The Equilibrium Measure for a Nonlocal Dislocation Energy

Maria Giovanna Mora, Luca Rondi, Lucia Scardia

Research output: Contribution to journalArticle

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.
LanguageEnglish
Pages136-158
Number of pages23
JournalCommunications on Pure and Applied Mathematics
Volume72
Issue number1
Early online date17 Jul 2018
DOIs
Publication statusPublished - Jan 2019

Fingerprint

Equilibrium Measure
Dislocation
Minimizer
Energy
Vertical
Semicircle Law
Interaction
One Dimension
Logarithmic
Tend
kernel

ASJC Scopus subject areas

  • Mathematics(all)
  • Applied Mathematics

Cite this

Mora, Maria Giovanna ; Rondi, Luca ; Scardia, Lucia. / The Equilibrium Measure for a Nonlocal Dislocation Energy. In: Communications on Pure and Applied Mathematics. 2019 ; Vol. 72, No. 1. pp. 136-158.
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The Equilibrium Measure for a Nonlocal Dislocation Energy. / Mora, Maria Giovanna; Rondi, Luca; Scardia, Lucia.

In: Communications on Pure and Applied Mathematics, Vol. 72, No. 1, 01.2019, p. 136-158.

Research output: Contribution to journalArticle

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