Partial differential equations

Cover......Page 1
Title Page......Page 4
Copyright......Page 5
Profile of A.N. Milgram......Page 6
Foreword......Page 8
Preface......Page 10
Acknowledgments......Page 11
Contents......Page 12
Part I. Hyperbolic and Parabolic Equations, by Farrz JoHN......Page 16
1. Equations of Hyperbolic and Parabolic Types......Page 18
2.1. The one-dimensional wave equation......Page 21
2.2. The initial value problem for the wave equation in three-space......Page 27
2.3. Analysis of the solution......Page 30
2.4. The method of descent......Page 33
2.5. The inhomogencous wave equation......Page 34
2.6. The Cauchy problem for general initial surfaces......Page 35
2.7. Energy integrals and a priori estimates......Page 41
2.8. The general linear equation with the wave operator as principal part......Page 48
2.9. Mixed problems......Page 52
3.1. Notation......Page 55
3.2. Relations between partial derivatives on a surface......Page 57
3.3. Free surfaces. Characteristic matrix......Page 60
3.4. Cauchy's problem. The uniqueness theorem of Holmgren......Page 62
3.5. Propagation of discontinuities......Page 70
4. Linear Hyperbolic Differential Equations......Page 79
4.1. Solution of the homogeneous equation with constant coefficients by Fourier transform......Page 81
4.2. Extension to hyperbolic systems of homogeneous equations with constant coefficients......Page 86
4.3. Method of decomposition into plane waves......Page 88
4.4. A priori estimates......Page 91
4.5. The general linear strictly hyperbolic equation with constant principal part......Page 95
4.6. First-order systems with constant principal part......Page 99
4.7. Symmetric hyperbolic systems with variable coefficients......Page 104
5.1. Parabolic equations in general......Page 111
5.2. The heat equation. Maximum principle......Page 113
5.3. Solution of the initial value problem......Page 115
5.4. Smoothness of solutions......Page 118
5.5. The boundary initial value problem for a rectangle......Page 121
6. Approximation of Solutions of Partial Differential Equations by the Method of Finite Differences......Page 125
6.1. Solution of parabolic equations......Page 126
6.2. Stability of difference schemes for other types of equations......Page 132
Bibliography......Page 140
1.1. Introduction......Page 148
1. Elliptic Equations and Their Solutions......Page 150
1.2. Linear elliptic equations......Page 151
1.3. Smoothness of solutions......Page 152
1.4. Unique continuation......Page 156
1.5. Boundary conditions......Page 158
Appendix I. Elliptic versus Strongly Elliptic......Page 160
Appendix II. "Weak Equals Strong"......Page 161
2.2. Statement and proof of the maximum principle......Page 167
2.3. Applications to the Dirichlet problem......Page 169
2.4. Applications to the generalized Neumann problem......Page 171
2.5. Solution of the Dirichlet problem by finite differences......Page 172
2.6. Solution of the difference equation by iterations......Page 175
2.7. A maximum principle for gradients......Page 177
2.8. Carleman's unique continuation theorem......Page 179
3.1. Periodic solutions......Page 181
3.2. The Hilbert spaces Hi......Page 182
3.3. Structure of the spaces H.......Page 184
3.4. Basic inequalities......Page 187
3.5. Differentiability theorem......Page 191
3.6. Solution of the equation Lit =f......Page 192
Appendix I. The Projection 'T'heorem......Page 194
Appendix II. The Fredholm-Riesz-Schauder Theory......Page 200
4.2. Interior regularity......Page 207
4.3. The spaces H' and H.......Page 209
4.4. Some lemmas in Ho......Page 210
4.5. The generalized Dirichlet problem......Page 213
4.6. Existence of weak solutions......Page 215
4.7. Regularity at the boundary......Page 217
4.8. Inequalities in a half-cube......Page 219
Appendix. Analyticity of Solutions......Page 224
5.1. Fundamental solutions. Parametrix......Page 228
5.2. Some function spaces......Page 233
5.3. Fundamental inequalities......Page 237
5.4. Local existence theorem......Page 245
5.5. Interior Schauder type estimates.......Page 248
5.6. Estimates up to the boundary......Page 252
5.7. Applications to the Dirichlet problem......Page 254
5.8. Smoothness of strong solutions......Page 257
Appendix I. Proofs of the Fundamental Inequalities......Page 259
Appendix II. Proofs of the Interpolation Lemmas......Page 267
6. Function Theoretical Approach......Page 271
6.1. Complex notation......Page 272
6.2. Beltrami equation......Page 274
6.3. A representation theorem......Page 276
6.4. Consequences of the representation theorem......Page 278
6.5. Two boundary value problems......Page 280
Appendix. Properties of the Beltrami Equation. Privaloff 's Theorem......Page 284
7.1. Boundary value problems.......Page 299
7.2. Methods of solution......Page 301
7.3. Examples......Page 303
Bibliography......Page 308
Supplement I. Eigenvalue Expansions, by Lnxs GARDING......Page 318
Supplement II. Parabolic Equations, by A. N. MILGRA.tit......Page 344
Index......Page 358
Back Cover......Page 361