On the Strong Attraction Limit for a Class of Nonlocal Interaction Energies

TL;DR

This paper investigates the behavior of nonlocal energy minimizers with competing power law interactions as the attraction strength becomes very large, using Gamma-convergence to characterize the limit and exploring symmetry properties.

Contribution

It introduces a Gamma-convergence framework for analyzing the strong attraction limit of nonlocal energies with power law kernels and characterizes the minimizers in this regime.

Findings

Limit minimizers characterized by isodiametric capacity problem

Evidence of symmetry-breaking in high dimensions

Analysis of the strong attraction limit via Gamma-convergence

Abstract

This note concerns the problem of minimizing a certain family of non-local energy functionals over measures on $\mathbb{R}^n$, subject to a mass constraint, in a strong attraction limit. In these problems, the total energy is an integral over pair interactions of attractive-repulsive type. The interaction kernel is a sum of competing power law potentials with attractive powers $\alpha \in (0, \infty)$ and repulsive powers associated with Riesz potentials. The strong attraction limit $\alpha \to \infty$ is addressed via Gamma-convergence, and minimizers of the limit are characterized in terms of an isodiametric capacity problem. We also provide evidence for symmetry-breaking in high dimensions.