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

This note concerns the problem of minimizing a certain family of non-local energy functionals over measures on Rn, 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 α∈(0,∞) and repulsive powers associated with Riesz potentials. The strong attraction limit α→∞ 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 of minimizers in high dimensions.