### Abstract

We study the Ginzburg-Landau energy of superconductors with a term a_{ε} modelling the pinning of vortices by impurities in the limit of a large Ginzburg-Landau parameter κ=1/ε. The function a_{ε} is oscillating between 1/2 and 1 with a scale which may tend to 0 as κ tends to infinity. Our aim is to understand that in the large κ limit, stable configurations should correspond to vortices pinned at the minimum of a_{ε} and to derive the limiting homogenized free-boundary problem which arises for the magnetic field in replacement of the London equation. The method and techniques that we use are inspired from those of Sandier and Serfaty, Annales Scientifiques de l'ENS (to appear) (in which the case a_{ε}≡1 was treated) and based on energy estimates, convergence of measures and construction of approximate solutions. Because of the term a_{ε}(x) in the equations, we also need homogenization theory to describe the fact that the impurities, hence the vortices, form a homogenized medium in the material.

Original language | English (US) |
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Pages (from-to) | 339-372 |

Number of pages | 34 |

Journal | Journal des Mathematiques Pures et Appliquees |

Volume | 80 |

Issue number | 3 |

DOIs | |

State | Published - Apr 2001 |

### Keywords

- Ginzburg-Landau
- Homogenization
- Pinning
- Superconductivity

### ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics

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## Cite this

*Journal des Mathematiques Pures et Appliquees*,

*80*(3), 339-372. https://doi.org/10.1016/S0021-7824(00)01180-6