A nonlocal isoperimetric problem with density perimeter

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 $$|x|^p$$ 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 $$\gamma \rightarrow 0$$ limit corresponds, under a suitable rescaling, to a small mass $$m=|\Omega |\rightarrow 0$$ limit when $$pd-\alpha +1$$ .