The ellipse law: Kirchhoff meets dislocations

J.A. Carrillo, J Mateu, Maria Giovanna Mora, Luca Rondi, Lucia Scardia, J Verdera

Research output: Contribution to journalArticle

Abstract

In this paper we consider a nonlocal energy Iα whose kernel is obtained by adding to the Coulomb potential an anisotropic term weighted by a parameter α∈R . The case α = 0 corresponds to purely logarithmic interactions, minimised by the circle law; α = 1 corresponds to the energy of interacting dislocations, minimised by the semi-circle law. We show that for α∈(0,1) the minimiser is the normalised characteristic function of the domain enclosed by the ellipse of semi-axes 1−α−−−−−√ and 1+α−−−−−√ . This result is one of the very few examples where the minimiser of a nonlocal anisotropic energy is explicitly computed. For the proof we borrow techniques from fluid dynamics, in particular those related to Kirchhoff’s celebrated result that domains enclosed by ellipses are rotating vortex patches, called Kirchhoff ellipses.
Original languageEnglish
Pages (from-to)1-18
JournalCommunications in Mathematical Physics
Early online date24 Apr 2019
DOIs
Publication statusE-pub ahead of print - 24 Apr 2019

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Mathematical Physics

Cite this

Carrillo, J. A., Mateu, J., Mora, M. G., Rondi, L., Scardia, L., & Verdera, J. (2019). The ellipse law: Kirchhoff meets dislocations. Communications in Mathematical Physics, 1-18. https://doi.org/10.1007/s00220-019-03368-w

The ellipse law: Kirchhoff meets dislocations. / Carrillo, J.A.; Mateu, J; Mora, Maria Giovanna; Rondi, Luca; Scardia, Lucia; Verdera, J.

In: Communications in Mathematical Physics, 24.04.2019, p. 1-18.

Research output: Contribution to journalArticle

Carrillo, JA, Mateu, J, Mora, MG, Rondi, L, Scardia, L & Verdera, J 2019, 'The ellipse law: Kirchhoff meets dislocations', Communications in Mathematical Physics, pp. 1-18. https://doi.org/10.1007/s00220-019-03368-w
Carrillo JA, Mateu J, Mora MG, Rondi L, Scardia L, Verdera J. The ellipse law: Kirchhoff meets dislocations. Communications in Mathematical Physics. 2019 Apr 24;1-18. https://doi.org/10.1007/s00220-019-03368-w
Carrillo, J.A. ; Mateu, J ; Mora, Maria Giovanna ; Rondi, Luca ; Scardia, Lucia ; Verdera, J. / The ellipse law: Kirchhoff meets dislocations. In: Communications in Mathematical Physics. 2019 ; pp. 1-18.
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