Global Minimizers of a Large Class of Anisotropic Attractive-Repulsive Interaction Energies in 2D

TL;DR

This paper characterizes the global energy minimizers for a broad class of anisotropic Riesz-type interaction energies in 2D, revealing explicit elliptical supports, phase transitions, and complex behaviors including zigzag patterns.

Contribution

It provides explicit formulas for minimizers, characterizes parameter ranges for different configurations, and extends results to the logarithmic case, revealing new phenomena in anisotropic interaction energies.

Findings

Explicit elliptical support formulas for minimizers.

Identification of parameter ranges with different minimizer behaviors.

Discovery of zigzag and non-collapse phenomena in minimizers.

Abstract

We study a large family of Riesz-type singular interaction potentials with anisotropy in two dimensions. Their associated global energy minimizers are given by explicit formulas whose supports are determined by ellipses under certain assumptions. More precisely, by parameterizing the strength of the anisotropic part we characterize the sharp range in which these explicit ellipse-supported configurations are the global minimizers based on linear convexity arguments. Moreover, for certain anisotropic parts, we prove that for large values of the parameter the global minimizer is only given by vertically concentrated measures corresponding to one dimensional minimizers. We also show that these ellipse-supported configurations generically do not collapse to a vertically concentrated measure at the critical value for convexity, leading to an interesting gap of the parameters in between. In this intermediate range, we conclude by infinitesimal concavity that any superlevel set of any local minimizer in a suitable sense does not have interior points. Furthermore, for certain anisotropic parts, their support cannot contain any vertical segment for a restricted range of parameters, and moreover the global minimizers are expected to exhibit a zigzag behavior. All these results hold for the limiting case of the logarithmic repulsive potential, extending and generalizing previous results in the literature. Various examples of anisotropic parts leading to even more complex behavior are numerically explored.