On the perfect superconducting solution for a generalized Ginzburg-Landau equation

摘要

We study a generalized Ginzburg-Landau equation that models a sample formedof a superconducting/normal junction and which is not submitted to an appliedmagnetic field. We prove the existence of a unique positive (and bounded)solution of this equation. In the particular case when the domain is the entireplane, we determine the explicit expression of the solution (and we find thatit satisfies a Robin (de Gennes) boundary condition on the boundary of thesuperconducting side). Using the result of the entire plane, we determine forthe case of general domains, the asymptotic behavior of the solution for largevalues of the Ginzburg-Landau parameter. The main tools are Hopf's Lemma, theStrong Maximum Principle, elliptic estimates and Agmon type estimates.