# On the homogenization of a Signorini-type problem
in a domain with inclusions

Sara Monsurrò∗†, Carmen Perugia‡,† and Federica Raimondi∗†

###### Abstract

In this paper we investigate the effect of a Signorini-type interface
condition on the asymptotic behaviour, as $\varepsilon$ tends to zero, of
problems posed in $\varepsilon$-periodic domains with inclusions. The
Signorini-type condition is expressed in terms of two complementary equalities
involving the jump of the solution on the interface and its conormal
derivative via a parameter $\gamma$. Our problem models the heat exchange in a
medium hosting an $\varepsilon$-periodic array of thermal conductors in
presence of impurities distributed on some regions of the interface. Different
limit problems are obtained according to different values of $\gamma$.
††Keywords: Homogenization, Domains with inclusions, Imperfect interface,
Signorini-type condition, Variational inequality, Unfolding method. ††MSC:
35B27, 35J25, 35J20, 35J60, 35R35.

* Università degli Studi di Salerno, Dipartimento di Matematica, Via Giovanni Paolo II 132, 84084 Fisciano (SA), Italy. Email<EMAIL_ADDRESS><EMAIL_ADDRESS>
$\ddagger$ Università degli Studi del Sannio, Dipartimento di Scienze e
Tecnologie, Via dei Mulini 74, 82100 Benevento, Italy. Email:
<EMAIL_ADDRESS>
$\dagger$ Member of the Gruppo Nazionale per l’Analisi Matematica, la
Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta
Matematica (INdAM).

##  1 Introduction

The Signorini problem was presented for the first time in [39] in the context
of linear elasticity, and, eventually, it was given a rigorous mathematical
justification in [22]. It concerns the equilibrium configuration of an elastic
body on a rigid frictionless surface, subject only to its mass forces.
Signorini-type boundary conditions appear in many problems in applied
mathematics deriving from engineering and physics. We can think, for instance,
to lubrication and filtration processes, hydrodynamics, plasticity, crack
theory, optimal control problems, etc.
The weak formulations of such kind of problems involve variational
inequalities corresponding to nonlinear free boundary-value problems. For
mathematical and historical backgrounds we refer to Duvaut and Lions [19],
Kinderleherer and Stampacchia [27] and Brezis [4]. The unique solvability of
these inequalities has been investigated in [29] and [40] (see also [28]).
Successively, this type of problems have been studied in different geometrical
settings. For instance, the homogenization of boundary value problems
involving Signorini boundary conditions has been treated in [23], [24] and
[30] in domains with oscillating boundaries in both linear and nonlinear
cases. In [7], [14] and [37] perforated domains have been considered. Cracks
and layers were taken into account in the works of [10] and [25].

In our paper we consider an open bounded domain $\Omega$ of $\mathbb{R}^{N}$
made up of two disjoint components $\Omega^{\varepsilon}_{1}$ and
$\Omega^{\varepsilon}_{2}$ separated by a Lipschitz-continuous interface
$\Gamma^{\varepsilon}$, where $\varepsilon$ is a small positive parameter
tending to zero. In this domain we deal with the following problem:

(1.1)
$\left\\{\begin{array}[]{lllll}\displaystyle-\operatorname{div}(A^{\varepsilon}\nabla
u^{\varepsilon}_{1})=f&\hbox{ in }\Omega_{1}^{\varepsilon},\\\\[5.69054pt]
\displaystyle-\operatorname{div}(A^{\varepsilon}\nabla
u^{\varepsilon}_{2})=f&\hbox{ in }\Omega_{2}^{\varepsilon},\\\\[5.69054pt]
\displaystyle(A^{\varepsilon}\nabla
u^{\varepsilon}_{1})\cdot\nu_{1}^{\varepsilon}=-(A^{\varepsilon}\nabla
u^{\varepsilon}_{2})\cdot\nu_{2}^{\varepsilon}&\hbox{ on
}\Gamma^{\varepsilon},\\\\[5.69054pt] \displaystyle[u^{\varepsilon}]\geq
0,\quad(A^{\varepsilon}\nabla
u^{\varepsilon}_{1})\cdot\nu_{1}^{\varepsilon}+\varepsilon^{\gamma}h^{\varepsilon}[u^{\varepsilon}]\geq
0&\hbox{ on }\Gamma^{\varepsilon},\\\\[5.69054pt]
\displaystyle[u^{\varepsilon}]((A^{\varepsilon}\nabla
u^{\varepsilon}_{1})\cdot\nu_{1}^{\varepsilon}+\varepsilon^{\gamma}h^{\varepsilon}[u^{\varepsilon}])=0&\hbox{
on }\Gamma^{\varepsilon},\\\\[5.69054pt] \displaystyle
u^{\varepsilon}_{1}=0&\hbox{ on }\partial\Omega,\end{array}\right.$

where $\nu_{i}^{\varepsilon}$ is the unit external normal vector to
$\Omega_{i}^{\varepsilon}$, $i=1,2$ and $[\cdot]$ is the jump of the solution
on the interface. In (1.1), the coefficients $A^{\varepsilon}$ and
$h^{\varepsilon}$ satisfy suitable hypotheses of uniform ellipticity,
boundedness and periodicity, $\gamma\in\mathbb{R}$ and $f\in L^{2}(\Omega)$.
Problem (1.1) can modelize the heat exchange in a medium hosting an
$\varepsilon$-periodic array of thermal conductors. Actually, in nature the
presence of impurities distributed in some regions of the interface may give
rise to a discontinuity of the temperature field. This is why the Signorini
interface condition in the fourth and fifth lines of (1.1) seems more
realistic to describe such real phenomena. Indeed, it means that one can
distinguish two a priori unknown subsets of $\Gamma^{\varepsilon}$ where the
couple $u^{\varepsilon}=(u^{\varepsilon}_{1},u^{\varepsilon}_{2})$ satisfies
the two alternative equalities:

$[u^{\varepsilon}]=0\qquad\text{or}\qquad(A^{\varepsilon}\nabla
u^{\varepsilon}_{1})\cdot\nu_{1}^{\varepsilon}+\varepsilon^{\gamma}h^{\varepsilon}[u^{\varepsilon}]=0.$

From a physical point of view there exist a zone of $\Gamma^{\varepsilon}$ of
perfect conduction i.e. absence of jump, and one characterized by an imperfect
contact transmission condition. Here, the external thermal flux is
proportional to the jump through a coefficient depending on the period
$\varepsilon$ and on the parameter $\gamma$ representing the order of
magnitude of the resistance with respect to $\varepsilon$.

Our aim is to investigate the effect of the Signorini interface condition on
the asymptotic behaviour, as $\varepsilon$ tends to zero, of problem (1.1).

We prove that the thermal properties of the limit problem vary according to
the different values of the parameter $\gamma$. For $\gamma<-1$ and
$-1<\gamma<1$ the homogenized problems are linear and standard (see [17],
[18], [31]) meaning that the Signorini-type interface condition in the
$\varepsilon$-problem does not affect the limit behaviour.

More precisely, if $\gamma<-1$, we get the classical homogenized problem of
Bensoussan-Lions-Papanicolau in a fixed domain, see [3]. Actually, the limit
problem behaves as in presence, at $\varepsilon$-level, of the only
transmission condition on the interface of type $[u^{\varepsilon}]=0$. Hence,
the part of $\Gamma^{\varepsilon}$ where the two temperature fields differ is
such small that it does not affect the asymptotic behaviour and we do not have
a variational inequality anymore.

On the contrary, when $\gamma\in]-1,1[$, we obtain the same limit behaviour of
[17] and [18] where, at $\varepsilon$-level, the only condition
$(A^{\varepsilon}\nabla
u^{\varepsilon}_{1})\cdot\nu_{1}^{\varepsilon}+\varepsilon^{\gamma}h^{\varepsilon}[u^{\varepsilon}]=0$
is assumed on the interface. More in particular, we get the classical
homogenized problem of Cioranescu-S.J. Paulin in the case of a perforated
domain with Neumann condition on the boundary of the holes, see [12]. Indeed,
the flux related to $u^{\varepsilon}_{2}$ asymptotically vanishes and the
effective diffusion of the first phase is the one obtained when there is no
material occupying $\Omega^{\varepsilon}_{2}$. However, in the limit problem
one has $f$ and not $\theta_{1}f$ ($\theta_{1}$ being the proportion of the
material occupying $\Omega_{1}^{\varepsilon}$) meaning that the heat exchange
takes into account also the source term inside the inclusions.

The most interesting cases are $\gamma=-1$ and $\gamma=1$ since the two
regions of $\Gamma^{\varepsilon}$ where the Signorini conditions are satisfied
at $\varepsilon$-level are both significant at the limit.

For $\gamma=-1$, we obtain a nonlinear homogenized problem (cf. (2.10)) where
the effective diffusion term takes into account the presence of the contact
barrier. Indeed, the associated nonlinear operator is defined through the
solution of a cell problem consisting in a variational inequality with a jump
on the interface $\Gamma$ (cf. (2.15)-(2.16)). Here, the nonnegativity
constraint of the jump on $\Gamma^{\varepsilon}$ gives rise to a nonnegative
jump on the interface $\Gamma$ in the cell problem.

At last, when $\gamma=1$, the Signorini-type interface condition at
$\varepsilon$-level leads to an obstacle-type homogenized problem (cf.
(2.23)). Unlike previous cases, at the limit we get two different temperature
fields. In some sense, as in the case $\gamma\in]-1,1[$, the part of the
interface where the condition $(A^{\varepsilon}\nabla
u^{\varepsilon}_{1})\cdot\nu_{1}^{\varepsilon}+\varepsilon^{\gamma}h^{\varepsilon}[u^{\varepsilon}]=0$
is assumed prevails on the one where $[u^{\varepsilon}]=0$ holds. However, the
gap between the two temperatures on $\Gamma^{\varepsilon}$ is so big that,
even if the flux related to $u^{\varepsilon}_{2}$ asymptotically vanishes as
for $\gamma\in]-1,1[$, the material occupying the inclusions contributes to
the heat exchange. Indeed, the initial jump between the two temperature fields
does not vanish but, at the limit, it reflects on the whole domain. As a
consequence, the homogenized problem consists in a variational inequality with
a lower order term depending on the physical properties of the interface and
the gap between the two temperature fields.

The main difficulty of this paper consists in the choice of appropriate test
functions allowing to identify the different limit problems. Indeed, only for
$\gamma<-1$ it is possible to use quite standard arguments. For $\gamma=-1$,
the required test functions lead to handle products of two weakly convergent
sequences. Lower semi-continuity arguments, suitable in the framework of
variational inequalities, allow us to manage this issue. Finally, in the cases
$\gamma\in]-1,1[$ and $\gamma=1$ we need to consider more regular test
functions together with opportune density results. Moreover, when $\gamma=1$,
we take advantage of lower semi-continuity arguments again.

The homogenization of boundary value problems involving this kind of composite
domains was performed at first by using the method of Tartar of oscillating
test functions in [31] and [32] for $\gamma\leq-1$ and in [18] for
$\gamma>-1$. Eventually, the same results were obtained by means of the
unfolding method in [17] and [16] in the linear and nonlinear case,
respectively. Further problems involving two-component domains with jump
conditions have been treated in [1], [2], [5], [6], [20], [36], [38] (and the
references therein) for what concerns the asymptotic behaviour, and in [21],
[33], [34] and [35] as for control problems.

The paper is organized into several sections. In Section 2 we set the problem
by describing the two component domain $\Omega$ and the functional framework.
Then, we state our main theorem describing the homogenization result. In
Section 3 we show existence, uniqueness and a uniform a priori estimate of the
solution to the fine-scale problem. In Section 4 we briefly recall the
definition and the properties of the unfolding operators in domains with
inclusions by adopting the same notations as in [16]. Finally, in Section 5 we
detail the proof of the main result according to the different values of the
parameter $\gamma$.

##  2 Setting of the problem and main result

###  2.1 The two-component domain

Let $\Omega$ be a bounded open set in $\mathbb{R}^{N}$, $N\geq 2$, with a
Lipschitz-continuous boundary $\partial\Omega$. We denote by
$Y\doteq\prod_{i=1}^{N}[0,l_{i}[$ the reference cell, with
$l_{i}>0,i=1,...,N$. We assume that $Y\doteq Y_{1}\cup\overline{Y_{2}}$, where
$Y_{1}$ and $Y_{2}$ are two disjoint connected open sets such that
$Y_{2}\neq\emptyset$, $\overline{Y_{2}}\subset Y$ and the common boundary
$\Gamma\doteq\partial Y_{2}$ is Lipschitz-continuous.
For any $k\in\mathbb{Z}^{N}$, let $k_{l}\doteq(k_{1}l_{1},...,k_{N}l_{N})$ and

$Y^{k}\doteq k_{l}+Y,\qquad Y^{k}_{i}\doteq k_{l}+Y_{i},\qquad i=1,2.$

Given a positive parameter $\varepsilon$ taking values in a sequence
converging to zero, we set:

$J_{\varepsilon}\doteq\\{k\in\mathbb{Z}^{N}|\
\varepsilon\overline{Y^{k}_{2}}\subset\Omega\\}$

and

$\Omega_{2}^{\varepsilon}\doteq\bigcup_{k\in J_{\varepsilon}}\varepsilon
Y^{k}_{2},\qquad\Omega_{1}^{\varepsilon}\doteq\Omega\setminus\overline{\Omega_{2}^{\varepsilon}},\qquad\Gamma^{\varepsilon}\doteq\partial\Omega_{2}^{\varepsilon}.$

Therefore, one has
$\Omega=\Omega_{1}^{\varepsilon}\cup\Omega_{2}^{\varepsilon}\cup\Gamma^{\varepsilon}$.
Moreover, for the sake of simplicity, we suppose that the second component
doesn’t meet the boundary of the domain.

Throughout the paper

  * •

$\displaystyle\mathcal{M}_{E}(v)$ is the average on $E$ of a function $v\in
L^{1}(E)$, where $E$ is an open subset of $\mathbb{R}^{N-1}$ or
$\mathbb{R}^{N}$;

  * •

$\displaystyle\chi_{\omega}$ is the characteristic function of any open set
$\omega$ of $\mathbb{R}^{N}$;

  * •

$\sim$ is the zero extension to the whole of $\Omega$ of functions defined on
$\Omega^{\varepsilon}_{1}$ or $\Omega^{\varepsilon}_{2}$;

  * •

$\theta_{i}\doteq\frac{|Y_{i}|}{|Y|}$, $i=1,2$;

  * •

$C$ denotes different positive constants independent of $\varepsilon$;

  * •

$\mathcal{M}(\alpha,\beta,Y)$ is the set of the $Y$-periodic matrix fields
$A=(a_{i,j})_{1\leq i,j\leq N}\in(L^{\infty}(Y))^{N\times N}$ such that
$(A(x)\lambda,\lambda)\geq\alpha|\lambda|^{2}$ and
$|A(x)\lambda|\leq\beta|\lambda|,\ \forall\lambda\in\mathbb{R}^{N}$ and a.e.
in $Y$, with $\alpha,\beta\in\mathbb{R}$, $0<\alpha<\beta$;

  * •

$\\{e_{1},...,e_{N}\\}$ is the canonical basis of $\mathbb{R}^{N}$;

  * •

$[v]$ denotes the jump through $\Gamma^{\varepsilon}$ of a function
$v=(v_{1},v_{2})$ where $v_{i}$ is defined in $\Omega_{i}^{\varepsilon}$,
$i=1,2$.

###  2.2 The problem

In the domain previously described, we consider the following problem:

(2.1)
$\left\\{\begin{array}[]{lllll}\displaystyle-\operatorname{div}(A^{\varepsilon}\nabla
u^{\varepsilon}_{1})=f&\hbox{ in }\Omega_{1}^{\varepsilon},\\\\[5.69054pt]
\displaystyle-\operatorname{div}(A^{\varepsilon}\nabla
u^{\varepsilon}_{2})=f&\hbox{ in }\Omega_{2}^{\varepsilon},\\\\[5.69054pt]
\displaystyle(A^{\varepsilon}\nabla
u^{\varepsilon}_{1})\cdot\nu_{1}^{\varepsilon}=-(A^{\varepsilon}\nabla
u^{\varepsilon}_{2})\cdot\nu_{2}^{\varepsilon}&\hbox{ on
}\Gamma^{\varepsilon},\\\\[5.69054pt] \displaystyle[u^{\varepsilon}]\geq
0,\quad(A^{\varepsilon}\nabla
u^{\varepsilon}_{1})\cdot\nu_{1}^{\varepsilon}+\varepsilon^{\gamma}h^{\varepsilon}[u^{\varepsilon}]\geq
0&\hbox{ on }\Gamma^{\varepsilon},\\\\[5.69054pt]
\displaystyle[u^{\varepsilon}]((A^{\varepsilon}\nabla
u^{\varepsilon}_{1})\cdot\nu_{1}^{\varepsilon}+\varepsilon^{\gamma}h^{\varepsilon}[u^{\varepsilon}])=0&\hbox{
on }\Gamma^{\varepsilon},\\\\[5.69054pt] \displaystyle
u^{\varepsilon}_{1}=0&\hbox{ on }\partial\Omega,\end{array}\right.$

where $\gamma\in\mathbb{R}$ and $\nu_{i}^{\varepsilon}$ is the unit external
normal vector to $\Omega_{i}^{\varepsilon}$, $i=1,2$.

Our aim is to study the asymptotic behavior, as $\varepsilon$ goes to zero, of
problem (2.1) under the following assumptions on the data: (A$1$) For any
$\varepsilon>0$, the coeffficients matrix $A^{\varepsilon}(x)\doteq
A\left(\frac{x}{\varepsilon}\right)$ a.e. in $\Omega$, where $A$ belongs to
$\mathcal{M}(\alpha,\beta,Y)$; (A$2$) $\gamma\leq 1$; (A$3$) $f\in
L^{2}(\Omega)$; (A$4$) $h^{\varepsilon}$ is defined by
$h^{\varepsilon}(x)\doteq h(\frac{x}{\varepsilon})$ a.e. on
$\Gamma^{\varepsilon}$, where $h$ is a $Y$-periodic function in
$L^{\infty}(\Gamma)$ such that

$\hbox{there exists }h_{0}\in\mathbb{R}:0<h_{0}<h(y)\hbox{ a.e. on }\Gamma.$

Let us remark that we assume hypothesis (A$2$) since, in [26], H.C. Hummel
showed that one cannot expect uniformly bounded solutions when $\gamma>1$.

###  2.3 The functional framework and the weak formulation

Let $V^{\varepsilon}\doteq\\{v\in H^{1}(\Omega_{1}^{\varepsilon})\ |\
v=0\hbox{ on }\partial\Omega\\}$ endowed with the norm

$\|v\|_{V^{\varepsilon}}\doteq\|\nabla v\|_{L^{2}(\Omega_{1}^{\varepsilon})}.$

It is known (see for instance [12, Lemma $1$]) that a Poincaré inequality in
$V^{\varepsilon}$ holds with a constant $C_{P}$ independent of $\varepsilon$,
that is

(2.2) $\|v\|_{L^{2}(\Omega^{\varepsilon}_{1})}\leq C_{P}\|\nabla
v\|_{L^{2}(\Omega^{\varepsilon}_{1})}\quad\forall v\in V^{\varepsilon}.$

Consequently, the norm in $V^{\varepsilon}$ is equivalent to the one in
$H^{1}(\Omega_{1}^{\varepsilon})$ via a constant independent of $\varepsilon$.
Let $H^{\varepsilon}_{\gamma}$ be the space defined by

$H^{\varepsilon}_{\gamma}\doteq\\{v=(v_{1},v_{2})|\ v_{1}\in V^{\varepsilon},\
v_{2}\in H^{1}(\Omega_{2}^{\varepsilon})\\},$

which, after the identification $\nabla v:=\widetilde{\nabla
v_{1}}+\widetilde{\nabla v_{2}}$, is a Hilbert space equipped with the norm

$\|v\|^{2}_{H^{\varepsilon}_{\gamma}}\doteq\|\nabla
v_{1}\|^{2}_{L^{2}(\Omega_{1}^{\varepsilon})}+\|\nabla
v_{2}\|^{2}_{L^{2}(\Omega_{2}^{\varepsilon})}+\varepsilon^{\gamma}\|[v]\|^{2}_{L^{2}(\Gamma^{\varepsilon})}.$

######  Proposition 2.1 ([15][17][31]).

Let $\gamma\leq 1$. There exist two positive constants $C_{1}$ and $C_{2}$,
independent of $\varepsilon$, such that

$\forall v\in H^{\varepsilon}_{\gamma},\qquad
C_{1}\|v\|^{2}_{V^{\varepsilon}\times
H^{1}(\Omega_{2}^{\varepsilon})}\leq\|v\|^{2}_{H^{\varepsilon}_{\gamma}}\leq
C_{2}(1+\varepsilon^{\gamma-1})\|v\|^{2}_{V^{\varepsilon}\times
H^{1}(\Omega_{2}^{\varepsilon})}.$

Therefore, if $v^{\varepsilon}=(v^{\varepsilon}_{1},v^{\varepsilon}_{2})$ is a
bounded sequence in $H^{\varepsilon}_{\gamma}$, then

$\begin{array}[]{lll}\displaystyle\|v^{\varepsilon}_{1}\|_{H^{1}(\Omega^{\varepsilon}_{1})}\leq
C,\\\\[8.53581pt]
\displaystyle\|v^{\varepsilon}_{2}\|_{H^{1}(\Omega^{\varepsilon}_{2})}\leq
C,\\\\[8.53581pt] \displaystyle\|[v]\|_{L^{2}(\Gamma^{\varepsilon})}\leq
C\varepsilon^{-\frac{\gamma}{2}},\end{array}$

with $C$ positive constant independent of $\varepsilon$.

Let us define the following closed and convex subset of
$H^{\varepsilon}_{\gamma}$:

(2.3) $\displaystyle K^{\varepsilon}_{\gamma}\doteq\\{v\in
H^{\varepsilon}_{\gamma}\ |\ [v]\geq 0\hbox{ on }\Gamma^{\varepsilon}\\}.$

Hence, the weak formulation of problem (2.1) is given by the variational
inequality

(2.4) $\left\\{\begin{array}[]{llll}\displaystyle\text{Find
}u^{\varepsilon}\in K^{\varepsilon}_{\gamma}\text{ such that }\\\\[5.69054pt]
\displaystyle\int_{\Omega_{1}^{\varepsilon}}A^{\varepsilon}\nabla
u^{\varepsilon}_{1}\nabla(v_{1}-u^{\varepsilon}_{1})dx+\int_{\Omega_{2}^{\varepsilon}}A^{\varepsilon}\nabla
u^{\varepsilon}_{2}\nabla(v_{2}-u^{\varepsilon}_{2})dx+\varepsilon^{\gamma}\int_{\Gamma^{\varepsilon}}h^{\varepsilon}[u^{\varepsilon}]([v]-[u^{\varepsilon}])d\sigma\\\\[14.22636pt]
\displaystyle\geq\int_{\Omega_{1}^{\varepsilon}}f(v_{1}-u^{\varepsilon}_{1})dx+\int_{\Omega_{2}^{\varepsilon}}f(v_{2}-u^{\varepsilon}_{2})dx,\quad\forall
v\in K^{\varepsilon}_{\gamma}.\end{array}\right.$

Indeed, by multiplying the first equation in problem (2.1) by
$u^{\varepsilon}$ and integrating by parts, one obtains

(2.5) $\displaystyle\int_{\Omega_{1}^{\varepsilon}}A^{\varepsilon}\nabla
u^{\varepsilon}_{1}\nabla
u^{\varepsilon}_{1}\,dx+\int_{\Omega_{2}^{\varepsilon}}A^{\varepsilon}\nabla
u^{\varepsilon}_{2}\nabla
u^{\varepsilon}_{2}\,dx-\int_{\Gamma^{\varepsilon}}(A^{\varepsilon}\nabla
u^{\varepsilon})\cdot\nu_{1}^{\varepsilon}[u^{\varepsilon}]d\sigma=\int_{\Omega_{1}^{\varepsilon}}fu^{\varepsilon}_{1}\,dx+\int_{\Omega_{2}^{\varepsilon}}fu^{\varepsilon}_{2}\,dx.$

Thus, adding and subtracting
$\varepsilon^{\gamma}\displaystyle\int_{\Gamma^{\varepsilon}}h^{\varepsilon}[u^{\varepsilon}]^{2}\,d\sigma$,
we get

(2.6)
$\begin{array}[]{ll}\displaystyle\int_{\Omega_{1}^{\varepsilon}}A^{\varepsilon}\nabla
u^{\varepsilon}_{1}\nabla
u^{\varepsilon}_{1}\,dx+\int_{\Omega_{2}^{\varepsilon}}A^{\varepsilon}\nabla
u^{\varepsilon}_{2}\nabla
u^{\varepsilon}_{2}\,dx-\int_{\Gamma^{\varepsilon}}((A^{\varepsilon}\nabla
u^{\varepsilon})\cdot\nu_{1}^{\varepsilon}+\varepsilon^{\gamma}h^{\varepsilon}[u^{\varepsilon}])[u^{\varepsilon}]d\sigma+\varepsilon^{\gamma}\int_{\Gamma^{\varepsilon}}h^{\varepsilon}[u^{\varepsilon}]^{2}\,d\sigma\\\\[14.22636pt]
\displaystyle=\int_{\Omega_{1}^{\varepsilon}}fu^{\varepsilon}_{1}\,dx+\int_{\Omega_{2}^{\varepsilon}}fu^{\varepsilon}_{2}\,dx,\end{array}$

which leads to

(2.7) $\displaystyle\int_{\Omega_{1}^{\varepsilon}}A^{\varepsilon}\nabla
u^{\varepsilon}_{1}\nabla
u^{\varepsilon}_{1}\,dx+\int_{\Omega_{2}^{\varepsilon}}A^{\varepsilon}\nabla
u^{\varepsilon}_{2}\nabla
u^{\varepsilon}_{2}\,dx+\varepsilon^{\gamma}\int_{\Gamma^{\varepsilon}}h^{\varepsilon}[u^{\varepsilon}]^{2}\,d\sigma=\int_{\Omega_{1}^{\varepsilon}}fu^{\varepsilon}_{1}\,dx+\int_{\Omega_{2}^{\varepsilon}}fu^{\varepsilon}_{2}\,dx,$

in view of the fifth condition in (2.1).
On the other hand, by multiplying the first equation in problem (2.1) by a
function $v\in K^{\varepsilon}_{\gamma}$, integrating by parts and taking into
account the fourth condition in (2.1), one has

(2.8)
$\begin{array}[]{lll}\displaystyle\int_{\Omega_{1}^{\varepsilon}}A^{\varepsilon}\nabla
u^{\varepsilon}_{1}\nabla
v_{1}\,dx+\int_{\Omega_{2}^{\varepsilon}}A^{\varepsilon}\nabla
u^{\varepsilon}_{2}\nabla
v_{2}\,dx+\varepsilon^{\gamma}\int_{\Gamma^{\varepsilon}}h^{\varepsilon}[u^{\varepsilon}][v]\,d\sigma\\\\[14.22636pt]
\displaystyle=\int_{\Omega}fv\,dx+\int_{\Gamma^{\varepsilon}}((A^{\varepsilon}\nabla
u^{\varepsilon})\cdot\nu_{1}^{\varepsilon}+\varepsilon^{\gamma}h^{\varepsilon}[u^{\varepsilon}])[v]\,d\sigma\geq\int_{\Omega_{1}^{\varepsilon}}fv_{1}\,dx+\int_{\Omega_{2}^{\varepsilon}}fv_{2}\,dx.\end{array}$

Finally, the variational formulation (2.4) is obtained by subtracting (2.7)
from (2.8).

In Section 3 we will prove that, for any fixed $\varepsilon$, problem (2.4)
admits a unique solution.

###  2.4 The main result

The following theorem, which is the main result of this paper, shows that the
asymptotic behaviour of the weak solution of problem (2.1) varies according to
the different values of the parameter $\gamma$.

######  Theorem 2.2.

Under assumptions $(\textbf{A}1)-(\textbf{A}4)$, let
$u^{\varepsilon}=(u^{\varepsilon}_{1},u^{\varepsilon}_{2})$ be the weak
solution of problem (2.1).

  * $\blacktriangleright$

Case $\gamma<1$
There exists $u_{1}\in H^{1}_{0}(\Omega)$ such that

(2.9)
$\displaystyle\widetilde{u^{\varepsilon}_{i}}\rightharpoonup\theta_{i}u_{1}\quad\hbox{
weakly in }L^{2}(\Omega),\ i=1,2,$

where $u_{1}$ is the unique solution of problem

(2.10)
$\left\\{\begin{array}[]{llll}\displaystyle-\operatorname{div}(A^{0}_{\gamma}(\nabla
u_{1}))=f&\hbox{ in }\Omega,\\\\[5.69054pt] \displaystyle u_{1}=0&\hbox{ on
}\partial\Omega.\end{array}\right.$

    * $\bullet$

If $\gamma<-1$,

$A^{0}_{\gamma}(\nabla u_{1})=A^{0}_{\gamma}\nabla u_{1}$

where the constant matrix $A^{0}_{\gamma}$ is given by

(2.11) $A^{0}_{\gamma}=A^{1}_{\gamma}+A^{2}_{\gamma}$

being

(2.12)
$A^{l}_{\gamma}\doteq\left\\{\theta_{l}\mathcal{M}_{Y_{l}}\left(a_{ij}-\sum_{k=1}^{N}a_{ik}\frac{\partial\overline{\chi}_{j}}{\partial
y_{k}}\right)\right\\}_{N\times N},\quad l=1,2$

with
$\overline{\chi}=\left(\overline{\chi}_{1},\ldots,\overline{\chi}_{N}\right)$,
and $\overline{\chi}_{j}$ is the solution of the cell problem

(2.13)
$\left\\{\begin{array}[]{llll}\displaystyle-\operatorname{div}(A\nabla(\overline{\chi}_{j}-y_{j}))=0\quad\hbox{
in }Y,\\\\[5.69054pt] \displaystyle\overline{\chi}_{j}\quad
Y-periodic,\quad\mathcal{M}_{\Gamma}(\overline{\chi}_{j})=0.\end{array}\right.$

Moreover it holds

(2.14) $A^{\varepsilon}\widetilde{\nabla u^{\varepsilon}_{l}}\rightharpoonup
A^{l}_{\gamma}\nabla u_{1}\quad\hbox{ weakly in }(L^{2}(\Omega))^{N},\quad
l=1,2.$

    * $\bullet$

If $\gamma=-1$,

(2.15)
$A^{0}_{-1}(\zeta)\doteq\sum_{i=1}^{2}\frac{1}{|Y|}\int_{Y_{i}}A(y)\left(\zeta+\nabla_{y}\widehat{\chi}_{i}(y,\zeta)\right)dy,\quad\zeta\in\mathbb{R}^{N},$

with $({\widehat{\chi}}_{1}(\cdot,\zeta),{\widehat{\chi}}_{2}(\cdot,\zeta))$
solution of the following cell problem:

(2.16) $\left\\{\begin{array}[]{llll}\displaystyle\text{Find
}({\widehat{\chi}}_{1}(\cdot,\zeta),{\widehat{\chi}}_{2}(\cdot,\zeta))\in\overline{W}\text{
such that }\\\\[5.69054pt]
\displaystyle\frac{1}{|Y|}\int_{Y_{1}}A(y)(\zeta+\nabla_{y}{\widehat{\chi}}_{1}(y,\zeta))(\nabla_{y}z_{1}-\nabla_{y}{\widehat{\chi}}_{1}(y,\zeta))dy\\\\[14.22636pt]
\displaystyle+\frac{1}{|Y|}\int_{Y_{2}}A(y)(\zeta+\nabla_{y}{\widehat{\chi}}_{2}(y,\zeta))(\nabla_{y}z_{2}-\nabla_{y}{\widehat{\chi}}_{2}(y,\zeta))dy\\\\[14.22636pt]
\displaystyle+\frac{1}{|Y|}\int_{\Gamma}h(y)({\widehat{\chi}}_{1}(y,\zeta)-{\widehat{\chi}}_{2}(y,\zeta))((z_{1}-z_{2})-({\widehat{\chi}}_{1}(y,\zeta)-{\widehat{\chi}}_{2}(y,\zeta)))\,dxd\sigma_{y}\geq
0,\\\\[14.22636pt] \displaystyle\forall(z_{1},z_{2})\in
H^{1}_{per}(Y_{1})\times H^{1}(Y_{2}):\ z_{1}\geq z_{2}\hbox{ on
}\Gamma,\end{array}\right.$

where

(2.17) $\displaystyle\overline{W}\doteq\\{(z_{1},z_{2})\in
H^{1}_{per}(Y_{1})\times H^{1}(Y_{2})|\mathcal{M}_{\Gamma}(z)=0\hbox{ and
}z_{1}\geq z_{2}\hbox{ on }\Gamma\\}.$

Moreover it holds

(2.18) $A^{\varepsilon}\widetilde{\nabla
u^{\varepsilon}_{l}}\rightharpoonup\theta_{l}\mathcal{M}_{Y_{l}}\left(A(y)\left(\nabla
u_{1}+\nabla_{y}\widehat{\chi}_{l}(y,\nabla u_{1})\right)\right)\quad\hbox{
weakly in }(L^{2}(\Omega))^{N},\quad l=1,2.$

    * $\bullet$

If $\gamma\in]-1,1[$,

$A^{0}_{\gamma}(\nabla u_{1})=A^{0}_{\gamma}\nabla u_{1}$

where the constant matrix $A^{0}_{\gamma}$ is given by

(2.19)
$A^{0}_{\gamma}\doteq\left\\{\theta_{1}\mathcal{M}_{Y_{1}}\left(a_{i,j}-\sum_{k=1}^{N}a_{i,k}\frac{\partial\chi_{j}}{\partial
y_{k}}\right)\right\\}_{N\times N},$

with $\chi=(\chi_{1},\ldots,\chi_{N})$, being $\chi_{j}$ the solution of the
cell problem

(2.20)
$\left\\{\begin{array}[]{llll}\displaystyle-\operatorname{div}(A(y)\nabla(\chi_{j}-y_{j}))=0&\hbox{
in }Y_{1},\\\\[5.69054pt]
\displaystyle(A(y)\nabla(\chi_{j}-y_{j}))\cdot\nu_{1}=0&\hbox{ on
}\Gamma,\\\\[5.69054pt] \displaystyle\chi_{j}\quad
Y-periodic,\qquad\mathcal{M}_{Y_{1}}(\chi_{j})=0.\end{array}\right.$

Moreover it holds

(2.21) $\left\\{\begin{array}[]{ll}A^{\varepsilon}\widetilde{\nabla
u^{\varepsilon}_{1}}\rightharpoonup A^{0}_{\gamma}\nabla u_{1}&\hbox{ weakly
in }(L^{2}(\Omega))^{N},\\\ A^{\varepsilon}\widetilde{\nabla
u^{\varepsilon}_{2}}\rightharpoonup 0&\hbox{ weakly in
}(L^{2}(\Omega))^{N}.\end{array}\right.$

  * $\blacktriangleright$

Case $\gamma=1$
There exist $(u_{1},u_{2})\in H^{1}_{0}(\Omega)\times L^{2}(\Omega)$ such that

(2.22)
$\displaystyle\widetilde{u^{\varepsilon}_{i}}\rightharpoonup\theta_{i}u_{i}\quad\hbox{
weakly in }L^{2}(\Omega),\ i=1,2,$

and $(u_{1},u_{2})$ is the unique solution of problem

(2.23) $\left\\{\begin{array}[]{lll}\displaystyle\hbox{Find }(u_{1},u_{2})\in
K\hbox{ s.t. }\\\\[5.69054pt] \displaystyle\int_{\Omega}A^{0}_{1}\nabla
u_{1}(\nabla\varphi_{1}-\nabla
u_{1})\,dx+\frac{\mathcal{M}_{\Gamma}(h)}{|Y|}\int_{\Omega}(u_{1}-u_{2})[(\varphi_{1}-\varphi_{2})-(u_{1}-u_{2})]\,dx\\\\[14.22636pt]
\displaystyle\geq\theta_{1}\int_{\Omega}f(\varphi_{1}-u_{1})\,dx+\theta_{2}\int_{\Omega}f(\varphi_{2}-u_{2})\,dx,\
\forall(\varphi_{1},\varphi_{2})\in K,\end{array}\right.$

where $K\doteq\\{(\varphi_{1},\varphi_{2})\in H^{1}_{0}(\Omega)\times
L^{2}(\Omega)\,|\,\varphi_{1}\geq\varphi_{2}\hbox{ in }\Omega\\}$ and the
constant matrix $A^{0}_{1}$ is as in (2.19).
Moreover, it holds (2.21) with $\gamma=1$.

######  Remark 2.3.

We explicitly observe that the most interesting cases are $\gamma=-1$ and
$\gamma=1$. Actually, for $\gamma=-1$, the homogenized problem (2.10) is
nonlinear. In particular, the associated operator is defined through the
solution of a cell problem consisting in a variational inequality presenting a
jump on the interface $\Gamma$ (cf. (2.15)-(2.16)). Here, the positivity
constraint of the jump on $\Gamma^{\varepsilon}$ gives rise to a positive jump
on $\Gamma$ in the cell problem.
When $\gamma=1$, the Signorini-type interface condition at $\varepsilon$ level
leads to an obstacle-type homogenized problem.
In the remaining cases the homogenized problems are linear and standard (see
[17], [18], [31]). Hence, the Signorini-type interface condition in the
$\varepsilon$-problem does not affect the limit behaviour.

##  3 Existence and uniqueness result

We prove here the existence and the uniqueness of the solution of problem
(2.4), for any fixed $\varepsilon$, together with some uniform a priori
estimates.

######  Theorem 3.1.

Under assumptions $(\textbf{A}1)-(\textbf{A}4)$, for any fixed $\varepsilon$,
there exists a unique solution $u^{\varepsilon}\in K^{\varepsilon}_{\gamma}$
of problem (2.4) satisfying

(3.1) $\|u^{\varepsilon}\|_{H^{\varepsilon}_{\gamma}}\leq C,$

with $C$ positive constant independent of $\varepsilon$.

###### Proof.

For any fixed $\varepsilon$, we want to apply [28, Proposition $2.5$ (pag.
179), Theorems $8.2-8.3$ (pag. $248$)]. To this aim, we consider the operator

$\mathcal{A}_{\varepsilon}:H^{\varepsilon}_{\gamma}\rightarrow(H^{\varepsilon}_{\gamma})^{\prime}$

such that

(3.2)
$\displaystyle\langle\mathcal{A}_{\varepsilon}(u),v\rangle_{(H^{\varepsilon}_{\gamma})^{\prime},H^{\varepsilon}_{\gamma}}\doteq\int_{\Omega_{1}^{\varepsilon}}A^{\varepsilon}\nabla
u_{1}\nabla v_{1}dx+\int_{\Omega_{2}^{\varepsilon}}A^{\varepsilon}\nabla
u_{2}\nabla
v_{2}dx+\varepsilon^{\gamma}\int_{\Gamma^{\varepsilon}}h^{\varepsilon}[u][v]d\sigma,$

where $(H^{\varepsilon}_{\gamma})^{\prime}$ denotes the dual space of
$H^{\varepsilon}_{\gamma}$.
Hence, problem (2.4) can be rewritten as

(3.3) $\left\\{\begin{array}[]{lll}\displaystyle\text{Find }u^{\varepsilon}\in
K^{\varepsilon}_{\gamma}\text{ such that }\\\\[5.69054pt]
\displaystyle\langle\mathcal{A}_{\varepsilon}(u^{\varepsilon}),v-u^{\varepsilon}\rangle_{(H^{\varepsilon}_{\gamma})^{\prime},H^{\varepsilon}_{\gamma}}\geq\int_{\Omega_{1}^{\varepsilon}}f(v_{1}-u^{\varepsilon}_{1})dx+\int_{\Omega_{2}^{\varepsilon}}f(v_{2}-u^{\varepsilon}_{2})dx,\quad\forall
v\in K^{\varepsilon}_{\gamma}.\end{array}\right.$

The hemicontinuity, the equi-boundedness and the strict monotonicity of
$\mathcal{A}_{\varepsilon}$ are immediate consequences of $(\textbf{A}1)$ and
$(\textbf{A}4)$. Therefore, by [28, Proposition $2.5$ (pag. 179)] one has that
$\mathcal{A}_{\varepsilon}$ is pseudo-monotone.
Moreover, it is easy to check that $\mathcal{A}_{\varepsilon}$ is also
equicoercive. Hence, the existence and the uniqueness of the solution follow
from [28, Theorems $8.2-8.3$].

Finally, we prove the a priori estimates by choosing $v=0$ as test function in
(3.3). One gets

$\langle\mathcal{A}_{\varepsilon}(u^{\varepsilon}),-u^{\varepsilon}\rangle_{(H^{\varepsilon}_{\gamma})^{\prime},H^{\varepsilon}_{\gamma}}\geq\int_{\Omega_{1}^{\varepsilon}}f(-u^{\varepsilon}_{1})dx+\int_{\Omega_{2}^{\varepsilon}}f(-u^{\varepsilon}_{2})dx.$

Using again $(\textbf{A}1)$ and $(\textbf{A}4)$ and applying the Holder
inequality, we have

(3.4) $\displaystyle\alpha\|(\nabla u^{\varepsilon}_{1},\nabla
u^{\varepsilon}_{2})\|^{2}_{L^{2}(\Omega_{1}^{\varepsilon})\times
L^{2}(\Omega_{2}^{\varepsilon})}+\varepsilon^{\gamma}h_{0}\|[u^{\varepsilon}]\|^{2}_{L^{2}(\Gamma^{\varepsilon})}\leq\langle\mathcal{A}_{\varepsilon}(u^{\varepsilon}),u^{\varepsilon}\rangle_{(H^{\varepsilon}_{\gamma})^{\prime},H^{\varepsilon}_{\gamma}}\leq(\|u^{\varepsilon}_{1}\|_{L^{2}(\Omega_{1}^{\varepsilon})}+\|u^{\varepsilon}_{2}\|_{L^{2}(\Omega_{2}^{\varepsilon})})\|f\|_{L^{2}(\Omega)}.$

In view of (2.2) and Proposition 2.1, the above inequality leads to

$\displaystyle\alpha\|(\nabla u^{\varepsilon}_{1},\nabla
u^{\varepsilon}_{2})\|^{2}_{L^{2}(\Omega_{1}^{\varepsilon})\times
L^{2}(\Omega_{2}^{\varepsilon})}+\varepsilon^{\gamma}h_{0}\|[u^{\varepsilon}]\|^{2}_{L^{2}(\Gamma^{\varepsilon})}\leq
C(C_{p},C_{1})\|u^{\varepsilon}\|_{H^{\varepsilon}_{\gamma}},$

with $C$ positive constant independent of $\varepsilon$. This gives the
result. ∎

##  4 The periodic unfolding method

In this section we recall the definitions and the properties of the periodic
unfolding operators for two-component domains introduced by P. Donato, K.H. Le
Nguyen and R. Tardieu in [17]. Throughout the paper, we use the same notation
of [16].

Let $z\in\mathbb{R}^{N}$, we denote by $[z]_{Y}$ its integer part such that
$z-[z]_{Y}$ belongs to $Y$ and set $\\{z\\}_{Y}\doteq z-[z]_{Y}$. Then, for
every positive $\varepsilon$,

$x=\varepsilon\left(\left[\frac{x}{\varepsilon}\right]_{Y}+\left\\{\frac{x}{\varepsilon}\right\\}_{Y}\right)\
\ \forall x\in\mathbb{R}^{N}.$

We introduce the following sets:

$\widehat{J}_{\varepsilon}\doteq\\{k\in\mathbb{Z}^{N}|\ \varepsilon
Y^{k}\subset\Omega\\},\ \
\widehat{\Omega}_{\varepsilon}\doteq\hbox{interior}\left\\{\bigcup_{k\in\widehat{J}_{\varepsilon}}\varepsilon(k_{l}+\overline{Y})\right\\},\
\ \Lambda_{\varepsilon}\doteq\Omega\setminus\widehat{\Omega}_{\varepsilon},$

and

$\widehat{\Omega}_{i}^{\varepsilon}\doteq\bigcup_{k\in\widehat{J}_{\varepsilon}}\varepsilon
Y^{k}_{i},\qquad\Lambda_{i}^{\varepsilon}\doteq\Omega_{i}^{\varepsilon}\setminus\widehat{\Omega}_{i}^{\varepsilon},\quad
i=1,2,\qquad\widehat{\Gamma}^{\varepsilon}\doteq\partial\widehat{\Omega}_{2}^{\varepsilon}.$

######  Definition 4.1.

For any Lebesgue-measurable function $\phi$ on $\Omega^{\varepsilon}_{i}$,
$i=1,2$, we set

(4.1)
$\mathcal{T}^{\varepsilon}_{i}(\phi)(x,y)\doteq\begin{cases}\phi\left(\varepsilon\left[\frac{x}{\varepsilon}\right]_{Y}+\varepsilon
y\right)&\hbox{ a.e. for }(x,y)\in\widehat{\Omega}_{\varepsilon}\times
Y_{i},\\\ 0&\hbox{ a.e. for }(x,y)\in\Lambda_{\varepsilon}\times
Y_{i}.\end{cases}$

We observe that $\mathcal{T}^{\varepsilon}_{1}$ is the unfolding operator
defined in [11] for perforated domains, while $\mathcal{T}^{\varepsilon}_{2}$
is the one introduced in [17] for the inclusions. If $\phi$ is defined in
$\Omega$, we simply write $\mathcal{T}^{\varepsilon}_{i}(\phi)$ instead of
$\mathcal{T}^{\varepsilon}_{i}(\phi_{|\Omega_{i}^{\varepsilon}})$, $i=1,2$.

We now recall the main properties of the unfolding operators.

######  Proposition 4.2 ([11][17]).

Let $p\in[1,+\infty[$ and $i=1,2$.

  1. 1.

$\mathcal{T}^{\varepsilon}_{i}$ is a linear and continuous operator from
$L^{p}(\Omega^{\varepsilon}_{i})$ to $L^{p}(\Omega\times Y_{i})$.

  2. 2.

$\mathcal{T}^{\varepsilon}_{i}(\phi\psi)=\mathcal{T}^{\varepsilon}_{i}(\phi)\mathcal{T}^{\varepsilon}_{i}(\psi)$
for every $\phi,\psi\in L^{p}(\Omega^{\varepsilon}_{i})$.

  3. 3.

Let $\phi\in L^{p}(Y_{i})$ be a $Y$-periodic function and set
$\phi_{\varepsilon}(x)=\phi(\frac{x}{\varepsilon})$.
Then,

$\mathcal{T}^{\varepsilon}_{i}(\phi_{\varepsilon})(x,y)=\begin{cases}\phi(y)&\hbox{
a.e. for }(x,y)\in\widehat{\Omega}_{\varepsilon}\times Y_{i},\\\ 0&\hbox{ a.e.
for }(x,y)\in\Lambda_{\varepsilon}\times Y_{i},\end{cases}$

and

$\mathcal{T}^{\varepsilon}_{i}(\phi_{\varepsilon})\rightarrow\phi\quad\hbox{
strongly in }L^{p}(\Omega\times Y_{i}).$

  4. 4.

For all $\phi\in L^{1}(\Omega^{\varepsilon}_{i})$, one has

$\frac{1}{|Y|}\int_{\Omega\times
Y_{i}}\mathcal{T}^{\varepsilon}_{i}(\phi)(x,y)dxdy=\int_{\widehat{\Omega}^{\varepsilon}_{i}}\phi(x)dx=\int_{\Omega^{\varepsilon}_{i}}\phi(x)dx-\int_{\Lambda^{\varepsilon}_{i}}\phi(x)dx.$

  5. 5.

$\|\mathcal{T}^{\varepsilon}_{i}(\phi)\|_{L^{p}(\Omega\times
Y_{i})}\leq|Y|^{\frac{1}{p}}\|\phi\|_{L^{p}(\Omega^{\varepsilon}_{i})}$ for
every $\phi\in L^{p}(\Omega^{\varepsilon}_{i})$.

  6. 6.

For $\phi\in L^{p}(\Omega)$,
$\mathcal{T}^{\varepsilon}_{i}(\phi)\rightarrow\phi\quad\hbox{strongly in
}L^{p}(\Omega\times Y_{i})$.

  7. 7.

Let $\\{\phi_{\varepsilon}\\}$ be a sequence in $L^{p}(\Omega)$ such that
$\phi_{\varepsilon}\rightarrow\phi\ \hbox{strongly in }L^{p}(\Omega)$. Then

$\mathcal{T}^{\varepsilon}_{i}(\phi_{\varepsilon})\rightarrow\phi\quad\hbox{strongly
in }L^{p}(\Omega\times Y_{i}).$

  8. 8.

Let $\\{\phi_{\varepsilon}\\}$ be a sequence in
$L^{p}(\Omega_{i}^{\varepsilon})$ such that
$\|\phi_{\varepsilon}\|_{L^{p}(\Omega_{i}^{\varepsilon})}\leq C$.
If
$\mathcal{T}^{\varepsilon}_{i}(\phi_{\varepsilon})\rightharpoonup\widehat{\phi}\hbox{
weakly in }L^{p}(\Omega\times Y_{i})$, then

$\widetilde{\phi_{\varepsilon}}\rightharpoonup\theta_{i}\mathcal{M}_{Y_{i}}(\widehat{\phi})\quad\hbox{weakly
in }L^{p}(\Omega).$

  9. 9.

If $\phi\in W^{1,p}(\Omega_{i}^{\varepsilon})$, then
$\nabla_{y}[\mathcal{T}^{\varepsilon}_{i}(\phi)]=\varepsilon\mathcal{T}^{\varepsilon}_{i}(\nabla\phi)$
and $\mathcal{T}^{\varepsilon}_{i}(\phi)\in L^{p}(\Omega,W^{1,p}(Y_{i}))$.

We give below the main propositions concerning the jump on the interface and
some convergence results.
Arguing as in the proof of Lemma 2.14 of [17], we can easily obtain the
following result:

######  Lemma 4.3.

Let $u^{\varepsilon}\in H^{\varepsilon}_{\gamma}$ and $h$ satisfy (A$4$). It
holds

(4.2)
$\varepsilon\int_{\Gamma^{\varepsilon}}h^{\varepsilon}(u^{\varepsilon}_{1}-u^{\varepsilon}_{2})^{2}d\sigma_{x}\geq\frac{1}{|Y|}\int_{\Omega\times\Gamma}h(y)(\mathcal{T}^{\varepsilon}_{1}(u^{\varepsilon}_{1})-\mathcal{T}^{\varepsilon}_{2}(u^{\varepsilon}_{2}))^{2}dxd\sigma_{y}.$

Moreover, one has:

######  Lemma 4.4.

[17] Let $u^{\varepsilon}\in H^{\varepsilon}_{\gamma}$ and $h$ satisfy (A$4$).
If $\phi\in\mathcal{D}(\Omega)$, then for $\varepsilon$ small enough one has

(4.3)
$\varepsilon\int_{\Gamma^{\varepsilon}}h^{\varepsilon}(u^{\varepsilon}_{1}-u^{\varepsilon}_{2})\phi
d\sigma_{x}=\frac{1}{|Y|}\int_{\Omega\times\Gamma}h(y)(\mathcal{T}^{\varepsilon}_{1}(u^{\varepsilon}_{1})-\mathcal{T}^{\varepsilon}_{2}(u^{\varepsilon}_{2}))\mathcal{T}^{\varepsilon}_{1}(\phi)dxd\sigma_{y}.$

We note that in the right hand sides of (4.2) and (4.3), the traces are well
defined in view of Proposition 4.29.

######  Theorem 4.5.

[8][11][16][17] Let $\gamma\in\mathbb{R}$ and $u^{\varepsilon}$ be a bounded
sequence in $H^{\varepsilon}_{\gamma}$. Then,

(4.4) $\begin{array}[]{lll}\displaystyle\|\mathcal{T}^{\varepsilon}_{1}(\nabla
u^{\varepsilon}_{1})\|_{L^{2}(\Omega\times Y_{1})}\leq C,\\\\[2.84526pt]
\displaystyle\|\mathcal{T}^{\varepsilon}_{2}(\nabla
u^{\varepsilon}_{2})\|_{L^{2}(\Omega\times Y_{2})}\leq C,\\\\[2.84526pt]
\displaystyle\|\mathcal{T}^{\varepsilon}_{1}(u^{\varepsilon}_{1})-\mathcal{T}^{\varepsilon}_{2}(u^{\varepsilon}_{2})\|_{L^{2}(\Omega\times\Gamma)}\leq
C\varepsilon^{\frac{1-\gamma}{2}},\end{array}$

and there exist a subsequence (still denoted by $\varepsilon$), $u_{1}\in
H^{1}_{0}(\Omega)$ and $\widehat{u}_{1}\in L^{2}(\Omega,H^{1}_{per}(Y_{1}))$
with $\mathcal{M}_{\Gamma}(\widehat{u}_{1})=0$ a.e. in $\Omega$ such that

(4.5)
$\left\\{\begin{array}[]{ll}\displaystyle\mathcal{T}^{\varepsilon}_{1}(u^{\varepsilon}_{1})\rightarrow
u_{1}&\hbox{ strongly in }L^{2}(\Omega,H^{1}(Y_{1})),\\\\[2.84526pt]
\displaystyle\mathcal{T}^{\varepsilon}_{1}(\nabla
u^{\varepsilon}_{1})\rightharpoonup\nabla
u_{1}+\nabla_{y}\widehat{u}_{1}&\hbox{ weakly in }L^{2}(\Omega\times
Y_{1}).\end{array}\right.$

Moreover, if $\gamma\leq 1$, there exist a subsequence (still denoted by
$\varepsilon$), $u_{2}\in L^{2}(\Omega)$ and $\overline{u}_{2}\in
L^{2}(\Omega,H^{1}(Y_{2}))$ with $\mathcal{M}_{\Gamma}(\overline{u}_{2})=0$
a.e. in $\Omega$ such that

(4.6)
$\left\\{\begin{array}[]{ll}\displaystyle\mathcal{T}^{\varepsilon}_{2}(u^{\varepsilon}_{2})\rightharpoonup
u_{2}&\hbox{ weakly in }L^{2}(\Omega,H^{1}(Y_{2})),\\\\[2.84526pt]
\displaystyle\mathcal{T}^{\varepsilon}_{2}(\nabla
u^{\varepsilon}_{2})\rightharpoonup\nabla_{y}\overline{u}_{2}&\hbox{ weakly in
}L^{2}(\Omega\times Y_{2}).\end{array}\right.$

Furthermore, if $\gamma<1$, then

$u_{1}=u_{2}.$

In addition, for $\gamma\leq-1$, there exists $\xi_{\Gamma}\in L^{2}(\Omega)$
such that, set

(4.7) $\widehat{u}_{2}\doteq\overline{u}_{2}-y_{\Gamma}\nabla
u_{1}-\xi_{\Gamma}\in L^{2}(\Omega,H^{1}(Y_{2}))$

with $y_{\Gamma}\doteq y-\mathcal{M}_{\Gamma}(y)$, one has:

  * •

if $\gamma<-1$,

(4.8) $\widehat{u}_{1}=\widehat{u}_{2}+\xi_{\Gamma}\quad\hbox{ on
}\Omega\times\Gamma;$

  * •

if $\gamma=-1$,

$\displaystyle\frac{\mathcal{T}^{\varepsilon}_{1}(u^{\varepsilon}_{1})-\mathcal{T}^{\varepsilon}_{2}(u^{\varepsilon}_{2})}{\varepsilon}\rightharpoonup\widehat{u}_{1}-\widehat{u}_{2}\quad\hbox{
weakly in }L^{2}(\Omega\times\Gamma).$

##  5 Proof of the homogenization result

In this section we prove the convergence results stated in Theorem 2.2. The
proof differs according to the values of the parameter $\gamma$.

For $\gamma<-1$, by choosing suitable test functions, it is possible to adapt
the arguments of [17] leading to the classical homogenized problem of
Bensoussan-Lions-Papanicolau in a fixed domain, see [3].

For $\gamma=-1$, the above mentioned arguments do not apply. Indeed, in this
case, the choice of the test functions leads to handle products of two weakly
convergent sequences. We can overcome this difficulty by using lower semi-
continuity arguments, due to the fact that we are dealing with variational
inequalities. In addition, the constraint imposed on the chosen test functions
gives rise to a nonlinear homogenized system where the cell problem consists
in a variational inequality.

For the cases $\gamma\in]-1,1[$ and $\gamma=1$, in order to analyse the
asymptotic behaviour of the problem at microscopic level, we need to consider
more regular test functions.
As far as it concerns the homogenized problem, in the case $\gamma\in]-1,1[$,
it turns out to be the classical limit problem of Cioranescu-S.J.Paulin in
perforated domains, see [12].
While, when $\gamma=1$, again taking advantage of lower semi-continuity
arguments, we obtain an obstacle-type homogenized problem.

###  5.1 The case $\gamma<-1$

Let $u^{\varepsilon}$ be the weak solution of problem (2.1). Theorem 4.5, for
the case $\gamma<-1$, ensures the existence of $u_{1}\in H^{1}_{0}(\Omega)$,
$\widehat{u}_{1}\in L^{2}(\Omega,H^{1}_{per}(Y_{1}))$ with
$\mathcal{M}_{\Gamma}(\widehat{u}_{1})=0$ a.e. in $\Omega$ and
$\widehat{u}_{2}\in L^{2}(\Omega,H^{1}(Y_{2}))$ such that, up to a
subsequence,

(5.1)
$\left\\{\begin{array}[]{lllll}\displaystyle\widetilde{u^{\varepsilon}_{i}}\rightharpoonup\theta_{i}u_{1}&\hbox{
weakly in }L^{2}(\Omega),\ i=1,2,\\\\[8.53581pt]
\displaystyle\mathcal{T}^{\varepsilon}_{1}(u^{\varepsilon}_{1})\rightarrow
u_{1}&\hbox{ strongly in }L^{2}(\Omega,H^{1}(Y_{1})),\\\\[8.53581pt]
\displaystyle\mathcal{T}^{\varepsilon}_{2}(u^{\varepsilon}_{2})\rightharpoonup
u_{1}&\hbox{ weakly in }L^{2}(\Omega,H^{1}(Y_{2})),\\\\[8.53581pt]
\displaystyle\mathcal{T}^{\varepsilon}_{1}(\nabla
u^{\varepsilon}_{1})\rightharpoonup\nabla
u_{1}+\nabla_{y}\widehat{u}_{1}&\hbox{ weakly in }L^{2}(\Omega\times
Y_{1}),\\\\[8.53581pt] \displaystyle\mathcal{T}^{\varepsilon}_{2}(\nabla
u^{\varepsilon}_{2})\rightharpoonup\nabla
u_{1}+\nabla_{y}\widehat{u}_{2}&\hbox{ weakly in }L^{2}(\Omega\times
Y_{2}).\end{array}\right.$

We first take $\varphi\in\mathcal{D}(\Omega)$ and use
$(u^{\varepsilon}_{1}+\varphi,u^{\varepsilon}_{2}+\varphi)\in
K^{\varepsilon}_{\gamma}$ as test function in (2.4), obtaining

(5.2) $\displaystyle\int_{\Omega_{1}^{\varepsilon}}A^{\varepsilon}\nabla
u^{\varepsilon}_{1}\nabla\varphi
dx+\int_{\Omega_{2}^{\varepsilon}}A^{\varepsilon}\nabla
u^{\varepsilon}_{2}\nabla\varphi dx\geq\int_{\Omega_{1}^{\varepsilon}}f\varphi
dx+\int_{\Omega_{2}^{\varepsilon}}f\varphi
dx,\quad\forall\varphi\in\mathcal{D}(\Omega).$

By (A$1$) and (3.1) one has, for $i=1,2$,

(5.3) $\int_{\Lambda_{i}^{\varepsilon}}A^{\varepsilon}\nabla
u^{\varepsilon}_{i}\nabla\varphi dx\leq\beta\|\nabla
u^{\varepsilon}_{i}\|_{L^{2}(\Omega_{i}^{\varepsilon})}\|\nabla\varphi\|_{L^{2}(\Lambda_{i}^{\varepsilon})}\leq\beta
C\|\nabla\varphi\|_{L^{2}(\Lambda_{i}^{\varepsilon})}\rightarrow
0,\quad\varepsilon\rightarrow 0,$

in view of the equicontinuity of the integral. Consequently, by Proposition
4.24, inequality (5.2) can be rewritten as

(5.4) $\begin{array}[]{cc}\displaystyle\frac{1}{|Y|}\int_{\Omega\times
Y_{1}}\mathcal{T}^{\varepsilon}_{1}(A^{\varepsilon})\mathcal{T}^{\varepsilon}_{1}(\nabla
u^{\varepsilon}_{1})\mathcal{T}^{\varepsilon}_{1}(\nabla\varphi)dxdy+\frac{1}{|Y|}\int_{\Omega\times
Y_{2}}\mathcal{T}^{\varepsilon}_{2}(A^{\varepsilon})\mathcal{T}^{\varepsilon}_{2}(\nabla
u^{\varepsilon}_{2})\mathcal{T}^{\varepsilon}_{2}(\nabla\varphi)dxdy\\\\[14.22636pt]
\displaystyle\geq\int_{\Omega_{1}^{\varepsilon}}f\varphi
dx+\int_{\Omega_{2}^{\varepsilon}}f\varphi
dx,\quad\forall\varphi\in\mathcal{D}(\Omega).\end{array}$

We pass to the limit, as $\varepsilon$ tends to zero, in the previous
inequality by using Proposition 4.23 and (5.1), we get

(5.5) $\displaystyle\frac{1}{|Y|}\int_{\Omega\times Y_{1}}A(y)(\nabla
u_{1}+\nabla_{y}\widehat{u}_{1})\nabla\varphi\,dxdy+\frac{1}{|Y|}\int_{\Omega\times
Y_{2}}A(y)(\nabla
u_{1}+\nabla_{y}\widehat{u}_{2})\nabla\varphi\,dxdy\geq\int_{\Omega}f\varphi\,dx,\quad\forall\varphi\in\mathcal{D}(\Omega).$

The validity of the previous inequality for every function in
$\mathcal{D}(\Omega)$, together with density arguments, gives

(5.6) $\displaystyle\frac{1}{|Y|}\int_{\Omega\times Y_{1}}A(y)(\nabla
u_{1}+\nabla_{y}\widehat{u}_{1})\nabla\varphi\,dxdy+\frac{1}{|Y|}\int_{\Omega\times
Y_{2}}A(y)(\nabla
u_{1}+\nabla_{y}\widehat{u}_{2})\nabla\varphi\,dxdy=\int_{\Omega}f\varphi\,dx,\quad\forall\varphi\in
H^{1}_{0}(\Omega).$

We now take $w\in\mathcal{D}(\Omega)$, $\Psi\in H^{1}_{per}(Y)$ and set
$\Psi^{\varepsilon}(x)\doteq\Psi(\frac{x}{\varepsilon})$. Then, we use
$(u^{\varepsilon}_{1}+\varepsilon
w\Psi^{\varepsilon},u^{\varepsilon}_{2}+\varepsilon w\Psi^{\varepsilon})\in
K^{\varepsilon}_{\gamma}$ as test function in (2.4), we have

(5.7) $\displaystyle\int_{\Omega_{1}^{\varepsilon}}A^{\varepsilon}\nabla
u^{\varepsilon}_{1}\nabla(\varepsilon
w\Psi^{\varepsilon})\,dx+\int_{\Omega_{2}^{\varepsilon}}A^{\varepsilon}\nabla
u^{\varepsilon}_{2}\nabla(\varepsilon
w\Psi^{\varepsilon})\,dx\geq\varepsilon\int_{\Omega_{1}^{\varepsilon}}fw\Psi^{\varepsilon}\,dx+\varepsilon\int_{\Omega_{2}^{\varepsilon}}fw\Psi^{\varepsilon}\,dx.$

Arguing as in [17], we pass to the limit by unfolding and get

(5.8) $\displaystyle\frac{1}{|Y|}\int_{\Omega\times Y_{1}}A(y)(\nabla
u_{1}+\nabla_{y}\widehat{u}_{1})\nabla_{y}\Phi\,dxdy+\frac{1}{|Y|}\int_{\Omega\times
Y_{2}}A(y)(\nabla u_{1}+\nabla_{y}\widehat{u}_{2})\nabla_{y}\Phi\,dxdy\geq 0,$

where $\Phi(x,y)=w(x)\Psi(y)$. By density, (5.8) holds for every $\Phi\in
L^{2}(\Omega;H^{1}_{per}(Y))$.
Taking into account (4.8) of Theorem 4.5, we set

(5.9)
$\displaystyle\widehat{u}\doteq\left\\{\begin{array}[]{ll}\displaystyle\widehat{u}_{1}&\hbox{
in }Y_{1},\\\\[5.69054pt] \displaystyle\widehat{u}_{2}+\xi_{\Gamma}&\hbox{ in
}Y_{2},\end{array}\right.$

and extend it by periodicity to a function still denoted by $\widehat{u}$,
which belongs to $L^{2}(\Omega,H^{1}_{per}(Y))$ and is such that
$\mathcal{M}_{\Gamma}(\widehat{u})=0$ a.e. in $\Omega$, in view of (4.7).
Hence, problem (5.8) can be written as follows:

(5.10) $\displaystyle\frac{1}{|Y|}\int_{\Omega\times Y}A(y)(\nabla
u_{1}+\nabla_{y}\widehat{u})\nabla_{y}\Phi\,dxdy\geq 0,\quad\forall\Phi\in
L^{2}(\Omega;H^{1}_{per}(Y)),$

which implies

(5.11) $\displaystyle\frac{1}{|Y|}\int_{\Omega\times Y}A(y)(\nabla
u_{1}+\nabla_{y}\widehat{u})\nabla_{y}\Phi\,dxdy=0,\quad\forall\Phi\in
L^{2}(\Omega;H^{1}_{per}(Y)).$

By summing up problems (5.6) and (5.11), we obtain

(5.12) $\left\\{\begin{array}[]{lll}\displaystyle\hbox{ Find
}(u_{1},\widehat{u})\in H^{1}_{0}(\Omega)\times
L^{2}(\Omega,H^{1}_{per}(Y)),\hbox{ with
}\mathcal{M}_{\Gamma}(\widehat{u})=0\hbox{ a.e. in }\Omega,\ \hbox{ s.t.
}\\\\[8.53581pt] \displaystyle\frac{1}{|Y|}\int_{\Omega\times Y}A(y)(\nabla
u_{1}+\nabla_{y}\widehat{u})(\nabla\varphi+\nabla_{y}\Phi)\,dxdy=\int_{\Omega}f\varphi\,dx,\\\\[14.22636pt]
\displaystyle\forall\varphi\in H^{1}_{0}(\Omega),\ \forall\Phi\in
L^{2}(\Omega;H^{1}_{per}(Y)).\end{array}\right.$

The above problem is the one obtained in the classical work [9] dealing with
the periodic unfolding method for fixed domains. Hence $(u_{1},\widehat{u})$
is unique and convergences (5.1) hold for the whole sequences. Moreover, if we
define $A^{0}_{\gamma}$ as in (2.11), by classical arguments $u_{1}$ results
the unique solution of problem (2.10).

###  5.2 The case $\gamma=-1$

Let $u^{\varepsilon}$ be the weak solution of problem (2.1). Theorem 4.5, for
$\gamma=-1$, assures that there exist $u_{1}\in H^{1}_{0}(\Omega)$,
$\widehat{u}_{1}\in L^{2}(\Omega,H^{1}_{per}(Y_{1}))$ with
$\mathcal{M}_{\Gamma}(\widehat{u}_{1})=0$ a.e. in $\Omega$ and
$\widehat{u}_{2}\in L^{2}(\Omega,H^{1}(Y_{2}))$ such that convergences (5.1)
hold, together with

(5.13)
$\displaystyle\frac{\mathcal{T}^{\varepsilon}_{1}(u^{\varepsilon}_{1})-\mathcal{T}^{\varepsilon}_{2}(u^{\varepsilon}_{2})}{\varepsilon}\rightharpoonup\widehat{u}_{1}-\widehat{u}_{2}\quad\hbox{
weakly in }L^{2}(\Omega\times\Gamma),$

up to a subsequence. We take $\varphi,w_{i}\in\mathcal{D}(\Omega)$, for
$i=1,2$, $\Psi_{1}\in H^{1}_{per}(Y_{1})$ and $\Psi_{2}\in H^{1}(Y_{2})$
(extended by $Y$-periodicity to $\mathbb{R}^{N}$) such that

(5.14) $\displaystyle w_{1}(x)\Psi_{1}(y)\geq w_{2}(x)\Psi_{2}(y)\hbox{ a.e.
in }\Omega\times\Gamma.$

Set $\Psi_{i}^{\varepsilon}(x)\doteq\Psi_{i}(\frac{x}{\varepsilon})$, for
$i=1,2$, one has $w_{1}(x)\Psi_{1}^{\varepsilon}(x)\geq
w_{2}(x)\Psi_{2}^{\varepsilon}(x)$ almost everywhere on
$\Gamma^{\varepsilon}$, so that one can choose $(\varphi+\varepsilon
w_{1}\Psi_{1}^{\varepsilon},\varphi+\varepsilon
w_{2}\Psi_{2}^{\varepsilon})\in K^{\varepsilon}_{\gamma}$ as test function in
(2.4) getting

(5.15)
$\begin{array}[]{lll}\displaystyle\int_{\Omega_{1}^{\varepsilon}}A^{\varepsilon}\nabla
u^{\varepsilon}_{1}(\nabla\varphi+\nabla(\varepsilon
w_{1}\Psi_{1}^{\varepsilon})-\nabla
u^{\varepsilon}_{1})dx+\int_{\Omega_{2}^{\varepsilon}}A^{\varepsilon}\nabla
u^{\varepsilon}_{2}(\nabla\varphi+\nabla(\varepsilon
w_{2}\Psi_{2}^{\varepsilon})-\nabla u^{\varepsilon}_{2})dx\\\\[14.22636pt]
\displaystyle+\int_{\Gamma^{\varepsilon}}h^{\varepsilon}[u^{\varepsilon}](w_{1}\Psi_{1}^{\varepsilon}-w_{2}\Psi_{2}^{\varepsilon})\,d\sigma\\\\[14.22636pt]
\displaystyle\geq\int_{\Omega_{1}^{\varepsilon}}f(\varphi+\varepsilon
w_{1}\Psi_{1}^{\varepsilon}-u^{\varepsilon}_{1})dx+\int_{\Omega_{2}^{\varepsilon}}f(\varphi+\varepsilon
w_{2}\Psi_{2}^{\varepsilon}-u^{\varepsilon}_{2})dx+\varepsilon^{-1}\int_{\Gamma^{\varepsilon}}h^{\varepsilon}[u^{\varepsilon}]^{2}\,d\sigma.\end{array}$

Let us observe that by (4.2)

(5.16)
$\varepsilon^{-1}\int_{\Gamma^{\varepsilon}}h^{\varepsilon}[u^{\varepsilon}]^{2}\,d\sigma\geq\frac{\varepsilon^{-2}}{|Y|}\int_{\Omega\times\Gamma}h(y)(\mathcal{T}^{\varepsilon}_{1}(u^{\varepsilon}_{1})-\mathcal{T}^{\varepsilon}_{2}(u^{\varepsilon}_{2}))^{2}dxd\sigma_{y}.$

Hence, arguing analogously to the previous case and taking into account the
lower semi-continuity of the norm with respect to the weak convergence, we
pass to the limit by unfolding thanks to (5.1) and (5.13). Set
$\Phi_{i}(x,y)=w_{i}(x)\Psi_{i}(y)$, for $i=1,2$, by density arguments one
gets

(5.17) $\begin{array}[]{lll}\displaystyle\frac{1}{|Y|}\int_{\Omega\times
Y_{1}}A(y)(\nabla u_{1}+\nabla_{y}\widehat{u}_{1})(\nabla(\varphi-
u_{1})+\nabla_{y}(\Phi_{1}-\widehat{u}_{1}))\,dxdy\\\\[14.22636pt]
\displaystyle+\frac{1}{|Y|}\int_{\Omega\times Y_{2}}A(y)(\nabla
u_{1}+\nabla_{y}\widehat{u}_{2})(\nabla(\varphi-
u_{1})+\nabla_{y}(\Phi_{2}-\widehat{u}_{2}))\,dxdy\\\\[14.22636pt]
\displaystyle+\frac{1}{|Y|}\int_{\Omega\times\Gamma}h(y)(\widehat{u}_{1}-\widehat{u}_{2})((\Phi_{1}-\widehat{u}_{1})-(\Phi_{2}-\widehat{u}_{2})\,dxd\sigma_{y}\geq\int_{\Omega}f(\varphi-
u_{1})\,dx,\\\\[14.22636pt] \displaystyle\forall\varphi\in H^{1}_{0}(\Omega),\
\forall\Phi_{1}\in L^{2}(\Omega;H^{1}_{per}(Y_{1})),\ \forall\Phi_{2}\in
L^{2}(\Omega;H^{1}(Y_{2})):\Phi_{1}\geq\Phi_{2}\hbox{ in
}\Omega\times\Gamma,\end{array}$

in view of (5.14).
Let us consider the following spaces:

(5.18)
$\displaystyle\mathcal{B}\doteq\\{v=(v_{1},\widehat{v}_{1},\widehat{v}_{2})\in
H^{1}_{0}(\Omega)\times L^{2}(\Omega,W_{per}(Y_{1}))\times
L^{2}(\Omega,H^{1}(Y_{2}))\\},$

equipped with the norm

(5.19) $\displaystyle\|v\|^{2}_{\mathcal{B}}\doteq\|\nabla
v_{1}+\nabla_{y}\widehat{v}_{1}\|^{2}_{L^{2}(\Omega\times Y_{1})}+\|\nabla
v_{1}+\nabla_{y}\widehat{v}_{2}\|^{2}_{L^{2}(\Omega\times
Y_{2})}+\|\widehat{v}_{1}-\widehat{v}_{2}\|^{2}_{L^{2}(\Omega\times\Gamma)},\quad
v=(v_{1},\widehat{v}_{1},\widehat{v}_{2})\in\mathcal{B}$

see [16, Lemma $5.3$], and

(5.20)
$\displaystyle\mathcal{W}\doteq\\{(v_{1},\widehat{v}_{1},\widehat{v}_{2})\in\mathcal{B}\,|\,\widehat{v}_{1}\geq\widehat{v}_{2}\hbox{
in }\Omega\times\Gamma\\}.$

Now, let us observe that

(5.21) $\displaystyle\widehat{u}_{1}\geq\widehat{u}_{2}\quad\hbox{ a.e. in
}\Omega\times\Gamma.$

Indeed, since $u^{\varepsilon}_{1}\geq u^{\varepsilon}_{2}$ a.e. on
$\Gamma^{\varepsilon}$, one deduces that
$\mathcal{T}^{\varepsilon}_{1}(u^{\varepsilon}_{1})\geq\mathcal{T}^{\varepsilon}_{2}(u^{\varepsilon}_{2})$
a.e. in $\Omega\times\Gamma$, hence (5.21) follows from convergence (5.13).
Thus, $(u_{1},\widehat{u}_{1},\widehat{u}_{2})$ satisfies

(5.22) $\left\\{\begin{array}[]{lll}\displaystyle\hbox{ Find
}(u_{1},\widehat{u}_{1},\widehat{u}_{2})\in\mathcal{W}\ \hbox{ s.t.
}\\\\[8.53581pt] \displaystyle\frac{1}{|Y|}\int_{\Omega\times
Y_{1}}A(y)(\nabla u_{1}+\nabla_{y}\widehat{u}_{1})(\nabla(\varphi-
u_{1})+\nabla_{y}(\Phi_{1}-\widehat{u}_{1}))\,dxdy\\\\[14.22636pt]
\displaystyle+\frac{1}{|Y|}\int_{\Omega\times Y_{2}}A(y)(\nabla
u_{1}+\nabla_{y}\widehat{u}_{2})(\nabla(\varphi-
u_{1})+\nabla_{y}(\Phi_{2}-\widehat{u}_{2}))\,dxdy\\\\[14.22636pt]
\displaystyle+\frac{1}{|Y|}\int_{\Omega\times\Gamma}h(y)(\widehat{u}_{1}-\widehat{u}_{2})((\Phi_{1}-\widehat{u}_{1})-(\Phi_{2}-\widehat{u}_{2})\,dxd\sigma_{y}\geq\int_{\Omega}f(\varphi-
u_{1})\,dx,\\\\[14.22636pt] \displaystyle\forall\varphi\in H^{1}_{0}(\Omega),\
\forall\Phi_{1}\in L^{2}(\Omega;H^{1}_{per}(Y_{1})),\ \forall\Phi_{2}\in
L^{2}(\Omega;H^{1}(Y_{2})):\Phi_{1}\geq\Phi_{2}\hbox{ in
}\Omega\times\Gamma.\end{array}\right.$

Taking into account our assumptions, by standard arguments on variational
inequalities, one also gets that problem (5.22) admits a unique solution.
Therefore convergences (5.1) and (5.13) hold for the whole sequences.

In order to describe the homogenized problem satisfied by $u_{1}$, we replace
the test function $\varphi$ by $\varphi+u_{1}$ and, in view of (5.21),
$\Phi_{i}$ by $\widehat{u}_{i}$, for $i=1,2$, in (5.22):

(5.23) $\begin{array}[]{llll}\displaystyle\frac{1}{|Y|}\int_{\Omega\times
Y_{1}}A(y)(\nabla
u_{1}+\nabla_{y}\widehat{u}_{1})\nabla\varphi\,dxdy+\frac{1}{|Y|}\int_{\Omega\times
Y_{2}}A(y)(\nabla
u_{1}+\nabla_{y}\widehat{u}_{2})\nabla\varphi\,dxdy\geq\int_{\Omega}f\varphi\,dx,\quad\forall\varphi\in
H^{1}_{0}(\Omega).\end{array}$

Now, taking $\varphi=-\varphi$ as test function we obtain

(5.24) $\displaystyle\frac{1}{|Y|}\int_{\Omega\times Y_{1}}A(y)(\nabla
u_{1}+\nabla_{y}\widehat{u}_{1})\nabla\varphi\,dxdy+\frac{1}{|Y|}\int_{\Omega\times
Y_{2}}A(y)(\nabla
u_{1}+\nabla_{y}\widehat{u}_{2})\nabla\varphi\,dxdy=\int_{\Omega}f\varphi\,dx,\quad\forall\varphi\in
H^{1}_{0}(\Omega).$

Lastly, by taking $\varphi=u_{1}$ as test function in (5.22), one gets

(5.25) $\begin{array}[]{ll}\displaystyle\frac{1}{|Y|}\int_{\Omega\times
Y_{1}}A(y)(\nabla
u_{1}+\nabla_{y}\widehat{u}_{1})(\nabla_{y}\Phi_{1}-\nabla_{y}\widehat{u}_{1})\,dxdy+\frac{1}{|Y|}\int_{\Omega\times
Y_{2}}A(y)(\nabla
u_{1}+\nabla_{y}\widehat{u}_{2})(\nabla_{y}\Phi_{2}-\nabla_{y}\widehat{u}_{2})\,dxdy\\\\[14.22636pt]
\displaystyle+\frac{1}{|Y|}\int_{\Omega\times\Gamma}h(y)(\widehat{u}_{1}-\widehat{u}_{2})((\Phi_{1}-\widehat{u}_{1})-(\Phi_{2}-\widehat{u}_{2}))\,dxd\sigma_{y}\geq
0,\\\\[14.22636pt] \displaystyle\forall\Phi_{1}\in
L^{2}(\Omega;H^{1}_{per}(Y_{1})),\ \forall\Phi_{2}\in
L^{2}(\Omega;H^{1}(Y_{2})):\Phi_{1}\geq\Phi_{2}\hbox{ in
}\Omega\times\Gamma.\end{array}$

To conclude, observe that classical arguments on variational inequalities give
that the solution of the cell problem (2.16) exists and it is unique, by
endowing the space $H^{1}_{per}(Y_{1})\times H^{1}(Y_{2})$ with the equivalent
norm

(5.26) $\displaystyle\|(z_{1},z_{2})\|^{2}_{H^{1}_{per}(Y_{1})\times
H^{1}(Y_{2})}\doteq\|\nabla z_{1}\|^{2}_{L^{2}(Y_{1})}+\|\nabla
z_{2}\|^{2}_{L^{2}(Y_{2})}+\|z_{1}-z_{2}\|^{2}_{L^{2}(\Gamma)}.$

Then, by the uniqueness of the solution of (2.16) written for $\zeta=\nabla
u_{1}$, we obtain that $\widehat{u}_{1}$ and $\widehat{u}_{2}$ in (5.25) can
be expressed as

(5.27) $\widehat{u}_{1}(x,y)=\widehat{\chi}_{1}(y,\nabla u_{1}(x))\quad\hbox{
and }\quad\widehat{u}_{2}(x,y)=\widehat{\chi}_{2}(y,\nabla u_{1}(x)).$

Finally, set $A^{0}_{-1}$ as in (2.15), by (5.24) and (5.27) the homogenized
problem satisfied by $u_{1}$ can be rewritten in the following form:

(5.28) $\displaystyle\int_{\Omega}A^{0}_{-1}(\nabla
u_{1})\nabla\varphi\,dx=\int_{\Omega}f\varphi\,dx,\quad\forall\varphi\in
H^{1}_{0}(\Omega),$

which gives problem (2.10). The uniqueness of solutions of problems (2.16) and
(5.22) easily implies the uniqueness of problem (2.10).

###  5.3 The case $\gamma\in]-1;1[$

Let $u^{\varepsilon}$ be the weak solution of problem (2.1). Theorem 4.5, for
$\gamma\in]-1;1[$, assures that there exist $u_{1}\in H^{1}_{0}(\Omega)$,
$\widehat{u}_{1}\in L^{2}(\Omega,H^{1}_{per}(Y_{1}))$ with
$\mathcal{M}_{\Gamma}(\widehat{u}_{1})=0$ a.e. in $\Omega$ and
$\overline{u}_{2}\in L^{2}(\Omega,H^{1}(Y_{2}))$ with
$\mathcal{M}_{\Gamma}(\overline{u}_{2})=0$ a.e. in $\Omega$ such that
convergences (5.1) hold, up to a subsequence, where the last convergence is
equivalent to

(5.29) $\displaystyle\mathcal{T}^{\varepsilon}_{2}(\nabla
u^{\varepsilon}_{2})\rightharpoonup\nabla_{y}\overline{u}_{2}\quad\hbox{
weakly in }L^{2}(\Omega\times Y_{2}),$

in view of (4.7).

We want to prove that $\nabla_{y}\overline{u}_{2}=0$ a.e. in $\Omega\times
Y_{2}$. To this aim, we take $w\in\mathcal{D}(\Omega)$, $\Psi_{2}\in
C^{1}_{c}(\overline{Y}_{2})$, set
$\Psi_{2}^{\varepsilon}(x)\doteq\Psi_{2}(\frac{x}{\varepsilon})$ and extend it
by periodicity. From the boundedness of $w$ and $\Psi_{2}$, there exists
$M\geq 0$ such that

(5.30)
$\displaystyle\|w\|_{L^{\infty}(\Omega)}\|\Psi_{2}\|_{L^{\infty}(\overline{Y}_{2})}\leq
M.$

Hence, $u^{\varepsilon}_{1}-u^{\varepsilon}_{2}+\varepsilon
w\Psi_{2}^{\varepsilon}+\varepsilon M\geq 0$ a.e. on $\Gamma^{\varepsilon}$,
so that $(u^{\varepsilon}_{1},u^{\varepsilon}_{2}-\varepsilon
w\Psi_{2}^{\varepsilon}-\varepsilon M)\in K^{\varepsilon}_{\gamma}$.
Therefore, we can use it as test function in (2.4) getting

(5.31) $\displaystyle-\int_{\Omega_{2}^{\varepsilon}}A^{\varepsilon}\nabla
u^{\varepsilon}_{2}\nabla(\varepsilon
w\Psi_{2}^{\varepsilon})\,dx+\varepsilon^{\gamma+1}\int_{\Gamma^{\varepsilon}}h^{\varepsilon}[u^{\varepsilon}](w\Psi_{2}^{\varepsilon}+M)\,d\sigma\geq-\varepsilon\int_{\Omega_{2}^{\varepsilon}}f(w\Psi_{2}^{\varepsilon}+M)\,dx.$

By Lemma 4.4 and (4.5), one has

(5.32)
$\begin{array}[]{ll}\displaystyle\varepsilon^{\gamma+1}\int_{\Gamma^{\varepsilon}}h^{\varepsilon}[u^{\varepsilon}]w\Psi_{2}^{\varepsilon}\,d\sigma=\varepsilon^{\gamma}\frac{1}{|Y|}\int_{\Omega\times\Gamma}h(y)(\mathcal{T}^{\varepsilon}_{1}(u^{\varepsilon}_{1})-\mathcal{T}^{\varepsilon}_{2}(u^{\varepsilon}_{2}))\mathcal{T}^{\varepsilon}_{2}(w)\Psi_{2}(y)dxd\sigma_{y}\\\\[14.22636pt]
\displaystyle\leq
C\varepsilon^{\gamma}\|\mathcal{T}^{\varepsilon}_{1}(u^{\varepsilon}_{1})-\mathcal{T}^{\varepsilon}_{2}(u^{\varepsilon}_{2})\|_{L^{2}(\Omega\times\Gamma)}\|\mathcal{T}^{\varepsilon}_{2}(w)\Psi_{2}(y)\|_{L^{2}(\Omega\times\Gamma)}\leq
C\varepsilon^{\gamma}\varepsilon^{\frac{1-\gamma}{2}}=C\varepsilon^{\frac{1+\gamma}{2}}.\end{array}$

On the other hand

(5.33)
$\begin{array}[]{ll}\displaystyle\varepsilon^{\gamma+1}M\int_{\Gamma^{\varepsilon}}h^{\varepsilon}[u^{\varepsilon}]\,d\sigma\leq
C\varepsilon^{\gamma+1}\|[u^{\varepsilon}]\|_{L^{2}(\Gamma^{\varepsilon})}|\Gamma^{\varepsilon}|^{\frac{1}{2}}\leq
C\varepsilon^{\frac{1+\gamma}{2}}.\end{array}$

Since $\gamma>-1$, by passing to the limit in (5.32) and (5.33), as
$\varepsilon$ tends to zero, we obtain

(5.34) $\displaystyle\lim_{\varepsilon\rightarrow
0}\varepsilon^{\gamma+1}\int_{\Gamma^{\varepsilon}}h^{\varepsilon}[u^{\varepsilon}](w\Psi_{2}^{\varepsilon}+M)\,d\sigma=0$

and, thus, (5.31) gives

(5.35) $\displaystyle\frac{1}{|Y|}\int_{\Omega\times
Y_{2}}A\nabla_{y}\overline{u}_{2}\nabla_{y}\Phi_{2}\,dxdy\leq 0,$

where $\Phi_{2}(x,y)=w(x)\Psi_{2}(y)$. By density, (5.35) holds for every
$\Phi_{2}\in L^{2}(\Omega;H^{1}(Y_{2}))$.
Now, by taking $\Phi_{2}=\overline{u}_{2}$ in (5.35) and using (A$1$), one
obtains that

(5.36) $\displaystyle\nabla_{y}\overline{u}_{2}=0\quad\hbox{ a.e. in
}\Omega\times Y_{2}.$

Arguing as for the case $\gamma<-1$, we take $\varphi\in\mathcal{D}(\Omega)$
and use $(u^{\varepsilon}_{1}+\varphi,u^{\varepsilon}_{2}+\varphi)\in
K^{\varepsilon}_{\gamma}$ as test function in (2.4) obtaining

(5.37) $\displaystyle\frac{1}{|Y|}\int_{\Omega\times Y_{1}}A(\nabla
u_{1}+\nabla_{y}\widehat{u}_{1})\nabla\varphi\,dxdy=\int_{\Omega}f\varphi\,dx,\quad\forall\varphi\in
H^{1}_{0}(\Omega),$

in view of (5.36).
Let $C^{\infty}_{per}(Y_{1}\cup\Gamma)\doteq\\{v_{|Y_{1}\cup\Gamma}\,|\,v\in
C^{\infty}_{per}(Y)\\}$. It is easily seen that
$\overline{C^{\infty}_{per}(Y_{1}\cup\Gamma)}=H^{1}_{per}(Y_{1})$. Hence, if
we take $w\in\mathcal{D}(\Omega)$ and $\Psi_{1}\in
C^{\infty}_{per}(Y_{1}\cup\Gamma)$, there exists $M\geq 0$ such that

(5.38)
$\displaystyle\|w\|_{L^{\infty}(\Omega)}\|\Psi_{1}\|_{L^{\infty}(Y_{1}\cup\Gamma)}\leq
M.$

Thus, set $\Psi_{1}^{\varepsilon}(x)\doteq\Psi_{1}(\frac{x}{\varepsilon})$, we
can choose $(u^{\varepsilon}_{1}+\varepsilon
w\Psi_{1}^{\varepsilon}+\varepsilon M,u^{\varepsilon}_{2})\in
K^{\varepsilon}_{\gamma}$ as test function in (2.4) getting:

(5.39) $\displaystyle\int_{\Omega_{1}^{\varepsilon}}A^{\varepsilon}\nabla
u^{\varepsilon}_{1}\nabla(\varepsilon
w\Psi_{1}^{\varepsilon})\,dx+\varepsilon^{\gamma+1}\int_{\Gamma^{\varepsilon}}h^{\varepsilon}[u^{\varepsilon}](w\Psi_{1}^{\varepsilon}+M)\,d\sigma\geq\varepsilon\int_{\Omega_{1}^{\varepsilon}}f(w\Psi_{1}^{\varepsilon}+M)\,dx.$

By unfolding and arguing as in (5.33) and (5.34), one has

(5.40) $\displaystyle\frac{1}{|Y|}\int_{\Omega\times Y_{1}}A(y)(\nabla
u_{1}+\nabla_{y}\widehat{u}_{1})\nabla_{y}\Phi_{1}\,dxdy\geq 0,$

where $\Phi_{1}(x,y)=w(x)\Psi_{1}(y)$. By density, the previous inequality
holds for every $\Phi_{1}\in L^{2}(\Omega;H^{1}_{per}(Y_{1}))$, which implies

(5.41) $\displaystyle\frac{1}{|Y|}\int_{\Omega\times Y_{1}}A(y)(\nabla
u_{1}+\nabla_{y}\widehat{u}_{1})\nabla_{y}\Phi_{1}\,dxdy=0,\quad\forall\Phi_{1}\in
L^{2}(\Omega;H^{1}_{per}(Y_{1})).$

Finally, we add (5.37) and (5.41), and obtain the following problem:

(5.42) $\left\\{\begin{array}[]{lll}\displaystyle\hbox{ Find
}(u_{1},\widehat{u}_{1})\in H^{1}_{0}(\Omega)\times
L^{2}(\Omega,H^{1}_{per}(Y_{1})),\hbox{ with
}\mathcal{M}_{\Gamma}(\widehat{u}_{1})=0\hbox{ a.e. in }\Omega,\ \hbox{ s.t.
}\\\\[14.22636pt] \displaystyle\frac{1}{|Y|}\int_{\Omega\times
Y_{1}}A(y)(\nabla
u_{1}+\nabla_{y}\widehat{u}_{1})(\nabla\varphi+\nabla_{y}\Phi_{1})\,dxdy=\int_{\Omega}f\varphi\,dx,\\\\[14.22636pt]
\displaystyle\forall\varphi\in H^{1}_{0}(\Omega),\quad\forall\Phi_{1}\in
L^{2}(\Omega;H^{1}_{per}(Y_{1})).\end{array}\right.$

The above problem is the one obtained in the classical work [8] dealing with
the periodic unfolding method for perforated domains. Hence
$(u_{1},\widehat{u}_{1})$ is unique and convergences (5.1) and (5.29) (where
$\nabla_{y}\overline{u}_{2}\equiv 0$) hold for the whole sequences. Moreover,
if we define $A^{0}_{\gamma}$ as in (2.19), by classical arguments $u_{1}$
results the unique solution of problem (2.10).

###  5.4 The case $\gamma=1$

Let $u^{\varepsilon}$ be the weak solution of problem (2.1). Arguing as in the
case $\gamma\in]-1;1[$, by Theorem 4.5 we get again (5.36), even in the case
$\gamma=1$. We deduce also the existence of $u_{1}\in H^{1}_{0}(\Omega)$,
$\widehat{u}_{1}\in L^{2}(\Omega,H^{1}_{per}(Y_{1}))$ with
$\mathcal{M}_{\Gamma}(\widehat{u}_{1})=0$ a.e. in $\Omega$ and ${u}_{2}\in
L^{2}(\Omega)$ such that, up to a subsequence,

(5.43)
$\left\\{\begin{array}[]{lllll}\displaystyle\widetilde{u^{\varepsilon}_{i}}\rightharpoonup\theta_{i}u_{i}&\hbox{
weakly in }L^{2}(\Omega),\ i=1,2,\\\\[8.53581pt]
\displaystyle\mathcal{T}^{\varepsilon}_{1}(u^{\varepsilon}_{1})\rightarrow
u_{1}&\hbox{ strongly in }L^{2}(\Omega,H^{1}(Y_{1})),\\\\[8.53581pt]
\displaystyle\mathcal{T}^{\varepsilon}_{2}(u^{\varepsilon}_{2})\rightharpoonup
u_{2}&\hbox{ weakly in }L^{2}(\Omega,H^{1}(Y_{2})),\\\\[8.53581pt]
\displaystyle\mathcal{T}^{\varepsilon}_{1}(\nabla
u^{\varepsilon}_{1})\rightharpoonup\nabla
u_{1}+\nabla_{y}\widehat{u}_{1}&\hbox{ weakly in }L^{2}(\Omega\times
Y_{1}),\\\\[8.53581pt] \displaystyle\mathcal{T}^{\varepsilon}_{2}(\nabla
u^{\varepsilon}_{2})\rightharpoonup 0&\hbox{ weakly in }L^{2}(\Omega\times
Y_{2}),\end{array}\right.$

and

(5.44) $\displaystyle\frac{1}{|Y|}\int_{\Omega\times Y_{1}}A(y)(\nabla
u_{1}+\nabla_{y}\widehat{u}_{1})\nabla_{y}\Phi_{1}\,dxdy=0,\quad\forall\Phi_{1}\in
L^{2}(\Omega;H^{1}_{per}(Y_{1})).$

By standard arguments, it follows that

(5.45) $\widehat{u}_{1}(x,y)=-\sum_{j=1}^{N}\frac{\partial u_{1}}{\partial
x_{j}}(x)\chi_{j}(y)$

where $\chi_{j}$, for $j=1,...,N$, are the unique solutions of the cell
problems (2.20).

Now, we take $\varphi_{i}\in\mathcal{D}(\Omega)$, $i=1,2$, such that
$\varphi_{1}\geq\varphi_{2}$ in $\Omega$. By choosing
$(\varphi_{1},\varphi_{2})\in K^{\varepsilon}_{\gamma}$ as test function in
(2.4) we get

(5.46)
$\begin{array}[]{ll}\displaystyle\int_{\Omega_{1}^{\varepsilon}}A^{\varepsilon}\nabla
u^{\varepsilon}_{1}\nabla(\varphi_{1}-u^{\varepsilon}_{1})dx+\int_{\Omega_{2}^{\varepsilon}}A^{\varepsilon}\nabla
u^{\varepsilon}_{2}\nabla(\varphi_{2}-u^{\varepsilon}_{2})dx+\varepsilon\int_{\Gamma^{\varepsilon}}h^{\varepsilon}[u^{\varepsilon}][\varphi]d\sigma\\\\[8.53581pt]
\displaystyle\geq\int_{\Omega_{1}^{\varepsilon}}f(\varphi_{1}-u^{\varepsilon}_{1})dx+\int_{\Omega_{2}^{\varepsilon}}f(\varphi_{2}-u^{\varepsilon}_{2})dx+\varepsilon\int_{\Gamma^{\varepsilon}}h^{\varepsilon}([u^{\varepsilon}])^{2}d\sigma.\end{array}$

We want to pass to the limit by unfolding in the previous inequality. To this
aim let us observe that by (4.2)

(5.47)
$\varepsilon\int_{\Gamma^{\varepsilon}}h^{\varepsilon}[u^{\varepsilon}]^{2}\,d\sigma\geq\frac{1}{|Y|}\int_{\Omega\times\Gamma}h(y)(\mathcal{T}^{\varepsilon}_{1}(u^{\varepsilon}_{1})-\mathcal{T}^{\varepsilon}_{2}(u^{\varepsilon}_{2}))^{2}dxd\sigma_{y}.$

Hence, by (5.47) and the lower semi-continuity of the norm with respect to the
weak convergence, we obtain

(5.48) $\begin{array}[]{lll}\displaystyle\frac{1}{|Y|}\int_{\Omega\times
Y_{1}}A(y)(\nabla u_{1}+\nabla_{y}\widehat{u}_{1})(\nabla\varphi_{1}-\nabla
u_{1}-\nabla_{y}\widehat{u}_{1})\,dxdy+\frac{1}{|Y|}\int_{\Omega\times\Gamma}h(y)(u_{1}-u_{2})(\varphi_{1}-\varphi_{2})\,dxdy\\\\[14.22636pt]
\displaystyle\geq\theta_{1}\int_{\Omega}f(\varphi_{1}-u_{1})\,dx+\theta_{2}\int_{\Omega}f(\varphi_{2}-u_{2})\,dx+\frac{1}{|Y|}\int_{\Omega\times\Gamma}h(y)(u_{1}-u_{2})^{2}\,dxdy.\end{array}$

If we write (5.44) for $\Phi_{1}=\widehat{u}_{1}$ and we add it to (5.48), we
get

(5.49) $\begin{array}[]{lll}\displaystyle\frac{1}{|Y|}\int_{\Omega\times
Y_{1}}A(y)(\nabla u_{1}+\nabla_{y}\widehat{u}_{1})(\nabla\varphi_{1}-\nabla
u_{1})\,dxdy+\frac{1}{|Y|}\int_{\Omega\times\Gamma}h(y)(u_{1}-u_{2})((\varphi_{1}-\varphi_{2})-(u_{1}-u_{2}))\,dxdy\\\\[14.22636pt]
\displaystyle\geq\theta_{1}\int_{\Omega}f(\varphi_{1}-u_{1})\,dx+\theta_{2}\int_{\Omega}f(\varphi_{2}-u_{2})\,dx.\end{array}$

We observe that

(5.50) $\displaystyle u_{1}\geq u_{2}\quad\hbox{ a.e. in }\Omega.$

Indeed, since $u^{\varepsilon}_{1}\geq u^{\varepsilon}_{2}$ a.e. on
$\Gamma^{\varepsilon}$, one deduces that
$\mathcal{T}^{\varepsilon}_{1}(u^{\varepsilon}_{1})\geq\mathcal{T}^{\varepsilon}_{2}(u^{\varepsilon}_{2})$
a.e. in $\Omega\times\Gamma$, hence (5.50) follows from convergences (5.43).

Hence, in view of (5.45), (5.50) and the fact that $A^{0}_{1}$ is given by
(2.19), set $K\doteq\\{(\varphi_{1},\varphi_{2})\in H^{1}_{0}(\Omega)\times
L^{2}(\Omega)\,|\,\varphi_{1}\geq\varphi_{2}\hbox{ in }\Omega\\}$, we get that
$(u_{1},u_{2})$ is solution of the following problem:

(5.51) $\left\\{\begin{array}[]{lll}\displaystyle\hbox{ Find }(u_{1},u_{2})\in
K\hbox{ s.t. }\\\\[8.53581pt] \displaystyle\int_{\Omega}A^{0}_{1}\nabla
u_{1}(\nabla\varphi_{1}-\nabla
u_{1})\,dx+\frac{\mathcal{M}_{\Gamma}(h)}{|Y|}\int_{\Omega}(u_{1}-u_{2})[(\varphi_{1}-\varphi_{2})-(u_{1}-u_{2})]\,dx\\\\[14.22636pt]
\displaystyle\geq\theta_{1}\int_{\Omega}f(\varphi_{1}-u_{1})\,dx+\theta_{2}\int_{\Omega}f(\varphi_{2}-u_{2})\,dx,\quad\forall(\varphi_{1},\varphi_{2})\in
K.\end{array}\right.$

Let us denote by $\mathcal{A}$ the operator

$\mathcal{A}:K\rightarrow K^{\prime}$

given by

(5.52)
$\left\langle\mathcal{A}(u),v\right\rangle_{K^{\prime},K}\doteq\int_{\Omega}A^{0}_{1}\nabla
u_{1}\nabla
v_{1}\,dx+\frac{\mathcal{M}_{\Gamma}(h)}{|Y|}\int_{\Omega}(u_{1}-u_{2})(v_{1}-v_{2})dx.$

By $(\textbf{A}4)$ and the fact that
$A^{0}_{1}\in\mathcal{M}(\alpha,\frac{\beta^{2}}{\alpha},\Omega)$ (see [13]),
one has

(5.53)
$\begin{array}[]{ll}\displaystyle\left\langle\mathcal{A}(w)-\mathcal{A}(v),w-v\right\rangle=\int_{\Omega}A^{0}_{1}\nabla(w_{1}-v_{1})\nabla(w_{1}-v_{1})\,dx+\frac{\mathcal{M}_{\Gamma}(h)}{|Y|}\int_{\Omega}[(w_{1}-w_{2})-(v_{1}-v_{2})]^{2}\,dx\\\\[14.22636pt]
\displaystyle\geq\alpha\|w_{1}-v_{1}\|_{H^{1}_{0}(\Omega)}^{2}+h_{0}\|(w_{1}-w_{2})-(v_{1}-v_{2})\|_{L^{2}(\Omega)}^{2}>0,\quad\hbox{
if }w\neq v.\end{array}$

From this and [28, Thereom 8.3 (pag. $248$)], we deduce that the couple
$(u_{1},u_{2})$ is unique. Hence, by (5.45), convergences (5.43) hold for the
whole sequences.

## Acknowledgement

This work was supported by the project PRIN2022 D53D23005580006 “Elliptic and
parabolic problems, heat kernel estimates and spectral theory” and by the
projects GNAMPA2023 (CUP E53C22001930001) “Problemi misti locali e non locali
con coefficienti singolari” and “Analisi asintotica di problemi al contorno in
strutture sottili o rugose”.

## References

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