Basic linear partial differential equations

Content: 
Editors
Page ii

Edited by
Page iii

Copyright page
Page iv

Preface
Pages ix-xi

Notation
Pages xiii-xvii

1 The Basic Examples of Linear PDEs
Pages 3-13

2 Existence and Smoothness of Solutions Not Submitted to Side Conditions
Pages 14-21

3 Analyticity of Solutions
Pages 22-25

4 Fundamental Solutions of Ordinary Differential Equations
Pages 26-33

5 Fundamental Solutions of the Cauchy-Riemann Operator
Pages 34-40

6 Fundamental Solutions of the Heat and of the Schrödinger Equations
Pages 41-46

7 Fundamental Solutions of the Wave Equation
Pages 47-58

8 More on the Supports and Singular Supports of the Fundamental Solutions of the Wave Equation
Pages 59-67

9 Fundamental Solutions of the Laplace Equation
Pages 68-76

10 Green\'s Formula. The Mean Value Theorem and the Maximum Principle for Harmonic Functions. The Poisson Formula. Harnack\'s Inequalities
Pages 77-86

11 The Cauchy Problem for Linear Ordinary Differential Equations
Pages 89-95

12 The Cauchy Problem for Linear Partial Differential Equations. Preliminary Observations
Pages 96-101

13 The Global Cauchy Problem for the Wave Equation. Existence and Uniqueness of the Solutions
Pages 102-110

14 Domain of Influence, Propagation of Singularities, Conservation of Energy
Pages 111-118

15 Hyperbolic First-Order Systems with Constant Coefficients
Pages 119-131

16 Strongly Hyperbolic First-Order Systems in One Space Dimension
Pages 132-141

17 The Cauchy–Kovalevska Theorem. The Classical and Abstract Versions
Pages 142-155

18 Reduction of Higher Order Systems to First-Order Systems
Pages 156-160

19 Characteristics. Invariant Form of the Cauchy–Kovalevska Theorem
Pages 161-173

20 The Abstract Version of the Holmgren Theorem
Pages 174-180

21 The Holmgren Theorem
Pages 181-186

22 The Dirichlet Problem. The Variational Form
Pages 189-200

23 Solution of the Weak Problem. Coercive Forms. Uniform Ellipticity
Pages 201-209

24 A More Systematic Study of the Sobolev Spaces
Pages 210-223

25 Further Properties of the Spaces H S
Pages 224-236

26 Traces in Hm(Ω)
Pages 237-248

27 Back to the Dirichlet Problem. Regularity up to the Boundary
Pages 249-258

28 A Weak Maximum Principle
Pages 259-267

29 Application: Solution of the Classical Dirichlet Problem
Pages 268-277

30 Theory of the Laplace Equation: Superharmonic Functions and Potentials
Pages 278-293

31 Laplace Equation and the Brownian Motion
Pages 294-305

32 Dirichlet Problems in the Plane. Conformal Mappings
Pages 306-313

33 Approximation of Harmonic Functions by Harmonic Polynomials in Three Space. Spherical Harmonics
Pages 314-321

34 Spectral Properties and Eigenfunction Expansions
Pages 322-331

35 Approximate Solutions to the Dirichlet Problem. The Finite Difference Method
Pages 332-346

36 Girding\'s Inequality. Dirichlet Problem for Higher Order Elliptic Equations
Pages 347-353

37 Neumann Problem and Other Boundary Value Problems (Variational Form)
Pages 354-366

38 Indications on the General Lopatinski Conditions
Pages 367-377

39 Functions and Distributions Valued in Banach Spaces
Pages 381-390

40 Mixed Problems. Weak Form
Pages 391-400

41 Energy Inequalities. Proof of Theorem 40.I: Existence and Uniqueness of the Weak Solution to the Parabolic Mixed Problem
Pages 401-407

42 Regularity of the Weak Solution with Respect to the Time Variable
Pages 408-415

43 The Laplace Transform
Pages 416-423

44 Application of the Laplace Transform to the Solution of Parabolic Mixed Problems
Pages 424-435

45 Rudiments of Continuous Semigroup Theory
Pages 436-448

46 Application of Eigenfunction Expansion to Parabolic and to Hyperbolic Mixed Problems
Pages 449-457

47 An Abstract Existence and Uniqueness Theorem for a Class of Hyperbolic Mixed Problems. Energy Inequalities
Pages 458-464

Bibliography
Pages 465-466

Index
Pages 467-470