Calculus of Variations and Geometric Measure Theory

V. Agostiniani - M. Fogagnolo - L. Mazzieri

Minkowski Inequalities via Nonlinear Potential Theory

created by fogagnolo on 21 Jun 2019
modified on 09 Mar 2022


Published Paper

Inserted: 21 jun 2019
Last Updated: 9 mar 2022

Journal: Archive for Rational Mechanics and Analysis
Year: 2019
Doi: 10.1007/s00205-022-01756-6

ArXiv: 1906.00322 PDF


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.