This book presents the mathematical study of vortices of the two-dimensional Ginzburg-Landau model. It acts a guide to the various branches of Ginzburg-Landau studies, provides context for the study of vortices, and presents a list of open problems in the field.

More than ten years have passed since the book of F. Bethuel, H. Brezis and F. H´ elein, which contributed largely to turning GinzburgLandau equations from a renowned physics model into a large PDE research ?eld, with an ever-increasing number of papers and research directions (the number of published mathematics papers on the subject is certainly in the several hundreds, and that of physics papers in the thousands). Having ourselves written a series of rather long and intricately - terdependent papers, and having taught several graduate courses and mini-courses on the subject, we felt the need for a more uni?ed and self-contained presentation. The opportunity came at the timely moment when Ha¨ ?m Brezis s- gested we should write this book. We would like to express our gratitude towards him for this suggestion and for encouraging us all along the way. As our writing progressed, we felt the need to simplify some proofs, improvesomeresults,aswellaspursuequestionsthatarosenaturallybut that we had not previously addressed. We hope that we have achieved a little bit of the original goal: to give a uni?ed presentation of our work with a mixture of both old and new results, and provide a source of reference for researchers and students in the ?eld.

Describes essential mathematical techniques and tools for analyzing the Ginzburg--Landau functional Presents an introduction to the theory with results and current research Acts a guide to the various branches of Ginzburg-Landau studies and provides context for the study of vortices Includes list of open problems

Klappentext

With the discovery of type-II superconductivity by Abrikosov, the prediction of vortex lattices, and their experimental observation, quantized vortices have become a central object of study in superconductivity, superfluidity, and Bose--Einstein condensation. This book presents the mathematics of superconducting vortices in the framework of the acclaimed two-dimensional Ginzburg-Landau model, with or without magnetic field, and in the limit of a large Ginzburg-Landau parameter, kappa.

This text presents complete and mathematically rigorous versions of both results either already known by physicists or applied mathematicians, or entirely new. It begins by introducing mathematical tools such as the vortex balls construction and Jacobian estimates. Among the applications presented are: the determination of the vortex densities and vortex locations for energy minimizers in a wide range of regimes of applied fields, the precise expansion of the so-called first critical field in abounded domain, the existence of branches of solutions with given numbers of vortices, and the derivation of a criticality condition for vortex densities of non-minimizing solutions. Thus, this book retraces in an almost entirely self-contained way many results that are scattered in series of articles, while containing a number of previously unpublished results as well.

The book also provides a list of open problems and a guide to the increasingly diverse mathematical literature on Ginzburg--Landau related topics. It will benefit both pure and applied mathematicians, physicists, and graduate students having either an introductory or an advanced knowledge of the subject.

Inhalt
Physical Presentation of the ModelCritical Fields.- First Properties of Solutions to the Ginzburg-Landau Equations.- The Vortex-Balls Construction.- Coupling the Ball Construction to the Pohozaev Identity and Applications.- Jacobian Estimate.- The Obstacle Problem.- Higher Values of the Applied Field.- The Intermediate Regime.- The Case of a Bounded Number of Vortices.- Branches of Solutions.- Back to Global Minimization.- Asymptotics for Solutions.- A Guide to the Literature.- Open Problems.