\documentclass[reqno]{amsart}
\usepackage{hyperref}

\AtBeginDocument{{\noindent\small
{\em Electronic Journal of Differential Equations},
Vol. 2006(2006), No. 156, pp. 1--7.\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 2006 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2006/156\hfil Elasto-plastic torsion problem]
{Elasto-plastic torsion problem as an infinity Laplace's equation}

\author[A. Addou, A. Lidouh, B. Seddoug \hfil EJDE-2006/156\hfilneg]
{Ahmed Addou, Abdeluaab Lidouh, Belkassem Seddoug}  % in alphabetical order

\address{Ahmed Addou \newline
Universit\'{e} Mohammed premier, Facult\'{e} des sciences,
 Oujda, Maroc}
\email{addou@sciences.univ-oujda.ac.ma}

\address{Abdeluaab Lidouh \newline
Universit\'{e} Mohammed premier, Facult\'{e} des sciences,
 Oujda, Maroc}
\email{lidouh@sciences.univ-oujda.ac.ma}

\address{Belkassem Seddoug \newline
Universit\'{e} Mohammed premier, Facult\'{e} des sciences,
 Oujda, Maroc}
\email{seddougbelkassem@yahoo.fr}


\thanks{Submitted October 11, 2006. Published December 18, 2006.}
\subjclass[2000]{35J70, 35J85, 74C05}
\keywords{Infinity Laplace equation; elasto-plastic torsion problem;
\hfill\break\indent variational inequality}

\begin{abstract}
 In this paper, we study a perturbed infinity Laplace's equation,
 the perturbation corresponds to an Leray-Lions operator with
 no coercivity assumption. We consider the case where data are
 distributions or $L^{1}$ elements.
 We show that this problem has an unique solution which is the solution to
 the variational inequality arising in the elasto-plastic torsion problem,
 associated with and operator $A$.
\end{abstract}

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

\section{Introduction}

Given a bounded open subset $\Omega $ of
$\mathbb{R}^N$, $N\geq 1$,
we consider the  Dirichlet Problem
\begin{equation}
\begin{gathered}
Au-\Delta _{\infty }u=f\quad \text{in }\Omega , \\
u=0\quad\text{on }\partial \Omega ,
\end{gathered} \label{Pinfty}
\end{equation}
where $\Delta _{\infty }u=u_{x_{i}}u_{x_{j}}u_{x_{i}x_{j}}$
(see \cite{BDM}), $f$ in $L^{1}(\Omega )$ or $W^{-1,p'}(\Omega )$
and $A$ is a
Leray-Lions operator with no coercivity assumption, i.e.
\begin{equation*}
Av=-\mathop{\rm div}(a(x,\nabla v(x)))
\end{equation*}
where $a:\Omega \times \mathbb{R}^{N}\to \mathbb{R}^{N}$
is a Caratheodory function satisfying the following
assumptions:

For almost every $x\in \Omega $ and for all $\xi ,\eta \in
\mathbb{R}
^{N}$, $(\xi \neq \eta )$, one has:
\begin{gather}
a(x,\xi )\xi \geq 0 , \label{coercivity}\\
 \vert a(x,\xi ) \vert \leq \beta  \big[ h(x)+ \vert \xi
 \vert ^{p-1} \big] , \label{croissance}\\
 \big[ a(x,\xi )-a(x,\eta ) \big]  ( \xi -\eta  ) >0
\label{strictmonot}
\end{gather}
with $1<p<+\infty $, $\beta >0$, $h\in L^{p'}(\Omega )$
($p'$ denotes the conjugate exponent of $p$, i.e:
$\frac{1}{p}+\frac{1}{p'}=1$).

By a solution to \ref{Pinfty} we will mean a variational
solution in the sense which extends that given in (\cite{BDM}) and
(\cite{Rosset}), that is, a function $u$ which is the limit of the sequence
$(u_{n})$ of solutions to the Dirichlet problems
\begin{equation*}
\begin{gathered}
Au_{n}-\Delta _{n}u_{n}=f\quad\text{in }\Omega , \\
u_{n}=0\quad\text{on }\partial \Omega ,
\end{gathered}
\end{equation*}
as $n\to \infty $, where $\Delta _{n}$ is the $n$-Laplacian
operator ($\Delta _{n}v=\mathop{\rm div}( \vert \nabla v \vert ^{n-2}\nabla v)$.

We show that in the variational case ($f\in W^{-1,p'}(\Omega )$),
the sequence $(u_{n})$ converges to the unique solution to the variational
inequality
\begin{gather*}
 \langle Au,v-u \rangle \geq  \langle f,v-u \rangle ,\text{
for all }v\in \mathcal{K},\\
u\in \mathcal{K}.
\end{gather*}
Where $\mathcal{K}$ is the bounded convex cone of $W_{0}^{1,p}(\Omega )$
defined as:
\begin{equation*}
\mathcal{K}= \{ v\in W_{0}^{1,p}(\Omega ):  \vert \nabla
v(x) \vert \leq 1\text{ a.e. in }\Omega  \} ,
\end{equation*}
and in the case $f\in L^{1}(\Omega )$, the sequence $(u_{n})$
converges to the unique solution to the problem
\begin{gather*}
  \langle Au,T_{k}(v-u) \rangle \geq \int_{\Omega }
fT_{k}(v-u)dx ,\quad \text{ for all }v\in \mathcal{K}, \\
u\in \mathcal{K},\quad \text{ for all }k>0.
\end{gather*}
Where $T_{k}:\mathbb{R}\to\mathbb{R}$ is the cut function defined
as
\begin{equation*}
T_{k}(s)= \begin{cases}
s &\text{if  } \vert s \vert \leq k \\
k\mathop{\rm sign}(s) &\text{if } \vert s \vert > k.
\end{cases}
\end{equation*}
here $ \langle .,. \rangle $ denotes the duality pairing between
$W^{-1,p'}(\Omega )$ and $W_{0}^{1,p}(\Omega )$.

Our approach is also inscribed among the techniques of ``the increase of
power'', first introduced by  Boccardo and  Murat in \cite{bocc1}, where
they approached the problem
\begin{gather*}
 \langle Au,v-u \rangle \geq  \langle f,v-u \rangle ,\quad
\text{for all }v\in \mathcal{K}_{0}, \\
u\in \mathcal{K}_{0}= \{ v\in W_{0}^{1,p}(\Omega ):  \vert
v(x) \vert \leq 1\text{ a.e. in }\Omega  \} ,
\end{gather*}
by the sequence of the Dirichlet equations
\begin{gather*}
Au_{n}- \vert u_{n} \vert ^{n-1}u_{n}=f\quad\text{in } D'(\Omega ), \\
u_{n}\in W_{0}^{1,p}(\Omega )\cap L^{n}(\Omega ),
\end{gather*}
where $f\in W^{-1,p'}(\Omega )$ and $A$ is modelled on the $
p$-Laplacian.

Then in \cite{Dall},  Dall'Aglio and  Orsina generalized this result
by considering increasing powers depending of a certain Caratheodory
function satisfying the sign condition and an integrability assumption.

Then finally in \cite{Meskine} the authors extended this result to the
case where increasing powers are multiplied by a quantity depending on the
gradient and verifying adequate conditions, they examine the two cases, $f$
in $L^{1}(\Omega )$ and in $W^{-1,p'}(\Omega )$.

In this paper we examine the case where the increasing powers carry on the
gradients and not on quantities independent of the gradient.

\section{The variational case}

Let $f\in W^{-1,p'}(\Omega )$, $1<p<+\infty $. For all integer
$n\geq p$, we consider the Dirichlet problem
\begin{equation}
\begin{gathered}
Au_{n}-\Delta _{n}u_{n}=f\quad\text{in }\Omega , \\
u_{n}\in W_{0}^{1,n}(\Omega ).
\end{gathered}   \label{Pn}
\end{equation}

It is known \cite{Lions1,Lions2} that, under assumptions
\eqref{coercivity}--\eqref{strictmonot}, the problem \eqref{Pn} has an
unique solution $u_{n}$, in the following sense:
\begin{equation}
\forall v\in W_{0}^{1,n}(\Omega ):\int_{\Omega } [ a(x,\nabla
u_{n})\nabla v+ \vert \nabla u_{n} \vert ^{n-2}\nabla u_{n}\nabla v
 ] dx= \langle f,v \rangle .  \label{Pn2}
\end{equation}

In the sequel $W_{0}^{1,p}(\Omega )$ is equipped with its usual
norm
$$
\|v\|_{W_{0}^{1,p}(\Omega )}= \Big[\int_{\Omega}|\nabla v|^{p}dx \Big]^{1/p}
$$
Let us now, state our first main result.

\begin{theorem}\label{theorem1}
Let $f\in W^{-1,p'}(\Omega )$, $1<p<+\infty $.
Under assumptions \eqref{coercivity}--\eqref{strictmonot}, if $u_{n}$
designates the solution to the problem \eqref{Pn}, then the
sequence $(u_{n})$ converges strongly in $W_{0}^{1,p}(\Omega )$, to the
unique solution $u$ to the problem
\begin{equation}
\begin{gathered}
 \langle Au,v-u \rangle \geq  \langle f,v-u \rangle ,
\quad \text{for all }v\in \mathcal{K}, \\
u\in \mathcal{K}.
\end{gathered}  \label{P}
\end{equation}
\end{theorem}

\subsection*{Proof of Theorem \ref{theorem1}}

\subsection*{A priori estimate}
With $u_{n}$ as a test function in (\ref{Pn2}), we get
\[
\int_{\Omega }a(x,\nabla u_{n})\nabla u_{n}dx+\int_{\Omega} \vert \nabla
u_{n} \vert ^{n}dx = \langle f,u_{n} \rangle
\leq  \Vert f \Vert _{-1,p'} \Vert u_{n} \Vert _{1,p}
\]
hence
\begin{equation}
\int_{\Omega } \vert \nabla u_{n} \vert ^{n}dx\leq c \Vert
u_{n} \Vert _{1,p}\quad \text{for all }n\geq p  \label{estim1}\,.
\end{equation}
In the sequel $c,c_{1},c_{2}\dots $. designate arbitrary constants.

 From (\ref{estim1}), and by splitting
$\int_{\Omega } \vert \nabla u_{n} \vert ^{p}dx$  as
\[
\int_{\Omega } \vert \nabla u_{n} \vert ^{p}dx
=\int_{ [ \vert \nabla u_{n} \vert \leq 1 ] } \vert \nabla
u_{n} \vert ^{p}dx+\int_{ [  \vert \nabla u_{n} \vert >1
 ] } \vert \nabla u_{n} \vert ^{p}dx,
\]
one deduces that
\begin{equation*}
\int_{\Omega } \vert \nabla u_{n} \vert ^{p}dx\leq  \vert
\Omega  \vert +c [ \int_{\Omega } \vert \nabla u_{n} \vert
^{p}dx ] ^{\frac{1}{p}}\quad \text{for all }n\geq p
\end{equation*}
and so
\begin{equation}
\int_{\Omega } \vert \nabla u_{n} \vert ^{p}dx\leq c\quad \text{for all
}n\geq p\,.  \label{estim2}
\end{equation}
Thereafter,
\begin{equation}
\int_{\Omega } \vert \nabla u_{n} \vert ^{n}dx\leq c\quad
\forall n\quad\text{and}\quad
\int_{\Omega } \vert \nabla u_{n} \vert^{q}dx\leq c\quad
\forall q,\;\forall n\geq q.  \label{estim3}
\end{equation}
Therefore, one can construct a subsequence, still denoted by $(u_{n})_{n}$,
such that
\begin{equation}
u_{n} \rightharpoonup u \quad \text{weakly in
$W_{0}^{1,q}(\Omega)$ and uniformly in $\bar{\Omega}$},
\label{weakcv}
\end{equation}
for some $u\in W_{0}^{1,q}(\Omega )\cap L^{\infty }(\Omega )$, for all $q>1$.
More precisely, we have
\begin{equation}
u\in W_{0}^{1,\infty }(\Omega )\quad \text{and}\quad
 \Vert \nabla u \Vert_{\infty }\leq 1.  \label{u_prop}
\end{equation}
Indeed, from (\ref{estim3}) and (\ref{weakcv}), one has
\[
 \Vert \nabla u \Vert _{\infty }
=\lim_{q \to \infty } \Vert \nabla u \Vert _{q}
\leq \lim_{q \to \infty } \big( \liminf_{n\to \infty } \Vert
\nabla u_{n} \Vert _{q} \big) \\
\leq \lim_{q \to \infty }c^{\frac{1}{q}}=1.
\]

\subsection*{Almost everywhere convergence of gradients}
With $v=u_{n}-u$, as a test function in (\ref{Pn2}), and using the fact that
\[
\nabla u_{n} ( \nabla u_{n}-\nabla u ) \geq 0
\] in the set $ \{  \vert \nabla u_{n} \vert \geq  \vert \nabla
u \vert  \}$,
one has
\begin{equation}
 \langle Au_{n},u_{n}-u \rangle +\int_{ \{  \vert \nabla
u_{n} \vert < \vert \nabla u \vert  \} } \vert \nabla
u_{n} \vert ^{n-2}\nabla u_{n} ( \nabla u_{n}-\nabla u )
dx\leq \varepsilon _{n},  \label{ineg1}
\end{equation}
We will denote by $\varepsilon _{n}$ any quantity which converges to zero as
$n$ tends to infinity.

Let $\varepsilon >0$, for the second term on the left in (\ref{ineg1}),
 one puts
\begin{equation*}
A_{1}= \{  \vert \nabla u_{n} \vert < \vert \nabla
u \vert \text{ and } \vert \nabla u_{n} \vert \leq
1-\varepsilon  \} ,\quad
A_{2}= \{ 1-\varepsilon < \vert
\nabla u_{n} \vert < \vert \nabla u \vert  \}
\end{equation*}
and so we have
\begin{equation}
\int_{A_{1}} \vert \nabla u_{n} \vert ^{n-2}\nabla
u_{n} ( \nabla u_{n}-\nabla u ) dx=\sigma
_{n,\varepsilon}, \label{ineg2}
\end{equation}
where $\sigma _{n,\varepsilon}$  denotes a quantity depending on
$n$ and $\varepsilon$, such that, for any fixed $\varepsilon >0$,
$\sigma _{n,\varepsilon} \to 0$, as $n  \to \infty$,
and which may change from line to line.
Also
\begin{equation}
\begin{aligned}
&\int_{A_{2}} \vert \nabla u_{n} \vert ^{n-2}\nabla u_{n} (
\nabla u_{n}-\nabla u ) dx \\
&=\int_{A_{2}} \vert \nabla u_{n} \vert ^{n-2} (  \vert \nabla u_{n} \vert
^{2}- \vert \nabla u \vert ^{2} ) dx
+\int_{A_{2}} \vert \nabla u_{n} \vert ^{n-2}\nabla u (
\nabla u-\nabla u_{n} ) dx   \\
&= q_{n}+I_{n},
\end{aligned}\label{ineg3}
\end{equation}
where the quantity $I_{n}$ is nonnegative, and
$q_{n}\in [-2\varepsilon|\Omega|,0]$.
Combining (\ref{ineg1}), (\ref{ineg2}) and (\ref{ineg3}), one gets
\begin{equation*}
 \langle Au_{n},u_{n}-u \rangle \leq
\sigma_{n,\varepsilon}+2\varepsilon|\Omega|,\forall \varepsilon
> 0
\end{equation*}
On the other hand, $\langle Au,u_{n}-u \rangle  \to 0$, as
$n  \to \infty $, so that
\begin{equation*}
0\leq \langle Au_{n}-Au,u_{n}-u \rangle \leq
\sigma_{n,\varepsilon}+2\varepsilon|\Omega|,\forall \varepsilon
> 0.
\end{equation*}
Passing to the limit as $n  \to \infty$, for any fixed
$\varepsilon$, one has
\begin{equation*}
0\leq \liminf_{n \to\infty}\langle Au_{n}-Au,u_{n}-u
\rangle \leq \limsup_{n \to\infty}\langle Au_{n}-Au,u_{n}-u
\rangle \leq 2\varepsilon|\Omega|\quad \forall \varepsilon
> 0.
\end{equation*}
By the arbitrariness of $\varepsilon$ (and since $\langle
Au_{n}-Au,u_{n}-u \rangle$ does not depend on $\varepsilon$) it
follows that
\begin{equation}
 \langle Au_{n}-Au,u_{n}-u \rangle \to
0\quad \text{as }n\to \infty .
\end{equation}
Which implies, thanks to \eqref{strictmonot}, that (for a subsequence),
\begin{equation*}
 ( a(x,\nabla u_{n})-a(x,\nabla u) )  ( \nabla u_{n}-\nabla
u ) \to 0\text{ a.e. in }\Omega .
\end{equation*}
For a fixed $k>1$, we put
\begin{equation*}
X=\bigcap_{q\in \mathbb{N}}\bigcup_{n\geq q} \{
 \vert \nabla u_{n} \vert \geq k \} ,\text{ and its
complement }Y=\bigcup_{q\in
\mathbb{N}}\bigcap_{n\geq q} \{  \vert \nabla
u_{n} \vert <k \} ,
\end{equation*}
for all $x\in Y$, the sequence $ ( \nabla u_{n}(x) ) $ is bounded
in $\mathbb{R}^{N}$, so
\begin{equation*}
\nabla u_{n}(x)\to \xi
\]
 for a subsequence and some $\xi \in \mathbb{R}^{N}$,
while \eqref{strictmonot} and the continuity of $a(x,.)$, implies
that
$\xi =\nabla u(x)$, we can then conclude that
\begin{equation*}
\nabla u_{n}(x)\to \nabla u(x)\quad \text{for all }x\in Y.
\end{equation*}
To show the almost everywhere convergence of
$ (\nabla u_{n} ) $, it suffices to prove that
$\mathop{\rm meas}(X)=0$.
In deed, from (\ref{estim3}), one has
\begin{equation}
\mathop{\rm meas} \{  \vert \nabla u_{n} \vert \geq
k \} =\int_{ \{  \vert \nabla u_{n} \vert \geq
k \} }1dx\leq \frac{c}{k^{n}}\,.  \label{ineg4}
\end{equation}
Since $X\subset \bigcup_{n\geq q} \{  \vert
\nabla u_{n} \vert \geq k \} $, for all $q$, one deduces that
\begin{equation*}
\mathop{\rm meas}(X)\leq \sum_{n\geq q}\mathop{\rm meas} \{  \vert
\nabla
u_{n} \vert \geq k \} \to 0 \quad\text{as }
q\to \infty .
\end{equation*}

\subsection*{Strong convergence in $W_{0}^{1,p} ( \Omega  ) $}
Thanks to Vitali's theorem, it suffices to show the equi-integrability of
$ (  \vert \nabla u_{n} \vert ^{p} ) $ in $L^{1}(\Omega )$,
what follows from (\ref{estim3}) with $q=p+1$.

Indeed for a measurable subset $E$ of $\Omega $, one has
\[
\int_{E} \vert \nabla u_{n} \vert ^{p}dx
\leq \Big(
\int_{E} \vert \nabla u_{n} \vert ^{p+1}dx \Big) ^{\frac{p}{p+1}
} \Big( \int_{E}1\,dx \Big) ^{\frac{1}{p+1}}
\leq c \big( \mathop{\rm meas}(E) \big) ^{\frac{1}{p+1}}.
\]

\subsection*{The function $u$ is solution to problem \eqref{P}}

Let $v\in \mathcal{K}$ and $0<\theta <1$, taking $z=u_{n}-\theta T_{k}(v)$
as a test function in (\ref{Pn2}), one gets
\begin{equation*}
 \langle Au_{n},z \rangle +\int_{\Omega } \vert \nabla
u_{n} \vert ^{n-2}\nabla u_{n}\nabla zdx= \langle f,z \rangle
\end{equation*}
While noticing that
\begin{equation*}
\int_{ \{  \vert \nabla u_{n} \vert \geq \theta  \vert
\nabla T_{k}(v) \vert  \} } \vert \nabla u_{n} \vert
^{n-2}\nabla u_{n} ( \nabla u_{n}-\theta \nabla T_{k}(v) ) dx\geq 0
\end{equation*}
one has
\begin{equation*}
 \langle Au_{n},z \rangle +\int_{ \{  \vert \nabla
u_{n} \vert <\theta  \vert \nabla T_{k}(v) \vert  \}
} \vert \nabla u_{n} \vert ^{n-2}\nabla u_{n}\nabla zdx\leq
 \langle f,z \rangle
\end{equation*}
Passing to the limit as $n\to \infty$, and using
standard result about Caratheodory functions satisfying
(\ref{croissance}), one gets
\begin{equation*}
 \langle Au,u-\theta T_{k}(v) \rangle \leq  \langle f,u-\theta
T_{k}(v) \rangle
\end{equation*}
The result is then obtained while passing to the limit as
$\theta \to 1$ and $k\to \infty $.

\section{The case $f$ $\in L^{1}(\Omega )$}

In this section, we suppose that $f\in L^{1}(\Omega )$, as in the previous
section. Now we prove our second main result.

\begin{theorem} \label{theorem2}
Let $f\in L^{1}(\Omega )$, $1<p<+\infty $. Under assumptions
\eqref{coercivity}--\eqref{strictmonot}, if $u_{n}$ $(n>N)$ designates the
solution to the problem \eqref{Pn}, then the sequence $(u_{n})$
converges strongly in $W_{0}^{1,p}(\Omega )$, to the unique solution $u$ to
the problem
\begin{equation}
\begin{gathered}
 \langle Au,T_{k}(v-u) \rangle \geq \int_{\Omega }
fT_{k}(v-u)dx \quad \text{ for all }v\in \mathcal{K}, \\
u\in \mathcal{K},\quad \text{for all }k>0.
\end{gathered}\label{P'}
\end{equation}
\end{theorem}

\subsection*{Proof of Theorem \ref{theorem2}}

According to the previous section, it is clear that the estimate
\eqref{estim3} permits to show that the sequence $(u_{n})$ converges in
$W_{0}^{1,p}(\Omega )$ and uniformly in $\bar{\Omega}$ (for a subsequence) to
$u$ satisfying (\ref{u_prop}).

We are  going to prove (\ref{estim3}) and the fact that $u$ is
the solution to \eqref{P'}.

\subsection*{A priori estimate}

With $u_{n}$ $(n>N)$ as a test function in (\ref{Pn2}), we get
\[
\int_{\Omega }a(x,\nabla u_{n})\nabla u_{n}dx+\int_{\Omega
} \vert \nabla
u_{n} \vert ^{n}dx = \int_{\Omega }fu_{n}dx
\leq  \Vert f \Vert _{1} \Vert u_{n} \Vert_{\infty }
\]
Let $q>N$ (fixed), by splitting $\int_{\Omega } \vert \nabla
u_{n} \vert ^{q}dx$ as
\begin{equation*}
\int_{\Omega } \vert \nabla u_{n} \vert ^{q}dx=\int_{ \{
 \vert \nabla u_{n} \vert <1 \} } \vert \nabla
u_{n} \vert ^{q}dx+\int_{ \{  \vert \nabla u_{n} \vert
\geq 1 \} } \vert \nabla u_{n} \vert ^{q}dx
\end{equation*}
and using Sobolev's inequality \cite{Adams}, one has
\begin{equation}
\int_{\Omega } \vert \nabla u_{n} \vert ^{q}dx\leq c\quad
\forall n\geq q;  \label{estim4}
\end{equation}
therefore,
\begin{equation*}
\int_{\Omega } \vert \nabla u_{n} \vert ^{n}dx\leq c\quad
\forall n>N\,.
\end{equation*}
It follows that the estimate (\ref{estim4}) holds for all $q>1$, what leads
to the estimate (\ref{estim3}).

\subsection*{The function $u$ is solution to problem \eqref{P'}}

Let $v\in \mathcal{K}$ and $0<\theta <1$, taking $z=T_{k}(u_{n}-\theta v)$
as a test function in (\ref{Pn2}), one gets
\begin{equation*}
 \langle Au_{n},z \rangle +\int_{\Omega } \vert
\nabla u_{n} \vert ^{n-2}\nabla u_{n}\nabla zdx=\int_{\Omega
} fzdx
\end{equation*}
While noticing that
\begin{equation*}
\int_{ \{  \vert \nabla u_{n} \vert \geq \theta  \vert
\nabla v \vert  \} } \vert \nabla u_{n} \vert
^{n-2}\nabla u_{n}\nabla T_{k}(u_{n}-\theta v)dx\geq 0
\end{equation*}
one has
\begin{equation*}
 \langle Au_{n},z \rangle +\int_{ \{  \vert
\nabla u_{n} \vert <\theta  \vert \nabla v \vert
 \} } \vert \nabla u_{n} \vert ^{n-2}\nabla
u_{n}\nabla zdx\leq \int_{\Omega } fz\,dx
\end{equation*}
Passing to the limit as $n\to \infty $, one gets
\begin{equation*}
 \langle Au,T_{k}(u-\theta v) \rangle \leq \int_{\Omega }
fT_{k}(u-\theta v)\,dx
\end{equation*}
The result is obtained when passing to the limit as
$\theta \to 1$.

\begin{remark} \label{rmk3.2} \rm
Since $u \in W_{0}^{1,\infty}(\Omega)$, the problem can be
formulated in this space by choosing $\mathcal{K}= \{ v\in
W_{0}^{1,\infty}(\Omega ):  \| \nabla v(x) \|_{\infty} \leq 1 \}$,
what permits to write the problem \eqref{P'} without truncation
operator, and simplify the proof of the step \textit{The function
u is solution to the problem }\eqref{P'}. But traditionally (see
for example \cite{Kind}), the elasto-plastic torsion problem is
written with $\mathcal{K}= \{ v\in W_{0}^{1,p}(\Omega ):
 \vert \nabla v(x) \vert \leq 1\text{ a.e. in }\Omega
 \}$, it's why we have done this choice.
\end{remark}


\subsection*{Acknowledgement}
The authors would like to thank the anonymous referee for his/her
interesting remarks.

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