### 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 |

State | Published - May 2000 |

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### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Annales Scientifiques de l'Ecole Normale Superieure*,

*33*(4), 561-592.

**A rigorous derivation of a free-boundary problem arising in superconductivity.** / Sandier, Étienne; Serfaty, Sylvia.

Research output: Contribution to journal › Article

*Annales Scientifiques de l'Ecole Normale Superieure*, vol. 33, no. 4, pp. 561-592.

}

TY - JOUR

T1 - A rigorous derivation of a free-boundary problem arising in superconductivity

AU - Sandier, Étienne

AU - Serfaty, Sylvia

PY - 2000/5

Y1 - 2000/5

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=0003275448&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0003275448&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:0003275448

VL - 33

SP - 561

EP - 592

JO - Annales Scientifiques de l'Ecole Normale Superieure

JF - Annales Scientifiques de l'Ecole Normale Superieure

SN - 0012-9593

IS - 4

ER -