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The Ellipse Law: Kirchhoff Meets Dislocations

Communications in Mathematical Physics volume 373, pages 507–524 (2020)Cite this article

Abstract

In this paper we consider a nonlocal energy Iα whose kernel is obtained by adding to the Coulomb potential an anisotropic term weighted by a parameter \({\alpha \in \mathbb{R}}\). The case α = 0 corresponds to purely logarithmic interactions, minimised by the circle law; α = 1 corresponds to the energy of interacting dislocations, minimised by the semi-circle law. We show that for \({\alpha \in (0, 1)}\) the minimiser is the normalised characteristic function of the domain enclosed by the ellipse of semi-axes \({\sqrt{1-\alpha}}\) and \({\sqrt{1+\alpha}}\). This result is one of the very few examples where the minimiser of a nonlocal anisotropic energy is explicitly computed. For the proof 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.

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Acknowledgements

JAC was partially supported by the Royal Society via a Wolfson Research Merit Award and by the EPSRC under the Grant EP/P031587/1. MGM and LR are partly supported by GNAMPA–INdAM. MGM acknowledges support by the European Research Council under Grant No. 290888. LR acknowledges support by the Università di Trieste through FRA 2016. LS acknowledges support by the EPSRC under the Grant EP/N035631/1. JM and JV acknowledge support by the Spanish projects MTM2013-44699 (MINECO) and MTM2016-75390 (MINECO), 2014SGR75 (Generalitat de Catalunya) and FP7-607647 (European Union).

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Authors and Affiliations

  1. Department of Mathematics, Imperial College London, London, United Kingdom

    J. A. Carrillo

  2. Department de Matemàtiques, Universitat Autònoma de Barcelona, Catalonia, Spain

    J. Mateu & J. Verdera

  3. Dipartimento di Matematica, Università di Pavia, Pavia, Italy

    M. G. Mora

  4. Dipartimento di Matematica, Università di Milano, Milano, Italy

    L. Rondi

  5. Department of Mathematics, Heriot-Watt University, Edinburgh, United Kingdom

    L. Scardia

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  1. J. A. Carrillo
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  2. J. Mateu
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  3. M. G. Mora
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  4. L. Rondi
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  5. L. Scardia
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  6. J. Verdera
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Correspondence to J. A. Carrillo or M. G. Mora.

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Carrillo, J.A., Mateu, J., Mora, M.G. et al. The Ellipse Law: Kirchhoff Meets Dislocations. Commun. Math. Phys. 373, 507–524 (2020). https://doi.org/10.1007/s00220-019-03368-w

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