A family of explicit minimizers for interaction energies

TL;DR

This paper derives explicit formulas for the unique minimizers of certain power-law interaction energies in various dimensions, using PDE techniques and dimension reduction methods.

Contribution

It provides explicit formulas for the energy minimizers in odd and even dimensions, employing novel PDE and dimension reduction approaches.

Findings

Explicit minimizers for odd dimensions derived from PDE methods.

Minimizers in even dimensions obtained via dimension reduction from higher dimensions.

Unique minimizers characterized up to translation for specified parameters.

Abstract

In this paper we consider the minimizers of the interaction energies with the power-law interaction potentials $W({\bf x}) = \frac{|{\bf x}|^a}{a} - \frac{|{\bf x}|^b}{b}$ in $d$ dimensions. For odd $d$ with $(a,b)=(3,2-d)$ and even $d$ with $(a,b)=(3,1-d)$, we give the explicit formula for the unique energy minimizer up to translation. For the odd dimensions, the key observation is that successive Laplacian of the Euler-Lagrange condition gives a local partial differential equation for the minimizer. For the even dimensions $d$, the minimizer is given as the projection and rescaling of the previously constructed minimizer in dimension $d+1$ via a new lemma on dimension reduction.