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

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.