We study the Ginzburg-Landau energy of superconductors with a term aε modelling the pinning of vortices by impurities in the limit of a large Ginzburg-Landau parameter κ=1/ε. The function aε 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 aε 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 aε≡1 was treated) and based on energy estimates, convergence of measures and construction of approximate solutions. Because of the term aε(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.
ASJC Scopus subject areas
- Applied Mathematics