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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 1b LIJ fA 'b I*:], -,"F, An ) C AS 0 - 4 A _: -9 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 COPYING AND REPRINTING. Individual readers of this publication, and nonprofit li- braries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication (including abstracts) is permitted only under license from the American Mathematical Society. Requests for such permission should be addressed to the Manager of Editorial Services, American Mathematical Society, P.O. Box 6248, Providence, Rhode Island 02940-6248. 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Printed in the United States of America The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. This publication was typeset using AMS-TEX, the American Mathematical Society's TEX macro system. 1098765432 1 979695949392 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 canonical 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 problems. Rather, we first use subharmonic functions of conformal mappings to prove the existence of solutions for the harmonic function Dirichlet boundary 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 problems. 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 discussion of this connection in all the chapters. Moreover, we have written two appendices to give a brief introduction to the theory of quasiconformal mappings 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 chapters. We begin by giving the convergence theorems for sequences of analytic and harmonic functions. Then we introduce several properties of subharmonic 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