The equilibrium measure for a nonlocal dislocation energy

Maria Giovanna Mora, Luca Rondi, Lucia Scardia

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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 languageEnglish
Pages (from-to)136-158
Number of pages23
JournalCommunications on Pure and Applied Mathematics
Issue number1
Early online date17 Jul 2018
Publication statusPublished - 18 Nov 2018


ASJC Scopus subject areas

  • Mathematics(all)
  • Applied Mathematics

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