Author | Pacard, Frank. author |
---|---|
Title | Linear and Nonlinear Aspects of Vortices [electronic resource] : The Ginzburg-andau Model / by Frank Pacard, Tristan Riviรจre |
Imprint | Boston, MA : Birkhรคuser Boston : Imprint: Birkhรคuser, 2000 |
Connect to | http://dx.doi.org/10.1007/978-1-4612-1386-4 |
Descript | X, 342 p. online resource |
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 Hรถlder 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 pohozaevโs 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 set of 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. Hรฉlein -- 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