Stable nucleation for the Ginzburg-Landau system with an applied magnetic field

We consider the Ginzburg-Landau system with an applied magnetic field and analyze the behavior of solutions when the domain is a cylinder (of radius (r) over bar) and the applied field is parallel to the axis. It is shown that there is an upper critical value (h) over bar such that, if the modulus of the applied field is greater than (h) over bar, the normal (nonsuperconducting) state tin which the order parameter is identically zero) is stable and if the modulus of the applied field is slightly below (h) over bar, the normal state is unstable. In addition, it is shown that there is a positive lower critical value (h) under bar less than or equal to (h) over bar such that the normal state is unstable if the modulus of the applied held is less than (h) under bar and stable if the modulus is slightly above (h) under bar. In the case of type-II materials for which the Ginzburg-Landau constant kappa is large, it is shown that there is a discrete set of radii B(kappa) such that if (r) over bar is not an element of B(kappa) and kappa (r) over bar is sufficiently large, then for each applied field of modulus slightly less than (h) over bar (or slightly more than (h) under bar) there is precisely one small superconducting solution (up to a gauge transformation) which is stable. Moreover for this solution, the complex-valued order parameter psi is zero only on the axis of the cylinder, and its winding number is proportional to the product of kappa(2) and the cross-sectional area of the cylinder. In addition, the solution exhibits "surface superconductivity" as predicted by the physicists DE GENNES and ST. JAMES.