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 byH! - for ! 2 .0; 1/ - the !-fractional perimeter and by J ! - for ! 2 .!d; 0/ - the !-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! and J ! , up to a suitable additive renormalization diverging when ! ! 0, belong to a continuous one-parameter family of functionals, which for ! D 0 gives back a new object we refer to as 0-fractional perimeter. All the convergence results with respect to the parameter ! and to the renormalization procedures are obtained in the framework of Ä-convergence. As a byproduct of our analysis, we obtain the isoperimetric inequality for the 0-fractional perimeter.