Break of radial symmetry for a class of attractive-repulsive interaction energy minimizers

TL;DR

This paper proves that for certain attractive-repulsive interaction potentials, energy minimizers are necessarily non-radially symmetric, using a novel approach that combines energy bounds and measure construction.

Contribution

It introduces a new method to rigorously demonstrate the break of radial symmetry in energy minimizers for a class of interaction potentials.

Findings

Energy minimizers are non-radially symmetric for specified potentials.

Minimizers are H"older continuous functions.

The approach combines lower bounds and measure construction.

Abstract

Break of radial symmetry for interaction energy minimizers is a phenomenon where a radial interaction potential whose associated energy minimizers are never radially symmetric. Numerically, it has been frequently observed for various types of interaction potentials, however, rigorous justification of this phenomenon was only done in very limited cases. We propose a new approach to prove the break of radial symmetry, by using a lower bound for the energy in the class of radial probability measures, combining with the construction of a probability measure whose energy is lower than this lower bound. In particular, we prove that for a class of interaction potentials that are repulsive at short distance and attractive at long distance, every energy minimizer is necessarily a H\"older continuous function which is not radially symmetric.