Stability of the ball for attractive-repulsive energies

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 larg...

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Bibliographic Details
Published inarXiv.org
Main Authors Bonacini, Marco, Cristoferi, Riccardo, Topaloglu, Ihsan
Format Paper Journal Article
LanguageEnglish
Published Ithaca Cornell University Library, arXiv.org 11.01.2024
Subjects
Mathematics - Analysis of PDEs
Mathematics - Functional Analysis
Mathematics - Mathematical Physics
Metric space
Perturbation
Physics - Mathematical Physics
Stability
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Summary: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.
ISSN:2331-8422
DOI:10.48550/arxiv.2207.00388