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This 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 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.
List of contents
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 .- 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.
About the author
Marisabina Russo's warm, distinctive picture books usually mirror everyday scenarios from her own family's life. The Big Brown Box was inspired by her three children's tendency to entertain themselves-and each other-with common household objects (which included many a cardboard box).
Ms. Russo has written and illustrated many picture books for children, and has illustrated several by other authors. She and her husband and children live in Westchester County, New York.
Summary
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 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.
