Inserted: 21 jun 2019
Last Updated: 9 mar 2022
Journal: Archive for Rational Mechanics and Analysis
In this paper, we prove an exended version of the Minkowski Inequality, holding for any smooth bounded subset $\Omega\subset\mathbb R^n$, $n\geq 3$. Our proof relies on the discovery of effective monotonicity formulas along the level set flow of the $p$-capacitary potentials associated with $\Omega$, in the limit as $p\to1^+$. These formulas also testify the existence of a link between the monotonicity formulas derived by Colding and Minicozzi for the level sets flow of Green's functions and the monotonicity formulas employed by Huisken, Ilmanen and several other authors in studying the geometric implications of the Inverse Mean Curvature Flow.