Basic Linear Partial Differential Equations
Published on 8. August 1975
470 pages
978-0-08-088025-9 (ISBN)
Description
Basic Linear Partial Differential Equations
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Francois Treves
Basic Linear Partial Differential Equations
Book
06/1975
Academic Press
€79.86
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Content
- Front Cover
- Basic Linear Partial Differential Equations
- Copyright Page
- Contents
- Preface
- Notation
- Chapter I. The Basic Examples of Linear PDEs and Their Fundamental Solutions
- 1. The Basic Examples of Linear PDEs
- 2 . Existence and Smoothness of Solutions Not Submitted to Side Conditions
- 3. Analyticity of Solutions
- 4. Fundamental Solutions of Ordinary Differential Equations
- 5. Fundamental Solutions of the Cauchy-Riemann Operator
- 6. Fundamental Solutions of the Heat and of the Schrödinger Equations
- 7. Fundamental Solutions of the Wave Equation
- 8. More on the Supports and Singular Supports of the Fundamental Solutions of the Wave Equation
- 9. Fundamental Solutions of the Laplace Equation
- 10. Green's Formula. The Mean Value Theorem and the Maximum Principle for Harmonic Functions. The Poisson Formula. Harnack's Ineqalities
- Chapter II. The Cauchy Problem
- 11. The Cauchy Problem for Linear Ordinary Differential Equations
- 12. The Cauchy Problem for Linear Partial Differential Equations. Preliminary Observations
- 13. The Global Cauchy Problem for the Wave Equation. Existence and Uniqueness of the Solutions
- 14. Domain of Influence, Propagation of Singularities, Conservation of Energy
- 15. Hyperbolic First-Order Systems with Constant Coefficients
- 16. Strongly Hyperbolic First-Order Systems in One Space Dimension
- 17. The Cauchy-Kovalevska Theorem. The Classical and Abstract Versions
- 18. Reduction of Higher Order Systems to First-Order Systems
- 19. Characteristics. Invariant Form of the Cauchy-Kovalevska Theorem
- 20. The Abstract Version of the Holmgren Theorem
- 21. The Holmgren Theorem
- Chapter III. Boundary Value Problems
- 22. The Dirichlet Problem. The Variational Form
- 23. Solution of the Weak Problem. Coercive Forms. Uniform Ellipticity
- 24. A More Systematic Study of the Sobolev Spaces
- 25. Further Properties of the Spaces Hs
- 26. Traces in Hm(O)
- 27. Back to the Dirichlet Problem. Regularity up to the Boundary
- 28. A Weak Maximum Principle
- 29. Application: Solution of the Classical Dirichlet Problem
- 30. Theory of the Laplace Equation: Superharmonic Functions and Potentials
- 31. Laplace Equation and the Brownian Motion
- 32. Dirichlet Problems in the Plane. Conformal Mappings
- 33. Approximation of Harmonic Functions by Harmonic Polynomials in Three Space. Spherical Harmonics
- 34. Spectral Properties and Eigenfunction Expansions
- 35. Approximate Solutions to the Dirichlet Problem. The Finite Difference Method
- 36. Gårding's Inequality. Dirichlet Problem for Higher Order Elliptic Equations
- 37. Neumann Problem and Other Boundary Value Problems (Variational Form)
- 38. Indications on the General Lopatinski Conditions
- Chapter IV. Mixed Problems and Evolution Equations
- 39. Functions and Distributions Valued in Banach Spaces
- 40. Mixed Problems. Weak Form
- 41. Energy Inequalities. Proof of Theorem 40.1 : Existence and Uniqueness of the Weak Solution to the Parabolic Mixed Problem
- 42. Regularity of the Weak Solution with Respect to the Time Variable
- 43. The Laplace Transform
- 44. Application of the Laplace Transform to the Solution of Parabolic Mixed Problems
- 45. Rudiments of Continuous Semigroup Theory
- 46. Application of Eigenfunction Expansion to Parabolic and to Hyperbolic Mixed Problems
- 47. An Abstract Existence and Uniqueness Theorem for a Class of Hyperbolic Mixed Problems. Energy Inequalities
- Bibliography
- Index
- Pure and Applied Mathematics
