Ginzburg-Landau Vortices: v. 13 (Progress in Nonlinear Differential Equations and Their Applications) - Hardcover

Bethuel, Fabrice; Brezis, Haim; Helein, Frederic

Hardcover

ISBN 10: 3764337230 ISBN 13: 9783764337230

Publisher: Birkhauser Verlag AG, 1994

Synopsis

The mathematics in this book apply directly to classical problems in superconductors, superfluids and liquid crystals. It should be of interest to mathematicians, physicists and engineers working on modern materials research. The text is concerned with the study in two dimensions of stationary solutions uE of a complex valued Ginzburg-Landau equation involving a small parameter E. Such problems are related to questions occuring in physics, such as phase transistion phenomena in superconductors and superfluids. The parameter E has a dimension of a length, which is usually small. Thus, it should be of interest to study the asymptotics as E tends to zero. One of the main results asserts that the limit u* of minimizers uE exists. Moreover, u* 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 led 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* 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* has infinite energy one can think of u* as having "less" infinite energy than any other map u with u=g on the boundary. The material presented in this book covers mostly recent and original results by the authors. It assumes a moderate knowledge of nonlinear functional analysis, partial differential equations and complex functions. It is designed for researchers and graduate students alike and can be used as a one-semester text.

About this title

Review

"The three authors are well-known excellent specialists in nonlinear functional analysis and partial differential equations and the material presented in the book covers some of their recent and original results. The book is written in a very clear and readable style with many examples."

--ZAA

"...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."

--ZAMP

From the Back Cover

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)

Edition details

• Publisher: Birkhauser Verlag AG
• Publication date: 1994
• ISBN 10: 3764337230
• ISBN 13: 9783764337230
• Binding: Hardcover
• Language: English
• Number of pages: 192