Partial differential equations in action: From modelling to theory

Title Page......Page 2
Copyright Page......Page 3
Preface......Page 4
Table of Contents......Page 7
1.1 Mathematical Modelling......Page 14
1.2 Partial Differential Equations......Page 15
1.3 Well Posed Problems......Page 18
1.4 Basic Notations and Facts......Page 20
1.5 Smooth and Lipschitz Domains......Page 23
1.6 Integration by Parts Formulas......Page 24
2.1.1 Introduction......Page 26
2.1.2 The conduction of heat......Page 27
2.1.3 Well posed problems (n = 1)......Page 29
2.1.4 A solution by separation of variables......Page 32
2.1.5 Problems in dimension n > 1......Page 40
2.2.1 Integral method......Page 43
2.2.2 Maximum principles......Page 44
2.3.1 Invariant transformations......Page 47
2.3.2 Fundamental solution (n = 1)......Page 49
2.3.4 Fundamental solution (n > 1)......Page 55
2.4 Symmetric Random Walk (n = 1)......Page 56
2.4.1 Preliminary computations......Page 57
2.4.2 The limit transition probability......Page 60
2.4.3 From random walk to Brownian motion......Page 62
2.5.1 Random walk with drift......Page 65
2.5.2 Pollution in a channel......Page 67
2.5.3 Random walk with drift and reaction......Page 70
2.6.1 The symmetric case......Page 71
2.7 An Example of Reaction−Diffusion (n = 3)......Page 75
2.8.1 The homogeneous case......Page 81
2.8.2 Existence of a solution......Page 82
2.8.3 The non homogeneous case. Duhamel’s method......Page 84
2.8.4 Maximum principles and uniqueness......Page 87
2.9.2 An evolution model for the price S......Page 90
2.9.3 The Black-Scholes equation......Page 93
2.9.4 The solutions......Page 96
2.9.5 Hedging and self-financing strategy......Page 101
2.10.1 Nonlinear diffusion. The porous medium equation......Page 103
2.10.2 Nonlinear reaction. Fischer’s equation......Page 106
3.1 Introduction......Page 115
3.2 Well Posed Problems. Uniqueness......Page 116
3.3.1 Discrete harmonic functions......Page 118
3.3.2 Mean value properties......Page 122
3.3.3 Maximum principles......Page 123
3.3.4 The Dirichlet problem in a circle. Poisson’s formula......Page 126
3.3.5 Harnack’s inequality and Liouville’s theorem......Page 130
3.3.6 A probabilistic solution of the Dirichlet problem......Page 131
3.3.7 Recurrence and Brownian motion......Page 136
3.4.1 The fundamental solution......Page 137
3.4.2 The Newtonian potential......Page 139
3.4.3 A divergence-curl system. Helmholtz decomposition formula......Page 141
3.5.1 An integral identity......Page 145
3.5.2 The Green function......Page 146
3.5.3 Green’s representation formula......Page 148
3.5.4 The Neumann function......Page 150
3.6 Uniqueness in Unbounded Domains......Page 152
3.7 Surface Potentials......Page 154
3.7.1 The double and single layer potentials......Page 155
3.7.2 The integral equations of potential theory......Page 159
4.1 Introduction......Page 169
4.2.1 Pollution in a channel......Page 170
4.2.2 Distributed source......Page 172
4.2.3 Decay and localized source......Page 173
4.2.4 Inflow and outflow characteristics. A stability estimate......Page 175
4.3.1 A macroscopic model......Page 177
4.3.2 The method of characteristics......Page 178
4.3.3 The green light problem......Page 181
4.3.4 Traffic jam ahead......Page 185
4.4.1 The method of characteristics revisited......Page 187
4.4.2 Definition of integral solution......Page 190
4.4.3 The Rankine-Hugoniot condition......Page 192
4.4.4 The entropy condition......Page 196
4.4.5 The Riemann problem......Page 198
4.4.6 Vanishing viscosity method......Page 199
4.4.7 The viscous Burger equation......Page 202
4.5.1 Characteristics......Page 205
4.5.2 The Cauchy problem......Page 207
4.5.3 Lagrange method of first integrals......Page 215
4.5.4 Underground flow......Page 218
4.6.1 Characteristic strips......Page 220
4.6.2 The Cauchy Problem......Page 223
5.1.1 Types of waves......Page 234
5.1.2 Group velocity and dispersion relation......Page 236
5.2.1 The model......Page 239
5.2.2 Energy......Page 241
5.3.1 Initial and boundary conditions......Page 242
5.3.2 Separation of variables......Page 244
5.4.1 The homogeneous equation......Page 249
5.4.2 Generalized solutions and propagation of singularities......Page 253
5.4.3 The fundamental solution......Page 257
5.4.4 Non homogeneous equation. Duhamel’s method......Page 259
5.4.5 Dissipation and dispersion......Page 260
5.5.1 Classification......Page 262
5.5.2 Characteristics and canonical form......Page 265
5.6 Hyperbolic Systems with Constant Coefficients......Page 270
5.7.1 Special solutions......Page 274
5.7.2 Well posed problems. Uniqueness......Page 276
5.8.1 Small vibrations of an elastic membrane......Page 279
5.8.2 Small amplitude sound waves......Page 283
5.9.1 Fundamental solution (n = 3) and strong Huygens’ principle......Page 287
5.9.2 The Kirchhoff formula......Page 290
5.9.3 Cauchy problem in dimension 2......Page 292
5.9.4 Non homogeneous equation. Retarded potentials......Page 294
5.10.1 A model for surface waves......Page 295
5.10.2 Dimensionless formulation and linearization......Page 299
5.10.3 Deep water waves......Page 301
5.10.4 Interpretation of the solution......Page 303
5.10.5 Asymptotic behavior......Page 305
5.10.6 The method of stationary phase......Page 306
6.1 Motivations......Page 315
6.2 Norms and Banach Spaces......Page 320
6.3 Hilbert Spaces......Page 324
6.4.1 Projections......Page 329
6.4.2 Bases......Page 333
6.5.1 Linear operators......Page 339
6.5.2 Functionals and dual space......Page 341
6.5.3 The adjoint of a bounded operator......Page 344
6.6.1 Bilinear forms and the Lax-Milgram Theorem......Page 347
6.6.2 Minimization of quadratic functionals......Page 352
6.6.3 Approximation and Galerkin method......Page 353
6.7.1 Compactness......Page 356
6.7.2 Weak convergence and compactness......Page 357
6.7.3 Compact operators......Page 361
6.8.1 Solvability for abstract variational problems......Page 363
6.8.2 Fredholm’s Alternative......Page 367
6.9.1 Spectrum of a matrix......Page 369
6.9.2 Separation of variables revisited......Page 370
6.9.3 Spectrum of a compact self-adjoint operator......Page 371
6.9.4 Application to abstract variational problems......Page 373
7.1 Distributions. Preliminary Ideas......Page 380
7.2 Test Functions and Mollifiers......Page 382
7.3 Distributions......Page 386
7.4.1 The derivative in the sense of distributions......Page 390
7.4.2 Gradient, divergence, laplacian......Page 392
7.5.1 Multiplication. Leibniz rule......Page 395
7.5.2 Composition......Page 397
7.5.3 Division......Page 398
7.5.4 Convolution......Page 399
7.6.1 Tempered distributions......Page 401
7.6.2 Fourier transform in S\'......Page 404
7.6.3 Fourier transform in L2......Page 406
7.7.1 An abstract construction......Page 407
7.7.2 The space H1 (Ω)......Page 409
7.7.3 The space H10 (Ω)......Page 412
7.7.4 The dual of H10 (Ω)......Page 414
7.7.5 The spaces Hm (Ω), m>1......Page 416
7.7.6 Calculus rules......Page 417
7.7.7 Fourier Transform and Sobolev Spaces......Page 418
7.8.1 Local approximations......Page 419
7.8.2 Estensions and global approximations......Page 420
7.9.1 Traces of functions in H1 (Ω)......Page 424
7.9.2 Traces of functions in Hm (Ω)......Page 427
7.9.3 Trace spaces......Page 428
7.10.1 Rellich’s theorem......Page 431
7.10.2 Poincaré’s inequalities......Page 432
7.10.3 Sobolev inequality in Rn......Page 433
7.10.4 Bounded domains......Page 435
7.11.1 Functions with values in Hilbert spaces......Page 437
7.11.2 Sobolev spaces involving time......Page 438
8.1 Elliptic Equations......Page 444
8.2 The Poisson Problem......Page 446
8.3.2 Dirichlet conditions......Page 448
8.3.3 Neumann, Robin and mixed conditions......Page 452
8.4.1 Dirichlet problem......Page 457
8.4.2 Neumann, Robin and mixed problems......Page 460
8.4.3 Eigenvalues of the Laplace operator......Page 464
8.4.4 An asymptotic stability result......Page 466
8.5.1 Basic assumptions......Page 467
8.5.2 Dirichlet problem......Page 468
8.5.3 Neumann problem......Page 474
8.5.4 Robin and mixed problems......Page 476
8.5.5 Weak Maximum Principles......Page 478
8.6 Regularity......Page 480
8.7 Equilibrium of a plate......Page 486
8.8 A Monotone Iteration Scheme for Semilinear Equations......Page 488
8.9.1 Structure of the problem......Page 491
8.9.2 Existence and uniqueness of an optimal pair......Page 493
8.9.3 Lagrange multipliers and optimality conditions......Page 494
8.9.4 An iterative algorithm......Page 496
9.1 Parabolic Equations......Page 505
9.2.1 The Cauchy-Dirichlet problem......Page 506
9.2.2 Faedo-Galerkin method (I)......Page 509
9.2.3 Solution of the approximate problem......Page 510
9.2.4 Energy estimates......Page 511
9.2.5 Existence, uniqueness and stability......Page 513
9.2.6 Regularity......Page 516
9.2.7 The Cauchy-Neuman problem......Page 518
9.2.8 Cauchy-Robin and mixed problems......Page 520
9.2.9 A control problem......Page 522
9.3.1 Weak formulation of initial value problems......Page 525
9.3.2 Faedo-Galerkin method (II)......Page 527
9.4.1 Hyperbolic Equations......Page 530
9.4.2 The Cauchy-Dirichlet problem......Page 531
9.4.3 Faedo-Galerkin method (III)......Page 533
9.4.4 Solution of the approximate problem......Page 534
9.4.5 Energy estimates......Page 535
9.4.6 Existence, uniqueness and stability......Page 538
A.1 Fourier coefficients......Page 543
A.2 Expansion in Fourier series......Page 546
B.1.1 A counting problem......Page 549
B.1.2 Measures and measurable functions......Page 551
B.1.3 The Lebesgue integral......Page 553
B.1.4 Some fundamental theorems......Page 554
B.1.5 Probability spaces, random variables and their integrals......Page 555
Cylindrical coordinates......Page 557
Spherical coordinates......Page 558
Identities......Page 559
Partial Differential Equations......Page 560
Numerical Analysis......Page 561
Stochastic Processes and Finance......Page 562
Index......Page 563