\documentclass[reqno]{amsart}

\AtBeginDocument{{\noindent\small
2004-Fez conference on Differential Equations and Mechanics \newline
{\em Electronic Journal of Differential Equations},
Conference 11, 2004, pp. 71--80.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or 
http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu  (login: ftp)}
\thanks{\copyright 2004 Texas State University - San Marcos.}
\vspace{9mm}}
\setcounter{page}{71}

\begin{document}

\title[\hfilneg EJDE/Conf/11 \hfil Asymptotic behavior of a non-Newtonian flow]
{Asymptotic behavior of a non-Newtonian flow with stick-slip condition}

\author[F. Boughanim, M. Boukrouche,  H. Smaoui\hfil EJDE/Conf/11\hfilneg]
{Fouad Boughanim, Mahdi Boukrouche,  Hassan Smaoui}  % in alphabetical order

\address{Fouad Boughanim \hfill\break
E.N.S.A.M D\'epartement Math\'ematiques-Informatique, Mekn\`es, Morocco}
\email{fboughanim@yahoo.fr}

\address{Mahdi Boukrouche \hfill\break
CNRS-UMR 5585, E.A.N St-Etienne, France}
\email{Mahdi.boukrouche@univ-etienne.fr}

\address{Hassan Smaoui \hfill\break
E.N.S.A.M D\'epartement Math\'ematiques-Informatique, Mekn\`es, 
Morocco}
\email{hassan.smaoui@usa.com}


\date{}
\thanks{Published October 15, 2004.}
\subjclass[2000]{35A15, 35B40, 35B45, 76A05, 76D05}
\keywords{Non-newtonian fluid; power law; stick-slip condition; 
\hfill\break\indent
asymptotic analysis}


\begin{abstract}
 This paper concerns the asymptotic behavior of  solutions of the
 3D non-newtonian fluid flow with slip condition (Tresca's type)
 imposed in a part of the boundary domain. Existence of at least
 one weak solution is proved. We study the limit when the thickness
 tends to zero and we prove a convergence theorem for velocity
 and pressure in appropriate functional spaces.
 The limit of slip condition is obtained. Besides, the uniqueness
 of the velocity and the pressure limits are also proved.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{remark}[theorem]{Remark}

\section{Introduction}

In the case of polymer fluids the no slip condition on the fluid-solid
interface is not always satisfied. This boundary condition is sometimes
overpassed and we must deal with slip at the wall. This phenomenon
has been studied in a lot of mechanical papers related to non newtonian 
fluids
(see \cite{Jac,Pit}). For polymer fluids, slip at the wall
is not surprising :
entangled polymer have a mixed fluid and solid dynamic behavior.

We consider the incompressible isothermal viscous flow of a non 
newtonian
fluid through a thin slab. The viscosity of fluid follows the power law
(see \cite{Bir}).
On the part of the boundary we consider the stick-slip condition given 
by
Tresca law. We suppose that the flow is steady and the Reynolds number 
is
proportional to $\varepsilon^{-\gamma}$. The inertia effects are
neglected, this condition is proved in \cite{Boug} for different cases
corresponding to various values of $\gamma$ and of the power $r$ of the
Carreau law. It is know that for polymer (non newtonian) flow
through a thin slab the Hele-Shaw equation is used. Our goal is to give
mathematical foundation for the nonlinear averaged momentum equation 
with
stick-slip condition.

Let $\omega$ be a bounded open set of $\mathbb{R}^2$ with sufficiently 
smooth
boundary. The domain is thin slab defined by:
$$
\Omega_\varepsilon=\{(x_1,x_2,x_3)\in\mathbb{R} ^3,(x_1,x_2)\in\omega, 
\hbox{ and }
0< x_3<\varepsilon h(x_1,x_2)\}
$$
Where $h:\omega\longrightarrow \mathbb{R}^*_+$, is a $C^1$.
The incompressibility equation is

\begin{equation}
\mathop{\rm div}v^\varepsilon=v^\varepsilon_{i,i}=0\quad \mbox{in }
\Omega_\varepsilon. \label{eq1}
\end{equation}
For simplicity we take the constant density $\rho=1$. Then
the equation of motion is
\begin{equation}
-\sigma^\varepsilon_{ij,j}=f_i \quad\mbox{in }
\Omega_\varepsilon\quad (i,j=1,2,3). \label{eq2}
\end{equation}
The constitutive law is
\begin{equation}
\sigma^\varepsilon=-p^\varepsilon
I+2\eta_0( D_{II})^{r-2\over 2}D(v^\varepsilon),
\label{eq3}
\end{equation}
where $v^\varepsilon=(v_i^\varepsilon)$ represents the velocity field,
$\sigma^\varepsilon=\sigma^\varepsilon_{ij}$ the stress tensor,
$f=(\hat f(x_1,x_2), f_3(x_1,x_2))$ the body forces, $D=D_{ij}$ the 
rate
of strain tensor, given by
$D_{ij}(v^\varepsilon)={1\over 
2}(v^\varepsilon_{i,j}+v^\varepsilon_{j,i})$,
$D_{II}=D_{ij}(v^\varepsilon)D_{ij}(v^\varepsilon)$, $p^\varepsilon$ 
denotes the
pressure, $\eta_0$ the viscosity and $r>1$ the power of law which
may represent a pseudo-plastic fluid if $1<r<2$,
a dilatant fluid if $r>2$.

We consider the following conditions on
$\partial\Omega_\varepsilon=\omega\cup\Gamma_a^\varepsilon$:
\begin{itemize}
\item On $\omega$:  $v^\varepsilon.n=0$ and
\begin{equation}
 \begin{gathered}
    |\tau^\varepsilon_t| < g(x_1,x_2)\; \Rightarrow\;
v^\varepsilon_t(x_1,x_2)=0\\
|\tau^\varepsilon_t| = g(x_1,x_2)\; \Rightarrow \; \exists\,
\lambda\geq 0 \hbox{ such that }
v^\varepsilon_t=-\lambda\tau^\varepsilon_t
\end{gathered} \label{eq4}
\end{equation}

\item On $\Gamma_a^\varepsilon$:
$v^\varepsilon=0$
\end{itemize}
Here $n=(n_i)$ is the unit outward normal to 
$\partial\omega_\varepsilon$,
and
\begin{gather*}
v^\varepsilon_t=v^\varepsilon - v^\varepsilon_n n, \quad
v^\varepsilon_n= v^\varepsilon_i n_i\\
\tau^\varepsilon_{ti}=\sigma^\varepsilon_{ij}n_j-
\sigma^\varepsilon_n n_i,\quad
\sigma_N^\varepsilon=\sigma^\varepsilon_{ij}n_in_j
\end{gather*}
are, respectively, the tangential velocity, normal velocity the 
components of
tangential stress tensor and the normal stress. $g(x_1,x_2)$ is a 
positive
function in $L^{\infty}(\omega)$ and $f$ in $L^{r'}(\Omega)$.
We use the re-scaling
$z={x_3\over\varepsilon}$ and the notation
$v(\varepsilon)(x_1,x_2,z)=v^\varepsilon(x_1,x_2,\varepsilon z)$,
$p(\varepsilon)(x_1,x_2,z)=p^\varepsilon(x_1,x_2,\varepsilon z)$.
 Hence, $v(\varepsilon)$ is sequence of functions defined on fixed
domain $\Omega$, then the system (\ref {eq1})-(\ref {eq4}) can be 
written
\begin{equation} \label{eq5}
\begin{gathered}
-\varepsilon^\gamma \mathop{\rm div}{}_\varepsilon(| 
D_\varepsilon(v(\varepsilon))|
^{r-2}D_\varepsilon(v(\varepsilon)))+ \nabla_\varepsilon p(\varepsilon) 
=f
\quad\mbox{in } \Omega\\
\mathop{\rm div}{}_\varepsilon(v(\varepsilon))=0\quad \mbox{in } \Omega
\end{gathered}
\end{equation}
On $\Gamma_a$: $v(\varepsilon)=0$\\
On $\omega$: $v(\varepsilon).n=0$ and
\begin{gather*}
|\tau_t(\varepsilon)| < g(x_1,x_2)\; \Rightarrow\;
v_t(\varepsilon)(x_1,x_2)=0\\
|\tau_t(\varepsilon)| = g(x_1,x_2)\; \Rightarrow \; \exists\,
\lambda\geq 0 \quad\hbox{such that }
v_t(\varepsilon)=-\lambda\tau_t(\varepsilon)\,.
\end{gather*}
Here $\nabla_\varepsilon$, $D_\varepsilon$,
$\mathop{\rm div}_\varepsilon$ are the corresponding rescaled 
differential
operators defined by
\begin{gather*}
(\nabla_\varepsilon v)_{i,j}={\partial v_i\over \partial x_j}\quad 
\hbox{for }
i=1,2,3;\; j=1,2;\\
(\nabla_\varepsilon v)_{i,3}={1\over\varepsilon}{\partial v_i\over 
\partial z}\
\quad \hbox{for } i=1,2,3;\\
D_\varepsilon(v)={1\over 2}((\nabla_\varepsilon v)+(\nabla_\varepsilon 
v)^t),
\quad \mathop{\rm div}{}_\varepsilon=\nabla_\varepsilon\,.
\end{gather*}
Our main aim in this paper is to prove the existence of weak solution
$(v(\varepsilon), p(\varepsilon))$ of problem (\ref{eq5}) and to study 
the
limit when the small thickness of the slab tends to zero.

\section{functional framework and existence}

To formulate the notion of weak solution of the problem (\ref{eq5}),
we recall some Sobolev spaces
\begin{gather*}
W^{1,r}(\Omega)=\{v\in L^r(\Omega)\hbox{ and } {\partial
v\over\partial x_i}\in L^r(\Omega),\;i=1,2,3\},\\
V^{r}=\{v\in (W^{1,r}(\Omega))^3,\; v=0 \hbox{ on } \Gamma_a,\; v.n=0
\hbox{ on } \omega\},\\
 V^{r}_{\rm div}=\{v\in V^{r},\mathop{\rm div}(v)=0\hbox{ in } 
\Omega\}.
\end{gather*}
On $V^r$, we define the functional\ $j:V^r\to \mathbb{R},\ \
v\mapsto \int_\omega g(s)| v_t(s)| ds$. Note that $j$ is continuous 
convex, but non
differentiable. The problem (\ref{eq6}) has a  variational formulation
 (see \cite{Duv}) written as follos:\\
Find $(v(\varepsilon),p(\varepsilon))\in V_{\rm div}^r\times 
L_0^{r'}(\Omega)$
 such that
\begin{equation}
\begin{aligned}
&\varepsilon^\gamma\int_\Omega|D_\varepsilon(v(\varepsilon))|^{r-2}
D_\varepsilon(v(\varepsilon))D_\varepsilon(w-v(\varepsilon))dx
+\int_\Omega p(\varepsilon)div_\varepsilon(w)dx + j(w) - 
j(v(\varepsilon)) \\
&\geq \int_\Omega f(w-v(\varepsilon))dx,\quad  \forall w\in V^r
\end{aligned} \label{eq6}
\end{equation}
For $v\in W^{1,r}(\Omega)$, we define the functional
\begin{equation}
F_r^\varepsilon(v)={{\varepsilon^\gamma}\over r}\int_\Omega | D(v)|^r 
dx
- \int_\Omega fv\, dx. \label{eq7}
\end{equation}
Note that $F_r^\varepsilon$ is Gateaux-differentiable and strictly 
convex,
and that for every $v,w$ in $W^{1,r}(\Omega)$,
\begin{equation}
\langle DF_r^\varepsilon(v),w\rangle
=\varepsilon^\gamma\int_\Omega | D(v)|^{r-2}D(v)D(w)dx -
 \int_\Omega fw\,dx, \label{eq8}
\end{equation}
where $\langle\cdot,\cdot\rangle$ denotes duality of pairing
$W^{-1,r'}(\Omega)\times W^{1,r}(\Omega)$, and $r'$ is the conjugate 
number
of $r$,(${1\over r}+{1\over {r'}}=1$).

The operators $DF_r^\varepsilon$ is strictly monotone bounded coercive 
and
hemicontinuous (see \cite {Ode}). Now, we introduce an auxiliary 
problem.\\
Find $v(\varepsilon)\in V_{div}^r$  such that
\begin{equation}
\langle DF_r^\varepsilon(v(\varepsilon)),w-v(\varepsilon)\rangle
+j(w) - j(v(\varepsilon))
 \geq \int_\Omega f(w-v(\varepsilon))dx,\quad  \forall w\in 
V_{div}^r\,.
 \label{eq9}
\end{equation}
Then we define the associated minimization problem:
Find $u\in V_{\rm div}^r$ such that
\begin{equation} \label{M}
\phi(u)={\inf_{w\in V^r_{div}}}[\phi(w)],
\end{equation}
where $\phi(w)=F_r^\varepsilon(w)+j(w)$.

\begin{lemma} \label{lem2.1}
Problems \eqref{eq9} and \eqref{M} are equivalent.
\end{lemma}

\begin{proof}
The proof use the monotonicity of the operators $DF_r^\varepsilon$ and
the convexity of $j$ (see \cite{Ode}).
\end{proof}


\begin{theorem}
For $r>1$ and fixed $\varepsilon$,
\eqref{eq5} has a unique solution
$(v(\varepsilon),p(\varepsilon))$ in $V_{\rm div}^r\times 
L_0^{r'}(\Omega)$.
\end{theorem}

\begin{proof}
 From nonlinear operator theory we deduce that (\ref{eq9})
has a unique solution $v(\varepsilon)\in V_{div}^r$
(see \cite{Lio,Ode}).
Using Lemma 2.1 we have that $v(\varepsilon)$ is also a unique solution
of the problem \eqref{M}.

 Let $Y=L^r(\omega)\times L^r(\Omega),$ we introduce the
following indicator functionals
\begin{gather*}
w\in L^r(\Omega);\quad \psi(w)=\begin{cases}
0&\mbox{if } w=0\\
+\infty &\mbox{otherwise}
\end{cases}\\
\upsilon\in V^r;\quad \mathcal{G}(\upsilon)
=(\upsilon/\omega,\mathop{\rm div}(\upsilon))\in Y,\\
q=(q_1,q_2)\in Y;\quad  \varphi(q)=j(q_1)\ +\ \psi(q_2).
\end{gather*}
The unique solution $v(\varepsilon)$ of (\ref{eq9}), satisfies
\begin{equation}
 v(\varepsilon):=\inf_{w\in 
V^r}[F_r^\varepsilon(w)+\varphi(\mathcal{G}(w))] %\mathcal{M'}
\label{eq10}
\end{equation}
The existence of pressure $p$ is assured by using the
dual problem. The problem dual to \eqref{eq10} can be written as
\begin{equation}
p^*=(p^*_1,p^*_2)\in Y;\quad p*:={\sup_{q^*\in Y^*}
[-F_r^{\varepsilon^*}
(\mathcal{G}^*(q^*))-\varphi^*(-q^*)]} %\qquad(\mathcal{M'})^*
\label{eqM*}
\end{equation}
$v(\varepsilon)$ and $p^*$ are solutions of \eqref{eq10} and \eqref{eqM*} 
verify
the following extremality relation (see \cite{Eke}):
\begin{equation}
F_r^\varepsilon(v(\varepsilon))+\varphi(\mathcal{G}(v(\varepsilon)))+
F_r^{\varepsilon^*}(\mathcal{G}^*(p^*))+
\varphi^*(-p^*)=0 \label{eq11}
\end{equation}\\
where $F_r^{\varepsilon^*}$, $\varphi^*$ denote the conjugates 
functionals
of $F_r^\varepsilon$ and $\varphi$ defined by
\begin{equation}
F_r^{\varepsilon^*}(\mathcal{G}^*(p^*))=\sup_{u\in 
V^r}\{<\mathcal{G}^*(p^*),u>-F(u)\}
\label{eq12}
\end{equation}
\begin{equation}
\begin{aligned}
\varphi^*(-q^*)
&=\sup_{q\in Y}\{\langle -q^*,q\rangle -\varphi(q)\}\\
&=\sup_{q_1\in L^r(\omega)}\{\langle -q_1^*,q_1\rangle-j(q_1)\}
+\sup_{q_2\in L^r(\Omega)}\{\langle -p_2^*,q_2\rangle - \psi(q_2)\}
\end{aligned} \label{eq13}
\end{equation}
Observing that $\sup_{q_2\in L^r(\Omega)}\{\langle-p_2^*,q_2\rangle- 
\psi(q_2)\}=0$
and replacing in (\ref{eq11}) $q$ by $\mathcal{G}(u),$ we obtain that
$v(\varepsilon)$ satisfies
$$
F_r^\varepsilon(v(\varepsilon))-F_r^\varepsilon(u)+j(\mathcal{G}_1(v(\varepsilon)))-
j(\mathcal{G}_1(u)) +\psi(\mathcal{G}_2(v(\varepsilon)))-
\langle p^*_2,div(u)\rangle \leq 0,\quad \forall u\in V^r.
$$
 Finally, we get the result by using lemma 2.1.
\end{proof}

\section{Convergence results} 

The limit model is obtained thanks to an asymptotic analysis. The used 
techniques
emanate from the ones used in homogenization. In this section we adopt
the following notation:
\begin{gather*}
v(\varepsilon)=(\hat v(\varepsilon),v_3(\varepsilon)),\quad
 \hat v(\varepsilon)=(v_1(\varepsilon),v_2(\varepsilon)),\\
x=(x',z),\quad x'=(x_1,x_2),\quad \delta={r-\gamma\over{r-1}}.
\end{gather*}
Also, we  use the functional space
 $$
 \chi^r=\{v\in L^r(\Omega)\hbox{ and } {\partial v\over\partial z}\in 
L^r(\Omega)\}.
$$
and we will need the following results given by lemmas 3.1 and  3.2.
For $v^\varepsilon\in (L^r(\Omega_\varepsilon))^3$, $1\leq r<+\infty$,
we have for every $v\in (L^r(\Omega_\varepsilon))^3$:
$\| v^\varepsilon \|_{(L^r(\Omega_\varepsilon))^3}
=\varepsilon^{1\over r}\| v(\varepsilon)\|_{(L^r(\Omega))^3}$.

\begin{lemma}[Poincar\'e inequality] \label{lem3.1}
For $v\in (W_0^{1,r}(\Omega_\varepsilon))^3$, $1< r<+\infty$,
\begin{equation}
\| v^\varepsilon\|_{(L^r(\Omega_\varepsilon))^3}\leq \varepsilon \|
{\partial v^\varepsilon\over \partial
x_3}\|_{(L^r(\Omega_\varepsilon))^3}.
\label{eq14}
\end{equation}
\end{lemma}

\begin{lemma}[Korn inequality] \label{lem3.2}
For $v\in (W_0^{1,r}(\Omega_\varepsilon))^3$,
$1< r<+\infty,$
\begin{equation}
\|\nabla v^\varepsilon\|_{(L^r(\Omega_\varepsilon))^9}
\leq C.\| D(v^\varepsilon)\|_{(L^r(\Omega_\varepsilon))^9}.
\label{eq15}
\end{equation}
where $C$ is a positive constant independent of $\varepsilon$ and 
$v^\varepsilon$.
\end{lemma}

The proof of the above lemma can be found in \cite{Kon} and
\cite{Mos}.

\begin{proposition} \label{prop3.1}
Let $(v(\varepsilon),p(\varepsilon))$, be a sequence solution to 
problem
(\ref{eq6}), we have
\begin{gather}
\hat u(\varepsilon)=\varepsilon^{-\delta}\hat v(\varepsilon) 
\rightharpoonup
\hat u\quad \hbox{in } (\chi^r)^2, \label{eq16} \\
u_3(\varepsilon)=\varepsilon^{-\delta}\hat v_3(\varepsilon) 
\rightharpoonup 0
\quad \hbox{in }\chi^r, \label{eq17} \\
p(\varepsilon) \rightharpoonup p(x')\quad  \hbox{in }L_0^{r'}(\Omega). 
\label{eq18}
\end{gather}
\end{proposition}

\begin{proof} Estimates for velocity and pressure are obtained
from (\ref{eq6}), by using the Poincar\'e and Korn inequalities.
Taking $w=-v(\varepsilon)$ in (\ref{eq6}) we obtain with Schwartz
inequality
\begin{equation}
\varepsilon^\gamma\int_\Omega| D_\varepsilon(v(\varepsilon))|^r
dx\leq c\|v(\varepsilon)\|_{L^r(\Omega)}. \label{eq19}
\end{equation}
Using (\ref{eq14}) and (\ref{eq15}), we deduce
$\| v(\varepsilon)\|_{(L^r(\Omega))^3}\leq \| {\partial 
v(\varepsilon)\over\partial
z}\|_{(L^r(\Omega))^3}$
and
$$
\| v(\varepsilon)\|_{(L^r(\Omega))^3}\leq C\varepsilon
\| D_\varepsilon(v(\varepsilon))
\|_{(L^r(\Omega))^{3\times 3}}
$$
which with (\ref{eq19}) give
\begin{equation}
\begin{gathered}
\| v(\varepsilon)\|_{(L^r(\Omega))^3}\leq C.\varepsilon^{\delta};\ \ \
\| \nabla_{x'} v(\varepsilon) \|_{(L^r(\Omega))^{3\time 2}}
\leq C.\varepsilon^{\delta-1},\\
\big\| {\partial v(\varepsilon)\over\partial z}
\big\|_{(L^r(\Omega))^3}\leq C.\varepsilon^{\delta}.
\end{gathered}\label{eq20}
\end{equation}
We have, with incompressibility condition,
for any $\varphi\in W_0^{1,r'}(\Omega)$,
$$
\int_\Omega div_\varepsilon(v(\varepsilon))\varphi dx'dz=
\int_\Omega div_{x'}v(\varepsilon) \nabla_{x'}\varphi dx'dz\ +
\ {1\over\varepsilon}\int_{\Omega} {\partial v_3(\varepsilon)\over 
\partial z}
\varphi dx=0
$$
using Green's formula, we obtain
$$
| \int_{\Omega} {\partial v_3(\varepsilon)\over \partial z}\varphi
dx|\leq \varepsilon\| v(\varepsilon)\|_{L^r(\Omega)}
\|\varphi\|_{W_0^{1,r'}(\Omega)}
$$
which with (\ref{eq20}) give
\begin{equation}
\| {\partial v_3(\varepsilon)\over \partial z}\|_{W^{-1,r}(\Omega)}
\leq C\varepsilon^{\delta+1}.
\label{eq20'}
\end{equation}
Let $\varphi\in (W_0^{1,r}(\Omega)^3$, multiplying, the first equation
of (\ref{eq5}), by $\varphi$ and integrating over $\Omega$, we obtain
with Green's formula and Schwartz inequality
$$
\langle \nabla_\varepsilon p(\varepsilon),\varphi\rangle
\leq\varepsilon^\gamma(\int_\Omega| D_\varepsilon (v(\varepsilon))
|^{r})^{1\over{r'}}\| D_\varepsilon (\varphi)\|_{(L^r(\Omega))^3}\ +\
C\| \varphi \|_{L^r(\Omega)}.
$$
Using the inequality $\| 
D_\varepsilon(\varphi)\|_{(L^r(\Omega))^{3\times 3}}
\leq C.{1\over \varepsilon}.\| \nabla\varphi\|_{(L^r(\Omega))^{3\times 
3}}$
and (\ref{eq20}) we deduce
$$
\langle \nabla_\varepsilon p(\varepsilon),\varphi\rangle
\leq C\varepsilon^{\gamma-1}\| D_\varepsilon (v(\varepsilon))\|^{r\over 
{r'}}
\|\varphi\|_{W_0^{1,r}(\Omega)}+\ C\|\varphi \|_{W_0^{1,r}(\Omega)}
$$
 which gives
\begin{equation}
\| p(\varepsilon)\|_{L_0^{r'}(\Omega)}\leq C;\quad \hbox{and}\quad
\big\| {\partial p(\varepsilon)\over\partial 
z}\big\|_{W^{-1,r'}(\Omega)}
\leq C\varepsilon.
\label{eq21}
\end{equation}
Finally (\ref{eq16}), (\ref{eq17}) and (\ref{eq18}) are direct 
consequence
of the {\it a priori} estimates (\ref{eq20}), (\ref{eq20'}) and 
(\ref{eq21}).
\end{proof}


\begin{proposition} \label{prop3.2}
The function $\overline{u}(x')$ defined by
$\overline{u}(x')=\int_0^{h(x')}\hat u(x',z)dz$ satisfies
\begin{equation}
\begin{gathered}
\mathop{\rm div}{}_{x'}(\overline{u}(x'))=0\quad  \hbox{in } \omega,\\
\nu.\overline{u}(x')=0\quad \hbox{on } \partial\omega.
\end{gathered}\label{eq22}
\end{equation}
where $\nu$ is the unit outward normal to $\partial\omega$.
\end{proposition}

\begin{proof}
Let $\varphi\in C^\infty_0(\omega)$, using the
incompressibility condition and Green's formula, we obtain
\begin{align*}
&\int_\Omega\nabla_\varepsilon.u(\varepsilon)(x',z)\varphi(x')dx'dz\\
&= -\int_\Omega u(\varepsilon)(x',z)\nabla_{x'}\varphi(x')dx'dz
+\int_{\partial\Omega}u(\varepsilon)(x',z).n\ \varphi(x')d\gamma,
\end{align*}
as $u(\varepsilon).n=0$ on $\partial\Omega$, we deduce
\begin{equation}
\nabla_{x'}.(\int_0^{h(x')}\hat u(\varepsilon)(x',z)dz)=0. \label{eq23}
\end{equation}
\begin{align*}
\|\int_0^{h(x')}\hat u(\varepsilon)(x',z)dz\|^r_{(L^r(\omega))^2}
&=\int_\omega| \int_0^{h(x')}\hat u(\varepsilon)(x',z)dz|^rdx'\\
&\leq C(\int_\Omega|\hat u(\varepsilon)(x',z)|^rdx'dz).
\end{align*}
This implies
\begin{equation}
\big\| \int_0^{h(x')}\hat u(\varepsilon)(x',z)dz
 \big\|_{(L^r(\omega))^2}\leq C\,. \label{eq24}
\end{equation}
Let $\varphi\in W^{1,r'}(\omega)$, multiplying (\ref{eq23}) by 
$\varphi$ and
integrating over $\omega$, we obtain with Green's formula
\begin{gather*}
| \int_{\partial\omega}\nu.(\int_0^{h(x')}\hat 
u(\varepsilon)(x',z)dz).\varphi(x')d\gamma|
\leq C.\|\varphi\|_{W^{1,r'}(\omega)},\\
\| \nu.(\int_0^{h(x')}\hat u(\varepsilon)(x',z)dz)
\|_{W^{-{1\over r},r}(\omega)}\leq C.
\end{gather*}
Finally, we prove (\ref{eq22}) passing to the limit in (\ref{eq23})
\end{proof}

\section{Limit problem}
\begin{theorem} \label{thm4.1}
The cluster point $(\hat u(x',z),p(x'))$
defined by \eqref{eq16} and \eqref{eq18}  verify the following limit 
problem
\begin{equation}
\begin{gathered}
\frac{-1}{2^{r/2}}\frac{\partial}{\partial z}
\big(\big|\frac{\partial {\hat u}}{\partial z} \big|^{r-2}
\frac{\partial{\hat u}}{\partial z} \big)
+\nabla_{x'}p=\hat f\quad \hbox{in } W^{-1,r'}(\Omega),\\
\big|\frac{\partial{\hat u}}{\partial z}\big|
<(g(x'))^{\frac{1}{r-1}} \;\Rightarrow\; \hat u(x',0)=0,\\
\big|\frac{\partial{\hat u}}{\partial z}\big|
=(g(x'))^{\frac{1}{r-1}} \;\Rightarrow\; \exists\,
\lambda\geq 0 \hbox{ such that } \hat u(x',0)=\lambda\hat\tau(x').
\end{gathered}\label{eq26}
\end{equation}
where  $\hat\tau(x')= |{\partial{\hat u}\over \partial z}(x',0)|^{r-2}
{\partial{\hat u}\over \partial z}(x',0)$.
\end{theorem}

\begin{proof}
To linearize the problem we use the Minty's lemma
(see\cite{Eke}), we obtain that (\ref{eq6}) is equivalent to
\begin{equation}
\begin{aligned}
&\varepsilon^\gamma\int_\Omega| D_\varepsilon(w)|^{r-2}D_\varepsilon(w)
D_\varepsilon(w-v(\varepsilon))dx +\int_\Omega p(\varepsilon)
\mathop{\rm div}{}_{\varepsilon}(w)dx
+ j(w) - j(v(\varepsilon)) \\
&\geq \int_\Omega f(w-v(\varepsilon))dx, \quad \forall w\in\ V^r
\end{aligned} \label{eq27}
\end{equation}
as $u(\varepsilon)=\varepsilon^{-\delta}v(\varepsilon)$, we take in
(\ref{eq27}), $w=(\hat w,w_3)=\varepsilon^\delta w$ and we divide the
inequality by $\varepsilon^\delta$, we obtain that $u(\varepsilon)$
satisfies
\begin{equation}
\begin{aligned}
&\varepsilon^r\int_\Omega|
D_\varepsilon(w)|^{r/2}D_\varepsilon(w)D_\varepsilon(w-u(\varepsilon))dx
+\int_\Omega p(\varepsilon)\mathop{\rm div}{}_{\varepsilon}(w)dx
+ j(w) - j(u(\varepsilon))\\
& \geq \int_\Omega f(w-u(\varepsilon))dx,\quad \forall w\in  V^r
\end{aligned} \label{eq28}
\end{equation}
Using proposition 3.1, we pass to the limit in the first term of
(\ref{eq28}), and we obtain that
$$
\varepsilon^r\int_\Omega| D_\varepsilon(w)|^{r/2}D_\varepsilon(w)
D_\varepsilon(w-u(\varepsilon))dx
$$
converges to
$$
\int_\Omega\Big( {1\over 2}\sum_{i=1}^2
({\partial w_i\over\partial z})^2 + ({\partial w_3\over\partial 
z})^2\Big)^{r-2\over 2}
\Big({1\over 2}\sum_{i=1}^2 {\partial w_i\over\partial z}
{\partial (w_i-u_i)\over\partial z} + {\partial w_3\over\partial z}
{\partial (w_3-u_3)\over\partial z}\Big)dx.
$$
With (\ref{eq12}) we have that $u_3=0$, then we can choose $w_3=0$,
using (\ref{eq16}), (\ref{eq18}) and the fact $j$ is convex and 
continuous,
when passing to limit in (\ref{eq28}), we get to $\hat u$ and
$p$ are the solution of the following inequality
\begin{equation}
\begin{aligned}
&{1\over{2^{r/2}}}\int_\Omega|{\partial{\hat w}\over
\partial z}|^{r-2}{\partial{\hat w}\over \partial
z}{\partial{(\hat w-\hat u)}\over \partial z}dx +
\langle \nabla_{x'}p,\hat w-\hat u\rangle + j(\hat w) - j(\hat u)\\
& \geq \int_\Omega\hat f(\hat w-\hat u)dx,\quad \forall \hat w\in V^r
\end{aligned}\label{eq29}
\end{equation}
Using again Minty's lemma, the last inequality is equivalent to
\begin{equation}
\begin{aligned}
&{1\over{2^{r/2}}}\int_\Omega|{\partial{\hat u}\over
\partial z}|^{r-2}{\partial{\hat u}\over \partial
z}{\partial{(\hat w-\hat u)}\over \partial z}dx\ +\
\langle \nabla_{x'}p,\hat w-\hat u\rangle
+ j(\hat w) - j(\hat u)\\
& \geq \int_\Omega\hat f(\hat w-\hat u)dx,\quad \forall \hat w\in V^r
\end{aligned}\label{eq30}
\end{equation}
Now, we use the density result shown in (\cite{Bouk}): there exists a 
sequence
of functions in $V^r$ which has $\hat u$ as a limit in $\chi^r$, then
we can take in (\ref{eq30})
$\hat w=\hat u \pm \varphi$, where $\varphi\in (W_0^{1,r}(\Omega))^2$,
and we get the first equation of (\ref{eq26}).

 From (\ref{eq30}), the first equation of (\ref{eq26}) and by using
Green's formula, we show that $\hat u$ satisfies
\begin{equation}
\int_\omega (g|\hat w| - \hat\tau\hat w)d\Gamma -
\int_\omega (g|\hat u| - \hat\tau\hat u)d\Gamma
\geq 0,\quad \forall \hat w\in (W^{1,r}(\Omega))^2.
\label{eq31}
\end{equation}
Choosing $\hat w$ in (\ref{eq31}) such that
$\hat w=\pm\lambda \hat w,$ where $\lambda\geq 0$, we obtain that
for all $\hat w\in (W_0^{1,r}(\Omega))^2$,
\begin{gather}
\int_\omega (g|\hat w| \pm \hat\tau\hat w)d\Gamma \geq 0, 
\label{eq32}\\
\int_\omega (g|\hat u| - \hat\tau\hat u)d\Gamma \leq 0\,. \label{eq33}
\end{gather}
Now, we introduce the functional space
$$
\mathcal{I}=\{\psi\in(W^{1-{1\over r},r}(\partial\Omega)),
\hbox{ whith a compact support on } \omega\}.
$$
 From (\ref{eq32}), we deduce that the function
$\hat w\in \mathcal{I}$, $\hat w \to \int_\omega\hat\tau\hat 
w\,d\Gamma$
is continuous for the topology induced by $(L^r(\omega))^2$.
Since $g$ is in $L^\infty(\omega)$ and
strictly positive,  with (\ref{eq32}) we have
$$
\int_\omega (g^{-1}\hat\tau)(g\hat w)\,d\Gamma\leq
\int_\omega g|\hat w|\,d\Gamma=\| g\hat w\|.$$
Since $\mathcal{I}$ is dense in $L^1(\omega)$, we get
\begin{equation}
g^{-1}\hat\tau\in L^\infty(\omega)\quad \hbox{and}\quad  |\hat\tau|\leq 
g
\hbox{ a.e. on }\omega.
\label{eq34}
\end{equation}
Which with (\ref{eq34}) and using inequality (\ref{eq33}) implies the 
boundary
conditions on $\omega$.
\end{proof}

\begin{theorem} \label{thm4.2}
The limit problem \eqref{eq26}
 has a unique solution $(\hat u(x',z),p(x'))$ in
$(\chi^r)^2\times L_0^{r'}(\omega)$.
\end{theorem}

\begin{proof}
The uniqueness will be proven by using proposition 3.2. As
usual, we assume that the problem (\ref{eq26}) has at least two 
solutions
$(\hat u_1,p_1)$ and $(\hat u_2,p_2)$. Integrating the first equation
with respect to $z$, we obtain
\begin{gather}
{1\over{2^{r/2}}}|{\partial\hat u_1\over \partial z}|^{r-2}
{\partial\hat u_1\over \partial z}=\tau_1(x')-z(\hat f - 
\nabla_{x'}p_1)
\label{eq35}\\
{1\over{2^{r/2}}}|{\partial\hat u_2\over \partial z}|^{r-2}
{\partial\hat u_2\over \partial z}=\tau_2(x')-z(\hat f - 
\nabla_{x'}p_2)
\label{eq36}
\end{gather}
where $\tau_i(x')=|{\partial\hat u_i\over \partial z}(x',0)|^{r-2}
{\partial\hat u_i\over \partial z}(x',0)$, $i=1,2$.
We consider the function $\eta_r$ defined by
$\xi\in\mathbb{R}^2$, $\eta_r(\xi)=| \xi|^{r-2}\xi$, which satisfies
(see \cite{Ode}, \cite{San}),
\begin{align*}
&\big(\eta_r({\partial\hat u_2\over\partial z})-
\eta_r({\partial\hat u_1\over\partial z}),
{\partial\hat u_2\over\partial z}-
{\partial\hat u_1\over\partial z}\big)\\
&\geq \begin{cases}
({1\over 2})^{r-1} \big|{\partial\hat u_2\over\partial z}
-{\partial\hat u_1\over\partial z}\big|^r& \mbox{if }r\geq 2\\
(r-1)\big(|{\partial\hat u_2\over\partial z}| +
|{\partial\hat u_1\over\partial z}|\big)^{r-2}
\big|{\partial\hat u_2\over\partial z}-{\partial\hat u_1\over\partial 
z}\big|^2
&\mbox{if } 1<r< 2\,. \end{cases}
\end{align*}
Let $\beta(x')=\tau_2(x')-\tau_1(x'),$ we have
\begin{equation}
\begin{aligned}
&\big(\beta(x') +z\nabla_{x'}(p_2-p_1),{\partial\over\partial z}(\hat 
u_2-\hat u_1)
\big)\\
&\geq \begin{cases}
({1\over 2})^{r-1}
\big|{\partial\hat u_2\over\partial z}-{\partial\hat u_1\over\partial 
z}\big|^r
& \mbox{if }r\geq 2\\
(r-1)\big(|{\partial\hat u_2\over\partial z}| +
|{\partial\hat u_1\over\partial z}|\big)^{r-2}
\big|{\partial\hat u_2\over\partial z}-{\partial\hat u_1\over\partial 
z}\big|^2
&\mbox{if } 1<r< 2
\end{cases}
\end{aligned}\label{eq37}
\end{equation}
Using the boundary conditions on $\omega$, we get
\begin{equation}
\int_\omega\big(\beta (x'),\hat{u}_2(x',0)-\hat{u}_1(x',0)\big)dx'\geq 
0
\label{eq38}
\end{equation}
Integrating (\ref {eq37}) over $\Omega$, from Green's formula 
proposition 3.2
and (\ref {eq38}), we have
\begin{gather}
| {\partial\hat u_2\over\partial z}-{\partial\hat u_1\over\partial 
z}|^r=0
\quad \hbox{if }r\geq 2\quad \hbox{and}\\
(|{\partial\hat u_2\over\partial z}| + |{\partial\hat u_1\over\partial 
z}|)^{r-2}
| {\partial\hat u_2\over\partial z}-{\partial\hat u_1\over\partial 
z}|^2=0\quad
\hbox{if }1<r< 2\,.
\end{gather}
Finally, since $\hat u_1(x',h(x'))=u_2(x',h(x'))=0$, we deduce that
$\hat u_2=\hat u_1$ a.e.\ in  $\Omega$ and
$p_2=p_1$  a.e.\  on $\omega$.
\end{proof}


\begin{theorem} \label{thm4.3}
Let $\delta>0$ and $0<\delta\leq h(x')\leq 1$. Then
$p(x')$, $\hat\tau(x')$, and $\bar{\hat{u}}(x')=\int_0^{h(x')} 
\hat{u}(x',z)\,dz$
satisfy $p(x')\in W^{1,r'}(\omega)$ and
\begin{gather*}
\mathop{\rm div}{}_{x'}(\bar{\hat{u}}(x'))=0\quad \hbox{in } \omega,\\
\nu.\bar{\hat{u}}(x')=0\quad \hbox{on } \partial\omega.
\end{gather*} %\label{eq39}
where
\begin{align*}
\bar{\hat{u}}(x')
&=\int_0^{h(x')}\big(\int_0^z| \hat\tau(x')-\gamma(x')\xi|^{r'-2}
(\hat\tau(x')-\gamma(x')\xi)d\xi\big) dz\\
&\quad +\ h(x')\int_0^{h(x')} | \gamma(x')\xi-\hat\tau(x')|^{r'-2}
(\gamma(x')\xi-\hat\tau(x'))d\xi.
\end{align*}
and $\gamma(x')=2^{r/2}(\hat f(x')-\nabla_{x'}p(x'))$.
\end{theorem}

\begin{proof}
As a weak limit $\hat u\in (\chi^r)^2$, and then
$$|{\partial\hat u\over \partial z}|^{r-2}
{\partial\hat u\over \partial z}\ \in\ (L^{r'}(\Omega)^2.$$

Taking the test function $\hat \phi(x',z)=\mathcal{G}(x')\psi(z)$,
for all $\mathcal{G} \in\ (C_0^\infty(\omega))^2$, and for fixed
$\psi\in\ C_0^\infty(0,\delta)$, $\psi \geq 0$, $\int_0^\delta 
\psi(z)\,dz=1$,
 we have by first equation of (\ref{eq26}),
\begin{equation}
{1\over 2^{r/2}}\int_\Omega \hat w
{\partial \hat u\over\partial z}\,dx'dz-\int_\omega p(x')
\mathop{\rm div}{}_{x'}\mathcal{G}(x')
=\int_\omega\hat f(x')\mathcal{G}(x')\,dx'. \label{eq40}
\end{equation}
Since $f$ is regular, (\ref{eq40}) shows that $\nabla_{x'}p\in 
L^{r'}(\Omega)$,
and we have
\[
\frac{-1}{2^{r/2}}\frac{\partial}{\partial z}
\big(\big|\frac{\partial {\hat u}}{\partial z} \big|^{r-2}
 \frac{\partial{\hat u}}{\partial z} \big) +\nabla_{x'}p
 =\hat f\quad\hbox{in } L^{r'}(\Omega).
\] % {eq41}
Integrating this equation twice with respect to $z$, we obtain
$$
\hat u(x',z)=\int_0^z| \hat\tau(x')-\gamma(x')\xi|^{r'-2}
\left(\hat\tau(x')-\gamma(x')\xi\right)d\xi\ +\ \hat u(x',0).
$$
Since $\hat u(x',h(x'))=0$, we get
$$
\hat u(x',0)=\int_0^{h(x')}| \gamma(x')\xi-\hat\tau(x')|^{r'-2}
(\gamma(x')\xi-\hat\tau(x'))d\xi.
$$
Finally, we obtain the result for proposition 3.2.
\end{proof}

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\end{document}