Monotonicity formulas in potential theory

Abstract

Using the electrostatic potential $u$ due to a uniformly charged body $\Omega \subset {\mathbb{R}}^n$, $n\ge 3$, we introduce a family of monotone quantities associated with the level set flow of $u$. The derived monotonicity formulas are exploited to deduce a new quantitative version of the classical Willmore inequality.

This is a new version of the paper, with new proofs and enhanced results. For a detailed description of these changes we refer the reader to the Added note after the Introduction