\documentclass[reqno]{amsart}
\AtBeginDocument{{\noindent\small
{\em Electronic Journal of Differential Equations},
Vol. 2005(2005), No. 43, pp. 1--5.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu (login: ftp)}
\thanks{\copyright 2005 Texas State University - San Marcos.}
\vspace{9mm}}
\begin{document}
\title[\hfilneg EJDE-2005/43\hfil Uniqueness of critical points]
{Uniqueness of critical points for semi-linear Dirichlet problems
in convex domains}
\author[J. Arango\hfil EJDE-2005/43\hfilneg]
{Jaime Arango}
\address{Jaime Arango\hfill\break
Universidad del Valle\\
Departamento de Matem\'aticas\\
A. A. 25 360 Cali, Colombia}
\email{jarango@univalle.edu.co}
\date{}
\thanks{Submitted February 3, 2005. Published April 7, 2005.}
\subjclass[2000]{74K15, 35J05, 65M06}
\keywords{Critical points; semi linear Dirichlet problems}
\begin{abstract}
We establish sufficient conditions for the existence
of a unique critical point for the solution to the semi linear
elliptic problem $ \Delta u = f(u)+w $ with zero Dirichlet
boundary condition.
\end{abstract}
\maketitle
\numberwithin{equation}{section}
%\def\N{\mathbb{N}}
%\def\Q{\mathbb{Q}}
%\def\C{\mathbb{C}}
%\def\Z{\mathbb{Z}}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\section{Introduction}
Regarding the St. Vennat elastic torsion problem
\begin{equation} \label{torsion}
\begin{gathered}
\Delta u = w \quad \text{in } \Omega,\\
u=0 \quad \text{on } \partial \Omega,
\end{gathered}
\end{equation}
in a convex domain $\Omega$,
Makar-Limanov \cite{makar} proved that for a solution $u$,
the function $z= \sqrt{-u}$ is concave provided $w $ is a positive constant.
From this, it follows immediately that $u$ has exactly one critical
point (point of vanishing gradient), which turns out to be an absolute
minimum of $u$ on $\Omega$.
The elastic torsion problem \eqref{torsion} is a classical issue
in PDE with references dating back to St. Vennat (1856). In
addition to its importance in elasticity, it arises in fluid
mechanics, where it describes the steady unidirectional flow of a
viscous fluid down a pipe of cross section $\Omega$, the pressure
of the gradient along the pipe being constant. It appears also in
connection with vortex streets (see \cite{keady1,keady2}) and
isoperimetrical inequalities \cite{payne}.
An important question related to problem \eqref{torsion} is to determine
the minimum of $u$ and its location in $\Omega$. This task can
be greatly simplified if it is assured -- and here lies the
importance of Makar-Limanov's result -- that $u$ has a single
minimum. In this paper we consider the following generalization of
the above problem,
\begin{equation} \label{deflection}
\begin{gathered}
\Delta u = f(u)+w \quad \text{in } \Omega,\\
u=0 \quad \text{on } \partial \Omega,
\end{gathered}
\end{equation}
and the question of whether $u$ possesses a unique critical point.
Addressing this question, Kawohl \cite{kawohl} shows the concavity
of $g \circ u$ (for an appropriated choice of $g$). Yet Kawohl's
results are strongly based on conditions on the second derivative
of the nonlinearity $f$. In a more recent paper Ma \cite{ma}
showed, in a slightly different context, the convexity of the
solution $u$ for specific nonlinearities.
The aim of this paper is to show that the solution to \eqref{deflection}
possesses a unique critical point (a fortiori an absolute minimum)
provided the following assumptions hold:
\begin{itemize}
\item[(A1)] $ \Omega $ is convex planar domain with a $C^2$ smooth
boundary $\partial \Omega$ having a positive curvature.
\item[(A2)] $w >0$, $f \in C^{\infty}(\mathbb{R},\mathbb{R})$, $f(0)=0$,
$f'(x)>0$ for $x<0$.
\end{itemize}
Our approach is strongly based in the geometrical properties of
eigenfunctions of some Laplace-related operators, and is very
closed to the one used by Cabr\'e and Chanillo in \cite{cabre}.
Before going into details, we remark that the solution of
\eqref{deflection} could be interpreted as the deflection of a
membrane fixed in its border hanging under the force $f(u)+w $,
where $w >0$ is a constant proportional to the density of the
membrane. Having this in mind it seems very plausible that $u$ has
a single minimum.
\section{Preliminaries}
Existence, uniqueness and regularity of solutions to semilinear
elliptic differential equation of second order are relatively
well understood issues in the theory of PDE. We refer the reader
to the classical work of Gilbarg and Trudinger \cite{gilbarg} for
a thorough treatment of these topics. For instance, the existence
of a negative solutions $u$ to the problem \eqref{deflection}
follows from standard techniques of upper and lower solutions,
whereas the uniqueness of negative solutions can be obtained via
maximum principle. We limit ourselves to analyze negative
solutions to \eqref{deflection} for they plausibly model the
deflections of membranes under its own weight. Therefore until
further notice, all solutions to \eqref{deflection} we consider
are supposed to be negative on $\Omega$. Regarding the regularity,
it can be shown that a solution $u$ to the problem
\eqref{deflection} satisfies
$u \in C^{\infty}(\Omega)\cap C^{1}(\overline{\Omega})$.
To begin the discussion of critical points let us suppose $x_m \in
\Omega$ is an absolute minimum of a solution to
\eqref{deflection}. Then for all $x\in \Omega$ we have
\begin{equation}
\label{ineqhopf} 0\le \Delta u (x_m) = f(u(x_m))+w \le f(u(x))+w
=\Delta u (x),
\end{equation}
the last inequality being a consequence of assumption (A2).
We go on by proving that the solution $u$ has no critical points on
$\partial\Omega$. The argument is based on Hopf's boundary point lemma.
For the reader's convenience we present here a slightly modification
of Hopf's result (see\cite{protter} and \cite{gilbarg}):
\begin{lemma}[Hopf, 1952] \label{lem1}
Let $G\subset \mathbb{R}^2$ be a bounded region. Further assume that
$u \in C^2(G) \cap C(\overline{G})$ satisfies $\Delta u \ge 0$
and $u \le 0$ in $G$. If $G$ fulfills an interior sphere condition
\footnote{There is a ball $B\subset G$ with $x_0\in \partial G$}
at $x_0\in \partial G$, and $u \in C^1(G \cup \{x_0\})$,
with $u(x)< u(x_0)$, for $x \in G$, then for every
direction $\nu$ pointing into an interior sphere we have
\[
\frac{\partial u}{\partial \nu} (x_0)<0.
\]
\end{lemma}
We remark that the region $\Omega$ along with every point $x_0\in
\partial \Omega$ and the solution $u$ to \eqref{deflection},
satisfy the assumptions of Hopf's lemmas. Therefore it may be
concluded that there are no critical points of $u$ at the border
$\partial \Omega$. The reader may have noticed that the border
$\partial \Omega$ is a level set of $u$. Unfortunately we cannot
repeat the same argument for any level set of $u^{-1}(c)$ since
there is no guaranty that such level sets fulfill the interior
sphere condition.
In a very influential paper, Cheng \cite{cheng} studies the topology
of particular level sets, more precisely he studies
the set $ v^{-1}(0)$, called nodal lines of $v$, when the real valued
function $v$ is a solution to an elliptic equation on a Riemann manifold.
The following summarizes Cheng's results when $\Omega$ is a bounded domain
of $\mathbb{R}^2$:
\begin{theorem}[Cheng, 1976] \label{thm1}
\label{chgthm} Let $h$ be $C^{\infty}$ on $\Omega$. If $v$ satisfies
\begin{equation} \label{chg}
\left( \Delta + h(x)\right) v=0 \quad \text{in } \Omega,
\end{equation}
then for all $p \in \Omega$, there exist a spherical harmonic
$P_N$ in $\mathbb{R}^2$ of degree $N \ge 1$, and a $C^1$ diffeomorphism $\Phi$
from a neighborhood of $p$ in $\Omega$ onto a neighborhood of $\Phi(p)=0$
in $\mathbb{R}^2$, such that
\[
v(x)= P_N(\Phi(x)).
\]
\end{theorem}
For the proof we refer the reader to \cite{cheng}.
Roughly speaking, Theorem \ref{chgthm} says that the nodal lines of $v$
are locally diffeomorphic to the nodal lines of spherical harmonics
of degree $N$ at the origin, for some $N\ge 1$. Now, as it is well known,
the nodal lines of spherical harmonics are a system of equiangular
rays at the origin. Accordingly, if $v$ satisfies (\ref{chg}),
then following are true:
\begin{itemize}
\item [(a)] The critical points on the nodal lines are isolated.
\item[(b)] The nodal lines meet only at critical points forming
there an equiangular system of more than three angles.
\end{itemize}
\section{Main results}
Let $\tau\in S^1\subset \mathbb{R}^2$ and assume $u$ solves problem
\eqref{deflection}. We shall consider the nodal lines of
$u_{\tau}$ defined by (cf. Cabr\'e and Chanillo \cite{cabre})
\[
u_{\tau}(x) = D u(x) \cdot \tau, \quad x\in \overline{\Omega }.
\]
Let us write $J (x)\equiv D^2u(x)$. A regular point $p\in \overline{\Omega}$ of the nodal lines $u_{\tau}$ satisfies $J (p)\, \tau \ne 0$, and the nodal lines can be locally parametrized at $p$ by the ODE
\begin{equation}
\label{nli}
\dot{x}(t) = R\,J (x(t))\, \tau,
\qquad x(0) = p,
\end{equation}
where $R$ is the $\frac{\pi}{2}$ rotation $2\times 2$-matrix.
\begin{lemma} \label{notangency}
Fix $\tau\in S^1$ and denote by $U$ the nodal lines of $u_{\tau}$.
If $p\in U\cap\partial\Omega$, then $p$ is a regular point of $U$ and $U$
is nowhere tangent to $\partial\Omega$. Moreover, $U$ split $\Omega$ in
a finite number of connected subregions.
\end{lemma}
\begin{proof}
Since $\partial\Omega$ has a positive curvature (assumption (A1)),
for all $p \in \partial\Omega$ we have
$|J (p)\, \alpha\cdot \alpha|>0$,
where $\alpha$ is any unitary tangent vector to $\partial \Omega$ at $p$.
On the one hand, $Du(p)$ is orthogonal to $\alpha$; on the other hand
$Du(p)$ is orthogonal to $\tau$ since $p\in U$, and therefore $\alpha$
and $\tau$ are collinear and
$|J (p)\, \tau\cdot\tau|>0$.
Thus $J (p)\, \tau \ne 0$, and this implies that $p$ is a regular
point of $U$.
Let us locally parametrize $U$ at $p\in U\cap\partial\Omega$ by
ODE \eqref{nli}. If $U$ is tangent to $\partial\Omega$ at $p$ then
$ R\, \dot{x}(0)\cdot \tau = 0$. From this it follows
$-J (p)\, \tau\cdot\tau=0 $ which is a contradiction.
\end{proof}
We remark that that $u_{\tau}$ satisfies
\begin{equation} \label{eveqn}
( \Delta + h(x))u_{\tau}=0 \quad \text{in } \Omega,
\quad \text{where}quad h(x)\equiv - f'(u(x)).
\end{equation}
Since any solution of \eqref{deflection} is $C^{\infty}$ we have,
in view of assumption (A2), that $h$ is $C^{\infty}$ and
negative on $\Omega$. Notice that $u_{\tau}$ cannot vanish on
the border $\partial G$ of a sub domain $G\subset \subset \Omega$.
For in that case, $u_{\tau}$ satisfies elliptic equation
\eqref{eveqn} in $G$ and the boundary condition
$u_{\tau}\big|_{\partial G}=0$, then a straightforward
application of the maximum principle (recall that $h$ is
negative) yields $u_{\tau}\equiv 0$ on $G$, and a fortiori,
$u_{\tau}\equiv 0$ on the whole domain $\bar{\Omega}$ which
contradicts Hopf's lemma.
\begin{lemma} \label{ncp}
There are no critical points of $u_{\tau}$ along the nodal lines
$u_{\tau}(x)=0$. Moreover, the nodal lines $u_{\tau}(x)=0$ reduces to
the trace of a single $C^{\infty}$ curve which intersects
$\partial \Omega$ in exactly two points.
\end{lemma}
\begin{proof} Let us denote by $U$ the nodal lines of of $u_{\tau}$ and
let $G$ be one of the subregions in which $U$ splits $\Omega$. We may rule
out the case $G\subset \subset \Omega$, hence $\partial G$ contains a
non-trivial connected subset of $\partial \Omega$.
By assumption (A1) and lemma \ref{notangency} there are exactly two
points on $\partial \Omega$ having a unit normal direction orthogonal
to a fixed $\tau\in S^1$. Hence, $U$ intersects $\partial \Omega$
in exactly two points.
Now suppose there is a critical point of $u_{\tau}$ along $U$.
By remark (b) there are at least four different regions, say
$G_j$ for $j=1, \dots 4$, in which in which $U$ splits $\Omega$.
Since for any $j=1, \dots 4$ we may rule out the case
$G_j\subset \subset \Omega$, we conclude that $U$ intersect
$\partial \Omega$ in at least four different points, which is
again a contradiction.
If $U$ contains no critical points of $u_{\tau}$, then by theorem \ref{chg},
$U$ is locally diffeomorphic to the nodal lines of an spherical harmonic
of degree 1. Therefore, $U$ is either diffeomorphic to a circle or is the
trace of a single $C^{\infty}$ curve intersecting $\partial \Omega$ in
exactly two points. But the first case does not apply since this would
imply $G\subset \subset \Omega$, where $G$ is the region bounded by $U$.
\end{proof}
We can rephrase Lemma \ref{ncp} as follows:
\[
u_{\tau}(q)=0 \quad \text{implies} \quad J (q)\, \tau \neq 0.
\]
Next observe that any critical point of $u$ belongs to any of the nodal
lines $u_{\tau}(x)=0$ for $\tau \in S^1$. Hence
\[
Du(q)=0 \quad \text{implies} \quad J (q)\, \tau \neq 0 \quad
\text{for all }\tau \in S^1 .
\]
In other words, if $q$ is a critical point of $u$ then the $2\times 2$
matrix $J (q)$ is nonsingular. We have thus proved the following result.
\begin{lemma} \label{finite}
The negative solution $u$ to \ref{deflection} possesses a finite number
of critical points.
\end{lemma}
By lemma \ref{ncp} the nodal lines are in fact the trace of a single
$C^{\infty}$ curve, thus it makes senses to to write nodal line instead of
nodal lines. For now on we shall adhere to this renaming.
Suppose a nodal line $Du(x)\cdot \tau=0$ passes through
$ p \in \partial \Omega$, i. e., $Du(p)\cdot \tau=0 $. It follows then
$ \tau =\pm R\,\mathbf{n}(p)$, where $\mathbf{n}(p)$ is the unit normal
outward directions at $p\in \partial \Omega$. In view of
\eqref{nli}, the nodal line passing through $ p \in \partial \Omega$
can be parametrized by the solution to the initial value ODE problem
\begin{equation} \label{ode}
\dot{x}(t) = R\,J (x(t))\, R\, \mathbf{n}(p),
\quad x(0) = p.
\end{equation}
We are now in a position to show the main result of this paper.
\begin{theorem} \label{thm2}
The negative solution $u$ to \ref{deflection} possesses a single critical
point which is an absolute minimum to $u$ in $\Omega$.
\end{theorem}
\begin{proof}
We begin by recalling that the solutions to \eqref{ode} have a uniformly
continuous dependence on $p \in \partial \Omega$. Furthermore, notice that
any level set meets all critical points, thus by lemma \ref{finite} it
makes sense to define $f(p)$, for $p \in \partial \Omega$,
to be the first critical point met by the solutions to (\ref{ode}).
By a standard continuity argument $f$ must be constant on the whole
border $\partial \Omega$.
By lemma \ref{ncp} any nodal line intersects $\partial \Omega$ in exactly
two points, hence, for a given $p \in \partial \Omega$ there exists a
unique $\bar{p} \in \partial \Omega$, with $\bar{p} \neq p$, such that
$\bar{p}$ and $ p$ belong to the same nodal line. It is clear that
$f(p) \neq f(\bar{p})$ unless $u$ possesses a single critical point.
\end{proof}
\subsection*{Acknowledgments}
The author would like to thank professor G. Keady at the
University of West Australia for pointing out the important references
on the torsion problem,
also to professor O. Perdomo at Universidad del Valle for several
helpful comments concerning Cheng's theorem,
The author also wants to thank Universidad del Valle for financially
supporting this investigation.
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\end{document}