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

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 ${\mathbb{R}}^N$.

This is a post-peer-review, pre-copyedit version of an article published in Pure and Applied Analysis

References in corpus (3)

• Existence of Ground States of Nonlocal-Interaction Energies
• Classifying minimum energy states for interacting particles: Regular simplices
• Stability of the ball for attractive-repulsive energies

Cited by in corpus (1)

• Continuity of critical points for 1-dimensional non-local energies