Radial symmetry for p-harmonic functions in exterior and punctured domains
Giorgio Poggesi · arXiv (Cornell University) · 2018
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 $$ | \ abla u| = c \\mbox{ on } \\Gamma. $$ 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 $$ - \\Delta_p u = K \\, \\delta_0 \\, \\mbox{ in } \\Omega , \\, u=c \\, \\mbox{ on } \\Gamma, \\; \\; \\; \\; \\; \\; \\; \\; 1<p<N, $$ $$ | \ abla u| = 1 \\mbox{ on } \\Gamma , $$ in a bounded star-shaped domain $\\Omega$.