| Preface | 7 |
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| Table of Contents | 13 |
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| 1. Introduction | 18 |
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| 1.1 The Green Representation Formula | 18 |

| 1.2 Boundary Potentials and Calder´ons Projector | 20 |

| 1.3 Boundary Integral Equations | 27 |

| 1.4 Exterior Problems | 30 |

| 1.5 Remarks | 36 |

| 2. Boundary Integral Equations | 41 |
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| 2.1 The Helmholtz Equation | 41 |

| 2.2 The Lam´e System | 61 |

| 2.3 The Stokes Equations | 78 |

| 2.4 The Biharmonic Equation | 95 |

| 2.5 Remarks | 107 |

| 3. Representation Formulae, Local Coordinates and Direct Boundary Integral Equations | 111 |
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| 3.1 Classical Function Spaces and Distributions | 111 |

| 3.2 Hadamards Finite Part Integrals | 117 |

| 3.3 Local Coordinates | 124 |

| 3.4 Short Excursion to Elementary Differential Geometry | 127 |

| 3.5 Distributional Derivatives and Abstract Greens Second Formula | 142 |

| 3.6 The Green Representation Formula | 146 |

| 3.7 Greens Representation Formulae in Local Coordinates | 151 |

| 3.8 Multilayer Potentials | 155 |

| 3.9 Direct Boundary Integral Equations | 161 |

| 3.10 Remarks | 173 |

| 4. Sobolev Spaces | 175 |
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| 4.1 The Spaces Hs(O) | 175 |

| 4.2 The Trace Spaces Hs(G) | 185 |

| 4.3 The Trace Spaces on an Open Surface | 205 |

| 4.4 The Weighted Sobolev Spaces Hm(Oc | .)and Hm(IRn .) |

| 5.1 Partial Differential Equations of Second Order | 211 |

| 5.2 Abstract Existence Theorems for Variational Problems | 234 |

| 5.3 The FredholmNikolski Theorems | 242 |

| 5.4 G°ardings Inequality for Boundary Value Problems | 259 |

| 5.5 Existence of Solutions to Strongly Elliptic Boundary Value Problems | 275 |

| 5.6 Solutions of Certain Boundary Integral Equations and Associated Boundary Value Problems | 281 |

| 5.7 Partial Differential Equations of Higher Order | 309 |

| 5.8 Remarks | 315 |

| 6. Introduction to Pseudodifferential Operators | 319 |
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| 6.1 Basic Theory of Pseudodifferential Operators | 319 |

| 6.2 Elliptic Pseudodifferential Operators on O IRn | 342 |

| 6.3 Review on Fundamental Solutions | 362 |

| 7. Pseudodifferential Operators as Integral Operators | 369 |
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| 7.1 Pseudohomogeneous Kernels | 369 |

| 7.2 Coordinate Changes and Pseudohomogeneous Kernels | 410 |

| 8. Pseudodifferential and Boundary Integral Operators | 429 |
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| 8.1 Pseudodifferential Operators on Boundary Manifolds | 430 |

| 8.2 Boundary Operators Generated by Domain Pseudodifferential Operators | 437 |

| 8.3 Surface Potentials on the Plane IRn 1 | 439 |

| 8.4 Pseudodifferential Operators with Symbols of Rational Type | 462 |

| 8.5 Surface Potentials on the Boundary Manifold G | 483 |

| 8.6 Volume Potentials | 492 |

| 8.7 Strong Ellipticity and Fredholm Properties | 495 |

| 8.8 Strong Ellipticity of Boundary Value Problems and Associated Boundary Integral Equations | 501 |

| 8.9 Remarks | 507 |

| 9. Integral Equations on G IR3 Recast as Pseudodifferential Equations | 509 |
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| 9.1 Newton Potential Operators for Elliptic Partial Differential Equations and Systems | 515 |

| 9.2 Surface Potentials for Second Order Equations | 523 |

| 9.3 Invariance of Boundary Pseudodifferential Operators | 540 |

| 9.4 Derivatives of Boundary Potentials | 551 |

| 9.5 Remarks | 563 |

| 10. Boundary Integral Equations on Curves in IR2 | 564 |
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| 10.1 Representation of the basic operators for the 2D Laplacian in terms of Fourier series | 565 |

| 10.2 The Fourier Series Representation of Periodic Operators A Lmc(G) | 571 |

| 10.3 Ellipticity Conditions for Periodic Operators on G | 577 |

| 10.4 Fourier Series Representation of some Particular Operators | 589 |

| 10.5 Remarks | 606 |

| A. Differential Operators in Local Coordinates with Minimal Differentiability | 608 |
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| References | 614 |
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| Index | 628 |