# The $0$-fractional perimeter between fractional perimeters and Riesz potentials

created by ponsiglio on 14 Jun 2019
modified by novaga on 26 May 2020

[BibTeX]

Accepted Paper

Inserted: 14 jun 2019
Last Updated: 26 may 2020

Journal: Ann. Scuola Normale Sup.Pisa
Year: 2019

ArXiv: 1906.06303 PDF

Abstract:

This paper provides a unified point of view on fractional perimeters and Riesz potentials. Denoting by $H^{\sigma}$ - for ${\sigma}\in (0,1)$ - the ${\sigma}$-fractional perimeter and by $J^{\sigma}$ - for ${\sigma}\in (-d,0)$ - the ${\sigma}$-Riesz energies acting on characteristic functions, we prove that both functionals can be seen as limits of renormalized self-attractive energies as well as limits of repulsive interactions between a set and its complement.

We also show that the functionals $H^{\sigma}$ and $J^{\sigma}$, up to a suitable additive renormalization diverging when ${\sigma}\to 0$, belong to a continuous one-parameter family of functionals, which for ${\sigma}=0$ gives back a new object we refer to as $0$-fractional perimeter. All the convergence results with respect to the parameter ${\sigma}$ and to the renormalization procedures are obtained in the framework of $\Gamma$-convergence. As a byproduct of our analysis, we obtain the isoperimetric inequality for the $0$-fractional perimeter.