The ellipse law: Kirchhoff meets dislocations

TL;DR

This paper introduces a nonlocal energy model interpolating between Coulomb and dislocation interactions, revealing that minimizers for certain parameters are elliptical domains, connecting vortex dynamics and dislocation theory.

Contribution

It explicitly computes minimizers of a nonlocal energy for a range of parameters, linking vortex ellipse solutions to dislocation models through fluid dynamics techniques.

Findings

Minimizers are elliptical domains for  in (0,1)

Explicit formula for minimizers involving ellipse characteristic functions

Establishes a connection between vortex dynamics and dislocation models

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.