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 (*)
		1.6 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