Minimizers of 3D anisotropic interaction energies

TL;DR

This paper investigates the minimizers of 3D anisotropic Riesz-type interaction energies, revealing explicit solutions for convex cases and phenomena like collapse into lower-dimensional measures for less singular potentials.

Contribution

It extends previous 2D results to 3D, providing explicit formulas for minimizers and analyzing collapse phenomena in anisotropic Riesz energies.

Findings

Explicit formulas for minimizers with convex potentials are ellipsoids.

Less singular potentials can cause collapse into lower-dimensional measures.

Partial results on intermediate singularities and collapse behavior.

Abstract

We study a large family of axisymmetric Riesz-type singular interaction potentials with anisotropy in three dimensions. We generalize some of the results of our recent work in two dimensions to the present setting. For potentials with linear interpolation convexity, their associated global energy minimizers are given by explicit formulas whose supports are ellipsoids. We show that for less singular anisotropic Riesz potentials, the global minimizer may collapse into one or two dimensional concentrated measures which minimize restricted isotropic Riesz interaction energies. Some partial aspects of these questions are also tackled in the intermediate range of singularities in which one dimensional vertical collapse is not allowed. Collapse to lower dimensional structures is proved at the critical value of the convexity but not necessarily to vertically or horizontally concentrated measures, leading to interesting open problems.