Explicit minimisers for anisotropic Coulomb energies in 3D (Q6065306)

This article is concerned with the study of the qualitative properties of the nonlocal interaction energy \(I:\mathcal{P}({\mathbb R}^3)\to \mathbb{R}\) given by \[ I(\mu)=\int_{{\mathbb R}^3}(W\ast \mu)(x)d\mu(x)+\int_{{\mathbb R}^3}|x|^2 d\mu(x), \] where \(\mathcal{P}({\mathbb R}^3)\) is the set of all probability measures on \({\mathbb R}^3\) and the potentrial \(W\) assumes the form \(W(x)=|x|^{-1}\Psi(x/|x|)\), where \(\Psi\) is even, smooth and strictly positive. It is obtained that the minimizer of the above functional is either the normalised characteristic function of an ellipsoid or a measure supported on a two-dimensional ellipse. Such a classification depends on the Fourier transform of \(\Psi\). It is also proven in the article that the minimizer is always an ellipsoid if the Fourier transform is strictly positive, while the minimizer can take either of the above forms when the Fourier transform is degenerate.