Focusing on the archetypes of linear partial differential equations, this text for upper-level undergraduates and graduate students employs nontraditional methods to explain classical material. Topics include the Cauchy problem, boundary value problems, and mixed problems and evolution equations. Nearly 400 exercises enable students to reconstruct proofs. 1975 edition. Front Cover 1 Basic Linear Partial Differential Equations 4 Copyright Page 5 Contents 6 Preface 10 Notation 14 Chapter I. The Basic Examples of Linear PDEs and Their Fundamental Solutions 22 1. The Basic Examples of Linear PDEs 24 2 . Existence and Smoothness of Solutions Not Submitted to Side Conditions 35 3. Analyticity of Solutions 43 4. Fundamental Solutions of Ordinary Differential Equations 47 5. Fundamental Solutions of the Cauchy鈥揜iemann Operator 55 6. Fundamental Solutions of the Heat and of the Schr枚dinger Equations 62 7. Fundamental Solutions of the Wave Equation 68 8. More on the Supports and Singular Supports of the Fundamental Solutions of the Wave Equation 80 9. Fundamental Solutions of the Laplace Equation 89 10. Green鈥檚 Formula. The Mean Value Theorem and the Maximum Principle for Harmonic Functions. The Poisson Formula. Harnack鈥檚 Ineqalities 98 Chapter II. The Cauchy Problem 108 11. The Cauchy Problem for Linear Ordinary Differential Equations 110 12. The Cauchy Problem for Linear Partial Differential Equations. Preliminary Observations 117 13. The Global Cauchy Problem for the Wave Equation. Existence and Uniqueness of the Solutions 123 14. Domain of Influence, Propagation of Singularities, Conservation of Energy 132 15. Hyperbolic First-Order Systems with Constant Coefficients 140 16. Strongly Hyperbolic First-Order Systems in One Space Dimension 153 17. The Cauchy鈥揔ovalevska Theorem. The Classical and Abstract Versions 163 18. Reduction of Higher Order Systems to First-Order Systems 177 19. Characteristics. Invariant Form of the Cauchy鈥揔ovalevska Theorem 182 20. The Abstract Version of the Holmgren Theorem 195 21. The Holmgren Theorem 202 Chapter III. Boundary Value Problems 208 22. The Dirichlet Problem. The Variational Form 210 23. Solution of the Weak Problem. Coercive Forms. Uniform Ellipticity 222 24. A More Systematic Study of the Sobolev Spaces 231 25. Further Properties of the Spaces Hs 245 26. Traces in Hm(惟) 258 27. Back to the Dirichlet Problem. Regularity up to the Boundary 270 28. A Weak Maximum Principle 280 29. Application: Solution of the Classical Dirichlet Problem 289 30. Theory of the Laplace Equation: Superharmonic Functions and Potentials 299 31. Laplace Equation and the Brownian Motion 315 32. Dirichlet Problems in the Plane. Conformal Mappings 327 33. Approximation of Harmonic Functions by Harmonic Polynomials in Three Space. Spherical Harmonics 335 34. Spectral Properties and Eigenfunction Expansions 343 35. Approximate Solutions to the Dirichlet Problem. The Finite Difference Method 353 36. G氓rding鈥檚 Inequality. Dirichlet Problem for Higher Order Elliptic Equations 368 37. Neumann Problem and Other Boundary Value Problems (Variational Form) 375 38. Indications on the General Lopatinski Conditions 388 Chapter IV. Mixed Problems and Evolution Equations 400 39. Functions and Distributions Valued in Banach Spaces 402 40. Mixed Problems. Weak Form 412 41. Energy Inequalities. Proof of Theorem 40.1 : Existence and Uniqueness of the Weak Solution to the Parabolic Mixed Problem 422 42. Regularity of the Weak Solution with Respect to the Time Variable 429 43. The Laplace Transform 437 44. Application of the Laplace Transform to the Solution of Parabolic Mixed Problems 445 45. Rudiments of Continuous Semigroup Theory 457 46. Application of Eigenfunction Expansion to Parabolic and to Hyperbolic Mixed Problems 470 47. An Abstract Existence and Uniqueness Theorem for a Class of Hyperbolic Mixed Problems. Energy Inequalities 479 Bibliography 486 Index 488 Pure and Applied Mathematics 492
