Characterization of balls as minimizers of an endpoint Gagliardo seminorm on the boundary

TL;DR

This paper establishes a sharp inequality linking the perimeter of a domain to the endpoint Gagliardo seminorm of its boundary's normal vector, proving that balls uniquely minimize this quantity, with implications for nonlocal geometry.

Contribution

The paper introduces a new sharp inequality connecting perimeter and Gagliardo seminorm of boundary normals, and proves that balls are the unique minimizers, using Bessel potentials and monotonicity formulas.

Findings

Balls are the unique minimizers of the inequality.

The inequality relates perimeter to the Gagliardo seminorm of boundary normals.

The results have implications for nonlocal minimal surfaces.

Abstract

Given a bounded $C^2$ domain $\Omega\subset{\mathbb R}^d$ with $d\geq3$, we prove a sharp inequality which relates the perimeter of ${\partial\Omega}$ to the endpoint Gagliardo seminorm in $W^{r,2}({\partial\Omega})$, corresponding to $r=0$, of the normal vector field on ${\partial\Omega}$. The proof of the inequality relies on the use of Bessel potentials and a monotonicity formula; we also show that balls are the unique minimizers. For $1/2<r<1$, the Gagliardo seminorm of the normal vector field on ${\partial\Omega}$ is related to a fractional second fundamental form which arises in the study of nonlocal perimeters and nonlocal minimal surfaces.