Monotonicity formulas in potential theory

Using the electrostatic potential $u$ due to a uniformly charged body $\Omega\subset\mathbb R^n$, $n\geq 3$, 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.
Comment: 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