Ginzburg-Landau Vortices: Modern Birkhäuser Classics
Autor Fabrice Bethuel, Haïm Brezis, Frédéric Héleinen Limba Engleză Paperback – 5 oct 2017
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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Specificații
ISBN-13: 9783319666723
ISBN-10: 331966672X
Pagini: 159
Ilustrații: XXIX, 159 p. 5 illus., 1 illus. in color.
Dimensiuni: 155 x 235 mm
Greutate: 0.34 kg
Ediția:1st ed. 2017
Editura: Springer International Publishing
Colecția Birkhäuser
Seria Modern Birkhäuser Classics
Locul publicării:Cham, Switzerland
ISBN-10: 331966672X
Pagini: 159
Ilustrații: XXIX, 159 p. 5 illus., 1 illus. in color.
Dimensiuni: 155 x 235 mm
Greutate: 0.34 kg
Ediția:1st ed. 2017
Editura: Springer International Publishing
Colecția Birkhäuser
Seria Modern Birkhäuser Classics
Locul publicării:Cham, Switzerland
Cuprins
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.
Textul de pe ultima copertă
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.
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)
- Alexander Mielke, Zeitschrift für angewandte Mathematik und Physik 46(5)
Caracteristici
Affordable, softcover reprint of a classic textbook Authors are well-known specialists in nonlinear functional analysis and partial differential equations Written in a clear, readable style with many examples