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arXiv · 1703.07013

The ellipse law: Kirchhoff meets dislocations

J. A. Carrillo·J. Mateu·M. G. Mora·L. Rondi·L. Scardia·J. Verdera
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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.

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J. A. Carrillo, J. Mateu, M. G. Mora, L. Rondi, L. Scardia, J. Verdera. 2017-03-21. The ellipse law: Kirchhoff meets dislocations. https://arxiv.org/abs/1703.07013

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