Explicit minimisers for anisotropic Coulomb energies in 3D

TL;DR

This paper characterizes the minimizers of a broad class of anisotropic Coulomb energies in three dimensions, revealing conditions under which they are ellipsoids or supported on ellipses, with explicit examples of dimensionality loss.

Contribution

It provides a complete characterization of minimizers for anisotropic Coulomb energies in 3D, depending on the Fourier transform of the potential, including explicit examples.

Findings

Minimizers are ellipsoids when the Fourier transform is strictly positive.

Degenerate Fourier transforms can lead to minimizers supported on ellipses.

Explicit example demonstrates loss of dimensionality in minimizers.

Abstract

In this paper we consider a general class of anisotropic energies in three dimensions and give a complete characterisation of their minimisers. We show that, depending on the Fourier transform of the interaction potential, the minimiser is either the normalised characteristic function of an ellipsoid or a measure supported on a two-dimensional ellipse. In particular, it is always an ellipsoid if the transform is strictly positive, while when the Fourier transform is degenerate both cases can occur. Finally, we show an explicit example where loss of dimensionality of the minimiser does occur.