Linear and Nonlinear Aspects of Vortices

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.- 2.2.1 Rescaled Schauder estimates.- 2.2.2 Mapping properties of the Laplacian in the injectivity range.- 2.2.3 Mapping properties of the Laplacian in the surjectivity range.- 2.3 Applications to nonlinear problems.- 2.3.1 Minimal surfaces with one catenoidal type end.- 2.3.2 Semilinear elliptic equations with isolated singularities.- 2.3.3 Singular perturbations for the Liouville equation.- 3 The Ginzburg-Landau Equation in ?.- 3.1 Radially symmetric solution on ?.- 3.2 The linearized operator about the radially symmetric solution.- 3.2.1 Definition.- 3.2.2 Explicit solutions of the homogeneous problem.- 3.3 Asymptotic behavior of solutions of the homogeneous problem.- 3.3.1 Classification of all possible asymptotic behaviors at 0.- 3.3.2 Classification of all possible asymptotic behaviors at ?.- 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.- 4.1.1 Higher eigenfrequencies.- 4.1.2 Lower eigenfrequencies.- 4.2 Function spaces.- 4.3 A right inverse for L? in B1 \ {0}.- 4.3.1 Higher eigenfrequencies.- 4.3.2 Lower eigenfrequencies.- 5 Families of Approximate Solutions with Prescribed Zero Set.- 5.1 The approximate solution ?.- 5.1.1 Notation.- 5.1.2 The approximate solution near the zeros.- 5.1.3 The approximate solution away from the zeros.- 5.2 A 3N dimensional family of approximate solutions.- 5.2.1 Definition of the 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.- 6.4.1 The interior Dirichlet to Neumann mapping.- 6.4.2 The exterior Dirichlet to Neumann mapping.- 6.4.3 Gluing together the two 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.- 7.3.1 Estimates of Q1.- 7.3.2 Estimates of Q2.- 7.3.3 Estimates of Q3.- 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.- 9.2.1 Notation.- 9.2.2 The comparison argument in the case of radially symmetric data.- 9.3 ?-conformal vector fields.- 9.4 Conservation laws.- 9.4.1 Comparing solutions through a Pohozaev type formula: the general case.- 9.4.2 Conservation laws for Ginzburg-Landau equation.- 9.4.3 The Pohozaev formula.- 9.4.4 Integration of the Pohozaev formula.- 9.5 Uniqueness results.- 9.5.1 A few uniqueness results.- 9.5.2 Uniqueness results for semilinear elliptic problems.- 9.6 Dealing with general nonlinearities.- 9.6.1 A Pohozaev formula for general nonlinearities.- 9.6.2 Uniqueness results for general nonlinearities.- 9.6.3 More about the quantities involved in the Pohozaev identity.- 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.- 10.2.1 Preliminary results.- 10.2.2 The proof of Theorem 10.1.- 11 Solving Uniqueness Questions.- 11.1 Statement of the uniqueness result.- 11.2 Proof of the uniqueness result.- 11.2.1 Geometric modification of the family ??.- 11.2.2 Estimating the L2 norm of $$ |{{u}_{\varepsilon }} - |{{\tilde{\upsilon }}_{\varepsilon }}| $$.- 11.2.3 Pointwise estimates for $$ u - {{\tilde{\upsilon }}_{\varepsilon }} $$and $$ |{{u}_{\varepsilon }}| - |{{\tilde{\upsilon }}_{\varepsilon }}| $$.- 11.2.4 Final arguments to prove that u? = ??.- 11.3 A conjecture of F. Bethuel, H. Brezis and F. Hélein.- 12 Towards Jaffe and Taubes Conjectures.- 12.1 Statement of the result.- 12.1.1 Preliminary remarks.- 12.1.2 The uniqueness result.- 12.2 Gauge invariant Ginzburg-Landau critical points with one zero.- 12.3 Proof of Theorem 12.2.- 12.3.1 The Coulomb gauge.- 12.3.2 Preliminary results.- 12.3.3 The Pohozaev formula.- 12.3.4 The end of the proof.- References.- Index of Notation.