On the existence of minimizing sets for a weakly-repulsive non-local energy

TL;DR

This paper investigates conditions under which non-local interaction energies have minimizers that are characteristic functions of sets or discrete points, providing existence results and characterizations for specific kernels and configurations.

Contribution

It establishes the existence of minimizers as characteristic functions for certain kernels and describes conditions for minimizers to be discrete point masses on regular polygons.

Findings

Minimizers are characteristic functions of sets for small or all masses under certain kernels.

Power-law kernels satisfy the regularity assumptions for minimizer existence.

Minimizers can be Dirac masses at vertices of regular polygons under specific conditions.

Abstract

We consider a non-local interaction energy over bounded densities of fixed mass $m$. We prove that under certain regularity assumptions on the interaction kernel these energies admit minimizers given by characteristic functions of sets when $m$ is sufficiently small (or even for every $m$, in particular cases). We show that these assumptions are satisfied by particular interaction kernels in power-law form, and give a certain characterization of minimizing sets. Finally, following a recent result of Davies, Lim and McCann, we give sufficient conditions on the interaction kernel so that the minimizer of the energy over probability measures is given by Dirac masses concentrated on the vertices of a regular $(N+1)$-gon of side length 1 in $\mathbb{R}^N$.