Inserted: 20 mar 2017
Last Updated: 20 mar 2017
In this paper we consider a nonlocal energy $I_\alpha$ whose kernel is obtained by adding to the Coulomb potential an anisotropic term weighted by a parameter $\alpha\in \mathbb R$. The case $\alpha=0$ corresponds to purely logarithmic interactions, minimised by the celebrated circle law for a quadratic confinement; $\alpha=1$ corresponds to the energy of interacting dislocations, minimised by the semi-circle law. We show that for $\alpha\in (0,1)$ the minimiser can be computed explicitly and is the normalised characteristic function of the domain enclosed by an ellipse. To prove our result we borrow techniques from fluid dynamics, in particular those related to Kirchhoff's celebrated result that domains enclosed by ellipses are rotating vortex patches, called Kirchhoff ellipses. Therefore we show a surprising connection between vortices and dislocations.