Optimal harvesting policy for the Beverton--Holt model

1.   Introduction

A Reinforced material is a composite building material consisting of two or more materials with different properties. The main objective of studies of reinforced materials is the prediction of their macroscopic behavior from the properties of their individual components as well as from their microstructural characteristics.

The theory of ideal fiber-reinforced composites was initiated by Adkins and Rivlin [4] who studied the deformation of a structure reinforced with thin, flexible and inextensible cords, which lie parallel and close together in smooth surfaces. This theory was further developed by the authors in [44], [1], [2], [3], [45].

The homogenization of elastic materials reinforced with highly contrasted inclusions has been considered by several authors in the two last decades (see for instance [6], [10], [17], [21], [18], and the references therein). The main result is that the materials obtained by the homogenization procedure have new elastic properties.

The homogenization of structures reinforced with fractal inclusions has been considered by various authors, among which [39], [31], [40], [41], [12], [42], [13], and [14]. Lancia, Mosco and Vivaldi studied in [31] the homogenization of transmission problems across highly conductive layers of iterated fractal curves. In [40], Mosco and Vivaldi dealt with the asymptotic behavior of a two-dimensional membrane reinforced with thin polygonal strips of large conductivity surrounding a pre-fractal curve obtained after $n$-iterations of the contractive similarities of the Sierpinski gasket. In [39], they considered an analogous problem with the Koch curve. The same authors considered in [41] a two-dimensional domain reinforced by an increasing number of thin conductive fibers developing a fractal geometry and studied the spectral asymptotic properties of conductive layered-thin-fibers of fractal nature in [42].

The homogenization of insulating fractal surfaces of Koch type approximated by three-dimensional insulating layers has been performed by Capitanelli et al. in [12], [13], and [14]. Due to the physical characteristics of the inclusions, singular energy forms containing fractal energies are obtained in these articles as the limit of non-singular full-dimensional energies. On the other hand, the effective properties of elastic materials fixed on rigid thin self-similar micro-inclusions disposed along two and three dimensional Sierpinski carpet fractals have been recently obtained in [20].

In the present work, we consider the deformation of a three-dimensional elastic material reinforced with highly contrasted thin vertical strips constructed on horizontal iterated Sierpinski gasket curves. Our main purpose is to describe the macroscopic behavior of the composite as the width of the strips tends to zero, their material coefficients tend to infinity, and the sequence of the iterated Sierpinski gasket curves converges to the Sierpinski gasket in the Hausdorff metric.

The asymptotic analysis of problems of this kind was previousely studied in [11], [26], [9], and [5], where the authors considered media comprising low dimensional thin inclusions or thin layers of higher conductivity or higher rigidity. The limit problems consist in second order transmission problems. Problems involving thin highly conductive fractal inclusions have been addressed in a series of papers (see for instance [39], [31], [12], [14], and [19]). The obtained mathematical models are elliptic or parabolic boundary value problems involving transmission conditions of order two on the interfaces. The homogenization of three-dimensional elastic materials reinforced by highly rigid fibers with variable cross-section, which may have fractal geometry, has been carried out in [21]. The authors showed that the geometrical changes induced by the oscillations along the fiber-cross-sections can provide jumps of displacement fields or stress fields on interfaces, including fractal ones, due to local concentrations of elastic rigidities. Note that the numerical approximation of second order transmission problems across iterated fractal interfaces has been considered in some few papers among which [32] and [15].

Let us first consider the points $A_{1} = \left( 0,0\right)$, $A_{2} = \left( 1,0\right)$ and $A_{3} = \left( 1/2,\sqrt{3}/2\right)$ of the $xy$-plane. Let $\mathcal{V}_{0} = \left\{ A_{1}\text{, }A_{2}\text{, }A_{3}\right\}$ be the set of vertices of the equilateral triangle $A_{1}A_{2}A_{3}$ of side one. We define inductively

Let us set

The Sierpinski gasket, which is denoted here by $\Sigma$, is then defined (see for instance [30]) as the closure of the set $\mathcal{V}_{\infty }$, that is,

We define the graph $\Sigma _{h} = \left( \mathcal{V}_{h},S_{h}\right)$, where $S_{h}$ is the set of edges $\left[ p,q\right]$; $p,q\in \mathcal{V} _{h}$, such that $\left\vert p-q\right\vert = 2^{-h}$, where $\left\vert p-q\right\vert$ is the Euclidian distance between $p$ and $q$. The graph $\Sigma _{h}$ is then the standard aproximation of the Sierpinski gasket, which means that the sequence $\left( \Sigma _{h}\right) _{h}$ converges, as $h$ tends to $\infty$, to the Sierpinski gasket $\Sigma$ in the Hausdorff metric.

The edges which belong to $S_{h}$ can be rearranged as $S_{h}^{k}$; $k = 1,2,...,N_{h}$, where $N_{h} = 3^{h+1}$.

Let $\omega$ be a bounded domain in $\mathbb{R}^{2}$ with Lipschitz continuous boundary $\partial \omega$ such that $\Sigma \subset \overline{ \omega }$ and

Let $\left( \varepsilon _{h}\right) _{h\in \mathbb{N}}$ be a sequence of positive numbers, such that

We define

and set

where $I_{h} = \left\{ 1,2,...,N_{h}\right\}$. Denoting $\left\vert T_{h}\right\vert$ the $2$-dimensional measure of $T_{h}$, one can see that

Let $\Omega = \omega \times \left( -1,1\right)$. We suppose that $\Omega \backslash T_{h}$ is the reference configuration of a linear, homogeneous and isotropic elastic material with Lamé coefficients $\mu >0$ and $\lambda \geq 0$. This means that the deformation tensor $e\left( u\right) = \left( e_{ij}\left( u\right) \right) _{i,j = 1,2,3}$, with $e_{ij}\left( u\right) = \dfrac{1}{2}\left( \dfrac{\partial u_{i}}{\partial x_{j}}+\dfrac{ \partial u_{j}}{\partial x_{i}}\right)$ for some displacement $u$, is linked to the stress tensor $\sigma \left( u\right) = \left( \sigma _{ij}\left( u\right) \right) _{i,j = 1,2,3}$ through Hooke's law

where the summation convention with respect to repeated indices has been used and will be used in the sequel, and $\delta _{ij}$ denotes Kronecker's symbol. We suppose that $T_{h}$ is the reference configuration of a linear, homogeneous and isotropic elastic material with $\sigma ^{h}\left( u\right) = \left( \sigma _{ij}^{h}\left( u\right) \right) _{i,j = 1,2,3}$:

with

where $\lambda _{0}$ and $\mu _{0}$ are positive constants and

The special scaling (10) and (11) of the Lamé -coefficients depend on the structural constants of $T_{h}$. The choice of $\eta _{h}$ is dictated by the lower bound inequality of assertion 3 of Proposition 6, which will play a crucial role in the asymptotic behavior of the energy $F_{h}$.

We suppose that a perfect adhesion occurs between $\Omega \backslash T_{h}$ and $T_{h}$ along their common interfaces. We suppose that the material in $\Omega$ is submitted to volumic forces with density $f\in L^{2}$ $\left( \Omega ,\mathbb{R}^{3}\right)$ and is held fixed on $\partial \Omega$. We define the energy functional $F_{h}$ on $L^{2}\left( \Omega ,\mathbb{R} ^{3}\right)$ through

where $ds$ is the one-dimensional Lebesgue measure on the line segments $S_{h}^{k}$; $k = 1,2,...,N_{h}$. The equilibrium state in $\Omega$ is described by the minimization problem

We use $\Gamma$-convergence methods (see for instance [5] and [16]) in order to describe the asymptotic behavior of problem (13) as $h$ goes to $\infty$. According to the critical term

which is associated with the size of the boundary layers taking place in the neighbourhoods of the fractal strips, we prove that if $\gamma \in \left( 0,+\infty \right)$ then the effective energy of the composite is given by

where $\overline{v} = \left( v_{1},v_{2}\right)$, $\mathcal{L}_{\Sigma }\left( \overline{v}\right)$ is a quadratic measure-valued gradient form supported on $\Sigma$ (see Proposition 1 in the next Section), $\mathcal{H}^{d}$ is the $d$-dimensional Hausdorff measure; $d$ being the fractal dimension of $\Sigma$ with

$\mathcal{D}_{\Sigma ,\mathcal{E}}$ is the domain of the energy supported on the fractal $\Sigma$ (see (27) in the next Section), and

where $\kappa = \dfrac{3\mu +\lambda }{\mu +\lambda }$ and $n\left( s\right)$ is the unit normal on $s\in \Sigma$.

The effective energy (15) contains new degrees of freedom implying nonlocal effects associated with thin boundary layer phenomena taking place near the fractal strips and a singular energy term supported on the Sierpinski gasket $\Sigma$. The equilibrium of the fractal $\Sigma$ is asymptotically described by a generalized Laplace equation which is related to the discontinuity of the effective stresses through the following relations (see Corollary 1):

where $\Delta _{\Sigma }$ is the Laplace operator on the Sierpinski gasket, that is, the second order operator in $L_{\mathcal{H}^{d}}^{2}\left( \Sigma , \mathbb{R}^{2}\right)$ defined by the form $\mathcal{E}_{\Sigma }$ in Lemma 2.1 in the next Section under the Dirichlet condition $V_{\alpha } = 0$ on $\mathcal{V}_{0} = \Sigma \cap \partial \omega$; $\alpha = 1,2$, $\mu _{0}$ is the effective shear modulus of the material occupying the fractal $\Sigma$,

is the jump of $\sigma _{\alpha 3}|_{x_{3} = 0}$ on $\Sigma \cap \partial \mathcal{T}_{m}$; $\left\{ \mathcal{T}_{m}\right\} _{m\in \mathbb{N}}$ being the network of the interiors of the triangles which are contained in the Sierpinski gasket $\Sigma$ (see Figure 2).

If $\gamma = +\infty$ then, for every $\left( u,v\right) \in H_{0}^{1}\left( \Omega ,\mathbb{R}^{3}\right) \times \mathcal{D}_{\Sigma ,\mathcal{E}}\times L_{\mathcal{H}^{d}}^{2}\left( \Sigma \right)$, $F_{\infty }\left( u,v\right) <\infty \Rightarrow u = v$ on $\Sigma$. In this case the energy supported on the structure is given by

If $\gamma = 0$ the displacements $u$ and $v$ are independent. In this case the effective energy of the structure turns out to be

The paper is organized as follows: in Section 2 we introduce the energy form and the notion of a measure-valued local energy on the Sierpinski gasket $\Sigma$. Section 3 is devoted to compactness results which is useful for the proof of the main result. In Section 4 we formulate the main result of this work. Section 5 is consacred to the proof of the main result. This proof is developed in 3 Subsections: in the first Subsection we study the boundary layers at the interface matrix/strips, in the second Subsection we establish the first condition of the $\Gamma$-convergence property, and in the third Subsection we prove the second condition of the $\Gamma$ -convergence property.

2.   The energy form on the Sierpinski gasket

In this Section we introduce the energy form and the notion of a measure-valued local energy (or Lagrangian) on the Sierpinski gasket. For the definition and properties of Dirichlet forms and measure energies we refer to [24], [35], and [37].

For any function $w:\mathcal{V}_{\infty }\longrightarrow \mathbb{R}^{2}$ we define

Let us define the energy

with domain $\mathcal{D}_{\infty } = \left\{ z:\mathcal{V}_{\infty }\longrightarrow \mathbb{R}^{2}:\mathcal{E}_{\Sigma }\left( z\right) <\infty \right\}$. Every function $z\in \mathcal{D}_{\infty }$ can be uniquely extended to be an element of $C\left( \Sigma ,\mathbb{R}^{2}\right)$, still denoted $z$. Let us set

where $\mathcal{E}_{\Sigma }\left( z\right) = \mathcal{E}_{\Sigma }\left( z\mid _{\mathcal{V}_{\infty }}\right)$. Then $\mathcal{D\subset }C\left( \Sigma ,\mathbb{R}^{2}\right) \subset L_{\mathcal{H}^{d}}^{2}\left( \Sigma , \mathbb{R}^{2}\right)$. We define the space $\mathcal{D}_{\mathcal{E}}$ as

where $\left\Vert .\right\Vert _{\mathcal{D}_{\mathcal{E}}}$ is the intrinsic norm

The space $\mathcal{D}_{\mathcal{E}}$ is injected in $L_{\mathcal{H} ^{d}}^{2}\left( \Sigma ,\mathbb{R}^{2}\right)$ and is an Hilbert space with the scalar product associated to the norm (26).Let us now define the space

We denote $\mathcal{E}_{\Sigma }\left( .,.\right)$ the bilinear form defined on $\mathcal{D}_{\Sigma ,\mathcal{E}}\times \mathcal{D}_{\Sigma ,\mathcal{E}}$ by

One can see that

where

The form $\mathcal{E}_{\Sigma }\left( .,.\right)$ is a closed Dirichlet form in the Hilbert space $L_{\mathcal{H}^{d}}^{2}\left( \Sigma ,\mathbb{R} ^{2}\right)$ and, according to [25,Theorem 4.1], $\mathcal{E} _{\Sigma }\left( .,.\right)$ is a local regular Dirichlet form in $L_{ \mathcal{H}^{d}}^{2}\left( \Sigma ,\mathbb{R}^{2}\right)$, which means that

1. (local property) $w,z\in \mathcal{D}_{\Sigma ,\mathcal{E}}$ with supp$\left[ w\right]$ and supp$\left[ z\right]$ are disjoint compact sets $\Longrightarrow$ $E_{\Sigma }\left( w,z\right) = 0$,

2. (regularity) $\mathcal{D}_{\Sigma ,\mathcal{E}}\cap C_{0}\left( \Sigma ,\mathbb{R}^{2}\right)$ is dense both in $C_{0}\left( \Sigma ,\mathbb{R} ^{2}\right)$ (the space of functions of $C\left( \Sigma ,\mathbb{R} ^{2}\right)$ with compact support) with respect to the uniform norm and in $\mathcal{D}_{\Sigma ,\mathcal{E}}$ with respect to the intrinsic norm (26).

The second property implies that $\mathcal{D}_{\Sigma ,\mathcal{E}}$ is not trivial (that is $\mathcal{D}_{\Sigma ,\mathcal{E}}$ is not made by only the constant functions). Moreover, every function of $\mathcal{D}_{\Sigma , \mathcal{E}}$ possesses a continuous representative. Indeed, according to [36,Theorem 6.3. and example 7$_{1}$], the space $\mathcal{D}_{\Sigma ,\mathcal{E}}$ is continuously embedded in the space $C^{\beta }\left( \Sigma ,\mathbb{R}^{2}\right)$ of Hölder continuous functions with $\beta = \ln \dfrac{5}{3}/\ln 4$.

Now, applying [29,Chap. 6], we have the following result:

Lemma 2.1. There exists a unique self-adjoint operator ${\bf{ \Delta }}_{\Sigma }$ on $L_{\mathcal{H}^{d}}^{2}\left( \Sigma ,\mathbb{R} ^{2}\right)$ with domain

dense in $L_{\mathcal{H}^{d}}^{2}\left( \Sigma ,\mathbb{R}^{2}\right)$, such that, for every $w\in \mathcal{D}_{\Delta _{\Sigma }}$ and $z\in \mathcal{D}_{\Sigma ,\mathcal{E}}$,

Let us consider the sequence $\left( \nu _{h}\right) _{h}$ of measures defined by

where $Card\left( \mathcal{V}_{h}\right) ) = \dfrac{3^{h+1}+3}{2}$ is the number of verticies of $\mathcal{V}_{h}$ and $\delta _{p}$ is the Dirac measure at the point $p$. We have the following result:

Lemma 2.2. The sequence $\left( \nu _{h}\right) _{h}$ weakly converges in $C\left( \Sigma \right) ^{\ast }$ to the measure

where $C\left( \Sigma \right) ^{\ast }$ is the topological dual of the space $C\left( \Sigma \right)$.

Proof. Let $\varphi \in C\left( \Sigma \right)$. Then, according to the ergodicity result of [22,Theorem 6.1],

We note that the approximating form $\mathcal{E}_{\Sigma }^{h}\left( .,.\right)$ can be written as

where $\nu _{h}$ is the measure defined in (31) and

where $\varkappa$ is the unique positive number for which the sequence $\left( \mathcal{E}_{\Sigma }^{h}\left( .,.\right) \right) _{h}$ has a non trivial limit (see [38] for more details). We note that, according to equality (22), $\varkappa = \ln \dfrac{5}{3}/\ln 2$. We have the following result:

Proposition 1. For every $w,z\in \mathcal{D}_{\Sigma ,\mathcal{E}}$, the sequence of measures $\left( \mathcal{L}_{\Sigma }^{h}\left( w,z\right) \right) _{h}$ defined by

weakly converges in the topological dual $C\left( \Sigma ,\mathbb{R} ^{2}\right) ^{\ast }$ of the space $C\left( \Sigma ,\mathbb{R}^{2}\right)$ to a signed finite Radon measure $\mathcal{L}_{\Sigma }\left( w,z\right)$ on $\Sigma$, called Lagrangian measure on $\Sigma$. Moreover,

Proof. The proof follows the lines of the proof of [23,Proposition 2.3] for the von Koch snowflake. Let us set, for every $w\in \mathcal{D}_{\Sigma , \mathcal{E}}$, $\mathcal{L}_{\Sigma }^{h}\left( w\right) = \mathcal{L} _{\Sigma }^{h}\left( w,w\right)$. We deduce from (23), (29), and (32) that $\left( \mathcal{L}_{\Sigma }^{h}\left( w\right) \left( \Sigma \right) \right) _{h}$ is a uniformly bounded sequence. Then, observing that, for every $w\in \mathcal{D}_{\Sigma , \mathcal{E}}$ and every $\varphi e_{1}\in \mathcal{D}_{\Sigma ,\mathcal{E} }\cap C\left( \Sigma ,\mathbb{R}^{2}\right)$, with $e_{1} = \left( 1,0\right)$,

we deduce, taking into account the regularity of the form $\mathcal{E} _{\Sigma }\left( .,.\right)$, that

On the other hand, according to [33,Proposition 1.4.1], the energy form $\mathcal{E}_{\Sigma }\left( w\right)$, which is a Dirichlet form of diffusion type, admits the following integral representation:

where $\mathcal{L}_{\Sigma }\left( w\right)$ is a positive Radon measure which is uniquely determined by the relation

Thus, combining with (34), the sequence $\left( \mathcal{L} _{\Sigma }^{h}\left( w\right) \right) _{h}$ converges in the sense of measures to the measure $\mathcal{L}_{\Sigma }\left( w\right)$. Now, observing that

we deduce that the sequence $\left( \mathcal{L}_{\Sigma }^{h}\left( w,z\right) \right) _{h}$ weakly converges in $C\left( \Sigma ,\mathbb{R} ^{2}\right) ^{\ast }$ to the measure $\mathcal{L}_{\Sigma }\left( w,z\right)$.

3.   Compactness results

In this Section we establish the compactness results which is very useful for the proof of the main homogenization result.

Lemma 3.1. For every sequence $\left( u_{h}\right) _{h}$; $u_{h}\in H_{0}^{1}\left( \Omega ,\mathbb{R}^{3}\right) \cap H^{1}\left( T_{h},\mathbb{ R}^{3}\right)$, such that $\sup_{h}F_{h}\left( u_{h}\right) <+\infty$, we have

1. $\sup_{h}\left\Vert u_{h}\right\Vert _{H_{0}^{1}\left( \Omega ,\mathbb{ R}^{3}\right) }<+\infty$,

2. $\dfrac{1}{\left\vert T_{h}\right\vert }\int\nolimits_{T_{h}}\left \vert u_{h}\right\vert ^{2}dsdx_{3}\leq C\left\{ \left\Vert u_{h}\right\Vert _{L^{2}\left( \Omega ,\mathbb{R}^{3}\right) }^{2}-\dfrac{2^{h}}{3^{h+1}}\ln \varepsilon _{h}\right\}$, where $C$ is a positive constant independent of $h$.

Proof. 1. Observing that

we have, using Korn's inequality (see for instance [43]), that

2. Let $n^{k}$ be the unit normal to $S_{h}^{k}$; $k\in I_{h}$. Then $n^{k} = \pm \left( -\sqrt{3}/2,1/2\right)$, $n^{k} = \pm \left( \sqrt{3} /2,1/2\right)$ or $n^{k} = \pm \left( 0,1\right)$. Let us denote $s,s^{\perp }$ the local coordinates defined by

by

and by

where the symbol $\perp$ represents the direction normal to the edge $S_{h}^{k}$. Let $s_{h}^{k} = \left( s_{1h}^{k},s_{2h}^{k}\right)$ denotes the center of $S_{h}^{k}$; $k\in I_{h}$, in the new coordinates. We define, for $\theta \in \left[ 0,2\pi \right)$ and $r\in \left( 0,\varepsilon _{h}/2\right)$,

Then, according to (36), we have, for $r_{1}\leq r_{2}<\varepsilon _{h}/2$ and $\theta \in \left[ 0,2\pi \right)$,

Solving the Euler equation of the following one dimensional minimization problem:

we deduce that, for every $\theta \in \left[ 0,2\pi \right)$,

Then, using (41) and (42), we obtain that

Let us define

We deduce from the inequality (43) that, for $r_{1}\leq r_{2}<\varepsilon _{h}/2$,

Observing that, for $k\in I_{h}$ and $\theta _{0}$ fixed in $\left[ 0,2\pi \right)$,

we deduce that

where $x^{\prime }\left( s\right) = \left( x_{1}\left( s\right) ,x_{2}\left( s\right) \right)$, $C_{h}^{k}$ is the cylinder of radius $\varepsilon _{h}$ around the edge $S_{h}^{k}$. This estimate implies that

Now, using (45) and (47), we deduce, by setting $m_{h} = \dfrac{1}{\varepsilon _{h}}\dfrac{2^{h}}{3^{h+1}}$, that, for $\theta _{0} = 0$ and $\pi$, and for every $r_{2}\in \left[ a_{h},b_{h}\right]$; $a_{h} = \dfrac{1}{4\sqrt{3}}\left( \dfrac{2}{3}\right) ^{h/2}$ and $b_{h} = 2a_{h}$,

Integrating with respect to $r_{2}$ over the interval $\left[ a_{h},b_{h} \right]$, we obtain that

Let $\mathcal{M}\left( \mathbb{R}^{3}\right)$ be the space of Radon measures on $\mathbb{R}^{3}$. We have the following result:

Lemma 3.2. Let $u_{h}\in L^{2}\left( \Omega ,\mathbb{R}^{3}\right) \cap L^{2}\left( T_{h},\mathbb{R}^{3}\right)$, such that

Then, there exists a subsequence of $\left( u_{h}\right) _{h}$, still denoted $\left( u_{h}\right) _{h}$, such that

with $v\left( s,0\right) \in L_{\mathcal{H}^{d}}^{2}\left( \Sigma , \mathbb{R}^{3}\right)$.

Proof. Let us consider the sequence of Radon measures $\left( \vartheta _{h}\right) _{h}$ on $\mathbb{R}^{3}$ defined by

Let $x_{h}^{k} = \left( x_{1h}^{k},x_{2h}^{k}\right)$ denotes the center of $S_{h}^{k}$; $k\in I_{h}$, in Cartesian coordinates. Then, using the ergodicity result of [22,Theorem 6.1], we have, for every $\varphi \in C_{0}\left( \mathbb{R}^{3}\right)$,

from which we deduce that $\vartheta _{h}\overset{\ast }{\underset{ h\rightarrow \infty }{\rightharpoonup }}\vartheta$, with

Let $u_{h}\in L^{2}\left( \Omega ,\mathbb{R}^{3}\right) \cap L^{2}\left( T_{h},\mathbb{R}^{3}\right)$, such that

As $\vartheta _{h}$ $\left( \mathbb{R}^{3}\right) = \dfrac{1}{\left\vert T_{h}\right\vert }\int\nolimits_{T_{h}}dsdx_{3} = 1$, we have

from which we deduce that the sequence $\left( u_{h}\vartheta _{h}\right) _{h}$ is uniformly bounded in variation, hence $\ast$-weakly relatively compact. Possibly passing to a subsequence, we can suppose that the sequence $\left( u_{h}\vartheta _{h}\right) _{h}$ converges to some $\chi$. Let $\varphi \in C_{0}\left( \mathbb{R}^{3},\mathbb{R}^{3}\right)$. Then, using Fenchel's inequality (also known as the Fenchel-Young inequality, see for instance [7]), we have

As the left hand side of this inequality is bounded, we deduce that

from which we deduce, according to Riesz' representation theorem, that there exists $v$ such that $\chi = v\left( s,x_{3}\right) \vartheta$ and $v\left( s,0\right) \in L_{\mathcal{H}^{d}}^{2}\left( \Sigma ,\mathbb{R}^{3}\right)$.

Proposition 2. Let $\left( u_{h}\right) _{h}$; $u_{h}\in H_{0}^{1}\left( \Omega ,\mathbb{R}^{3}\right) \cap H^{1}\left( T_{h},\mathbb{R}^{3}\right)$, be a sequence, such that $\sup_{h}F_{h}\left( u_{h}\right) <+\infty$. There exists a subsequence, still denoted $\left( u_{h}\right) _{h}$, such that

1. $u_{h}\underset{h\rightarrow \infty }{\rightharpoonup }u$ $H_{0}^{1}\left( \Omega ,\mathbb{R}^{3}\right)$-weak,

2. If $\gamma \in \left( 0,+\infty \right)$ then

with $v\left( s,0\right) \in L_{\mathcal{H}^{d}}^{2}\left( \Sigma ,\mathbb{R} ^{3}\right)$.

3. If $\gamma \in \left( 0,+\infty \right)$ then, with $\eta _{h}$ given in (11), we have $\overline{v}\left( s,0\right) \in \mathcal{D} _{\Sigma ,\mathcal{E}}$ and

Proof. 1. Thanks to Lemma 3.1$_{1}$, one immediately obtains that, up to some subsequence, $u_{h}\underset{h\rightarrow \infty }{\rightharpoonup }u$ $\ H_{0}^{1}\left( \Omega ,\mathbb{R}^{3}\right)$-weak.

2. If $\gamma \in \left( 0,+\infty \right)$ then, according to Lemma 3.1$_{2}$ and Lemma 3.2, one has, up to some subsequence,

with $v\left( s,0\right) \in L_{\mathcal{H}^{d}}^{2}\left( \Sigma ,\mathbb{R} ^{3}\right)$.

3. One can easily check that

Computing the strain tensor in the local coordinates (37) and (38), observing that for $S_{h}^{k}\perp \left( -\sqrt{3}/2,1/2\right)$ or $S_{h}^{k}\perp \left( \sqrt{3}/2,1/2\right)$ the covariant derivative $\dfrac{\partial u_{\alpha ,h}^{k}}{\partial s^{\perp }} = 0$ on $S_{h}^{k}$; $\alpha = 1,2$, we obtain

For $S_{h}^{k}\perp \left( 0,1\right)$, since $\dfrac{\partial u_{\alpha ,h}^{k}}{\partial x_{2}} = 0$ on $S_{h}^{k}$; $\alpha = 1,2$, we have

According to (48) and (10), we deduce from (49) and (50) that

Let us set $\overline{u}_{h} = \left( u_{1,h},u_{2,h}\right)$ and $\widetilde{ \overline{u}}_{h} = \dfrac{1}{2\varepsilon _{h}}\int_{-\varepsilon _{h}}^{\varepsilon _{h}}\overline{u}_{h}\left( .,x_{3}\right) dx_{3}$. We introduce the harmonic extension of $\widetilde{\overline{u}}_{h}\mid _{ \mathcal{V}_{h}}$ obtained by the decimation procedure (see for instance [30,Proposition 1] and [8,Corollary1]):

We define the function $H_{h+1}\widetilde{\overline{u}}_{h}:\mathcal{V} _{h+1}\longrightarrow \mathbb{R}^{2}$ as the unique minimizer of the problem

Then $\mathcal{E}_{\Sigma }^{h+1}\left( H_{h+1}\widetilde{\overline{u}} _{h}\right) = \mathcal{E}_{\Sigma }^{h}\left( \widetilde{\overline{u}} _{h}\right)$. For $m>h$, we define the function $H_{m}\widetilde{\overline{u }}_{h}$ from $\mathcal{V}_{m}$ into $\mathbb{R}^{2}$ by

For every $m>h$ we have $H_{m}\widetilde{\overline{u}}_{h}\mid _{\mathcal{V} _{h}} = \widetilde{\overline{u}}_{h}\mid _{\mathcal{V}_{h}}$ and

Now we define, for a fixed $h\in \mathbb{N}$, the function $H\widetilde{ \overline{u}}_{h}$ on $\mathcal{V}_{\infty }$ as follows. For $p\in \mathcal{ V}_{\infty }$, we choose $m\geq h$ such that $p\in \mathcal{V}_{m}$ and set

As $\underset{h}{\sup }\int_{T_{h}}\sigma _{ij}^{h}\left( u_{h}\right) e_{ij}\left( u_{h}\right) dsdx_{3}\mathcal{<\infty }$, we have, according to (51) and (53),

from which we deduce, using Section $2$, that $H\widetilde{\overline{u}}_{h}$ has a unique continuous extension on $\Sigma$, still denoted $H\widetilde{ \overline{u}}_{h}$, and that the sequence $\left( H\widetilde{\overline{u}} _{h}\right) _{h}$ is bounded in $\mathcal{D}_{\Sigma ,\mathcal{E}}$. Therefore, there exists a subsequence, still denoted $\left( H\widetilde{ \overline{u}}_{h}\right) _{h}$, weakly converging to some $\overline{u} ^{\ast }\in \mathcal{D}_{\Sigma ,\mathcal{E}}$, with

On the other hand, using Lemma 3.2, we have, for every $\varphi \in C_{0}\left( \Sigma ,\mathbb{R}^{2}\right)$,

which implies that $\overline{u}^{\ast }\left( s\right) = \overline{v}\left( s,0\right)$. Therefore $\overline{v}\left( s,0\right) \in \mathcal{D} _{\Sigma ,\mathcal{E}}$ and, according to (51) and (56),

4.   The main result

In this Section we state the main result of this work. According to Proposition 2 we introduce the following topology $\tau$:

Definition 4.1. We say that a sequence $\left( u_{h}\right) _{h}$; $u_{h}\in H_{0}^{1}\left( \Omega ,\mathbb{R}^{3}\right) \cap H^{1}\left( T_{h},\mathbb{ R}^{3}\right)$, $\tau$-converges to $\left( u,v\right)$ if

with $v\equiv v\left( s,0\right) \in L_{\mathcal{H}^{d}}^{2}\left( \Sigma , \mathbb{R}^{3}\right)$.

Our main result in this work reads as follows:

Theorem 4.2. If $\gamma \in \left( 0,+\infty \right)$ then

1. ($\lim \sup$ inequality) for every $\left( u,v\right)$ $\in H_{0}^{1}\left( \Omega ,\mathbb{R}^{3}\right) \times \mathcal{D}_{\Sigma , \mathcal{E}}\times L_{\mathcal{H}^{d}}^{2}\left( \Sigma \right)$ there exists a sequence $\left( u_{h}\right) _{h}$; $u_{h}\in H_{0}^{1}\left( \Omega ,\mathbb{R}^{3}\right) \cap H^{1}\left( T_{h},\mathbb{R}^{3}\right)$ , such that $\left( u_{h}\right) _{h}$ $\tau$-converges to $\left( u,v\right)$ and

where $F_{\infty }$ is the functional defined in (15),

2. ($\lim \inf$ inequality) for every sequence $\left( u_{h}\right) _{h}$ ; $u_{h}\in H_{0}^{1}\left( \Omega ,\mathbb{R}^{3}\right) \cap H^{1}\left( T_{h},\mathbb{R}^{3}\right)$, such that $\left( u_{h}\right) _{h}$ $\tau$ -converges to $\left( u,v\right)$, we have $\overline{v}\in \mathcal{D} _{\Sigma ,\mathcal{E}}$ and

Before proving this Theorem, let us write the homogenized problem obtained at the limit as $h\longrightarrow \infty$.

Corollary 1. Problem (13) admits a unique solution $U_{h}$ which, under the hypothesis of Theorem 4.2, $\tau$-converges to $\left( U,V\right) \in H_{0}^{1}\left( \Omega ,\mathbb{R}^{3}\right) \times \mathcal{D}_{\Delta _{\Sigma }}\times L_{\mathcal{H}^{d}}^{2}\left( \Sigma \right)$ solution of the problem

Proof. One can easily check that problem (13) has a unique solution $U_{h}\in H_{0}^{1}\left( \Omega ,\mathbb{R}^{3}\right) \cap H^{1}\left( T_{h},\mathbb{R}^{3}\right)$. Now, observing that

we deduce, using the fact that $\underset{h\rightarrow \infty }{\lim }\eta _{h} = +\infty$, the Korn inequality, and the Poincaré inequality, that

from which we deduce that $\sup_{h}F_{h}\left( U_{h}\right) <+\infty$. Then, using Proposition 2 and Theorem 4.2, we deduce, according to [16,Theorem 7.8]), that the sequence $\left( U_{h}\right) _{h}$ $\tau$-converges to the solution $\left( U,V\right)$ of the problem

where $V = H_{0}^{1}\left( \Omega ,\mathbb{R}^{3}\right) \times \mathcal{D} _{\Delta _{\Sigma }}\times L_{\mathcal{H}^{d}}^{2}\left( \Sigma \right)$. On the other hand, according to [27,Theorem 6], the trace of $\xi \in H^{1}\left( \Omega ,\mathbb{R}^{3}\right)$ on $\omega \cap \Sigma$ exists for $\mathcal{H}^{d}$-almost-every $x\in \omega \cap \Sigma$ and belongs to the Besov space $B_{d}^{2}\left( \Sigma ,\mathbb{R}^{3}\right)$ of functions $\psi :\Sigma \longrightarrow \mathbb{R}^{3}$ such that

Then, according to Lemma 2.1, we obtain from (58), using for example [46,Theorems 3.1 and 3.3], that $\overline{v}\in \mathcal{D}_{\Delta _{\Sigma }}$ and for every $\left( \xi ,\zeta \right) \in H_{0}^{1}\left( \Omega ,\mathbb{R}^{3}\right) \times \mathcal{D}_{\Sigma ,\mathcal{E}}\times L_{\mathcal{H}^{d}}^{2}\left( \Sigma \right)$,

$B_{-d}^{2}\left( \Sigma ,\mathbb{R}^{3}\right)$ being the dual space of $B_{d}^{2}\left( \Sigma ,\mathbb{R}^{3}\right)$ (see [28,p. 291]). Since $\left( V_{1},V_{2}\right) \in D_{\Sigma ,\mathcal{E}}\subset L_{ \mathcal{H}^{d}}^{2}\left( \Sigma ,\mathbb{R}^{2}\right)$, the trace of $U$ on $\Sigma$ belongs to $B_{d}^{2}\left( \Sigma ,\mathbb{R}^{3}\right) \subset L_{\mathcal{H}^{d}}^{2}\left( \Sigma ,\mathbb{R}^{3}\right)$, and, according to Lemma 2.1, $\Delta _{\alpha ,\Sigma }$; $\alpha = 1,2$, is a second order operator in $L_{\mathcal{H}^{d}}^{2}\left( \Sigma \right)$ defined by the form $\mathcal{E}_{\Sigma }$ under the Dirichlet condition $V_{\alpha } = 0$ on $\mathcal{V}_{0}$; $\alpha = 1,2$, the transmission condition

in problem (57) is well posed.

5.   Proof of the main result

The proof of Theorem 4.2 is given in three steps.

5.1.   Step 1: Boundary layers

We consider here a local problem associated with boundary layers in the vicinity of the strips. We denote $w^{m}$; $m = 1,2$, the solution of the following boundary value problem:

where $\mathbb{R}^{2+} = \left\{ y = \left( y_{1},y_{2}\right) \in \mathbb{R}^{2} \text{; }y_{2}>0\right\}$ and $e_{m} = \left( \delta _{1m},\delta _{2m}\right)$; $m = 1,2$. The displacement $w^{m}$; $m = 1,2$, which belongs to the space $H_{loc}^{1}\left( \mathbb{R}^{2+},\mathbb{R}^{2}\right)$, is given (see for instance [34] and [18]) by

and

where

One can check that $w^{m}\left( y\right)$; $m = 1,2$, is also the solution of problem (61) posed in the half-plane $\mathbb{R}^{2-}$:

We introduce the scalar problem

The solution of (65) is given by

Observe that $w\left( y\right)$ is also the solution of problem (65) posed in the half-plane $\mathbb{R}^{2-}$. We now state the following preliminary result in this section:

Proposition 3. ([18,Proposition 7]). One has

1. $\underset{R\rightarrow \infty }{\lim }\dfrac{1}{\ln R} \int\nolimits_{B\left( 0,R\right) \cap {\bf{R}}^{2\pm }}\sigma _{ij}\left( w^{m}\right) e_{ij}\left( w^{l}\right) dy = \delta _{ml}\dfrac{ 2\mu \pi }{\left( 1+\kappa \right) \left( \ln 2\right) ^{2}}$; $m,l = 1,2$,

2. $\underset{R\rightarrow \infty }{\lim }\dfrac{1}{\ln R} \int\nolimits_{B\left( 0,R\right) \cap {\bf{R}}^{2\pm }}\left\vert \nabla w\right\vert ^{2}dy = \dfrac{\pi }{\left( \ln 2\right) ^{2}}$, where $B\left( 0,R\right)$ is a disc of radius $R$ centred at the origin.

Let $r_{h}$ be a positive parameter, such that

We define the rotation $\mathcal{R}\left( x_{h}^{k}\right)$; $x_{h}^{k} = \left( x_{1h}^{k},x_{2h}^{k}\right)$ being the center of $S_{h}^{k}$ in Cartesian coordinates, by

where $n^{k}$ is the unit normal on $S_{h}^{k}$ and $Id_{\mathbb{R}^{3}}$ is the $3\times 3$ identity marix. Let $\varphi _{h}^{k}$; $k\in I_{h}$, be the truncation function defined on $\mathbb{R}^{2}$ by

where $R_{h}^{k}\left( x\right) = \sqrt{\left( \left( x^{\prime }-x_{h}^{k}\right) .n^{k}\right) ^{2}+x_{3}^{2}}$ with $x^{\prime } = \left( x_{1},x_{2}\right)$. We define, for $k\in I_{h}$,

and the cylinder

We then set

We define, the function $w_{h}^{mk}\left( x\right)$; $k\in I_{h}$ and $m = 1,2,3$, by

and

where $e_{m} = \left( \delta _{1m},\delta _{2m},\delta _{3m}\right)$; $m = 1,2,3$. We define the local perturbation $w_{\varepsilon }^{m}$; $m = 1,2,3$, on $\Omega$ by

We have the following result:

Lemma 5.1. If $\gamma \in \left( 0,+\infty \right)$ then, for every $\Phi \in C^{1}\left( \overline{\Omega },\mathbb{R}^{3}\right)$, we have

where $A\left( s\right)$ is the material matrix defined in (17).

Proof. Let us introduce the change of variables

on $Z_{h}^{k}$; $k\in I_{h}$. Then, using the smoothness of $\Phi$ and Proposition 3, we have

where $B =$Diag$\left( 1,\dfrac{2}{\left( 1+\kappa \right) },\dfrac{2}{\left( 1+\kappa \right) }\right)$ and $\mathcal{R}\left( s\right)$ is the rotation matrix defined by $\mathcal{R}\left( s\right)$ $= Id_{\mathbb{R} ^{3}}$ on the faces of $\Sigma$ which are perpendicular to the vectors $\pm \left( 0,1\right)$, by $\mathcal{R}\left( s\right) = \left( \begin{array}{ccc} 1/2 & \sqrt{3}/2 & 0 \\ -\sqrt{3}/2 & 1/2 & 0 \\ 0 & 0 & 1 \end{array} \right)$ on the faces of $\Sigma$ which are perpendicular to the vectors $\pm \left( -\sqrt{3}/2,1/2\right)$, and $\left( \begin{array}{ccc} -1/2 & \sqrt{3}/2 & 0 \\ \sqrt{3}/2 & 1/2 & 0 \\ 0 & 0 & 1 \end{array} \right)$ on the faces of $\Sigma$ which are perpendicular to the vectors $\pm \left( \sqrt{3}/2,1/2\right)$. Then observing that

we have the result.

5.2.   Step 2: Lim-sup inequality

In this Subsection we prove the lim-sup condition of the $\Gamma$ -convergence property stated in Theorem 4.2. Let $p_{h}^{k} = \left( p_{h,1}^{k},p_{h,2}^{k}\right)$, $q_{h}^{k} = \left( q_{h,1}^{k},q_{h,2}^{k}\right)$ be the extremities of the line segment $S_{h}^{k}$. Let $v\in C_{c}^{1}\left( \omega ,\mathbb{R}^{3}\right)$. Then, we build the following sequence:

for every $x^{\prime }\in \omega$, where, using the local coordinates (37) for $S_{h}^{k}\perp \left( -\sqrt{3}/2,1/2\right)$,

using the local coordinates (38) for $S_{h}^{k}\perp \left( \sqrt{3 }/2,1/2\right)$,

and, using the local coordinates (39) for $S_{h}^{k}\perp \left( 0,1\right)$,

Let us now introduce the intervals $J_{h}^{p_{h}^{k}}$ and $J_{h}^{q_{h}^{k}}$ centred at the points $p_{h}^{k}$ and $q_{h}^{k}$ respectively, such that

where ${\bf{s}}_{h} = \left( \begin{array}{c} s_{h} \\ s_{h} \end{array} \right)$, such that $\underset{h\rightarrow \infty }{\lim }2^{h}s_{h} = 0$. Let $\psi _{h}^{k}$ be a $C_{c}^{\infty }\left( S_{h}^{k}\cup J_{h}^{p_{h}^{k}}\cup J_{h}^{q_{h}^{k}}\right)$ test-function, such that

We define the test-function $v_{h}$ by

We have the following convergences:

Lemma 5.2. We have

1. $v_{h}\dfrac{\boldsymbol{1}_{T_{h}}\left( x\right) }{\left\vert T_{h}\right\vert }dsdx_{3}\overset{\ast }{\underset{h\rightarrow \infty }{ \rightharpoonup }}v\boldsymbol{1}_{\Sigma }\left( s\right) \dfrac{d\mathcal{H }^{d}\left( s\right) }{\mathcal{H}^{d}\left( \Sigma \right) }$,

2. $\underset{h\rightarrow \infty }{\lim }\int_{T_{h}}\sigma _{ij}^{h}\left( v_{h}\right) e_{ij}\left( v_{h}\right) dsdx_{3}\mathcal{ = \mu }_{0}\underset{h\rightarrow \infty }{\lim }\left( \dfrac{5}{3}\right) ^{h} \underset{\underset{\left\vert p-q\right\vert = 2^{-h}}{p,q\in \mathcal{V}_{h} }}{\underset{\alpha = 1,2}{\sum \limits}}\left\vert v_{\alpha }\left( p\right) -v_{\alpha }\left( q\right) \right\vert ^{2}$.

Proof. 1. Let $\varphi \in C_{0}\left( \mathbb{R}^{3},\mathbb{R}^{3}\right)$. We have

where $C$ is a positive constant independent of $h$. On the one hand we have

On the other hand, since

and $\left\vert p_{h}^{k}-q_{h}^{k}\right\vert = 2^{-h}$, we have

2. Computing tensors in local coordinates (37) and (38), we obtain, for $S_{h}^{k}\perp \left( -\sqrt{3}/2,1/2\right)$ or $S_{h}^{k}\perp \left( \sqrt{3}/2,1/2\right)$,

and if $S_{h}^{k}\perp \left( 0,1\right)$,

Thus, according to (77)-(78), we obtain on each $S_{h}^{k}$; $k\in I_{h}$,

which implies that

We prove here the lim-sup condition of the $\Gamma$-convergence property stated in Theorem 4.2$_{1}$.

Proposition 4. If $\gamma \in \left( 0,+\infty \right)$ then, for every $\left( u,v\right)$ $\in H_{0}^{1}\left( \Omega ,\mathbb{R}^{3}\right) \times \mathcal{D}_{\Sigma ,\mathcal{E}}\times L_{\mathcal{H}^{d}}^{2}\left( \Sigma \right)$, there exists a sequence $\left( u_{h}\right) _{h}$, such that $u_{h}\in H_{0}^{1}\left( \Omega ,\mathbb{R}^{3}\right) \cap H^{1}\left( T_{h},\mathbb{R}^{3}\right)$, $\left( u_{h}\right) _{h}$ $\tau$ -converges to $\left( u,v\right)$, and

Proof. Let $\left( u,v\right)$ $\in H_{0}^{1}\left( \Omega ,\mathbb{R}^{3}\right) \times \mathcal{D}_{\Sigma ,\mathcal{E}}\times L_{\mathcal{H}^{d}}^{2}\left( \Sigma \right)$. Let $\left( u_{n},v_{n}\right) _{n}$ be a sequence in the space $C_{c}^{1}\left( \Omega ,\mathbb{R}^{3}\right) \times \left( C_{c}^{1}\left( \Omega ,\mathbb{R}^{3}\right) \cap \mathcal{D}_{\Sigma , \mathcal{E}}\times L_{\mathcal{H}^{d}}^{2}\left( \Sigma \right) \right)$ such that $u_{n}\underset{n\rightarrow \infty }{\longrightarrow }u$ $\ H^{1}\left( \Omega ,\mathbb{R}^{3}\right)$-strong, $\overline{v}_{n} \underset{n\rightarrow \infty }{\longrightarrow }\overline{v}$ strongly with respect to the norm (26), and $v_{3,n}\underset{n\rightarrow \infty } {\longrightarrow }v_{3}$ strongly with respect to the norm of $L_{\mathcal{H} ^{d}}^{2}\left( \Sigma \right)$. We define the sequence $\left( u_{n,h}^{0}\right) _{h,n}$ by

where $v_{n,h}$ is the test-function (81) associated with $v_{n}$, and $w_{h}^{m}$ is the perturbation defined in (76). Then $u_{n,h}^{0}\in H_{0}^{1}\left( \Omega ,\mathbb{R}^{3}\right) \cap H^{1}\left( T_{h},\mathbb{R}^{3}\right)$ and, using Lemma 5.1, Lemma 5.2, and the fact that the measure $\left\vert Z_{h}\right\vert$ of the set $Z_{h}$ tends to zero as $h$ tends to $\infty$, that $\left( u_{n,h}^{0}\right) _{h}$ $\tau$-converges to $\left( u_{n},v_{n}\right)$ as $h$ tends to $\infty$.

We have

We immediately obtain

Using Lemma 5.1, it follows that

and, using Lemma 5.2 and Proposition 1, we obtain

This yields

The continuity of $F_{\infty }$ implies that $\lim_{n\rightarrow \infty }\lim_{h\rightarrow \infty }F_{h}\left( u_{n,h}^{0}\right) = F_{\infty }\left( u,v\right)$. Then, using the diagonalization argument of [5,Corollary 1.18], we prove the existence of a sequence $\left( u_{h}\right) _{h} = \left( u_{n\left( h\right) ,h}^{0}\right) _{h}$: $\underset {h\rightarrow \infty }{\lim }n\left( h\right) = +\infty$, such that

5.3.   Step 3: Lim-inf inequality

In this Subsection we prove the second assertion of Theorem 4.2.

Proposition 5. If $\gamma \in \left( 0,+\infty \right)$, then, for every sequence $\left( u_{h}\right) _{h}$, such that $u_{h}\in H_{0}^{1}\left( \Omega ,\mathbb{R}^{3}\right) \cap H^{1}\left( T_{h},\mathbb{ R}^{3}\right)$ and $\left( u_{h}\right) _{h}$ $\tau$-converges to $\left( u,v\right)$, we have $\overline{v}\in \mathcal{D}_{\Sigma ,\mathcal{E}}$ and

Proof. Let $\left( u_{h}\right) _{h}$; $u_{h}\in H_{0}^{1}\left( \Omega ,\mathbb{R} ^{3}\right) \cap H^{1}\left( T_{h},\mathbb{R}^{3}\right)$, such that $\left( u_{h}\right) _{h}$ $\tau$-converges to $\left( u,v\right)$. We suppose that $\sup_{h}F_{h}\left( u_{h}\right) <+\infty$, otherwise the $\lim \inf$ inequality is trivial. Then, owing to Proposition 2 and Proposition 1, we have that $\overline{v}\in \mathcal{D}_{\Sigma , \mathcal{E}}$ and

Let $\left( u_{n},v_{n}\right) _{n}$ $\subset C_{c}^{1}\left( \Omega , \mathbb{R}^{3}\right) \times \left( C_{c}^{1}\left( \Omega ,\mathbb{R} ^{3}\right) \cap \mathcal{D}_{\Sigma ,\mathcal{E}}\times L_{\mathcal{H} ^{d}}^{2}\left( \Sigma \right) \right)$, such that

$\overline{v}_{n}\underset{n\rightarrow \infty }{\longrightarrow }\overline{v }$ strongly with respect to the norm (26), and $v_{3,n}\underset{ n\rightarrow \infty }{\longrightarrow }v_{3}$ strongly with respect to the norm of $L_{\mathcal{H}^{d}}^{2}\left( \Sigma \right)$. Let $\left( u_{n,h}^{0}\right) _{h,n}$ be the corresponding sequence defined in (82). We have from the definition of the subdifferentiability of convex functionals

Due to the structure of the sequence $\left( u_{n,h}^{0}\right) _{h}$, we have

Since $\left\vert Z_{h}\right\vert$ tends to zero as $h$ tends to $\infty$, it follows that

Using the definition of the perturbation $w_{h}^{m}$ and the expressions (62), (63) and (66), we get

where $C_{n}^{m}$ is a positive constant which may depend of $n$. Then, using Lemma 5.1, we obtain that

We deduce from (84) that

On the other hand, as $\left\vert Z_{h}\right\vert$ tends to zero as $h$ tends to $\infty$, we have

We deduce from (86)-(93) that

Letting $n$ tend to $\infty$ in the right hand side of the above inequality, we deduce that

which is equivalent to

This ends the proof of Theorem 4.2.