|wε-β|pε(x){1mε(x)σ(x)}}dx,for z∈Ω,β∈ℝ, the infimum being taken over wε∈W1,pε(·)(Khz).We assume that the local characteristics ofΩ are such that(C.3.3)
for anyx∈Ω and any a∈ℝn, there is a continuous function T(x,a) and a real number γ=γ0 (0<γ0<p-) such that, for any {pε}(ε>0)⊂𝔏p0(·)ε,
(77)limh→0lim¯ε→0h-ncpε(·)ε,h(x,a)=limh→0lim_ε→0h-ncpε(·)ε,h(x,a)=T(x,a).(C.3.4)
for anyx∈Ω and any β∈ℝ, there is a continuous function b(x,β) and a real number γ=γ1 (0<γ1<p-) such that, for any {pε}(ε>0)⊂𝔏p0(·)ε,
(78)limh→0lim¯ε→0h-nbpε(·)ε,h(x,β)=limh→0lim_ε→0h-nbpε(·)ε,h(x,β)=b(x,β).Remark 21.
It is crucial in conditions (C.3.3) and (C.3.4) that the limit functionsT(x,b) and b(x,β) do not depend on the particular choice of the sequence {pε}(ε>0)⊂𝔏p0(·)ε. It is proved in [26] that these assumptions are fulfilled for periodic and locally periodic media.Remark 22.
Contrary to the standard growth setting as considered in [20, 32], the local characteristic bpε(·)ε,h(z;β) is not homogeneous with respect to the parameter β. This induces the appearance of a nonlinear function b(x,u) in the homogenized functional (see Theorem 23 below).The main result of the section is the following.Theorem 23.
Letuε be a solution of (72). Assume that pε∈𝔏p0(·)ε and conditions (A.3.4)–(A.3.5), (K.1)-(K.2), and (C.3.1)–(C.3.4) are satisfied. Then uε, solution of the variational problem (72), converges strongly in Lp0(·)(Ωfε) to u, solution of the following variational problem:
(79)Jhom[u]⟶min,u∈W01,p0(·)(Ω),
the homogenized functional Jhom:W01,p0(·)(Ω)→ℝ∪{+∞} being defined by
(80)Jhom[u]≝{∫ΩG0(x,u,∇u)dxifu∈W01,p0(·)(Ω),+∞otherwise,
where
(81)G0(x,u,∇u)≝T(x,∇u)+ρ(x)σ(x)|u|σ(x)G0(x,u,∇u)+b(x,u)-g(x)ρ(x)u.
Moreover, for any smooth function ζ in Ω, we have
(82)limε→0∫Ωmε{1pε(x)|uε|σ(x)-2(u(x)uε-|uε|2)+1σ(x)|uε|σ(x)}ζ(x)dx=∫Ωb(x,u)ζ(x)dx.The scheme of the proof of Theorem23 (see [26] for more details) is similar to the scheme of the proof of Theorems 17 and 18.


### 7.2. Periodic Examples

Theorem23 of Section 7.1 provides sufficient conditions for the existence of the homogenized functional (80) and for the convergence of minimizers of the variational problem (72) to the minimizer of the homogenized variational problem (79) under conditions (A.3.4)-(A.3.5), (K.1)-(K.2), and (C.3.1)–(C.3.4). It is important to show that the “intersection” of these conditions is not empty. The goal of this section is to prove that for periodic media all the conditions of the above-mentioned theorem are satisfied and to compute the coefficients of the homogenized functional (80) either in an explicit form or as usually by the solution of a corresponding cell problem.LetΩ be a bounded domain in ℝn (n≥2) with Lipschitz boundary. We assume that in the standard periodic cell Y≝]0,1[n, there is an obstacle M⊂Y with Lipschitz boundary ∂M. We assume that this geometry is repeated periodically in the whole ℝn. The geometric structure within the domain Ω is then obtained by intersecting the ε-multiple of this geometry with Ω. Let {xi,ε} be an ε-periodic grid in Ω. Then we define Ωmε as the union of sets Miε⊂Kεi (i=1,2,…,Nε) obtained from εM by translations of vectors ε∑j=1nkjej, that is,
(83)Ωmε=⋃iNε‍Miε,Ωfε=Ω∖Ωmε¯,
where Kεi is the cube centered at the point xi,ε and of length ε, kj∈ℤ, {ej}j=1n is the canonical basis of ℝn, and Nε→+∞ as ε→0.Letp0=p0(x) be a log-Hölder continuous function such that
(84)2<p-≡minx∈Ω¯p0(x)≤p0(x)≤maxx∈Ω¯p0(x)≡p+<+∞inΩ¯.
Let {pε}(ε>0)⊂𝔏p0(·)ε be a sequence defined by
(85)pε(x)≝p0(x)+dε(x),
where the function dε is such that dε=o(1) as ε→0. The asymptotic behavior of dε will be specified below. On the space W1,pε(·)(Ω), we define the functional Jε:W1,pε(·)(Ω)→ℝ∪{+∞},
(86)Jε[u]≝{∫Ω{Kε(x)pε(x)|∇u|pε(x)+1σ(x)|u|σ(x)-gε(x)u{Kε(x)pε(x)}}dxifu∈W1,pε(·)(Ω),+∞otherwise,
where
(87)Kε(x)={kfinΩfε,kmεp0(x)inΩmε,
the function σ satisfies conditions (A.3.4) with σ->2, (A.3.5), the function gε is defined by gε(x)≝1fε(x)g(x), g∈C(Ω). Here kf and km are strictly positive constants independent of ε.Consider the following variational problem:(88)Jε[uε]⟶min,uε∈W01,pε(·)(Ω).
We aim to study the asymptotic behavior of uε the solution of (88).To formulate the main result of this section, we introduce some notations. We denote byua=ua(x,y) the unique solution in W#1,p0(·)(F) of the following cell problem:
(89)divy{kf|∇yua|p0(x)-2∇yua}=0inF,(kf|∇yua|p0(x)-2∇yua-a,ν→)=0on∂M,y⟶ua(y)Y-periodic,
where F=Y∖M¯, ν→ is the outward normal vector to ∂M, and a∈ℝn. We denote by wβ=wβ(x,y) the unique solution in W#1,p0(·)(M) of the following cell problem:
(90)-divy{kmd(x)|∇ywβ|p0(x)-2∇ywβ}+|wβ|σ(x)-2wβ=0inM,wβ(y)=βon∂M,y⟶wβ(y)Y-periodic.
Notice that in (89) and (90) x is a parameter. Regularity results for ua and wβ are thus easily deduced from [33] and [34]. We also introduce the homogenized functional Jhom:W1,p0(·)(Ω)→ℝ∪{+∞}:
(91)Jhom[u]≝{∫Ω{T(x,∇u)+ρ⋆σ(x)|u|σ(x)+b(x,u)-g(x)ρ⋆u{ρ⋆σ(x)}}dxifu∈W01,p0(·)(Ω),+∞otherwise,the following results hold.Theorem 24.
Letuε be a solution of (88). Assume that
(92)limε→0ε-dε(·)=d(·)
uniformly in Ω. Then uε converges strongly in Lp0(·)(Ωfε) to u the solution of the variational problem:
(93)Jhom[u]⟶min,u∈W01,p0(·)(Ω),
where
(94)ρ⋆≝measF,(95)T(x,a)≝1p0(x)∫F|∇yua(x,y)-a|p0(x)dy,(96)b(x,β)≝∫ℳ{1p0(x)[βwβ|wβ|σ(x)-2-|wβ|σ(x)]+1σ(x)|wβ|σ(x)}dy.Theorem 25.
Letuε be a solution of (88). Assume that, for any x∈Ω,
(97)limε→0ε-dε(x)=+∞.
Then uε converges strongly in Lp0(·)(Ωfε) to u the solution of the variational problem (93), where ρ⋆ and the function T(x,a) are given in (94), (95), and
(98)b(x,β)≝measMσ(x)|β|σ(x).Theorem 26.
Letuε be a solution of (88). Assume that, for any x∈Ω,
(99)limε→0ε-dε(x)=0.
Then uε converges strongly in Lp0(·)(Ωfε) to u the solution of the variational problem (93), where ρ⋆ and the function T(x,a) are given in (94), (95), and b(x,β)=0.Remark 27.
Notice that ifKε(x)=1fεkf+1mεkmεpε(x) then Theorem 24 holds true with d(x)≡1.


## 7.1. Statement of the Problem and the Main Result

LetΩ=Ωfε∪Ωmε¯ be a bounded domain of ℝn (n≥2) with Lipschitz boundary ∂Ω. Here {Ωmε}(ε>0) is a family of open subsets in Ω. We assume that the set Ωmε is distributed in an asymptotically regular way in Ω; moreover, for the sake of simplicity, we suppose that Ωmε∩∂Ω=∅.Remark 20.
In the framework of the method of local energy characteristics presented in the section, we do not specify the geometrical structure of the setΩmε. Generally speaking, it may consist of Nε (Nε→+∞ as ε→0) small isolated components such that their diameters go to zero as ε→0 or it may be defined as fibres becoming more and more dense as ε→0, such that the diameters of the fibers go to zero as ε→0.We assume that the family of functions{pε}(ε>0) belongs to the class 𝔏p0(·)ε. Let σ∈C(Ω¯) be such that(A.3.4)
there exist two real numbersσ- and σ+ such that the function σ is bounded in the following sense:
(69)0<σ-≡minx∈Ω¯σ(x)≤σ(x)≤maxx∈Ω¯σ(x)≡σ+<minx∈Ω¯p0(x)nn-p0(x)inΩ¯,(A.3.5)
the functionσ satisfies the log-Hölder continuity property.Let us now define the variational problem under consideration. To this end, we consider the functionalJε:W1,pε(·)(Ω)→ℝ∪{+∞},
(70)Jε[u]≝{∫ΩGε(x,u,∇u)dxifu∈W1,pε(·)(Ω),+∞otherwise,
where
(71)Gε(x,u,∇u)≝ϰε(x)|∇u|pε(x)+1σ(x)|u|σ(x)-gε(x)uwithϰε(x)≝Kε(x)pε(x).
Here the function gε is defined by gε(x)=def1fε(x)g(x), g∈C(Ω). We denote by 1kε the characteristic function of the set Ωkε, k=f,m. Kε is a measurable function in Ω such that(K.1)
there exists a real numberk0 such that 0<k0≤Kε(x)≤k0-1 in Ωfε;(K.2)
for anyε>0, there exists a real number kε such that supx∈ΩmεKε(x)=kε>0 and kε→0 as ε→0.We consider the following variational problem:(72)Jε[uε]⟶min,uε∈W01,pε(·)(Ω).
It is known from [1] that for each ε>0, there exists a unique solution uε∈W01,pε(·)(Ω) of the variational problem (72). We aim to study the asymptotic behavior of the family {uε} as ε→0, bearing in mind that the geometry of Ω=Ωfε∪Ω¯mε depends on ε. So, we have to specify this geometry. Most of the papers dealing with homogenization assume that Ω is a periodic repetition of a standard cell. This classical periodicity assumption is here substituted by an abstract one covering a variety of concrete behaviors including periodicity and almost periodicity. We thus make the following assumptions: (C.3.1)
the local concentration of the setΩfε has a positive continuous limit; that is, the indicator of Ωfε converges weakly in L2(Ω) to a continuous positive limit. This implies that there exists a continuous positive function ρ=ρ(x) such that
(73)limh→0limε→0h-nmeas(Khx∩Ωfε)=ρ(x)for any open cubeKhx centered at x∈Ω with lengths equal to h>0;(C.3.2)
for any{pε}(ε>0)⊂𝔏p0(·)ε, there is a constant Cpε≥0 such that if the function pε⋆ is defined by pε⋆(x)=pε(x)-Cpε in Ω, then:(i)
the sequence{pε⋆}(ε>0) belongs to 𝔏p0(·)ε, that is, limε→0Cpε=0;(ii)
there exists a family of extension operatorsPε:W1,pε⋆(·)(Ωfε)→W1,pε⋆(·)(Ω) such that, for any vε∈W1,pε(·)(Ωfε),
(74)Pεvε=vεinΩfε,∥Pεvε∥W1,pε⋆(·)(Ω)≤Φ(∥vε∥W1,pε(·)(Ωfε)),whereΦ=Φ(t) is a strictly monotone continuous function in ℝ+ such that Φ(0)=0 and Φ(t)→+∞ as t→+∞.We also impose several conditions on the local characteristic of the setΩfε and Ωmε associated to the functional (70). Let Khz be an open cube centered at z∈Ω with lengths equal to h (0<ε≪h≪1). We introduce the following functionals:(i)
the functionalcpε(·)ε,h associated to the energy in Ωfε is defined in Ω×ℝn by
(75)cpε(·)ε,h(z;a)≝infvε∫Khz∩Ωfε{ϰε(x)|∇vε(x)|pε(x)+h-pε(x)-γ|vε(x)-(x-z,a)|pε(x)}dx,
for z∈Ω, a∈ℝn, where γ>0 and the infimum is taken over vε∈W1,pε(·)(Khz∩Ωfε).(ii)
The functionalbpε(·)ε,h associated to the energy exchange between Ωfε and Ωmε is defined in Ω×ℝn by
(76)bpε(·)ε,h(z;β)≝infwε∫Khz{ϰε(x)|∇wε|pε(x)+1mε(x)σ(x)|wε|σ(x)+h-pε+h-pε(x)-γ1fε(x)|wε-β|pε(x){1mε(x)σ(x)}}dx,for z∈Ω,β∈ℝ, the infimum being taken over wε∈W1,pε(·)(Khz).We assume that the local characteristics ofΩ are such that(C.3.3)
for anyx∈Ω and any a∈ℝn, there is a continuous function T(x,a) and a real number γ=γ0 (0<γ0<p-) such that, for any {pε}(ε>0)⊂𝔏p0(·)ε,
(77)limh→0lim¯ε→0h-ncpε(·)ε,h(x,a)=limh→0lim_ε→0h-ncpε(·)ε,h(x,a)=T(x,a).(C.3.4)
for anyx∈Ω and any β∈ℝ, there is a continuous function b(x,β) and a real number γ=γ1 (0<γ1<p-) such that, for any {pε}(ε>0)⊂𝔏p0(·)ε,
(78)limh→0lim¯ε→0h-nbpε(·)ε,h(x,β)=limh→0lim_ε→0h-nbpε(·)ε,h(x,β)=b(x,β).Remark 21.
It is crucial in conditions (C.3.3) and (C.3.4) that the limit functionsT(x,b) and b(x,β) do not depend on the particular choice of the sequence {pε}(ε>0)⊂𝔏p0(·)ε. It is proved in [26] that these assumptions are fulfilled for periodic and locally periodic media.Remark 22.
Contrary to the standard growth setting as considered in [20, 32], the local characteristic bpε(·)ε,h(z;β) is not homogeneous with respect to the parameter β. This induces the appearance of a nonlinear function b(x,u) in the homogenized functional (see Theorem 23 below).The main result of the section is the following.Theorem 23.
Letuε be a solution of (72). Assume that pε∈𝔏p0(·)ε and conditions (A.3.4)–(A.3.5), (K.1)-(K.2), and (C.3.1)–(C.3.4) are satisfied. Then uε, solution of the variational problem (72), converges strongly in Lp0(·)(Ωfε) to u, solution of the following variational problem:
(79)Jhom[u]⟶min,u∈W01,p0(·)(Ω),
the homogenized functional Jhom:W01,p0(·)(Ω)→ℝ∪{+∞} being defined by
(80)Jhom[u]≝{∫ΩG0(x,u,∇u)dxifu∈W01,p0(·)(Ω),+∞otherwise,
where
(81)G0(x,u,∇u)≝T(x,∇u)+ρ(x)σ(x)|u|σ(x)G0(x,u,∇u)+b(x,u)-g(x)ρ(x)u.
Moreover, for any smooth function ζ in Ω, we have
(82)limε→0∫Ωmε{1pε(x)|uε|σ(x)-2(u(x)uε-|uε|2)+1σ(x)|uε|σ(x)}ζ(x)dx=∫Ωb(x,u)ζ(x)dx.The scheme of the proof of Theorem23 (see [26] for more details) is similar to the scheme of the proof of Theorems 17 and 18.


## 7.2. Periodic Examples

Theorem23 of Section 7.1 provides sufficient conditions for the existence of the homogenized functional (80) and for the convergence of minimizers of the variational problem (72) to the minimizer of the homogenized variational problem (79) under conditions (A.3.4)-(A.3.5), (K.1)-(K.2), and (C.3.1)–(C.3.4). It is important to show that the “intersection” of these conditions is not empty. The goal of this section is to prove that for periodic media all the conditions of the above-mentioned theorem are satisfied and to compute the coefficients of the homogenized functional (80) either in an explicit form or as usually by the solution of a corresponding cell problem.LetΩ be a bounded domain in ℝn (n≥2) with Lipschitz boundary. We assume that in the standard periodic cell Y≝]0,1[n, there is an obstacle M⊂Y with Lipschitz boundary ∂M. We assume that this geometry is repeated periodically in the whole ℝn. The geometric structure within the domain Ω is then obtained by intersecting the ε-multiple of this geometry with Ω. Let {xi,ε} be an ε-periodic grid in Ω. Then we define Ωmε as the union of sets Miε⊂Kεi (i=1,2,…,Nε) obtained from εM by translations of vectors ε∑j=1nkjej, that is,
(83)Ωmε=⋃iNε‍Miε,Ωfε=Ω∖Ωmε¯,
where Kεi is the cube centered at the point xi,ε and of length ε, kj∈ℤ, {ej}j=1n is the canonical basis of ℝn, and Nε→+∞ as ε→0.Letp0=p0(x) be a log-Hölder continuous function such that
(84)2<p-≡minx∈Ω¯p0(x)≤p0(x)≤maxx∈Ω¯p0(x)≡p+<+∞inΩ¯.
Let {pε}(ε>0)⊂𝔏p0(·)ε be a sequence defined by
(85)pε(x)≝p0(x)+dε(x),
where the function dε is such that dε=o(1) as ε→0. The asymptotic behavior of dε will be specified below. On the space W1,pε(·)(Ω), we define the functional Jε:W1,pε(·)(Ω)→ℝ∪{+∞},
(86)Jε[u]≝{∫Ω{Kε(x)pε(x)|∇u|pε(x)+1σ(x)|u|σ(x)-gε(x)u{Kε(x)pε(x)}}dxifu∈W1,pε(·)(Ω),+∞otherwise,
where
(87)Kε(x)={kfinΩfε,kmεp0(x)inΩmε,
the function σ satisfies conditions (A.3.4) with σ->2, (A.3.5), the function gε is defined by gε(x)≝1fε(x)g(x), g∈C(Ω). Here kf and km are strictly positive constants independent of ε.Consider the following variational problem:(88)Jε[uε]⟶min,uε∈W01,pε(·)(Ω).
We aim to study the asymptotic behavior of uε the solution of (88).To formulate the main result of this section, we introduce some notations. We denote byua=ua(x,y) the unique solution in W#1,p0(·)(F) of the following cell problem:
(89)divy{kf|∇yua|p0(x)-2∇yua}=0inF,(kf|∇yua|p0(x)-2∇yua-a,ν→)=0on∂M,y⟶ua(y)Y-periodic,
where F=Y∖M¯, ν→ is the outward normal vector to ∂M, and a∈ℝn. We denote by wβ=wβ(x,y) the unique solution in W#1,p0(·)(M) of the following cell problem:
(90)-divy{kmd(x)|∇ywβ|p0(x)-2∇ywβ}+|wβ|σ(x)-2wβ=0inM,wβ(y)=βon∂M,y⟶wβ(y)Y-periodic.
Notice that in (89) and (90) x is a parameter. Regularity results for ua and wβ are thus easily deduced from [33] and [34]. We also introduce the homogenized functional Jhom:W1,p0(·)(Ω)→ℝ∪{+∞}:
(91)Jhom[u]≝{∫Ω{T(x,∇u)+ρ⋆σ(x)|u|σ(x)+b(x,u)-g(x)ρ⋆u{ρ⋆σ(x)}}dxifu∈W01,p0(·)(Ω),+∞otherwise,the following results hold.Theorem 24.
Letuε be a solution of (88). Assume that
(92)limε→0ε-dε(·)=d(·)
uniformly in Ω. Then uε converges strongly in Lp0(·)(Ωfε) to u the solution of the variational problem:
(93)Jhom[u]⟶min,u∈W01,p0(·)(Ω),
where
(94)ρ⋆≝measF,(95)T(x,a)≝1p0(x)∫F|∇yua(x,y)-a|p0(x)dy,(96)b(x,β)≝∫ℳ{1p0(x)[βwβ|wβ|σ(x)-2-|wβ|σ(x)]+1σ(x)|wβ|σ(x)}dy.Theorem 25.
Letuε be a solution of (88). Assume that, for any x∈Ω,
(97)limε→0ε-dε(x)=+∞.
Then uε converges strongly in Lp0(·)(Ωfε) to u the solution of the variational problem (93), where ρ⋆ and the function T(x,a) are given in (94), (95), and
(98)b(x,β)≝measMσ(x)|β|σ(x).Theorem 26.
Letuε be a solution of (88). Assume that, for any x∈Ω,
(99)limε→0ε-dε(x)=0.
Then uε converges strongly in Lp0(·)(Ωfε) to u the solution of the variational problem (93), where ρ⋆ and the function T(x,a) are given in (94), (95), and b(x,β)=0.Remark 27.
Notice that ifKε(x)=1fεkf+1mεkmεpε(x) then Theorem 24 holds true with d(x)≡1.


## 8. Homogenization of a Class of Quasilinear Parabolic Equations with Nonstandard Growth

### 8.1. Statement of the Problem and Assumptions

In this section, we describe a mesoscopic double porosity model in a periodic fractured medium. We consider a reservoirΩ⊂ℝn (n≥2) to be a bounded connected domain with a periodic structure. More precisely, we will scale this periodic structure by a parameter ε which represents the ratio of the cell size to the size of the whole region Ω, and we will assume that ε is a parameter tending to zero. Let Y≝]0,1[n represent the mesoscopic domain of the basic cell of a fractured porous medium. For the sake of simplicity and without loss of generality, we assume that Y is made up of two homogeneous porous media M⋐Y and F corresponding to parties of the mesoscopic domain occupied by the matrix block and the fracture, respectively. Thus Y=M∪Γm,f∪F, where Γm,f denotes the interface between the two media and the subscripts m and f refer to the matrix and fracture, respectively. Let Ωiε with i=m or f denotes the open set filled with the porous medium i. Then Ω=Ωmε∪Γm,fε∪Ωfε, where Γm,fε=∂Ωmε∩∂Ωfε. For the sake of simplicity, we will assume that ∂Ω∩Ωmε=∅.Let us introduce the nonstandard growth function used in this section. We assume that a family of continuous functionspε=pε(x), ε>0, is defined in Ω¯ and satisfies the following conditions: (A.4.1)
functionspε are bounded from below such that: pε(x)≥2inΩ¯;(A.4.2)
the functionpε(x) satisfies the log-Hölder continuity property;(A.4.3)
the functionγε(x)≝pε(x)-2 converges uniformly to zero in Ω¯.Now let us introduce the permeability coefficient and the porosity of the porous mediumΩ. We set
(100)kε(x)=kmε21mε(x)+kf1fε(x),ωε(x)=ωm1mε(x)+ωf1fε(x),
where kf is the permeability or the hydraulic conductivity of fissures, km is the permeability or the hydraulic conductivity of blocks, ωf is the porosity of fissures, and ωm is the porosity of blocks; 1fε=1fε(x) and 1mε=1mε(x) denote the characteristic functions of the sets Ωfε and Ωmε, respectively. Here 0<kf,km,ωf,ωm<+∞.We consider the following initial boundary value problem for the functionuε:Q↦ℝ:
(101)ωε(x)∂uε∂t-div(kε(x)∇uε|∇uε|pε(x)-2)=g(t,x)inQ,uε=0on(0,T)×∂Ω,uε(0,x)=u0(x)inΩ,
where Q denotes the cylinder (0,T)×Ω, T>0 is given, and g,u0 are given functions.For simplicity and without loss of generality, we restrict the presentation to a homogeneous Dirichlet boundary condition on∂Ω, but it is easy to see that all results also hold for other boundary conditions.Throughout the section,C will denote a generic positive constant, independent of ε, and may take different values for different occurrences.


### 8.2. Preliminary Results

The goal of this section is to obtaina priori estimates for the solution uε of problem (101). We start by formulating the existence and uniqueness result for (101). It is given by the following theorem (see [35]).Theorem 28.
Letg∈C(0,T;L2(Ω)) and u0∈H2(Ω). Then, for any ε>0, there exists a unique solution uε=uε(t,x) of the boundary value problem (101) in the space L∞(0,T;W1,pε(·)(Ω)). Furthermore, this solution satisfies the following a priori estimates, for a.e. t∈(0,T):
(102)∥uε(t)∥L2(Ω)2+∫0t‍dt∫Ω|∂uε∂t|2dx+∫Ωkε(x)|∇uε|pε(x)dx≤C,(103)∥uε(t+δt)-uε(t)∥L2(Ω)≤C(δt)κwith0<κ<1,
where δt>0 is a time step which tends to zero.We study the asymptotic behavior of the solutionuε of problem (101) as ε→0. For this it is convenient to introduce the following notation:
(104)uε={ρεinΩfε,σεinΩmε,
and to rewrite (101) separately in the domains Ωfε,Ωmε with appropriate interface conditions. Namely, in the domain Ωfε (101) reads
(105)ωf∂ρε∂t-div(kf∇ρε|∇ρε|pε(x)-2)=g(t,x)in(0,T)×Ωfε,kf∇ρε|∇ρε|pε(x)-2·ν→=kmεpε(x)∇σε|∇σε|pε(x)-2·ν→on(0,T)×Γm,fε,ρε=0on(0,T)×∂Ω,ρε(0,x)=u0(x)inΩfε,
where ν→ is the outward normal vector to Γmfε. In the domain Ωmε (101) reads
(106)ωm∂σε∂t-div(kmε2∇σε|∇σε|pε(x)-2)=g(t,x)in(0,T)×Ωmε,σε=ρεon(0,T)×Γmfε,σε(0,x)=u0(x)inΩmε.To establish some preliminary compactness result, first we notice that thea priori estimate (102), conditions (A.4.1), (A.4.3) along with (110), and inequalities (106), and (107) imply the bound for a.e. t∈(0,T):
(107)∥uε(t)∥L2(Ω)+∫0t‍dt∫Ω|∂uε∂t|2dx+∫Ωfε|∇uε|2dx+ε2∫Ωmε|∇uε|2dx≤C.
Therefore, from (105), (106) and (107), we have for a.e. t∈(0,T):
(108)∥ρε(t)∥L2(Ωfε)+∫0t‍dt∫Ωfε|∂ρε∂t|2dx+∫Ωfε|∇ρε|2dx≤C,∥σε(t)∥L2(Ωmε)+∫0t‍dt∫Ωmε|∂σε∂t|2dx+ε2∫Ωmε|∇σε|2dx≤C.Now using the bounds (108), along with the extension result [30], it is easy to prove the following compactness result.Lemma 29.
Letuε=〈ρε,σε〉 be the solution of problem (101). Then there exists a subsequence, still denoted by {uε}ε>0, and functions uf=uf(t,x),vf=vf(x,y),um=um(t,x,y) such that(a.1)
uf∈H1(0,T;L2(Ω))∩L∞(0,T;H1(Ω)), vf∈L2(Ω;H#1(F)∖ℝ),(a.2)
um∈H1(0,T;L2(Ω×Y))∩L∞(0,T;L2(Ω;H1(M))), with um(t,x,y)=uf(t,x) for y∈∂M,(b)
for anyt∈(0,T), ρε⇀2suf and σε⇀2sum,(c)
for anyφ∈L∞(0,T;L2(Ω;C#(Y))),
(109)∫0T‍dt∫Ωfε∂uε∂tφ(t,x,xε)dx⟶∫0T‍dt∫Ω×F∂uf∂tφ(t,x,y)dxdy,∫0T‍dt∫Ωmε∂uε∂tφ(t,x,xε)dx⟶∫0T‍dt∫Ω×M∂um∂tφ(t,x,y)dxdy,(d)
for a.e.t∈(0,T), ∇ρε⇀2s(∇xuf+∇yvf)1f(y) and ∇σε⇀2s∇yum1m(y).


### 8.3. Homogenization Results

In this section, we formulate the main results of the section. We give homogenization results for the problem (101). The convergence of the homogenization process is obtained by combining the technique of two-scale convergence and the variational homogenization method (see, e.g., [9, 16, 27] and the references therein).The idea of the proof is the following. First we will reduce our parabolic problem to an elliptic one depending on the time variable as a parameter. Then we introduce a functional corresponding to this elliptic problem and study the minimization problem for it in the limit of smallε. Then we obtain the limit functional corresponding to the homogenized problem. Regarding the variational technique, it is worth to mention one trick used in the paper. In order to obtain the lower bound for the original functional, we first replace the original exponent pε(x) by a new one p0=2 and consider the corresponding family of auxiliary functionals. Then the lower semicontinuity property of convex functionals with respect to the two-scale convergence implies the desired inequality. Finally, it is not difficult to show that the limit functional for the auxiliary family does not exceed the limiting functional for the original one.Now we are in position to formulate the first homogenization result of the section.Theorem 30.
Letuε=uε(t,x) be the solution of the boundary value problem (101) and let conditions (A.4.1)–(A.4.3) be satisfied. Moreover, we assume that there exists a function α∈C(Ω¯) such that uniformly in x∈Ω¯,
(110)limε→0ε-γε(x)=α(x).
Then, for a.e. t∈(0,T), uε two-scale converges to u*∈L2((0,T)×Ω×Y) such that
(111)u*(t,x,y)={uf(t,x)inQ×F,um(t,x,y)inQ×M,
where the couple (uf,um)∈L2(0,T;H1(Ω))×L∞(0,T;L2(Ω;H1(M))) is the unique solution of the homogenized problem
(112)ωf|F|∂uf∂t-divx(K*∇xuf)=S(x,um)inQ,uf=0on(0,T)×∂Ω,uf(0,x)=u0(x)inΩ,ωm∂um∂t-K~*(x)Δyum=g(t,x)inQ×M,um(t,x,y)=uf(t,x)onQ×∂M,um(0,x,y)=u0(x)inΩ×M,
where |F| is the measure of the set F and K*={kij*} is the homogenized permeability tensor defined by
(113)kij*=kf∫F(e→i+∇ywi)·(e→j+∇ywj)dy
with {e→1,e→2,…,e→n} the canonical basis of ℝn and wi being the unique solution in H#1(F)∖ℝ of
(114)-kfΔywi=0inF,(e→i+∇ywi)·ν→=0on∂M,y⟶wi(x,y)Y-periodic,
where ν→=ν→(y) is the outer normal vector at ∂M, and the coefficient K~* in the local problem is given by
(115)K~*(x)=α(x)km,
and the effective source term S(x,um) is given by
(116)S(x,um)=|F|g(t,x)-K~*(x)∫∂M(∇yum·ν→)dsy.Remark 31.
The source term which appears in the right-hand side of the first equation in (112) is well defined, since um∈L∞(0,T;L2(Ω;H1(M))), and it follows from the third equation of (112) that
(117)-Δyum∈L2(0,T;H-1/2(∂M)),
which allows one to define (∇yum·ν→) as an element of L2(0,T;H-1/2(∂M)).The Scheme of the Proof of Theorem30 Is as Follows.We consider our parabolic boundary value problem (101) as an elliptic one depending on the time variable t as a parameter. Namely, we consider the following boundary value problem, for a.e. t∈]0,T[,
(118)-div(kε(x)∇uε|∇uε|pε(x)-2)=GεinQ,uε=0on]0,T[×∂Ω,
where the function Gε,
(119)Gε=Gε(t,x)≝g(t,x)-ωε(x)∂uε∂t(t,x),
is considered as a given function. Then, for any Δt⊂[0,T], uε minimizes the functional
(120)Jε[u]≝∫Δtdt∫Ω{kε(x)pε(x)|∇u|pε(x)-Gεu}dx
over u∈L∞(0,T;W1,pε(·)(Ω)). Using the two-scale convergence arguments, we obtain the “limsup”-inequality for the functional Jε. The “liminf”-inequality is obtained in two steps. First, we introduce the auxiliary functional
(121)J~ε[u]≝∫Δtdt∫Ω{k~ε(x)pε(x)|∇u|2-Gεu}dxwithk~ε(x)=α(x)kmε21mε(x)+kf1fε(x)
and obtain the inequality
(122)lim_ε→0J~ε[uε]≥Jhom[uf,um],
where
(123)Jhom[uf,um]=∫Δtdt∫Ω{{-|F|(g(t,x)-ωf∂uf∂t)uf}12(K*∇xuf·∇xuf)=∫Ω{12(·∇xuf)-|F|(g(t,x)-ωf∂uf∂t)uf}dx+∫Δtdt∫Ω×M{α(x)km2|∇yum|2+∫Δtdt-(g(t,x)-ωm∂um∂t)um}dxdy.Then in the second step, we obtain the desired “liminf”-inequality for the initial functional. This completes the proof of Theorem 30.The macroscopic model corresponding to the second situation is given by the following convergence result.Theorem 32.
Letuε=〈ρε,σε〉 be the solution of the boundary value problem (101) and let conditions (A.4.1)–(A.4.3) be satisfied. Moreover, we assume that for any x∈Ω¯,
(124)limε→0ε-γε(x)=+∞.
Then, for a.e. t∈(0,T), uε converges in L2(Ω), as ε→0, to u0∈L2(0,T;H1(Ω)), the solution of
(125)(ωf|F|+ωm|M|)∂u0∂t-divx(K*∇xu0)=g(t,x)inQ,u0=0on(0,T)×∂Ω,u0(0,x)=u0(x)inΩ,
where K*={kij*} is the homogenized permeability tensor defined in (113)-(114).The proof of Theorem32 is similar to that of Theorem 30.Remark 33.
Notice that the structure of the limit problem depends crucially on the rate of convergence of(pε(·)-2) to zero. The critical rate of convergence is
(126)(pε(·)-2)~1|lnε|.
More precisely, if lim¯ε→0|lnε|(pε(x)-2)<+∞, then the limit model is of a double porosity type. If limε→0|lnε|(pε(x)-2)=+∞, then in the limit we obtain a single porosity model.The proofs of Theorems30 and 32 are given in [21].


## 8.1. Statement of the Problem and Assumptions

In this section, we describe a mesoscopic double porosity model in a periodic fractured medium. We consider a reservoirΩ⊂ℝn (n≥2) to be a bounded connected domain with a periodic structure. More precisely, we will scale this periodic structure by a parameter ε which represents the ratio of the cell size to the size of the whole region Ω, and we will assume that ε is a parameter tending to zero. Let Y≝]0,1[n represent the mesoscopic domain of the basic cell of a fractured porous medium. For the sake of simplicity and without loss of generality, we assume that Y is made up of two homogeneous porous media M⋐Y and F corresponding to parties of the mesoscopic domain occupied by the matrix block and the fracture, respectively. Thus Y=M∪Γm,f∪F, where Γm,f denotes the interface between the two media and the subscripts m and f refer to the matrix and fracture, respectively. Let Ωiε with i=m or f denotes the open set filled with the porous medium i. Then Ω=Ωmε∪Γm,fε∪Ωfε, where Γm,fε=∂Ωmε∩∂Ωfε. For the sake of simplicity, we will assume that ∂Ω∩Ωmε=∅.Let us introduce the nonstandard growth function used in this section. We assume that a family of continuous functionspε=pε(x), ε>0, is defined in Ω¯ and satisfies the following conditions: (A.4.1)
functionspε are bounded from below such that: pε(x)≥2inΩ¯;(A.4.2)
the functionpε(x) satisfies the log-Hölder continuity property;(A.4.3)
the functionγε(x)≝pε(x)-2 converges uniformly to zero in Ω¯.Now let us introduce the permeability coefficient and the porosity of the porous mediumΩ. We set
(100)kε(x)=kmε21mε(x)+kf1fε(x),ωε(x)=ωm1mε(x)+ωf1fε(x),
where kf is the permeability or the hydraulic conductivity of fissures, km is the permeability or the hydraulic conductivity of blocks, ωf is the porosity of fissures, and ωm is the porosity of blocks; 1fε=1fε(x) and 1mε=1mε(x) denote the characteristic functions of the sets Ωfε and Ωmε, respectively. Here 0<kf,km,ωf,ωm<+∞.We consider the following initial boundary value problem for the functionuε:Q↦ℝ:
(101)ωε(x)∂uε∂t-div(kε(x)∇uε|∇uε|pε(x)-2)=g(t,x)inQ,uε=0on(0,T)×∂Ω,uε(0,x)=u0(x)inΩ,
where Q denotes the cylinder (0,T)×Ω, T>0 is given, and g,u0 are given functions.For simplicity and without loss of generality, we restrict the presentation to a homogeneous Dirichlet boundary condition on∂Ω, but it is easy to see that all results also hold for other boundary conditions.Throughout the section,C will denote a generic positive constant, independent of ε, and may take different values for different occurrences.


## 8.2. Preliminary Results

The goal of this section is to obtaina priori estimates for the solution uε of problem (101). We start by formulating the existence and uniqueness result for (101). It is given by the following theorem (see [35]).Theorem 28.
Letg∈C(0,T;L2(Ω)) and u0∈H2(Ω). Then, for any ε>0, there exists a unique solution uε=uε(t,x) of the boundary value problem (101) in the space L∞(0,T;W1,pε(·)(Ω)). Furthermore, this solution satisfies the following a priori estimates, for a.e. t∈(0,T):
(102)∥uε(t)∥L2(Ω)2+∫0t‍dt∫Ω|∂uε∂t|2dx+∫Ωkε(x)|∇uε|pε(x)dx≤C,(103)∥uε(t+δt)-uε(t)∥L2(Ω)≤C(δt)κwith0<κ<1,
where δt>0 is a time step which tends to zero.We study the asymptotic behavior of the solutionuε of problem (101) as ε→0. For this it is convenient to introduce the following notation:
(104)uε={ρεinΩfε,σεinΩmε,
and to rewrite (101) separately in the domains Ωfε,Ωmε with appropriate interface conditions. Namely, in the domain Ωfε (101) reads
(105)ωf∂ρε∂t-div(kf∇ρε|∇ρε|pε(x)-2)=g(t,x)in(0,T)×Ωfε,kf∇ρε|∇ρε|pε(x)-2·ν→=kmεpε(x)∇σε|∇σε|pε(x)-2·ν→on(0,T)×Γm,fε,ρε=0on(0,T)×∂Ω,ρε(0,x)=u0(x)inΩfε,
where ν→ is the outward normal vector to Γmfε. In the domain Ωmε (101) reads
(106)ωm∂σε∂t-div(kmε2∇σε|∇σε|pε(x)-2)=g(t,x)in(0,T)×Ωmε,σε=ρεon(0,T)×Γmfε,σε(0,x)=u0(x)inΩmε.To establish some preliminary compactness result, first we notice that thea priori estimate (102), conditions (A.4.1), (A.4.3) along with (110), and inequalities (106), and (107) imply the bound for a.e. t∈(0,T):
(107)∥uε(t)∥L2(Ω)+∫0t‍dt∫Ω|∂uε∂t|2dx+∫Ωfε|∇uε|2dx+ε2∫Ωmε|∇uε|2dx≤C.
Therefore, from (105), (106) and (107), we have for a.e. t∈(0,T):
(108)∥ρε(t)∥L2(Ωfε)+∫0t‍dt∫Ωfε|∂ρε∂t|2dx+∫Ωfε|∇ρε|2dx≤C,∥σε(t)∥L2(Ωmε)+∫0t‍dt∫Ωmε|∂σε∂t|2dx+ε2∫Ωmε|∇σε|2dx≤C.Now using the bounds (108), along with the extension result [30], it is easy to prove the following compactness result.Lemma 29.
Letuε=〈ρε,σε〉 be the solution of problem (101). Then there exists a subsequence, still denoted by {uε}ε>0, and functions uf=uf(t,x),vf=vf(x,y),um=um(t,x,y) such that(a.1)
uf∈H1(0,T;L2(Ω))∩L∞(0,T;H1(Ω)), vf∈L2(Ω;H#1(F)∖ℝ),(a.2)
um∈H1(0,T;L2(Ω×Y))∩L∞(0,T;L2(Ω;H1(M))), with um(t,x,y)=uf(t,x) for y∈∂M,(b)
for anyt∈(0,T), ρε⇀2suf and σε⇀2sum,(c)
for anyφ∈L∞(0,T;L2(Ω;C#(Y))),
(109)∫0T‍dt∫Ωfε∂uε∂tφ(t,x,xε)dx⟶∫0T‍dt∫Ω×F∂uf∂tφ(t,x,y)dxdy,∫0T‍dt∫Ωmε∂uε∂tφ(t,x,xε)dx⟶∫0T‍dt∫Ω×M∂um∂tφ(t,x,y)dxdy,(d)
for a.e.t∈(0,T), ∇ρε⇀2s(∇xuf+∇yvf)1f(y) and ∇σε⇀2s∇yum1m(y).


## 8.3. Homogenization Results

In this section, we formulate the main results of the section. We give homogenization results for the problem (101). The convergence of the homogenization process is obtained by combining the technique of two-scale convergence and the variational homogenization method (see, e.g., [9, 16, 27] and the references therein).The idea of the proof is the following. First we will reduce our parabolic problem to an elliptic one depending on the time variable as a parameter. Then we introduce a functional corresponding to this elliptic problem and study the minimization problem for it in the limit of smallε. Then we obtain the limit functional corresponding to the homogenized problem. Regarding the variational technique, it is worth to mention one trick used in the paper. In order to obtain the lower bound for the original functional, we first replace the original exponent pε(x) by a new one p0=2 and consider the corresponding family of auxiliary functionals. Then the lower semicontinuity property of convex functionals with respect to the two-scale convergence implies the desired inequality. Finally, it is not difficult to show that the limit functional for the auxiliary family does not exceed the limiting functional for the original one.Now we are in position to formulate the first homogenization result of the section.Theorem 30.
Letuε=uε(t,x) be the solution of the boundary value problem (101) and let conditions (A.4.1)–(A.4.3) be satisfied. Moreover, we assume that there exists a function α∈C(Ω¯) such that uniformly in x∈Ω¯,
(110)limε→0ε-γε(x)=α(x).
Then, for a.e. t∈(0,T), uε two-scale converges to u*∈L2((0,T)×Ω×Y) such that
(111)u*(t,x,y)={uf(t,x)inQ×F,um(t,x,y)inQ×M,
where the couple (uf,um)∈L2(0,T;H1(Ω))×L∞(0,T;L2(Ω;H1(M))) is the unique solution of the homogenized problem
(112)ωf|F|∂uf∂t-divx(K*∇xuf)=S(x,um)inQ,uf=0on(0,T)×∂Ω,uf(0,x)=u0(x)inΩ,ωm∂um∂t-K~*(x)Δyum=g(t,x)inQ×M,um(t,x,y)=uf(t,x)onQ×∂M,um(0,x,y)=u0(x)inΩ×M,
where |F| is the measure of the set F and K*={kij*} is the homogenized permeability tensor defined by
(113)kij*=kf∫F(e→i+∇ywi)·(e→j+∇ywj)dy
with {e→1,e→2,…,e→n} the canonical basis of ℝn and wi being the unique solution in H#1(F)∖ℝ of
(114)-kfΔywi=0inF,(e→i+∇ywi)·ν→=0on∂M,y⟶wi(x,y)Y-periodic,
where ν→=ν→(y) is the outer normal vector at ∂M, and the coefficient K~* in the local problem is given by
(115)K~*(x)=α(x)km,
and the effective source term S(x,um) is given by
(116)S(x,um)=|F|g(t,x)-K~*(x)∫∂M(∇yum·ν→)dsy.Remark 31.
The source term which appears in the right-hand side of the first equation in (112) is well defined, since um∈L∞(0,T;L2(Ω;H1(M))), and it follows from the third equation of (112) that
(117)-Δyum∈L2(0,T;H-1/2(∂M)),
which allows one to define (∇yum·ν→) as an element of L2(0,T;H-1/2(∂M)).The Scheme of the Proof of Theorem30 Is as Follows.We consider our parabolic boundary value problem (101) as an elliptic one depending on the time variable t as a parameter. Namely, we consider the following boundary value problem, for a.e. t∈]0,T[,
(118)-div(kε(x)∇uε|∇uε|pε(x)-2)=GεinQ,uε=0on]0,T[×∂Ω,
where the function Gε,
(119)Gε=Gε(t,x)≝g(t,x)-ωε(x)∂uε∂t(t,x),
is considered as a given function. Then, for any Δt⊂[0,T], uε minimizes the functional
(120)Jε[u]≝∫Δtdt∫Ω{kε(x)pε(x)|∇u|pε(x)-Gεu}dx
over u∈L∞(0,T;W1,pε(·)(Ω)). Using the two-scale convergence arguments, we obtain the “limsup”-inequality for the functional Jε. The “liminf”-inequality is obtained in two steps. First, we introduce the auxiliary functional
(121)J~ε[u]≝∫Δtdt∫Ω{k~ε(x)pε(x)|∇u|2-Gεu}dxwithk~ε(x)=α(x)kmε21mε(x)+kf1fε(x)
and obtain the inequality
(122)lim_ε→0J~ε[uε]≥Jhom[uf,um],
where
(123)Jhom[uf,um]=∫Δtdt∫Ω{{-|F|(g(t,x)-ωf∂uf∂t)uf}12(K*∇xuf·∇xuf)=∫Ω{12(·∇xuf)-|F|(g(t,x)-ωf∂uf∂t)uf}dx+∫Δtdt∫Ω×M{α(x)km2|∇yum|2+∫Δtdt-(g(t,x)-ωm∂um∂t)um}dxdy.Then in the second step, we obtain the desired “liminf”-inequality for the initial functional. This completes the proof of Theorem 30.The macroscopic model corresponding to the second situation is given by the following convergence result.Theorem 32.
Letuε=〈ρε,σε〉 be the solution of the boundary value problem (101) and let conditions (A.4.1)–(A.4.3) be satisfied. Moreover, we assume that for any x∈Ω¯,
(124)limε→0ε-γε(x)=+∞.
Then, for a.e. t∈(0,T), uε converges in L2(Ω), as ε→0, to u0∈L2(0,T;H1(Ω)), the solution of
(125)(ωf|F|+ωm|M|)∂u0∂t-divx(K*∇xu0)=g(t,x)inQ,u0=0on(0,T)×∂Ω,u0(0,x)=u0(x)inΩ,
where K*={kij*} is the homogenized permeability tensor defined in (113)-(114).The proof of Theorem32 is similar to that of Theorem 30.Remark 33.
Notice that the structure of the limit problem depends crucially on the rate of convergence of(pε(·)-2) to zero. The critical rate of convergence is
(126)(pε(·)-2)~1|lnε|.
More precisely, if lim¯ε→0|lnε|(pε(x)-2)<+∞, then the limit model is of a double porosity type. If limε→0|lnε|(pε(x)-2)=+∞, then in the limit we obtain a single porosity model.The proofs of Theorems30 and 32 are given in [21].



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*Source: 693529-2013-04-15.xml*