Conformal Mappings and Boundary Value Problems

313 Pages·1992·3.673 MB·English

by  Guo-Chun Wen

Translations of
.MATHEMATICAL
MONOGRAPHS
Volume 106
Conformal Mappings and
Boundary Value Problems
Guo-Chun Wen
American Mathematical Society
Conformal Mappings and
Boundary Value Problems
10 6
Translations of Mathematical Monographs
Translations of
MATHEMATICAL
MONOGRAPHS
Volume 106
Conformal Mappings and
Boundary Value Problems
Guo-Chun Wen
IIT01 %III o American Mathematical Society
Providence, Rhode Island
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Translated from the Chinese by Kuniko Weltin
1991 Mathematics Subject Classification. Primary 30C35, 30E25, 30D99;
Secondary 30C20, 30C60, 30C7S.
Library of Congress Cataloging-in-Publication Data
Wen, Guo Chun.
[Kung hsing ying she yU pien chih wen t' i. English]
Conformal mappings and boundary value problems/Guo-Chun Wen.
p. cm.-(Translations of mathematical monographs; v. 106)
Translation of. Kung hsing ying she yU pien chih wen t' i.
Includes bibliographical references and index.
ISBN 0-8218-4562-4
1. Conformal mapping. 2. Boundary value problems. I. Title. II. Series.
QA360.W4613 1992 92-14225
515.9-dc20 CIP
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Conformal Mappings and Boundary Value Problems
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Contents
Preface vii
CHAPTER 1. Some Properties of Analytic and Harmonic Functions
1
§ 1. The convergence of sequences of analytic functions
1
§2. The convergence of sequences of harmonic functions 6
§3. Some properties of subharmonic functions 12
§4. The Dirichlet problem for analytic and harmonic functions 15
CHAPTER 2. Conformal Mappings of Simply Connected Domains 25
§ 1. The fundamental theorem for conformal mappings on simply
connected domains 25
§2. Boundary correspondence theorems for conformal mappings 32
§3. The distortion theorem and estimates of coefficients for
univalent functions 41
§4. The convergence of conformal mappings for sequences of
simply connected domains 57
§5. The representations of conformal mappings on polygonal
domains 65
§6. The representations of conformal mappings with orthogonal
polynomials 80
CHAPTER 3. Conformal Mappings of Multiply Connected Domains 95
§ 1. A general discussion of conformal mappings between multiply
connected domains 95
§2. Conformal mappings for domains with parallel slits 102
§3. Conformal mappings for domains with spiral slits 108
§4. The convergence of conformal mappings of sequences of
multiply connected domains 112
§5. Conformal mappings from multiply connected domains onto
circular domains 118
§6. Mappings from multiply connected domains onto strips 128
CHAPTER 4. Applications of Integrals of the Cauchy Type to
Boundary Value Problems 137
§ 1. Integrals of the Cauchy type and their limiting values 137
§2. The Riemann boundary value problem for analytic functions 149
§3. The Hilbert boundary value problem for analytic functions on
simply connected domains 154
V
vi CONTENTS
§4. Piecewise continuous boundary value problems for analytic
functions 162
§5. Mixed boundary value problems for analytic and harmonic
functions 174
CHAPTER 5. The Hilbert Boundary Value Problem for Analytic
Functions on Multiply Connected Domains 189
§ 1. Formulation of the Hilbert boundary value problem on
multiply connected domains 189
§2. Uniqueness of the solution to the Hilbert boundary value
problem 192
§ 3. A priori estimates of solutions to the Hilbert problem for
analytic functions 195
§4. Solvability of the Hilbert boundary value problem for analytic
functions 204
§ 5. Integral representations of solutions to the Hilbert boundary
value problem for analytic functions 211
§ 6. Composite boundary value problems for analytic functions on
multiply connected domains 216
CHAPTER 6. Basic Boundary Value Problems for Harmonic
Functions 231
§ 1. Uniqueness of solutions to boundary value problems for
harmonic functions 231
§2. The first and second boundary value problems for harmonic
functions 237
§3. The third boundary value problem for harmonic functions and
its generalizations 247
§4. Irregular oblique derivative boundary value problems for
harmonic functions 254
§ 5. Properties of biharmonic functions and the basic boundary
value problem 265
APPENDIX 1. A Brief Introduction to Quasiconformal Mappings 273
§ 1. Continuously differentiable transformations and
K-quasi-mappings 273
§2. The relationship between quasiconformal mappings and partial
differential equations 278
APPENDIX 2. Some Connections between Integral Equations and
Boundary Value Problems 285
§ 1. The relationship between characteristic equations and Riemann
boundary value problems 285
§2. Solving the Dirichlet problem by the method of integral
equations 291
§3. The regularization and fundamental theorem of singular
integral equations 293
References 299
Subject Index 301
Preface
Conformal mapping and boundary value problems are two major branches
of complex function theory. The former is the geometric theory of analytic
functions and the latter is the analytic theory governing the close relationship
between the abstract theory and many concrete problems. Moreover, there
is also an intimate relationship between conformal mappings and boundary
value problems.
The first three chapters of this book deal with the existence and uniqueness
of conformal mappings from multiply connected domains to various canoni-
cal domains, as well as certain properties of univalent functions and explicit
representation formulas of conformal mappings. The methods used to arrive
at these results are diverse. For example, to get the existence of conformal
mappings from multiply connected domains to canonical domains, we use
the method of locating extrema, the continuity method, and the modified
Dirichlet problem for analytic functions. To get the explicit representation
formulas of the conformal mappings from polygons bounded by straight lines
or circular arcs to the upper half-plane or unit disc, we use the method of
analytic continuation. In addition, the use of the convergence properties of
sequences of analytic functions is among the main ingredients in the proof
of many theorems concerning conformal mappings.
The remaining three chapters give a fairly detailed introductory account
of the basic boundary value problems for analytic functions on multiply
connected domains (the Riemannian boundary value problem, the Hilbert
boundary value problem, etc.), as well as those for harmonic functions (the
first, second, and third mixed boundary value problems and the irregular
oblique derivative boundary value problem). Here, we have not used the
method of integral equations which is common in the study of these prob-
lems. Rather, we first use subharmonic functions of conformal mappings to
prove the existence of solutions for the harmonic function Dirichlet bound-
ary value problem on multiply connected domains. Then, on the basis of
a priori estimates for the solutions of these boundary value problems, we
use the continuity method and related methods to prove the solvability of
the aforementioned analytic function Hilbert boundary value problem. We
vii
viii PREFACE
also give the integral representation formulas for the solutions of these prob-
lems. As for the Riemann boundary value problem for analytic functions,
we have followed the usual method of using integrals of the Cauchy type.
The existence of solutions for certain boundary value problems for harmonic
functions is based on the solvability of these problems for analytic functions.
We also discuss a new kind of mixed boundary value problem which consists
of generalizing, in the third boundary value problem, the boundary condition
involving the directional derivative.
In order to make clear the connection between the theory of conformal
mappings and certain boundary value problems, we have included a discus-
sion of this connection in all the chapters. Moreover, we have written two
appendices to give a brief introduction to the theory of quasiconformal map-
pings and the connection between boundaries and singular integral equations.
We have also included some exercises for the benefit of the readers.
The writing of this book was based on the author's many years of expe-
rience in teaching and scientific research. Its content has been used many
times as a text for upper division undergraduate courses as well as elemen-
tary and advanced graduate courses at Peking University. The constructive
suggestions and valuable advice that the author has received over the years
have resulted in many revisions. He should especially thank the following
individuals for significant improvements in both content and exposition of
the present text: Professors Zhuang Zi-Tai and Lu Jian-Ke, and Messieurs
Dai Zhong-Wei, Chen Fang-Quan, Huang Sha, Tian Mao-Ying, and Wu Zhi-
Jian. Due to the author's limitations, there are bound to be many defects in
the book, and the author would welcome the criticisms of the readers.
Guo-Chun Wen
Peking University
May, 1984
CHAPTER 1
Some Properties of Analytic
and Harmonic Functions
The content of this chapter provides the basic tools for the discussions of
conformal mappings and boundary value problems in the subsequent chap-
ters. We begin by giving the convergence theorems for sequences of ana-
lytic and harmonic functions. Then we introduce several properties of sub-
harmonic functions and use them to discuss the solvability of the Dirichlet
boundary value problem for harmonic and analytic functions.
§1. The convergence of sequences of analytic functions
It is well known that for any bounded sequence of numbers {z} in the
z-plane, we can always find a convergent subsequence {znk 1. We now ask
whether or not one can draw the same conclusion for a sequence of analytic
functions {fn(z)} in a domain D; in other words, what conditions must be
placed on {f(z)}, such that we can choose a subsequence {f(z)} which
converges to an analytic function in D ?
To answer this question, we need two definitions.
DEFINITION 1.1. Suppose A = {f(z)} is a family of functions on a set
E in the z-plane. If there exists a positive number M for all functions
f(z) in A such that If (z) I < M (for z E E), then this family of functions
A is said to he uniformly bounded on E. If A is defined in a domain D
and is uniformly bounded on every closed set E c D, then A is said to be
uniformly bounded on closed sets in D. *
DEFINITION 1.2. Suppose A = {f(z)} is a family of analytic functions
in a domain D. If every sequence of functions {fn(z)} in A contains a
subsequence Ifn (z)} which uniformly converges to an analytic function on
k
closed sets in D, then the family A is said to be sequentially compact in
D. ({fn (z)} is said to be uniformly convergent on closed sets in D, if it is
k
* Translator's note. This terminology is not commonly used in the English literature. A
more idiomatic rendition would have been "uniformly bounded on closed subsets in D," but
this would be confused with uniform boundedness on the closed subsets of D, whereas the
author's intention seems to be uniform boundedness on those sets closed in C which happen
to be inside D .
1