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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