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MARC 21

Ginzburg-Landau Vortices
Tag	Description
020	$a9783319666730
082	$a515.353
099	$aOnline resource: Birkhäuser
100	$aBethuel, Fabrice.$d1963-
245	$aGinzburg-Landau Vortices$h[EBook]$cby Fabrice Bethuel, Haim Brezis, Frederic Helein.
260	$aCham$bBirkhäuser$c2017.
300	$aXXIX, 159 pages, 1 illus.$bonline resource.
336	$atext
338	$aonline resource
440	$aModern Birkhäuser Classics,$x2197-1803
505	$aIntroduction -- Energy Estimates for S1-Valued Maps -- A Lower Bound for the Energy of S1-Valued Maps on Perforated Domains -- Some Basic Estimates for u? -- Toward Locating the Singularities: Bad Discs and Good Discs -- An Upper Bound for the Energy of u? away from the Singularities -- u?_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?u? -- Non-Minimizing Solutions of the Ginzburg-Landau Equation -- Open Problems.
520	$aThis book is concerned with the study in two dimensions of stationary solutions of u? of a complex valued Ginzburg-Landau equation involving a small parameter ?. Such problems are related to questions occurring in physics, e.g., phase transition phenomena in superconductors and superfluids. The parameter ? has a dimension of a length which is usually small.? Thus, it is of great interest to study the asymptotics as ? tends to zero. One of the main results asserts that the limit u-star of minimizers u? 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 f?r angewandte Mathematik und Physik 46(5).
520	$aIntroduction -- Energy estimates for S1-valued maps -- A lower bound for the energy of S1-valued maps on perforated domains -- Some basic estimates for ue -- Towards locating the singularities: bad discs and good discs -- An upper bound for the energy of ue away from the singularities -- ue converges: u* is born! -- u* coincides with THE canonical harmonic map having singularities (aj) -- The configuration (aj) minimized the renormalized energy W -- Some additional properties of ue -- Non minimizing solutions of the Ginzburg-Landau equation -- Open problems.
538	$aOnline access to this digital book is restricted to subscription institutions through IP address (only for SISSA internal users)
700	$aBrezis, Haim.$eauthor.
700	$aHelein, Frederic.$eauthor.
710	$aSpringerLink (Online service)
830	$aModern Birkh?ser Classics,
856	$uhttp://dx.doi.org/10.1007/978-3-319-66673-0