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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2011 (2011), No. 67, pp. 1--15.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2011 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2011/67\hfil $p$-capacity of the singular sets]
{Vanishing $p$-capacity of singular sets for $p$-harmonic functions}

\author[T. Sato, T. Suzuki, F. Takahashi\hfil EJDE-2011/67\hfilneg]
{Tomohiko Sato, Takashi Suzuki, Futoshi Takahashi}
% in alphabetical order

\address{Tomohiko Sato \newline
Department of Mathematics, Faculty of Science \\
Gakushuin University \\
1-5-1 Mejiro, Toshima-ku \\
Tokyo, 171-8588, Japan}
\email{tomohiko.sato@gakushuin.ac.jp}

\address{Takashi Suzuki \newline
Division of Mathematical Science \\
Department of System Innovation \\
Graduate School of Engineering Science \\
Osaka University \\
Machikaneyamacho 1-3 \\
Toyonakashi, 560-8531, Japan}
\email{suzuki@sigmath.es.osaka-u.ac.jp}

\address{Futoshi Takahashi \newline
Department of Mathematics \\
Graduate School of Science \\
Osaka City University \\
Sugimoto 3-3-138, Sumiyoshiku \\
Osakashi, 535-8585, Japan}
\email{futoshi@sci.osaka-cu.ac.jp}

\thanks{Submitted April 4, 2011. Published May 18, 2011.}
\subjclass[2000]{35B05, 35B45, 35J15, 35J70}
\keywords{$p$-harmonic function; capacity; singular set;
removable singularity; \hfill\break\indent weak Sobolev space}

\begin{abstract}
 In this article, we study a counterpart of the removable
 singularity property of $p$-harmonic functions.
 It is shown that $p$-capacity of the singular set of any
 $p$-harmonic function vanishes, and such function is always
 weakly $N(p-1)/(N-p)$-integrable. Several related results
 are also shown.
\end{abstract}

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

\section{Introduction}

This article estimates the size of the
singular sets and the local behavior of solutions
to some (quasilinear) elliptic equations of second order.
The equations to be treated here are general enough
to include those studied by Serrin
in his milestone paper \cite{Serrin1}.
The size of singular sets is measured by the capacity, and
the local behavior of the solution
is described by the weak $L^q$ norm, for appropriate $q$.

First, given $1 \le p < N$, we put
\[
K^p=\{ f\in L^{p^\ast}(\mathbb{R}^N, \mathbb{R}) : \nabla f
\in L^p(\mathbb{R}^N, \mathbb{R}^N)\},
\]
where $\frac{1}{p^\ast}=\frac{1}{p}-\frac{1}{N}$. Also define
\[
\operatorname{Cap}_p(A) = \inf \big\{ \int_{\mathbb{R}^N} |
 \nabla f|^p dx :   f \ge 0, \; f \in K^p, \; A \subset \{f(x)
 \ge 1 \}^{\circ} \big\},
\]
where $A \subset \mathbb{R}^N$ is a given subset,
and $B^{\circ}$ indicates the interior of the set
$B\subset \mathbb{R}^N$.
The operator $\operatorname{Cap}_p(A)$ is called
{\it $p$-capacity} of $A$ in short, and provides an outer
measure to $\mathbb{R}^N$; see \cite{Evans-Gariepy} for more
information.

Next, given an open set $\Omega$ in $\mathbb{R}^N$ with $N \ge 3$
and $1 < q < \infty$, the weak $L^q$ space on $\Omega$,
denoted by $L^q_w(\Omega)$, is defined by
\[
L^q_w(\Omega) = \{ u \in L^1_{\rm loc}(\Omega) :
 \| u \|_{L^q_w(\Omega)} < +\infty \}
\]
and
\[
\| u \|_{L^q_w(\Omega)} = \sup
  \{ | K|^{-1 + 1/q} \int_K | u| dx : K \subset \Omega
\text{ compact}\},
\]
where $| K|$ indicates the $N$-dimensional Lebesgue measure of $K$.
Thus, we obtain $| x|^{-\alpha} \in L^{N/\alpha}_w(B)$ and
$| x|^{-\alpha} \notin L^{N/\alpha}(B)$ for $0<\alpha<N$ and
$B = B_1(0)$.  This $L^q_w$-space is sometimes called
the Marcinkiewicz space or Lorentz $L^{q,\infty}$ space;
see \cite{Veron, Ziemer}.

The singular set is indicated by a closed set $\Sigma \subset \Omega$
in this paper.  In the linear case, we study the second order
elliptic operator of divergence form defined on
$\Omega\setminus \Sigma$; i.e.,
\[
Lu = \sum_{i,j=1}^N D_i (a_{ij}(x) D_j u) + c(x) u
\]
satisfying the strict ellipticity condition
\begin{equation}
\sum_{i,j=1}^N a_{ij}(x) \xi_i \xi_j \geq \delta| \xi |^2
 \label{se}
\end{equation}
for any $x \in \Omega \setminus \Sigma$ and
$\xi = (\xi_1, \dots,\xi_N) \in \mathbb{R}^N$,
where $\delta>0$ is a constant and $a_{ij}(x)=a_{ji}(x), c(x)$
are bounded measurable functions.

The function $u = u(x)$ discussed in the following theorem is
defined on $\Omega \setminus \Sigma$, and is locally H\"older
continuous there by the result of DeGiorgi, Nash,
and Moser \cite{Gilbarg-Trudinger}.
The crucial assumption  is as  follows:
\begin{itemize}
\item[(A)] There is $s_0 > 0$ such that
$\Omega_{s_0} \subset \subset \Omega$, $\Omega_{s_0}$ has
a Lipschitz boundary, and $\Omega_s$ is open for any $s \ge s_0$,
where $\Omega_s = \{ x \in \Omega \setminus \Sigma :
 | u(x)| > s \} \cup \Sigma$.
\end{itemize}
This means that $\Sigma$ is an actual singular set of $u=u(x)$,
 and henceforth $u=u(x)$ is identified with a function defined
on $\Omega$, taking $| u|=+\infty$ on $\Sigma$ unless otherwise
stated.

\begin{theorem} \label{thm:1}
Let $c(x) \ge 0$ a.e. $x \in \Omega \setminus \Sigma$, and
$u=u(x) \in H^1_{\rm loc}(\Omega \setminus \Sigma)$ be a solution to
\[
Lu = 0 \quad \text{in} \; \Omega \setminus \Sigma
\]
satisfying {\rm (A)}.  Then, it holds that
$\operatorname{Cap}_2(\Sigma) = 0$ and $u \in L^{N/(N-2)}_w(\Omega)$.
\end{theorem}

There is an analogous result for the parabolic
equation \cite{Sakaguchi-Suzuki}, i.e., the blow-up set
$D(t)=\{ x\in \Omega \mid u(x,t)=+\infty\}\subset \subset \Omega$
of the solution $u=u(x,t)$ to the differential inequality
$u_t - \Delta u \ge 0$ is negligible with respect to the
$N$-dimensional Lebesgue measure for a.e. $t$.
See also \cite{Suzuki-Takahashi} for further developments on
this subject.

The next theorem is the simplest form of our result on
the quasilinear case.  Here, we obtain
$u \in C^{1,\alpha}_{\rm loc}(\Omega \setminus \Sigma)$
by a theorem of Tolksdorf, DiBenedetto, and
Lewis \cite{Tolksdorf, DiBenedetto, Lewis}, and therefore,
$\partial \Omega_{s_0}$ is smooth if $s_0$ is a regular value of $u$.

\begin{theorem}  \label{thm:2}
Let $1<p<N$ and $u=u(x) \in W^{1,p}_{\rm loc}(\Omega \setminus \Sigma)$
be a solution to
\begin{equation}
\operatorname{div}( | \nabla u|^{p-2}\nabla u) = 0 \quad
\text{in }\Omega \setminus \Sigma
 \label{p-lap}
\end{equation}
satisfying {\rm (A)}.  Then
$\operatorname{Cap}_p(\Sigma) = 0$ and
$u \in L^{N(p-1)/(N-p)}_w(\Omega)$.
\end{theorem}


\begin{remark} \label{rmk1} \rm
Henceforth, $B^m_r(z)$ denotes the $m$-dimensional ball
centered at $z$ with radius $r$.
We have two typical examples related to Theorem \ref{thm:2}.
First, if $\Omega = B^N_1(0)$ and $1<p<N$, then
$u(x) = | x|^{(p-N)/(p-1)}$ is a solution to \eqref{p-lap}
for $\Sigma=\{ 0\}$, and actually,
it holds that $\operatorname{Cap}_p(\Sigma) = 0$ and
$u \in L^{N(p-1)/(N-p)}_w(\Omega)$ if $\frac{2N}{N+1} < p < N$.

Next, if $p<m<N$ is an integer,
$\Omega = B^m_1(0) \times \mathbb{R}^{N-m}$, and
\[
\tilde{x} = (x_1,\dots,x_m, \underbrace{0,\dots,0}_{N-m})
\]
for $x=(x_1, \dots, x_m, x_{m+1}, \dots, x_N)$, then
$u(x) = | \tilde{x}|^{(p-m)/(p-1)}$ is a solution to
\eqref{p-lap} for $\Sigma = \{ 0 \} \times \mathbb{R}^{N-m}$.
In this example, the assumption (A) does not hold in the strict sense,
because $\Omega_{s}$ is not bounded for any $s\geq 0$.
But, if $H^{N-m}(A)$ denotes the $(N-m)$-dimensional Hausdorff
measure of $A$, then it holds that
$H^{N-m}(\Sigma\cap B^{N}_R(0)) < +\infty$ for any $R>0$.
This implies $\operatorname{Cap}_m(\Sigma\cap B^{N}_R(0)) = 0$
from the general theory and hence $\operatorname{Cap}_m(\Sigma) = 0$.
More precisely, we have
\[
\operatorname{Cap}_p(A) \le C H^{N-p}(A)
\]
and $H^{N-p}(A) < +\infty$ implies $\operatorname{Cap}_p(A) = 0$
for any $1<p<N$.
It holds also that
\[
 u \in L^{m(p-1)/(m-p)}_w(\Omega) \subset L^{N(p-1)/(N-p)}_w(\Omega)
\]
because $m(p-1)/(m-p) \ge N(p-1)/(N-p)$ by $p < m \le N$.
\end{remark}

\begin{remark} \label{rmk2} \rm
The above theorems may be compared to the removable
singularity property studied by many authors. First,
from the classical theorem by Carleson \cite[p.88]{Carleson}
 if $u=u(x) \in L^{\infty}_{\rm loc}(\Omega)$ is a solution to
\[
\Delta u = 0 \quad \text{in }\Omega \setminus \Sigma,
\]
then the set $\Sigma$ is removable (that is, there exists
a harmonic function $\tilde{u}$ defined in $\Omega$ such
that $\tilde{u} = u$ on $\Omega \setminus \Sigma$)
if and only if $\operatorname{Cap}_2(\Sigma) = 0$.

This fact is extended to $p$-harmonic functions.
Indeed, in  \cite[Theorem 7.36]{HKM},
Heinonen, Kilpel\"ainen, and Martio considered
a more general equation
\begin{equation}
\operatorname{div} A(x, \nabla u) = 0 \label{qle}
\end{equation}
with the vector function $A$ satisfying the growth condition
$A(x, \xi) \simeq | \xi|^{p-1}$.  They proved that
if $u \in L^{\infty}_{\rm loc}(\Omega)$ is a solution of
 \eqref{qle} in $\Omega \setminus \Sigma$, then
$\Sigma$ is removable if and only if
$\operatorname{Cap}_p(\Sigma) = 0$.

These results are concerned with bounded solutions,
while  Serrin \cite{Serrin1} proved the following theorem concerning
$\theta$-integrable solutions
(and \cite{Serrin2} for the linear case):
Let $u$ be a continuous solution to the quasilinear elliptic
equation of divergence form
\begin{equation}
    \operatorname{div} A(x, u, \nabla u) = B(x,u,\nabla u)
    \label{QL}
\end{equation}
in $\Omega \setminus \Sigma$, where $A$ and $B$ satisfy
certain structural conditions admitting the $p$-Laplace
equation as a typical example. First,
if $u \in W^{1,p}_{\rm loc}(\Omega \setminus \Sigma)$
is a weak solution to \eqref{QL} then $u$ is locally
 H\"older continuous in $\Omega\setminus \Sigma$.
One of the main theorem of \cite{Serrin1}, now says that
if $\operatorname{Cap}_q(\Sigma) = 0$ for $1 < p \le q \le N$
and $u \in L^\theta(\Omega \setminus \Sigma)$ for
$\theta > q(p-1)/(q-p)$, then $\Sigma$ is removable; that is,
 there is continuous $\tilde{u}$ defined on all of $\Omega$ such
that $\tilde{u} = u$ on $\Omega \setminus \Sigma$.

In the other result of \cite{Serrin1}, if $u=u(x)$ is a solution
to \eqref{QL} in $\Omega \setminus \{ 0 \}$ with
$B \equiv 0$ and $1<p<N$, and satisfies $u \ge L$ for some $L >0$,
then either $\Sigma=\{ 0\}$ is removable or
$u(x) \simeq | x|^{(p-N)/(p-1)}\to +\infty$ as $| x| \to 0$.
We see that $u \in L^{N(p-1)/(N-p)}_w(\Omega)$ holds in the
latter case.

The singular set $\Sigma$ of our theorems are not removable.
However, this set must be small measured by the capacity, just
because it is an actual singular set of the solution. The
solution, on the other hand, is neither locally bounded in
$\Omega$ nor $\theta$-integrable in  $\Omega \setminus \Sigma$ for
some $\theta$ from the results quoted above, but still obeys a
profile of weak integrability in $\Omega$.

This weak integrability is slightly worse than the condition for
which Serrin's removability theorem holds, and is just the same as
the one of the fundamental solution to the $p$-harmonic equation.
\end{remark}

\begin{remark} \label{rmk3} \rm
 The solution in our theorems is assumed to be only in
$W^{1,p}_{\rm loc}(\Omega \setminus \Sigma)$.  In contrast with
this, if there is $u=u(x) \in W^{1,p}(\Omega_0 \setminus \Sigma)$
satisfying (A), then it follows that $\operatorname{Cap}_p(\Sigma)
= 0$, where $\Omega_0=\Omega_{s_0}$. In other words, under the
cost of global $p$-integrability on $\Omega_0\setminus \Sigma$
with its first derivatives, this $u$ does not need to be a
solution to any equation to infer $\operatorname{Cap}_p(\Sigma) =
0$. Here, $\Gamma_0=\partial\Omega_{0}$ may not be Lipschitz
continuous.

In fact, since $| u|=+\infty$ on $\Sigma$, we obtain  $\min
\{| u|,s \} = s$ on $\Sigma$ for $s > s_0$.  Now, we
define $f_s=f_s(x)\in K^p$ by
\[
f_s(x) =  \begin{cases} \frac{1}{s-s_0} \big( \min \{| u(x)|, s
\}-s_0\big) & x\in \Omega_{0} \\
0 & x\in \Omega_{0}^c.
\end{cases}
\]
Then $\Sigma \subset \{ x \in \mathbb{R}^N : \ f_s(x)
= 1 \}^{\circ}$ and
\[
\nabla f_s =   \begin{cases}
        \frac{1}{s-s_0} \nabla |u| & \text{on }
\Omega_{0} \setminus \Omega_s \\
        0 & \text{on }\Omega_s\cup\Omega_{0}^c,
\end{cases}
\]
which implies
\begin{align*}
\operatorname{Cap}_p(\Sigma)
&\leq \int_{\mathbb{R}^N} | \nabla f_s|^p dx
 = \frac{1}{(s-s_0)^p} \int_{\Omega_{0} \setminus \Omega_s} | \nabla u|^p dx \\
&\leq \frac{1}{(s-s_0)^p} \int_{\Omega_0 \setminus \Sigma}
 | \nabla u |^p dx = o(1)
\end{align*}
as $s \to +\infty$ by $u \in W^{1,p}(\Omega_0 \setminus \Sigma)$.

Here, we note two properties related to the above consideration.
First, any function in $W^{1,p}(\Omega_0\setminus \Sigma)$
is identified with the one in $W^{1,p}(\Omega_0)$
if $\operatorname{Cap}_p(\Sigma)=0$, and therefore,
each $u\in W^{1,p}(\Omega_0\setminus \Sigma)$ satisfying
(A) (with $\Gamma_0=\partial\Omega_{0}$ not necessarily
 Lipschitz continuous) belongs to $W^{1,p}(\Omega_0)$.
Next, $\operatorname{Cap}_p(\Sigma) = 0$ follows  from
\begin{equation}
\int_{\Omega_{0} \setminus \Omega_s} | \nabla u|^p dx
= o(s^p) \quad \text{as }s \to +\infty
\label{energy_growth}
\end{equation}
if $u\in W^{1,p}_{\rm loc}(\Omega\setminus \Sigma)$ satisfies (A).
This fact is often used in the rest of the present paper.

If the solution $u=u(x)$ is sufficiently smooth on
$\Omega \setminus \Sigma$, our theorems have a simple proof
using classical co-area formula, Sard's lemma, and isoperimetric
inequality.  This argument is described in \S 2 for the reader's
convenience.  In the general case without regularity,
we follow the argument of Talenti \cite{Talenti} to compensate
the lack of smoothness of the solution.  See \S 3.
\end{remark}

\section{Regular case}

In the regular case, there is a transparent proof of
Theorem \ref{thm:1}.  This section is devoted to the
description of the main idea of the proof, restricted to this case.
Thus, we treat the solution $u=u(x)$ to $\Delta u = 0$ in
$\Omega \setminus \Sigma$ satisfying (A).

Since $u$ is smooth in $\Omega\setminus \Sigma$ in this case,
we may assume that $s_0>0$ is a regular value of
$| u|=| u|(x)$ by Sard's lemma.  Let $\Omega_0 = \Omega_{s_0}$.
Then $\Gamma_0=\partial\Omega_{0}$ is smooth and the disjoint
union of the boundaries of
$\Omega_0^\pm=\{ x\in \Omega_0 \setminus\Sigma \mid \pm u(x)>s_0
\}\cup \Sigma$.  We obtain
$u \in H^1_{\rm loc}(\overline{\Omega}_0 \setminus \Sigma)$ and
\begin{equation}
\Delta u = 0, \quad | u| > s_0 \quad \text{in }
\Omega_0 \setminus \Sigma, \quad | u| = s_0 \quad \text{on }\Gamma_0.
    \label{eq:lap}
\end{equation}
Furthermore, for any $s > s_0$,
\begin{equation}
    \varphi_s = (\operatorname{sgn}u) \cdot
\max \{ s-| u|, 0\}
    \label{phi_s}
\end{equation}
satisfies $\varphi_s \in H^1(\overline{\Omega}_0 \setminus \Sigma)$,
$\operatorname{supp}\varphi_s \subset \overline{\Omega_0}\setminus
\Sigma$,
\[
 \varphi_s |_{\Gamma_0} = (\operatorname{sgn}u) \cdot (s-s_0), \quad
\varphi_s = 0 \quad \text{on }\Omega_s \setminus \Sigma,
\]
and
\[
        \nabla \varphi_s =  \begin{cases}
        - (\nabla u) \quad & \text{on }
\Omega_0 \setminus \overline{\Omega}_s  \\
        0 \quad & \text{on }\Omega_s \setminus \Sigma.
    \end{cases}
\]
Testing this on \eqref{eq:lap}, we obtain
\begin{equation}
    \int_{\Omega_0 \setminus \Omega_s} | \nabla u|^2 dx
= (s - s_0) K = o(s^2)
    \label{BP1}
\end{equation}
as $s \to +\infty$, where
\[
 K = -\int_{\Gamma_0} (\operatorname{sgn}u)
 \frac{\partial u}{\partial \nu} dH^{N-1}
\]
and $\nu$ is the outer unit normal to $\Gamma_0$.
Since $\Gamma_0$ is smooth, the above $K>0$ is defined
in the classical sense.
This implies $\operatorname{Cap}_2(\Sigma) = 0$ by \eqref{BP1}.
See \eqref{energy_growth} of Remark \ref{rmk3}.

Next, differentiating both sides of \eqref{BP1}, we have
\[
- \frac{d}{ds} \int_{\Omega_s \setminus \overline{\Omega}_{s'}}
| \nabla u|^2 dx = \frac{d}{ds} \int_{\Omega_0 \setminus \Omega_s}
| \nabla u|^2 dx =K
\]
for $s \in (s_0, s')$, where $s' > s_0$ is arbitrary.
 Since $u=u(x)$ is smooth on $\Omega \setminus \Sigma$,
Sard's lemma guarantees that the set of critical values
 of $u$ has the one-dimensional Lebesgue measure $0$.
Then, from the co-area formula, we obtain
\begin{equation}
K = - \frac{d}{ds} \int_{\Omega_s \setminus \overline{\Omega}_{s'}}
| \nabla u|^2 dx
    = \int_{\{ |u| = s \}} | \nabla u| dH^{N-1}
\quad \text{a.e. }s \in (s_0, s').
    \label{co-area}
\end{equation}

We apply the co-area formula also to
$\mu(s) = | \Omega_s| = \int_{\Omega_s} dx$.
Again, Sard's lemma assures
\begin{equation}
    -\mu'(s) = \int_{\{ |u| = s \}} | \nabla u|^{-1} dH^{N-1} \quad
\text{a.e. }s > s_0.
    \label{d_ds_mu}
\end{equation}
By \eqref{co-area}, \eqref{d_ds_mu}, and
the Schwarz inequality
\[
\Big( \int_{\{ |u| = s \}} dH^{N-1} \Big)^2
    \le \int_{\{ |u| = s \}} | \nabla u | dH^{N-1} \cdot
\int_{\{ |u| = s \}} | \nabla u |^{-1} dH^{N-1},
\]
now we obtain
\begin{equation}
    H^{N-1}(\{ |u| = s \})^2 \le K \cdot (-\mu'(s)) \quad \text{a.e. }
s \in (s_0, s').
    \label{level}
\end{equation}

The classical isoperimetric inequality in $\mathbb{R}^N$,
on the other hand, implies
\[
 N C_N^{1/N} H^N(\Omega_s)^{(N-1)/N} \le H^{N-1}(\{ |u| = s \}),
\]
where $C_N$ is the volume of $N$-dimensional unit ball.
Combining this with \eqref{level}, it follows that
\[
N^2 C_N^{2/N} \mu(s)^{2(N-1)/N} \le K \cdot (-\mu'(s));
\]
that is,
\begin{equation}
    C(N,K) \le \mu(s)^{-2(N-1)/N} \cdot (-\mu'(s)) \quad \text{a.e. }
s \in (s_0, s')
    \label{diff_ineq}
\end{equation}
for $C(N,K) = N^2 C_N^{2/N} K^{-1}$.

If we define
\begin{equation}
\phi(\mu) = \frac{N}{N-2} \mu^{-(N-2)/N},
 \label{phi}
\end{equation}
then
\[
\frac{d}{ds} \phi(\mu(s)) =  \mu(s)^{-2(N-1)/N} \cdot (-\mu'(s)),
\]
and therefore, \eqref{diff_ineq} is written as
\[
C(N,K) \le \frac{d}{ds} \phi(\mu(s)) \quad \text{a.e. }
s\in (s_0 , s').
\]
Integrating both sides from $s_0$ to $s'$ and rewriting $s'$ to $s$,
we obtain
\begin{equation}
\begin{gathered}
 C(N,K)(s - s_0) + \phi(\mu(s_0)) \le \phi(\mu(s)), \\
 \phi(\mu(s))^{-1} \le \{ C(N,K)(s - s_0) + \phi(\mu(s_0)) \}^{-1}.
 \end{gathered} \label{daha}
\end{equation}
Here, we used
\[
\int_{s_0}^{s'} \frac{d}{ds} \phi(\mu(s)) ds \leq \phi(\mu(s'))
- \phi(\mu(s_0)),
\]
assured by the fact that $s \mapsto \phi(\mu(s))$ is non-decreasing.
We note that the distribution function $\mu=\mu(s)$ is not
necessarily absolutely continuous in $s$ even if $u=u(x)$
is smooth in $x$.  More precisely, it is only right-continuous
and even discontinuous points can arise.

Multiplying both sides by $s$ in \eqref{daha}, now we have
\begin{gather*}
 \big( \frac{N-2}{N} \big) s \mu(s)^{(N-2)/N}
\le \frac{s}{C(N,K)(s-s_0) + \phi(\mu(s_0))} \\
 s^{N/(N-2)} \mu(s)\le \big( \frac{N}{N-2} \big)
^{N/(N-2)}s^{N/(N-2)}\{C(N,K)(s-s_0)
+ \phi(\mu(s_0)) \}^{-N/(N-2)}
\end{gather*}
for $s > s_0$, and therefore,
\[
s^{N/(N-2)} \mu(s) = O(1) \quad \text{as }s \to +\infty.
\]
This implies $u \in L^{N/(N-2)}_w(\Omega)$.
See \cite{folland}.

\section{Irregular case}

In the irregular case, we use the co-area formula
and the isoperimetric inequality associated with the perimeter.
Such tools were adopted by  Talenti \cite{Talenti} in the
proof of his comparison theorem to overcome the lack of
smoothness of the solution.  To begin with, we collect several
facts concerning the perimeter used in later arguments.

First, the co-area formula to functions of bounded variation
 Fleming and Rishel \cite{Fleming-Rishel} is applicable to
$u \in W^{1,1}_{\rm loc}(\Omega \setminus \Sigma)$,
and it holds that
\begin{equation}
-\frac{d}{ds} \int_{\Omega_s \setminus \overline{\Omega}_{s'}}
| \nabla u| dx  = P( \Omega_s ) \quad \text{a.e. }s_0 < s < s'.
    \label{Fleming_Rishel}
\end{equation}
The right-hand side abbreviates $P(\Omega_s, \mathbb{R}^N)$,
where for the measurable set $E \subset \mathbb{R}^N$
and an open set $U \subset \mathbb{R}^N$, $P(E,U)$
denotes DeGiorgi's perimeter of $E$ in $U$; i.e.,
\[
 P(E,U) = \sup \big\{ \int_E \operatorname{div}\vec{g} dx :
 \vec{g} \in C_0^{\infty}(U, \mathbb{R}^N),
 \max_{x \in U} : \vec{g}(x)| \le 1 \big\}.
\]

A measurable set $E \subset \mathbb{R}^N$ satisfying
$P(E) < +\infty$ is called a Caccioppoli set, or a set of
finite perimeter in $\mathbb{R}^N$.  It is a set whose indicator
function has a bounded total variation  on $\mathbb{R}^N$.
See \cite{Giusti}.  DeGiorgi's isoperimetric inequality
is concerned with these Caccioppoli sets in $\mathbb{R}^N$.
More precisely, if $E$ is such a set, then
\begin{equation}
    N C_N^{1/N} | E|^{(N-1)/N} \le P(E).
    \label{DeGiorgi_isoperimetric}
\end{equation}

Finally, we use the general trace lemma. See
\cite[Chapter I, Theorem 1.2]{Temam} or
\cite[Lemma 1.2.2]{Sohr} for the proof.

\begin{lemma} \label{lem:3}
If $\Omega \subset \mathbb{R}^N$ ($N \ge 2$) is a bounded
domain with Lipschitz boundary $\partial \Omega$, $1<q<\infty$,
 $q' = \frac{q}{q-1}$,
\[
E_q(\Omega) = \{ \vec{v} \in L^q(\Omega)^N :
\operatorname{div} \; \vec{v} \in L^q(\Omega) \},
\]
and
\[
\| \vec{v} \|_{E_q} =
\Big(\| \vec{v} \|_q^q + \| \operatorname{div} \; \vec{v} \|_q^q
\Big)^{1/q},
\]
then there is a bounded linear operator, called the generalized
normal component trace,
\[
 \Gamma_{\nu}: \vec{v}\in E_q(\Omega) \mapsto
\Gamma_{\nu} \vec{v} \in W^{-1/q,q}(\partial \Omega)
 = \big( W^{1/q,q'}(\partial \Omega) \big)^{*}
\]
such that $g\mapsto \langle g, \Gamma_{\nu} \vec{v}\rangle$
is compatible to the functional
\[
g\in W^{1/q,q'}(\partial \Omega) \mapsto \int_{\partial \Omega} g(x)
\nu(x) \cdot \vec{v}(x) dH^{N-1}
\]
defined for $\vec{v} \in C^{\infty}(\overline{\Omega})^N$ and
the exterior unit normal $\nu$.  It holds that
\[
( \vec{v}, \nabla \varphi)+(\operatorname{div}\vec{v}, \varphi)
=\langle \Gamma_\nu\vec{v}, \gamma_0\varphi\rangle
\]
for $\vec{v}\in E_q(\Omega)$ and $\varphi\in W^{1,q'}(\Omega)$, where
\[
 \gamma_0:W^{1,q'}(\Omega)\to W^{1/q, q'}(\partial\Omega)
\]
is the usual trace operator.
\end{lemma}

\subsection*{Proof of Theorem \ref{thm:1}}
 As in the previous section, we note that
$u \in H^1_{\rm loc}(\overline{\Omega}_0 \setminus \Sigma)$ satisfies
\[
Lu = 0, \; | u|> s_0 \quad \text{in } \Omega_0 \setminus \Sigma,
\quad | u| = s_0 \quad \text{on }\Gamma_0
 \]
for $\Omega_0 = \Omega_{s_0}$ and $\Gamma_0=\partial\Omega_0$.
Since $s_0>0$, this $\Gamma_0$ is the disjoint union of
the Lipschitz boundaries of
$\Omega_0^\pm=\{ x\in \Omega_0\setminus \Sigma : \pm u(x)>s_0 \}
\cup\Sigma$, and testing this by
$\varphi_s = \varphi_s(x)$ defined in \eqref{phi_s} is permitted.
 We obtain
\begin{equation}
\sum_{i,j=1}^N\int_{\Omega_0 \setminus \Omega_s} a_{ij} D_j u  D_i u dx = (s-s_0) K - \int_{\Omega_0 \setminus \Omega_s} c | u| (s-|u|) dx,
    \label{LBP1}
\end{equation}
where
\[
K = - \langle \frac{\partial u}{\partial \nu_L},
\operatorname{sgn}u \rangle_{H^{-1/2}(\Gamma_0), H^{1/2}(\Gamma_0)}
\]
and
\[
\frac{\partial u}{\partial \nu_L}
= \sum_{i,j=1}^N \nu_i a_{ij} D_j u\in H^{-1/2}(\Gamma_0)
\equiv W^{-1/2, 2}(\Gamma_0)
\]
is the general trace of
\[
\vec{v}=\Big( \sum_{j=1}^Na_{ij}D_ju\Big)_{i=1, \dots, N}
\in E_{2, \rm loc}(\overline{\Omega_0}\setminus\Sigma).
\]
We emphasize that $\operatorname{sgn}u=\pm 1$ exclusively on
each component of $\Gamma_0$, because $u=u(x)$ is continuous
in $\Omega\setminus\Sigma$.  Using \eqref{se} and $c \ge 0$,
we obtain
\[
\delta \int_{\Omega_0 \setminus \Omega_s} | \nabla u|^2 dx
\le (s-s_0) K = o(s^2)
\]
as $s \to +\infty$, and hence $\operatorname{Cap}_2 (\Sigma) = 0$.

To show $u \in L^{N/(N-2)}_w(\Omega)$, we use the fact that
\[
 g_s(x)= c(x) | u(x)|(s-| u(x)|)
\]
is non-negative and non-decreasing in $s$ for each
$x\in \Omega_0 \setminus \Omega_s$.  The set
$\Omega_0 \setminus \Omega_s$ is also non-decreasing in $s$,
and therefore, the function
\[
s \mapsto  I(s)=\int_{\Omega_0 \setminus \Omega_s} c | u|
(s-|u|) dx
\]
is non-decreasing.  Thus, differentiating both sides
of \eqref{LBP1}, we obtain
\begin{equation}
\begin{aligned}
\frac{d}{ds} \int_{\Omega_0 \setminus \Omega_s}
\sum_{i,j=1}^Na_{ij} D_j u D_i u dx
& =  -\frac{d}{ds} \int_{\Omega_s \setminus \overline{\Omega}_{s'}}
  \sum_{i,j=1}^Na_{ij} D_j u D_i u dx \\
&\leq K \quad \text{a.e. }s \in (s_0, s'),
\end{aligned}\label{LBP2}
\end{equation}
where $s' > s_0$ is arbitrary.

The next lemma is a key ingredient of the proof,
where $\mu(s) = |\Omega_s|$.

\begin{lemma}
It holds that
\begin{equation}
-\frac{d}{ds} \int_{\Omega_s \setminus \overline{\Omega}_{s'}}
| \nabla u| dx \leq (-\mu'(s))^{1/2}
\Big( -\frac{d}{ds} \int_{\Omega_s \setminus
\overline{\Omega}_{s'}} \delta^{-1} \sum_{i,j=1}^Na_{ij} D_j u
D_i u dx \Big)^{1/2} \label{Prop3.2}
\end{equation}
a.e. $s \in (s_0, s')$.
 \label{prop:3.2}
\end{lemma}

\begin{proof}

First, the mapping
\[
s\in (s_0,s') \mapsto \int_{\Omega_s \setminus
\overline{\Omega}_{s'}} | \nabla u| dx
\]
is non-increasing.
Given $0<h \ll 1$ in $s < s + h < s'$, we take its differential
quotient.  In fact, by the Schwarz inequality and \eqref{se}, we obtain
\begin{align*}
&\frac{1}{h} \big[ \int_{\Omega_s \setminus \overline{\Omega}_{s'}}
|\nabla u| dx - \int_{\Omega_{s+h} \setminus \overline{\Omega}_{s'}}
|\nabla u| dx \big] \\
&= \frac{1}{h} \int_{\Omega_s \setminus \Omega_{s+h}} | \nabla u| dx \\
&  \le  \Big( \frac{1}{h} \int_{\Omega_s \setminus \Omega_{s+h}} dx
 \Big)^{1/2} \Big( \frac{1}{h} \int_{\Omega_s \setminus \Omega_{s+h}}
| \nabla u|^2 dx \Big)^{1/2} \\
& \le   \Big( \frac{\mu(s) - \mu(s+h)}{h} \Big)^{1/2}
    \Big(\frac{1}{h} \int_{\Omega_s \setminus
\Omega_{s+h}} \delta^{-1}\sum_{i,j=1}^Na_{ij} D_j u  D_i u dx \Big)^{1/2} \\
& = \big(-\mu'(s) \big)^{1/2}
 \Big( -\frac{d}{ds} \int_{\Omega_s \setminus \overline{\Omega}_{s'}}
\delta^{-1} \sum_{i,j=1}^Na_{ij} D_j u  D_i u dx \Big)^{1/2} + o(1)
\end{align*}
as $h \downarrow 0$, and hence \eqref{Prop3.2} follows.
\end{proof}

Now, we continue the proof of Theorem \ref{thm:1}.  It holds that
\begin{equation}
N C_N^{1/N}  \mu(s)^{(N-1)/N} \le P(\Omega_s) = -\frac{d}{ds} \int_{\Omega_s \setminus \overline{\Omega}_{s'}} | \nabla u| dx \quad \text{a.e. $s \in (s_0, s')$}
    \label{DG-FR}
\end{equation}
by \eqref{DeGiorgi_isoperimetric}-\eqref{Fleming_Rishel}.
 Combining this with \eqref{Prop3.2}, we obtain
\[
N^2 C_N^{2/N} \leq \mu(s)^{-2(N-1)/N} (-\mu'(s))
\Big( -\frac{d}{ds} \int_{\Omega_s \setminus \overline{\Omega}_{s'}}
\delta^{-1} \sum_{i,j=1}^Na_{ij} D_j u  D_i u dx\Big)
\]
for a.e. $s \in (s_0, s')$.  Then \eqref{LBP2} guarantees
\[
C(N,K) \le \frac{d}{ds} \phi(\mu(s)) \quad \text{a.e. }
s \in (s_0, s')
\]
for $\phi=\phi(\mu)$ defined by \eqref{phi}, where
$C(N,K) = \delta N^2 C_N^{2/N} K^{-1}$.

At this stage, we can follow the argument in the previous section,
and obtain $u \in L^{N/(N-2)}_{w}(\Omega)$.


\subsection*{Proof of Theorem \ref{thm:2}}
Testing
\[
\operatorname{div}\big( | \nabla u|^{p-2} \nabla u\big) = 0,
\quad | u| > s_0 \quad \text{in }\Omega_0 \setminus \Sigma,
 \quad | u | = s_0 \quad \text{on }\Gamma_0
\]
by $\varphi_s=\varphi_s(x)$ of \eqref{phi_s} is permitted
similarly, and then we obtain
\begin{equation}
\int_{\Omega_0 \setminus \Omega_s} | \nabla u|^p dx
= (s-s_0) K = o(s^p) \quad \text{as }s \to +\infty,
\label{TestPB}
\end{equation}
where
\[
K = - \langle |\nabla u|^{p-2} \frac{\partial u}{\partial \nu},
\operatorname{sgn}u \rangle_{W^{-1/p',p'}(\Gamma_0),
W^{1-1/p,p}(\Gamma_0)}
\]
for $\frac{1}{p'} + \frac{1}{p} = 1$.
Thus, it holds that $\operatorname{Cap}_p(\Sigma)=0$.
Differentiating \eqref{TestPB} with respect to $s$, on the other hand,
we obtain also
\begin{equation}
-\frac{d}{ds} \int_{\Omega_s \setminus \overline{\Omega}_{s'}}
| \nabla u|^p dx = \frac{d}{ds} \int_{\Omega_0 \setminus \Omega_s}
| \nabla u|^p dx = K \quad \text{a.e. }s \in (s_0, s'),
    \label{PBP1}
\end{equation}
for $s' > s_0$ arbitrarily fixed.
Then, the following lemma takes place
of Lemma \ref{prop:3.2}, which is proven
by H\"older's inequality instead of the Schwarz inequality.

\begin{lemma}
It holds that
\[
-\frac{d}{ds} \int_{\Omega_s \setminus \overline{\Omega}_{s'}}
| \nabla u| dx \le
(-\mu'(s))^{1/p'} \Big( -\frac{d}{ds} \int_{\Omega_s \setminus
\overline{\Omega}_{s'}} | \nabla u|^p dx \Big)^{1/p}
\]
for a.e. $s \in (s_0, s')$.
 \label{prop:3.3}
\end{lemma}

Inequality \eqref{DG-FR}, on the other hand, is derived
from DeGiorgi's isoperimetric inequality and Fleming-Rishel's
co-area formula.  This inequality, therefore, is applicable even
to this case, and we obtain
\[
N C_N^{1/N}  \mu(s)^{(N-1)/N} \le (-\mu'(s))^{1/p'}
\Big( -\frac{d}{ds} \int_{\Omega_s \setminus \overline{\Omega}_{s'}}
| \nabla u|^p dx \Big)^{1/p},
\]
and hence
\[
N^{p'}C_N^{p'/N} \le  \mu(s)^{-p'(N-1)/N} (-\mu'(s))
\Big( -\frac{d}{ds} \int_{\Omega_s \setminus \overline{\Omega}_{s'}}
| \nabla u|^p dx \Big)^{p'/p}
 \]
for a.e. $s \in (s_0, s')$.
Combining this with \eqref{PBP1}, we have
\[
N^{p'}C_N^{p'/N} \le \mu(s)^{-\frac{p'(N-1)}{N}} K^{p'/p}
\cdot (-\mu'(s));
\]
i.e.,
\begin{equation}
C(N,K) \le \frac{d}{ds} \phi(\mu(s)) \quad \text{a.e. }
s \in (s_0,s')
 \label{this},
\end{equation}
for
\begin{equation}
\phi(\mu) = \frac{N(p-1)}{N-p} \mu^{-\frac{N-p}{N(p-1)}}
\quad \text{and} \quad
C(N,K) = N^{p'}C_N^{p'/N} K^{-p'/p}.
 \label{phi-2}
\end{equation}

Integrating \eqref{this} from $s_0$ to $s'$, rewriting $s'$ to $s$,
and noting the monotonicity of $s \mapsto \phi(\mu(s))$, we obtain
\begin{gather*}
C(N,K)(s-s_0) + \phi(\mu(0)) \le \phi(\mu(s)), \\
\phi(\mu(s))^{-1} \le \big( C(N,K)(s-s_0) + \phi(\mu(0)) \big)^{-1}.
\end{gather*}
Multiplying both sides by $s$, now we have
\begin{gather*}
\frac{N-p}{N(p-1)} \cdot s\mu(s)^{(N-p)/N(p-1)}
\leq \frac{s}{C(N,K)(s-s_0) + \phi(\mu(0))} \\
 \Big(\frac{N-p}{N(p-1)} \Big)^{\frac{N(p-1)}{N-p}}
 \cdot s^{\frac{N(p-1)}{N-p}} \mu(s)
\leq s^{\frac{N(p-1)}{N-p}}\{C(N,K)(s-s_0)
 + \phi(\mu(0))\}^{-\frac{N(p-1)}{N-p}}.
\end{gather*}
This implies
\begin{equation}
s^{\frac{N(p-1)}{N-p}} \mu(s) = O(1) \quad \text{as }
s \to \infty,
 \label{conclude}
\end{equation}
and hence $u \in L^{N(p-1)/(N-p)}_w(\Omega)$.

\section{Generalizations}

This section is devoted to several generalizations.
First, we show the following result.

\begin{theorem} \label{thm:6}
Regardless of the sign of $c=c(x)$, it holds that
$\operatorname{Cap}_2(\Sigma)=0$ and
$\log( 1 + |u|) \in L_w^{N/(N-2)}(\Omega)$ in Theorem \ref{thm:1}.
\end{theorem}

\begin{proof}
 The second term of the right-hand side of \eqref{LBP1},
\[
-\int_{\Omega_0\setminus \Omega_s}c| u|( s-|u|)dx = -I(s)
 \]
is estimated from above by
\[
-I(s) \le \| c_-\|_\infty\int_{\Omega_0\setminus\Omega_s}| u|
(s-|u|)dx
\leq s^2\| c_-\|_\infty\int_{\Omega_0\setminus
\Omega_s}\frac{| u|}{s}dx
\]
where $c(x) = c_+(x) - c_-(x)$, $c_{\pm} \ge 0$.

Recall that by assumption (A), $\Omega_0$ is a compact subset
of $\Omega$.
Since
\[
\int_{\Omega_0\setminus \Omega_s}\frac{| u|}{s}dx
= \int_{\Omega_0}I_{\Omega_s^{c}} (x) \frac{| u|}{s}dx
\]
and
\[
\big| I_{\Omega_s^{c}} (x) \frac{| u|}{s} \big|
\le 1 \in  L^1(\Omega_0), \quad
I_{\Omega_s^{c}} (x) \frac{| u(x) |}{s} \to 0 \quad \text{a.e. }
 x \in \Omega_0
\]
as $s \to \infty$, where $I_A$ is the indicator function of a set $A$,
we obtain
\[
\int_{\Omega_0\setminus \Omega_s}\frac{| u|}{s}dx = o(1)
\]
from the dominated convergence theorem.
Going back to \eqref{LBP1}, we have
\[
\int_{\Omega_0\setminus \Omega_s}| \nabla u|^2dx=o(s^2).
 \]
Hence $\operatorname{Cap}_2(\Sigma)=0$.
Also we have
\[
\int_{\Omega_0\setminus \Omega_s}c| u|(s-| u|)dx=o(s^2).
\]
Since $c=c(x)\in L^\infty(\Omega\setminus \Sigma)$,
the above $I=I(s)$ is a function of bounded variation in $s$.
 Given $0<h\ll 1$ in $s<s+h<s'$, we obtain
\begin{align*}
&  -\Big( \frac{I(s+h)-I(s)}{h} \Big) \\
&  =-\frac{1}{h}\Big( \int_{\Omega_0\setminus \Omega_{s+h}}c| u|
 ( s+h-| u|)dx - \int_{\Omega_0\setminus \Omega_s}c| u | ( s-| u|)dx \Big)
 \\
&=-\frac{1}{h}\int_{\Omega_s\setminus \Omega_{s+h}}c| u|
( s+h-| u|)dx-\int_{\Omega_0\setminus \Omega_s}c| u| dx \\
&  \leq \| c_-\|_\infty\Big( \int_{\Omega_s\setminus \Omega_{s+h}}
| u| dx+\int_{\Omega_0\setminus \Omega_s}| u| dx\Big) \\
&=\| c_-\|_\infty\int_{\Omega_0\setminus \Omega_{s+h}}| u| dx\\
&\leq \| c_-\|_\infty |\Omega_0| (s+h).
\end{align*}
Therefore,
\[
 \frac{d}{ds}\int_{\Omega_0\setminus \Omega_s}
\sum_{i,j=1}^Na_{ij}D_juD_iu\,dx
\leq K+\| c_-\|_\infty | \Omega_0| s\quad \text{a.e. }
s\in (s_0, s').
\]
Using this instead of \eqref{LBP2}, we obtain
\begin{gather*}
 \phi(\mu(s))^{-1}\leq C\left(\log (s+1) \right)^{-1}, \\
 \left( \log (s+1) \right)^{N/(N-2)}\mu(s)=O(1) \quad
\text{as }s\to +\infty
\end{gather*}
for $\phi=\phi(\mu)$ defined by \eqref{phi}.
The last equality implies that
$\log( 1 + |u|) \in L_w^{N/(N-2)}(\Omega)$.
\end{proof}


Similar results are also valid to the problems formulated
by Serrin \cite{Serrin1}.  Treating a simple case, we take
the mappings $A: \Omega \setminus \Sigma \times \mathbb{R} \times
\mathbb{R}^N \to \mathbb{R}^N$ and
$B: \Omega \setminus \Sigma \times \mathbb{R} \times \mathbb{R}^N
\to \mathbb{R}$ such that $x \mapsto A(x, z, \xi)$ and
$x \mapsto B(x, z, \xi)$ are measurable for each
$(z, \xi) \in \mathbb{R} \times \mathbb{R}^N$ and
$(z,\xi)\mapsto A(x,z,\xi)$ and $(z, \xi)\mapsto B(x,z,\xi)$
are continuous in $(z, \xi)$ for a.e.
$x \in \Omega \setminus \Sigma$.  We assume the ellipticity
\[
 A(x, z, \xi) \cdot \xi \ge \delta | \xi|^p
\]
and the growth rates
\begin{equation}
\begin{gathered}
 | A(x,z,\xi)| \le \Lambda | \xi|^{p-1}, \\
 | B(x,z,\xi)| \le a(x) | \xi|^{p-1} + b(x) | z|^{p-1}
\end{gathered} \label{gr}
\end{equation}
for $(z, \xi) \in \mathbb{R} \times \mathbb{R}^N$ and a.e.
$x \in \Omega \setminus \Sigma$, where $\delta>0$, $1<p<N$,
and $\Lambda >0$ are constants, and
$a,b \in L^{\infty}_{\rm loc}(\Omega \setminus \Sigma)$.
As Serrin \cite{Serrin1} proved among other things,
in this case the solution
$u=u(x)\in W^{1,p}_{\rm loc}(\Omega \setminus \Sigma)$ to
\begin{equation}
\operatorname{div}A(x, u, \nabla u) = B(x, u, \nabla u) \quad \text{in $\Omega \setminus \Sigma$}
\label{eq:AB}
\end{equation}
is locally H\"older continuous.  Then, we obtain the following
result.

\begin{theorem} \label{thm:7}
If $u \in W^{1,p}_{\rm loc}(\Omega \setminus \Sigma)$
is a solution to  \eqref{eq:AB} satisfying {\rm (A)},
then $\operatorname{Cap}_p(\Sigma) = 0$ and
$u \in L^{N(p-1)/(N-p)}_w(\Omega)$, provided that
\begin{equation}
(\operatorname{sgn}z) \cdot B(x, z, \xi) \le 0
 \label{sign}
\end{equation}
for any $(\xi, z)\in \mathbb{R}^N\times \mathbb{R}$ and a.e.
$x\in \Omega \setminus \Sigma$.

In the other case without  \eqref{sign}, we obtain
$\operatorname{Cap}_p(\Sigma)=0$ if the second relation
of \eqref{gr} is slightly strengthened; i.e.,
any $\varepsilon>0$ admits $C_\varepsilon>0$ such that
\begin{equation}
| B(x,z,\xi)| \leq \varepsilon| \xi|^{p-1}+C_\varepsilon| z|^{p-1}
 \label{gr2}
\end{equation}
for $(z, \xi) \in \mathbb{R} \times \mathbb{R}^N$ and a.e.
$x \in \Omega \setminus \Sigma$.  If $\varepsilon=0$
is attained in {\rm \eqref{gr2}}, then
$\log(1 + |u|) \in L_w^{\frac{N(p-1)}{N-p}}(\Omega)$
follows furthermore.
\end{theorem}

Now we check several key points.


\noindent (1) Testing
\begin{gather*}
\operatorname{div}A(x, u, \nabla u) = B(x, u, \nabla u), \quad
| u| > s_0 \quad \text{in }\Omega_0 \setminus \Sigma \\
 | u| = s_0 \quad \text{on }\Gamma_0
\end{gather*}
with $\varphi_s = \varphi_s(x)$ of \eqref{phi_s}, we obtain
\begin{equation}
\int_{\Omega_0 \setminus \Omega_s} A(x, u, \nabla u)
\cdot \nabla u dx
 = (s-s_0) K + \int_{\Omega_0 \setminus \Omega_s} B(x, u,
\nabla u) (\operatorname{sgn}u)(s - | u|) dx,
\label{TestAB}
\end{equation}
where $K = - \langle \Gamma_{\nu} A(x,u,\nabla u),
\operatorname{sgn}u \rangle_{W^{-1/p',p'}(\Gamma_0),
W^{1-1/p,p}(\Gamma_0)}$.
 If \eqref{sign} holds, then the second term of the right-hand
side of \eqref{TestAB} is non-positive, and this implies
\[
\delta \int_{\Omega_0 \setminus \Omega_s}
|\nabla u|^p dx \le (s-s_0) K = o(s^p) \quad \text{as }
s \to +\infty.
\]
In the other case of \eqref{gr2}, we obtain
\begin{align*}
& \delta\int_{\Omega_0\setminus \Omega_s}| \nabla u|^pdx \\
& \leq (s-s_0)K+ \varepsilon\cdot s\int_{\Omega_0\setminus \Omega_s}
 | \nabla u|^{p-1}dx +C_\varepsilon\int_{\Omega_0\setminus \Omega_s}
 | u|^{p-1}( s-| u|)dx \\
& \leq o(s^p)+\varepsilon s| \Omega_0|^{1/p}
 \Big(\int_{\Omega_0\setminus \Omega_s}| \nabla u|^pdx\Big)^{(p-1)/p} \\
& \leq o(s^p)+ \frac{\delta}{2}\int_{\Omega_0\setminus \Omega_s}|
\nabla u|^pdx+C\varepsilon^ps^p.
\end{align*}
Here, as before, we estimate
\[
\int_{\Omega_0\setminus \Omega_s}| u|^{p-1}(s-|u|)dx
\le s^p \int_{\Omega_0\setminus \Omega_s}
\big( \frac{\vert u \vert}{s} \big)^{p-1} dx = o(s^p)
\]
by the dominated convergence theorem.
Then $\operatorname{Cap}_p(\Sigma)=0$ follows.
\medskip

\noindent (2) When \eqref{sign} holds, we differentiate
\eqref{TestAB} in $s$, using the monotonicity of
\[
s \mapsto I(s)=\int_{\Omega_0 \setminus \Omega_s} B(x, u, \nabla u)
(\operatorname{sgn}u)(s - | u|) dx.
\]
Then
\begin{equation}
\begin{aligned}
&\frac{d}{ds} \int_{\Omega_0 \setminus \Omega_s} A(x, u, \nabla u)
 \cdot \nabla u dx \\
& = -\frac{d}{ds} \int_{\Omega_s \setminus \overline{\Omega}_{s'}}
 A(x, u, \nabla u) \cdot \nabla u dx \\
& = K + \frac{d}{ds} \int_{\Omega_0 \setminus \Omega_s}
 B(x, u, \nabla u) (\operatorname{sgn}u) (s - | u|) dx \\
& \le K \quad \text{a.e. $s \in (s_0, s')$}.
 \end{aligned}   \label{ABP1}
\end{equation}
Even in the other case without \eqref{sign}, $s\mapsto I(s)$
is a function of bounded variation, and it holds that
\[
\frac{I(s+h)-I(s)}{h}\leq \int_{\Omega_0\setminus \Omega_{s+h}}|
B(x,u, \nabla u)| dx
\]
for $s<s+h<s'$.  Therefore, if $\varepsilon=0$ is attained
in \eqref{gr2}, we obtain
\begin{equation}
\begin{aligned}
\frac{d}{ds} \int_{\Omega_0 \setminus \Omega_s} A(x, u, \nabla u)
\cdot \nabla u dx
&\leq K+C_0\int_{\Omega_0\setminus \Omega_s}| u|^{p-1}dx \\
& \leq  K+C_0| \Omega_0| s^{p-1} \quad \text{a.e. }s \in (s_0, s').
\end{aligned} \label{ABP1'}
\end{equation}

\noindent (3) We establish Talenti's inequality; i.e.,
\[
-\frac{d}{ds} \int_{\Omega_s \setminus \overline{\Omega}_{s'}}
  | \nabla u| dx
 \le (-\mu'(s))^{1/p'}
\Big( -\frac{d}{ds} \int_{\Omega_s \setminus \overline{\Omega}_{s'}}
\delta^{-1}A(x,u,\nabla u) \cdot \nabla u dx \Big)^{1/p}
\]
for a.e. $s \in (s_0, s')$ and then combine this
with DeGiorgi's isoperimetric inequality and Fleming-Rishel
formula \eqref{DG-FR}; i.e.,
\begin{equation}
N^{p'} C_N^{p'/N} \le  \mu(s)^{-\frac{p'(N-1)}{N}} (-\mu'(s))
\Big( -\frac{d}{ds} \int_{\Omega_s \setminus \overline{\Omega}_{s'}}
 \delta^{-1}A(x, u, \nabla u) \cdot \nabla u dx \Big)^{p'/p}
\label{ABP2}
\end{equation}
for a.e. $s \in (s_0, s')$, where $\frac{1}{p'}
+\frac{1}{p}=1$ and $\mu(s) = | \Omega_s|$.
\medskip

 \noindent(4)
If \eqref{ABP1} is available, then \eqref{ABP2} implies
\[
\delta^{p'/p} N^{p'}C_N^{p'/N} \le \mu(s)^{-p'(N-1)/N} K^{p'/p}
\cdot (-\mu'(s)).
\]
In this case, we obtain
\[
\frac{d}{ds}\varphi(\mu(s))\geq c \quad \text{a.e. }
s\in (s_0, s')
 \]
for $\phi=\phi(\mu)$ defined by \eqref{phi-2} with a constant $c>0$,
and we end up with \eqref{conclude}.

In the other case of \eqref{ABP1'}, we obtain
\[
\frac{d}{ds}\varphi(\mu(s))\geq \frac{c}{1+s} \quad \text{a.e. }
s\in (s_0, s').
 \]
Then
\[
( 1+\log s)^{\frac{N(p-1)}{N-p}}\mu(s)=O(1) \quad
\text{as }s\to+\infty.
 \]
This implies $\log(1 + |u|) \in L_w^{\frac{N(p-1)}{N-p}}(\Omega)$.

\begin{thebibliography}{00}

\bibitem{Carleson} L. Carleson:
\emph{Selected problems on exceptional sets,}
\newblock Van Nostrand Math. Studies no.13. Van Nostrand, Princeton, NJ. (1967)

\bibitem{DiBenedetto} E. DiBenedetto:
\emph{$C^{1+\alpha}$ local regularity of weak solutions of degenerate elliptic equations,}
\newblock Nonlinear Anal. T. M. A. {\bf 7} (1983) 827-850.

\bibitem{Evans-Gariepy} L. C. Evans, and R. F. Gariepy:
\emph{Measure theory and fine properties of functions,}
\newblock Studies in Advanced Mathematics. CRC press, Boca Raton, FL. (1992)

\bibitem{Fleming-Rishel} W. Fleming, and R. I. Rishel:
\emph{An integral formula for total gradient variations,}
\newblock Arch. Math. {\bf 11} (1960) 218-222.

\bibitem{folland} G.B. Folland:
\emph{Real Analysis, Modern Techniques and Their Applications,}
\newblock Wiley Interscience Publication, New York, (1984).

\bibitem{Gilbarg-Trudinger} D. Gilbarg, and N. S. Trudinger:
\emph{Elliptic partial differential equations of second order,} second edition.
\newblock Springer-Verlag, Berlin-New York, (1983).

\bibitem{Giusti} E. Giusti:
\emph{Minimal surfaces and functions of bounded variation,}
\newblock Birkh\"auser, Boston, (1984).

\bibitem{HKM} J. Heinonen, T. Kilpel\"ainen, and O. Martio:
\emph{Nonlinear potential theory of degenerate elliptic equations,}
\newblock Oxford Mathematical Monographs. Clarendon press. (1992)

\bibitem{Lewis} J. Lewis:
\emph{Regularity of the derivatives of solutions to certain degenerate elliptic equations,}
\newblock Indiana Univ. Math. J. {\bf 32} (1983) 849-858.

\bibitem{Sakaguchi-Suzuki} S. Sakaguchi, and T. Suzuki:
\emph{Interior imperfect ignition cannot occur on a set of positive measure,}
\newblock Arc. Rational Mech. Anal. {\bf 142} (1998) 143-153.

\bibitem{Serrin1} J. Serrin:
\emph{Local behavior of solutions of quasilinear equations,}
\newblock Acta Math. {\bf 111} (1964) 247-302.

\bibitem{Serrin2} J. Serrin:
\emph{Removable singularities of solutions of elliptic equations,}
\newblock Arc. Rational Mech. Anal. {\bf 17} (1964) 67-78.

\bibitem{Sohr} H. Sohr:
\emph{The Navier-Stokes equations: an elementary functional analytic approach,}
\newblock Birkh\"auser, Boston, Basel, Berlin, (2001).

\bibitem{Suzuki-Takahashi} T. Suzuki, and F. Takahashi:
\emph{Capacity estimates for the blow-up set of parabolic equations,}
\newblock Math. Zeit. {\bf 259}, no.4, (2008) 867-878.

\bibitem{Talenti} G. Talenti:
\emph{Elliptic equations and rearrangements,}
\newblock Ann. Scuola Norm. Sup. Pisa Cl. Sci. {\bf 3} (1976) 697-718.

\bibitem{Temam} R. Temam:
\emph{Navier-Stokes equations,}
\newblock North-Holland, Amsterdam, (1977).

\bibitem{Tolksdorf} P. Tolksdorf:
\emph{Regularity for a more general class of quasilinear elliptic equations,}
\newblock J. Diff. Eq. {\bf 51} (1984) 126-150.

\bibitem{Veron} L. V\'eron:
\emph{Singularities of solutions of second order quasilinear equations,}
\newblock Pitman Research Notes in Math. Ser. 353. Addison-Wesley-Longman, (1996).

\bibitem{Ziemer} W. P. Ziemer:
\emph{Weakly differentiable functions; Sobolev spaces and functions of bounded variation,}
\newblock Graduate Texts in Math. 120. Springer-Verlag, (1989).

\end{thebibliography}

\end{document}