Ginzburg-Landau Vortices

ISBN-13: 9783319666723 / Angielski / Miękka / 2017 / 159 str.

Fabrice Bethuel; Haim Brezis; Frederic Helein

Opis

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.

Spis treści

Introduction.- Energy Estimates for S¹-Valued Maps.- A Lower Bound for the Energy of S¹-Valued Maps on Perforated Domains.- Some Basic Estimates for u_ɛ.- Toward Locating the Singularities: Bad Discs and Good Discs.- An Upper Bound for the Energy of u_ɛ away from the Singularities.- u_{ɛ_n}: u-star is Born! - u-star Coincides with THE Canonical Harmonic Map having Singularities (a_j).- The Configuration (a_j) Minimizes the Renormalization Energy W.- Some Additional Properties of u_ɛ.- Non-Minimizing Solutions of the Ginzburg-Landau Equation.- Open Problems.

Blurb

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 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)