The Ginzburg--Landau equations for superconducting films and the Meissner effect

The existence of the solutions at the lowest energy level to the Ginzburg--Landau equations describing superconducting films under the influence of an external magnetic field is proved. The condition obtained for maintaining superconducting states does not involve the Ginzburg--Landau parameter, and, more surprisingly, in this condition both the external magnetic field and the thickness of the film play equal parts. The asymptotic behavior of these solutions for small or large values of the external field is also studied, which qualitatively verifies the Meissner effect and quantitatively reproduces the solution found from the classical London theory