Boundary Integral Equations

Preface to the Second Edition
Preface to the First Edition
Acknowledgements
Table of Contents
1. Introduction
1.1 The Green Representation Formula
1.2 Boundary Potentials and Calderón’s Projector
1.3 Boundary Integral Equations
1.3.1 The Dirichlet Problem
1.3.2 The Neumann Problem
1.4 Exterior Problems
1.4.1 The Exterior Dirichlet Problem
1.4.2 The Exterior Neumann Problem
1.5 Remarks
2. Boundary Integral Equations
2.1 The Helmholtz Equation
2.1.1 Low Frequency Behaviour
2.2 The Lamé System
2.2.1 The Interior Displacement Problem
2.2.2 The Interior Traction Problem
2.2.3 Some Exterior Fundamental Problems
2.2.4 The Incompressible Material
2.3 The Stokes Equations
2.3.1 Hydrodynamic Potentials
2.3.2 The Stokes Boundary Value Problems
2.3.3 The Incompressible Material — Revisited
2.4 The Biharmonic Equation
2.4.1 Calderón’s Projector
2.4.2 Boundary Value Problems and Boundary Integral Equations
2.5 Remarks
3. Representation Formulae, Local Coordinates and Direct Boundary Integral Equations
3.1 Classical Function Spaces and Distributions
3.2 Hadamard’s Finite Part Integrals
3.3 Local Coordinates
3.4 Short Excursion to Elementary Differential Geometry
3.4.1 Second Order Differential Operators in Divergence Form
3.5 Distributional Derivatives and Abstract Green’s Second Formula
3.6 The Green Representation Formula
3.7 Green’s Representation Formulae in Local Coordinates
3.8 Multilayer Potentials
3.9 Direct Boundary Integral Equations
3.9.1 Boundary Value Problems
3.9.2 Transmission Problems
3.10 Remarks
4. Sobolev Spaces
4.1 The Spaces Hs(Ω)
4.2 The Trace Spaces Hs(Γ)
4.2.1 Trace Spaces for Periodic Functions on a Smooth Curve in IR
4.2.2 Trace Spaces on Curved Polygons in IR
4.3 The Trace Spaces on an Open Surface
4.4 The Weighted Sobolev Spaces Hm(Ωc; λ) and Hm(IRn; λ)
4.5 Function Spaces H( div ,Ω) and H( curl,Ω)
5. Variational Formulations
5.1 Partial Differential Equations of Second Order
5.1.1 Interior Problems
5.1.2 Exterior Problems
5.1.3 Transmission Problems
5.2 Abstract Existence Theorems for Variational Problems
5.2.1 The Lax–Milgram Theorem
5.3 The Fredholm–Nikolski Theorems
5.3.1 Fredholm’s Alternative
5.3.2 The Riesz–Schauder and the Nikolski Theorems
5.3.3 Fredholm’s Alternative for Sesquilinear Forms
5.3.4 Fredholm Operators
5.4 Gårding’s Inequality for Boundary Value Problems
5.4.1 Gårding’s Inequality for Second Order Strongly Elliptic Equations in Ω
5.4.2 The Stokes System
5.4.3 Gårding’s Inequality for Exterior Second Order Problems
5.4.4 Gårding’s Inequality for Second Order Transmission Problems
5.5 Existence of Solutions to Strongly Elliptic Boundary Value Problems
5.5.1 Interior Boundary Value Problems
5.5.2 Exterior Boundary Value Problems
5.5.3 Transmission Problems
5.6 Solutions of Certain Boundary Integral Equations and Associated Boundary Value Problems
5.6.1 The Generalized Representation Formula for Second Order Systems
5.6.2 Continuity of Some Boundary Integral Operators
5.6.3 Continuity Based on Finite Regions
5.6.4 Continuity of Hydrodynamic Potentials
5.6.5 The Equivalence Between Boundary Value Problems and Integral Equations
5.6.6 Variational Formulation of Direct Boundary Integral Equations
5.6.7 Positivity and Contraction of Boundary Integral Operators
5.6.8 The Solvability of Direct Boundary Integral Equations
5.6.9 Positivity of the Boundary Integral Operators of the Stokes System
5.7 Partial Differential Equations of Higher Order
5.8 Remarks
5.8.1 Assumptions on Γ
5.8.2 Higher Regularity of Solutions
5.8.3 Mixed Boundary Conditions and Crack Problem
6. Electromagnetic Fields
6.1 Introduction
6.2 Maxwell Equations
6.3 Constitutive Equations
6.4 Time Harmonic Fields
6.4.1 Plane waves
6.5 Electromagnetic potentials
6.6 Transmission and Boundary Conditions
6.7 Boundary Value Problems
6.7.1 Scattering problems
6.7.2 Eddy current problems
6.8 Uniqueness
6.8.1 The cavity problem
6.8.2 Exterior problems
6.8.3 The transmission problem
6.9 Representation Formulae
6.10 Boundary Integral Equations for Electromagnetic fields
6.10.1 The Calderon projector and the capacity operators
6.10.2 Weak solutions for a fundamental problem
6.10.2.1 Interior Dirichlet problem in Ω.
6.10.2.2 A reduction to boundary integral equations.
6.11 Application of the Electromagnetic Potentials to Eddy Current Problems
6.11.1 The ’(A, ϕ) − (A) − (ψ)’ formulation in the bounded domain
6.11.2 The ’(A, ϕ) − (ψ)’ formulation in an unbounded domain
6.11.3 Electric field in the dielectric domain ΩD.
6.11.4 Vector potentials — revisited
6.12 Applications of boundary integral equations to scattering problems
6.12.1 Scattering by a perfect electric conductor, EFIE and MFIE
6.12.2 Scattering by a dielectric body
6.12.3 Scattering by objects with impedance boundary conditions
7. Introduction to Pseudodifferential Operators
7.1 Basic Theory of Pseudodifferential Operators
7.2 Elliptic Pseudodifferential Operators on Ω ⊂ IRn
7.2.1 Systems of Pseudodifferential Operators
7.2.2 Parametrix and Fundamental Solution
7.2.3 Levi Functions for Scalar Elliptic Equations
7.2.4 Levi Functions for Elliptic Systems
7.2.5 Strong Ellipticity and Gårding’s Inequality
7.3 Review on Fundamental Solutions
7.3.1 Local Fundamental Solutions
7.3.2 Fundamental Solutions in IRn for Operators with Constant
Coefficients
7.3.3 Existing Fundamental Solutions in Applications
8. Pseudodifferential Operators as Integral Operators
8.1 Pseudohomogeneous Kernels
8.1.1 Integral Operators as Pseudodifferential Operators of Negative Order
8.1.2 Non–Negative Order Pseudodifferential Operators as Hadamard Finite Part Integral Operators
8.1.3 Parity Conditions
8.1.4 A Summary of the Relations between Kernels and Symbols
8.2 Coordinate Changes and Pseudohomogeneous Kernels
8.2.1 The Transformation of General Hadamard Finite Part Integral Operators under Change of Coordinates
8.2.2 The Class of Invariant Hadamard Finite Part Integral Operators under Change of Coordinates
9. Pseudodifferential and Boundary Integral Operators
9.1 Pseudodifferential Operators on Boundary Manifolds
9.1.1 Ellipticity on Boundary Manifolds
9.1.2 Schwartz Kernels on Boundary Manifolds
9.2 Boundary Operators Generated by Domain Pseudodifferential Operators
9.3 Surface Potentials on the Plane IRn−1
9.4 Pseudodifferential Operators with Symbols of Rational Type
9.5 Surface Potentials on the Boundary Manifold Γ
9.6 Volume Potentials
9.7 Strong Ellipticity and Fredholm Properties
9.8 Strong Ellipticity of Boundary Value Problems and Associated Boundary Integral Equations
9.8.1 The Boundary Value and Transmission Problems
9.8.2 The Associated Boundary Integral Equations of the First Kind
9.8.3 The Transmission Problem and Gårding’s inequality
9.9 Remarks
10. Integral Equations on Γ ⊂ IR3 Recast as
Pseudodifferential Equations
10.1 Newton Potential Operators for Elliptic Partial Differential Equations and Systems
10.1.1 Generalized Newton Potentials for the Helmholtz Equation
10.1.2 The Newton Potential for the Lamé System
10.1.3 The Newton Potential for the Stokes System
10.2 Surface Potentials for Second Order Equations
10.2.1 Strongly Elliptic Differential Equations
10.2.2 Surface Potentials for the Helmholtz Equation
10.2.3 Surface Potentials for the Lamé System
10.2.4 Surface Potentials for the Stokes System
10.3 Invariance of Boundary Pseudodifferential Operators
10.3.1 The Hypersingular Boundary Integral Operators for the Helmholtz Equation
10.3.2 The Hypersingular Operator for the Lamé System
10.3.3 The Hypersingular Operator for the Stokes System
10.4 Derivatives of Boundary Potentials
10.4.1 Derivatives of the Solution to the Helmholtz Equation
10.4.2 Computation of Stress and Strain on the Boundary for the Lamé System
10.5 Remarks
11. Boundary Integral Equations on Curves in IR2
11.1 Representation of the basic operators for the 2D–Laplacian in terms of Fourier series
11.2 The Fourier Series Representation of Periodic Operators A ∈ L m cl(Γ)
11.3 Ellipticity Conditions for Periodic Operators on Γ
11.3.1 Scalar Equations
11.3.2 Systems of Equations
11.3.3 Multiply Connected Domains
11.4 Fourier Series Representation of some Particular Operators
11.4.1 The Helmholtz Equation
11.4.2 The Lamé System
11.4.3 The Stokes System
11.4.4 The Biharmonic Equation
11.5 Remarks
12. Remarks on Pseudodifferential Operators
Related to the Time Harmonic Maxwell
Equations
12.1 Introduction
12.2 Symbols of P and the corresponding Newton potentials
12.3 Representation formulae
12.4 Symbols of the Electromagnetic Boundary Potentials
12.5 Symbols of boundary integral operators
12.6 Symbols of the Capacity Operators
12.7 Boundary Integral Operators for the Fundamental Boundary Value Problems
12.8 Coerciveness and Strong Ellipticity
12.9 Gårding’s inequality for the sesquilinear form A in (6.12.23)
12.10 Existence Theorem 6.12.6 revisited
12.11 Concluding Remarks
A. Differential Operators in Local Coordinates
with Minimal Differentiability
B. Vector Identities, Integration Formulae and
Surface Differential Operators
References
Index