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XVIII Table of Contents

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
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9. Integral Equations on Γ ⊂ IR 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