Accepted Paper

Inserted: 4 dec 2016
Last Updated: 7 nov 2017

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

Abstract:

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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• Semicircle-Mora-Rondi-Scardia.pdf