\magnification = \magstephalf \hsize=14truecm \hoffset=1truecm \parskip=5pt \nopagenumbers \input amssym.def % The R for Real nunbers. \font\eightrm=cmr8 \font\eighti=cmti8 \font\eightbf=cmbx8 \headline={\ifnum\pageno=1 \hfill\else% {\tenrm\ifodd\pageno\rightheadline \else \leftheadline\fi}\fi} \def\rightheadline{EJDE--1998/34\hfil Symmetry and convexity of level sets \hfil\folio} \def\leftheadline{\folio\hfil Edi Rosset \hfil EJDE--1998/34} \voffset=2\baselineskip \vbox {\eightrm\noindent\baselineskip 9pt % Electronic Journal of Differential Equations, Vol. {\eightbf 1998}(1998) No.~34, pp. 1--12.\hfill\break ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu \hfil\break ftp 147.26.103.110 or 129.120.3.113 (login: ftp)} \footnote{}{\vbox{\hsize=10cm\eightrm\noindent\baselineskip 9pt % 1991 {\eighti Subject Classification:} 35J70, 35B05. \hfil\break {\eighti Key words and phrases:} $\infty$-Laplace equation, $p$-Laplace equation. \hfil\break \copyright 1998 Southwest Texas State University and University of North Texas.\hfil\break Submitted July 23, 1998. Published December 9, 1998.\hfil\break Partially supported by Fondi MURST.} } \bigskip\bigskip \centerline{SYMMETRY AND CONVEXITY OF LEVEL SETS OF SOLUTIONS} \centerline{TO THE INFINITY LAPLACE'S EQUATION} \medskip \centerline{Edi Rosset} \bigskip\bigskip {\eightrm\baselineskip=10pt \narrower \centerline{\eightbf Abstract} We consider the Dirichlet problem $$\displaylines{ -\Delta_\infty u=f(u) \quad \hbox{in }\Omega\,,\cr u=0\quad \hbox{on }\partial\Omega\,,} $$ where $\Delta_\infty u=u_{x_i}u_{x_j}u_{x_ix_j}$ and $f$ is a nonnegative continuous function. We investigate whether the solutions to this equation inherit geometrical properties from the domain $\Omega$. We obtain results concerning convexity of level sets and symmetry of solutions. \bigskip} \bigbreak \centerline{\bf 1. Introduction} \medskip\nobreak Given a bounded domain $\Omega\subset{\Bbb R}^n$, we consider the following Dirichlet problem for the $\infty$-Laplace operator $$ \displaylines{ \hfill -\Delta_\infty u=f(u) \quad \hbox{in }\Omega\,,\hfill \llap{($D_\infty$)}\cr u=0 \quad \hbox{on }\partial\Omega\,,\cr} $$ where $\Delta_\infty u=u_{x_i}u_{x_j}u_{x_ix_j}$ and $f$ is a nonnegative continuous function. We investigate whether the solutions to $(D_\infty)$ inherit geometrical properties from the domain $\Omega$. By a solution to $(D_\infty)$ we will mean a variational solution in a sense which extends that given in [B-D-M], that is, roughly speaking, a function which is the limit of a sequence of solutions to the Dirichlet problems for the $p$-Laplace operator $$ \displaylines{ \hfill -\Delta_p u=f(u) \quad \hbox{in }\Omega\,,\hfill \llap{($D_p$)}\cr u=0 \quad \hbox{on }\partial\Omega\,,\cr} $$ as $p\rightarrow\infty$ (see Definition 2.1 below). When $\Omega$ is a convex domain, we prove that the restriction of any solution $u$ of $(D_\infty)$ to the convex ring $\Omega\setminus\Omega_{s_M}$, where $\Omega_{s_M}=\{x\in\Omega : d(x,\partial\Omega)>s_M\}$, has convex level sets, preserves the symmetries of $\Omega$, and is uniquely determined (see Theorem 2.5 and Corollary 2.6). Here, the number $s_M$ is determined by $f$ and the maximum $M$ of $u$ in $\Omega$ only. If, for instance, $f$ is strictly positive at $M$, then $\Omega_{s_M}=\emptyset$. Notice that by symmetry, we mean not only a reflection but any orthogonal transformation. When $\Omega$ is a ball $B_R$, any solution to $(D_\infty)$ is radially symmetric, has a very simple representation, and coincides with the distance function from $\partial \Omega$ in the annulus $\{s_M<|x|2$ and $f\in C^1$ (see [Br-H]). For $p>2$ and $f$ changing sign, Brock has established a partial form of symmetry, the so-called {\it local symmetry in every direction}, and symmetry results under some growth conditions on $f$ in neighborhoods of its zero points, via continuous Steiner symmetrization (see [Br1], [Br2]). The incompleteness of the result of Theorem 2.5 is due to the fact that a variational solution $u$ to $(D_\infty)$ may be sensitive to the behaviour of $f$ outside its range, through the influence of $f$ on the sequence of solutions $u_{p_k}$ to $(D_{p_k})$ converging to $u$. In Section 3 we provide an Example which illustrates this phenomenon. In Section 4 we propose an alternative definition of solution which we have called a {\it tame variational solution} (see Definition 4.1), which prevents the occurrence of the ``improper'' solutions which may be introduced by the limit process described above. We show that any {\it tame variational solution} $u$ has convex level sets, preserves the symmetries of the convex domain $\Omega$ and, when $\Omega=B_R$, then either $u=U$ or $u$ is a truncation of $U$, where $U(x)=R-|x|$ (Theorem 4.3 and Theorem 4.4). \bigbreak \centerline{\bf 2. Statements and proofs}\medskip\nobreak Let us recall some facts about the case $f=f(x)$, which stem from results in [B-D-M] and [J]. Given a bounded domain $\Omega\subset{\Bbb R}^n$ and a bounded nonnegative continuous function $f$ defined in $\Omega$, $f\not\equiv 0$, let $u_p\in W^{1,p}_0$ be the unique weak solution to $$ \displaylines{ \hfill -\Delta_p u=f \quad \hbox{in }\Omega\,,\hfill\llap{(2.1)}\cr u=0 \quad \hbox{on }\partial\Omega\,.\cr} $$ Then there exists a unique function $u_\infty\in W^{1,\infty} (\Omega)\cap C_0(\bar \Omega)$ such that $$ u_p\rightarrow u_\infty\quad\hbox{weakly in } W^{1,m}(\Omega), \forall m>1,\hbox{ and uniformly in } \bar\Omega\,. $$ The function $u_\infty$ obtained by this limit process is called a {\it variational solution\/} to $$ \displaylines{ \hfill -\Delta_\infty u=f \quad \hbox{in }\Omega\,,\hfill\llap{(2.2)}\cr u=0 \quad \hbox{on }\partial\Omega\,,\cr} $$ and is characterized by the following two conditions: \item{$i)$} The function $u_\infty$ solves the maximum problem $$ J_\infty(u_\infty)=\max_{\cal K} J_\infty,\eqno(P_\infty) $$ where $J_\infty(\varphi)=\int_\Omega f\varphi$, and $$ {\cal K}=\{\varphi\in W^{1,\infty}(\Omega)\cap C_0(\bar\Omega) \,:\,\|\nabla \varphi\|_\infty=1\} $$ \item{$ii)$} The function $u_\infty$ is a viscosity solution to $$\Delta_\infty u=0,\quad\hbox{in the interior of } \{f=0\}.\eqno(2.3)$$ Next let us consider the case $f=f(u)$. Let $f:{\Bbb R}\rightarrow{\Bbb R}$ be a continuous nonnegative function such that the Dirichlet problem $(D_p)$ is solvable in $W^{1,p}(\Omega)$ for $p$ large enough, say $p\geq\bar p$. Let $u_p$ be a solution to $(D_p)$, for $p\geq\bar p$. Let us assume that $f$ is bounded or, more generally, that $f(u)=O(u^s)$ as $u\rightarrow\infty$, for some $s>0$. From the weak formulation of $(D_p)$ and the H\"older and Poincar\'e inequalities, it follows easily that $\|\nabla u_p\|_m$ is bounded uniformly in $p$, for any $m>1$. Therefore, one can construct a sequence $p_k\rightarrow\infty$, such that $$ u_{p_k}\rightarrow u,\quad\hbox{weakly in } W^{1,m}(\Omega), \forall m>1,\hbox{ and uniformly in } \bar\Omega,\eqno(2.4) $$ for some $u\in W^{1,\infty}(\Omega)\cap C_0(\bar \Omega)$. In view of the above arguments, we give the following definition.\smallskip \noindent {\bf Definition 2.1.} A function $u\in W^{1,\infty}(\Omega) \cap C_0(\bar\Omega)$ is called a {\it variational solution} to $(D_\infty)$ if there exists a sequence $u_{p_k}$ of solutions to $(D_{p_k})$, with $p_k\rightarrow\infty$, such that $(2.4)$ holds. \smallskip Let us notice that if $u$ is a variational solution to $(D_\infty)$, then $\|u_{p_k}\|_\infty$ is uniformly bounded, so that, by the continuity of $f$, there exists a positive constant $K$ such that $\|f(u_{p_k})\|_\infty\leq K$. Therefore, by the H\"older and Poincar\'e inequalities, we have $$\|\nabla u_{p_k}\|_m\leq C^{1/({p_k}-1)}K^{1/({p_k}-1)}|\Omega|^{ {1 \over m}+{1 \over n({p_k}-1)}}\eqno(2.5)$$ and $$\|\nabla u\|_\infty=\lim_{m\rightarrow\infty}\|\nabla u\|_m \leq\lim_{m\rightarrow\infty}\left(\liminf_{k\rightarrow\infty} \|\nabla u_{p_k}\|_m\right)=\lim_{m\rightarrow\infty} |\Omega|^{1/m}=1. \eqno(2.6)$$ Since $f\geq 0$, we have $u_p\geq 0$ and therefore $u\geq 0$. From $(2.6)$ and from $u|_{\partial\Omega}\equiv 0$ it follows that $u$ is Lipschitz continuous with Lipschitz constant $L\leq 1$, and $u(x)\leq d(x,\partial\Omega)$. Summarizing, we have $$ \displaylines{\hfill \|\nabla u\|_\infty\leq1\,,\hfill\llap{(2.7)} \cr \hfill 0\leq u\leq U\,,\hfill\llap{(2.8)}\cr } $$ where $$U(x)=d(x,\partial\Omega).\eqno(2.9)$$ Given a variational solution $u$ to $(D_\infty)$, $u=\lim_{k\rightarrow\infty}u_{p_k}$, let us define $$ \displaylines{ \hfill E_p=\int_\Omega |\nabla u_p|^p=\int_\Omega(f\circ u_p)u_p\,, \hfill \llap{(2.10)} \cr \hfill E_\infty=\int_\Omega(f\circ u)u=\lim_{k\rightarrow\infty}E_{p_k}\,, \hfill \llap{(2.11)}\cr \hfill f^*=f\circ u\,,\hfill\llap{(2.12)}\cr \hfill \Omega_0^*=\{x\in\Omega : u(x)\in int{\{f=0\}}\}\,. \hfill\llap{(2.13)} \cr} $$ \proclaim Lemma 2.2. Let $u$ be a variational solution to $(D_\infty)$. If $f^*\not\equiv 0$ then $u\not\equiv 0$ and $E_\infty>0$. \par \noindent{\bf Proof.} Let us see that $u\equiv 0$ implies $f^*\equiv 0$. If $u\equiv 0$, then there are two cases: either $f(0)=0$ or $f(0)>0$. In the former case $f^*\equiv 0$, whereas in the latter case, by the continuity of $f$, we have $f(u_{p_k})\geq \delta$ for $k\geq\bar k$, for some $\bar k\in{\Bbb N}$, $\delta>0$. Let $v_p$ be the solution to $$ \displaylines{ -\Delta_p v_p=\delta \quad \hbox{in }\Omega\,,\cr v_p=0 \quad \hbox{on }\partial\Omega\,.\cr} $$ By the comparison principle for the $p$-Laplace operator (see [T]), we have $u_{p_k}\geq v_{p_k}$. Moreover from $i)$ it follows easily that $v_p\rightarrow v_\infty=U$ (see [B-D-M]), so that $u\geq U$, contradicting $u\equiv 0$. Let $f^*\not\equiv 0$, so that $u\not\equiv 0$. Let us assume, by contradiction, that $0=E_\infty=\int_{\{f^*>0\}}f^*u$. Since $u\geq 0$, we have $u\equiv 0$ in $\{f^*>0\}$, that is: $f(u(x))>0$ implies $u(x)=0$. Therefore, denoting $M=\max_\Omega u$, we have $f(t)=0$ for every $t\in(0,M]$. From the continuity of $u$ it follows that $f(0)=0$, that is $f^*\equiv 0$, contradicting the hypothesis. \hfill$\diamondsuit$ \proclaim Proposition 2.3. Let $u$ be a variational solution to $(D_\infty)$ such that $f^*\not\equiv 0$. Then, \item{$i^*)$} the function $u$ solves the maximum problem $$J^*_\infty(u)=\max_{\cal K} J^*_\infty,\eqno(P^*_\infty)$$ where $J^*_\infty(\varphi)=\int_\Omega f^*\varphi$ and $${\cal K}=\{\varphi\in W^{1,\infty}(\Omega)\cap C_0(\bar\Omega) : \|\nabla \varphi\|_\infty=1\}\,$$ and \item{$ii^*)$} the function $u$ is a viscosity solution of $$\Delta_\infty u=0\quad \hbox{in } \Omega_0^*\,.\eqno(2.14)$$ \par \noindent{\bf Proof.} From the definition of weak solution to $(D_p)$ and from H\"older inequality, we have $$\int_\Omega(f\circ u_p)\varphi= \int_\Omega |\nabla u_p|^{p-2}\nabla u_p\cdot \nabla\varphi \leq E_p^{(p-1)/p}\|\nabla\varphi\|_p,$$ for any $\varphi\in W^{1,p}_0(\Omega)$. Hence, for any $\varphi\in W^{1,\infty}(\Omega)\cap C_0(\bar\Omega)$, $\varphi\not\equiv 0$, we have $${\int_\Omega(f\circ u)\varphi \over \|\nabla\varphi\|_\infty} =\lim_{k\rightarrow\infty} {\int_\Omega(f\circ u_{p_k})\varphi \over \|\nabla\varphi\|_{p_k}} \leq \lim_{k\rightarrow\infty} E_{p_k}^{(p_k-1)/p_k}=E_\infty=J^*_\infty(u).\eqno(2.15)$$ Substituting $\varphi=u$ in the above inequality and noting that $E_\infty>0$ by Lemma 2.2, we have $\|\nabla u\|_\infty\geq 1$. From $(2.7)$ it follows that $\|\nabla u\|_\infty=1$, that is, $u\in \cal K$, and $i^*)$ follows immediately from $(2.15)$. In order to verify $ii^*)$, let us consider any $x\in\Omega_0^*$. Since $u_{p_k}$ converges uniformly to $u$, there exist a neighborhood $V$ of $x$ and an index $\bar k$ such that $f\circ u_{p_k}\equiv 0$ in $V$ for every $k\geq\bar k$. For any $p>1$, let $v_p$ be the unique solution to $$\displaylines{ \Delta_p v_p=0 \quad \hbox{in } V\,,\cr v_p=u \quad \hbox{on }\partial V\,.\cr} $$ It is well known (see [J]) that $v_p$ converges uniformly to the unique viscosity solution $v_\infty$ of $$\displaylines{ \Delta_\infty v_\infty=0\quad \hbox{in }V\,,\cr v_\infty=u \quad \hbox{on }\partial V\,.\cr} $$ On the other hand, applying the comparison principle for the $p$-Laplace operator (see [T]) to the functions $u_{p_k}$, $v_{p_k}$ in $V$, we have that $\lim_{k\rightarrow\infty} \max_V |u_{p_k}-v_{p_k}|=0$, so that $u_\infty=v_\infty$, and $ii^*)$ follows. \hfill$\diamondsuit$ \proclaim Corollary 2.4. In the hypotheses of Proposition 2.3, we have $$u(x)=U(x), \quad\forall x\in \overline{\{f^*>0\}}.\eqno(2.16)$$ \par \noindent{\bf Proof.} Substituting $U\in \cal K$ in $(P^*_\infty)$, we have $$\int_{\{f^*>0\}}(u-U)f^*\geq 0,$$ so that $(2.16)$ follows from $(2.8)$. \hfill$\diamondsuit$ \smallskip Let us introduce the following notation: $$\displaylines{ \Omega_t=\{x\in\Omega : d(x,\partial\Omega)>t\}=\{U>t\},\cr \Omega_{r,s}=\{x\in\Omega : r0\}\cap(0,M)\right)\,.\hfill \llap{(2.18)}\cr} $$ Then $00$, \item{$\beta$)}$f(M)=0$, $s_M=M$, \item{$\gamma$)} $f(M)=0$, $s_Mt\}$ are convex for every $t\in[0,s_M)$; case $\delta$) cannot occur. If, moreover, $\Omega$ is invariant with respect to an orthogonal transformation $T$, then if either $\alpha)$ or $\beta)$ occurs, then $u$ is symmetric with respect to $T$; if $\gamma$) occurs, then $u_{|\Omega\setminus\Omega_{s_M}}$ is symmetric with respect to $T$. \par \noindent{\bf Proof.} Let $x_0\in\Omega$ be a point where $u$ attains its maximum $M$, and let $s_M$ be as defined in $(2.18)$. Let $(c_i,d_i)$, $i\in I_M$, be the connected components of $\{f>0\}\cap(0,M)$. For any half line $r$ having origin at $x_0$, let us denote $S_r=r\cap \bar \Omega$. We have $u(S_r)=[0,M]$. From the convexity of $\Omega$, it follows easily that for every $d$, $0\leq d0\}$ there exists a unique $x\in S_r$ such that $u(x)=l=U(x)$. Since this fact holds for any half line $r$ having origin at $x_0$, we have $U(x_0)\geq s_M$ and $u=U$ in $\Omega_{c_i,d_i}$ for every $i\in I_M$. The connected components of $\Omega_{0,s_M}\setminus \cup_{i\in I_M}\overline{ \Omega_{c_i, d_i}}$ are convex rings $A_j=\Omega_{a_j, b_j}$, $j\in J_M$, where $a_ja_j$ in $A_j$, let us introduce, for any $p>1$, the unique solution $v_p$ to $$\displaylines{ \Delta_p v_p=0 \quad \hbox{in } A_j\,,\cr v_p=u_p \quad \hbox{on }\partial A_j\,,\cr} $$ and the unique solution $w_p$ to $$ \displaylines{ \Delta_p w_p=0 \quad \hbox{in } A_j\,,\cr w_p=U \quad \hbox{on } \partial A_j\,.\cr} $$ From the fact that $u_{p_k}\rightarrow u=U$ on $\partial A_j$ and from the comparison principle for the $p$-Laplace operator (see [T]), we see that for any $\epsilon>0$ there exists $k_\epsilon$ such that $$u_{p_k}\geq v_{p_k}\geq w_{p_k}-\epsilon,\qquad\hbox{in }A_j\eqno(2.19)$$ for $k\geq k_\epsilon$. Moreover, $w_p$ converges uniformly to the unique viscosity solution $w_\infty$ to $$ \displaylines{ \Delta_\infty w_\infty=0 \quad \hbox{in } A_j\,,\cr w_\infty=U \quad \hbox{on }\partial A_j\,,\cr} $$ (see [J]), so that the Harnack inequality for the $\infty$-Laplace operator (see [L-M]) implies that $w_\infty(A_j)\subset(a_j,b_j)$. Passing to the limit as $k\rightarrow\infty$ in $(2.19)$, we have $u\geq w_\infty>a_j$ in $A_j$. Let us distinguish two cases: $f(M)>0$ and $f(M)=0$. In the former case we have $f^*(x_0)>0$, so that there exists a neighborhood $V$ of $x_0$ where $f^*$ is positive, and, by $(2.16)$, $u=U$ in $V$. Hence $x_0$ has to be a point of local maximum for $U$, and, since $\Omega$ is convex, $x_0$ is a point of absolute maximum for $U$. Indeed, otherwise, let $w$ be a point of absolute maximum for $U$ and let us consider the segment $L$ joining $x_0$ and $w$. By the convexity of $\Omega$, we have that $U(z)>U(x_0)$, for any point $z\in L$, $z\neq x_0$, contradicting that $x_0$ is a point of local maximum for $U$. Hence $U(x_0)=R$, where $$R=\max_{x\in\Omega}d(x,\partial\Omega)\eqno(2.20)$$ is the radius of the largest ball contained in $\Omega$, and $s_M=M=u(x_0)=U(x_0)=R$. Moreover $u=U$ in $\Omega_{c_i,d_i}$ for every $i\in I_M$ and $\Omega_0^*=\cup_{j\in J_M}A_j$. If $f(M)=0$, then $M\leq R$ by $(2.8)$, $\cup_{j\in J_M}A_j\subset\Omega_0^*$, and, by the continuity of $u$ and by $(2.18)$, $u\equiv s_M$ on $\partial\Omega_{s_M}$ and $u\geq s_M$ in $\Omega_{s_M}$. From the convexity of $\Omega$ it follows that $\Omega_t$ is convex for every $t\in {\Bbb R}$. Collecting the previous results, we have: $ub_j$ in $\Omega_{b_j}$ for every $j\in J_M$; $ud_i$ in $\Omega_{d_i}$ for every $i\in I_M$ such that $d_i\neq s_M$; $u\geq s_M$ in $\Omega_{s_M}$. It follows easily that $u>0$ in $\Omega$, so that the level set $\{u>0\}=\Omega$ is convex, and that if $t\in(0,s_M)\cap\overline{\{f>0\}}$, then $\{u>t\}=\Omega_t$ is convex. If $ t\in(0,s_M)\setminus\overline{\{f>0\}}$, then $t\in(a_j,b_j)$ for some $j\in J_M$, and, by $ii^*)$, $u$ is the viscosity solution in the convex ring $A_j$ of the capacitary problem $$ \displaylines{ \Delta_\infty u=0 \quad \hbox{in }A_j\,,\cr u=a_j\quad \hbox{on }\{U=a_j\}\,,\cr u=b_j \quad \hbox{on } \{U=b_j\}\cr} $$ for the $\infty$-Laplace operator. From the previous results, $\{u>t\}=\overline{\Omega_{b_j}}\cup \{x\in A_j : u(x)>t\}$, which is convex since $u|_{\overline{ A_j}}$ can be obtained as the uniform limit, as $p\rightarrow\infty$, of the solutions $u_p$ to the $p$-capacitary problem (see [J]) $$ \displaylines{ \Delta_p u=0 \quad \hbox{in } A_j\,,\cr u=a_j\quad \hbox{on }\{U=a_j\}\,,\cr u=b_j\quad \hbox{on }\{U=b_j\},\cr} $$ for which Lewis ([L]) established convexity of level sets. If $\Omega$ is invariant with respect to an orthogonal transformation $T$, the function $U$ and the sets $\Omega_t$ are invariant with respect to $T$. If $v$ is a viscosity solution to $$ \displaylines{ \Delta_\infty v=0 \quad \hbox{in }\Omega_{r,s},\cr \hfill v=r\quad \hbox{on } \{U=r\}, \hfill\llap{(2.21)}\cr v=s\quad \hbox{on } \{U=s\},\cr} $$ then also $v\circ T$ solves $(2.21)$ since the $\infty$-Laplace operator is invariant under orthogonal transformations. By the uniqueness of the viscosity solution to $(2.21)$, established by Jensen ([J]), it follows that $v\circ T=v$. Hence $u_{|\Omega\setminus\Omega_{s_M}}$ is invariant with respect to $T$. If $\alpha)$ occurs, then $s_M=M=R$, so that $\Omega_{s_M}=\emptyset$, and convexity of all the level sets and symmetry of $u$ with respect to $T$ follow. If $\beta)$ occurs, then $u\equiv s_M$ in $\Omega_{s_M}$, and again convexity of all the level sets and symmetry of $u$ with respect to $T$ follow. Let us assume that $f(M)=0$ and $s_Ms_M$ in $\Omega_{s_M}\setminus \bar V$. In the former case, we have a contradiction with $w_p=u>s_M$ on $\partial V$, whereas in the latter case we have $u>s_M$ in all of $\Omega_{s_M}$. Finally, if case $\delta$) occurs, we have that $\Omega_{s_M}\subset\Omega_0^*$, and by $ii^*)$, $\Delta_\infty u=0$ in $\Omega_{s_M}$. Hence $u\equiv s_M$ in $\Omega_{s_M}$, contradicting $s_M0\}}$. Moreover, the values of $u$ are uniquely determined in $\Omega\setminus \Omega_{s_M}$. More precisely, $$u=U\qquad\hbox{in } \cup_{i\in I_M}\overline{\Omega_{c_i, d_i}},\eqno(2.22)$$ and $u$ is the viscosity solution to $$ \displaylines{ \Delta_\infty u=0 \quad \hbox{in } A_j\,,\cr \hfill u=a_j \quad \hbox{on } \{U=a_j\},\hfill \llap{(2.23)}\cr u=b_j\quad \hbox{on } \{U=b_j\},\cr} $$ where $(c_i,d_i)$, $i\in I_M$, are the connected components of $\{f>0\}\cap(0,M)$, $A_j=\Omega_{a_j, b_j}$, $j\in J_M$, and where $(a_j,b_j)$ are the connected components of $int(\{f=0\})\cap(0,s_M)$. Moreover, if $\beta)$ occurs, then $u\equiv s_M$ in $\Omega_{s_M}$. \par \noindent {\bf Remark.} Let us notice that if $s_M=M$, that is, if either $\alpha)$ or $\beta)$ occurs, then $u$ is determined in all of $\Omega$. \proclaim Theorem 2.7 (Spherical symmetry and representation of the solutions when $\Omega=B_R$). Let $\Omega=B_R$ and let $u\in W^{1,\infty}(B_R) \cap C_0(\bar B_R)$ be a variational solution to $(D_\infty)$ such that $f^*\not\equiv 0$. Then $u$ is radially symmetric and radially non-increasing. Furthermore, $M\in\overline{\{f>0\}}$, and case $\delta$) cannot occur. If $\alpha)$ occurs, then $M=R$ and $u=U$. If $\beta)$ occurs, then $$ u(x)=\cases{ U(x)\equiv R-|x| & if $R-s_M\leq |x|\leq R$,\cr s_M=M & if $|x|\leq R-s_M$.\cr} \eqno(2.24) $$ If $\gamma$) occurs, then $$ u(x)=\cases{ U(x)\equiv R-|x| & if $R-s_M\leq |x|\leq R$,\cr \lambda(R-s_M-|x|)+s_M & if $R-s_M-{M-s_M \over \lambda}\leq|x|\leq R-s_M$,\cr M & if $|x|\leq R-s_M-{M-s_M \over \lambda}$,\cr} \eqno(2.25) $$ for some $\lambda\in[{M-s_M \over R-s_M},1]$. Here $U(x)=d(x,\partial\Omega)=R-|x|$, as defined in $(2.9)$. \par \noindent{\bf Proof.} In view of Theorem 2.5, it only remains to prove that $u=U$ in $A_j$ for every $j\in J_M$ and that, if $\gamma$) occurs, then $(2.25)$ holds for some $\lambda\in[{M-s_M \over R-s_M},1]$. Since $A_j=\{R-b_j<|x|s_M$ in $\Omega_{s_M}$, the representation $(2.25)$ follows immediately. \hfill$\diamondsuit$ \bigbreak \centerline{\bf 3. An Example}\medskip\nobreak Let us show, by the following Example, that, when case $\gamma$) occurs, there may be nontrivial variational solutions to $(D_\infty)$ such that $f^*\equiv 0$. \smallskip \noindent {\bf Example.} Let $\Omega=B_R$, $f(t)=(t-M)\chi_{(M,\infty)}$, with $0M$, for every $p>2$. Let $r_p\in(0,R)$ be such that $u_p(r_p)=M$. Then $$ u_p=\cases{u_p^- & in $r_p<|x|M \quad \hbox{in } B_{r_p}\,,\cr u_p^+=M \quad \hbox{on } |x|=r_p\,,\cr} $$ and the following transmission condition holds $$u_p^-,_r(r_p)=u_p^+,_r(r_p).\eqno(3.1)$$ An easy calculation gives $$u_p^-=M\left({R^{p-n \over p-1}-r^{p-n \over p-1} \over R^{p-n \over p-1}-r_p^{p-n \over p-1}}\right).$$ Let $w_p=-(u_p^+-M)$. Then $w_p$ is a negative radial solution to $$ \displaylines{ -\Delta_p w_p=w_p \quad \hbox{in }B_{r_p}\,,\cr w_p=0 \quad \hbox{on } |x|=r_p\,,\cr} $$ or, equivalently, $$\displaylines{ \hfill (w_p,_r)^{p-1},_r+{n-1 \over r}(w_p,_r)^{p-1} +w_p=0\,,\hfill \llap{(3.2)}\cr \hfill w_p(r_p)=0\,,\hfill \llap{(3.3)} \cr \hfill w_p,_r(0)=0\,. \hfill \llap{(3.4)}\cr} $$ From now on, let $n=1$, so that the second term in $(3.2)$ disappears and $(3.2)$ is an autonomous nonlinear equation. Substituting $w_p,_r=y$, thinking of $y$ as a function of $w$, integrating $(3.2)$, and imposing $(3.4)$, we have $$w_p,_r=\left({p \over p-1}\right)^{1/p} \left({c_p^2-w_p^2 \over 2}\right)^{1/p},\eqno(3.5)$$ where $c_p=-w_p(0)=M_p-M$. By integrating over $(0,r)$ and changing variable, we have $$ \left({p \over 2(p-1)}\right)^{1/p}r= \int_0^r{w_p,_r \over (c_p^2-w_p^2)^{1/p}}dr= \int_{-c_p}^{w_p(r)}{dw \over (c_p^2-w^2)^{1/p}}. $$ By imposing the transmission conditions $(3.1)$ and $(3.3)$, we easily have $$\displaylines{ \hfill \left({p \over p-1}\right){c_p^2 \over 2}= \left({M \over R-r_p}\right)^p\,,\hfill \llap{(3.6)}\cr \hfill \left({p \over 2(p-1)}\right)^{1/p}r_p= c_p^{p-2 \over p}\int_0^1 {dz \over (1-z^2)^{1/p}}\,. \hfill \llap{(3.7)} \cr} $$ Solving $(3.6)$ in $c_p$ and substituting in $(3.7)$, we are led to find $r_p\in(0,R)$ satisfying the equation $$g_p(x)=\gamma_p,\eqno(3.8)$$ where $$\displaylines{ g_p(x)=x^{1 / p}(R-x)^{p-2 \over 2p}\,,\cr \gamma_p=\left({2(p-1) \over p}\right)^{1/(2p)} M^{p-2 \over 2p}\left(\int_0^1 {dz \over (1-z^2)^{1/p}} \right)^{1/p}.\cr} $$ We have $g_p(0)=g_p(R)=0$, $g_p'(x)={1 \over p} (R-x)^{-{p+2 \over 2p}}x^{{1 \over p}-1}(R-{p \over 2}x)$. Hence $x_p={2R \over p}$ is the unique point where $g_p$ attains its maximum $$g_p(x_p)=\left({2R \over p}\right) ^{1/p}\left(R-{2R \over p}\right)^{p-2 \over 2p} $$ over the interval $[0,R]$. Notice that $\gamma_p\rightarrow \sqrt M$, whereas $g_p(x_p)\rightarrow \sqrt R>\sqrt M$, as $p\rightarrow\infty$. Therefore, for $p$ sufficiently large, there are exactly two points $r'_p$, $r''_p$ in $(0,R)$, with $r'_p0$, let us define $$ f_M(t)=\cases{ f(t) & $0\leq t\leq M$,\cr 0 & $ t\geq M$, \cr} \eqno(4.1) $$ if $f(M)=0$, and $f_M=f$ otherwise. Let us consider the Dirichlet problem $$ \displaylines{ \hfill -\Delta_p v=f_M(v) \quad \hbox{in }\Omega\,,\hfill \llap{($\tilde D_p$)}\cr v=0 \quad \hbox{on } \partial\Omega\,.\cr} $$ \noindent {\bf Definition 4.1.} A function $u\in W^{1,\infty}(\Omega) \cap C_0(\bar\Omega)$ such that $M=\max_\Omega u$ is called a {\it tame variational solution} to $(D_\infty)$ if there exists a sequence $u_{p_k}$ of solutions to $(\tilde D_{p_k})$, with $p_k\rightarrow\infty$, such that $(2.4)$ holds. \smallskip \noindent {\bf Remark.} It is clear, from the preceding arguments, that tame variational solutions are variational solutions. Of course, there are either functions $f$ for which variational solutions which are not tame do exist (see, for instance, Section 3), or functions $f$ for which every variational solution is tame (for instance $f$ strictly positive in some interval $[0, L)$ and vanishing outside). \smallskip Since the above definition precludes case $\gamma$), the following results follow easily from Theorem 2.5, Corollary 2.6 and Theorem 2.7. \proclaim Lemma 4.2. Let $u$ be a tame variational solution to $(D_\infty)$. Then $u\equiv 0$ if and only if $f^*\equiv 0$. If $u\not\equiv 0$ then $E_\infty>0$. \par \proclaim Theorem 4.3. Let $\Omega\subset{\Bbb R}^n$ be a convex domain. Let $u\in W^{1,\infty}(\Omega) \cap C_0(\bar \Omega)$ be a tame variational solution to $(D_\infty)$. Then the level sets $\{u>t\}$ are convex, $s_M=M\in\overline{\{f>0\}}$, and $u$ is uniquely determined in all of $\Omega$ by (2.22)--(2.23) and by $u\equiv s_M$ in $\Omega_{s_M}$. If, moreover, $\Omega$ is invariant with respect to an orthogonal transformation $T$, then $u$ is symmetric with respect to $T$. \par \proclaim Theorem 4.4. Let $\Omega=B_R$ and let $u\in W^{1,\infty}(B_R) \cap C_0(\bar B_R)$ be a nontrivial tame variational solution to $(D_\infty)$. Then $u$ is radially symmetric and radially non-increasing. Moreover, $s_M=M\in\overline{\{f>0\}}$, and $u$ is given by $(2.24)$. \par \bigbreak \centerline{\bf References}\nobreak \item{\eightrm [A]} G. Aronsson, Extension of functions satisfying Lipschitz conditions, {\it Ark. Mat.} {\bf 6}, (1967), 551--561. \item{\eightrm [B-D-M]} T. Bhattacharya, E. DiBenedetto, J. Manfredi, Limits as $p\rightarrow\infty$ of $\Delta_p (u)=f$ and related extremal problems, {\it Rend. Sem. Mat. Univ. Pol. Torino, Fascicolo Speciale Nonlinear PDE's}, (1989), 15--68. \item{\eightrm [Br1]} F. Brock, Radial symmetry for nonnegative solutions of semilinear elliptic equations involving the $p$-Laplacian, {\it Progress in P.D.E.} Pont-a-Mousson, (1997), {\bf I}, eds. H. Amann et al., 46--57. \item{\eightrm [Br2]} F. Brock, Continuous rearrangement and symmetry of solutions of elliptic problems, habilitation thesis, Leipzig, (1998). \item{\eightrm [Br-H]} F. Brock, A. Henrot, A symmetry result for an overdetermined elliptic problem using continuous rearrangement and domain derivative, preprint. \item{\eightrm [D-P]} L. Damascelli, F. Pacella, Monotonicity and symmetry of solutions of $p$-Laplace equations, $1