Abstract
In this paper we characterize 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 semicircle law, a well‐known measure that also arises as the minimizer 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 minimizer of a nonlocal energy is explicitly computed and the only one in the case of anisotropic kernels.
Original language | English |
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Pages (from-to) | 136-158 |
Number of pages | 23 |
Journal | Communications on Pure and Applied Mathematics |
Volume | 72 |
Issue number | 1 |
Early online date | 17 Jul 2018 |
DOIs | |
Publication status | Published - Jan 2019 |
ASJC Scopus subject areas
- General Mathematics
- Applied Mathematics