Anisotropic fractional perimeters

TL;DR

This paper investigates the behavior of anisotropic fractional perimeters and seminorms as the fractional parameter approaches 1, establishing convergence results and inequalities related to convex bodies and their moment bodies.

Contribution

It introduces convergence results for anisotropic fractional perimeters and seminorms to their classical counterparts, extending previous isotropic results to anisotropic settings.

Findings

Convergence of anisotropic fractional perimeter to anisotropic perimeter as s approaches 1

Establishment of anisotropic fractional Sobolev inequalities

Minimizers of anisotropic fractional isoperimetric inequality converge to the moment body of K

Abstract

The anisotropic fractional $s$-perimeter with respect to a convex body $K$ in $\rn$ is shown to converge to the anisotropic perimeter with respect to the moment body of $K$ as $s\to 1^-$. A corresponding result is established for anisotropic fractional $s$-seminorms on $BV(\rn)$ (generalizing results of Bourgain, Brezis & Mironescu and D\'avila). The minimizers of the anisotropic fractional $s$-isoperimetric inequality with respect to $K$ are shown to converge to the moment body of $K$. Anisotropic fractional Sobolev inequalities are established.