The equilibrium measure for a nonlocal dislocation energy
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
Cite
@article{arxiv.1612.01137,
title = {The equilibrium measure for a nonlocal dislocation energy},
author = {Maria Giovanna Mora and Luca Rondi and Lucia Scardia},
journal= {arXiv preprint arXiv:1612.01137},
year = {2017}
}
Comments
16 pages. Accepted version, to appear on Comm. Pure Appl. Math