Preface ....................................................... VII
1. Introduction ................................................. 1
1.1. The Green Representation Formula ........................ 1
1.2. Boundary Potentials and Calderon's Projector ............ 3
1.3. Boundary Integral Equations ............................ 10
1.3.1. The Dirichlet Problem ........................... 11
1.3.2. The Neumann Problem ............................. 12
1.4. Exterior Problems ...................................... 13
1.4.1. The Exterior Dirichlet Problem .................. 13
1.4.2. The Exterior Neumann Problem .................... 15
1.5. Remarks ................................................ 19
2. Boundary Integral Equations ................................. 25
2.1. The Helmholtz Equation ................................. 25
2.1.1. Low Frequency Behaviour ......................... 31
2.2. The Lame System ........................................ 45
2.2.1. The Interior Displacement Problem ............... 47
2.2.2. The Interior Traction Problem ................... 55
2.2.3. Some Exterior Fundamental Problems .............. 56
2.2.4. The Incompressible Material ..................... 61
2.3. The Stokes Equations ................................... 62
2.3.1. Hydrodynamic Potentials ......................... 65
2.3.2. The Stokes Boundary Value Problems .............. 66
2.3.3. The Incompressible Material - Revisited ......... 75
2.4. The Biharmonic Equation ................................ 79
2.4.1. Calderon's Projector ............................ 83
2.4.2. Boundary Value Problems and Boundary
Integral Equations .............................. 85
2.5. Remarks ................................................ 91
3. Representation Formulae ..................................... 95
3.1. Classical Function Spaces and Distributions ............ 95
3.2. Hadamard's Finite Part Integrals ...................... 101
3.3. Local Coordinates ..................................... 108
3.4. Short Excursion to Elementary Differential Geometry ... 1ll
3.4.1. Second Order Differential Operators
in Divergence Form ............................. 119
3.5. Distributional Derivatives and Abstract Green's
Second Formula ........................................ 126
3.6. The Green Representation Formula ...................... 130
3.7. Green's Representation Formulae in Local
Coordinates ........................................... 135
3.8. Multilayer Potentials ................................. 139
3.9. Direct Boundary Integral Equations .................... 145
3.9.1. Boundary Value Problems ........................ 145
3.9.2. Transmission Problems .......................... 155
3.10.Remarks ............................................... 157
4. Sobolev Spaces ............................................. 159
4.1. The Spaces HS(Ω) ...................................... 159
4.2. The Trace Spaces Нs(Г) ................................ 169
4.2.1. Trace Spaces for Periodic Functions on
a Smooth Curve in IR2 .......................... 181
4.2.2. Trace Spaces on Curved Polygons in IR2 ......... 185
4.3. The Trace Spaces on an Open Surface ................... 189
4.4. Weighted Sobolev Spaces ............................... 191
5. Variational Formulations ................................... 195
5.1. Partial Differential Equations of Second Order ........ 195
5.1.1. Interior Problems .............................. 199
5.1.2. Exterior Problems .............................. 204
5.1.3. Transmission Problems .......................... 215
5.2. Abstract Existence Theorems for Variational
Problems .............................................. 218
5.2.1. The Lax-Milgram Theorem ........................ 219
5.3. The Fredholm-Nikolski Theorems ........................ 226
5.3.1. Fredholm's Alternative ......................... 226
5.3.2. The Riesz-Schauder and the Nikolski Theorems ... 235
5.3.3. Fredholm's Alternative for Sesquilinear
Forms .......................................... 240
5.3.4. Fredholm Operators ............................. 241
5.4. Garding's Inequality for Boundary Value Problems ...... 243
5.4.1. Garding's Inequality for Second Order
Strongly Elliptic Equations in Ω .............. 243
5.4.2. The Stokes System .............................. 250
5.4.3. Garding's Inequality for Exterior Second
Order Problems ................................. 254
5.4.4. Garding's Inequality for Second Order
Transmission Problems .......................... 259
5.5. Existence of Solutions to Boundary Value Problems ..... 259
5.5.1. Interior Boundary Value Problems ............... 260
5.5.2. Exterior Boundary Value Problems ............... 264
5.5.3. Transmission Problems .......................... 264
5.6. Solution of Integral Equations via Boundary Value
Problems .............................................. 265
5.6.1. The Generalized Representation Formula for
Second Order Systems ........................... 265
5.6.2. Continuity of Some Boundary Integral
Operators ...................................... 267
5.6.3. Continuity Based on Finite Regions ............. 270
5.6.4. Continuity of Hydrodynamic Potentials .......... 272
5.6.5. The Equivalence Between Boundary Value
Problems and Integral Equations ................ 274
5.6.6. Variational Formulation of Direct Boundary
Integral Equations ............................. 277
5.6.7. Positivity and Contraction of Boundary
Integral Operators ............................. 287
5.6.8. The Solvability of Direct Boundary Integral
Equations ...................................... 291
5.6.9. Positivity of the Boundary Integral
Operators of the Stokes System ................. 292
5.7. Partial Differential Equations of Higher Order ........ 293
5.8. Remarks ............................................... 299
5.8.1. Assumptions on Г ............................... 299
5.8.2. Higher Regularity of Solutions ................. 299
5.8.3. Mixed Boundary Conditions and Crack Problem .... 300
6. Introduction to Pseudodifferential Operators ............... 303
6.1. Basic Theory of Pseudodifferential Operators .......... 303
6.2. Elliptic Pseudodifferential Operators on Ω ⊂ IRn ...... 326
6.2.1. Systems of Pseudodifferential Operators ........ 328
6.2.2. Parametrix and Fundamental Solution ............ 331
6.2.3. Levi Functions for Scalar Elliptic Equations ... 334
6.2.4. Levi Functions for Elliptic Systems ............ 341
6.2.5. Strong Ellipticity and Garding's Inequality .... 343
6.3. Review on Fundamental Solutions ....................... 346
6.3.1. Local Fundamental Solutions .................... 347
6.3.2. Fundamental Solutions in IRn for Operators
with Constant Coefficients ..................... 348
6.3.3. Existing Fundamental Solutions in
Applications ................................... 352
7. Pseudodifferential Operators as Integral Operators ......... 353
7.1. Pseudohomogeneous Kernels ............................. 353
7.1.1. Integral Operators as Pseudodifferential
Operators of Negative Order .................... 356
7.1.2. Non-Negative Order Pseudodifferential
Operators as Hadamard Finite Part Integral
Operators ...................................... 380
7.1.3. Parity Conditions .............................. 389
7.1.4. A Summary of the Relations between Kernels
and Symbols .................................... 392
7.2. Coordinate Changes and Pseudohomogeneous Kernels ...... 394
7.2.1. The Transformation of General Hadamard
Finite Part Integral Operators under Change
of Coordinates ................................. 397
7.2.2. The Class of Invariant Hadamard Finite Part
Integral Operators under Change of
Coordinates .................................... 404
8. Pseudodifferential and Boundary Integral Operators ......... 413
8.1. Pseudodifferential Operators on Boundary Manifolds .... 414
8.1.1. Ellipticity on Boundary Manifolds .............. 418
8.1.2. Schwartz Kernels on Boundary Manifolds ......... 420
8.2. Boundary Operators Generated by Domain
Pseudodifferential Operators .......................... 421
8.3. Surface Potentials on the Plane IRn-1 ................. 423
8.4. Pseudodifferential Operators with Symbols of
Rational Type ......................................... 446
8.5. Surface Potentials on the Boundary Manifold Г ......... 467
8.6. Volume Potentials ..................................... 476
8.7. Strong Ellipticity and Fredholm Properties ............ 479
8.8. Strong Ellipticity of Boundary Value Problems
and Associated Boundary Integral Equations ............ 485
8.8.1. The Boundary Value and Transmission Problems ... 485
8.8.2. The Associated Boundary Integral Equations
of the First Kind .............................. 488
8.8.3. The Transmission Problem and Garding's
inequality ..................................... 489
8.9. Remarks ............................................... 491
9. Integral Equations on Г ⊂ IR3 Recast as
Pseudodifferential Equations ............................... 493
9.1. Newton Potential Operators for Elliptic Partial
Differential Equations and Systems .................... 499
9.1.1. Generalized Newton Potentials for the
Helmholtz Equation ............................. 502
9.1.2. The Newton Potential for the Lame System ....... 505
9.1.3. The Newton Potential for the Stokes System ..... 506
9.2. Surface Potentials for Second Order Equations ......... 507
9.2.1. Strongly Elliptic Differential Equations ....... 510
9.2.2. Surface Potentials for the Helmholtz
Equation ....................................... 514
9.2.3. Surface Potentials for the Lame System ......... 519
9.2.4. Surface Potentials for the Stokes System ....... 524
9.3. Invariance of Boundary Pseudodifferential Operators ... 524
9.3.1. The Hypersingular Boundary Integral
Operators for the Helmholtz Equation ........... 525
9.3.2. The Hypersingular Operator for the Lame
System ......................................... 531
9.3.3. The Hypersingular Operator for the Stokes
System ......................................... 535
9.4. Derivatives of Boundary Potentials .................... 535
9.4.1. Derivatives of the Solution to the Helmholtz
Equation ....................................... 541
9.4.2. Computation of Stress and Strain on the
Boundary for the Lame System ................... 543
9.5. Remarks ............................................... 547
10.Boundary Integral Equations on Curves in IR2 ............... 549
10.1.Fourier Series Representation of the Basic
Operators ............................................. 550
10.2.The Fourier Series Representation of Periodic
Operators А ∈ L (Г) ................................. 556
10.3.Ellipticity Conditions for Periodic Operators on Г .... 562
10.3.1.Scalar Equations ............................... 563
10.3.2.Systems of Equations ........................... 568
10.3.3.Multiply Connected Domains ..................... 572
10.4.Fourier Series Representation of some Particular
Operators ............................................. 574
10.4.1.The Helmholtz Equation ......................... 574
10.4.2.The Lame System ................................ 578
10.4.3.The Stokes System .............................. 581
10.4.4.The Biharmonic Equation ........................ 582
10.5.Remarks .................................................. 591
A. Differential Operators in Local Coordinates with Minimal
Differentiability .......................................... 593
References .................................................... 599
Index ......................................................... 613
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