TOPOLOGICAL METHODS FOR THE GINZBURG-LANDAU EQUATIONS

We consider the complex-valued Ginzburg-Landau equation on a two-dimen sional domain Omega, with boundary data g, such that \g\ = 1, -Delta u = 1/epsilon(2) u(1-\u\(2)), u=g. We develop a variational framework f or this equation: in particular we show that the topology of the level sets is related to a finite dimensional functional, the renormalized energy. As an application, we prove a multiplicity result of solutions for the equation, when epsilon is small and the winding number of g i s larger or equal to 2. (C) Elsevier, Paris.