Suboptimal boundary controls for elliptic equation in critically perforated domain

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Suboptimal boundary controls for elliptic equation in critically perforated domain

Ciro D’Apice

, Umberto De Maio

, Peter I. Kogut

aUniversità di Salerno, Dipartimento di Ingegneria dell’Informazione e Matematica Applicata, via Ponte don Melillo, 84084 Fisciano (SA), Italy bUniversità degli Studi di Napoli “Federico II”, Dipartimento di Matematica e Applicazioni “R. Caccioppoli”, Complesso Monte S. Angelo,

via Cintia, 80126 Napoli, Italy

cDepartment of Differential Equations, Dnipropetrovsk National University, Naukova str., 13, 49050 Dnipropetrovsk, Ukraine Received 3 January 2007; received in revised form 10 July 2007; accepted 16 July 2007

Available online 17 October 2007

Abstract

In this paper we study the asymptotic behaviour, asεtends to zero, of a class of boundary optimal control problemsPε, set in ε-periodically perforated domain. The holes have a critical size with respect toε-sized mesh of periodicity. The support of controls is contained in the set of boundaries of the holes. This set is divided into two parts, on one part the controls are of Dirichlet type;

on the other one the controls are of Neumann type. We show that the optimal controls of the homogenized problem can be used as suboptimal ones for the problemsPε.

©2007 Elsevier Masson SAS. All rights reserved.

MSC:35B27; 49J20; 49J27

Keywords:Optimal control; Homogenization; Perforated domain; Variational convergence; Measure approach

1. Introduction

LetΩ⊂Rⁿ,n2, be a bounded open domain, and letεbe a small positive parameter. To define a perforated domain Ωε, we introduce the following sets: Y = [−1/2,+1/2)ⁿ; Qand K are compact subsets of Y such that 0∈intK∩∂Q,

Θ_ε=

k=(k₁, k₂, . . . , k_n)∈Zⁿ: (εY+εk)∩Ω= ∅

; (1.1)

Yε=

k∈Θε

ε(Y+k)

; Tε=

k∈Θε

{ε^n/(n−1)Q+εk}; (1.2)

S_ε=

ε^n/(n−2)K∩∂(ε^n/(n−1)Q), n3,

exp(−1/ε²)K∩∂(ε^n/(n−1)Q), n=2, (1.3)

* Corresponding author.

E-mail addresses:[email protected] (C. D’Apice), [email protected] (U. De Maio), [email protected] (P.I. Kogut).

0294-1449/$ – see front matter ©2007 Elsevier Masson SAS. All rights reserved.

doi:10.1016/j.anihpc.2007.07.001

Γ_ε^D=

k∈Θ_ε

{S_ε+εk}

∩Ω, Γ_ε^N= [∂T_ε\Γ_ε^D] ∩Ω. (1.4)

Then we setΩ_ε=Ω\T_ε. The principal feature of the perforated domainΩ_ε is the fact that the size of the holes Q_ε+εkand their boundariesΓ_ε^D,Γ_ε^Nare not proportional to the size of the periodicity cellεY. InΩ_ε we consider the following boundary value problem:

−y_ε+y_ε=f_ε, inΩ_ε,

∂νyε= −k0yε+pε, onΓ_ε^N, y_ε=u_ε, onΓ_ε^D,

y_ε=0, onΣ_ε=∂Ω∩∂Ω_ε,

⎫⎪

⎪⎬

⎪⎪

⎭

(1.5)

wheref_ε∈L²(Ω)is a given function,k₀is a positive constant,∂_ν=∂/∂νis the outward normal derivative.

In (1.5)u_εandp_εare the control functions which act on the system through the set of boundaries of the holes. We say that the control functionsu_εandp_εare admissible if the following conditions hold:

p_ε∈L²(Γ_ε^N), u_ε∈U_ε=

a|Γ_ε^D: a∈H₀¹(Ω)∩H²(Ω),aH²(Ω)C0

. (1.6)

Then the optimal control problem Pε can be formulated as follows: Given zε ∈ L²(Ω), C0 >0, find a triple (u⁰_ε, p⁰_ε, y_ε⁰)∈Ξ_εsuch that

I_ε(u⁰_ε, p_ε⁰, y_ε⁰)= inf

(uε,pε,yε)∈Ξε

I_ε(u_ε, p_ε, y_ε), (1.7)

where the cost functionalI_εand the set of admissible tripletsΞ_εare defined as I_ε=

Ω_ε

|∇y_ε|²dx+

Ω_ε

|y_ε−z_ε|²dx+

Γ_ε^N

p_ε²dHⁿ⁻¹+β(ε)

Γ_ε^D

u²_εdHⁿ⁻¹, (1.8)

Ξ_ε=

(u_ε, p_ε, y_ε)∈H¹(Γ_ε^D)×L²(Γ_ε^N)×H¹(Ω_ε;Σ_ε): (u_ε, p_ε, y_ε)satisfies (1.5)–(1.6)

. (1.9)

HereH¹(Ω_ε;Σ_ε)= {y_ε∈H¹(Ω_ε): y_ε=0 onΣ_ε},β(ε)=ε^n/(2−n)ifn3, andβ(ε)=ε²exp(ε⁻²)ifn=2.

The asymptotic analysis of the boundary value problems in perforated domain with small holes (without controls) has been widely studied by many authors. We mainly could mention Cioranescu and Donato and Murat and Zuazua [7], Cioranescu and Saint Jean Paulin [10], Cioranescu and Murat [9], Dal Maso and Murat [14], Marchenko and Khruslov [23], Zhikov and Kozlov and Oleinik [29], Scrypnik [27]. It is well known the interesting effect of ho- mogenization of the Poisson equation with (zero) Dirichlet conditions on the boundary of the holes, when a "strange term" appears in the limit equation (see [9,23]). Another effect of homogenization of the same equation with a crit- ical size of the holes, when nonhomogeneous Neumann conditions on the boundary of the holes are assumed, was studied by Conca and Donato [11]. In this case some constant that is proportional to the limit of the total flux of the solution through the boundary of the holes, appears in the limit equation. Cardone and D’Apice and De Maio in [5]

and Corbo Esposito and D’Apice and Gaudiello in [12] examined the same equation with mixed boundary conditions on the holes. As proved in [12], in the context of perforated domains with a rather simple geometry of the holes, an interference phenomenon in the homogenization of such boundary value problems is present.

Optimal control problems in perforated domains have been the object of intensive research in the past years [8,18, 24,26]. The numerical computation of such problems is very complicated through thick perforations ofΩ_ε. Therefore, the asymptotic analysis is one of the main approaches to the study of optimization problems in perforated domains.

The goal of this paper is to obtain an appropriate approximation for the optimal solutions to the problemPεfor small enough values ofε. Using the ideas of theΓ-convergence theory and the concept of the variational convergence of constrained minimization problems (see [2,3,19,20]), we show that the homogenized problem for the original one can be recovered in the following analytical form:

Ω

(∇y· ∇ϕ) dx+ρ^∗

Ω

(y−a)ϕ dμ^∗+

1+k₀|∂Q|H Ω

yϕ dx

=

Ω

f ϕ dx+ |∂Q|H

Ω

pϕ dx, ∀ϕ∈H₀¹(Ω)∩L²(Ω, dμ^∗). (1.10)

y∈H₀¹(Ω), p∈L²(Ω), a∈H²(Ω)∩H₀¹(Ω), (1.11)

y−a∈L²(Ω, dμ^∗), aH²(Ω)C₀, (1.12)

I0(a, p, y)=

Ω

|∇y|²dx+

Ω

|y−z^∂|²dx+ρ^∗

Ω

(y−a)²dμ^∗

+ |∂Q|H

Ω

p²dx+ |K∩∂Λ|H

Ω

a²dx−→inf. (1.13)

Here the parametersρ^∗,|∂Q|H,|K∩∂Λ|H ∈Rand the Borel measureμ^∗are coming from the geometry of control zones. In contrast toPε, the limit control problem (1.10)–(1.13) contains two independent distributed control func- tions. We show that this problem has a unique optimal solution, derive the corresponding optimality conditions, and establish that the optimal solution for the homogenized problem can be used as a suboptimal control for the original one.

2. Preliminaries and notation

Throughout the paper we suppose thatΩ is a measurable set in the sense of Jordan; the small parameterεvaries in a strictly decreasing sequence of positive numbers which converges to 0;QandKare compact subsets ofY such that 0∈intK∩∂Q; the setQhas Lipschitz boundary∂Q, intQis a strongly connected set,Q⊂ {x=(x₁, . . . , x_n)∈ Rⁿ:x₁0}, and its boundary∂Qcontains the origin;A=B(0, r0)is an open ball centered at the origin with a radius r0<1/2, so thatAY andKA(see Fig. 1 for 2-d example);C0>0 is a constant independent ofε; the functions f_ε∈L²(Ω),z_ε∈L²(Ω)are such thatf_ε f andz_ε→z^∂ inL²(Ω)asε→0. For any subsetE⊂Ω we denote by

|E|itsn-dimensional Lebesgue measureLⁿ(E), whereas|∂E|H denotes the(n−1)-dimensional Hausdorff measure of manifold∂EonRⁿ. We suppose that the setsK∩∂Q^ςand∂Q\(K∩∂Q^ς)have nonzero capacity for anyς >1, whereQ^ς= {ς x,∀x=(x1, . . . , xn)∈Q}is the homothetic stretching ofQby a factor ofς. Hence|K∩∂Q^ς|H=0 for allς >1.

LetMb(Ω)be the space of bounded Borel measures onΩ with values in[0,+∞]. LetM⁺₀(Ω)be the cone of all nonnegative Borel measuresμonΩ such thatμ(B)=0 for every setB⊆Ω with cap(B, Ω)=0, andμ(B)= inf{μ(U ): Uquasi open, B⊆U}for every Borel setB⊆Ω. Note that ifμ∈M⁺₀(Ω), then the functions ofH¹(Ω) are definedμ-almost everywhere and areμ-measurable inΩ, hence the spaceH¹(Ω)∩L²(Ω, dμ)is well defined.

In view of [21] (see Theorems 1.1 and 2.2, Chapter IV), the following result can be easily proved:for every fixed εand for any control functionsuε∈H¹(Γ_ε^D)andpε∈L²(Γ_ε^N)there exists a unique functionyε=yε(uε, pε)such that

Fig. 1. Example of perforation scheme.

y_ε−a_ε∈H¹(Ω_ε;Γ_ε^D∪Σ_ε),

Ωε

(∇y_ε· ∇ϕ) dx+

Ωε

y_εϕ dx+k₀

Γ_ε^N

y_εϕ dHⁿ⁻¹

=

Ωε

f_εϕ dx+

Γ_ε^N

p_εϕ dHⁿ⁻¹ ∀ϕ∈H¹(Ω_ε;Γ_ε^D∪Σ_ε), (2.1)

y_εH¹(Ω_ε)C

f_εL²(Ω)+D₁(ε)u_εH¹(Γ_ε^D)+D₂(ε)p_εL²(Γ_ε^N)

, (2.2)

whereC,D₁(ε), andD₂(ε)are some positive constants,Cis independent ofε, and the functiona_ε∈H₀¹(Ω)∩H²(Ω) is such thata_εH²(Ω)C₀anda_ε|Γ_ε^D=u_ε. In the sequel we call the functiony_εweak solution to the problem (1.5) and identifyy_εwith its quasi continuous representative [14].

Let{(uⁿ_ε, pⁿ_ε, y_εⁿ)} ⊂Ξ_εbe a minimizing sequence for the problemPε. Then, using the compact embedding result H^3/2(Γ_ε^D) →H¹(Γ_ε^D)which impliesuⁿ_ε →u⁰_ε=a_ε⁰|Γ_ε^D strongly inH¹(Γ_ε^D)and the direct method of Calculus of variation, we come to the following conclusion (for details see [16], [13]):

Theorem 2.1.For everyεthere exists a unique solution(u⁰_ε, p⁰_ε, y_ε⁰)∈Ξ_εof the optimal control problemPε. 3. On formulation of the homogenization problem

We begin this section with the description of the geometry of the perforated domain Ωε. We describe the class of admissible solutions to problemPε in the terms of singular periodic Borel measures onRⁿ. To do so, we use the approach of Zhikov, Bouchitté and Fragala (see [1,28]).

Let us denote byK^λ andQ^h the homothetic contractions of the setsK andQatλ⁻¹ andh⁻¹ times. In what follows it is assumed that 0< λh <1. Let the setsΓ^λ,handΛ^λ,hbe defined as follows:

Γ^λ,h=K^λ∩∂Q^h, Λ^λ,h=∂Q^h\Γ^λ,h. (3.1)

Letμ^λ,handν^λ,hbe the normalized periodic Borel measures onRⁿwith the periodicity cellYsuch thatμ^λ,his con- centrated onΓ^λ,h,ν^λ,his concentrated onΛ^λ,h, and both these measures are proportional to the(n−1)-dimensional Hausdorff measure. Since these measures are concentrated and uniformly distributed on the corresponding sets, it follows thatμ^λ,h(Y\Γ^λ,h)=0.

For any functionϕ∈C^∞(Rⁿ)we have

Γ^λ,h

ϕ dHⁿ⁻¹= |Γ^λ,h|H

Y

ϕ dμ^λ,h=λⁿ⁻¹|K∩∂Q^h/λ|H

Y

ϕ dμ^λ,h, (3.2)

Λ^λ,h

ϕ dHⁿ⁻¹= |Λ^λ,h|H

Y

ϕ dν^λ,h=

|∂Q^h|H− |Γ^λ,h|H Y

ϕ dν^λ,h

=

hⁿ⁻¹|∂Q|H−λⁿ⁻¹|K∩∂Q^h/λ|H Y

ϕ dν^λ,h. (3.3)

We introduce also the scaling measures μ^λ,h_ε and ν_ε^λ,h by setting μ^λ,h_ε (B) = εⁿμ^λ,h(ε⁻¹B), ν^λ,h_ε (B) = εⁿν^λ,h(ε⁻¹B)for every Borel setB⊂Rⁿ, and relate the parametersλ,h, andεby the rule

h(ε)=ε^n/(n−1), λ(ε)=ε^n/(n−2) ifn3, and λ(ε)=exp(−ε⁻²) ifn=2. (3.4) Then

εYdμ^λ,h_ε =εⁿ

Ydμ^λ,h=εⁿ,

εYdν_ε^λ,h=εⁿ

Ydν^λ,h=εⁿ. It means that the measuresμ^λ,h_ε andν_ε^λ,hweakly converge to the Lebesgue measure:dμ^λ,h_ε dx,dν^λ,h_ε dx, that is for everyϕ∈C₀^∞(Rⁿ)we have

εlim→0

Rⁿ

ϕ dμ^λ,h_ε =

Rⁿ

ϕ dx, lim

ε→0

Rⁿ

ϕ dν^λ,h_ε =

Rⁿ

ϕ dx. (3.5)

Remark 3.1.It is easy to see that the scaling measuresμ^λ,h_ε andν_ε^λ,hbelong to the classM⁺₀(Ω). Hence the spaces H₀¹(Ω)∩H²(Ω)∩L²(Ω, dμ^λ,h_ε )andH¹(Ω;Σ_ε)∩L²(Ω, dν_ε^λ,h)are well defined [14].

Now we turn back to the definition of the set of admissible solutions of the problemPε(see (1.9)). We see that Γ_ε^D=

k∈Θ_ε

K^λ(ε)∩∂Q^h(ε)+εk

, Γ_ε^N=

k∈Θ_ε

∂Q^h(ε)\(K^λ(ε)∩∂Q^h(ε))+εk .

Then, using properties (3.2)–(3.3) and setting σ (ε)=

h(ε)ⁿ⁻¹|∂Q|H−λ(ε)ⁿ⁻¹|K∩∂Q^h(ε)/λ(ε)|H

, (3.6)

the term

Γ_ε^Np_εϕ dHⁿ⁻¹of the integral identity (2.1) can be rewritten in the form

Γ_ε^N

p_εϕ dHⁿ⁻¹=

k∈Θ_ε

∂Q^h(ε)\Γ^λ(ε),h(ε)+εk

p_εϕ dHⁿ⁻¹

=σ (ε)

k∈Θε

ε(Y+k)

ˆ

pεϕ dν^λ(ε),h(ε)(x/ε)=ε⁻ⁿσ (ε)

Ω

ˆ

pεϕ dν_ε^λ,h (3.7)

for every functionϕ∈C₀^∞(Rⁿ;Σ_ε∪Γ_ε^D)= {ψ∈C^∞₀ (Rⁿ): ψ=0 onΣ_ε∪Γ_ε^D}.

Herepˆε is a function ofL²(Ω, dν_ε^λ,h)taking the same values aspε∈L²(Γ_ε^N)onΓ_ε^N. It is clear that for every boundary controlpε∈L²(Γ_ε^N), one can find a functionpˆε∈L²(Ω, dν_ε^λ,h)such thatpε= ˆpεonΓ_ε^N. Hence

Γ_ε^N

p²_εdHⁿ⁻¹=ε⁻ⁿσ (ε)

Ω

ˆ

p²_εdν^λ,h_ε , (3.8)

where

ε⁻ⁿσ (ε)=

|∂Q|H−ε^n/(n−2)|K∩∂Q^h(ε)/λ(ε)|H, ifn3,

|∂Q|H−_ε¹2exp(−_ε¹2)|K∩∂Q^h(ε)/λ(ε)|H, ifn=2. (3.9) By analogy we obtain

k₀

Γ_ε^N

y_εϕ dHⁿ⁻¹=ε⁻ⁿσ (ε)k₀

Ω

˘

y_εϕ dν_ε^λ,h, (3.10)

Γ_ε^D

u²_εdHⁿ⁻¹=ε⁻ⁿλ(ε)ⁿ⁻¹|K∩∂Q^h(ε)/λ(ε)|H

Ω

a_ε²dμ^λ,h_ε . (3.11)

Herey˘_ε∈H¹(Ω;Σ_ε)is an extension of the weak solutiony_εto the problem (1.5) to the whole of domainΩ, and the functiona_ε∈H₀¹(Ω)∩H²(Ω)∩L²(Ω, dμ^λ,h_ε )is a prototype of the Dirichlet controlu_ε∈U_ε(see (1.6)).

Remark 3.2.In view of our initial suppositions, the measureν_ε^λ,his supported on the set with nonzero capacity for everyε >0. Since every elementvof the spaceH¹(Ω;Σ_ε)∩L²(Ω, dν_ε^λ,h)can be interpreted as a quasi continuous function, it is reasonable to suppose that for every elementv∈H¹(Ω), one can find a sequence{v_k∈C(Ω)}k∈Nsuch that

sup

k∈Nlim sup

ε→0

Vk

(v−v_k)²dν_ε^λ,h<+∞ and lim

k→∞cap(Vk, Ω)=0,

whereVk = {x∈Ω: v=v_k}. We assume that the same property is valid for the elements of the space H₀¹(Ω)∩ L²(Ω, dμ^λ,h_ε ).

As a result, we can reformulate the original optimal control problem Pε (1.7)–(1.9) as follows: Find some (a⁰_ε, p⁰_ε, y_ε⁰)∈Xεsuch that

Iˆ_ε(a⁰_ε, p_ε⁰, y⁰_ε)= inf

(aε,pε,yε)∈ ˆΞε

Iˆ_ε(a_ε, p_ε, y_ε), (3.12)

Xε=

H₀¹(Ω)∩H²(Ω)∩L²(Ω, dμ^λ,h_ε )

×L²(Ω, dν_ε^λ,h)×

H¹(Ω;Σε)∩L²(Ω, dν_ε^λ,h) , where

Iˆ_ε(a_ε, p_ε, y_ε)=

Ω

χ_ε|∇ ˘y_ε|²dx+

Ω

|χ_εy˘_ε−z^∂_ε|²dx

+ε⁻ⁿσ (ε)

Ω

p_ε²dν_ε^λ,h+ |K∩∂Q^h(ε)/λ(ε)|H

Ω

a²_εdμ^λ,h_ε , (3.13)

ˆ Ξ_ε=

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩

(a_ε, p_ε, y_ε)

˘

yε−aε∈H¹(Ω;Γ_ε^D∪Σε),aH²(Ω)C0,

Ωχε(∇ ˘yε· ∇ϕ) dx+

Ωχεy˘εϕ dx +k₀ε⁻ⁿσ (ε)

Ωy˘_εϕdν_ε^λ,h−

Ωχ_εf_εϕ dx

=ε⁻ⁿσ (ε)

Ωp_εϕ dν_ε^λ,h, ∀ϕ∈H¹(Ω;Γ_ε^D∪Σ_ε).

⎫⎪

⎪⎪

⎬

⎪⎪

⎪⎭

(3.14)

We denote withPˆεthe optimal control problem (3.12)–(3.14). It is clear thatPˆεhas a unique solution(a_ε⁰, p_ε⁰, y_ε⁰)for everyε[16,22]. This solution can be viewed as a prototype of the optimal triplet toPε-problem. Moreover, in this case a priori norm estimate (2.2) takes the form

˘y_εH¹_(Ω,χεdx)+

ε⁻ⁿσ (ε)y_εL2(Ω,dνε^λ,h)Cˆ

ε⁻ⁿσ (ε)p_εL2(Ω,dν^λ,hε )+ f_εL²_(Ω) +

|K∩∂Q^h(ε)/λ(ε)|Ha_εH1

0(Ω)∩L²(Ω,dμ^λ,h_ε )

. (3.15)

To end this section, we list some auxiliary results that will be useful in the sequel. Let ς (ε)=h(ε)/λ(ε)=

exp(_n2−⁻3nⁿ+2lnε) ifn3, ε²exp(_ε¹2) ifn=2.

(3.16) Thenς (ε)∈(1,+∞)∀εand limε→0ς (ε)= +∞. We are interested in the limit behaviour of the sequence {|K∩

∂Q^ς(ε)|H}asε→0. We recall that the setQ^ς(ε)= {ς (ε)x,∀x=(x1, . . . , xn)∈Q}is the homothetic stretching of Qby a factor ofς (ε).

Proposition 3.3.There exists an open coneΛ⊂ {x∈Rⁿ: x₁>0}such that

εlim→0

K∩∂Q^ς(ε)

H= |K∩∂Λ|H. (3.17)

Proof. Indeed, by the initial assumptions, the origin is a Lipschitz point of the boundary∂Qand intQis a strongly connected set in the classical sense. Hence, there is a neighbourhood U(0)such thatU(0)∩intQis a convex set [4,15]. ThenΛ= {x∈t[U(0)∩intQ] ∀t∈(0,+∞)}is a nonempty open cone.

Assume that the origin does not belong to a smooth part of the boundary∂Q. Then the inclusionK∩Λ⊂K∩Q^{ς (ε)} holds true for εsmall enough, and it immediately implies the existence of a valueε₀>0 such that |K∩∂Λ|H =

|K∩∂Q^ς(ε)|H ∀ε < ε₀. If a part of the boundary∂Qcontaining the origin is smooth, then it follows that there is a neighbourhoodU(0)of the origin such thatU(0)∩∂Qis the graph of a smooth function whose epigraph contains U(0)∩Q. So that, we may always suppose that there is a functionΨ: Rⁿ⁻¹→R satisfyingΨ ∈C₀^∞(Rⁿ⁻¹)and x₁=Ψ (x₂, . . . , x_n)for everyx=(x₁, x₂, . . . , x_n)∈U(0)∩∂Q.

LetΛ= {x∈Rⁿ: x₁>0}. Then∂Λ= {x∈Rⁿ: x₁=0}and we deduce:x=(x₁, x₂, . . . , x_n)∈K∩ς (ε)∂Qfor small enoughεif and only ifx∈ς (ε)(U(0)∩∂Q). Hence

x₁= 1 ς (ε)Ψ

ς (ε)x₂, . . . , ς (ε)x_n

for everyx∈K∩ς (ε)∂Q.

Since limε→0ς⁻¹(ε)Ψ (ς (ε)x₂, . . . , ς (ε)x_n)=0 by the definition of the Hausdorff measureHⁿ⁻¹, we immediately obtain the required result. 2

Remark 3.4.As follows from Proposition 3.3 and its proof, the coneΛcan be recovered in an explicit form in the case when the origin belongs to a smooth part of the boundary∂Q. Moreover, in view of (3.6) we have limε→0σ (ε)/εⁿ=

|∂Q|H.

In a similar way the following statement can be proved:

Proposition 3.5.Let{ρ_ε ∈R}ε>0be a sequence of numbers such that

ρ_ε= |A\Q^{ς (ε)}|/|A| ∀ε >0. (3.18)

Then the sequence{ρ_ε}ε>0is monotone and there exists a valueρ^∗∈ [1/2,1)such thatlimε→0ρ_ε=ρ^∗. 4. Convergence in the variable spaceXε

Let us recall the main types of convergence in variable spaces occurring in the homogenization theory (see [28]).

We cite them with respect to the family of the periodic Borel measureμ^λ,h_ε . Here the parametersλ=λ(ε)andh=h(ε) are defined by (3.4). Let{v_ε^λ,h∈L²(Ω, dμ^λ,h_ε )}be a bounded sequence, i.e. lim sup_ε→0

Ω(v^λ,h_ε )²dμ^λ,h_ε <+∞.

1. The weak convergencev^λ,h_ε vinL²(Ω, dμ^λ,h_ε )means thatv∈L²(Ω)and limε→0

Ωv_ε^λ,hϕ dμ^λ,h_ε =

Ωvϕ dx for anyϕ∈C₀^∞(Ω);

2. The strong convergence v^λ,h_ε →v in L²(Ω, dμ^λ,h_ε ) means that v∈L²(Ω) and limε→0

Ωv^λ,h_ε z^λ,h_ε dμ^λ,h_ε =

Ωvz dxifz^λ,h_ε zinL²(Ω, dμ^λ,h_ε ).

The following properties of the convergence in variable spaces hold:

(a) Compactness criterium: if a sequence is bounded inL²(Ω, dμ^λ,h_ε ), then this sequence is compact in the sense of the weak convergence;

(b) Property of lower semicontinuity: ifv_ε^λ,h vinL²(Ω, dμ^λ,h_ε ), then lim inf

ε→0

Ω

(v^λ,h_ε )²dμ^λ,h_ε

Ω

v²dx;

(c) Criterium of strong convergence:v_ε^λ,h→vif and only ifv^λ,h_ε vand limε→0

Ω(v^λ,h_ε )²dμ^λ,h_ε =

Ωv²dx;

(d) Sinceμ^λ,h_ε dx, it follows that limε→0

Ωϕ dμ^λ,h_ε =

Ωϕ dx ∀ϕ∈C(Ω)and lim sup_ε→0μ^λ,h_ε (F )|F|for any compact setF ⊂Ω.

We begin with the following concept:

Definition 4.1.Let{v_ε^λ,h∈H₀¹(Ω)∩L²(Ω, dμ^λ,h_ε )}be a bounded sequence. We say that this sequence converges weakly inH₀¹(Ω)∩L²(Ω, dμ^λ,h_ε )tov∈H¹(Ω)if

v_ε^λ,h v inH₀¹(Ω) and v^λ,h_ε v inL²(Ω, dμ^λ,h_ε ). (4.1) In order to check the correctness of this definition we make use of the following auxiliary statements:

Lemma 4.2.Ifv∈H₀¹(Ω), thenlimε→0

Ωvϕ dμ^λ,h_ε =

Ωvϕ dx∀ϕ∈C₀^∞(Ω).

Proof. Ifv∈C₀(Ω), then the weak convergenceμ^λ,h_ε dxof the measures immediately implies this relation. Let vbe an arbitrary element ofH₀¹(Ω). Then for everyδ >0 there exist a setA^δ⊂Ω and a functionv^δ∈C₀(Ω)such

that cap(A^δ, Ω) < δ, andv^δ=vonΩ\A^δ. In what follows, we suppose that the setA^δis closed. We now consider the following estimate:

Ω

vϕ dμ^λ,h_ε −

Ω

vϕ dx

Ω

(v−v^δ)ϕ dμ^λ,h_ε +

Ω

v^δϕ dμ^λ,h_ε −

Ω

v^δϕ dx +

Ω

(v−v^δ)ϕ dx

=J₁+J₂+J₃.

Owing to the weak convergenceμ^λ,h_ε dx, we haveJ2→0 asε→0. By Lusin’s Theorem we may suppose that there is a constant d₁>0 such thatv−v^δL²(Ω)d₁δ. HenceJ₃d₁ϕL²(Ω)δ. As for the value J₁, we note that each of the measure μ^λ,h_ε is supported on a set with nonzero capacity. So, there is a constantd2>0 such that v−v^δ_L2(Ω,dμ^λ,h_ε )d2for anyδ >0 small enough (see Remark 3.2). It implies the following estimate:

J₃=

Ω

(v−v^δ)ϕ dμ^λ,h_ε

d₂ϕC(Ω)

μ^λ,h_ε (A^δ)1/2

. As a result, we have:

(i) lim sup_ε→0μ^λ,h_ε (A^δ)|A^δ| by property (d); (ii) |A^δ|d₃cap^n/(n−2)(A^δ) by the properties of capacity (see [15]); (iii) cap(A^δ) < δby the initial assumption. Hence, summing up all estimates that were obtained before, we conclude|

Ωvϕ dμ^λ,h_ε −

Ωvϕ dx|dδfor anyδ >0 small enough (here the constantd does not depend onδ).

This completes the proof. 2

We now consider a more delicate situation.

Lemma 4.3. Let {v_ε^λ,h∈H₀¹(Ω)∩L²(Ω, dμ^λ,h_ε )} and v∈L²(Ω) be such that v_ε^λ,h v in H₀¹(Ω)and hence v_ε^λ,h→vinL²(Ω). Then

εlim→0

Ω

v_ε^λ,hdμ^λ,h_ε −

Ω

v_ε^λ,hdx

=0. (4.2)

Proof. As in the previous lemma, we introduce two functionsv˜_ε^λ,h∈C(Ω)andv˜∈C(Ω)such thatv˜^λ,h_ε =v_ε^λ,hand

˜

v=vquasi everywhere. Let us partition the setΩ into cubesεYwith edgesεand denote these cubes withεY^j. Then there are pointsx_j^λ,h∈εY^jsuch that

Ω

˜

v_ε^λ,hdμ^λ,h_ε =

εY^j

˜

v_ε^λ,h(x) dμ^λ,h_ε +

Ω∩εY^j

˜

v_ε^λ,h(x) dμ^λ,h_ε

=

˜

v_ε^λ,h(x_j^λ,h)

εY^j

dμ^λ,h_ε +

Ω∩εY^j

˜

v^λ,h_ε (x) dμ^λ,h_ε ,

where the second sum is calculated over the set of the “boundary” cubes. By the definition of the measureμ^λ,h_ε , we have

εY^jdμ^λ,h_ε =εⁿ

Ydμ^λ,h_ε =εⁿ. Hence

Ω

˜

v_ε^λ,hdμ^λ,h_ε =

˜

v_ε^λ,h(x_j^λ,h)εⁿ+

Ω∩εY^j

˜

v_ε^λ,h(x) dμ^λ,h_ε . (4.3)

It is clear that an analogous representation takes place for the second term in (4.2), namely,

Ω

˜

v_ε^λ,hdx=

˜ v_ε^λ,h(x_j)

εY^j

dx+

Ω∩εY^j

˜

v_ε^λ,h(x) dx

=

˜

v_ε^λ,h(x_j)εⁿ+

Ω∩εY^j

˜

v^λ,h_ε (x) dx, (4.4)