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\def\rightheadline{EJDE--1994/09\hfil A Rad\'o type theorem for 
$p$-harmonic functions \hfil\folio}
\def\leftheadline{\folio\hfil Tero Kilpel\"ainen
 \hfil EJDE--1994/09}

\def\pretitle{\vbox{\eightrm\noindent\baselineskip 9pt %
Electronic Journal of Differential Equations\hfil\break
Vol. {\eightbf 1994}(1994), No. 09, pp. 1-4. Published December 6, 1994.
\hfil\break
ISSN 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu 
\hfil\break ftp (login: ftp) 147.26.103.110 or 129.120.3.113 
\bigskip} }

\topmatter

\title A Rad\'o type theorem for $p$-harmonic functions in the plane
 \endtitle
\thanks \noindent
{\eighti1991 Mathematics Subject Classifications:} 35J60, 35B60, 31C45, 
30C62.  \hfil\break
{\eighti Key words and phrases:} $p$-harmonic functions, $p$-Laplacian, 
removable sets.
\hfil\break
\copyright 1994 Southwest Texas State University  and
University of North Texas.\hfil\break
Submitted: September 29, 1994.\hfil\break
\endthanks

\author Tero Kilpel\"ainen \endauthor
\address  Department of Mathematics\newline
 University of Jyv\"askyl\"a \newline
 P.O. Box 35, 40351 Jyv\"askyl\"a\newline
 Finland\newline \newline
 E-mail: TeroK\@ math.jyu.fi
\endaddress

\redefine\div{\operatorname{div}}
\abstract
We show that if $u\in C^1(\Omega)$ satisfies the 
$p$-Laplace equation 
$$\div(|\nabla u|^{p-2}\nabla u)=0$$ 
in $\Omega\setminus \{x\:u(x)=0\}$,
then $u$ is a solution to the $p$-Laplacian in 
the whole $\Omega\subset\Bbb R^2$.
\endabstract

\endtopmatter

\define\loc{\operatorname{loc}}

\document
Throughout this paper we let $\Omega$ be an open set in $\Bbb R^n$, 
$n\ge 2$ and  $1<p< \infty$ a fixed number.
The divergence form differential operator 
$$
 \Delta_p u=\div(|\nabla u|^{p-2}\nabla u)
$$
is called the $p$-Laplacian, and a continuous function 
$u\in W^{1,p}_{\loc}(\Omega)$ is termed to be $p$-{\it harmonic\/} in $\Omega$ if
$$
\Delta_p u=0
$$
in $\Omega$ in the sense of distributions.

In this note we prove the following theorem:

\proclaim{\bf 1. Theorem}
Let $n=2$ and $u\in C^1(\Omega)$. 
If $u$ is $p$-harmonic\ in $\Omega\setminus\{x\:u(x)=0\}$,
then $u$ is $p$-harmonic\ in $\Omega$.
\endproclaim

Theorem 1 can be regarded as an extension of
Rad\'o's classical theorem on analytic functions that states that
a continuous complex valued function is analytic as soon
as it is analytic outside the preimage of a point. For ordinary
harmonic functions (i.e\. $p=2$)  in any dimension
this extension is due to Kr\'al \cite{5}.
We also discuss the nonlinear case in higher dimensions, but so far we
have not been able to treat it completely. Our argument in the plane
relies on the intimate connection between quasiregular
mappings and planar $p$-harmonic\ functions; therefore it cannot be generalized
for higher dimensions nor for more general equations in the plane.

Clearly the conclusion of the theorem fails to hold 
if we only assumed that
$u$ be Lipschitz (for example, let $u$ be the positive part of
the first coordinate of $x$). 
Moreover, for general quasilinear equations with
measurable coefficients the assumption that $u\in C^1(\Omega)$ does not
make sense, for then  solutions  are usually only H\"older continuous.
 Removability of the zero
sets was
studied in \cite{4} for more general nonlinear equations; there
the focus was on the rate of the decay, not on the smoothness
properties.

\subheading{2. The singular set}
We first show that in the plane the set where the gradient vanishes
is removable for $C^1$-smooth $p$-harmonic\ functions.

\proclaim{\bf 3. Proposition\/}
Let $n=2$ and $u\in C^1(\Omega)$. If $u$ is $p$-harmonic\ in
$$
\Omega\setminus\{x\in\Omega\:\nabla u(x)=0\}\,,
$$
then $u$ is $p$-harmonic\ in $\Omega$.
\endproclaim

\demo{Proof}
Let $E=\{x\in \Omega\: \nabla u(x)=0\}$. It is well known (see
\cite{1} or \cite{7}) that the complex gradient
$f=u_{x_1}-i\,u_{x_2} =(\partial_1 u,-\partial_2u)$ of the $p$-harmonic\ function $u$
is quasiregular in $\Omega\setminus E$, i.e\. 
$f\in W^{1,2}_{\loc}(\Omega\setminus E)$ and
$$
 ||f'(x)||^2\le (p-1+ \frac 1{p-1}) J(x,f)
$$
for a.e\. $x\in\Omega\setminus E$; here $||f'(x)||$ is the sup-norm of
the formal derivative $f'(x)$ and $J(x,f)$ is the Jacobian determinant
of $f$.
 Since $f$ is continuous in $\Omega$, Rad\'o's theorem for quasiregular
mappings \cite{4, 14.47} ensures us that $f$ is indeed quasiregular
in the whole $\Omega$. Since nonconstant quasiregular mappings are discrete
(see \cite{6, Ch\. IV} or \cite{9, Thm II.6.3}), 
we are free to assume that $E$ is a discrete
set in $\Omega$. In particular, the $1$-dimensional measure of $E$ is zero,
whence $E$ is removable for Lipschitz continuous $p$-harmonic\ functions by
\cite{2, 4.5}. Thus $u$ is $p$-harmonic\ in $\Omega$.
\enddemo

\subheading{4. Remarks} a) If $1<p\le2$, we can give a more elementary proof
without appealing to the deep result on discreteness of
quasiregular mappings. Indeed, the proof of Rad\'o's theorem for 
quasiregular mappings reveals to us that the set $E$ above is of $2$-capacity
zero, hence of $p$-capacity zero if $1<p\le 2$; thereby $E$ is
removable for bounded $p$-harmonic functions. The case $p>2$ is different,
for then not even single points are removable for bounded $p$-harmonic\
functions in general.

b) If $n\ge3$ and $p\not=2$, then it is not known whether the set
$\{\nabla u=0\}$ can contain interior points, even if we knew 
{\it a priori\/} that $u$ is nonconstant and $p$-harmonic\ in $\Omega$.


\subheading{5. Nonvanishing gradient}
Next we treat the case where the gradient of $u$ does not vanish.
It appears to be simpler than the general case and our proof
works in all dimensions.
We are going to employ a change of variables argument. 
For the reader's convenience we state a removability result
for  slightly more general equations of the $p$-Laplacian type:

\proclaim{\bf 6. Lemma\/}
Suppose that $\Cal A\:\Omega\times\Bbb R^n\to\Bbb R^n$ is a continuous mapping such that
$$
\Cal A(x,\xi)\cdot\xi\approx|\xi|^p
$$
with equivalence constants independent of $x$. If $H$ is a hyperplane in
$\Bbb R^n$ and $u\in C^1(\Omega)$ is a function that satisfies the equation
$$
\div\Cal A(x,\nabla u)=0\tag{7}
$$
in $\Omega\setminus H$, then $u$ verifies equation (7) in $\Omega$.
\endproclaim

Lemma 6 was proven by Martio in \cite{8, 2.22} for $p=n$,
but his proof can be extended verbatim to cover all values of $p$.

We now return to our issue and show:

\proclaim{\bf 8. Proposition\/}
Suppose that $u\in C^1(\Omega)$ is such that $\nabla u\not=0$ in $\Omega$.
If $u$ is $p$-harmonic\ in $\Omega\setminus\{x\:u(x)=0\}$,
then $u$ is $p$-harmonic\ in $\Omega$.
\endproclaim

\demo{Proof}
Since the problem is local we may assume that $|\nabla u|$ is bounded
away from zero in $\Omega$. Hence the set
$$
 S=\{x\in \Omega\: u(x)=0\}
$$
is a regular $C^1$-hypersurface.
By localizing further, if necessary, we find an open neighborhood
$G$ of $S$ in $\Omega$ and a bilipschitz diffeomorphism $g$ from $G$ onto
an open set $V$ in $\Bbb R^n$ such that $S$ is mapped into the hyperplane
$$
 H=\{(x_1,x_2,\dots,x_n)\in\Bbb R^n\: x_n=0\}\,.
$$
If $f=g^{-1}$, then the pull-back of the $p$-Laplacian under $f$ is
$\div\Cal A =0$, where
$$
 \Cal A(x,\xi)= J(x,f) f'(x)^{-1}|{f'(x)^{-1}}^*\xi|^{p-2}{f'(x)^{-1}}^*\xi\,;
$$
here $B^*$ stands for the transpose of the matrix $B$. Then it is easily
checked that $\Cal A$ satisfies the assumptions of Lemma 6
(cf\. \cite{3, 14.90}). Moreover, we have that $u$ is $p$-harmonic\ in $G$
if and only if $v=u\circ f$ satisfies the equation
$$
\div\Cal A(x,\nabla v)=0\tag{9}
$$
in $V$ (see \cite{3, 13.2 \& 14.92}). Since $v$ satisfies equation
(9) in $V\setminus H$ we conclude from Lemma 6
that $v$ is a solution of (9) in $V$. Consequently, $u$
is $p$-harmonic\ in $G$ and hence in $\Omega$ as desired.
\enddemo

\subheading{Proof of Theorem 1}
By Proposition 8 $u$ is $p$-harmonic in 
$$
 \Omega\setminus\{x\:u(x)=0\text{ and }\nabla u(x)=0\}\,.
$$
Hence the theorem follows from Proposition 3.



\Refs
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\by B. Bojarski and T. Iwaniec
\paper $p$-harmonic equation and quasiregular mappings
\book Partial differential equations
\bookinfo Banach Center Publications Vol 19
\publ PWN-Polish Scientific Publishers
\publaddr Warsaw\yr1987\pages 25--38
\endref

\ref\no 2 % {\bf HK}
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elliptic equations
\jour Ark. Mat.
\vol 26
\pages 87--105
\yr 1988
\endref

\ref\no 3 % {\bf HKM}
\by J. Heinonen, T. Kilpel\"ainen, and O. Martio
\book Nonlinear potential theory of degenerate
elliptic equations
\publ Oxford University Press\publaddr Oxford\yr 1993
\endref

\ref\no 4 % {\bf KKM}
\by T. Kilpel\"ainen, P. Koskela, and O. Martio
\paper On the fusion problem for degenerate
elliptic equations
\jour Comm. P.D.E.\toappear
\endref


\ref\no 5 % {\bf K}
\by J. Kr\'al
\paper Some extension results concerning harmonic functions
\jour J. London Mat. Soc\yr 1983\vol 28\pages 62--70
\endref

\ref\no 6 % {\bf LV}
\by O. Lehto and K.I. Virtanen
\book Quasiconformal mappings in the plane
\bookinfo(2nd ed.)
\publ Springer-Verlag\publaddr Berlin\yr 1973
\endref

\ref\no 7% {\bf M}
\by J. J. Manfredi
\paper $p$-harmonic functions in the plane
\jour Proc. Amer. Math. Soc.
\yr 1988\pages 473--479\vol 103
\endref

\ref\no 8 % {\bf OM}
\by O. Martio
\paper Counterexamples for unique continuation
\jour Manuscripta Math.
\yr 1988\vol 60 \pages 21--47
\endref

\ref\no 9 % {\bf R}
\by Yu. G. Reshetnyak
\book Space mappings with bounded distortion
\bookinfo Translations of Mathematical monographs, Vol. 73
\publ American Mathematical Society
\yr 1989
\endref
\endRefs

\enddocument
\bye