Linear and Nonlinear Aspects of Vortices: The Ginzburg-Andau Model

ISBN-13: 9780817641337 / Angielski / Twarda / 2000 / 342 str.

Frank Pacard; Tristan Riviere; Tristan Riviere

Opis

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 Holder 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."

Recenzje

"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." (Vicentiu D.Radulescu, 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

Spis treści

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 \ .- 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 $$ |{_{\varepsilon }} - |{{\tilde{\upsilon }}_{\varepsilon }}| $$.- 11.2.3 Pointwise estimates for $$ u - {{\tilde{\upsilon }}_{\varepsilon }} $$and $$ |{_{\varepsilon }}| - |{{\tilde{\upsilon }}_{\varepsilon }}| $$.- 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.