Nonlinear Functional Analysis and Its Applications

to the Poisson Equation.- 19.12. Application to the Eigenvalue Problem for the Laplace Equation.- 19.13. The Inequality of Poincaré and the Compactness Theorem of Rellich.- 19.14. Functions of Self-Adjoint Operators.- 19.15. Application to the Heat Equation.- 19.16. Application to the Wave Equation.- 19.17. Semigroups and Propagators, and Their Physical Relevance.- 19.18. Main Theorem on Abstract Linear Parabolic Equations.- 19.19. Proof of Theorem 19.D.- 19.20. Monotone Operators and the Main Theorem on Linear Nonexpansive Semigroups.- 19.21. The Main Theorem on One-Parameter Unitary Groups.- 19.22. Proof of Theorem 19.E.- 19.23. Abstract Semilinear Hyperbolic Equations.- 19.24. Application to Semilinear Wave Equations.- 19.25. The Semilinear Schrödinger Equation.- 19.26. Abstract Semilinear Parabolic Equations, Fractional Powers of Operators, and Abstract Sobolev Spaces.- 19.27. Application to Semilinear Parabolic Equations.- 19.28. Proof of Theorem 19.I.- 19.29. Five General Uniqueness Principles and Monotone Operators.- 19.30. A General Existence Principle and Linear Monotone Operators.- 20 Difference Methods and Stability.- 20.1. Consistency, Stability, and Convergence.- 20.2. Approximation of Differential Quotients.- 20.3. Application to Boundary Value Problems for Ordinary Differential Equations.- 20.4. Application to Parabolic Differential Equations.- 20.5. Application to Elliptic Differential Equations.- 20.6. The Equivalence Between Stability and Convergence.- 20.7. The Equivalence Theorem of Lax for Evolution Equations.- Linear Monotone Problems.- 21 Auxiliary Tools and the Convergence of the Galerkin Method for Linear Operator Equations.- 21.1. Generalized Derivatives.- 21.2. Sobolev Spaces.- 21.3. The Sobolev Embedding Theorems.- 21.4. Proof of the Sobolev Embedding Theorems.- 21.5. Duality in B-Spaces.- 21.6. Duality in H-Spaces.- 21.7. The Idea of Weak Convergence.- 21.8. The Idea of Weak* Convergence.- 21.9. Linear Operators.- 21.10. Bilinear Forms.- 21.11. Application to Embeddings.- 21.12. Projection Operators.- 21.13. Bases and Galerkin Schemes.- 21.14. Application to Finite Elements.- 21.15. Riesz-Schauder Theory and Abstract Fredholm Alternatives.- 21.16. The Main Theorem on the Approximation-Solvability of Linear Operator Equations, and the Convergence of the Galerkin Method.- 21.17. Interpolation Inequalities and a Convergence Trick.- 21.18. Application to the Refined Banach Fixed-Point Theorem and the Convergence of Iteration Methods.- 21.19. The Gagliardo-Nirenberg Inequalities.- 21.20. The Strategy of the Fourier Transform for Sobolev Spaces.- 21.21. Banach Algebras and Sobolev Spaces.- 21.22. Moser-Type Calculus Inequalities.- 21.23. Weakly Sequentially Continuous Nonlinear Operators on Sobolev Spaces.- 22 Hilbert Space Methods and Linear Elliptic Differential Equations.- 22.1. Main Theorem on Quadratic Minimum Problems and the Ritz Method.- 22.2. Application to Boundary Value Problems.- 22.3. The Method of Orthogonal Projection, Duality, and a posteriori Error Estimates for the Ritz Method.- 22.4. Application to Boundary Value Problems.- 22.5. Main Theorem on Linear Strongly Monotone Operators and the Galerkin Method.- 22.6. Application to Boundary Value Problems.- 22.7. Compact Perturbations of Strongly Monotone Operators, Fredholm Alternatives, and the Galerkin Method.- 22.8. Application to Integral Equations.- 22.9. Application to Bilinear Forms.- 22.10. Application to Boundary Value Problems.- 22.11. Eigenvalue Problems and the Ritz Method.- 22.12. Application to Bilinear Forms.- 22.13. Application to Boundary-Eigenvalue Problems.- 22.14. Gårding Forms.- 22.15. The Gårding Inequality for Elliptic Equations.- 22.16. The Main Theorems on Gårding Forms.- 22.17. Application to Strongly Elliptic Differential Equations of Order 2m.- 22.18. Difference Approximations.- 22.19. Interior Regularity of Generalized Solutions.- 22.20. Proof of Theorem 22.H.- 22.21. Regularity of Generalized Solutions up to the Boundary.- 22.22. Proof of Theorem 22.I.- 23 Hilbert Space Methods and Linear Parabolic Differential Equations.- 23.1. Particularities in the Treatment of Parabolic Equations.- 23.2. The Lebesgue Space Lp(0, T; X) of Vector-Valued Functions.- 23.3. The Dual Space to Lp(0, T; X).- 23.4. Evolution Triples.- 23.5. Generalized Derivatives.- 23.6. The Sobolev Space WP1 (0, T; V, H).- 23.7. Main Theorem on First-Order Linear Evolution Equations and the Galerkin Method.- 23.8. Application to Parabolic Differential Equations.- 23.9. Proof of the Main Theorem.- 24 Hilbert Space Methods and Linear Hyperbolic Differential Equations.- 24.1. Main Theorem on Second-Order Linear Evolution Equations and the Galerkin Method.- 24.2. Application to Hyperbolic Differential Equations.- 24.3. Proof of the Main Theorem.