A nonlocal isoperimetric problem with density perimeter

Abstract

We consider the minimization of an energy functional given by the sum of a density perimeter and a nonlocal interaction of Riesz type with exponent alpha, under volume constraint, where the strength of the nonlocal interaction is controlled by a parameter gamma. We show that for a wide class of density functions the energy admits a minimizer for any value of gamma. Moreover these minimizers are bounded. For monomial densities of the form vertical bar x vertical bar(p) we prove that when gamma is sufficiently small the unique minimizer is given by the ball of fixed volume. In contrast with the constant density case, here the gamma -> 0 limit corresponds, under a suitable rescaling, to a small mass m = vertical bar Omega vertical bar -> 0 limit when p < d - alpha + 1, but to a large mass m -> infinity for powers p > d - alpha + 1.