Linear and Nonlinear Aspects of Vortices

Equations of the GinzburgLandau 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.

Klappentext

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.

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

Inhalt
1 Qualitative Aspects of Ginzburg-Landau Equations.- 1.1 The integrable case.- 1.2 The strongly repulsive case.- 1.3 The existence result.- 1.4 Uniqueness results.- 2 Elliptic Operators in Weighted Hölder Spaces.- 2.1 Function spaces.- 2.2 Mapping properties of the Laplacian.- 2.3 Applications to nonlinear problems.- 3 The Ginzburg-Landau Equation in ?.- 3.1 Radially symmetric solution on ?.- 3.2 The linearized operator about the radially symmetric solution.- 3.3 Asymptotic behavior of solutions of the homogeneous problem.- 3.4 Bounded solution of the homogeneous problem.- 3.5 More solutions to the homogeneous equation.- 3.6 Introduction of the scaling factor.- 4 Mapping Properties of L?.- 4.1 Consequences of the maximum principle in weighted spaces.- 4.2 Function spaces.- 4.3 A right inverse for L? in B1 \ {0}.- 5 Families of Approximate Solutions with Prescribed Zero Set.- 5.1 The approximate solution ?.- 5.2 A 3N dimensional family of approximate solutions.- 5.3 Estimates.- 5.4 Appendix.- 6 The Linearized Operator about the Approximate Solution ?.- 6.1 Definition.- 6.2 The interior problem.- 6.3 The exterior problem.- 6.4 Dirichlet to Neumann mappings.- 6.5 The linearized operator in all ?.- 6.6 Appendix.- 7 Existence of Ginzburg-Landau Vortices.- 7.1 Statement of the result.- 7.2 The linear mapping DM(0,0,0).- 7.3 Estimates of the nonlinear terms.- 7.4 The fixed point argument.- 7.5 Further information about the branch of solutions.- 8 Elliptic Operators in Weighted Sobolev Spaces.- 8.1 General overview.- 8.2 Estimates for the Laplacian.- 8.3 Estimates for some elliptic operator in divergence form.- 9 Generalized Pohozaev Formula for ?-Conformal Fields.- 9.1 The Pohozaev formula in the classical framework.- 9.2 Comparing Ginzburg-Landau solutions using pohozaev's argument.- 9.3 ?-conformal vector fields.- 9.4 Conservation laws.- 9.5 Uniqueness results.- 9.6 Dealing with general nonlinearities.- 10 The Role of Zeros in the Uniqueness Question.- 10.1 The zero setof solutions of Ginzburg-Landau equations.- 10.2 A uniqueness result.- 11 Solving Uniqueness Questions.- 11.1 Statement of the uniqueness result.- 11.2 Proof of the uniqueness result.- 11.3 A conjecture of F. Bethuel, H. Brezis and F. Hélein.- 12 Towards Jaffe and Taubes Conjectures.- 12.1 Statement of the result.- 12.2 Gauge invariant Ginzburg-Landau critical points with one zero.- 12.3 Proof of Theorem 12.2.- References.- Index of Notation.