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Vortices in the magnetic Ginzburg-Landau model. (English) Zbl 1112.35002

Progress in Nonlinear Differential Equations and Their Applications 70. Basel: Birkhäuser (ISBN 978-0-8176-4316-4/hbk; 978-0-8176-4550-2/ebook). xii, 322 p. (2007).
This book deals with the mathematical study of the two-dimensional Ginzburg-Landau model with magnetic field. This important model was introduced by Ginzburg and Landau in the 1950s as a phenomenological model to describe superconductivity consisting in then complete loss of resistivity of certain metals and alloys at very low temperatures. The Ginzburg-Landau model allows to predict the possibility of a mixed state in type II superconductors where triangular lattices appear. These vortices can be described as a quantized amount of vorticity of the superconducting current localized near a point.
Chapter 1 presents the plan of the book. The goal of the authors consists in describing, through a mathematical analysis, in the asymptotic limit (\(e\) small), the minimizers of the Gibbs energy Ge and their critical points in terms of their vortices. This includes the determination of their precise optimal vortex-locations.
Chapter 2 starts with an heuristic presentation of the model.
Chapter 3 describes the basic mathematical results on the Ginzburg-Landau equation (existence of solutions, a priori estimates,…).
Chapter 4 presents the “ball construction method” allowing to obtain universal lower bounds for Ginzburg-Landau energies in terms of cortices and their degrees.
The second part of the book (chapters 7 through 12) presents results obtained through minimization and the third part (chapter 13) contains results for nonminimizing solutions.
All parts of this interesting book are clearly and rigorously written. A consistent bibliography is given and several open problems are detailed. This work has to be recommended.

MSC:

35-02 Research exposition (monographs, survey articles) pertaining to partial differential equations
82-02 Research exposition (monographs, survey articles) pertaining to statistical mechanics
58E50 Applications of variational problems in infinite-dimensional spaces to the sciences
82D55 Statistical mechanics of superconductors
35B25 Singular perturbations in context of PDEs
35B27 Homogenization in context of PDEs; PDEs in media with periodic structure
35B44 Blow-up in context of PDEs
35J60 Nonlinear elliptic equations
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