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9783319666723: Ginzburg-Landau Vortices (Modern Birkhäuser Classics)

Sinopsis

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 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.

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De la contraportada

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 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)



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Fabrice Bethuel
Publicado por Birkh?user, 2017
ISBN 10: 331966672X ISBN 13: 9783319666723
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Bethuel, Fabrice (Author)/ Brezis, Haïm (Author)/ Hélein, Frédéric (Author)
Publicado por Birkhäuser, 2017
ISBN 10: 331966672X ISBN 13: 9783319666723
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Fabrice Bethuel|Haïm Brezis|Frédéric Hélein
Publicado por Springer International Publishing, 2017
ISBN 10: 331966672X ISBN 13: 9783319666723
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Kartoniert / Broschiert. Condición: New. Dieser Artikel ist ein Print on Demand Artikel und wird nach Ihrer Bestellung fuer Sie gedruckt. Affordable, softcover reprint of a classic textbookAuthors are well-known specialists in nonlinear functional analysis and partial differential equationsWritten in a clear, readable style with many examplesThis book is concerned . Nº de ref. del artículo: 155914773

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