Linear and Nonlinear Aspects of Vortices The Ginzburg-andau Model
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 G...
| Main Authors: | , |
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| Format: | eBook |
| Language: | English |
| Published: |
Boston, MA
Birkhäuser
2000, 2000
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| Edition: | 1st ed. 2000 |
| Series: | Progress in Nonlinear Differential Equations and Their Applications
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| Subjects: | |
| Online Access: | |
| Collection: | Springer Book Archives -2004 - Collection details see MPG.ReNa |
Table of Contents:
- 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
- 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
- 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 ?