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Birkhäuser

Vortices in the Magnetic Ginzburg-Landau Model

  • Textbook
  • © 2007

Overview

Authors:
  1. Etienne Sandier

    1. Département de mathématiques, Université Paris-XII Val-de-Marne, Créteil Cedex, France

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  2. Sylvia Serfaty

    1. Courant Institute of Mathematical Sciences, New York University, New York, USA

    View author publications

    You can also search for this author in PubMed   Google Scholar

  • Describes essential mathematical techniques and tools for analyzing the Ginzburg--Landau functional
  • Presents an introduction to the theory with results and current research
  • Acts a guide to the various branches of Ginzburg-Landau studies and provides context for the study of vortices
  • Includes list of open problems

Part of the book series: Progress in Nonlinear Differential Equations and Their Applications (PNLDE, volume 70)

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Table of contents (15 chapters)

  1. Front Matter

    Pages i-xii
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  2. Introduction

    Pages 1-24

  3. Physical Presentation of the Model—Critical Fields

    Pages 25-38

  4. First Properties of Solutions to the Ginzburg-Landau Equations

    Pages 39-58

  5. The Vortex-Balls Construction

    Pages 59-81

  6. Coupling the Ball Construction to the Pohozaev Identity and Applications

    Pages 83-115

  7. Jacobian Estimate

    Pages 117-125

  8. The Obstacle Problem

    Pages 127-154

  9. Higher Values of the Applied Field

    Pages 155-163

  10. The Intermediate Regime

    Pages 165-206

  11. The Case of a Bounded Number of Vortices

    Pages 207-218

  12. Branches of Solutions

    Pages 219-242

  13. Back to Global Minimization

    Pages 243-251

  14. Asymptotics for Solutions

    Pages 253-282

  15. A Guide to the Literature

    Pages 283-297

  16. Open Problems

    Pages 299-302

  17. Back Matter

    Pages 303-322
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About this book

More than ten years have passed since the book of F. Bethuel, H. Brezis and F. H´ elein, which contributed largely to turning Ginzburg–Landau equations from a renowned physics model into a large PDE research ?eld, with an ever-increasing number of papers and research directions (the number of published mathematics papers on the subject is certainly in the several hundreds, and that of physics papers in the thousands). Having ourselves written a series of rather long and intricately - terdependent papers, and having taught several graduate courses and mini-courses on the subject, we felt the need for a more uni?ed and self-contained presentation. The opportunity came at the timely moment when Ha¨ ?m Brezis s- gested we should write this book. We would like to express our gratitude towards him for this suggestion and for encouraging us all along the way. As our writing progressed, we felt the need to simplify some proofs, improvesomeresults,aswellaspursuequestionsthatarosenaturallybut that we had not previously addressed. We hope that we have achieved a little bit of the original goal: to give a uni?ed presentation of our work with a mixture of both old and new results, and provide a source of reference for researchers and students in the ?eld.

Reviews

"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 the complete loss of resistivity of certain metals and alloys at very low temperatures...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."
—Zentralblatt MATH

"In conclusion, this book is an excellent, up-to-the-minute presentation of the current state of the mathematics of vortices in Ginzburg-Landau models. It also represents a tour de force of mathematical analysis, revealing a fascinating and intricate picture of a physical model which may have been unexpected based on heuristic considerations. I strongly recommend this book to researchers who are interested in vortices (andother quantized singularities) as these methods will continue to be instrumental in forthcoming research in the field. One could also find interesting material to supplement a graduate coursc in variational methods or PDEs."
—SIAM Review

Authors and Affiliations

  • Département de mathématiques, Université Paris-XII Val-de-Marne, Créteil Cedex, France

    Etienne Sandier

  • Courant Institute of Mathematical Sciences, New York University, New York, USA

    Sylvia Serfaty

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