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arXiv · 2003.13776

Explicit minimisers of some nonlocal anisotropic energies: a short proof

J. Mateu·M. G. Mora·l. Rondi·L. Scardia·J. Verdera
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Abstract

In this paper we consider nonlocal energies defined on probability measures in the plane, given by a convolution interaction term plus a quadratic confinement. The interaction kernel is $-\log|z|+α\, x^2/|z|^2, \; z=x+iy,$ with $-1 < α< 1.$ This kernel is anisotropic except for the Coulombic case $α=0.$ We present a short compact proof of the known surprising fact that the unique minimiser of the energy is the normalised characteristic function of the domain enclosed by an ellipse with horizontal semi-axis $\sqrt{1-α}$ and vertical semi-axis $\sqrt{1+α}.$ Letting $α\to 1^-$ we find that the semicircle law on the vertical axis is the unique minimiser of the corresponding energy, a result related to interacting dislocations, and previously obtained by some of the authors. We devote the first sections of this paper to presenting some well-known background material in the simplest way possible, so that readers unfamiliar with the subject find the proofs accessible

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J. Mateu, M. G. Mora, l. Rondi, L. Scardia, J. Verdera. 2020-10-09. Explicit minimisers of some nonlocal anisotropic energies: a short proof. https://doi.org/10.1070/im9048

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