Explicit minimisers of some nonlocal anisotropic energies: a short proof

In this paper we consider non-local energies defined on probability measures in the plane, given by a convolution interaction term plus a quadratic confinement. The anisotropy is given by a parameter a, with -1<1. The kernel is anisotropic except for the Coulomb case a=0. We present a short compact proof of the known surprising fact that the unique minimizer of the energy is the normalized characteristic function of the domain enclosed by an ellipse whose horizontal semi-axis and vertical semi-axis can be explicitly computed. Letting a go to -1, we find that the semicircle law on the vertical axis is the unique minimizer of th corresponding energy, a result related to interacting dislocations, and previously obtained by some of the authors. We devote the first sections of this paper to presenting some well-known background material in the simplest way possible, so that readers unfamiliar with the subject find the proofs accessible.