Existence of Ground States of Nonlocal-Interaction Energies

TL;DR

This paper determines the conditions under which nonlocal-interaction energies possess ground states, linking their existence to the stability of pairwise potentials, using calculus of variations.

Contribution

It provides a sharp criterion for the existence of ground states in nonlocal-interaction energies based on stability conditions.

Findings

Established a necessary and sufficient condition for ground state existence.

Connected ground state existence to the stability of interaction potentials.

Applied calculus of variations to analyze energy minimizers.

Abstract

We investigate which nonlocal-interaction energies have a ground state (global minimizer). We consider this question over the space of probability measures and establish a sharp condition for the existence of ground states. We show that this condition is closely related to the notion of stability (i.e. $H$-stability) of pairwise interaction potentials. Our approach uses the direct method of the calculus of variations.