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Partial differential equations

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Group theory --- Semigroups of operators --- Nonlinear operators --- Congresses --- Semigroups of operators - Congresses --- Nonlinear operators - Congresses

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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 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 S1 with prescribed boundary condition g.  Topological obstructions imply that every map u into S1 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(5).

Mathematics. --- Partial differential equations. --- Mathematical physics. --- Partial Differential Equations. --- Mathematical Applications in the Physical Sciences. --- Singularities (Mathematics) --- Superconductors --- Physical mathematics --- Physics --- Mathematics --- Geometry, Algebraic --- Superconducting materials --- Superconductive devices --- Cryoelectronics --- Electronics --- Solid state electronics --- Materials --- Differential equations, partial. --- Partial differential equations

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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 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 S1 with prescribed boundary condition g.  Topological obstructions imply that every map u into S1 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(5).

Partial differential equations --- Differential equations --- Mathematics --- differentiaalvergelijkingen --- wiskunde