## 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