Variational Techniques for Ginzburg-Landau Energies

Ginzburg-Landau-type problems are variational problems which consider a Dirichlet-type energy posed on complex-valued functions, penalized by a potential term which has a well in the unit circle of the complex plane. The denomination comes from the physical model of superconductivity of Ginzburg and Landau. They are phase-transition-type models in the sense that they describe the state of the material according to different "phases" which can coexist in a sample and be separated by various types of interfaces. We start by presenting the physical model (readers familiar with it may wish to skip the next two sections and go straight to the section "The simplified model").