\magnification = \magstephalf
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\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 $d$, $0\leq d__0\}$ 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
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conditions, {\it Ark. Mat.} {\bf 6}, (1967), 551--561.
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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).
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overdetermined elliptic problem using continuous rearrangement and domain
derivative, preprint.
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