#  Homogenization of a nonlinear elliptic problem
with imperfect rough interface

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

###### Abstract

This paper deals with an elliptic problem with a nonlinear lower order term
set in an open bounded cylinder of $\mathbb{R}^{N}$, $N\geq 2$, divided into
two connected components by an imperfect rough interface. More precisely, we
assume that at the interface the flux is continuous and proportional, via a
nonlinear rule, to the jump of the solution.

According to the amplitude of the interface oscillations and the
proportionality coefficient, we derive different homogenized problems.

††Keywords: Homogenization, nonlinear elliptic problems, rough interface††MSC:
35B27, 35J60, 35J66, 35Q92

* 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

This paper concerns a nonlinear elliptic problem with a lower order term
defined in an open bounded cylinder $Q=\omega\times]-\ell,\ell[$ of
$\mathbb{R}^{N}$, with $N\geq 2$, $\ell>0$ and $\omega$ bounded smooth domain
of $\mathbb{R}^{N-1}$. The cylinder is divided into two connected components
$Q^{+}_{\varepsilon}$ and $Q^{-}_{\varepsilon}$ by a rough surface
$\Gamma^{\varepsilon}$, $\varepsilon$ being a positive parameter tending to
zero. This common interface is represented by the graph of a quickly
oscillating function $\varepsilon$-periodic in the first $N-1$ variables,
having oscillations’ amplitude of order $\varepsilon^{k}$, with $k>0$. This
means that, as $\varepsilon$ goes to zero, the interface turns into a flat
surface.

Given $f\in L^{2}(Q)$, we consider the following problem:

(1.1)
$\left\\{\begin{array}[]{llll}\displaystyle-\textnormal{div}(D^{\varepsilon}\nabla
u_{\varepsilon})+h_{1}(u_{\varepsilon})=f&\textnormal{in}\;Q\setminus\Gamma^{\varepsilon},\\\\[5.69054pt]
\displaystyle(D^{\varepsilon}\nabla
u_{\varepsilon})^{+}\cdot\nu_{\varepsilon}=(D^{\varepsilon}\nabla
u_{\varepsilon})^{-}\cdot\nu_{\varepsilon}&\textnormal{on}\;\Gamma^{\varepsilon},\\\\[5.69054pt]
\displaystyle(D^{\varepsilon}\nabla
u_{\varepsilon})^{+}\cdot\nu_{\varepsilon}=-\varepsilon^{\gamma}h_{2}([u_{\varepsilon}])&\textnormal{on}\;\Gamma^{\varepsilon},\\\\[5.69054pt]
\displaystyle u_{\varepsilon}=0&\textnormal{on}\;\partial
Q,\end{array}\right.$

where $\nu_{\varepsilon}$ is the unit outward normal to $Q^{+}_{\varepsilon}$
and $[\cdot]$ is the jump of the solution across $\Gamma^{\varepsilon}$. We
assume that the matrix $D^{\varepsilon}$ is $\varepsilon$-periodic, uniformly
elliptic and bounded, and the nonlinear functions $h_{1}$ and $h_{2}$ satisfy
some natural regularity and growth conditions. For more details about the
setting of the problem we refer the reader to Section 2.

We are interested in studying the asymptotic behaviour, as $\varepsilon$ tends
to zero, of the solutions of problem (1.1). The values of the parameter
$\gamma\in\mathbb{R}$ in the jump condition and the parameter $k$ in the
amplitude of the oscillations play a crucial role and give rise to three
different limit cases denoted by A), B) and C). In all of them, we obtain a
homogenized elliptic equation with a nonlinear lower order term.

In cases A) and B), the limit domain is a cylinder made up of two components
$Q^{+}$ and $Q^{-}$ separated by a flat interface $\Gamma_{0}$. In particular,
in case A), we obtain an effective imperfect transmission problem where the
flux is proportional, by means of a nonlinear function, to the jump of the
solution via different constants whose values depend on $k$ and $\gamma$. In
case B), one obtains a homogenized problem equivalent to two independent
problems set in $Q^{+}$ and $Q^{-}$, with homogeneous Neumann boundary
conditions on $\Gamma_{0}$.

Finally, in case C), we find a homogenized Dirichlet elliptic problem in the
whole cylinder $Q$.

In [14], the authors study, in the same domain $Q$, the homogenization of the
stationary heat diffusion with a discontinuity of the temperature field
through the interface modeled by a jump of the solution linearly proportional
to the heat flux (see also [2], [3], [7], [15], [16], [21], [25], and the
references therein for physical justifications of the jump condition). The
homogenization of a related eigenvalue problem has been considered in [4]. In
[12], it is analyzed the parabolic version of [14]. The homogenization for the
elliptic problem of [14] with an additional singular lower order term is
treated in [11]. Nonlinear lower order and interfacial terms, as in our case,
appear in [13] where, by using the unfolding operator, it is investigated the
effective thermal conduction in a two component composite with inclusions.

On the other hand, as evidenced in [1], nonlinear interface conditions also
appear in the modeling of the electrical conduction in biological tissues, as
a consequence of an interfacial resistance property of the cell membranes.
Moreover, nonlinear reaction and jump terms, as in (1.1), occur very often
when studying some metabolic processes arising in biological cells, in which
biochemical species can diffuse and react either in the cytosol or in the
various organelles which are present in the cytoplasm. These species can even
react on the organelles’ membrane which is not perfectly smooth but it
contains some irregularities which play an important role mainly in problems
related to enzymatic kinetic. A precise mathematical model for such processes
in the non-stationary case and in a two component composite with inclusions is
analyzed in [17] and [18]. In [10], the authors study some nonlinear models
involving adsorption and reactions arising in porous media. The homogenization
of nonlinear problems modelling the chemical reactive flows through the
exterior of a domain containing periodically distributed reactive solid grains
is considered in [9]. In order to model some reaction-diffusion processes in
porous media, in [6], it has been analyzed the macroscopic behavior of the
solution of a parabolic and coupled system of three partial differential
equations depending on a small positive parameter $\varepsilon$ related to the
size and periodicity of the structure. For reaction-diffusion equations we can
refer, for instance, to [17]$\div$ [20], [22] $\div$ [24] and to the papers
quoted therein.

Our work is organized as follows.

In Section 2 we describe the two component composite cylinder with rough
interface, introduce the problem together with its functional setting and
state the main result.

Section 3 is devoted to some technical tools related to the geometry of the
domain. At first we adapt to our context certain estimates contained in [8].
Then, we recall some compactness results of [14]. Successively, we generalize
and unify some convergence results proved in [14]. This argument allows us to
develop the homogenization process without the introduction of the different
extension operators needed in [14]. Finally, we show an useful convergence
result related only to the geometry of the interface and to the parameter
$\gamma$.

The existence and uniqueness of a weak solution of problem (1.1) and some a
priori estimates are proved in Section 4.

At last, in Section 5, we carry out the homogenization process.

##  2 Setting of the problem and main result

Throughout the paper, we use the notation given in [14] to specify the
structure of $Q$ and to define the functional spaces involved.
Let $N\geq 2$ and

  * $\bullet$

$Y=]0,1[^{N}$ be the volume reference cell,

  * $\bullet$

$Y^{\prime}=]0,1[^{N-1}$ be the surface reference cell,

  * $\bullet$

$\mathcal{M}_{E}(v)$ be the average on $E$ of any function $v\in L^{1}(E)$,

  * $\bullet$

$\chi_{E}$ be the characteristic function of any open set
$E\subseteq\mathbb{R}^{N}$.

Our domain is characterized by a rough interface described by means of the
graph of a function $g$ satisfying the following assumption:

Ag) The function $g:Y^{\prime}\rightarrow\mathbb{R}$ is positive, periodic,
Lipschitz continuous.

We set

(2.1) $\displaystyle\bar{g}=\max g.$

Given a positive parameter $\varepsilon$ converging to zero, $k>0$, and set
$x^{\prime}=(x_{1},...,x_{N-1})$, the graph

$\Gamma^{\varepsilon}=\left\\{x\in Q\ |\
x_{N}=\varepsilon^{k}g\left(\frac{x^{\prime}}{\varepsilon}\right)\right\\}$

represents an oscillating (rough) interface which divides the set $Q$ in two
subdomains.
More precisely, we denote by

$Q_{\varepsilon}^{+}=\left\\{x\in Q\ |\
x_{N}>\varepsilon^{k}g\left(\frac{x^{\prime}}{\varepsilon}\right)\right\\}$

the upper part of $Q$ and by

$Q_{\varepsilon}^{-}=\left\\{x\in Q\ |\
x_{N}<\varepsilon^{k}g\left(\frac{x^{\prime}}{\varepsilon}\right)\right\\}$

its lower part (see Figure 1).

$\ell$$-\ell$$\varepsilon^{k}\bar{g}$$0$$Q$$\varepsilon$$Q_{\varepsilon}^{-}$$Q_{\varepsilon}^{+}$$\Gamma^{\varepsilon}$
Figure 1: The cylinder $Q$ with rough interface $\Gamma^{\varepsilon}$

As observed in [14], in the case $0<k<1$, one has a fastly oscillating
interface. When $k=1$, the interface $\Gamma^{\varepsilon}$ is obtained by
homothetic dilatation of the fixed function $y_{N}=g(y^{\prime})$ in
$\mathbb{R}^{N}$, hence, it presents a self-similar geometry. Finally, the
flat case is obtained for $k>1$.
In view of (2.1), the oscillating surface is contained in the cylinder

(2.2) $\displaystyle S_{\varepsilon}=\omega\times[0,\varepsilon^{k}\bar{g}]$

and, as $\varepsilon$ goes to zero,

(2.3) $\displaystyle|S_{\varepsilon}|\to 0.$

######  Remark 2.1.

The cylindrical domain can be generalized to an arbitrary smooth domain $Q$,
where in any point of $\partial Q\cup\\{x|x_{N}=0\\}$ the normal to $\partial
Q$ is not parallel to the $N$-th coordinate vector.

Now, we state the specific hypotheses satisfied by the matrix
$D^{\varepsilon}$ and the nonlinear lower order and flux terms. Thus, let us
consider the following assumptions:

(A$1$) For any $\varepsilon>0$, the coeffficients matrix $D^{\varepsilon}$ is
defined by $D^{\varepsilon}(x)=D\left(\frac{x}{\varepsilon}\right)$, where $D$
is $Y$-periodic, symmetric and satisfies

$\displaystyle(D(x)\lambda,\lambda)\geq\alpha|\lambda|^{2},\qquad|D(x)\lambda|\leq\beta|\lambda|,\qquad\forall\lambda\in\mathbb{R}^{N}\hbox{
and a.e. in }Y,$

with $\alpha,\beta\in\mathbb{R}$, $0<\alpha<\beta$.

(A$2$) The function $h_{1}:\mathbb{R}\rightarrow\mathbb{R}$ is such that

  * 1.

$h_{1}$ is continuous;

  * 2.

$h_{1}$ is increasing and $h_{1}(0)=0$;

  * 3.

there exist a positive constant $C$ and an exponent $q_{1}$, with

(2.4) $1\leq q_{1}<2\,\text{ if }N\in\\{2,3\\}\quad\text{and}\quad 1\leq
q_{1}\leq\dfrac{N+2}{N-2}=\dfrac{2^{*}}{(2^{*})^{\prime}}\text{ if }N>3$

such that

(2.5) $|h_{1}(z)|\leq C\left(1+|z|^{q_{1}}\right).$

(A$3$) The flux term $h_{2}:\mathbb{R}\rightarrow\mathbb{R}$ is such that

  * 1.

$h_{2}$ is continuous;

  * 2.

$h_{2}$ is an increasing function;

  * 3.

there exists a positive constant $C$ such that

(2.6) $zh_{2}(z)\geq Cz^{2},\quad\forall s\in\mathbb{R};$

  * 4.

there exist a positive constant $C$ and an exponent $q_{2}$, with

(2.7) $1\leq q_{2}<2\,\text{ if }N\in\\{2,3\\}\quad\text{and}\quad 1\leq
q_{2}\leq\dfrac{N}{N-2}\text{ if }N>3$

such that

(2.8) $|h_{2}(z)|\leq C\left(1+|z|^{q_{2}}\right).$

######  Remark 2.2.

Let us observe that assumption (A$2$) implies

$z\,h_{1}(z)\geq 0\quad\text{ for all }z\in\mathbb{R}.$

We now precise the functional framework. To this aim, for any function $v$
defined on $Q$, we set

$v_{\varepsilon}^{+}=v_{|Q_{\varepsilon}^{+}},\quad
v_{\varepsilon}^{-}=v_{|Q_{\varepsilon}^{-}}.$

Hence, we introduce the Hilbert space

$W_{0}^{\varepsilon}=\\{v\in L^{2}(Q)\ |\ v_{\varepsilon}^{+}\in
H^{1}(Q_{\varepsilon}^{+}),v_{\varepsilon}^{-}\in
H^{1}(Q_{\varepsilon}^{-}),v=0\hbox{ on }\partial Q\\},$

equipped with the norm

(2.9) $\|v\|_{W_{0}^{\varepsilon}}=\|\nabla
v\|_{L^{2}(Q\setminus\Gamma^{\varepsilon})},$

where

(2.10) $\nabla v=\widetilde{\nabla v_{\varepsilon}^{+}}+\widetilde{\nabla
v_{\varepsilon}^{-}}.$

In view of (2.10), we identify $\nabla v$ with the absolute continuous part of
the gradient of $v$.

Moreover, to describe the limit domain, we set

$Q^{+}=\\{x\in Q\ |\ x_{N}>0\\},\quad Q^{-}=\\{x\in Q\ |\
x_{N}<0\\},\quad\Gamma_{0}=\\{x\in Q\ |\ x_{N}=0\\},$

and

$v^{+}=v_{|Q^{+}},\quad v^{-}=v_{|Q^{-}},$

for any function $v$ defined on $Q$ (see Figure 2).

$\ell$$-\ell$$Q$$0$$Q^{-}$$Q^{+}$$\Gamma_{0}$ Figure 2: The cylinder $Q$ with
flat interface $\Gamma_{0}$

Observe that

(2.11)
$\displaystyle\chi_{Q_{\varepsilon}^{\pm}}\rightarrow\chi_{Q^{\pm}}\quad\text{
strongly in }L^{p}(Q),\,1\leq p<+\infty\,\text{ and weakly* in
}L^{\infty}(Q).$

Then, we define the Hilbert space

$W_{0}^{0}=\\{v\in L^{2}(Q)\ |\ v^{+}\in H^{1}(Q^{+}),v^{-}\in
H^{1}(Q^{-}),v=0\hbox{ on }\partial Q\\},$

endowed with the norm

(2.12) $\|v\|_{W_{0}^{0}}=\|\nabla v\|_{L^{2}(Q\setminus\Gamma_{0})},$

with $\nabla v$ defined similarly to (2.10). Let us observe that (2.9) and
(2.12) are norms, as a consequence of the Poincaré inequality. In particular,
we explictly observe that, for any $v\in W^{\varepsilon}_{0}$, there exists a
positive constant C, independent of $\varepsilon$, such that

(2.13) $\|v\|_{L^{2}(Q)}\leq C\|\nabla
v\|_{L^{2}(Q\setminus\Gamma^{\varepsilon})}.$

The variational formulation of problem (1.1) is then given by

(2.14) $\left\\{\begin{array}[]{l}\displaystyle\text{Find }u_{\varepsilon}\in
W_{0}^{\varepsilon}\text{ such that }\\\\[14.22636pt]
{\displaystyle\int_{Q\setminus\Gamma^{\varepsilon}}D^{\varepsilon}\nabla
u_{\varepsilon}\nabla\varphi\,dx}+\varepsilon^{\gamma}{\displaystyle\int_{\Gamma^{\varepsilon}}h_{2}(u^{+}_{\varepsilon}-u^{-}_{\varepsilon})(\varphi_{\varepsilon}^{+}-\varphi_{\varepsilon}^{-})\,d\sigma}+{\displaystyle\int_{Q}h_{1}(u_{\varepsilon})\varphi\,dx}={\displaystyle\int_{Q}f\varphi\,dx},\\\\[14.22636pt]
\text{ for all }\varphi\in W_{0}^{\varepsilon}.\end{array}\right.$

Under our assumptions the above integrals make sense and problem (2.14) admits
a unique solution (see Section 4 for details).

###  2.1 Statement of the main result

In order to describe the asymptotic behaviour of the sequence of solutions of
problem (2.14), let us introduce the homogenized tensor $D^{0}$ (see also [5])
defined by

(2.15) $\displaystyle
D^{0}\lambda=\mathcal{M}_{Y}(D(\nabla\omega_{\lambda})),$

with $\omega_{\lambda}\in H^{1}(Y)$ solution, for any
$\lambda\in\mathbb{R}^{N}$ of

(2.16)
$\left\\{\begin{array}[]{ll}\displaystyle-\hbox{div}\left(D\nabla\omega_{\lambda}\right)=0&\text{
in }Y,\\\\[5.69054pt] \displaystyle\omega_{\lambda}-\lambda\cdot y&Y-\text{
periodic},\\\\[5.69054pt] \mathcal{M}_{Y}(w_{\lambda}-\lambda\cdot
y)=0.\par\end{array}\right.$

The main result of our paper is given in the following theorem whose proof is
contained in Section 5:

######  Theorem 2.3.

Under assumptions $(\textbf{A}_{g})$, $(\textbf{A}1)$, $(\textbf{A}2)$,
$(\textbf{A}3)$ and given $f\in L^{2}(Q)$, let $u_{\varepsilon}$ be the
solution of problem (2.14). Then, there exists a function $u\in W^{0}_{0}$
such that

(2.17) $\left\\{\begin{array}[]{lll}\displaystyle u_{\varepsilon}\rightarrow
u&\hbox{ strongly in }L^{2}(Q),\\\\[5.69054pt]
\displaystyle\chi_{Q^{+}_{\varepsilon}}\nabla
u_{\varepsilon}\rightharpoonup\chi_{Q^{+}}\nabla u&\text{ weakly in
}(L^{2}(Q))^{N},\\\\[5.69054pt] \displaystyle\chi_{Q^{-}_{\varepsilon}}\nabla
u_{\varepsilon}\rightharpoonup\chi_{Q^{-}}\nabla u&\text{ weakly in
}(L^{2}(Q))^{N}.\end{array}\right.$

__Case A): $(k\geq 1$ and $\gamma=0)$ or $(0<k<1$ and $\gamma=1-k)$

The limit function $u\in W^{0}_{0}$ is the unique solution of problem

(2.18) $\left\\{\begin{array}[]{ll}\displaystyle-\textnormal{div}(D^{0}\nabla
u)+h_{1}(u)=f&\textnormal{ in }Q\setminus\Gamma_{0},\\\\[5.69054pt]
\displaystyle(D^{0}\nabla u)^{+}\cdot\nu=(D^{0}\nabla
u)^{-}\cdot\nu&\textnormal{ on }\Gamma_{0},\\\\[5.69054pt]
\displaystyle(D^{0}\nabla u)^{+}\cdot\nu=-{G}\,h_{2}(u^{+}-u^{-})&\textnormal{
on }\Gamma_{0},\\\\[5.69054pt] \displaystyle u=0&\textnormal{ on }\partial
Q,\end{array}\right.$

where $\nu$ is the unit outward normal to $Q^{+}$ and

(2.19)
$\displaystyle{G}=\left\\{\begin{array}[]{ll}\displaystyle\mathcal{M}_{Y^{\prime}}\left(\sqrt{1+\left|\nabla_{y^{\prime}}g(y^{\prime})\right|^{2}}\right)&\hbox{
if }k=1,\gamma=0,\\\\[5.69054pt] \displaystyle 1&\hbox{ if
}k>1,\gamma=0,\\\\[5.69054pt]
\displaystyle\mathcal{M}_{Y^{\prime}}\left(\left|\nabla_{y^{\prime}}g(y^{\prime})\right|\right)&\hbox{
if }0<k<1,\gamma=1-k.\end{array}\right.$

__Case B): $(k\geq 1$ and $\gamma>0)$ or $(0<k<1$ and $\gamma>1-k)$ The limit
function $u\in W^{0}_{0}$ is the unique solution of problem

(2.20) $\left\\{\begin{array}[]{ll}\displaystyle-\textnormal{div}(D^{0}\nabla
u)+h_{1}(u)=f&\textnormal{ in }Q\setminus\Gamma_{0},\\\\[5.69054pt]
\displaystyle(D^{0}\nabla u)^{-}\cdot\nu=(D^{0}\nabla
u)^{+}\cdot\nu=0&\textnormal{ on }\Gamma_{0},\\\\[5.69054pt] \displaystyle
u=0&\textnormal{ on }\partial Q,\end{array}\right.$

where $\nu$ is the unit outward normal to $Q^{+}$.

__Case C): $({k\geq 1}$ and ${\gamma<0})$ or $({0<k<1}$ and ${\gamma<1-k})$

The limit function $u$ belongs to $H^{1}_{0}(Q)$ and it is the unique solution
of problem

(2.21) $\left\\{\begin{array}[]{ll}\displaystyle-\textnormal{div}(D^{0}\nabla
u)+h_{1}(u)=f&\textnormal{ in }Q,\\\\[5.69054pt] \displaystyle
u=0&\textnormal{ on }\partial Q.\end{array}\right.$

######  Remark 2.4.

We observe that in the three limit problems we always get a homogenized
elliptic equation with a nonlinear lower order term. The main differences are
due to the interfacial contribution. In particular, in case A), we obtain an
effective imperfect transmission problem where the flux is proportional, by
means of a nonlinear function, to the jump of the solution via different
constants whose values depend on $k$ and $\gamma$. In case B), the flat
surface $\Gamma_{0}$ is an isolating interface. Indeed, one obtains a
homogenized problem equivalent to two independent problems set in $Q^{+}$ and
$Q^{-}$, with homogeneous Neumann boundary conditions on $\Gamma_{0}$ (see
also [14]). Finally, in case C), the presence of the interface is neglectful
since we find a homogenized Dirichlet elliptic problem in the whole cylinder
$Q$.

##  3 Some preliminary and compactness results

This section is devoted to some tools, related to the geometry of the domain,
needed to study the asymptotic behaviour of problem (2.14).

We start stating a generalized version of a lemma proved in [8] (see also
[14]). The proof can be easily obtained following along the lines the one of
Lemma 1 in [8], with opportune modifications.

######  Lemma 3.1.

Let $p\geq 1$. For any $w_{1}\in W^{1,p}(Q^{-}_{\varepsilon})$, $w_{2}\in
W^{1,p}(Q^{+}_{\varepsilon})$ and $w_{3}\in W^{1,p}(S_{\varepsilon})$ it holds

(3.1)
$\displaystyle\left\|w_{1}\left(x^{\prime},\varepsilon^{k}g\left(\frac{x^{\prime}}{\varepsilon}\right)\right)-w_{1}\left(x^{\prime},0\right)\right\|_{L^{2}(\omega)}\leq
C{\varepsilon^{k/p}}\|w_{1}\|_{W^{1,p}(Q^{-}_{\varepsilon})},$ (3.2)
$\displaystyle\left\|w_{2}\left(x^{\prime},\varepsilon^{k}g\left(\frac{x^{\prime}}{\varepsilon}\right)\right)-w_{2}\left(x^{\prime},\varepsilon^{k}\bar{g}\right)\right\|_{L^{2}(\omega)}\leq
C{\varepsilon^{k/p}}\|w_{2}\|_{W^{1,p}(Q^{+}_{\varepsilon})},$

and

(3.3)
$\displaystyle\left\|w_{3}\left(x^{\prime},\varepsilon^{k}\bar{g}\right)-w_{3}\left(x^{\prime},0\right)\right\|_{L^{2}(\omega)}\leq
C{\varepsilon^{k/p}}\|w_{3}\|_{W^{1,p}(S_{\varepsilon})},$

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

Now, we recall a compactness result given in [14].

######  Proposition 3.2.

Let $\\{w_{\varepsilon}\\}$ be a family of functions in $W_{0}^{\varepsilon}$
such that

(3.4) $\displaystyle\|w_{\varepsilon}\|_{W_{0}^{\varepsilon}}\leq C,$

with $C$ positive constant independent of $\varepsilon$. Then, there exist a
subsequence (still denoted $\varepsilon$) and a function $w$ in $W_{0}^{0}$
such that

(3.5) $\left\\{\begin{array}[]{lll}\displaystyle w_{\varepsilon}\rightarrow
w&\text{ strongly in }L^{2}(Q)\text{ and a.e. in }Q,\\\\[5.69054pt]
\displaystyle\chi_{Q^{+}_{\varepsilon}}\nabla
w_{\varepsilon}\rightharpoonup\chi_{Q^{+}}\nabla w&\text{ weakly in
}(L^{2}(Q))^{N},\\\\[5.69054pt] \displaystyle\chi_{Q^{-}_{\varepsilon}}\nabla
w_{\varepsilon}\rightharpoonup\chi_{Q^{-}}\nabla w&\text{ weakly in
}(L^{2}(Q))^{N}.\end{array}\right.$

In [14] the authors prove some convergences for the sequence of solutions
$u_{\varepsilon}$ of problem (2.14) by using different extension operators
according to the values of $k$.

In the next proposition, inspired by the argument used in [14] for $0<k<1$, we
give a unified proof of these convergences valid for $k>0$ and for any
uniformly bounded sequence $w_{\varepsilon}$ in $W_{0}^{\varepsilon}$. This
consents to develop the homogenization process without the introduction of the
extension operators used in [14] for the case $k\geq 1$, and allow us to treat
all the cases in a unique way.

######  Proposition 3.3.

Under the hypotheses of Proposition 3.2, there exists a subsequence (still
denoted $\varepsilon$) such that

(3.6) $\displaystyle
w^{\pm}_{\varepsilon}\left(\cdot,\varepsilon^{k}g\left(\frac{\cdot}{\varepsilon}\right)\right)\,\rightarrow\,w^{\pm}(\cdot,0)\quad\text{
strongly in }L^{2}(\omega).$

###### Proof.

To obtain the desired convergences we show that

(3.7)
$\displaystyle\left\|w^{+}_{\varepsilon}\left(\cdot,\varepsilon^{k}g\left(\frac{\cdot}{\varepsilon}\right)\right)-w^{+}\left(\cdot,0\right)\right\|_{L^{2}(\omega)}\to
0,$

and

(3.8)
$\displaystyle\left\|w^{-}_{\varepsilon}\left(\cdot,\varepsilon^{k}g\left(\frac{\cdot}{\varepsilon}\right)\right)-w^{-}\left(\cdot,0\right)\right\|_{L^{2}(\omega)}\to
0,$

up to a subsequence.

We proceed into three steps. In the first two steps we show (3.7), while (3.8)
is proved in the last one.

Step 1. Let us prove that there exists a suitable extension operator denoted
by $\check{w}^{+}_{\varepsilon}$ such that

(3.9) $\displaystyle\check{w}^{+}_{\varepsilon}\to w^{+}\quad\text{ strongly
in }L^{2}(Q^{+}).$

To this aim, we restrict the function $w^{+}_{\varepsilon}$ to the set
$Q^{+}\setminus S_{\varepsilon}$ and define its extension
$\check{w}^{+}_{\varepsilon}$ to $Q^{+}$ as follows:

(3.10)
$\displaystyle\check{w}^{+}_{\varepsilon}=\left\\{\begin{array}[]{ll}\displaystyle
w^{+}_{\varepsilon}&\text{ in }Q^{+}\setminus S_{\varepsilon},\\\\[5.69054pt]
\displaystyle Pw^{+}_{\varepsilon}&\text{ in
}S_{\varepsilon},\end{array}\right.$

where $P$ denotes the usual extension operator in Sobolev spaces such that

(3.11) $\displaystyle\|\check{w}^{+}_{\varepsilon}\|_{H^{1}(Q^{+})}\leq
C\|w^{+}_{\varepsilon}\|_{H^{1}(Q^{+}\setminus S_{\varepsilon})},$

with $C$ positive constant independent of $\varepsilon$. Due to the Sobolev
embeddings and (3.4), inequality (3.11) implies

(3.12) $\displaystyle\|\check{w}^{+}_{\varepsilon}\|_{L^{2^{*}}(Q^{+})}\leq
C,$

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

One has

(3.13)
$\begin{array}[]{c}\displaystyle\int_{Q^{+}}\left|\check{w}^{+}_{\varepsilon}-w^{+}\right|^{2}\,dx=\int_{Q^{+}}\left|\chi_{Q^{+}\setminus
S_{\varepsilon}}\check{w}^{+}_{\varepsilon}+\chi_{S_{\varepsilon}}\check{w}^{+}_{\varepsilon}-w^{+}\right|^{2}\,dx\\\\[14.22636pt]
\displaystyle\leq C\left(\int_{Q^{+}}\left|\chi_{Q^{+}\setminus
S_{\varepsilon}}\check{w}^{+}_{\varepsilon}-w^{+}\right|^{2}\,dx+\int_{Q^{+}}\left|\chi_{S_{\varepsilon}}\check{w}^{+}_{\varepsilon}\right|^{2}\,dx\right)\\\\[14.22636pt]
\displaystyle\leq
C\left(\int_{Q^{+}}\left|w^{+}_{\varepsilon}-w^{+}\right|^{2}\,dx+\int_{Q^{+}}\left|\chi_{S_{\varepsilon}}\check{w}^{+}_{\varepsilon}\right|^{2}\,dx\right).\end{array}$

In view of (3.5), the first integral on the right hand side of (3.13)
converges to zero. Concerning the second integral, by Hölder inequality, we
get

$\displaystyle\int_{Q^{+}}\left|\chi_{S_{\varepsilon}}\check{w}^{+}_{\varepsilon}\right|^{2}\,dx\leq\left|S_{\varepsilon}\right|^{\frac{1}{(2^{*}/2)^{\prime}}}\left\|\check{w}^{+}_{\varepsilon}\right\|^{2}_{L^{2^{*}}(Q^{+})}.$

Thus, this integral goes to zero too, in view of (2.3) and (3.12). This
concludes the proof of (3.9).

Step 2. Here we prove (3.7).
Let us observe that

(3.14)
$\begin{array}[]{c}\displaystyle\left\|w^{+}_{\varepsilon}\left(\cdot,\varepsilon^{k}g\left(\frac{\cdot}{\varepsilon}\right)\right)-w^{+}\left(\cdot,0\right)\right\|_{L^{2}(\omega)}\\\\[14.22636pt]
\displaystyle\leq\left\|w^{+}_{\varepsilon}\left(\cdot,\varepsilon^{k}g\left(\frac{\cdot}{\varepsilon}\right)\right)-w_{\varepsilon}^{+}\left(\cdot,\varepsilon^{k}\bar{g}\right)\right\|_{L^{2}(\omega)}+\displaystyle\left\|w_{\varepsilon}^{+}\left(\cdot,\varepsilon^{k}\bar{g}\right)-w^{+}\left(\cdot,0\right)\right\|_{L^{2}(\omega)}.\end{array}$

By (3.2), the first term on the right hand side of (3.14) tends to zero. Let
us prove that the second term converges to zero as well. One has

(3.15)
$\begin{array}[]{c}\displaystyle\left\|w_{\varepsilon}^{+}\left(\cdot,\varepsilon^{k}\bar{g}\right)-w^{+}\left(\cdot,0\right)\right\|_{L^{2}(\omega)}\\\\[14.22636pt]
\displaystyle\leq\left\|w_{\varepsilon}^{+}\left(\cdot,\varepsilon^{k}\bar{g}\right)-w^{+}\left(\cdot,\varepsilon^{k}\bar{g}\right)\right\|_{L^{2}(\omega)}+\left\|w^{+}\left(\cdot,\varepsilon^{k}\bar{g}\right)-w^{+}\left(\cdot,0\right)\right\|_{L^{2}(\omega)}.\end{array}$

Let us first show that

(3.16)
$\begin{array}[]{c}\displaystyle\left\|w_{\varepsilon}^{+}\left(\cdot,\varepsilon^{k}\bar{g}\right)-w^{+}\left(\cdot,\varepsilon^{k}\bar{g}\right)\right\|_{L^{2}(\omega)}=\left\|\check{w}_{\varepsilon}^{+}\left(\cdot,\varepsilon^{k}\bar{g}\right)-w^{+}\left(\cdot,\varepsilon^{k}\bar{g}\right)\right\|_{L^{2}(\omega)}\\\\[14.22636pt]
\displaystyle\leq\left\|\check{w}_{\varepsilon}^{+}\left(\cdot,0\right)-w^{+}\left(\cdot,0\right)\right\|_{L^{2}(\omega)}+\left\|\check{w}_{\varepsilon}^{+}\left(\cdot,\varepsilon^{k}\bar{g}\right)-w^{+}\left(\cdot,\varepsilon^{k}\bar{g}\right)-\check{w}_{\varepsilon}^{+}\left(\cdot,0\right)+w^{+}\left(\cdot,0\right)\right\|_{L^{2}(\omega)}\to
0.\end{array}$

Indeed, by (3.4), (3.9) and (3.11) the difference
$(w^{+}-\check{w}^{+}_{\varepsilon})$ converges to zero weakly in
$H^{1}(Q^{+})$. Thus, by the compactness of the trace it holds

(3.17)
$\displaystyle\left\|\check{w}_{\varepsilon}^{+}\left(\cdot,0\right)-w^{+}\left(\cdot,0\right)\right\|_{L^{2}(\omega)}\to
0,$

up to a subsequence.

Furthermore, by (3.3) applied to the function
$(\check{w}^{+}_{\varepsilon}-w^{+})$, one also has

(3.18)
$\displaystyle\left\|\check{w}_{\varepsilon}^{+}\left(\cdot,\varepsilon^{k}\bar{g}\right)-w^{+}\left(\cdot,\varepsilon^{k}\bar{g}\right)-\check{w}_{\varepsilon}^{+}\left(\cdot,0\right)+w^{+}\left(\cdot,0\right)\right\|_{L^{2}(\omega)}\to
0.$

Combining (3.17) and (3.18) we get (3.16).

Moreover, by the continuity of trace arguments, as observed in [14], we get

(3.19)
$\displaystyle\left\|w^{+}\left(\cdot,\varepsilon^{k}\bar{g}\right)-w^{+}\left(\cdot,0\right)\right\|_{L^{2}(\omega)}\to
0.$

Putting together (3.15), (3.16) and (3.19) we obtain

(3.20)
$\begin{array}[]{c}\displaystyle\left\|w_{\varepsilon}^{+}\left(\cdot,\varepsilon^{k}\bar{g}\right)-w^{+}\left(\cdot,0\right)\right\|_{L^{2}(\omega)}\to
0.\end{array}$

The previous arguments, together with (3.14) and (3.20), entail the claimed
convergence (3.7).

Step 3. The proof of (3.8) is easier since there is no need to use extension
operators. Indeed,

(3.21)
$\begin{array}[]{c}\displaystyle\left\|w^{-}_{\varepsilon}\left(\cdot,\varepsilon^{k}g\left(\frac{\cdot}{\varepsilon}\right)\right)-w^{-}\left(\cdot,0\right)\right\|_{L^{2}(\omega)}\\\\[14.22636pt]
\displaystyle\leq\left\|w^{-}_{\varepsilon}\left(\cdot,\varepsilon^{k}g\left(\frac{\cdot}{\varepsilon}\right)\right)-w_{\varepsilon}^{-}\left(\cdot,0\right)\right\|_{L^{2}(\omega)}+\displaystyle\left\|w_{\varepsilon}^{-}\left(\cdot,0\right)-w^{-}\left(\cdot,0\right)\right\|_{L^{2}(\omega)}.\end{array}$

By (3.1) it holds

(3.22)
$\displaystyle\left\|w^{-}_{\varepsilon}\left(\cdot,\varepsilon^{k}g\left(\frac{\cdot}{\varepsilon}\right)\right)-w_{\varepsilon}^{-}\left(\cdot,0\right)\right\|_{L^{2}(\omega)}\to
0.$

On the other hand, by (2.13), (3.4) and the compactness of the trace operator,
we get

(3.23)
$\displaystyle\left\|w_{\varepsilon}^{-}\left(\cdot,0\right)-w^{-}\left(\cdot,0\right)\right\|_{L^{2}(\omega)}\to
0,$

up to a subsequence.
From (3.21), (3.22) and (3.23), one obtains (3.8).

This concludes the proof of the claimed convergence results. ∎

######  Remark 3.4 ([14]).

We observe that the interface integral in (2.14), in the coordinates
$x^{\prime}$, reads as follows

$\begin{array}[]{c}{\displaystyle\int_{\Gamma^{\varepsilon}}h_{2}(u^{+}_{\varepsilon}-u^{-}_{\varepsilon})(\varphi_{\varepsilon}^{+}-\varphi_{\varepsilon}^{-})\,d\sigma}=\\\\[14.22636pt]
{\displaystyle\int_{\omega}h_{2}\left(u^{+}_{\varepsilon}\left(x^{\prime},\varepsilon
g\left(\frac{x^{\prime}}{\varepsilon}\right)\right)-u^{-}_{\varepsilon}\left(x^{\prime},\varepsilon
g\left(\frac{x^{\prime}}{\varepsilon}\right)\right)\right)}\times\\\\[14.22636pt]
\displaystyle{\left(\varphi_{\varepsilon}^{+}\left(x^{\prime},\varepsilon
g\left(\frac{x^{\prime}}{\varepsilon}\right)\right)-\varphi_{\varepsilon}^{-}\left(x^{\prime},\varepsilon
g\left(\frac{x^{\prime}}{\varepsilon}\right)\right)\right)}\times{\displaystyle\sqrt{1+\varepsilon^{2(k-1)}(\left|\nabla_{y^{\prime}}g(y^{\prime})\right|^{2})_{|y^{\prime}=x^{\prime}/\varepsilon}}\,dx^{\prime}}.\end{array}$

Due to Remark 3.4, to study the limit behaviour of problem (2.14), we need the
following convergence result related only to the geometry of the interface and
to the parameter $\gamma$.

######  Proposition 3.5.

Under the assumption (Ag) it holds

$\displaystyle\varepsilon^{\gamma}\sqrt{1+\varepsilon^{2(k-1)}(\left|\nabla_{y^{\prime}}g(y^{\prime})\right|^{2})_{|y^{\prime}=x^{\prime}/\varepsilon}}\,\to\,G\quad\text{
weakly * in }L^{\infty}(Y^{\prime}),$

where

(3.24)
$\displaystyle{G}=\left\\{\begin{array}[]{ll}\displaystyle\mathcal{M}_{Y^{\prime}}\left(\sqrt{1+\left|\nabla_{y^{\prime}}g(y^{\prime})\right|^{2}}\right)&\hbox{
if }k=1\hbox{ and }\gamma=0,\\\\[5.69054pt] \displaystyle 1&\hbox{ if
}k>1\hbox{ and }\gamma=0,\\\\[5.69054pt]
\displaystyle\mathcal{M}_{Y^{\prime}}\left(\left|\nabla_{y^{\prime}}g(y^{\prime})\right|\right)&\hbox{
if }0<k<1\hbox{ and }\gamma=1-k,\end{array}\right.$

while for ( $k\geq 1\hbox{ and }\gamma>0$) or ( $0<k<1\hbox{and }\gamma>1-k$),

(3.25)
$\displaystyle\varepsilon^{\gamma}\sqrt{1+\varepsilon^{2(k-1)}(\left|\nabla_{y^{\prime}}g(y^{\prime})\right|^{2})_{|y^{\prime}=x^{\prime}/\varepsilon}}\,\to\,0\quad\text{
weakly * in }L^{\infty}(Y^{\prime}).$

###### Proof.

For $k=1$, one has

$\displaystyle\sqrt{1+(\left|\nabla_{y^{\prime}}g(y^{\prime})\right|^{2})_{|y^{\prime}=x^{\prime}/\varepsilon}}\,\to\,\displaystyle\mathcal{M}_{Y^{\prime}}\left(\sqrt{1+\left|\nabla_{y^{\prime}}g(y^{\prime})\right|^{2}}\right)\quad\text{
weakly * in }L^{\infty}(Y^{\prime}).$

Therefore, for $\gamma=0$ we get the first condition in (3.24), while for
$\gamma>0$ we obtain (3.25).
For $k>1$

$\displaystyle\sqrt{1+\varepsilon^{2(k-1)}(\left|\nabla_{y^{\prime}}g(y^{\prime})\right|^{2})_{|y^{\prime}=x^{\prime}/\varepsilon}}\,\to\,1\quad\text{
weakly * in }L^{\infty}(Y^{\prime}).$

Thus, we easily get the second condition in (3.24) for $\gamma=0$ and (3.25)
for $\gamma>0$.
On the other hand, for $0<k<1$, we can write

(3.26)
$\displaystyle\sqrt{1+\varepsilon^{2(k-1)}(\left|\nabla_{y^{\prime}}g(y^{\prime})\right|^{2})_{|y^{\prime}=x^{\prime}/\varepsilon}}=\varepsilon^{k-1}\sqrt{\varepsilon^{2(1-k)}+(\left|\nabla_{y^{\prime}}g(y^{\prime})\right|^{2})_{|y^{\prime}=x^{\prime}/\varepsilon}}$

where

(3.27)
$\displaystyle\sqrt{\varepsilon^{2(1-k)}+(\left|\nabla_{y^{\prime}}g(y^{\prime})\right|^{2})_{|y^{\prime}=x^{\prime}/\varepsilon}}\,\to\,\mathcal{M}_{Y^{\prime}}\left(\left|\nabla_{y^{\prime}}g(y^{\prime})\right|\right)\quad\text{
weakly * in }L^{\infty}(Y^{\prime}).$

By combining (3.26) and (3.27) we get the third condition in (3.24) for
$\gamma=1-k$ and (3.25) for $\gamma>1-k$. ∎

##  4 Existence and uniqueness result

Here, we aim to prove the existence and uniqueness of the solution of problem
(2.14), for any fixed $\varepsilon$, together with some uniform a priori
estimates.

Let us start by proving that all the terms in (2.14) make sense.

######  Proposition 4.1.

Under assumptions (Ag), $(\textbf{A}2)$ and $(\textbf{A}3)$, suppose that $r$
and $\bar{r}$ are such that

(4.1) $\left\\{\begin{array}[]{ll}\displaystyle i)\ \text{ if }N=2,\quad
r,\,\bar{r}\in[1,+\infty),\\\\[5.69054pt] \displaystyle ii)\ \text{ if
}N>2,\quad
r=\dfrac{2(N-1)}{N-2},\,\bar{r}=2^{*}=\dfrac{2N}{N-2},\end{array}\right.$

and

(4.2) $\displaystyle r\geq q_{2}\,r^{\prime},\qquad\bar{r}\geq
q_{1}\bar{r}^{\prime},$

where $r^{\prime}$ and $\bar{r}^{\prime}$ are the coniugate exponents of $r$
and $\bar{r}$, respectively.

Then, for any fixed $\varepsilon$, the following maps are bounded and
continuous:

(4.3) $\begin{array}[]{ll}\displaystyle i)\ H_{1}:w_{\varepsilon}\in
W_{0}^{\varepsilon}\rightarrow H_{1}(w_{\varepsilon})\in
L^{\frac{\bar{r}}{q_{1}}}(Q),\\\\[5.69054pt] \displaystyle ii)\
H_{2}:w_{\varepsilon}\in W_{0}^{\varepsilon}\rightarrow
H_{2}(w_{\varepsilon})\in
L^{\frac{r}{q_{2}}}(\Gamma^{\varepsilon}),\end{array}$

where $H_{1}(w_{\varepsilon})(x)=h_{1}(w_{\varepsilon}(x))$,
$H_{2}(w_{\varepsilon})(x)=h_{2}(w_{\varepsilon}^{+}(x)-w_{\varepsilon}^{-}(x))$.

Moreover for every $w_{\varepsilon}$ and $v_{\varepsilon}$ in
$W_{0}^{\varepsilon}$, we get

(4.4) $\left\\{\begin{array}[]{l}\displaystyle
h_{1}(w_{\varepsilon})\,v_{\varepsilon}\in L^{1}(Q),\\\\[5.69054pt]
\displaystyle
h_{2}(w^{+}_{\varepsilon}-w^{-}_{\varepsilon})(v^{+}_{\varepsilon}-v^{-}_{\varepsilon})\in
L^{1}(\Gamma^{\varepsilon}).\end{array}\right.$

###### Proof.

Let us consider the lower order term. If $N>2$, the embeddings
$H^{1}(Q_{\varepsilon}^{\pm})\subset L^{2^{*}}(Q_{\varepsilon}^{\pm})$ are
continuous. Hence, we get $w_{\varepsilon}\in L^{2^{*}}(Q)$. By
$(\textbf{A}2)_{1,3}$ and the property of the Nemytskii operator associated to
the function $h_{1}$ we get that
$H_{1}(w_{\varepsilon})(x)=h_{1}(w_{\varepsilon}(x))$ is a bounded and
continuous map from $W_{0}^{\varepsilon}$ to $L^{\frac{2^{*}}{q_{1}}}(Q)$. By
(2.4) the first relation in (4.4) holds for $N>2$. For $N=2$ we argue in a
similar way as above by using the continuous embeddings
$H^{1}(Q_{\varepsilon}^{\pm})\subset L^{\bar{r}}(Q_{\varepsilon}^{\pm})$ for
any $\bar{r}\in[1,+\infty)$ and again (2.4) for $N=2$.

For the interface term, let

$j:w_{\varepsilon}\in
W^{\varepsilon}_{0}\,\to\,w_{\varepsilon}^{+}-w_{\varepsilon}^{-}\in
L^{r}(\Gamma^{\varepsilon})$

and

$j_{2}:\overline{w}_{\varepsilon}\in
L^{r}(\Gamma^{\varepsilon})\,\to\,j_{2}(\overline{w}_{\varepsilon})\in
L^{\frac{r}{q_{2}}}(\Gamma^{\varepsilon})$

be defined by
$j_{2}(\overline{w}_{\varepsilon})(x)=h_{2}(\overline{w}_{\varepsilon}(x))$.
By the linear continuity of the inclusions $H^{1}(Q_{\varepsilon}^{+})\subset
L^{r}(\Gamma^{\varepsilon})$ and $H^{1}(Q_{\varepsilon}^{-})\subset
L^{r}(\Gamma^{\varepsilon})$, $j$ is bounded and continuous. Moreover,
assumption $(\textbf{A}3)_{1,4}$ and the property of the Nemytskii operator
associated to the function $h_{2}$ imply that $j_{2}$ is bounded and
continuous from $L^{r}(\Gamma^{\varepsilon})$ to
$L^{\frac{r}{q_{2}}}(\Gamma^{\varepsilon})$. Hence,
$H_{2}\left(w_{\varepsilon}\right)=\left(j_{2}\circ j\right)(w_{\varepsilon})$
is a bounded and continuous map from $W^{\varepsilon}_{0}$ into
$L^{\frac{r}{q_{2}}}(\Gamma^{\varepsilon})$. Thus, by (2.7) and (4.2) the
second relation in (4.4) is true. ∎

At this point, we prove the main result of this section.

######  Theorem 4.2.

Under assumptions $(\textbf{A}_{g})$, $(\textbf{A}1)$, $(\textbf{A}2)$,
$(\textbf{A}3)$ and given $f\in L^{2}(Q)$, for any fixed $\varepsilon$, there
exists a unique solution $u_{\varepsilon}\in W^{\varepsilon}_{0}$ of problem
(2.14) satisfying

(4.5) $\left\\{\begin{array}[]{ll}\displaystyle i)\ \|\nabla
u_{\varepsilon}\|_{L^{2}(Q\setminus\Gamma^{\varepsilon})}\leq
C,\\\\[5.69054pt] \displaystyle ii)\
\|u^{+}_{\varepsilon}-u^{-}_{\varepsilon}\|_{L^{2}(\Gamma^{\varepsilon})}\leq
C\varepsilon^{-\frac{\gamma}{2}},\\\\[5.69054pt] \end{array}\right.$

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

###### Proof.

The proof follows the same arguments of Theorem 3.1 in [13]. We start proving,
for any fixed $\varepsilon$, the existence of a solution of the variational
problem (2.14) by using the Minty-Browder theorem for the operator
$\mathcal{G}:u_{\varepsilon}\in
W^{\varepsilon}_{0}\to\mathcal{G}(u_{\varepsilon})\in(W^{\varepsilon}_{0})^{\prime}$
where

(4.6)
$\displaystyle\langle\mathcal{G}\left(u_{\varepsilon}\right),\varphi_{\varepsilon}\rangle_{(W^{\varepsilon}_{0})^{\prime},W^{\varepsilon}_{0}}={\displaystyle\int_{Q\setminus\Gamma^{\varepsilon}}D^{\varepsilon}\nabla
u_{\varepsilon}\nabla\varphi_{\varepsilon}\,dx}+{\displaystyle\int_{\Gamma^{\varepsilon}}h_{2}(u_{\varepsilon}^{+}-u_{\varepsilon}^{-})(\varphi_{\varepsilon}^{+}-\varphi_{\varepsilon}^{-})\,d\sigma}+{\displaystyle\int_{Q}h_{1}(u_{\varepsilon})\varphi_{\varepsilon}\,dx}-{\displaystyle\int_{Q}f\varphi_{\varepsilon}\,dx}.$

The continuity of $\mathcal{G}$ follows directly from assumption
$(\textbf{A}1)$ and Proposition 4.1. Moreover hypotheses $(\textbf{A}1)$,
$(\textbf{A}2)_{2}$ and $(\textbf{A}3)_{2}$, assure the monotonicity of
$\mathcal{G}$. It remains to prove the coercivness of $\mathcal{G}$. To this
aim let us observe that by $(\textbf{A}1)$, $(\textbf{A}3)_{3}$, Remark 2.2,
(2.9) and (2.13), it holds

(4.7)
$\begin{array}[]{c}\displaystyle\langle\mathcal{G}\left(u_{\varepsilon}\right),u_{\varepsilon}\rangle\geq\alpha\left\|\nabla
u_{\varepsilon}\right\|^{2}_{L^{2}(Q\setminus\Gamma^{\varepsilon})}+C\left\|u_{\varepsilon}^{+}-u_{\varepsilon}^{-}\right\|^{2}_{L^{2}(\Gamma^{\varepsilon})}-\|f\|_{L^{2}(Q)}\|u_{\varepsilon}\|_{L^{2}(Q)}\\\\[5.69054pt]
\displaystyle\geq
C_{1}\|u_{\varepsilon}\|^{2}_{W^{\varepsilon}_{0}}-C_{2}\|u_{\varepsilon}\|_{W^{\varepsilon}_{0}},\end{array}$

with $C_{1}$ and $C_{2}$ positive constants. The above inequality yields that
the operator $\mathcal{G}$ is coercive. Hence, by the Minty-Browder theorem,
$\mathcal{G}$ is surjective, so that there exists at least one function
$u_{\varepsilon}\in W^{\varepsilon}_{0}$ such that
$\mathcal{G}(u_{\varepsilon})=0$. This means that $u_{\varepsilon}$ is a
solution of problem (2.14). As for the uniqueness, let us suppose that problem
(2.14) admits another solution $\bar{u}_{\varepsilon}$. Choosing as test
function $\varphi_{\varepsilon}=u_{\varepsilon}-\bar{u}_{\varepsilon}$ in
(2.14) written in correspondence of the solutions $u_{\varepsilon}$ and
$\bar{u}_{\varepsilon}$, respectively, we get

$\begin{array}[]{c}{\displaystyle\int_{Q\setminus\Gamma^{\varepsilon}}D^{\varepsilon}\nabla
u_{\varepsilon}\nabla(u_{\varepsilon}-\bar{u}_{\varepsilon})\,dx}+{\displaystyle\int_{\Gamma^{\varepsilon}}h_{2}(u_{\varepsilon}^{+}-u_{\varepsilon}^{-})((u_{\varepsilon}^{+}-\bar{u}_{\varepsilon}^{+})-(u_{\varepsilon}^{-}-\bar{u}_{\varepsilon}^{-}))\,d\sigma}\\\\[14.22636pt]
{\displaystyle+\int_{Q}h_{1}(u_{\varepsilon})(u_{\varepsilon}-\bar{u}_{\varepsilon})\,dx}={\displaystyle\int_{Q}f(u_{\varepsilon}-\bar{u}_{\varepsilon})\,dx}.\end{array}$

and

$\begin{array}[]{c}{\displaystyle\int_{Q\setminus\Gamma^{\varepsilon}}D^{\varepsilon}\nabla\bar{u}_{\varepsilon}\nabla(u_{\varepsilon}-\bar{u}_{\varepsilon})\,dx}{\displaystyle\int_{\Gamma^{\varepsilon}}h_{2}(\bar{u}_{\varepsilon}^{+}-\bar{u}_{\varepsilon}^{-})((u_{\varepsilon}^{+}-\bar{u}_{\varepsilon}^{+})-(u_{\varepsilon}^{-}-\bar{u}_{\varepsilon}^{-}))\,d\sigma}\\\\[14.22636pt]
{\displaystyle+\int_{Q}h_{1}(\bar{u}_{\varepsilon})(u_{\varepsilon}-\bar{u}_{\varepsilon})\,dx}={\displaystyle\int_{Q}f(u_{\varepsilon}-\bar{u}_{\varepsilon})\,dx}\end{array}$

Then, we subtract and obtain

(4.8)
$\begin{array}[]{l}{\displaystyle\int_{Q\setminus\Gamma^{\varepsilon}}D^{\varepsilon}(\nabla
u_{\varepsilon}-\nabla\bar{u}_{\varepsilon})(\nabla
u_{\varepsilon}-\nabla\bar{u}_{\varepsilon})\,dx}\\\\[14.22636pt]
{\displaystyle+\int_{\Gamma^{\varepsilon}}\left[h_{2}(u_{\varepsilon}^{+}-u_{\varepsilon}^{-})-h_{2}(\bar{u}_{\varepsilon}^{+}-\bar{u}_{\varepsilon}^{-})\right]((u_{\varepsilon}^{+}-\bar{u}_{\varepsilon}^{+})-(u_{\varepsilon}^{-}-\bar{u}_{\varepsilon}^{-}))\,d\sigma}\\\\[11.38109pt]
+{\displaystyle\int_{Q}\left[h_{1}(u_{\varepsilon})-h_{1}(\bar{u}_{\varepsilon})\right](u_{\varepsilon}-\bar{u}_{\varepsilon})\,dx}=0.\end{array}$

By assumption $(\textbf{A}1)$, the monotonicity of $h_{1}$ and $h_{2}$,
equality (4.8) gives

$\displaystyle\nabla(u_{\varepsilon}-\bar{u}_{\varepsilon})=0\quad\text{ in
}Q\setminus\Gamma^{\varepsilon}.$

Then, by using Poincaré inequality (2.13) we get
$u_{\varepsilon}=\bar{u}_{\varepsilon}$ which ends the proof of uniqueness.

In order to prove uniform estimates in (4.5), let us take $u_{\varepsilon}$ as
test function in (2.14) and obtain

${\displaystyle\int_{Q\setminus\Gamma^{\varepsilon}}D^{\varepsilon}\nabla
u_{\varepsilon}\nabla
u_{\varepsilon}\,dx}+\varepsilon^{\gamma}{\displaystyle\int_{\Gamma^{\varepsilon}}h_{2}(u^{+}_{\varepsilon}-u^{-}_{\varepsilon})(u_{\varepsilon}^{+}-u_{\varepsilon}^{-})\,d\sigma}+{\displaystyle\int_{Q}h_{1}(u_{\varepsilon})u_{\varepsilon}\,dx}={\displaystyle\int_{Q}fu_{\varepsilon}\,dx}.$

By $(\textbf{A}1)$, $(\textbf{A}3)_{3}$ and Remark 2.2 we get

$\begin{array}[]{c}\displaystyle\alpha\|\nabla
u_{\varepsilon}\|^{2}_{L^{2}(Q\setminus\Gamma^{\varepsilon})}+C\,\varepsilon^{\gamma}\|u^{+}_{\varepsilon}-u^{-}_{\varepsilon}\|^{2}_{L^{2}(\Gamma^{\varepsilon})}\leq\|f\|_{L^{2}(Q)}\|u_{\varepsilon}\|_{L^{2}(Q)},\end{array}$

which implies the claimed result in view of (2.13). ∎

##  5 Homogenization

Let us start with some preliminary convergence results for the sequence of
solutions of problem (2.14) needed for the proof of our main result. They
follow by (2.9), Proposition 3.2 and the a priori estimate (4.5)i in Theorem
4.2.

######  Corollary 5.1.

Let $u_{\varepsilon}$ be a solution of problem (2.14). Then, there exist a
subsequence (still denoted $\varepsilon$) and a function $u\in W^{0}_{0}$ such
that

$\left\\{\begin{array}[]{lll}\displaystyle u_{\varepsilon}\rightarrow u&\text{
strongly in }L^{2}(Q)\text{ and a.e. in }Q,\\\\[5.69054pt]
\displaystyle\chi_{Q^{+}_{\varepsilon}}\nabla
u_{\varepsilon}\rightharpoonup\chi_{Q^{+}}\nabla u&\text{ weakly in
}(L^{2}(Q))^{N},\\\\[5.69054pt] \displaystyle\chi_{Q^{-}_{\varepsilon}}\nabla
u_{\varepsilon}\rightharpoonup\chi_{Q^{-}}\nabla u&\text{ weakly in
}(L^{2}(Q))^{N}.\end{array}\right.$

######  Remark 5.2.

Since $\chi_{Q^{+}_{\varepsilon}}$ and $\chi_{Q^{+}_{\varepsilon}}$ converge
a.e. to $\chi_{Q^{+}}$ and $\chi_{Q^{-}}$, respectively, as an immediate
consequence of Corollary 5.1 one has

$\left\\{\begin{array}[]{ll}\displaystyle\chi_{Q^{+}_{\varepsilon}}u_{\varepsilon}\rightarrow\chi_{Q^{+}}u&\text{
strongly in }L^{2}(Q),\\\\[5.69054pt]
\displaystyle\chi_{Q^{-}_{\varepsilon}}u_{\varepsilon}\rightarrow\chi_{Q^{-}}u&\text{
strongly in }L^{2}(Q).\end{array}\right.$

By arguing as in Proposition 3.1 of [14], we get the convergence result below.

######  Corollary 5.3.

Let $k>0$ and $u_{\varepsilon}$ be the solution of problem (2.14). Then, there
exists a subsequence (still denoted $\varepsilon$) such that

(5.1) $\left\\{\begin{array}[]{lll}\displaystyle i)\
\chi_{Q^{+}_{\varepsilon}}D^{\varepsilon}\nabla
u_{\varepsilon}\rightharpoonup\chi_{Q^{+}}D^{0}\nabla u&\text{ weakly in
}(L^{2}(Q))^{N},\\\\[5.69054pt] \displaystyle ii)\
\chi_{Q^{-}_{\varepsilon}}D^{\varepsilon}\nabla
u_{\varepsilon}\rightharpoonup\chi_{Q^{-}}D^{0}\nabla u&\text{ weakly in
}(L^{2}(Q))^{N}.\end{array}\right.$

###  5.1 Proof of Theorem 2.3

At first we consider cases A) and B). To this aim, observe that given
$\varphi\in W^{0}_{0}$ there exist $\psi_{1}$ and $\psi_{2}$ in $H^{1}_{0}(Q)$
such that $\varphi^{+}=\psi_{1|Q^{+}}$ and $\varphi^{-}=\psi_{2|Q^{-}}$.
Set
$\varphi_{\varepsilon}=\chi_{Q^{+}_{\varepsilon}}\psi_{1}+\chi_{Q^{-}_{\varepsilon}}\psi_{2}$,
one has that $\varphi_{\varepsilon}\in W^{\varepsilon}_{0}$. By construction
and using (2.11), it holds

(5.2) $\displaystyle\varphi_{\varepsilon}\rightarrow\varphi\quad\text{
strongly in }L^{2^{*}}(Q).$

Now, let us take $\varphi_{\varepsilon}$ as test function in the variational
equality (2.14). We get

(5.3)
${\displaystyle\int_{Q\setminus\Gamma^{\varepsilon}}D^{\varepsilon}\nabla
u_{\varepsilon}\nabla\varphi_{\varepsilon}\,dx}+\varepsilon^{\gamma}{\displaystyle\int_{\Gamma^{\varepsilon}}h_{2}(u^{+}_{\varepsilon}-u^{-}_{\varepsilon})(\varphi_{\varepsilon}^{+}-\varphi_{\varepsilon}^{-})\,d\sigma}+{\displaystyle\int_{Q}h_{1}(u_{\varepsilon})\varphi_{\varepsilon}\,dx}={\displaystyle\int_{Q}f\varphi_{\varepsilon}\,dx}.$

We would like to pass to the limit in (5.3).

By Corollary 5.3, we have, up to a subsequence,

(5.4)
$\begin{array}[]{c}\displaystyle\int_{Q\setminus\Gamma^{\varepsilon}}D^{\varepsilon}\nabla
u_{\varepsilon}\nabla\varphi_{\varepsilon}\,dx=\displaystyle\int_{Q}\chi_{Q^{+}_{\varepsilon}}D^{\varepsilon}\nabla
u_{\varepsilon}\nabla\psi_{1}\,dx+\int_{Q}\chi_{Q^{-}_{\varepsilon}}D^{\varepsilon}\nabla
u_{\varepsilon}\nabla\psi_{2}\,dx\\\\[14.22636pt]
\displaystyle\rightarrow\int_{Q}\chi_{Q^{+}}D^{0}\nabla
u\nabla\psi_{1}\,dx+\int_{Q}\chi_{Q^{-}}D^{0}\nabla
u\nabla\psi_{2}\,dx=\int_{Q\setminus\Gamma_{0}}D^{0}\nabla
u\nabla\varphi\,dx.\end{array}$

Moreover, (5.2) gives

(5.5) $\displaystyle\int_{Q}f\varphi_{\varepsilon}\,dx\ \rightarrow\
\int_{Q}f\varphi\,dx.$

In order to study the other two terms, at first, let us consider more regular
functions $\psi_{1}$ and $\psi_{2}$ in $\mathcal{D}(\Omega)$. Hence, (5.2)
becomes

(5.6) $\displaystyle\varphi_{\varepsilon}\rightarrow\varphi\quad\text{
strongly in }L^{p}(Q),\ \text{ for any }p\geq 1.$

Concerning the lower order term, by $(\textbf{A}2)_{1,3}$, Corollary 5.1, the
properties of the Nemytskii operator associated to function $h_{1}$ and (5.6)
written for $p=\left(\frac{2}{q_{1}}\right)^{\prime}$, we get, up to a
subsequence,

(5.7) $\displaystyle\int_{Q}h_{1}(u_{\varepsilon})\varphi_{\varepsilon}\,dx\
\rightarrow\ \int_{Q}h_{1}(u)\varphi\,dx.$

Now, let us treat the interface integral. By Remark 3.4, we have that

(5.8)
$\begin{array}[]{c}\displaystyle\varepsilon^{\gamma}{\displaystyle\int_{\Gamma^{\varepsilon}}h_{2}(u^{+}_{\varepsilon}-u^{-}_{\varepsilon})(\varphi_{\varepsilon}^{+}-\varphi_{\varepsilon}^{-})\,d\sigma}\\\\[14.22636pt]
{\displaystyle=\varepsilon^{\gamma}\int_{\omega}h_{2}\left(u^{+}_{\varepsilon}\left(x^{\prime},\varepsilon
g\left(\frac{x^{\prime}}{\varepsilon}\right)\right)-u^{-}_{\varepsilon}\left(x^{\prime},\varepsilon
g\left(\frac{x^{\prime}}{\varepsilon}\right)\right)\right)}\times\\\\[14.22636pt]
\displaystyle{\left(\psi_{1}\left(x^{\prime},\varepsilon
g\left(\frac{x^{\prime}}{\varepsilon}\right)\right)-\psi_{2}\left(x^{\prime},\varepsilon
g\left(\frac{x^{\prime}}{\varepsilon}\right)\right)\right)}\times{\displaystyle\sqrt{1+\varepsilon^{2(k-1)}(\left|\nabla_{y^{\prime}}g(y^{\prime})\right|^{2})_{|y^{\prime}=x^{\prime}/\varepsilon}}\,dx^{\prime}.}\end{array}$

In view of (4.5)i, by Proposition 3.3 one has, up to a subsequence,

(5.9) $\displaystyle u^{\pm}_{\varepsilon}\left(\cdot,\varepsilon
g\left(\frac{\cdot}{\varepsilon}\right)\right)\ \rightarrow\
u^{\pm}(\cdot,0)\,\text{ strongly in }L^{2}(\omega).$

Thus, by $({\bf A}3)_{1,4}$, (5.9) and again the properties of the Nemytskii
operator associated to $h_{2}$, we get

(5.10) $\displaystyle h_{2}\left(u^{+}_{\varepsilon}\left(\cdot,\varepsilon
g\left(\frac{\cdot}{\varepsilon}\right)\right)-u^{-}_{\varepsilon}\left(\cdot,\varepsilon
g\left(\frac{\cdot}{\varepsilon}\right)\right)\right)\ \rightarrow\
h_{2}\left(u^{+}(\cdot,0)-u^{-}(\cdot,0)\right)\text{ strongly in
}L^{\frac{2}{q_{2}}}(\omega).$

On the other hand, in view of Lemma 3.1 and since $\psi_{i}\in\mathcal{D}(Q)$,
for every $p\geq 1$ we get

(5.11) $\displaystyle\left\|\psi_{i}\left(x^{\prime},\varepsilon
g\left(\frac{x^{\prime}}{\varepsilon}\right)\right)-\psi_{i}\left(x^{\prime},0\right)\right\|_{L^{p}(\omega)}\leq
C\varepsilon^{\frac{k}{p}}\|\psi_{i}\|_{W^{1,p}(Q)},$

i.e.

(5.12) $\displaystyle\psi_{i}\left(x^{\prime},\varepsilon
g\left(\frac{x^{\prime}}{\varepsilon}\right)\right)\ \to\
\psi_{i}\left(x^{\prime},0\right)\quad\text{ strongly in }L^{p}(\omega),$

for any fixed $i\in\\{1,2\\}$.

Hence, we are able to pass to the limit in (5.8) obtaining different
convergence results according to the values of $k$ and $\gamma$.

By Proposition 3.5, (5.8), (5.10) and (5.12) written for
$p=\left(\frac{2}{q_{2}}\right)^{\prime}$, in case A) we have, up to a
subsequence,

(5.13)
${\displaystyle\varepsilon^{\gamma}\int_{\Gamma^{\varepsilon}}h_{2}(u^{+}_{\varepsilon}-u^{-}_{\varepsilon})(\varphi_{\varepsilon}^{+}-\varphi_{\varepsilon}^{-})\,d\sigma}\\\
\rightarrow\displaystyle
G\;\int_{\omega}h_{2}(u^{+}\left(x^{\prime},0\right)-u^{-}\left(x^{\prime},0\right))(\psi_{1}\left(x^{\prime},0\right)-\psi_{2}\left(x^{\prime},0\right))\,dx^{\prime},$

for any $\psi_{1}$ and $\psi_{2}$ in $\mathcal{D}(Q)$, where $G$ is defined in
(3.24), while in case B) it holds

(5.14)
${\displaystyle\varepsilon^{\gamma}\int_{\Gamma^{\varepsilon}}h_{2}(u^{+}_{\varepsilon}-u^{-}_{\varepsilon})(\varphi_{\varepsilon}^{+}-\varphi_{\varepsilon}^{-})\,d\sigma}\
\rightarrow\ 0.$

Now, in both cases, we fix $i\in\\{1,2\\}$ and take $\psi_{i}\in
H_{0}^{1}(Q)$. Then, there exists a sequence
$\\{{\psi_{i}}_{n}\\}\subset\mathcal{D}(Q)$ such that

$\displaystyle{\psi_{i}}_{n}\rightarrow\psi_{i}\quad\text{ strongly in
}H_{0}^{1}(Q),$

which implies

(5.15) $\displaystyle{\psi_{i}}_{n}\rightarrow\psi_{i}\quad\text{ strongly in
}L^{p}(Q),\text{ for any }p\in[1,2^{*}].$

Set $\varphi_{n}=\chi_{Q^{+}}{\psi_{1}}_{n}+\chi_{Q^{-}}{\psi_{2}}_{n}$, it
holds

(5.16) $\displaystyle\varphi_{n}\rightarrow\varphi\quad\text{ strongly in
}L^{2^{*}}(Q).$

We observe that if $N>2$ the embeddings $H^{1}(Q^{\pm})\subset
L^{2^{*}}(Q^{\pm})$ are continuous, which give $u\in L^{2^{*}}(Q)$. Therefore,
by $(\textbf{A$2$})_{3}$ and (5.16) we obtain

(5.17) $\displaystyle\int_{Q}h_{1}(u)\varphi_{n}\,dx\ \rightarrow\
\int_{Q}h_{1}(u)\varphi\,dx.$

For $N=2$ we argue as above using the continuous embeddings
$H^{1}(Q^{\pm})\subset L^{\bar{r}}(Q^{\pm})$ for any $\bar{r}\in[1,+\infty)$
and again $(\textbf{A$2$})_{3}$.

Regarding the interface term

$G\int_{\omega}h_{2}\left(u^{+}(x^{\prime},0)-u^{-}(x^{\prime},0)\right)({\psi_{1}}_{n}(x^{\prime},0)-{\psi_{2}}_{n}(x^{\prime},0))\,dx^{\prime},$

the linear continuity of the inclusions $H^{1}(Q^{\pm})\subset
L^{r}(\Gamma_{0})$, where

$\left\\{\begin{array}[]{ll}\displaystyle\hbox{ if }N=2,\quad
r\in[1,+\infty),\\\\[5.69054pt] \displaystyle\hbox{ if }N>2,\quad
r=\dfrac{2(N-1)}{N-2},\end{array}\right.$

yields that, for $i\in\\{1,2\\}$, ${\psi_{i}}_{n}|_{\Gamma_{0}}$ is in
$L^{r}(\Gamma_{0})$. Thus, ${\psi_{i}}_{n}(\cdot,0)$ belongs to
$L^{r}(\omega)$.
By $(\textbf{A$3$})_{4}$ we always have $r^{\prime}\leq\dfrac{r}{q_{2}}$ since
$\dfrac{r}{r^{\prime}}=\dfrac{N}{N-2}$. Hence, we get

$\begin{array}[]{c}\displaystyle
G\int_{\omega}h_{2}\left(u^{+}(x^{\prime},0)-u^{-}(x^{\prime},0)\right)({\psi_{1}}_{n}(x^{\prime},0)-{\psi_{2}}_{n}(x^{\prime},0))\,dx^{\prime}\\\\[14.22636pt]
\displaystyle\rightarrow\
G\int_{\omega}h_{2}\left(u^{+}(x^{\prime},0)-u^{-}(x^{\prime},0)\right)(\psi_{1}(x^{\prime},0)-\psi_{2}(x^{\prime},0))\,dx^{\prime},\end{array}$

for $\psi_{1}$ and $\psi_{2}$ in $H^{1}_{0}(Q)$ where $G$ is defined in (3.24)
of Proposition 3.5.

The above results allow us to have, by a density argument, the limit problems
(2.18) and (2.20).

Now, it remains to treat case C). To this aim, let us prove that

(5.18) $u^{+}_{|\Gamma_{0}}=u^{-}_{|\Gamma_{0}}=0.$

For $k\geq 1$ and $\gamma<0$, by Remark 3.4, Proposition 3.3 and (4.5), we
have, up to a subsequence,

(5.19)
$\begin{array}[]{c}{\displaystyle\int_{\Gamma^{\varepsilon}}(u^{+}_{\varepsilon}-u^{-}_{\varepsilon})^{2}\,d\sigma}\\\\[14.22636pt]
{\displaystyle=\int_{\omega}\left(u^{+}_{\varepsilon}\left(x^{\prime},\varepsilon
g\left(\frac{x^{\prime}}{\varepsilon}\right)\right)-u^{-}_{\varepsilon}\left(x^{\prime},\varepsilon
g\left(\frac{x^{\prime}}{\varepsilon}\right)\right)\right)^{2}}\times{\displaystyle\sqrt{1+\varepsilon^{2(k-1)}(\left|\nabla_{y^{\prime}}g(y^{\prime})\right|^{2})_{|y^{\prime}=x^{\prime}/\varepsilon}}\,dx^{\prime}}\\\\[14.22636pt]
\displaystyle\rightarrow\left\\{\begin{array}[]{ll}{\displaystyle\int_{\omega}\left(u^{+}(x^{\prime},0)-u^{-}(x^{\prime},0)\right)^{2}\,dx^{\prime}}&\hbox{
if }k>1,\\\\[8.53581pt]
{\displaystyle\mathcal{M}_{Y^{\prime}}\left(\sqrt{1+(\left|\nabla_{y^{\prime}}g(y^{\prime})\right|^{2})}\right)\int_{\omega}\left(u^{+}(x^{\prime},0)-u^{-}(x^{\prime},0)\right)^{2}\,dx^{\prime}}&\hbox{
if }k=1\end{array}\right.\end{array}$

and

(5.20)
${\displaystyle\int_{\Gamma^{\varepsilon}}(u^{+}_{\varepsilon}-u^{-}_{\varepsilon})^{2}\,d\sigma}\leq\varepsilon^{-\gamma}\
\rightarrow\\\ 0.$

Thus, (5.18) holds by uniqueness.

For $0<k<1$ and $\gamma<1-k$, again by Remark 3.4, Proposition 3.3 and (4.5),
one obtains, up to a subsequence,

(5.21)
$\begin{array}[]{c}{\displaystyle\varepsilon^{1-k}\int_{\Gamma^{\varepsilon}}(u^{+}_{\varepsilon}-u^{-}_{\varepsilon})^{2}\,d\sigma}=\\\\[14.22636pt]
{\displaystyle\int_{\omega}\left(u^{+}_{\varepsilon}\left(x^{\prime},\varepsilon
g\left(\frac{x^{\prime}}{\varepsilon}\right)\right)-u^{-}_{\varepsilon}\left(x^{\prime},\varepsilon
g\left(\frac{x^{\prime}}{\varepsilon}\right)\right)\right)^{2}}\times{\displaystyle\sqrt{\varepsilon^{2(1-k)}+(\left|\nabla_{y^{\prime}}g(y^{\prime})\right|^{2})_{|y^{\prime}=x^{\prime}/\varepsilon}}\,dx^{\prime}}\\\\[14.22636pt]
\displaystyle\rightarrow\mathcal{M}_{Y^{\prime}}\left(\left|\nabla_{y^{\prime}}g(y^{\prime})\right|\right)\displaystyle\int_{\omega}\left(u^{+}(x^{\prime},0)-u^{-}(x^{\prime},0)\right)^{2}\,dx^{\prime},\end{array}$

and

(5.22)
${\displaystyle\varepsilon^{1-k}\int_{\Gamma^{\varepsilon}}(u^{+}_{\varepsilon}-u^{-}_{\varepsilon})^{2}\,d\sigma}\leq\varepsilon^{1-k-\gamma}\
\rightarrow\ 0.$

Hence, again we get (5.18) by uniqueness.

Thus, we can choose a test function $\varphi\in H_{0}^{1}(Q)$ in (5.3). As a
consequence, the interface term disappears and one easily obtains the limit
problem (2.21).

In conclusion, let us observe that since the limit problems have a unique
solution, the convergences in (2.17) hold for the whole sequence. $\Box$

## References

  * [1] M. Amar, D. Andreucci, R. Bisegna, and R. Gianni, Evolution and memory effects in the homogenization limit for electrical conduction in biological tissues, Math. Models Methods Appl. Sci., Vol.14 (2004), 1261–1295.
  * [2] M. Amar, D. Andreucci, and C. Timofte, Heat conduction in composite media involving imperfect contact and perfectly conductive inclusions, Math. Methods Appl. Sci., Vol.45 (2022), 11355–11379.
  * [3] M. Amar, D. Andreucci, and C. Timofte, Asymptotic analysis for non-local problems in composites withdifferent imperfect contact conditions, Appl. Anal., https://doi.org/10.1080/00036811.2022.2120867.
  * [4] J. Avila, S. Monsurrò, and F. Raimondi, Homogenization of an eigenvalue problem through rough surfaces, submitted.
  * [5] A. Bensoussan, J.-L. Lions, and G. Papanicolaou, “Asymptotic analysis for periodic structures”, North-Holland Publishing Company, Netherlands, 1978.
  * [6] G. Cardone, C. Perugia, and C. Timofte, Homogenization results for a coupled system of reaction diffusion equations, Nonlinear Anal., Vol.188 (2019), 236–264.
  * [7] H. S. Carslaw, and J. C. Jaeger, “Conduction of Heat in Solids”, At the Clarendon Press, Oxford, 1947.
  * [8] G. A. Chechkin, A. Friedman, and A. L. Piatnitski, The boundary-value problem in domains with very rapidly oscillating boundary, J. Math. Anal. Appl., Vol.231 (1999), 213–234.
  * [9] C. Conca, J. I. Diaz, A. Linan, and C. Timofte, Homogenization in Chemical Reactive Flows, Electron. J. Diff. Equ., Vol.40 (2004), 1–22.
  * [10] C. Conca, J. I. Diaz, and C. Timofte, Effective Chemical Processes in Porous Media, Math. Models Methods Appl. Sci., Vol.13 (2003), 1–26.
  * [11] P. Donato, and D. Giachetti,Existence and homogenization for a singular problem through rough surfaces, SIAM J. Math. Anal., Vol.48 (2016), 4047–4086.
  * [12] P. Donato, E.C. Jose, and D. Onofrei, Asymptotic analysis of a multiscale parabolic problem with a rough fast oscillating interface, Arch. Appl. Mech., Vol.89 (2019), 437–465.
  * [13] P. Donato, and K.H. Le Nguyen, Homogenization of diffusion problems with a nonlinear interfacial resistance, Nonlinear Diff. Equ. Appl., Vol.22 (2015), 1345–1380.
  * [14] P. Donato, and A. Piatnitski, On the effective interfacial resistance through rough surfaces, Comm. Pure and Appl. Math., Vol.9 (2010), 1295–1310.
  * [15] L. Faella, S. Monsurrò, and C. Perugia, Homogenization of imperfect transmission problems: the case of weakly converging data, Differ. Integral Equ., Vol.31 (2018), 595–620.
  * [16] L. Faella, S. Monsurró, and C. Perugia, Exact controllability for evolutionary imperfect transmission problems, J. Math. Pures Appl., Vol.122 (2019), 235–271.
  * [17] M. Gahn, M. Neuss-Radu, and P. Knabner, Homogenization of reaction-diffusion processes in a two-component porous medium with nonlinear flux conditions at the interface, SIAM J. Appl. Math., Vol.76 (2016), 1819–1843.
  * [18] M. Gahn, M. Neuss-Radu, and P. Knabner, Derivation of an effective model for metabolic processes in living cells including substrate channeling, Vietnam J. Math., Vol.45 (2017), 265–293.
  * [19] U. Hornung, W. Jäger, and A. Mikelić, Reactive transport through an array of cells with semi-permeable membranes, ESAIM Math. Model. Numer. Anal., Vol.28 (1994), 59–94.
  * [20] K. Kumar, M. Neuss-Radu, and I.S. Pop, Homogenization of pore scale model for precipitation and dissolution in porous media, IMA J. Appl. Math., Vol.81(2016), 877–897.
  * [21] S. Monsurrò, and C. Perugia, Homogenization and exact controllability for problems with imperfect interface, Netw. Heterog. Media, Vol.14 (2019), 411–444.
  * [22] A. Muntean, and M. Neuss-Radu, A multiscale Galerkin approach for a class of nonlinear coupled reaction-diffusion systems in complex media, J. Math. Anal. Appl., Vol.371 (2010), 705–718.
  * [23] M. A. Peter, and M. Böhm, Different choices of scaling in homogenization of diffusion and interfacial exchange in a porous medium, Math. Meth. Appl. Sci., Vol.31 (2008), 1257–1282.
  * [24] M. Radu-Neuss, and W. Jäger, Effective transmission conditions for reactions-diffusion processes in domains separated by an interface, SIAM J. Math. Anal., Vol.39 (2007), 687–720.
  * [25] F. Raimondi, Homogenization of a class of singular elliptic problems in two-component domains, Asymptot. Anal., Vol.132 (2023), 1–27.