Overview

The mathematics in this book apply directly to classical problems in superconductors, superfluids and liquid crystals. It should be of interest to mathematicians, physicists and engineers working on modern materials research. The text is concerned with the study in two dimensions of stationary solutions uE of a complex valued Ginzburg-Landau equation involving a small parameter E. Such problems are related to questions occuring in physics, such as phase transistion phenomena in superconductors and superfluids. The parameter E has a dimension of a length, which is usually small. Thus, it should be of interest to study the asymptotics as E tends to zero. One of the main results asserts that the limit u* of minimizers uE exists. Moreover, u* 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 led 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* 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* has infinite energy one can think of u* as having ""less"" infinite energy than any other map u with u=g on the boundary. The material presented in this book covers mostly recent and original results by the authors. It assumes a moderate knowledge of nonlinear functional analysis, partial differential equations and complex functions. It is designed for researchers and graduate students alike and can be used as a one-semester text.

Full Product Details

Author:   Fabrice Bethuel ,  Haim Brezis ,  Frederic Helein
Publisher:   Birkhauser Boston Inc
Imprint:   Birkhauser Boston Inc
Edition:   1994 ed.
Volume:   13
Dimensions:   Width: 15.50cm , Height: 1.00cm , Length: 23.50cm
Weight:   0.630kg
ISBN:  

9780817637231

ISBN 10:   0817637230
Pages:   162
Publication Date:   28 March 1994
Audience:   College/higher education ,  Professional and scholarly ,  Postgraduate, Research & Scholarly ,  Professional & Vocational
Format:   Paperback
Publisher's Status:   Active
Availability:   In Print  
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Table of Contents

I. Energy estimates for S1-valued maps.- 1. An auxiliary linear problem.- 2. Variants of Theorem I.1.- 3. S1-valued harmonic maps with prescribed isolated singularities. The canonical harmonic map.- 4. Shrinking holes. Renormalized energy.- II. A lower bound for the energy of S1-valued maps on perforated domains.- III. Some basic estimates for u?.- 1. Estimates when G=BR and g(x)=x/|x|.- 2. An upper bound for E? (u?).- 3. An upper bound for $$ \frac{1}{{{\varepsilon^2}}}{\smallint_G}{\left( {{{\left| {{u_{\varepsilon }}} \right|}^2} - 1} \right)^2} $$.- 4. $$ \left| {{u_e}} \right| \geqslant \frac{1}{2} $$ on “good discs”.- IV. Towards locating the singularities: bad discs and good discs.- 1. A covering argument.- 2. Modifying the bad discs.- V. An upper bound for the energy of u? away from the singularities.- 1. A lower bound for the energy of u? near aj.- 2. Proof of Theorem V.l.- VI. u?n converges: u? is born!.- 1. Proof of Theorem VI.1.- 2. Further properties of u? : singularities have degree one and they are not on the boundary.- VII. u? coincides with THE canonical harmonic map having singularities (aj).- VIII. The configuration (aj) minimizes the renormalized energy W.- 1. The general case.- 2. The vanishing gradient property and its various forms.- 3. Construction of critical points of the renormalized energy.- 4. The case G=B1 and $$ g\left( \theta \right) = {e^{{i\theta }}} $$.- 5. The case G=B1 and $$ g\left( \theta \right) = {e^{{i\theta }}} $$ with d?.- IX. Some additional properties of u?.- 1. The zeroes of u?.- 2. The limit of $$ \left\{ {{E_{\varepsilon }}\left( {{u_{\varepsilon }}} \right) - \pi d\left| {\log \varepsilon } \right|} \right\} $$ as $$ \varepsilon \to 0 $$.- 3. $$ {\smallint_G}{\left| {\nabla \left| {{u_{\varepsilon }}}\right|} \right|^2} $$ remains bounded as $$ \varepsilon \to 0 $$.- 4. The bad discs revisited.- X. Non minimizing solutions of the Ginzburg-Landau equation.- 1. Preliminary estimates; bad discs and good discs.- 2. Splitting $$ \left| {\nabla {v_{\varepsilon }}} \right| $$.- 3. Study of the associated linear problems.- 4. The basic estimates: $$ {\smallint_G}{\left| {\nabla {v_{\varepsilon }}} \right|^2} \leqslant C\left| {\log \;\varepsilon } \right| $$ and $$ {\smallint_G}{\left| {\nabla {v_{\varepsilon }}} \right|^p} \leqslant {C_p} $$ for p

Reviews

The three authors are well-known excellent specialists in nonlinear functional analysis and partial differential equations and the material presented in the book covers some of their recent and original results. The book is written in a very clear and readable style with many examples. --ZAA ...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. --ZAMP

The three authors are well-known excellent specialists in nonlinear functional analysis and partial differential equations and the material presented in the book covers some of their recent and original results. The book is written in a very clear and readable style with many examples. <p>--ZAA <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>--ZAMP

The three authors are well-known excellent specialists in nonlinear functional analysis and partial differential equations and the material presented in the book covers some of their recent and original results. The book is written in a very clear and readable style with many examples. --ZAA ...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. --ZAMP

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