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\markboth{\hfil $p$ HARMONIC FUNCTIONS \hfil EJDE--1994/03}% 
{EJDE--1994/03\hfil J.L. Lewis\hfil}
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\begin{document}
\ifx\Box\undefined \newcommand{\Box}{\diamondsuit}\fi
\title{\vspace{-1in}\parbox{\linewidth}{\footnotesize\noindent
{\sc  Electronic Journal of Differential Equations}\newline
Vol. {\bf 1994}(1994), No. 03, pp. 1-4. Published July 6, 1994.\newline
ISSN 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu
\newline ftp (login: ftp) 147.26.103.110 or 129.120.3.113}
 \vspace{\bigskipamount} \\
On Critical Points of $p$ Harmonic Functions in the Plane
\thanks{ {\em 1991 Mathematics Subject Classifications:}
35J70, 35B05.\newline\indent
{\em Key words and phrases:} $p$ harmonic functions,
 $p$ Laplacian, quasiregular mappings.
\newline\indent
\copyright 1994 Southwest Texas State University  and University of 
North Texas.\newline\indent
Submitted: March 30, 1994.\newline\indent
Supported in part by a grant from the NSF.} }
\date{}
\author{John L. Lewis}

\maketitle

\begin{abstract} We show that if $ u $ is a $p$ harmonic  function,
 $  1 < p < \infty, $  in the unit disk and equal to a polynomial $P $
of positive degree on the boundary of 
this disk, then $ \nabla u $ has at most deg$\,P - 1$
zeros in the unit disk.   \end{abstract}

In this note we prove the following theorem.
\begin{theorem}  Given $ p, 1 < p < \infty, $ let $ u $ be a real valued 
weak solution to 
\eqn* \begin{equation} \nabla \cdot ( | \nabla u |^{p-2}  \nabla u ) = 0 
 \end{equation}
in  $ D = \{ ( x_1, x_2 ) : x_1^2 + x_2^2 < 1 \} \subset 
{\bf R^2 } $  with $ u = P $   on $ \partial \, D $ 
where $ P $ is a real polynomial in $ x_1, x_2  $  of degree $ m \geq  1$.
Then   $ \nabla u $ has at most $ m - 1 $  zeros  in $ D $ 
counted according to multiplicity. \end{theorem}

In (*), \,  $ \nabla \cdot $ denotes the divergence operator while $ \nabla u $ 
 denotes the gradient of $u. $
The above theorem answers a question in the affirmative first posed by 
D. Khavinson in connection with determining the extremal functions for 
certain linear functionals in the Bergman space of $ p $ th power 
integrable analytic functions on $D, \,  1 < p < \infty$. 
We note that the differential operator in (*) is often called the 
$ p $ Laplacian and it is well known (see [GT]) that solutions to 
this equation  are 
infinitely differentiable (in fact real analytic) at each point where
$ \nabla u \not = 0 $  while (*) is degenerate elliptic at each point 
where $ \nabla u = 0$.  The above theorem appears to be the first of its kind 
to establish independent of $ p $ and the structure constants for 
 the $ p $ Laplacian,  a bound 
($m$ - 1) for the number of  points in $ D $ where 
(*) degenerates.  Because of this independence we conjecture that 
our theorem  also remains true for $ p = \infty$  and the so called 
$ \infty $ Laplacian (see [BBM] or [J] for definitions). Finally we 
remark  that in [Al] a result, in the same spirit as ours, is obtained
for smooth linear equations whose matrix of coefficients has determinant 
one. 
 
\section*{Proof of main theorem.}
 Consider the strong solutions,  $v = v( \cdot, \epsilon, p ), $ to 
\eqn{**} \begin{equation}   
\nabla \cdot ( ( \epsilon + |\nabla v|^2 )^{\frac{p}{2} - 1} \nabla v ) 
 =  0  \end{equation}
in $ D, $ with $ v = P$ on $ \partial D$.  
We note that (**) implies 
\eqn{0}
\begin{equation}
Lv = (p - 2 )\sum_{j,k = 1}^2 \,  v_{x_j x_k } v_{x_j} v_{x_k} + 
( \epsilon + | \nabla v |^2 ) \, \Delta v = 0 
\end{equation} 
at each point of  $ D. $ Here $ \Delta $ denotes the Laplacian. 
From (0) and elliptic theory it 
 follows that  $ v ( \cdot, \epsilon ) $ is unique  and  infinitely 
differentiable in the 
closed unit disk ($ v \in C^{\infty } (\bar{D})$).  
Indeed this statement follows easily from Schauder's theorem (see 
[GT], ch 6)
and induction once $ C^{1, \alpha} $ regularity of $ v $ in 
$ \bar{D} $ is established ( for $ C^{1, \alpha} $ regularity of 
$v$ see [L]). 

Next we introduce complex notation. Let $ z = x_1 + i x_2$,  $i = 
\sqrt{ - 1}$, and put
 $ g_z = \frac12( g_{x_1} - i g_{x_2})$, $g_{\bar{z}} = 
\frac12(g_{x_1} + i g_{x_2})$. as usual
and note from (0) as in [GT, ch 11, section 2] or [IM], 
that if $f(z)  = f( z, \epsilon, p ) = v_z ( z )$,
then $ f $ is  quasiregular in $ D $ with $ k = | 1 - 2/p |$. 
 That is $ f $ 
is a  sense preserving mapping of  $ D $ and
\eqn{1} 
\begin{equation}
| f_{\bar{z}} | \, \leq  \, | 1 - 2/p | \, | f_z |
\end{equation} 
at each point of $ D$.  
From  the factorization theorem for quasiregular mappings (see 
 [A, ch  V ] ) 
we find  that 
 $ f = g \circ h $ 
where  $ g $ is analytic in $ h ( D ) $
 and $ h $ is a QC mapping of $ {\bf R^2 } $ onto itself (i.e. a 
quasiregular homeomorphism of ${\bf R^2 })$.
Using this factorization, the argument principle 
for analytic functions,  and $ C^1$ smoothness of $ f $ in 
$ \bar{D}, $ it follows that we can calculate the 
number of zeros of  $ f $ counted according to multiplicity  
inside a contour $ \Gamma \subset \bar{D} $ with $ f \not = 0 $ 
on $ \Gamma $ (i.e the number of zeros of $g$ counted according 
to multiplicity inside $ h ( \Gamma ))$  by calculating 
\eqn{2}
\begin{equation}
 ( 2 \pi i )^{-1} \,  \int_{\Gamma}  \frac{d \log f( z ( t ))}{dt} \,  dt 
\end{equation}
 where 
$ \log f $ denotes 
a continuous branch of the logarithm of $f$  on $ \Gamma $ and we assume $ z = z(t) $ is 
a piecewise smooth parametrization of $\Gamma$ .
Now we can write $ x_1, x_2 $ in terms of $ z, \bar{z} $ in the usual 
way and thus regard  $ P $ as a function of $ z, \bar{z}$.  If $ z = 
e^{i \theta}, \theta $ real, we note first that 
$ \bar{z} = z^{ - 1 } $  and second that 
\[ P_\theta ( z ) = i z P_z - i \bar{z} P_{\bar{z}} \]
is identically equal to a rational function of degree at most $ 2 m $ 
on $ \partial D. $ 
To construct $ \Gamma $ let $ z_j = e^{i \theta_j }, j = 1, 2, 
\dots n  $ be the distinct zeros of $ \frac{\partial P }{\partial \theta } $ on 
$ \partial D. $   From our note we 
have  $ n  \leq 2m. $  For 
small $ \de > 0 $ let $ D ( z_j, \de ) = \{ z : | z - z_j| < \de \}$
for $ 1 \leq j \leq n $. Then for $ \de $ small enough, clearly 
$ \partial D \setminus \cup_{i = 1}^{n} D ( z_j, \de ) $ consists of 
$ n   $ closed arcs, say $ \cup_{i=1}^{n} \ga_i, $ oriented 
counterclockwise, as seen from the origin. Let $ C_j $ be the arc of $ \partial 
D( z_j, \de )$ that lies 
inside the unit circle for $  1 \leq j \leq n $  oriented counterclockwise
as seen from the origin. We put $ \Gamma  = ( \cup C_j ) \cup ( \cup \ga_j )$.  and shall show that the integral in (2) is $ \leq m - 1$.  
To this end, let $ \ga \in \{ \ga_j \} $ and note that 
if $ z = e^{i \theta}, $ then 
$ P_\theta = 2 \mbox{ Re } ( i z v_z )$.  Since $ P_\theta  $ does not 
change sign on $ \ga $ it follows  that the image of $ \ga $ under 
$ z f = z v_z $ lies inside a 
halfplane whose boundary contains 0. Thus a continuous 
argument of $ z f $   can  change 
by at most $ \pi $ on $ \ga $ and so  
\eqn{3}
\begin{equation}
\left|   \mbox{ Re } \left[ ( 2 \pi i )^{-1}  \int_{\ga} 
 \frac{d \log [ z (t ) f( z ( t ))]}{dt} \,   dt \right] 
 \right| \leq 1/2. 
\end{equation}
 Next we consider  $ C_k \in \{ C_j \} $.  Recall that $ v \in C^{\infty}
 (\bar{D}) $. If $ v_z ( z_k ) \not = 0 $ then clearly 
\eqn{4}
\begin{equation}
\left|  ( 2 \pi i )^{-1} \,  \int_{C_k} 
 \frac{d \log [ z (t ) f( z ( t ))]}{dt}\,  dt  \right|  \rar 0  
\end{equation}
as $ \de \rar 0 $.  Otherwise, let  $ l > 1 $ be 
the largest  positive integer such that all homogeneous 
Taylor polynomials  of $v - v( z_k ) $ 
about $ z_k $ of degree less than $ l $ are identically 0
 and let $ Q $ be the homogeneous Taylor polynomial 
of degree $l $ about $ z_k $  corresponding to $ v - v( z_k ) $.  Using (0) and   continuity of the derivatives of 
$v$ in $ \bar{D}$ 
  we see that 
for $ z \in D \cap D( z_k, \de )$
\eqn{5}
\begin{equation} 
0 = L v (z) =  O ( | z - z_k |^{ 3l - 4 } )  + \epsilon \, \Delta Q (z) 
\end{equation}
as $ z \rar z_k,  $   
Now  $ \Delta Q $ is either a homogeneous 
polynomial of degree $ l - 2 $ or $ \Delta Q \equiv 0 $. Dividing (5) by 
$ | z - z_k |^{l - 2} $  and taking a limit as $ z \rar z_k $ we conclude that 
the second possibility must occur. Thus   $ Q $ is   harmonic 
 and so \, 
$ Q = \mbox{ Re }[ c ( z - z_k)^{l} ]  $ for some complex $ c$.  From 
this fact we conclude first that for a continuous branch 
of $ \log f $ on $ C_k $ we 
have 
\[ \log ( i z f(z) ) = \log[iz Q_z(z)] + o(1), \mbox{ as }  \de \rar 0  \mbox{ for } z \in C_k,  
\]
where the $ o(1) $ term is independent of $ z \in C_k$.
Second we conclude 
\eqn{6} 
\begin{equation}
  ( 2 \pi i )^{-1}   \int_{C_k} 
 \frac{d \log [ z (t ) f( z ( t ))]}{dt} \,  dt   \rar - (l - 1)/2 
\end{equation}  
as $ \de \rar 0$. Since the integral in (2) must be a nonnegative 
integer we see from (3) and (6) that for $ \de $ sufficiently small 
\eqn{7} 
\begin{equation}
  ( 2 \pi i )^{-1} \int_{\Gamma} 
 \frac{d \log [ f( z ( t ))]}{dt}\,  dt \,   \leq  \, m  - 1 
\end{equation}
since there are at most $ 2  m  $ members of $ \{\ga_j \} $ and the argument 
of $ z $ changes by $ 2 \pi $ as we go around $ \Gamma$. 

Finally, $ v, v_z $ considered as functions of $ \epsilon $ converge uniformly 
on compact subsets of $ D $ 
to $ u, u_z, $ for a fixed $p $ as $ \epsilon \rar 0$.  These facts follow from 
the uniqueness of $ u $ as a solution to the $ p $ Laplacian and 
$ C^{1,\alpha } $ regularity of $ u, v $ (with constants 
independent of $ \epsilon $). 
Moreover from (1) it follows that 
$ u_z $ is  quasiregular in $ D $  with $ k = | 1 - 2/p | $ 
(again see [IM] for these facts). 
From these observations, (7), and another winding number argument we find 
that if  $ u_z \not = 0 $ on $ \{ z : | z | = r \} $ 
 for some $r,  0 < r < 1, $  then   $ u_z $ has at most $ m - 1 $ zeros 
in $ \{ z : | z | < r \}$.  Hence our theorem is true. 
 $ \Box $ 

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L.V. Ahlfors, {\em Quasiconformal Mappings,}
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\bibitem[Al]{}
 G. Alessandrini,{\em Critical points of solutions 
of elliptic equations in two variables,} Ann. Scuola Norm. Sup. Pisa Cl. 
Sci. 4 {\bf 14}(1987), 229-256.   
\bibitem[BBM]{}
 T. Bhattacharya, E. DiBenedetto, and J. Manfredi, 
{\em Limits as $ p \rightarrow \infty $ of $ \Delta_p u_p = f $ 
and related extremal problems,} Rend. Sem. Mat. Univ. Pol. torino, 
Fascicolo Speciale  (1989) Nonlinear PDE's, \, 15-68.  
\bibitem[GT]{} 
 D. Gilbarg and N. Trudinger, {\em Elliptic partial differential 
equations of second order, } Springer Verlag, New York, 1977. 
\bibitem[IM]{}
 T. Iwaniec and J. Manfredi, {\em Regularity of $p$ harmonic 
functions on the plane}, Revista Matematica Iberoamericana {\bf 5 }
(1989), 1-19. 
\bibitem[J]{}
 R. Jensen, \, {\em  Uniqueness of Lipschitz extensions: minimizing
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\bibitem[ L]{}
 G. M. Lieberman, {\em Boundary regularity for solutions of 
degenerate elliptic equations,} Nonlinear Anal. {\bf 12 }(1988), 1203-1219.

\end{thebibliography}

{\sc John L Lewis\newline
Department of Mathematics, University of Kentucky, 
  Lexington, Kentucky 40506-0027}\newline
E-mail address: john@ms.uky.edu 

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