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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 semicircle law. We show that for α∈(0,1) the minimiser is the normalised characteristic function of the domain enclosed by the ellipse of semiaxes 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 language  English 

Pages (fromto)  507524 
Number of pages  18 
Journal  Communications in Mathematical Physics 
Volume  373 
Issue number  2 
Early online date  24 Apr 2019 
DOIs  
Publication status  Published  31 Jan 2020 
ASJC Scopus subject areas
 Statistical and Nonlinear Physics
 Mathematical Physics
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 1 Finished

Dislocation Patterns Beyond Optimality
Scardia, L.
Engineering and Physical Sciences Research Council
1/10/16 → 30/09/18
Project: Research council