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 |z| + αx²/|z|², z = x + iy, with −1 < α < 1. This kernel is anisotropic except for the Coulomb case α = 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 √1 − α and vertical semi-axis √1 + α. Letting α → 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.