Linear and Nonlinear Aspects of Vortices: The Ginzburg-andau Model [Paperback]

Aimed at mathematicians, physicists, engineers, and grad students, this monograph will be useful for the nonlinear analysis of problems arising in geometry or mathematical physics. The material presented covers recent and original results by the authors, and serves as an excellent classroom text or a valuable self-study resource.

Equations of the GinzburgLandau vortices have particular applications to a number of problems in physics, including phase transition phenomena in superconductors, superfluids, and liquid crystals.  Building on the results presented by Bethuel, Brazis, and Helein, this current work further analyzes Ginzburg-Landau vortices with a particular emphasis on the uniqueness question.

The authors begin with a general presentation of the theory and then proceed to study problems using weighted H?lder spaces and Sobolev Spaces. These are particularly powerful tools and help us obtain a deeper understanding of the nonlinear partial differential equations associated with Ginzburg-Landau vortices. Such an approach sheds new light on the links between the geometry of vortices and the number of solutions.

Aimed at mathematicians, physicists, engineers, and grad students, this monograph will be useful in a number of contexts in the nonlinear analysis of problems arising in geometry or mathematical physics. The material presented covers recent and original results by the authors, and will serve as an excellent classroom text or a valuable self-study resource.

1 Qualitative Aspects of Ginzburg-Landau Equations.- 1.1 The integrable case.- 1.2 The strongly repulsive case.- 1.3 The existence result.- 1.4 Uniqueness results.- 2 Elliptic Operators in Weighted H?lder Spaces.- 2.1 Function spaces.- 2.2 Mapping properties of the Laplacian.- 2.2.1 Rescaled Schauder estimates.- 2.2.2 Mapping properties of the Laplacian in the injectivity range.- 2.2.3 Mapping properties of the Laplacian ilӎ