The ellipse law: Kirchhoff meets dislocations
File(s)
Published version
Author(s)
Type
Journal Article
Abstract
In this paper we consider a nonlocal energyIαwhose kernel is obtained by addingto the Coulomb potential an anisotropic term weighted by a parameterα∈R. The caseα= 0corresponds to purely logarithmic interactions, minimised by the circle law;α= 1 correspondsto the energy of interacting dislocations, minimised by the semi-circle law. We show that forα∈(0,1) the minimiser is the normalised characteristic function of the domain enclosed bytheellipseof semi-axes√1−αand√1 +α. This result is one of the very few examples wherethe minimiser of a nonlocal anisotropic energy is explicitly computed. For the proof we borrowtechniques from fluid dynamics, in particular those related to Kirchhoff’s celebrated result thatdomains enclosed by ellipses are rotating vortex patches, calledKirchhoff ellipses.
Date Issued
2019-04-24
Date Acceptance
2019-01-05
Citation
Communications in Mathematical Physics, 2019, 373, pp.507-524
ISSN
0010-3616
Publisher
Springer (part of Springer Nature)
Start Page
507
End Page
524
Journal / Book Title
Communications in Mathematical Physics
Volume
373
Copyright Statement
© The Author(s) 2019. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution,
and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source,
provide a link to the Creative Commons license, and indicate if changes were made.
and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source,
provide a link to the Creative Commons license, and indicate if changes were made.
Sponsor
Engineering & Physical Science Research Council (EPSRC)
Identifier
https://link.springer.com/article/10.1007%2Fs00220-019-03368-w
Grant Number
EP/P031587/1
Subjects
0101 Pure Mathematics
0105 Mathematical Physics
0206 Quantum Physics
Mathematical Physics
Publication Status
Published
Date Publish Online
2019-04-24