*Published Paper*

**Inserted:** 1 jul 2020

**Last Updated:** 1 apr 2021

**Journal:** Calc. Var.

**Volume:** 60

**Pages:** 1

**Year:** 2021

**Doi:** 10.1007/s00526-020-01865-8

**Notes:**

This is a post-peer-review, pre-copyedit version of an article published in Calc. Var. The final authenticated version is available online.

**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 $

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 $p<d-\alpha+1$, but to a large mass $m\to\infty$ for powers $p>d-\alpha+1$.

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