This book presents a systematic exposition of the theory of conformal mappings, boundary value problems for analytic and harmonic functions, and the relationship between the two subjects. It is suitable for use as an undergraduate or graduate level textbook, and exercises are included.
The first three chapters recount existence and uniqueness theorems of conformal mappings from simply and multiply connected domains to standard domains, some properties of analytic functions, harmonic functions and schlicht meromorphic functions, and representations of conformal mappings. In the remaining three chapters, the basic boundary value problems for analytic and harmonic functions are discussed in detail, including some new methods and results obtained by the author. For example, the Riemann-Hilbert boundary value problem with piecewise continuous coefficients in a multiply connected domain is covered in chapter five, and some irregular oblique derivative problems are treated in chapter six.
Readership: Graduate students as well as experts in theoretical and mathematical physics, differential and integral equations and mathematical analysis.
Author(s): Guo-Chun Wen
Series: Translations of Mathematical Monographs, Vol. 106
Publisher: American Mathematical Society
Year: 1992
Language: English
Pages: C+viii+303+B
Cover
S Title
Translations of Mathematical Monographs 106
Conformal Mappings and Boundary Value Problems
Copyright (c) 1992 by the American Mathematical Society
ISBN 0-8218-4562-4
QA360.W4613 1992 515.9-dc20
LCCN 92-14225 CIP
Contents
Preface
CHAPTER 1 Some Properties of Analytic and Harmonic Functions
§1. The convergence of sequences of analytic functions
§2. The convergence of sequences of harmonic functions
§3. Some properties of subharmonic functions
§4. The Dirichlet problem for analytic and harmonic functions
Exercises
CHAPTER 2 Conformal Mappings of Simply Connected Domains
§1. The fundamental theorem for conformal mappings on simply connected domains
§2. Boundary correspondence theorems for conformal mappings
§3. The distortion theorem and estimates of coefficients for univalent functions
§4. The convergence of conformal mappings for sequences of simply connected domains
§5. The representations of conformal mappings on polygonal domains
§6. The representations of conformal mappings with orthogonal polynomials
Exercises
CHAPTER 3 Conformal Mappings of Multiply Connected Domains
§1. A general discussion of conformal mappings between multiply connected domains
§2. Conformal mappings for domains with parallel slits
§3. Conformal mappings for domains with spiral slits
§4. The convergence of conformal mappings of sequences of multiply connected domains
§5. Conformal mappings from multiply connected domains onto circular domains
§6. Mappings from multiply connected domains onto strips
Exercises
CHAPTER 4 Applications of Integrals of the Cauchy Type to Boundary Value Problems
§1. Integrals of the Cauchy type and their limiting values
I. Integrals of the Cauchy type and the Plemelj formulas
II. Holder continuity of the boundary values of integrals of the Cauchy type
§2. The Riemann boundary value problem for analytic functions
§3. The Hilbert boundary value problem for analytic functions on simply connected domains
§4. Piecewise continuous boundary value problems for analytic functions
I. Piecewise continuous Riemann boundary value problems for analytic functions on closed curves.
II. The Riemann boundary value problem for analytic functions on nonclosed curves.
III. The piecewise continuous Hilbert boundary value problem for analytic functions.
§5. Mixed boundary value problems for analytic and harmonic functions
I. Mixed boundary value problems for analytic functions
II. Mixed boundary value problems for harmonic functions.
Exercises
CHAPTER 5 The Hilbert Boundary Value Problem for Analytic Functions on Multiply Connected Domains
§1. Formulation of the Hilbert boundary value problem on multiply connected domains
§2. Uniqueness of the solutionto the Hilbert boundary value problem
§3. A priori estimates of solutions to the Hilbert problem for analytic functions
I. A priori estimates of solutions to Problem D for analytic functions on the unit disc
II. A priori estimates for solutions to Problem B for analytic functions on multiply connected domains
§4. Solvability of the Hilbert boundary value problem for analytic functions
§5. Integral representations of solutions to the Hilbert boundary value problem for analytic functions
§6. Composite boundary value problems for analytic functions on multiply connected domains
Exercises
CHAPTER 6 Basic Boundary Value Problems for Harmonic Functions
§1. Uniqueness of solutions to boundary value problems for harmonic functions
§2. The first and second boundary value problems for harmonic functions
I. The uniqueness and integral representation of the solution to Problem I.
II. The existence and integral representation of the solution to Problem II.
§3. The third boundary value problem for harmonic functions and its generalizations
§4. Irregular oblique derivative boundary value problems for harmonic functions
§5. Properties of biharmonic functions and the basic boundary value problem
Exercises
APPENDIX 1 A Brief Introduction to Quasiconformal Mappings
§1. Continuously differentiable transformations and K-quasi-mappings
§2. The relationship between quasiconformal mappings and partial differential equations
APPENDIX 2 Some Connections between Integral Equations and Boundary Value Problems
§1. The relationship between characteristic equations and Riemann boundary value problems
§2. Solving the Dirichlet problem by the method of integral equations
§3. The regularization and fundamental theory of singular integral equations
References
Subject Index
Back Cover
