The equilibrium measure for an anisotropic nonlocal energy

J. A. Carrillo, Joan Mateu, M. G. Mora, Luca Rondi, Lucia Scardia, Joan Verdera

Research output: Contribution to journalArticlepeer-review

Abstract

In this paper we characterise the minimisers of a one-parameter family of nonlocal and anisotropic energies Iα defined on probability measures in Rn, with n≥3. The energy Iα consists of a purely nonlocal term of convolution type, whose interaction kernel reduces to the Coulomb potential for α=0 and is anisotropic otherwise, and a quadratic confinement. The two-dimensional case arises in the study of defects in metals and has been solved by the authors by means of complex-analysis techniques. We prove that for α∈(−1,n−2], the minimiser of Iα is unique and is the (normalised) characteristic function of a spheroid. This result is a paradigmatic example of the role of the anisotropy of the kernel on the shape of minimisers. In particular, the phenomenon of loss of dimensionality, observed in dimension n=2, does not occur in higher dimension at the value α=n−2 corresponding to the sign change of the Fourier transform of the interaction potential.
Original languageEnglish
Article number109
JournalCalculus of Variations and Partial Differential Equations
Volume60
Issue number3
Early online date11 May 2021
DOIs
Publication statusPublished - Jun 2021

ASJC Scopus subject areas

  • Analysis
  • Applied Mathematics

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