The Equilibrium Measure for a Nonlocal Dislocation Energy. Issue 1 (17th July 2018)

• Record Type: Journal Article
• Authors: Mora, Maria Giovanna; Rondi, Luca; Scardia, Lucia
• Is Part Of: Communications on pure and applied mathematics. Volume 72:Issue 1(2019:Jan.)
• Page Start: 136
• Page End: 158
• Publication Date: 2018-07-17
• Subjects: Mathematics -- Periodicals; Mechanics -- Periodicals; Mathématiques -- Périodiques; Mécanique -- Périodiques; 510.5
• DOI: 10.1002/cpa.21762
• Languages: English
• ISSNs: 0010-3640

Abstract

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. © 2018 Wiley Periodicals, Inc.