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arXiv · 1612.01137

The equilibrium measure for a nonlocal dislocation energy

Maria Giovanna Mora·Luca Rondi·Lucia Scardia
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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.

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Maria Giovanna Mora, Luca Rondi, Lucia Scardia. 2016-12-04. The equilibrium measure for a nonlocal dislocation energy. https://arxiv.org/abs/1612.01137

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