Stability of the ball for attractive-repulsive energies

TL;DR

This paper investigates the stability of the ball as a minimizer for a class of attractive-repulsive energies with power-law kernels, analyzing conditions under which the ball is stable or unstable based on mass.

Contribution

It characterizes the mass ranges for stability and shows the stability implies local minimality in Hausdorff distance but not in L^1 sense.

Findings

Large masses lead to positive second variation (stability)

Small masses lead to negative second variation (instability)

Stability implies local minimality in Hausdorff distance

Abstract

We consider a class of attractive-repulsive energies, given by the sum of two nonlocal interactions with power-law kernels, defined over sets with fixed measure. It has recently been proved by R. Frank and E. Lieb that the ball is the unique (up to translation) global minimizer for sufficiently large mass. We focus on the issue of the stability of the ball, in the sense of the positivity of the second variation of the energy with respect to smooth perturbations of the boundary of the ball. We characterize the range of masses for which the second variation is positive definite (large masses) or negative definite (small masses). Moreover, we prove that the stability of the ball implies its local minimality among sets sufficiently close in the Hausdorff distance, but not in $L^1$-sense.