# Resolution of a nonlinear elasticity problem subject to friction laws.

Mahdi Boukrouche

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## To cite this version:

Mahdi Boukrouche. Resolution of a nonlinear elasticity problem subject to friction laws.. Journal of Fixed Point Theory and Applications, 2024, 10.1007/s11784-024-01144-5. http://09-04935014

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Submitted on 7 Feb 2025

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# Resolution of a nonlinear elasticity problem subject to friction laws

Mahdi Boukrouche

**Abstract.** We consider a problem in a bounded domain, with Dirichlet condition on one part of its boundary and on other parts non-linear slip conditions governed by Coulomb friction law and Fourier law. We assume that the problem is also governed by a particular constitutive law of elasticity system with a strongly nonlinear strain tensor given by $\sigma_{ij} = \sum_{k,h=1}^{3} a_{ijkh} E_{hk}(\nabla u)$ where $u$ is a displacement of a substance, $(a_{ijkh})_{1\le i,j,k,h\le 3}$ are the coefficients of elasticity and $E_{hk}$ are the components of the nonlinear deformation tensor of St Venant $E(\nabla u) = \frac{1}{2}(\nabla u + \nabla u + \nabla u \nabla u)$. The functional framework leads to use Sobolev spaces with variable exponent. The formulation of the problem leads to a variational inequality, for which we prove, by Schauder fixed point theorem, an existence solution.

**Mathematics Subject Classification (2010).** Primary 35R35; Secondary 74B20, 70F40, 46E30.

**Keywords.** Nonlinear elasticity problem, Coulomb friction law, Fourier law, Sobolev spaces with variable exponent, Variational problem, existence.

## 1. Description of the problem

The study of partial differential equations in problems with variable exponents comes from the theory of nonlinear elasticity, elastic mechanics, fluid dynamics, electrorheological fluids, image processing, etc. (see [1, 4, 10, 16, 18]).

Let $\Omega$ be a connected open bounded domain of $\mathbb{R}^3$ with Lipschitz boundary $\partial\Omega$. To a given field of displacement $u$, we associate a nonlinear deformation tensor of St Venant $E$ defined [9] by

$$E(\nabla u) = \frac{1}{2}(\nabla u + \nabla u + \nabla u \nabla u),$$
where $^T\nabla u$ is the transposed matrix of $\nabla u$, the components of $E(\nabla u)$ are given by:

$$E_{kh}(\nabla u) = \frac{1}{2} \left( \frac{\partial u_k}{\partial x_h} + \frac{\partial u_h}{\partial x_k} + \sum_{m=1}^{3} \frac{\partial u_m}{\partial x_k} \frac{\partial u_m}{\partial x_h} \right) \quad 1 \le k, h \le 3. \quad (1.1)$$

The corresponding nonlinear constraints tensor $\sigma(u) = (\sigma_{ij}(u))_{1\le i,j\le 3}$ is then given by:

$$\sigma_{ij}(u) = \sum_{k,h=1}^{3} a_{ijkh} E_{kh}(\nabla u) \quad 1 \le i, j \le 3, \quad (1.2)$$

which describes a nonlinear relation with the deformation tensor $E_{kh}$. The coefficients of elasticity $a_{ijkh}$ for $1 \le i, j, h, k \le 3$ (see also [9]), satisfy the following symmetried properties:

$$a_{ijkh} = a_{jikh} = a_{ikhk}, \quad \text{for all } 1 \le i, j, k, h \le 3. \quad (1.3)$$

The aim of this paper is to prove the existence of a weak solution for the following nonlinear problem, encountered in the theory of nonlinear elasticity:

Let $\Gamma_0$ be a bounded domain in $\mathbb{R}^2$ situated in the plane of equation $x_3 = 0$. We suppose that $\Gamma_0$ represents the lower surface of the domain occupied by the substance. The upper surface $\Gamma_1$ is defined by

$$\{(x', x_3) \in \mathbb{R}^2 \times \mathbb{R}, x_3 = h(x') \text{ and } x' \in \Gamma_0\},$$

where $h$ is a function defined and bounded on $\Gamma_0$. We study the displacement of a substance in

$$\Omega = \{(x', x_3) \in \mathbb{R}^3 : (x', 0) \in \Gamma_0 \text{ and } 0 < x_3 < h(x')\}$$

the boundary $\partial\Omega = \overline{\Gamma_0} \cup \overline{\Gamma_1} \cup \overline{\Gamma_L}$, where $\Gamma_L$ is the lateral surface of $\Omega$ with measure $|\Gamma_L| > 0$.

The Einstein's convention, which consists of making the sum on the repeated indices, will be used unless otherwise stated.

The outer normal unit vector on $\partial\Omega$ is denoted by $\nu = (\nu_1, \nu_2, \nu_3)$. The normal and the tangential components $u_\nu$ and $u_\tau$, of the displacement variable $u$, are defined by

$$u_\nu = u \cdot \nu = u_i \nu_i, \quad u_{\tau_i} = u_i - u_\nu \nu_i, \quad 1 \le i \le 3. \quad (1.4)$$

The normal and tangential components $\sigma_\nu$ and $\sigma_\tau$ of the strain tensor, are defined by

$$\sigma_\nu = (\sigma \cdot \nu) \cdot \nu = \sigma_{ij} \nu_i \nu_j, \quad \sigma_{\tau_i} = \sigma_{ij} \nu_j - \sigma_\nu \nu_i \quad 1 \le i, j \le 3. \quad (1.5)$$

We assume that the equation of motion is given by

$$-\frac{\partial}{\partial x_j} \sigma_{ij}(u(x)) = f_i(x), \quad 1 \le i, j \le 3, \quad x \in \Omega, \quad (1.6)$$

where $f = (f_1, f_2, f_3)$ represents a mass density of the external forces.
The boundary conditions are given by

$$ \sigma_{\tau}(u) + gu = 0, \text{ and } u \cdot \nu = 0 \text{ on } \Gamma_1, \qquad (1.7) $$

where g is a given function.

$$ u = 0 \text{ on } \Gamma_L, \qquad (1.8) $$

$$ u \cdot \nu = 0 \text{ on } \Gamma_0, \qquad (1.9) $$

we assume that $u \cdot \tau \neq 0$ so there is a tangential force exerted by the surface $\Gamma_0$ on the substance. This tangential force can not exceed a certain threshold $\mu|\sigma_\nu(u)|$. Coulomb law assumes that this function $\mu$ is fixed positive and known

$$ |\sigma_{\tau}(u)| \leq \mu|\sigma_{\nu}(u)| \text{ on } \Gamma_0, \qquad (1.10) $$

$|\sigma_{\tau}(u)|$ and $|\sigma_{\nu}(u)|$ are respectively the modulus of the tangential and the normal constraints defined on $\Gamma_0$ by (1.5).

As long as the tangential constraint $\sigma_{\tau}(u)$ has not reached the threshold $\mu|\sigma_{\nu}(u)|$, the substance moves with a given displacement $s$, which is the displacement of the lower surface $\Gamma_0$ (adhesion). When the threshold is reached, the substance and the surface move tangentially relative to each other and there is proportional sliding. What can be summarized as follows [8]

$$ \left. \begin{array}{l} |\sigma_{\tau}(u)| < \mu|\sigma_{\nu}(u)| \Rightarrow u_{\tau} = s, \\ |\sigma_{\tau}(u)| = \mu|\sigma_{\nu}(u)| \Rightarrow \exists \lambda \geq 0 \text{ such as } u_{\tau} = s - \lambda\sigma_{\tau}(u), \end{array} \right\} \text{on } \Gamma_0 \qquad (1.11) $$

where the positive real $\lambda$ is unknown.

This problem (1.6)-(1.11) models the behavior of a heterogeneous material with Fourier friction condition (1.7), Coulomb friction free boundary condition (1.11) and the homogenous Dirichlet condition (1.8).

Similar work can be seen in [2] where the problem was with the same equation of motion (1.6) but with Dirichlet conditions on two parts of the boundary and Tresca friction law on the other. Also In [14] only Dirichlet condition on all the boundary was considered.

Our present problem (1.6)-(1.11) is new, and presents several mathematical difficulties to overcome, so it can be considered as a generalisation of [2] and of [14]. So the rest of this paper is devoted to the study of this highly non-linear problem (1.6)-(1.11) and to the proof of existence of its variational problem 1. In Section 2 we introduce the functional framework and derive the variational problem 1 of this strong problem (1.6)-(1.11), and check (see Remark 2.1) that all its terms are well defined. To solve the variational problem 1, we first consider in Section 3 Tresca problem 2 as an auxiliary problem of problem 1. To solve Tresca problem 2 we begin by an approximate problem 3, then in Section 4 we deduce from the approximate problem 3, the finite dimensional problem 4. In Subsection 4 we establish in Theorem 4.1 the existence of a solution to the finite dimensional problem 4, and in Theorem 4.2 we prove its uniqueness. In Subsection 4.2 we establish in Lemma 4.3 some estimates and in Lemma 4.4 and Lemma 4.5 some limits needed. In Section
5 we obtain in Theorem 5.1 the existence and uniqueness to the solution of the approximate problem 3, then in Theorem 5.2 we deduce the existence and uniqueness to the solution of the Tresca problem 2. Finally in Theorem 5.3 by Schauder fixed point theorem we obtain an existence result to Problem 1.

## 2. Variational formulation of the problem

We consider the functional framework of the considered problem (1.6)-(1.11), using Lebesgue and Sobolev spaces with variable exponent see for example [5]. However it is not necessary to use this notion of Lebesgue and Sobolev spaces with variable exponents, to study this problem. But we see it as a good generalization to the same study with Lebesgue and Sobolev spaces with fixed exponents.

Thus for definitions and properties of Lebesgue and Sobolev spaces with variable exponents (see for example [5, 6, 7, 11, 12, 17] for the proofs and more details).

Let $p(\cdot) : \Omega \to [1, +\infty]$ be a measurable function, called the variable exponent. We adopt the following needed notations:

$$
\begin{align*}
C_+(\bar{\Omega}) &= \{p(\cdot) \in C(\bar{\Omega}), \quad p(x) \ge 1 \quad \forall x \in \bar{\Omega}\} \\
p^- &= \operatorname*{ess\,inf}_{x \in \bar{\Omega}} p(x), \quad p^+ = \operatorname*{ess\,sup}_{x \in \bar{\Omega}} p(x), \quad p' = \frac{p}{p-1} \\
V(\Omega) &= \{\varphi \in (W_{\Gamma_L}^{1,p(\cdot)}(\Omega))^3 : \varphi \cdot \nu = 0 \text{ on } \Gamma_0 \cup \Gamma_1\} \\
W_{\Gamma_L}^{1,p(\cdot)}(\Omega) &= \{\varphi \in W^{1,p(\cdot)}(\Omega) : \varphi = 0 \text{ on } \Gamma_L\} \\
\mathbf{W}_{\Gamma_L}^{p(\cdot)}(\Omega) &= V(\Omega) \cap (W^{2,p(\cdot)}(\Omega))^3
\end{align*}
$$

The Problem (1.6)-(1.11) leads (see for example [8]) to the following variational problem :

**Problem 1.** Find $u \in \mathbf{W}_{\Gamma_L}^{p(\cdot)}(\Omega)$, $p(\cdot) > 3$, such that

$$
\int_{\Omega} a_{ijkh} E_{kh}(\nabla u) \frac{\partial(\varphi_i - u_i)}{\partial x_j} dx + \int_{\Gamma_0} \mu |\sigma_{\nu}(u)| (|\varphi - s| - |u - s|) dx' \\
+ \int_{\Gamma_1} g u_i (\varphi_i - u_i) dS \geq \int_{\Omega} f_i (\varphi_i - u_i) dx, \forall \varphi \in V(\Omega)
$$

**Remark 2.1.** We first check that all terms of the above variational inequality are well defined. Indeed as

$$E_{kh}(\nabla u) = d_{kh}(u) + \frac{1}{2} \frac{\partial u_m}{\partial x_k} \frac{\partial u_m}{\partial x_h}$$

thus taking $u \in \mathbf{W}_{\Gamma_L}^{p(\cdot)}(\Omega)$ we get $\frac{\partial u_m}{\partial x_k} \frac{\partial u_m}{\partial x_h} \in W^{1,p(\cdot)}(\Omega) \times W^{1,p(\cdot)}(\Omega)$, then with $p(x) > 3$ for $x \in \Omega$, and by the continuous injection from $W^{1,p(\cdot)}(\Omega)$ into $L^{\infty}(\Omega)$ we get $E_{kh}(\nabla u) \in W^{1,p(\cdot)}(\Omega)$, so $E_{kh}(\nabla u) \in L^{p(\cdot)}(\Omega)$, and as
$p(x) > p'(x)$ for $x \in \Omega$, so $E_{kh}(\nabla u) \in L^{p'(\cdot)}(\Omega)$, thus $E_{kh}(\nabla u) \frac{\partial(\varphi_i - u_i)}{\partial x_j} \in L^1(\Omega)$ then the first term in the above variational inequality is well defined just with $a_{ijkh} \in L^\infty(\Omega)$. Note that if $u \in W^{1,p(\cdot)}(\Omega)$ only we can't define the first term of the above variational inequality. Also $u \in W_{\Gamma_L}^{p(\cdot)}(\Omega)$, with $a_{ijkh} \in L^\infty(\Omega)$, the trace on $\Gamma_0$ of $\sigma_{ij}(u) = a_{ijkh}E_{kh}(\nabla u)$ is not defined! So we take $a_{ijkh} \in W^{1,p^+}(\Omega) \cap L^\infty(\Omega)$, thus $\sigma_\nu(u) \in W^{1-\frac{1}{p(\cdot)},p(\cdot)}(\Gamma_0) = W^{\frac{1}{p(\cdot)},p(\cdot)}(\Gamma_0)$, then taking $\mu \in L_+^\infty(\Gamma_0)$, so the integral on $\Gamma_0$, in the above variational inequality, can be defined. Also as $(\varphi_i - u_i) \in W^{1,p(\cdot)}(\Omega)$ its traces on $\Gamma_1$, is in $W^{-\frac{1}{p(\cdot)},p(\cdot)}(\Gamma_1)$ then taking $g \in L^\infty(\Gamma_1)$, the integral on $\Gamma_1$ can be defined. Taking $f \in (L^{p'(\cdot)}(\Omega))^3$ the last integral on $\Omega$ can be defined. $\square$

So we give now the following assumptions (H):

$$f \in (L^{p'(\cdot)}(\Omega))^3, \qquad (2.1)$$

$$g \in L^\infty(\Gamma_1), \qquad (2.2)$$

$$\mu \in L_+^\infty(\Gamma_0), \qquad (2.3)$$

$$a_{ijkh} \in W^{1,p^+}(\Omega) \cap L^\infty(\Omega) \text{ for } 1 \le i, j, k, h \le 3,$$

$$\exists(\alpha_0, \beta) \in \mathbb{R}_+^* \times \mathbb{R}_+^* \text{ such that } \alpha_0 \le a_{ijkh}(x) \le \beta \text{ a.e. in } \Omega. \qquad (2.4)$$

To solve Problem 1, we first consider the following Tresca Problem 2, where we only change Coulomb functional :

$$\varphi \to \int_{\Gamma_0} \mu |\sigma_\nu(u)| (|\varphi - s| - |u - s|) dx'$$

in Problem 1 by Tresca functional :

$$\varphi \to \int_{\Gamma_0} \ell(|\varphi - s| - |u - s|) dx'$$

where $x \to \ell(x)$ is just a given function on $\Gamma_0$. Then we must prove in Subsection 5.1 and Subsection 5.2 the existence and uniqueness of the solution to Tresca Problem 2, then in Subsection 5.3 we will apply Schauder fixed point theorem to solve Problem 1.

### 3. Tresca problem

**Problem 2.** Find $u \in W_{\Gamma_L}^{p(\cdot)}(\Omega)$, $p(\cdot) > 3$, such as

$$\int_{\Omega} a_{ijkh} E_{kh}(\nabla u) \frac{\partial}{\partial x_j} (\varphi_i - u_i) dx + \int_{\Gamma_0} \ell(|\varphi - s| - |u - s|) dx' \\ + \int_{\Gamma_1} g u_i (\varphi_i - u_i) dS \ge \int_{\Omega} f_i (\varphi_i - u_i) dx, \forall \varphi \in V(\Omega)$$
with

$$ \ell \in W_{+}^{-\frac{1}{p'(\cdot)}, p'(\cdot)}(\Gamma_0) $$

To study this problem 2, let $\varepsilon > 0$, we introduce the functional $\psi_{\varepsilon}$ defined for $\ell \in W_{+}^{-\frac{1}{p'(\cdot)}, p'(\cdot)}(\Gamma_0)$ by

$$ v \to \psi_{\varepsilon}(v) = \int_{\Gamma_0} \ell \sqrt{\varepsilon^2 + |v-s|^2} dx' $$

which is Gâteaux differentiable on $V(\Omega)$, and his Gâteaux differential $\psi'_{\varepsilon}$ is defined by

$$ v \to < \psi'_{\varepsilon}(u_{\varepsilon}), v > = \int_{\Gamma_0} \ell \frac{(u_{\varepsilon} - s) \cdot v}{\sqrt{\varepsilon^2 + |u_{\varepsilon} - s|^2}} dx'. $$

Then we set the approximate problem:

**Problem 3.** Find $u_{\varepsilon} \in W_{\Gamma_L}^{p(\cdot)}(\Omega)$, $p(\cdot) > 3$, such as

$$ \int_{\Omega} a_{ijkh} E_{kh}(\nabla u_{\varepsilon}) \frac{\partial v_i}{\partial x_j} dx + < \psi'_{\varepsilon}(u_{\varepsilon}), v > + \int_{\Gamma_1} g u_{\varepsilon} \cdot v dS \\ = \int_{\Omega} f \cdot v dx, \forall v \in V(\Omega) $$

This approximate problem 3 is linear in $v$, thus we deduce in the next section the following finite dimension problem 4 associated to problem 3. We prove in Theorem 4.1 and Theorem 4.2, existence and uniqueness of the solution of the finite dimension problem 4, then we establish in Lemma 4.3 some estimates on $\|\nabla u_{n\varepsilon}\|_{L^p(\Omega)^9}$ and on $\|E_{hk}(\nabla u_{n\varepsilon})\|_{L^p(\Omega)}$, then we justify how to pass to the limit, in Lemma 4.4 and Lemma 4.5, and how to obtain in Subsection 5.1 and Subsection 5.2 that the limit of $u_{n\varepsilon}$, when $n \to +\infty$ then when $\varepsilon \to 0$ be the unique solution of Tresca problem 2.

## 4. Finite dimension problem

Since $W_{\Gamma_L}^{p(\cdot)}(\Omega)$ is separable (see for example Theorem 8.1.6 page 249 in [5]) then, there exists a countable family $(w_j)_{j \in \mathbb{N}^*}$ dense in $W_{\Gamma_L}^{p(\cdot)}(\Omega)$. Let $V_n = \text{Vect}\{w_j, j = 1, \dots, n\}$ be the vector space generated by the first $n$ functions of this family. So the dimension of $V_n \le n$, $V_n \subset V_{n+1}$, $\forall n \in \mathbb{N}^*$, and their union is dense in $W_{\Gamma_L}^{p(\cdot)}(\Omega)$. Thus we set the following finite dimensional

**Problem 4.** For $n \in \mathbb{N}^*$ and $\varepsilon \in \mathbb{R}_+^*$, find $u_{n\varepsilon} \in V_n$, with $p(\cdot) > 3$, such as

$$ \int_{\Omega} a_{ijkh} E_{kh}(\nabla u_{n\varepsilon}) \frac{\partial v_i}{\partial x_j} dx + < \psi'_{\varepsilon}(u_{n\varepsilon}), v > + \int_{\Gamma_1} g u_{n\varepsilon} \cdot v dS \\ = \int_{\Omega} f \cdot v dx, \forall v \in V_n. $$
### 4.1. The solution to Problem 4

**Theorem 4.1.** Let assumptions (H) hold, $\ell \in W_+^{-\frac{1}{p'(\cdot)}, p'(\cdot)}(\Gamma_0)$ and $g$ such as:

$$ \int_{\Gamma_1} gv^2 dS \geq \int_{\Gamma_0} \ell |v| dx' \quad \forall v \in V_n \quad (4.1) $$

then there exists a solution of the finite dimensional problem 4.

*Proof.* Let $T: V_n \to V'_n$, where $V'_n$ is the topological dual of $V_n$, defined by:
For $u$ and $v$ in $V_n$,

$$ \langle T(u), v \rangle = \int_{\Omega} a_{ijkh} E_{hk}(\nabla(u)) \frac{\partial v_i}{\partial x_j} dx + \langle \psi'_{\varepsilon}(u), v \rangle + \int_{\Gamma_1} g u \cdot v dS \quad (4.2) $$

We must prove that $T$ is continuous and coercive then, see [13] for every $\bar{f} \in V'_n$ there exists $u_{n\varepsilon} \in V_n$ such as

$$ T(u_{n\varepsilon}) = \bar{f}. $$

Taking the application $V_n \to \mathbb{R}$ defined by

$$ v \mapsto \langle f, v \rangle = \int_{\Omega} f(x) \cdot v(x) dx $$

which is linear and continuous, thus this application is in $V'_n$, so we obtain

$$ \langle T(u_{n\varepsilon}), v \rangle = \langle f, v \rangle \quad \forall v \in V_n $$

which is the variational equality in Problem 4. So we must prove that $T$ is continuous and coercive. Let $u$ and $\bar{u}$ in $V_n$ satisfying (4.2) thus

$$ \langle T(u) - T(\bar{u}), v \rangle = \int_{\Omega} a_{ijkh} (E_{hk}(\nabla(u) - E_{hk}(\nabla(\bar{u}))) \frac{\partial v_i}{\partial x_j} dx \\ + \langle \psi'_{\varepsilon}(u) - \psi'_{\varepsilon}(\bar{u}), v \rangle + \int_{\Gamma_1} g(u - \bar{u}) \cdot v dS $$

then taking $v = u - \bar{u}$ we obtain

$$ \langle T(u) - T(\bar{u}), u - \bar{u} \rangle = \int_{\Omega} a_{ijkh} (E_{hk}(\nabla u) - E_{hk}(\nabla \bar{u})) \frac{\partial (u_i - \bar{u}_i)}{\partial x_j} dx \\ + \int_{\Gamma_1} g (u^2 - \bar{u}^2) dS + \langle \psi'_{\varepsilon}(u) - \psi'_{\varepsilon}(\bar{u}), u - \bar{u} \rangle \quad (4.3) $$

as $p(\cdot) > 3$, so $E_{hk}$ is bounded in $L^{p'}(\Omega)$ by a constant $C_1 > 0$ for $1 \le h, k \le 3$,

$$ \left| \int_{\Omega} a_{ijkh} (E_{hk}(\nabla u) - E_{hk}(\nabla \bar{u})) \frac{\partial (u_i - \bar{u}_i)}{\partial x_j} dx \right| \le 2C_1 \beta \|u - \bar{u}\|_{V(\Omega)} \quad (4.4) $$

and

$$ \left| \int_{\Gamma_1} g (u^2 - \bar{u}^2) dS \right| = \left| \int_{\Gamma_1} g (u + \bar{u})(u - \bar{u}) dS \right| \\ \le \|g\|_{L^{\infty}(\Gamma_1)} \|u + \bar{u}\|_{L^{p'(\cdot)}(\Gamma_1)} \|u - \bar{u}\|_{L^{p(\cdot)}(\Gamma_1)} $$
by the continuous injection from $V(\Omega)$ into $L^{p(\cdot)}(\Gamma_1)$ there exists a constant $C_2 > 0$ and as $u$ and $\bar{u}$ are in $V_n$ so $\|u + \bar{u}\|_{L^{p'(\cdot)}(\Gamma_1)}$ bounded by a constant $C_3 > 0$, thus

$$ \left| \int_{\Gamma_1} g(u^2 - \bar{u}^2) dS \right| \le C_2 C_3 \|g\|_{L^\infty_+(\Gamma_1)} \|u - \bar{u}\|_{V(\Omega)} \quad (4.5) $$

moreover

$$ \begin{aligned} \langle \psi'_{\varepsilon}(u) - \psi'_{\varepsilon}(\bar{u}), u - \bar{u} \rangle = & \int_{\Gamma_0} \frac{\ell(u - \bar{u}) \cdot (u - s)}{\sqrt{\varepsilon^2 + |u - s|^2}} dx' \\ & - \int_{\Gamma_0} \frac{\ell(u - \bar{u}) \cdot (\bar{u} - s)}{\sqrt{\varepsilon^2 + |\bar{u} - s|^2}} dx' \end{aligned} \quad (4.6) $$

thus

$$ \begin{aligned} |\langle \psi'_{\varepsilon}(u) - \psi'_{\varepsilon}(\bar{u}), u - \bar{u} \rangle| & \le \left| \int_{\Gamma_0} \frac{\ell(u - \bar{u}) \cdot (u - s)}{\sqrt{\varepsilon^2 + |u - s|^2}} dx' \right| \\ & \quad + \left| \int_{\Gamma_0} \frac{\ell(u - \bar{u}) \cdot (\bar{u} - s)}{\sqrt{\varepsilon^2 + |\bar{u} - s|^2}} dx' \right| \\ & \le 2 \int_{\Gamma_0} \ell |u - \bar{u}| dx' \\ & \le 2 \|\ell\|_{W_+^{-\frac{1}{p'(\cdot)}, p'(\cdot)}(\Gamma_0)} \|u - \bar{u}\|_{W_-^{\frac{1}{p'(\cdot)}, p(\cdot)}(\Gamma_0)} \end{aligned} $$

and with the continuous injection of $V(\Omega)$ into $W_{\frac{1}{p'(\cdot)}, p(\cdot)}^{\frac{1}{p'(\cdot)}, p(\cdot)}(\Gamma_0)$ there exists a constant $C_4 > 0$ we obtain

$$ | \langle \psi'_{\varepsilon}(u) - \psi'_{\varepsilon}(\bar{u}), u - \bar{u} \rangle | \le 2C_4 \|\ell\|_{W_+^{-\frac{1}{p'(\cdot)}, p'(\cdot)}(\Gamma_0)} \|u - \bar{u}\|_{V(\Omega)} \quad (4.7) $$

then from (4.3), (4.4), (4.5) and (4.7) we obtain

$$ \|T(u) - T(\bar{u})\|_{V_n'} = \sup_{V_n} \langle T(u) - T(\bar{u}), u - \bar{u} \rangle \le C_5 \|u - \bar{u}\|_{V(\Omega)} \quad (4.8) $$

where $C_5 = 2C_1\beta + C_2C_3\|g\|_{L^\infty_+(\Gamma_1)} + 2C_4\|\ell\|_{W_+^{-\frac{1}{p'(\cdot)}, p'(\cdot)}(\Gamma_0)}$, thus $T$ is continuous.

Now let $u \in V_n$ so

$$ \langle T(u), u \rangle = \int_{\Omega} a_{ijkh} E_{hk}(\nabla u) \frac{\partial u_i}{\partial x_j} dx + \int_{\Gamma_1} g u^2 dS + \langle \psi'_{\varepsilon}(u), u \rangle $$

as

$$ | \langle \psi'_{\varepsilon}(u), u \rangle | \le \int_{\Gamma_0} \ell |u| dx' $$

using (4.1) with $v = u$ we get

$$ \int_{\Gamma_1} g u^2 dS + \langle \psi'_{\varepsilon}(u), u \rangle \ge \int_{\Gamma_1} g u^2 dS - \int_{\Gamma_0} \ell |u| dx' \ge 0 $$
then using [15] we have

$$<T(u), u> \geq \int_{\Omega} a_{ijkl} E_{hk}(\nabla u) \frac{\partial u_i}{\partial x_j} dx \geq 81\alpha_0 \|\nabla u\|_{(L^p(\Omega))}^{p(\cdot)} \quad (4.9)$$

for $\|\nabla u\|_{(L^p(\Omega))}^9 \leq 1$, so as $p^+ \geq p(\cdot)$ then

$$<T(u), u> \geq 81\alpha_0 \|\nabla u\|_{(L^p(\Omega))}^{p(\cdot)} \geq 81\alpha_0 \|\nabla u\|_{(L^p(\Omega))}^{p^+}$$

for $\|\nabla u\|_{(L^p(\Omega))}^9 \geq 1$, so as $p^- \leq p(\cdot)$ then

$$<T(u), u> \geq 81\alpha_0 \|\nabla u\|_{(L^p(\Omega))}^{p(\cdot)} \geq 81\alpha_0 \|\nabla u\|_{(L^p(\Omega))}^{p^-}$$

so

$$\frac{\langle T(v), v \rangle_{V'_n, V_n}}{\|v\|_{V_n}} \geq 81\alpha_0 \frac{\|\nabla u\|_{(L^p(\Omega))}^{p^-}}{\|v\|_{V_n}} = 81\alpha_0 \left( \frac{\|\nabla u\|_{(L^p(\Omega))}^{p^-}}{\|v\|_{V_n}} \right)^{p^-} (\|v\|_{V_n})^{p^- - 1}$$

or

$$\frac{\langle T(v), v \rangle_{V'_n, V_n}}{\|v\|_{V_n}} \geq 81\alpha_0 \frac{\|\nabla u\|_{(L^p(\Omega))}^{p^+}}{\|v\|_{V_n}} = 81\alpha_0 \left( \frac{\|\nabla u\|_{(L^p(\Omega))}^{p^+}}{\|v\|_{V_n}} \right)^{p^+} (\|v\|_{V_n})^{p^+ - 1}$$

as

$$0 < \frac{\|\nabla u\|_{L^p(\Omega)}}{\|v\|_{V_n}} \leq 1$$

thus

$$\frac{\langle T(v), v \rangle_{V'_n, V_n}}{\|v\|_{V_n}} \to +\infty \quad \text{when } \|v\|_{V_n} \to +\infty$$

then we deduce that $T$ is coercive. Then there exists a solution of the finite dimensional problem 4. □

To prove the uniqueness of the above solution to the finite dimensional problem 4, we use a theorem in [14] given also with its proof in [2] as requested by a referee.

**Theorem 4.2.** With the assumptions of theorem 4.1. Then the solution of the finite dimensional problem 4 is unique.

*Proof.* Suppose now that the variational equality of the finite dimensional problem 4 has two solutions $u_{n\epsilon}$ and $w_{n\epsilon}$ thus

$$\int_{\Omega} a_{ijkl} E_{kh}(\nabla u_{n\epsilon}) \frac{\partial v_i}{\partial x_j} dx + \langle \psi'_{\epsilon}(u_{n\epsilon}), v \rangle + \int_{\Gamma_1} g u_{n\epsilon} \cdot v dS \\ = \int_{\Omega} f \cdot v dx, \forall v \in V_n$$

$$\int_{\Omega} a_{ijkl} E_{kh}(\nabla w_{n\epsilon}) \frac{\partial v_i}{\partial x_j} dx + \langle \psi'_{\epsilon}(w_{n\epsilon}), v \rangle + \int_{\Gamma_1} g w_{n\epsilon} \cdot v dS \\ = \int_{\Omega} f \cdot v dx, \forall v \in V_n$$
by substitution

$$ \int_{\Omega} a_{ijkh}(E_{kh}(\nabla u_{n\epsilon}) - E_{kh}(\nabla w_{n\epsilon})) \frac{\partial v_i}{\partial x_j} dx + \langle \psi'_{\epsilon}(u_{n\epsilon}) - \psi'_{\epsilon}(w_{n\epsilon}), v \rangle + \int_{\Gamma_1} g(u_{n\epsilon} - w_{n\epsilon}) \cdot v dS = 0 \quad \forall v \in V_n $$

taking $v = u_{n\epsilon} - w_{n\epsilon}$ we obtain

$$ \int_{\Omega} a_{ijkh}(E_{kh}(\nabla u_{n\epsilon}) - E_{kh}(\nabla w_{n\epsilon})) \frac{\partial(u_{n\epsilon} - w_{n\epsilon})_i}{\partial x_j} dx + \langle \psi'_{\epsilon}(u_{n\epsilon}) - \psi'_{\epsilon}(w_{n\epsilon}), u_{n\epsilon} - w_{n\epsilon} \rangle + \int_{\Gamma_1} g(u_{n\epsilon} - w_{n\epsilon})^2 dS = 0 \quad (4.10) $$

from (4.6) with $\bar{u} = w_{n\epsilon}$ and let the function $y_i \mapsto \varphi(y_i) = \frac{y_i - s_i}{\sqrt{\epsilon^2 - (y_i - s_i)^2}}$ satisfies $\varphi'(y_i) > 0$, $y = u$ or $\bar{u}$. We can check that

$$ \langle \psi'_{\epsilon}(u) - \psi'_{\epsilon}(\bar{u}), u - \bar{u} \rangle = \int_{\Gamma_0} \ell(u - \bar{u}) \cdot (\varphi(u) - \varphi(\bar{u})) dx' $$

Then as $\ell \ge 0$, so

$$ \langle \psi'_{\epsilon}(u_{n\epsilon}) - \psi'_{\epsilon}(w_{n\epsilon}), u_{n\epsilon} - w_{n\epsilon} \rangle \ge 0. $$

Then from (4.10) we get

$$ \int_{\Omega} a_{ijkh}(E_{kh}(\nabla u_{n\epsilon}) - E_{kh}(\nabla w_{n\epsilon})) \frac{\partial(u_{n\epsilon} - w_{n\epsilon})_i}{\partial x_j} dx < 0 $$

but from (2.4) $a_{ijkh} \ge 0$ for $1 \le i, j, h, k \le 3$, so from [14] see also [2] $E_{kh}$ is monotone for $1 \le h, k \le 3$ thus

$$ \int_{\Omega} a_{ijkh}(E_{kh}(\nabla u_{n\epsilon}) - E_{kh}(\nabla w_{n\epsilon})) \frac{\partial(u_{n\epsilon} - w_{n\epsilon})_i}{\partial x_j} dx \ge 0 $$

we can conclude that

$$ \int_{\Omega} a_{ijkh}(E_{kh}(\nabla u_{n\epsilon}) - E_{kh}(\nabla w_{n\epsilon})) \frac{\partial(u_{n\epsilon} - w_{n\epsilon})_i}{\partial x_j} dx = 0. $$

As from (2.4) $a_{ijkh} \ge 0$ for $1 \le i, j, h, k \le 3$, and $E_{kh}$ is monotone then we deduce that

$$ a_{ijkh}(E_{kh}(\nabla u_{n\epsilon}) - E_{kh}(\nabla w_{n\epsilon})) \frac{\partial(u_{n\epsilon} - w_{n\epsilon})_i}{\partial x_j} = 0 \quad \text{in } L^1(\Omega) $$

thus either

$$ \frac{\partial(u_{n\epsilon} - w_{n\epsilon})_i}{\partial x_j} = 0 \quad \text{in } L^p(\Omega) \quad \text{for } 1 \le i, j \le 3, \quad (4.11) $$

or

$$ E_{kh}(\nabla u_{n\epsilon}) - E_{kh}(\nabla w_{n\epsilon}) = 0 \quad \text{in } L^{p'}(\Omega) \quad \text{for } 1 \le k, h \le 3. \quad (4.12) $$
using naw Young inequality we get

$$ \|\nabla u_{n\varepsilon}\|_{(L^{p(\cdot)}(\Omega))^9} \le \left( \frac{C_P}{81\alpha_0} \|f\|_{(L^{p'(\cdot)}(\Omega))^3} \right)^{\frac{p'_{+}}{p_{-}}} = \mathcal{A}_1, $$

and from (4.13) we deduce (4.14), see (1.1), which close this proof. □

**Lemma 4.4.** With the assumptions of Theorem 4.1, we obtain the following limits when $n \to +\infty$, :

$$ u_{\varepsilon n} \to u_{\varepsilon} \text{ (weakly) in } W_{\Gamma_L}^{1,p(\cdot)}(\Omega), \quad (4.15) $$

$$ u_{n\varepsilon} \to u_{\varepsilon} \quad (\text{strongly}) \quad \text{in } L^{p(\cdot)}(\Omega), \quad (4.16) $$

$$ u_{\varepsilon n} \to u_{\varepsilon} \quad \text{in } L^{p(\cdot)}(\Gamma_1), \quad (4.17) $$

$$ u_{n\varepsilon} \to u_{\varepsilon} \quad (\text{in } L^2(\Gamma_0)), \quad (4.18) $$

$$ E_{kh}(\nabla u_{\varepsilon n}) \to E_{kh}(\nabla u_{\varepsilon}) \quad \text{in } L^{p'(\cdot)}(\Omega). \quad (4.19) $$

*Proof.* Using the estimate (4.13), Poincaré inequality, we get $(u_{\varepsilon n})_{n \in \mathbb{N}^*}$ is bounded in $W_{\Gamma_L}^{1,p(\cdot)}(\Omega)$, so there exists a subsequence denoted again $(u_{\varepsilon n})_{n \in \mathbb{N}^*}$ such that (4.15) hold, thus with the compact injection from $W_{\Gamma_L}^{1,p(\cdot)}(\Omega)$ to $L^{p(\cdot)}(\Omega)$, we deduce (4.16).

where $\frac{1}{p} < s < 1$, we obtain

$$ u_{n\varepsilon} \to u_{\varepsilon} \quad (\text{strongly}) \text{ in } W_{\Gamma_L}^{s,p}(\Omega) $$

now by the trace theorem from $W_{\Gamma_L}^{s,p}(\Omega)$ to $W^{s-\frac{1}{p},p}(\Gamma_0)$ and $W^{s-\frac{1}{p},p}(\Gamma_1)$ we get (4.17) and (4.18). Also from the estimate (4.14) we obtain $E_{kh}(\nabla u_{\varepsilon n})$ is bounded in $L^{p(\cdot)}(\Omega)$. As $p(\cdot) > 3$ then $p(\cdot) > p'(\cdot)$ thus $E_{kh}(\nabla u_{\varepsilon n})$ is also bounded in $L^{p'(\cdot)}(\Omega)$. So there exists $\zeta_{kh}^{\varepsilon} \in L^{p'(\cdot)}(\Omega)$ such that, for a subsequence

$$ E_{kh}(\nabla u_{\varepsilon n}) \to \zeta_{kh}^{\varepsilon} \quad \text{in } L^{p'(\cdot)}(\Omega) \quad (4.20) $$

then it remain to prove that

$$ \zeta_{kh}^{\varepsilon} = E_{kh}(\nabla u_{\varepsilon}). $$

Indeed let $v \in V(\Omega)$, as $V(\Omega)$ is separable then there exists a sequence $(v_n)_{n \in \mathbb{N}^*}$ such that

$$ v_n \to v \quad \text{in} \quad V(\Omega) \quad (4.21) $$

$$ \nabla v_n \to \nabla v \quad \text{strongly in } (L^{p(\cdot)}(\Omega))^9 \quad (4.22) $$

using in Problem 4 with $v = v_n$ we get
$$ 
\begin{aligned}
& \int_{\Omega} a_{ijkh} E_{kh}(\nabla u_{n\epsilon}) \frac{\partial v_{ni}}{\partial x_j} dx + \langle \psi'_{\epsilon}(u_{n\epsilon}), v_n \rangle + \int_{\Gamma_1} g u_{n\epsilon} \cdot v_n dS \\
& = \int_{\Omega} f \cdot v_n dx, \quad \forall v \in V_n \quad (4.23)
\end{aligned}
$$

passing to the limit in (4.23) for $n \to +\infty$ using (4.20), (4.22), (4.21), (4.17) and (4.18) we obtain

$$ 
\begin{aligned}
& \int_{\Omega} a_{ijkh} \zeta_{kh}^{\epsilon} \frac{\partial v_i}{\partial x_j} dx + \langle \psi'_{\epsilon}(u_{\epsilon}), v \rangle + \int_{\Gamma_1} g u_{\epsilon} \cdot v \, dS \\
& = \int_{\Omega} f \cdot v \, dx, \quad \forall v \in V(\Omega) \quad (4.24)
\end{aligned}
$$

it remain now to prove :

$$ \int_{\Omega} a_{ijkh} E_{kh}(\nabla u_{\epsilon}) \frac{\partial v_i}{\partial x_j} dx = \int_{\Omega} a_{ijkh} \zeta_{kh}^{\epsilon} \frac{\partial v_i}{\partial x_j} dx \quad \forall v \in V(\Omega). \quad (4.25) $$

Let

$$ 
\begin{aligned}
& \int_{\Omega} a_{ijkh} (E_{kh}(\nabla u_{n\epsilon}) - E_{kh}(\nabla v_n)) \left( \frac{\partial u_{n\epsilon_i}}{\partial x_j} - \frac{\partial v_{n_i}}{\partial x_j} \right) dx \\
& = A_{1,n} - A_{2,n} - A_{3,n} + A_{2,n} \quad (4.26)
\end{aligned}
$$

we passe first to the limite for $n \to +\infty$ on

$$ A_{1,n} = \int_{\Omega} a_{ijkh} E_{kh}(\nabla u_{n\epsilon}) \frac{\partial u_{n\epsilon_i}}{\partial x_j} dx $$

as

$$ 
\begin{aligned}
A_{1,n} = \int_{\Omega} a_{ijkh} E_{kh}(\nabla u_{n\epsilon}) \frac{\partial u_{n\epsilon i}}{\partial x_j} dx = \int_{\Omega} f \cdot u_{n\epsilon} dx - \langle \psi'_{\epsilon}(u_{n\epsilon}), u_{n\epsilon} \rangle \\
- \int_{\Gamma_1} g u_{n\epsilon} \cdot u_{n\epsilon} dS
\end{aligned}
$$

from (4.16)-(4.18) we obtain

$$ \lim_{n \to +\infty} A_{1,n} = \int_{\Omega} f \cdot u_{\epsilon} dx - \langle \psi'_{\epsilon}(u_{\epsilon}), u_{\epsilon} \rangle - \int_{\Gamma_1} g u_{\epsilon} \cdot u_{\epsilon} dS $$

and from (4.24)

$$ \int_{\Omega} f \cdot u_{\epsilon} dx - \langle \psi'_{\epsilon}(u_{\epsilon}), u_{\epsilon} \rangle - \int_{\Gamma_1} g u_{\epsilon} \cdot u_{\epsilon} dS = \int_{\Omega} a_{ijkh} \zeta_{kh}^{\epsilon} \frac{\partial u_{\epsilon i}}{\partial x_j} dx $$

then we obtain that

$$ \lim_{n \to +\infty} A_{1,n} = \int_{\Omega} a_{ijkh} \zeta_{kh}^{\epsilon} \frac{\partial u_{\epsilon i}}{\partial x_j} dx. \quad (4.27) $$
From (4.20) and (4.22)

$$ \lim_{n \to +\infty} A_{2,n} = \lim_{n \to +\infty} \int_{\Omega} a_{ijkh} E_{kh}(\nabla u_{n\varepsilon}) \frac{\partial v_{ni}}{\partial x_j} dx = \int_{\Omega} a_{ijkh} \zeta_{kh}^{\varepsilon} \frac{\partial v_i}{\partial x_j} dx, $$

using now (4.22) (for $E_{kh}(\nabla v_n)$ and (4.15) (for $\frac{\partial u_{n\varepsilon i}}{\partial x_j}$) to get

$$ \begin{aligned} \lim_{n \to +\infty} A_{3,n} &= \lim_{n \to +\infty} \int_{\Omega} a_{ijkh} E_{kh}(\nabla v_n) \frac{\partial u_{n\varepsilon i}}{\partial x_j} dx \\ &= \int_{\Omega} a_{ijkh} E_{kh}(\nabla v) \frac{\partial u_{\varepsilon i}}{\partial x_j} dx, \end{aligned} $$

using (4.22) we get also

$$ \begin{aligned} \lim_{n \to +\infty} A_{4,n} &= \lim_{n \to +\infty} \int_{\Omega} a_{ijkh} E_{kh}(\nabla v_n) \frac{\partial v_{ni}}{\partial x_j} dx \\ &= \int_{\Omega} a_{ijkh} E_{kh}(\nabla v) \frac{\partial v_i}{\partial x_j} dx. \end{aligned} $$

Using now the monotony of $E_{kh}$ for $1 \le h, k \le 3$ (see [14] see also [2])
to conclude from (4.26) that

$$ A_{1,n} - A_{2,n} - A_{3,n} + A_{4,n} \ge 0 $$

then for $n \to +\infty$ we obtain

$$ \int_{\Omega} a_{ijkh} (\zeta_{hk}^{\varepsilon} - E_{kh}(\nabla v)) \left( \frac{\partial u_{\varepsilon i}}{\partial x_j} - \frac{\partial v_i}{\partial x_j} \right) dx \ge 0 \quad \forall v \in V(\Omega). $$

Choosing now $v = u_{\varepsilon} + \frac{1}{m}w$ with $w \in V(\Omega)$ and $m \in \mathbb{N}^*$ we get

$$ -\frac{1}{m} \int_{\Omega} a_{ijkh} \left( \zeta_{hk}^{\varepsilon} - E_{kh} \left( \nabla u_{\varepsilon} - \frac{1}{m} \nabla w_i \right) \right) \frac{\partial w_i}{\partial x_j} dx \ge 0 \quad \forall w \in V(\Omega). $$

or

$$ \int_{\Omega} a_{ijkh} \left( \zeta_{hk}^{\varepsilon} - E_{kh} \left( \nabla u_{\varepsilon} - \frac{1}{m} \nabla w \right) \right) \frac{\partial w_i}{\partial x_j} dx \le 0 \quad \forall w \in V(\Omega). \quad (4.28) $$

For $m \to +\infty$, $u_{\varepsilon} + \frac{1}{m}w \to u_{\varepsilon}$ in $V(\Omega)$, and as $E_{kh}$ is bounded and continuous function (for k, h = 1 to 3) see [2], then with Lebesgue dominated convergence theorem we have $E_{kh}(\nabla u_{\varepsilon} + \frac{1}{m}\nabla w) \to E_{kh}(\nabla u_{\varepsilon})$ in $L^{p'(\cdot)}(\Omega)$. Thus passing to the limite for $m \to +\infty$, in (4.28) to get

$$ \int_{\Omega} a_{ijkh} (\zeta_{hk}^{\varepsilon} - E_{kh}(\nabla u_{\varepsilon})) \frac{\partial w_i}{\partial x_j} dx \le 0 \quad \forall w \in V(\Omega). $$

Changing $w$ by $-w$ we obtain

$$ \int_{\Omega} a_{ijkh} (\zeta_{hk}^{\varepsilon} - E_{kh}(\nabla u_{\varepsilon})) \frac{\partial w_i}{\partial x_j} dx = 0 \quad \forall w \in V(\Omega). $$
So (4.25) is proved, thus (4.19) is proved, moreover $w$ in $V(\Omega)$ so $\frac{\partial w_i}{\partial x_j}$ in $L^{p(\cdot)}(\Omega)$ and $a_{ijkh}$ in $L^\infty(\Omega)$ thus $\zeta_{hk}^\varepsilon - E_{kh}(\nabla u_\varepsilon) = 0$ in $L^{p'(\cdot)}(\Omega)$. Then we can conclude the proof of this Lemma 4.4. $\square$

Moreover as the estimate in Lemma 4.3 not depends neither on $n$ nor on $\varepsilon$, then we have also :

**Lemma 4.5.** With the assumptions of Theorem 4.1, we obtain the following limits when $\varepsilon \to 0$, :

$$u_{\varepsilon} \to u \text{ (weakly) in } W_{\Gamma_L}^{1,p(\cdot)}(\Omega), \quad (4.29)$$

$$u_{\varepsilon} \to u \quad (\text{strongly}) \quad \text{in } L^{p(\cdot)}(\Omega), \quad (4.30)$$

$$u_{\varepsilon} \to u \quad \text{in } L^{p(\cdot)}(\Gamma_1), \quad (4.31)$$

$$u_{\varepsilon} \to u \quad (\text{in } L^2(\Gamma_0)), \quad (4.32)$$

$$E_{kh}(\nabla u_{\varepsilon}) \to E_{kh}(\nabla u) \quad \text{in} \quad L^{p'(\cdot)}(\Omega). \quad (4.33)$$

# 5. Existence results

## 5.1. The solution to the approximate problem

**Theorem 5.1.** With the assumptions of Theorem 4.1, there exists a unique solution to problem 3.

*Proof.* Using Lemma 4.4, we can now pass to the limit for $n \to +\infty$, in finite dimensional problem 4, to obtain

$$\int_{\Omega} a_{ijkh} E_{kh}(\nabla u_{\varepsilon}) \frac{\partial v_i}{\partial x_j} dx + \langle \psi'_{\varepsilon}(u_{\varepsilon}), v \rangle + \int_{\Gamma_1} g u_{\varepsilon} \cdot v \, dS \\ = \int_{\Omega} f \cdot v \, dx, \forall v \in V(\Omega)$$

then using a regularity result see [3] we obtain that $u_{\varepsilon} \in W_{\Gamma_L}^{p(\cdot)}(\Omega)$, $p(\cdot) > 3$, thus $u_{\varepsilon}$ is a solution of the approximate Problem 3.

We prove also the uniqueness of this solution of the approximate Problem 3, with the same proof of the uniqueness to the solution to Problem 4 see the proof in theorem 4.2. $\square$
## 5.2. The solution to Tresca problem

**Theorem 5.2.** With the assumptions of Theorem 4.1, there exists a unique solution to problem 2.

*Proof.* By definition of subdifferential of $\psi_{\varepsilon}$:

$$ \psi_{\varepsilon}(v + u_{\varepsilon}) - \psi_{\varepsilon}(u_{\varepsilon}) \geq \langle \psi'_{\varepsilon}(u_{\varepsilon}), v \rangle $$

The variational equality in problem 3 becomes

$$ \int_{\Omega} a_{ijkh} E_{kh}(\nabla u_{\varepsilon}) \frac{\partial v_i}{\partial x_j} dx + \psi_{\varepsilon}(v + u_{\varepsilon}) - \psi_{\varepsilon}(u_{\varepsilon}) \\ + \int_{\Gamma_1} g u_{\varepsilon} \cdot v \, dS \geq \int_{\Omega} f \cdot v \, dx $$

Passing now to the limit for $\varepsilon \to 0$ using Lemma 4.5, then taking

$$ v = \varphi - u $$

we get

$$ \int_{\Omega} a_{ijkh} E_{kh}(\nabla u) \frac{\partial}{\partial x_j} (\varphi_i - u_i) dx + \int_{\Gamma_0} \ell(|\varphi - s| - |u - s|) dx' \\ + \int_{\Gamma_1} g u \cdot (\varphi - u) dS \geq \int_{\Omega} f \cdot (\varphi - u) dx, \forall \varphi \in V(\Omega) $$

by regularity $u \in W_{\Gamma_L}^{p(\cdot)}(\Omega)$, $p(\cdot) > 3$, thus $u$ is a solution to Tresca problem 2.

Let $u_1, u_2$ two solutions to Problem 2 taking $\varphi = u_2$ in the variational inequality of Problem 2 where $u_1$ is solution then $\varphi = u_1$ in Problem 2 where $u_2$ is solution by adding them we obtain

$$ \int_{\Omega} a_{ijkh} (E_{kh}(\nabla u_1) - E_{kh}(\nabla u_2)) \frac{\partial}{\partial x_j} (u_{i1} - u_{i2}) dx \leq 0 $$

then we use the same arguments as in the proof of Theorem 4.2, to conclude the uniqueness. $\square$

## 5.3. Existence for the variational problem 1.

**Theorem 5.3.** With the assumptions in Theorem 4.1, there exists at least one solution to the variational of problem 1.

*Proof.* Let $\Psi$ from $\ell \in W_{+}^{-\frac{1}{p'(\cdot)}, p'(\cdot)}(\Gamma_0)$ to $\Psi(\ell) = \mu|\sigma_{\nu}(u_{\ell})|$, where $u_{\ell}$ is the unique solution of problem 2.

As $u_{\ell} \in W_{\Gamma_L}^{p(\cdot)}(\Omega)$ then $E_{hk}(\nabla u_{\ell}) \in W_{\Gamma_L}^{1, p(\cdot)}(\Omega)$ for $1 \le h, k \le 3$, with (2.4) we obtain that $\sigma_{ij}(u_{\ell}) \in W_{\Gamma_L}^{1, p(\cdot)}(\Omega)$, thus using the continuous injection from $W_{\Gamma_L}^{1, p(\cdot)}(\Omega)$ into $W^{-\frac{1}{p'(\cdot)}, p'(\cdot)}(\Gamma_0)$, we obtain that $\sigma_{\nu}(u_{\ell}) \in W^{-\frac{1}{p'(\cdot)}, p'(\cdot)}(\Gamma_0)$, moreover from (2.3) $\mu \in L_{+}^{\infty}(\Gamma_0)$ then

$$ \Psi(\ell) = \mu|\sigma_{\nu}(u_{\ell})| \in W_{+}^{-\frac{1}{p'(\cdot)}, p'(\cdot)}(\Gamma_0). $$
Now let us show that $\Psi$ is continuous. So let $(\ell_\eta)_{\eta \in \mathbb{N}}$ such that $\ell_\eta \in W_+^{-\frac{1}{p'(\cdot)},p'(\cdot)}(\Gamma_0)$ which strongly converges to $\ell$ in $W_+^{-\frac{1}{p'(\cdot)},p'(\cdot)}(\Gamma_0)$. We must check that $\Psi(\ell_\eta)$ strongly converges to $\Psi(\ell)$.

As $u_\ell$ is solution of:

$$ \int_{\Omega} a_{ijkh} E_{kh}(\nabla u_{\ell}) \frac{\partial}{\partial x_j} (\varphi_i - u_{i\ell}) dx + \int_{\Gamma_0} \ell(|\varphi - s| - |u_\ell - s|) dx' \\ + \int_{\Gamma_1} g u_\ell \cdot (\varphi - u_\ell) dS \geq \int_{\Omega} f \cdot (\varphi - u_\ell) dx, \forall \varphi \in V(\Omega) \quad (5.1) $$

and $u_{\ell_\eta}$ is solution of:

$$ \int_{\Omega} a_{ijkh} E_{kh}(\nabla u_{\ell_\eta}) \frac{\partial}{\partial x_j} (\varphi_i - u_{i\ell_\eta}) dx + \int_{\Gamma_0} \ell_\eta(|\varphi - s| - |u_{\ell_\eta} - s|) dx' \\ + \int_{\Gamma_1} g u_{\ell_\eta} \cdot (\varphi - u_{\ell_\eta}) dS \geq \int_{\Omega} f \cdot (\varphi - u_{\ell_\eta}) dx, \forall \varphi \in V(\Omega) \quad (5.2) $$

taking in (5.1) $\varphi = u_{\ell_\eta}$

$$ \int_{\Omega} a_{ijkh} E_{kh}(\nabla u_{\ell}) \frac{\partial}{\partial x_j} (u_{i\ell_\eta} - u_{i\ell}) dx + \int_{\Gamma_0} \ell(|u_{\ell_\eta} - s| - |u_\ell - s|) dx' \\ + \int_{\Gamma_1} g u_\ell \cdot (u_{\ell_\eta} - u_\ell) dS \geq \int_{\Omega} f \cdot (u_{\ell_\eta} - u_\ell) dx, \quad (5.3) $$

then in (5.2) $\varphi = u_\ell$

$$ \int_{\Omega} a_{ijkh} E_{kh}(\nabla u_{\ell_\eta}) \frac{\partial}{\partial x_j} (u_{i\ell} - u_{i\ell_\eta}) dx + \int_{\Gamma_0} \ell_\eta(|u_\ell - s| - |u_{\ell_\eta} - s|) dx' \\ + \int_{\Gamma_1} g u_{\ell_\eta} \cdot (u_\ell - u_{\ell_\eta}) dS \geq \int_{\Omega} f \cdot (u_\ell - u_{\ell_\eta}) dx, \quad (5.4) $$

from (5.3) and (5.4) by subtraction we obtain

$$ \int_{\Omega} a_{ijkh} [E_{kh}(\nabla u_{\ell_\eta}) - E_{kh}(\nabla u_\ell)] \frac{\partial}{\partial x_j} (u_{i\ell_\eta} - u_{i\ell}) dx \\ + \int_{\Gamma_1} g (u_{\ell_\eta} - u_\ell)^2 dS \leq \\ \int_{\Gamma_0} (\ell_\eta - \ell) (|u_{\ell_\eta} - s| - |u_\ell - s|) dx' \quad (5.5) $$

then using the monotony of $E_{kh}$ for $1 \le h, k \le 3$ see [2] and the hypothesis that $a_{ijkh}$ is positive for $1 \le i, j, h, k \le 3$ we have the left side of (5.5) is positive and
$$ 
\begin{aligned}
0 \le \int_{\Gamma_0} (\ell_{\eta} - \ell)(|u_{\ell_{\eta}} - s| - |u_{\ell} - s|)dx' &\le \int_{\Gamma_0} |\ell_{\eta} - \ell| |u_{\ell_{\eta}} - u_{\ell}|dx' \\
&\le \|u_{\ell_{\eta}} - u_{\ell}\|_{W_{+}^{\frac{1}{p'(\cdot)},p'(\cdot)}(\Gamma_0)} \|\ell_{\eta} - \ell\|_{W_{+}^{-\frac{1}{p'(\cdot)},p'(\cdot)}(\Gamma_0)}
\end{aligned}
\quad (5.6) $$

then from (5.5) and (5.6), we deduce

$$ 
\begin{aligned}
\int_{\Omega} a_{ijkl} [E_{kh}(\nabla u_{\ell_{\eta}}) - E_{kh}(\nabla u_{\ell})] \frac{\partial}{\partial x_j} (u_{i\ell_{\eta}} - u_{i\ell}) dx \\
\le \|u_{\ell_{\eta}} - u_{\ell}\|_{W_{+}^{-\frac{1}{p'(\cdot)},p'(\cdot)}(\Gamma_0)} \|\ell_{\eta} - \ell\|_{W_{+}^{-\frac{1}{p'(\cdot)},p'(\cdot)}(\Gamma_0)}
\end{aligned}
\quad (5.7) $$

We deduce that taking $l_{\eta}$ strongly converges to $\ell$ in $W_{+}^{-\frac{1}{p'(\cdot)},p'(\cdot)}(\Gamma_0)$, we obtain

$$ [E_{kh}(\nabla u_{\ell_{\eta}}) - E_{kh}(\nabla u_{\ell})] \frac{\partial}{\partial x_j} (u_{i\ell_{\eta}} - u_{i\ell}) \to 0 \quad \text{in } L^1(\Omega) \\ \text{for } 1 \le i, j, k, h \le 3 $$

and as $u_{\ell_{\eta}}$ and $u_{\ell}$ are in $W_{\Gamma_L}^{p(\cdot)}(\Omega)$ then $\nabla u_{\ell_{\eta}}$ and $\nabla u_{\ell}$ are in $(W_{\Gamma_L}^{1,p(\cdot)}(\Omega))^9$ so if $\|\ell_{\eta} - \ell\|_{W_{+}^{-\frac{1}{p'(\cdot)},p'(\cdot)}(\Gamma_0)} \to 0$ we deduce that either

$$ \frac{\partial}{\partial x_j} (u_{i\ell_{\eta}} - u_{i\ell}) \to 0 \quad \text{in } W_{\Gamma_L}^{1,p(\cdot)}(\Omega) \quad \text{for } 1 \le i, j \le 3, \quad (5.8) $$

or

$$ (E_{kh}(\nabla u_{\ell_{\eta}}) - E_{kh}(\nabla u_{\ell})) \to 0 \quad \text{in } (W_{\Gamma_L}^{1,p(\cdot)}(\Omega))' \quad \text{for } 1 \le k, h \le 3 $$

but using that $E_{kh}$ is continuous for $1 \le k, h \le 3$ see [2], thus we obtain from (5.8) that

$$ (E_{kh}(\nabla u_{\ell_{\eta}}) - E_{kh}(\nabla u_{\ell})) \to 0 \quad \text{in } W_{\Gamma_L}^{1,p(\cdot)}(\Omega) \quad \text{for } 1 \le k, h \le 3 \quad (5.9) $$

Moreover

$$ 
\begin{aligned}
\|\Psi(l_{\eta}) - \Psi(l)\|_{W_{+}^{-\frac{1}{p'(\cdot)},p'(\cdot)}(\Gamma_0)} &= \|\mu(|\sigma_{\nu}(u_{l_{\eta}})| - |\sigma_{\nu}(u_l)|)\|_{W_{+}^{-\frac{1}{p'(\cdot)},p'(\cdot)}(\Gamma_0)} \\
&\le A \sum_{i,j=1}^{3} \|\mu(\sigma_{ij}(u_{l_{\eta}}) - \sigma_{ij}(u_l))\|_{W_{\Gamma_L}^{1,p(\cdot)}(\Omega)}, \\
&\le 9A\beta \|\mu\|_{L^{\infty}(\Gamma_0)} \sum_{h,k=1}^{3} \|E_{kh}(u_{l_{\eta}}) - E_{kh}(u_l)\|_{W_{\Gamma_L}^{1,p(\cdot)}(\Omega)}.
\end{aligned}
\quad (5.10) $$
where A depends on the continuous injection constant, from $W_{\Gamma_L}^{1,p(\cdot)}(\Omega)$ into
$W_+^{-\frac{1}{p'(\cdot)},p'(\cdot)}(\Gamma_0)$. Then from (5.9) (5.10) we deduce that

$$ \|\ell_{\eta} - \ell\|_{W_{+}^{-\frac{1}{p'(\cdot)},p'(\cdot)}(\Gamma_0)} \to 0 \implies \Psi(l_{\eta}) \to \Psi(l) \quad \text{in } W_{+}^{-\frac{1}{p'(\cdot)},p'(\cdot)}(\Gamma_0). $$

This proves that the operator $\Psi$ is continuous. Now by Schauder Fixed Point theorem we deduce that the operator $\Psi$ has a fixed point

$$ l^* = \Psi(l^*) = \mu|\sigma_{\nu}(u_{l^*})| \in W_{+}^{-\frac{1}{p'(\cdot)},p'(\cdot)}(\Gamma_0) $$

then $u_{l^*}$ is a solution of the variational problem 1. □

## References

[1] E.Azroul, M.B.Benboubker, and M.Rhoudaf, Entropy solution for some $p(x)$-Quasilinear problems with right-hand side measure, Afr. diaspora J. Math. Vol.13, No 2, (2012), p.23-44,

[2] M.Boukrouche, B.Merouani, and F.Zoubai, On a nonlinear elasticity problem with friction and Sobolev spaces with variable exponents, Fixed Point Theory and Algorithms Sci.Eng. 2022, Paper No. 14, 16p (2022),

[3] M.Boukrouche and F.Saidi, Non-isothermal lubrication problem with Tresca fluid-solid interface law. Part I, Nonlinear Analysis: Real World Applications 7, (2006), p.1145 1166,

[4] Y.Chen, S.Levine, and M.Rao, Variable exponent, linear growth functionals in image restoration, SIAM J. Appl. Math., 66, (2006), p.1383-1406,

[5] L.Diening, P.Harjulehto, P.Hästö, and M.Ruzicka, Lebesgue and Sobolev spaces with variable exponents. Lecture Notes in Mathematics, vol. 2017, Springer Verlag Heidelberg, 2011,

[6] L. Diening, Riesz potential and Sobolev embeddings on generalized Lebesgue and Sobolev spaces $L^{p(\cdot)}$ and $W^{k,p(\cdot)}$, Math. Nachr. 268, (2004), p.31-43,

[7] L.Diening, P.Hästö, and A.Nekvinda, Open problems in variable exponent Lebesgue and Sobolev spaces. In: FSDONA04 Proceedings (Milovy, Czech Republic), (2004), p.38-58,

[8] G. Duvaut, and J.L.Lions, Les inéquations en mécanique et physique, Dunod, Gauthiers-Villars, Paris, (1972).

[9] P.G. Ciarlet, Mathematical elasticity, Vol.I: Three-Dimensional Elasticity, North-Holland, P. G., Amsterdam (1988),

[10] A.El Hachimi, and S.Maatouk, Existence of periodic solutions for some quasi-linear parabolic problems with variable exponents, Arab. J. Math.6, (2017), p.263-280,
DOI 10.1007/s40065-017-0178-0

[11] X. Fan, J. Shen, and D. Zhao, Sobolev embedding theorems for spaces $W^{k,p(\cdot)}(\Omega)$, J. Math. Anal. Appl. 262, (2001), p.749-760,

[12] X.Fan, and D.Zhao, On the spaces $L^{p(x)}(\Omega)$ and $W^{k,p(\cdot)}(\Omega)$, J. Math. Anal. Appl.263, (2001), p.424-446,
[13] J.Leray, and J.L.Lions, Quelques résultats de Višik sur les problèmes elliptiques non linéaires par les méthodes de Minty-Browder, Bulletin de la S.M.F., tome 93 (1965), p.97-107,

[14] B.Merouani, and F.Zoubai, A nonlinear elasticity system in Sobolev spaces with variable exponents, Bull. Math. Soc. Sci. Math. Roumanie, Tome 64 (112) No.1,(2021), p.17-33,

[15] J.Tinsley Oden, Existence Theorems for a Class of Problems in Nonlinear Elasticity. Journal of Mathematical Analysis and applications 69, (1979), p.51-83,

[16] M. Růžička, Electrorheological Fluids: Modeling and Mathematical Theory, Lecture Notes in Math., vol. 1784, Springer-Verlag, Berlin, 2000.

[17] J. Yao, Solutions for Neumann boundary value problems involving $p(x)$-Laplace operators. Nonlinear Anal. 68, 1271-1283 (2008).

[18] V.V. Zhikov, Averaging of functionals of the calculus of variations and elasticity theory, *Izv. Math. USSR* 29, (1987), pp.33–36.

Mahdi Boukrouche

Jean Monnet University, Institut Camille Jordan

UMR 5208, 42023 Saint-Etienne, France. Phone +33651193290

e-mail: mahdi.boukrouche@univ-st-etienne.fr