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.- 2.3 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.

“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