Inserted: 4 dec 2016
Last Updated: 4 dec 2016
The aim of this paper is to 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. This result gives a positive answer to the conjecture that positive dislocations tend to form vertical walls.
Keywords: edge dislocations, nonlocal interaction, semi-circle law