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

## TL;DR

This paper unifies fractional perimeters and Riesz potentials through a limit framework, introducing the new concept of the $0$-fractional perimeter and establishing its properties via $ ext{Gamma}$-convergence.

## Contribution

It introduces the $0$-fractional perimeter as a new object connecting fractional perimeters and Riesz potentials, with a comprehensive limit and convergence analysis.

## Key findings

- Established the $0$-fractional perimeter as a limit of fractional perimeters and Riesz energies.
- Proved the $0$-fractional perimeter satisfies an isoperimetric inequality.
- Developed a continuous family of functionals connecting different fractional regimes.

## 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 {\it $0$-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.