The Boundary Value Problems of Mathematical Physics

Title page
Introduction
Basic Notation
CHAPTER 1 Preliminary Considerations
1. Normed Spaces and Hilbert Spaces
2. Some Properties of Linear Functionals and Bounded Linear Operators in Hilbert Space
3. Unbounded Operators
4. Generalized Derivatives and Averages
5. Definition of the Spaces W^l_m(Ω) and W?(Ω)
6. The Spaces ?(Ω) and W¹₂(Ω) and Their Basic Properties
7. Multiplicative Inequalities for Elements of W?(Ω) and W¹_m(Ω)
8. Embedding Theorems for the Spaces W?(Ω) and W¹_m(Ω)
Supplements and Problems
CHAPTER II Equations of Elliptic Type
1. Posing of Boundary Value Problems. Description of the Basic Material of the Chapter
2. Generalized Solutions in W¹₂(Ω). The First (Energy) Inequality
3. Solvability of the Dirichlet Problem in the Space W¹₂(Ω). Three Theorems of Fredholm
4. Expansion in Eigenfunctions of Symmetric Operators
5. The Second and Third Boundary Value Problems
6. The Second Fundamental Inequality for Elliptic Operators
7. Solvability of the Dirichlet Problem in W²₂(Ω)
8. Approximate Methods of Solving Boundary Value Problems
Supplements and Problems
CHAPTER III Equations of Parabolic Type
1. Posing Initial-Boundary Value Problems and the Cauchy Problem
2. First Initial-Boundary Value Problem for the Heat Equation
3. First Initial-Boundary Value Problem for General Parabolic Equations
4. Other Boundary Value Problems. The Method of Fourier and Laplace. The Second Fundamental Inequality
5. The Method of Rothe
Supplements and Problems
CHAPTER IV Equations of Hyperbolic Type
1. General Considerations. Posing the Fundamental Problems
2. The Energy Inequality. Finiteness of the Speed of Propagation of Perturbations. Uniqueness Theorem for Solutions in W²₂
3. The First Initial-Boundary Value Problem. Solvability in W¹₂(Q_T)
4. On the Smoothness of Generalized Solutions
5. Other Initial-Boundary Value Problems
6. The Functional Method of Solving Initial-Boundary Value Problems
7. The Methods of Fourier and Laplace
Supplements and Problems
CHAPTER V Some Generalizations
1. Elliptic Equations of Arbitrary Order. Strongly Elliptic Systems
2. Strongly Parabolic and Strongly Hyperbolic Systems
3. Schrôdinger-Type Equations and Related Equations
4. Diffraction Problems
Supplements and Problems
CHAPTER VI The Method of Finite Differences
1. General Description of the Method. Some Principles of Constructing Convergent Difference Schemes
2. The Fundamental Difference Operators and Their Properties
3. Interpolations of Grid Functions. The Elementary Embedding Theorems
4. General Embedding Theorems
5. The Finite-Difference Method of Fourier
6. The Elementary Equations
7. The Dirichlet Problem for General Elliptic Equations of Second Order
8. The Neumann Problem and Third Boundary Value Problem for Elliptic Equations
9. Equations of Parabolic Type
10. Equations of Hyperbolic Type
11. Strong Convergence, Systems, Diffraction Problems
12. Approximation Methods
Supplements and Problems
Bibliography
Index