\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 $$\label{a1} \begin{gathered} -\Delta u=f|u|^q \quad\text{in }\Omega,\\ u\big|_{\partial\Omega}=0, \end{gathered}$$ 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 $$\label{a2} \inf_{v\in H_0^{1}(\Omega),\, f\in\mathfrak{F}(f_0)} \int_\Omega \Big(\frac{q+1}{2} |Dv|^2-v|v|^q f\Big) dx,$$ where, denoting by $|A|$ the Lebesgue measure of a set $A$, $$\label{a5} \mathfrak{F}(f_0)=\{f\in L^\infty(\Omega): |\{f\geq c\}| =|\{f_0\geq c\}|\;\forall c\in\mathbb{R}\};$$ 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 $$E(f)=\inf_{v\in H_0^1(\Omega)} \int_\Omega\Big(\frac{q+1}{2} |Dv|^2-v|v|^q f\Big) dx.$$ 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 $$\label{a6} \overline{f}=\phi(u_{\overline{f}}).$$ 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, $00$. 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 0c_1\},\quad D_2=\{u_{\overline{f}}>c_2\},\quad \dots, \quad D_m=\{u_{\overline{f}}>c_m\}, $$for suitable constants c_1>c_2>\dots>c_m>0. We show regularity of \partial D_i for each i proving that |Du_{\overline{f}}|>0 in \partial D_i. Following the method used in \cite{CKT}, we consider $$\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,$$ where v\in H_0^{1}(\Omega), \overline{f}_t is a family of functions such that \overline{f}_t\in\mathfrak{F}(f_0) with \overline{f}_0=\overline{f}, and s\in \mathbb{R}. We have$$ E(s,t)\geq E(0,0)=0\quad \forall s, t. Therefore, (s,t)=(0,0) is a minimum point; it follows that \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. 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)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}, -\infty0\}. LetJ=\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 $$\label{2.1} D_2(t)=K\cup\{\phi(\xi,\beta): \xi \in J, \, \beta < g(\xi, t)\},$$ where$g: J\times (-t_0,t_0)\to\mathbb{R}$,$t_0>0$, is a function such that $$\label{c1} g(\xi, t),\; g_t(\xi, t),\; g_{tt}(\xi, t)\in C(\overline{J})\quad \forall t\in (-t_0,t_0)$$ and $$\label{2.2} g(\xi, 0)\equiv 0\quad \forall \xi\in J.$$ 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$g$: $$\label{2.3} \int_J\Big(g(\xi, t) |\gamma'| +\frac{1}{2}\ \big(g(\xi, t)\big)^2 \langle \mathbf{N},\mathbf{N}^\bot\rangle \Big) d\xi=0 \quad \forall t\in (-t_0,t_0).$$ Moreover, we calculate the derivatives of$A(t)$, that we will use later. $$\label{2.9} \begin{gathered} A'(t)=\int_J \big(g_t(\xi, t) |\gamma'(\xi)|+g(\xi, t)g_t(\xi, t) \langle \mathbf{N}',\mathbf{N}^\bot\rangle \big) d\xi=0; \\ A''(t)=\int_J \big(g_{tt}(\xi, t) |\gamma'(\xi)|+\big(g(\xi, t)g_{tt}(\xi, t)+g^2_t(\xi, t)\big) \langle \mathbf{N}',\mathbf{N}^\bot\rangle \big) d\xi=0. \end{gathered}$$ 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$v\in H_0^{1}(\Omega), the conditions \eqref{1.4} becomes \label{a8} \begin{aligned} &\int_\Omega\big( |Dv|^2-qu_{\overline{f}}\ v^2|u_{\overline{f}}|^{q-2}\overline{f}\big) dx\cdot\int_\gamma g_{t}^2(\gamma^{-1},0) |Du_{\overline{f}}| d\sigma \\ &\geq b_2c_2^q\Big(\int_\gamma g_{t}(\gamma^{-1},0)\ v d\sigma\Big)^2. \end{aligned} \end{lemma} \begin{proof} We calculate the second derivative of the functional \eqref{1.3}, with respect tos$. 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 $$\label{2.5} \frac{\partial^2 E}{\partial s^2} (0,0) = (q+1)\int_\Omega\big( |Dv|^2-qu_{\overline{f}}\ v^2|u_{\overline{f}}|^{q-2}\overline{f}\big) dx.$$ Before calculating the second derivative of$E$with respect to$t, we rewrite \eqref{1.3} in the form \begin{align*} E(s,t)&=\int_{\Omega}\frac{q+1}{2} \ |Du_{\overline{f}}+sDv|^2dx-b_1\int_{D_1}(u_{\overline{f}}+sv)|u_{\overline{f}}+sv|^q dx\\ &\quad -b_2\int_{D_2(t)}(u_{\overline{f}}+sv)|u_{\overline{f}}+sv|^q dx-b_3\int_{D_3}(u_{\overline{f}}+sv)|u_{\overline{f}}+sv|^q dx-\eta. \end{align*} We observe that, ifF:\mathbb{R}^2\to\mathbb{R}$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$v\big(\phi(\xi, g(\xi,t))\big)=v. Moreover \begin{align*} \frac{\partial^2 E }{\partial t^2} & = -b_2\int_J |u_{\overline{f}}+ sv|^q\Big\{\big[g_{tt}(\xi,t)(u_{\overline{f}}+sv) +(q+1)g^2_{t}(\xi,t)\langle Du_{\overline{f}}+sDv,\mathbf{N}\rangle \big]\\ &\quad\times J(\xi, g(\xi,t)) + g^2_{t}(\xi,t)(u_{\overline{f}}+sv)\langle\mathbf{N}', \mathbf{N}^{\bot}\rangle \Big\}d\xi, \end{align*} where we have used that \begin{gather*} \frac{\partial }{\partial t}\ u_{\overline{f}}\big(\phi(\xi, g(\xi,t))\big) = \langle Du_{\overline{f}}\big(\phi(\xi, g(\xi,t))\big), \mathbf{N}\rangle g_{t}(\xi,t),\\ \frac{\partial }{\partial t} J(\xi,g(\xi,t)) =g_t\langle\mathbf{N}',\mathbf{N}^\bot\rangle . \end{gather*} We note that, whent=0$, $$u_{\overline{f}}\big(\phi(\xi,g(\xi,t))\big) =u_{\overline{f}}(\gamma(\xi))=c_2,$$$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} witht=0we find \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} 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, \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} 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} onlyg(\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 $$\int_\Omega\Big( |Dv|^2-qu_{\overline{f}}\ v^2|u_{\overline{f}}|^{q-2}\overline{f}\Big) dx\cdot\int_\gamma h^2|Du_{\overline{f}}|\,d\sigma \geq b_2c_2^q\Big(\int_\gamma h(v-a)\,d\sigma\Big)^2.$$ \end{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 $$\label{3.1} \sum_{k=1}^\infty \int_{\gamma(J_k)} \frac{1}{|Du_{\overline{f}}| }\,d\sigma =\infty.$$ Let$V$be a smooth radial function in$\mathbb{R}^2$, decreasing in$|x|, defined by $$\begin{cases} V(x)=2, &|x| =0\\ 10 in B_{r_m}, \gamma(J_l)\subset B_{r_m} for all l\geq m and v_k(x)-1 \to 1 as x\to0 and \eqref{3.1}, we have$$ \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. 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