Homogenization of <span class="mathjax-formula">$p$</span>-Laplacian in Perforated Domain

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Homogenization of p-Laplacian in perforated domain

B. Amaziane

^a,^∗

, S. Antontsev

^b

, L. Pankratov

^a,c

, A. Piatnitski

^d,e

aLaboratoire de Mathématiques et leur Applications, CNRS-UMR5142, Université de Pau, Av. de l’Université, 64000 Pau, France bCMAF, Universidade de Lisboa, Av. Prof. Gama Pinto, 2, 1649-003, Lisboa, Portugal

cDepartment of Mathematics, B.Verkin Institut for Low Temperature Physics and Engineering, 47, av. Lenin, 61103, Kharkov, Ukraine dNarvik University College, Postbox 385, Narvik, 8505, Norway

eLebedev Physical Institute RAS, leninski prospect 53, Moscow, 119991, Russia Received 4 February 2007; accepted 10 June 2009

Available online 4 July 2009

Abstract

We study the homogenization of the following nonlinear Dirichlet variational problem:

inf Ω^ε

1

p_ε(x)|∇u|^p^ε^(x)+ 1

p_ε(x)|u|^p^ε^(x)−f (x)u

dx:u∈W₀^1,p^ε⁽^·⁾ Ω^ε

in a perforated domainΩ^ε=Ω\F^ε⊂Rⁿ, n2, where εis a small positive parameter that characterizes the scale of the microstructure. The non-standard exponentp_ε(x)is assumed to be an oscillating continuous function inΩ¯ such that, for any ε >0, 1< pε(x)ninΩ; for anyx, y∈Ω,|pε(x)−pε(y)|ωε(|x−y|)with lim_τ_→₀ωε(τ )ln(1/τ )=0; and converges uniformly inΩto a functionp₀which satisfies the same properties. Moreover, we assume thatp_ε(x)p₀(x)inΩ. Denotingu^ε a minimizer in the above variational problem, without any periodicity assumption, for a large range of perforated domains we find, by means of the variational homogenization technique, the global behavior ofu^εasεtends to zero. It is shown thatu^εextended by zero inF^ε, converges weakly inW^1,p⁰⁽^·⁾(Ω)to the solution of the following nonlinear variational problem:

min Ω

1

p₀(x)|∇u|^p⁰^(x)+ 1

p₀(x)|u|^p⁰^(x)+c(x, u)−f (x)u

dx:u∈W₀^1,p⁰⁽^·⁾(Ω)

,

where the functionc(x, u)is defined in terms of the local characteristic ofΩ^ε. This result is then illustrated with a periodic and a non-periodic examples.

Résumé

Nous étudions l’homogénéisation du problème variationnel de Dirichlet nonlinéaire suivant : inf

Ω^ε 1

p_ε(x)|∇u|^p^ε^(x)+ 1

p_ε(x)|u|^p^ε^(x)−f (x)u

dx:u∈W₀^1,p^ε⁽^·⁾ Ω^ε

* Corresponding author.

E-mail addresses:brahim.amaziane@univ-pau.fr (B. Amaziane), antontsevsn@mail.ru (S. Antontsev), leonid.pankratov@univ-pau.fr (L. Pankratov), andrey@sci.lebedev.ru (A. Piatnitski).

doi:10.1016/j.anihpc.2009.06.004

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dans un domaine perforéΩ^ε=Ω\F^ε⊂Rⁿ,n2, oùε >0 est un petit paramètre qui caractérise la taille des perforations.

La fonction puissancepε(x)est nonstandard et supposée être une fonction continue et oscillante dansΩ. Elle vérifie, pour tout¯ ε >0, 1< p_ε(x)ndansΩ, pour toutx, y∈Ω,|p_ε(x)−p_ε(y)|ω_ε(|x−y|)avec lim_τ_→₀ω_ε(τ )ln(1/τ )=0 ; et elle est uniformément convergente dansΩvers une fonctionp₀qui vérifie les mêmes propriétés. De plus, on suppose quepε(x)p₀(x) dansΩ. On noteu^ε une solution du problème de minimisation variationnel ci-dessus, sans hypothèse de périodicité et pour différents milieux perforés, on trouve le problème limite décrivant le comportement global deu^ε lorsqueε tend vers zéro, en utilisant la technique de l’homogénéisation variationnelle. On montre queu^ε, prolongée par zéro dansF^ε, converge faiblement dansW^1,p⁰⁽^·⁾(Ω), quandεtend vers zéro, vers la solutionudu problème variationel nonlinéaire suivant :

min Ω

1

p₀(x)|∇u|^p⁰^(x)+ 1

p₀(x)|u|^p⁰^(x)+c(x, u)−f (x)u

dx:u∈W₀^1,p⁰⁽^·⁾(Ω)

,

où la fonctionc(x, u)est définie à partir des caractéristiques géométriques locales du domaineΩ^ε. Enfin, nous présentons deux exemples, un périodique et l’autre nonpériodique, pour illustrer les résultats obtenus.

MSC:35B40; 35J60; 46E35; 74Q05; 76M50

Keywords:Homogenization; Nonlinear; Non-standard growth

1. Introduction

In this paper we study the homogenization of the following nonlinear problem:

−div∇u^ε^p^ε^(x)⁻²∇u^ε

+u^ε^p^ε^(x)⁻²u^ε=f (x) inΩ^ε, u^ε∈W₀^1,p^ε⁽^·⁾ Ω^ε

, (1.1)

whereεis a small positive parameter,Ω^ε=Ω\F^ε is a perforated domain inRⁿ (n2) withΩ being a bounded Lipschitz domain, and pε is a smooth positive oscillating function in Ω satisfying some conditions which will be specified in Section 3, and uniformly converging inΩ to a smooth functionp₀.f is a given function. Equations of such type are calledp_ε(x)-Laplacian equations with non-standard growth conditions.

In recent years, there has been an increasing interest in the study of such equations (in the case where there is no dependence on the small parameter) motivated by their applications to the mathematical modeling in continuum mechanics. These equations arise, for example, from the modeling of non-Newtonian fluids with thermo-convective effects (see, e.g., [7,9]), the modeling of electro-rheological fluids (see, e.g., [30,31]), the thermistor problem (see, e.g., [39]), the problem of image recovery (see, e.g., [24]), and the motion of a compressible fluid in a heterogeneous anisotropic porous medium obeying to the nonlinear Darcy law (see, e.g., [8,11]).

Eq. (1.1) is an idealized model for a variety of interesting physical problems; we motivate our work by describing one of them. We consider a steady flow of a compressible barotropic gaz through a porous medium. The nonlinear Darcy law with the continuity equation lead to the equation given by [10]

−div

K(x)|∇u|^p(x)⁻²∇u

+R(x)|u|^p(x)⁻²u=f (x, t ). (1.2)

ustands for the fluid pressure,f is a source term andK,p,Rare characteristic functions of the heterogeneous porous medium. For more details on the formulation of such problems see for instance [10,13]. We refer to [10,11,17,18] and the references therein for a detailed analysis of such equations.

In the present paper we deal with the Dirichlet boundary value problem for the nonlinear equation (1.1). More precisely, we consider the corresponding variational problem:

inf

Ω^ε

1

p_ε(x)|∇u|^p^ε^(x)+ 1

p_ε(x)|u|^p^ε^(x)−f (x)u

dx: u∈W₀^1,p^ε⁽^·⁾ Ω^ε

. (1.3)

The homogenization of the Dirichlet boundary value problem was studied for the first time in [25] and then it was revisited by many authors (see, e.g., [12,15,16,20,26,33], and the references therein). Note also that the homogenization of nonlinear elliptic equations is a long-standing problem and a number of methods have been developed. There is an extensive literature on this subject. We will not attempt a review of the literature here, but merely mention a few references, see for instance [2,14,16,29], and the references therein. Let us mention that the homogenization problems for

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the Lagrangians with variable exponents were first studied in [22,34–37] (see also the book [38]) which focus on the variational functionals with non-standard growth conditions. In particular, the homogenization and Γ-convergence problems for Lagrangians with variable rapidly oscillating exponentsp(x)were considered in [35,36]. Variational functionals with non-standard growth conditions have also been considered in the book [14], namely Chapter 21 of this book focuses on theΓ-convergence of such functionals inL^p spaces. The Dirichlet homogenization problem and related questions for Lagrangians ofp_ε(x)growth inW^1,p^ε⁽^·⁾(Ω^ε), whereΩ^ε is a perforated domain, have been studied recently in [3–6].

Following the approach developed in [20], instead of a classical periodicity assumption on the structure of the perforated domainΩ^ε, we impose certain conditions on the so-calledlocal energy characteristicsassociated with the boundary value problem (1.1). It will be shown that the asymptotic behavior, asε→0, of the solutionu^εis described by the following variational problem:

inf

Ω

1

p₀(x)|∇u|^p⁰^(x)+ 1

p₀(x)|u|^p⁰^(x)+c(x, u)−f (x)u

dx: u∈W₀^1,p⁰⁽^·⁾(Ω)

, (1.4)

where the functionc(x, u)is calculated by the local energy characteristic ofΩ^ε.

The proof of the main result is based on the variational homogenization technique which is nowadays widely used in the homogenization theory (see, e.g., [14,26,38] and the references therein). Let us also mention that another non-periodic homogenization approach was proposed recently in [28] for nonlinear monotone operators.

The paper is organized as follows. In Section 2, for the sake of completeness, we recall the definition and the main results on the Lebesgue and Sobolev spaces with variable exponents which will be used in the sequel. In Section 3 we state the problem and formulate the main result which will be proved in Section 4. Two examples of periodic and locally periodic structures are considered in Section 5.

2. Sobolev spaces with variable exponents

In this section we introduce the function spaces used throughout the paper and describe their basic properties, see for instance [19,21,27,32].

We assume thatΩ is a bounded Lipschitz domain inRⁿand the functionp(x)satisfies the following conditions:

1< p⁽⁻⁾=inf

Ω p(x)p(x)sup

Ω

p(x)=p⁽⁺⁾<+∞ withp⁽⁺⁾n. (2.1)

For allx, y∈Ω,

p(x)−p(y)ω

|x−y|

with lim

τ→0ω(τ )ln 1

τ =0. (2.2)

1. ByL^p(^·⁾(Ω)we denote the space of measurable functionsf inΩ such that A_p(_·₎(f )=

Ω

f (x)^p(x)dx <+∞.

The spaceL^p(^·⁾(Ω)equipped with the norm fL^p(^·⁾(Ω)=inf

λ >0:A_p(_·₎ f

λ 1

(2.3) becomes a Banach space.

2. The following inequalities hold

⎧⎪

⎨

⎪⎩ min

f^p_L⁽_p(·)⁻⁾_(Ω),f^p_L⁽_p(·)⁺⁾_(Ω)

A_p(_·₎(f )max

f^p_L⁽_p(·)⁻⁾_(Ω),f^p_L⁽_p(·)⁺⁾_(Ω) , min

A

p(−)1

p(·) , A

p(+)1

p(·)

fL^p(^·⁾(Ω)max A

p(−)1

p(·) , A

p(+)1

p(·)

.

(2.4)

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3. Letf∈L^p(^·⁾(Ω),g∈L^q(^·⁾(Ω)with 1

p(x)+ 1

q(x)=1, 1< p⁽⁻⁾p(x)p⁽⁺⁾<∞, 1< q⁽⁻⁾q(x)q⁽⁺⁾<+∞. Then the Hölder’s inequality holds

Ω

|f g|dx2f_L^p(·)(Ω)g_L^q(·)(Ω). (2.5)

4. According to (2.5), for every 1q=const< p⁽⁻⁾p(x) <+∞

fL^q(Ω)Cf_L^p(·)(Ω) with the constantC=21

L

p(·) p(·)−q(Ω)

. (2.6)

It is straightforward to check that for domainsΩsuch that measΩ <+∞, 1_L^p(·)(Ω)2 max

[measΩ]^2/p⁽⁻⁾,[measΩ]^1/2p⁽⁺⁾

. (2.7)

5. The spaceW^1,p(^·⁾(Ω),p(·)∈ [p⁽⁻⁾, p⁽⁺⁾] ⊂ ]1,+∞[, is defined by W^1,p(^·⁾(Ω)=

f ∈L^p(^·⁾(Ω): |∇f| ∈L^p(^·⁾(Ω) .

If condition (2.2) is satisfied,W₀^{1, p(}^·⁾(Ω)is the closure of the setC₀^∞(Ω)with respect to the norm ofW^{1, p(}^·⁾(Ω).

If the boundary ofΩ is Lipschitz-continuous andp(x)satisfies (2.2), thenC₀^∞(Ω)is dense inW₀^{1, p(}^·⁾(Ω). The norm in the spaceW₀^1,p(^·⁾is defined by

u_W^1,p(·)

0 =

i

D_iuL^p(^·⁾(Ω)+ uL^p(^·⁾(Ω).

If the boundary ofΩ is Lipschitz andp∈C⁰(Ω), then the norm · _W^1,p(·)

0 (Ω)is equivalent to the norm u_W^1,p(x)

0 (Ω)=

i

DiuL^p(^·⁾(Ω). (2.8)

6. Ifp∈C⁰(Ω), thenW^1,p(^·⁾(Ω)is separable and reflexive.

7. Ifp, q∈C⁰(Ω), p_∗(x)=

p(x)n

n−p(x) ifp(x) < n,

+∞ ifp(x) > n, and 1< q(x)sup

Ω

q(x) <inf

Ω p_∗(x), then the embeddingW₀^1,p(^·⁾(Ω) →L^q(^·⁾(Ω)is continuous and compact.

8. Friedrich’s inequality is valid in the following form: ifp(x)satisfies conditions (2.1)–(2.2), then there exists a constantC >0 such that for everyf ∈W₀^1,p(^·⁾(Ω)

fL^p(·)(Ω)C∇fL^p(·)(Ω). (2.9)

3. Statement of the problem and the main result

LetΩ be a bounded domain inRⁿ (n2) with sufficiently smooth boundary. LetF^ε be a closed subset inΩ. Here εis a small parameter characterizing the scale of the microstructure. We assume thatF^ε is distributed in an asymptotically regular way in Ω, i.e., for any ball V (y, r)of radius r centered aty∈Ω andε >0 small enough (εε₀(r)),V (y, r)∩F^ε= ∅andV (y, r)∩(Ω\F^ε)= ∅. We set

Ω^ε=Ω\F^ε. (3.1)

Letp_ε=p_ε(x)be a continuous function defined in the domainΩ. We assume that, for anyε >0, it satisfies the following conditions:

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(i) this function is bounded in the following sense:

1<p⁽⁻⁾p⁽_ε⁻⁾≡min

x∈Ω

p_ε(x)p_ε(x)max

x∈Ω

p_ε(x)≡p_ε⁽⁺⁾^p⁽⁺⁾n inΩ; (3.2) (ii) for anyx, y∈Ω, we have

pε(x)−pε(y)ωε

|x−y|

with lim

τ→0ωε(τ )ln 1

τ =0; (3.3)

(iii) the functionp_εconverges uniformly inΩ to a functionp₀, i.e.,

εlim→0p_ε−p₀_C0(Ω)=0, (3.4)

where the limit functionp₀is assumed to be bounded in the sense of the condition (2.1) and satisfies (2.2);

(iv) the functionp_εis such that

p_ε(x)p₀(x) inΩ. (3.5)

We consider the following variational problem:

min

J^ε[u]: u∈W₀^1,p^ε⁽^·⁾ Ω^ε

, J^ε[u] =

Ω^ε

1

p_ε(x)|∇u|^p^ε^(x)+ 1

p_ε(x)|u|^p^ε^(x)−f (x)u

dx, (3.6)

wheref ∈C¹(Ω). It is known from [1,10,11,17] that, for eachε >0, there exists a unique solution (minimizer) u^ε∈W^1,p^ε⁽^·⁾(Ω^ε)of problem (3.6). Let us extendu^εinF^εby zero (keeping for it the same notation). Then we obtain the family{u^ε} ⊂W^1,p^ε⁽^·⁾(Ω). We study the asymptotic behavior ofu^εasε→0.

Instead of the classical periodicity assumption on the microstructure of the perforated domainΩ^ε, we impose certain conditions on the local energy characteristic of the setF^ε. To this end we introduceK_h^zan open cube centered atz∈Ω with length equal toh(0< εh <1) and we set

c^ε,h(z, b)=inf

v^ε

K_h^z

1

p_ε(x)∇v^ε^p^ε^(x)+h⁻^p⁽⁺⁾⁻^γS

v^ε−b

dx, (3.7)

whereγ >0, S

v^ε−b

=v^ε−b^p^ε^(x)+v^ε−b^p⁰^(x), (3.8)

and the infimum is taken overv^ε∈W^1,p^ε⁽^·⁾(Ω)that equal zero inF^ε. We assume that:

(C.1) there exists a continuous functionc(x, b)such that for anyx∈Ω, and anyb∈R, and a certainγ=γ₀>0,

hlim→0lim

ε→0h⁻ⁿc^ε,h(z, b)=lim

h→0lim

ε→0

h⁻ⁿc^ε,h(z, b)=c(x, b); (C.2) there exists a constantAindependent ofεsuch that, for anyx∈Ω,

hlim→0lim

ε→0h⁻ⁿc^ε,h(z, b)A

1+ |b|^p⁰^(x) .

The examples of the functionsp_ε(x)and the domainsΩ^ε which satisfy all the above conditions, will be given in Section 5.

The main result of the paper is the following.

Theorem 3.1. Let conditions(i)–(iv)on the functionp_ε and conditions(C.1)–(C.2)on the local characteristic be satisfied. Thenu^εthe solution(minimizer)of the variational problem(3.6) (extended by zero inF^ε)converges weakly inW^1,p⁰⁽^·⁾(Ω)touthe solution(minimizer)of

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inf

Ω

1

p0(x)|∇u|^p⁰^(x)+ 1

p0(x)|u|^p⁰^(x)+c(x, u)−f (x)u

dx: u∈W₀^1,p⁰⁽^·⁾(Ω)

. (3.9)

Remark 1. The condition (C.1) and the definition of the local energy characteristic c^ε,h(z, b) imply that meas[F^ε∩K_h^x] =o(hⁿ)for sufficiently smallε(εε(h)), uniformly with respect to˜ z∈Ω. Therefore, measF^ε→0 asε→0.

Notation.In what followsC,C₁,C₂, etc. are generic constants independent ofε.

4. Proof of Theorem 3.1

It follows from (3.6), (2.4), and the regularity properties of the functionsf,p_εthat u^ε

W^{1,pε (·}⁾(Ω^ε)C. (4.1)

We extendu^ε by zero to the setF^ε and consider{u^ε}as a sequence in the spaceW^1,p^ε⁽^·⁾(Ω). It follows from (4.1) that u^ε

W^{1,pε (·}⁾(Ω)C. (4.2)

Condition (iv) and (4.2) immediately imply that u^ε

W^1,p⁰^(·)(Ω)C. (4.3)

Hence, one can extract a subsequence{u^ε, ε=εk→0}that converges weakly to a functionu∈W^1,p⁰⁽^·⁾(Ω). We will show thatu=u(x)is the solution of the variational problem (3.9). The proof will be done in two mains steps.

4.1. Step 1. Upper bound

Let{x^α}be a periodic grid inΩ with a periodh=h−h¹⁺^{γ /}^p⁽⁺⁾ (εh1, 0< γ <p⁽⁺⁾). Let us cover the domainΩ by the cubes K_h^α of lengthh >0 centered at the pointsx^α. We associate with this covering a partition of unity{ϕα}: 0ϕα(x)1;ϕα(x)=0 forx /∈K_h^α;ϕα(x)=1 forx∈K_h^α\

β=αK_h^β;

αϕα(x)=1 forx∈Ω;

|∇ϕα(x)|Ch⁻¹⁻^{γ /p}⁽⁺⁾.

Now letv_α^ε=v^ε_α(x)be a function minimizing the functional (3.7)–(3.8) withb=b_αandz=x^α, whereb_α will be specified later. It follows from condition (C.1) that, ash→0,

εlim→0

K^α_h∩Ω^ε

1

p_ε(x)∇v_α^ε^p^ε^(x)dx=O hⁿ

; lim

ε→0

K_h^α∩Ω^ε

S v_α^ε−bα

dx=O

hⁿ⁺^p⁽⁺⁾⁺^γ

. (4.4)

Moreover, condition (iv) implies that

εlim→0

K^α_h∩Ω^ε

∇v^ε_α^p⁰^(x)dx=O hⁿ

ash→0. (4.5)

Denote byK_h^α andΠ_h^α the cube of lengthh centered at the pointx^α, and the setK_h^α\K_h^α, respectively. It follows from condition (C.1) of Theorem 3.1 that, ash→0,

εlim→0

Π_h^α∩Ω^ε

1

pε(x)∇v_α^ε^p^ε^(x)dx=o hⁿ

; lim

ε→0

Π_h^α∩Ω^ε

S

v_α^ε−b_α dx=o

hⁿ⁺^p⁽⁺⁾⁺^γ

. (4.6)

Moreover, condition (iv) implies that

εlim→0

Π_h^α∩Ω^ε

∇v_α^ε^p⁰^(x)dx=o hⁿ

ash→0. (4.7)

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Fig. 1. The setB^α(ε, h;ϑ)and the functionv^ε_α.

Fig. 2. The functionV_α^ε.

Now letwbe a smooth function inΩ such thatw(x)=0 on∂Ω and letKθ denotes a subset of the cubesK_h^α coveringΩsuch that|w(x)|> θ >0 for anyx∈K_h^α. We set

bα=w x^α

forK_h^α∈Kθ and bα=1 forK_h^α∈/Kθ. For anyK_h^α, we also define the set (see Fig. 1)

B^α(ε, h;ϑ )=

x∈K_h^α: v^ε_α(x)signb_α|b_α| −ϑ

(4.8) and the function (see Fig. 2)

V_α^ε(x)=

v_α^ε(x) inB^α(ε, h;ϑ );

b^ϑ_α ≡(|b_α| −ϑ )signb_α inK_h^α\B^α(ε, h;ϑ ), (4.9) where 0< ϑθ/21.

Now let us estimate measB^α(ε, h;ϑ ). Forεsufficiently small, from (4.5), we have ϑ^p⁽⁻⁾measB^α(ε, h;ϑ )

B^α(ε,h;ϑ)∩Ω^ε

v_α^ε−b_α^p^ε^(x)dx

K_h^α∩Ω^ε

v^ε_α−b_α^p^ε^(x)dxChⁿ⁺^p⁽⁺⁾⁺^γ. We setϑ=h. Then

εlim→0measB^α(ε, h;ϑ )=O

hⁿ⁺⁽^p⁽⁺⁾⁻^p⁽⁻⁾⁾⁺^γ

=o hⁿ

ash→0. (4.10)

In the domainΩ^ε we introduce the function w^ε_h(x)=w(x)+

α

w(x) b^ϑ_α

V_α^ε(x)−b^ϑ_α

ϕ_α(x). (4.11)

From the definition of the functions{ϕ_α}and (4.9) we have thatw_h^ε∈W₀^1,p^ε⁽^·⁾(Ω^ε).

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Sinceu^εis the solution of the variational problem (3.6) then we have J^ε

u^ε J^ε

w_h^ε

. (4.12)

Let us estimate the right-hand side of the inequality (4.12). It is clear that J^ε

w_h^ε

α

K_h^α∩Ω^ε F_ε

x, w^ε_h,∇w^ε_h

dx+

α,β

(K_h^α∩K_h^β)∩Ω^ε

F_ε

x, w^ε_h,∇w_h^εdx, (4.13)

where

F_ε(x, u,∇u)= 1

p_ε(x)|∇u|^p^ε^(x)+ 1

p_ε(x)|u|^p^ε^(x)−f (x)u. (4.14)

First, we consider the second term on the right-hand side of (4.13). It follows from the definition of the partition of unity{ϕ_α}that for any intersectionK_h^α∩K_h^β the number of terms in the sum overα, βis finite and does not depend onε. Then to estimate the second term on the right-hand side of (4.13) it is sufficient to consider the following integral:

j^ε w_h^ε

=

1 p_ε

∇

w+ w b_α^ϑ

V_α^ε−b^ϑ_α

ϕα ^p^ε+ 1 p_ε

w+ w b^ϑ_α

V_α^ε−b^ϑ_α ϕα

^p^ε

−f (x)

w+ w b^ϑ_α

V_α^ε−b^ϑ_α

ϕα dx

≡j^ε₁ w^ε_h

+j^ε₂ w_h^ε

+j^ε₃ w^ε_h

. (4.15)

For the first term on the right-hand side of (4.15) we have j^ε₁

w^ε_h

=

1 p_ε(x)

∇

w+ w b^ϑ_α

V_α^ε−b^ϑ_α

ϕ_α ^p^ε^(x)dx C₁

|∇w|^p^ε^(x)dx+C₁

(K_h^α∩K^β_h)∩Ω^ε

∇w 1 b^ϑ_α

V_α^ε−b^ϑ_α ϕ_α

^p^ε^(x)dx +C₁

1 pε(x)

w

b^ϑ_α∇v_α^εϕ_α

^p^ε^(x)dx+C₁

w b_α^ϑ

V_α^ε−b^ϑ_α

∇ϕ_α

^p^ε^(x)dx. (4.16)

First, it is clear that

εlim→0

|∇w|^p^ε^(x)dx=o hⁿ

ash→0. (4.17)

For the second term on the right-hand side of (4.16), from (4.6), we have, ash→0,

εlim→0

∇w 1 b^ϑ_α

V_α^ε−b^ϑ_α ϕ_α

^p^εdxC₂lim

ε→0

v^ε_α−b_α^p^εdx=o hⁿ

. (4.18)

For the third term on the right-hand side of (4.16), from (4.6), we have, ash→0,

εlim→0

1 p_ε

w b^ϑ_α∇v_α^εϕα

^p^εdxC3lim

ε→0

(K^α_h∩K_h^β)∩Ω^ε

1

p_ε∇v^ε_α^p^εdx=o hⁿ

. (4.19)

Finally, for the fourth term on the right-hand side of (4.16), from (4.6) and the properties ofϕ_α, we have

(9)

εlim→0

w b^ϑ_α

V_α^ε−b^ϑ_α

∇ϕ_α ^p^ε^(x)dx C₄h⁻^p⁽⁺⁾⁻^γ lim

ε→0

v_α^ε−b_α^p^ε^(x)dx=o hⁿ

ash→0. (4.20)

Thus, from (4.15)–(4.20) we get

hlim→0lim

ε→0j^ε₁ w_h^ε

=0. (4.21)

In a similar way we can estimate the integralsj^ε₂[w_h^ε],j^ε₃[w^ε_h]. Therefore, for the second term on the right-hand side of (4.13), we get

hlim→0lim

ε→0

α,β

_F_ε

x, w_h^ε,∇w_h^εdx=0. (4.22)

Consider now the first term on the right-hand side of (4.13). First, let us denote:

B^α₁(ε, h)=

K_h^α∩Ω^ε

∩B^α(ε, h;ϑ ) and B^α₂(ε, h)=

K_h^α∩Ω^ε

\B₁^α(ε, h), (4.23)

where the setB^α(ε, h;ϑ )is defined in (4.8) withϑ=h. Thenw^ε_h(x)=w(x)inB₂^α(ε, h)and

B^α2(ε,h) F_ε

x, w_h^ε,∇w^ε_h dx=

B^α2(ε,h)

F_ε(x, w,∇w) dx=

B^α2(ε,h)

F₀(x, w,∇w) dx+I_F^ε, (4.24) where

F₀(x, w,∇w)= 1

p₀(x)|∇w|^p⁰^(x)+ 1

p₀(x)|w|^p⁰^(x)−f (x)w (4.25)

and I_F^ε=

B^α2(ε,h)

F_ε(x, w,∇w)−^F0(x, w,∇w)

dx. (4.26)

Moreover, it follows from (3.4) that

εlim→0

I_F^ε=0. (4.27)

Therefore, from (4.24)–(4.27) and the regularity properties of the functionsw,f, we have

εlim→0

B^α2(ε,h) F_ε

x, w_h^ε,∇w^ε_h dx

K_h^α

F₀(x, w,∇w) dx+o hⁿ

ash→0. (4.28)

Let us consider now the integral over the setB^α₁(ε, h)(K_h^α∈Kθ). In the setB^α₁(ε, h)the functionw_h^εhas the form:

w^ε_h(x)=w(x)+w(x) b_α^ϑ

v^ε_α−b_α^ϑ

inB₁^α(ε, h). (4.29)

Therefore, we have

B^α1(ε,h) F_ε

x, w_h^ε,∇w^ε_h dx=

B^α1(ε,h)

1

p_ε∇w^ε_h^p^εdx+

B1^α(ε,h)

1

p_εw^ε_h^p^ε−w_h^εf

dx. (4.30)

Now it follows from the regularity properties of the functionsw,f, the estimate for the measure of the setB^α(ε, h;ϑ ) (see (4.10)) and the boundedness of the functionv_α^ε on the setB^α(ε, h;ϑ )that