## Boundary Integral EquationsThis book is devoted to the mathematical foundation of boundary integral equations. The combination of ?nite element analysis on the boundary with these equations has led to very e?cient computational tools, the boundary element methods (see e.g., the authors [139] and Schanz and Steinbach (eds.) [267]). Although we do not deal with the boundary element discretizations in this book, the material presented here gives the mathematical foundation of these methods. In order to avoid over generalization we have con?ned ourselves to the treatment of elliptic boundary value problems. The central idea of eliminating the ?eld equations in the domain and - ducing boundary value problems to equivalent equations only on the bou- ary requires the knowledge of corresponding fundamental solutions, and this idea has a long history dating back to the work of Green [107] and Gauss [95, 96]. Today the resulting boundary integral equations still serve as a major tool for the analysis and construction of solutions to boundary value problems. |

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### Contents

58 Remarks | 299 |

583 Mixed Boundary Conditions and Crack Problem | 300 |

6 Introduction to Pseudodifferential Operators | 303 |

62 Elliptic Pseudodifferential Operators on 𝛺 C IR | 326 |

621 Systems of Pseudodifferential Operators | 328 |

622 Parametrix and Fundamental Solution | 331 |

623 Levi Functions for Scalar Elliptic Equations | 334 |

624 Levi Functions for Elliptic Systems | 341 |

Boundary Integral Equations | 25 |

211 Low Frequency Behaviour | 31 |

22 The Lamé System | 45 |

221 The Interior Displacement Problem | 47 |

222 The Interior Traction Problem | 55 |

223 Some Exterior Fundamental Problems | 56 |

224 The Incompressible Material | 61 |

23 The Stokes Equations | 62 |

231 Hydrodynamic Potentials | 65 |

232 The Stokes Boundary Value Problems | 66 |

233 The Incompressible Material Revisited | 75 |

24 The Biharmonic Equation | 79 |

241 Calderóns Projector | 83 |

242 Boundary Value Problems and Boundary Integral Equations | 85 |

25 Remarks | 91 |

3 Representation Formulae | 95 |

32 Hadamards Finite Part Integrals | 101 |

33 Local Coordinates | 108 |

34 Short Excursion to Elementary Differential Geometry | 111 |

341 Second Order Differential Operators in Divergence Form | 119 |

35 Distributional Derivatives and Abstract Greens Second Formula | 126 |

36 The Green Representation Formula | 130 |

37 Greens Representation Formulae in Local Coordinates | 135 |

38 Multilayer Potentials | 139 |

39 Direct Boundary Integral Equations | 145 |

392 Transmission Problems | 155 |

310 Remarks | 157 |

4 Sobolev Spaces | 159 |

42 The Trace Spaces HГ | 169 |

421 Trace Spaces for Periodic Functions on a Smooth Curve in IR2 | 181 |

422 Trace Spaces on Curved Polygons in IR2 | 185 |

43 The Trace Spaces on an Open Surface | 189 |

44 Weighted Sobolev Spaces | 191 |

5 Variational Formulations | 195 |

511 Interior Problems | 199 |

512 Exterior Problems | 204 |

513 Transmission Problems | 215 |

52 Abstract Existence Theorems for Variational Problems | 218 |

521 The LaxMilgram Theorem | 219 |

53 The FredholmNikolski Theorems | 226 |

532 The RieszSchauder and the Nikolski Theorems | 235 |

533 Fredholms Alternative for Sesquilinear Forms | 240 |

534 Fredholm Operators | 241 |

54 Gardings Inequality for Boundary Value Problems | 243 |

542 The Stokes System | 250 |

543 Gardings Inequality for Exterior Second Order Problems | 254 |

544 Gardings Inequality for Second Order Transmission Problems | 259 |

551 Interior Boundary Value Problems | 260 |

552 Exterior Boundary Value Problems | 264 |

56 Solutions of Integral Equations via Boundary Value Problems | 265 |

562 Continuity of Some Boundary Integral Operators | 267 |

563 Continuity Based on Finite Regions | 270 |

564 Continuity of Hydrodynamic Potentials | 272 |

565 The Equivalence Between Boundary Value Problems and Integral Equations | 274 |

566 Variational Formulation of Direct Boundary Integral Equations | 277 |

567 Positivity and Contraction of Boundary Integral Operators | 287 |

568 The Solvability of Direct Boundary Integral Equations | 291 |

569 Positivity of the Boundary Integral Operators of the Stokes System | 292 |

57 Partial Differential Equations of Higher Order | 293 |

625 Strong Ellipticity and Gardings Inequality | 343 |

63 Review on Fundamental Solutions | 346 |

631 Local Fundamental Solutions | 347 |

632 Fundamental Solutions in IR for Operators with Constant Coefficients | 348 |

633 Existing Fundamental Solutions in Applications | 352 |

7 Pseudodifferential Operators as Integral Operators | 353 |

711 Integral Operators as Pseudodifferential Operators of Negative Order | 356 |

712 NonNegative Order Pseudodifferential Operators as Hadamard Finite Part Integral Operators | 380 |

713 Parity Conditions | 389 |

714 A Summary of the Relations between Kernels and Symbols | 392 |

72 Coordinate Changes and Pseudohomogeneous Kernels | 394 |

721 The Transformation of General Hadamard Finite Part Integral Operators under Change of Coordinates | 397 |

722 The Class of Invariant Hadamard Finite Part Integral Operators under Change of Coordinates | 404 |

8 Pseudodifferential and Boundary Integral Operators | 413 |

81 Pseudodifferential Operators on Boundary Manifolds | 414 |

811 Ellipticity on Boundary Manifolds | 418 |

812 Schwartz Kernels on Boundary Manifolds | 420 |

82 Boundary Operators Generated by Domain Pseudodifferential Operators | 421 |

83 Surface Potentials on the Plane IR | 423 |

84 Pseudodifferential Operators with Symbols of Rational Type | 446 |

85 Surface Potentials on the Boundary Manifold Г | 467 |

86 Volume Potentials | 476 |

87 Strong Ellipticity and Fredholm Properties | 479 |

88 Strong Ellipticity of Boundary Value Problems and Associated Boundary Integral Equations | 485 |

882 The Associated Boundary Integral Equations of the First Kind | 488 |

883 The Transmission Problem and Gardings inequality | 489 |

89 Remarks | 491 |

9 Integral Equations on Γ IR3 Recast as Pseudodifferential Equations | 493 |

91 Newton Potential Operators for Elliptic Partial Differential Equations and Systems | 499 |

911 Generalized Newton Potentials for the Helmholtz Equation | 502 |

912 The Newton Potential for the Lamé System | 505 |

913 The Newton Potential for the Stokes System | 506 |

92 Surface Potentials for Second Order Equations | 507 |

921 Strongly Elliptic Differential Equations | 510 |

922 Surface Potentials for the Helmholtz Equation | 514 |

923 Surface Potentials for the Lamé System | 519 |

924 Surface Potentials for the Stokes System | 524 |

931 The Hypersingular Boundary Integral Operators for the Helmholtz Equation | 525 |

932 The Hypersingular Operator for the Lamé System | 531 |

933 The Hypersingular Operator for the Stokes System | 535 |

941 Derivatives of the Solution to the Helmholtz Equation | 541 |

942 Computation of Stress and Strain on the Boundary for the Lamé System | 543 |

95 Remarks | 547 |

10 Boundary Integral Equations on Curves in IR˛ | 548 |

101 Fourier Series Representation of the Basic Operators | 550 |

102 The Fourier Series Representation of Periodic Operators | 556 |

103 Ellipticity Conditions for Periodic Operators on Г | 562 |

1031 Scalar Equations | 563 |

1032 Systems of Equations | 568 |

1033 Multiply Connected Domains | 572 |

104 Fourier Series Representation of some Particular Operators | 574 |

1042 The Lamé System | 578 |

1043 The Stokes System | 581 |

1044 The Biharmonic Equation | 582 |

105 Remarks | 591 |

A Differential Operators in Local Coordinates with Minimal Differentiability | 593 |

References | 599 |

613 | |

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### Common terms and phrases

boundary conditions boundary integral equations boundary integral operators boundary potentials boundary value problems Cauchy data classical coefficients consider constant continuous corresponding defined definition denote derivatives Dirichlet problem domain double layer potential eigensolutions elliptic system exists extension conditions exterior Dirichlet finite part integral Fredholm Fredholm operator fundamental solution Gĺrding's inequality given Green's formula Helmholtz equation Hence Hilbert space hypersingular implies IR'n jump relations Lamé system Laplacian Lemma linear Lipschitz domain mapping properties matrix Moreover Neumann problem Newton potential norm obtain parity conditions principal symbol Proof properly supported pseudodifferential operators pseudohomogeneous rational type representation formula respectively right-hand side satisfies scalar Schwartz kernel Section sesquilinear form singular Sobolev spaces Stokes system strongly elliptic supp symbols of rational tangential differential operators transmission problems Tricomi conditions variational formulation XL XL

### References to this book

Applied Functional Analysis: Main Principles and Their Applications Eberhard Zeidler Limited preview - 1995 |

Applied Functional Analysis: Main Principles and Their Applications Eberhard Zeidler No preview available - 1995 |