\documentstyle[twoside]{article}
\input amssym.def % used for R in Real numbers
\pagestyle{myheadings}
\markboth{\hfil Radial and nonradial Minimizers \hfil EJDE--1996/03}%
{EJDE--1996/03\hfil Orlando Lopes\hfil}
\begin{document}
\title{\vspace{-1in}\parbox{\linewidth}{\footnotesize\noindent
{\sc Electronic Journal of Differential Equations},
Vol.\ {\bf 1996}(1996), No.\ 3, pp. 1--14. \newline
ISSN: 1072-6691. URL: http://ejde.math.swt.edu (147.26.103.110)\newline
telnet (login: ejde), ftp, and gopher access:
ejde.math.swt.edu or ejde.math.unt.edu}
\vspace{\bigskipamount} \\
Radial and Nonradial Minimizers for Some Radially Symmetric Functionals
\thanks{ {\em 1991 Mathematics Subject Classifications:}
35J20, 49J10.\newline\indent
{\em Key words and phrases:} Variational Problems, Radial and Nonradial
Minimizers.
\newline\indent
\copyright 1996 Southwest Texas State University and University of
North Texas.\newline\indent
Submitted December 6, 1995. Published February 29, 1996.} }
\date{}
\author{Orlando Lopes}
\maketitle
\begin{abstract}
In a previous paper we have considered the functional
$$ V(u) = \frac12\int_{\Bbb R^N} |{\rm\ grad}\, u(x)|^2\, dx +
\int_{\Bbb R^N}F(u(x))\,dx $$
subject to
$$ \int_{\Bbb R^N} G(u(x))\, dx = \lambda > 0\,,$$
where $u(x) = (u_1(x) , \ldots, u_K(x))$ belongs to
$H^1_K (\Bbb R^N) = H^1 (\Bbb R^N) \times\cdots\times H^1(\Bbb R^N)$
$(K$ times) and $|{\rm\ grad}\, u(x)|^2$ means
$ \sum^K_{i=1}|{\rm\ grad}\, u_i (x)|^2$.
We have shown that, under some technical assumptions and except for a
translation in the space variable $x$, any global minimizer is radially
symmetric.
In this paper we consider a similar question except that the integrals in
the definition of the functionals are taken on some set $\Omega$ which is
invariant under rotations but not under translations, that is, $\Omega$ is
either a ball, an annulus or the exterior of a ball. In this case we show
that for the minimization problem without constraint, global minimizers
are radially symmetric. However, for the constrained problem, in general,
the minimizers are not radially symmetric. For instance, in the case of
Neumann boundary conditions, even local minimizers are not radially
symmetric (unless they are constant). In any case, we show that the
global minimizers have a symmetry of codimension at most one.
We use our method to extend a very well known result of Casten and
Holland to the case of gradient parabolic systems.
The unique continuation principle for elliptic systems plays a crucial
role in our method.
\end{abstract}
\newcommand{\dis}{\displaystyle}
\newcommand{\grad}{\mathop{\rm grad}}
\newcommand{\pa}{\partial}
\section*{I. Introduction}
In a previous paper ([1]) we have shown that, under some technical
assumptions and except for a translation in the space variable, any
global minimizer of the functional
\renewcommand{\theequation}{I.\arabic{equation}}
\setcounter{equation}{0}
\begin{equation}
V(u) = \frac{1}{2} \int_{\Bbb R^N} |\grad u(x)|^2\, dx + \int_{\Bbb R^N}
F(u(x))\,dx
\end{equation}
subject to
\begin{equation}
\int_{\Bbb R^N} G(u(x))\, dx = \lambda > 0
\end{equation}
is radially symmetric.
In (I.1) and (I.2) $u(x) = (u_1(x) , \ldots, u_K(x))$ belongs to the
space $H^1_K (\Bbb R^N) = H^1 (\Bbb R^N) \times \cdots \times H^1
(\Bbb R^N)$ $(K$ times) and $|\grad u(x)|^2$ means
$\sum^K_{i=1} |\grad u_i (x)|^2$.
In this paper we consider a similar problem except that the integrals
in the definition of the functionals are taken on a set $\Omega$ which is
rotation invariant but not translation invariant (that is, $\Omega$ is
either a ball or an annulus or the exterior of a ball).
In this case we show that, for the unconstrained minimization
problem with either Dirichlet or Neumann boundary conditions,any global
minimizer is radially symmetric. However, for the constrained
minimization problem with Neumann boundary conditions, even a local
minimizer is not radially symmetric (unless it is a constant
function).
In the case of a constrained minimization problem with Dirichlet
boundary condition, there are examples for which the global minimizer
is radially symmetric and examples for which the global minimizer is
not radially symmetric.
In all cases we show that the global minimizers have a symmetry of
codimension at most one.
\section*{ II. The Unconstrained Problem}
\renewcommand{\theequation}{II.\arabic{equation}}
We consider the functional
\setcounter{equation}{0}
\begin{equation}
W(u) = \frac{1}{2} \int_\Omega |\grad u(x)|^2\, dx +
\int_\Omega F(x,u(x))\,dx
\end{equation}
Here $u(x) = (u_1(x), u_2(x), \ldots, u_K (x))$ is a $K$-vector valued
function defined for $x \in \Omega \subset \Bbb R^N$. The function
$u(\cdot)$
will be taken in the space $H(\Omega)$ which is the Cartesian product of
$K$ factors each of which is either $H^1_0(\Omega)$ or $H^1(\Omega)$.
Our assumptions are the following
\begin{description}
\item{$H_1)$} $\Omega$ is a $C^2$ (bounded or unbounded) open connected
set of $\Bbb R^N$ which is symmetric with respect to the hyperplane
$x_1=0$.
\item{$H_2)$} $F(x,u)$ is a real valued function defined for $(x,u) \in
\Omega \times \Bbb R^K$, which is continuous with respect to $(x,u)$
together with their first and second derivatives with respect to
$u$.
\item{$H_3$)} $F(-x_1, x_2,\ldots, x_N,u) = F(x_1,x_2,\ldots,x_N,u)$ for $x \in
\Omega$ and $u \in \Bbb R^K$.
\item{$H_4$)} $W(u)$ is well defined for $u \in H(\Omega)$.
\item{$H_5$)} If $u\in H(\Omega)$ minimizes $W$ then it satisfies the Euler
system
\begin{equation}
-\Delta u(x) + \grad F(x,u(x)) = 0
\end{equation}
together with boundary conditions and $u \in L^\infty (\Omega)$.
\end{description}
\paragraph{Remarks.}
\begin{description}
\item{II.3.} Assumptions $H_4$ and $H_5$ are satisfied if $F(x,u)$ and
its derivatives with respect to $u$ satisfy certain growth conditions on
$u$ and depend
on $x$ in a convenient way (see [2] for a discussion of them).
\item{II.4.} The boundary condition for the i-component $u_{i}(x)$ of the
minimizer $u(x)$ is either Dirichlet or Neumann boundary condition,
depending on whether the i-component of the space $H(\Omega )$ is
$H_{0}^{1}(\Omega )$ or $H^{1}(\Omega )$.
\end{description}
\paragraph{Theorem II.5.}
Under assumptions $H_1-H_5$, if $u(\cdot)$ minimizes $W$
(given by II.1) in the space $H(\Omega)$ then $u(-x_1,x_2,\ldots
, x_N) =
u(x_1,x_2,\ldots,x_N)$ for $x \in \Omega$.
\paragraph{Proof.} We define the sets $\Omega_l = \{x\in \Omega : x_1 \leq
0\}$ and $\Omega_r = \{x \in \Omega : x_1 \geq 0\}$.
Let $u \in H(\Omega)$ be a minimizer of $W$. We claim that
\setcounter{equation}{5}
\begin{eqnarray}
\lefteqn{\frac12 \int_{\Omega_l} |\grad u(x)|^2\, dx + \int_{\Omega_l}
F(x,u(x))\, dx =} \\
&& \frac{1}{2} \int_{\Omega_r} |\grad u(x)|^2\, dx + \int_{\Omega_r}
F(x,u(x))\, dx . \nonumber
\end{eqnarray}
Let us denote by $A$ and $B$ the left and the right side of II.6,
respectively, and, by contradiction, suppose $A < B$. We define $U(x)$
in the following way:
\begin{equation}
U(x) = \left\{ \begin{array}{ll}
u(x) & \mbox{ if $x \in \Omega_l$}\\
u(x') & \mbox{ if $x \in \Omega_ri$} \end{array}
\right. \end{equation}
where $x'$ denotes the reflection of $x$ with respect to the
hyperplane $x_1 = 0$. Clearly $U \in H(\Omega)$; moreover we have $W(U)
= A + A < A + B = W(u)$, a contradiction. This proves II.6.
Keeping the definition II.7 for $U(\cdot)$, from II.6 we conclude that
$W(U) = A + A = A + B = W(u)$ and this means that both $u$ and $U$
are minimizers and so, by assumption $H_5$, they satisfy the Euler
systems:
\begin{eqnarray}
- \Delta u (x) + \grad F (u(x)) &=& 0 \\
- \Delta U (x) + \grad F (U(x)) &=& 0\,.
\end{eqnarray}
If we define $z(x) = u(x) - U(x)$ and we subtract II.9 from II.8
then, from assumptions $H_2$ and $H_5$, we see that $z(x)$ satisfies
a linear system of the form
\[
- \Delta z + A(x) z = 0\,,
\]
where $A(x)$ is a $K \times K$ matrix whose entries belong to
$L_\infty (\Omega)$ and, since $z = 0$ in $\Omega_l$, from the unique
continuation principle ([3]), we conclude that $z(x)$ vanishes in
$\Omega$ and this proves the theorem. $\spadesuit$
\paragraph{Corollary II.10.} If $\Omega$ is either a ball or an annulus
or the exterior of a ball centered at the origin and
$F(x,u) = F(|x|, u)$, where $|x| = (x^2_1 +
\cdots + x^2_N)^{1/2}$, then any minimizer for $W$ is radially
symmetric.
\paragraph{Proof.} We apply theorem II.5 for any hyperplane passing
through the origin.
\paragraph{Corollary II.10.} has been proved in [1] for the case of
Dirichlet boundary conditions assuming that $F$ does not depend on $|x|$.
Theorem II.5 has been proved in [4] in the case of Dirichlet
boundary condition and assuming
that $F$ does not depend on $x$ and that $\Omega$ is convex in
the $x_1$-direction (that is, if two points $A$ and $B$ belong to
$\Omega$ and the segment $AB$ is parallel to $x_1$, then the segment
$AB$ is contained in $\Omega$). The method of the proof of theorem II.5
is, basically, the method used in [1] and [4].
\paragraph{Remarks.}
\begin{description}
\item{II.11.} In the scalar case $(K=1)$, if $\Omega$ and $F$ are as in
corollary II.10, then any {\it local } minimizer for $W$ is radially
symmetric ([5]).
The proof of this statement depends on the maximum principle and so
it can be extended to the system case provided $\grad F(u)$ satisfies a
cooperative condition.
\item{II.12.} Corollary II.10 holds for functionals given by
$\int_\Omega \varphi (|x|,u(x), |\grad u(x)|)\,dx$ provided that the
corresponding Euler system is elliptic nondegenerate and the
minimizer is regular enough.
\item{II.13.} If we add to II.1 a term $\int_{\pa\Omega} H(u(x))\,dS$ involving an
integral on the boundary, then we can prove corollary II.10 for other
boundary conditions.
\end{description}
\section*{ III. Nonradial Minimizers}
\renewcommand{\theequation}{III.\arabic{equation}}
\setcounter{equation}{0}
In this part we consider the functional
\begin{equation}
V(u) = \frac{1}{2} \int_\Omega |\grad u(x)|^2\, dx +
\int_\Omega F(u(x))\, dx
\end{equation}
subject to
\begin{equation}
\int_\Omega G(u(x))\, dx = \lambda.
\end{equation}
The function $u(x) = (u_1(x), \ldots, u_K (x))$ will be taken in the
space $H^1_K (\Omega) = H^1 (\Omega) \times \cdots \times H^1 (\Omega)$
($K$ times) and by $|\grad u(x)|^2$ we mean
$\sum^K_{i=1}|\grad u_i (x)|^2$.
The first results about the break of symmetry of minimizers for the
problem III.1 -- III.2 are due to M. Esteban ([6], [7]) and V.
Coti-Zelati-M. Esteban ([8]) for the scalar case. If $\Omega$ is either
the exterior of a ball centered at the origin or the ball itself,
$F(u) = u^2$, $ G(u) = |u|^p$, $ 2 < p < {\frac{2N}{N-2}}$
and $ \lambda > 0$,
then the global minimizers for III.1 -- III.2 are not radially symmetric ([6],
[7]). A similar statement holds for the annulus ([8]).
In this paper the set $\Omega \subset \Bbb R^N$ will be either a ball with radius $R$
centered at the origin or the annulus $\{x \in \Bbb R^N : 0 < R < |x| <
R_1\}$ ($R_1$ is allowed to be $+\infty$) and our assumptions are the
following:
\begin{description}
\item{$H_1$)} $F, G : \Bbb R^K \rightarrow \Bbb R$ are $C^2$ functions.
\item{$H_2$)} $|F'(u)|, \ |G'(u)| \leq $ \ const. \ $|u|^{p-1}$ for $|u|$
large with $p < {\frac{2N}{N-2}}$ ($p$ finite if $N = 2)$.
\item{$H_3$)} In the case $R_1 = + \infty$ we also assume that
$F(0) = G(0)=0$ and $F'(0) = G'(0)= 0$.
\end{description}
\paragraph {Theorem III.3.}
Suppose $\Omega , F(u)$ and $G(u)$ satisfy the assumptions above
and let $u \in H^2_K (\Omega)$ be a radially symmetric non-constant
solution of the system
\setcounter{equation}{3}
\begin{eqnarray}
- \Delta u + \grad H(u) &=& 0 \\
\frac{\pa u}{\pa n} &=& 0 \quad {\rm on} \quad \pa\Omega\,,
\end{eqnarray}
where $H(u) = F(u) + \alpha G(u), \ \alpha =$ constant. Suppose also that
grad\, $G(u(\cdot)) \not\equiv 0$ (a manifold condition) and that
$u(\cdot)$ satisfies III.2.
Then $u(\cdot)$ is not a local minimizer for the problem III.1 -
III.2.
\paragraph{Proof.} Due to the growth and regularity assumptions on $F(u)$
and $G(u)$ we know that $u\in W^{3,q}(\Omega )$, $2\leq q < \infty $.
We consider the quadratic functional
\begin{equation}
Q(h)=\int_\Omega |\grad h(x)|^2\,dx +\int_\Omega \langle h(x),H''(u(x))h(x) \rangle \,dx
\end{equation}
subject to the linear constraint
\begin{equation}
\int_\Omega \langle \grad G(u(x)),h(x)\rangle \,dx=0,
\end{equation}
for $h\in H_{K}^{1}(\Omega )$, where $H''(u(x))$ denotes the Hessian
matrix of $H(u)$ at $u=u(x)$.
Denoting by ${\cal F}(\Omega )$ the set of the $C^{\infty }$ functions
defined in $\Omega$ with values in $R^K$ and having support contained in
a ball centered at the origin (of course this last demand makes sense
only if $\Omega$ is the exterior of a ball),our first claim is that there
is an element $k\in {\cal F}(\Omega)$ satisfying III.7 such that
$Q(k) < 0$.
We start by showing that there is an element $k\in H_{K}^{1}(\Omega )$
satisfying III.7 such that $Q(k) < 0$.
In fact, if we define $h_{0}(x)={\frac{\pa u}{\pa x_{1}}=
\frac{x_{1}}{r}}u'(r)$ and we differentiate III.4 with respect to $x_{1}$
we get
\begin{equation}
-\Delta h_0 +H''(u(x))h_0=0
\end{equation}
and taking the scalar product of III.8 with $h_{0}(x)$ and integrating
we get $Q(h_{0})=0$ ( we have used that $h_{0}$ vanishes on $\pa\Omega$,
due to III.5).
Moreover, since the function \ ${\frac{1}{r}} \langle \grad
G(u(r)),u'(r)\rangle $ depends just on $r$ and $x=(x_{1},x_{2},\ldots,x_N)
\rightarrow x_1$ is an odd function, we have $$\int_\Omega
\langle \grad G(u(r)),h_0(x)\rangle \,dx=0$$ and this means that $h_0$
satisfies III.7.
Next we claim that, in despite of III.8, $h_0$ is not a critical point for
the problem III.6-III.7 for $h$ in the space $H^1_K(\Omega)$. In fact,
if it was, then for some real number $\beta$ and any $\varphi \in H_{K}^{1}
(\Omega )$ we would have
\begin{eqnarray*}
\int_\Omega \langle \grad h_0(x),\varphi (x) \rangle \,dx
+\int_\Omega \langle \varphi (x),H''(u(x))h_{0}(x)\rangle\,dx &&\\
+ \beta \int_\Omega \langle \grad \,G(u(x)),\varphi (x) \rangle \,dx &=& 0,
\end{eqnarray*}
and, since $h_{0}(x)$ is regular enough, an integration by parts would
give
$$-\Delta h_{0}(x) + H''(u(x))h_{0}(x) +\beta \grad \,G(u(x))=0$$
(which holds with $\beta =0$ in view of III.8) and
${\frac {\pa h_{0}}{\pa n }}=0$ on $\pa\Omega$. But, from the definition
of $h_{0}(x)$ and the boundary condition $u'(R)=0$, we see that
${\frac {\pa h_{0}}{\pa n }}=0$ on $\pa\Omega$ implies $u''(R)=0$.
But since III.4 is a second order ordinary differential system, if we
have $u'(R)=0=u''(R)$ then, by uniqueness, $u(r)$ is constant and this
contradiction shows that $h_{0}$ is not a critical point for III.6-III.7;
in particular, it is not a minimizer and this implies that there is a
$k\in H_{K}^{1}(\Omega)$ satisfying III.7 such that $Q(k)<0$.
The fact that $k$ can be taken in ${\cal F}(\Omega)$ follows from the
following remark: if $g\in L_{K}^{2}(\Omega )$ and $ g\not\equiv 0$
(in our case $g(x)=\grad G(u(x))$) then the set of the elements
$\psi \in {\cal F}(\Omega )$ such that $\int_\Omega
\langle g(x),\psi (x) \rangle \,dx=0$
is dense in the set of the elements $\psi \in H_{K}^{1}(\Omega)$ such
that $\int_ \Omega\langle g(x),\psi (x)\rangle\,dx=0$.
In order to prove the remark we fix an element $\varphi\in{\cal F}(\Omega)$
such that $\int_\Omega \langle g(x),\varphi (x)
\rangle \,dx \neq 0$; if $f\in H_{K}^{1}(\Omega )$ is such that
$\int_\Omega \langle g(x),f(x) \rangle =0$ and $f_{n}\in
{\cal F}(\Omega)$ is a sequence converging to $f$ in $H_{K}^{1}(\Omega )$
and we define $\hat{f}_{n} = f_{n} + \epsilon _{n}\varphi$,
where $\epsilon_{n}$ is chosen in such way that
$\int_\Omega \langle g(x),\hat{f}_{n}(x) \rangle \,dx=0$,
then $\hat{f}_{n}$ converges to $f$ in $H_{K}^{1}(\Omega )$ because
$\epsilon _{n}$ tends to zero.
Now, if $k\in {\cal F}(\Omega )$ is as above, we can construct a smooth
admissible curve that is tangent to $k$ at $u$. In fact, if $\varphi \in
{\cal F}(\Omega )$ is a fixed element such that
$\int_\Omega \langle \grad \,G(u(x)),\varphi (x) \rangle\,dx \neq 0$ and
we define the function
$$S(s,t)={\int_\Omega} G(u(x) +s\varphi (x) +tk(x))\,dx -\lambda\,,$$
we have $S(0,0)=0$ and
$${\frac{\pa S}{\pa s}}(0,0) ={\dis \int_\Omega}
\langle \grad G(u(x)),\varphi (x) \rangle \,dx \neq 0\,.$$
Hence, by the implicit
function theorem, there is $C^{2}$ function $s(t)$ defined for $t$ in
some open interval $J$ containing $t=0$ such that $s(0)=0$ and
\begin{equation}
\int_\Omega G(u(x) +s(t)\varphi (x) + tk(x))\,dx = \lambda,
\end{equation}
for $t$ in the interval $J$. So, if we define $h(t,x)=u(x) +s(t)\varphi
(x) +tk(x)$ we have ${\dis \frac{\pa h}{\pa t}}
(0,x)=k(x)$ and differentiating III.9 twice with respect to $t$ and
setting $t=0$ we get
\begin{equation}
\int_\Omega (\langle \grad G(u(x)), \frac{\pa ^{2} h}{\pa t^{2}}
(0,x) \rangle +\langle k(x),G''(u(x))k(x))\rangle )\,dx=0.
\end{equation}
Now, a short computation shows that ${\frac {d}{dt}}V(h(t,x))\bigl| _
{t=0} =0$ ( of course this is a consequence of the fact that $u$ is a
critical point for III.1-III.2 and that the curve $h(t,.)$ is admissible
in the sense of III.9). Furthermore,
\begin{eqnarray*}
\frac {d^{2}}{dt^2}V(h(t,x))\biggl| _{t=0}
&=&\int_\Omega (\langle \grad k(x),\grad \,k(x)\rangle +\langle \grad \,
u(x),\frac {\pa^{2}h(0,x)}{\pa t^{2}} \rangle \\
&&+\langle \grad F(u(x)),\frac {\pa^{2}h(0,x)}{\pa t^{2}}\rangle +
\langle k(x),F''(u(x))k(x)\rangle )\,dx \\
&=& \int_\Omega (\langle \grad \,k(x),\grad \,k(x)\rangle -\langle \Delta
u(x),\frac {\pa^{2}h(0,x)}{\pa t^{2}}\rangle \\
&&+ \langle \grad F(u(x)),\frac {\pa^{2}h(0,x)}{\pa t^{2}}\rangle
+ \langle k(x), F''(u(x))k(x)\rangle )\,dx\\
&=& \int_\Omega |\grad \,k(x)|^2\,dx +
\int_\Omega \langle k(x),H''(u(x))k(x)\rangle \,dx \\
&=&Q(k)<0\,.
\end{eqnarray*}
(we have performed an integration by parts and have used that
${\frac{\pa u}{\pa n }}=0$ in $\pa\Omega$;we have also used III.4
and III.10).
The conclusion is this: under the assumptions of the theorem, we were able
to find a curve $h(t,x)$ such that $h(0,x)=u(x)$,
$$\int_\Omega G(h(t,x))\,dx =\lambda,\quad \frac {dV(h(t))}{dt}
\biggl| _{t=0}=0\mbox{ and } \frac {d^{2}V(h(t))}{dt^{2}}\biggl| _{t=0} <0
\,.$$
Clearly this implies that $u$ is not a local minimizer for III.1-III.2
and the theorem is proved. $\spadesuit$
\paragraph {Remarks.}
\begin{description}
\item{III.11.} For bounded $\Omega$ the existence of global minimizer for
III.1 - III.2 in the subcritical case follows from standard arguments.
In a forthcoming paper we will discuss the same question for the
exterior domain.
\item{III.12.} Theorem III.3 with the same proof also holds in the case of
several constraints.
\item{III.13.} The stability and the un-stability of standing waves for the
Schrodinger equation in the exterior of a ball, with Neumann
boundary conditions, have been studied in [9]. For particular
nonlinearities it has been shown that radially symmetric standing
waves are unstable. Theorem III.3 is, perhaps, an indication that
those waves are unstable for more general nonlinearities.
If $\Omega$ is the exterior of a ball then the case $u =$ constant
cannot occur. If $\Omega$ is bounded and $F(u)$ as a function from
$\Bbb R^K$ into $\Bbb R$ has a global minimum at $u = u_0$ and $\lambda = $
(meas. $\Omega) \ G(u_0)$, then $u=u_0$ is a global minimum for III.1 -
III.2. This means that if $\Omega$ is bounded the case $u =$ constant
may occur. \end{description}
Next we give an example for which the global minimizer is not a
constant.
If we take $F(u,v) = a u v^2 + (a+1)u^3 - (4+a)u$, $G(u,v) = u^2+v^2$,
$\lambda = 1$ and $\Omega$ is either a ball or an annulus with, say,
measure one, then the minimum of III.1 - III.2 on the set of the
constant functions is achieved at $u = 1$ and $v = 0$. In order to
analyze whether this function is a local minimizer for III.1 - III.2 we
have to consider the quadratic form
\[
Q(h,k) = \int_\Omega (|\grad h(x)|^2 + |\grad k(x)|^2)\, dx +
\int_\Omega((8a-1) h^2 (x) + k^2(x))\, dx
\]
subject to $\int_\Omega h(x)\, dx = 0$. If we take $h=\varphi_2$ and
$k=0$, where $\varphi_2$ is the eigenfunction corresponding to the
second eigenvalue of $-\Delta$ on $\Omega$ with Neumann boundary
condition, we have $Q(\varphi_2, 0) = 8a-1 + \lambda_2$ and so, if we
choose a in such way that $8a-1 + \lambda_2 < 0$ then $u = 1$ and $v=0$ is
not a local minimum for III.1 - III.2. If $N \leq 3$ and $F(u,v)$ and
$G(u,v)$ are as above then $V(u,v)$ is bounded below on the admissible
set and then, by the classical methods, $V(u,v)$ has minimizer and,
according with the previous argument, this minimizer cannot be a constant
and so, by theorem III.3, it is not radially symmetric.
If we consider the unconstrained problem
\setcounter{equation}{13}
\begin{equation}
V(u) = \frac{1}{2} \int_\Omega |\grad u(x)|^2 \, dx + \int_\Omega
F(u(x))\, dx
\end{equation}
where $\Omega$ is either a ball or an annulus centered at the origin
then, the same argument presented in the proof of theorem III.3,
shows that a local minimizer for $V(u)$ in $H^1_K (\Omega) = H^1(\Omega)
\times \cdots \times H^1 (\Omega) (K$ times) is either a constant or not
radially symmetric.
If $u \in H^1_K (\Omega)$ is a {\it global} minimizer for III.14 then,
according with corollary II.10, $u$ has to be radially symmetric and
so it has to be constant.
In the scalar case $(K = 1)$ it has been known since 1978 ([10]) that
local minimizers for III.12 in $H^1(\Omega)$ are constant if either
$\Omega$ is bounded and convex (with smooth boundary) or $\Omega$ is an
annulus. This result is false if, for instance, $\Omega$ consists of two
balls joined by a thin channel ([11]) or it is shaped like a dumbbell
([12]).
Next we show that in the case $\Omega$ is bounded and convex, then the
same result holds for the system case. The proof is basically the
same as in the scalar case but a slight change is required to avoid
the maximum principle. If $\Omega$ is an annulus we do not know how to
do that (unless $\grad F(u)$ satisfies a cooperative condition).
\paragraph {Theorem III.15.}
Let $F : \Bbb R^K \rightarrow \Bbb R$ be a $C^2$ function
whose first derivatives are bounded by $c|u|^{p-1}$ for $u$ large
and $p < \frac{2N}{N-2}$. If $\Omega$ is bounded, convex and has a
smooth boundary then any local minimizer of $V$ given by III.14 in the
space $H^1_K(\Omega)$ is constant.
\paragraph{Proof.} Let $u$ be a local minimizer of $V$ in the space
$H^1_K(\Omega)$; then $u$ satisfies the elliptic system
\setcounter{equation}{15}
\begin{equation}
- \Delta u + \grad F(u) = 0, \quad \frac{\pa u}{\pa n } = 0 \qquad {\rm
on} \qquad \pa\Omega.
\end{equation}
In view of the assumptions on $F(u)$ we see that $u \in W^{3,q}_K
(\Omega), \ 1 \leq q < \infty$. If we differentiate III.16 with respect
to $x_i$, take the scalar product of the resulting equation with
$\frac{\pa u}{\pa x_i}$ and integrate we get:
\begin{equation}
\sum^N_{i=1} V'' (u) \left(\frac{\pa u}{\pa x_i}, \frac{\pa u}{\pa
x_i} \right) = \sum^K_{j=1} \int_{\pa\Omega} \langle\grad u_j (x),
\frac{\pa}{\pa n } \grad u_j (x) \rangle\,dS
\end{equation}
If $x$ is an arbitrary point in $\pa\Omega$ then, without loss we
generality, we can assume that $x$ is the origin of the coordinate
system. We suppose that $x_N = g(x_1, \ldots, x_{N-1})$ is a $C^2$
function whose graph describes the boundary of $\Omega$ in some
neighborhood of the origin and it is such that at the origin the $-x_N$
axis in the outward normal direction. Then according with [10], page
269, equation 12, we have
\begin{equation}
\langle \grad u_j(0), \frac{\pa}{\pa n } \grad u_j (0) \rangle = -
\sum^{N-1}_{i=1} \sum^{N-1}_{k=1} \frac{\pa^2 g(0)}{\pa x_i \pa x _k}
\frac{\pa u_j(0)}{\pa x _i} \frac{\pa u_j(0)}{\pa x _k} .
\end{equation}
\indent For sake of completeness we reproduce here the argument
presented in [10].
Since ${\dis \frac {\pa u_{j}(0)}{\pa x_{N}}}=
{\dis \frac { \pa u_{j}(0)}{\pa n}}=0$ we have
\begin{equation}
\langle \grad u_{j}(0),\frac {\pa \grad u_{j}(0)}{\pa n}
\rangle = - \sum_{i=1}^{N-1}\frac{\pa u_{j}(0)}{\pa x_{i}}
\frac {\pa ^{2}u_{j}(0)}{\pa x_{i}x_{N}}.
\end{equation}
\indent Now ${\dis \frac { \pa u_{j}}{\pa n}}=0$ on
$\pa \Omega$ is equivalent to
\begin{eqnarray}
\lefteqn{\sum_{i=1}^{N-1}\frac {\pa u_{j}}{\pa x_{i}}(x_{1},...,x_{N-1},
g(x_{1},...,x_{N-1})) \frac {\pa g}{\pa x_{i}}(x_{1},...,x_{N-1}) -}\\
&& \frac {\pa u_{j}}{\pa x_{N}}(x_{1},...,x_{N-1},
g(x_{1},...,x_{N-1}))=0\quad j=1,...,K. \nonumber ,
\end{eqnarray}
If we differentiate III.20 with respect to $x_{k}$, $k=1,\ldots,N-1$ and
take in account that $\frac {\pa g(0)}{\pa x_{i}}=0$,
$i=1,\ldots,N-1$ (because $x_{N}=0$ is tangent to the graph of
$g(x_1,\ldots,x_{N-1})$ at the origin) we get
\begin{equation}
\sum_{i=1}^{N-1} \frac {\pa u_{j}(0)}{\pa x_{i}}
\frac{\pa ^{2}u_j(0)}{\pa x_{k} \pa x_{N}} -
\frac {\pa^2u_j(0)}{\pa x _k \pa x _N}=0
\end{equation}
$k=1,\ldots ,N-1$.Substituting expressions III.21 for ${\dis
\frac {\pa ^{2}u_j(0)}{\pa x_{k} \pa x_{N}}}$ into III.19
we get III.18.
Since $\Omega$ is convex the right hand side of III.18 is nonpositive.
In order to prove the theorem we have to show that there is an
element $h \in H^1_K (\Omega)$ such that $V''(u) (h,h) < 0$, unless $u$
is constant. So, assume $V''(u) (h,h) \geq 0$ for any $h \in
H^1_K(\Omega)$. Since $\dis \sum^N_{i=1} V''(u) \left( \frac{\pa
u}{\pa x _i}, \frac{\pa u}{\pa x _i} \right) \leq 0$ we must have
$V''(u) \left( {\dis\frac{\pa u}{\pa x _i}, \frac{\pa u}{\pa x _i}} \right)
= 0, i=1, \ldots, N$, and ${\dis\frac{\pa}{\pa n } \frac{\pa u}{\pa x _i}}
= 0, \ i=1, \ldots, N$ because the quadratic functional $V''(u)(h,h)$
has a minimum at $h = {\dis\frac{\pa u}{\pa x _i}}$.
For a compact smooth hypersurface we know that there is a point where
the Gauss-Kronecker curvature is strictly positive; this means that
for an open set of the boundary the Hessian matrix of the function
$g(x_1, \ldots, x_{N-1})$ is positive definite and then, from III.18, we
conclude that $\frac{\pa u}{\pa x _i} = 0$, $i = 1, \ldots, N$, in
an open set of the boundary. But, since
$$- \Delta \frac{\pa u}{\pa x_i} + F''(u(x)) \frac{\pa u}{\pa x _i} = 0
\quad\mbox{and}\quad
\frac{\pa}{\pa n } \left( \frac{\pa u}{\pa x _i} \right) = 0\,,$$
from the unique continuation principle ([3]) we get $\frac{\pa
u}{\pa x_i} = 0$ everywhere in $\Omega, i = 1, \ldots, N$, and then $u$
is constant and the theorem is proved. $\spadesuit$
Next we make some remarks about the global minimizer in the case of
Dirichlet boundary condition. So we take $K=1$ (the scalar case) and
we consider III.1 - III.2 in the space $H^1_0(\Omega)$.
If $\Omega$ is the exterior of a bounded domain then there is no global
minimizer ([6]).
If $\Omega$ is a ball centered at the origin and $F(u)$ and $G(u)$ are
even functions, then a global minimizer cannot change sign and so,
thanks to a theorem of Gidas, Ni and Nirenberg ([13]), a global
minimizer has to be radially symmetric.
If $\Omega$ is the annulus $\{0< R_2\leq |x| \leq R_1 < \infty \}
$ and we take
$F(u) \equiv 0$ and $G(u) = |u|^p$, then for $p=2$ the global
minimizer in $H^1_0(\Omega)$ is the first eigenfunction which, by
uniqueness, is radially symmetric; however, for $N\geq 3$, there is a
$p_0, 2 < p_0 < \frac{2N}{N-2}$, such that for $p_0 < p <
\frac{2N}{N-2}$ the global minimizer is not radially symmetric
([14]).
\section*{IV. Codimension One Symmetry of Minimizers}
\renewcommand{\theequation}{IV.\arabic{equation}}
In this final section we consider the functional
\setcounter{equation}{0}
\begin{equation}
E(u) = \frac{1}{2} \int_\Omega |\grad u (x)|^2dx + \int_\Omega F(r, u(x)) dx
\end{equation}
subject to
\begin{equation}
\int_\Omega G(r, u(x)) dx = \lambda >0\,.
\end{equation}
As before, $u(x) = (u_1(x),\ldots, u_K(x))$ is a vector-valued function,
$|\grad u(x)|^2$ means $|\grad u_1(x)|^2 +\ldots+ |\grad u_K(x)|^2$ and
$r = (x^2_1+\ldots+x^2_N)^{1/2}$.
Problem IV.1 - IV.2 will considered in the space $H(\Omega)$ as in
section II, that is, $H(\Omega)$ is the Cartesian product of $K$
factors, each of which is either $H^1(\Omega)$ or $H^1_0(\Omega)$ and our
assumptions are the following:
\begin{description}
\item{$H_1$)} $\Omega$ is either a ball or an annulus or the exterior of a ball
centered at the origin.
\item{$H_2$)} $F(r,u)$ and $G(r,u)$ are real valued functions defined for
$(x,u) \in \Omega \times \Bbb R^K$ which are continuous with respect to
$(x,u)$ together with their first and second derivatives with
respect to $u$.
\item{$H_3$)} $E(u)$ is well defined for $u \in H(\Omega)$ in the admissible
set.
\item{$H_4$)} if $u \in H(\Omega)$ minimizes $E(u)$ on the admissible set,
then it satisfies the Euler system
\[
-\Delta u(x) + \grad F(r, u(x)) + \alpha \grad G(r, u(x)) = 0,
\]
for some constant $\alpha$, together with boundary conditions
(see remark II.4) and $u \in L^\infty(\Omega)$.
\end{description}
\paragraph{ Theorem IV.3.}
Under assumptions $H_1-H_4$, if $u \in
H(\Omega)$ is a global minimizes for IV.1 - IV.2, then there is line $L$
through the origin such that $u(\cdot)$ is symmetric with respect to any
hyperplane containing $L$.
\paragraph {Remark IV.4.}
The motivation for theorem IV.3 is that for real valued positive
functions $u(x)$ defined on the exterior of a ball, it is possible to
define a function $u^*(x)$ that has the symmetry mentioned in the
theorem and behaves like the symmetrization for positive functions
defined in $\Bbb R^N$ (see [9], proposition I.4).
For the proof of theorem IV.3 we need the following
\paragraph{Lemma IV.5.}
Let $\Omega$ be a rotation invariant subset of
$\Bbb R^N$ and let $h$ be an element of $L^1(\Omega)$. Then for any
subspace $S \subset \Bbb R^N$ of
codimension 2 there is a hyperplane $P$ containing $S$ such that
$$\int_{\Omega+} h(x)dx = \int_{\Omega-} h(x)dx\,,$$
where $\Omega_{+}$ and $\Omega_{-}$ are the intersections of $\Omega$ with
the half-spaces determined by $P$ ( we will say that the hyperplane $P$
splits the integral $ \int_{\Omega}h(x)\,dx$ in the
middle).
\paragraph{Proof.} We assume that assume the orthogonal subspace to $S$
is spanned by $e_1 =(1,0,\ldots,0)$ and $e_2 = (0,1,\ldots,0)$. We define
$e(\theta) = (\cos \theta , \sin \theta , \ldots, 0), \ 0\leq \theta \leq \pi$,
we denote by $P(\theta )$ the hyperplane spanned by $\{e(\theta )\}\cup S$ and by
$P_+(\theta )$ and $P_{-}(\theta )$ the half spaces of the vectors $x$ such
that $\langle x, e(\theta ) \rangle \geq 0$ and $\langle x, e(\theta )\rangle
\leq 0$, respectively.
Next we let
\[
g(\theta ) = \int_{\Omega+(\theta )} h(x) dx - \int_{\Omega_{-}(\theta )} h(x) dx
\]
where $\Omega_+ (\theta ) = \Omega \cap P_+ (\Omega)$ and
$\Omega_- (\theta ) = \Omega \cap P_- (\theta )$.
Then $g(\theta )$ is a single valued continuous function for $\theta $ in
$[0,\pi]$ and $g(\pi) = - g(0)$; this implies that $g(\theta )$ vanishes
somewhere and the lemma is proved.
\paragraph{Remark IV.6.}
Clearly lemma IV.5 holds also if $S$ has codimension greater than
two.
\paragraph{Proof of Theorem IV.3.}
Let $u \in H(\Omega)$ be as in the theorem; we define $h(x) = G(r,
u(x))$ and we start with any line $L_0$ through the origin. According
with lemma IV.5, we know that there is a hyperplane $P_1$ containing
$L_0$ that splits the constraint in the middle. We denote by $L_1$
the line orthogonal to $P_1$ through the origin and by $N_1$ its
corresponding unit vector. Using lemma IV.5 for $L_1$, we construct a
hyperplane $P_2$ containing $L_1$ that splits the constraint in the
middle. We denote by $L_2$ the line orthogonal to $P_2$ through the
origin and by $N_2$ the corresponding unit vector. Next we use lemma
IV.5 for the subspace $S_2$ spanned by $N_1$ and $N_2$. With this procedure
we construct mutually orthogonal hyperplanes $P_1, P_2,\ldots,P_{N-1}$
containing the the origin such that each one splits the constraint
in the middle. We may assume that the coordinate system $(x_1,
x_2,\ldots, x_N)$ is such that $P_i$ is the hyperplane $x_i=0$.
Now, arguing exactly as in the proof of theorem II.4, we see that
any global minimizer of IV.1-IV.2 is symmetric with respect to any
hyperplane containing the origin that splits the constraint in the
middle and so, $u(-x_1,x_2,\ldots, x_N) = u(x_1, x_2,\ldots, x_N)$,
$u(x_1,-x_2, \ldots, x_N) = u(x_1, x_2, \ldots, x_N)$ and then \newline
$u(-x_1, -x_2,\ldots,
x_N) = u(x_1, x_2, \ldots, x_N)$. From this last equality we see that
any hyperplane containing the $x_N-$axis splits the constraint in the
middle and this proves the theorem.
\paragraph {Acknowledgment.} We thank Prof. F. Mercury for having
taught Gauss-Kronecker curvature to us and Prof. O. Kavian for
reference [14].
\begin{thebibliography}{99}
\item O. Lopes, Radial Symmetry of Minimizers for Some Translation
and Rotation Invariant Functionals, J. Diff. Eqns. (to appear).
\item O. Kavian, Introduction \`a la Theorie des Points Critiques,
Mathematiques et Applications (SMAI), Springer-Verlag, 1993.
\item L. Hormander, The Analysis of Linear Partial Differential
Operators, vol. III, Springer-Verlag.
\item K. Benmlik, Etude Qualitative de Certain Problemes
Semi-Lineaires Elliptique, Th\`ese, Universit\'e Henry Poincar\'e,
Nancy I, 1994.
\item G. Sweers, A sign-changing global minimizer on a convex domain,
C. Bandle et allii, Progress in PDE's, Pitman Research Notes in
Mathematics Series, 266(1992), 251-258.
\item M. Esteban, Nonsymmetric ground states of symmetric
variational problems, Comm. Pure and Appl. Math., vol. XLIV(1991),
259-274.
\item M. Esteban, Rupture de sym\'etrie pour des probl\`emes de
Neumann sur-lin\'eaires dans les ouverts ext\'erieur, C.R.A.S. 308,
S\'erie I, (1989), 281-286.
\item V. Coti-Zelati-M. Esteban, Symmetry breaking and multiple
solutions for a Neumann problem in an exterior domain Proc. Royal
Soc. Edinburgh, vol. 116A(1990), 327-339.
\item M. Esteban and W. Strauss, Nonlinear bound states outside an
insulated sphere, Comm. Partial Diff. Eqns., 19(1994), No. 1-2, 177-197.
\item R. Casten and C. Holland, Instability results for a
reaction-diffusion equation with Neumann boundary conditions, J.
Diff. Eqns., 27(1978), 266-273.
\item J. Hale and J. Vegas, A nonlinear parabolic equation with
varying domain, Arch. Rat. Mech. Anal., 86(1984), No. 2,
99-123.
\item H. Matano, Asymptotic behavior and stability of solutions of
semilinear diffusion equations, Publ. Res. Inst. Math. Sci. Kyoto
Univ., 15(1979), 401-454.
\item B. Gidas, W.N. Ni and L. Nirenberg, Symmetry of Positive
Solutions of Nonlinear Elliptic Equations in $\Bbb R^N$, in Math. Anal.
Appl., part I, L. Nachbin (ed.) Academic Press, 1981.
\item O. Kavian, Personal Communication, 1995.
\end{thebibliography}
\bigskip
{\sc Orlando Lopes\newline
IMECC-UNICAMP - CP 6065 \newline
13081-970, Campinas, SP, Brazil \newline}
E-mail: lopes@ime.unicamp.br
\end{document}