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Volume 9, Issue 1

# Homogenization of a net of periodic critically scaled boundary obstacles related to reverse osmosis “nano-composite” membranes

Jesús Ildefonso Díaz
/ David Gómez-Castro
• Instituto de Matemática Interdisciplinar, Universidad Complutense de Madrid, Plaza de Ciencias 3, 28040 Madrid, Spain; and Departamento de Matemática Aplicada, E.T.S. de Ingeniería – ICAI, Universidad Pontificia de Comillas
• orcid.org/0000-0001-8360-3250
• Email
• Other articles by this author:
/ Alexander V. Podolskiy
/ Tatiana A. Shaposhnikova
Published Online: 2018-10-21 | DOI: https://doi.org/10.1515/anona-2018-0158

## Abstract

One of the main goals of this paper is to extend some of the mathematical techniques of some previous papers by the authors showing that some very useful phenomenological properties which can be observed at the nano-scale can be simulated and justified mathematically by means of some homogenization processes when a certain critical scale is used in the corresponding framework. Here the motivating problem in consideration is formulated in the context of the reverse osmosis. We consider, on a part of the boundary of a domain $\mathrm{\Omega }\subset {ℝ}^{n}$, a set of very small periodically distributed semipermeable membranes having an ideal infinite permeability coefficient (which leads to Signorini-type boundary conditions) on a part ${\mathrm{\Gamma }}_{1}$ of the boundary. We also assume that a possible chemical reaction may take place on the membranes. We obtain the rigorous convergence of the problems to a homogenized problem in which there is a change in the constitutive nonlinearities. Changes of this type are the reason for the big success of the nanocomposite materials. Our proof is carried out for membranes not necessarily of radially symmetric shape. The definition of the associated critical scale depends on the dimension of the space (and it is quite peculiar for the special case of $n=2$). Roughly speaking, our result proves that the consideration of the critical case of the scale leads to a homogenized formulation which is equivalent to having a global semipermeable membrane, at the whole part of the boundary ${\mathrm{\Gamma }}_{1}$, with a “finite permeability coefficient of this virtual membrane”, which is the best we can get, even if the original problem involves a set of membranes of any arbitrary finite permeability coefficients.

MSC 2010: 35B27; 76M50; 35M86; 35J87

## 1 Introduction and statement of results

We present new results concerning the asymptotic behavior, as $\epsilon \to 0$, of the solution ${u}_{\epsilon }$ of a family of boundary value problems formulated in a cavity (or plant) represented by a bounded domain $\mathrm{\Omega }\subset {ℝ}^{n}$ in which a linear diffusion equation is satisfied. The boundary $\partial \mathrm{\Omega }$ is split into two regions. On one of the regions, homogeneous Dirichlet conditions are specified. On the other one, some small subsets ${G}_{\epsilon }$ are ε-periodically distributed and some unilateral boundary conditions are specified on them. We also assume that a possible “reaction” may take place on a net ${G}_{\epsilon }$ of small pieces of the boundary given by the periodic repetition of a rescaled particle ${G}_{0}$.

There are several relevant problems in a wide spectrum of applications leading to such type of formulations, ranging from water and wastewater treatment to food and textile engineering as well as pharmaceutical and biotechnology applications (for a recent review see [29]). One of them concerns the reverse osmosis when we apply it, for instance, to desalination processes (see, e.g., [24] and the references therein). Without intending to use here a “realistic model”, we shall present an over-simplified formulation that, nonetheless, preserves most of the mathematical difficulties concerning the passing to the limit as $\epsilon \to 0$. Some examples of more complex formulations covering different aspects of the problems considered here can be found, for instance, in [17] and the many references therein.

We start by recalling that, roughly speaking, semipermeable membranes allow the passing of a certain type of molecules (the so-called “solvents”) but block another type of molecules (the “solutes”). The solvents flow from the region of smaller concentration of solutes to the region of higher concentration (the difference of the concentrations produces the phenomenon known as osmotic pressure). Nevertheless, by creating a very high pressure it is possible to produce an inverse flow, such as the one being used in desalination plants: it is the so-called “reverse osmosis”. Since in many cases the semipermeable membrane contains some chemical products (e.g., polyamides; see [18]), our formulation will contain also a nonlinear kinetic reaction term in the flux given by a continuous nondecreasing function $\sigma \left(s\right)$. Let us call ${w}_{\epsilon }$ the solvent concentration corresponding to the membrane periodicity scale ε. Let us modulate the intensity of the reaction in terms of a factor ${\epsilon }^{-k}$, with k to be analyzed later. So, for a critical value of the solvent concentration ψ (associated to the osmotic pressure) the flux (including the reaction kinetic term) is an incoming flux with respect to the solvents plant Ω if the concentration of the solvent molecules $w\left(x\right)$ on the semipermeable membrane ${G}_{\epsilon }\subset \partial \mathrm{\Omega }$ is smaller than or equal to this critical value, but it remains isolated (with no boundary flow, excluding the reaction term, on the membrane, i.e. when the concentration is $w\left(x\right)<\psi$). So, if ν is the exterior unit normal vector to the membrane surface, we have

${\partial }_{\nu }{w}_{\epsilon }+{\epsilon }^{-k}\sigma \left(\psi -{w}_{\epsilon }\right)=0$${\partial }_{\nu }{w}_{\epsilon }+{\epsilon }^{-k}\sigma \left(\psi -{w}_{\epsilon }\right)=-{\epsilon }^{-k}\mu \left(\psi -{w}_{\epsilon }\right)$

for some parameter $\mu >0$ called the “finite permeability coefficient of the membrane” (usually, in practice, μ takes big values). We assume a simplified linear diffusion equation on the solvent concentration

and some boundary conditions on the rest of the boundary $\partial \mathrm{\Omega }$. For instance, we can distinguish some subregions where Dirichlet or Neumann types of boundary conditions hold, and so, if we introduce the partition $\partial \mathrm{\Omega }={\mathrm{\Gamma }}_{1}\cup {\mathrm{\Gamma }}_{2}$ and assume that, in fact, ${G}_{\epsilon }\subset {\mathrm{\Gamma }}_{1}$, then we can imagine that

and

Figure 1 presents a simplified case of the above-mentioned framework.

Figure 1

A simple illustration of a plant with a reverse osmosis membrane.

We are especially interested in the study of new behaviors arising in the reverse osmosis membranes having a periodicity ε of the order of nanometers (see, e.g., [4] and its references). Mathematically, we shall give a sense to those extremely small scales by demanding that the diameter of these subsets included in ${G}_{\epsilon }$ is of order ${a}_{\epsilon }$, where ${a}_{\epsilon }\ll \epsilon$.

Sometimes it is interesting to consider semipermeable membranes with an “infinite permeability coefficient” (formally $\mu =+\mathrm{\infty }$, but only for the case ${w}_{\epsilon }\left(x\right)=\psi$), and thus ψ becomes an obstacle which is periodically repeated in ${G}_{\epsilon }$. By following the approach presented in [16], this can be formulated as

${\partial }_{\nu }{w}_{\epsilon }+{\epsilon }^{-k}\sigma \left(\psi -{w}_{\epsilon }\right)=0$${\partial }_{\nu }{w}_{\epsilon }+{\epsilon }^{-k}\sigma \left(\psi -{w}_{\epsilon }\right)\le 0$$\left({w}_{\epsilon }-\psi \right)\left({\partial }_{\nu }{w}_{\epsilon }+{\epsilon }^{-k}\sigma \left(\psi -{w}_{\epsilon }\right)\right)=0$

Now, to carry out our mathematical treatment, it is quite convenient to work with the new unknown

${u}_{\epsilon }\left(x\right):=\psi -{w}_{\epsilon }\left(x\right),$

and thus if we assume (again for simplicity) that $h=g=0$ and $f:=-F$, we simplify the formulation to arrive at the following formulation which will be the object of study in this paper:

$\left\{\begin{array}{cccc}& -\mathrm{\Delta }{u}_{\epsilon }=f\left(x\right),\hfill & & \hfill x\in \mathrm{\Omega },\\ & {u}_{\epsilon }\ge 0,{\partial }_{\nu }{u}_{\epsilon }+{\epsilon }^{-k}\sigma \left({u}_{\epsilon }\right)\ge 0,{u}_{\epsilon }\left({\partial }_{\nu }{u}_{\epsilon }+{\epsilon }^{-k}\sigma \left({u}_{\epsilon }\right)\right)=0,\hfill & & \hfill x\in {G}_{\epsilon },\\ & {\partial }_{\nu }{u}_{\epsilon }=0,\hfill & & \hfill x\in {\mathrm{\Gamma }}_{1}\setminus {\overline{G}}_{\epsilon },\\ & {u}_{\epsilon }=0,\hfill & & \hfill x\in {\mathrm{\Gamma }}_{2}.\end{array}$(1.1)

Notice that in the reaction kinetics we made emerge a re-scaling factor $\beta \left(\epsilon \right):={\epsilon }^{-k}$, where $k\in ℝ$. The relation between the exponent k and the diameter of the chemical particles (which we shall assume to be given by ${a}_{\epsilon }={C}_{0}{\epsilon }^{\alpha }$, where ${C}_{0}>0$ and $\alpha >1$) will be discussed later. This relation will depend on the dimension of the space $n\ge 3$. The case $n=2$ is rather special and will require a different treatment: we shall assume that ${a}_{\epsilon }={C}_{0}\epsilon {e}^{-{\alpha }^{2}/\epsilon }$ and $\beta \left(\epsilon \right)={e}^{{\alpha }^{2}/\epsilon }$.

Homogenization results for boundary value problems with alternating types of boundary conditions, including Robin-type conditions, were widely considered in the literature. We refer, for instance, to the papers [34, 8, 5, 1] which already contain an extensive bibliography on the subject. Huge attention was drawn to similar homogenization problems but in domains perforated by the tiny sets on which some nonlinear Robin-type condition is specified on their boundaries. Some pioneering works in this direction are the papers by Kaizu [25, 26] (see also [7]). There he investigated all the possible relations between parameters except one: the case of the “critical” relation between parameters α and $\beta \left(\epsilon \right)$, i.e. $\alpha =k=\frac{n}{n-2}$. Later on, this critical case was considered in [22] for $n=3$ and for the sets ${G}_{\epsilon }$ given by balls. It seems that it was in the paper [22] where the effect of “nonlinearity change due to the homogenization process” was discovered for the first time. After that, by using some different methods of proof, the critical case was solved for $n\ge 3$ in [35] (see also [21]). The consideration of the case $n=2$ and for arbitrary shaped domains ${G}_{\epsilon }$ was carried out in [33]. More recently, many results concerning the asymptotic behavior of solutions of problems similar to (1.1) were published in the literature [35, 23, 19, 20, 9, 11, 10, 12]. Nevertheless, in all the above-mentioned works the particles (or perforations, according to the physical model used as motivation of the mathematical formulation) of subsets ${G}_{\epsilon }$ where assumed to be balls (having a critical radius). We also mention here the paper [14] that describes the asymptotic behavior of some related problem for the case of arbitrary shaped sets ${G}_{\epsilon }$ and for $n\ge 3$. One of the main goals of this paper is to extend some of the techniques of [14] to problem (1.1), where the periodically distributed reactions arise merely on some part of the global boundary $\partial \mathrm{\Omega }$ for the critical scaled case. This is the case for which some phenomenological properties which arise at the nano-scale can be simulated and justified by means of homogenization processes.

## Case $n\ge 3$.

In order to present the main results of this paper, and their application to the reverse osmosis framework, we need to introduce some auxiliary notations. We start by considering the case $n\ge 3$. We assume that Ω is a bounded domain in ${ℝ}^{n}\cap \left\{{x}_{1}>0\right\}$, $n\ge 3$, with a piecewise-smooth boundary $\partial \mathrm{\Omega }$ that consists of two parts ${\mathrm{\Gamma }}_{1}$ and ${\mathrm{\Gamma }}_{2}$, with the property that

${\mathrm{\Gamma }}_{1}=\partial \mathrm{\Omega }\cap \left\{x\in {ℝ}^{n}:{x}_{1}=0\right\}\ne \mathrm{\varnothing }.$

We consider a model ${G}_{0}$ such that $\overline{{G}_{0}}\subset \left\{x\in {ℝ}^{n}:{x}_{1}=0,|x|<\frac{1}{4}\right\}$ with $\overline{{G}_{0}}$ being diffeomorphic to a ball. We define $\delta B=\left\{x:{\delta }^{-1}x\in B\right\}$, $\delta >0$. Let

$\stackrel{~}{{G}_{\epsilon }}=\bigcup _{j\in {ℤ}^{\prime }}\left({a}_{\epsilon }{G}_{0}+\epsilon j\right)=\bigcup _{j\in {ℤ}^{\prime }}{G}_{\epsilon }^{j},$

where

${ℤ}^{\prime }=\left\{0\right\}×{ℤ}^{n-1}$

and

(1.2)

A justification of the above choice of exponent k can be found, for instance, in [28] (see also [32]). We define the net of sets ${G}_{\epsilon }$ as the union of sets ${G}_{\epsilon }^{j}\subset \stackrel{~}{{G}_{\epsilon }}$ such that $\overline{{G}_{\epsilon }^{j}}\subset {\mathrm{\Gamma }}_{1}$ and $\rho \left(\partial {\mathrm{\Gamma }}_{1},\overline{{G}_{\epsilon }^{j}}\right)\ge 2\epsilon$, i.e.

${G}_{\epsilon }=\bigcup _{j\in {\mathrm{Υ}}_{\epsilon }}{G}_{\epsilon }^{j},$

where

${\mathrm{Υ}}_{\epsilon }=\left\{j\in {ℤ}^{\prime }:\rho \left(\partial {\mathrm{\Gamma }}_{1},\overline{{G}_{\epsilon }^{j}}\right)\ge 2\epsilon \right\}.$

Figure 2

Domain Ω when $n\ge 3$.

Notice that $|{\mathrm{Υ}}_{\epsilon }|\cong d{\epsilon }^{1-n}$, $d=\mathrm{const}>0$. Later it will be useful to observe that if we denote by ${T}_{r}\left({x}_{0}\right)$ the ball in ${ℝ}^{n}$ of radius r centered at a point ${x}_{0}$, and if we define the boundary points

and the set ${T}_{\epsilon /4}^{j}={T}_{\epsilon /4}\left({P}_{\epsilon }^{j}\right)$ , then we have $\overline{{G}_{\epsilon }^{j}}\subset {T}_{\epsilon /4}^{j}$. In this geometrical setting (see Figure 2), the so-called “strong formulation” of the problem for which we want to study the asymptotic behavior of its solutions is the following:

$\left\{\begin{array}{cccc}& -\mathrm{\Delta }{u}_{\epsilon }=f\left(x\right),\hfill & & \hfill x\in \mathrm{\Omega },\\ & {u}_{\epsilon }\ge 0,{\partial }_{\nu }{u}_{\epsilon }+{\epsilon }^{-k}\sigma \left({u}_{\epsilon }\right)\ge 0,{u}_{\epsilon }\left({\partial }_{\nu }{u}_{\epsilon }+{\epsilon }^{-k}\sigma \left({u}_{\epsilon }\right)\right)=0,\hfill & & \hfill x\in {G}_{\epsilon },\\ & {\partial }_{\nu }{u}_{\epsilon }=0,\hfill & & \hfill x\in {\mathrm{\Gamma }}_{1}\setminus {\overline{G}}_{\epsilon },\\ & {u}_{\epsilon }=0,\hfill & & \hfill x\in {\mathrm{\Gamma }}_{2},\end{array}$(1.3)

where $\sigma :ℝ\to ℝ$ is a locally Hölder continuous nondecreasing function and, at most, super-linear at infinity, i.e. such that

(1.4)

for all $t,s\ge 0$, where ${k}_{1},{K}_{1},{K}_{2}>0$, $\sigma \left(0\right)=0$. In problem (1.3), $\nu =\left(-1,0,\mathrm{\dots },0\right)$ is the unit outward normal vector Ω at $\left\{{x}_{1}=0\right\}$ and ${\partial }_{\nu }u=-\frac{\partial u}{\partial {x}_{1}}$ is the normal derivative of u at this part of the boundary.

#### Example 1.1.

Notice that examples of such functions cover

$\sigma \left(s\right)=\sqrt{s}.$

Furthermore, the behavior at infinity may be superlinear as, for example, in

$\sigma \left(s\right)=\left\{\begin{array}{cc}\sqrt{s},\hfill & 0\le s\le {s}_{0},\hfill \\ \sqrt{{s}_{0}}+{\left(s-{s}_{0}\right)}^{2},\hfill & s>{s}_{0}.\hfill \end{array}$

The weak formulation of (1.3) is the following definition (see, e.g., [16]).

#### Definition 1.2.

We say that ${u}_{\epsilon }$ is a weak solution of (1.1) if

and

${\int }_{\mathrm{\Omega }}\nabla {u}_{\epsilon }\nabla \left(\phi -{u}_{\epsilon }\right)dx+{\epsilon }^{-k}{\int }_{{G}_{\epsilon }}\sigma \left({u}_{\epsilon }\right)\left(\phi -{u}_{\epsilon }\right)𝑑{x}^{\prime }\ge {\int }_{\mathrm{\Omega }}f\left(\phi -{u}_{\epsilon }\right)𝑑x$(1.5)

for all $\phi \in {K}_{\epsilon }$.

By ${H}^{1}\left(\mathrm{\Omega },{\mathrm{\Gamma }}_{2}\right)$ we denote the closure in ${H}^{1}\left(\mathrm{\Omega }\right)$ of the set of infinitely differentiable functions in $\overline{\mathrm{\Omega }}$, vanishing on the boundary ${\mathrm{\Gamma }}_{2}$.

It is well known (see, e.g., some references in [10]) that problem (1.5) has a unique weak solution ${u}_{\epsilon }\in {K}_{\epsilon }$. From (1.5) we immediately deduce that

${\parallel \nabla {u}_{\epsilon }\parallel }_{{L}_{2}\left(\mathrm{\Omega }\right)}\le K,$(1.6)

where, here and below, the constant K is independent of ε. Hence, there exists a subsequence (denoted as the original sequence by ${\stackrel{~}{u}}_{\epsilon }$) such that, as $\epsilon \to 0$, we have

${u}_{\epsilon }⇀{u}_{0}$${u}_{\epsilon }\to {u}_{0}$(1.7)

By using the monotonicity of the function $\sigma \left(u\right)$ one can show (see some references in [10]) that ${u}_{\epsilon }$ satisfies the following “very weak formulation”:

${\int }_{\mathrm{\Omega }}\nabla \phi \nabla \left(\phi -{u}_{\epsilon }\right)dx+{\epsilon }^{-k}{\int }_{{G}_{\epsilon }}\sigma \left(\phi \right)\left(\phi -{u}_{\epsilon }\right)𝑑{x}^{\prime }\ge {\int }_{\mathrm{\Omega }}f\left(\phi -{u}_{\epsilon }\right)𝑑x,$(1.8)

where φ is an arbitrary function from ${K}_{\epsilon }$.

The main goal of this paper is to consider this critical relation between parameters. This scale is characterized by the fact that the resulting homogenized problem will contain a so-called “strange term” expressing the fact that the character of some nonlinearity arising in the homogenized problem differs from the original nonlinearity appearing in the boundary condition of (1.3). Still focusing first on the case $n\ge 3$, one can show that the critical scale of the size of the holes is given by (see, e.g., the arguments used in [28, 10])

$\alpha =\frac{n-1}{n-2}.$

The appropriate scaling of the reaction term so that both the diffusive and nonlinear characters are preserved at the limit is, as usual, driven by ${\epsilon }^{k}\sim |{G}_{\epsilon }|$. Therefore,

$k=\frac{n-1}{n-2}.$

In the present paper, we construct a homogenized problem with a nonlinear Robin-type boundary condition that contains a new nonlinear term, and prove the corresponding theorem stating that the solution of the original problem converges, as $\epsilon \to 0$, to the solution of the homogenized problem.

We point out that the main difficulties to get a homogenized problem associated to (1.3) come from the following different aspects:

• (i)

The low differentiability assumed on the function σ (since it is non-Lipschitz continuous at $u=0$ and it has quadratic growth at infinity).

• (ii)

The unilateral formulation of the boundary conditions on ${G}_{\epsilon }$.

• (iii)

The general shape assumed on the sets ${G}_{\epsilon }$.

• (iv)

The critical scale of the sets ${G}_{\epsilon }^{j}$.

Some of those difficulties where already in the previous short presentation paper by the authors [13], but only for $n=2$, without (ii) and by assuming that σ is Lipschitz continuous. Our main goal is to extend our techniques to the above-mentioned more general framework.

To build the homogenized problem we still need to introduce some “capacity-type” auxiliary problems. Given $u\in ℝ$, for $y\in {\left({ℝ}^{n}\right)}^{+}={ℝ}^{n}\cap \left\{{y}_{1}>0\right\}$ we introduce the new auxiliary function $\stackrel{^}{w}\left(y;u\right)$, depending also on ${G}_{0}$ and σ, as the (unique) solution of the problem

(1.9)

Remember that ${C}_{0}>0$ was given in the structural assumption (1.2). The existence and uniqueness of $\stackrel{^}{w}\left(y;u\right)$ is given in Lemma 2.2 below. Let us introduce also the auxiliary function $\stackrel{^}{\kappa }\left(y\right)$, $y\in {\left({ℝ}^{n}\right)}^{+},$ as the unique solution of the problem

We then define, for $u\in ℝ$, the possibly nonlinear function

${H}_{{G}_{0}}\left(u\right):={\int }_{{G}_{0}}{\partial }_{\nu }\stackrel{^}{w}\left(u,{y}^{\prime }\right)𝑑{y}^{\prime }={C}_{0}{\int }_{{G}_{0}}\sigma \left(u-\stackrel{^}{w}\left(u,{y}^{\prime }\right)\right)𝑑{y}^{\prime }$(1.10)

and the scalar

${\lambda }_{{G}_{0}}:={\int }_{{G}_{0}}{\partial }_{\nu }\stackrel{^}{\kappa }\left({y}^{\prime }\right)𝑑{y}^{\prime },$(1.11)

where in both definitions ${y}^{\prime }=\left(0,{y}_{2},\mathrm{\dots },{y}_{n}\right)$. Notice that $\stackrel{^}{\kappa }\left(y\right)$ can be extended by symmetry to ${ℝ}^{n}\setminus {\overline{G}}_{0}$ as a harmonic function. Moreover, by the maximum principle, $\stackrel{^}{\kappa }\left(y\right)$ reaches its maximum in ${G}_{0}$, and so by the strong maximum principle ${\partial }_{\nu }\stackrel{^}{\kappa }=-{\partial }_{{x}_{1}}\stackrel{^}{\kappa }>0$. Then we know that

${\lambda }_{{G}_{0}}>0.$

Some properties of the function ${H}_{{G}_{0}}$ will be presented later. In particular, in Lemma 2.8 below we will show that ${H}_{{G}_{0}}$ is always a Lipschitz continuous function (even if σ is merely Hölder continuous) such that

$0\le {H}_{{G}_{0}}^{\prime }\le {\lambda }_{{G}_{0}}.$(1.12)

The following theorem gives a description of the limiting function ${u}_{0}$ obtained in (1.7).

#### Theorem 1.3.

Let $n\mathrm{\ge }\mathrm{3}$, let $\alpha \mathrm{=}k\mathrm{=}\frac{n\mathrm{-}\mathrm{1}}{n\mathrm{-}\mathrm{2}}$ and let ${u}_{\epsilon }$ be a weak solution of problem (1.3). Then the function ${u}_{\mathrm{0}}$ defined in (1.7) is a weak solution of the following problem:

$\left\{\begin{array}{cccc}& -\mathrm{\Delta }{u}_{0}=f,\hfill & & \hfill x\in \mathrm{\Omega },\\ & {\partial }_{\nu }{u}_{0}+{C}_{0}^{n-2}{H}_{{G}_{0}}\left({u}_{0,+}\right)-{\lambda }_{{G}_{0}}{C}_{0}^{n-2}{u}_{0,-}=0,\hfill & & \hfill x\in {\mathrm{\Gamma }}_{1},\\ & {u}_{0}=0,\hfill & & \hfill x\in {\mathrm{\Gamma }}_{2},\end{array}$

where ${H}_{{G}_{\mathrm{0}}}$ is defined by (1.10) and ${\lambda }_{{G}_{\mathrm{0}}}$ by (1.11). Here, as usual, ${u}_{\mathrm{0}\mathrm{,}\mathrm{+}}\mathrm{:=}\mathrm{max}\mathit{}\mathrm{\left\{}{u}_{\mathrm{0}}\mathrm{,}\mathrm{0}\mathrm{\right\}}$ and ${u}_{\mathrm{0}\mathrm{,}\mathrm{-}}\mathrm{=}\mathrm{max}\mathit{}\mathrm{\left\{}\mathrm{-}{u}_{\mathrm{0}}\mathrm{,}\mathrm{0}\mathrm{\right\}}$, so that ${u}_{\mathrm{0}}\mathrm{=}{u}_{\mathrm{0}\mathrm{,}\mathrm{+}}\mathrm{-}{u}_{\mathrm{0}\mathrm{,}\mathrm{-}}$.

## Case $n=2$.

As mentioned before (see also [13]), the case $n=2$ requires to introduce some slight changes. The domain is given now in the following way: We consider Ω to be a bounded domain in ${ℝ}^{2}\cap \left\{{x}_{2}>0\right\}$, the boundary of which consists of two parts $\partial \mathrm{\Omega }={\mathrm{\Gamma }}_{1}\cup {\mathrm{\Gamma }}_{2}$ and ${\mathrm{\Gamma }}_{1}=\partial \mathrm{\Omega }\cap \left\{{x}_{2}=0\right\}=\left[-l,l\right]$, $l>0$, ${\mathrm{\Gamma }}_{2}=\partial \mathrm{\Omega }\cap \left\{{x}_{2}>0\right\}$. We set

${Y}_{1}=\left\{\left({y}_{1},0\right):-\frac{1}{2}<{y}_{1}<\frac{1}{2}\right\},{\stackrel{^}{l}}_{0}=\left\{\left({y}_{1},0\right):-{l}_{0}<{y}_{1}<{l}_{0}\right\}\subset {Y}_{1},{l}_{0}\in \left(0,\frac{1}{2}\right).$

For a small parameter $\epsilon >0$ and $0<{a}_{\epsilon }<\epsilon$, we introduce the sets

${\stackrel{~}{G}}_{\epsilon }=\bigcup _{j\in {ℤ}^{\prime }}\left({a}_{\epsilon }{\stackrel{^}{l}}_{0}+\epsilon j\right)=\bigcup _{j\in {ℤ}^{\prime }}{l}_{\epsilon }^{j},$

where ${ℤ}^{\prime }$ is a set of vectors $j=\left({j}_{1},0\right)$ and ${j}_{1}$ is a whole number. Set

${\mathrm{Υ}}_{\epsilon }=\left\{j\in {ℤ}^{\prime }:\overline{{l}_{\epsilon }^{j}}\subset \left\{x=\left({x}_{1},0\right):{x}_{1}\in \left[-l+2\epsilon ,l-2\epsilon \right]\right\}\right\}.$

Consider ${Y}_{\epsilon }^{j}=\epsilon {Y}_{0}+\epsilon j$ and

${l}_{\epsilon }=\bigcup _{j\in {\mathrm{Υ}}_{\epsilon }}{l}_{\epsilon }^{j}.$

It is easy to see that $\overline{{l}_{\epsilon }^{j}}\subset {Y}_{\epsilon }^{j}$. Set ${\gamma }_{\epsilon }={\mathrm{\Gamma }}_{1}\setminus \overline{{l}_{\epsilon }}$. Note that for for all $j\in {ℤ}^{\prime }$ we have $|{l}_{\epsilon }^{j}|=2{a}_{\epsilon }{l}_{0}$ and $|{l}_{\epsilon }|\cong d{a}_{\epsilon }{\epsilon }^{-1}$; see Figure 3.

Figure 3

Domain Ω in two dimensions.

The formulation of the problem starts by searching

(1.13)

being a solution to the following variational inequality:

${\int }_{\mathrm{\Omega }}\nabla {u}_{\epsilon }\nabla \left(\phi -{u}_{\epsilon }\right)dx+{e}^{\frac{{\alpha }^{2}}{\epsilon }}{\int }_{{G}_{\epsilon }}\sigma \left({u}_{\epsilon }\right)\left(\phi -{u}_{\epsilon }\right)𝑑{x}_{1}\ge {\int }_{\mathrm{\Omega }}f\left(\phi -{u}_{\epsilon }\right)𝑑x,$(1.14)

where φ is an arbitrary function from ${K}_{\epsilon }$. This time, for simplicity, we assume merely that the function $\sigma :ℝ\to ℝ$ is continuously differentiable, $\sigma \left(0\right)=0$ and there exist positive constants ${k}_{1}$, ${k}_{2}$ such that the following condition is satisfied:

(1.15)

(more general terms $\sigma \left(u\right)$ where considered in [13], but without the Signorini-type constraints). Therefore, we have

Note that (1.13), (1.14) is a weak formulation of the following strong formulation of the problem:

$\left\{\begin{array}{cccc}\hfill -\mathrm{\Delta }{u}_{\epsilon }& =f,\hfill & & \hfill x\in \mathrm{\Omega },\\ \hfill {u}_{\epsilon }& \ge 0,{\partial }_{{x}_{2}}{u}_{\epsilon }-\beta \left(\epsilon \right)\sigma \left({u}_{\epsilon }\right)\ge 0,{u}_{\epsilon }\left({\partial }_{{x}_{2}}{u}_{\epsilon }-\beta \left(\epsilon \right)\sigma \left({u}_{\epsilon }\right)\right)=0,\hfill & & \hfill x\in {l}_{\epsilon },\\ \hfill {\partial }_{{x}_{2}}{u}_{\epsilon }& =0,\hfill & & \hfill x\in {\gamma }_{\epsilon },\\ \hfill {u}_{\epsilon }& =0,\hfill & & \hfill x\in {\mathrm{\Gamma }}_{1}.\end{array}$(1.16)

Our homogenized result in this case is the following theorem.

#### Theorem 1.4.

Let ${a}_{\epsilon }\mathrm{=}{C}_{\mathrm{0}}\mathit{}\epsilon \mathit{}{e}^{\mathrm{-}{\alpha }^{\mathrm{2}}\mathrm{/}\epsilon }$, $\beta \mathit{}\mathrm{\left(}\epsilon \mathrm{\right)}\mathrm{=}{e}^{{\alpha }^{\mathrm{2}}\mathrm{/}\epsilon }$, $\alpha \mathrm{\ne }\mathrm{0}$, ${C}_{\mathrm{0}}\mathrm{>}\mathrm{0}$ and let ${u}_{\epsilon }$ be a solution to problem (1.16). Then there exists a subsequence such that ${u}_{\epsilon }\mathrm{\to }{u}_{\mathrm{0}}$, strongly in ${L}^{\mathrm{2}}\mathit{}\mathrm{\left(}\mathrm{\Omega }\mathrm{\right)}$ and weakly in ${H}^{\mathrm{1}}\mathit{}\mathrm{\left(}\mathrm{\Omega }\mathrm{,}{\mathrm{\Gamma }}_{\mathrm{1}}\mathrm{\right)}$ as $\epsilon \mathrm{\to }\mathrm{0}$, and the function ${u}_{\mathrm{0}}\mathrm{\in }{H}^{\mathrm{1}}\mathit{}\mathrm{\left(}\mathrm{\Omega }\mathrm{,}{\mathrm{\Gamma }}_{\mathrm{1}}\mathrm{\right)}$ is a weak solution to the following boundary value problem:

$\left\{\begin{array}{cccc}\hfill -\mathrm{\Delta }{u}_{0}& =f,\hfill & & \hfill x\in \mathrm{\Omega },\\ \hfill {\partial }_{{x}_{2}}{u}_{0}& =\frac{\pi }{{\alpha }^{2}}\left({H}_{{l}_{0}}\left({u}_{0,+}\right)-{u}_{0,-}\right),\hfill & & \hfill x\in {\mathrm{\Gamma }}_{1},\\ \hfill {u}_{0}& =0,\hfill & & \hfill x\in {\mathrm{\Gamma }}_{2},\end{array}$

where ${H}_{{l}_{\mathrm{0}}}\mathit{}\mathrm{\left(}u\mathrm{\right)}$ verifies the functional equation

$\pi {H}_{{l}_{0}}\left(u\right)=2{l}_{0}{\alpha }^{2}{C}_{0}\sigma \left(u-{H}_{{l}_{0}}\left(u\right)\right).$(1.17)

The special case of dimension $n=2$ is illustrative in order to get a complete identification of the function ${H}_{{G}_{0}}$. Curiously enough, the characterization condition for ${H}_{{G}_{0}}$ and ${H}_{{l}_{0}}$ (see, e.g., the functional equations (1.10) and (1.17), respectively) is quite related (but not exactly the same) to the one obtained in [10], although the problem under consideration in that paper was not the same as problem (1.3). It was shown in [10] that if $H:ℝ\to ℝ$ is the solution of a problem

$H\left(u\right)=C\sigma \left(u-H\left(u\right)\right),$

then H is given by

$H\left(u\right)={\left(I+{\left(C\sigma \right)}^{-1}\right)}^{-1}\left(u\right)$(1.18)

for $u\in ℝ$. When the Signorini condition is included, σ can be generalized to the maximal monotone graph

$\stackrel{~}{\sigma }\left(u\right)=\left\{\begin{array}{cc}\sigma \left(u\right),\hfill & u>0,\hfill \\ \left(-\mathrm{\infty },0\right],\hfill & u=0,\hfill \\ \mathrm{\varnothing },\hfill & u<0.\hfill \end{array}$

It was proven in [10] that the corresponding zero-order term is given by

$\stackrel{~}{H}\left(u\right)=\left\{\begin{array}{cc}H\left(u\right),\hfill & u>0,\hfill \\ u,\hfill & u\le 0.\hfill \end{array}$

This behavior is interesting because $\stackrel{~}{H}$ matches (1.18) formally. The maximal monotone graph

$\gamma \left(u\right)=\left\{\begin{array}{cc}0,\hfill & u>0,\hfill \\ \left(-\mathrm{\infty },0\right],\hfill & u=0,\hfill \\ \mathrm{\varnothing },\hfill & u<0,\hfill \end{array}$

has, formally, the inverse

${\gamma }^{-1}\left(u\right)=\left\{\begin{array}{cc}\mathrm{\varnothing },\hfill & u>0,\hfill \\ \left[0,+\mathrm{\infty }\right),\hfill & u=0,\hfill \\ 0,\hfill & u<0.\hfill \end{array}$

In particular,

${H}_{\gamma }\left(u\right):={\left(I+{\left(C\gamma \right)}^{-1}\right)}^{-1}\left(u\right)=\left\{\begin{array}{cc}\mathrm{\varnothing },\hfill & u>0,\hfill \\ \left[0,+\mathrm{\infty }\right),\hfill & u=0,\hfill \\ u,\hfill & u<0.\hfill \end{array}$

In this way, it is clear that

$\stackrel{~}{H}\left(u\right)=\left\{\begin{array}{cc}H\left(u\right),\hfill & u>0,\hfill \\ {H}_{\gamma }\left(u\right),\hfill & u<0.\hfill \end{array}$

For the case $n=3$ we will make further comments in this direction in Section 6.1.

Coming back to the framework of the semipermeable membranes problems, what we can conclude is that the homogenization of a set of periodic semipermeable membranes with an “infinite permeability coefficient”, in the critical case, leads to a homogenized formulation which is equivalent to having a global semipermeable membrane, at ${\mathrm{\Gamma }}_{1}$, with a “finite permeability coefficient of this virtual membrane” ${\mu }_{\mathrm{\infty }}$, which is the best we can get, even if the original problem involves a finite permeability μ. Indeed, this comes from the properties of the function ${H}_{{G}_{0}}$ (which can also be computed for the case of a microscopic membrane with finite permeability), for instance, on the subpart of the boundary $\left\{x\in {\mathrm{\Gamma }}_{1}:{w}_{0}\left(x\right)\le \psi \right\}$ (with ${w}_{0}\left(x\right):=\psi -{u}_{0}\left(x\right)$, i.e. where ${u}_{0}\left(x\right)\le 0$). Moreover, by using that ${H}_{{G}_{0}}\left(0\right)=0$ and the decomposition ${u}_{0}={u}_{0,+}-{u}_{0,-}$, we know now that ${\partial }_{\nu }{w}_{0}={\lambda }_{{G}_{0}}{C}_{0}^{n-2}{w}_{0}$, and thus, for the case of a microscopic finite permeability membrane, it is not difficult to show that we get a homogenized permeability membrane coefficient given by ${\lambda }_{{G}_{0}}{C}_{0}^{n-2}$, which is larger than μ (see more details in Section 6.1 below).

The plan of the rest of the paper is the following: Part I (containing Sections 26) is devoted to the study of the case $n\ge 3$ and contains also some comments and possible extensions (for instance, we give some link between the present homogenization results and the homogenization of some problems involving non-local fractional operators). Part II (containing Section 7) is devoted to the proof of the convergence result for $n=2$.

## 2.1 On the auxiliary function $\stackrel{^}{\kappa }$

By using the method of sub- and supersolutions as in [14], the following result holds.

#### Lemma 2.1.

There exists a unique solution $\stackrel{\mathrm{^}}{\kappa }\mathrm{\in }\mathrm{X}$ of problem (1.9), where

$𝕏=\left\{v\in {L}_{\mathrm{loc}}^{2}\left(\overline{{\left({ℝ}^{n}\right)}^{+}}\right):\nabla v\in {L}^{2}{\left({\left({ℝ}^{n}\right)}^{+}\right)}^{n},|v|\le \frac{K}{{|y|}^{n-2}}\right\}$

such that

and

We define also the family of auxiliary functions

${\stackrel{^}{\kappa }}_{\epsilon }^{j}=\kappa \left(\frac{x-{P}_{\epsilon }^{j}}{{a}_{\epsilon }}\right).$

## 2.2 On the auxiliary function $\stackrel{^}{w}$

Arguing as above, in terms similar to the paper [14], we get the following lemma.

#### Lemma 2.2.

Let σ be a maximal monotone graph. There exists a unique solution $\stackrel{\mathrm{^}}{w}\mathit{}\mathrm{\left(}u\mathrm{,}\mathrm{\cdot }\mathrm{\right)}\mathrm{\in }\mathrm{X}$ of problem (1.9). Furthermore, the following assertions hold:

• If $u\ge 0$ , then $0\le \stackrel{^}{w}\left(u,y\right)\le u\stackrel{^}{\kappa }\left(y\right)\le u$.

• If $u\le 0$ , then $0\ge \stackrel{^}{w}\left(u,y\right)\ge u\stackrel{^}{\kappa }\left(y\right)\ge u$.

Hence,

Concerning the dependence with respect to the parameter u, we start by considering the case in which σ is a maximal monotone graph associated to a continuous function.

#### Lemma 2.3.

Let σ be a nondecreasing continuous function such that $\sigma \mathit{}\mathrm{\left(}\mathrm{0}\mathrm{\right)}\mathrm{=}\mathrm{0}$. Then

$|\stackrel{^}{w}\left({u}_{1},y\right)-\stackrel{^}{w}\left({u}_{2},y\right)|\le |{u}_{1}-{u}_{2}|$

for all ${u}_{\mathrm{1}}\mathrm{,}{u}_{\mathrm{2}}\mathrm{\in }\mathrm{R}$.

#### Proof.

Let $\stackrel{^}{w}\left({u}_{1},y\right)$, $\stackrel{^}{w}\left({u}_{2},y\right)$ be two solutions of problem (1.9) with parameters ${u}_{1},{u}_{2}\in ℝ$. Consider the function $v=w\left({u}_{1},y\right)-w\left({u}_{2},y\right)$. This function is a solution of the following exterior problem:

(2.1)

First consider the case ${u}_{1}>{u}_{2}$. If we choose ${v}^{-}$ as a test function in the integral identity for problem (2.1), we arrive at

${\int }_{{\left({ℝ}^{n}\right)}^{+}}{|\nabla {v}^{-}|}^{2}𝑑x+{C}_{0}{\int }_{{G}_{0}}\left(\sigma \left({u}_{1}-\stackrel{^}{w}\left({u}_{1},y\right)\right)-\sigma \left({u}_{2}-\stackrel{^}{w}\left({u}_{2},y\right)\right)\right){v}^{-}𝑑s=0.$

The second integral in the obtained expression can be nonzero only if $v<0$, i.e. $\stackrel{^}{w}\left({u}_{1},y\right)-\stackrel{^}{w}\left({u}_{2},y\right)<0$. By combining this inequality with the condition ${u}_{1}>{u}_{2}$, we get ${u}_{1}-\stackrel{^}{w}\left({u}_{1},y\right)>{u}_{2}-\stackrel{^}{w}\left({u}_{2},y\right)$. This inequality and the monotonicity of the function σ imply that the second integral is non-negative. Hence, two integrals must be equal to zero, so ${v}^{-}=0$ ${\mathcal{ℒ}}^{n-1}$-a.e. in ${G}_{0}$ and ${v}^{-}=c$ in ${\left({ℝ}^{n}\right)}^{+}$. But we have $v\to 0$ as $|y|\to \mathrm{\infty }$, and hence $c=0$, i.e. $\stackrel{^}{w}\left({u}_{1},y\right)-\stackrel{^}{w}\left({u}_{2},y\right)\ge 0$.

One can construct a function $\phi \left(r\right)\in {C}_{0}^{\mathrm{\infty }}\left(ℝ\right)$ such that $\phi =0$ if $|r|>1$, and $\phi =1$ if $|r|<0.5$. We take ${\left({u}_{1}-{u}_{2}-v\right)}^{-}\phi \left(\rho \left(x,{G}_{0}\right)/R\right)$ as a test function in the integral identity for problem (2.1) and obtain

$-{\int }_{{\left({ℝ}^{n}\right)}^{+}}\nabla v\nabla {\left({u}_{1}-{u}_{2}-v\right)}^{-}\phi \left(\rho \left(x,{G}_{0}\right)/R\right)dx-{\int }_{{\left({ℝ}^{n}\right)}^{+}}\frac{{\phi }^{\prime }\left(\rho \left(x,{G}_{0}\right)/R\right)}{R}{\left({u}_{1}-{u}_{2}-v\right)}^{-}\nabla v\nabla \rho dx$$+{C}_{0}{\int }_{{G}_{0}}\left(\sigma \left({u}_{1}-\stackrel{^}{w}\left({u}_{1},y\right)\right)-\sigma \left({u}_{2}-\stackrel{^}{w}\left({u}_{2},y\right)\right)\right){\left({u}_{1}-{u}_{2}-v\right)}^{-}𝑑s$$={I}_{1,R}+{I}_{2,R}+{I}_{3}=0.$

Since σ is monotone ${I}_{3}\le 0$.

For the first integral we have

${I}_{1,R}\le -{\int }_{{\mathcal{𝒟}}_{1,R}}{|\nabla {\left({u}_{1}-{u}_{2}-v\right)}^{-}|}^{2}dx\le 0,$

where ${\mathcal{𝒟}}_{1,R}=\left({\left({ℝ}^{n}\right)}^{+}\right)\cap \left\{x\in {ℝ}^{n}:\rho \left(x,{G}_{0}\right). We have that

${I}_{1}=-{\int }_{{\left({ℝ}^{n}\right)}^{+}}{|\nabla {\left({u}_{1}-{u}_{2}-v\right)}^{-}|}^{2}dx=\underset{R\to \mathrm{\infty }}{lim}{I}_{1,R}\le 0.$

For the second integral we derive the estimation

where ${\mathcal{𝒟}}_{2,R}=\left({\left({ℝ}^{n}\right)}^{+}\right)\cap \left\{x\in {ℝ}^{n}:\frac{R}{2}<\rho \left(x,{G}_{0}\right). Thereby, as $R\to \mathrm{\infty }$, we have ${I}_{1}+{I}_{3}=0$, ${I}_{1}\le 0$, ${I}_{3}\le 0$, and so

${I}_{1}=0,{I}_{3}=0.$

Taking into account that , we derive from the last corollary that ${\left({u}_{1}-{u}_{2}-v\right)}^{-}\equiv 0$, i.e. $0\le v<{u}_{1}-{u}_{2}$ in ${\left({ℝ}^{n}\right)}^{+}$. Moreover, we have that $v<{u}_{1}-{u}_{2}$ ${\mathcal{ℒ}}^{n-1}$-a.e. in ${G}_{0}$.

The case ${u}_{1}<{u}_{2}$ is analogous to the one above, so we have ${u}_{1}-{u}_{2}\le v\le 0$ in ${\left({ℝ}^{n}\right)}^{+}$ and ${\mathcal{ℒ}}^{n-1}$-a.e. in ${G}_{0}$. This concludes the proof. ∎

#### Lemma 2.4.

Let ${u}_{\mathrm{1}}\mathrm{>}{u}_{\mathrm{2}}$. Then

$0\le \stackrel{^}{w}\left({u}_{1},y\right)-\stackrel{^}{w}\left({u}_{2},y\right)\le \left({u}_{1}-{u}_{2}\right)\stackrel{^}{\kappa }\left(y\right).$(2.2)

#### Proof.

The functions ${\phi }_{1}\left(y\right)=\stackrel{^}{w}\left({u}_{1},y\right)-\stackrel{^}{w}\left({u}_{2},y\right)$ and ${\phi }_{2}\left(y\right)=\left({u}_{1}-{u}_{2}\right)\stackrel{^}{\kappa }\left(y\right)$ can be extended (by symmetry) as harmonic functions to ${ℝ}^{n}\setminus \overline{{G}_{0}}$ such that ${lim}_{|y|\to \mathrm{\infty }}\left({\phi }_{2}-{\phi }_{1}\right)=0$ and ${\phi }_{2}-{\phi }_{1}\ge 0$ on ${G}_{0}$. The comparison principle proves the result. ∎

A more regular dependence with respect to u can also be proved under additional regularity of the function σ.

#### Lemma 2.5 (Differentiable dependence of solutions).

Suppose that $\sigma \mathrm{\in }{\mathcal{C}}^{\mathrm{1}}$ and ${\sigma }^{\mathrm{\prime }}\mathrm{\ge }{k}_{\mathrm{1}}\mathrm{>}\mathrm{0}$. Then the map $u\mathrm{\in }\mathrm{R}\mathrm{↦}\stackrel{\mathrm{^}}{w}\mathit{}\mathrm{\left(}u\mathrm{,}\mathrm{\cdot }\mathrm{\right)}\mathrm{\in }{L}_{\mathrm{loc}}^{\mathrm{2}}\mathit{}\mathrm{\left(}K\mathrm{\right)}$ is differentiable for every smooth bounded set K such that ${G}_{\mathrm{0}}\mathrm{\subset }K\mathrm{\subset }\overline{{\mathrm{\left(}{\mathrm{R}}^{n}\mathrm{\right)}}^{\mathrm{+}}}$. Furthermore, if we define

$\stackrel{^}{W}\left(u,y\right)=\frac{\partial \stackrel{^}{w}\left(u,y\right)}{\partial u},$

then

${\int }_{{\left({ℝ}^{n}\right)}^{+}}\nabla \stackrel{^}{W}\left(u,y\right)\nabla \phi dy={C}_{0}{\int }_{{G}_{0}}{\sigma }^{\prime }\left(u-\stackrel{^}{w}\left(u,y\right)\right)\left(1-\stackrel{^}{W}\left(u,y\right)\right)\phi \left(y\right)𝑑{S}_{y}$(2.3)

for $\phi \mathrm{\in }{C}_{c}^{\mathrm{\infty }}\mathit{}\mathrm{\left(}\overline{{\mathrm{\left(}{\mathrm{R}}^{n}\mathrm{\right)}}^{\mathrm{+}}}\mathrm{\right)}$, $\mathrm{\nabla }\mathit{}\stackrel{\mathrm{^}}{W}\mathrm{\in }{L}^{\mathrm{2}}\mathit{}{\mathrm{\left(}\mathrm{\Omega }\mathrm{\right)}}^{n}$ and $\mathrm{0}\mathrm{\le }\stackrel{\mathrm{^}}{W}\mathit{}\mathrm{\left(}u\mathrm{,}y\mathrm{\right)}\mathrm{\le }\stackrel{\mathrm{^}}{\kappa }\mathit{}\mathrm{\left(}y\mathrm{\right)}$. In particular,

$0\le \frac{\partial \stackrel{^}{w}}{\partial u}\le \stackrel{^}{\kappa }\left(y\right).$

#### Proof.

Considering the difference of two solutions, we obtain

${\int }_{{\left({ℝ}^{n}\right)}^{+}}\nabla \frac{\stackrel{^}{w}\left(u+h,y\right)-\stackrel{^}{w}\left(u,y\right)}{h}\nabla \phi dy={C}_{0}{\int }_{{G}_{0}}\frac{\sigma \left(u+h-\stackrel{^}{w}\left(u+h,y\right)\right)-\sigma \left(u-\stackrel{^}{w}\left(u,y\right)\right)}{h}\phi 𝑑{S}_{y}$$={C}_{0}{\int }_{{G}_{0}}{\sigma }^{\prime }\left({\xi }_{h}\left(y\right)\right)\left(1-\frac{\stackrel{^}{w}\left(u+h,y\right)-\stackrel{^}{w}\left(u,y\right)}{h}\right)\phi \left(y\right)𝑑{S}_{y}$

for some ${\xi }_{h}$ in between $u+h-\stackrel{^}{w}\left(u+h,y\right)$ and $u-\stackrel{^}{w}\left(u,y\right)$. From this we obtain

${\int }_{{\left({ℝ}^{n}\right)}^{+}}\nabla \frac{\stackrel{^}{w}\left(u+h,y\right)-\stackrel{^}{w}\left(u,y\right)}{h}\nabla \phi dy+{C}_{0}{\int }_{{G}_{0}}{\sigma }^{\prime }\left({\xi }_{h}\left(y\right)\right)\frac{\stackrel{^}{w}\left(u+h,y\right)-\stackrel{^}{w}\left(u,y\right)}{h}\phi \left(y\right)𝑑{S}_{y}$$={C}_{0}{\int }_{{G}_{0}}{\sigma }^{\prime }\left({\xi }_{h}\left(y\right)\right)\phi \left(y\right)𝑑{S}_{y}.$

Taking $\phi =\frac{\stackrel{^}{w}\left(u+h,y\right)-\stackrel{^}{w}\left(u,y\right)}{h}$, and using the fact that $\stackrel{^}{w}\left(u,y\right)$ can be bounded and ${\sigma }^{\prime }$ is continuous, we obtain

${\parallel \nabla \left(\frac{\stackrel{^}{w}\left(u+h,y\right)-\stackrel{^}{w}\left(u,y\right)}{h}\right)\parallel }_{{L}^{2}{\left(\mathrm{\Omega }\right)}^{n}}^{2}+{k}_{1}{C}_{0}{\parallel \frac{\stackrel{^}{w}\left(u+h,y\right)-\stackrel{^}{w}\left(u,y\right)}{h}\parallel }_{{L}^{2}\left({G}_{0}\right)}^{2}\le C$

for h small. Thus $\frac{\stackrel{^}{w}\left(u+h,y\right)-\stackrel{^}{w}\left(u,y\right)}{h}$ admits a weak limit as $h\to 0$ in ${H}^{1}\left(K\right)$; let it be denoted by $\stackrel{^}{W}\left(u,y\right)$. Thus, up to a subsequence, it admits a pointwise limit and strong limit in ${L}^{2}\left(K\right)$. It is clear that

By passing to the limit for $\phi \in {\mathcal{𝒞}}_{c}^{\mathrm{\infty }}\left(\overline{{\left({ℝ}^{n}\right)}^{+}}\right)$ fixed, we characterize (2.3). From (2.2) we deduce that, for $h>0$,

$0\le \frac{\stackrel{^}{w}\left(u+h,y\right)-\stackrel{^}{w}\left(u,y\right)}{h}\le \kappa \left(y\right).$

As $h\to 0$, we deduce the result $0\le \stackrel{^}{W}\left(u,y\right)\le \kappa \left(y\right)$. ∎

#### Remark 2.6.

Notice that $\stackrel{^}{W}$ is the unique solution of

$\left\{\begin{array}{cccc}& -\mathrm{\Delta }\stackrel{^}{W}=0,\hfill & & \hfill y\in {\left({ℝ}^{n}\right)}^{+},\\ & {\partial }_{\nu }\stackrel{^}{W}+{C}_{0}{\sigma }^{\prime }\left(u-\stackrel{^}{w}\right)\stackrel{^}{W}={C}_{0}{\sigma }^{\prime }\left(u-\stackrel{^}{w}\right),\hfill & & \hfill y\in {G}_{0},\\ & {\partial }_{\nu }\stackrel{^}{W}=0,\hfill & & \hfill y\notin {G}_{0},{y}_{1}=0,\\ & \stackrel{^}{W}\to 0,\hfill & & \hfill |y|\to +\mathrm{\infty }.\end{array}$

#### Remark 2.7.

Assume $\sigma \left(u\right)=\mu u$. Then ${\sigma }^{\prime }\left(u-\stackrel{^}{w}\right)\equiv \mu$, and $\stackrel{^}{W}$ does not depend on μ. Therefore, $\stackrel{^}{w}\left(x,u\right)=u\stackrel{^}{W}\left(x\right)$. Furthermore,

$H\left(u\right)={\lambda }_{\mu }u.$

## 2.3 On the regularity of the function H

#### Lemma 2.8.

Let σ be a maximal monotone graph. The function ${H}_{{G}_{\mathrm{0}}}\mathit{}\mathrm{\left(}u\mathrm{\right)}$ defined by (1.10) is nondecreasing and Lipschitz continuous of constant ${\lambda }_{{G}_{\mathrm{0}}}$ given by (1.11), i.e. if ${u}_{\mathrm{1}}\mathrm{>}{u}_{\mathrm{2}}$, then

$0\le H\left({u}_{1}\right)-H\left({u}_{2}\right)\le {\lambda }_{{G}_{0}}\left({u}_{1}-{u}_{2}\right),$(2.4)

i.e. in the notation of weak derivatives,

#### Remark 2.9.

Notice that $\stackrel{^}{\kappa }$ (and thus ${\lambda }_{{G}_{0}}$) does not depend on σ, but only on ${G}_{0}$.

#### Proof.

First, let σ be smooth and ${\sigma }^{\prime }\ge {k}_{1}>0$. Again, let $\stackrel{^}{W}=\frac{\partial \stackrel{^}{w}}{\partial u}$. Taking derivatives in (1.10), we have that

${H}^{\prime }\left(u\right)={C}_{0}{\int }_{{G}_{0}}{\sigma }^{\prime }\left(u-\stackrel{^}{w}\right)\left(1-\stackrel{^}{W}\right)𝑑{y}^{\prime }.$

Since $\stackrel{^}{W}\le 1$, we have that ${H}^{\prime }\ge 0$. Using $\stackrel{^}{\kappa }$ as a test function in (2.3), we obtain that

${H}^{\prime }\left(u\right)={C}_{0}{\int }_{{G}_{0}}{\sigma }^{\prime }\left(u-\stackrel{^}{w}\right)\left(1-\stackrel{^}{W}\right)𝑑{y}^{\prime }$$={\int }_{{G}_{0}}\left({\sigma }^{\prime }\left(u-\stackrel{^}{w}\right)\left(1-\stackrel{^}{W}\right)\right)\stackrel{^}{\kappa }𝑑{y}^{\prime }$$={\int }_{{\left({ℝ}^{n}\right)}^{+}}\nabla \stackrel{^}{W}\nabla \stackrel{^}{\kappa }dy$$={\int }_{{G}_{0}}\stackrel{^}{W}\left({\partial }_{\nu }\stackrel{^}{\kappa }\right)𝑑{y}^{\prime }$$\le {\int }_{{G}_{0}}\left({\partial }_{\nu }\stackrel{^}{\kappa }\right)𝑑{y}^{\prime }$$=\lambda ,$

using the facts that ${\partial }_{\nu }\stackrel{^}{\kappa }\ge 0$ and $0\le \stackrel{^}{W}\le 1$.

If σ is a general maximal monotone graph, estimate (2.4) is maintained by approximation by a smooth sequence of the function ${\sigma }_{k}$. ∎

## 3.1 On the auxiliary function ${w}_{\epsilon }^{j}$

We introduce a function ${w}_{\epsilon }^{j}\left(u,x\right)$ as a solution of the boundary value problem

$\left\{\begin{array}{cccc}\hfill \mathrm{\Delta }{w}_{\epsilon }^{j}& =0,\hfill & & \hfill x\in {\left({T}_{\epsilon /4}^{j}\right)}^{+},\\ \hfill {\partial }_{\nu }{w}_{\epsilon }^{j}& ={\epsilon }^{-k}\sigma \left(u-{w}_{\epsilon }^{j}\right),\hfill & & \hfill x\in {G}_{\epsilon }^{j},\\ \hfill {\partial }_{\nu }{w}_{\epsilon }^{j}& =0,\hfill & & \hfill x\in {\left({T}_{\epsilon /4}^{j}\right)}^{0}\setminus \overline{{G}_{\epsilon }^{j}},\\ \hfill {w}_{\epsilon }^{j}& =0,\hfill & & \hfill x\in {\left(\partial {T}_{\epsilon /4}^{j}\right)}^{+},\end{array}$(3.1)

where $u\in ℝ$ is a parameter. We will compare this auxiliary function with the functions

${\stackrel{^}{w}}_{\epsilon }^{j}\left(u,x\right)=\stackrel{^}{w}\left(u,\frac{x-{P}_{\epsilon }^{j}}{{a}_{\epsilon }}\right).$

The function ${w}_{\epsilon }^{j}\in {H}_{0}^{1}\left({T}_{\epsilon /4}^{j}\right)$ is a weak solution of problem (3.1) if it satisfies the integral identity

${\int }_{{\left({T}_{\epsilon /4}^{j}\right)}^{+}}\nabla {w}_{\epsilon }^{j}\nabla \phi dx-{\epsilon }^{-k}{\int }_{{G}_{\epsilon }^{j}}\sigma \left(u-{w}_{\epsilon }^{j}\right)\phi 𝑑{x}^{\prime }=0$

for an arbitrary function $\phi \in {H}_{0}^{1}\left({T}_{\epsilon /4}^{j}\right)$.

From the uniqueness of problem (3.1) and the method of sub- and supersolutions, we have the following lemma.

#### Lemma 3.1.

The function ${w}_{\epsilon }^{j}$ satisfies the following estimations:

• (i)

If $u\ge 0$ , then $0\le {w}_{\epsilon }^{j}\left(u,x\right)\le {\stackrel{^}{w}}_{\epsilon }^{j}\left(u,x\right)\le u$.

• (ii)

If $u\le 0$ , then $u\le {\stackrel{^}{w}}_{\epsilon }^{j}\left(u,x\right)\le {w}_{\epsilon }^{j}\left(u,x\right)\le 0$.

#### Remark 3.2.

Since σ is monotone, from the previous result we obtain

$|\sigma \left(u-{w}_{\epsilon }^{j}\right)|\le |\sigma \left(u\right)|.$(3.2)

We define the function

${W}_{\epsilon }=\left\{\begin{array}{cc}{w}_{\epsilon }^{j}\left(u,x\right),\hfill & x\in {\left({T}_{\epsilon /4}^{j}\right)}^{+},j\in {\mathrm{Υ}}_{\epsilon }\hfill \\ 0,\hfill & x\in {\left({ℝ}^{n}\right)}^{+}\setminus \bigcup _{j\in {\mathrm{Υ}}_{\epsilon }}\overline{{\left({T}_{\epsilon /4}^{j}\right)}^{+}}.\hfill \end{array}$(3.3)

Notice that ${W}_{\epsilon }\in {H}^{1}\left(\mathrm{\Omega },{\mathrm{\Gamma }}_{2}\right)$ for all $u\in ℝ$. The following lemma proposes an estimate of the introduced function and its gradient.

#### Lemma 3.3.

The following estimations for the function ${W}_{\epsilon }$, which was defined in (3.3), are valid:

$\left\{\begin{array}{cc}\hfill {\parallel \nabla {W}_{\epsilon }\parallel }_{{L}_{2}\left(\mathrm{\Omega }\right)}^{2}& \le K|u||\sigma \left(u\right)|,\hfill \\ \hfill {\parallel {W}_{\epsilon }\parallel }_{{L}_{2}\left(\mathrm{\Omega }\right)}^{2}& \le K{\epsilon }^{2}|u||\sigma \left(u\right)|.\hfill \end{array}$(3.4)

#### Proof.

From the weak formulation of problem (3.1) for $j\in {\mathrm{Υ}}_{\epsilon }$ we know

${\int }_{{\left({T}_{\epsilon /4}^{j}\right)}^{+}}\nabla {w}_{\epsilon }^{j}\nabla \phi dx-{\epsilon }^{-k}{\int }_{{G}_{\epsilon }^{j}}\sigma \left(u-{w}_{\epsilon }^{j}\right)\phi 𝑑{x}^{\prime }=0.$

We take $\phi ={w}_{\epsilon }^{j}$ as a test function in this expression and obtain

${\int }_{{\left({T}_{\epsilon /4}^{j}\right)}^{+}}{|\nabla {w}_{\epsilon }^{j}|}^{2}𝑑x-{\epsilon }^{-k}{\int }_{{G}_{\epsilon }^{j}}\sigma \left(u-{w}_{\epsilon }^{j}\right){w}_{\epsilon }^{j}𝑑{x}^{\prime }=0.$

Then we can transform the obtained relation to the following expression:

${\int }_{{\left({T}_{\epsilon /4}^{j}\right)}^{+}}{|\nabla {w}_{\epsilon }^{j}|}^{2}𝑑x+{\epsilon }^{-k}{\int }_{{G}_{\epsilon }^{j}}\sigma \left(u-{w}_{\epsilon }^{j}\right)\left(u-{w}_{\epsilon }^{j}\right)𝑑{x}^{\prime }={\epsilon }^{-k}{\int }_{{G}_{\epsilon }^{j}}\sigma \left(u-{w}_{\epsilon }^{j}\right)u𝑑{x}^{\prime }.$

By using the monotonicity of $\sigma \left(u\right)$, we derive the following inequality:

${\int }_{{\left({T}_{\epsilon /4}^{j}\right)}^{+}}{|\nabla {w}_{\epsilon }^{j}|}^{2}𝑑x+{\epsilon }^{-k}{\int }_{{G}_{\epsilon }^{j}}\sigma \left(u-{w}_{\epsilon }^{j}\right)\left(u-{w}_{\epsilon }^{j}\right)𝑑{x}^{\prime }\le {\epsilon }^{-k}{\int }_{{G}_{\epsilon }^{j}}|\sigma \left(u-{w}_{\epsilon }^{j}\right)||u|𝑑{x}^{\prime }.$

Due to the monotonicity of σ and (3.2), we have that

${\parallel \nabla {w}_{\epsilon }^{j}\parallel }_{{L}_{2}\left({\left({T}_{\epsilon /4}^{j}\right)}^{+}\right)}^{2}\le {\epsilon }^{-k}{\int }_{{G}_{\epsilon }^{j}}|\sigma \left(u-{w}_{\epsilon }^{j}\right)||u|𝑑{x}^{\prime }\le {k}_{2}|u||\sigma \left(u\right)|{\epsilon }^{-k}|{G}_{\epsilon }^{j}|.$

Hence, the following estimate is valid:

${\parallel \nabla {w}_{\epsilon }^{j}\parallel }_{{L}_{2}\left({\left({T}_{\epsilon /4}^{j}\right)}^{+}\right)}^{2}\le K|u||\sigma \left(u\right)|{\epsilon }^{n-1}.$

Adding over all cells, we get

${\parallel \nabla {W}_{\epsilon }\parallel }_{{L}_{2}\left(\mathrm{\Omega }\right)}^{2}\le K|u||\sigma \left(u\right)|.$

Friedrich’s inequality implies

${\parallel {w}_{\epsilon }^{j}\parallel }_{{L}_{2}\left({\left({T}_{\epsilon /4}^{j}\right)}^{+}\right)}^{2}\le K{\epsilon }^{2}{\parallel \nabla {w}_{\epsilon }^{j}\parallel }_{{L}_{2}\left({\left({T}_{\epsilon /4}^{j}\right)}^{+}\right)}^{2}.$

Summing over all cells and using the obtained estimations, we derive

${\parallel {W}_{\epsilon }\parallel }_{{L}_{2}\left(\mathrm{\Omega }\right)}^{2}\le K{\epsilon }^{2}{\parallel \nabla {W}_{\epsilon }\parallel }_{{L}_{2}\left(\mathrm{\Omega }\right)}^{2}\le K{\epsilon }^{2}|u||\sigma \left(u\right)|,$

which concludes the proof. ∎

Hence, as $\epsilon \to 0$ we have

${W}_{\epsilon }⇀0$${W}_{\epsilon }\to 0$

## 3.2 The comparison between ${w}_{\epsilon }^{j}$ and ${\stackrel{^}{w}}_{\epsilon }^{j}$

As an immediate consequence of Lemma 3.1 we have the following lemma.

#### Lemma 3.4.

For all $u\mathrm{\in }\mathrm{R}$ and a.e. $x\mathrm{\in }{\mathrm{\left(}{T}_{\epsilon \mathrm{/}\mathrm{4}}^{j}\mathrm{\right)}}^{\mathrm{+}}$, we have

$|{w}_{\epsilon }^{j}\left(u,x\right)|\le |{\stackrel{^}{w}}_{\epsilon }^{j}\left(u,x\right)|.$

The following lemma gives an estimate of the proximity of the functions ${w}_{\epsilon }^{j}$ and $\stackrel{^}{w}$.

#### Lemma 3.5.

For the introduced functions ${w}_{\epsilon }^{j}\mathit{}\mathrm{\left(}u\mathrm{,}x\mathrm{\right)}$ and $\stackrel{\mathrm{^}}{w}\mathit{}\mathrm{\left(}y\mathrm{,}u\mathrm{\right)}$ following estimations hold:

${\parallel \nabla \left({v}_{\epsilon }^{j}\left(u,x\right)\right)\parallel }_{{L}_{2}\left({\left({T}_{\epsilon /4}^{j}\right)}^{+}\right)}^{2}\le K{|u|}^{2}{\epsilon }^{n},$${\parallel {v}_{\epsilon }^{j}\left(u,x\right)\parallel }_{{L}_{2}\left({\left({T}_{\epsilon /4}^{j}\right)}^{+}\right)}^{2}\le K{|u|}^{2}{\epsilon }^{n+2},$

where ${v}_{\epsilon }^{j}\mathit{}\mathrm{\left(}u\mathrm{,}x\mathrm{\right)}\mathrm{=}{w}_{\epsilon }^{j}\mathit{}\mathrm{\left(}u\mathrm{,}x\mathrm{\right)}\mathrm{-}{\stackrel{\mathrm{^}}{w}}_{\epsilon }^{j}\mathit{}\mathrm{\left(}u\mathrm{,}x\mathrm{\right)}$.

#### Proof.

The function ${v}_{\epsilon }^{j}$ is a solution to the following boundary value problem:

$\left\{\begin{array}{cccc}\hfill \mathrm{\Delta }{v}_{\epsilon }^{j}& =0,\hfill & & \hfill x\in {\left({T}_{\epsilon /4}^{j}\right)}^{+}\setminus \overline{{G}_{\epsilon }^{j}},\\ \hfill {\partial }_{\nu }{v}_{\epsilon }^{j}& ={\epsilon }^{-k}\left(\sigma \left(u-{\stackrel{^}{w}}_{\epsilon }^{j}\right)-\sigma \left(u-{w}_{\epsilon }^{j}\right)\right),\hfill & & \hfill x\in {G}_{\epsilon }^{j},\\ \hfill {\partial }_{\nu }{v}_{\epsilon }^{j}& =0,\hfill & & \hfill x\in {\left({T}_{\epsilon /4}^{j}\right)}^{0}\setminus \overline{{G}_{\epsilon }^{j}},\\ \hfill {v}_{\epsilon }^{j}& =-{\stackrel{^}{w}}_{\epsilon }^{j},\hfill & & \hfill x\in {\left(\partial {T}_{\epsilon /4}^{j}\right)}^{+}.\end{array}$

Applying the comparison principle, we have $|{v}_{\epsilon }^{j}\left(u,x\right)|\le |{\stackrel{^}{w}}_{\epsilon }^{j}\left(u,x\right)|$ a.e. in ${\left({T}_{\epsilon /4}^{j}\right)}^{+}$. We take ${v}_{\epsilon }^{j}$ as a test function in the corresponding weak solution integral expression for the above problem:

${\parallel \nabla {v}_{\epsilon }^{j}\parallel }_{{L}_{2}\left({\left({T}_{\epsilon /4}^{j}\right)}^{+}\right)}^{2}+{\epsilon }^{-k}{\int }_{{G}_{\epsilon }^{j}}\left(\sigma \left(u-{w}_{\epsilon }^{j}\right)-\sigma \left(u-{\stackrel{^}{w}}_{\epsilon }^{j}\right)\right){v}_{\epsilon }^{j}𝑑{x}^{\prime }=-{\int }_{{\left(\partial {T}_{\epsilon /4}^{j}\right)}^{+}}{\partial }_{\nu }{v}_{\epsilon }^{j}{\stackrel{^}{w}}_{\epsilon }^{j}ds.$(3.5)

We transform the right-hand side expression of the inequality in the following way:

$-{\int }_{{\left(\partial {T}_{\epsilon /4}^{j}\right)}^{+}}{\partial }_{\nu }{v}_{\epsilon }^{j}{\stackrel{^}{w}}_{\epsilon }^{j}ds=-{\int }_{{\left({T}_{\epsilon /4}^{j}\right)}^{+}\setminus \overline{{\left({T}_{\epsilon /8}^{j}\right)}^{+}}}\nabla {v}_{\epsilon }^{j}\nabla {\stackrel{^}{w}}_{\epsilon }^{j}dx+{\int }_{{\left(\partial {T}_{\epsilon /8}^{j}\right)}^{+}}{\partial }_{\nu }{v}_{\epsilon }^{j}{\stackrel{^}{w}}_{\epsilon }^{j}ds.$

Let us estimate the obtained terms. We can extend ${v}_{\epsilon }^{j}$ by the symmetry ${v}_{\epsilon }^{j}\left(x,u\right)={v}_{j}^{\epsilon }\left(-x,u\right)$ for $x\in {\left({T}_{\epsilon /4}^{j}\right)}^{-}$, which is harmonic in ${T}_{\epsilon /4}^{j}\setminus \overline{{G}_{0}}$. By using some estimates on the derivatives of harmonic functions and the maximum principle, for $\stackrel{~}{x}\in \partial {T}_{\epsilon /8}^{j}$, we get

$|{\partial }_{{x}_{i}}{v}_{\epsilon }^{j}\left(\stackrel{~}{x}\right)|\le \frac{1}{|{T}_{\epsilon /16}\left(\stackrel{~}{x}\right)|}|{\int }_{{T}_{\epsilon /16}\left(\stackrel{~}{x}\right)}\frac{\partial {v}_{\epsilon }^{j}}{\partial {x}_{i}}𝑑x|=\frac{K}{{\epsilon }^{n}}|{\int }_{\partial {T}_{\epsilon /16}\left(\stackrel{~}{x}\right)}{v}_{\epsilon }^{j}{\nu }_{i}𝑑x|\le K|u|$

since, on $\partial {T}_{\epsilon /16}\left(\stackrel{~}{x}\right)$, from the maximum principle we have

$|{v}_{\epsilon }^{j}|\le |{\stackrel{^}{w}}_{\epsilon }^{j}|\le \frac{K|u|}{{|\left(x-{P}_{\epsilon }^{j}\right)/{a}_{\epsilon }|}^{n-2}}=K|u|{\epsilon }^{2-n}{a}_{\epsilon }^{n-2}=K|u|{\epsilon }^{2-n}{\epsilon }^{n-1}=K|u|\epsilon .$

This last estimate implies that $|\nabla {v}_{\epsilon }^{j}\left(\stackrel{~}{x}\right)|\le K$ for $\stackrel{~}{x}\in \partial {T}_{\epsilon /8}^{j}$. Therefore, we get an estimate of the second term:

$|{\int }_{{\left(\partial {T}_{\epsilon /8}^{j}\right)}^{+}}{\partial }_{\nu }{v}_{\epsilon }^{j}{\stackrel{^}{w}}_{\epsilon }^{j}ds|\le K|u|\underset{\partial {T}_{\epsilon /8}^{j}}{\mathrm{max}}|{\stackrel{^}{w}}_{\epsilon }^{j}||\partial {T}_{\epsilon /8}^{j}|\le K{|u|}^{2}{\epsilon }^{n}.$

Then we estimate the first term:

$|{\int }_{{\left({T}_{\epsilon /4}^{j}\right)}^{+}\setminus \overline{{\left({T}_{\epsilon /8}^{j}\right)}^{+}}}\nabla {v}_{\epsilon }^{j}\nabla \stackrel{^}{w}dx|\le K{|u|}^{2}\epsilon {a}_{\epsilon }^{n-2}=K{|u|}^{2}{\epsilon }^{n}.$

Combining the obtained estimates and using the properties of the function σ, we get

${\parallel \nabla {v}_{\epsilon }^{j}\parallel }_{{L}_{2}\left({\left({T}_{\epsilon /4}^{j}\right)}^{+}\right)}^{2}\le K{|u|}^{2}{\epsilon }^{n}.$

Friedrich’s inequality implies that

${\parallel {v}_{\epsilon }^{j}\parallel }_{{L}_{2}\left({\left({T}_{\epsilon /4}^{j}\right)}^{+}\right)}^{2}\le K{|u|}^{2}{\epsilon }^{n+2}.$

This concludes the proof. ∎

#### Lemma 3.6.

We have that1

$\frac{1}{|{G}_{\epsilon }^{j}|}{\int }_{{G}_{\epsilon }^{j}}{|{v}_{\epsilon }^{j}\left(u,{x}^{\prime }\right)|}^{2}𝑑{x}^{\prime }\le \epsilon .$

#### Proof.

From (3.5), using (1.4), we deduce that

${k}_{1}{a}_{\epsilon }^{-1}{\int }_{{G}_{\epsilon }^{j}}{|{v}_{\epsilon }^{j}|}^{2}𝑑{x}^{\prime }\le K{|u|}^{2}{\epsilon }^{n}.$

Thus,

$\frac{1}{|{G}_{\epsilon }^{j}|}{\int }_{{G}_{\epsilon }^{j}}{|{v}_{\epsilon }^{j}\left(u,{x}^{\prime }\right)|}^{2}𝑑{x}^{\prime }\le K{a}_{\epsilon }^{1-n}{\int }_{{G}_{\epsilon }^{j}}{|{v}_{\epsilon }^{j}\left(u,{x}^{\prime }\right)|}^{2}𝑑{x}^{\prime }$$\le K{a}_{\epsilon }^{2-n}{a}_{\epsilon }^{-1}{\int }_{{G}_{\epsilon }^{j}}{|{v}_{\epsilon }^{j}\left(u,{x}^{\prime }\right)|}^{2}𝑑{x}^{\prime }$$\le K{|u|}^{2}{a}_{\epsilon }^{2-n}{\epsilon }^{n}$$=K{|u|}^{2}\epsilon .$

This completes the proof. ∎

## 3.3 Convergence to the “strange term”

The following result plays a crucial role in the proof of Theorem 1.3.

#### Lemma 3.7.

Let H be the function defined by (1.10), and let φ be an arbitrary function in ${C}^{\mathrm{\infty }}\mathit{}\mathrm{\left(}\mathrm{\Omega }\mathrm{\right)}$. Then for any test function $h\mathrm{\in }{H}^{\mathrm{1}}\mathit{}\mathrm{\left(}\mathrm{\Omega }\mathrm{,}{\mathrm{\Gamma }}_{\mathrm{2}}\mathrm{\right)}$ we have

$|\sum _{j\in {\mathrm{Υ}}_{\epsilon }}{\int }_{{\left(\partial {T}_{\epsilon /4}^{j}\right)}^{+}}{\partial }_{{\nu }_{x}}{\stackrel{^}{w}}_{\epsilon }^{j}\left(\phi \left({\stackrel{~}{P}}_{\epsilon }^{j}\right),s\right)h\left(s\right)𝑑s+{C}_{0}^{n-2}{\int }_{{\mathrm{\Gamma }}_{1}}H\left(\phi \left({x}^{\prime }\right)\right)h\left({x}^{\prime }\right)𝑑{x}^{\prime }|\to 0$(3.6)

as $\epsilon \mathrm{\to }\mathrm{0}$, where ${\stackrel{\mathrm{~}}{P}}_{\epsilon }^{j}\mathrm{\in }{G}_{\epsilon }^{j}$ and $\nu \mathrm{=}\mathrm{\left(}\mathrm{-}\mathrm{1}\mathrm{,}\mathrm{0}\mathrm{,}\mathrm{\dots }\mathrm{,}\mathrm{0}\mathrm{\right)}$ is the unit outward normal to ${\mathrm{\left(}\mathrm{\partial }\mathit{}{T}_{\epsilon \mathrm{/}\mathrm{4}}^{j}\mathrm{\right)}}^{\mathrm{+}}$.

#### Proof.

Consider the cylinder

${Q}_{\epsilon }^{j}=\left\{x\in {ℝ}^{n}:0<{x}_{1}<\epsilon ,-\frac{\epsilon }{2}<{x}_{i}-{\left({\stackrel{~}{P}}_{\epsilon }^{j}\right)}_{i}<\frac{\epsilon }{2},i=2,\mathrm{\dots },n\right\}.$

We define the auxiliary function ${\theta }_{\epsilon }^{j}$ as the unique solution to the following boundary value problem:

(3.7)

The constant ${\mu }_{\epsilon }^{j}$ is defined from the solvability condition for problem (3.7):

${\mu }_{\epsilon }^{j}=-{C}_{0}^{n-2}H\left(\phi \left({\stackrel{~}{P}}_{\epsilon }^{j}\right)\right).$

We take ${\theta }_{\epsilon }^{j}$ as a test function in the integral identity associated to problem (3.7) and obtain

${\int }_{{Y}_{\epsilon }^{j}}{|\nabla {\theta }_{\epsilon }^{j}|}^{2}𝑑x=-{\mu }_{\epsilon }^{j}{\int }_{{\gamma }_{\epsilon }^{j}}{\theta }_{\epsilon }^{j}𝑑{x}^{\prime }+{\int }_{{S}_{\epsilon /4}^{j,+}}{\partial }_{\nu }\stackrel{^}{w}{\theta }_{\epsilon }^{j}ds.$

Using the embedding theorems, we obtain the estimate

${\int }_{{\gamma }_{\epsilon }^{j}}|{\theta }_{\epsilon }^{j}|𝑑{x}^{\prime }\le K{\epsilon }^{\left(n-1\right)/2}{\parallel {\theta }_{\epsilon }^{j}\parallel }_{{L}_{2}\left({\gamma }_{\epsilon }^{j}\right)}\le K{\epsilon }^{n/2}{\parallel \nabla {\theta }_{\epsilon }^{j}\parallel }_{{L}_{2}\left({Y}_{\epsilon }^{j}\right)}.$(3.8)

Taking into account that

$\underset{{\left(\partial {T}_{\epsilon /4}^{j}\right)}^{+}}{\mathrm{max}}|{\partial }_{\nu }{\stackrel{^}{w}}_{\epsilon }^{j}\left(\phi \left({\stackrel{~}{P}}_{\epsilon }^{j}\right),x\right)|\le K\frac{{a}_{\epsilon }^{-1}}{{|\frac{x-{P}_{\epsilon }^{j}}{{a}_{\epsilon }}|}^{n-1}}=K{a}_{\epsilon }^{n-2}{\epsilon }^{1-n}\le K$

and using some estimates proved in [31], we derive

${\int }_{{\left(\partial {T}_{\epsilon /4}^{j}\right)}^{+}}|\left({\partial }_{\nu }{\stackrel{^}{w}}_{\epsilon }^{j}\left(\phi \left({\stackrel{~}{P}}_{\epsilon }^{j}\right),s\right)\right){\theta }_{\epsilon }^{j}|𝑑s\le K{\int }_{{\left(\partial {T}_{\epsilon /4}^{j}\right)}^{+}}|{\theta }_{\epsilon }^{j}|𝑑s$$\le K{\epsilon }^{\left(n-1\right)/2}{\parallel {\theta }_{\epsilon }^{j}\parallel }_{{L}_{2}\left({\left(\partial {T}_{\epsilon /4}^{j}\right)}^{+}\right)}$$\le K{\epsilon }^{\left(n-1\right)/2}\left\{{\epsilon }^{-1/2}{\parallel {\theta }_{\epsilon }^{j}\parallel }_{{L}_{2}\left({Y}_{\epsilon }^{j}\right)}+\sqrt{\epsilon }{\parallel \nabla {\theta }_{\epsilon }^{j}\parallel }_{{L}_{2}\left({Y}_{\epsilon }^{j}\right)}\right\}$$\le K{\epsilon }^{n/2}{\parallel \nabla {\theta }_{\epsilon }^{j}\parallel }_{{L}_{2}\left({Y}_{\epsilon }^{j}\right)}.$(3.9)

From the above estimates (3.8) and (3.9) we get

${\parallel \nabla {\theta }_{\epsilon }^{j}\parallel }_{{L}_{2}\left({Y}_{\epsilon }^{j}\right)}^{2}\le K{\epsilon }^{n}.$(3.10)

From estimate (3.10) it follows that

$\sum _{j\in {\mathrm{Υ}}_{\epsilon }}{\parallel {\theta }_{\epsilon }^{j}\parallel }_{{L}_{2}\left({Y}_{\epsilon }^{j}\right)}^{2}\le K\epsilon .$

Adding all the above integral identities for problem (3.7), we derive that for $h\in {H}^{1}\left(\mathrm{\Omega }\right)$ the following inequality holds:

$|\sum _{j\in {\mathrm{Υ}}_{\epsilon }}{\int }_{{\left(\partial {T}_{\epsilon /4}^{j}\right)}^{+}}{\partial }_{\nu }{\stackrel{^}{w}}_{\epsilon }^{j}\left(\phi \left({\stackrel{~}{P}}_{\epsilon }^{j}\right),s\right)h𝑑s+{C}_{0}^{n-2}{\int }_{{\mathrm{\Gamma }}_{1}}H\left(\phi \left({x}^{\prime }\right)\right)h𝑑{x}^{\prime }|$$\le |\sum _{j\in {\mathrm{Υ}}_{\epsilon }}{\int }_{{\stackrel{^}{Y}}_{\epsilon }^{j}}\nabla {\theta }_{\epsilon }^{j}\nabla hdx|+|\sum _{j\in {\mathrm{Υ}}_{\epsilon }}{\mu }_{\epsilon }^{j}{\int }_{{\gamma }_{\epsilon }^{j}}h𝑑{x}^{\prime }+{C}_{0}^{n-2}{\int }_{{\mathrm{\Gamma }}_{1}}H\left(\phi \left({x}^{\prime }\right)\right)h𝑑{x}^{\prime }|.$(3.11)

Let us estimate the terms on the right-hand side of inequality (3.11). By using estimate (3.10), we get the following inequality for the first term:

$|\sum _{j\in {\mathrm{Υ}}_{\epsilon }}{\int }_{{Y}_{\epsilon }^{j}}\nabla {\theta }_{\epsilon }^{j}\nabla hdx|\le K\sqrt{\epsilon }{\parallel h\parallel }_{{H}^{1}\left(\mathrm{\Omega }\right).}$(3.12)

Set

${\stackrel{^}{\gamma }}_{\epsilon }^{j}={\left({Q}_{\epsilon }^{j}\right)}^{0},{\mathrm{\Gamma }}_{1}^{\epsilon }=\bigcup _{j\in {\mathrm{Υ}}_{\epsilon }}{\stackrel{^}{\gamma }}_{\epsilon }^{j}.$

Then we have

$|\sum _{j\in {\mathrm{Υ}}_{\epsilon }}{\mu }_{\epsilon }^{j}{\int }_{{\gamma }_{\epsilon }^{j}}h𝑑{x}^{\prime }+{C}_{0}^{n-2}{\int }_{{\mathrm{\Gamma }}_{1}}H\left(\phi \left({x}^{\prime }\right)\right)h𝑑{x}^{\prime }|$$\le |{C}_{0}^{n-2}\sum _{j\in {\mathrm{Υ}}_{\epsilon }}\left({\int }_{{\gamma }_{\epsilon }^{j}}H\left(\phi \left({\stackrel{~}{P}}_{\epsilon }^{j}\right)\right)h𝑑{x}^{\prime }-{\int }_{{\stackrel{^}{\gamma }}_{\epsilon }^{j}}H\left(\phi \left({x}^{\prime }\right)\right)h𝑑{x}^{\prime }\right)|$$\le {C}_{0}^{n-2}|\sum _{j\in {\mathrm{Υ}}_{\epsilon }}{\int }_{{\stackrel{^}{\gamma }}_{\epsilon }^{j}}\left(H\left(\phi \left({\stackrel{~}{P}}_{\epsilon }^{j}\right)\right)-H\left(\phi \left({x}^{\prime }\right)\right)\right)h𝑑{x}^{\prime }|+{C}_{0}^{n-2}|\sum _{j\in {\mathrm{Υ}}_{\epsilon }}\left({\int }_{{\gamma }_{\epsilon }^{j}}H\left(\phi \left({\stackrel{~}{P}}_{\epsilon }^{j}\right)\right)h𝑑{x}^{\prime }-{\int }_{{\stackrel{^}{\gamma }}_{\epsilon }^{j}}H\left(\phi \left({\stackrel{~}{P}}_{\epsilon }^{j}\right)\right)h𝑑{x}^{\prime }\right)|.$

Let us estimate the terms on the right-hand side of the obtained inequality. For the first term we have

$|\sum _{j\in {\mathrm{Υ}}_{\epsilon }}{\int }_{{\stackrel{^}{\gamma }}_{\epsilon }^{j}}\left(H\left(\phi \left({\stackrel{~}{P}}_{\epsilon }^{j}\right)\right)-H\left(\phi \left({x}^{\prime }\right)\right)\right)h𝑑{x}^{\prime }|\le K{\parallel h\parallel }_{{L}_{2}\left({\mathrm{\Gamma }}_{1}^{\epsilon }\right)}\underset{\begin{array}{c}j\in {\mathrm{Υ}}_{\epsilon }\\ {x}^{\prime }\in {\stackrel{^}{\gamma }}_{\epsilon }^{j}\end{array}}{\mathrm{max}}|H\left(\phi \left({\stackrel{~}{P}}_{\epsilon }^{j}\right)\right)-H\left(\phi \left({x}^{\prime }\right)\right)|$$\le K{\parallel h\parallel }_{{L}_{2}\left({\mathrm{\Gamma }}_{1}^{\epsilon }\right)}\underset{\begin{array}{c}j\in {\mathrm{Υ}}_{\epsilon }\\ {x}^{\prime }\in {\stackrel{^}{\gamma }}_{\epsilon }^{j}\end{array}}{\mathrm{max}}|\phi \left({\stackrel{~}{P}}_{\epsilon }^{j}\right)-\phi \left({x}^{\prime }\right)|$$\le K\epsilon {\parallel h\parallel }_{{H}^{1}\left(\mathrm{\Omega }\right)}.$

By using the continuity in ${L}^{2}$-norm on the hyperplanes of the functions from ${H}^{1}\left(\mathrm{\Omega }\right)$, we estimate the second term:

$|\sum _{j\in {\mathrm{Υ}}_{\epsilon }}\left({\int }_{{\gamma }_{\epsilon }^{j}}H\left(\phi \left({\stackrel{~}{P}}_{\epsilon }^{j}\right)\right)h𝑑{x}^{\prime }-{\int }_{{\stackrel{^}{\gamma }}_{\epsilon }^{j}}H\left(\phi \left({\stackrel{~}{P}}_{\epsilon }^{j}\right)\right)h𝑑{x}^{\prime }\right)|\le K\sqrt{\epsilon }{\parallel h\parallel }_{{H}^{1}\left(\mathrm{\Omega }\right)}.$

Hence we have

$|\sum _{j\in {\mathrm{Υ}}_{\epsilon }}{\mu }_{\epsilon }^{j}{\int }_{{\gamma }_{\epsilon }^{j}}h𝑑{x}^{\prime }+{C}_{0}^{n-2}{\int }_{{\mathrm{\Gamma }}_{1}}H\left(\phi \left({x}^{\prime }\right)\right)h𝑑{x}^{\prime }|\le K\sqrt{\epsilon }{\parallel h\parallel }_{{H}^{1}\left(\mathrm{\Omega }\right)}.$(3.13)

Combining estimates (3.12) and (3.13), we conclude the proof. ∎

## 4.1 The auxiliary function ${\kappa }_{\epsilon }^{j}$

We introduce the function ${\kappa }_{\epsilon }^{j}$ as the unique solution of the following problem:

$\left\{\begin{array}{cccc}\hfill \mathrm{\Delta }{\kappa }_{\epsilon }^{j}& =0,\hfill & & \hfill x\in {\left({T}_{\epsilon /4}^{j}\right)}^{+}\setminus \overline{{G}_{\epsilon }^{j}},\\ \hfill {\kappa }_{\epsilon }^{j}& =1,\hfill & & \hfill x\in {G}_{\epsilon }^{j},\\ \hfill {\partial }_{\nu }{\kappa }_{\epsilon }^{j}& =0,\hfill & & \hfill x\in {\left(\partial {T}_{\epsilon /4}^{j}\right)}^{0}\setminus \overline{{G}_{\epsilon }^{j}},\\ \hfill {\kappa }_{\epsilon }^{j}& =0,\hfill & & \hfill x\in \partial {T}_{\epsilon /4}^{j},\end{array}$

and then we define

${\kappa }_{\epsilon }=\left\{\begin{array}{cc}{\kappa }_{\epsilon }^{j}\left(x\right),\hfill & x\in {\left({T}_{\epsilon /4}^{j}\right)}^{+},j\in {\mathrm{Υ}}_{\epsilon },\hfill \\ 0,\hfill & x\in {ℝ}^{n}\setminus \bigcup _{j\in {\mathrm{Υ}}_{\epsilon }}\overline{{\left({T}_{\epsilon /4}^{j}\right)}^{+}}.\hfill \end{array}$

It is easy to see that ${\kappa }_{\epsilon }\in {H}^{1}\left(\mathrm{\Omega }\right)$ and

(4.1)

## 4.2 Estimate of the difference between ${\kappa }_{\epsilon }^{j}$ and ${\stackrel{^}{\kappa }}_{\epsilon }^{j}$

#### Lemma 4.1.

Let ${\kappa }_{\epsilon }^{j}$ and $\stackrel{\mathrm{^}}{\kappa }$ be as above. Then

$\sum _{j\in {\mathrm{Υ}}_{\epsilon }}{\parallel {\kappa }_{\epsilon }^{j}-{\stackrel{^}{\kappa }}_{\epsilon }^{j}\parallel }_{{H}^{1}\left({\left({T}_{\epsilon /4}^{j}\right)}^{+}\right)}^{2}\le K\epsilon .$

#### Proof.

The function ${v}_{\epsilon }^{j}={\kappa }_{\epsilon }^{j}-\stackrel{^}{\kappa }$ satisfies the following problem:

$\left\{\begin{array}{cccc}\hfill \mathrm{\Delta }{v}_{\epsilon }^{j}& =0,\hfill & & \hfill x\in {\left({T}_{\epsilon /4}^{j}\right)}^{+},\\ \hfill {v}_{\epsilon }^{j}& =0,\hfill & & \hfill x\in {G}_{\epsilon }^{j},\\ \hfill {\partial }_{\nu }{v}_{\epsilon }^{j}& =0,\hfill & & \hfill x\in {\left({T}_{\epsilon }^{j}\right)}^{0}\setminus \overline{{G}_{\epsilon }^{j}},\\ \hfill {v}_{\epsilon }^{j}& =-{\stackrel{^}{\kappa }}_{\epsilon }^{j},\hfill & & \hfill x\in {\left(\partial {T}_{\epsilon /4}^{j}\right)}^{+}.\end{array}$

We take ${v}_{\epsilon }^{j}$ as a test function in an integral identity for the above problem:

${\parallel \nabla {v}_{\epsilon }^{j}\parallel }_{{L}_{2}\left({\left({T}_{\epsilon /4}^{j}\right)}^{+}\right)}=-{\int }_{\partial {\left({T}_{\epsilon /4}^{j}\right)}^{+}}{\partial }_{\nu }{v}_{\epsilon }^{j}\stackrel{^}{\kappa }ds.$

We transform the right-hand side expression of the identity in the following way:

$-{\int }_{\partial {\left({T}_{\epsilon /4}^{j}\right)}^{+}}{\partial }_{\nu }{v}_{\epsilon }^{j}\stackrel{^}{\kappa }ds=-{\int }_{{\left({T}_{\epsilon /4}^{j}\right)}^{+}\setminus \overline{{T}_{\epsilon /8}^{j}}}\nabla {v}_{\epsilon }^{j}\nabla \stackrel{^}{\kappa }dx+{\int }_{\partial {T}_{\epsilon /8}^{j}}{\partial }_{\nu }{v}_{\epsilon }^{j}\stackrel{^}{\kappa }ds.$

For an arbitrary point ${x}_{0}\in \partial {T}_{\epsilon /8}^{j}$ we have

$|{\partial }_{{x}_{i}}{v}_{\epsilon }^{j}\left({x}_{0}\right)|\le \frac{1}{|{T}_{\epsilon /16}\left({x}_{0}\right)|}|{\int }_{{T}_{\epsilon /16}\left({x}_{0}\right)}\frac{\partial {v}_{\epsilon }^{j}}{\partial {x}_{i}}𝑑x|=\frac{K}{{\epsilon }^{n}}|{\int }_{\partial {T}_{\epsilon /16}\left({x}_{0}\right)}{v}_{\epsilon }^{j}{\nu }_{i}𝑑x|\le K.$

The last estimate implies that $|\nabla {v}_{\epsilon }^{j}\left(\stackrel{~}{x}\right)|\le K$ for $\stackrel{~}{x}\in \partial {T}_{\epsilon /8}^{j}$. Therefore, we can estimate the second term in the following way:

$|{\int }_{\partial {T}_{\epsilon /8}^{j}}{\partial }_{\nu }{v}_{\epsilon }^{j}\stackrel{^}{\kappa }ds|\le K\underset{\partial {T}_{\epsilon /8}^{j}}{\mathrm{max}}|\stackrel{^}{\kappa }||\partial {T}_{\epsilon /8}^{j}|\le K{\epsilon }^{n}.$

Then we estimate the first term:

$|{\int }_{{\left({T}_{\epsilon /4}^{j}\right)}^{+}\setminus \overline{{T}_{\epsilon /8}^{j}}}\nabla {v}_{\epsilon }^{j}\nabla \stackrel{^}{\kappa }dx|\le K\epsilon {a}_{\epsilon }^{n-2}=K{\epsilon }^{n}.$

By combining acquired estimations, we derive

${\parallel \nabla {v}_{\epsilon }^{j}\parallel }_{{L}_{2}\left({\left({T}_{\epsilon /4}^{j}\right)}^{+}\right)}^{2}\le K{\epsilon }^{n}$

Friedrich’s inequality implies

${\parallel {v}_{\epsilon }^{j}\parallel }_{{L}_{2}\left({\left({T}_{\epsilon /4}^{j}\right)}^{+}\right)}^{2}\le K{\epsilon }^{n+2}.$

This concludes the proof. ∎

## 4.3 Convergence to the “strange term”

#### Lemma 4.2.

Let ${\lambda }_{{G}_{\mathrm{0}}}$ be given by (1.11). Then for all functions $h\mathrm{\in }{H}^{\mathrm{1}}\mathit{}\mathrm{\left(}\mathrm{\Omega }\mathrm{,}{\mathrm{\Gamma }}_{\mathrm{2}}\mathrm{\right)}$ we have

$|\sum _{j\in {\mathrm{Υ}}_{\epsilon }}{\int }_{{\left(\partial {T}_{\epsilon /4}^{j}\right)}^{+}\cap \mathrm{\Omega }}{\partial }_{{\nu }_{x}}\stackrel{^}{\kappa }{}_{\epsilon }{}^{j}\left(s\right)h\left(s\right)ds+{C}_{0}^{n-2}{\lambda }_{{G}_{0}}{\int }_{{\mathrm{\Gamma }}_{1}}hd{x}^{\prime }|\to 0$

as $\epsilon \mathrm{\to }\mathrm{0}$, where ν is the unit outward normal to $\mathrm{\partial }\mathit{}{T}_{\epsilon \mathrm{/}\mathrm{4}}^{j}\mathrm{\cap }\mathrm{\Omega }$.

#### Proof.

By analogy with the proof of Lemma 3.7, we define the function ${\theta }_{\epsilon }^{j}$ as a solution to the following boundary value problem:

$\left\{\begin{array}{cccc}\hfill \mathrm{\Delta }{\theta }_{\epsilon }^{j}& =0,\hfill & & \hfill {Y}_{\epsilon }^{j},\\ \hfill {\partial }_{\nu }{\theta }_{\epsilon }^{j}& =-{\partial }_{\nu }{\stackrel{^}{\kappa }}_{\epsilon }^{j},\hfill & & \hfill x\in {\left(\partial {T}_{\epsilon /4}^{j}\right)}^{+},\\ \hfill \frac{\partial {\theta }_{\epsilon }^{j}}{\partial {x}_{1}}& =\mu ,\hfill & & \hfill x\in {\gamma }_{\epsilon }^{j},\\ \hfill \frac{\partial {\theta }_{\epsilon }^{j}}{\partial {x}_{1}}& =0,\hfill & & \hfill \partial {Q}_{\epsilon }^{j}\setminus \left({\left(\partial {T}_{\epsilon /4}^{j}\right)}^{+}\cup {\gamma }_{\epsilon }^{j}\right),\\ \hfill {〈{\theta }_{\epsilon }^{j}〉}_{{Y}_{\epsilon }^{j}}& =0.\hfill \end{array}$(4.2)

The constant μ is defined from the solvability condition for problem (4.2):

$\mu =-{C}_{0}^{n-2}\lambda .$

By using the same technique as in the proof of the Lemma 3.6, we have

${\parallel \nabla {\theta }_{\epsilon }^{j}\parallel }_{{L}_{2}\left({Y}_{\epsilon }^{j}\right)}^{2}\le K{\epsilon }^{n},\sum _{j\in {\mathrm{Υ}}_{\epsilon }}{\parallel {\theta }_{\epsilon }^{j}\parallel }_{{L}_{2}\left({Y}_{\epsilon }^{j}\right)}^{2}\le K\epsilon .$

Summing up all integral identities for problem (4.2), we derive that for the arbitrary function from ${H}^{1}\left(\mathrm{\Omega }\right)$ the following inequality is true:

$|\sum _{j\in {\mathrm{Υ}}_{\epsilon }}{\int }_{{\left(\partial {T}_{\epsilon /4}^{j}\right)}^{+}}\left({\partial }_{\nu }{\stackrel{^}{\kappa }}_{\epsilon }^{j}\right)h𝑑s+{C}_{0}^{n-2}\lambda {\int }_{{\mathrm{\Gamma }}_{1}}h𝑑{x}^{\prime }|$$\le |\sum _{j\in {\mathrm{Υ}}_{\epsilon }}{\int }_{{\stackrel{^}{Y}}_{\epsilon }^{j}}\nabla {\theta }_{\epsilon }^{j}\nabla hdx|+|\sum _{j\in {\mathrm{Υ}}_{\epsilon }}\mu {\int }_{{\gamma }_{\epsilon }^{j}}h𝑑{x}^{\prime }+{C}_{0}^{n-2}\lambda {\int }_{{\mathrm{\Gamma }}_{1}}h𝑑{x}^{\prime }|.$(4.3)

Let us estimate the terms on the right-hand side of inequality (4.3). By using estimate (3.10), we get following estimation of the first term:

$|\sum _{j\in {\mathrm{Υ}}_{\epsilon }}{\int }_{{Y}_{\epsilon }^{j}}\nabla {\theta }_{\epsilon }^{j}\nabla hdx|\le K\sqrt{\epsilon }{\parallel h\parallel }_{{H}^{1}\left(\mathrm{\Omega }\right)}.$(4.4)

Then we have

$|\sum _{j\in {\mathrm{Υ}}_{\epsilon }}\mu {\int }_{{\gamma }_{\epsilon }^{j}}h𝑑{x}^{\prime }+{C}_{0}^{n-2}\lambda {\int }_{{\mathrm{\Gamma }}_{1}}h𝑑{x}^{\prime }|\le |{C}_{0}^{n-2}\lambda \sum _{j\in {\mathrm{Υ}}_{\epsilon }}\left({\int }_{{\gamma }_{\epsilon }^{j}}h𝑑{x}^{\prime }-{\int }_{{\stackrel{^}{\gamma }}_{\epsilon }^{j}}h𝑑{x}^{\prime }\right)|.$

By using the continuity in ${L}_{2}$-norm on the hyperplanes of the functions from ${H}^{1}\left(\mathrm{\Omega }\right)$, we estimate the second term:

$|\sum _{j\in {\mathrm{Υ}}_{\epsilon }}\left({\int }_{{\gamma }_{\epsilon }^{j}}h𝑑{x}^{\prime }-{\int }_{{\stackrel{^}{\gamma }}_{\epsilon }^{j}}h𝑑{x}^{\prime }\right)|\le K\sqrt{\epsilon }{\parallel h\parallel }_{{H}^{1}\left(\mathrm{\Omega }\right)}.$

Hence we have

$|\sum _{j\in {\mathrm{Υ}}_{\epsilon }}{\mu }_{\epsilon }{\int }_{{\gamma }_{\epsilon }^{j}}h𝑑{x}^{\prime }+{C}_{0}^{n-2}{\int }_{{\mathrm{\Gamma }}_{1}}h𝑑{x}^{\prime }|\le K\sqrt{\epsilon }{\parallel h\parallel }_{{H}^{1}\left(\mathrm{\Omega }\right)}.$(4.5)

Combining estimations (4.4) and (4.5), we conclude the proof. ∎

## 5 Proof of Theorem 1.3

For different reasons it is convenient to introduce some new notation: instead of using the decomposition ${u}_{0}={u}_{0,+}-{u}_{0,-}$ mentioned in Section 1 (see the statement of Theorem 1.3), we shall use the alternative decomposition ${u}_{0}={u}_{0}^{+}+{u}_{0}^{-}$ (i.e. ${u}_{0}^{+}={u}_{0,+}$, but ${u}_{0}^{-}=-{u}_{0,-}$).

#### Proof.

Let $\phi \left(x\right)$ be an arbitrary function from ${C}_{0}^{\mathrm{\infty }}\left(\mathrm{\Omega }\right)$. We choose a point ${\stackrel{^}{P}}_{\epsilon }^{j}\in \overline{{G}_{\epsilon }^{j}}$ such that

$\underset{x\in \overline{{G}_{\epsilon }^{j}}}{\mathrm{min}}{\phi }^{+}\left(x\right)={\phi }^{+}\left({\stackrel{^}{P}}_{\epsilon }^{j}\right),$

where ${\phi }^{+}=\mathrm{max}\left\{0,\phi \left(x\right)\right\}$ and ${\phi }^{-}\left(x\right)=\phi \left(x\right)-{\phi }^{+}\left(x\right)$. Define the function

${\mathcal{𝒲}}_{\epsilon }\left({\phi }^{+},x\right)=\left\{\begin{array}{cc}{w}_{\epsilon }^{j}\left({\phi }^{+}\left({\stackrel{^}{P}}_{\epsilon }^{j}\right),x\right),\hfill & x\in {\left({T}_{\epsilon /4}^{j}\right)}^{+},j\in {\mathrm{Υ}}_{\epsilon }\hfill \\ 0,\hfill & x\in {ℝ}^{n}\setminus \bigcup _{j\in {\mathrm{Υ}}_{\epsilon }}\overline{{\left({T}_{\epsilon /4}^{j}\right)}^{+}}.\hfill \end{array}$

From estimates (3.4) we conclude that

${\mathcal{𝒲}}_{\epsilon }\left({\phi }^{+},x\right)⇀0$(5.1)

in ${H}^{1}\left(\mathrm{\Omega },{\mathrm{\Gamma }}_{2}\right)$ as $\epsilon \to 0$. We set

$v={\phi }^{+}-{\mathcal{𝒲}}_{\epsilon }\left({\phi }^{+},x\right)+\left(1-{\kappa }_{\epsilon }\right){\phi }^{-}$

as a test function in the integral inequality (1.8), where φ is an arbitrary function from ${C}_{0}^{\mathrm{\infty }}\left(\mathrm{\Omega }\right)$. Notice that $v\in {K}_{\epsilon }$. Indeed, according to Lemma 3.1 and using that ${\kappa }_{\epsilon }\equiv 1$ in ${G}_{\epsilon }$, we have for all $x\in {G}_{\epsilon }^{j}$ that

$v={\phi }^{+}-{\mathcal{𝒲}}_{\epsilon }\left({\phi }^{+},x\right)+\left(1-{\kappa }_{\epsilon }\right){\phi }^{-}\ge {\phi }^{+}\left({\stackrel{^}{P}}_{\epsilon }^{j}\right)-{w}_{\epsilon }^{j}\left({\phi }^{+}\left({\stackrel{^}{P}}_{\epsilon }^{j}\right),x\right)\ge 0.$

Hence we get

${\int }_{\mathrm{\Omega }}\left\{\nabla \left({\phi }^{+}-{\mathcal{𝒲}}_{\epsilon }\left({\phi }^{+},x\right)+\left(1-{\kappa }_{\epsilon }\right){\phi }^{-}\right)\nabla \left({\phi }^{+}-{\mathcal{𝒲}}_{\epsilon }\left({\phi }^{+},x\right)+\left(1-{\kappa }_{\epsilon }\right){\phi }^{-}-{u}_{\epsilon }\right)\right\}𝑑x$$+{\epsilon }^{-k}\sum _{j\in {\mathrm{Υ}}_{\epsilon }}{\int }_{{G}_{\epsilon }^{j}}\sigma \left({\phi }^{+}-{w}_{\epsilon }^{j}\left({\phi }^{+}\left({\stackrel{^}{P}}_{\epsilon }^{j}\right),x\right)\right)\left({\phi }^{+}-{w}_{\epsilon }^{j}\left({\phi }^{+}\left({\stackrel{^}{P}}_{\epsilon }^{j}\right),x\right)-{u}_{\epsilon }\right)𝑑{x}^{\prime }$$\ge {\int }_{\mathrm{\Omega }}f\left({\phi }^{+}-{\mathcal{𝒲}}_{\epsilon }\left({\phi }^{+},x\right)+\left(1-{\kappa }_{\epsilon }\right){\phi }^{-}-{u}_{\epsilon }\right)𝑑x.$

Considering the first integral of the right-hand side of the inequality above, we have

${\int }_{\mathrm{\Omega }}\nabla \left(\phi -{\mathcal{𝒲}}_{\epsilon }\left({\phi }^{+},x\right)-{\kappa }_{\epsilon }{\phi }^{-}\right)\nabla \left(\phi -{\mathcal{𝒲}}_{\epsilon }\left({\phi }^{+},x\right)-{\kappa }_{\epsilon }{\phi }^{-}-{u}_{\epsilon }\right)dx$$={\int }_{\mathrm{\Omega }}\nabla \phi \nabla \left(\phi -{\mathcal{𝒲}}_{\epsilon }\left({\phi }^{+}\left({\stackrel{^}{P}}_{\epsilon }^{j}\right),x\right)-{\kappa }_{\epsilon }{\phi }^{-}-{u}_{\epsilon }\right)dx-{\int }_{\mathrm{\Omega }}\nabla {\mathcal{𝒲}}_{\epsilon }\left({\phi }^{+},x\right)\nabla \left(\phi -{\mathcal{𝒲}}_{\epsilon }\left({\phi }^{+},x\right)-{\kappa }_{\epsilon }{\phi }^{-}-{u}_{\epsilon }\right)dx$$-{\int }_{\mathrm{\Omega }}\nabla \left({\kappa }_{\epsilon }{\phi }^{-}\right)\nabla \left(\phi -{\mathcal{𝒲}}_{\epsilon }\left({\phi }^{+},x\right)-{\kappa }_{\epsilon }{\phi }^{-}-{u}_{\epsilon }\right)dx$$=\sum _{i=1}^{3}{J}_{\epsilon }^{i}.$(5.2)

By using (4.1) and (5.1), we have

$\underset{\epsilon \to 0}{lim}{J}_{\epsilon }^{1}={\int }_{\mathrm{\Omega }}\nabla \phi \nabla \left(\phi -{u}_{0}\right)dx.$(5.3)

Then we proceed by transforming ${J}_{\epsilon }^{2}$ in the following way:

${J}_{\epsilon }^{2}=-\sum _{j\in {\mathrm{Υ}}_{\epsilon }}{\int }_{{\left({T}_{\epsilon /4}^{j}\right)}^{+}}\nabla {w}_{\epsilon }^{j}\left({\phi }^{+}\left({\stackrel{^}{P}}_{\epsilon }^{j}\right),x\right)\cdot \nabla \left(\phi -{w}_{\epsilon }^{j}\left({\phi }^{+}\left({\stackrel{^}{P}}_{\epsilon }^{j}\right),x\right)-{\kappa }_{\epsilon }{\phi }^{-}-{u}_{\epsilon }\right)dx$$=-\sum _{j\in {\mathrm{Υ}}_{\epsilon }}{\int }_{{\left({T}_{\epsilon /4}^{j}\right)}^{+}}\left\{\nabla \left({w}_{\epsilon }^{j}\left({\phi }^{+}\left({\stackrel{^}{P}}_{\epsilon }^{j}\right),x\right)-{\stackrel{^}{w}}_{\epsilon }^{j}\left({\phi }^{+}\left({\stackrel{^}{P}}_{\epsilon }^{j}\right),x\right)\right)\nabla \left(\phi -{w}_{\epsilon }^{j}\left({\phi }^{+}\left({\stackrel{^}{P}}_{\epsilon }^{j}\right),x\right)-{\kappa }_{\epsilon }{\phi }^{-}-{u}_{\epsilon }\right)\right\}𝑑x$$-\sum _{j\in {\mathrm{Υ}}_{\epsilon }}{\int }_{{\left({T}_{\epsilon /4}^{j}\right)}^{+}}\nabla {\stackrel{^}{w}}_{\epsilon }^{j}\left({\phi }^{+}\left({\stackrel{^}{P}}_{\epsilon }^{j}\right),x\right)\nabla \left(\phi -{w}_{\epsilon }^{j}\left({\phi }^{+}\left({\stackrel{^}{P}}_{\epsilon }^{j}\right),x\right)-{\kappa }_{\epsilon }{\phi }^{-}-{u}_{\epsilon }\right)dx$$={I}_{\epsilon }^{1}+{I}_{\epsilon }^{2}.$(5.4)

Lemma 3.5 implies that

(5.5)

By using Green’s formula, we have the following decomposition of the second integral:

${I}_{\epsilon }^{2}=-\sum _{j\in {\mathrm{Υ}}_{\epsilon }}{\int }_{{\left(\partial {T}_{\epsilon /4}^{j}\right)}^{+}}\left\{{\partial }_{\nu }{\stackrel{^}{w}}_{\epsilon }^{j}\left({\phi }^{+}\left({\stackrel{^}{P}}_{\epsilon }^{j}\right),x\right)\left({\phi }^{+}-{w}_{\epsilon }^{j}\left({\phi }^{+}\left({\stackrel{^}{P}}_{\epsilon }^{j}\right),x\right)-{u}_{\epsilon }\right)\right\}𝑑s$$-{\epsilon }^{-k}\sum _{j\in {\mathrm{Υ}}_{\epsilon }}{\int }_{{G}_{\epsilon }^{j}}\left\{\sigma \left({\phi }^{+}\left({\stackrel{^}{P}}_{\epsilon }^{j}\right)-{\stackrel{^}{w}}_{\epsilon }^{j}\left({\phi }^{+}\left({\stackrel{^}{P}}_{\epsilon }^{j}\right),x\right)\right)\left({\phi }^{+}-{w}_{\epsilon }^{j}\left({\phi }^{+}\left({\stackrel{^}{P}}_{\epsilon }^{j}\right),x\right)-{u}_{\epsilon }\right)\right\}𝑑{x}^{\prime }$$={\mathcal{ℐ}}_{\epsilon }^{1}+{\mathcal{ℐ}}_{\epsilon }^{2}.$(5.6)

From Lemma 3.7 we have

$\underset{\epsilon \to 0}{lim}{\mathcal{ℐ}}_{\epsilon }^{1}={C}_{0}^{n-2}{\int }_{\mathrm{\Omega }}H\left({\phi }^{+}\left(x\right)\right)\left({\phi }^{+}-{u}_{0}\right)𝑑x.$(5.7)

Combining (5.4)–(5.7) implies that

$\underset{\epsilon \to 0}{lim}{J}_{\epsilon }^{2}={C}_{0}^{n-2}{\int }_{\mathrm{\Omega }}H\left({\phi }^{+}\left(x\right)\right)\left({\phi }^{+}-{u}_{0}\right)𝑑x+\underset{\epsilon \to 0}{lim}{\mathcal{ℐ}}_{\epsilon }^{2}.$(5.8)

Now we consider the third term of identity (5.2). By using the fact that

$\nabla \left({\kappa }_{\epsilon }^{j}{\phi }^{-}\right)\cdot \nabla {\rho }_{\epsilon }=\left(\nabla {\kappa }_{\epsilon }^{j}\cdot \nabla {\rho }_{\epsilon }\right){\phi }^{-}+{\kappa }_{\epsilon }^{j}\nabla {\phi }^{-}\cdot \nabla {\rho }_{\epsilon }$$=\nabla {\kappa }_{\epsilon }^{j}\cdot \nabla \left({\phi }^{-}{\rho }_{\epsilon }\right)-\left(\nabla {\kappa }_{\epsilon }^{j}\cdot \nabla {\phi }^{-}\right){\rho }_{\epsilon }+{\kappa }_{\epsilon }^{j}\nabla {\phi }^{-}\cdot \nabla {\rho }_{\epsilon }$$=\nabla {\stackrel{^}{\kappa }}_{\epsilon }^{j}\cdot \nabla \left({\phi }^{-}{\rho }_{\epsilon }\right)-\nabla \left({\stackrel{^}{\kappa }}_{\epsilon }^{j}-{\kappa }_{\epsilon }^{j}\right)\cdot \nabla \left({\phi }^{-}{\rho }_{\epsilon }\right)-\left(\nabla {\kappa }_{\epsilon }^{j}\cdot \nabla {\phi }^{-}\right){\rho }_{\epsilon }+{\kappa }_{\epsilon }^{j}\nabla {\phi }^{-}\cdot \nabla {\rho }_{\epsilon }.$

we deduce that

${J}_{\epsilon }^{3}=-\sum _{j\in {\mathrm{Υ}}_{\epsilon }}{\int }_{{\left({T}_{\epsilon /4}^{j}\right)}^{+}}\nabla {\stackrel{^}{\kappa }}_{\epsilon }^{j}\nabla \left({\phi }^{-}\left(\phi -{\mathcal{𝒲}}_{\epsilon }\left({\phi }^{+},x\right)-{\kappa }_{\epsilon }{\phi }^{-}-{u}_{\epsilon }\right)\right)dx$$-\sum _{j\in {\mathrm{Υ}}_{\epsilon }}{\int }_{{\left({T}_{\epsilon /4}^{j}\right)}^{+}}\nabla \left({\kappa }_{\epsilon }^{j}-{\stackrel{^}{\kappa }}_{\epsilon }^{j}\right)\nabla \left({\phi }^{-}\left(\phi -{\mathcal{𝒲}}_{\epsilon }\left({\phi }^{+},x\right)-{\kappa }_{\epsilon }{\phi }^{-}-{u}_{\epsilon }\right)\right)dx$$+{\int }_{\mathrm{\Omega }}\left(\nabla {\kappa }_{\epsilon }\cdot \nabla {\phi }^{-}\right)\left({\phi }^{-}\left(\phi -{\mathcal{𝒲}}_{\epsilon }\left({\phi }^{+},x\right)-{\kappa }_{\epsilon }{\phi }^{-}-{u}_{\epsilon }\right)\right)𝑑x$$-{\int }_{\mathrm{\Omega }}{\kappa }_{\epsilon }\nabla {\phi }^{-}\cdot \nabla \left({\phi }^{-}\left(\phi -{\mathcal{𝒲}}_{\epsilon }\left({\phi }^{+},x\right)-{\kappa }_{\epsilon }{\phi }^{-}-{u}_{\epsilon }\right)\right)dx$$={\mathcal{𝒬}}_{\epsilon }^{1}+{\mathcal{𝒬}}_{\epsilon }^{2}+{\mathcal{𝒬}}_{\epsilon }^{3}+{\mathcal{𝒬}}_{\epsilon }^{4}.$

Lemma 4.1 implies that ${\mathcal{𝒬}}_{\epsilon }^{2}\to 0$ as $\epsilon \to 0$. Then we use (1.6), (1.7), (4.1) and (5.1) to derive that ${\mathcal{𝒬}}_{\epsilon }^{3}\to 0$ and ${\mathcal{𝒬}}_{\epsilon }^{4}\to 0$ as $\epsilon \to 0$. Then we transform ${\mathcal{𝒬}}_{\epsilon }^{1}$ using the Green’s formula:

${\mathcal{𝒬}}_{\epsilon }^{1}=-\sum _{j\in {\mathrm{Υ}}_{\epsilon }}{\int }_{{\left(\partial {T}_{\epsilon /4}^{j}\right)}^{+}}\left({\partial }_{\nu }{\stackrel{^}{\kappa }}_{\epsilon }^{j}\right){\phi }^{-}\left(\phi -{\mathcal{𝒲}}_{\epsilon }\left({\phi }^{+},x\right)-{\kappa }_{\epsilon }{\phi }^{-}-{u}_{\epsilon }\right)𝑑s-\sum _{j\in {\mathrm{Υ}}_{\epsilon }}{\int }_{{G}_{\epsilon }^{j}}\left({\partial }_{\nu }{\stackrel{^}{\kappa }}_{\epsilon }^{j}\right){\phi }^{-}\left(\phi -{\mathcal{𝒲}}_{\epsilon }\left({\phi }^{+},x\right)-{\kappa }_{\epsilon }{\phi }^{-}-{u}_{\epsilon }\right)𝑑{x}^{\prime }.$

By using the fact that ${\partial }_{\nu }\stackrel{^}{\kappa }\ge 0$ and ${\phi }^{-}{\mathcal{𝒲}}_{\epsilon }\left({\phi }^{+},x\right)\le 0$, ${\phi }^{-}{u}_{\epsilon }\le 0$ a.e. in ${G}_{\epsilon }$, we have that

$-\sum _{j\in {\mathrm{Υ}}_{\epsilon }}{\int }_{{G}_{\epsilon }^{j}}\left({\partial }_{\nu }{\stackrel{^}{\kappa }}_{\epsilon }^{j}\right){\phi }^{-}\left(\phi -{\mathcal{𝒲}}_{\epsilon }\left({\phi }^{+},x\right)-{\kappa }_{\epsilon }{\phi }^{-}-{u}_{\epsilon }\right)𝑑{x}^{\prime }\le 0.$

Hence we conclude

${\mathcal{𝒬}}_{\epsilon }^{1}\le -\sum _{j\in {\mathrm{Υ}}_{\epsilon }}{\int }_{\partial {T}_{\epsilon /4}^{j}\cap \mathrm{\Omega }}\left({\partial }_{\nu }{\stackrel{^}{\kappa }}_{\epsilon }^{j}.\right){\phi }^{-}\left(\phi -{u}_{\epsilon }\right)ds.$

Lemma 4.2 implies

$\underset{\epsilon \to 0}{lim}{J}_{\epsilon }^{3}=\underset{\epsilon \to 0}{lim}{\mathcal{𝒬}}_{\epsilon }^{1}\le \lambda {C}_{0}^{n-2}{\int }_{\mathrm{\Omega }}{\phi }^{-}\left(\phi -{u}_{\epsilon }\right)𝑑x.$(5.9)

From (5.2), (5.3), (5.8) and (5.9) we derive that

${\int }_{\mathrm{\Omega }}\nabla \phi \nabla \left(\phi -{u}_{0}\right)dx+{C}_{0}^{n-2}{\int }_{{\mathrm{\Gamma }}_{1}}H\left({\phi }^{+}\right)\left(\phi -{u}_{0}\right)𝑑{x}^{\prime }+\lambda {C}_{0}^{n-2}{\int }_{{\mathrm{\Gamma }}_{1}}{\phi }^{-}\left(\phi -{u}_{0}\right)𝑑{x}^{\prime }$$+\underset{\epsilon \to 0}{lim}\left\{{\epsilon }^{-k}\sum _{j\in {\mathrm{Υ}}_{\epsilon }}{\int }_{{G}_{\epsilon }^{j}}\sigma \left({\phi }^{+}-{\mathcal{𝒲}}_{\epsilon }\left({\phi }^{+},x\right)\right)\left({\phi }^{+}-{\mathcal{𝒲}}_{\epsilon }\left({\phi }^{+},x\right)-{u}_{\epsilon }\right)d{x}^{\prime }$$-{\epsilon }^{-k}\sum _{j\in {\mathrm{Υ}}_{\epsilon }}{\int }_{{G}_{\epsilon }^{j}}\sigma \left({\phi }^{+}\left({\stackrel{^}{P}}_{\epsilon }^{j}\right)-{\stackrel{^}{w}}_{\epsilon }^{j}\left({\phi }^{+}\left({\stackrel{^}{P}}_{\epsilon }^{j}\right),x\right)\right)\left({\phi }^{+}-{\mathcal{𝒲}}_{\epsilon }\left({\phi }^{+},x\right)-{u}_{\epsilon }\right)d{x}^{\prime }\right\}$$\ge {\int }_{\mathrm{\Omega }}f\left(\phi -{u}_{0}\right)𝑑x.$(5.10)

We first notice that ${\phi }^{+}-{w}_{\epsilon }^{j}\left({\phi }^{+}\left({\stackrel{^}{P}}_{\epsilon }^{j}\right),x\right)\ge {\phi }^{+}\left({\stackrel{^}{P}}_{\epsilon }^{j}\right)-{\stackrel{^}{w}}_{\epsilon }^{j}\left({\phi }^{+}\left({\stackrel{^}{P}}_{\epsilon }^{j},x\right)\right)$, and so

$\left\{\sigma \left({\phi }^{+}-{w}_{\epsilon }^{j}\left({\phi }^{+}\left({\stackrel{^}{P}}_{\epsilon }^{j}\right),x\right)\right)-\sigma \left({\phi }^{+}\left({\stackrel{^}{P}}_{\epsilon }^{j}\right)-{\stackrel{^}{w}}_{\epsilon }^{j}\left({\phi }^{+}\left({\stackrel{^}{P}}_{\epsilon }^{j}\right),x\right)\right)\right\}{u}_{\epsilon }\ge 0.$

Thus we only need to study the term

${\epsilon }^{-k}{\int }_{{G}_{\epsilon }^{j}}\left\{\sigma \left({\phi }^{+}-{w}_{\epsilon }^{j}\left({\phi }^{+}\left({\stackrel{^}{P}}_{\epsilon }^{j}\right),x\right)\right)-\sigma \left({\phi }^{+}\left({\stackrel{^}{P}}_{\epsilon }^{j}\right)-{\stackrel{^}{w}}_{\epsilon }^{j}\left({\phi }^{+}\left({\stackrel{^}{P}}_{\epsilon }^{j}\right),x\right)\right)\right\}\left({\phi }^{+}-{w}_{\epsilon }^{j}\left({\phi }^{+}\left({\stackrel{^}{P}}_{\epsilon }^{j}\right),x\right)\right)𝑑{x}^{\prime }.$

On the other hand,

$|{\phi }^{+}-{\mathcal{𝒲}}_{\epsilon }\left({\phi }^{+},x\right)|\le |{\phi }^{+}|\le K,$${w}_{\epsilon }^{j}\left({\phi }^{+}\left({\stackrel{^}{P}}_{\epsilon }^{j}\right),x\right)\le {\stackrel{^}{w}}_{\epsilon }^{j}\left({\phi }^{+}\left({\stackrel{^}{P}}_{\epsilon }^{j}\right),x\right)\le |{\phi }^{+}|\le K.$

Since σ is Hölder continuous and $\phi \in {\mathcal{𝒞}}^{1}\left(\overline{\mathrm{\Omega }}\right)$, we have that, for a.e. $x\in {G}_{\epsilon }^{j}$,

$|\sigma \left({\phi }^{+}-{\mathcal{𝒲}}_{\epsilon }\left({\phi }^{+},x\right)\right)-\sigma \left({\phi }^{+}\left({\stackrel{^}{P}}_{\epsilon }^{j}\right)-{\mathcal{𝒲}}_{\epsilon }\left({\phi }^{+},x\right)\right)|\le K\sum _{i=1}^{2}{|{\phi }^{+}\left(x\right)-{\phi }^{+}\left({\stackrel{^}{P}}_{\epsilon }^{j}\right)|}^{{\rho }_{i}}$$\le K\sum _{i=1}^{2}{|x-{\stackrel{^}{P}}_{\epsilon }^{j}|}^{{\rho }_{i}}$$=K\sum _{i=1}^{2}{a}_{\epsilon }^{{\rho }_{i}}.$

By using the same reasoning, estimate (3.4) implies that

$|{\epsilon }^{-k}\sum _{j\in {\mathrm{Υ}}_{\epsilon }}{\int }_{{G}_{\epsilon }^{j}}\left(\sigma \left({\phi }^{+}-{\mathcal{𝒲}}_{\epsilon }\left({\phi }^{+},x\right)\right)-\sigma \left({\phi }^{+}\left({\stackrel{^}{P}}_{\epsilon }^{j}\right)-{\mathcal{𝒲}}_{\epsilon }\left({\phi }^{+},x\right)\right)\right)\left({\phi }^{+}-{w}_{\epsilon }^{j}\left({\phi }^{+}\left({\stackrel{^}{P}}_{\epsilon }^{j}\right),x\right)\right)𝑑{x}^{\prime }|$$\le {K}_{2}{\epsilon }^{-k}|{G}_{\epsilon }|\sum _{i=1}^{2}{a}_{\epsilon }^{{\rho }_{i}}\le K\sum _{i=1}^{2}{a}_{\epsilon }^{{\rho }_{i}}.$(5.11)

Then by Lemma 3.5 we have that

${\epsilon }^{-k}|\sum _{j\in {\mathrm{Υ}}_{\epsilon }}{\int }_{{G}_{\epsilon }^{j}}\left(\sigma \left({\phi }^{+}\left({\stackrel{^}{P}}_{\epsilon }^{j}\right)-{\mathcal{𝒲}}_{\epsilon }\left({\phi }^{+},x\right)\right)-\sigma \left({\phi }^{+}\left({\stackrel{^}{P}}_{\epsilon }^{j}\right)-{\stackrel{^}{w}}_{\epsilon }^{j}\right)\right)\left({\phi }^{+}-{\mathcal{𝒲}}_{\epsilon }\left({\phi }^{+},x\right)\right)𝑑{x}^{\prime }|$$\le K{\epsilon }^{-k}\sum _{j\in {\mathrm{Υ}}_{\epsilon }}\sum _{i=1}^{2}{\int }_{{G}_{\epsilon }^{j}}{|v\left({\phi }^{+}\left({\stackrel{^}{P}}_{\epsilon }^{j}\right),x\right)|}^{{\rho }_{i}}𝑑{x}^{\prime }$$\le K{\epsilon }^{-k}\sum _{j\in {\mathrm{Υ}}_{\epsilon }}|{G}_{\epsilon }^{j}|\sum _{i=1}^{2}\frac{1}{|{G}_{\epsilon }^{j}|}{\int }_{{G}_{\epsilon }^{j}}{|v\left({\phi }^{+}\left({\stackrel{^}{P}}_{\epsilon }^{j}\right),x\right)|}^{{\rho }_{i}}𝑑{x}^{\prime },$

which, by applying the ${L}^{2}\left({G}_{0}\right)\to {L}^{{\rho }_{i}}\left({G}_{0}\right)$ embedding for $0<{\rho }_{i}\le 2$, can be estimated as

$\mathrm{\cdots }\le K{\epsilon }^{-k}\sum _{j\in {\mathrm{Υ}}_{\epsilon }}|{G}_{\epsilon }^{j}|\sum _{i=1}^{2}{\left(\frac{1}{|{G}_{\epsilon }^{j}|}{\int }_{{G}_{\epsilon }^{j}}{|{v}_{\epsilon }^{j}\left({\phi }^{+}\left({\stackrel{^}{P}}_{\epsilon }^{j}\right),x\right)|}^{2}𝑑{x}^{\prime }\right)}^{\frac{{\rho }_{i}}{2}}.$

By Lemma 3.6 and using $0<\rho \le 2$, we obtain

$\mathrm{\cdots }\le K{\epsilon }^{-k}\sum _{j\in {\mathrm{Υ}}_{\epsilon }}|{G}_{\epsilon }^{j}|\sum _{i=1}^{2}{\epsilon }^{\frac{{\rho }_{i}}{2}}\le K{\epsilon }^{-k}|{\mathrm{Υ}}_{\epsilon }||{G}_{\epsilon }^{j}|\sum _{i=1}^{2}{\epsilon }^{\frac{{\rho }_{i}}{2}}\le K{\epsilon }^{-k+1-n+k\left(n-1\right)}\sum _{i=1}^{2}{\epsilon }^{\frac{{\rho }_{i}}{2}}=K\sum _{i=1}^{2}{\epsilon }^{\frac{{\rho }_{i}}{2}}\to 0.$(5.12)

Combining these estimates with (5.10), we derive, since $\rho >0$, that

${\int }_{\mathrm{\Omega }}\nabla \phi \nabla \left(\phi -{u}_{0}\right)dx+{C}_{0}^{n-2}{\int }_{{\mathrm{\Gamma }}_{1}}H\left({\phi }^{+}\right)\left(\phi -{u}_{0}\right)𝑑{x}^{\prime }+\lambda {C}_{0}^{n-2}{\int }_{{\mathrm{\Gamma }}_{1}}{\phi }^{-}\left(\phi -{u}_{0}\right)\ge {\int }_{\mathrm{\Omega }}f\left(\phi -{u}_{0}\right)𝑑x$(5.13)

holds for any $\phi \in {H}^{1}\left(\mathrm{\Omega },{\mathrm{\Gamma }}_{2}\right)$.

Finally, given $\psi \in {H}^{1}\left(\mathrm{\Omega },{\mathrm{\Gamma }}_{2}\right)$, we consider the test function $\phi ={u}_{0}±\delta \psi$, $\delta >0$ in (5.13) and we pass to the limit as $\delta \to 0$. By doing so, we get that ${u}_{0}$ satisfies the integral condition

${\int }_{\mathrm{\Omega }}\nabla {u}_{0}\nabla \psi dx+{C}_{0}^{n-2}{\int }_{{\mathrm{\Gamma }}_{1}}H\left({u}_{0}^{+}\right)\psi 𝑑{x}^{\prime }+\lambda {C}_{0}^{n-2}{\int }_{{\mathrm{\Gamma }}_{1}}{u}_{0}^{-}\psi 𝑑{x}^{\prime }={\int }_{\mathrm{\Omega }}f\psi 𝑑x$

for any $\psi \in {H}^{1}\left(\mathrm{\Omega },{\mathrm{\Gamma }}_{2}\right)$. This concludes the proof. ∎

## 6.1 Extension to the case of σ as a maximal monotone graph

In [10], the authors showed that a similar problem, although restricted to the case of spherical particles distributed through the whole domain, could be treated in the general framework of maximal monotone graphs σ, which allow for a common roof between the Dirichlet, Neumann and Signorini boundary conditions and many more. We have restricted ourselves here to the case of Hölder continuous σ (see (1.4)), but this condition is only used at the very end, in estimates (5.11) and (5.12) to compute the last term of (5.10). The superlinearity condition is only used to obtain Lemma 3.6. These seem to be only technical difficulties, and can probably be avoided. Let us introduce what results can be expected if these problems could be circumvented.

## Maximal monotone graph of ${ℝ}^{2}$.

A monotone graph of ${ℝ}^{2}$ is a map (or operator) $\sigma :D\left(\sigma \right)\subset ℝ\to \mathcal{𝒫}\left(ℝ\right)\setminus \left\{\mathrm{\varnothing }\right\}$ such that

The set $D\left(\sigma \right)$ is called domain of the multivalued operator σ. Some authors define maximal monotone graphs as maps $\sigma :ℝ\to \mathcal{𝒫}\left(ℝ\right)$ and define $D\left(\sigma \right)=\left\{x\in \mathrm{\Omega }:\sigma \left(x\right)\ne \mathrm{\varnothing }\right\}$.

A monotone graph σ is extended by another monotone graph $\stackrel{~}{\sigma }$ if $D\left(\sigma \right)\subset D\left(\stackrel{~}{\sigma }\right)$ and $\sigma \left(x\right)\subset \stackrel{~}{\sigma }\left(x\right)$ for all $x\in D\left(\sigma \right)$. A monotone graph is called maximal if it admits no proper extension; for further references, see [2].

## Definition of solution.

The solution ${u}_{\epsilon }$ is also well-defined, although the set ${K}_{\epsilon }$ must now be written as

We will have that the integral condition (1.8) turns into

${\int }_{\mathrm{\Omega }}\nabla \phi \nabla \left(\phi -{u}_{\epsilon }\right)dx+{\epsilon }^{-k}{\int }_{{G}_{\epsilon }}\xi \left(x\right)\left(\phi -{u}_{\epsilon }\right)𝑑{x}^{\prime }\ge {\int }_{\mathrm{\Omega }}f\left(\phi -{u}_{\epsilon }\right)𝑑x$

for all $\phi \in {K}_{\epsilon }$ and $\xi \in {L}^{2}\left({G}_{\epsilon }\right)$ such that $\xi \left({x}^{\prime }\right)\in \sigma \left(\phi \left({x}^{\prime }\right)\right)$ for a.e. ${x}^{\prime }\in {G}_{\epsilon }$. The existence and uniqueness of this solution follows as in [10] and the references therein.

## The auxiliary functions.

The equation of $\stackrel{^}{w}$ is well-defined when σ is a maximal monotone graph. As we have proved in this paper, the estimate $0\le {H}^{\prime }\le {\lambda }_{{G}_{0}}$ is independent of σ, and so H is Lipschitz continuous for any maximal monotone graph σ.

## Signorini boundary conditions.

This is the case under study in this paper. Nonetheless, let us study in the general setting. For this kind of boundary condition, we need to consider the following maximal monotone graph:

$\stackrel{~}{\sigma }\left(s\right)=\left\{\begin{array}{cc}\sigma \left(s\right),\hfill & s>0,\hfill \\ \left(-\mathrm{\infty },0\right],\hfill & s=0,\hfill \\ \mathrm{\varnothing },\hfill & s<0,\hfill \end{array}$(6.1)

and $D\left(\sigma \right)=\left[0,+\mathrm{\infty }\right)$. Let us compute ${\stackrel{~}{H}}_{{G}_{0}}$ in this setting:

• For $u<0$, we can see what happens explicitly. We have that $u\le \stackrel{^}{w}\left(u,\cdot \right)\le 0$. Thus $u-\stackrel{^}{w}\le 0$. Since $D\left(\sigma \right)=\left[0,+\mathrm{\infty }\right)$, we must have that $\stackrel{^}{w}\left(u,y\right)=u$ on ${G}_{0}$. But then $\stackrel{^}{w}\left(u,y\right)=u\stackrel{^}{\kappa }\left(y\right)$. Hence ${\stackrel{~}{H}}_{{G}_{0}}\left(u\right)={\lambda }_{{G}_{0}}u$ when $u<0$.

• When $u>0$, we have that $0\le u-\stackrel{^}{w}\left(u,\cdot \right)$. Thus only the values of σ affect ${H}_{{G}_{0}}\left(u\right)$.

We conclude that

${\stackrel{~}{H}}_{{G}_{0}}\left(u\right)=\left\{\begin{array}{cc}{H}_{{G}_{0}}\left(u\right),\hfill & u>0,\hfill \\ {\lambda }_{{G}_{0}}u,\hfill & u\le 0.\hfill \end{array}$

The computations with maximal monotone graphs yield precisely Theorem 1.3. Notice that the bound on ${\stackrel{~}{H}}^{\prime }$ given by (1.12) is sharp.

## Dirichlet boundary conditions.

In this case, we would have $D\left(\sigma \right)=\left\{0\right\}$ and

$\stackrel{~}{\sigma }\left(0\right)=\left(-\mathrm{\infty },+\mathrm{\infty }\right).$

By the same reasoning, we have that

${\stackrel{~}{H}}_{{G}_{0}}\left(u\right)={\lambda }_{{G}_{0}}u$

for all $u\in ℝ$. In this case of Dirichlet boundary conditions, the critical case generates a linear term in the homogenized equation. This type of phenomena was already noticed by Cioranescu and Murat [6].

## Cases of finite and infinite permeable coefficient.

The Signorini boundary condition imposed as a maximal monotone graph (6.1) is the extreme case of infinite permeability, aiming to represent the behavior of very large finite permeability given by a reaction term of the form

${\stackrel{~}{\sigma }}_{\mu }\left(u\right)=\left\{\begin{array}{cc}\sigma \left(u\right),\hfill & u>0,\hfill \\ \mu u,\hfill & u\le 0,\hfill \end{array}$

where μ is very large. As in Remark 2.7, it is easy to show that the corresponding kinetic will be of the form

${\stackrel{~}{H}}_{\mu }\left(u\right)=\left\{\begin{array}{cc}H\left(u\right),\hfill & u>0,\hfill \\ {\lambda }_{\mu }u,\hfill & u\le 0.\hfill \end{array}$

Moreover, since we have proven that the Signorini boundary condition is an extremal case (i.e. ${\stackrel{~}{H}}^{\prime }\left(u\right)\le {\lambda }_{{G}_{0}}$), we have that ${\lambda }_{\mu }\le {\lambda }_{{G}_{0}}$.

## 6.2 On the super-linearity condition

The condition

$|\sigma \left(s\right)-\sigma \left(t\right)|\ge {k}_{1}|s-t|$

is only used in the proof of Lemma 3.6. However, it is our belief that this condition can be removed and that the result can still be obtained. We provide here a proof for $n=3$ and a ball ${G}_{0}$.

We define the auxiliary function ${w}_{\epsilon }$ as the unique solution of

$\left\{\begin{array}{cccc}\hfill -\mathrm{\Delta }{w}_{\epsilon }^{j}& =0,\hfill & & \hfill {T}_{\epsilon }^{0},\\ \hfill {w}_{\epsilon }& =0,\hfill & & \hfill {G}_{\epsilon }^{0},\\ \hfill {w}_{\epsilon }& =1,\hfill & & \hfill \partial {T}_{\epsilon }^{0},\end{array}$

where ${T}_{\epsilon }^{0}$ is given by

${T}_{\epsilon }^{0}=\left\{x\in {ℝ}^{3}:\frac{{x}_{1}^{2}}{1-{\left({a}_{\epsilon }{\epsilon }^{-1}\right)}^{2}}+{x}_{2}^{2}+{x}_{3}^{2}<{\epsilon }^{2}\right\}.$

Using prolate ellipsoidal coordinates, we can give an explicit expression of ${w}_{\epsilon }$. These coordinates are given by

${x}_{1}={a}_{\epsilon }\mathrm{sinh}\psi \mathrm{cos}{\theta }_{1},$${x}_{2}={a}_{\epsilon }\mathrm{cosh}\psi \mathrm{sin}{\theta }_{1}\mathrm{cos}{\theta }_{2},$${x}_{3}={a}_{\epsilon }\mathrm{cosh}\psi \mathrm{sin}{\theta }_{1}\mathrm{sin}{\theta }_{2},$

where $0\le \psi <\mathrm{\infty }$, $0\le {\theta }_{1}\le \pi$ and $0\le {\theta }_{2}\le 2\pi$. By defining $\sigma =\mathrm{sinh}\psi$, it can be proven through symmetry that ${w}_{\epsilon }\left(x\right)={V}_{\epsilon }\left(\sigma \right)$. Furthermore, ${V}_{\epsilon }$ is the unique solution of the one-dimensional problem

$\left\{\begin{array}{cccc}& \frac{d}{d\sigma }\left(\left(1+{\sigma }^{2}\right)\frac{dV}{d\sigma }\right)=0,\hfill & & \hfill \sigma \in \left(0,\sqrt{{\left({a}_{\epsilon }^{-1}\epsilon \right)}^{2}-1}\right),\\ & V\left(0\right)=0,\hfill \\ & V\left(\sqrt{{\left({a}_{\epsilon }^{-1}\epsilon \right)}^{2}-1}\right)=1.\hfill \end{array}$

Integrating this simple one-dimensional boundary value problem, we obtain

$V\left(\sigma \right)=\frac{\mathrm{arctan}\sigma }{\mathrm{arctan}\left(\mathrm{sinh}\sqrt{{\left({a}_{\epsilon }^{-1}\epsilon \right)}^{2}-1}\right)}$

since we can recover from the change in variable

$\sigma =\mathrm{sinh}\psi =\sqrt{\frac{{|x|}^{2}-{a}_{\epsilon }^{2}+\sqrt{{\left({a}_{\epsilon }^{2}-{|x|}^{2}\right)}^{2}+4{x}_{1}^{2}{a}_{\epsilon }^{2}}}{2{a}_{\epsilon }^{2}}}.$

Due to mirror symmetry it is clear that ${{\partial }_{{x}_{1}}{w}_{\epsilon }|}_{\left\{{x}_{1}=0\right\}}=0$. Thus we have

${\int }_{{\left({T}_{\epsilon }^{0}\right)}^{+}}\nabla {w}_{\epsilon }\nabla \left({u}^{2}\right)dx={\int }_{{\left(\partial {T}_{\epsilon }^{0}\right)}^{+}}{\partial }_{{x}_{1}}{w}_{\epsilon }{u}^{2}dS-{\int }_{{G}_{\epsilon }^{0}}{\partial }_{{x}_{1}}{w}_{\epsilon }{u}^{2}dS.$

Using the explicit expression of ${w}_{\epsilon }$, we can compute that

${{\partial }_{\nu }{w}_{\epsilon }|}_{{\left(\partial {T}_{\epsilon }^{0}\right)}^{+}}\sim {a}_{\epsilon }{\epsilon }^{-2},$${{\partial }_{\nu }{w}_{\epsilon }|}_{{G}_{\epsilon }^{0}}\sim -\frac{1}{\sqrt{{a}_{\epsilon }^{2}-{|x|}^{2}}}.$

Now let

${T}_{\epsilon }^{j}={P}_{\epsilon }^{j}+{T}_{\epsilon }^{0},$${T}_{\epsilon }=\bigcup _{j\in {\mathrm{Υ}}_{\epsilon }}{T}_{\epsilon }^{j},$

Summing over ${\mathrm{Υ}}_{\epsilon }$, we deduce that

${\int }_{{\left({T}_{\epsilon }\right)}^{+}}\nabla {W}_{\epsilon }\nabla \left({u}^{2}\right)ds=\sum _{j\in {\mathrm{Υ}}_{\epsilon }}{\int }_{{\left(\partial {T}_{\epsilon }^{j}\right)}^{+}}{\partial }_{\nu }{w}_{\epsilon }^{j}{u}^{2}ds-{\int }_{{G}_{\epsilon }}{\partial }_{{x}_{1}}{W}_{\epsilon }{u}^{2}ds.$(6.2)

It is easy to prove that

${\int }_{{\left(\partial {T}_{\epsilon }^{j}\right)}^{+}}{\partial }_{\nu }{w}_{\epsilon }^{j}{h}^{2}ds\le K\sum _{j\in {\mathrm{Υ}}_{\epsilon }}{\parallel h\parallel }_{{H}^{1}\left({\left({T}_{\epsilon }^{j}\right)}^{+}\right)}^{2}$(6.3)

for any $h\in {H}^{1}\left(\mathrm{\Omega }\right)$. We now apply that

${\parallel u\parallel }_{{L}^{2}\left({G}_{\epsilon }\right)}^{2}\le K\left({\epsilon }^{-1}{\parallel u\parallel }_{{L}^{2}\left({T}_{\epsilon }^{+}\right)}+\epsilon {\parallel \nabla u\parallel }_{{L}^{2}\left({T}_{\epsilon }^{+}\right)}^{2}\right).$(6.4)

With (6.2)–(6.4) we can prove Lemma 3.6 for ${k}_{1}=0$.

## 6.3 Connections to fractional operators

Let us consider a domain $\mathrm{\Omega }={\mathrm{\Omega }}^{\prime }×\left(0,+\mathrm{\infty }\right)$, where ${\mathrm{\Omega }}^{\prime }\subset {ℝ}^{n-1}$ is a smooth bounded domain. Then ${\mathrm{\Gamma }}_{1}={\mathrm{\Omega }}^{\prime }$ and ${\mathrm{\Gamma }}_{2}=\partial \mathrm{\Omega }×\left(0,+\mathrm{\infty }\right)$. The related problem

$\left\{\begin{array}{cccc}& -\mathrm{\Delta }{u}_{\epsilon }=0,\hfill & & \hfill \left(x,y\right)\in {\mathrm{\Omega }}^{\prime }×\left(0,+\mathrm{\infty }\right),\\ & {\partial }_{\nu }{u}_{\epsilon }+{\epsilon }^{-k}\sigma \left({u}_{\epsilon }\right)={\epsilon }^{-k}{g}_{\epsilon },\hfill & & \hfill x\in {G}_{\epsilon },\\ & {\partial }_{\nu }{u}_{\epsilon }=0,\hfill & & \hfill x\in {\mathrm{\Omega }}^{\prime }\setminus {\overline{G}}_{\epsilon },\\ & {u}_{\epsilon }=0,\hfill & & \hfill \left(x,y\right)\in \partial {\mathrm{\Omega }}^{\prime }×\left(0,+\mathrm{\infty }\right),\\ & {u}_{\epsilon }\to 0,\hfill & & \hfill |y|\to +\mathrm{\infty },\end{array}$(6.5)

is very relevant because it can be linked to the study of the fractional Laplacian ${\left(-\mathrm{\Delta }\right)}^{1/2}$. In fact, the boundary conditions on ${\mathrm{\Omega }}^{\prime }$ can be written compactly as

${\partial }_{\nu }{u}_{\epsilon }+{\epsilon }^{-k}{\chi }_{{G}_{\epsilon }}\sigma \left({u}_{\epsilon }\right)={\epsilon }^{-k}{\chi }_{{G}_{\epsilon }}{g}_{\epsilon },x\in {\mathrm{\Omega }}^{\prime },$(6.6)

where χ is the indicator function. This boundary condition can be written as an equation of ${\mathrm{\Omega }}^{\prime }$ not involving the interior part of the domain, ${\mathrm{\Omega }}^{\prime }×\left(0,+\mathrm{\infty }\right)$, by understanding the normal derivative of problem (6.5) as the fractional Laplace operator ${\left(-\mathrm{\Delta }\right)}^{1/2}$ in ${\mathrm{\Omega }}^{\prime }$ (see [3, 15] and the references therein). Then (6.6) can be written as

${\left(-\mathrm{\Delta }\right)}^{\frac{1}{2}}{u}_{\epsilon }+{\epsilon }^{-k}{\chi }_{{G}_{\epsilon }}\sigma \left({u}_{\epsilon }\right)={\epsilon }^{-k}{\chi }_{{G}_{\epsilon }}{g}_{\epsilon },x\in {\mathrm{\Omega }}^{\prime }.$(6.7)

Thus the study of the limit of (6.5) will provide a homogenization result for (6.7). By applying similar techniques to this paper and previous results in the literature [12], the homogenized problem

${\left(-\mathrm{\Delta }\right)}^{\frac{1}{2}}{u}_{0}+CH\left(x,{u}_{0}\right)=Ch,x\in {\mathrm{\Omega }}^{\prime },$

is expected, where H and h will depend on σ and ${g}_{\epsilon }$. This could provide some new results of critical size homogenization for the fractional Laplacian (in the spirit of the important work [3], where some random aspects on the net, and for a general fractional power of the Laplacian, are also considered).

## 7 Proof of Theorem 1.4

It is well known that problem (1.13), (1.14) has a unique weak solution ${u}_{\epsilon }\in {H}^{1}\left(\mathrm{\Omega },{\mathrm{\Gamma }}_{2}\right)$. By using (1.14) and condition (1.15), which was set on the function σ, we get the following estimates:

${\parallel \nabla {u}_{\epsilon }\parallel }_{{L}_{2}\left(\mathrm{\Omega }\right)}\le K,$${e}^{\frac{{\alpha }^{2}}{\epsilon }}{\parallel {u}_{\epsilon }\parallel }_{{L}_{2}\left({G}_{\epsilon }\right)}^{2}\le {K}_{1}.$

Here and below, the constants K and ${K}_{1}$ are independent of ε.

Hence there exists a subsequence (denote as the original sequence ${u}_{\epsilon }$) such that, as $\epsilon \to 0$, we have

${u}_{\epsilon }⇀{u}_{0}$${u}_{\epsilon }\to {u}_{0}$

We introduce auxiliary functions ${w}_{\epsilon }^{j}$ and ${q}_{\epsilon }^{j}$ as weak solutions to the following problems:

$\left\{\begin{array}{cccc}\hfill \mathrm{\Delta }{w}_{\epsilon }^{j}& =0,\hfill & & \hfill x\in {T}_{\epsilon /4}^{j}\setminus \overline{{T}_{{a}_{\epsilon }}^{j}},\\ \hfill {w}_{\epsilon }^{j}& =1,\hfill & & \hfill x\in \partial {T}_{{a}_{\epsilon }}^{j},\\ \hfill {w}_{\epsilon }^{j}& =0,\hfill & & \hfill x\in \partial {T}_{\epsilon /4}^{j},\end{array}$

and

$\left\{\begin{array}{cccc}\hfill \mathrm{\Delta }{q}_{\epsilon }^{j}& =0,\hfill & & \hfill x\in {T}_{\epsilon }^{j}\setminus \overline{{l}_{\epsilon }^{j}},\\ \hfill {q}_{\epsilon }^{j}& =1,\hfill & & \hfill x\in {l}_{\epsilon }^{j},\\ \hfill {q}_{\epsilon }^{j}& =0,\hfill & & \hfill x\in \partial {T}_{\epsilon /4}^{j}.\end{array}$(7.1)

Figure 4

Domain ${T}_{\epsilon }^{+,j}\setminus \overline{{T}_{{a}_{\epsilon }}^{j}}$ and ${l}_{\epsilon }^{j}$.

Note that ${w}_{\epsilon }^{j}$ and ${q}_{\epsilon }^{j}$ are also a solutions of the following boundary value problems in the domains ${\left({T}_{\epsilon /4}^{j}\right)}^{+}\setminus \overline{{T}_{{a}_{\epsilon }}^{j}}$ and ${\left({T}_{\epsilon /4}^{j}\right)}^{+}$ (see Figure 4), respectively, where ${\left({T}_{r}^{j}\right)}^{+}={T}_{r}^{j}\cap \left\{{x}_{2}>0\right\}$,

$\left\{\begin{array}{cccc}\hfill \mathrm{\Delta }{w}_{\epsilon }^{j}& =0,\hfill & & \hfill x\in {\left({T}_{\epsilon /4}^{j}\right)}^{+}\setminus \overline{{T}_{{a}_{\epsilon }}^{j}},\\ \hfill {w}_{\epsilon }^{j}& =0,\hfill & & \hfill x\in \partial {T}_{\epsilon /4}^{j}\cap \left\{{x}_{2}>0\right\},\\ \hfill {w}_{\epsilon }^{j}& =1,\hfill & & \hfill x\in \partial {T}_{{a}_{\epsilon }}^{j}\cap \left\{{x}_{2}>0\right\},\\ \hfill {\partial }_{{x}_{2}}{w}_{\epsilon }^{j}& =0,\hfill & & \hfill x\in \left\{{x}_{2}=0\right\}\cap \left({T}_{\epsilon /4}^{j}\setminus \overline{{T}_{{a}_{\epsilon }}^{j}}\right),\end{array}$$\left\{\begin{array}{cccc}\hfill \mathrm{\Delta }{q}_{\epsilon }^{j}& =0,\hfill & & \hfill x\in {\left({T}_{\epsilon /4}^{j}\right)}^{+},\\ \hfill {q}_{\epsilon }^{j}& =0,\hfill & & \hfill x\in {\left(\partial {T}_{\epsilon /4}^{j}\right)}^{+},\\ \hfill {q}_{\epsilon }^{j}& =1,\hfill & & \hfill x\in {l}_{\epsilon }^{j},\\ \hfill {\partial }_{{x}_{2}}{q}_{\epsilon }^{j}& =0,\hfill & & \hfill x\in \left({T}_{\epsilon /4}^{j}\cap \left\{{x}_{2}=0\right\}\right)\setminus \overline{{l}_{\epsilon }^{j}},\end{array}$

where $j\in {\mathrm{Υ}}_{\epsilon }$ and ${l}_{\epsilon }^{j}={a}_{\epsilon }{\stackrel{^}{l}}_{0}+\epsilon j$. Define

${W}_{\epsilon }\left(x\right)=\left\{\begin{array}{cc}{w}_{\epsilon }^{j}\left(x\right),\hfill & x\in {\left({T}_{\epsilon /4}^{j}\right)}^{+}\setminus \overline{{T}_{{a}_{\epsilon }}^{j}},j\in {\mathrm{Υ}}_{\epsilon },\hfill \\ 1,\hfill & x\in {\left({T}_{{a}_{\epsilon }}^{j}\right)}^{+},\hfill \\ 0,\hfill & x\in {ℝ}_{+}^{2}\setminus \bigcup _{j\in {\mathrm{Υ}}_{\epsilon }}{\overline{{T}_{\epsilon /4}^{j}}}^{+},\hfill \end{array}$(7.2)

where ${ℝ}_{+}^{2}=\left\{{x}_{2}>0\right\}$, and

${Q}_{\epsilon }\left(x\right)=\left\{\begin{array}{cc}{q}_{\epsilon }^{j}\left(x\right),\hfill & x\in {\left({T}_{\epsilon /4}^{j}\right)}^{+},j\in {\mathrm{Υ}}_{\epsilon },\hfill \\ 0,\hfill & x\in {ℝ}_{+}^{2}\setminus \bigcup _{j\in {\mathrm{Υ}}_{\epsilon }}{\overline{{T}_{\epsilon /4}^{j}}}^{+}.\hfill \end{array}$(7.3)

We have ${W}_{\epsilon },{Q}_{\epsilon }\in {H}_{0}^{1}\left(\mathrm{\Omega }\right)$ and

#### Lemma 7.1.

Let ${W}_{\epsilon }$ be a function defined by formula (7.2), and let ${Q}_{\epsilon }$ be a function defined by formula (7.3). Then

${\parallel {W}_{\epsilon }-{Q}_{\epsilon }\parallel }_{{H}^{1}\left(\mathrm{\Omega }\right)}\le K\sqrt{\epsilon }.$

#### Proof.

Note that for an arbitrary function $\psi \in {H}^{1}\left({T}_{\epsilon /4}^{j}\right)$ such that $\psi =0$ on ${l}_{\epsilon }^{j}$, we have

${\int }_{{\left({T}_{\epsilon /4}^{j}\right)}^{+}}\nabla {q}_{\epsilon }^{j}\nabla \psi d{x}_{1}d{x}_{2}=0.$

We consider $\psi ={w}_{\epsilon }^{j}-{q}_{\epsilon }^{j}$ as a test function in the obtained equality and get

${\int }_{{\left({T}_{\epsilon /4}^{j}\right)}^{+}}\nabla {q}_{\epsilon }^{j}\nabla \left({w}_{\epsilon }^{j}-{q}_{\epsilon }^{j}\right)d{x}_{1}d{x}_{2}=0.$(7.4)

${\int }_{{\left({T}_{\epsilon /4}^{j}\right)}^{+}}\nabla {w}_{\epsilon }^{j}\nabla \left({w}_{\epsilon }^{j}-{q}_{\epsilon }^{j}\right)d{x}_{1}d{x}_{2}={\int }_{\partial {T}_{{a}_{\epsilon }}^{j}\cap \left\{{x}_{2}>0\right\}}{\partial }_{\nu }{w}_{\epsilon }^{j}\left({w}_{\epsilon }^{j}-{q}_{\epsilon }^{j}\right)𝑑s.$(7.5)

By subtracting (7.4) from (7.5), we derive

${\int }_{{\left({T}_{\epsilon /4}^{j}\right)}^{+}}{|\nabla \left({w}_{\epsilon }^{j}-{q}_{\epsilon }^{j}\right)|}^{2}𝑑{x}_{1}𝑑{x}_{2}={\int }_{\partial {T}_{{a}_{\epsilon }}^{j}\cap \left\{{x}_{2}>0\right\}}{\partial }_{\nu }{w}_{\epsilon }^{j}\left({w}_{\epsilon }^{j}-{q}_{\epsilon }^{j}\right)𝑑s.$(7.6)

Note that

${w}_{\epsilon }^{j}\left(x\right)=\frac{\mathrm{ln}\left(4r/\epsilon \right)}{\mathrm{ln}\left(4{a}_{\epsilon }/\epsilon \right)}\mathit{ }\text{and}\mathit{ }{\partial }_{\nu }{w}_{\epsilon }^{j}=-\frac{1}{{a}_{\epsilon }\mathrm{ln}\left(4{a}_{\epsilon }/\epsilon \right)}.$

Hence, (7.6) implies that

${\parallel \nabla \left({w}_{\epsilon }^{j}-{q}_{\epsilon }^{j}\right)\parallel }_{{L}^{2}\left({\left({T}_{\epsilon /4}^{j}\right)}^{+}\right)}^{2}\le \frac{1}{{a}_{\epsilon }|\mathrm{ln}\left(4{a}_{\epsilon }/\epsilon \right)|}{\int }_{\partial {T}_{{a}_{\epsilon }}^{j}\cap \left\{{x}_{2}>0\right\}}|{w}_{\epsilon }^{j}-{q}_{\epsilon }^{j}|𝑑s$$=\frac{1}{|\mathrm{ln}\left(4{a}_{\epsilon }/\epsilon \right)|}{\int }_{\partial {T}_{1}^{j}\cap \left\{{y}_{2}>0\right\}}|{w}_{\epsilon }^{j}-{q}_{\epsilon }^{j}|𝑑{s}_{y}\equiv {J}_{\epsilon }.$

Given that ${w}_{\epsilon }^{j}-{q}_{\epsilon }^{j}=0$ if $y\in {\stackrel{^}{l}}_{0}$ and using the embedding theorem, we get

${J}_{\epsilon }\le \frac{K}{|\mathrm{ln}\left(4{a}_{\epsilon }/\epsilon \right)|}{\left({\int }_{{\left({T}_{1}^{j}\right)}^{+}}{|{\nabla }_{y}\left({w}_{\epsilon }^{j}-{q}_{\epsilon }^{j}\right)|}^{2}𝑑y\right)}^{1/2}\le K\epsilon {\parallel \nabla \left({w}_{\epsilon }^{j}-{q}_{\epsilon }^{j}\right)\parallel }_{{L}^{2}\left({\left({T}_{\epsilon /4}^{j}\right)}^{+}\right)}.$

From here we derive the estimate

${\parallel \nabla \left({w}_{\epsilon }^{j}-{q}_{\epsilon }^{j}\right)\parallel }_{{L}^{2}\left({\left({T}_{\epsilon /4}^{j}\right)}^{+}\right)}\le K\epsilon .$

From this estimation it follows that

${\parallel {W}_{\epsilon }-{Q}_{\epsilon }\parallel }_{{H}^{1}\left(\mathrm{\Omega }\right)}\le K\sqrt{\epsilon }.$

This concludes the proof. ∎

We introduce the function $m\left(y\right)\in {H}^{1}\left({\left({T}_{1}^{0}\right)}^{+}\right)$ as the weak solution to the following boundary value problem:

$\left\{\begin{array}{cccc}\hfill {\mathrm{\Delta }}_{y}m& =0,\hfill & & \hfill y\in {T}_{1}^{0}\cap \left\{{y}_{2}>0\right\}={\left({T}_{1}^{0}\right)}^{+},\\ \hfill {\partial }_{{y}_{2}}m& =1,\hfill & & \hfill y\in {\stackrel{^}{l}}_{0},\\ \hfill {\partial }_{\nu }m& =\frac{2{l}_{0}}{\pi },\hfill & & \hfill y\in \partial {T}_{1}^{0}\cap \left\{{y}_{2}>0\right\}={\left(\partial {T}_{1}^{0}\right)}^{+},\\ \hfill {\partial }_{{y}_{2}}m& =0,\hfill & & \hfill y\in \partial {\left({T}_{1}^{0}\right)}^{+}\setminus {\stackrel{^}{l}}_{0}\cup {\left(\partial {T}_{1}^{0}\right)}^{+}.\end{array}$

Consider

${m}_{\epsilon }^{j}\left(x\right)=\epsilon m\left(\frac{x-{P}_{\epsilon }^{j}}{{a}_{\epsilon }}\right),x\in {\left({T}_{{a}_{\epsilon }}^{j}\right)}^{+}.$

The function ${m}_{\epsilon }^{j}\left(x\right)$ verifies the problem

$\left\{\begin{array}{cccc}\hfill {\mathrm{\Delta }}_{x}{m}_{\epsilon }^{j}& =0,\hfill & & \hfill x\in {\left({T}_{{a}_{\epsilon }}^{j}\right)}^{+},\\ \hfill {\partial }_{\nu }{m}_{\epsilon }^{j}& =\frac{\epsilon {a}_{\epsilon }^{-1}2{l}_{0}}{\pi },\hfill & & \hfill x\in {\left(\partial {T}_{{a}_{\epsilon }}^{j}\right)}^{+},\\ \hfill {\partial }_{{x}_{2}}{m}_{\epsilon }^{j}& =\epsilon {a}_{\epsilon }^{-1},\hfill & & \hfill x\in \left\{{x}_{2}=0:|x-{P}_{\epsilon }^{j}|\le {a}_{\epsilon }{l}_{0}\right\}={l}_{\epsilon }^{j},\\ \hfill {\partial }_{{x}_{2}}{m}_{\epsilon }^{j}& =0\hfill & & \hfill \text{on the rest of the boundary}.\end{array}$

#### Lemma 7.2.

Let $n\mathrm{=}\mathrm{2}$ and let ${h}_{\epsilon }\mathrm{\in }{H}^{\mathrm{1}}\mathit{}\mathrm{\left(}\mathrm{\Omega }\mathrm{,}{\mathrm{\Gamma }}_{\mathrm{2}}\mathrm{\right)}$ be a bounded sequence. Then the following estimate holds:

$|\frac{2{l}_{0}\epsilon }{\pi {a}_{\epsilon }}\sum _{j\in {\mathrm{Υ}}_{\epsilon }}{\int }_{{\left(\partial {T}_{{a}_{\epsilon }}^{j}\right)}^{+}}{h}_{\epsilon }𝑑s-\frac{\epsilon }{{a}_{\epsilon }}{\int }_{{l}_{\epsilon }}{h}_{\epsilon }𝑑{x}_{1}|\le K\sqrt{\epsilon }.$

#### Proof.

We have that

$|\frac{2{l}_{0}\epsilon {a}_{\epsilon }^{-1}}{\pi }{\int }_{{\left(\partial {T}_{{a}_{\epsilon }}^{j}\right)}^{+}}{h}_{\epsilon }𝑑s-\epsilon {a}_{\epsilon }^{-1}{\int }_{{l}_{\epsilon }^{j}}{h}_{\epsilon }𝑑{x}_{1}|=|{\int }_{{\left({T}_{{a}_{\epsilon }}^{j}\right)}^{+}}{\nabla }_{x}{m}_{\epsilon }^{j}\nabla {h}_{\epsilon }dx|\le {\parallel {\nabla }_{x}{m}_{\epsilon }^{j}\parallel }_{{L}^{2}\left({\left({T}_{{a}_{\epsilon }}^{j}\right)}^{+}\right)}{\parallel \nabla {h}_{\epsilon }\parallel }_{{L}^{2}\left({\left({T}_{{a}_{\epsilon }}^{j}\right)}^{+}\right)}.$(7.7)

Due to the fact that

${\parallel {\nabla }_{x}{m}_{\epsilon }^{j}\parallel }_{{L}^{2}\left({\left({T}_{{a}_{\epsilon }}^{j}\right)}^{+}\right)}^{2}={\epsilon }^{2}{\parallel {\nabla }_{y}m\left(y\right)\parallel }_{{L}^{2}\left({\left({T}_{1}^{0}\right)}^{+}\right)}^{2}\le K{\epsilon }^{2},$

we have

$\sum _{j\in {\mathrm{Υ}}_{\epsilon }}{\parallel {\nabla }_{x}{m}_{\epsilon }^{j}\parallel }_{{L}^{2}\left({\left({T}_{{a}_{\epsilon }}^{j}\right)}^{+}\right)}^{2}\le K\epsilon .$(7.8)

From (7.7), (7.8) we derive

$|{e}^{{\alpha }^{2}/\epsilon }\frac{\pi }{2{l}_{0}}{\int }_{{l}_{\epsilon }}{h}_{\epsilon }𝑑{x}_{1}-{e}^{{\alpha }^{2}/\epsilon }\sum _{j\in {\mathrm{Υ}}_{\epsilon }}{\int }_{\partial {T}_{{a}_{\epsilon }}^{j}\cap \left\{{x}_{2}>0\right\}}{h}_{\epsilon }𝑑s|\le {\delta }^{-1}\sum _{j\in {\mathrm{Υ}}_{\epsilon }}{\parallel {\nabla }_{x}{m}_{\epsilon }^{j}\parallel }_{{L}^{2}\left({\left({T}_{{a}_{\epsilon }}^{j}\right)}^{+}\right)}^{2}+\delta {\parallel \nabla {h}_{\epsilon }\parallel }_{{L}^{2}\left(\mathrm{\Omega }\right)}^{2}\le K\sqrt{\epsilon }$

if $\delta =\sqrt{\epsilon }$. ∎

#### Proof of Theorem 1.4.

First of all, equation (1.17) has a unique solution $H\left(u\right)$ that is a Lipschitz continuous function in $ℝ$ and satisfies

$\left(H\left(u\right)-H\left(v\right)\right)\left(u-v\right)\ge {\stackrel{~}{k}}_{1}|u-v|,|H\left(u\right)|\le |u|,$(7.9)

for all $u,v\in ℝ$ and a certain constant ${\stackrel{~}{k}}_{1}>0$.

We take $v=\psi -{Q}_{\epsilon }\left(H\left({\psi }^{+}\right)+{\psi }^{-}\right)$ as a test function in (1.14), where $\psi \in {C}^{\mathrm{\infty }}\left(\overline{\mathrm{\Omega }}\right)$, $\psi \left(x\right)=0$ in the neighborhood of ${\mathrm{\Gamma }}_{2}$ and $H\left(u\right)$ satisfies the functional equation (1.17). Note that from (7.1), (7.3) and (7.9) we have $v\ge 0$ on ${l}_{\epsilon }$, so $v\in {K}_{\epsilon }$. Hence we get

${\int }_{\mathrm{\Omega }}\nabla \left(\psi -{Q}_{\epsilon }\left(H\left({\psi }^{+}\right)+{\psi }^{-}\right)\right)\nabla \left(\psi -{Q}_{\epsilon }\left(H\left({\psi }^{+}\right)+{\psi }^{-}\right)-{u}_{\epsilon }\right)dx+{e}^{\frac{{\alpha }^{2}}{\epsilon }}{\int }_{{l}_{\epsilon }}\sigma \left({\psi }^{+}-H\left({\psi }^{+}\right)\right)\left({\psi }^{+}-H\left({\psi }^{+}\right)-{u}_{\epsilon }\right)𝑑{x}_{1}$$\ge {\int }_{\mathrm{\Omega }}f\left(\psi -{Q}_{\epsilon }\left(H\left({\psi }^{+}\right)+{\psi }^{-}\right)-{u}_{\epsilon }\right)𝑑x.$(7.10)

We rewrite inequality (7.10) in the following way:

${\int }_{\mathrm{\Omega }}\nabla \left(\psi -{W}_{\epsilon }\left(H\left({\psi }^{+}\right)+{\psi }^{-}\right)\right)\nabla \left(\psi -{Q}_{\epsilon }\left(H\left({\psi }^{+}\right)+{\psi }^{-}\right)-{u}_{\epsilon }\right)dx$$-{\int }_{\mathrm{\Omega }}\nabla \left(\left({Q}_{\epsilon }-{W}_{\epsilon }\right)\left(H\left({\psi }^{+}\right)+{\psi }^{-}\right)\right)\nabla \left(\psi -{Q}_{\epsilon }\left(H\left({\psi }^{+}\right)+{\psi }^{-}\right)-{u}_{\epsilon }\right)dx$$+{e}^{\frac{{\alpha }^{2}}{\epsilon }}{\int }_{{l}_{\epsilon }}\sigma \left({\psi }^{+}-H\left({\psi }^{+}\right)\right)\left({\psi }^{+}-H\left({\psi }^{+}\right)-{u}_{\epsilon }\right)𝑑{x}_{1}$$\ge {\int }_{\mathrm{\Omega }}f\left(\psi -{Q}_{\epsilon }\left(H\left({\psi }^{+}\right)+{\psi }^{-}\right)-{u}_{\epsilon }\right)𝑑x.$(7.11)

From the fact that ${Q}_{\epsilon }⇀0$ as $\epsilon \to 0$ weakly in ${H}^{1}\left(\mathrm{\Omega },{\mathrm{\Gamma }}_{2}\right)$, we have

$\underset{\epsilon \to 0}{lim}{\int }_{\mathrm{\Omega }}f\left(\psi -{Q}_{\epsilon }\left(H\left({\psi }^{+}\right)+{\psi }^{-}\right)-{u}_{\epsilon }\right)𝑑x={\int }_{\mathrm{\Omega }}f\left(\psi -{u}_{0}\right)𝑑x,$(7.12)$\underset{\epsilon \to 0}{lim}\nabla \psi \nabla \left(\psi -{Q}_{\epsilon }\left(H\left({\psi }^{+}\right)+{\psi }^{-}\right)-{u}_{\epsilon }\right)dx={\int }_{\epsilon \to 0}\nabla \psi \nabla \left(\psi -{u}_{0}\right)dx.$(7.13)

Lemma 7.1 implies that

$\underset{\epsilon \to 0}{lim}{\int }_{\mathrm{\Omega }}\nabla \left(\left({Q}_{\epsilon }-{W}_{\epsilon }\right)\left(H\left({\psi }^{+}\right)+{\psi }^{-}\right)\right)\nabla \left(\psi -{Q}_{\epsilon }\left(H\left({\psi }^{+}\right)+{\psi }^{-}\right)-{u}_{\epsilon }\right)dx=0.$(7.14)

Consider the remaining integrals in (7.11). Set

${I}_{\epsilon }\equiv -{\int }_{\mathrm{\Omega }}\nabla \left({W}_{\epsilon }\left(H\left({\psi }^{+}\right)+{\psi }^{-}\right)\right)\nabla \left(\psi -{Q}_{\epsilon }\left(H\left({\psi }^{+}\right)+{\psi }^{-}\right)-{u}_{\epsilon }\right)dx$$=-{\int }_{\mathrm{\Omega }}\nabla {W}_{\epsilon }\nabla \left\{\left(H\left({\psi }^{+}\right)+{\psi }^{-}\right)\left(\psi -{Q}_{\epsilon }\left(H\left({\psi }^{+}\right)+{\psi }^{-}\right)-{u}_{\epsilon }\right)\right\}dx+{\alpha }_{\epsilon },$

where ${\alpha }_{\epsilon }\to 0$ as $\epsilon \to 0$.

It is easy to see that

${I}_{\epsilon }=-{\int }_{\mathrm{\Omega }}\nabla {W}_{\epsilon }\nabla \left\{\left(H\left({\psi }^{+}\right)+{\psi }^{-}\right)\left(\psi -{Q}_{\epsilon }\left(H\left({\psi }^{+}\right)+{\psi }^{-}\right)-{u}_{\epsilon }\right)\right\}dx$$=-\sum _{j\in {\mathrm{Υ}}_{\epsilon }}{\int }_{{\left({T}_{\epsilon /4}^{j}\right)}^{+}\setminus \overline{{T}_{{a}_{\epsilon }}^{j}}}\nabla {w}_{\epsilon }^{j}\nabla \left\{\left(H\left({\psi }^{+}\right)+{\psi }^{-}\right)\left(\psi -{q}_{\epsilon }^{j}\left(H\left({\psi }^{+}\right)+{\psi }^{-}\right)-{u}_{\epsilon }\right)\right\}dx+\stackrel{~}{{\alpha }_{\epsilon }}$$=-\sum _{j\in {\mathrm{Υ}}_{\epsilon }}{\int }_{\partial {T}_{\epsilon /4}^{j}\cap \left\{{x}_{2}>0\right\}}{\partial }_{\nu }{w}_{\epsilon }^{j}\left(H\left({\psi }^{+}\right)+{\psi }^{-}\right)\left(\psi -{u}_{\epsilon }\right)𝑑s$$-\sum _{j\in {\mathrm{Υ}}_{\epsilon }}{\int }_{\partial {T}_{{a}_{\epsilon }}^{j}\cap \left\{{x}_{2}>0\right\}}{\partial }_{\nu }{w}_{\epsilon }^{j}\left(H\left({\psi }^{+}\right)+{\psi }^{-}\right)\left({\psi }^{+}-H\left({\psi }^{+}\right)-{u}_{\epsilon }\right)𝑑s+\stackrel{~}{{\alpha }_{\epsilon }},$(7.15)

where $\stackrel{~}{{\alpha }_{\epsilon }}\to 0$, $\epsilon \to 0$.

Since

${{\partial }_{\nu }{w}_{\epsilon }^{j}|}_{\partial {T}_{\epsilon /4}^{j}}=\frac{4}{\epsilon \mathrm{ln}\left(4{a}_{\epsilon }/\epsilon \right)}=\frac{4}{-{\alpha }^{2}+\epsilon \mathrm{ln}\left(4{C}_{0}\right)},$

using the results of [27], we derive

$-\underset{\epsilon \to 0}{lim}\sum _{j\in {\mathrm{Υ}}_{\epsilon }}{\int }_{\partial {T}_{\epsilon /4}^{j}\cap \left\{{x}_{2}>0\right\}}{\partial }_{\nu }{w}_{\epsilon }^{j}\left(H\left({\psi }^{+}\right)+{\psi }^{-}\right)\left(\psi -{u}_{\epsilon }\right)𝑑s$$=\underset{\epsilon \to 0}{lim}\frac{4}{{\alpha }^{2}-\epsilon \mathrm{ln}\left(4{C}_{0}\right)}\sum _{j\in {\mathrm{Υ}}_{\epsilon }}{\int }_{\partial {T}_{\epsilon /4}^{j}\cap \left\{{x}_{2}>0\right\}}\left(H\left({\psi }^{+}\right)+{\psi }^{-}\right)\left(\psi -{u}_{\epsilon }\right)𝑑s$$=\frac{\pi }{{\alpha }^{2}}{\int }_{{\mathrm{\Gamma }}_{2}}\left(H\left({\psi }^{+}\right)+{\psi }^{-}\right)\left(\psi -{u}_{0}\right)𝑑s.$(7.16)

Let us find the limit of the expression

$-\sum _{j\in {\mathrm{Υ}}_{\epsilon }}{\int }_{\partial {T}_{{a}_{\epsilon }}^{j}\cap \left\{{x}_{2}>0\right\}}{\partial }_{\nu }{w}_{\epsilon }^{j}\left(H\left({\psi }^{+}\right)+{\psi }^{-}\right)\left({\psi }^{+}-H\left({\psi }^{+}\right)-{u}_{\epsilon }\right)𝑑s+{e}^{\frac{{\alpha }^{2}}{\epsilon }}{\int }_{{l}_{\epsilon }}\sigma \left({\psi }^{+}-H\left({\psi }^{+}\right)\right)\left({\psi }^{+}-H\left({\psi }^{+}\right)-{u}_{\epsilon }\right)𝑑{x}_{1}$$=\sum _{j\in {\mathrm{Υ}}_{\epsilon }}\frac{{\left({\alpha }^{2}{C}_{0}\right)}^{-1}{e}^{{\alpha }^{2}/\epsilon }}{1-\epsilon {\alpha }^{-2}\mathrm{ln}\left(4{C}_{0}\right)}{\int }_{\partial {T}_{{a}_{\epsilon }}^{j}\cap \left\{{x}_{2}>0\right\}}\left(H\left({\psi }^{+}\right)+{\psi }^{-}\right)\left({\psi }^{+}-H\left({\psi }^{+}\right)-{u}_{\epsilon }\right)𝑑s$$+{e}^{{\alpha }^{2}/\epsilon }{\int }_{{l}_{\epsilon }}\sigma \left({\psi }^{+}-H\left({\psi }^{+}\right)\right)\left({\psi }^{+}-H\left({\psi }^{+}\right)-{u}_{\epsilon }\right)𝑑{x}_{1}$

$={e}^{{\alpha }^{2}/\epsilon }{\int }_{{l}_{\epsilon }}\sigma \left({\psi }^{+}-H\left({\psi }^{+}\right)\right)\left({\psi }^{+}-H\left({\psi }^{+}\right)-{u}_{\epsilon }\right)𝑑{x}_{1}$$-\frac{{e}^{{\alpha }^{2}/\epsilon }}{{\alpha }^{2}{C}_{0}}\sum _{j\in {\mathrm{Υ}}_{\epsilon }}{\int }_{\partial {T}_{{a}_{\epsilon }}^{j}\cap \left\{{x}_{2}>0\right\}}\left(H\left({\psi }^{+}\right)+{\psi }^{-}\right)\left({\psi }^{+}-H\left({\psi }^{+}\right)-{u}_{\epsilon }\right)𝑑s+{\stackrel{^}{\alpha }}_{\epsilon }$$\equiv {D}_{\epsilon }+{\stackrel{^}{\alpha }}_{\epsilon },$(7.17)

where ${\stackrel{^}{\alpha }}_{\epsilon }\to 0$ as $\epsilon \to 0$.

To conclude the proof we will estimate the limit of ${D}_{\epsilon }$. We have

${D}_{\epsilon }=\left\{\frac{\pi }{2{l}_{0}{\alpha }^{2}{C}_{0}}{e}^{{\alpha }^{2}/\epsilon }{\int }_{{l}_{\epsilon }}\left(H\left({\psi }^{+}\right)+{\psi }^{-}\right)\left({\psi }^{+}-H\left(\psi \right)-{u}_{\epsilon }\right)d{x}_{1}$$-\frac{{e}^{{\alpha }^{2}/\epsilon }}{{\alpha }^{2}{C}_{0}}\sum _{j\in {\mathrm{Υ}}_{\epsilon }}{\int }_{\partial {T}_{{a}_{\epsilon }}^{j}\cap \left\{{x}_{2}>0\right\}}\left(H\left({\psi }^{+}\right)+{\psi }^{-}\right)\left({\psi }^{+}-H\left({\psi }^{+}\right)-{u}_{\epsilon }\right)ds\right\}$$+{e}^{{\alpha }^{2}/\epsilon }{\int }_{{l}_{\epsilon }}\left\{\sigma \left({\psi }^{+}-H\left({\psi }^{+}\right)\right)-\frac{\pi }{2{l}_{0}{\alpha }^{2}{C}_{0}}H\left({\psi }^{+}\right)\right\}\left({\psi }^{+}-H\left({\psi }^{+}\right)-{u}_{\epsilon }\right)𝑑{x}_{1}$$-\frac{\pi }{2{l}_{0}{\alpha }^{2}{C}_{0}}{e}^{{\alpha }^{2}/\epsilon }{\int }_{{l}_{\epsilon }}{\psi }^{-}\left({\psi }^{+}-H\left(\psi \right)-{u}_{\epsilon }\right)𝑑{x}_{1}$$={\mathcal{𝒥}}_{\epsilon }^{1}+{\mathcal{𝒥}}_{\epsilon }^{2}+{\mathcal{𝒥}}_{\epsilon }^{3}.$(7.18)

Lemma 7.2 implies that

$|{\mathcal{𝒥}}_{\epsilon }^{1}|\le K\sqrt{\epsilon }.$(7.19)

Then ${\mathcal{𝒥}}_{\epsilon }^{2}$ vanishes due to equation (1.17). Using that ${u}_{\epsilon }\ge 0$, ${\psi }^{-}\le 0$ on ${l}_{\epsilon }$ and the fact that ${\psi }^{-}\left({\psi }^{+}-H\left({\psi }^{+}\right)\right)\equiv 0$, we have

${\mathcal{𝒥}}_{\epsilon }^{3}\le 0.$(7.20)

Hence we have that ${lim}_{\epsilon \to 0}{D}_{\epsilon }\le 0$ and

$\underset{\epsilon \to 0}{lim}\left({e}^{\frac{{\alpha }^{2}}{\epsilon }}{\int }_{{l}_{\epsilon }}\sigma \left({\psi }^{+}-H\left({\psi }^{+}\right)\right)\left({\psi }^{+}-H\left({\psi }^{+}\right)-{u}_{\epsilon }\right)𝑑{x}_{1}+{I}_{\epsilon }\right)\le \frac{\pi }{{\alpha }^{2}}{\int }_{{\mathrm{\Gamma }}_{2}}\left(H\left({\psi }^{+}\right)+{\psi }^{-}\right)\left(\psi -{u}_{0}\right)𝑑s.$(7.21)

Therefore, from (7.10)–(7.21) we conclude that ${u}_{0}\in {H}^{1}\left(\mathrm{\Omega },{\mathrm{\Gamma }}_{2}\right)$ satisfies the inequality

${\int }_{\mathrm{\Omega }}\nabla \psi \nabla \left(\psi -{u}_{0}\right)dx+\frac{\pi }{{\alpha }^{2}}{\int }_{{\mathrm{\Gamma }}_{1}}\left(H\left({\psi }^{+}\right)+{\psi }^{-}\right)\left(\psi -{u}_{0}\right)𝑑{x}_{1}\ge {\int }_{\mathrm{\Omega }}f\left(\psi -{u}_{0}\right)𝑑x$

for any $\psi \in {H}^{1}\left(\mathrm{\Omega },{\mathrm{\Gamma }}_{2}\right)$, where $H\left(u\right)$ satisfies the functional equation (1.17). This concludes the proof. ∎

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## Footnotes

• 1

The function ${v}_{\epsilon }^{j}$ in this section depends on u.

Accepted: 2018-07-18

Published Online: 2018-10-21

Published in Print: 2019-03-01

Funding Source: Ministerio de Ciencia e Innovación

Award identifier / Grant number: MTM 2014-57113-P

Award identifier / Grant number: MTM2017-85449-P

The research of the first and second authors was partially supported by the projects ref. MTM 2014-57113-P and MTM2017-85449-P of the DGISPI (Spain).

Citation Information: Advances in Nonlinear Analysis, Volume 9, Issue 1, Pages 193–227, ISSN (Online) 2191-950X, ISSN (Print) 2191-9496,

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