Variational techniques for elliptic partial differential equations theoretical tools and advanced applications
Cover -- Half Title -- Title Page -- Copyright Page -- Dedication -- Contents -- Preface -- Authors -- Part I: Fundamentals -- 1. Distributions -- 1.1 The test space -- 1.2 Distributions -- 1.3 Distributional differentiation -- 1.4 Convergence of distributions -- 1.5 A fundamental solution (*) -- 1....
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| Sprache: | eng |
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Boca Raton, London, New York
CRC Press, Taylor & Francis Group
2019
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Tag hinzufügen | Zusammenfassung: | Cover -- Half Title -- Title Page -- Copyright Page -- Dedication -- Contents -- Preface -- Authors -- Part I: Fundamentals -- 1. Distributions -- 1.1 The test space -- 1.2 Distributions -- 1.3 Distributional differentiation -- 1.4 Convergence of distributions -- 1.5 A fundamental solution (*) -- 1.6 Lattice partitions of unity -- 1.7 When the gradient vanishes (*) -- 1.8 Proof of the variational lemma (*) -- Final comments and literature -- Exercises -- 2. The homogeneous Dirichlet problem -- 2.1 The Sobolev space H1(O) -- 2.2 Cuto and molli cation -- 2.3 A guided tour of mollification (*) -- 2.4 The space H10(O) -- 2.5 The Dirichlet problem -- 2.6 Existence of solutions -- Final comments and literature -- Exercises -- 3. Lipschitz transformations and Lipschitz domains -- 3.1 Lipschitz transformations of domains -- 3.2 How Lipschitz maps preserve H1 behavior (*) -- 3.3 Lipschitz domains -- 3.4 Localization and pullback -- 3.5 Normal elds and integration on the boundary -- Final comments and literature -- Exercises -- 4. The nonhomogeneous Dirichlet problem -- 4.1 The extension theorem -- 4.2 The trace operator -- 4.3 The range and kernel of the trace operator -- 4.4 The nonhomogeneous Dirichlet problem -- 4.5 General right-hand sides -- 4.6 The Navier-Lamé equations (*) -- Final comments and literature -- Exercises -- 5. Nonsymmetric and complex problems -- 5.1 The Lax-Milgram lemma -- 5.2 Convection-di usion equations -- 5.3 Complex and complexified spaces -- 5.4 The Laplace resolvent equations -- 5.5 The Ritz-Galerkin projection (*) -- Final comments and literature -- Exercises -- 6. Neumann boundary conditions -- 6.1 Duality on the boundary -- 6.2 Normal components of vector fields -- 6.3 Neumann boundary conditions -- 6.4 Impedance boundary conditions -- 6.5 Transmission problems (*) -- 6.6 Nonlocal boundary conditions (*) -- 6.7 Mixed boundary conditions (*) -- Final comments and literature -- Exercises -- 7. Poincar e inequalities and Neumann problems -- 7.1 Compactness -- 7.2 The Rellich-Kondrachov theorem -- 7.3 The Deny-Lions theorem -- 7.4 The Neumann problem for the Laplacian -- 7.5 Compact embedding in the unit cube -- 7.6 Korn's inequalities (*) -- 7.7 Traction problems in elasticity (*) -- Final comments and literature -- Exercises -- 8. Compact perturbations of coercive problems -- 8.1 Self-adjoint Fredholm theorems -- 8.2 The Helmholtz equation -- 8.3 Compactness on the boundary -- 8.4 Neumann and impedance problems revisited -- 8.5 Kirchho plate problems (*) -- 8.6 Fredholm theory: the general case -- 8.7 Convection-diffusion revisited -- 8.8 Impedance conditions for Helmholtz (*) -- 8.9 Galerkin projections and compactness (*) -- Final comments and literature -- Exercises -- 9. Eigenvalues of elliptic operators -- 9.1 Dirichlet and Neumann eigenvalues -- 9.2 Eigenvalues of compact self-adjoint operators -- 9.3 The Hilbert-Schmidt theorem -- 9.4 Proof of the Hilbert-Schmidt theorem (*) -- 9.5 Spectral characterization of Sobolev spaces -- 9.6 Classical Fourier series -- 9.7 Steklov eigenvalues (*) -- 9.8 A glimpse of interpolation (*) -- Final comments and literature -- Exercises -- Part II: Extensions and Applications -- 10. Mixed problems -- 10.1 Surjectivity -- 10.2 Systems with mixed structure -- 10.3 Weakly imposed Dirichlet conditions -- 10.4 Saddle point problems -- 10.5 The mixed Laplacian -- 10.6 Darcy flow -- 10.7 The divergence operator -- 10.8 Stokes flow -- 10.9 Stokes-Darcy flow -- 10.10 Brinkman flow -- 10.11 Reissner-Mindlin plates -- Final comments and literature -- Exercises -- 11. Advanced mixed problems -- 11.1 Mixed form of reaction-diffusion problems -- 11.2 More inde nite problems -- 11.3 Mixed form of convection-di usion problems 11.4 Double restrictions -- 11.5 A partially uncoupled Stokes-Darcy formulation -- 11.6 Galerkin methods for mixed problems -- Final comments and literature -- Exercises -- 12. Nonlinear problems -- 12.1 Lipschitz strongly monotone operators -- 12.2 An embedding theorem -- 12.3 Laminar Navier-Stokes flow -- 12.4 A nonlinear diffusion problem -- 12.5 The Browder-Minty theorem -- 12.6 A nonlinear reaction-diffusion problem -- Final comments and literature -- Exercises -- 13. Fourier representation of Sobolev spaces -- 13.1 The Fourier transform in the Schwartz class -- 13.2 A first mix of Fourier and Sobolev -- 13.3 An introduction to H2 regularity -- 13.4 Topology of the Schwartz class -- 13.5 Tempered distributions -- 13.6 Sobolev spaces by Fourier transforms -- 13.7 The trace space revisited -- 13.8 Interior regularity -- Final comments and literature -- Exercises -- 14. Layer potentials -- 14.1 Green's functions in free space -- 14.2 Single and double layer Yukawa potentials -- 14.3 Properties of the boundary integral operators -- 14.4 The Calderón calculus -- 14.5 Integral form of the layer potentials -- 14.6 A weighted Sobolev space -- 14.7 Coulomb potentials -- 14.8 Boundary-field formulations -- Final comments and literature -- Exercises -- 15. A collection of elliptic problems -- 15.1 T-coercivity in a dual Helmholtz equation -- 15.2 Diffusion with sign changing coefficient -- 15.3 Dependence with respect to coefficients -- 15.4 Obstacle problems -- 15.5 The Signorini contact problem -- 15.6 An optimal control problem -- 15.7 Friction boundary conditions -- 15.8 The Lions-Stampacchia theorem -- 15.9 Maximal dissipative operators -- 15.10 The evolution of elliptic operators -- Final comments and literature -- Exercises -- 16. Curl spaces and Maxwell's equations -- 16.1 Sobolev spaces for the curl -- 16.2 A first look at the tangential trace -- 16.3 Curl-curl equations -- 16.4 Time-harmonic Maxwell's equations -- 16.5 Two de Rham sequences -- 16.6 Maxwell eigenvalues -- 16.7 Normally oriented trace fields -- 16.8 Tangential trace spaces and their rotations -- 16.9 Tangential definition of the tangential traces -- 16.10 The curl-curl integration by parts formula -- Final comments and literature -- Exercises -- 17. Elliptic equations on boundaries -- 17.1 Surface gradient and Laplace-Beltrami operator -- 17.2 The Poincar e inequality on a surface -- 17.3 More on boundary spaces -- Final comments and literature -- Exercises -- Appendix A: Review material -- A.1 The divergence theorem -- A.2 Analysis -- A.3 Banach spaces -- A.4 Hilbert spaces -- Appendix B: Glossary -- B.1 Commonly used terms -- B.2 Some key spaces -- Bibliography -- Index |
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| Beschreibung: | Literaturverzeichnis: Seite 479-487 |
| Beschreibung: | xxii, 492 Seiten Illustrationen, Diagramme 24 cm |
| ISBN: | 9781138580886 978-1-138-58088-6 |