On representation of boundary integrals involving the mean curvature for mean-convex domains

TL;DR

This paper derives a new representation formula for boundary integrals involving the mean curvature of mean-convex domains, expressing them through volume integrals and defect measures on the ridge set, enhancing understanding of geometric properties.

Contribution

It introduces a novel representation formula for boundary integrals of mean curvature functions in mean-convex domains, linking boundary and volume measures.

Findings

Representation formula involving volume integrals and defect measures

Applicable to domains with $C^{2,1}$ boundary and non-increasing functions

Provides insights into geometric analysis of mean-convex domains

Abstract

Given a mean-convex domain $\Omega\subset \R^n$ with boundary of class $C^{2,1}$, we provide a representation formula for a boundary integral of the type \[ \int_{\partial \Omega} f(k(x)) \, d\mathcal{H}^{n-1} \] where $k\geq 0$ is the mean curvature of $\partial \Omega$ and $f$ is non-increasing and sufficiently regular, in terms of volume integrals and defect measure on the ridge set.