On the shape of Meissner solutions to the 2-dimensional Ginzburg-Landau system

This paper concerns the asymptotic behavior of the stable solution (f(lambda),Q(lambda)) of the full Meissner state equation for a two-dimensional superconductor with penetration depth lambda and Ginzburg-Landau parameter kappa, and subjected to an applied magnetic field H-e. It is known that the solution is stable if the minimum value of vertical bar f(lambda)(x)vertical bar(2)-vertical bar Q(lambda)(x)vertical bar(2) is larger than 1/3, and the solution loses its stability when the minimum value reached 1/3. It has been conjectured that the location of the minimum points of vertical bar f(lambda)(x)vertical bar(2)-vertical bar Q(lambda)(x)vertical bar(2) has connection with the location of vortex nucleation of the superconductor. In this paper, we prove that if the penetration depth lambda is small, the solution (f(lambda),Q(lambda)) exhibits boundary layer behavior, and (1-f(lambda),Q(lambda)) exponentially decays in the normal direction away from the boundary. Moreover, the minimum points of vertical bar f(lambda)(x)vertical bar(2)-vertical bar Q(lambda)(x)vertical bar(2) locate near the set S(He), which is determined by the applied magnetic field He and the geometry of the domain. In the special case where the applied magnetic field He is constant, the minimum points of vertical bar f(lambda)(x)vertical bar(2)-vertical bar Q(lambda)(x)vertical bar(2) locate near the maximum points of the curvature of the domain boundary.