Calculus of Variations and Geometric Measure Theory

J. Mateu - M. G. Mora - L. Rondi - L. Scardia - J. Verdera

Stability of ellipsoids as the energy minimisers of perturbed Coulomb energies

created by mora on 28 Dec 2021


Submitted Paper

Inserted: 28 dec 2021
Last Updated: 28 dec 2021

Year: 2021


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.

Keywords: Potential theory, nonlocal interactions, global minimisers, Coulomb kernel