## 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| + αx^{2}/|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 language | English |
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Pages (from-to) | 468-482 |

Number of pages | 15 |

Journal | Izvestiya: Mathematics |

Volume | 85 |

Issue number | 3 |

DOIs | |

Publication status | Published - 1 Jun 2021 |

## Keywords

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

## ASJC Scopus subject areas

- General Mathematics