Explicit minimisers of some nonlocal anisotropic energies: a short proof
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 $-\log|z|+α\, x^2/|z|^2, \; z=x+iy,$ 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 $\sqrt{1-α}$ and vertical semi-axis $\sqrt{1+α}.$ Letting $α\to 1^-$ 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