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 α$$\alpha $$, under volume constraint, where the strength of the nonlocal interaction is controlled by a parameter γ$$\gamma $$. We show that for a wide class of density functions the energy admits a minimizer for any value of γ$$\gamma $$. Moreover these minimizers are bounded. For monomial …