Explicit minimisers of some nonlocal anisotropic energies: a short proof
Abstract
In this paper we consider nonlocal energies defined on probability measures in the plane, given by a convolution interaction term plus a quadratic confinement. The interaction kernel is -|z|+α\, x2/|z|2, \; z=x+iy, with -1 < α< 1. This kernel is anisotropic except for the Coulombic case α=0. We present a short compact proof of the known surprising fact that the unique minimiser of the energy is the normalised characteristic function of the domain enclosed by an ellipse with horizontal semi-axis 1-α and vertical semi-axis 1+α. Letting α 1- we find that the semicircle law on the vertical axis is the unique minimiser of the 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