Calculus of Variations and Geometric Measure Theory

S. Borghini - G. Mascellani - L. Mazzieri

Some Sphere Theorems in Linear Potential Theory

created by mascellani on 17 Nov 2017
modified by mazzieri on 21 Feb 2023


Published Paper

Inserted: 17 nov 2017
Last Updated: 21 feb 2023

Journal: Transactions of the American Mathematical Society
Year: 2017

ArXiv: 1705.09940 PDF


In this paper we analyze the capacitary potential due to a charged body in order to deduce sharp analytic and geometric inequalities, whose equality cases are saturated by domains with spherical symmetry. In particular, for a regular bounded domain $\Omega \subset \mathbb{R}^n$, $n\geq 3$, we prove that if the mean curvature $H$ of the boundary obeys the condition $ - \bigg[ \frac{1}{\text{Cap}(\Omega)} \bigg]^{\frac{1}{n-2}} \leq \frac{H}{n-1} \leq \bigg[ \frac{1}{\text{Cap}(\Omega)} \bigg]^{\frac{1}{n-2}} $, then $\Omega$ is a round ball.