A characterization of balls through optimal concavity for potential functions

In this short note two unconventional overdetermined problems are considered. Let p ∈ (1, n); firstly, the following is proved: if Ω is a bounded domain in R^n whose p-capacitary potential function u has two homotetic convex level sets, then Ω is a ball. Then, as an application, we obtain the follow- ing: if Ω is a convex domain in Rn whose p-capacitary potential function u is (1 − p)/(n − p)-concave (i.e. u(1−p)/(n−p) is convex), then Ω is a ball.