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\markboth{\hfil Removable singular sets \hfil EJDE--1999/04}
{EJDE--1999/04\hfil Lihe Wang \& Ning Zhu \hfil}
\begin{document}
\title{\vspace{-1in}\parbox{\linewidth}{\footnotesize\noindent
{\sc  Electronic Journal of Differential Equations},
Vol. {\bf 1999}(1999), No.~04, pp. 1--5. \newline
ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu
\newline ftp  ejde.math.swt.edu \quad ftp ejde.math.unt.edu (login: ftp)}
 \vspace{\bigskipamount} \\
%
 Removable singular sets of fully nonlinear elliptic equations 
\thanks{ {\em 1991 Mathematics Subject Classifications:} 35B65.
\hfil\break\indent
{\em Key words and phrases:} Nonlinear PDE, Monge-Ampere Equation, 
Removable singularity.
\hfil\break\indent
\copyright 1999 Southwest Texas State University  and University of
North Texas. \hfil\break\indent
Submitted March 17, 1998. Published February 17, 1999. \hfil\break\indent
Partially supported  by NSF grant DMS-9801374 and a Sloan
Foundation Fellowship} }
\date{}
%
\author{Lihe Wang \& Ning Zhu}
\maketitle

\begin{abstract} 
In this paper we consider fully nonlinear elliptic equations,
including the  Monge-Ampere equation and the Weingarden  equation. 
We assume that
\begin{eqnarray*}
&F(D^2u, x) = f(x) \quad x \in \Omega\,,&\\
&u(x) = g(x) \quad x\in \partial \Omega & 
\end{eqnarray*}
has a solution $u$ in $C^2(\Omega) \cap C(\bar {\Omega} )$, and
\begin{eqnarray*}
&F(D^2v(x), x) = f(x) \quad x\in \Omega\setminus S\,,&\\
&v(x)= g(x)\quad x\in \partial \Omega &
\end{eqnarray*}
has a solution $v$ in 
$C^2(\Omega\setminus S ) \cap \mbox{Lip}(\Omega) \cap C(\bar {\Omega})$.
We prove that under certain conditions on $S$ and $v$, 
the singular set $S$ is removable; i.e., $u=v$. 
\end{abstract}


\newtheorem{theorem}{Theorem}
\newtheorem{lemma}{Lemma}
\newtheorem{corollary}{Corollary}


\section{Introduction} 
Removability of singularities of solutions to
elliptic equations has studied extensible.
Known results include the fact that isolated singularities of bounded harmonic 
functions are removable. J\"orgens \cite{J} stated the related
result that the isolated singularity of the Monge-Ampere equation,
in two dimensions, is removable if the solution is $C^1$ along a curve 
passing though the singularity. J\"orgens' result was extended in 1995 
by Beyerstedt \cite{B} who considered isolated singularity
for general equations in $n$-dimensions.

In this paper, we  use rather elementary tools to prove removability of singular
sets in arbitrary dimensions
. Our result  for the Monge-Ampere 
equation is optimal, as shown by the examples in \cite{C}.

\paragraph{The Maximum Principle.}
In this paper, we use a generalized version of the Aleksandroff Maximum 
Principle (see Lemma 2 below). Let us start out with the following lemma.

\begin{lemma}
Let $B= \{ x: \Gamma v(x) = v(x) \} $, where 
$$\Gamma u(x) = \sup \{ w(x): w \mbox{ is convex and } w\le v \mbox{ on } 
\bar {\Omega}\} \,.$$
If $ v\in \mbox{\rm Lip}(\Omega)$ and  $v|_{\partial \Omega} \ge 0 $, 
then $ \{ p: |p|< M/D \}$ is contained in the set 
$$\{p: p \mbox{ is normal of the tangent plane of }  z(x)=v(x)  
\mbox{  at  some } x_0\in B \}\,. $$
\end{lemma}

\paragraph{Proof.}
For each $p$ satisfying  $ |p|\le M/D$, suppose that $v$ take its minimum at  
$x_0$, and $v(x_0) = -M$.
Consider the plane $ \pi $ defined by
$$
x_{n+1} = -M + p \cdot ( x-x_0)\,.
$$
When $ x\in \partial \Omega $, we have 
\begin{eqnarray*}
x_{n+1} & \le & -M + |p\cdot (x-x_0)|\\
&\le & -M + |p|D \le 0\,.
\end{eqnarray*}
But $ \min_{\
partial \Omega} v(x) \ge 0 $, so that, 
$ -M*p\cdot (x-x_0)|_{\partial \Omega} \le v(x) |_{\partial \Omega}$.
We can take $ M_0\le -M $ such that for all $ x\in \bar {\Omega} $
we have 
$$
M_0 + p\cdot (x-x_0) \le v(x) 
$$
and for all $ M'> M_0 $, there exist $ x_1 \in \bar{\Omega} $, such 
that 
$$
M' + p\cdot (x_1-x_0) > v(x_1)\,. 
$$
We can also prove that the set 
$$
G= \{ x : M_0 + p\cdot (x-x_0) = v(x) \}
$$
satisfies $ G \subset B $. In fact, if there is a point $ y\in G $
with $ y\notin B $, then
 $\Gamma v(y)< v(y) = M_0 + p\cdot ( y-x_0)   $.
The set  
$ G_1 = \{ y: \Gamma v(y) < v(y), y \in \bar{\Omega }\} $
is open in $ \bar{ \Omega }$. Since $ v(y) \ge v(y),
y\in G_1 $, we can take 
 $$
\Gamma' v(x) = \left\{ 
\begin{array}{ll}
\Gamma v(x)  & x \notin G_1\\
M_0+p\cdot (x-x_0) & x\in \bar{ G}_1\,.
\end{array}
\right.
 $$
Then $ \Gamma ' v $ is convex, and $ \Gamma' v \le v, \Gamma' v(x)>
\Gamma v(x) $ for $ x\in G_1  $,  which is a contradiction to the
definition of $ \Gamma v $.
Therefore,  $ G\subset B $ and the present proof is complete. 

\begin{lemma}
For $ u\in \mbox{\rm Lip} (\Omega )$, $u|_{\partial \Omega} \ge 0$,  and
$\min_{\bar{\Omega}}u = M <0 $,  there is a constant $ C $
depending only on the domain $\Omega$ and $ n $, such that 
 $$
-\min_{\bar{\Omega}} u \le C\bigg[ \big( 
\int _{B\setminus S} \det D^2u(x) \, dx \big)^{1/n}
+ \big( \mbox{\rm meas} \{ \nabla u(x)| x\in S\cap B \} \big)^{1/n}
\bigg]\,,
 $$
where $ B $ is the set  $\{ x : \Gamma u(x)= u(x) \} $,  
$ S =\{ x : D^2u(x)
 \mbox{does not exist }\} $,
and $ \nabla u(x_0)$ denotes all $ p\in {\mathbb R}^n $ satisfying
 $$
p\cdot (x-x_0) + u(x_0) \le u(x)\,. 
 $$
\end{lemma}

\paragraph{Proof.}
By Lemma 1, we have 
\begin{eqnarray*}
-\min_{\bar{\Omega}} u &\le & \frac D{K_n^{1/n}}\left[
\mbox{meas} \{ p: p \mbox{ is normal to the tangent plane at }
x\in \{ \Gamma u = u\} \} \right]^{1/n}
\\
&=&  \frac D{K_n^{1/n}}\big( \mbox{ meas} \{ 
\nabla u(x) :x\in \{ \Gamma u =u\}, D^2u(x) \mbox{ exits } \}\big)^{1/n}
 \\
&&+ \mbox{meas}\{ \nabla u(x) : x\in \{ \Gamma u=u\}, 
D^2u(x) \mbox{ does not exist }\}^{1/n}\\
&=& \frac D{K_n^{1/n}}\big(\int_{\{ \Gamma u=u \} \setminus S}
\det D^2 u \, dx \big)^{1/n}
\\
&& + \frac D{K_n^{1/n}}\big( \mbox{ meas} \{ \nabla u(x):
x\in \{ \Gamma u = u\}, D^2u(x) \mbox{ does not exist} \}
\big)^{1/n}
\end{eqnarray*}
where $ D= \mbox{ dim } \Omega$, and $K_n $ is the volume of the unit ball in
${\mathbb R^n}$.


\section {Main Theorem}

Using the Lemmas 1 and 2, we can prove the following theorem.

\begin{theorem}
Let $ F(A, x) $  be a function defined on a convex cone $C$ of
symmetric matrices
$S^n$, which satisfies the following conditions:
\begin{enumerate}
\item  For $A$ and $B$ in $C$  with $ A> B $, 
		$F(A, x) > F(B, x) $.
\item 
The equation  
\begin{eqnarray*}
&F(D^2u(x), x) =0 \quad x\in \Omega\,,& \\
&u(x) = g(x) \quad x\in \partial \Omega &
\end{eqnarray*}
has a solution  $u$ in $C^2(\Omega) \cap C(\bar{\Omega}) $.
\end{enumerate}
%
Also assume that $v\in C^2(\Omega \setminus S) \cap \mbox{\rm 
Lip}
(\Omega)\cap C(\bar{\Omega})$ is a solution to
\begin{eqnarray*}
&F(D^2v(x), x) = 0 \quad x\in \Omega \setminus S\,,&\\
&v(x) = g(x) \quad x\in \partial \Omega\,, &
\end{eqnarray*}
where $ S\subset \subset \Omega $ satisfies 
\begin{enumerate} 
\item	The dimension of $S$ is $l$ with $l < n $.
\item	For every $ x\in S $, there are $ l+1 $ independent $ C^2 $
curves $ \{ r_{xi}\} $ through $ x$, with $i\in \{1,2,\cdots ,l+1\} $, 
such that $ v(r_{xi}) \in C^1 $.
\end{enumerate}
Then  $ v$ is in $C^2 $,  
satisfies the equation in $\Omega$, and
$ u(x)=v(x)$.
\end{theorem}
		
\paragraph{Proof.}
Let $w(x)= u(x)-v(x)$.
Then $ w(x)|_{x\in \partial \Omega } = 0 $.
Suppose $ \min_{\bar{\Omega}} w <0 $. Then 
\begin{eqnarray*}
-\inf_{\bar{\Omega}} w &\le & C\bigg[ \int _{\{\Gamma
w=w\}\setminus S} \det (D^2w(x))\, dx \bigg] ^{1/n}\\
&& +C\left[\mbox{meas}\{ \nabla w(x) : x\in S 
    \cap \{ \Gamma w=w\}\}\right]^{1/n}\,.
\end{eqnarray*}
If there is $ x_0 \in \{ \Gamma w=w \} \setminus S $ 
such that $ \det (D^2 w(x
_0)) \neq 0 $, then by the
convexity of $ \Gamma w, D^2 w(x_0)\ge D^2\Gamma w(x_0) \ge 0 $.
So $ D^2w(x_0) > 0, $ or $ D^2u(x_0)> D^2v(x_0) $.
By the structure conditions on $ F $ we have
$$
0=F(D^2u(x_0), x_0) > F(D^2v(x_0), x_0) =0
$$
which is a contradiction.

Next, for $ x_0 \in S\cap \{ \Gamma w=w\} $, 
there are $ l+1 $ independent $ C^2 $ curves through $x_0 $
satisfying $ v(r_{x_0i}(t)) \in C^1$, with $i=1,2,\cdots, l+1$.
 Without loss of generality, we can assume that 
 $r_{x_0i}(0)=x_0$ for $i=1,2
,\cdots, l+1$.
Then for any $ p\in \{ \nabla w(x_0) \}=
\{ p: w(x_0)+ p\cdot (x-x_0) \le w(x) \}$
we have 
$$
p \cdot \frac d{dt} \big( r_{x_0i}(0) \big) =c_i(x_0) \quad
\mbox{for } i=1,2,\cdots, l+1\,.
$$
Since $ r_{x_0i} (t) $ are independent, we obtain that 
$ \{ \nabla w(x_0) \}$ is a subset in the $n-(l+1)$
dimensional space. 
We have that
\begin{eqnarray*}
\lefteqn{ \mbox{meas}_n \{ \nabla w(x) : x\in S\cap \{ \Gamma w=w\} \}  }\\
&\le& \mbox{meas}_n [\{ x\in S\cap \{ \Gamma w=w\}\}
\times \{ \nabla w
 (x)\}]\,.
\end{eqnarray*}
From 
$$\mbox{dim} S + \mbox{ dim } \{ \nabla w \} =
l+(n-l-1)=n-1 < n \,,$$ 
and the boundedness of $ \|\nabla w(x)\| $ and of $ S $, 
we conclude that
$$
\mbox{meas}\{ \nabla w(x) | x\in S\cap \{ \Gamma w = w \}\} =0\,,
 $$
which implies that
$$
- \inf_{\bar{\Omega}}w \le 0\,,
$$
which, in turn, allows	us to see that $w\ge 0 $ or $ u\ge v$.
In a similar way, we can prove that 
$$
u\le v\,.
$$
Thus $ u=v $. This completes the present proof. \medskip
		

For the Monge-Ampere equation, we have the following corollary

\begin{corollary}
Suppose that 
\begin{eqnarray*}
&\det (D^2u(x)) =f(x) \quad x\in \Omega\,, &\\
&u(x)=g(x) \quad x\in \partial \Omega &
\end{eqnarray*}
has a convex solution  $ u \in C^2(\Omega) \cap
C(\bar{\Omega}) $, and suppose that 
\begin{eqnarray*}
&\det (D^2v(x)) = f(x) \quad x\in \Omega \setminus S\,,&\\
&v(x) = g(x) \quad x \in \partial \Omega 
\end{eqnarray*}
has  a convex solution $ v\in C^2(\Omega \setminus S) \cap 
\mbox{\rm Lip}(\Omega)\cap C(\bar{\Omega}
)$. 
Also assume that $S\subset \subset \Omega $  satisfies
\begin{enumerate}
\item The dimension of $ S $ is $l$ with $l < n $.
\item For every $ x\in S $, there are $ l+1 $ independent 
$C^2 $ curves $ \{ r_{xi}\} $ through $ x$, with $i\in \{1,2,\cdots l+1\} $, 
such that $ v(r_{xi}) \in C^1 $.
\end{enumerate}
 
Then  $ v$ is in $C^2 $, satisfies the above equations in 
$\Omega$, and $ u(x)=v(x)$.
\end{corollary}

\paragraph{Remark} It is straight forward to prove this Corollary
the above equation with a $\nabla u$ term added.

\begin{thebibliography}{99}

\bibitem{B}  R. Beyerstedt, {\em Removable singularities of solutions to
elliptic Monge-Amp\'ere equations}, Math. Z. 208 (1991),
no. 3, 363--373.

\bibitem{C} L. Caffarelli, {\em A note on the degeneracy of convex
solutions to Monge-Ampere equation}, Comm. Partial Diff. Eqns.,
 18(1993), 7-8, 1213-1217.

\bibitem{D}
Dong Guang-chang, {\em Second Order Linear Partial Differential Equations},
Zhejiang University Press 1992.

\bibitem{J}  K. J\"orgens, {\em Harmonische Abbildungen und die
Differentialgleichung $rt-s^2=1$}, Math. Ann. 129 (1955), 330--344.

\bibitem{SW}
Schulz and Wang, {\em Isolated Singularities for Monge-Amp\'ere Equations},
proceeding of AMS, vol 123, No 12(1995), 3705-3708.

\end{thebibliography} \bigskip

{\sc Lihe Wang }\\
Department of Mathematics, 
University of Iowa \\
Iowa City, IA 52242,    USA \\
E-mail address:  lwang@math.uiowa.edu \medskip

{\sc Ning Zhu}\\
Department of Mathematics, Suzhou University\\
Suzhou 215006, China

\end{document}