Ginzburg-Landau Vortices / by Fabrice Bethuel, Haïm Brezis, Frédéric Hélein.

Format:

Book

Author/Creator:

Bethuel, Fabrice., Author.

Brezis, Haïm., Author.

Hélein, Frédéric., Author.

Series:

Modern Birkhäuser Classics, 2197-1803

Language:

English

Subjects (All):

Differential equations, Partial.

Mathematical physics.

Partial Differential Equations.

Mathematical Applications in the Physical Sciences.

Local Subjects:

Partial Differential Equations.

Mathematical Applications in the Physical Sciences.

Physical Description:

1 online resource (XXIX, 159 p. 1 illus.)

Edition:

1st ed. 2017.

Place of Publication:

Cham : Springer International Publishing : Imprint: Birkhäuser, 2017.

Summary:

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).

Contents:

Introduction

Energy Estimates for S1-Valued Maps

A Lower Bound for the Energy of S1-Valued Maps on Perforated Domains

Some Basic Estimates for uɛ

Toward Locating the Singularities: Bad Discs and Good Discs

An Upper Bound for the Energy of uɛ away from the Singularities

uɛ_n: u-star is Born! - u-star Coincides with THE Canonical Harmonic Map having Singularities (aj)

The Configuration (aj) Minimizes the Renormalization Energy W

Some Additional Properties of uɛ

Non-Minimizing Solutions of the Ginzburg-Landau Equation

Open Problems.

Notes:

"Reprint of the 1994 Edition."

Includes bibliographical references and index.

ISBN:

3-319-66673-8