Fabrice Bethuel, Haim Brezis, Frederic Helein Ginzburg-Landau Vortices

Annotation

This book is concerned with the study in two dimensions of stationary solutions of uE of a complex valued Ginzburg-Landau equation involving a small parameter E . Such problems are related to questions occurring in physics, e.g., phase transition phenomena in superconductors and superfluids. The parameter E has a dimension of a length which is usually small. Thus, it is of great interest to study the asymptotics as E tends to zero. One of the main results asserts that the limit u-star of minimizers uE 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.