Existence of Compactly Supported Global Minimisers for the Interaction Energy

TL;DR

This paper proves the existence of compactly supported global minimisers for particle interaction models with certain potentials, including power-law and Morse types, under near-optimal conditions.

Contribution

It establishes the existence of global minimisers for non-H-stable potentials, extending classical results to a broader class of interaction potentials.

Findings

Global minimisers have compact support under specified conditions.

Existence results include power-law and Morse potentials.

Support of local minimisers is also shown to be compact.

Abstract

The existence of compactly supported global minimisers for continuum models of particles interacting through a potential is shown under almost optimal hypotheses. The main assumption on the potential is that it is catastrophic, or not H-stable, which is the complementary assumption to that in classical results on thermodynamic limits in statistical mechanics. The proof is based on a uniform control on the local mass around each point of the support of a global minimiser, together with an estimate on the size of the "gaps" it may have. The class of potentials for which we prove existence of global minimisers includes power-law potentials and, for some range of parameters, Morse potentials, widely used in applications. We also show that the support of local minimisers is compact under suitable assumptions.