\def\eqn#1{\def\theequation{#1}} \documentstyle[twoside]{article} \pagestyle{myheadings} \markboth{\hfil $p$ HARMONIC FUNCTIONS \hfil EJDE--1994/03}% {EJDE--1994/03\hfil J.L. Lewis\hfil} \newtheorem{theorem}{Theorem} \newcommand{\rar}{\rightarrow} \newcommand{\de}{\delta} \newcommand{\ga}{\gamma} \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* $$\nabla \cdot ( | \nabla u |^{p-2} \nabla u ) = 0$$ 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{**} $$\nabla \cdot ( ( \epsilon + |\nabla v|^2 )^{\frac{p}{2} - 1} \nabla v ) = 0$$ in $D,$ with $v = P$ on $\partial D$. We note that (**) implies \eqn{0} $$Lv = (p - 2 )\sum_{j,k = 1}^2 \, v_{x_j x_k } v_{x_j} v_{x_k} + ( \epsilon + | \nabla v |^2 ) \, \Delta v = 0$$ 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} $$| f_{\bar{z}} | \, \leq \, | 1 - 2/p | \, | f_z |$$ 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} $$( 2 \pi i )^{-1} \, \int_{\Gamma} \frac{d \log f( z ( t ))}{dt} \, dt$$ 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} $$\left| \mbox{ Re } \left[ ( 2 \pi i )^{-1} \int_{\ga} \frac{d \log [ z (t ) f( z ( t ))]}{dt} \, dt \right] \right| \leq 1/2.$$ 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} $$\left| ( 2 \pi i )^{-1} \, \int_{C_k} \frac{d \log [ z (t ) f( z ( t ))]}{dt}\, dt \right| \rar 0$$ 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} $$0 = L v (z) = O ( | z - z_k |^{ 3l - 4 } ) + \epsilon \, \Delta Q (z)$$ 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} $$( 2 \pi i )^{-1} \int_{C_k} \frac{d \log [ z (t ) f( z ( t ))]}{dt} \, dt \rar - (l - 1)/2$$ 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} $$( 2 \pi i )^{-1} \int_{\Gamma} \frac{d \log [ f( z ( t ))]}{dt}\, dt \, \leq \, m - 1$$ 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}