Minimizers for an aggregation model with attractive-repulsive interaction

TL;DR

This paper explicitly solves a minimization problem for probability measures with attractive-repulsive interactions, revealing conditions under which minimizers concentrate on lower-dimensional spheres, using convexity estimates on hypergeometric functions.

Contribution

It provides an explicit solution to a class of minimization problems and characterizes the shape of minimizers in different parameter regimes, extending previous work.

Findings

Minimizers are uniform on spheres in certain regimes.

Concentration occurs on lower-dimensional sets.

Convexity estimates on hypergeometric functions are used in the proof.

Abstract

We solve explicitly a certain minimization problem for probability measures involving an interaction energy that is repulsive at short distances and attractive at large distances. We complement earlier works by showing that part of the remaining parameter regime all minimizers are uniform distributions on a surface of a sphere, thus showing concentration on a lower dimensional set. Our method of proof uses convexity estimates on hypergeometric functions.