# Some Sphere Theorems in Linear Potential Theory

## TL;DR

This paper establishes sharp geometric inequalities related to the capacitary potential of charged bodies, characterizing when domains are spherical based on boundary mean curvature conditions.

## Contribution

It proves new inequalities linking mean curvature and capacity, with equality cases characterized by spherical symmetry in linear potential theory.

## Key findings

- Domains with boundary mean curvature bounds are necessarily spherical.
- Sharp inequalities are derived connecting capacity and geometric properties.
- Equality cases correspond to perfect spheres.

## Abstract

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.