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 $-\log|z|+\alpha x^2/|z|^2$, $z=x+iy$, 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 $\sqrt{1-\alpha}$ and vertical semi-axis $\sqrt{1+\alpha}$. Letting $\alpha \to 1^-$, 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.