Communications in Mathematical Physics

, Volume 367, Issue 1, pp 317–349 | Cite as

On the First Critical Field in the Three Dimensional Ginzburg–Landau Model of Superconductivity

  • Carlos RománEmail author


The Ginzburg–Landau model is a phenomenological description of superconductivity. A crucial feature of type-II superconductors is the occurrence of vortices, which appear above a certain value of the strength of the applied magnetic field called the first critical field. In this paper we estimate this value, when the Ginzburg–Landau parameter is large, and we characterize the behavior of the Meissner solution, the unique vortexless configuration that globally minimizes the Ginzburg–Landau energy below the first critical field. In addition, we show that beyond this value, for a certain range of the strength of the applied field, there exists a unique Meissner-type solution that locally minimizes the energy.


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I am very grateful to my former Ph.D. advisors Etienne Sandier and Sylvia Serfaty for suggesting the problem and for useful comments. I also thank the anonymous referees for helpful comments. Most of this work was done while I was a Ph.D. student at the Jacques-Louis Lions Laboratory of the Pierre and Marie Curie University, supported by a public grant overseen by the French National Research Agency (ANR) as part of the “Investissements d’Avenir” program (reference: ANR-10-LABX-0098, LabEx SMP). Part of this work was supported by the German Science Foundation DFG in the context of the Emmy Noether junior research group BE 5922/1-1.

Conflict of interest

The author declares that they have no conflict of interest.


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© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Mathematisches InstitutUniversität LeipzigLeipzigGermany

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