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Explicit minimizers of some non-local anisotropic energies: A short proof

Izvestiya Mathematics, ISSN: 1468-4810, Vol: 85, Issue: 3, Page: 468-482
2021
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  • Citations
    3
    • Citation Indexes
      3
  • Social Media
    17
    • Shares, Likes & Comments
      17
      • Facebook
        17

Article Description

In this paper we consider non-local energies defined on probability measures in the plane, given by a convolution interaction term plus a quadratic confinement. The interaction kernel is − log |z| + αx/|z|, z = x + iy, with −1 < α < 1. This kernel is anisotropic except for the Coulomb case α = 0. We present a short compact proof of the known surprising fact that the unique minimizer of the energy is the normalized characteristic function of the domain enclosed by an ellipse with horizontal semi-axis √1 − α and vertical semi-axis √1 + α. Letting α → 1, we find that the semicircle law on the vertical axis is the unique minimizer 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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