Bethuel / Brezis / Hélein Ginzburg-Landau Vortices

1. Auflage 2017
ISBN: 978-3-319-66673-0
Verlag: Springer International Publishing
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 166 Seiten, eBook

Reihe: Modern Birkhäuser Classics

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

Zielgruppe

Research

Autoren/Hrsg.

Bethuel, Fabrice

Brezis, Haïm

Hélein, Frédéric

Inhaltsverzeichnis

Introduction.- Energy Estimates for S 1 -Valued Maps.- A Lower Bound for the Energy of S 1 -Valued Maps on Perforated Domains.- Some Basic Estimates for u? .- Toward Locating the Singularities: Bad Discs and Good Discs.- An Upper Bound for the Energy of u? away from the Singularities.- u ?_ n : u -star is Born! - u -star Coincides with THE Canonical Harmonic Map having Singularities (a j ).- The Configuration (a j ) Minimizes the Renormalization Energy W .- Some Additional Properties of  u? .- Non-Minimizing Solutions of the Ginzburg-Landau Equation.- Open Problems.