\documentclass[reqno]{amsart} \AtBeginDocument{{\noindent\small {\em Electronic Journal of Differential Equations}, Vol. 2005(2005), No. 09, pp. 1--16.\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 2005 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2005/09\hfil Nonlinear Neumann problems] {Nonlinear Neumann problems on bounded Lipschitz domains} \author[A. Siai \hfil EJDE-2005/09\hfilneg]{Abdelmajid Siai} \address{Abdelmajid Siai \hfill\break Institut Pr\'{e}paratoire aux \'{E}tudes d'Ing\'{e}nieurs de Nabeul - 8050, Nabeul, Tunisie} \email{abdelmejid.s@gnet.tn} \date{} \thanks{Submitted December 29, 2004. Published January 12, 2005.} \thanks{Partially supported by DGRST and DAAD} \subjclass[2000]{35J60, 35J70, 47J05} \keywords{Nonlinear Neumann problem; m-completely accretive operator} \begin{abstract} We prove existence and uniqueness, up to a constant, of an entropy solution to the nonlinear and non homogeneous Neumann problem \begin{gather*} -\mathop{\rm div}[ \mathbf{a}(.,\nabla u)] +\beta (u)=f \quad\mbox{ in } \Omega \\ \frac{\partial u}{\partial \nu _{\mathbf{a}}}+\gamma (\tau u)=g \quad \mbox{on } \partial \Omega\,. \end{gather*} Our approach is based essentially on the theory of m-accretive operators in Banach spaces, and in order preserving properties. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{Definition}[theorem]{Definition} \allowdisplaybreaks \section{Introduction} Let $\Omega $ be a connected open bounded set in $\mathbb{R}^{N}$, $N\geq 3$, with a connected Lipschitz boundary $\partial \Omega $. Let $(f,g)\in L^{1}(\Omega )\times L^{1}(\partial \Omega )$ satisfy the condition $\int_{\Omega }f(y)dy+\int_{\partial \Omega }g(y)d\sigma (y)=0$ which is necessary and sufficient for solving classical Neumann problem in smooth bounded domains \cite{F}. Let $\mathbf{a}(x,\xi )$ be an operator of Leray-Lions type, in the sense that $(x,\xi )\mapsto \mathbf{a}(x,\xi )$ is a Carath\'{e}odory function from $\Omega \times \mathbb{R}^{N}$ to $\mathbb{R}^{N}$, $\langle \mathbf{a}(x,\xi _{1})-\mathbf{a}(x,\xi _{2}), \xi _{1}-\xi_{2}\rangle >0$, if $\xi _{1}\neq \xi _{2}$ and there exist some constants $p>1$, $C_{1}$, $C_{2}>0$ and a function $h_{0}\in L^{p^{\prime }}(\Omega )$, $p^{\prime }=\frac{p}{p-1}$, such that $\langle \mathbf{a}(x,\xi ),\xi \rangle \geq C_{1}|\xi |^{p}$ and $|\mathbf{a}(x,\xi )|\leq C_{2}(h_{0}(x)+|\xi |^{p-1})$, for a.e. $x\in \Omega $ and all $\xi \in \mathbb{R}^{N}$, (see \cite{LL}). We discuss existence and uniqueness of a solution $u$ for the nonlinear and non homogeneous Neumann problem \begin{equation} \begin{gathered} -\mathop{\rm div}[ \mathbf{a}(.,\nabla u)] +\beta (u)=f \quad \mbox{in }\Omega \\ \frac{\partial u}{\partial \nu _{\mathbf{a}}}+\gamma (\tau u)=g \quad\mbox{ on } \partial \Omega , \end{gathered} \label{1} \end{equation} The trace $\tau u$ on $\partial \Omega $ is taken in the sense of \cite{AMST}. The gradient $\nabla u$ is defined by mean of truncating, in the sense of \cite{BBGGPV}, $\nabla u=DT_{k}u$ on every set $\{|u|\leq k\}$, $k>0$, where $T_{k}(r)=\max \{-k,\min (k,r)\}$, $r\in \mathbb{R}$. The normal derivative $\frac{\partial u}{\partial \nu _{\mathbf{a}}}$ related to the operator $\mathbf{a}$ may be interpreted as the trace of the inner product in $\mathbb{R}^{N}$ $\langle \mathbf{a}(.,\nabla u),\nu \rangle $, where $\nu $ is the outward normal derivative vector field. More precisely $\langle \mathbf{a}(.,DT_{k}u),\nu \rangle $ represents a.e. on $\partial \Omega $ an element of the dual space of $W^{1-\frac{1}{p},p}(\partial \Omega )\cap L^{\infty }(\partial \Omega )$ (see \cite{CaF}), but this interpretation is not essential to our approach, since it does not appear explicitly in the definitions, given later, for weak solutions as well as for entropy solutions. For a sake of simplicity, $\beta ,\gamma $ are taken as continuous non decreasing real functions everywhere defined on $\mathbb{R}$, with $\beta(0)=\gamma (0)=0$. We may extend our approach to the case where $\beta $, $\gamma $ are maximal monotone graphs in $\mathbb{R}^{2}$ with some compatibility conditions on their domains, as given in \cite{S1}. We prove existence and uniqueness up to a constant, of an entropy solution $u $ to the problem \eqref{1}, in the sense that $u:\Omega \to \mathbb{R}$ is measurable, $DT_{k}u\in L^{p}(\Omega )$, for every $k>0$, $\beta (u)\in L^{1}(\Omega )$, $\gamma (\tau u)\in L^{1}(\partial \Omega )$, and for every $\varphi \in \mathcal{C}_{0}^{\infty }(\mathbb{R}^{N})$, $u$ satisfies \begin{equation} \int_{\Omega }\langle \mathbf{a}(.,\nabla u),DT_{k}(u-\varphi )\rangle \leq \int_{\Omega }( f-\beta (u)) T_{k}(u-\varphi )+\int_{\partial \Omega }( g-\gamma (\tau u)) T_{k}(\tau u-\varphi). \label{2} \end{equation} We cannot expect better result for uniqueness, since in the particular case where $\beta =\gamma =0$, if $u$ is a solution, then it is so for $u+c$, where $c$ is an arbitrary real constant. Later on, for uniqueness, we will take in \eqref{2} the test function $\varphi$ in a class larger than $\mathcal{C}_{0}^{\infty }(\mathbb{R}^{N})$. To the best of our knowledge, the Non homogeneous case $g\neq 0$, with a double nonlinearity $\beta (u)$ and $\gamma (\tau u)$, even in the Quasilinear case where $\mathop{\rm div}[\mathbf{a}(x,\nabla u)]=\Delta u$, is new. The homogeneous case $g=0$ has been discussed by many authors. See e.g. \cite{B}, \cite{BrS}. For the nonlinear problem, with particular $\beta$ and $\gamma $, we refer the reader to \cite{AMST} for the case $\beta (u)=u$, and to \cite{P} for $\beta =0$ and $\gamma (\tau u)=\lambda \tau u$. In all these approaches, the boundary condition is a part of the definition of the operator's domain. This is no longer possible in our situation. For this reason, to investigate the non homogeneous quasilinear Neumann problem in a half-space, we used in \cite{S1} a matrix operator $A$ on a product space. This had been extended in \cite{S2} to the problem, \begin{equation} \begin{gathered} -\mathop{\rm div}[ \mathbf{a}(.,\nabla u)] +\beta (u)=f\quad \mbox{in } \mathbb{R}^{N}\setminus \partial \Omega \\ \big[ \frac{\partial u}{\partial \nu _{\mathbf{a}}}\big] +\gamma (\tau u)=g\quad\mbox{on } \partial \Omega \\ [ u] =0 \quad \mbox{on } \partial \Omega \,. \end{gathered} \label{3} \end{equation} Where $\Omega $ is given as previously, $[\frac{\partial u}{\partial \nu _{\mathbf{a}}}] $ and $[u] $ are respectively the jump of the normal derivative $\frac{\partial u}{\partial \nu _{\mathbf{a}}}$ and of the trace $\tau u$ across $\partial \Omega $. In the present, $X_{1}=L^{1}(\Omega )\times L^{1}(\partial \Omega )$ and $A$ is an operator related to the problem \begin{equation} \begin{gathered} -\mathop{\rm div}[ \mathbf{a}(.,\nabla u)] =f\quad \text{in }\Omega \\ \frac{\partial u}{\partial \nu _{\mathbf{a}}}=g\quad \text{on }\partial \Omega , \end{gathered} \label{4} \end{equation} in the sense that $A(u,\tau u)=(f,g)$, if $u$ is an entropy solution to % \eqref{4}. $A_{1}$ is the restriction of $A$ to $X_{1}$. Our approach is based essentially on the theory of m-accretive operators in Banach spaces and the following order preserving properties: If $F_{i}=(f_{i},g_{i})\in L^{1}(\Omega )\times L^{1}(\partial \Omega )$, $i=1,2$, satisfy the conditions $\int_{\Omega }f_{i}(y)dy+\int_{\partial \Omega }g_{i}(y)d\sigma (y)=0$, $A(u_{i},\tau u_{i})=(f_{i},g_{i})$ and $\varphi =\mathop{\rm sign}_{0}(u_{1}-u_{2})$ and $\psi =\mathop{\rm sign}_{0}(\tau u_{1}-\tau u_{2})$, then \begin{equation} \int_{\Omega }(f_{1}-f_{2})\varphi +\int_{\Omega \cap \{ u_{1}=u_{2}\} }| f_{1}-f_{2}| +\int_{\partial \Omega }(g_{1}-g_{2})\psi +\int_{\partial \Omega \cap \{ \tau u_{1}=\tau u_{2}\} }| g_{1}-g_{2}| \geq 0 \label{5} \end{equation} If in addition, $(u_{i},\tau u_{i})\in X_{1}$, $i=1,2$ and $A_{1}(u_{i},\tau u_{i})=(f_{i},g_{i})$, then for every $\varphi \in \mathop{\rm sign}% (u_{1}-u_{2}) $ and every $\psi \in \mathop{\rm sign}(\tau u_{1}-\tau u_{2})$, we have \begin{equation} \int_{\Omega }(f_{1}-f_{2})\varphi +\int_{\partial \Omega }(g_{1}-g_{2})\psi \geq 0\text{,} \label{6} \end{equation} where \begin{equation*} \mathop{\rm sign}(r)= \begin{cases} r/| r| \quad \mbox{if } \quad r\neq 0 \\ [-1,1] \quad \mbox{if } \quad r=0, \end{cases} \quad \mathop{\rm sign _{0}}(r)= \begin{cases} r/|r| \quad \mbox{if } \quad r\neq 0 \\ 0 \quad \mbox{if } \quad r=0. \end{cases} \end{equation*} Note that the main difficulty of the problem here as well as in \cite{S2} is that the domain of the operator $A$ in not necessary included in $% L^{1}+L^{\infty }$. The inequality \eqref{5} is applied to the proof of uniqueness for the nonlinear perturbation $( \beta ( u) ,\gamma ( \tau u) ) $, in the problem % \eqref{1} which leads to the uniqueness of the solution $u$ up to a constant, while \eqref{6} is applied to the proof of existence of a solution to \eqref{1}, and mainly when $\beta $ and $\gamma $ are possibly, mulivalued maximal monotone graphs in $\mathbb{R}^{2}$. Following \cite{B} and \cite{BrS}, the inequalities \eqref{5} and \eqref{6} may be interpreted as properties of maximum principle type or order preserving properties in the sense of \cite{BC} related to an operator whose domain is not necessary included in $L^{1}(\Omega )$. For the sequel, we proceed as follows: In Section 2, we collect some properties of functional spaces and traces. In Section 3, we prove that operator $A$ is one-to-one (modulo constants) and $A_{1}$ is m-completely accretive on $X_{1}$. Section 4 is devoted to establish order preserving properties \eqref{5} and \eqref{6}. In Section 5, we discuss existence and uniqueness for the entropy solution to \eqref{1}. \section{Preliminaries and notations} Let $\mathcal{M}(\Omega )$ be the space of classes of Borel measurable real valued functions on $\Omega $, equipped with the topology of the convergence in measure \begin{equation*} d(f,g)=\int_{\Omega }\frac{| f-g| (x)}{1+| f-g| (x)}dx . \end{equation*} For $r>0$, we consider the functional $\mathcal{N}_{r}$ and the Marcinkiewicz $\text{space }M^{r}(\Omega )$, \begin{equation*} \mathcal{N}_{r}(u)=\big[ \underset{\lambda >0}{\sup }\lambda ^{r}| \{ | u| >\lambda \} | \big] ^{1/r}, \quad\mbox{if $u\in \mathcal{M}$ and } M^{r}(\Omega )=\{u\in \mathcal{M}: \mathcal{N}_{r}(u)<\infty \} \end{equation*} If $r>1$ and $\mathcal{B}$ is the Borel family of subsets of $\Omega $ or $\partial \Omega $, then $\mathcal{N}_{r}$ is equivalent to the norm, \begin{equation*} \| u\| _{M^{r}}=\sup_{K\in \mathcal{B},\;|K| >0} \frac{1}{| K| ^{\frac{1}{ r^{\prime}}}}\int_{K}| u(x)| d\mu (x),\quad \frac{1}{r}+\frac{1}{r^{\prime}}=1. \end{equation*} $(M^{r}(\Omega ),\| \text{ }\| _{M^{r}})$, (respect: $(M^{r}(\partial \Omega )$)$,\| \text{ }\| _{M^{r}})$, is a Banach space and the inclusion $M^{r}\subset L^{q}$ holds for all $r,q$, $1\leq q0$, $\mathcal{N}% _{r}(u+v)\leq 2^{1/r}[ \mathcal{N}_{r}(u)+\mathcal{N}_{r}(v)]$, if $00. \label{7} \end{equation} and the set $\mathcal{T}^{1,p}(\Omega )$ is given by \begin{equation*} \mathcal{T}^{1,p}(\Omega )=\{u\in \mathcal{M}(\Omega )\text{, such that } DT_{k}u\in L^{p}(\Omega )\text{, for every }k>0\}. \end{equation*} Note that in view of the identity $\mathbf{1}_{\{ | u|0$ and $\tilde{\tau}u=\tau u$ a.e. on $\partial \Omega $. \item[(ii)] If $u\in \mathcal{T}_{\mathrm{tr}}^{1,p}(\Omega )$, then $\tau( T_{k}u) =T_{k}( \tilde{\tau}u) $, for all $k>0$. \item[(iii)] $\widetilde{\tau }(u-\varphi )=\widetilde{\tau }u-\tau \varphi $, if $\varphi \in W^{1,p}(\Omega )$ and $u\in \mathcal{T}_{\mathrm{tr}% }^{1,p}(\Omega )$. \end{itemize} In the sequel, $\widetilde{\tau }u$ is noted simply $\tau u$. \begin{lemma} \label{lem2.1} Let be given $\delta >0$ and $p$ such that $10, \label{9} \end{equation} then we have \begin{itemize} \item[(i)] $u\in M^{p_{1}}(\Omega )$ and there exists a constant $C_{3}=C_{3}(N,p,\Omega ,\delta )$ such that \begin{equation} | \{ | u| >k\} | \leq C_{3}k^{-p_{1}}\quad \text{for every }k>0, \label{10} \end{equation} \item[(ii)] $\nabla u\in M^{p_{2}}(\Omega )$ and there exists a constant $C_{4}=C_{4}(N,p,\Omega ,\delta )$ such that \begin{equation} | \{ | \nabla u| >k\} | \leq C_{4}k^{-p_{2}},\quad \text{for every }k>0. \label{11} \end{equation} \end{itemize} \end{lemma} \begin{proof} (i) We denote by $\overline{v}=\frac{1}{| \partial \Omega | }\int_{\partial \Omega }v$ the mean of any measurable function $v$, when it exits and we select $k_{0}>2| \overline{u}| $. If $k\geq k_{0}$, then, \begin{align*} | \{ | u| \geq k\} | &=| \{ |T_{k}u| =k\} | \\ &\leq | \{ | T_{k}u| +\frac{k}{2}\geq k+| \overline{u}| \} |\\ &\leq \big|\big\{ | T_{k}u| +\frac{k}{2}\geq k+| \overline{T_{k}u}| \big\} \big| \\ &\leq \big| \big\{ | T_{k}u-\overline{T_{k}u}| \geq \frac{k}{2}\big\} \big| \\ &\leq \big( \frac{2}{k}\| T_{k}u-\overline{T_{k}u}\| _{p^{\ast }}\big) ^{p^{\ast }}, \end{align*} where $p^{\ast }=\frac{Np}{N-p}$. The last estimation follows from H\"{o}lder inequality. Indeed if $\Omega_{k}'=\{ | T_{k}u-\overline{T_{k}u}| \geq \frac{k}{2}\} $, then \[ \frac{k}{2}| \Omega _{k}'| \leq \int_{\Omega_{k}'}| T_{k}u-\overline{T_{k}u}| \leq \| T_{k}u-\overline{T_{k}u}\| _{p^{\ast }}| \Omega _{k}'| ^{1-\frac{1}{p^{\ast }}} \] Then applying \cite[page 191]{Zi}, there exists a constant $C=C(N,p,\Omega )$ such that \[ | \{ | u| \geq k\} | \leq C( \| DT_{k}u\| _{p}) ^{p^{\ast }}( \frac{k}{2}) ^{-p^{\ast }}\leq 2^{p^{\ast }}C\delta ^{\frac{p\ast }{p}}k^{\frac{p^{\ast }}{p}% -p^{\ast }},\quad\text{if }k\geq k_{0}. \] Hence, \eqref{10} follows if we select for example, $C_{3}=\max \{2^{p^{\ast }}C\delta ^{\frac{p\ast }{p}},k_{0}^{p_{1}}| \Omega | \}$. \noindent (ii) The same proof of \cite[lemma 4.2]{BBGGPV} apply here, taking into account that\ the constant $C_{4}$ depends on $\Omega $. \end{proof} \section{Accretive operators and entropy solutions} We define the Banach spaces, $X_{r}=L^{r}(\Omega )\times L^{r}(\partial \Omega )$, $r\geq 1$, $X_{\infty }=L^{\infty }( \Omega ) \times L^{\infty }(\partial \Omega )$ and the measure space $\mathcal{X}=(\Omega \cup \partial \Omega $, $\mathcal{B}_{\Omega }\cup \mathcal{B}_{\partial \Omega }$, $dx\oplus d\sigma )$. For $( U,V) \in X_{r}\times X_{r^{\prime}}$, $\frac{1}{r}+\frac{1}{r^{\prime}% }=1$, $U=( u_{1},u_{2}) $, $V=( v_{1},v_{2}) $, we use the notation \begin{equation*} UV=( u_{1}v_{1},u_{2}v_{2}) \quad \text{and}\quad \int_{\mathcal{X}}UV =\int_{\Omega }u_{1}v_{1}+\int_{\partial \Omega }u_{2}v_{2} \end{equation*} The spaces $X_{r}$, $r\geq 1$, and $X_{\infty }$ are equipped respectively with the norms \begin{equation*} \| F\| _{r}=\Big[ \int_{\mathcal{X}}( | f| ^{r},| g| ^{r}) \Big] ^{1/r}= \Big[ \int_{\Omega }| f(x)|^{r}dx+\int_{\partial \Omega }| g(x)| ^{r}d\sigma (x)\Big]^{1/r}, \end{equation*} for $F=(f,g)\in X_{r}$ and \begin{equation*} \| F\| _{\infty }=\mathop{\rm ess\,sup}_{x\in \Omega } | f(x)| +\mathop{\rm ess\,sup}_{x\in \partial \Omega } | g(x)|, \end{equation*} for $F=(f,g)\in X_{\infty }$. Let us recall the classical sets \begin{gather*} \mathcal{P}_{0}=\{p:\mathbb{R}\to \mathbb{R}, p\text{ Lipschitz, odd, non decreasing and }p^{\prime}\text{ has a compact support} \}, \\ \mathcal{J}_{0}=\{j:\mathbb{R}\to \mathbb{R}, j\text{ is convex, lower semi-continuous, with }\min j=j(0)=0\}. \end{gather*} \begin{Definition} \rm If $A_{1}$ is a mapping from $D(A_{1})\subset $ $X_{1}$ to $X_{1}$, then $A_{1}$ is said to be is \textbf{m-accretive in } $X_{1}$, if the resolvent $J_{\lambda }^{A_{1}}=( I+\lambda A_{1}) ^{-1}$ satisfies, \[ J_{\lambda }^{A_{1}}\text{ is a contraction everywhere defined in $X_{1}$, for every }\lambda >0. \] \end{Definition} $X_{1}=L^{1}(\mathcal{X})$ is a normal Banach space in the sense of \cite[page 53]{BC}. If $U_{i}\in D(A_{1})$, $F_{i}\in X_{1}$, are given such that, $A_{1}U_{i}=F_{i}$, $i=1,2$ and $p\in \mathcal{P}_{0}$, then \begin{equation*} (A_{1}U_{1}-A_{1}U_{2})p(U_{1}-U_{2})\in L^{1}(\mathcal{X}). \end{equation*} Therefore, the condition \begin{equation*} \int_{\mathcal{X}}( (A_{1}U_{1}-A_{1}U_{2})p(U_{1}-U_{2})) ^{+} \geq \int_{\mathcal{X}}( (A_{1}U_{1}-A_{1}U_{2})p(U_{1}-U_{2})) ^{-} \end{equation*} is equivalent to \begin{equation*} \int_{\mathcal{X}}(A_{1}U_{1}-A_{1}U_{2})p(U_{1}-U_{2})\geq 0\text{,} \end{equation*} This leads to the next definition \cite[proposition 2.2]{BC}. \begin{Definition} \rm $A_{1}$ is \textbf{m-completely accretive} in $X_{1}$, if $A_{1}$ is m-accretive and verifies one of the following equivalent conditions, \begin{equation} \int_{\mathcal{X}}j(J_{\lambda }^{A_{1}}U_{1}-J_{\lambda }^{A_{1}}U_{2})\leq \int_{\mathcal{X}}j(U_{1}-U_{2}),\text{ for all }U_{1},U_{2}\in X_{1}\text{, }\lambda >0\text{ and }j\in \mathcal{J}_{0}. \label{12} \end{equation} \begin{equation} \int_{\mathcal{X}}(A_{1}U_{1}-A_{1}U_{2})p(U_{1}-U_{2})\geq 0\text{, for all }U_{1},U_{2}\in D(A_{1})\text{ and }p\in \mathcal{P}_{0}. \label{13} \end{equation} \end{Definition} As a consequence, if $A_{1}( u_{i},\tau u_{i}) =( f_{i},g_{i}) $, $i=1,2$, then by selecting $p(r)=mT_{\frac{1}{m}}( r) $ in \eqref{13}, $r\in \mathbb{R}$, and let $m\to +\infty $, we obtain the next particular order preserving property for $A_{1}$, \begin{equation} \int_{\Omega }(f_{1}-f_{2})\mathop{\rm sign}{}_{0}(u_{1}-u_{2}) +\int_{\partial \Omega }(g_{1}-g_{2})\mathop{\rm sign}{}_{0}(\tau u_{1} -\tau u_{2})\geq 0 \label{14} \end{equation} If $A_{1}$ is m-completely accretive in $X_{1}$, we know from \cite[proposition 3.7]{BC}, (see also \cite{B}), that the Yosida approximation $A_{1,\lambda }=\dfrac{I-J_{\lambda }^{A_{1}}}{\lambda } =A_{1}J_{\lambda }^{A_{1}}$ satisfies, for every $U\in D(A_{1})$, \begin{equation} A_{1,\lambda }\text{ is m-completely accretive, Lipschitz with coefficient } \frac{2}{\lambda }\text{ and }\lim_{\lambda \downarrow 0} A_{1,\lambda }U=A_{1}U. \label{15} \end{equation} The operator $\mathbf{a}$ of Leray-Lions type is defined as follows, \begin{enumerate} \item[(H1)] $\mathbf{a}:\Omega \times \mathbb{R}^{N}\to \mathbb{R}^{N}$, $(x,\xi )\mapsto \mathbf{a}(x,\xi )$ is a Carath\'{e}odory function in the sense that, $\mathbf{a}$ is continuous in $\xi $, for almost every $x\in \Omega $ and measurable in $x$ for every $\xi \in \mathbb{R}^{N}$. \item[(H2)] There exist $p$, $10$, so that, \begin{equation*} \langle \mathbf{a}(x,\xi ),\xi \rangle \geq C_{1}| \xi | ^{p}, \text{ for a.e } x\in \mathbb{R}^{N}\text{ and every }\xi \in \mathbb{R}^{N}. \end{equation*} \item[(H3)] $\langle \mathbf{a}(x,\xi _{1})-\mathbf{a}(x,\xi _{2}),\xi _{1}-\xi _{2}\rangle >0$, if $\xi _{1}\neq \xi _{2}$, for a.e. $x\in \Omega $. \item[(H4)] There exists some $h_{0}\in L^{p^{\prime}}(\Omega )$, $p'=\frac{p}{p-1}$ and $C_{2}>0$, such that, \begin{equation*} | \mathbf{a}(x,\xi )| \leq C_{2}(h_{0}(x)+| \xi | ^{p-1}),\text{ for a.e } x\in \Omega \text{ and every }\xi \in \mathbb{R}^{N}. \end{equation*} \end{enumerate} \begin{Definition} If ${u}$ is any measurable function on $\Omega $, then $u$ is a weak solution to the problem \eqref{4}, if $u\in \mathcal{T}_{\rm tr}^{1,p}(\Omega )$, $\mathbf{a}(.,\nabla u)\in L^{1}(\Omega )$ and for every $v\in \mathcal{C}_{0}^{\infty }(\mathbb{R}^{N})$, \[ \int_{\Omega }\langle \mathbf{a}(.,\nabla u),Dv\rangle =\int_{\mathbb{R}^{N}}fv+\int_{\partial \Omega }g.\tau v, \] \end{Definition} It is well known, from \cite{Se}, that uniqueness of weak solutions for degenerate elliptic equations, fails to be always true, then following \cite {BBGGPV}, (see \cite{M}, for example, for another type of solutions, the renormalized solutions). \begin{Definition} \rm $u$ is said to be an entropy solution to \eqref{4}, if $u\in \mathcal{T}_{\rm tr}^{1,p}(\Omega )$ and $u$ satisfies, \begin{equation} \int_{\Omega }\mathbf{a}(.,Du)DT_{k}(u-\varphi )\leq \int_{\Omega }fT_{k}(u-\varphi )+\int_{\partial \Omega }gT_{k}(\tau u-\varphi ) \label{16} \end{equation} for every $\varphi \in \mathcal{T}_{\rm tr}^{1,p}(\Omega )\cap L^{\infty }(\Omega )$. \end{Definition} We notice that if we set $K=k+\| \varphi \| _{\infty }$, then $\{| u-\varphi | \leq k\}\subset \{| u| \leq K\}$, thus $\mathbf{1}_{\{| u-\varphi | \leq k\}}.\nabla u=\mathbf{1}_{\{| u-\varphi | \leq k\}}.DT_{K}u=\mathbf{1}_{\{| u-\varphi | \leq k\}}.Du$, since $T_{K}u=u$ on the set $\{| u-\varphi | \leq k\}$. We can prove easily as in \cite{BBGGPV} that if $u$ is an entropy solution of \eqref{4}, then $u$ is a weak solution. To discuss uniqueness, for the problem \eqref{4}, the test functions $% \varphi $ must be taken in a class, larger than $\mathcal{C}_{0}^{\infty }(% \overline{\Omega })$ and that contains $T_{k}u$. In \cite{BBGGPV}, the class $\mathcal{T}_{\text{0}}^{1,p}(\Omega )\cap L^{\infty }(\Omega )$ is well adapted to the problem with Dirichlet condition on $\partial \Omega $. In \cite{S2} this extension is obtained directly from \cite{BBGGPV}, in the class $\mathcal{T}^{1,p}(\mathbb{R}^{N})\cap \mathcal{L}_{0}(\mathbb{R} ^{N})\cap L^{\infty }(\mathbb{R}^{N})$, since we have the identity $\mathcal{T}_{0}^{1,p}(\mathbb{R}^{N})\cap \mathcal{L}_{0}(\mathbb{R}^{N}) =\mathcal{T}^{1,p}(\mathbb{R}^{N})\cap \mathcal{L}_{0}(\mathbb{R}^{N})$, where $\mathcal{L}_{0}(\mathbb{R}^{N})=\{u\in \mathcal{M}(\mathbb{R}^{N})\text{ s.t. }| \{ | u| >k\} | <+\infty, \text{ for every }k>0\}$. In the present, we need the next lemma. \begin{lemma} \label{lem3.1} (i) If $\varphi \in \mathcal{T}_{\rm tr}^{1,p}(\Omega )\cap L^{\infty }(\Omega )$, then, there exists a sequence $(\varphi _{n})_{n}$ in $\mathcal{C}_{0}^{\infty }(\overline{\Omega })$, $n\in \mathbb{N}$, such that $T_{k}(u-\varphi _{n})$ converges a.e. on $\Omega $ to $T_{k}(u-\varphi )$, $\tau T_{k}(u-\varphi _{n})$ converges a.e. on $\partial \Omega $ to $\tau T_{k}(u-\varphi )$ and $DT_{k}(u-\varphi _{n})$ converges weakly in $( L^{p}(\Omega )) ^{N}$ to $DT_{k}(u-\varphi)$, for every $k>0$. \noindent(ii)In particular, $u\in \mathcal{T}_{\rm tr}^{1,p}(\Omega )$ is an entropy solution to \eqref{4}, if and only if $u$ satisfies \eqref{16}, for every $\varphi \in \mathcal{T}_{\rm tr}^{1,p}(\Omega ) \cap L^{\infty }(\Omega )$. \end{lemma} \begin{proof} (i) Let $(\theta _{n})_{n}$ a regularizing sequence in $\mathbb{R}^{N}$, $\theta _{n}\in \mathcal{C}_{0}^{\infty }(\mathbb{R}^{N})$ and $\phi _{n,m}=( \varphi .\mathbf{1}_{B(0,m)}) \ast \theta_{n}$, $m,n\in \mathbb{N}$. By the diagonal process, there exists a sequence $\varphi _{n}$ that converges a.e. on $\Omega $ to $\varphi $. $T_{k}(u-\varphi )$ and $T_{k}(u-\varphi _{n})$ are in $W^{1,1}(\Omega )$, we assume that $T_{k}(u-\varphi _{n})$ converges a.e. on $\Omega $ to $T_{k}(u-\varphi )$ and the same type of convergence for the trace on $\partial \Omega $. For the weak convergence of the sequence $DT_{k}(u-\varphi _{n})$, it is sufficient to prove that \[ \int_{\Omega }\langle DT_{k}(u-\varphi _{n}),\psi \rangle \to \int_{\Omega }\langle DT_{k}(u-\varphi ),\psi \rangle \text{, for every }\psi \in [ \mathcal{C}_{0}^{\infty }(\overline{% \Omega })] ^{N}. \] By the divergence theorem, we have, \[ \int_{\Omega }\mathop{\rm div}[ T_{k}(u-\varphi _{n})\psi ] =\int_{\partial \Omega }T_{k}(u-\varphi _{n})(x)\langle \psi (x),\nu (x)\rangle d\sigma (x) \] Where $\nu (x)$ is the outward normal vector in $x\in \partial \Omega $. Thus, \begin{align*} &\int_{\Omega }\langle DT_{k}(u-\varphi _{n}),\psi \rangle\\ &=-\int_{\Omega }[ T_{k}(u-\varphi _{n})\mathop{\rm div}\psi ] +\int_{\partial \Omega }T_{k}(u-\varphi _{n})(x)\langle \psi (x),\nu (x)\rangle d\sigma (x) \end{align*} The lemma is proved by applying dominated convergence in the last two integrals and then again the divergence theorem in the opposite sense. Part (ii) is an immediate consequence of part (i). \end{proof} For the sequel, the operators $A$ and $A_{1}$ are given as follows, \begin{equation} \begin{aligned} &\mbox{$(u,\tau u)\in D(A)$, if $u\in \mathcal{T}_{\rm tr}^{1,p}(\Omega )$, $\tau u\in L^{1}(\partial \Omega )$ and there exists} \\ &\mbox{$(f,g)\in X_{1}$ such that, $u$ is an entropy solution for \eqref{4}.} \\ &A_{1} \mbox{is the restriction of $A$ to } X_{1}. \end{aligned}\label{17} \end{equation} \begin{theorem} \label{thm3.1} (i) $A_{1}$ is m-completely accretive in $X_{1}$.\\ (ii) $A$ is one-to-one. \end{theorem} \begin{proof} (i). In view of (H3) the inequality \eqref{13} is satisfied. Hence, $A_{1}$ is completely accretive in $X_{1}$. Now, we prove that $I+A_{1}$ is onto from $X_{1}$ to $X_{1}$. \noindent \textbf{Step 1:} Construction of an approximate sequence. First, we consider the reflexive Banach space \[ E=W^{1,p}(\Omega )\times L^{p}(\partial \Omega ), \quad \| U\|_{E}=( \| u\| _{W^{1,p}(\Omega )}^{p}+\| v\| _{L^{p}(\partial \Omega )}^{p}) ^{1/p}. \] and we define a subspace $X_{0}$ and an operator $A_{0}$ as follows, $X_{0}=\{(u,v)\in E: v=\tau u\}$, and $U=(u,\tau u)\in X_{0}$ is a solution to $A_{0}(u,\tau u)=(f,g)\in E'$ if, \[ \int_{\Omega }\langle \mathbf{a}(.,Du),Dv\rangle =\int_{\Omega }fv+\int_{\partial \Omega }g.\tau v,\quad \text{ for every } V=(v,\tau v)\in X_{0}. \] Next, we consider, the convex functional \[ \Phi _{n}(u,v)=\frac{1}{2}\Big[ \int_{\Omega }u^{2}(x)dx+\int_{\partial \Omega }v^{2}(x)d\sigma (x)\Big] +\frac{1}{np}\int_{\Omega }| u| ^{p}dx\text{; }U=(u,v)\in X_{0}, \] The real mapping $t\mapsto \langle A_{0}(u+tv),w\rangle $ is continuous, for all $u,v,w\in E$, then $A_{0}$ is monotone and Hemi-continuous (see \cite{Br}, \cite{Li}), thus it is Pseudo-monotone. It is also coercive in the sense that \[ \underset{\| U\| _{E}\to +\infty ,U\in X_{0}}{\lim }% \dfrac{\langle A_{0}U,U\rangle +\Phi _{n}(U)}{\| U\| _{E}}=+\infty . \] Let $F=(f,g)$ be given in $X_{1}$ and $F_{n}=(f_{n},g_{n})=(T_{n}f,T_{n}g)$. Then $F_{n}\in X_{1}\cap X_{\infty }\subset E'$, $\| f_{n}\| _{1}\leq \| f\| _{1}$ and $\| g_{n}\| _{1}\leq \| g\| _{1}$, $f_{n}$ converges to $f$ in $L^{1}(\Omega ) $, and $g_{n}$ converges to $g$ in $L^{1}(\partial \Omega )$. By \cite[corollary 30]{Br}, there exists $U_{n}\in X_{0}$ that satisfies, for all $V\in X_{0}$, \[ \Phi _{n}(V)-\Phi _{n}(U_{n})\geq \langle F_{n}-A_{0}U_{n},V-U_{n}\rangle _{E'\times E} \] Thus, $F_{n}-A_{0}U_{n}=\partial \Phi _{n}(U_{n})\in E'$, $\partial \Phi _{n}$ is the subdifferential of $\Phi _{n}$, $\partial \Phi _{n}$ is univalued here. In other words, $U_{n}=(u_{n},\tau u_{n})\in X_{0}$ and $U_{n}$ satisfies, \begin{equation} \int_{\Omega }u_{n}v+\int_{\partial \Omega }\tau u_{n}\tau v+\int_{\Omega }\langle \mathbf{a}(.,Du_{n}),Dv\rangle +\frac{1}{n}\int_{\Omega }| u_{n}| ^{p-2}u_{n}v=\int_{\Omega }f_{n}.v+\int_{\partial \Omega }g_{n}\tau v, \label{18} \end{equation} for every $V=(v,\tau v)\in X_{0}$. \noindent \textbf{Step 2:} we claim that the sequence $( \frac{1}{n}| u_{n}| ^{p-1}) _{n}$converges to $0$ in $L^{1}(\Omega )$. If we take $v=T_{k}( u_{n}) $ in \eqref{18}, then $v\in W^{1,p}(\Omega )$, and we obtain, \begin{equation} \frac{C_{1}}{k}\int_{\{ | u_{n}| t\} \subset \{ | u_{n}|>k\} \cup \{ | u_{m}| >k\} \cup \{ | u_{n}| \leq k,|u_{m}| \leq k,| u_{n}-u_{m}| >t\} \] Since $( u_{n}) _{n}$ is uniformly bounded in the Marcinkiewicz space $M^{p_{1}}$, then, for every $\varepsilon >0$, there exists $k_{0}$ such that $| \{ | u_{n}|>k\} | <\varepsilon $ and $| \{ |u_{m}| >k\} |<\varepsilon $, if $k>k_{0}$. Next, if we select some $k>k_{0}$, since $(T_{k}u_{n})_{n}$, is bounded in $W^{1,p}\mathbb{(}\Omega \mathbb{)}$, we assume then, up to a subsequence, that $(T_{k}u_{n})_{n}$ is a Cauchy sequence in $L^{1}(\Omega )$ and in measure. Then we have $| \{ | u_{n}| \leq k,| u_{m}| \leq k,| u_{n}-u_{m}| >t\} | <\varepsilon $, if $m$, $n$ are sufficiently large. Hence, up to a subsequence, $( u_{n}) _{n}$ converges in measure and a.e. to some $u$ and $u\in M^{p_{1}}(\Omega )$. \noindent\textbf{Step 4:} Convergence of the sequence $( Du_{n}) _{n}$. If $k,l,t,\varepsilon $ are positive real numbers, we have the inclusions, \begin{align*} \{ | Du_{n}-Du_{m}| >t\} \subset &\{ | u_{n}-u_{m}|>k\} \cup \{ | Du_{n}|>l\} \cup \{ | Du_{m}| >l\} \\ &\cup \{ | Du_{n}| \leq l,| Du_{m}| \leq l,| u_{n}-u_{m}| \leq k,|Du_{n}-Du_{m}|>t\} . \end{align*} We proceed, first, with the last term in the previous inclusion Let a compact $\mathbf{K}$ and a function $\mu $ be given as follows, \begin{gather*} \mathbf{K}=\{ (\xi ,\zeta )\in \mathbb{R}^{N}\times \mathbb{R}^{N}: | \xi | \leq l,| \zeta | \leq l,| \xi -\zeta | \geq t\},\\ \mu (x)=\min_ {(\xi ,\zeta )\in \mathbf{K}} \langle \mathbf{a}(x,\xi )-\mathbf{a}(x,\zeta ),\xi -\zeta \rangle \end{gather*} We derive from (H1) and (H3), that $\mu $ is defined for a.e. $x\in \Omega $ and is positive. Thus $| \{ \mu=0\} | =0$, and there exists $\eta >0$, such that for every measurable subset $S$ of $\Omega $, if $\int_{S}\mu <\eta $, then $| S| <\varepsilon $. By applying this last statement to, \[ S=\{ | Du_{n}| \leq l,| Du_{m}| \leq l,| u_{n}-u_{m}| \leq k,| Du_{n}-Du_{m}|>t\} , \] Since, \[ \int_{S}\mu \leq \int_{S}\langle \mathbf{a}(x,Du_{n})-\mathbf{a}(x,Du_{m}), Du_{n}-Du_{m}\rangle \leq k\| F\| _{1}, \] then $|S| <\varepsilon $, if $k$ is small enough. Next, if $k$ is fixed small enough, from the step 3, we have, $| \{ | u_{n}-u_{m}| >k\} | <\varepsilon $, if $m,n$ are sufficiently large. According to \eqref{19} and \eqref{11}, $( Du_{n}) _{n}$ is uniformly bounded in the Marcinkiewicz space $M^{p_{2}}$, $p_{2}=\frac{N(p-1)}{N-1}$. Hence $| \{ | Du_{n}|>l\} | <\varepsilon $ and $| \{ | Du_{m}|>l\} | <\varepsilon$, for $l$ sufficiently large. Then, we may assume that, $Du_{n}$ converges in measure and a.e. on $\Omega$, if $n\to +\infty$ to some $V$ and $| V| \in M^{p_{2}}(\mathbb{R}^{N})$. We claim that $u\in \mathcal{T}^{1,p}(\Omega )$ and $\nabla u=V$. For a fixed $k>0$, on one hand $T_{k}u_{n}$ is converging to $T_{k}u$ by dominated convergence, and therefore $DT_{k}u_{n}$ is converging to $DT_{k}u$ in $\mathcal{D}'(\mathbb{R}^{N})$, on the other hand, $( DT_{k}u_{n}) _{n}$ is bounded in $L^{p}(\Omega )$, thus, $DT_{k}u_{n}$ is converges weakly to a $V_{k}$ in $L^{p}$, therefore also in $\mathcal{D}'(\mathbb{R}^{N})$. By uniqueness, $DT_{k}u=V_{k}\in L^{p}(\Omega )$ and $DT_{k}u_{n}$ converges weakly in $L^{p}(\Omega )$ to $DT_{k}u$. Consequently, $u\in \mathcal{T}^{1,p}(\Omega )$. Next, we prove that \begin{equation} DT_{k}u_{n}\text{ converges in measure and a.e. on $\Omega$ to $\nabla u$, as $n\to +\infty$ and $k\to +\infty$.} \label{20} \end{equation} Since $T_{k+\alpha }\circ T_{k}=T_{k}$, for every $k>0$ and $\alpha >0$, then, by the same arguments as for $( Du_{n}) _{n}$, we obtain from \eqref{20} that $( DT_{k}u_{n}) _{n}$ converges in measure to some $v_{k}$, then claim that $(DT_{k}u_{n}) _{n}$ converges weakly to $v_{k}$, since that leads to $v_{k}=DT_{k}u$. Indeed, if $\varepsilon >0$, and $\varphi \in L^{p'}(\Omega )$, then for every $k>0$, we can select two positive constants $c_{k}$ and $\eta >0$ such that, for every $n\in \mathbb{N}$ and every measurable subset $S\subset \Omega $, we have \[ \| DT_{k}u_{n}\| _{p}\leq c_{k},\quad\text{and}\quad \| \varphi\| _{L^{p'}(S)}\leq \frac{\varepsilon }{4c_{k}},\quad \text{if }| S| \leq \eta \] By Fatou lemma, we have $\| v_{k}\| _{p}\leq c_{k}$. Next, if we set, \[ \lambda =\frac{\varepsilon }{2| \Omega | ^{\frac{1}{p}}\| \varphi \| _{L^{p'}(\Omega )}}, \quad S_{\lambda }=\{ | DT_{k}u_{n}-v_{k}| >\lambda \} \] and $n_{0}$ such that $| S_{\lambda }| \leq \eta$ for $n\geq n_{0}$, then \begin{align*} | \int_{\Omega }( DT_{k}u_{n}-v_{k}) \varphi | &\leq \int_{S_{\lambda }}| DT_{k}u_{n}-v_{k}| | \varphi | +\int_{\Omega \backslash S_{\lambda }}| DT_{k}u_{n}-v_{k}| |\varphi | \\ &\leq 2c_{k}\| \varphi \| _{L^{p'}(S_{\lambda })}+\lambda | \Omega | ^{\frac{1}{p}}\| \varphi \| _{L^{p'}(\Omega )}\leq \varepsilon \,. \end{align*} Thus, $DT_{k}u_{n}$ converges in measure and a.e to $DT_{k}u$ , as $n\to +\infty $. Consequently, for every $k>0$, there exists some $n_{k}\in \mathbb{N}$, such that, $d(DT_{k}u_{n_{k}},DT_{k}u)\leq \frac{1}{k}$, where $d$ is the metric on $\mathcal{M}$. On the other hand, \begin{equation} d(DT_{k}u,\nabla u)=\int_{\Omega }\frac{| DT_{k}u-\nabla u| }{1+| DT_{k}u-\nabla u| } \leq | \{| u| >k\}| \to 0,\quad\text{as }k\to +\infty . \label{21} \end{equation} Therefore, we assume the subsequence $( DT_{k}u_{n_{k}}) _{k}$ converges in measure and a.e. to $\nabla u$, as $k\to +\infty $. But for a.e. $x\in \Omega $, if $k>| u(x)| $ and $k'>|u(x)| $, we have \begin{align*} &| DT_{k}u_{n}(x)-DT_{k'}u_{m}(x)| \\ &\leq |DT_{k}u_{n}(x)-Du(x)| +| Du(x)-DT_{k'}u_{m}(x)| \\ &= | DT_{k}u_{n}(x)-DT_{k}u(x)| +| DT_{k'}u(x)-DT_{k'}u_{m}(x)| \leq \varepsilon\, \end{align*} if $m$ and $n$ are sufficiently large, At last by the same argument as in \eqref{21}, we conclude that, up to a subsequence $DT_{k}u_{n}$ converges in measure and a.e. to $Du_{n}$, if $k\to +\infty $. Since $( Du_{n}) _{n}$ converges in measure and a.e. to $\nabla u$, we conclude that $DT_{k}u_{n}$ converges in measure and a.e. to $\nabla u$, as $n,k\to +\infty $. This completes the proof of \eqref{20}. Applying classical arguments for Carath\'{e}odory functions, we assume that the sequence $( \mathbf{a}(.,Du_{n})) _{n}$ converges in measure to $\mathbf{a}(.,Du)$. >From (H4) and the fact that $| Du_{n}| ^{p-1}$ a.e. uniformly bounded in the Marcinkiewicz space $M^{\frac{N}{N-1}}(\Omega )$, we deduce that $( \mathbf{a}(.,Du_{n})) _{n}$ is equi-integrable on $\Omega $. Hence, $\mathbf{a}(.,Du_{n})$ converges in $L^{1}(\Omega )$ to $\mathbf{a}(.,Du)$. \noindent\textbf{Step 5:} Convergence of the trace. We prove that $(\tau u_{n})_{n}$ converges to some $w\in \mathcal{M}(\partial \Omega )$, that $u\in \mathcal{T}_{\rm tr}^{1,p}(\Omega )$ and $w=\tau u$, $d\sigma$ a.e. Since the trace operator is completely continuous from $W^{1,p}(\Omega )$ to $L^{p}(\partial \Omega )$, we assume that $T_{k}\tau u_{n}=\tau T_{k}u_{n}\to \tau T_{k}u$, a.e. on $S_{k}=\{x\in \partial \Omega $; $| T_{k}u| 0$. Thus $\tau u_{n}$ converges a.e. to $w$ on $\partial \Omega $, $w=\tau T_{k}u$, a.e. on $S_{k}$, for every $k>0$. On the other hand, $DT_{k}u_{n}$ converges weakly in $L^{p}$ and in measure to $DT_{k}u$, we deduce also that $DT_{k}u_{n}$ converges to $DT_{k}u$ in $L^{1}(\Omega )$. We summarize, $DT_{k}u_{n}$ $DT_{k}u$, we deduce that $L^{1}(\Omega ).$% \begin{gather*} \mbox{$(u_{n})_{n}$ converges in measure to some $u$},\\ \mbox{$DT_{k}u_{n}$ converges to $DT_{k}u$ in $L^{1}(\Omega )$}, \\ \mbox{$\tau u_{n}$ converges a.e. to $w$ on $\partial \Omega $.} \end{gather*} We conclude, as defined in \eqref{8}, that $u\in \mathcal{T}_{\text{tr}}^{1,p}(\Omega )$ and $\tau u=w$. \noindent\textbf{Step 6:} It remains to prove that $u$ is an entropy solution to \begin{gather*} u-\mathop{\rm div}[ \mathbf{a}(.,Du)] =f \quad\mbox{in $\Omega $} \\ \tau u+\frac{\partial u}{\partial \nu _{\mathbf{a}}}=g\quad\mbox{on } \partial \Omega \end{gather*} If $\varphi \in \mathcal{C}_{0}^{\infty }(\overline{\Omega })$, and $v=T_{k}(u_{n}-\varphi )$ in \eqref{18}, then we have \begin{align*} &\int_{\Omega }\langle \mathbf{a}(.,Du_{n}),DT_{k}(u_{n}-\varphi )\rangle\\ &=\int_{\Omega }( f_{n}-u_{n}-\frac{1}{n}| u_{n}| ^{p-2}u_{n}) T_{k}(u_{n}-\varphi ) +\int_{\partial \Omega }( g_{n}-\tau u_{n}) \tau T_{k}(u_{n}-\varphi ), \end{align*} $u_{n}\to u$ a.e. on $\Omega $ and $\tau u_{n}\to \tau u$ a.e. on $\partial \Omega $. For the left member, we notice first that $D( u-\varphi ) =0$, a.e. on the set $\{ | u-\varphi | =k\} $. Then $DT_{k}(u_{n}-\varphi )=D(u_{n}-\varphi )\mathbf{1}_{\{ | u_{n}-\varphi | 0$, then by taking $T_{k+h}(u-T_{k}u)$ as test functions in \eqref{18}, and applying (H2), we obtain \[ C_{1}\int_{\{ h\leq | u| \leq k+h\} }| Du| ^{p}\leq k\int_{\{ h\leq | u| \leq k+h\} }| f| +k\int_{\{ h\leq | \tau u| \leq k+h\} }| g| \leq k\| F\| _{1}. \] In particular \eqref{9} while taking $h=0$. We deduce, then from \eqref{11} and the condition $\tau u\in L^{1}(\partial \Omega )$ in \eqref{17} that, \begin{equation} \lim_{h\to +\infty } \int_{\{ h\leq | u| \leq k+h\} }| Du| ^{p}=0. \label{22} \end{equation} Next, if $u_{1},u_{2}\in \mathcal{T}_{\rm tr}^{1,p}(\Omega )$ are two entropy solutions to \eqref{4} with the same data $(f,g)$, by taking the same decomposition as in \cite{BBGGPV}, for a fixed $k$, \begin{gather*} S_{1}( h) =\{ | u_{1}-u_{2}| \leq k\} \cap [ \{ | u_{1}| 0\,. \] It arises from (H3), that $Du_{1}=Du_{2}$, a.e. in $\Omega $. \end{proof} \section{An order preserving property} \begin{theorem} \label{thm4.1} (i) If $u_{1},u_{2}\in \mathcal{T}_{\rm tr}^{1,p}(\Omega )$ are entropy solutions for \begin{equation} \begin{gathered} -\mathop{\rm div} [ \mathbf{a}(.,Du_{i})] =f_{i}, \quad\mbox{in }\Omega \\ \frac{\partial u_{i}}{\partial \nu _{a}}=g_{i}\quad\mbox{on } \partial \Omega, \end{gathered} \label{23} \end{equation} $i=1,2$, and $\varphi =\mathop{\rm sign}_{0}(u_{1}-u_{2})$, $\psi =\mathop{\rm sign}_{0}(\tau u_{1}-\tau u_{2})$, then we have the following order preserving property: \begin{equation} \int_{\Omega \cap \{ u_{1}-u_{2}\} }| f_{1}-f_{2}| +\int_{\partial \Omega \cap \{ \tau u_{1}-\tau u_{2}\} }| g_{1}-g_{2}| +\int_{\Omega }(f_{1}-f_{2})\varphi +\int_{\partial \Omega }(g_{1}-g_{2})\psi \geq 0 \label{24} \end{equation} (ii) If furthermore, $U_{i}=(u_{i},\tau u_{i})\in Dom(A_{1})$, then for every $\varphi \in \mathop{\rm sign}(u_{1}-u_{2})$ and $\psi \in \mathop{\rm sign}(\tau u_{1}-\tau u_{2})$, we have \begin{equation} \int_{\Omega }(f_{1}-f_{2})\varphi +\int_{\partial \Omega }(g_{1}-g_{2})\psi \geq 0\,. \label{25} \end{equation} \end{theorem} \begin{proof} (i) If $U_{i}=( u_{i},\tau u_{i}) \in D(A)$, is an entropy solution for the problem \eqref{23} with data $F_{i}=(f_{i},g_{i})\in X_{1}$, $i=1,2$, then from Theorem \ref{thm3.1}, there exist $V_{i,n}=(v_{i,n},\tau v_{i,n})\in X_{1}$ such that $V_{i,n}$ is an entropy solution for, \begin{gather*} \frac{1}{n}v_{n,i}-\mathop{\rm div} [ \mathbf{a}(.,Dv_{n,i})] =f_{i} \quad\mbox{in } \Omega \\ \frac{1}{n}\tau v_{n,i}+\frac{\partial v_{n,i}}{\partial \nu _{\mathbf{a}}} =g_{i}\quad\mbox{on } \partial \Omega\,,\quad {n\in \mathbb{N}}. \end{gather*} By taking $\varphi =0$ in the entropy condition and applying (H2), we have \[ \frac{C_{1}}{k}\int_{\{ | u_{n,i}|