Asymptotics of the $s$-perimeter as $s\searrow 0$

TL;DR

This paper investigates the behavior of the s-perimeter of sets within a domain as s approaches zero, establishing conditions for the existence of the limit and providing explicit formulas and counterexamples.

Contribution

It offers a comprehensive analysis of the asymptotic behavior of the s-perimeter as s tends to zero, including necessary and sufficient conditions and explicit measure-based formulations.

Findings

Identifies conditions for the existence of the s-perimeter limit as s approaches zero.

Provides explicit formulas for the limit in terms of Lebesgue measures.

Constructs examples where the limit does not exist.

Abstract

We deal with the asymptotic behavior of the $s$-perimeter of a set $E$ inside a domain $\Omega$ as $s\searrow0$. We prove necessary and sufficient conditions for the existence of such limit, by also providing an explicit formulation in terms of the Lebesgue measure of $E$ and $\Omega$. Moreover, we construct examples of sets for which the limit does not exist.