Boundary Integral Equations

Table of Contents
Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . VII
1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 The Green Representation Formula . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Boundary Potentials and Calder´on’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 Lam´e 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 Calder´on’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 . . . . . . . 111
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 H s (Ω). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
4.2 The Trace Spaces H s (Γ ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
4.2.1 Trace Spaces for Periodic Functions on a Smooth
Curve in IR
2
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181
4.2.2 Trace Spaces on Curved Polygons in IR
2
. . . . . . . . . . . . . 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 G˚arding’s Inequality for Boundary Value Problems . . . . . . . . . 243
5.4.1 G˚arding’s Inequality for Second Order Strongly
Elliptic Equations in Ω . . . . . . . . . . . . . . . . . . . . . . . . . . . 243
5.4.2 The Stokes System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250
5.4.3 G˚arding’s Inequality for Exterior Second Order
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254
5.4.4 G˚arding’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 Γ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 ⊂ IR
n
. . . . . . . . . . . 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 G˚arding’s Inequality . . . . . . . . . . 343
6.3 Review on Fundamental Solutions . . . . . . . . . . . . . . . . . . . . . . . . 346
6.3.1 Local Fundamental Solutions . . . . . . . . . . . . . . . . . . . . . . 347
6.3.2 Fundamental Solutions in IR
n
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 IR
n − 1
. . . . . . . . . . . . . . . . . . . . . 423
8.4 Pseudodifferential Operators with Symbols of Rational Type . 446
8.5 Surface Potentials on the Boundary Manifold Γ . . . . . . . . . . . . 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 G˚arding’s inequality . . 489
8.9 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 491
9. Integral Equations on Γ ⊂ IR
3
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 Lam´e 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 Lam´e 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 Lam´e 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 Lam´e System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 543
9.5 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 547
10. Boundary Integral Equations on Curves in IR
2
. . . . . . . . . . . . 549
10.1 Fourier Series Representation of the Basic Operators . . . . . . . . 550
10.2 The Fourier Series Representation of Periodic Operators
A ∈ L m
c
(Γ ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 556
10.3 Ellipticity Conditions for Periodic Operators on Γ . . . . . . . . . . 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 Lam´e 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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