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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Superconductivity near the normal state in a half-plane under the action of a perpendicular electric current and an induced magnetic field


Authors: Yaniv Almog, Bernard Helffer and Xing-Bin Pan
Journal: Trans. Amer. Math. Soc. 365 (2013), 1183-1217
MSC (2010): Primary 82D55, 35B25, 35B40, 35Q55
Published electronically: August 9, 2012
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Abstract: We consider the linearization of the time-dependent Ginzburg-Landau system near the normal state. We assume that a constant magnetic field and an electric current are applied through the sample, which captures half of the plane, inducing thereby a linearly varying magnetic field. In the limit of small normal conductivity we prove that if the electric current is lower than some critical value, the normal state loses its stability. For currents stronger than this critical value, the normal state is stable. To obtain this stability result we analyze both the spectrum and the pseudo-spectrum of the linearized operator. The critical current tends, in this small conductivity limit, to another critical current which had been obtained for a reduced model which neglects magnetic field effects.


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Additional Information

Yaniv Almog
Affiliation: Department of Mathematics, Louisiana State University, Baton Rouge, Louisiana 70803
Email: almog@math.lsu.edu

Bernard Helffer
Affiliation: Laboratoire de Mathématiques, Université Paris-Sud 11 et CNRS, Bât 425, 91 405 Orsay Cedex, France
Email: Bernard.Helffer@math.u-psud.fr

Xing-Bin Pan
Affiliation: Department of Mathematics and Center for PDE, East China Normal University, Shanghai 200062, People’s Republic of China
Email: xbpan@math.ecnu.edu.cn

DOI: http://dx.doi.org/10.1090/S0002-9947-2012-05572-3
PII: S 0002-9947(2012)05572-3
Keywords: Superconductivity, critical current, critical magnetic field
Received by editor(s): July 19, 2010
Received by editor(s) in revised form: February 20, 2011
Published electronically: August 9, 2012
Article copyright: © Copyright 2012 American Mathematical Society