\documentclass[reqno]{amsart}
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\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2015 (2015), No. 229, pp. 1--5.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2015 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2015/229\hfil Porosity of the free boundary]
{Porosity of the free boundary for singular $p$-parabolic obstacle problems}

\author[A. Lyaghfouri \hfil EJDE-2015/229\hfilneg]
{Abdeslem Lyaghfouri}

\address{Abdeslem Lyaghfouri \newline
American University of Ras Al Khaimah,
Department of Mathematics and Natural Sciences,
Ras Al Khaimah, UAE}
\email{abdeslem.lyaghfouri@aurak.ac.ae}

\thanks{Submitted June 2, 2015. Published September 10, 2015.}
\subjclass[2010]{35K59, 35K67, 35K92, 35R35}
\keywords{Singular $p$-parabolic obstacle problem; free boundary; porosity}

\begin{abstract}
 In this article we establish the exact growth of the solution to
 the singular quasilinear p-parabolic obstacle problem near the free
 boundary from which follows its porosity.
\end{abstract}

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

\section{Introduction}\label{S:intro}

Let  $\Omega$ be an open bounded domain of
$\mathbb{R}^n$, $n\geq 2$, $T>0$.
We consider the  problem: Find 
$u\in L^p(0,T; W^{1,p}(\Omega))$  such that:
\begin{itemize}
\item[(i)] $u\geq 0$ in $\Omega_T=\Omega\times(0,T)$,
\item[(ii)] $L_p(u)=u_t-\Delta_p u=-f(x)$ in $\{u>0\}$,
\item[(iii)] $u=g$ on 
$\partial_p\Omega_T=(\Omega\times\{0\})\cup(\partial\Omega\times(0,T))$,
\end{itemize}
where $p>1$, $\Delta_p$ is the $p$-Laplacian defined by 
$\Delta_p u=\operatorname{div}\big(|\nabla u|^{p-2}\nabla u\big)$,
and $f$, $g$ are functions defined in $\Omega_T$ and satisfying for
two positive constants $\lambda_0$ and  $\Lambda_0 $
\begin{equation}\label{e1.1}
 \lambda_0 \leq f \leq \Lambda_0 \quad \text{a.e. in }\Omega_T.
\end{equation}
Moreover we assume that
\begin{gather}
f\text{ is non-increasing in }t. \label{e-1.2}\\
g(x,0)=0\quad \text{ a.e. in }\Omega.\label{e1.3}\\
g\text{ is non-decreasing in }t. \label{e1.4}
\end{gather}
The variational formulation of the above problem is:
Find 
\[
u\in K_g=\{v\in V^{1,p}(\Omega_T)/v=g\text{ on } \partial_p\Omega_T,\quad
v\geq0 \text{ a.e. in } \Omega_T\}
\]
such that for all $ h>0 $  and $t<T-h$:
\begin{equation}\label{eP}
\int_\Omega \partial_t u_h(v-u)dx
+\int_\Omega \big(|\nabla u|^{p-2}\nabla u\big)_h.\nabla(v-u)dx
+\int_\Omega f_h (v-u)dx\geq 0,
\end{equation}
a.e. in $t\in(0,T)$,  and for all $v\in K_g$,
where 
\[
V^{1,p}(\Omega_T)=L^\infty(0,T; L^1(\Omega))\cap L^p(0,T; W^{1,p}(\Omega)),
\]
and $v_h$ is the Steklov average of a function $v$ defined by
\[
v_h(x,t)=\frac{1}{h}\int_t^{t+h}v(x,s)ds,\quad\text{ if }  t\in(0,T-h]\quad
v_h(x,t)=0,\quad\text{if }  t>T-h\,.
\]

Let us recall the following existence and uniqueness theorem of the
solution of the problem \eqref{eP} \cite{S}.

\begin{theorem}\label{thm1.1} 
Assume that $f$ and $g$ satisfy \eqref{e1.1}--\eqref{e1.4}. Then there
exists a unique solution $u$ of the problem \eqref{eP} which satisfies
\begin{gather}
 0\leq u \leq M=\| g\|_{\infty,\Omega_T} \quad \text{in }\Omega_T, \label{e1.5} \\
 u_t \geq 0\quad \text{in } \Omega_T.  \nonumber \\
  f \chi_{\{u>0\}} \leq  \Delta_p u-u_t\leq f \quad \text{a.e.  in }\Omega_T.
\label{e1.6}
\end{gather}
\end{theorem}

\begin{remark}\label{rem1.1} \rm
We deduce from \eqref{e1.5}--\eqref{e1.6} (see \cite[Theorems 7 and 8]{Ch})
that 
$u\in C_{loc}^{0,\alpha}(\Omega_T)\cap C_{x,loc}^{1,\alpha}(\Omega_T)$
for some $\alpha\in(0,1)$.
 \end{remark}

The main result of this article is as follows.

\begin{theorem}\label{thm1.3} 
Assume that $1<p<2$ and that $f$ and $g$ satisfy \eqref{e1.1}--\eqref{e1.4},
and let $u$ be the solution of \eqref{eP}.  
Then for every compact set $K\subset\Omega_T$,
the intersection  $(\partial\{u>0\})\cap K\cap\{t=t_0\} $ is porous 
in $\mathbb{R}^{n}$ with porosity constant depending only on 
 $n$, $p$, $\lambda_0$, $\Lambda_0$, $M$, and
$\operatorname{dist}(K,\partial_p \Omega_T)$.
\end{theorem}

We recall that a set $E\subset \mathbb{R}^n$ is
called  porous  with porosity $\delta$, if there is an $r_0>0$
such that for all $x\in E$ and all $r\in (0,r_0)$, there exists 
$y\in \mathbb{R}^n$  such that 
\[
 B_{\delta r}(y)\subset B_{ r}(x)\setminus E.
\]

A porous set has  Hausdorff dimension not  exceeding
$n-c\delta^n$, where $c=c(n) >0$ is a  constant depending  only
on $n$. In particular a porous set has Lebesgue measure zero.

Theorem \ref{thm1.3} extends the result established in \cite{S}
in the quasilinear degenerate and linear cases $p\geq 2$.
The proof is based on the exact growth of the solution of
the problem \eqref{eP} near the free boundary which is given by the
next theorem.

\begin{theorem}\label{thm1.4} 
Assume that $1<p<2$ and that $f$ and $g$ satisfy \eqref{e1.1}--\eqref{e1.4},
and let $u$ be the solution of the problem \eqref{eP}.  
Then there exists two positive constants $c_0=c_0(n, p, \lambda_0)$ and 
$C_0=C_0(n, p, \lambda_0, \Lambda_0, M)$
such that for every compact set $K\subset\Omega_T$, 
$(x_0,t_0)\in (\partial\{u>0\})\cap K$,
the following estimates hold
\begin{equation}\label{e1.7}
c_0 r^q\leq \sup_{B_r(x_0)} u(.,t_0)\leq C_0 r^q,
\end{equation}
where $q=p/(p-1)$ is the conjugate of $p$.
\end{theorem}

Since the proof of Theorem \ref{thm1.3} relies on the one of Theorem \ref{thm1.4},
it will be enough to prove the latter one. On the other hand we observe that the left hand side
inequality in \eqref{e1.7} was established in \cite[Lemma 2.1]{S}  for any $p>1$,
while the right hand side inequality in \eqref{e1.7} was established only for
$p\geq 2$. In the next section, we shall
establish the second inequality for a class of functions in the singular case
i.e. for $1<p<2$. Then the right hand side inequality will follow
exactly as in \cite{S} and we refer the reader to that reference for the details.
Hence the proof of Theorem \ref{thm1.3} will follow.

For similar results in the quasilinear elliptic case, we refer to
\cite{KKPS,CL1,CL2}, respectively for the $p$-obstacle problem,
the $A$-obstacle problem, and the $p(x)$-obstacle problem. 
For the obstacle problem for a
class of heterogeneous quasilinear elliptic operators with variable growth, 
we refer to \cite{CLRT}.

\section{A  class of functions on the unit cylinder}\label{sec2}

In this section, we assume that $1<p<2$ and consider the family
$\mathcal{F}=\mathcal{F}(p,n,M,\Lambda_0)$ of functions $u$ defined
on the unit cylinder $Q_1=B_1\times(-1,1)$ by
$u\in\mathcal{F}$ if it satisfies
\begin{gather}
 u\in W^{1,p}(Q_1), \quad  0\leq -u_t+\Delta_p u \leq \Lambda_0\quad \text{in } Q_1,
\label{e2.1} \\
 0\leq u \leq M  \quad \text{in } Q_1, \label{e2.2}\\
 u(0,0)=0, \label{e2.3}\\
 u_t\geq 0 \quad \text{ in } Q_1. \label{e2.4}
\end{gather}

The following theorem  gives the growth
of the  elements of the family $ \mathcal{F}$.
This completes a result proved in \cite{S}
for the degenerate case $p\geq 2$.

\begin{theorem}\label{thm2.1} 
There exists a positive  constant
$C=C(p,n,M,\Lambda_0)$ such that for every $u\in \mathcal{F}$, we have
$$ 
u(x,t)\leq Cd(x,t) \quad \forall (x,t)\in Q_{1/2}
$$
where $d(x,t)=\sup\{r:Q_r(x,t)\subset \{u>0\}\}$ for 
$(x,t)\in \{u>0\}$, and
$d(x,t)=0$ otherwise, and where $Q_r(x,t)=B_r(x)\times(s-r^q,s+r^q)$.
\end{theorem}

To prove Theorem \ref{thm2.1}, we need to introduce some
notation inspired from \cite{S}.
For a nonnegative  bounded function  $u$, we define the  quantities
\[
Q_r^-=B_r\times(-r^q,0),\quad 
S(r,u) = \sup_{(x,t)\in Q_r^-} u(x,t).
\]
Also  for $u\in\mathcal{F}$  define the set
\begin{equation}\label{e2.5}
\mathbb{M}(u) = \{ j\in \mathbb{N}\cup\{0\}: A S( 2^{-j-1},u)\geq S( 2^{-j},u) \}
\end{equation}
where $A= 2^q \max\big(1,1/ C_0\big) $  and
$C_0$ is the constant in \eqref{e1.7}.
As in \cite{S}, we first show a weaker version of the inequality.

\begin{lemma}\label{lem2.2} 
There exists  a constant $C_1=C_1(p,n,M,\Lambda_0)$ such that
$$ 
S( 2^{-j-1},u) \leq C_1 2^{-qj} \quad
 \forall u\in \mathcal{F} , \; \forall j\in \mathbb{M}(u).
$$
\end{lemma}

\begin{proof} We argue by contradiction and assume that:
for all $k\in  \mathbb{N}$ there exist $u_k \in \mathcal{F}$ and
$j_k\in \mathbb{M}(u_k)$  such that
\begin{equation}\label{e2.6}
 S( 2^{-j_k-1},u_k) \geq k 2^{-qj_k}.
\end{equation}
Let $\alpha_k=2^{-pj_k}(S(2^{-j_k-1},u_k))^{2-p}$,
and let 
\[
v_k(x,t)=\frac{u_k(2^{-j_k}x,\alpha_k t)}{S(2^{-j_k-1},u_k)}
\]
for $(x,t)\in Q_1$.
First we observe that since $u(0,0)=0$ and $u$ is continuous, we have
$\alpha_k\to 0$ as $k\to\infty$. Moreover, we have
\begin{equation}\label{e2.7}
\begin{gathered}
\nabla v_k(x,t)=\frac{2^{-j_k}}{S( 2^{-j_k-1},u_k)}
\nabla u_k(2^{-j_k}x,\alpha_k t) \\
v_{kt}(x,t)=\frac{\alpha_k}{S( 2^{-j_k-1},u_k)} u_{kt}(2^{-j_k}x,\alpha_k t)
=\Big(\frac{2^{-qj_k}}{S(2^{-j_k-1},u_k)}\Big)^{p-1}
u_{kt}(2^{-j_k}x,\alpha_k t)
\end{gathered}
\end{equation}
and
\begin{equation}\label{e2.8}
\begin{aligned}
&\Delta_p v_k(x,t) \\
&=\operatorname{div}\big(|\nabla v_k|^{p-2}\nabla v_k\big) \\
&=\Big(\frac{2^{-j_k}}{S(2^{-j_k-1},u_k)}\Big)^{p-1}\operatorname{div}
 \big(|\nabla u_k(2^{-j_k}x,\alpha_k t)|^{p-2}\nabla u_k(2^{-j_k}x,\alpha_k t)\big) \\
&=2^{-j_k}\Big(\frac{2^{-j_k}}{S(2^{-j_k-1},u_k)}\Big)^{p-1}
 \Delta_p u_k(2^{-j_k}x,\alpha_k t) \\
&=\Big(\frac{2^{-qj_k}}{S(2^{-j_k-1}},u_k)\Big)^{p-1}
 \Delta_p u_k(2^{-j_k}x,\alpha_k t).
\end{aligned}
\end{equation}
We deduce from \eqref{e2.7}--\eqref{e2.8} that
\begin{equation}\label{e2.9}
v_{kt}-\Delta_p v_k(x,t) 
= \Big(\frac{2^{-qj_k}}{S(2^{-j_k-1},u_k)}\Big)^{p-1}(u_{kt}
-\Delta_p u_k)(2^{-j_k}x,\alpha_k t).
\end{equation}
Combining \eqref{e2.1}--\eqref{e2.6} and \eqref{e2.9}, we obtain
\begin{gather}
0 \leq -v_{kt}+\Delta_p v_k\leq \frac{\Lambda_0}{k^{p-1}} \quad \text{in } Q_1, 
 \label{e2.10} \\
0\leq v_k\leq\frac {S( 2^{-j_k},u_k)}{S( 2^{-j_k-1},u_k)} \leq A\quad \text{in } 
 Q_1^-, \label{e2.11}\\
v_{kt}\geq 0\quad \text{in } Q_1^-, \label{e2.12}\\
{\sup_{Q_{1/2}^-} } v_k=1, \label{e2.13}\\
v_k(0,t)=0\quad\forall t\in(-1,0). \label{e2.14}
\end{gather}
Taking into account \eqref{e2.10}--\eqref{e2.11}, and using 
\cite[Theorem 1.1]{L} and \cite[Theorem 1]{Ch},
we deduce that $v_k$ is locally uniformly bounded in $L^\infty(Q_1)$ 
independently of $k$.
Therefore we obtain from \cite[Theorems 7 and 8]{Ch}, that $v_k$ is
uniformly bounded in $C^{0,\alpha}(\overline{Q_{3/4}})$ and in
$C_x^{1,\alpha}(\overline{Q_{3/4}})$ independently of $k$, for a constant
$\alpha=\alpha(n, p, A, \Lambda_0)\in(0,1)$.
It follows then from Ascoli-Arzella's theorem that there exists
a subsequence, still denoted by $v_k$, and a function 
$v\in C^{0,\alpha}(\overline{Q_{3/4}})\cap C_x^{1,\alpha}(\overline{Q_{3/4}})$ 
such that $ v_k \to v$  and $ \nabla v_k \to \nabla v$ uniformly in 
$\overline{Q_{3/4}}$. Moreover, using
\eqref{e2.10}--\eqref{e2.14}, we see that $v$ satisfies
\begin{gather*}
v_t-\Delta_p v =0 \quad \text{in } Q_{3/4}^-,\quad  v, v_t\geq 0\quad 
\text{in } Q_{3/4}^-,\\
\sup_{x\in Q_{1/2}^-} v(x,t)=1,\quad v(0,t)=0\quad\forall t\in(-3/4,0).
\end{gather*}
We discuss two cases:
\smallskip

\noindent\textbf{Case 1:} for all $(x,t)\in Q_{3/4}^-$  $v(x,t)=0$.
In particular we have $v\equiv 0$ in $Q_{1/2}^-$ which contradicts the fact that
${\sup_{x\in Q_{1/2}^-} v(x)=1}$.
\smallskip

\noindent\textbf{Case 2:} 
There exists $(x_0,t_0)\in Q_{3/4}^-$ such that $v(x_0,t_0)>0$.
Since $v(.,t_0)$ is not identically zero and $v(0,t_0/2)=0$, we get from the strong
maximum principle (see \cite{N}) that $v(x,t_0/2)=0$ for all $x\in B_{3/4}$.
By the monotonicity of $v$ with respect to $t$ and the fact that $v$ is nonnegative, 
we have necessarily $v(x,t)=0$ for all
$(x,t)\in B_{3/4}\times(-3/4,t_0/2)$, which is in contradiction
with the fact that $v(x_0,t_0)>0$.
\end{proof}

\begin{proof}[Proof of Theorem \ref{thm2.1}]
 Using Lemma \ref{lem2.2}, the proof follows exactly as
the one of \cite[Theorem 2.2]{S}.
\end{proof}

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\end{document}