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Table of Contents XVII

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
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6.2 Elliptic Pseudodifferential Operators on Ω ⊂ IR ........... 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
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6.3.2 Fundamental Solutions in IR 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