On the strong attraction limit for a class of nonlocal interaction energies. (September 2020) - Burchard, Almut; Choksi, Rustum; Hess-Childs, Elias
Abstract: This note concerns the problem of minimizing a certain family of non-local energy functionals over measures on 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 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.