Asymptotics of the $s$-fractional Gaussian perimeter as $s\to 0^+$

TL;DR

This paper investigates the behavior of the renormalised $s$-fractional Gaussian perimeter of sets as the parameter $s$ approaches zero, revealing unique properties distinct from the Euclidean case.

Contribution

It provides the first analysis of the asymptotic behavior of the $s$-fractional Gaussian perimeter as $s$ tends to zero, highlighting differences from classical Euclidean perimeters.

Findings

Limit set function is not additive as $s o 0^+$.

Shape at infinity does not influence the limit due to finite Gaussian measure.

Contrasts with Euclidean case where shape at infinity matters.

Abstract

We study the asymptotic behaviour of the renormalised $s$-fractional Gaussian perimeter of a set $E$ inside a domain $\Omega$ as $s\to 0^+$. Contrary to the Euclidean case, as the Gaussian measure is finite, the shape of the set at infinity does not matter, but, surprisingly, the limit set function is never additive.