Stability of ellipsoids as the energy minimisers of perturbed Coulomb energies

TL;DR

This paper proves that under certain conditions, ellipsoids remain stable energy minimisers for perturbed Coulomb energies, extending the known stability of balls for the unperturbed case.

Contribution

It characterizes the minimizers of a class of perturbed Coulomb energies as ellipsoids, showing their stability as energy minimisers under specific perturbation conditions.

Findings

Minimizer is the normalized characteristic function of an ellipsoid.

Ellipsoids are stable energy minimisers under certain perturbations.

Results extend the stability of balls in Coulomb energy to ellipsoids with perturbations.

Abstract

In this paper we characterise the minimiser for a class of nonlocal perturbations of the Coulomb energy. We show that the minimiser is the normalised characteristic function of an ellipsoid, under the assumption that the perturbation kernel has the same homogeneity as the Coulomb potential, is even, smooth off the origin and sufficiently small. This result can be seen as the stability of ellipsoids as energy minimisers, since the minimiser of the Coulomb energy is the normalised characteristic function of a ball.