Minimizers of the Lawrence-Doniach energy in the small-coupling limit: finite width samples in a parallel field

In this paper we study the Lawrence-Doniach model for layered superconductors, for a sample with finite width subjected to a magnetic field parallel to the superconducting layers. We provide a rigorous analysis of the energy minimizers in the limit as the coupling between adjacent superconducting layers tends to zero. We identify a unique global minimizer of the Gibbs free energy in this regime ("vortex planes"), and reveal a sequence of first-order phase transitions by which Josephson vortices are nucleated via the boundary. The small coupling limit is studied via degenerate perturbation theory based on a Lyapunov-Schmidt decomposition which reduces the Lawrence-Doniach system to a finite-dimensional variational problem. Finally, a lower bound on the radius of validity of the perturbation expansion (in terms of various parameters appearing in the model) is obtained.