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

This paper provides a unified point of view on fractional perimeters and Riesz potentials. Denoting by $H^σ$ - for $σ\in (0,1)$ - the $σ$-fractional perimeter and by $J^σ$ - for $σ\in (-d,0)$ - the $σ$-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 $H^σ$ and $J^σ$\,, up to a suitable additive renormalization diverging when $σ\to 0$, belong to a continuous one-parameter family of functionals, which for $σ=0$ gives back a new object we refer to as {\it $0$-fractional perimeter}. All the convergence results with respect to the parameter $σ$ and to the renormalization procedures are obtained in the framework of $Γ$-convergence. As a byproduct of our analysis, we obtain the isoperimetric inequality for the $0$-fractional perimeter.

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