Explicit minimizers of some non-local 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 interaction kernel is - log vertical bar z vertical bar + ax(2)/vertical bar z vertical bar(2), z = x + iy, with -1 < alpha < 1. This kernel is anisotropic except for the Coulomb case alpha = 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 with horizontal semi-axis root 1 - alpha and vertical semi-axis root 1 + alpha. Letting alpha -> 1(-), we find that the semicircle law on the vertical axis is the unique minimizer 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.