Affordable, softcover reprint of a classic textbook
Authors are well-known specialists in nonlinear functional analysis and partial differential equations
Written in a clear, readable style with many examples
This book is concerned with the study in two dimensions of stationary solutions of u
e
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 u
e
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 S
1
with prescribed boundary condition g. Topological obstructions imply that every map u into S
1
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
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