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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.

Ginzburg-Landau Vortices

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 (a_j).- The Configuration (a_j) Minimizes the Renormalization Energy W.- Some Additional Properties of u .- Non-Minimizing Solutions of the Ginzburg-Landau Equation.- Open Problems.