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In this chapter, we establish the existence of multiple branches of stable solutions of () which have an arbitrary number of vortices , with both bounded and unbounded, but not too large, in a wide regime of applied fields. These solutions are obtained by minimizing the energy over subsets of the functional space which correspond, very roughly speaking, to configurations with vortices (or only allow for such when minimizing); the heart of the matter consists in proving that the minimum is achieved in the interior of , thus yielding locally minimizing solutions of the equations. These solutions turn out to be global energy minimizers in some narrow intervals of values of .

In this chapter, we establish which solutions, among the ones found in Theorem 11.1, minimize the energy globally. This of course depends on the value . As increases, we will see that the minimizers have one, then two, then more and more vortices, as predicted by the physics. This allows us to give precise expansions of the critical fields.

The problem we have dealt with until now was to understand the → 0 limits of the vorticity measures associated to minimizers of the Ginzburg- Landau functional. We now wish to derive a criticality condition for a limiting vorticity measure associated to a family (,) of solutions of the Ginzburg-Landau system () which are not necessarily minimizing.

Our goal here is to give a brief overview of results on Ginzburg-Landau, and point towards suitable references (in thematic, rather than chronological or hierarchical order). We apologize for not being able to be completely exhaustive.

The aim of this chapter is to provide one of the basic tools for the analysis of the Ginzburg-Landau functional in terms of vortices.