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 A-convergence. As a byproduct of our analysis, we obtain the isoperimetric inequality for the 0-fractional perimeter.