abstract
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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 $\gamma\to 0$ limit corresponds, under a suitable
rescaling, to a small mass $m=|\Omega|\to 0$ limit when $p