Asymptotics of non-local perimeters

TL;DR

This paper introduces a generalized non-local perimeter concept based on arbitrary measures, unifying existing notions like fractional and anisotropic perimeters, and studies their asymptotic behavior to recover known convergence results.

Contribution

It defines a broad class of non-local perimeters through arbitrary measures and analyzes their asymptotics, extending and unifying previous perimeter concepts.

Findings

Recovered convergence results for fractional perimeters

Extended asymptotic analysis to anisotropic fractional perimeters

Unified various perimeter notions under a single framework

Abstract

We introduce a notion of non-local perimeter which is defined through an arbitrary positive Borel measure on $\mathbb{R}^d$ which integrates the function $1\wedge |x|$. Such definition of non-local perimeter encompasses a wide range of perimeters which have been already studied in the literature, including fractional perimeters and anisotropic fractional perimeters. The main part of the article is devoted to the study of the asymptotic behaviour of non-local perimeters. As direct applications we recover well-known convergence results for fractional perimeters and anisotropic fractional perimeters