Monotonicity formulas in potential theory

TL;DR

This paper introduces new monotonicity formulas related to electrostatic potentials and uses them to derive a quantitative version of the Willmore inequality, advancing understanding in potential theory and geometric inequalities.

Contribution

It presents a novel family of monotone quantities linked to level set flows of electrostatic potentials, leading to a new quantitative inequality in geometric analysis.

Findings

Established a family of monotone quantities for potential level sets.

Derived a quantitative version of the Willmore inequality.

Applied monotonicity formulas to geometric inequalities in potential theory.

Abstract

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.