Calculus of Variations and Geometric Measure Theory
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M. G. Mora - L. Rondi - L. Scardia

The equilibrium measure for a nonlocal dislocation energy

created by mora on 04 Dec 2016
modified on 07 Nov 2017


Accepted Paper

Inserted: 4 dec 2016
Last Updated: 7 nov 2017

Journal: Comm. Pure Appl. Math.
Year: 2016


In this paper we characterise 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 semi-circle law, a well-known measure which also arises as the minimiser 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 minimiser of a nonlocal energy is explicitly computed and the only one in the case of anisotropic kernels.

Keywords: edge dislocations, nonlocal interaction, semi-circle law


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