| Description |
1 online resource |
| Contents |
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) |
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2.5 The Dirichlet problem2.6 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 |
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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 (*) |
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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 |
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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 (*) |
| Summary |
Variational Techniques for Elliptic Partial Differential Equations, intended for graduate students studying applied math, analysis, and/or numerical analysis, provides the necessary tools to understand the structure and solvability of elliptic partial differential equations. Beginning with the necessary definitions and theorems from distribution theory, the book gradually builds the functional analytic framework for studying elliptic PDE using variational formulations. Rather than introducing all of the prerequisites in the first chapters, it is the introduction of new problems which motivates the development of the associated analytical tools. In this way the student who is encountering this material for the first time will be aware of exactly what theory is needed, and for which problems. Features A detailed and rigorous development of the theory of Sobolev spaces on Lipschitz domains, including the trace operator and the normal component of vector fields An integration of functional analysis concepts involving Hilbert spaces and the problems which can be solved with these concepts, rather than separating the two Introduction to the analytical tools needed for physical problems of interest like time-harmonic waves, Stokes and Darcy flow, surface differential equations, Maxwell cavity problems, etc. A variety of problems which serve to reinforce and expand upon the material in each chapter, including applications in fluid and solid mechanics |
| Bibliography |
Includes bibliographical references and index |
| Notes |
Description based on print version record |
| Subject |
Differential equations, Elliptic.
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Differential equations, Partial.
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MATHEMATICS -- Calculus.
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MATHEMATICS -- Mathematical Analysis.
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MATHEMATICS -- Applied.
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MATHEMATICS -- Differential Equations.
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MATHEMATICS -- Number Systems.
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Differential equations, Elliptic
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Differential equations, Partial
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| Form |
Electronic book
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| Author |
Brown, Thomas S. (Mathematician), author.
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Hassell, Matthew E., author
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| LC no. |
2020693313 |
| ISBN |
9780429016196 |
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0429016190 |
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9780429016202 |
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0429016204 |
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0429507062 |
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9780429016189 |
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0429016182 |
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9780429507069 |
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