Ginzburg-Landau Vortices

  • Fabrice Bethuel
  • Haim Brezis
  • Frederic Helein

Part of the Modern Birkhäuser Classics book series (MBC)

Table of contents

  1. Front Matter
    Pages i-xxix
  2. Fabrice Bethuel, Haim Brezis, Frederic Helein
    Pages 1-30
  3. Fabrice Bethuel, Haim Brezis, Frederic Helein
    Pages 31-41
  4. Fabrice Bethuel, Haim Brezis, Frederic Helein
    Pages 42-47
  5. Fabrice Bethuel, Haim Brezis, Frederic Helein
    Pages 48-51
  6. Fabrice Bethuel, Haim Brezis, Frederic Helein
    Pages 52-56
  7. Fabrice Bethuel, Haim Brezis, Frederic Helein
    Pages 57-64
  8. Fabrice Bethuel, Haim Brezis, Frederic Helein
    Pages 65-75
  9. Fabrice Bethuel, Haim Brezis, Frederic Helein
    Pages 76-99
  10. Fabrice Bethuel, Haim Brezis, Frederic Helein
    Pages 100-106
  11. Fabrice Bethuel, Haim Brezis, Frederic Helein
    Pages 107-136
  12. Fabrice Bethuel, Haim Brezis, Frederic Helein
    Pages 137-141
  13. Back Matter
    Pages 142-159

About this book

Introduction

This book is concerned with the study in two dimensions of stationary solutions of uɛ of a complex valued Ginzburg-Landau equation involving a small parameter ɛ. Such problems are related to questions occurring in physics, e.g., phase transition phenomena in superconductors and superfluids. The parameter ɛ has a dimension of a length which is usually small.  Thus, it is of great interest to study the asymptotics as ɛ tends to zero.

One of the main results asserts that the limit u-star of minimizers uɛ exists. Moreover, u-star is smooth except at a finite number of points called defects or vortices in physics. The number of these defects is exactly the Brouwer degree – or winding number – of the boundary condition. Each singularity has degree one – or as physicists would say, vortices are quantized.

The singularities have infinite energy, but after removing the core energy we are lead to a concept of finite renormalized energy.  The location of the singularities is completely determined by minimizing the renormalized energy among all possible configurations of defects. 

The limit u-star can also be viewed as a geometrical object.  It is a minimizing harmonic map into S1 with prescribed boundary condition g.  Topological obstructions imply that every map u into S1 with u = g on the boundary must have infinite energy.  Even though u-star has infinite energy, one can think of u-star as having “less” infinite energy than any other map u with u = g on the boundary.

The material presented in this book covers mostly original results by the authors.  It assumes a moderate knowledge of nonlinear functional analysis, partial differential equations, and complex functions.  This book is designed for researchers and graduate students alike, and can be used as a one-semester text.  The present softcover reprint is designed to make this classic text available to a wider audience.

"...the book gives a very stimulating account of an interesting minimization problem. It can be a fruitful source of ideas for those who work through the material carefully."

- Alexander Mielke, Zeitschrift für angewandte Mathematik und Physik 46(5)



Keywords

Ginzburg-Landau Vortices Partial Differential Equations Phase Transition Phenomena Superconductors Superfluids Nonlinear Functional Analysis

Authors and affiliations

  • Fabrice Bethuel
    • 1
  • Haim Brezis
    • 2
  • Frederic Helein
    • 3
  1. 1.Laboratory Jacques-Louis LionsPierre and Marie Curie University Laboratory Jacques-Louis LionsParisFrance
  2. 2.Rutgers UniversityPiscatawayUSA
  3. 3.Université Paris Diderot - Paris 7ParisFrance

Bibliographic information

  • DOI https://doi.org/10.1007/978-3-319-66673-0
  • Copyright Information Springer International Publishing AG 2017
  • Publisher Name Birkhäuser, Cham
  • eBook Packages Mathematics and Statistics
  • Print ISBN 978-3-319-66672-3
  • Online ISBN 978-3-319-66673-0
  • Series Print ISSN 2197-1803
  • Series Online ISSN 2197-1811
  • About this book
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