Invitation to Partial Differential Equations

Invitation to Partial Differential Equations......Page 1
Contents......Page 3
Preface......Page 9
Selected notational conventions......Page 11
1.1. Definition and examples......Page 13
1.2. The total and the principal symbols......Page 14
1.3. Change of variables......Page 16
1.4. The canonical form of second order operators with constant coefficients......Page 18
1.5. Characteristics. Ellipticity and hyperbolicity......Page 19
1.6. Characteristics and the canonical form of 2nd order operators and 2nd order equations for n = 2......Page 21
1.7. The general solution of a homogeneous hyperbolic equation with constant coefficients for n = 2......Page 23
1.8. Appendix: Tangent and cotangent vectors......Page 24
1.9. Problems......Page 26
2.1. Vibrating string equation......Page 29
2.2. Unbounded string. The Cauchy problem. D'Alembert's formula......Page 35
2.3. A semi-bounded string. Re ection of waves from the end of the string......Page 39
2.4. A bounded string. Standing waves. The Fourier method (separation of variables method)......Page 41
2.5. Appendix: Calculus of variations and classical mechanics......Page 48
2.6. Problems......Page 55
3.1. Ordinary differential equations......Page 59
3.2. Formulation of the problem......Page 60
3.3. Basic properties of eigenvalues and eigenfunctions......Page 62
3.4. The short-wave asymptotics......Page 66
3.5. The Green function and completeness of the system of eigenfunctions......Page 68
3.6. Problems......Page 73
4.1. Motivation of the definition. Spaces of test functions.......Page 75
4.2. Spaces of distributions......Page 81
4.3. Topology and convergence in the spaces of distributions......Page 86
4.4. The support of a distribution......Page 89
4.5. Differentiation of distributions and multiplication by a smooth function......Page 93
4.6. A general notion of the transposed (adjoint) operator. Change of variables. Homogeneous distributions......Page 108
4.7. Appendix: Minkowski inequality......Page 112
4.8. Appendix: Completeness of distribution spaces......Page 115
4.9. Problems......Page 117
5.1. Convolution and direct product of regular functions......Page 119
5.2. Direct product of distributions......Page 121
5.3. Convolution of distributions......Page 124
5.4. Other properties of convolution. Support and singular support of a convolution......Page 128
5.5. Relation between smoothness of a fundamental solution and that of solutions of the homogeneous equation......Page 130
5.6. Solutions with isolated singularities. A removable singularity theorem for harmonic functions......Page 134
5.7. Estimates of derivatives for a solution of a hypoelliptic equation......Page 136
5.8. Fourier transform of tempered distributions......Page 139
5.9. Applying Fourier transform to nd fundamental solutions......Page 143
5.10. Liouville's theorem......Page 144
5.11. Problems......Page 146
6.1. Mean-value theorems for harmonic functions......Page 149
6.2. The maximum principle......Page 151
6.3. Dirichlet's boundary-value problem......Page 154
6.4. Hadamard's example......Page 155
6.5. Green's function: first steps......Page 158
6.6. Symmetry and regularity of Green's functions......Page 162
6.7. Green's function and Dirichlet's problem......Page 167
6.8. Explicit formulas for Green's functions......Page 171
6.9. Problems......Page 183
7.1. Physical meaning of the heat equation......Page 187
7.2. Boundary value problems for the heat and Laplace equations......Page 189
7.3. A proof that the limit function is harmonic.......Page 191
7.4. A solution of the Cauchy problem for the heat equation and Poisson's integral......Page 192
7.5. The fundamental solution for the heat operator. Duhamel's formula......Page 198
7.6. Estimates of derivatives of a solution of the heat equation......Page 201
7.7. Holmgren's principle. The uniqueness of solution of the Cauchy problem for the heat equation......Page 202
7.8. A scheme of solving the first and second initial-boundary value problems by the Fourier method......Page 205
7.9. Problems......Page 207
8.1. Spaces H s( )......Page 209
8.2. Spaces H s( )......Page 215
8.3. Dirichlet's integral. The Friedrichs inequality......Page 219
8.4. Dirichlet's problem (generalized solutions)......Page 221
8.5. Problems......Page 228
9.1. Symmetric and self-adjoint operators in Hilbert space......Page 231
9.2. The Friedrichs extension......Page 236
9.3. Discreteness of spectrum for the Laplace operator in a bounded domain......Page 241
9.4. Fundamental solution of the Helmholtz operator and the analyticity of eigenfunctions of the Laplace operator at the interior points. Bessel's equation......Page 242
9.5. Variational principle. The behavior of eigenvalues under variation of the domain. Estimates of eigenvalues......Page 249
9.6. Problems......Page 253
10.1. Physical problems leading to the wave equation......Page 257
10.2. Plane, spherical and cylindric waves......Page 262
10.3. The wave equation as a Hamiltonian system......Page 265
10.4. A spherical wave caused by an instant ash and a solution of the Cauchy problem for the 3-dimensional wave equation......Page 271
10.5. The fundamental solution for the three-dimensional wave operator and a solution of the non-homogeneous wave equation......Page 277
10.6. The two-dimensional wave equation (the descent method)......Page 279
10.7. Problems......Page 282
Chapter 11 Properties of the potentials and their calculations......Page 285
11.1. Definitions of potentials......Page 286
11.2. Functions smooth up to from each side, and their derivatives......Page 288
11.3. Jumps of potentials......Page 296
11.4. Calculating potentials......Page 298
11.5. Problems......Page 301
12.1. Characteristics as surfaces of jumps......Page 303
12.2. The Hamilton-Jacobi equation. Wave fronts, bicharacteristics and rays......Page 309
12.3. The characteristics of hyperbolic equations......Page 316
12.4. Rapidly oscillating solutions. The eikonal equation and the transport equations......Page 318
12.5. The Cauchy problem with rapidly oscillating initial data......Page 329
Problems......Page 335
Chapter 13 Answers and Hints. Solutions......Page 337
Bibliography......Page 361
Index......Page 363