Ginzburg-Landau Vortices

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 para...

Πλήρης περιγραφή

Λεπτομέρειες βιβλιογραφικής εγγραφής
Κύριοι συγγραφείς: Bethuel, Fabrice (Συγγραφέας), Brezis, Haim (Συγγραφέας), Helein, Frederic (Συγγραφέας)
Συγγραφή απο Οργανισμό/Αρχή: SpringerLink (Online service)
Μορφή: Ηλεκτρονική πηγή Ηλ. βιβλίο
Γλώσσα:English
Έκδοση: Cham : Springer International Publishing : Imprint: Birkhäuser, 2017.
Σειρά:Modern Birkhäuser Classics,
Θέματα:
Διαθέσιμο Online:Full Text via HEAL-Link
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100 1 |a Bethuel, Fabrice.  |e author. 
245 1 0 |a Ginzburg-Landau Vortices  |h [electronic resource] /  |c by Fabrice Bethuel, Haim Brezis, Frederic Helein. 
264 1 |a Cham :  |b Springer International Publishing :  |b Imprint: Birkhäuser,  |c 2017. 
300 |a XXIX, 159 p. 1 illus.  |b online resource. 
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490 1 |a Modern Birkhäuser Classics,  |x 2197-1803 
505 0 |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. 
520 |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). 
650 0 |a Mathematics. 
650 0 |a Partial differential equations. 
650 0 |a Mathematical physics. 
650 1 4 |a Mathematics. 
650 2 4 |a Partial Differential Equations. 
650 2 4 |a Mathematical Applications in the Physical Sciences. 
700 1 |a Brezis, Haim.  |e author. 
700 1 |a Helein, Frederic.  |e author. 
710 2 |a SpringerLink (Online service) 
773 0 |t Springer eBooks 
776 0 8 |i Printed edition:  |z 9783319666723 
830 0 |a Modern Birkhäuser Classics,  |x 2197-1803 
856 4 0 |u http://dx.doi.org/10.1007/978-3-319-66673-0  |z Full Text via HEAL-Link 
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950 |a Mathematics and Statistics (Springer-11649)