Radial symmetry for p-harmonic functions in exterior and punctured domains

Abstract

We prove symmetry for the p-capacitary potential satisfying $$ \Delta_p u = 0 \, \mbox{ in } \mathbb{R}^N \setminus \overline{\Omega} , \; u=1 \, \mbox{ on } \Gamma, \; \lim_{|x|\rightarrow \infty} u(x)=0 , \; \; \; \; \; \; \; \; 1<p<N, $$ under Serrin's overdetermined condition $$ | \nabla u| = c \mbox{ on } \Gamma. $$ Here $\Omega$ is any bounded domain on which no a priori assumption is made, and $\Gamma$ denotes its boundary. Our result improves on a work of Garofalo and Sartori, where the same conclusion was obtained when $\Omega$ is star-shaped. Our proof uses the maximum principle for an appropriate $P$-function, some integral identities, the isoperimetric inequality, and a Soap Bubble-type Theorem. We then treat the case $1<p=N$, improving previous results present in the literature. Finally, with analogous tools we give a new proof of symmetry for the interior overdetermined problem $$ - \Delta_p u = K \, \delta_0 \, \mbox{ in } \Omega , \, u=c \, \mbox{ on } \Gamma, \; \; \; \; \; \; \; \; 1<p<N, $$ $$ | \nabla u| = 1 \mbox{ on } \Gamma , $$ in a bounded star-shaped domain $\Omega$.