Stability and minimality of the ball for attractive-repulsive energies with perimeter penalization

TL;DR

This paper analyzes the stability and minimality of the ball shape for a class of nonlocal attractive-repulsive energies with perimeter penalization, identifying parameter regions where the ball remains stable or becomes unstable.

Contribution

It characterizes the stability and minimality regions of the ball under perimeter-perturbed nonlocal energies with explicit curves in parameter space.

Findings

Existence of stability/instability separation curves in parameter space.

Identification of at least two stable regions separated by an instability zone.

Precise description of stability regions for certain interaction kernels.

Abstract

We consider perimeter perturbations of a class of attractive-repulsive energies, given by the sum of two nonlocal interactions with power-law kernels, defined over sets with fixed measure. We prove that there exists curves in the perturbation-volume parameters space that separate stability/instability and global minimality/non-minimality regions of the ball, and provide a precise description of these curves for certain interaction kernels. In particular, we show that in small perturbation regimes there are (at least) two disconnected regions for the mass parameter in which the ball is stable, separated by an instability region.