Existence of minimizers for interaction energies with external potentials

Abstract

In this paper we study the existence of minimizers for interaction energies with the presence of external potentials. We consider a class of subharmonic interaction potentials, which include the Riesz potentials $|{\bf x}|^{-s},\,\max\{0,d-2\}<s<d$ and its anisotropic counterparts. The underlying space is taken as $\mathbb{R}^d$ or a half-space with possibly curved boundary. We give a sufficient and almost necessary condition for the existence of minimizers, as well as the uniqueness of minimizers. The proof is based on the observation that the Euler-Lagrange condition for the energy minimizer is almost the same as that for the maximizer of the height functional, defined as the essential infimum of the generated potential. We also give two complimentary results: a simple sufficient condition for the existence of minimizers for general interaction/external potentials, and a slight improvement to the known result on the existence of minimizers without external potentials.