Preface
Contents
1 Introduction
1.1 An Example
1.2 A Selection of Problems
1.2.1 Perturbation Problems for the Riemann Map
1.2.2 Linear Elliptic Boundary Value Problems
1.2.3 Eigenvalues Problems
1.2.4 Nonlinear Boundary Value Problems
1.2.5 Problems in Periodic Domains
1.2.6 Different Boundary Perturbations
1.2.7 Perturbation Results for Integral Operators
1.3 Structure of the Book
2 Preliminaries
2.1 Basic Notation
2.2 Preliminaries of Linear Functional Analysis
2.3 Spaces of Classically Differentiable Functions
2.4 Distributions and Weak Derivatives
2.5 Real Analytic Functions and Spaces of Real Analytic Functions
2.6 Spaces of Hölder and Lipschitz Continuous Functions
2.7 Coordinate Cylinders and Local Strict Hypographs
2.8 Tangent Space to a Local Strict Hypograph
2.9 Lipschitz Subsets of Rn
2.10 Elementary Inequalities on the Boundary of a LipschitzSubset of Rn
2.11 Schauder Spaces in Open Subsets of Rn
2.12 Composition of Functions in Schauder Spaces
2.13 Local Strict Hypographs of a Schauder Class
2.14 Extendibility of Functions of Schauder Spaces on an Open Subset of Class Cm,α
2.15 On the Extendibility of Continuous Functions to the Closure of Open Sets of Class C1
2.16 A Consequence of the Rule of Change of Variables for Diffeomorphisms
2.17 A Fundamental Inequality of the Unit Normal on the Boundary of a Set of Class C1,α
2.18 Existence of Tubular Neighborhoods of the Boundary of Bounded Open Sets
2.19 A Sufficient Condition for the Hölder Continuity of Continuously Differentiable Functions, in the Wake of the Work of CarloMiranda
2.20 Schauder Spaces on a Compact Manifold and on the Boundary of a Bounded Open Subset of Rn
2.21 Tangential Derivatives
2.22 Schauder Spaces in Open Subsets of Rn, a Case of a Negative Exponent
3 Preliminaries on Harmonic Functions
3.1 Basic Properties of Harmonic Functions
3.2 A Fundamental Solution for the Laplace Operator
3.3 Isolated Singularities of Harmonic Functions
3.4 Behavior at Infinity of Harmonic Functions
4 Green Identities and Layer Potentials
4.1 Green Identities for Bounded Domains
4.2 Green Identities for Harmonic Functions on Exterior Domains
4.3 Preliminaries on Singular Integrals and Layer Potentials
4.4 The Single Layer Potential
4.5 The Double Layer Potential
4.6 A Regularizing Property of the Double Layer Potential on the Boundary
5 Preliminaries on the Fredholm Alternative Principle
5.1 Fredholm Alternative
5.2 Fredholm Alternative in a Dual System
6 Boundary Value Problems and Boundary Integral Operators
6.1 The Geometric Setting
6.2 The Dirichlet and Neumann Boundary Value Problems
6.3 Uniqueness for the Interior and Exterior Dirichlet and Neumann Boundary…
6.4 The Boundary Integral Operators Associated to the Single and Double Layer Potentials
6.5 The Null Spaces of 12I+WΩ and 12I+WtΩ
6.6 The Null Spaces of -12I+WΩ and -12I+WtΩ
6.7 The Dirichlet Problem in Ω
6.8 The Dirichlet Problem in Ω-
6.9 The Neumann Problem in Ω and Ω-
6.10 Further Mapping Properties of VΩ
6.11 A Mixed Boundary Value Problem
6.12 The Operators I+λWΩ and I+λWtΩ
6.13 A Linear Transmission Problem
6.14 A Robin Problem
7 Poisson Equation and Volume Potentials
7.1 Preliminary Remarks on the Poisson Equation
7.2 Volume Potentials
7.2.1 Volume Potentials with Weakly Singular Kernels
7.2.2 Volume Potentials with Kernels Which are Weakly Singular Together with Their First OrderPartial Derivatives
7.2.3 Volume Potentials with Singular Kernels and with a Constant Density
7.2.4 Volume Potentials with Kernels Which are Weakly Singular and Which Have a Strong Singularity in the First Order Partial Derivatives
7.2.5 The Newtonian Potential in Schauder Spaces
7.2.6 Volume Potentials in Roumieu Classes
7.3 Boundary Value Problems for the Poisson Equation in Schauder Spaces
7.3.1 The Interior Dirichlet Problem for the Poisson Equation in Schauder Spaces
7.3.2 The Interior Neumann Problem for the Poisson Equation in Schauder Spaces
7.3.3 The Interior Robin Problem for the Poisson Equation in Schauder Spaces
8 A Dirichlet Problem in a Domain with a Small Hole
8.1 The Geometric Setting
8.2 A Dirichlet Problem for the Laplace Equation
8.3 Analysis for n≥3
8.4 Analysis for n=2
8.4.1 Analysis of System (8.32)
8.4.2 Analysis of System (8.33)
8.4.3 Real Analytic Representation of the Map ε→uε.
8.4.4 Some Remarks on the Logarithmic Behavior
8.5 How to Compute the Coefficients (in Dimension 2)
8.5.1 Series Expansions of (Φi[ε],Φo[ε]) and (Ψi[ε],Ψo[ε])
8.5.2 Series Expansion of uε
8.5.3 Principal Terms in the Series Expansion of uε
8.5.4 Series Expansion for the Energy of uε
8.5.5 Series Expansions in a Circular Annulus
9 Other Problems with Linear Boundary Conditions in a Domain with a Small Hole
9.1 The Geometric Setting
9.2 A Mixed Boundary Value Problem for the Laplace Equation
9.3 A Mixed Boundary Value Problem for the Poisson Equation
9.4 A Steklov Eigenvalue Problem
9.4.1 Some Basic Facts on Steklov Eigenvaluesand Eigenfunctions
9.4.2 Formulation of the Steklov Problem (9.31) in Terms of Integral Equations
9.4.3 Real Analytic Representations for the Simple Steklov Eigenvalues and Eigenfunctions
10 A Dirichlet Problem in a Domain with Two Small Holes
10.1 The Geometric Setting
10.2 A Dirichlet Problem in Ω(ε1,ε2)
10.3 Close and Moderately Close Holes in Dimension n≥3
10.3.1 Moderately Close Holes in Dimension n≥3
10.3.2 Close Holes in Dimension n≥3
10.4 Moderately Close Holes in Dimension n=2
10.4.1 Integral Representation of the Solution
10.4.2 Analysis of System (10.39)
10.4.3 Analysis of System (10.40)
10.4.4 The Auxiliary Functions HξΩ1, HξΩ2, and HΩox
10.4.5 Representation of uε1,ε2 in Terms of Analytic Maps
10.4.6 Asymptotic Behavior of uε1,ε2 as (ε1,ε2)→(0,0)
11 Nonlinear Boundary Value Problems in Domains witha Small Hole
11.1 The Geometric Setting
11.2 A Nonlinear Robin Problem
11.2.1 Formulation of a Nonlinear Robin Problem in Terms of Integral Equations
11.2.2 Formulation of Problems (11.1) and (11.2) in Terms of Integral Equations
11.2.3 Analytic Representation for the Family { u(ε,·)}ε]0,ε'[
11.2.4 Local Uniqueness of the Family { u(ε,·)}ε]0,ε0[
11.2.5 Analytic Representation for the Energy Integral of the Family { u(ε,·)}ε]0,ε'[
11.3 A Nonlinear Transmission Problem
11.3.1 Formulation of the Nonlinear Transmission Problem in Terms of Integral Equations
11.3.2 Analytic Representation for the Family of Solutions {(ui(ε,·),uo(ε,·))}ε]0,ε'[
11.3.3 A Property of Local Uniqueness for the Family of Solutions {(ui(ε,·),uo(ε,·))}ε]0,ε'[
11.3.4 Analytic Representation for the Energy Integrals of the Family of Solutions {(ui(ε,·),uo(ε,·))}ε]0,ε'[
12 Boundary Value Problems in Periodic Domains, a Potential Theoretic Approach
12.1 A Periodic Analog of the Fundamental Solution
12.2 Periodic Layer Potentials for the Laplace Equation
12.2.1 Geometric Setting
12.2.2 Definition and Properties of the Periodic Layer Potentials
12.3 Uniqueness Results for Periodic Boundary Value Problems
12.4 Mapping Properties of 12I+Wq, ΩQ and 12I+Wtq, ΩQ
12.5 Existence Results for Periodic Boundary Value Problems
13 Singular Perturbation Problems in Periodic Domains
13.1 Introduction
13.2 The Geometric Setting
13.3 Perturbed Problems in Periodic Domains
13.4 Preliminaries and Notation
13.5 Asymptotic Behavior of the Longitudinal Flow
13.5.1 Asymptotic Behavior of ΣII[ε]
13.6 A Singularly Perturbed Non-ideal Transmission Problem
13.6.1 Transmission Problems with Non-ideal ContactConditions
13.6.2 Formulation of the Singularly Perturbed Transmission Problem in Terms of Integral Equations
13.6.3 A Functional Analytic Representation Theorem for the Solutions of the Singularly Perturbed TransmissionProblem
13.6.4 A Functional Analytic Representation Theorem for the Effective Conductivity
13.7 Series Expansion for the Effective Conductivity
13.7.1 Preliminaries
13.7.2 Power Series Expansion for ρ(ε)1/r#
13.7.3 Power Series Expansions for ρ(ε)ε/r#
13.8 A Quasilinear Heat Transmission Problem
13.8.1 Introduction
13.8.2 An Equivalent Formulation of Problem (13.132)
13.8.3 Formulation of Problem (13.135) in Terms of Integral Equations
13.8.4 A Representation Theorem for the Family of Solutions of Problem (13.132)
Appendix A
A.1 The Homomorphism Theorem
A.2 The Inductive Topology
A.3 Lebesgue Number of an Open Cover
A.4 Perforated Connected Domains Are Connected
A.5 Measure Theory
A.6 Calculus in Banach Spaces and the Implicit Function Theorem
A.7 Composition Operators
A.8 Integral Operators with Real Analytic Kernel
A.9 Sard's Theorem
A.10 Theorem of Invariance of Domain
A.11 Mollifiers
A.12 The Partition of Unity
References
Index