Pinning phenomena in the Ginzburg-Landau model of superconductivity

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.