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

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.