Variational reduction for Ginzburg-Landau vortices

Abstract

Let Omega be a bounded domain with smooth boundary in R-2. We construct non-constant solutions to the complex-valued Ginzburg-Landau equation epsilon(2)Delta u + (1 - vertical bar u vertical bar(2))u = 0 in Omega, as epsilon -> 0, both under zero Neumann and Dirichlet boundary conditions. We reduce the problem of finding solutions having isolated zeros (vortices) with degrees +/- 1 to that of finding critical points of a small C-1-perturbation of the associated renormalized energy. This reduction yields general existence results for vortex solutions. In particular, for the Neumann problem, we find that if Omega is not simply connected, then for any k >= 1 a solution with exactly k vortices of degree one exists.