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

• Published in: Izvestiya. Mathematics Vol. 85; no. 3; pp. 468 - 482
• Main Authors: Mateu, J., Mora, M. G., Rondi, L., Scardia, L., Verdera, J.
• Format: Journal Article
• Language: English
• Published: Providence London Mathematical Society, Turpion Ltd and the Russian Academy of Sciences 01.06.2021; IOP Publishing
• Subjects: Characteristic functions; Convolution; Kernels; maximum principle; non-local interaction; Plemelj formula; potential theory
• ISSN: 1064-5632; 1468-4810
• DOI: 10.1070/IM9048

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 , , with . This kernel is anisotropic except for the Coulomb case . 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 and vertical semi-axis . Letting , 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.