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

Abstract

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