Ginzburg-Landau Vortices (eBook)

Fabrice Bethuel, Haïm Brezis, Frédéric Hélein (Autoren)

eBook Download: PDF

2017 | 1st ed. 2017
XXIX, 159 Seiten
Springer International Publishing (Verlag)
978-3-319-66673-0 (ISBN)

Infos

This book is concerned with the study in two dimensions of stationary solutions of u_ɛ 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 u_ɛ 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.

Inhalt

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.

Produkt-Details

• Erscheint lt. Verlag: 21.9.2017
• Reihe/Serie: Modern Birkhäuser Classics
• Reihe/Serie: Modern Birkhäuser Classics
• Zusatzinfo: XXIX, 159 p. 5 illus., 1 illus. in color.
• Verlagsort: Cham
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Mathematik
• Schlagworte: Ginzburg-Landau Vortices • nonlinear functional analysis • Partial differential equations • Phase Transition Phenomena • superconductors • Superfluids
• ISBN-10: 3-319-66673-8 / 3319666738
• ISBN-13: 978-3-319-66673-0 / 9783319666730