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Linear and Nonlinear Aspects of Vortices

The Ginzburg-andau Model

  • Frank Pacard
  • Tristan Rivière

Part of the Progress in Nonlinear Differential Equations and Their Applications book series (PNLDE, volume 39)

Table of contents

  1. Front Matter
    Pages i-x
  2. Frank Pacard, Tristan Rivière
    Pages 1-19
  3. Frank Pacard, Tristan Rivière
    Pages 21-49
  4. Frank Pacard, Tristan Rivière
    Pages 51-71
  5. Frank Pacard, Tristan Rivière
    Pages 73-101
  6. Frank Pacard, Tristan Rivière
    Pages 103-123
  7. Frank Pacard, Tristan Rivière
    Pages 125-149
  8. Frank Pacard, Tristan Rivière
    Pages 151-165
  9. Frank Pacard, Tristan Rivière
    Pages 167-190
  10. Frank Pacard, Tristan Rivière
    Pages 191-223
  11. Frank Pacard, Tristan Rivière
    Pages 225-252
  12. Frank Pacard, Tristan Rivière
    Pages 253-278
  13. Frank Pacard, Tristan Rivière
    Pages 279-327
  14. Back Matter
    Pages 329-342

About this book

Introduction

Equations of the Ginzburg–Landau 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 Hölder 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.

Keywords

Operator Sobolev space Vector field geometry linear optimization mathematical physics maximum principle model partial differential equation

Authors and affiliations

  • Frank Pacard
    • 1
  • Tristan Rivière
    • 2
    • 3
  1. 1.Département de MathématiquesUniversité de Paris XIICreteil CedexFrance
  2. 2.Department of MathematicsCourant Institute of Mathematical SciencesNew YorkUSA
  3. 3.CMLA, ENS-CACHANCentre National de la Recherche Scientifique 61CachanFrance

Bibliographic information