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

Jоan Mateu, Maria Giovanna Mora, Luca Rondi, Lucia Scardia, Jоan Verdera

Research output: Contribution to journalArticlepeer-review

Abstract

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| + αx2/|z|2, 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.

Original languageEnglish
Pages (from-to)468-482
Number of pages15
JournalIzvestiya: Mathematics
Volume85
Issue number3
DOIs
Publication statusPublished - 1 Jun 2021

Keywords

  • Maximum principle
  • Non-local interaction
  • Plemelj formula
  • Potential theory

ASJC Scopus subject areas

  • Mathematics(all)

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