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.

Introduction.- Energy Estimates for S1-Valued Maps.- A Lower Bound for the Energy of S1-Valued Maps on Perforated Domains.- Some Basic Estimates foru.- Toward Locating the Singularities: Bad Discs and Good Discs.- An Upper Bound for the Energy ofu 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 EnergyW.- Some Additional Properties ofu.- Non-Minimizing Solutions of the Ginzburg-Landau Equation.- Open Problems.