Vortices in the Magnetic Ginzburg-Landau Model

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. HŽ elein, which contributed largely to turning Ginzburg-Landau 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 Haš ? 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.

Physical Presentation of the Model Critical 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.