\documentclass[reqno]{amsart}
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\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2012 (2012), No. 145, pp. 1--6.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2012 Texas State University - San Marcos.}
\vspace{9mm}}
\begin{document}
\title[\hfilneg EJDE-2012/145\hfil Minimizers of a variational problem]
{Minimizers of a variational problem inherit the symmetry of the domain}
\author[M. Montenegro, E. Valdinoci \hfil EJDE-2012/145\hfilneg]
{Marcelo Montenegro, Enrico Valdinoci} % in alphabetical order
\address{Marcelo Montenegro \newline
Departamento de Matem\'atica \\
Universidade Estadual de Campinas - IMECC\\
Rua S\'ergio Buarque de Holanda, 651 \\
Campinas-SP, CEP 13083-859, Brazil}
\email{msm@ime.unicamp.br}
\address{Enrico Valdinoci \newline
Dipartimento di Matematica\\
Universit\`a di Roma Tor Vergata\\
Via della Ricerca Scientifica, 1\\
00133 Roma, Italy}
\email{enrico@math.utexas.edu}
\thanks{Submitted March 27, 2012. Published August 21, 2012.}
\subjclass[2000]{35A30, 47J30, 49K30, 35J85, 35J60, 58E05}
\keywords{Minimizers; PDEs; symmetry}
\begin{abstract}
We give a general framework under which the minimizers of a variational
problem inherit the symmetry of the ambient space. The main technique used
is the moving plane (or sliding) method.
\end{abstract}
\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{remark}[theorem]{Remark}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{example}[theorem]{Example}
\allowdisplaybreaks
\section{Introduction}
It is a hot topic in PDE to determine
wether or not a solution possess some kind of symmetry.
Besides the celebrated result of \cite{GNN},
much effort has been put in addressing this problem
in several situations, and many fundamental questions
are still open (see, among the others, for instance \cite{K0,K}
and references therein).
Indeed, the so-called moving plane (or sliding)
method has been widely used to prove radial
symmetry for positive solutions of elliptic equations. The classical
references are the papers \cite{GNN, GNN2}, where the authors
proved that positive solutions of the semilinear equation
$-\Delta u(x)+f(u(x))=0$ on a ball are radially
symmetric provided that $u=0$ on the boundary of the ball.
The same conclusion holds in $x\in \mathbb{R}^N$ if one assumes that $u$ decays to zero at infinity. They were also able to treat the equation with radially dependent
nonlinearity $-\Delta u(x)+f(|x|,u(x))=0$
provided $\partial f(r,u)/\partial r<0$ for every $r>0$.
Later, in \cite{lo2} and \cite{lo3} radial symmetry or partial symmetry
for global minimizers of functionals was considered, and the advantage of
the reflection method considered there relies in its simplicity and
generality. Indeed, there are at least three cases not covered by
\cite{GNN,GNN2}
which have been taken into account in \cite{lo2} and \cite{lo3}, which
also includes more general boundary conditions, no
requirement is taken on the sign
of the minimizer, and domains like the annulus may be treated as well.
The technique of reflecting the minimizers
has been also used in \cite{MM},
where the use of the unique
continuation principle of \cite{lo2,lo3}
was replaced by suitable regularity assumptions.
Another approach to prove symmetry lies in
the technique of symmetrization, which, for example, may be
used to show the symmetry of minimizers assuming
that the minimizer is positive and that the
nonlinearity is monotone with respect to $r$ (see~\cite{schaft})
The foliated Schwartz symmetrization can be used to
prove the axial symmetry of the minimizers without
any assumption on the sign of $u$ and of $\partial
f(r,u)/\partial r$ (see \cite{schaft,willem}).
We also refer to~\cite{B} for
further insight on symmetry problems.
In this note, which is very elementary in spirit,
we show that minimizers of variational problems
inherit the natural symmetries induced by the domain
and by the equation. For this,
the use of the maximum principle or of the moving plane method
is not necessary and so things are much
easier, and much more general, than in the case
of nonminimal solutions.
Indeed, we will revisit the approach of~\cite{lo2,lo3}
to obtain symmetry in a more
general setting. Our motivation also
comes from the paper \cite{KS} where
the
authors use a generalization of \cite{GNN} to obtain symmetry for
positive solutions of $-\Delta u + f(u)=0$ with Dirichlet boundary
condition on a domain that could be, for instance, a David star, a square,
a stellated cube or a Kepler's stella octangula. The general feature in common to
these domains is the so-called Steiner-symmetry,
which is one of the essential ingredients of~\cite{KS}.
Here we are able to treat non-Steiner-symmetric domains like
the five star pentagon and the Kepler-Poinsot polyhedron
(see, e.g., Example~\ref{EP}). The reader is
referred to~\cite{KS} for many pictures of such domains.
The method we use is somewhat classical,
and the basic idea is already sketched
on \cite[p.~19]{K0}, of which
we repeat here the very clear exposition:
\begin{quote}
Suppose $u$ minimizes a strictly convex functional $\mathcal{I}( v )$ on
a convex set of admissible functions $v$. Moreover $v$ is defined on a
symmetric set $\Omega$; i.e., $\Omega$ is invariant under some
group action. If $g$ is an element of the group,
$g(\Omega) =\Omega$ and consequently $u ( x ) = u(g(x))$;
i.e., $u$ is invariant under the group action; otherwise the convex
combination $w ( x ) =[u(x) + u ( g ( x ) ) ] / 2 $ would have
smaller ``energy" $ \mathcal{I} ( w ) <\mathcal{I}( u ) $, a contradiction.
\end{quote}
\section{Statements of results}
Now, we introduce formally our framework.
Given $n$, $m$, $\ell\in\mathbb{N}$, we define
$$
\mathbb{R}_\ell:=\mathbb{R}^{nm\ell}\times\dots\times
\mathbb{R}^{nm}\times\mathbb{R}^m=\big[\prod_{j=1}^\ell \mathbb{R}^{nmj}
\big]\times\mathbb{R}^m.
$$
For an open set $\Omega\subseteq\mathbb{R}^n$, we consider a measurable
function $\psi:\mathbb{R}_\ell\times\Omega\to\mathbb{R}$.
Let ${\mathcal{W}}(\Omega)$
be the set of functions from $\Omega\subseteq\mathbb{R}^n$ to $\mathbb{R}^m$
that are $\ell$-times
differentiable a.e. in $\Omega$.
For any $u\in {\mathcal{W}}(\Omega)$ and any $x\in\Omega$ we write
$$
\Psi[u](x):=\psi\big(D^\ell u(x), \dots, Du(x),u(x),x\big).
$$
We consider the set of admissible functions
$$
{\mathcal{A}}(\Omega):=\big\{ u\in {\mathcal{W}}(\Omega)
{\text{ s.t. }}\Psi[u]\in L^1(\Omega)\big\}
$$
and we define
$$
\mathcal{I}_\Omega[u]:=\begin{cases}
\int_\Omega \Psi[u](x)\,dx & \text{if }u\in{\mathcal{A}}(\Omega),\\
+\infty &\text{otherwise.}
\end{cases}
$$
Given $u$, $v\in {\mathcal{W}}(\Omega)$, $t\in(0,1)$ and $x\in\Omega$,
we consider the convex combination
$$
[u,v]_t(x):=tu(x)+(1-t)v(x).
$$
We take $ {\mathcal{S}}(\Omega)\subseteq {\mathcal{A}}(\Omega)$
such that
\begin{equation}\label{S1}
\text{if $u$ and $v\in {\mathcal{S}}(\Omega)$
then there exists $t\in(0,1)$
such that $[u,v]_t\in{\mathcal{S}}(\Omega)$.}
\end{equation}
We suppose that $\Psi$ is strictly convex in ${\mathcal{S}}(\Omega)$;
i.e., for every $u$, $v\in{\mathcal{S}}(\Omega)$,
if $t\in(0,1)$ is as in~\eqref{S1}, we have that
\begin{equation}\label{S2}
\begin{split}
&\Psi[[u,v]_t](x)\le t\Psi[u](x)+(1-t)\Psi[v](x) \text{ for every $x\in
\Omega$,}\\
&\text{ and if equality holds for
a.e. $x\in\Omega$ then $u=v$ a.e. in $\Omega$.}
\end{split}\end{equation}
Given a Lipschitz bijection $S:\overline\Omega\to\overline\Omega$,
we say that $S$ is a symmetry for $\mathcal{I}_\Omega$
in ${\mathcal{S}}(\Omega)$ if the following conditions hold:
\begin{equation}\label{US}
\text{if $u\in {\mathcal{S}}(\Omega)$ and $u_S(x):=u(S(x))$
for any $x\in\Omega$, we have that $u_S\in
{\mathcal{S}}(\Omega)$}
\end{equation}
and
\begin{equation}\label{US2}
\begin{split}
& \psi\big(D^\ell u_S(x), \dots, Du_S(x),u_S(x),x\big) \\
&=\psi\big(D^\ell u(S(x)), \dots, Du(S(x)),u(S(x)), S(x)\big)
\,|\det DS(x)| \quad \text{for a.e. $x\in\Omega$.}
\end{split}
\end{equation}
In this setting, minimizers inherit the symmetry of $S$:
\begin{theorem}\label{thm1}
Let $S:\overline\Omega\to\overline\Omega$
be a symmetry for $\mathcal{I}_\Omega$ in ${\mathcal{S}}(\Omega)$.
Assume that there exists $u^\star\in{\mathcal{S}}(\Omega)$
such that
$$
\mathcal{I}_\Omega[u^\star]=\inf_{ u\in{\mathcal{S}}(\Omega)}
\mathcal{I}_\Omega[u]<+\infty.
$$
Then $u^\star\big( S(x)\big)=u(x)$ for a.e. $x\in\Omega$.
\end{theorem}
\begin{proof}
The proof is a simple combination of two well-known principles.
The first principle is the fact that strictly
convex functionals attain at most one minimum. The second one is
that uniqueness implies symmetry with respect to every transformation
which leaves the functional values unchanged.
Here is the argument in detail.
By \eqref{US},
\begin{equation}\label{0.3bis}
u^\star_S\in {\mathcal{S}}(\Omega).
\end{equation}
Also, by \eqref{US2},
\begin{align*}
\mathcal{I}_\Omega[u^\star_S]
&=\int_\Omega
\psi\big(D^\ell u^\star_S (x), \dots,u^\star_S(x),x\big)\,dx\\
&=\int_\Omega \psi\big(D^\ell u^\star (S(x)), \dots,u^\star(S(x)),S(x)\big)\,
|\det DS(x)|\,dx\\
&= \int_\Omega \psi\big(D^\ell u^\star (y), \dots,u^\star(y),y\big)\,dy\\
&= \mathcal{I}_\Omega[u^\star].
\end{align*}
Therefore, by~\eqref{S1},~\eqref{S2}
and~\eqref{0.3bis}, there exists $t\in(0,1)$ such that the following
computation holds:
\begin{align*}
\mathcal{I}_\Omega[u^\star]
&= \inf_{ u\in{\mathcal{S}}(\Omega)}\mathcal{I}_\Omega[u]\\
&\leq \mathcal{I}_\Omega[[u^\star,u^\star_S]_t] \\
&= \int_\Omega \Psi[[u^\star,u^\star_S]_t](x)\,dx\\
&\leq \int_\Omega t\Psi[u^\star](x)+(1-t)\Psi[u^\star_S](x)\,dx\\
&= t\mathcal{I}_\Omega[u^\star]+(1-t)\mathcal{I}_\Omega[u^\star_S]\\
&= \mathcal{I}_\Omega[u^\star].
\end{align*}
Hence
$$ \Psi[[u^\star,u^\star_S]_t]=
t\Psi[u^\star](x)+(1-t)\Psi[u^\star_S](x)$$
a.e. $x\in\Omega$, so \eqref{S2} implies that $u^\star=u^\star_S$
a.e. in $\Omega$, as desired.
\end{proof}
\begin{remark}{\rm
Of course, given the simplicity of Theorem~\ref{thm1}
and of its proof, we cannot really claim any priority or originality in it,
but we think it could be useful to have
the result stated and understood in such a general form.}
\end{remark}
\begin{remark}{\rm
In most of the applications, the symmetry $S$ is
a rigid motion (in particular, a reflection
or a rotation), so $|\det DS|=1$. However,
we thought it was somewhat useful to speak about
more general type of symmetries (see also
the forthcoming Example~\ref{E1} where $|\det DS|\neq 1$).
}\end{remark}
\begin{remark}{\rm
The space ${\mathcal{S}}(\Omega)$ is designed
to include the boundary data (see the examples below).
}\end{remark}
\begin{remark}{\rm
In the particular case $\psi:=|\nabla u|^2 +G(u)$, the convexity
condition~\eqref{S2} boils down to the monotonicity of the
nonlinearity $G'(u)$, i.e. on a sign condition on the linear term $G''(u)$
driving the linearized equation. In this case,
this assumption reduces to the classical
one which makes the maximum principle hold.
}\end{remark}
Though the statement of Theorem~\ref{thm1}
is quite general, it may be useful
to give some very simple, not exhaustive,
but concrete, applications.
\begin{example}\label{E1}{\rm
Let $n=m=\ell=1$. We take the rectangle $\Omega:=[-1/2,1]\times[-1,1]$
and we define, for any $x=(x_1,x_2)\in\mathbb{R}^2$,
$$
S(x):=\begin{cases}
(-2x_1,x_2) & \text{if }x_1<0,\\
(-x_1/2,x_2) & \text{if } x_1\ge 0.
\end{cases}
$$
Let also
$$
a(x):=\begin{cases}
1 & \text{if } x_1<0,\\
2 & \text{if } x_1\ge 0.
\end{cases}
$$
We observe that, if $x_1\ne0$, then
\begin{equation}\label{11}
a(S(x))=\frac2{a(x)}.
\end{equation}
Given $(r,x)\in\mathbb{R}^2\times\Omega$, we define
$$
\psi(r,x):= a(x)r_1^2+\frac{r_2^2}{a(x)}.
$$
We also take $\bar u\in C^\infty(\mathbb{R})$ and
$u_o(x_1,x_2)= \bar u(x_2)$. We notice that
\begin{equation}\label{10}
u_o(S(x))=u_o(x).
\end{equation}
Thus, if we define
\begin{equation}\label{17}
{\mathcal{S}}(\Omega):=W^{1,2}_{u_o}(\Omega)=
\big\{u\in W^{1,2}(\Omega)
{\text{ s.t. }} u-u_o \in W^{1,2}_{0}(\Omega)
\big\},\end{equation}
we have that \eqref{US} holds, due to \eqref{10}.
Moreover, a careful computation shows that \eqref{US2}
is satisfied,
due to \eqref{11}.
Also, \eqref{S2} follows from the strict convexity of the maps $r_1\mapsto r_1^2$
and $r_2\mapsto r_2^2$; notice that if equality holds in~\eqref{S2}
then $\nabla u=\nabla v$, hence $u=v+c$, for some $c\in\mathbb{R}$. {F}rom
the boundary data in~\eqref{17}, we obtain that $c=0$,
and this shows that~\eqref{S2} is satisfied.
Then, Theorem~\ref{thm1} applies to this case.
We remark that, in this case, $S$ is not a rigid motion.
In fact, more general types of symmetries and domains (such
as the ones with the shape of a scamorza-cheese)
may be treated with the same idea; i.e., decomposing $S$
into a reflection and two dialations on the opposite
halfplane. Of course, the more complicated $\Omega$
and $S$ are, the more complicated needs to be
the function $\psi$ in order to satisfy
the invariance in~\eqref{S2}.}
\end{example}
\begin{example}{\rm
Let $\ell=1$, $n=2$ and $m=1$. We take $\Omega=[-1,1]\times
[-2,2]$ and we consider the odd reflection $S(x):=-x$.
Let $G\in C^\infty(\mathbb{R})$, $p\in(1,+\infty)$ and, for
every $r\in\mathbb{R}^2$ and $\tau\in\mathbb{R}$, we set
$$
\psi(r,\tau,x):= \frac{ |r|^p}p-\frac{ar_1^2}{2}+G(|x|^2)\tau
$$
and ${\mathcal{S}}(\Omega):=W^{1,p}_0(\Omega)$.
The corresponding PDE (in the weak sense) is
$$
\Delta_p u-a\partial_{11} u=G(|x|^2),
$$
where, as usual, $\Delta_p u:=\,{\rm div}\,(|\nabla u|^{p-2}
\nabla u)$ is the $p$-Laplace operator.
Then, Theorem~\ref{thm1} applies and it gives that
the minimal solution is odd.
The case of an even function $G(x_1,x_2)$ may be treated
as well. Notice that, in general, some conditions
are needed to ensure that the
functional $\mathcal{I}_\Omega$ is bounded from below
(but it is not the aim of this note to discuss such conditions,
since we just assume in Theorem~\ref{thm1} the existence of
a minimizer).
}\end{example}
\begin{example}\label{EP}{\rm
Let $\ell=1$, $n=2$ and $m=1$.
Given $G$, $H\in C^\infty(\mathbb{R})$,
$r\in\mathbb{R}^2$ and $\tau\in\mathbb{R}$,
we define
$$
\psi(r,\tau,x):= \frac{|r|^2}2+G(|x|^2) H(\tau).
$$
We observe that the associated PDE has the form
\begin{equation}\label{12}
\Delta u=G(|x|^2) H'(u).
\end{equation}
Let $R\in{\rm Mat}(n\times n)$ be the anticlockwise
rotation of angle $2\pi/5$ and let $\Omega$
be a regular five-pointed star (i.e., a
star pentagon) centered at the origin.
In this case, the setting of Theorem~\ref{thm1}
holds with $S(x):=Rx$, if $H$ is convex and $G\ge0$.
This gives that minimal solutions
of \eqref{12} in the five-pointed star domain
are symmetric under $2\pi/5$ rotations.
We remark that the five-pointed star domain is not
Steiner-symmetric, so it is not known in general whether
all the solutions endow the same symmetry
(see \cite{KS}).
}\end{example}
\subsection*{Acknowledgments}
Part of this work was developed while EV was visiting
the Universidade Estadual de Campinas, whose warm
hospitality and pleasant atmosphere was greatly appreciated.
MM has been partially supported by CNPq.
E.~V. has been supported by FIRB Analysis and Beyond
and GNAMPA Equazioni nonlineari su variet\`a:
propriet\`a qualitative e classificazione delle soluzioni.
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\end{document}