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The Equilibrium Measure for a Nonlocal Dislocation Energy
Author(s) -
Mora Maria Giovanna,
Rondi Luca,
Scardia Lucia
Publication year - 2019
Publication title -
communications on pure and applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 3.12
H-Index - 115
eISSN - 1097-0312
pISSN - 0010-3640
DOI - 10.1002/cpa.21762
Subject(s) - measure (data warehouse) , logarithm , mathematics , conjecture , anisotropy , dimension (graph theory) , dislocation , plane (geometry) , mathematical analysis , energy (signal processing) , geometry , pure mathematics , condensed matter physics , physics , quantum mechanics , statistics , database , computer science
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. © 2018 Wiley Periodicals, Inc.