Anisotropic fractional perimeters

Abstract

The anisotropic fractional $s$-perimeter with respect to a convex body $K$ in $\mathbb{R}^n$ is shown to converge as $s\to 1^-$ to the anisotropic perimeter with respect to the moment body of $K$. For anisotropic fractional $s$-seminorms on $BV(\mathbb{R}^n)$, the corresponding result is established (generalizing results of Bourgain, Brezis, and Mironescu and Dávila). Minimizers of the anisotropic fractional iso-perimetric inequality with respect to $K$ are shown to converge to the moment body of $K$ as $s\to 1^-$. Anisotropic fractional Sobolev inequalities are established.