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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2013 (2013), No. 108, pp. 1--10.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2013 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2013/108\hfil Symmetry and regularity]
{Symmetry and regularity of an optimization problem related
to a nonlinear BVP}

\author[C. Anedda, F. Cuccu\hfil EJDE-2013/108\hfilneg]
{Claudia Anedda, Fabrizio Cuccu}  % in alphabetical order

\address{Claudia Anedda \newline
Dipartimento di Matematica e Informatica,
Universit\'a di Cagliari,
Via Ospedale 72, 09124 Cagliari, Italy}
\email{canedda@unica.it}

\address{Fabrizio Cuccu \newline
Dipartimento di Matematica e Informatica,
Universit\'a di Cagliari,
Via Ospedale 72, 09124 Cagliari, Italy}
\email{fcuccu@unica.it}

\thanks{Submitted January 7, 2013. Published April 29, 2013.}
\subjclass[2000]{35J20, 35J60, 40K20}
\keywords{Laplacian; optimization problem; rearrangements;
 Steiner symmetry; \hfill\break\indent regularity}

\begin{abstract}
 We consider the functional
 $$
 f\mapsto\int_\Omega \big(\frac{q+1}{2} |Du_f|^2-u_f|u_f|^q f\big) dx,
 $$
 where $u_f$ is the unique nontrivial weak solution of the boundary-value
 problem
 $$
  -\Delta u=f|u|^q\quad \text{in }\Omega,\quad
  u\big|_{\partial\Omega}=0,
 $$
 where $\Omega\subset\mathbb{R}^n$ is a bounded smooth domain.
 We prove a result of Steiner symmetry preservation and, if $n=2$,
 we show the regularity of the level sets of minimizers.
\end{abstract}

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

\section{Introduction}

Let $\Omega$ be a bounded domain $\Omega\subset \mathbb{R}^n$ with 
smooth boundary.
We consider the  Dirichlet problem
\begin{equation}\label{a1}
 \begin{gathered}
  -\Delta u=f|u|^q \quad\text{in }\Omega,\\
  u\big|_{\partial\Omega}=0,
 \end{gathered}
\end{equation}
where $0\leq q<\min\{1,4/ n\}$ and $f$ is a nonnegative
bounded function non identically zero. We consider nontrivial 
solutions of \eqref{a1} in $H^1_0(\Omega)$. The equation \eqref{a1} is the Euler-Lagrange equation of the
integral functional
$$
v\mapsto\int_\Omega \Big(\frac{q+1}{2} |Dv|^2-v|v|^q f\Big) dx, \quad
 v\in H_0^{1} (\Omega).
$$
By using a standard compactness argument, it can be proved that there 
exists a nontrivial minimizer of the above functional.
This minimizer is a nontrivial solution of \eqref{a1}.

From the maximum principle, every nontrivial solution of \eqref{a1} 
is positive.  Then, by \cite[Theorem 3.2]{CPS}
the uniqueness of problem \eqref{a1} follows.
To underscore the dependence on $f$ of the solution of \eqref{a1}, 
we denote it by $u_f$. Moreover,
 $u_f\in W^{2,2}(\Omega)\cap C^{1,\alpha}(\overline{\Omega})$
for all $\alpha$, $ 0<\alpha<1$ (see \cite{GT,T}).

Let $f_0$ be a fixed bounded nonnegative function. We study the problem

\begin{equation}\label{a2}
\inf_{v\in H_0^{1}(\Omega),\, f\in\mathfrak{F}(f_0)}
\int_\Omega \Big(\frac{q+1}{2} |Dv|^2-v|v|^q f\Big) dx,
\end{equation}
where, denoting by $|A|$ the Lebesgue measure of a set $A$,
\begin{equation}\label{a5}
\mathfrak{F}(f_0)=\{f\in L^\infty(\Omega): |\{f\geq c\}|
=|\{f_0\geq c\}|\;\forall c\in\mathbb{R}\};
\end{equation}
here $\mathfrak{F}(f_0)$ is called class of rearrangements of 
$f_0$ (see \cite{K}).

Problems of this kind are not new; see for example
\cite{ATL,CGIKO,CGK,CPS}.
From the results in \cite{CPS} it follows that \eqref{a2} has 
a minimum and a representation formula. Let
\begin{equation}
E(f)=\inf_{v\in H_0^1(\Omega)}
\int_\Omega\Big(\frac{q+1}{2} |Dv|^2-v|v|^q f\Big)
dx.
\end{equation}
Renaming $q'$ the constant $q$ and putting $p=2$ and $q'=q+1$ 
in \cite{CPS} we have
$$
E(f)=\frac{q'-2}{2}\,I(f),
$$
where
$$
I(f)=\sup_{H^1_0(\Omega)}\frac{q'}{2-q'}\int_\Omega
\Big(\frac{2}{q'} f|v|^{q'}-|Dv|^2\Big) dx
$$
is defined in the same paper.
By \cite[Theorem 2.2]{CPS} it follows that there exist minimizers 
of $E(f)$ and that, if $\overline{f}$ is a minimizer, there exists 
an increasing function $\phi$ such that
\begin{equation}\label{a6}
\overline{f}=\phi(u_{\overline{f}}).
\end{equation}

We denote by $\operatorname{supp}f$ the support of $f$, and we call a level
set of $f$ the set  $\{x\in\Omega: f(x)>c\}$, for some
constant $c$.

In Section 2, we consider a Steiner symmetric domain $\Omega$ and $f_0$ 
bounded  and nonnegative, such that
$|\operatorname{supp}f_0|<|\Omega|$. Under these assumptions,
 we prove that the level sets  of the minimizer
$\overline{f}$ are Steiner symmetric
with respect to the same hyperplane of $\Omega$.
As a consequence, we have exactly one optimizer when $\Omega$ is a ball.


 Chanillo,  Kenig and To \cite{CKT} studied the regularity of the minimizers 
to the problem
$$
\lambda(\alpha, A)=\inf_{u\in H_0^1(\Omega),\, \|u\|_2, |D|=A}
\int_\Omega|Du|^2\,dx+\alpha\int_D u^2\,dx,
$$
where $\Omega\subset\mathbb{R}^2$ is a bounded domain, $0<A<|\Omega|$ 
and $\alpha>0$. In particular they prove that,
 if $D$ is a minimizer, then $\partial D$ is analytic.

In Section 3, following the ideas in \cite{CKT}, we give our main result. 
We restrict our attention to $\Omega\subset\mathbb{R}^2$.
Let $b_1,\ldots, b_m>0$ and $0<a_1<\cdots<a_m<|\Omega|$, $m\geq 2$,
be fixed.
Consider $f_0=b_1\chi_{G_1}+\cdots+b_m\chi_{G_m}$, where 
$|G_i|=a_i$  for all $i$, 
$G_i\subset G_{i+1}$, $i=1,\dots,m-1$.

We call $\eta$ the minimum in \eqref{a2}; i.e.,
$$
\eta=\int_\Omega \Big(\frac{q+1}{2} |D u_{\overline{f}}|^2
-u_{\overline{f}}|u_{\overline{f}}|^q\,\overline{f}\Big)dx,
$$
where $\overline{f}$ and $u_{\overline{f}}$ are, respectively, the
minimizing function and the corresponding solution of \eqref{a1}.

In this case \eqref{a6} becomes
$$
\overline{f}=\sum_{i=1}^mb_i\chi_{D_i},
$$
where
$$
D_1=\{u_{\overline{f}}>c_1\},\quad
D_2=\{u_{\overline{f}}>c_2\},\quad \dots, \quad
D_m=\{u_{\overline{f}}>c_m\},
$$
for suitable constants $c_1>c_2>\dots>c_m>0$.

We show regularity of $\partial D_i$ for each $i$ proving 
that $|Du_{\overline{f}}|>0$ in $\partial D_i$.
Following the method used in \cite{CKT}, we consider
\begin{equation}\label{1.3}
E(s,t)=\int_{\Omega}\Big(\frac{q+1}{2}
|Du_{\overline{f}}+sDv|^2-(u_{\overline{f}}+sv)
|u_{\overline{f}}+sv|^q{\overline{f}_t} \Big)dx-\eta,
\end{equation}
where $v\in H_0^{1}(\Omega)$, $\overline{f}_t$ is a family of
functions such that $\overline{f}_t\in\mathfrak{F}(f_0)$ with
$\overline{f}_0=\overline{f}$, and $s\in \mathbb{R}$.
We have
$$
E(s,t)\geq E(0,0)=0\quad \forall s, t.
$$ 
Therefore, $(s,t)=(0,0)$ is a minimum point; it follows that
\begin{equation}\label{1.4}
\begin{vmatrix} \frac{\partial^2 E}{\partial s^2} (0,0)
& \frac{\partial^2 E}{\partial s \partial t} (0,0) \\
\noalign{\vskip10pt}\frac{\partial^2 E}{\partial t \partial s} (0,0)
& \frac{\partial^2 E}{\partial t^2} (0,0)\end{vmatrix}
\geq 0.
\end{equation}
Expanding \eqref{1.4} in detail and using some lemmas 
from \cite{CKT} we prove that the boundaries
of level sets of $\overline{f}$ are regular.

\begin{theorem} \label{thm1}
Let $\Omega\subset\mathbb{R}^2$, $f_0=\sum_{i=1}^m b_i\chi_{G_i}$ with 
$m\geq 2$, and $\overline{f}=\sum_{i=1}^m b_i\chi_{D_i}$ a minimizer 
of \eqref{a2}.
Then $|Du_{\overline{f}}|>0$ on $\partial D_i$, $i=1,\ldots,m$.
\end{theorem}

\section{Symmetry}

In this section we consider Steiner symmetric domains.
We prove that, under suitable conditions on $f_0$ in \eqref{a5}, 
minimizers inherit Steiner symmetry.

\begin{definition} \rm
Let $P\subset\mathbb{R}^n$ be a hyperplane.
 We say that a set $A\subset\mathbb{R}^n$
is \emph{Steiner symmetric} relative to the hyperplane $P$ if
for every straight line $L$ perpendicular to $P$, the set
$A\cap P$ is either empty or a symmetric segment with respect
to $P$.
\end{definition}

To prove the symmetry, we need \cite[Theorem 3.6 and Corollary 3.9]{F}, 
that, for more convenience for the reader, we state here.
These results are related to the classical paper \cite{GNN}.

\begin{theorem}\label{t1}
Let $\Omega\subset\mathbb{R}^n$ be bounded, connected and Steiner symmetric
relative to the hyperplane $P$. Assume that $u:\overline{\Omega}
\to\mathbb{R}$ has the following properties:
\begin{itemize}
\item $u\in C(\overline{\Omega})\cap C^1(\Omega)$, $u>0$ in
$\Omega$, $u|_{\partial\Omega}=0$;

\item for all $\phi\in C^\infty_0(\Omega)$,
\begin{equation*}
\int_\Omega Du\cdot D\phi\,dx=\int_\Omega\phi F(u)\,dx,
\end{equation*}
where $F$ has a decomposition $F=F_1+F_2$ such that $F_1:
[0,\infty)\to\mathbb{R}$ is locally Lipschitz continuous, while
$F_2:[0,\infty)\to\mathbb{R}$ is non-decreasing and identically
$0$ on $[0,\epsilon]$ for some $\epsilon>0$.
\end{itemize}
Then $u$ is symmetric with respect to $P$ and 
$\frac{\partial u} {\partial\mathbf{v}}(x)<0$, where $\mathbf{v}$ 
is a unit vector orthogonal to $P$ and $x$ belongs to the part of 
$\Omega$ that lies in the halfspace (with origin in $P$) in which $\mathbf{v}$
points.
\end{theorem}

\begin{theorem}
Let $\Omega$ be Steiner symmetric and $f_0$ a bounded nonnegative function.
If $|\operatorname{supp}f_0|<|\Omega|$ and $\overline{f}\in\mathfrak{F}(f_0)$ 
is a minimizer of \eqref{a2}, then the level sets  of
$\overline{f}$ are Steiner symmetric with respect to the same hyperplane 
of $\Omega$.
\end{theorem}

\begin{proof}
Let $u=u_{\overline{f}}$ be the solution of \eqref{a2}.
Then $u\in C^0(\overline{\Omega})\cap C^1(\Omega)$ and
satisfies
\begin{equation*}
 \int_{\Omega}D u\cdot D\psi\,dx=\int_{\Omega}\psi fu^q\,dx\quad
 \forall \psi\in C_0^\infty(\Omega).
\end{equation*}
Since $u>0$ and since (from \eqref{a6}) $\overline{f}=\phi(u)$ with 
$\phi$ increasing function, it follows that $\phi(u)\equiv 0$ on 
$\{x\in\Omega: u(x)<d\}$ for some positive constant $d$.  
Then we have $u^q\overline{f}=F_1(u)+F_2(u)$ with $F_1(u)\equiv 0$ and 
$F_2(u)=\phi(u)u^q$.
From Theorem 2.1 and $\overline{f}=\phi(u)$ we have the assertion.
\end{proof}

\begin{remark} \rm
By this theorem, if $\Omega$  is an open ball and 
$|\operatorname{supp}f_0|<|\Omega|$, then $\overline{f}$
is radially symmetric and decreasing.
\end{remark}


\section{Regularity of the free boundaries}

In this section we prove the following result.

\begin{theorem}\label{thm6}
Let $\Omega\subset\mathbb{R}^2$, $f_0=\sum_{i=1}^m b_i\chi_{G_i}$ with 
$m\geq 2$, and
$\overline{f}=\sum_{i=1}^m b_i\chi_{D_i}$ a minimizer of \eqref{a2}.
Then $|Du_{\overline{f}}|>0$ on $\partial D_i$, $i=1,\ldots,m$.
\end{theorem}

We use the notation introduced in Section 1.
Without loss of generality we can assume $m=3$; the general case
easily follows. Let
$\overline{f}=b_1\chi_{D_1}+b_2\chi_{D_2}+b_3\chi_{D_3}$. We will
prove $|Du_{\overline{f}}|>0$ in $\partial D_2$; we omit the proof for $D_1$ and
$D_3$ because it is similar. We define the family $\overline{f}_t$
by replacing only the set $D_2$ by a family of domains $D_2(t)$.

First of all, we explain how to define the family $D_2(t)$.\\
In the sequel we use the notation introduced in \cite{CKT}, 
reorganized according to our needs.

We call a curve $\gamma:[a,b]\to\mathbb{R}$, $-\infty<a<b<\infty$, regular if:
\begin{itemize}
\item[(i)] it is simple, that is: if $a\leq x<y \leq b$ and $x\neq a$ or 
$y\neq b$, then $\gamma(x)\neq \gamma(y)$;

\item[(ii)] $\|\gamma\|_{C^2(a,b)}$ is finite;

\item[(iii)] $|\gamma'|$ is uniformly bounded away from zero.
\end{itemize}
If, in addiction, $\gamma(a)=\gamma(b)$, we say that the curve is closed and 
regular.
If the domain of $\gamma$ is $(a,b)$ we say that $\gamma$ is regular
 (respectively, closed and regular) if the continuous extension of
 $\gamma$ to $[a,b]$ is regular (respectively, closed and regular).

Now, we introduce the notation
$$
\mathcal{F}:=\partial D_2;\quad 
\mathcal{F}^*:=\mathcal{F}\cap\{|Du_{\overline{f}}|>0\}.
$$
Let $J=\cup_{k=1}^pJ_k$ be a finite union of open bounded intervals
$J_k\subset \mathbb{R}$, $\gamma= (\gamma_1, \gamma_2):J\to
\mathcal{F}^*$ a simple curve which is regular on each interval
$J_k$ and $\overline{\gamma(J)}\subset\mathcal{F}^*$. We suppose
that $\operatorname{dist}\big(\gamma(J_k), \gamma(J_h)\big)>0$  for
$1\leq h\neq k\leq p$. Assume also that $|\gamma'|\geq \theta$ on $J$. For
each $\xi \in J$, we denote by
 $\mathbf{N}(\xi)= \big(N_1(\xi), N_2(\xi)\big)$ the outward unit normal
 with respect to $D_2$ at $\gamma(\xi)$.
We also define the tangent vector to $\gamma$ 
$\mathbf{N}^\bot(\xi)=\big(-N_2(\xi),N_1(\xi)\big)$, and $\mathbf{N}'$
the first derivative of $\mathbf{N}$.

Reversing the direction of $\gamma$ if necessary, we will assume, without 
loss of generality, that $\gamma'$ and
$\mathbf{N}^\bot$ have the same direction; i.e.,
 $\angle \gamma', \mathbf{N}^\bot\rangle =|\gamma'|$.
We observe that, because $\gamma$ is $C^2$ and simple on
$\overline{J_k}$, for each $k$ there exists $\beta_k>0$ such that
the function 
$$
\phi_k:J_k \times[-\beta_k,
\beta_k]\to\mathbb{R}^2, \; (\xi,\beta)\mapsto
(x_1,x_2)=\phi_k(\xi, \beta)=\gamma(\xi)+\beta \mathbf{N}(\xi)
$$ 
is injective. 

 Because $\operatorname{dist}\big(\gamma(J_k), \gamma(J_h)\big)>0$
for all $h\neq k$, we can find a number $\beta_0>0$ and we can
paste together the functions $\phi_k$ to obtain a function $\phi$
injective on $J\times [-\beta_0, \beta_0]$. Choose $\beta_0$ such
that $\operatorname{dist}\big(\phi(J\times [-\beta_0, \beta_0]),\partial
D_1\big)>0$ and $\operatorname{dist}\big(\phi(J\times [-\beta_0,
\beta_0]),\partial D_3\big)>0$.

 Now, we define 
$$
K= D_2\setminus \phi\big(J\times (-\beta_0, 0]\big);
$$ 
for $t\in(-t_0,t_0)$ we define
\begin{equation}\label{2.1}
D_2(t)=K\cup\{\phi(\xi,\beta): \xi \in J, \, \beta < g(\xi, t)\},
\end{equation}
where $g: J\times (-t_0,t_0)\to\mathbb{R}$, $t_0>0$, is a function such that
\begin{equation}\label{c1}
g(\xi, t),\; g_t(\xi, t),\; g_{tt}(\xi, t)\in C(\overline{J})\quad
 \forall t\in (-t_0,t_0)
\end{equation}
and
\begin{equation}\label{2.2}
 g(\xi, 0)\equiv 0\quad \forall \xi\in J.
\end{equation}
We observe that $D_2(0)=D_2$. 
Next we compute the measure of $D_2(t)$.
Put $A(t)=|D_2(t)|$ and $A=|D_2(0)|=|D_2|$; we have  
$$
A(t)= |D_2| +\int_J \int_0^{g(\xi,t)} J(\xi, \beta) d\beta d\xi, 
$$
 where
\begin{align*}
J(\xi, \beta) 
&= \frac{\partial(x_1, x_2)}{\partial(\xi,\beta)}=\begin{vmatrix} \gamma_1'+\beta N_1' & N_1 \\
\noalign{\vskip10pt}\gamma_2'+\beta N_2' & N_2 \end{vmatrix}\\
&=  \big| -\langle\gamma', \mathbf{N}^\bot\rangle
-\beta\langle \mathbf{N}',\mathbf{N}^\bot\rangle\big|
=\big| |\gamma'|+\beta\langle \mathbf{N}',\mathbf{N}^\bot\rangle\big|.
\end{align*}
 We show that $ |\gamma'|+\beta\langle \mathbf{N}',\mathbf{N}^\bot\rangle\geq 0$.
Indeed, from the fact that
  $\|\gamma\|_{C^2(J)}<\infty$,
 we have $\|\langle\mathbf{N}',\mathbf{N}^\bot\rangle\|_{L^\infty(J)}<\infty$.
Substituting $t_0$ by a smaller positive number if necessary, we can assume that
  $$\| g\|_{L^\infty(J\times(-t_0,t_0))}<\beta_0$$
 and
  $$\|\langle\mathbf{N}',\mathbf{N}^\bot\rangle\|_{L^\infty(J)}\ \|g\|_{L^\infty(J\times(-t_0,t_0))}<\theta.$$
  Note that the first of these assumptions guarantees that $\partial D_2(t)$ has
  positive distance from $\partial D_1$ and $\partial D_3$.
We have
$$
|\beta| \left| \langle \mathbf{N}',\mathbf{N}^\bot\rangle\right|
\leq \|g\|_{L^\infty(J\times(-t_0,t_0))}
  \|\langle \mathbf{N}',\mathbf{N}^\bot\rangle\|_{L^\infty(J)}
\leq \theta\leq |\gamma'|
$$
for all $\xi\in J$ and
$|\beta|\leq \|g\|_{L^\infty(J\times(-t_0,t_0))}$.
Thus, $J(\xi,\beta) = |\gamma'|+\beta\langle \mathbf{N}',\mathbf{N}^\bot\rangle$.
Substituting into the formula for $A(t)$ we have
\begin{align*}
A(t)&=A+\int_J \int_0^{g(\xi, t)} 
 ( |\gamma'|+\beta   \langle \mathbf{N}',\mathbf{N}^\bot\rangle )d\beta d\xi\\
&= A+\int_J\Big(g(\xi, t)|\gamma'|+\frac{1}{2} (g(\xi, t))^2
 \langle \mathbf{N}',\mathbf{N}^\bot\rangle \Big)d\xi.
\end{align*}
To obtain $|D_2(t)|=|D_2|$ for all $t\in (-t_0,t_0)$, we find the further 
constraint on $g$:
\begin{equation}\label{2.3}
 \int_J\Big(g(\xi, t) |\gamma'| +\frac{1}{2}\ \big(g(\xi, t)\big)^2 
\langle \mathbf{N},\mathbf{N}^\bot\rangle \Big) d\xi=0
\quad \forall t\in (-t_0,t_0).
\end{equation}
Moreover, we calculate the derivatives of $A(t)$, that we will use later. 
\begin{equation}\label{2.9}
\begin{gathered}
A'(t)=\int_J \big(g_t(\xi, t)  |\gamma'(\xi)|+g(\xi, t)g_t(\xi, t)
 \langle \mathbf{N}',\mathbf{N}^\bot\rangle \big) d\xi=0;
\\
A''(t)=\int_J \big(g_{tt}(\xi, t)  |\gamma'(\xi)|+\big(g(\xi, t)g_{tt}(\xi,
t)+g^2_t(\xi, t)\big)
 \langle \mathbf{N}',\mathbf{N}^\bot\rangle \big) d\xi=0.
\end{gathered}
\end{equation}

 Once we have defined the family $D_2(t)$, we can go back to
the functional \eqref{1.3}.
The following lemma describes \eqref{1.4} with 
$\overline{f}_t=b_1\chi_{D_1}+b_2\chi_{D_2(t)}+b_3\chi_{D_3}$. We find
an inequality corresponding to \cite[(2.3) of Lemma 2.1]{CKT}.

\begin{lemma}\label{lem1} 
Let $\overline{f}_t=b_1\chi_{D_1}+b_2\chi_{D_2(t)}+b_3\chi_{D_3}$, where
the variation of domain $D_2(t)$ is described by \eqref{2.1} and 
$g: J\times (-t_0,t_0)\to\mathbb{R}$, $t_0>0$,
satisfies \eqref{c1}, \eqref{2.2} and \eqref{2.3}. Then, 
for all $v\in H_0^{1}(\Omega)$,
the conditions \eqref{1.4} becomes
\begin{equation}\label{a8}
\begin{aligned}
&\int_\Omega\big(
|Dv|^2-qu_{\overline{f}}\ v^2|u_{\overline{f}}|^{q-2}\overline{f}\big)
dx\cdot\int_\gamma g_{t}^2(\gamma^{-1},0)
|Du_{\overline{f}}| d\sigma \\
&\geq b_2c_2^q\Big(\int_\gamma g_{t}(\gamma^{-1},0)\ v d\sigma\Big)^2.
\end{aligned}
\end{equation}
\end{lemma}

\begin{proof}
We calculate the second derivative of  the
functional \eqref{1.3}, with respect to $s$.
We have
$$
\frac{\partial E}{\partial s} =(q+1)
\int_\Omega\big(\langle Du_{\overline{f}}+sDv,
Dv\rangle -v|u_{\overline{f}}+sv|^q \overline{f}_t\big)dx
$$ 
and
\begin{equation}  \label{2.5}
\frac{\partial^2 E}{\partial s^2} (0,0) = (q+1)\int_\Omega\big(
|Dv|^2-qu_{\overline{f}}\ v^2|u_{\overline{f}}|^{q-2}\overline{f}\big)
dx.
\end{equation}
Before calculating the second derivative of $E$ with respect to $t$, we rewrite
\eqref{1.3} in the form
\begin{align*}
E(s,t)&=\int_{\Omega}\frac{q+1}{2} \
|Du_{\overline{f}}+sDv|^2dx-b_1\int_{D_1}(u_{\overline{f}}+sv)|u_{\overline{f}}+sv|^q
dx\\
&\quad -b_2\int_{D_2(t)}(u_{\overline{f}}+sv)|u_{\overline{f}}+sv|^q
dx-b_3\int_{D_3}(u_{\overline{f}}+sv)|u_{\overline{f}}+sv|^q
dx-\eta.
\end{align*}
We observe that, if $F:\mathbb{R}^2\to\mathbb{R}$ is a
continuous function, then
$$
\int_{D_2(t)} F-\int_{D_2} F = \int_J \int_0^{g(\xi,t)}
 F\big(\phi(\xi, g(\xi,\beta))\big)J(\xi, \beta) d\beta d\xi;
$$
whence, from the Fundamental Theorem of Calculus,
$$
\frac{\partial}{\partial t}  \int_{D_2(t)} F
=\int_J g_t(\xi,t) F\big(\phi(\xi, g(\xi,t))\big)J(\xi,g(\xi,t)) d\xi.
$$
Using the above relation with $F= (u+sv)\big|u+sv\big|^q$, we have
$$
\frac{\partial E}{\partial t}= -b_2\int_J g_t(\xi,t)
\big(u_{\overline{f}}+sv\big)
 \big|u_{\overline{f}}+sv\big|^q J(\xi, g(\xi,t))\,d\xi,
$$
where, for simplicity of notation, we set
$u_{\overline{f}}\big(\phi(\xi, g(\xi,t))\big)=u_{\overline{f}}$
and $v\big(\phi(\xi, g(\xi,t))\big)=v$. Moreover
\begin{align*}
\frac{\partial^2 E }{\partial t^2}
& = -b_2\int_J |u_{\overline{f}}+
sv|^q\Big\{\big[g_{tt}(\xi,t)(u_{\overline{f}}+sv)
 +(q+1)g^2_{t}(\xi,t)\langle Du_{\overline{f}}+sDv,\mathbf{N}\rangle
 \big]\\
&\quad\times J(\xi, g(\xi,t))
 +  g^2_{t}(\xi,t)(u_{\overline{f}}+sv)\langle\mathbf{N}',
 \mathbf{N}^{\bot}\rangle \Big\}d\xi,
\end{align*}
where we have used  that
\begin{gather*}
\frac{\partial }{\partial t}\ u_{\overline{f}}\big(\phi(\xi, g(\xi,t))\big)
= \langle Du_{\overline{f}}\big(\phi(\xi, g(\xi,t))\big), \mathbf{N}\rangle
 g_{t}(\xi,t),\\
 \frac{\partial }{\partial t} J(\xi,g(\xi,t))
=g_t\langle\mathbf{N}',\mathbf{N}^\bot\rangle .
\end{gather*}
We note that, when $t=0$,
$$
u_{\overline{f}}\big(\phi(\xi,g(\xi,t))\big)
=u_{\overline{f}}(\gamma(\xi))=c_2,
$$
$Du_{\overline{f}}\big(\phi(\xi,g(\xi,0))\big)
=-|Du_{\overline{f}}\big(\gamma (\xi)\big)| \mathbf{N}(\xi)$
and $J(\xi, g(\xi,0))=J(\xi, 0)=|\gamma'(\xi)|$. Evaluating the above 
expression in $(0,0)$, we find
\begin{align*}
\frac{\partial^2 E }{\partial t^2} (0,0)
&=-b_2 c_2^{q+1}\int_J \Big[g_{tt}(\xi,0)|\gamma'(\xi)|
+g^2_{t}(\xi,0)\langle\mathbf{N}',\mathbf{N}^{\bot}\rangle\Big]d\xi\\
 &\quad + b_2 c_2^q(q+1)\int_J
 g^2_{t}(\xi,0)|Du_{\overline{f}}(\gamma(\xi))| |\gamma'(\xi)|d\xi.
\end{align*}
By using \eqref{2.9} with $t=0$ we find
\begin{equation}\label{a7}
\begin{aligned}
 \frac{\partial^2 E }{\partial t^2} (0,0)
&= b_2 c_2^q(q+1) \int_J  g^2_{t}(\xi,0)|Du_{\overline{f}}(\gamma(\xi))|\,
 |\gamma'(\xi)|d\xi \\
&=  b_2 c_2^q(q+1)\int_\gamma g_{t}^2(\gamma^{-1},0)
 |Du_{\overline{f}}|\,d\sigma.
\end{aligned}
\end{equation}
We also have
$$
\frac{\partial^2 E }{\partial s\partial t}= -b_2(q+1)\int_J g_{t}(\xi,t)
v|u_{\overline{f}}+sv|^q J(\xi, g(\xi,t))\,d\xi;
$$
that is,
\begin{equation}\label{2.7}
\begin{aligned}
\frac{\partial^2 E }{\partial s\partial t} (0,0)
&= -b_2c_2^q(q+1)\int_J g_{t}(\xi,0)\ v\big(\gamma(\xi)\big)  |\gamma'
(\xi)|\,d\xi\\
&= -b_2c_2^q(q+1)\int_\gamma g_{t}(\gamma^{-1},0)\
v\,d\sigma .
\end{aligned}
\end{equation}
Using \eqref{1.4} in the form 
$$
\frac{\partial^2 E }{\partial s^2}(0,0)
\frac{\partial^2 E }{\partial t^2} (0,0)\geq
\Big( \frac{\partial^2 E }{\partial s \partial t} (0,0)\Big)^2,
$$
and using  \eqref{2.5}, \eqref{a7} and \eqref{2.7} in this inequality,
 we obtain \eqref{a8}.
\end{proof}

 Note that in  inequality \eqref{a8} only $g(\gamma^{-1},0)$ appears.
 Moreover, $g(\gamma^{-1},0)$ has null integral on $\gamma$.
Indeed, differentiating \eqref{2.3} 
with respect to $t$ and putting $t=0$, we obtain
$$
\int_J g(\xi,0)|\gamma'| d\xi=0.
$$
Now a natural question arises: does inequality \eqref{a8} hold for any 
function $h$ with null integral on $\gamma$? The answer is contained in
the following result.

\begin{lemma}\label{lem2} 
Let $J$ and $\gamma$ be the same as described. Let
$h:\gamma\to\mathbb{R}$ bounded, continuous and such that
$\int_\gamma h\,d\sigma=0$. Then, for all $ v\in H_0^1(\Omega)$ and
for all $a\in \mathbb{R}$ we have
\begin{equation} 
\int_\Omega\Big(
|Dv|^2-qu_{\overline{f}}\ v^2|u_{\overline{f}}|^{q-2}\overline{f}\Big)
dx\cdot\int_\gamma h^2|Du_{\overline{f}}|\,d\sigma \geq
b_2c_2^q\Big(\int_\gamma h(v-a)\,d\sigma\Big)^2.
\end{equation}
\end{lemma}


The proof of the above lemma is similar to that of \cite[Lemma 2.2]{CKT};
we omit it.
The following lemma is an analogue to \cite[Lemma 3.1]{CKT}.

\begin{lemma}\label{lem3} 
Let $P$ be a point on $\mathcal{F}=\partial \{u_{\overline{f}}>c_2\}$.
Suppose that for all $k\in \mathbb{Z}^+$ there exist a positive
number $r_k$, a bounded open interval $J_k$ and a regular curve
$\gamma_k:J_k\to \mathcal{F}^*$ such that
$r_1>r_2>\dots\to 0$, $\overline{\gamma_k(J_k)}\subset
\mathcal{F}^* \cap B_{r_k} (P) \setminus \overline{B_{r_{k+1}}(P)}$.
Then we must have 
$$
\sum_{k=1}^\infty \int_{\gamma(J_k)}
\frac{1}{|Du_{\overline{f}}|}\,d\sigma< \infty.
$$
 \end{lemma}

\begin{proof}
Without loss of generality, we assume that $P$ is the origin.
 We suppose also that $J_k\cap J_h=\emptyset$ for all $k\neq h$,
and denote all $\gamma_k$ with $\gamma$. We define
$$
J_{k,m} =\begin{cases} 
J_k  \cup J_{k+1}\cup\cdots\cup J_m &\text{if } m\geq k, \\
\emptyset  &\text{otherwise.}
\end{cases}
$$
We suppose by contradiction that
\begin{equation} \label{3.1}
\sum_{k=1}^\infty  \int_{\gamma(J_k)} \frac{1}{|Du_{\overline{f}}| }\,d\sigma
=\infty.
\end{equation} 
Let $V$ be a smooth radial function in
$\mathbb{R}^2$, decreasing in $|x|$, defined by
$$
\begin{cases} 
V(x)=2,  &|x| =0\\
1<V(x)<2,  &0<|x|< 1/2\\
0<V(x)<1,  &1/2<|x|< 1\\  
V(x)=0, &|x|\geq1.
\end{cases}
$$
For all $k\in \mathbb{Z}^+$ we define $v_k(x)=V(\frac{x}{r_k})$.
Consider $k$ large enough such that $\operatorname{supp}\,v_k\subset\Omega$. 
Now we fix $k$; we have
$$
\begin{cases}
 v_k(x)-1=1, &|x|=0\\
0<v_k(x)-1<1, &0<|x|< r_k/2\\
-1<v_k(x)-1<0, &r_k/2<|x|< r_k\\ 
 v_k(x)-1=-1, &|x|\geq r_k.
\end{cases}
$$
Since $J_k$ and $|\gamma'|$ are bounded, $\gamma(J_k)$ is of finite length.
Moreover, $|D u_{\overline{f}}|$ is uniformly bounded away from 0
on $\gamma(J_k)$ since $\overline{\gamma(J_k)}\subset\mathcal{F}^*$.
Together with the fact that $\gamma(J_{1,k-1})\subset (B_{r_k})^C $,
we have
$$ 
-\infty< \int_{\gamma(J_{1,k-1})} \frac{v_k-1}{|Du_{\overline{f}}| } \,d\sigma 
= -\int_{\gamma(J_{1,k-1})} \frac{1}{|Du_{\overline{f}}| }\,d\sigma
<0.
$$ 
Choose $m$ such that $r_m<r_k/2$.
From the facts that $v_k(x)-1>0$ in $B_{r_m}$,
$\gamma(J_l)\subset  B_{r_m}$ for all $l\geq m$ and $v_k(x)-1
\to 1$ as $x\to0$ and \eqref{3.1}, we have
$$  
\int_{\gamma(J_{m,l})} \frac{v_k -1}{|Du_{\overline{f}}|}\,d\sigma
\to\infty \ \ \ \text{for }l\to\infty.
$$
Consequently, there must be a number $l\geq m$ such that
$$  
\int_{\gamma(J_{m,l-1})} \frac{v_k-1}{|Du_{\overline{f}}| }\,d\sigma \leq
 -\int_{\gamma(J_{1,k-1})} \frac{v_k-1}{|Du_{\overline{f}}| } \,d\sigma <
\int_{\gamma(J_{m,l})} \frac{v_k-1}{|Du_{\overline{f}}| } \,d\sigma.
$$
Choose a subinterval $J_l'\subset J_l$  such that
$$
\int_{\gamma(J_{m,l-1})} \frac{v_k-1}{|Du_{\overline{f}}| } \, d\sigma\ +
\int_{\gamma(J_{l}')} \frac{v_k-1}{|Du_{\overline{f}}| } \,d\sigma=
-\int_{\gamma(J_{1,k-1})} \frac{v_k-1}{|Du_{\overline{f}}| }\,d\sigma.
$$
Then we have
$$
\int_{\gamma(J^k)} \frac{v_k-1}{|Du_{\overline{f}}| }\,d\sigma=0,
$$ 
where $J^k=J_{1,k-1}\cup  J_{m,l-1}\cup J_l'$.

Now we can apply Lemma \ref{lem2} to $J^k$, $\gamma$, $v_k$, $a=1$ and
$h=\frac{v_k -1}{|Du_{\overline{f}}|}\ $ and, after rearranging, obtain
$$
\int_\Omega\left(
|Dv_k|^2-qu_{\overline{f}}\ v_k^2|u_{\overline{f}}|^{q-2}\overline{f}\right)
dx\geq b_2c_2^q\int_{\gamma(J^k)}\frac{(v_k -1)^2}{|Du_{\overline{f}}|}
\,d\sigma.
$$
We find that
$$
\int_\Omega\Big(|Dv_k|^2-qu_{\overline{f}}\ v_k^2|u_{\overline{f}}|^{q-2}
\overline{f}\Big) dx\leq \int_{B_1(0)}|DV|^2dx.
$$
By the above estimate, for a suitable constant $C$, we have
\begin{align*}
C\int_{B_1(0)} |D V|^2 dx
&\geq\int_{\gamma(J^k)} \frac{(v_k -1)^2}{|Du_{\overline{f}}|}\,d\sigma \\
& \geq \int_{\gamma(J_{1,k-1})} \frac{(v_k -1)^2}{|Du_{\overline{f}}|}
 \,d\sigma \\
&= \sum_{h=1}^{k-1} \int_{\gamma(J_h)} 
\frac{(v_k -1)^2}{|Du_{\overline{f}}|}\, d\sigma.
\end{align*}
Then, when $k\to \infty$, we have
$$
C\int_{B_1(0)} |D V|^2 dx\geq\sum_{h=1}^{\infty} \int_{\gamma(J_h)} 
\frac{(v_k -1)^2}{|Du_{\overline{f}}|}\,d\sigma
=+\infty
,$$
which is a contradiction.
So we must have  
$$
\sum_{k=1}^\infty \int_{\gamma(J_k)}
\frac{d\sigma}{|Du_{\overline{f}}|}\ < \infty,
$$ 
as desired.
\end{proof}

\begin{lemma}\label{lem4} 
Let $P$ be a point on $\mathcal{F}=\partial \{u_{\overline{f}}>c_2\}$.
Suppose that there are numbers $K\in\mathbb{Z}$ and $\overline{\sigma}>0$  
such that, for each $k\geq K$,
 there exists a regular curve $\gamma_k: J_k\to \mathcal{F}^*$
 with the following two properties:
\begin{gather*}
\overline{\gamma(J_k)}\subset\mathcal{F}^*\cap B_{2^{-k}} (P) 
\setminus \overline{B_{2^{-(k+1)}}(P)},\\
\mathcal{H}^1 (\gamma_k(J_k))=\int_{J_k}|\gamma'(\xi)| d\xi
>\overline{\sigma}\,2^{-k}.
\end{gather*}
 Then $|Du_{\overline{f}}(P)|>0$.
  \end{lemma}

For a proof of the above lemma, see \cite[Lemma 3.2]{CKT}.
From an intuitive point of view, this lemma says that, if
the set $\partial \{u_{\overline{f}}>c_2\}\cap\{|Du_{\overline{f}}|>0\}$ is big
enough around a point of $\partial\{u_{\overline{f}}>c_2\} $, then
$|Du_{\overline{f}}|>0$ at this point.

Now, we are able to prove our main theorem.

\begin{proof}[Proof of Theorem \ref{thm6}]
By using the previous Lemmas and superharmonicity of $u_{\overline{f}}$ 
the Theorem follows from the results of sections 5 and 6 in \cite{CKT}.
\end{proof}

\subsection*{Open problems}
The method used in this paper to prove regularity does not work when 
the number of level sets of $\overline f$ is infinite. 
Therefore it remains to study the boundaries of level sets of
$\overline f$ in the case of the rearrangement class $\mathfrak{F}(f_0)$ 
of a general function $f_0$.

We can obtain an analogous result to Lemma \ref{lem3} for the $p$-Laplacian 
operator, but we cannot go further because we lack a suitable regularity 
theory for the $p$-Laplacian operator and its solutions. 
We think that it is reasonable to guess that a regularity result of the 
type that we have proven in this work will hold for the situation with 
the $p$-Laplacian when $p < 2$.


\subsection*{Acknowledgments}
 The authors want to thank the anonymous referees for their valuable 
comments and suggestions.


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\end{document}