Partial Differential Equations: A Unified Hilbert Space Approach (De Gruyter Expositions in Mathematics 55)

Preface......Page 8
Contents......Page 12
Nomenclature......Page 16
1.1 Hilber tSpace......Page 20
1.2 Some Construction Principles of Hilbert Spaces......Page 21
1.2.1 Direct Sums of Hilbert Spaces......Page 22
1.2.2 Dual Spaces......Page 34
1.2.3 Tensor Products of Hilbert Spaces......Page 38
2.1 Sobolev Chains......Page 49
2.2 Sobolev Lattices......Page 75
2.3 Sobolev Lattices from Tensor Products of Sobolev Chains......Page 84
3.1.1 First Steps Towards a Solution Theory......Page 91
3.1.2 The Tarski–Seidenberg Theorem and some Consequences......Page 95
3.1.3 Regularity Loss (0,....,0)......Page 108
3.1.4 Classification of Partial Differential Equations......Page 109
3.1.5 The Classical Classification of Partial Differential Equations......Page 113
3.1.6 Elliptic, Parabolic, Hyperbolic?......Page 123
3.1.7 Evolutionary Expressions in Canonical Form......Page 126
3.1.8 Functions of [xxx]v and Convolutions......Page 133
3.1.9 Systems and Scalar Equations......Page 140
3.1.10 Causality......Page 144
3.1.11 Initial Value Problems......Page 161
3.1.12.1 Transport Equation......Page 175
3.1.12.2 Acoustics......Page 179
3.1.12.3 Thermodynamics......Page 189
3.1.12.4 Electrodynamics......Page 190
3.1.12.5 Elastodynamics......Page 200
3.1.12.6 Fluid Dynamics......Page 214
3.1.12.7 Quantum Mechanics......Page 217
3.2.1 Extension of the Solution Theory to H-∞(Dv)......Page 224
3.2.2.1 Helmholtz Equation in R3......Page 242
3.2.2.2 Helmholtz Equation in R2......Page 245
3.2.2.4 Wave Equation in R2 (Method of Descent)......Page 248
3.2.2.5 Plane Waves......Page 251
3.2.2.6 Linearized Navier–Stokes Equations......Page 252
3.2.2.7 Electro- and Magnetostatics......Page 254
3.2.2.8 Force-free Magnetic Fields......Page 255
3.2.2.9 Beltrami Field Expansions......Page 257
3.2.3 Convolutions in H-∞(Dv)H, v∈Rn+1......Page 262
3.2.4.1 An Integral Representation of the Solution of the Transport Equation......Page 267
3.2.4.2 Potentials, Single and Double Layers in R3......Page 268
3.2.4.3 Electro- and Magnetostatics (Biot–Savart’s Law)......Page 294
3.2.4.4 Potential Theory in R2......Page 297
3.2.4.5 Cauchy’s Integral Formula......Page 300
3.2.4.6 Integral Representations of Solutions of the Helmholtz Equation in R3......Page 304
3.2.4.7 Retarded Potentials......Page 315
3.2.4.8 Integral Representations of Solutions of the Time-Harmonic Maxwell Equations......Page 316
4.1.1 Polynomials of Commuting Operators......Page 322
4.1.2 Polynomials of Commuting, Selfadjoint Operators......Page 323
4.2.1 Classification of Operator Polynomials with Time Differentiation......Page 324
4.2.2 Causality of Evolutionary Problems......Page 327
4.2.3 Abstract Initial Value Problems......Page 335
4.2.4 Systems and Scalar Equations......Page 348
4.2.5 First-Order-in-Time Evolution Equations in Sobolev Lattices......Page 353
5.1.1 The Selfadjoint Laplace Operator......Page 357
5.1.2.1 Bounded Perturbations......Page 361
5.1.2.2 Relatively Bounded Perturbations (the Coulomb Potential)......Page 365
5.2 Heat Equation......Page 368
5.2.1 TheSelfadjointOperatorCase......Page 369
5.2.1.1 Prescribed Dirichlet and Neumann Boundary Data......Page 376
5.2.1.2 Transmission Initial Boundary Value Problem......Page 385
5.2.2 Stefan Boundary Condition......Page 388
5.2.3 LowerOrderPerturbations......Page 390
5.3 Acoustics......Page 391
5.3.1 Dirichlet and Neumann Boundary Condition......Page 393
5.3.2 Wave Equation......Page 396
5.3.3 ReversibleHeatTransport......Page 401
5.4 Electrodynamics......Page 404
5.4.1 The Electric Boundary Condition......Page 406
5.4.2 Some Decomposition Results......Page 410
5.4.3 TheExtendedMaxwellSystem......Page 413
5.4.4 The Vectorial Wave Equation for the Electromagnetic Field......Page 419
5.5 Elastodynamics......Page 426
5.5.1 The Rigid Boundary Condition......Page 428
5.5.2 Free Boundary Condition......Page 431
5.5.3 Shear andPressureWaves......Page 432
5.6 Plate Dynamics......Page 434
5.7 Thermo-Elasticity......Page 439
6 A "Royal Road" to Initial Boundary Value Problems......Page 445
6.1 A Class of Evolutionary Material Laws......Page 446
6.2.1 The Shape of Evolutionary Problems with Material Laws......Page 450
6.2.2 Some Special Cases......Page 456
6.2.3 Material Laws via Differential Equations......Page 459
6.2.4 CoupledSystems......Page 460
6.2.5 Initial Value Problems......Page 462
6.2.6 MemoryProblems......Page 466
6.3 Some Applications......Page 472
6.3.1 ReversibleHeatTransfer......Page 473
6.3.2 Models of Thermoelasticity......Page 474
6.3.3 Thermo-Piezo-Electro-Magnetism......Page 476
Conclusion......Page 478
Bibliography......Page 480
Index......Page 484