Abstract
In this paper we consider a nonlocal energy I _{α} 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 1α 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)  507–524 
Journal  Communications in Mathematical Physics 
Volume  373 
Early online date  24 Apr 2019 
DOIs  
Publication status  Epub ahead of print  24 Apr 2019 
ASJC Scopus subject areas
 Statistical and Nonlinear Physics
 Mathematical Physics
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Profiles

Lucia Scardia
 School of Mathematical & Computer Sciences  Associate Professor
 School of Mathematical & Computer Sciences, Mathematics  Associate Professor
Person: Academic (Research & Teaching)