First Properties of Solutions to the Ginzburg-Landau Equations

Abstract

In this chapter, we start to investigate the mathematical aspects of the Ginzburg-Landau energy and equations. Whereas the material in the first three sections is relatively easy or standard (existence of minimizers, regularity of solutions, apriori estimates …) and used throughout the later chapters, the material of the last two sections is more advanced, contains several results stated without proofs, and is only used in Chapter 5 and then Chapters 10 to 12. However, we feel that the material is important enough, like the uniqueness result of P. Mironescu (Theorem 3.2), or basic enough to deserve to be stated early on.