Existence results on ginzburg-landau equations

The present survey is essentially the contents of the lectures given in the CIMPA school and the ISFMA symposium on the topics of Ginzburg-Landau vortices held at Fudan Uinversity, in the November of 2002. It is a review of some contributions to the existence results of Ginzburg-Landau equations in recent years. In the first place we recall some known results concerning the asymptotic behavior of minimizers of a simplified model studied by Bethuel-Brezis-Hélein. In the second place we deal with the renormalized energy: we study the topological similarity between the level sets and using variational methods to prove the multiplicity of solutions with Dirichlet boundary conditions. Finally, existence results with other boundary conditions are presented.