The Soap Bubble Theorem and a $p$-Laplacian overdetermined problem

Abstract

We consider the $p$-Laplacian equation $-{\Delta }_{p}u=1$ for $1<p<2$, on a regular bounded domain $\Omega \subset {\mathbb{R}}^N$, with $N\ge 2$, under homogeneous Dirichlet boundary conditions. In the spirit of Alexandrov's Soap Bubble Theorem and of Serrin's symmetry result for the overdetermined problems, we prove that if the mean curvature $H$ of $\partial \Omega$ is constant, then $\Omega$ is a ball and the unique solution of the Dirichlet $p$-Laplacian problem is radial. The main tools used are integral identities, the $P$-function, and the maximum principle.

18 pages, 0 figures