\documentclass[reqno]{amsart}
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\AtBeginDocument{{\noindent\small {\em Electronic Journal of
Differential Equations}, Vol. 2006(2006), No. 122, pp. 1--4.\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 2006 Texas State University - San Marcos.}
\vspace{9mm}}
\begin{document}
\title[\hfilneg EJDE-2006/122\hfil Infinity-harmonic functions]
{A remark on $C^2$ infinity-harmonic functions}
\author[Y. Yu\hfil EJDE-2006/122\hfilneg]
{Yifeng Yu}
\address{ Yifeng Yu \newline
Department of Mathematics, University of Texas, Austin,
TX 78712, USA}
\email{yifengyu@math.utexas.edu}
\date{}
\thanks{Submitted June 15, 2006. Published October 6, 2006.}
\subjclass[2000]{35B38}
\keywords{Infinity Laplacian equation;
infinity harmonic function; \hfill\break\indent viscosity solutions}
\begin{abstract}
In this paper, we prove that any nonconstant, $C^2$ solution
of the infinity Laplacian equation $u_{x_i}u_{x_j}u_{x_ix_j}=0$
can not have interior critical points. This result was first
proved by Aronsson \cite{a2} in two dimensions. When the solution
is $C^4$, Evans \cite{e1} established a Harnack inequality
for $|Du|$, which implies that non-constant $C^4$ solutions
have no interior critical points for any dimension.
Our method is strongly motivated by the work in \cite{e1}.
\end{abstract}
\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{remark}[theorem]{Remark}
\section{Introduction}
In the 1960's, Aronsson introduced the notion of the absolutely
minimizing Lipschitz extension. Namely, $u\in W^{1,\infty}(\Omega)$
is said to be an \emph{absolutely minimizing Lipschitz extension} in
some bounded open subset $\Omega$ if for any open set
$V\subset\Omega$, we have that
$$
\sup_{x\ne y\in \partial V} {|u(x)-u(y)|\over |x-y|} =\sup_{x\ne
y\in \bar V} {|u(x)-u(y)|\over |x-y|}.
$$
The results in Crandall-Evans-Gariepy \cite{c-e-g} imply that the
above definition is in fact equivalent to say that for any open
set $V\subset \Omega$ and $v\in W^{1,\infty}(V)$,
$$
u|_{\partial V}=v|_{\partial V}\; \Rightarrow \;
||Du||_{L^{\infty}(V)}\leq ||Dv||_{L^{\infty}(V)}.
$$
The second characterization is what Jensen used in his influential
paper \cite{j1} where he proved that $u\in W^{1,\infty}(\Omega)$ is
an absolutely minimizing Lipschitz extension with given Lipschitz
continuous boundary date $g$ if and only if it is a viscosity
solution of the following infinity Laplacian equation.
\begin{gather*}
u_{x_i}u_{x_j}u_{x_ix_j}=0 \quad \text{in }\Omega \\
u|_{\partial \Omega}=g.
\end{gather*}
He also showed that the above infinity Laplacian equation has
a unique viscosity solution with given continuous boundary data. A direct
consequence is that absolute minimizing Lipschitz extension is unique with given
boundary data.
We also name a viscosity solution of the infinity
Laplacian equation as an infinity harmonic function. Recently, people have
tremendous interest in this
degenerate elliptic equation. The interested readers can find most
of relevant works in the note Crandall \cite{c1}.
The focus of this work is on classical solutions (i.e, $C^2$) of the
infinity Laplacian equation. As observed by Aronsson \cite{a2},
smooth solutions of the infinity Laplacian equation have some
special properties which
are in general not possessed by viscosity solutions. In our paper,
we study one of them, i.e, the non-vanishing gradient. From now on, we assume that
$\Omega$ is a connected bounded open set. By carefully studying the gradient flows
of $C^2$ solutions (note that
$|Du|$ is constant along the gradient flow of a $C^2$ solution $u$), Aronsson
proved in \cite{a2} that $|Du|$ will nowhere be zero unless $u$ is constant when
$n=2$.
Recently Jensen mentioned a simple proof of Aronsson's
result in a seminar talk. Using some elementary maximum principle
argument, Evans \cite{e1} extended Aronsson's result to $n\geq 3$ for $C^4$
infinity harmonic functions. In fact, Evans established a harnack inequality for
$|Du|$. We found that part of
Evans's argument can be interpreted in viscosity sense. From
that, we are able to establish a weak Hopf-type lemma for
$|Du|$ instead of the Harnack inequality, which is sufficient to
prove the following new result.
\begin{theorem} \label{thm1.1}
Let $\Omega$ be a connected open subset of $\mathbb{R}^n$. Assume that
$u$ is a $C^2$ solution of
$$
\Delta _{\infty}u=0 \quad \text{in }\Omega.
$$
If $Du(z)=0$ for some $z\in \Omega$, then $u\equiv u(z)$.
\end{theorem}
\begin{remark} \label{rmk1.2} \rm
In general, infinity harmonic functions might not be $C^2$. For
example, $u(x,y)=x^{4\over 3}-y^{4\over 3}$ is a $C^{1,{1\over 3}}$
infinity harmonic function in $\mathbb{R} ^2$. See Aronsson
\cite{a3}. It is clear that Theorem \ref{thm1.1} does not hold for this
non-classical solution since $(0,0)$ is its critical point. A main
open problem of the infinity laplacian equation is whether any
viscosity solution is $C^1$. Savin \cite{s1} proved the $C^1$
regularity when $n=2$. We just learned that in a forthcoming paper,
Evans and Savin \cite{e-s} proved the $C^{1, \alpha}$ regularity
when $n=2$. For higher dimensions, the regularity issue remains a
very challenging problem.
\end{remark}
\section{Proof of the main theorem}
In this section, we prove Theorem \ref{thm1.1}.
Following the notations in Evans \cite{e1}, we denote
$v(x)=|Du(x)|$. If $v(x)\ne 0$, set
$$
\nu ^{i}={u_{x_i}\over |Du|}={u_{x_i}\over v}\quad (1\leq i\leq n),
$$
and also write
$$
h_{ij}=\nu ^{i}\nu ^{j}.
$$
Then we have the following lemma.
\begin{lemma} \label{lem2.1}
If $v\ne 0$ in $\Omega$, then $v$ is a viscosity solution of
\begin{equation} \label{e2.1}
h_{ij}v_{x_ix_j}=-{|Dv|^2\over v} \quad \text{in $\Omega$}.
\end{equation}
\end{lemma}
\begin{proof}
First we want to remark that \eqref{e2.1} was derived in \cite{e1}
for $u\in C^3$. Since $u\in C^2$, we have that
$$
\nu ^{i} v_{x_i}=0 \quad \text{in }\Omega.
$$
Hence
$$
(h_{ij}v_{x_i})_{x_j}=0.
$$
Assume that for $x_0\in \Omega$ and $\phi \in C^2(\Omega)$,
$$
\phi (x)-v(x)>\phi (x_0)-v(x_0)=0,
$$
for $x\in \Omega\backslash \{x_0 \}$. Then a standard argument shows
that
$$
(h_{ij}\phi_{x_i})_{x_j}(x_0)\geq 0.
$$
Therefore, following the calculations in \cite{e1},
\begin{align*}
h_{ij}\phi_{x_ix_j}(x_0)
&\geq -\nu_{x_j}^{i}\nu ^{j}\phi_{x_i}(x_0)-\nu_{x_j}^{j}\nu ^{i}\phi_{x_i}(x_0) \\
&=-\nu_{x_j}^{i}\nu ^{j}v_{x_i}(x_0)-\nu_{x_j}^{j}\nu ^{i}v_{x_i}(x_0) \\
&=-\nu_{x_j}^{i}\nu ^{j}v_{x_i}(x_0) \\
&=-({u_{x_ix_j}\over v}-{u_{x_i}v_{x_j}\over v^{2}})\nu ^{j}v_{x_i}(x_0) \\
&=-{\nu ^{j}v_{x_i}u_{x_ix_j}\over v}(x_0)=-{|Dv(x_0)|^{2}\over v(x_0)}.
\end{align*}
Hence $v$ is a viscosity subsolution of \eqref{e2.1}. Similarly, we
can show that $v$ is a viscosity supersolution of \eqref{e2.1}.
\end{proof}
Next we prove a weak Hopf type Lemma.
\begin{lemma} \label{lem2.2}
Suppose that $v\ne 0$ in $\Omega$ and $\bar B_{r}(x_0)\subset \Omega $
for some $r>0$. Assume that
$\min_{\bar B_{r\over 2}(x_0)}v\geq \delta$.
Then there exists $\epsilon_{0}>0$ which only depends on $r$ and
$\delta$ such that if $0<\epsilon=\min_{\partial B_{r}(x_0)}v
<\epsilon _{0}$, then
$$
\epsilon=\min_{\partial B_{r}(x_0)}v>\min_{\bar \Omega}v.
$$
\end{lemma}
\begin{proof}
Choose $x_{\epsilon}\in \partial B_{r}(x_0)$ such that
$$
v(x_{\epsilon})=\epsilon=\min_{\partial B_{r}(x_0)}v.
$$
Let
$$
w_{\epsilon}=\log v-\log \epsilon.
$$
Since $\nu ^{i}v_{x_i}=0$, owing to Lemma \ref{lem2.1}, we discover that
$w_{\epsilon}$ is a viscosity solution of
$$
h_{ij}w_{\epsilon,x_ix_j}=-|Dw_{\epsilon}|^2.
$$
For $k>0$, denote
$$
f_{k}(x)=k(r^2-|x-x_0|^2).
$$
A simple calculation shows that if we choose $k=4/r^2$,
$$
h_{ij}f_{k,x_ix_j}>-|Df_{k}|^2 \quad \text{in } \{{r\over 2}\leq
|x-x_0|\leq r \}.
$$
Since $\min_{B_{r\over 2}(x_0)}{\log }v\geq {\log }\delta$, there
exists a $\epsilon_{0}$ depending only on $r$ and $\delta$ such that
if $\epsilon <\epsilon _{0}$, we have that
$$
w_{\epsilon}\geq f_{4/r^2} \quad \text{on }\partial B_{r\over
2}(x_0).
$$
Also,
$$
w_{\epsilon}\geq 0=f_{4/ r^2} \quad \text{on }
\partial B_{r}(x_0).
$$
Since $f_{4/r^2} $ is smooth, by comparison, we derive that
$$
w_{\epsilon}\geq f_{4/r^2} \quad \text{in }\{{r\over 2}\leq
|x-x_0|\leq r \}.
$$
In particular,
$$
{\partial w_{\epsilon}\over \partial n}(x_{\epsilon})\geq {\partial
f_{4/r^2}\over \partial n}(x_{\epsilon})={8\over r}>0,
$$
where $n$ is the inward normal vector of $\partial B_{r}(x_0)$ at
$x_{\epsilon}$. Hence Lemma \ref{lem2.2} holds.
\end{proof}
\begin{proof}[Proof of Theorem \ref{thm1.1}]
We argue by contradiction. If $u$ is
not constant, then there exists $x_0\in \Omega$ and $r>0$ such that
$v>0$ in $B_{r}(x_0)$ and
$$
\partial B_{r}(x_0)\cap \{x\in \Omega|\ v(x)=0\}\ne \Phi.
$$
For $\epsilon >0$, denote
\begin{gather*}
u_{\epsilon}(x,x_{n+1})=u(x)+\epsilon x_{n+1},
B_{r}(x_0,0)=\{(x,x_{n+1})\in \mathbb{R} ^n\times \mathbb{R}|\
|x-x_0|^2+x_{n+1}^{2}\leq r^2 \}.
\end{gather*}
Then we have that for any $\epsilon >0$,
\begin{gather*}
\min_{\partial B_r(x_0,0)}|Du_{\epsilon}|=\epsilon=\min_{\Omega
\times \mathbb{R}}|Du_{\epsilon}|,
\\
\min_{\bar B_{r\over 2}(x_0,0)}|Du_{\epsilon}|>\min_{B_{r\over
2}(x_0)}|Du|>0.
\end{gather*}
Since $u_{\epsilon}$ is a $C^2$ infinity harmonic function in
$\Omega \times \mathbb{R}$ and $|Du_{\epsilon}|>0$, applying
Lemma \ref{lem2.2} to $u_{\epsilon}$, we get contradiction for small $\epsilon$.
\end{proof}
\begin{remark} \label{rmk2.3} \rm
Evans \cite{e1} showed that if $u\in C^4$, then $z={|D|Du||\over
|Du|}$ is a subsolution of the following equation
\begin{equation} \label{e2.2}
-h_{ij}z_{x_ix_j}\leq -z^2+w_{x_i}z_{x_i},
\end{equation}
where $w=\log |Du|$. Owing to the quadratic term $z^2$, he is able
to derive that $z$ is locally bounded, which implies that $|Du|$
satisfies a Harnack inequality. Evans's proof also implies that the
only entirely $C^4$ solutions (i,e, $u\in C^{4}(\mathbb R ^n)$) in
$\mathbb{R}^n$ are linear functions.
Aronsson \cite{a2} proved this Liouville type theorem for $C^2$ solutions
when $n=2$. It is not clear to us whether $z$ is a viscosity
subsolution
of \eqref{e2.2} if we only assume that $u\in C^2$. If it is true,
we can show that the only entirely classical solutions (i.e, $u\in
C^2(\mathbb R ^n)$) in $\mathbb{R} ^n$ are linear functions.
\end{remark}
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\emph{Extension of functions satisfying Lipschitz conditions}, Ark.
Mat. 6 (1967), 551-561.
\bibitem{a2} G. Aronsson,
\emph{On the partial differential equation
$u_{x}^{2}u_{xx}+2u_{x}u_{y}u_{xy}+u_{y}^{2}u_{yy}=0$}, Ark. Mat. 7
1968 395--425 (1968).
\bibitem{a3} G. Aronsson,
\emph{On certain singular solutions of the partial differential
equation $u_{x}^{2}u_{xx}+2u_{x}u_{y}u_{xy}+u_{y}^{2}u_{yy}=0$},
Manuscripta Math. 47 (1984), no. 1-3, 133--151.
\bibitem{c1} M. G. Crandall, \emph{A visit with the $\infty$-Laplace equation},
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\bibitem{c-e-g} M. G. Crandall, L. C. Evans, R. Gariepy,
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\emph{Estimates for smooth absolutely minimizing Lipschitz
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\bibitem{e-s} L. C. Evans, O. Savin, in preparation.
\bibitem{s1} O. Savin,
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\bibitem{j1} R. Jensen, \emph{Uniqueness of Lipschitz extensions minimizing the
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\end{thebibliography}
\end{document}