Introduction to Partial Differential Equations

Preface......Page 7
Contents......Page 10
Notations......Page 13
 1.1 Partial Differential Equations......Page 15
 1.2 Example: d'Alembert's Wave Equation......Page 16
 1.3 Types of Equations......Page 17
 1.4 Well Posed Problems......Page 19
 1.5 Approaches......Page 20
 2.1 Real Numbers......Page 22
 2.2 Complex Numbers......Page 23
 2.3 Domains in mathbbRn......Page 25
 2.4 Differentiability......Page 26
 2.5 Ordinary Differential Equations......Page 28
 2.6 Vector Calculus......Page 31
 2.7 Exercises......Page 36
 3.1 Model Problem: Oxygen in the Bloodstream......Page 38
 3.2 Lagrangian Derivative and Characteristics......Page 40
 3.3 Higher-Dimensional Equations......Page 45
 3.4 Quasilinear Equations......Page 48
 3.5 Exercises......Page 54
 4.1 Model Problem: Vibrating String......Page 58
 4.2 Characteristics......Page 60
 4.3 Boundary Problems......Page 64
 4.4 Forcing Terms......Page 66
 4.5 Model Problem: Acoustic Waves......Page 72
 4.6 Integral Solution Formulas......Page 74
 4.7 Energy and Uniqueness......Page 80
 4.8 Exercises......Page 82
5 Separation of Variables......Page 87
 5.2 Helmholtz Equation......Page 88
 5.3 Circular Symmetry......Page 93
 5.4 Spherical Symmetry......Page 99
 5.5 Exercises......Page 105
 6.1 Model Problem: Heat Flow in a Metal Rod......Page 108
 6.2 Scale-Invariant Solution......Page 112
 6.3 Integral Solution Formula......Page 114
 6.4 Inhomogeneous Problem......Page 118
 6.5 Exercises......Page 120
 7.1 Inner Products and Norms......Page 122
 7.2 Lebesgue Integration......Page 125
 7.3 Lp Spaces......Page 127
 7.4 Convergence and Completeness......Page 130
 7.5 Orthonormal Bases......Page 133
 7.6 Self-adjointness......Page 136
 7.7 Exercises......Page 139
 8.1 Series Solution of the Heat Equation......Page 142
 8.2 Periodic Fourier Series......Page 145
 8.3 Pointwise Convergence......Page 149
 8.4 Uniform Convergence......Page 152
 8.5 Convergence in L2......Page 154
 8.6 Regularity and Fourier Coefficients......Page 156
 8.7 Exercises......Page 162
 9.1 Model Problem: The Laplace Equation......Page 165
 9.2 Mean Value Formula......Page 171
 9.3 Strong Principle for Subharmonic Functions......Page 175
 9.4 Weak Principle for Elliptic Equations......Page 177
 9.5 Application to the Heat Equation......Page 180
 9.6 Exercises......Page 184
 10.1 Test Functions and Weak Derivatives......Page 187
 10.2 Weak Solutions of Continuity Equations......Page 191
 10.3 Sobolev Spaces......Page 197
 10.4 Sobolev Regularity......Page 200
 10.5 Weak Formulation of Elliptic Equations......Page 204
 10.6 Weak Formulation of Evolution Equations......Page 206
 10.7 Exercises......Page 212
11 Variational Methods......Page 215
 11.1 Model Problem: The Poisson Equation......Page 216
 11.2 Dirichlet's Principle......Page 217
 11.3 Coercivity and Existence of a Minimum......Page 218
 11.4 Elliptic Regularity......Page 224
 11.5 Eigenvalues by Minimization......Page 227
 11.6 Sequential Compactness......Page 234
 11.7 Estimation of Eigenvalues......Page 237
 11.8 Euler-Lagrange Equations......Page 244
 11.9 Exercises......Page 247
 12.1 Model Problem: Coulomb's Law......Page 249
 12.2 The Space of Distributions......Page 252
 12.3 Distributional Derivatives......Page 255
 12.4 Fundamental Solutions......Page 258
 12.5 Green's Functions......Page 262
 12.6 Time-Dependent Fundamental Solutions......Page 267
 12.7 Exercises......Page 269
 13.1 Fourier Transform......Page 271
 13.2 Tempered Distributions......Page 277
 13.3 The Wave Kernel......Page 281
 13.5 Exercises......Page 283
 Erratum to:D. Borthwick, Introduction to Partial Differential Equations, Universitext, http://doi.org/10.1007/978-3-319-48936-0......Page 286
Appendix A Analysis Foundations......Page 287
References......Page 290
Index......Page 291