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.
The Equilibrium Measure for a Nonlocal Dislocation Energy / M.G. Mora, L. Rondi, L. Scardia. - In: COMMUNICATIONS ON PURE AND APPLIED MATHEMATICS. - ISSN 0010-3640. - 72:1(2019 Jan), pp. 136-158.
|Titolo:||The Equilibrium Measure for a Nonlocal Dislocation Energy|
|Parole Chiave:||equilibrium measure; nonlocal and anisotropic energies; dislocations|
|Settore Scientifico Disciplinare:||Settore MAT/05 - Analisi Matematica|
|Data di pubblicazione:||gen-2019|
|Digital Object Identifier (DOI):||http://dx.doi.org/10.1002/cpa.21762|
|Appare nelle tipologie:||01 - Articolo su periodico|