| We consider a coupled Ginzburg-Landau system, called the Lawrence-Doniach system, for layered superconductors such as the recently discovered high-temperature superconductors as well as the anisotropic Ginzburg-Landau system. First, we summarize results on the anisotropic Ginzburg-Landau system which are similar to those on the (isotropic) Ginzburg-Landau system. Next, we introduce the Lawrence-Doniach system and prove an existence result and a maximum principle for its solutions. Unlike the isotropic and anisotropic Ginzburg-Landau systems, higher regularity of the solutions to the Lawrence-Doniach system is not obvious. We develop integral representations of the magnetic vector potential of its solutions in a divergence free gauge and then improve the regularity of solutions in that gauge. The anisotropic Ginzburg-Landau model can be considered as an approximation of the Lawrence-Doniach model in a continuous limit. We prove this by showing that as the layer spacing tends to zero, the Lawrence-Doniach system converges to the anisotropic Ginzburg-Landau system in an appropriate sense. It was shown for the (isotropic) Ginzburg-Landau system that even at very low temperature a strong applied magnetic field will prevent a potentially superconducting material from actually entering the superconducting state. We analytically show the existence of a critical magnetic field, h¯, such that when the absolute value of an applied magnetic field is greater than h¯, the normal (nonsuperconducting) state is the only solution to the Lawrence-Doniach system. Finally we estimate h¯. As kappa → infinity, we derive h¯ = O(kappa) for both the Lawrence-Doniach system and the anisotropic Ginzburg-Landau system. |
