A characterization of balls through optimal concavity for potential functions

Abstract:

In this short note two unconventional overdetermined problems are considered. Let $p\in (1,n)$; first, the following is proved: if $\Omega$ is a bounded domain in $\mathbb {R}^n$ whose $p$-capacitary potential function $u$ has two homotetic convex level sets, then $\Omega$ is a ball. Then, as an application, we obtain the following: if $\Omega$ is a convex domain in $\mathbb {R}^n$ whose $p$-capacitary potential function $u$ is $(1-p)/(n-p)$-concave (i.e. $u^{(1-p)/(n-p)}$ is convex), then $\Omega$ is a ball.