Some Sphere Theorems in Linear Potential Theory

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 $\mathrm{\Omega }\subset {\mathbb{R}}^n$, $n\ge 3$, we prove that if the mean curvature $H$ of the boundary obeys the condition $$-[\frac{1}{\text{Cap}(\mathrm{\Omega })}{]}^{\frac{1}{n-2}}\le \frac{H}{n-1}\le [\frac{1}{\text{Cap}(\mathrm{\Omega })}{]}^{\frac{1}{n-2}},$$ then $\mathrm{\Omega }$ is a round ball.