The equilibrium measure for a nonlocal dislocation energy

TL;DR

This paper characterizes the equilibrium measure for a nonlocal, anisotropic energy describing positive dislocations, showing they tend to form vertical walls with the measure supported on the axis following a semi-circle law.

Contribution

It explicitly computes the minimizer of a nonlocal anisotropic energy, confirming the formation of vertical dislocation walls and providing a rare explicit solution.

Findings

Minimizer supported on the vertical axis

Distribution follows the semi-circle law

Dislocations tend to form vertical walls

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.