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Asymptotic Analysis 00 (2012) 1–12 1
DOI 10.3233/ASY-2012-1104
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46 46Homogenization results for elliptic problems
in periodically perforated domains with
mixed-type boundary conditions
Anca Capatinaa, ∗,H o r i aE n ea,band Claudia Timoftec
aInstitute of Mathematics of the Romanian Academy, Bucharest, Romania
E-mails: {anca.capatina, horia.ene}@imar.ro
bUniversity of Pite¸ sti, Pite¸ sti, România
cUniversity of Bucharest, Faculty of Physics, Bucharest–M˘ agurele, Romania
E-mail: [anonimizat]
Abstract. The asymptotic behaviour of a class of elliptic equations with highly oscillating coefficients, in a perforated periodic
domain, is analyzed. We consider, in each period, two types of holes and we impose, on their boundaries, a Signorini and, re-
spectively, a Neumann condition. Using the periodic unfolding method, we prove that the limit problem contains two additional
terms, a right-hand side term and a “strange” one.
Keywords: homogenization, periodic unfolding method, variational inequality, critical holes
1. Introduction
The goal of this paper is to analyze the effective behavior of a class of elliptic second-order equations
with highly oscillating coefficients, in a perforated periodic domain. We address here the case of an ε-
periodic perforated structure, with two holes of different sizes in each period. Depending on the boundary
interaction that take place at their surfaces, two distinct conditions, one of Signorini’s type and another
one of Neumann type, are imposed on the corresponding boundaries of the holes. On the exterior fixed
boundary of the perforated domain, an homogeneous Dirichlet condition is prescribed.
The main feature of this kind of problems is the existence of a critical size of the perforations that
separates different emerging phenomena as the small parameter εtends to zero. In this critical case, we
prove that the homogenized problem, obtained by the periodic unfolding method, contains two additional
terms coming from the particular geometry. These new terms, a right-hand side term and a “strange” one,
capture the two sources of oscillations involved in this problem, i.e. those arising from the special size
of the holes and those due to the periodic heterogeneity of the medium.
Similar problems were addressed in the literature. The homogenization of the Poisson equation with
a Dirichlet condition for perforated domains was treated by Cioranescu and Murat [13], putting into
evidence, in the case of critical holes, the appearance of a “strange” term. Their results were extended,
*Corresponding author: Anca Capatina, Institute of Mathematics of the Romanian Academy, P.O. Box 1-764, RO-70700
Bucharest, Romania. E-mail: [anonimizat].
0921-7134/12/$27.50 ©2012 – IOS Press and the authors. All rights reserved

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46 46using different techniques, to heterogeneous media by Ansini and Braides [1], Dal Maso and Murat [16]
and Cioranescu et al. [6]. Recently, Damlamian and Meunier [18] studied the periodic homogenization
for multivalued Leray–Lions operators in perforated domains.
The case of non homogeneous Neumann boundary conditions was considered, among others, by
Conca and Donato [14] and Onofrei [24].
For problems with Robin or nonlinear boundary conditions we refer, for instance, to [3,7–9]. Also, for
Signorini’s type problems we mention [15,22].
The homogenization of problems involving perforated domains with two kinds of holes of various
sizes, was recently considered by Cioranescu and Hammouda [12].
The non-standard feature of the problem we address in this paper is given, as already mentioned,
by the presence, in each period, of two holes of different sizes and with different conditions (2.1) 2,3
imposed on their boundaries. More precisely, we shall consider the case of Signorini and, respectively,
critical Neumann holes. The Signorini condition (2.1) 2(see [25]) implies that the variational formulation
(2.2) of our problem is expressed as an inequality, which creates further difficulties. The technique we
use in homogenizing our variational inequality leads to a more general result, stated in Lemma 3.1, with
possible applications to other situations.
Problems involving such boundary conditions arise in groundwater hydrology, chemical flows in me-
dia with semipermeable membranes, etc. For more details concerning the physical interpretation of the
above mentioned boundary conditions, the interested reader is referred to [19,21].
Our approach is based on the periodic unfolding method introduced, for fixed domains, by Cioranescu
et al. [4,5] and Damlamian [17]. These results were extended to perforated domains by Cioranescu et al.[9,10] and, further, by Cioranescu et al. [6], Onofrei [24] and Damlamian and Meunier [18] for the case
of small holes.
Finally, let us briefly discuss the structure of this paper. In Section 2, we give the geometrical setting
and the formulation of our problem. Section 3 is devoted to the proof of our main convergence result
stated in Theorem 3.1.
2. Setting of the problem
Let us begin this section by fixing the notation.
LetΩ ⊂R
n,n/greaterorequalslant3, be a bounded open set such that |∂Ω |=0a n dl e t Y=( −1
2,1
2)nbe the reference
cell.
Let/epsilon1be a real parameter taking values in a sequence of positive numbers converging to zero. We
shall consider an /epsilon1Yperiodic perforated structure with two kind of holes: some of size /epsilon1δ1and the other
ones of size /epsilon1δ2, with δ1andδ2depending on /epsilon1and going to zero as /epsilon1goes to zero. More precisely, we
consider two open sets BandTwith smooth boundaries such that B ⊂⊂Y,T ⊂⊂YandB ∩T=∅
and we denote the above mentioned holes by
B/epsilon1δ1=/uniondisplay
ξ ∈Zn/epsilon1(ξ+δ1B),
T/epsilon1δ2=/uniondisplay
ξ ∈Zn/epsilon1(ξ+δ2T).
LetYδ1δ2=Y \(δ1B ∪δ2T) be the part occupied by the material in the cell and suppose that it is
connected.

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46 46The perforated domain Ω/epsilon1,δ1δ2with holes of size of order /epsilon1δ1a n do fs i z eo fo r d e r /epsilon1δ2at the same time,
is defined by
Ω/epsilon1,δ1δ2=Ω \(B/epsilon1δ1 ∪T/epsilon1δ2)=/braceleftbigg
x ∈Ω/vextendsingle/vextendsingle/vextendsingle/braceleftbigg
x
/epsilon1/bracerightbigg
Y∈Yδ1δ2/bracerightbigg
.
LetA ∈L∞(Ω)n ×nbe aY-periodic symmetric matrix. We suppose that there exist two positive
constants αandβ, with 0 <α<β , such that
α |ξ |2/lessorequalslantA(y)ξ ·ξ/lessorequalslantβ |ξ |2∀ξ ∈Rn, ∀y ∈Y.
Moreover, we assume that Ais continuous at the point 0.
Given a Y-periodic function g ∈L2(∂T) and a function f ∈L2(Ω), we consider the problem
⎧
⎪⎪⎨
⎪⎪⎩−div/parenleftbigA/epsilon1∇u/epsilon1,δ1δ2/parenrightbig=f inΩ/epsilon1,δ1δ2,
u/epsilon1,δ1δ2/greaterorequalslant0, A/epsilon1∇u/epsilon1,δ1δ2 ·νB/greaterorequalslant0, u/epsilon1,δ1δ2A/epsilon1∇u/epsilon1,δ1δ2 ·νB=0o n ∂B/epsilon1δ1,
A/epsilon1∇u/epsilon1,δ1δ2 ·νT=g/epsilon1δ2 on∂T/epsilon1δ2,
u/epsilon1,δ1δ2=0o n ∂extΩ/epsilon1,δ1δ2,(2.1)
where
A/epsilon1(x)=A/parenleftbiggx
/epsilon1/parenrightbigg
and
g/epsilon1δ2(x)=g/parenleftbigg1
δ2/braceleftbiggx
/epsilon1/bracerightbigg
Y/parenrightbigg
a.e.x ∈∂T/epsilon1δ2.
In (2.1), νBandνTare the unit exterior normals to the set Band, respectively, T. Obviously, by
construction, they are also the unit exterior normals to B/epsilon1δ1and, respectively, T/epsilon1δ2.
In order to obtain a variational formulation of problem (2.1), we introduce the space
V/epsilon1
δ1δ2=/braceleftbigv ∈H1(Ω/epsilon1,δ1δ2) |v=0o n∂extΩ/epsilon1,δ1δ2/bracerightbig
and the convex set
K/epsilon1
δ1δ2=/braceleftbigv ∈V/epsilon1
δ1δ2|v/greaterorequalslant0o n∂B/epsilon1δ1/bracerightbig.
Therefore, the variational formulation of (2.1) is the following variational inequality:
(P/epsilon1,δ1δ2)⎧
⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎩Findu/epsilon1,δ1δ2 ∈K/epsilon1
δ1δ2such that/integraldisplay
Ω/epsilon1,δ1δ2A/epsilon1∇u/epsilon1,δ1δ2 ·( ∇v −∇u/epsilon1,δ1δ2)dx
/greaterorequalslant/integraldisplay
Ω/epsilon1,δ1δ2f/epsilon1(v −u/epsilon1,δ1δ2)dx+/integraldisplay
∂T/epsilon1δ2g/epsilon1δ2(v −u/epsilon1,δ1δ2)ds ∀v ∈K/epsilon1
δ1δ2.(2.2)

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46 46Classical results for variational inequalities (see, e.g., [2,20,26]) ensure the existence and the unique-
ness of a weak solution of the problem ( P/epsilon1,δ1δ2).
We are interested in obtaining the asymptotic behavior of the solution of problem (2.2) when
/epsilon1,δ1,δ2 →0.
The results of this paper will be obtained for the case in which
⎧
⎪⎪⎪⎨
⎪⎪⎪⎩k1=lim
/epsilon1 →0δn/2 −1
1
/epsilon1,0<k 1< ∞,
k2=lim
/epsilon1 →0δn −1
2
/epsilon1,0 <k 2< ∞,(2.3)
which means that we are dealing with the case of critical size, both for the Signorini holes and, respec-
tively, the Neumann ones. Due to (2.3), we shall write that /epsilon1 →0 instead ( /epsilon1,δ1,δ2) →( 0 ,0 ,0 ) .
As we shall see later (see Remark 3.2), the cases k1=0o rk2=0 are much simpler.
3. The homogenization result
In order to state the main convergence result of our paper, let us introduce the following functional
space
KB=/braceleftbigv ∈L2∗/parenleftbigRn/parenrightbig; ∇v ∈L2/parenleftbigRn/parenrightbig,v=ct. onB/bracerightbig,
where 2∗is the Sobolev exponent2n
n −2associated to 2.
Also, for i=1,n, let us consider χithe solution of the cell problem
χi ∈H1
per(Y),/integraldisplay
YA ∇(χi −yi) ·∇φdy=0 ∀φ ∈H1
per(Y) (3.1)
andθthe solution of the problem
θ ∈KB,θ(B)=1,/integraldisplay
Rn \BA(0) ∇θ ·∇vdz=0 ∀v ∈KBwithv(B)=0. (3.2)
The main result of our paper is stated in the following theorem.
Theorem 3.1. Letu/epsilon1,δ1δ2be the solution of the variational inequality (2.2). Under the above hypotheses ,
there exists u ∈H1
0(Ω)such that
T/epsilon1(u/epsilon1,δ1δ2)/arrowrighttophalfu weakly in L2/parenleftbigΩ;H1(Y)/parenrightbig, (3.3)
where u ∈H1
0(Ω)is the unique solution of the homogenized problem
u ∈H1
0(Ω),/integraldisplay
ΩAhom∇u ·∇ϕdx −k2
1/integraldisplay
Ωμu−ϕdx=/integraldisplay
Ωfϕdx +k2 |∂T |M∂T(g)/integraldisplay
Ωϕdx
∀ϕ ∈H1
0(Ω). (3.4)

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46 46In(3.4),Ahomis the classical homogenized matrix defined ,in terms of χisolution of (3.1), as
Ahom
ij =/integraldisplay
Y/parenleftBigg
aij(y) −n/summationdisplay
k=1aik(y)∂χj
∂yk(y)/parenrightBigg
dy
andμis the capacity of the set B,given by
μ=/integraldisplay
Rn \BA(0) ∇zθ ·∇zθdz,
where θverifies (3.2).
Remark 3.1. In the limit problem (3.4), we can see the presence of two extra terms, generated by the
suitable sizes of our holes. Also, let us notice in (3.4) the spreading effect of the unilateral condition
imposed on the boundary of the Signorini holes: the strange term, depending on the matrix A,c h a r g e s
only the negative part u−of the solution.
Remark 3.2. In the case k1=0, the extra term generated by the Signorini holes vanishes in the limit,
while for k2=0 the contribution of the Neumann holes disappears.
Proof of Theorem 3.1. It is well known that the variational inequality (2.2) is equivalent to the following
minimization problem
/braceleftbiggFindu/epsilon1,δ1δ2 ∈K/epsilon1
δ1δ2such that
J/epsilon1
δ1δ2(u/epsilon1,δ1δ2)/lessorequalslantJ/epsilon1
δ1δ2(v) ∀v ∈K/epsilon1
δ1δ2,(3.5)
where
J/epsilon1
δ1δ2(v)=1
2/integraldisplay
Ω/epsilon1,δ1δ2A/epsilon1 ∇v ·∇vdx −/integraldisplay
Ω/epsilon1,δ1δ2fvdx −/integraldisplay
∂T/epsilon1δ2g/epsilon1δ2vds. (3.6)
We begin our proof by showing that
lim sup
/epsilon1 →0J/epsilon1
δ1δ2(u/epsilon1,δ1δ2)/lessorequalslantJ0(ϕ) ∀ϕ ∈D(Ω), (3.7)
where
J0(ϕ)=1
2/integraldisplay
ΩAhom∇ϕ ·∇ϕdx+1
2k2
1/integraldisplay
Ωμ/parenleftbigϕ−/parenrightbig2dx
−/integraldisplay
Ωfϕdx+k2 |∂T |M∂T(g)/integraldisplay
Ωϕdx. (3.8)
Forϕ ∈D(Ω), we put
h/epsilon1(x)=ϕ(x) −/epsilon1n/summationdisplay
i=1∂ϕ
∂xi(x)χi/parenleftbiggx
/epsilon1/parenrightbigg
,

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46 46where each χiis the solution of the problem (3.1).
Let us take v/epsilon1δ1=h+
/epsilon1 −w/epsilon1δ1h−/epsilon1,w h e r e
w/epsilon1δ1(x)=1 −θ/parenleftbigg1
δ1/braceleftbiggx
/epsilon1/bracerightbigg
Y/parenrightbigg
∀x ∈Rn,
withθgiven by (3.2).
We have
J/epsilon1
δ1δ2(v/epsilon1δ1)=I1
/epsilon1 −I2
/epsilon1,
where
I1
/epsilon1=1
2/integraldisplay
Ω/epsilon1,δ1δ2A/epsilon1/parenleftbig∇h+
/epsilon1 −w/epsilon1δ1 ∇h−/epsilon1−h−/epsilon1∇w/epsilon1δ1/parenrightbig·/parenleftbig∇h+/epsilon1−w/epsilon1δ1 ∇h−/epsilon1−h−/epsilon1∇w/epsilon1δ1/parenrightbigdx,
I2
/epsilon1=/integraldisplay
Ω/epsilon1,δ1δ2f/epsilon1/parenleftbigh+/epsilon1−w/epsilon1δ1h−/epsilon1/parenrightbigdx+/integraldisplay
∂T/epsilon1δ2g/epsilon1/parenleftbigh+/epsilon1−w/epsilon1δ1h−/epsilon1/parenrightbigds.
Using the periodic unfolding operators T/epsilon1and T/epsilon1δ1introduced by [4,6] we get
T/epsilon1(A/epsilon1)(x,y)=A(y)i n Ω ×Y,
T/epsilon1δ1(w/epsilon1δ1)(x,z)= T/epsilon1(w/epsilon1δ1)(x,δ1z)=1 −θ(z)i nΩ ×Rn,
T/epsilon1δ1( ∇w/epsilon1δ1)(x,z)= −1
/epsilon1δ1∇zθ(z)i n Ω ×Rn.
Thus, we obtain
I1
/epsilon1=1
2/integraldisplay
Ω ×YT/epsilon1(A/epsilon1) T/epsilon1/parenleftbig∇h+/epsilon1·∇h+/epsilon1+(w/epsilon1δ1)2∇h−/epsilon1·∇h−/epsilon1/parenrightbigdxdy
+δn
1/integraldisplay
Ω ×RnT/epsilon1δ1(A/epsilon1)(x,y) T/epsilon1δ1(w/epsilon1δ1) T/epsilon1δ1/parenleftbigh−/epsilon1/parenrightbigT/epsilon1δ1/parenleftbig∇h−/epsilon1/parenrightbig·T/epsilon1δ1( ∇w/epsilon1δ1)dxdz
+δn
1
2/integraldisplay
Ω ×RnT/epsilon1δ1(A/epsilon1)/parenleftbigT/epsilon1δ1/parenleftbigh−/epsilon1/parenrightbig/parenrightbig2T/epsilon1δ1( ∇w/epsilon1δ1) ·T/epsilon1δ1( ∇w/epsilon1δ1)dxdz
=1
2/integraldisplay
Ω ×YA(y)/parenleftbigT/epsilon1( ∇h/epsilon1) ·T/epsilon1( ∇h/epsilon1)+ T/epsilon1/parenleftbig∇h−/epsilon1/parenrightbig·T/epsilon1( ∇h−/epsilon1) T/epsilon1/parenleftbigw2
/epsilon1δ1−1/parenrightbig/parenrightbigdxdy
−δn/2 −1
1
/epsilon1δn/2
1/integraldisplay
Ω ×RnA(δ1z)/parenleftbig1 −θ(z)/parenrightbigT/epsilon1δ1/parenleftbigh−/epsilon1/parenrightbigT/epsilon1δ1( ∇h/epsilon1) ·∇zθ(z)dxdz
+1
2/parenleftbiggδn/2 −1
1
/epsilon1/parenrightbigg2/integraldisplay
Ω ×RnA(δ1z)/parenleftbigT/epsilon1δ1/parenleftbigh−/epsilon1/parenrightbig/parenrightbig2∇zθ(z) ·∇zθ(z)dxdz. (3.9)
Obviously, we have
/braceleftbiggw/epsilon1δ1/arrowrighttophalf1 weakly in H1(Ω),
h/epsilon1 →ϕ strongly in H1(Ω).(3.10)

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46 46Taking into account the properties of the unfolding operator T/epsilon1and T/epsilon1δ1, we get the following conver-
gences
⎧
⎨
⎩T/epsilon1(h/epsilon1) →ϕ strongly in L2(Ω ×Y),
T/epsilon1( ∇h/epsilon1) →∇ xϕ+ ∇yϕ1 strongly in L2(Ω ×Y),
T/epsilon1δ1( ∇h/epsilon1) →∇ xϕ+ ∇yϕ1strongly in L2/parenleftbigΩ ×Rn/parenrightbig,(3.11)
where ϕ1=/summationtextn
i=1∂ϕ
∂xiχi.
Now, from (3.10), (3.11) and the fact that { ∇h−
/epsilon1}/epsilon1is bounded in ( L2(Ω))n, we can pass to the limit in
(3.9) and we obtain
lim
/epsilon1 →0I1
/epsilon1=1
2/integraldisplay
Ω ×YA( ∇ϕ+ ∇yϕ1) ·( ∇ϕ+ ∇yϕ1)dxdy
+1
2k2
1/integraldisplay
Ω ×(Rn \B)A(0)/parenleftbigϕ−/parenrightbig2∇zθ ·∇zθdxdz,
which, combined with (3.1), yields
lim
/epsilon1 →0I1
/epsilon1=1
2/integraldisplay
Ω ×YAhom∇ϕ ·∇ϕdxdy+1
2k2
1/integraldisplay
Ω ×(Rn \B)A(0)/parenleftbigϕ−/parenrightbig2∇zθ ·∇zθdxdz. (3.12)
Following the same technique as in [12], i.e. using the boundary unfolding operator Tb
/epsilon1δ2, we can pass
easily to the limit in I2
/epsilon1and we obtain
lim
/epsilon1 →0I2
/epsilon1=/integraldisplay
Ωfϕdx+k2 |∂T |M∂T(g)/integraldisplay
Ωϕdx. (3.13)
Therefore, putting together (3.12) and (3.13), we get
lim
/epsilon1 →0J/epsilon1
δ1δ2(v/epsilon1δ1)=J0(ϕ) ∀ϕ ∈D(Ω). (3.14)
From (3.5) and (3.14), it results (3.7).
In the sequel, we intent to prove that
lim inf
/epsilon1 →0J/epsilon1,δ1δ2(u/epsilon1,δ1δ2)/greaterorequalslantJ0(u). (3.15)
To this end, let us decompose our solution into its positive and, respectively, its negative part, i.e.
u/epsilon1,δ1δ2=u+
/epsilon1,δ1δ2−u−/epsilon1
,δ1δ2.
From the problem ( P/epsilon1,δ1δ2), it follows that there exists a constant Csuch that
/bardblu/epsilon1,δ1δ2 /bardblH1(Ω/epsilon1,δ1δ2)/lessorequalslantC. (3.16)

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46 46Sinceu/epsilon1,δ1δ2 ∈V/epsilon1
δ1δ2, we can assume that, up to a subsequence, there exists u ∈H1
0(Ω) such that
⎧
⎪⎨
⎪⎩T/epsilon1(u/epsilon1,δ1δ2)/arrowrighttophalfu weakly in L2/parenleftbigΩ;H1(Y)/parenrightbig,/vextenddouble/vextenddoubleu−
/epsilon1,δ1δ2−u−/vextenddouble/vextenddouble
L2(Ω/epsilon1,δ1δ2)→0,
T/epsilon1/parenleftbigu−
/epsilon1,δ1δ2/parenrightbig→u−strongly in L2(Ω ×Y).(3.17)
Obviously, we have
lim inf
/epsilon1 →0/integraldisplay
Ω/epsilon1,δ1δ2A/epsilon1∇u+/epsilon1
,δ1δ2·∇u+/epsilon1
,δ1δ2dx/greaterorequalslant/integraldisplay
ΩAhom∇u+·∇u+dx. (3.18)
In order to get (3.15), taking into account that the linear terms pass immediately to the limit, it remains
only to prove that
lim inf
/epsilon1 →0/integraldisplay
Ω/epsilon1,δ1δ2A/epsilon1∇u−/epsilon1
,δ1δ2·∇u−/epsilon1
,δ1δ2dx/greaterorequalslant/integraldisplay
ΩAhom∇u−·∇u−dx+k2
1/integraldisplay
Ωμ/parenleftbigu−/parenrightbig2dx. (3.19)
Since
/integraldisplay
Ω/epsilon1,δ1δ2A/epsilon1∇/parenleftbigu−/epsilon1
,δ1δ2−h/epsilon1w/epsilon1δ1/parenrightbig·∇/parenleftbigu−/epsilon1
,δ1δ2−h/epsilon1w/epsilon1δ1/parenrightbigdx/greaterorequalslant0,
we have
/integraldisplay
Ω/epsilon1,δ1δ2A/epsilon1∇u−/epsilon1
,δ1δ2·∇u−/epsilon1
,δ1δ2dx
/greaterorequalslant −/integraldisplay
Ω/epsilon1,δ1δ2A/epsilon1(h/epsilon1)2∇w/epsilon1δ1 ·∇w/epsilon1δ1dx −/integraldisplay
Ω/epsilon1,δ1δ2A/epsilon1(w/epsilon1δ1)2∇h/epsilon1 ·∇h/epsilon1dx
+2/integraldisplay
Ω/epsilon1,δ1δ2A/epsilon1h/epsilon1 ∇u−/epsilon1
,δ1δ2·∇w/epsilon1δ1dx+2/integraldisplay
Ω/epsilon1,δ1δ2A/epsilon1w/epsilon1δ1 ∇u−/epsilon1
,δ1δ2·∇h/epsilon1dx
−2/integraldisplay
Ω/epsilon1,δ1δ2A/epsilon1h/epsilon1w/epsilon1δ1 ∇w/epsilon1δ1 ·∇h/epsilon1dx
= −/parenleftbiggδn/2 −1
1
/epsilon1/parenrightbigg2/integraldisplay
Ω ×RnT/epsilon1δ1(A/epsilon1)/parenleftbigT/epsilon1δ1(h/epsilon1)/parenrightbig2∇zθ ·∇zθdxdz
−/integraldisplay
Ω ×YT/epsilon1(A/epsilon1) T/epsilon1( ∇h/epsilon1) ·T/epsilon1( ∇h/epsilon1)/parenleftbigT/epsilon1(w/epsilon1δ1)/parenrightbig2dxdy
−2δn/2 −1
1
/epsilon1/integraldisplay
Ω ×RnT/epsilon1δ1(A/epsilon1) T/epsilon1δ1(h/epsilon1)/bracketleftbigδn/2
1 T/epsilon1δ1/parenleftbig∇u−/epsilon1
,δ1δ2/parenrightbig/bracketrightbig·∇zθ(z)dxdz
+2/integraldisplay
Ω ×YT/epsilon1(A/epsilon1) T/epsilon1(w/epsilon1δ1) T/epsilon1/parenleftbig∇u−/epsilon1
,δ1δ2/parenrightbig·T/epsilon1( ∇h/epsilon1)dx
+2δn/2 −1
1
/epsilon1δn/2
1/integraldisplay
Ω ×RnT/epsilon1δ1(A/epsilon1) T/epsilon1δ1(h/epsilon1)/parenleftbig1 −θ(z)/parenrightbig∇zθ ·T/epsilon1δ1( ∇h/epsilon1)dxdz.

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46 46From (3.16) and (3.17), it follows that there exist u1 ∈L2(Ω;H1
per(Y)) and U1 ∈L2(Ω;L2
loc(Rn))
such that
/braceleftBiggT/epsilon1/parenleftbig∇u−
/epsilon1,δ1δ2/parenrightbig/arrowrighttophalf ∇u−+ ∇yu1weakly in L2(Ω ×Y),
δn/2
1 T/epsilon1δ1/parenleftbig∇u−/epsilon1
,δ1δ2/parenrightbig/arrowrighttophalf ∇z U1 weakly in L2/parenleftbigΩ ×Rn/parenrightbig.(3.20)
Therefore, we obtain
lim inf
/epsilon1 →0/integraldisplay
Ω/epsilon1,δ1δ2A/epsilon1∇u−/epsilon1
,δ1δ2·∇u−/epsilon1
,δ1δ2dx
/greaterorequalslant −k2
1/integraldisplay
Ωμϕ2dx −/integraldisplay
Ω ×YA( ∇ϕ+ ∇yϕ1) ·( ∇ϕ+ ∇yϕ1)dxdy
−2k1/integraldisplay
Ω ×(Rn \B)A0ϕ ∇z U1 ·∇zθdxdz
+2/integraldisplay
Ω ×YA/parenleftbig∇u−+ ∇yu1/parenrightbig·( ∇ϕ+ ∇yϕ1)dxdy ∀ϕ ∈H1
0(Ω). (3.21)
Since T/epsilon1δ1(u−/epsilon1
,δ1δ2)=0o nΩ ×B,w eh a v e U1=0o nΩ ×B. Therefore, W1= U1 −k1u−∈
L2(Ω;KB).
On the other hand, the “cell problem” (3.2), implies
−divz/parenleftbigA(0) ∇zθ/parenrightbig=0i n D/prime/parenleftbigΩ ×/parenleftbigRn\¯B/parenrightbig/parenrightbig
which, by Stokes formula, gives
/integraldisplay
Rn \BA(0) ∇zθ ·∇zψdz=ψ(B)/integraldisplay
∂BA(0) ∇zθ ·νBds ∀ψ ∈KB. (3.22)
For almost every x ∈Ω,W1(x, ·) ∈KB, so (3.22) gives
/integraldisplay
Rn \BA(0) ∇zθ ·∇zW1dz=W1(x,B)/integraldisplay
∂BA(0) ∇zθ ·νBds.
Now, since ∇zW1= ∇z U1and U1(x,B)=0, we obtain
/integraldisplay
Rn \BA(0) ∇zθ ·∇z U1dz= −k1u−/integraldisplay
∂BA(0) ∇zθ ·νBds= −k1u−μ
which gives
−2k1/integraldisplay
Ω ×(Rn \B)A(0)ϕ ∇z U1 ·∇zθdx=2k2
1/integraldisplay
Ωμu−ϕdx.
Taking ϕ=u−in (3.21) and using the fact that/summationtextn
i=1∂u−
∂xiχi=u1, we obtain (3.19).

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46 46Finally, from (3.7), (3.15) and by the density D(Ω)/arrowhookleft →H1
0(Ω) , we deduce
lim
/epsilon1 →0J/epsilon1
δ1δ2(u/epsilon1,δ1δ2)=J0(u)/lessorequalslantJ0(ϕ) ∀ϕ ∈H1
0(Ω). (3.23)
Asμis non-negative, by Lax–Milgram theorem, it follows that the minimum point for the func-
tional J0is unique. This means that the whole sequence T/epsilon1(u/epsilon1,δ1δ2)c o n v e r g e st o u.
Using a classical trick (see, e.g., [13,23]), one can prove that the functional
P(v)=1
2/integraldisplay
Ω/parenleftbigv−/parenrightbig2dx ∀v ∈H1
0(Ω)
is Fréchet (and thus Gâteaux) differentiable and its gradient is given by
P/prime(u) ·v= −/integraldisplay
Ωu−vdx ∀u,v ∈H1
0(Ω).
Therefore, the functional J0is Gâteaux differentiable on H1
0(Ω) and this ensures the equivalence of the
minimization problem
J0(u)=min
ϕ ∈H1
0(Ω)J0(ϕ) (3.24)
with the problem (3.4) and this ends the proof of Theorem 3.1. /fill50
Remark 3.3. From (3.16), it follows that there exists an extension /hatwideu/epsilon1,δ1δ2of our solution to the whole
ofΩ, positive on the Signorini holes (see [15]), such that
/hatwideu/epsilon1,δ1δ2/arrowrighttophalfu weakly in H1
0(Ω). (3.25)
For instance, in a first step we extend our solution inside the Signorini holes in such a way that
/braceleftbiggΔ/hatwideu/epsilon1,δ1δ2=0i n B/epsilon1δ1,
/hatwideu/epsilon1,δ1δ2=u/epsilon1,δ1δ2on∂B/epsilon1δ1,
and, then, we extend it further in a standard way (see, e.g., [14]) inside the Neumann holes. As a matter
of fact, the use of unfolding operators allows us to work without extending our solution to the whole
ofΩ.
The proof of Theorem 3.1 shows, in fact, that we obtained a slightly more general result, stated in the
following lemma.
Lemma 3.1. LetΩ ⊂Rnbe a bounded open set with |∂Ω |=0.Let us perforated it with critical
holesS/epsilon1.Then ,under our hypotheses on the matrix A,for any z/epsilon1 ∈H1
0(Ω)such that z/epsilon1=0onS/epsilon1and
z/epsilon1/arrowrighttophalfz weakly în H1
0(Ω),we have
lim inf
/epsilon1 →0/integraldisplay
ΩA/epsilon1∇z/epsilon1 ·∇z/epsilon1dx/greaterorequalslant/integraldisplay
ΩAhom∇z ·∇zdx+/angbracketleftbigμ,z2/angbracketrightbig.

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46 46Remark 3.4. We can treat in a similar manner the problem (2.1) for a general matrix Asatisfying the
usual conditions of boundedness and coercivity but we have to suppose, like in [6] or [18], that there
exist two matrix fields AandA0, such that
T/epsilon1/parenleftbigA/epsilon1/parenrightbig(x,y) →A(x,y)a . e . i n Ω ×Y
and
T/epsilon1δ1/parenleftbigA/epsilon1/parenrightbig(x,z) →A0(x,z)a . e . i n Ω ×/parenleftbigRn\B/parenrightbig.
The only difference is the fact that in this case the corresponding homogenized matrix, the cell problems
and the strange term μdepend also on x.
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