A nonlocal isoperimetric problem with density perimeter

TL;DR

This paper studies a nonlocal isoperimetric problem with density, proving existence and boundedness of minimizers, and characterizing their shape for small nonlocal interaction strength, especially for monomial densities.

Contribution

It establishes existence and boundedness of minimizers for a class of density functions and characterizes the minimizers for monomial densities at small interaction strength.

Findings

Minimizers exist for all interaction strengths in a wide class of densities.

Minimizers are bounded in size.

For monomial densities, the ball is the unique minimizer when the interaction parameter is small.

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 $p>d-\alpha+1$.