Vortices in Ginzburg-Landau billiards

We present an analysis of the Ginzburg-Landau equations for the description of a two-dimensional superconductor in a bounded domain. Using the propert ies of a particular integrability point of these equations which allows vor tex solutions, we obtain a closed expression for the energy of the supercon ductor. The role of the boundary of the system is to provide a selection me chanism for the number of vortices. A geometrical interpretation of these r esults is presented and they are applied to the analysis of the magnetizati on recently measured on small superconducting discs. Problems related to th e interaction and nucleation of vortices are discussed.