Abstract
In this paper we characterize the equilibrium measure for a nonlocal and anisotropic weighted energy describing the interaction of positive dislocations in the plane. We prove that the minimum value of the energy is attained by a measure supported on the vertical axis and distributed according to the semicircle law, a well‐known measure that also arises as the minimizer of purely logarithmic interactions in one dimension. In this way we give a positive answer to the conjecture that positive dislocations tend to form vertical walls. This result is one of the few examples where the minimizer of a nonlocal energy is explicitly computed and the only one in the case of anisotropic kernels.
Original language  English 

Pages (fromto)  136158 
Number of pages  23 
Journal  Communications on Pure and Applied Mathematics 
Volume  72 
Issue number  1 
Early online date  17 Jul 2018 
DOIs  
Publication status  Published  Jan 2019 
ASJC Scopus subject areas
 Mathematics(all)
 Applied Mathematics
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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)