### Abstract

We study the Ginzburg-Landau energy of superconductors submitted to a possibly non-uniform magnetic field, in the limit of a large Ginzburg-Landau parameter κ . We prove that the induced magnetic fields associated to minimizers of the energy-functional converge as κ→+∞ to the solution of a free-boundary problem. This free-boundary problem has a nontrivial solution only when the applied magnetic field is of the order of the "first critical field", i.e. of the order of logκ . In other cases, our results are contained in those we had previously obtained [15, 16, 14]. We also derive a convergence result for the density of vortices.

Original language | English (US) |
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Pages (from-to) | 561-592 |

Number of pages | 32 |

Journal | Annales Scientifiques de l'Ecole Normale Superieure |

Volume | 33 |

Issue number | 4 |

DOIs | |

State | Published - May 2000 |

### ASJC Scopus subject areas

- Mathematics(all)

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## Cite this

Sandier, É., & Serfaty, S. (2000). A rigorous derivation of a free-boundary problem arising in superconductivity.

*Annales Scientifiques de l'Ecole Normale Superieure*,*33*(4), 561-592. https://doi.org/10.1016/S0012-9593(00)00122-1