Linear and Nonlinear Aspects of Vortices

Equations of the Ginzburg–Landau vortices have particular applications to a number of problems in physics, including phase transition phenomena in superconductors, superfluids, and liquid crystals.  Building on the results presented by Bethuel, Brazis, and Helein, this current work further analyzes Ginzburg-Landau vortices with a particular emphasis on the uniqueness question.

The authors begin with a general presentation of the theory and then proceed to study problems using weighted Hölder spaces and Sobolev Spaces. These are particularly powerful tools and help us obtain a deeper understanding of the nonlinear partial differential equations associated with Ginzburg-Landau vortices. Such an approach sheds new light on the links between the geometry of vortices and the number of solutions.

Aimed at mathematicians, physicists, engineers, and grad students, this monograph will be useful in a number of contexts in the nonlinear analysis of problems arising in geometry or mathematical physics. The material presented covers recent and original results by the authors, and will serve as an excellent classroom text or a valuable self-study resource.

• Operator
• Sobolev space
• Vector field
• geometry
• linear optimization
• mathematical physics
• maximum principle
• model
• partial differential equation
• partial differential equations

“The present monograph contains new and deep original results in the geometrical theory of Ginzburg-Landau vortices. … This book is suitable for readers with a knowledge of nonlinear partial differential equations and should appeal to researchers interested in such diverse areas … . For all these reasons, the reviewer strongly believes that ‘Linear and Nonlinear Aspects of Vortices (The Ginzburg-Landau Model)’ is an outstanding contribution to the field and should be available in all mathematics and physics libraries.” (Vicenţiu D.Rădulescu, zbMATH 0948.35003, 2022)

"In the course of their argument, the authors traverse a broad range of nontrivial analysis: elliptic equations on weighted Holder and Sobolev spaces, radially symmetric solutions, gluing techniques, Pokhozhaev-type arguments for solutions of semilinear elliptic equations, and more. Clearly aimed at a research audience, this book provides a fascinating and original account of the theory of Ginzburg-Landau vortices."

--Mathematical Reviews

• Département de Mathématiques, Université de Paris XII, Creteil Cedex, France

  Frank Pacard

• Department of Mathematics, Courant Institute of Mathematical Sciences, New York, USA

  Tristan Rivière

• CMLA, ENS-CACHAN, Centre National de la Recherche Scientifique 61, Cachan, France

  Tristan Rivière

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