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|a 9783319666730
|9 978-3-319-66673-0
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|a 10.1007/978-3-319-66673-0
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|a MAT007000
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|a 515.353
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|a Bethuel, Fabrice.
|e author.
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|a Ginzburg-Landau Vortices
|h [electronic resource] /
|c by Fabrice Bethuel, Haim Brezis, Frederic Helein.
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|a Cham :
|b Springer International Publishing :
|b Imprint: Birkhäuser,
|c 2017.
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|a XXIX, 159 p. 1 illus.
|b online resource.
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|a text
|b txt
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|a computer
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|a online resource
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|a text file
|b PDF
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|a Modern Birkhäuser Classics,
|x 2197-1803
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|a 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.
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|a 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).
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|a Mathematics.
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|a Partial differential equations.
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|a Mathematical physics.
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|a Mathematics.
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|a Partial Differential Equations.
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|a Mathematical Applications in the Physical Sciences.
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|a Brezis, Haim.
|e author.
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|a Helein, Frederic.
|e author.
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|a SpringerLink (Online service)
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|t Springer eBooks
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|i Printed edition:
|z 9783319666723
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|a Modern Birkhäuser Classics,
|x 2197-1803
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|u http://dx.doi.org/10.1007/978-3-319-66673-0
|z Full Text via HEAL-Link
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|a ZDB-2-SMA
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|a Mathematics and Statistics (Springer-11649)
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