Stability of the ball for attractive-repulsive energies
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 -sense.
Keywords
Cite
@article{arxiv.2207.00388,
title = {Stability of the ball for attractive-repulsive energies},
author = {Marco Bonacini and Riccardo Cristoferi and Ihsan Topaloglu},
journal= {arXiv preprint arXiv:2207.00388},
year = {2024}
}
Comments
This is a post-peer-review, pre-copyedit version of an article published in the SIAM Journal on Mathematical Analysis. The final authenticated version is available online at: https://doi.org/10.1137/22M1506894