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Ginzburg-Landau Vortices


Ginzburg-Landau Vortices


Modern Birkhäuser Classics

von: Fabrice Bethuel, Haïm Brezis, Frédéric Hélein

74,89 €

Verlag: Birkhäuser
Format: PDF
Veröffentl.: 21.09.2017
ISBN/EAN: 9783319666730
Sprache: englisch

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Beschreibungen

<p>This book is concerned with the study in two dimensions of stationary solutions of u<sub>ɛ</sub> 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.</p> <p>One of the main results asserts that the limit u-star of minimizers u<sub>ɛ</sub> 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.</p> <p>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.</p><p></p><p></p><p></p><p></p>
<p>Introduction.- Energy Estimates for S<sup>1</sup>-Valued Maps.- A Lower Bound for the Energy of S<sup>1</sup>-Valued Maps on Perforated Domains.- Some Basic Estimates for <i>u</i><sub>ɛ</sub>.- Toward Locating the Singularities: Bad Discs and Good Discs.- An Upper Bound for the Energy of <i>u</i><sub>ɛ</sub> away from the Singularities.- <i>u</i><sub>ɛ_<i>n</i></sub>: <i>u</i>-star is Born! - <i>u</i>-star Coincides with THE Canonical Harmonic Map having Singularities (a<sub>j</sub>).- The Configuration (a<sub>j</sub>) Minimizes the Renormalization Energy <i>W</i>.- Some Additional Properties of <i>u</i><sub>ɛ</sub>.- Non-Minimizing Solutions of the Ginzburg-Landau Equation.- Open Problems.</p>
<p>This book is concerned with the study in two dimensions of stationary solutions of u<sub>ɛ</sub> 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.</p><p> </p><p>One of the main results asserts that the limit u-star of minimizers u<sub>ɛ</sub> 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.</p><p></p>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. <p></p><p></p><p>The limit u-star can also be viewed as a geometrical object.  It is a minimizing harmonic map into S<sup>1</sup> with prescribed boundary condition g.  Topological obstructions imply that every map u into S<sup>1</sup> 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.</p>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.<div><br/></div><div><p>"...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."</p> <p>- Alexander Mielke, <i>Zeitschrift für angewandte Mathematik und Physik 46</i>(5)</p></div><div><br/></div><div><br/></div>
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
<p>Affordable, softcover reprint of a classic textbook</p> <p>Authors are well-known specialists in nonlinear functional analysis and partial differential equations</p> <p>Written in a clear, readable style with many examples</p>

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