Variational Techniques for Elliptic Partial Differential Equations: Theoretical Tools and Advanced Applications

Description

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 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