Title: The ellipse law: Kirchhoff meets dislocations,
Abstract: 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 $\alpha\in \R$. 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 \emph{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 \emph{Kirchhoff ellipses}. Therefore we show a
surprising connection between vortices and dislocations.