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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 $
\begin{thebibliography}{MMM}
\bibitem[A]{}
L.V. Ahlfors, {\em Quasiconformal Mappings,}
Van Nostrand Company, Princeton, New Jersey, 1966.
\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
the sup norm of the gradient, } Arch. Rational Mech. Anal{ \bf 123}
(1993), 51-74.
\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
\end{document}