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 Hσ - for σ ∈ (0,1) - the σ-fractional perimeter and by Jσ - for σ ∈ (−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 Hσ and Jσ , up to a suitable additive renormalization diverging when σ → 0, belong to a continuous one-parameter family of functionals, which for σ = 0 gives back a new object we refer to as 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.