## Rigorous results on Ginzburg-Landau models in a film submitted to an exterior parallel magnetic field. I.(English)Zbl 0857.34006

Summary: This paper is the first part of a mathematical study of the Ginzburg-Landau equations for a film. We first give here mathematical definitions for the different critical fields. We improve results on the existence and uniqueness of bifurcating superconducting solutions. Then, we analyze different limit problems first introduced by physicists: in particular a limit constant model which will be seen in Part II as the starting point of the study of the asymptotic problem when $$\kappa$$ tends to 0 or when the width of the film tends to 0.

### MSC:

 34A12 Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions to ordinary differential equations 35Q60 PDEs in connection with optics and electromagnetic theory 78A35 Motion of charged particles 34C23 Bifurcation theory for ordinary differential equations 37G99 Local and nonlocal bifurcation theory for dynamical systems