Superconductors surrounded by normal materials

We consider a generalized Ginzburg–Landau energy functional modelling a superconductor surrounded by a material in the normal state. In this model, the order parameter is defined in the whole space. We derive existence of a global minimizer in weighted Sobolev spaces for both square-integrable and constant-applied magnetic fields. We then prove boundedness and classical elliptic estimates for the order parameter, in order to study the loss of superconductivity for high applied magnetic fields. In two dimensions for the general case and in three dimensions for the case of constant permeability, we show the existence of an upper critical field above which the only finite-energy weak solutions are the normal states. For the three-dimensional case, we show that as the applied field tends to infinity, finite-energy weak solutions tend to the normal state.