Closing the gap: Maz'ya-Shaposhnikova and asymptotics of fractional perimeters

TL;DR

This paper generalizes the Maz'ya-Shaposhnikova formula for fractional perimeters, linking classical norms and nonlocal perimeters through a new limiting functional, with extensions to metric measure spaces.

Contribution

It introduces a new framework for understanding the asymptotics of fractional perimeters and Gagliardo seminorms, extending classical results to broader function spaces and metric measure spaces.

Findings

Derived a generalized limit formula for fractional Gagliardo seminorms at s→0+

Connected classical L^2 norms with fractional perimeter asymptotics

Established Gamma-convergence of seminorms in weak-L^2 topology

Abstract

We prove a generalization of the Maz'ya-Shaposhnikova formula in the case $p=2$ for functions that may not belong to ${L^2}(\mathbb{R}^d)$ and, thus, might not vanish at infinity. By introducing a notion of mass at infinity, we explicitly characterize the limit as $s\to0^+$ of Gagliardo seminorms localized on a bounded Lipschitz domain $\Omega$. By `localized', we mean here that we account only for interactions involving at least one point in $\Omega$. The identified limiting functional provides a unifying framework to link the classical Maz'ya-Shaposhnikova formula and the asymptotics of nonlocal perimeters. On the one hand, it reduces to the classical $L^2$ norm for functions that are globally integrable on $\mathbb{R}^d$. On the other hand, it recovers the pointwise limit of $s$-fractional perimeters when evaluated on characteristic functions of sets. We further show that the same functional encodes the asymptotic behavior of Gagliardo seminorms in the sense of Gamma-convergence with respect to the weak-$L^2$ topology. Finally, we provide an extension to the setting of metric measure spaces.