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

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.