Some sphere theorems in linear potential theory

Borghini, Stefano; Mascellani, Giovanni; Mazzieri, Lorenzo

doi:10.1090/tran/7637

Some sphere theorems in linear potential theory
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by Stefano Borghini, Giovanni Mascellani and Lorenzo Mazzieri PDF

Trans. Amer. Math. Soc. 371 (2019), 7757-7790 Request permission

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 $\Omega \subset \mathbb {R}^n$, $n\geqslant 3$, we prove that if the mean curvature $H$ of the boundary obeys the condition \begin{equation*} - \bigg [ \frac {1}{\operatorname {Cap}(\Omega )} \bigg ]^{\frac {1}{n-2}} \!\! \leqslant \frac {H}{n-1} \leqslant \bigg [ \frac {1}{\operatorname {Cap}(\Omega )} \bigg ]^{\frac {1}{n-2}} , \end{equation*} then $\Omega$ is a round ball.

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