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

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- by Yaniv Almog, Bernard Helffer and Xing-Bin Pan PDF
- Trans. Amer. Math. Soc.
**365**(2013), 1183-1217 Request permission

## 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.## References

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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
- MR Author ID: 83860
- 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
- Received by editor(s): July 19, 2010
- Received by editor(s) in revised form: February 20, 2011
- Published electronically: August 9, 2012
- © Copyright 2012 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**365**(2013), 1183-1217 - MSC (2010): Primary 82D55, 35B25, 35B40, 35Q55
- DOI: https://doi.org/10.1090/S0002-9947-2012-05572-3
- MathSciNet review: 3003262