Complex Analytic Methods for Partial Differential Equations: An Introductory Text

COMPLEX ANALYTIC METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS AN INTRODUCTORY TEXT

Heinrich G. W. Begehr

Complex Analytic Methods for Partial Differential Equations

COMPLEX ANALYTIC METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS AN INTRODUCTORY TEXT

Heinrich G. W. Begehr Mathematik Freie Universitat Berlin

World Scientific

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Library of Congress Cataloging-in-Publication Data Begehr, Heinrich G. W. Complex analytic methods for partial differential equations : an introductory text / Heinrich G. W. Begehr. p. cm. Includes bibliographical references and indexes. ISBN 9810215509 1. Differential equations, Partial--Numerical solutions. 2. Functions of complex variables. 3. Mathematical analysis.

I. Title. QA377.B43

1994

515'.353--dc2O

94-34414

CIP

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In memory of my parents and my sister Edda Jaeger

Preface For more than one century complex analysis has fascinated mathematicians since Cauchy, WeierstraO and Rieniann had built up the field from their different points of view. Richness, beauty and fascination originate from the coincidence of analytic, algebraic and geometric methods. While the theory of meromorphic functions culminated in value distribution theory, for several decades geometrical function theory flourhished, and complex approximation theory was developed. For recent books in these topics see [Gaie80], [Laga86], [Pomm92] and also the survey article [Gaie90]. An excellent presentation of classical complex analysis including many historical facts and remarks are the two volumes [Remrn92]. 't'here are many other facets of complex analysis e.g. analytic number theory and complex ordinary differential equations. And there is the wide field of complex analysis of several variables. But complex analysis is not any more in the center of mathematical interest neither in one nor in several variables. Nevertheless there is a branch which just recently became quite active: complex analysis in partial differential equations. In the last

ten years more than 20 monographs appeared in this area as well as several collections of articles, see the list of references, part a. Although already classical via the theory of harmonic functions this area became very lifely through the investigations of I.N. VEKUA, N.I. MUSKHELISHVILi, L. BERS, F.D. GAKHOV, W. HAACK,

R.P. GILBERT and others. While some of these considerations develop the theory of boundary value problems for analytic functions others are concerned with building up some theories for classes of complex partial differential equations and systems. The present book gives some introduction in complex methods for partial differential equations and systems mainly of first and second order. Classical natural boundary value problems are considered, which in general are reduced to singular integral equations by utilizing proper integral representation formulas. The basic boundary conditions are of RIEMANN and RIEMANN-HILBERT type. Several generalizations and extensions are presented as e.g the POINCARE problem and discontinuous bound-

ary value problems. But in general we stay with stronger assumptions in order to keep the introductory character of the book. After the reader has become acquainted with the material he can pass to other secondary literature or even to original research papers. On the basis of a first course in complex analysis chapter 1 introduces the necessary background of function and potential theory. Properties of CAUCHY integrals, GREEN and NEUMANN functions and SCHWARZ operators, fundamental boundary value problems for analytic functions are extensively discussed. The RIEMANN mapping theorem often used to reduce boundary value problems for simply connected domains to the case of the unit disc is presented with proof. It also serves to motivate the introduction of the GREEN function. For multiply connected domains the BERGMAN kernel serves to find operators of SCHWARZ type. These representation formulas were recently developed by A. DZHURAEV. In chapter 2 beyond analytic functions solutions to nonlinear CAUCHY RIEMANN systems and later on in chapter 3 to generalized BELTRAMI equations are studied. The classical GAUSS theorem leads to a generalization of the CAUCHY representation for analytic functions, the

vii

so-called CAUCHY-POMPEIU formula. The area integral added here is the celebrated T-operator in the VEKUA theory of generalized analytic functions. Its z-derivative is a singular integral operator of CALDERON-ZYGMUND type which serves to transform boundary value problems for the functions just mentioned to singular integral equations. They can be solved and lead to a priori estimates of the solutions as well. This T-operator and its first order z-derivative are just two particular operators of a whole bunch of integral operators useful for first and higher order complex partial differential equations. They were only recently systematically worked out by G.N. HILE and the author after one and the other have occasionally been used before. The section on discontiunuous boundary value problems is technically involved and might be skipped at a first reading of the chapter. For nonlinear BELTRAMI equations the related integral equations become nonlinear. In chapter 4 as an example entire solutions are studied. In principle this leads to the solution of the RIEMANN boundary value problem for these nonlinear equations. Here as well pseudoparabolic equations as first order composite type systems are considered, where the methods developed before turn out to be useful, too. In the final chapter 5 some special boundary value problems for elliptic second and higher order equations are discussed in multiply connected domains and the unit disc, respectively. The singular integral operators involved are expressed by the BERGMAN and related kernel functions. They transform the problem to a singular integral equation to which the FREDHOLM alternative applies. Boundary value problems in the theory of analytic functions of several variables are difficult in principle because of the complicated structure of the integral representation formulas. In case when analytic functions satisfy some partial differential equations it is possible to solve boundary value problems. Some results of A. DZHURAEV and the author on first order systems in two complex variables with analytic coefficients are presented. Here the DOUGLIS algebra of hypercomplex variables proves to be useful. The SCHWARZ-POISSON formula which turns out as essential for the DIRICHLET problem for analytic functions in chapter 1 can be extended to several variables. This formula was just recently published by A. KUMAR and the author. Its deduction is included here, too.

The main parts of the first three chapters were distributed as Lecture Notes at the University of Assiut, Egypt in 1991 during a short time visitorship granted by the German Academic Exchange Service (DAAD Kurzzeitdozentur). Moreover, the material was presented in special courses at the Freie Universitat Berlin. I appreciated the support through DAAD and the scientific discussions with the colleagues and students in Assuit, in Qena and in Berlin very much. I am much indebted to my secretary Barbara M. Wengel for her careful preparation of the camera-ready copy of the manuscript. The figures were prepared by Ute Fuchs. Thanks to her as well as to the staff of World Scientific for their patience and cooperation. Berlin, May 1994

Heinrich Begehr

Table of Contents Preface ....................................................................... v

1. Function theoretical tools ............................................... I I.I. CA U CH Y integral ..................................................... 1

1.2. GREEN functions and SCHWARZ operators ........................... 18 1.3. RIEM A N N boundary value problem ................................... 38

1.4. RIEMANN-HILBERT boundary value problem .........................45

2. Inhomogeneous CAUCHY-RIEMANN systems ........................69 2.1. Integral representations .............................................. 69 2.2. Properties of integral operators ...................................... 74

3. Boundary value problems for generalized BELTRAMI equations .................................................................99

3.1. Generalized BELTRAMI equation ...................................... 99 3.2. RIEMA N N boundary value problem ................................. 114

3.3. RIEMANN-HILBERT boundary value problem ....................... 118

3.4. POINCARE boundary value problem ................................. 144 3.5. Discontinuous boundary value problems ............................. 149

4. Other equations and systems related to the BELTRAMI

equation .................................................................175 4.1. Initial boundary value problem for pseudoparabolic equations ....... 175 4.2. Initial boundary value problem for a composite type system ......... 184 4.3. Entire solution to nonlinear generalized BELTRAMI equations ........ 194

5. Higher order complex differential equations and equations in several complex variables ......................................... 203 5.1. Elliptic second order equations ..................................... 203

5.2. Higher order equations ............................................. 233 5.3. First order systems in two complex variables with analytic coefficients 239 5.4. The SCHWARZ-POISSON formula for polydiscs

...................... 252

References .............................................................257

Index of Names ....................................................... 267

Index of Subjects ..................................................... 269

Index of Corollaries, Definitions, Lemmas, Theorems ............. 273

ix

1. Function theoretical tools 1.1 CAUCHY integrals In this section the behaviour of CAUCHY integrals along smooth curves or systems of curves in the complex plane d' will be reported on. A smooth curve r is a closed or open JORDAN arc with continuously varying tangent. It thus may be represented as

r= {z:z=z(r),0