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Ginzburg-Landau Vortices
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| 概要 | 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).続きを見る |
| 目次 | 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.続きを見る |
| 冊子版へのリンク | https://hdl.handle.net/2324/1001646368 |
| 本文を見る | Springer Nature Mathematics and Statistics (R0) eBooks 2017 English/International: 2017 |
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| 登録日 | 2023.09.29 |
| 更新日 | 2024.01.30 |