TY - BOOK AU - Bethuel,Fabrice AU - Brezis,Haïm AU - Hélein,Frédéric ED - SpringerLink (Online service) TI - Ginzburg-Landau Vortices T2 - Modern Birkhäuser Classics, SN - 9783319666730 AV - QA370-380 U1 - 515.353 23 PY - 2017/// CY - Cham PB - Springer International Publishing, Imprint: Birkhäuser KW - Partial differential equations KW - Mathematical physics KW - Partial Differential Equations KW - Mathematical Applications in the Physical Sciences N1 - 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; Available to subscribing member institutions only. Доступно лише організаціям членам підписки N2 - 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) UR - https://doi.org/10.1007/978-3-319-66673-0 ER -