First Order Elliptic Systems: A Function Theoretic Approach (Mathematics in Science & Engineering, 163)

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Author: Robert P. Gilbert | James Buchanan

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First Order Elliptic Systems A FUNCTION THEORETIC APPROACH

This is Volume 163 in MATHEMATICS IN SCIENCE AND ENGINEERING A Series of Monographs and Textbooks Edited by RICHARD BELLMAN, University of Southern California The complete listing of books in this series is available from the Publisher upon request.

First Order Elliptic Systems A FUNCTION THEORETIC APPROACH ROBERT P. GILBERT Applied Mathematics Institute and Department of Mathematics University of Delaware Newark, Delaware

JAMES L. BUCHANAN Department of Mathematics U . S . Naval Academy Annapolis, Maryland

1983

ACADEMIC PRESS A Subsidiary of Harcourt Brace Jovanovich, Publishers New York London Paris San Diego San Francisco Szio Paulo Sydney Tokyo Toronto

COPYRIGHT @ 1983, BY ACADEMIC PRESS,INC.

ALL RIGHTS RESERVED. NO PART OF THIS PUBLICATION MAY BE REPRODUCED OR TRANSMITTED IN ANY FORM OR BY ANY MEANS, ELECTRONIC OR MECHANICAL, INCLUDING PHOTOCOPY, RECORDING, OR ANY INFORMATION STORAGE AND RETRIEVAL SYSTEM, WITHOUT PERMISSION IN WRlTINQ FROM THE PUBLISHER.

ACADEMIC PRESS,INC.

111 Fifth Avenue, New York. New York 10003

United Kingdom Edition published by ACADEMIC PRESS, INC. ( L O N D O N ) LTD. 24/28 Oval Road, London N W l IDX

Library of Congress Cataloging in Publication Data Gilbert, Robert P., Date First order elliptic systems. ) (Mathematics in' science and engineering ; Includes index. I . Differential equations, Elliptic. I . Buchanan, James. I I . Title. I l l . Series. QA377.6498 1982 515.3'53 82-8703 ISBN 0-12-283280-9 AACRZ

PRINTED IN THE UNITED STATES OF AMERICA

83 84 85 86

9876 5 43 2 1

J . L . Buchanan dedicates this book to his parents, Vivian and William, and R . P . Gilbert dedicates this book to his wife, Nancy.

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Contents

Preface

ix

0. Introduction

1

1. Elliptic Systems in the Plane I . Introduction 6 2. Hyperanalytic Functions II 24 3. Generalized Derivatives and the Hypercomplex Pompieu Opera tor 32 4. Generalized Hyperanalytic Functions and Liouville's Theorem 44 5 . Cauchy Representation for Generalized Hyperanalytic Functions 6. M-Analytic Functions 53 7. Approximate Solutions 57

2. Boundary Value Problems I . Introduction 61 2. The Plemlj Formulas 63 3. The Hilbert Problem for Hyperanalytic Functions 65 4. The Representation of a Piecewise Generalized Hyperanalytic Function in Terms of a Density 70 75 5 . The Hilbert Problem for Generalized Hyperanalytic Fulzctions 6 . The Hilbert Problem in the Purely Hypercomplex Case 83 87 7. The Riemann-Hilbert Problem for Hypercomplex Functions 8. The Representation of a Generalized Hyperanalyiic Funciion in Terms of a Real Density 92 9. The Riemann-Hilbert Problem for Generalized Hyperanalytic Functions 93 10. The Riemann-Hilbert Problem in the Purely Hypercomplex Case 102

3. Reductions to Hyperanalyticity 1 . Introduction 109 2. Similarity Principles

110

vii

viii 3. 4. 5. 6. 7.

CONTENTS

Global Similarity Principle 121 The Riemann-Hilbert Problem 133 Hyperanalytic Functions Having Distributional Boundary Data 136 Nonlinear Problems and Reductions to Linear Problems 140 Liouville's Theorem and the Similarity Principle for Pascali Systems

144

4. Function Theory over Clifford Algebras 1 . Introduction 150 2. Regular Functions 154 3. Hilbert Modules I65 4. Liouville's Theorem 172 5. a-Holomorphic Functions I74 6 . Generalized Regular Functions in R " 175 7. Overdetermined Elliptic Systems 199 8. Function Theory for Higher Order Elliptic Systems with Analytic Coeficients 207 9. Commutative Alternatives for Higher Dimensional Function Theory

210

5. Partial Differential Equations of Several Complex Variables 1 . Inhomogeneous Cauchy-Riemann Equations in Polycylinders 216 2. Inhomogeneous Cauchy-Riemann Systems for Several Unknowns 221 3. Existence Theorems for Solutions of Partial Differential Equations in Several Complex Variables 226 4. Real-Linear Equations in Two Complex Variables 231 5 . Nonhomogeneous Cauchy-Riemann Equations in Analytic Polyhedra 240 6 . Pluriharmonic Functions 253

Bibliography Index 275

269

Preface

In this volume, a successor to an earlier monograph in this series, we seek from among those systems of first order partial differential equations that are in some sense elliptic those that share common properties with the prototypical elliptic system, the Cauchy-Riemann equations. Our considerations will be dominated by the following questions concerning solutions to such systems and their similarity to analytic functions: (a) Do they possess integral representations analogous to the Cauchy integral formula? (b) Are the classical boundary value problems for analytic functionsthe Hilbert and Riemann-Hilbert problems-still appropriate? (c) Do they have the unique continuation property so that if all entries of a solution vector vanish on an open set, then the solution is identically zero? (d) Can the zeros common to all entries of a solution have an accumulation point within the domain? (e) Is Liouville’s theorem still valid? Most particularly, must an entire solution which vanishes at infinity vanish identically? (f) Can the notion of the order of a zero be extended? Questions (a) and (b) have affirmative answers, at least in the plane. However, as may be gathered from the way in which they are posed, the answers to the last four questions are all negative for elliptic systems in general. In fact, workers in this area have produced counterexamples to (c) and (e) and, consequently, to (d) and (f) (for references, see the introduction to Chapter 1). ix

X

PREFACE

Since the questions posed, if not always the means by which they are resolved, are elementary and may be of interest to anyone acquainted with analytic function theory, we have attempted to write this volume with both the general reader and the specialist in mind. For the nonspecialist we have endeavored to state results (the detailed proofs of which are beyond our purview) with sufficient clarity and explicitness so that immediate reference to the fundamental treatises in the field will not be necessary. On the other hand, it is hoped that the specialist will find the new results, reformulations, refinements, and extensions of familiar material contained in this monograph to be of interest. Our topic selection has been eclectic. We have confined ourselves only by the stricture that the subjects given exposition be in some sense “function theoretic.” Where considerations of space and time have prohibited full exposition, we have summarized results and approaches. We hope that this diversity will appeal to both constituencies. The book divides roughly into two parts. In the first three chapters we give answers, to the extent they are known, to the six questions posed above for elliptic systems in the plane. Inasmuch as some of these topics have been the subject of a recent major work “Elliptic Systems in the Plane” by W. L. Wendland, we note that the reader will encounter little redundancy. Whereas Professor Wendland’s approach derives primarily from the modern theory of partial differential equations, ours is predominantly function theoretic. In Chapter 4 we investigate certain sytems in higher dimensions. To the best of our knowledge this is the first time most of the subjects therein have appeared outside of research journals. In R“, our considerations, perforce, become more fundamental. Whereas in the plane many results may be inferred from mappings between solutions to elliptic systems and sets of analytic functions, no such expedient is available in higher dimensional space. The first problem to be confronted is supplanting the algebra 1,i. It turns out that there is no fully suitable replacement, for we must relinquish at least one of the following: (a) commutativity, (b) associativity, or (c) invertibility of all nonzero elements. The situation is further complicated by the difficulty that working with commutative algebras severely restricts the generality of the systems considered. In Chapter 4, the authors deal with the “quaternion” or “Clifford algebra” formulation, which surrenders (a) and (b) but allows relatively general systems to be treated; however, we consider briefly algebras that make alternative concessions. In Chapter 5 we take a somewhat different view of higher dimensional systems and explore the generalizations of the results of the first three chapters to functions of several complex variables. Finally, a word about applications. Higher order elliptic equations bear the same relation to elliptic systems that higher order ordinary differential

PREFACE

xi

equations bear to first order systems, that is, the former are reducible to the latter. Thus the pool of potential applications is vast, including subjects such as electrostatics, elasticity, and fluid dynamics. For a compendium of such applications, see the previously mentioned work by Wendland. There is good reason to believe that recently developed “symbolic manipulation” computer languages will make function theoretic methods useful tools in solving physical problems in the fields listed above and others. In closing, one of the authors (R.P.G.) would like to thank the National Science Foundation, which supported, in part, his efforts through grant MCS 78-02452. He also thanks his wife, Nancy, for providing a pleasant atmosphere in the home, where much of the writing took place, and for her encouragement and understanding during the sometimes difficult stages through which the manuscript had to evolve. Finally, both authors express their thanks to Ms. Janice Spry, senior secretary at the Applied Mathematics Institute, University of Delaware, who diligently and patiently typed the manuscript, completing this arduous task even after one of the authors had left for his sabbatical at Oxford and the other had returned to teaching at the Naval Academy.

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~

Introduction

The (2r

+ 2)

x (2r

UXb,

Y)

+

2) system of real equations

+ 4 x 3 Y)U,(X,

Y)

+ B(x, Y ) U ( X , Y ) = 0

(0.1)

is elliptic if the matrix A possesses no real eigenvalues. In the 2 x 2 case, solutions to (0.1) share many common properties with analytic functions. Indeed, under certain differentiability assumptions upon the coefficients, a 2 x 2 system of the form (0.1) may be put into the normal form

- + uw aw

az

+ bW

=

0,

where all quantities are now complex valued. Thus, in the absence of the lower order coefficients a and b, solutions to (0.2) are, in fact, analytic functions. Carleman (1933) showed that, even with lower order terms present, solutions to (0.2) must have isolated zeros of finite order. Later Haack (1952), Bers (1953), and I. N. Vekua (1952, 1954) independently arrived at a concise expression of the holomorphic nature of these solutions, the similarity principle

w(z, Z) = &)es(z*z), (0.3) where C#J is an analytic function and s is bounded. The similarity principle implies the Carleman theorem and other results such as Liouville’s theorem. Vekua’s formulation is based upon the Sobolev derivative and requires the least regularity for the coefficients a and b. In this monograph we shall usually be concerned with extensions of Vekua’s approach in various directions. 1

2

0. INTRODUCTION

In Chapter 1 we explore the extent to which solutions to the system (0.1) of dimension (2r + 2) bear resemblance to analytic functions. That no result as satisfactory as the similarity principle exists is shown by examples of PliS (1960, 1961), which indicate that solutions to (0.1) may vanish on arcs and even on open sets without vanishing identically. We shall restrict attention to those systems that are transformable to ( r + 1) x ( r + 1) complex systems of the form

Dw

+ A w + Btii = 0 ,

(0.4)

where

DW

:= w7

+ Qw,,

and the matrix Q is quasi-diagonal and nilpotent of order ( r + 1). Solutions to D w = 0, termed hyperanalytic functions by Douglis (1953, 1960), who introduced this normal form, give us a higher dimensional analog of analytic functions. There is, in fact, an invertible homomorphism between the space of hyperanalytic functions and the space of analytic vectors, and we utilize this mapping to help establish the function theory for hyperanalytic functions developed by Douglis (1953) and extended by Kuhn (1974). The last part of Chapter I is devoted to solutions of the general system (0.4), which we term generalized hyperanalytic functions. The properties of these functions have been investigated under various assumptions on the coefficients A and B by Bojarski (1966, 1958), Gilbert and Hile (1974, 1976), Gilbert and Wendland (1975), and Goldschmidt (1979, 1975, 1978a,b). We follow the approach of Bojarski (1966) with modifications and extensions. In the general case considered by Bojarski it is known that generalized analytic functions do not necessarily obey Liouville’s theorem because of a counterexample due to Habetha (1976). Thus it is worthwhile to consider the stronger function theory developed by Gilbert and Hile (1974) under the assumption that A and B have quasidiagonal form. Here results are obtained (excluding the similarity principle) that more closely resemble those of the Bers-Haack-Vekua formulation for r = 0. In Chapter 2 we treat boundary value problems for generalized hyperanalytic functions. Two boundary value problems for analytic functions, the Hilbert and Riemann-Hilbert problems, may also be posed for solutions to (0.4). The Hilbert problem is to. find a generalized hyperanalytic function w that is continuous in C\@, where 6 is a system of closed nonintersecting curves and satisfies the jump condition W+ -

Hw-

=

h

(0.6)

3

0. INTRODUCTION

across 6. The Riemann-Hilbert problem is to find a generalized hyperanalytic function that is continuous in @ and satisfies Re(Aw) = c

(0.7)

on &, the boundary of a. Here the matrices H and A and the vecwith det H and tors h and c are Holder continuous functions on det A # 0. The indices of the Hilbert and Riemann-Hilbert problems are, respectively, 1 KH := - A($ arg det H IT

and 1 KR := - A($ arg det 7T

K’X.

In the case of general (non-quasi-diagonal) coefficient matrices A , B, H , and A, Bojarski (1966) has employed the theory of singular integral equations to derive relations between the indices KH and KR and the number of independent solutions to the homogeneous (h, c = 0) Hilbert and Riemann-Hilbert problems and suitably chosen adjoint problems. Compatibility conditions on the data H, h, A, and c can also be phrased in terms of the adjoint solutions. We present a modification of Bojarski’s argument. When the various matrices involved are restricted to have quasi-diagonal form, Begehr and Gilbert (1977-1979) have shown that the sign of the index governs solvability. We also present this argument. In Chapter 3 some instances in which similarity principles, local or global, obtain for higher order elliptic systems are discussed. These all entail restriction of the coefficient matrices A and B in (0.4) to quasidiagonal or at least lower trianguiar form. Kiihn (1974) has shown that in the lower triangular case a generalized hyperanalytic function has a local similarity representation. From this representation it is possible to define the order of a zero. A global similarity principle is known only in the very restrictive case in which A is quasi-diagonal and B = 0. We present such a similarity principle, due to Kiihn (1974), along with refinements due to Begehr and Gilbert (1982). The notion of the order of a zero and a corresponding concept for a pole are utilized to derive further results on the Riemann-Hilbert problem. Also, in Chapter 3 the homomorphism between analytic vectors and hyperanalytic functions introduced in Chapter 1 is used to treat the Hilbert problem with distributional data specified on the real axis. Finally, we mention some nonlinear results for the Riemann-Hilbert problem.

4

0. INTRODUCTION

In Chapter 4 we explore function theory in R". Early results due to Hamilton (1853), Scheffers (1894), Ketchum (1928, 1929), Theodorescu (1931), and Fueter (1932) have been extended in the past two decades in various directions by Delanghe (1970a-d, 1972, 1973, 1976), Delanghe and Brackx (1980), Delanghe and Sommen (1980), Sommen (1980a,b), and Coroi-Nedelcu (1959, 1960, 1965a,b, 1967a,b), Hile (1976, 1977), Iftimie (1965, 1966), Goldschmidt (1980), Rosculet (1955, 1975), and Snyder (1968). This is still an active area of research and, as we shall see, most of the work done thus far concerns problems of integral representations and local behavior. Boundary value problems remain to be considered. There is no uniquely suitable replacement for the algebra { 1, i}; however considerations of generality and physical applications suggest that the quaternion or Clifford algebra formulation is often appropriate. Here the independent variable is z = xI

where e: = operator is

- 1,

+ xzez +

i > 1, and eiej

=

+ &en,

-ejei, and the Cauchy-Riemann

The absence of commutativity and the fact that the algebra (1, e2, ..., en} is not closed under multiplication create difficulties that do not arise in the plane. Nonetheless, a function theory of sorts is available. We give the formulation of Delanghe (1970a-d, 1972, 1973, 1976) for Taylor and Laurent series for solutions to sw = 0, and a Cauchy-Pompieu integral representation due to Hile (1976). We also give an exposition of generalizations of the theory of inhomogeneous equations of Chapter 1 due to Hile (1972) and Goldschmidt (1980). Included in Chapter 4 is a discussion of the work of Hile and h o t t er (1977a,b) on overdetermined systems that take the form

where the Pi(x) are m x n matrices with m > n. The system (0.8) is considered elliptic at the point x if

has a rank n for all 5 :=

(el, ...,

[k)

E Rk\{0}.

We give a unique

0. INTRODUCTION

5

continuation theorem of Hile and Protter for a class of systems of the form (0.8). Also in Chapter 4 we mention a function theory developed by Habetha (1973) for systems which are elliptic in the sense of Doughs and Nirenberg (1953). Finally, in this chapter we discuss some commutative alternatives to the Clifford algebra formulation that have been developed by Edenhoffer (1976a,b), Phipps (private communication), Rosculet (1955, 1975), Synder (1968), and other authors. The material in Chapter 5 presents first a synopsis of the material developed by Tutschke (1971-1974, 1977) concerning inhomogeneous Cauchy-Riemann equations in Cartesian product domains 8 = 8 , x x C @" of the form

(i = 1, ..., n ; j = 1, ..., m).Also, we make mention of the difficulties involved when the right-hand side of (0.9) depends upon Wj. In order to avoid the restraints of solving equations only in Cartesian product domains, the integral representations of Bergman-Weil and Bochner-Martinelli are generalized to the case of inhomogeneous equations. In particular, using the Bergman parameterization (1934a, 1936), we develop a Pompieu representation. Chapter 5 concludes with pseudoconvex domains and the generalized Dirichlet problem for plurisubharmonic functions. We prove the result, due to Bremermann (1959), that the upper envelope 4(z) of the class of plurisubharmonic functions in a strictly pseudoconvex domain exists, is plurisubharmonic (but not pluriharmonic), and assumes the given boundary data.

Elliptic Systems in the Plane

1. INTRODUCTION

The n x n first order system of partial differential equations u k , Y ) + A(x, y)u,(x, Y ) + H x , y)u(x,Y )

+ C ( x ,Y )

=

0

(1.1)

is elliptic in a domain 8 if the matrix A has no real eigenvalues at any point of 8. For real-valued matrices this can happen only if the system is of even dimension. Henceforth we denote the dimension of (1.1) by 2r + 2. It is well known that in the two-dimensional case (r = 0) the system ( 1 . 1 ) is transformable into the single complex equation w7 + a w

+ bE + f = 0,

w7 := t(w,

+ iw,,),

( 1 .2)

providing only that the elements of the matrix A are in C'@,the space of functions with Holder continuous derivatives of order a,0 < a < 1 (see I. N. Vekua, 1962). Since the principal part of (1.2) is the Cauchy-Riemann operator (gB = 0 implies that g is analytic), one would expect solutions of (1.2) to behave somewhat like analytic functions. This is indeed the case; when f = 0 solutions of (1.2) satisfy a similarity principle w(z) = +(z)es(z), where 4 is an analytic function and s is bounded and continuous. This result has immediate consequences. Whenf= 0, solutions of (1.2), called generalized analytic or pseudoanalytic functions, must have isolated zeros, and it is meaningful to consider the order of a zero. Also, entire 6

7

1. INTRODUCTION

generalized analytic functions satisfy a Liouville theorem. The above and many other results were established independently by Bers (1953), Haack (1952), and I. N. Vekua (1952, 1954). In the case of higher dimensions ( r > O), little can be said of elliptic systems in general. PliS (1960) has exhibited an elliptic system with nontrivial solutions vanishing on a line, i.e., there are systems whose solutions seem to have little resemblance to analytic functions. In this chapter, we transform the system (1.1) into a normal form. This gives us higher order analogs of the Cauchy-Riemann operator and analytic functions. Having investigated these, we develop a Bers-Vekua theory for the situation in which lower order terms are present. In view of the example cited above, we cannot obtain results of comparable strength to those of the case r = 0 without restricting the scope of the elliptic systems considered. We now proceed with the reduction to normal form. The particular form we shall attain is that of Douglis (1953) as modified by Kuhn (1974). Let R(x, y ) be the matrix that carries A(x, y ) into Jordan normal form. Then J := RAR-'

=

diag{Jo, 70, ..., J,,

Js},

where Jn := diag{h(x, Y ) ,

..., Ap(x, Y ) }

and

While this form can be achieved at each point (x, y ) , there is no assurance that the multiplicity of the eigenvalues, and hence the dimension of the subblocks of J , will remain constant throughout @. Moreover, the matrix R may not possess as much differentiability as the matrix A . For this, and subsequent statements without proof or citation, see Wendland (1979, p. 76ff). The following will simply be assumed: (a) The matrices R and A are in C'**(@). (b) The matrix J = RAR-' has subblocks whose dimensions are constant throughout 8. (c) The eigenvalues Aj(x) E Col(@).

If u := Ru, then (1.1) becomes u,

+ Ju, + lower order terms

=

0.

8

1. ELLIPTIC SYSTEMS IN THE PLANE

Next on our agenda is the investigation of solutions to u,

+ JU,

=

0.

(1.4)

This system consists of uncoupled subsystems of the form w,+J,w,=O,

1=1,

..., s.

(1 3

In terms of the operators

we can write (1.5) as

(1

- ih)w,

+ ( 1 + iX)w, + ie(w, - w,) = 0,

(1.6)

where

Since the subsystems of (1.4) are uncoupled, there is no loss of generality in considering the system (1.6) instead of (1.4). For future reference, we note that the requirement for the principal part of the transformed system as a whole (rather than just those of the separate subsystems to take the form (1.6)) is for A to have a single eigenvalue of multiplicity r + 1 and a single independent eigenvector. pair (A, i) We now use r + 1 to designate the dimension of (1.6). Since the form of (1.6) is invariant under complex conjugation, we may assume Im h(z) > 0 without loss of generality. However, we shall assume more; the system (1.6) will be taken to be uniformly elliptic: Im h(z) 5 c0 > 0, Dividing (1.6) by 1

-

z

E

a.

ih, we obtain

i (I - m e ) , ,

+ (-1I1 -+i hih + L 1 - ih e)w, = 0,

(1.8)

where I is the identity matrix. Since the linear transformation (1 + i{)/( 1 - it) maps Im { 5 c0 onto some disk 161 6 p < 1 , we conclude that 1 + ih(z) &(z) := 1 - ih(z) (1.9) satisfies Jijo(z)l 6 p < 1 in

for some constant p .

9

1. INTRODUCTION

Definition An n x n matrix A ajk

=

= (ajk);,k=l

is quasi-diagonal if

for k >j, for l S k + r n S j + r n S n .

a j + m.k + rn

Thus quasi-diagonal matrices are lower triangular matrices having identical elements along each diagonal. We shall adopt the following convention for quasi-diagonal matrices:

(a) a. denotes the element along the main diagonal. (b) aj, 1 s j s n - 1 , denotes the element along the jth diagonal below the main one. Two properties of quasi-diagonal matrices are easily seen: (1) if A and B are quasi-diagonal, then so is AB. Moreover, AB = BA . (2) if det A # 0, then A - ' exists and is quasi-diagonal. Clearly, det A # 0 if and only if a. # 0. The system (1.8) can now be written as wz

+

Q(z)w, = 0.

(1.10)

The coefficients of w, and w, in (1.8) are quasi-diagonal and hence so is Q(z). The element along the main diagonal is the function ijo given by (1.9). The first equation in (1.10) is wz

+ ijow, = 0.

This is the complex form of the Beltrami equations. Let us now assume that is a bounded domain enclosed by a finite number of piecewise smooth curves. Since A and hence ijo is in Ca(@) we can continue ijo so that it is in Ca(@) and vanishes outside of some =s p c 1 , it can sufficiently large circle. With this and the fact that be shown that the Beltrami system has a solution 5 E C'*"(@).See I. N. Vekua (1962, Chapter 11). Under the coordinate transformation 5 = ( ( z ) , (1.10) becomes This is the desired normal form. If we denote the coefficient of wl;by Q and return to using z as the independent variable, we have W,

+ QwZ = 0.

The matrix Q is quasi-diagonal with qo = 0.

(1.11)

10

1. ELLIPTIC SYSTEMS I N THE PLANE

The quasi-diagonal form of Q led Douglis (1953), who first studied the function theoretic properties of this system, to introduce the hypercom-

plex algebra.

Definition a := akek,where e is given by (1.7) and ak is a complex number, called a hypercomplex number. The identity matrix henceforth k ake will be denoted by 1; a, is called the complex part of a and the nilpotent part. Also,

and ak is called the kth component of a.

Note that e is nilpotent of order r + I , i.e., algebra is commutative. It is easy to verify that and

lab1 6 lallbl

la

+

er+l =

bl s la1

+

0 and that the

IbJ.

Also observe that if a. # 0, then the hypercomplex number a has an inverse

where A is the nilpotent part of a. In terms of the hypercomplex algebra, (1.1 1) can be written as (1.12)

Let

In view of the nilpotency of e, r

+

Dw = x , e k w k Z k=O

c2 r

r

k=l j=O

eJ+kqkwjz

(1.13)

Thus (1.12) can be written more simply as Dw = 0. The operator D is the higher order analog of the Cauchy-Riemann operator.

11

2. HYPERANALYTIC FUNCTIONS


Definition A hypercomplex function w E C'(@) that is a solution of = 0 is said to be hyperanalytic. The following two properties of hyperanalytic functions are easily verified:

Dw

+

(1) D(uu) = u Du u Du; hence uu is hyperanalytic. (2) If u = Xi=,, ekukand uo # 0, then D(u-'u) = 0.

Theorem 1.1 (Gilbert and Hile, 1974) If w(z) is hyperanalytic and w(z) 0 in a, then the zeros o f w ( z ) are isolated.

+

PROOFLet w(z) = X i = p e"w,(z), where w J z ) is the first component that is not identically zero. Since Dw = 0, it follows from (1.13) that wpz + 2;:; qp-,wrni = 0, and hence wpz = 0 since w, = 0 for m s p - 1 . Thus w pis analytic in @ and hence has isolated zeros. (Note that the poles are also isolated.) We now introduce the concept of a generating solution (see Douglis

(1953)). Let Bk(@) be the space of hypercomplex functions that have

bounded and continuous derivatives up to order k in 8. Let Bk9"(@)be the space of hypercomplex functions that are in Bk(@)and have a Holder continuous kth derivative with index a in (9. Definition (Douglis, 1953) A hypercomplex function t(z) will be called a generating solution for the operator D if (1) (2) (3)

t has the form t(z)

T E B'(C), and Dt = 0 in C.

=

z

+ Xi=, ekr,(z) := z +

T(z),

In order to exhibit such a function we first define the Pompieu operator for the domain @ (bounded) as

The following two properties of Jcyf will be needed: (1) Iff E B"*"(@)for some 0 < a < 1, n some p , 1 s p < 2, then J& E B"+'."(@). (2) a/az Jl&z) = f(z), z E 8 .

2

0, and f E Lp(@)for

See I. N. Vekua (1962) for a more comprehensive listing of results on this operator.

12


We now proceed to show the existence of a generating solution. We add the following to our list of assumptions: The q k , k = 1, ..., r , are in BoT"(@)for some a,0 < a < 1, and can be continued to C in such a way that they are in BoTa(C)and vanish outside of a sufficiently large circle. Let tO(Z) := z ,

ektk(z) E B'."(C).

From (1) above T := Also, from (1.13)

a

Dt = -t

az

+ 4 -azat k-l k=l

=

a

j=O

0.

Hence t(z) is a generating solution. Note that t(z) + E, E nilpotent, is also a generating solution. The following property of t will be used frequently in what ensues: For some constant M, (1.14) 1

is an alternative notation for (t(5) - t ( z ) ) - ' . This t(Z) inverse exists as long as 5 - z, the complex part of t(5) - t(z), is not zero. Inequality (1.14) follows easily from Recall that

t(5) -

Since T E B'*"(C), the result is a consequence of Taylor's theorem. Theorem 1.2 (Douglis, 1953) (Green's identity) Let @ be a bounded region whose boundary @ is a Jinite number of closed piecewise differentiable curves. Then i f u , u E C'(@) are hypercomplex functions,

2i

t,(u

Du

+ u Du) dx dy

=

uu dt(z),

13


where t is the generating solution for

(3

and

PROOF Under the above hypotheses Green's identity is valid; this implies that for any complex valued (and hence hypercomplex valued) function w 2i

w zdx dy

=

1. cv

w dz and 2i

]Icv

w, dx dy =

-

(1.15)

Define the adjoint operator

D + w = w7 + (qw),;

then from (1.15)

Note that since Dt

+ qt, = 0, we have = t: dZ + I , dz = t,(dz - q dZ).

= t,

dt

Thus, using the identity D + ( w , w 2 )= w 2 Dw, + wI D + w 2 ,we obtain

=

/J&u

= I]Jt,(u

Du

+ u D+(t,u))dx dy

Du + u Du)+ uu D't,) dx dy.

Since

a az

D + t , = -(t,)

a + -(qt,) az

a

= -(tz

az

a Dt + qt,) = az

=

0,

this gives the desired identity. Theorem 1.3 (Douglis, 1953) Let (3, 6, and t be as in the previous theorem. Let u E C'(@ be a hypercomplex function and let zo E (3. Then

'I

- 7r

t, Du dxdy. 8 t(z) - ~ ( z o )

(1.16)

14


PROOF Let

> 0, and define BE := @\Ss(zo),

E

{z : ( z - zol < E } and

is such that S,(zo) C invertible hypercomplex function u

Du

u-'

E

Du,and, in particular,

- t(zo))- I =

in Sc(zo). By Green's identity,

2i

II. tz

'D t(z) -

where S,(ZO) = note that for any

+ u Du-' = D(u-'u)= D(I) = 0.

Hence Du-' = -u-' D(t(z)

a. We

-(r(z)

-

t(zo))-' Dt(z)

b

=

o

4z) Lol=E u(z)

t(Z0) d x d y = . t

-

t(z) - t(Z0)'

Now t(z) = z + T ( z ) , where the nilpotent part T is in B',"(@). On the boundary of S,(zo) we have z = zo + E d e , and hence ( d t (z)l

=

liceiedo1

+ I(ieeieTz- iEe-"TZ) dfll G MIE do.

From (1.14),

Now

where

and hence IF(Zo)l

(MIE)(M~/E)~M~ = 2MIMZM3 3

where M 3 := S U ~ ~ lu(z) ~ - ~ u(zo)l ~ ~ + =, 0~ as Let A ( z , zo) denote T ( z ) - T(z0). Then

E

+0

since u is continuous.

Since dT

k+ I

Lk +d l ( L z )- z o

k+ 1

2-zo +(L)

-, dz

z-zo

15


we have

(1.17)

Thus

1

Z--20l=E

Hence

and as

E +

dt(2) = 27ri. t(Z) - f ( Z o )

0 we obtain

If u is hyperanalytic, then as an analog to Cauchy's integral formula we have Corollary 1.4 If u is hyperanalytic in

a, then (1.18)

From the Cauchy integral representation (1.18) we can derive the following useful representation of a hyperanalytic function in terms of r + 1 analytic functions. Once again, we employ the notation A(z, zo) := T(z) - T(zo). Note that

16


Thus

Definition For f E C(@), let

where A(z, zo) := T(z) - T(zo). Note that C is a linear operator over the hypercomplex numbers. We shall now continue with the development of hyperanalytic function theory. Most of the results stated are due either to Douglis (1953) or Kuhn (1974). A hypercomplex function each component of which is TERMINOLOGY an analytic function will still be called an analytic function. When emphasis or clarity demands it, we shall use the more cumbersome hypercomplex analytic function or complex analytic function.

Let us introduce the hypercomplex analog of the a/az partial derivative. Definition Let

a az

D := a ( z ) where

Note that Dt = 1.

+ P ( Z ) -aza,

17


and

=

Ckolfi +

1

Ar 4

ar+Y

az"'

Note that Nq = 0 since both q and A are nilpotent. (2) Dt = 1 implies that b A = 1 - p:

j=l

=

1 ajf 1 . aj+'f 1 ar+Y AJ-1 ( j - I)! azJ + j = o-j ! AJa azJaz +,upr. azr+' ,

C

C[Zolfi + aC [zolfi.

(3) Cf cg = r-k

CC-A k!l! r

k=O

k+lakfalg

--

azk az'

Corollary 1.6 C[zolf is hyperanalytic in any domain in which f is analytic.

The following computation will be used in several proofs: Lemma 1.7 Let (l/j!) A(z, zo)' :=

X;=j djl(z, zo) el,

1 s j s r. Zff

:=

18


x.[l=ofkekis analytic, then for n D"C[Zo]f =

=

0 , I , 2, ...,

if p ' e k + i:c 2 e k d m / f k - / . k

/

(m+n)

k=l /=I

k=O

m=l

PROOF From Theorem ( 1 3)(2),D"C[zolf

r

r

k

= C[zOlf('').Hence

I

after rearranging sums and relabeling indices. Note that this lemma says

((*'*)k

means the kth component of (.-.))

f I?), (b"C[zolf)I = f l + dldb"+l),

(~"c~z01f)= 0

and so forth, i.e., (D"C[zo]f ) k depends on f i only for j

S

k.

Theorem 1.8 (Dough, 1953; Kuhn, 1974) Let (8 be the domain of Theoand w E C'(@) that is hyperanalytic rem (Z.2). Then f o r each zo E in (8, there is a unique set of r + I (complex valued) analytic functions cfo~zol(z),...,f,[z~l(z))such that w(z) = C[zolf [zol(z), where f[zol = E.[l=,fk[zo1ek.

PROOF From (1.18) and (1.19) w(z)

=

C[zolf[zol where

This establishes existence. For uniqueness suppose that C[zolf = 0. Since by Lemma (1.7) k=O

k=l / = I

an easy induction argument shows f

=

0.

m=l

19


With Theorem (1.8), it is now easy to see that a hyperanalytic function defined on a domain @ that is not bounded, or otherwise does not satisfy the hypotheses of Green's theorem, still has the representation w = C[Zolf. Corollary 1.9 ( D o u g h , 1953; Kuhn, 1974) Let @ be a connected component of a domain @. Given a function w(z) which is hyperanalytic in @ and a point zo E 6,there is a function f [ z o ] analytic , in @, such that w(z) = ~ [ z o l f [ z o l .

PROOFBecause of connectivity, there is a Green's domain that contains zo and any other given point z I . From Theorem (1.8) there is a function f [ z o ] ,analytic in aI,such that w(z) = C[zo] f [ z o l .The uniqueness part of this theorem implies that f is independent of the choice of since all such domains must intersect in a neighborhood of zo. We now wish to investigate the convergence of the formal power series m

w(z) := where each c, :=

C cn(t(z) - t(zo))",

n=O

(1.20)

Xi=o cnkek.Also, let

S,(zo) := {z : Iz

- zol

< 4.

Since t(z) - t(zo) = C[zo](z- zo), Theorem (1.5) implies that the partial sums of (1.20) are N

This suggests the relation w(z) = C[zo]f.We verify this in the next theorem.

Theorem 1.10 ( D o u g h , 1953; Kiihn, 1974) Let p = mink Pk. Then S,(zo) is the largest disk in which w(z) converges absolutely. The series w(z) converges uniformly on compact subsets of S,(zo). Moreover, w(z) where f is defined by (1.21).

=

C[ZOlf,

20


PROOF Let Then

<m

as N + m when z E S,(zo), since all theft' converge absolutely in S,(zo). Thus w(z) converges absolutely in S,(zo). Since

the convergence is uniform on compact subsets of S,(zo). Finally, suppose that a

n=O

Icn(t(z) - t ( ~ o ) ) ~o. 4P

By virtue of Lemma (1.24) the proofs of these theorems, which are based on the Hadamard inequalities, are identical to those for the complex case (see I. N. Vekua, 1962). An n-dimensional version of Theorem (1.26) will be proved in Chapter 4. Corollary 1.27 Jcrfis a compact mapping of L2(@ into itsev.

PROOF By the theorem Ja maps sets of functions that are bounded in the L2(@)metric into sets that are bounded and equicontinuous in this metric. Hence the result follows from Arzela's theorem.

30


Theorem 1.28 Let @ be a bounded domain and A(z) E Lp(@)for some p > 2. Then for p / ( p - 1 ) G p' s 2 p / ( p - 2), J d A f ) is a compact operator on Lp'(@).Also, if n > - (2P i + p ' - : ) b1 n - l , P - 2 P

o 2. Then

(1.28)

PROOF From Corollary (1.23), w(z) = @(z) + Jcr(Dw),where @(z) is hyperanalytic in @ and continuous in @. Noting that Jw(Dw) is hyperanalytic in C\@ and vanishes at infinity, we conclude that @ has the given form. Corollary 1.32 If$ is hyperanalytic outside of a bounded domain @, continuous on and It-"(z)$(z)l remains bounded as Iz( +- m for some nonnegative integer K , then there are hypercomplex constants ak, k = 0 , ..., K , such that

m,

where

32


PROOFApply Theorem (1.30) to

4. GENERALIZED HYPERANALYTIC FUNCTIONS AND

LIOUVILLE'S THEOREM

In the next two sections we shall be concerned with deriving an analog of the Cauchy integral formula for (2r + 2) x (2r + 2) (real) elliptic systems. As we shall see there is a fundamental distinction between r = 0 and the general case for r > 0, namely, the absence of a Liouville theorem for the latter. First, let us return to the reduction to complex form begun in Section 1 and consider the case in which lower order terms are present. After transforming to Jordan normal form, system (1.1) of dimension 2r + 2 becomes u,

+ diag(Jo, To, ..., J,, S,)u, + Lu = F.

(1.30)

The first two terms separate into pairs of subsystems of dimension rj + 1 that have the form (see (1.6)) (1 - iAj)wjz+ (1

+ iAj)wiz + ie(wj, - wjT),

(1 - iXj)wj+l,z+ (1

+ i s ; i ) ~ ~++ie(wj+t,z ,,~ - W~+I,~).

where wj = (wjo, ..., wjr,)' (the prime indicates transpose). Consistency requires that wj+I = y.Thus (1.30) is reducible to a system of dimension r + 1 of the form (see (1.10)) w,

+

Qw, + A w

+ Bn

=

7.

(1.31)

The form of the matrix Q,which we shall call subblock quasi-diagonal,

33

4. GENERALIZED HYPERANALYTIC FUNCTIONS

is

0 .

(1.32)

Such matrices commute with other matrices of the same form and are nilpotent when the entire main diagonal vanishes. Let 0 be any one of the subblocks. For this discussion, each of the first ro + 1 entries in 0 will be considered to be a subblock. Let i3 denote the part of w that multiplies 2, in (1.31). Denote the domain of definition of (1.1) by a. If Go denotes the main diagonal element of Q, then as before we consider the Beltrami homeomorphism p = p(z), satisfying Pz

+ GOP,

=0

(recall that lQol lC@; + [P,(l

- B02,~1-1b2~-q01 + Q)16,).

The coefficient of i3, is nilpotent quasi-diagonal. Denoting it by Q, we have $2 + QCZ = [&(Z - zjoQ)] Di3, ( I .33) where

a ap

a

D := - + Q - .

aP

Let t(5) be the Douglis generating function for the operator D. From (1.28) we have

34


Changing variables yields

and thus

where f ( 7 ) := t(p(7)) and it is now convenient to write duTin lieu of the measure dh d p . Let i ( z ) denote the subblock quasi-diagonal matrix consisting of subblocks of the form t(p(z)), and also define

bw

=

wz

+

Qwz.

Note that from (1.33) D'i = 0. With T as defined, ( I .34) now has the form of (1.28) with 'iand 0 replacing t and D . It generally will be the situation that the necessity to distinguish between the subblock quasi-diagonal case and the quasi-diagonal (i.e., hypercomplex) case is transient. Consequently, we shall not distinguish between the two notationally, and we now revert to the notation t, D , Q , A, B, and f for T,D, Q , A , B, and 7 with the understanding that they are as defined above. When it is necessary to make a distinction we shall so indicate. We continue to use the term hyperanalytic in the subblock quasi-diagonal case. Definition A solution to (1.31) is a generalized hyperanalytic function.

Let us note that Theorem (1.25) remains valid for the subblock quasidiagonal case with the minor amendment that the Holder index in Theorem (1.25) (2) is now ( ( p - 2)/p) min 6, where the 6 range over the Holder indices of the Beltrami homeomorphisms of the various quasidiagonal subblocks. Note also that the product of a matrix and a vector is the hypercomplex product in this case. In general, we d o not assume the matrices A and B to have subblock quasi-diagonal form. However, the conclusion of Theorem (1.28) is valid

35


for arbitrary matrices A since lAfl

S

IAllfl, where

(1.35) When A has subblock quasi-diagonal form, this norm coincides with the hypercomplex norm. Note also that the product of a matrix and a vector is the hypercomplex product in this case. Let us consider the operator MW := w

+ Jc(Aw + BE) .

(1.36)

Observe that solutions to (1.3 1) satisfy integral equations of the form Mw = @

+ J&,

(1.37)

where is hyperanalytic, and, conversely, any solution to (1.37) satisfies (1.31). In the Bers-Vekua case (r = 0) Mw = 0 has only the trivial solution. This is a consequence of the fact that solutions to w7 + Aw + BE = 0 satisfy a Liouville theorem. That this is not in general the case for r 3 1 is shown by the following example. Example (Habetha, 1976) Let Q = 0. Then t ( z ) = z. The equation Mw = 0 where B = 0, A = 0 for IzI > 1 , and for IzI S 1 A :=

L-1

has continuous solution

( I - ?z2, z - Qz’z’)’ w = { (0,3z-’)’

0

J for (zI s 1 , for IzI > 1 ,

as can easily be verified from (1.28). Note also that w(w) = (0, 0)’ and thus Liouville’s theorem does not obtain. Thus in general we must assume that Mw = 0 has nontrivial solutions. Since this will entail substantive difficulties, let us first investigate a case in which Mw = 0 has only the trivial solution, namely, the situation in which A and B have subblock lower triangular form. We follow the work of Gilbert and Hile (1974). Note that this case contains the Bers-Vekua case r = 0. For the remainder of this section we work with the system Dw

+ C ek C ( A u w l + &Wl) r

k

k=O

1=0

=

0.

(1.38)

36


Because of the assumption of subblock lower triangularity it suffices to work with each subblock separately. Hence (1.38) represents the system for a subblock only. Moreover we may take the Beltrami homeomorphism associated with this subblock as the independent variable. Definition Let LP3”(C)be the space of functions w such that

( I ) w(z) is defined in @. (2) If C0 = {z : (zI s I}, then w(z) E LP.”(@,). (3) If w(”)(z):= I Z I - ~ w(l/z) for z E C0,then w(”)E LP(@,). We define the norm of LP*”(@) to be Iw,

@I,.

:= Iw,

@alp +

Iw(”), @0lP.

REMARKIf w is defined in an unbounded domain @ by setting w = 0 in @\a.

a, we extend it to

Theorem 1.33 (Gilbert and Hile, 1974) Let u E LP9’(C),where 2 < p < CQ. Then J u := Jcu satisfies (1) IJu(z)l 6 M ( q , P)l& @IP.Z1 z E @. (2) IJu(zJ - J U ( Z Z ) ~M ( q , P ) / v ,@lp.zIZ~ - z2I( P - 2)/2. (3) For any R > 1 there is a constant M ( q , p , R ) such that for IzI 3 R

IJWls M ( q , P , R)lu, @lP,zl~1‘2-p)’p. (4) D(Ju) = u in @.

PROOF (1) Let w(z) = Ju(z). Then

:= @(z)

+ G(Z),

5

=

5 + iq, z

=x

Since u E LP(C0),we have by Theorem (1.25)

l@(z)l

W q ,p)lv, @,Ip

for z E C.

+ iy.


37

By a change of variables,

Since

t5 is

bounded and (1.14) holds,

We can argue as in I. N. Vekua (1962) that the integral is bounded by

and hence we have

we have

38

1 . ELLIPTIC SYSTEMS IN THE PLANE

It can be shown (I. N. Vekua, 1962) that

by Holder's inequality. From (l),

where a = ( p - 2)/p. The function x"/(x - 1) takes its maximum in ( R , 03) at x = R , and thus lzl/(lzl - 1) s R / ( R - 1) for IzI 5 R > 1, from which it follows that

39


(4) The proof of this part is the same as that in Theorem (1.22) except that the use of Fubini’s theorem is now justified by the following computation:

as was shown in part (1). Since supp 4 is compact, we have the result. Corollary 1.34 If w E L;oc(C),u E LP-’(C),2 < p < in C, then w = @ + J u , where @ is hyperanalytic.

PROOF D(w

-

Ju)

= u -

03,

u = 0.

Theorem 1.35 (Gilbert and Hile, 1974) Let w , u , and wu E Du E Lp(@),2 < p < CQ. Then D(wu)

PROOF I f w

E C ’ ( @ ) ,4

and u = Dw

=

WDU

and Dw,

+ UDW.

E CA(@), then

= -

t,uw

04 dx dy

Hence we have the result when one of the functions is C’(@). Let be a bounded subdomain of (3 such that GI C @. By the corollary to Theorem (1.22) we have w =@

+ Jcv,(Dw),

u =9

+ Jcr,(Du)

in @’, where @ and 9 are hyperanalytic. From Theorem (1.23,

40


Ja,(Dw) and J@,(Du)are in Bo*(p-2)'p(@), and thus in BI, D(wu) = Q, Du

+ * Dw + D[J(sJ,(Dw)Ja,(DU)l. *

Hence it suffices to prove the product rule for D [ J C ~ , ( D- wJc@u)]. ) There exist JI, E Cl(8,) such that JI, + Ja,w in Lp(Bl),and by Theorem (1.20), Jol JI,(z) E C'(BI). From the above we know that for 4 E

CW,

By Theorem (1.25) (l), JC$#, + J@,(Dw)uniformly in @; also, by this theorem,

Theorem 1.36 (Gilbert and Hile, 1974) Let w E L,'oc(8), Dw E Lp(8), 2 < p < 0 3 , and w(z) = wo(z) + N ( z ) , where wo(z) is complex-valued and N ( z ) is nilpotent. Let the values of w o lie inside a bounded domain a*,and let f be a complex function that is analytic in a domain containing @*. Then in 0,Df(w(z)) = f'(w(z))(Dw)(z).

PROOFExpanding f ( w ( z ) ) into a power series and rearranging yield

If is a bounded subdomain of 8 such that @I C 0,then for 4 E C:.(SJ we have w = where Q, is hyperanalytic. + Jal(Dw) in There exist JI, E Cm(8Jsuch that JI, + Dw in LP(a1), and so by Theorem (1.25) the sequence w, := + Js,JIn converges uniformly to w in B l . If we write w, = (w,),,+ N,, where N , is nilpotent, then we have ( w J o E 0*for n sufficiently large, and

41


uniformly in 8. Thus

r

r

The next theorem establishes a Liouville-type theorem for solutions of (1.38). As we shall show, this prevents rank deficiencies in the operator w + J(Aw + B E ) .

Theorem 1.37 (Gilbert and Hile, 1974) Suppose that in (1.38) Akl,Bkl E LpS2(@), 2 < p < m, and let w be a solution of (1.43) which is continuous and bounded in @. Then w has theform w(z) = C exp w(z),

(1.39)

where C is a hypercomplex constant and w is a hypercomplex function in Bo*"(@),a = ( p - 2 ) / p . Moreover, w(z) = O(lzl-") as IzI + m.

PROOF If w = 0 in @, we obtain the result by setting C = 0, w(z) = 0. Hence if w := wkek, it suffices to assume that some wk is not identically zero. Let ko denote the smallest such k. Since w is continuous we have from (1.38) Dw E LP**(C). and bounded and Akl,Bkl E LP**(@), Since wk = 0, k < ko, and the koth component of the left side of (1.38) must be zero, we have

a az

-wko

+ A k o k o W k o + Bkokowko

= 0.

By the corresponding theorem for the complex case, wb is bounded away from zero, and hence ( E i = k o wkek-ko)-'exists and is bounded and continuous. Define u to be the function

42


Then u E Lps2(@), and

=

-Dw.

Define w := -J u , @ := w exp( - w ) rule and the chain rule, we have D@ = (Dw)exp( - w ) =

-wuexp(-w)

=

w exp(Ju). Employing the product

-

w(Dw) exp( - w )

+ wuexp(-o)

=

0.

Thus

is hyperanalytic in @. From Theorem (1.25), w is bounded, and O(lzl-"), a = ( p - 2 ) / p as IzI + m. Thus if w = wo + E(z), E(z) nilpotent, then

o =

is clearly bounded in @. A hypercomplex version of Liouville's theorem can be proved in the same way as in the complex case since we know that

for all n and any r > 0. Thus we conclude that @ is constant, which implies the result. Corollary 1.38 If w satisfies the conditions of the above theorem and w(zo) = 0 for some zo in C (including zo = m), then w is identically zero.

PROOFIf in the above theorem, w

= wo

+ E, E nilpotent, then

If C := Xi=o ckf?k , then co = 0. The result now follows by an induction argument. Let

43


Theorem 1.39 (Gilbert and Hile, 1974) If each Akl,Bkl is in L"*'(@),2 < p < m, then Q is a compact operator on Bo(C), and it maps this space into Bo9"(@),a = ( p - 2 ) / p . Moreover, at inJinify, lQw(z)l = O(lzl-") as 121 + m.

PROOFIf w

E

Bo(@),then u :=

c ek E ( A k I w +I Bk/wl) r

k

k=O

/=o

I

k

/

\

By (1) and (2), Qw(z) E B0**(@),and so any uniformly bounded family in Bo(C) is mapped into a uniformly bounded and equicontinuous family. Hence Q is compact by Arzela's theorem. That lQw(z)l = O(lzl-") as JzI + co is immediate from (3). Theorem 1.40 (Gilbert and Hile, 1974) If each A k l ,Bkl is in LpV2(@), 2< p < w, and u E Bo(@),then the equation

w + Q w = h has a unique solution in Bo(@).

PROOF We show that w + Qw = 0 has no nontrivial solutions. If w o is such a solution, then it vanishes at infinity by the preceding theorem. Thus w o is a solution to (1.38) that vanishes at infinity. Corollary (1.38) gives w o = 0. Theorem (1.40) permits us to give an analog to the "generating pairs" of Bers (1953). Suppose that w is continuous and bounded in @ and satisfies (1.3% with Akl,Bkl E LpS2(@), 2 < p < 03. By Corollary (1.23), w + Qw = Q, for some hyperanalytic function 0.Since w, Qw E Bo(C), Q, = c , a

44


constant, by Liouville's theorem. Thus w

+ Qw

= c = S

+i?,

(1.40)

where c' and c^ are real hypercomplex numbers (i.e., each coefficient of ek is real). By the previous theorem, there exist unique k, ii, E BOW such that k+Qk= 1

and

G+QR=i.

The unique solution to (1.40) in BO(@) is now easily seen to be w = + Fii,. Hence for a given set of Akl and Bklrevery solution of (1.38) must be of this form. The solutions k and ii, form the generating pair associated with the coefficients Akl,Bkl and Eq. (1.38). c'k

5. CAUCHY REPRESENTATION FOR GENERALIZED HYPERANALYTIC FUNCTIONS

We now return to the study of Eq. (1.31): Dw

+ Aw + BW

=

f,

where A and B are not assumed to have any diagonal form. As noted in the previous section, we can solve (1.31) by solving M w = h , where M is given by (1.36). Henceforth we assume without explicit mention that A and B are (r + 1 ) x (r + 1) matrices having elements in LP*'(@) for some p > 2. Let us consider the solvability of M w = h , where h E L2*'(@).We have

where A , B E LP7'(@),p > 2, and L ( z , 5) is the subblock quasi-diagonal matrix consisting of the subblocks of the form

The matrix L is in L"(@). Let w be an Lzg'(@) solution to M w = h , and define

wdz)

= w(z),

wz(z)

= (1/Z)W(1/Z)

(1.42)

45

5. CAUCHY REPRESENTATION

for z E C0 := {lzl : IzI s 1). Then

+ Z2B(5-')w2(5)1dcC

=

h2(~).

Here hi, j = 1, 2, are defined in the same manner as (1.42). We write the above more concisely as M ~ W:=

w

+ ffG(KI(z, 5)w(5) + KAz, 51W5)) dug = h9

where

L

are (2r + 2) x (2r + 2) matrices. Conversely, given an L'(@~)solution to (1.43), we can use (1.42) to define w and h and by reflection conclude that w satisfies Mw = h. Note also that w E L2(C0) if and only if

46


w E L2*'(C).Thus finding all L2*'(C)solutions to Mw = h is equivalent to finding all L2(Co)solutions to (1.42). Solving the latter problem is, as we shall see, easier by virtue of the fact that the integral term in Mo is a real-linear compact operator (by Theorem (1.26)) on the Hilbert space L2(Co).We now proceed to study solutions to Mow = h E L2(Co). We take as our inner product

where (wI, vI) denotes the Euclidean scalar product adjoint to Mo in this inner product is M,*v = v

-

/la

E:=o wljulj.The

(KK5, z)v(O + G ( 5 , z)v(O) d u t y

where primes indicate matrix transpose. By Corollary (1.27), .IG is a compact operator in L2(Co).Consequently, there is a finite number of functions vl, ..., vN, which are linearly independent over the reals, such that M$vJ = 0. Likewise there is a set of N functions w', and satisfy

..., wN that

are independent

MowJ = 0. We may choose both sets to be orthonormal:

[w', dl is

=

[v', vk] = 6jk.

The necessary and sufficient condition for Mow = h to have solutions j = 1,..., N .

[h,v'] = 0 , Let a 0 w := Mow

+

c [w, N

W'IV'.

I= 1

Lemma 1.41 k o w = 0 admits only the trivial solution.

PROOFIf wo is a solution of k o w o = 0, then [kOwo,v'] = [Mowo,v']

+ [w0, w']

=

[w', w'];

hence [wo, w'l = 0 f o r j = I , ..., N. This implies that Mowo = 0 and that wo is orthogonal to all of the wj. Hence wo = 0. Thus k o w = h has a unique solution for each h E L2(C,). This solution

47

5. CAUCHY REPRESENTATION

has the resolvent representation

Substituting (1.44) into $tow = h and noting that the resulting equation must be true for all h E L2(C0),we obtain the integral equations for r,

Also, the identity w = R[fiow]must be satisfied for all w E L*(C,). This yields the two additional integral equations

rr

2 '{W'CZ), 2

/=I

w1(7)} = 0.

(1.46)

48


In obtaining these relations we used w' = R[u'], which is a consequence of Elow' = v'. If solvability condition [h, v'] = 0 is met, then a solution of &low = h is also a solution of Mow = h since

+ [w, w'l

0 = [fiow, v'l = [Mow, v'l =

[w,Mo*v'] + [w, w']

=

[w, w'l,

which implies that f i o w = Mow. All solutions of Mow = h are then given by w = R[h] + EL, c'w', where each cI is an arbitrary real constant. We can perform a similar analysis for Mzv = h. Let R*[h] denote the resolvent of M,*v = Mov + [v, v'lw'. The resolvent kernels fig are TXz, 5) and T&, 5). Thus if w is a solution of Mow = h, [R[h]

N

+ I= I C'W',

Wk] = [R[h], Wk] +

since R*[Wk] = vk and [h, vk] = 0, k ck =

=

ck

1,

=

[h,R*[Wk]] +

..., N.

ck

=

ck

Hence

[w,Wk].

We also have the following regularity result: Theorem 1.42 (Goldschmidt, 1980) Suppose that Mow = h, where h E Co9@(@) for some 0 < /3 < 1. for a = min(p,((p - 2)/p)G), where G ranges over for the Beltrami homeomorphisms for the subblocks

w E L2(C0) and Then w E Co3m(@o) the Holder indices of Q.

PROOF Let P'W := - J J ~ ( K~, ( z , 5)w( 0 is in L2~'(@). If we substitute the right-hand side into (1.50) and change the order of integration, we obtain

where

The flj are called the generalized Cauchy kernels for A and B. Since ( z - T)rj(z, T ) E L**O(C)x L2(C), it is not difficult to see that

for some 0 s a < 1 . Hence the dominant singularity of the generalized Cauchy kernel is that of the hyperanalytic Cauchy kernel. We summarize. Theorem 1.45 (Bojarski, 1966; Gilbert and Hile, 1974) Let @ be a regular and continuous in domain and suppose w is a solution to (1.31) in @. Then

27ri

I

Q

fldz, 5)w(5)dt(5) - W z , 5 ) N J df(5)

(1.56)

Note that in view of Theorem (1.39)no w' terms appear in the representation (1.56) when the matrices A and B are lower triangular.

6. M-ANALYTIC FUNCTIONS

53


In this section we outline a somewhat different approach, due to Hile

(1977), to obtaining a function theory for elliptic systems. For a related approach see Horvath (1961). Rather than transforming to Douglis normal

form, we work directly from the homogeneous equation

fx + M f ,

(1.57)

0.

=

The matrix M is assumed to be constant and nonsingular; hence our considerations are somewhat less general than in preceding sections. The matrix M will in general be assumed to be complex-valued. The unknown complex-valued function f will be taken to be an m x s matrix. The assumption that the system (1 37) is elliptic requires M to have no real eigenvalues, i.e., det(xZ + y M ) # 0

for all (x, y ) E R2\{0, 0).

(1.58)

As the independent variables in this function theory we take the matrices Z := XZ

-

yM-’,

2

=

XZ

+ yM-’.

The matrices 2 and Z are invertible for all (x, y ) except (0, 0). Both the Euclidean norm IZI := (x2 + y

and the matrix norm IlAll = (Trace A*A)’/’ =

(1.59)

y

(5

i.j= I

Iaij)”’

(1.60)

will prove useful. Here A* is the conjugate transpose of A . The Cauchy-Riemann operator is

and its conjugate operator is

Thus the homogeneous system (1 37) may be written more concisely as

af/aZ

=

0.

(1.61)

Following Hile (1978), we term solutions to (1.61) M-analytic functions.

54


A function f is M-differentiable at 2-(x, y) if

exists. It can be shown that M-differentiability is equivalent to = 0. Each M-analytic function is the solution of a higher order elliptic differential equation. Let

af/a2

p(M) := M"

+ a,-IM"-' +

+

.**

uIM

+

uOZ

=

0

be the minimal monic polynomial for M. Since M has no real eigenvalues, p ( x ) has no real roots or, equivalently, q ( x , y) := y"p

(-);- c =

aj(-l)Jxjy"-j

j:o

has no real roots in R2\{(0, 0)). Sincef, = -M&, it follows that

a'f -- (-M)J-... a'f _

aYJ Consequently, f satisfies the differential equation ax'

Functions that are C1(@)satisfy the Cauchy-Pompieu representation w ( Z ) = Q-'

(S - Z ) - ' dS w(S)

where This can be shown by employing Green's theorem. The matrix Q that occurs instead of 27ri can be shown to commute with M. It can be found from the monic polynomial p. Let

Z(X - 5)"' I 5

@(x) =

be uniquely determined by

j=

55


Then Q = 27riP(M) (see Hile (1978)). In particular, if all of the eigenvalues Aj satisfy Im Aj > 0, then Q = 27rZ. If M2 = -Z, then Q = 27rM. Functions that are M-analytic in a disk IZ - Zol < p have Taylor series expansions (1.62)

that are uniformly convergent on subsets of the disk

lz - Zol < P/YlY2.

Here y1 and y2 are constants such that

llzll

YdZI and llz-’ll 6 Yzlzl-’ are valid for all Z. In general, y l , y2 are >1 . It can be shown by example that the power series (1.62) need not converge up to the first singularity off(see Hile (1978, p. 967)). A function that is M-analytic in the disk 0 < IZ - Zol < p has a Laurent series where An

{(SIS

:= Q-’

- Zo)--l d S f ( S ) , -

zol< p .

Convergence is uniform on compact subsets of 0 < IZ - Zol < p/yly2. There are instances in which the most straightforward reduction of a higher order elliptic equation to an elliptic system does not result in a nonsingular matrix M. For example, consider the fourth-order equation

A24(x, Y) = g(x, Y, On introducing the unknowns

v4,A49 V(A4)).

and taking into account the consistency conditions -au, =-

ay

we are led to the system u,

au2 ax’

-au4 =-

ay

+ Mu, = h ,

au, ax’

i7 1

56


where 0 M : = 0I

0 0 0 1 0

-0

1 0 0

0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 -.1

.

h :=

In the above example, M - ' does not exist; however Gilbert and Hile (1981) have developed a function theory for the case in which M possesses an appropriate generalized inverse. Definition A matrix M has a group inverse M" if

(1) MM#M = M , (2) M'MM" = M", and ( 3 ) MM# = M#M. It is the commutativity condition (3) which makes the group inverse particularly suitable among generalized inverses for the development of a function theory. We mention several properties of the group inverse. For proofs and further properties the reader is referred to the book by Ben-Israel and Greville (1980). (a) (b) (c) (d) (e)

MY exists if and only if rank M = rank M 2 and it is unique. If M-I exists, then M" = M - I . (Mk)" = (M")&,k an integer. (M")" = M . If M" exists and S is invertible, then (SMS-I)" = S-IM'S.

In what follows, we assume that the matrix M in (1.57) has a group inverse M' and that all of its nonzero eigenvalues have nonzero imaginary parts. Let E denote the idempotent matrix MM", and define Z = XE

-

yM",

2

=

XE

+ yM".

The correspondence with points in R2 is one-to-one. If Z # 0, then Z" exists and ZZ" = E. For Z , Z I , and Z 2 , we have the commutation relations ZM

=

MZ,

ZM"

=

ZlZZ = 2221, (Gilbert and Hile, 1981).

M'Z,

Z'M

=

ZI"Z2 = Z 2 Z I " ,

MZ",

Z"M" = M'Z",

Zl"Z2" = Z2#ZI'

57

7. APPROXIMATE SOLUTIONS

Define the operators

Thus fx + Mf, = 0 may be written as E af/aZ = 0 . We once again designate C' functions satisfying this equation as M-analytic. The Mdifferentiability condition is now

d f ( ~:= ) f ' ( ~ ):= lim (m>#[f(z + AZ) dz

AZ-0

-~

(z)I.

For M-differentiable functions we have F ' ( Z ) = E(aF/aZ)(Z).It can be shown that the powers of Z", n = 0 , 1, 2, ..., are M-analytic, as are the powers of Z"". Moreover, d -(z')" = -n(z")". dZ The Cauchy-Pompieu integral relation for w E Cl,x,(8) is now Ew(Z)

=

Q" lb(S- Z)" dS w(S) + 2Q"M' /ls(S

-

aw Z ) ' =(S) d t dq,

as

where S = E( - M"q and The matrix Q commutes with both M and M'. When all of the nonzero eigenvalues of M have positive imaginary parts, Q = 27riE. The only possible eigenvalues of Q are 0 and +27ri. Taylor and Laurent series representations can also be established for M-analytic functions in the generalized inverse case (see Gilbert and Hile (198 1 ) ) . Numerous other examples that lead to a system (1.57) having a group inverse M' are given by Gilbert and Hile (1981). These include the isotropic plate equations, the equations for plane anisotropic stress, and the polyharmonic equation, for example. 7. APPROXIMATE SOLUTIONS

The problem of approximating solutions to the general elliptic boundary value problem Ru :=

Du := u, ru =

9

+ Buy + Cu = f

in on

g,

8,

(1.63)

58


in which the Lopatinski condition is satisfied, is treated in the book by Wendland (1979). If the system (1.63) has nullity N > 0, then N linearly independent conditions A,u = cI ( I = 1, ..., N) may be added, so that the problem has a unique solution. The approximate solutions are sought in a finite dimensional subspace f i h , which depends on a parameter h > 0, and which as h + 0 approximates the domain of R. A direct approach to approximating solutions of (1.63) is to minimize the defect

d x ) := llDx - fllC.(@) + Irx

-

$IC'.Y&)

+

c lA/X N

I= 1

C/I

(1.64)

for x E fi (0< LY c 1, (Y fixed). Let U be a solution of the finite problem, for every h, > 0.If A, is continuous and let U h r h o f i ( h )be dense in C'*m(G) then it may be seen that limkol)u - qlc~.. = 0. This result on C'*a(G), follows directly from the a priori estimate

By using the LP-norm estimate (Wendland, 1979), Y(IlvxIlv(@)

+

l I V y l l L p ~ B 1+

IlVllLp(@,) (1.66)

a more practical minimization problem is arrived at, namely, that of minimizing *

PO(X) := l P X - fllcwi1

For the method of least squares, without weighted norms, one is led to minimize the quadratic functional (Wendland, 1979, p. 347)

I P X - flltw, + Irx - $IL(@+ Ic IArX - C/I2 = I N

P d X ) :=

over

fi.?

If U minimizes this problem, then an equivalent statement is

t H'@) is a Sobolev space of fractional order r .

59

7. APPROXIMATE SOLUTIONS

that the Gauss equations

N

hold for all to be

x

E

(1.68)

8 with U

E

8. The above scalar products are defined

( I .69)

Equations (1.68) are equivalent to a system of M linear equations for the coefficients y i of U := ZE, yixi(x,y ) , where the xi form a basis for

8.

Stephan and Wendland (1977) used the method of weighted squares to solve (1.68). Assuming that the nullspace of the adjoint operator R* is 0,

d(x) := IIDx

-f

N

l l t 2 ( ~+ h-'lrx - +ltw + I = I ~ A I X- cl12

for x E 8. If U E equations

(1.70)

is the unique solution of the associated Gauss

[U - u,XI := (D(U- u),Dx)o + h - ' (r(U - u),rx)o N

+ I2 (MU = I =

[U,xl

-

where L(x) :=

fTDx dx dy

- u))

L(x) = 0

- AIX for all

x E 8,

(1.71)

+ h-' f6 $ T r ds ~ + c cIAIx, I= I N

they show that there is an optimal order of convergence, namely, they establish that for t d s and 1 s s s m + 1, 0 s t s m, there exists a constant c that is independent of h and u such that the asymptotic

60


estimate IIU -

4I"W

~~s-'l141~~(~v)

(1.72)

holds, provided that the A, are continuous on H o ( @ ) . For 1 s s s m + 1 the least square approximations converge uniformly with provided I? satisfies a stronger condition given in the book by Fix and Strang (1973) (see Wendland (1979, p. 353)). For further details concerning the discretization of the equations the reader is directed to Wendland (1979, Chapter 8).

2 Boundary Value Problems

1. INTRODUCTION

In this chapter we generalize two classical boundary value problems for analytic functions to generalized hyperanalytic functions, i.e., solutions to

Dw

+ Aw + BW

=f

,

D

a

=-

az

+ Q -a.

az

As in Chapter 1 we take the dimension of system (2.1) to be r + 1. Let @ + be a multiply connected domain in C that is bounded by a finite number of closed, nonintersecting C',"curves @ k , k = 0 , ..., m , with a0containing the remainder of the curves in its interior, We denote the domains composing the complement of @+ by @ ik, = 0 , ..., m , with (3; being unbounded. Let @ - denote Up=o@; (see Fig. I). As is usual, we orient a0so that counterclockwise is positive, and thus clo.ckwise is positive for the other @ k . If f ( z ) is a function defined on C\@, then for r E @ we denote byf+(r) the limit off(z) (if it exists) as z --f r from within @+. We shall write

Because of the orientation given to 6, we also refer tof+(r) as the limit from the left. In an analogous manner we define the limit from the right as f-(r) = lim f(z). 2-7-

61

62

2. BOUNDARY VALUE PROBLEMS

Fig. 1

Let us now state the two problems that will be considered. Hilbert Problem (H) Let there be given an ( r + 1) x ( r + .l) matrix H and an ( r 1)-vector h that are Holder continuous on Find a solution to (2.1) in 8 ' U 8 - that is Holder continuous in @+ and @-, vanishes at infinity, and satisfies the jump condition

+

on

6.

a.

W+

- Hw-

=

h

(2.2)

Riemann-Hilbert Problem (RH) Let there be given an (r + 1) x ( r + 1) matrix function A and a real-valued ( r + 1)-vector q that are Find a Holder continuous on 8. Suppose moreover det A # 0 on solution to (2.1) that is Holder continuous in B+and satisfies the boundary condition

a.

on

(2.3)

Re(Aw+) = c

6.

REMARKIt is possible to generalize the Hilbert problem somewhat: (1) the requirement that the solution vanishes at 03 can be replaced by the requirement that it have a finite order of growth at m; (2) the jump condition could be taken to be W+ -

H ~ w -- HZiL

=

h.

The solution of the problems (H) and (RH) involves the extension of some of the fundamental results on the Cauchy integral to the hyperanalytic case. For a comprehensive exposition of these results in the analytic case the reader is referred to Muskhelishvili (1953) or Gakhov (1962).

63

2. PLEMW FORMULAS

2. THE PLEMW FORMULAS

Let us first show that we can define the singular Cauchy integral

in a principal value sense when 7 E 6 and p i s Holder continuous on 8. We assume that each curve composing @ is parameterized with respect to arc length from some fixed point on the curve. Let 1 , ( ~ )denote the segment of 8 lying in the closed disk {z : Iz - 71 =s E } , and let T ~ ( E )= , ~ 1, 2, denote the endpoints of 1 , ( ~ )(see Fig. 2). We now restrict attention to a particular quasi-diagonal subblock of the matrix 1. As in Chapter 1, Section 4, p will denote the Beltrami homeomorphism for this subblock, t(p(z)) the corresponding Douglis generating function, and i: the dimension. In view of (1.17)

Define for

5, 7 E

6

c

Fig. 2

P(5)

9

s=s.

5

= 7,

64


where s, is the value of the arc-length parameter corresponding to T and the overdots denote differentiation with respect to s. Since T, p , and T ( S ) all have Holder continuous first derivatives with respect to their arguments, we conclude that N(5, T ) is Holder continuous with respect to each argument separately. Moreover,

=

lim log(p(5) e-0

-

~ ( 7 )= ) 7ri.

Thus in view of the assumed Holder continuity of p ,

Note that the integrand of the last integral is weakly singular, and hence this term is defined in the usual sense. Hence we attach the following principal value sense to Sp:

If

then for

T

E

6,

With an analogous argument for limits from the right we obtain the hyperanalytic Plemlj formulas.

65

3. HYPERANALYTIC HILBERT PROBLEM

Moreover, @ is Holder continuous in @ + and @ - and has Holder continuous left- and right-hand limits.

PROOF Formulas (2.8) have already been established. The Holder continuity of @ follows easily from the Holder continuity of the analytic Cauchy integral (see Muskhelishvili (1953)) and the Holder continuity of the first derivatives of the Beltrami homeomorphisms. 3. THE HILBERT PROBLEM FOR HYPERANALYTIC FUNCTIONS

In this section we solve the problem (H) for solutions to

Dw

=

f,

(2.9)

. where f E LpV2(@), p > 2. We assume that the matrix H : = hkek is hypercomplex valued, i.e., that it has quasi-diagonal form. The general (non-quasi-diagonal) case will be treated in a later section. We take the domains @ + and @ - to be the image domains of the Beltrami homeomorphism. All results obtained can be transferred to the subblock quasidiagonal case by a change of variables (see Chapter 1 , Section 4), providing, of course, we assume that H has the same subblock quasi-diagonal form as Q. First let us define for hypercomplex f = fo + F, F nilpotent,

(2.10)

logf = log&

+

I

k= I

fofO.

(2.11)

66


Theorem 2.2

Let f and g be hypercomplex numbers. Then

( I ) ef+R = efeg, (2) log ef = elo& =

f 9 f O

# 0.

PROOF (1) and (2) can be obtained by collecting powers of the nilpotent parts. All of the other elementary properties of exponents and logarithms can be derived from (1) and (2).

As in the analytic case (cf. Muskhelishvili (1953) and Gakhov (1962)) we seek a canonical factorization of the matrix H. Definition H(T) = X+(T)t(T)KXO-(T), T E 8, is a canonical factorization of H if (1) X ( z ) is hyperanalytic and invertible in a+; (2) Xo(z) is hyperanalytic in 8 - and invertible in 8 - U (3) K is an integer.

{m};

Let us start by seeking a piecewise hyperanalytic function satisfying the jump condition x+(T)

- H(T)x-(T)

=

0,

T E

Note that for a quasi-diagonal matrix det H(T) hO(T) # 0. Taking logarithms, we obtain log

x,

-

log

x-

=

G. # 0 is equivalent to

log H .

(2.12)

However, an ambiguity arises if log H is not single-valued. Observe from (2.1 I) that all considerations concerning single-valuedness reduce to those of log ho. Thus log H is single-valued if the change in the argument of ho is zero after traversing any of the bounding curves @ k . Definition The index of problem (H) is K

1

.

:= - Aii arg ho =

27r

1

-27rl .

d log ho =

2 Ak,

k=O

where Ak

and

1 27r

:= - A &

arg ho,

6 k is traversed positively.

k = 0, ..., m ,

67

3. HYPERANALWIC HILBERT PROBLEM

Let

zk

be a fixed point in the interior of

a;,

k

=

1, ..., m. Also, let

m

The complex part of P ( z ) is np=l(z Let the origin of coordinates lie in a+.The complex part of t ( ~ ) - ~ P ( z ) H is ( zZ) - ~ & ~ = I ( Z - Zk)*' h ~ ( zand ) Aci z

n(z rn

- ~

I= I

- Z ~ ) % ~ ( Z )=

0,

k

=

0,

..., m.

Thus if Ho := log t ( ~ ) - ~ P ( z ) H (then z ) , log Ho is single-valued and Holder continuous on 6.If instead of (2.12) we write lOg(P(T)x+(T)) - lOg(t(TyX-(T)) = log H o ( T ) ,

(2.13)

then no ambiguity arises and we have as a consequence of the Plemlj formulas (2.8)

Thus H(T)= x + ( 7 ) f ( 7 ) K X ~ - (for 7)

T

E

6, where ,

ZEa-.

Let us now return to the Hilbert problem for solutions to Dw = f. Define for a given function w Z E

w.

The problem (H) is clearly equivalent to DiT

=x

z

E

a+u (3-,

68


U (3- and @(m) = 0. Since the Pompieu where @ is hyperanalytic in operator is continuous in @, the jump condition for (HI) may be written as @+(T) - t ( T ) " @ - ( T ) = X - ' ( T ) ~ ( T) J c $ + f + t(T)"J($-f (2.14)

It is now necessary to distinguish between two cases. K 3

0.

From Corollary (1.31), we have upon taking Cauchy integrals

and

+ JC$+L z E a - 9 := Ek .K!,: akt(z)kis an arbitrary polynomial.

where p,-'(z) tion to (HI) must have the form

+ f(Z)"J 2 , and H is a Holder continuous matrix satisfying det H # 0 on 6. Recall from Chapter 1 that

where the Ljare weakly singular, i.e.,

ILj(z,{)I

MI5 - z I - ~ ,

where M is a constant and 0 < a < 1. From this we immediately infer that the Plemlj formulas (2.8) are valid with

(2.34)

adz,5)

-

P(5),

z E 6.

The integral Sp is defined in the principal value sense. Suppose that w is a solution to the problem (H). Then it has the representation (2.28) in terms of the complex density p = w + - w - , where p must satisfy (2.29). If we substitute this representation into the jump condition w + - H w - = h , we obtain the singular integral equation pp := (1 +

+

H(T))p(T)

(1 - H(T))(sp)(T) N

=

2 h ( ~) 2(Z - H ( T ) ) ( ~ , ) (T )2(1 - H(7)) C CIW'(T), I= I

76


where

( m z ) :=

-1. lr j j p d z , 5)f(5)

+

Wz9

5)SCO)da,.

Let us compute the dominant part of this equation. For each subblock o f t we have ( p denotes the Beltrami homeomorphism)

where N(5, z) is defined by (2.5). Let

P(T) and ) 6 ( ~ )denote (s is arc length)

The Holder continuity of p then implies that

is weakly singular. As 5

+ T,

and thus the dominant part is (I +

H(T))~(T)

+ (I

1

1.

1

-p(5) d5. lrz '55 - r

- ~ ( 7 ) )

The adjoint to P p under the inner product

(2.35)

77

5. GENERALIZED HYPERANALYTIC HILBERT PROBLEM

dr ds

((7):= - -

Q

-.d5 ds

Let { ~ j } ; = and ~ {v j } I'j = l denote the solutions to P p = 0 and P*v = 0, respectively. From (2.35) the index of the real-linear singular integral operator P is K

Hence J

-

J*

= K

1

:= - A& arg det H . T

(see Muskhelishvili (1953) or N. P. Vekua (1967)).

Lemma 2.8 Let v be a solution to

c aIit'w*', N

P*V

=

(2.36)

I= I

corresponding to some choice of the set a := {al, ..., aN}of real constants. Here w*I is defined by (2.26). For each set a for which (2.36) has solutions and each u satisfying (2.36) for that choice of a , define

' I.(a;(&',

w*[a, vl(z) = -2T1

(3

z)(Z - H')iF

-

a;({,z)(Z

-

H')is)ds

- - C arw*'(z). I N 2 /=I

(2.37)

Then the necessary and sufficient condition f o r a solution to

C C~W' N

P/.L = 2h - 2(Z - H ) J f - 2(Z - H )

I= 1

(2.38)

to exist and correspond to a solution to (H) through the representation

78


is

+ 2 Re for all choices of a and

IIC(f,

w*[a, u1) duz = 0

(2.40)

Y.

PROOFNecessity: Let w be a solution to (H). Then by Theorem (2.6) there is a density p and set of real constants c := {cI, ..., cN}such that w has the representation (2.39). The jump condition w + - H w - = h implies that p and c satisfy (2.38). Thus for any Y that solves (2.36), 2h =

-

2(Z - H ) j f - 2(Z - H )

Re I&(Pp, F) ds

=

I

N

2 I= I

C,W/,

Re &(p,P*u)ds

the last equality being a consequence of (2.29). Hence

(2.41)

Observe that (z = z(s)) Re I&((Z - HI@,

ij)

ds

(2.42)

79


Also from the Plemlj formulas, w*,[a,vI(7)

c

l N 2 /=I

- -

U/W*'

c U/W*/(7)

l N

--

2 /=I

(2.43)

= ['-'iv.

Substituting this and (2.42) into (2.41), we obtain h

-

+ 2 Re which is (2.40).

(I

c N

-

H)

llc(f,

C I W ' , ( - i ) [ ' w * , [ a ,vl

w*[a, v1) dv

=

0,

Suficiency: Assume that (2.40) holds for all w*[a, v]. The solutions to (2.36) corresponding to a = 0 are the vj. From (2.43), w*,[O,d ] = ['-'iFj, and thus, taking into account (2.42), we conclude that (2.38) has solutions. If, for a fixed set c := { c I ,..., c ~ } po , is such a solution, then all solutions are given by p = po + X;= I bjpJ.For p and c to correspond through (2.39) to a piecewise generalized hyperanalytic function, the condition (2.29)

1 = 1, ...)N ,

80


is necessary and sufficient. Hence it is a question of whether the bj can be chosen so that J=

1

Re

I ,

(pj, i

dQ.[w*')bj = -Re -

2 Re

ll,(f,

w*')d u , ,

1 = 1, ..., N . (2.44)

This is possible if for each set of real constants {al, ..., aN}such that (2.45) it is also true that N

du, = 0.

(2.46)

Since (2.45) is the condition for (2.36) to have solutions, (2.46) is equivalent to

which is equivalent to (2.41), which is in turn equivalent to (2.40). Thus (2.40) implies that we can solve (2.44) to obtain a density p corresponding to a piecewise generalized hyperanalytic function w. Since p also satisfies (2.38), w satisfies the jump condition and hence is a solution to (H). The above lemma suggests that we associate with problem (H) the adjoint problem

(H*) Find solutions to

ze&

-

D*w* - A'w* - B'w* = 0

9

(2.47)

that vanish at infinity and satisfy the jump condition

['-'H'['w*,

- W? =

0.

(2.48)

Theorem 2.9 (Bojarski, 1966) The necessary and sufficient condition for (H) to have solutions is

I

Re h(h, i d,.r w * , )

+ 2 Re

llc(f,

w*)duz = 0

(2.49)

81


for every solution to (H*). Zf k is the number of real linearly independent solutions to the homogeneous ( h , f = 0) version of (H) and k* is the number of real linearly independent solutions to (H*), then

k

-

k*

=

(2.50)

K,

where K

1

= - A(&arg det H.

(2.51)

7r

REMARK On the basis of this theorem we designate K to be the index of the problem (H). Note that it differs (by a factor of 2(r + 1)) from the index for the case of quasi-diagonal H given in Section 3. This is a consequence of the fact that the present index (2.51) counts real linearly independent solutions, whereas the index of Section 3 counts independent hypercomplex solutions. Note also that the sign of K no longer determines whether the problem (H) is solvable without restriction on h and f as it does in the quasi-diagonal case (Section 3). This is attributable to two differences. First, even for the analytic case ( Q , A, B,f= 0), the Hilbert problem can be solved without restriction only when all of the subindices of H, i.e., the unique integers K~ occurring in the canonical factorization H = X , diag{zKo,..., z K r } X O -are , nonnegative. However, even this is not sufficient in the present case, because the presence of solutions to Mw = 0 can prevent the problem (H) from having solutions for arbitrary h and f even when all subindices are nonnegative (Buchanan, 1980). Hence it does not appear possible to infer k* = 0 from any property of the matrix H alone. PROOF Necessity: Suppose that w* is a solution to (H*). Let us show that w* is of the form w*[a, u ] (see (2.37)).From Theorem (2.6), w* has the representation (2.25) in terms of the density F = w*, - wT = w*, - ('-'H'('w*, = ('-'(Z - H')('w*,. Thus there is a set of constants c* := {cr, ..., cR} such that

Applying the Plemlj formulas,

-

-

N

a;((,T)(Z - H')('w:) ds + I = I c?w*'(T)

82


or

or

Hence w*(z) = w*[- 2c*, (2.8).

- it'w;

I.

Necessity now follows from Lemma

SufJiciency: From Lemma (2.8), for (H) to have solutions there must be a set c := { c l , ..., cN}such that (2.40) holds for all w*[a, v]. Since w*[a, v] is real-linear in [a, v], we can find such a c if

Re !-.((I

-

1 = 1,

H)w', dQfTiw*,[a,v]) = 0,

..., N ,

(2.52)

implies

From. (2.37) and the Plemlj formulas, w * _ [a ,Y](T) = i['-lH'F(T) for T E a. From (2.43), wT[a, v](T) = it'-%. Thus [ ' - ' H ' [ ' W * , [ Uv, ] ( ~-) w?[a,

vI(7) = [ ' - ' H ' ~ J ( T )-

~['-IH'F(T) = 0,

i.e., (2.48) is satisfied by any w*[a, v]. Since

I

I I

Re &((I - H)w', dQfTiw*,[a, v]) = Re &((I =

-

H)w',('i-*['-IJ) ds

Re & ( d , ~ i w '['-'(Z ,

it follows that (2.52) implies that the density [ ' - ' ( I Consequently, from Theorem (2.5),

-

- H')iF),

H')F satisfies (2.27).

is a solution to (2.47) for z $5 6. Hence if (2.52) is satisfied, then w*[a, v] is a solution to (H*). Consequently, from (2.49) we have (2.53). Therefore (H) has solutions. Finally, let us show that k - k* = K . We assume that h and f a r e

83

6. PURELY HYPERCOMPLEX HILBERT PROBLEM

identically zero. First let us count the number of independent choices of w*[a, v] that can be made in (2.40). The number of a's for which (2.36) has solutions is N - rl where rl is the rank of the J x N matrix

For each a let v[a] be a solution to (2.36). Then all solutions are given by u = u [ a ] + Z;ll djvJ. Thus there are N - rl + J* independent choices for w*[a, v]. Next we count the number of solutions to (H). From (2.40) there are N - r2 independent choices of c = { c ~ ,..., , CN} that correspond to solutions of (H), where r2 is the rank of the (N - rl + J*) x N matrix

For each choice of c we see from (2.44) that there are J - rl independent choices of b := {bl, ..., bJ} for which po + Z;=, bjpj corresponds to a solution to (H). Thus the number of solutions to (H) is k := N r2 + J - r l . In the first part of the proof of this theorem we showed that (2.52) holds if and only if w*[a, v] is a solution to (H*). Consequently, k* = N - rl + J* - r 2 , and thus k - k* = J - J* = K. 6. THE HILBERT PROBLEM IN THE PURELY HYPERCOMPLEX CASE

As indicated in the previous chapter, more can be said concerning solutions to (2.1) in the case r = 0 (one complex equation) and its higher dimensional extension, the purely quasi-diagonal case, than in the general non-quasi-diagonal case. This is also true of boundary value problems, as we shall now illustrate. We follow the work of Begehr and Gilbert (1979). We consider the problem DW

+ AW + Bw

=

0,

w+ - Hw- = h

w(w) = 0 ,

on

6,

where A, B, and H are now assumed to be hypercomplex numbers or, equivalently, quasi-diagonal matrices. Extension of the results obtained in this section to the subblock quasi-diagonal case poses no difficulty. Since it is desirable to introduce a generalization of the polynomials, we wish solutions to (H) to have integral orders at infinity. For this reason

84


we assume that A and B vanish outside of some bounded domain a* and A , B E Lp(@*)for some p > 2. The boundary matrix H is assumed to be Holder continuous on &. We begin with the canonical factorization H(T) = X + ( T ) ~ ( T ) ~ X of ~-(T) Section 3. If we let (2.54)

then we obtain the simpler problem Diit

+ Ait + s%= 0,

iit+(7) -

where

&(T) =

i t ( m ) = 0, 7E

f(7)Kit-(7= ) K(7),

6,

(H)

X - ' ( T ) ~ ( Tand )

At this point our considerations separate into three cases depending upon the sign of the index K .

n2

CASE(I) K = 0. If 6, and are the Cauchy kernels for A and then the Plemlj formulas imply that

3,

(2.55)

is a solution. Any solution to the homogeneous problem & = 0 would be a continuous solution to Dit + Ait + = 0 in @, and thus from Theorem (1.37) and the condition i t ( m ) = 0 we have it = 0. Hence (2.55) gives the general solution to the problem.

CASE(11) that

K

> 0. We seek first a special solution having the property lim t(z)"iit(z)= 0.

Z+=

For such a solution it, let

it,(z) :=

{

it(Z),

Z E @+,

t(z)Kit(z),

z

E @-.

6. PURELY HYPERCOMPLEX HILBERT PROBLEM

Then k, is a solution to the Hilbert problem

where

Thus our special solution is given by

where @')(z, 5) and @"(z, 5) are the generalized Cauchy kernels associated with A and B,.Thus the special solution to problem (H) is

To complete the solution to the problem we must characterize all solutions to the homogeneous problem (fi, h = 0). We introduce functions Fk and Gk,k = 0 , ..., K - 1 akin to the generating pairs introduced in Chapter 1. We take pk and 2 . k to be bounded and continuous solutions to w

+ Jc(Aw + B l f " t - " W ) = 0

such that Pk(..) = 1 and e k ( w ) = i. The existence and uniqueness of such functions may be established as in Chapter 1. For k = 0, ..., K - 1,

and define ck in the same manner. Note that the Fk and ck are solutions to the homogeneous problem (fi). They form a linearly independent set of the real hypercomplex functions, as can easily be seen from their different orders at infinity. Let us now show that every solution k of the homogeneous problem (fi) has the form

86


a*,and hence G(z)

Such a function G is hyperanalytic outside of o(lzl'-") at infinity for some o s I s K - I. Let wo := G

=

+ pkek),

- z(Akpk k=O

where the real hypercomplex numbers hk and If we choose

pk

are to be determined.

+ ipr= Iim t(z)"-'G(z), 2-=

then w must be O ( [ Z ] ' - ~at- ~03.) Proceeding sequentially, we conclude that with the choices

+ i&, for v

=

0,

..., I ,

=

lim f(Z)'-"

2-m

{

I

% -

2

k=u+l

(Akpk -k pkGk)

I

the function w o is O ( ~ Z . - " -Consequently, ~). Go :=

Z € {"07 tKWO,

@+,

z E @-,

defines a function continuous in C and vanishing at infinity, and consequently Go = 0 by Theorem (1.37). With GI as defined by (2.56), all solutions to the problem (fi) are given by

CASE(111)

K

< 0. We have DG

+ A @ + BF = 0, W + - tKG- = h .

G(w) =

Again we let GI :=

Then G , is a solution to

{

G(Z),

t(z)"G(z),

@+, z € @- .

Z €

0,

87

7. HYPERCOMPLEX RH PROBLEM

where

The solution, if any, is given by w(z) =

1

1.

2Tl

(v

fiY(Z,

KO;

5) dQ5 K(5) - f i Y Y Z , 5)

(2.57)

however, the condition Gz(z) = O((z("-')is not automatically met. In view of the assumption that A and B vanish in a neighborhood of infinity, we have (cf. (1.55))

for IzI sufficiently large. We have the convergent expansion

for

5 E 6 and IzI sufficiently large. Thus near infinity we may write

Hence the necessary and sufficient condition for (2.57) to give the solution to (HI) is

t(5)k6(5)d?(l;)= 0,

k = 0,

..., - K

-

1.

7. THE RIEMANN-HILBERT PROBLEM FOR HYPERANALYTIC FUNCTIONS

For a given real hypercomplex number a = a. > 0, we define a'/' to be the number b such that b2 = a. We can solve this equation to obtain

88


and so forth. Thus, providing we adopt the convention of taking the positive root for bo, a"' is uniquely determined. ' For the purposes of this section, it will suffice to take the domain 8 under consideration to be simply-connected. We define the Schwarz operator Sy, y a Holder continuous real hypercomplex function defined on a,to be the hyperanalytic function such that lim Re(Sy)(z) = y(7).

+T'Z

To show the existence of the Schwarz operator, we note that for any complex valued function ti, that is C 1 ( 8 + )and Holder continuous in we have the representation (cf. Haack and Wendland (1972))

where GI and G" are the first and second Green's functions for the ' and d,, denotes the differential in the normal direction. We domain 8 may normalize G" so that d,,G"(z, 7) = 1 for 7 E a. Thus the left-hand side of (2.58) depends upon Im E only through an imaginary constant. Hence we may take w = Sy to be defined successively by the equations (recall that w7 + qw, = 0)

k = 0, ..., r. Now let us consider the hypercomplex Riemann-Hilbert problem

Re { W w + (7))= Y ( 7 )

7

(2.60)

7 . HYPERCOMPLEX RH PROBLEM

89

where w is hyperanalytic and A is hypercomplex. We define the index of this problem to be 1 K := Ind 1:= - A& arg X. 2lr

As in the complex case (cf. Gakhov (1962)) we may solve this problem by seeking a regulurizingfuctor, by which we shall mean a real hyper) on @ that has a positive complex part and complex function p ( ~defined is such that p ( ~ ) x ( 7 )is the boundary value function of a function hyperanalytic in @+. Since the complex part p o of p is nonnegative, Ind p o = 0. Let us now find a regularizing factor for the problem (2.60). We assume that the origin of coordinates lies in @+. Let

If A = A.

+ L, L nilpotent, then -

Thus 0, and by a similar argument JI, are real hypercomplex functions. As T traverses @ the function 0 ( ~ )- $ ( T ) is single-valued. Let (2.61)

where

is easily seen to have the properties required of a regularizing factor.

90


With a regularizing factor p , we may now solve problem (2.60). We investigate the three cases separately. CASE(I) written as

K

=

0. The boundary condition for our problem can be Re{w+/px} = p-ly/Xx.

where

is given by (2.61) and c is a real hypercomplex constant.

CASE(11)

K

where

< 0. We have from the Cauchy representation

-

4 := exp(iS(8 - $)},

and

Consequently,

91

7 . HYPERCOMPLEX RH PROBLEM

Since R,(z) is hyperanalytic in

a+,we have

and hence

c- I

(2.63)

CASE(111)

K

< 0. We have

w+(z) = $+(z)(t(z)

-

t(O))"(S{p-'y/Ax) + ic)

(2.64)

as the solution, provided that the constant c can be chosen so that S ( ~ - ' A / A ~+} ic

has a zero of order at least

K.

Finally, for what ensues it is necessary to consider the problem Re[Ww-(dI = ~ ( 4 ,

171

= 1,

(2.65)

where w is to be hyperanalytic in the exterior of the unit disk Co and bounded at 03. Let NZ, 2)

= WU/Z,

l/zh

Then

and hence 9 is hyperanalytic in C,. The boundary condition is now Re [~ ( T )+C(711 = y (7). Thus the index of problem (2.65) is 1

-A& 2T

arg ho.

(2.66)

92


8. THE REPRESENTATION OF A GENERALIZED HYPERANALYTIC FUNCTION IN TERMS OF A REAL DENSITY

By conformal mapping the domain @+ can be bounded by a set of + C0 denotes circles with the exterior circle @o the unit circle. If q: this conformal mapping, then for w(q, ?j)= w(q(z), Tj(z)) E q . (2.1) becomes

which is of the. same form. The mapping q has Holder continuous first derivatives on @ by virtue of the continuity assumptions made concerning @ at the beginning of the chapter (see I. N. Vekua, 1962). Theorem 2.10 (Bojarski, 1966) Let w be a generalized hyperanalytic Then there is a real density function that is Holder continuous in p that is Holder continuous on @ and f o r which

w.

(2.67) The vector p is determined up to a cons?ant on the curves = 1 , ..., m and uniquely determined on a0.

k

6jkf o r

PROOF From the Cauchy representation, (2.68)

Fig. 3

93

9. GENERALIZED HYPERANALYTIC PROBLEM (RH)

As we saw in the preceding section, we can solve the boundary value problem (2.69) Re{iQ,-(T)} = Re(iw+(r)} for each of the simply connected domains @ ik, = 1, ..., m. The function Q, is to be hyperanalytic. For any given real hypercomplex constant c we can solve the boundary value problem Re{i@-(T)} = Re{i(w+(T)- c)}

(2.70)

on 8; (see (2.66)). We use Q, to denote the function on U7*o@I)k which is composed of the above solutions. Let 1

p := T ( W +

1

p := ,(w+

-

(.€-)

on

&,

-

a-

on

&.

- c)

k = l ,

..., m,

Subtracting the identity

from the Cauchy representation (2.68), we obtain the existence of the representation (2.67). To establish the assertions about uniqueness, suppose that

is identically zero for z E a+,where p is real-valued. From the Plemlj formulas, w- = - p on each @ k , k = 0, ..., m. Consequently, Re{iG-} = 0. From the preceding section this implies that p is-determined up to a constant vector ak on @ k , k = 1, ..., m. For a0we have the additional condition that G(m) = ic. This determines the one arbitrary constant in the representation (2.67). 9. THE RIEMANN-HILBERT PROBLEM FOR

GENERALIZED HYPERANALYTIC FUNCTIONS

We now consider the general case

Dw

+ Aw + BW

= f,

Re{Aw+} = y

z E @+, on

&

94


of the RiemannTHilbert problem. Here A and y are Holder continuous, det A # 0 on 8 , and we take A, B, and f E L p ( v )for some p > 2. For our purposes it is convenient to regard A, B, and f as extended to C and vanishing identically in @-. We seek solutions w that are Holder None of the matrices A, B, or A is assumed to have continuous in subblock quasi-diagonal form. Solutions to (2.1) have the resolvent representation

v.

w(z) = Wz) +

+ rAz,

/~$+(rAz, 5)@(5)

Om)due + c ckwI(z). N

I= 1

(2.71)

If we substitute the potential representation (2.67) of Theorem (2.10) into (2.71), then we obtain by manipulations similar to those of Theorem (2.6) w(z) =

1

/,

-l.r l (8(fl,(z,5) da5 - f12(z, 5) &&45) -

lr

//@+(fl~(z, Of(5) + fldz,

+ iC)

5>7(5>) d q + c c ~ w ' ( z ) , (2.72) I= N

I

where the real density p and the real constant vector C must satisfy Re /&(p

+ i?, i('w*') ds + Re //@+(f,w*') du, = 0

(2.73)

for 1 = 1, ..., N. This condition is also sufficient for a function w defined by (2.72) to be a generalized hyperanalytic function in @+. From the Plemlj formulas W + ( 7 ) = P(7)

1

+ 2 /(&fli(T, 5)6(5)

- flz(7, ()$({))P({)ds

+ @7)? + jf(7) + Ic C/W/(7), = N

I

7E 6,

where (I is the identity matrix) 1

1.

fi(~) := il + (flh, 5)5(5) + MT, 5>&3 l r @

3f(z) :=

lr

/J@+(fldz, 5)f(5) +

M Z ,

ds,

5)J;CO) dug.

In view of the Riemann-Hilbert boundary value condition, p must satisfy the singular integral equation


95

(2.74) The adjoint of P in the inner product s(i(p, v) ds is P*v := (Re A(r)’)v(r)

By computations similar to those carried out for the Hilbert problem, it can be shown that the dominant part of Pp is

The index of P is thus K

1 := - A(i arg(A(r)-’m). lr

* bases for solutions to P p = 0 and P*v = If {P’}~=: and { ~ j } ~ = : denote 0, respectively, then J - J* = K . In analogy to the case of the Hilbert problem, we consider solutions to

c a,Re{it’w*’). N

P*v

=

I= I

(2.75)

If v is such a solution, then for g as defined by (2.74)

N

+ I = I a,

I(.$?,

Re [ ’ w * ‘ ) ds.

(2.76)

96


Let us write i? = C ;:', f?ki?k,where ek = (0, we may write the above equation as

..., 0,

I , 0,

..., 0)'. Then

r r

N

(2.77)

Lemma 2.11 For w as given by (2.72) to be a solution to (RH) it is necessary and suficient that y , f, i?, and c := {c,, ..., cN}satisfy (2.77) for all a := {al, ..., aN}f o r which (2.75) has solutions and, f o r each a , for all u corresponding to that a.

PROOFWe have already shown necessity. For sufficiency observe that when a = 0, the corresponding v are the v J .Thus (2.76) immediately implies that (2.74) has solutions. If po is any particular solution, then all bj&. For (2.72) to give a solutions are of the form p = po + solution to (RH) we need show only that (2.73) holds since (2.74) guarantees that the boundary condition is satisfied. Thus we must choose the bj so that Re /&A,'

J

i('w*') ds

+ j2 bj /& = 1

Re(pj, i('w*I) ds

This can be accomplished, provided that

(2.79)

whenever the aj are such that

1.

Re(pj, it'

C aIw*' N

I= I

(2.80)

97


for j = 1, ..., J . Equation (2.80) implies that (2.75) has solutions corresponding to this choice of the a j . Consequently, N

P*v)ds

=

Re/($g, v ) d s ,

and thus (2.79) follows from (2.76). Theorem 2.12 (Bojarski, 1966) The necessary and suficient condition for (RH) to have solutions is

Re(y, iA’-’[’w*+) ds + 2 Re

If@+(

f, w*) d r , = 0

(2.81)

Re{A’-’[’wf} = 0.

(RH*)

for all solutions to the problem

D*w* - A’w*

-

B’w*

= 0,

PROOF Necessity: Suppose that w* is a solution to (RH*). Then

AI-l[rw*+ - iv for some real v, and from the Cauchy representation

+ Ic ctw*‘. = I N

The Plemlj formulas imply that 1 (;)it’-’A’v(T) = -2T

T)A([)’+ a;({, ~ ) % [ ) ‘ ) v ( ds () /.(a;({, @

+ Ic C?W*/(T), = I N

i.e.,

c(- 2 c 3 Re{i[’w*’}, N

=

I= I

and thus

c( 2c7) Re{i[’w*’}. N

P*v

=

I= 1

-

98


The result now follows from Lemma (2.11). Sufjciency: We must show that (2.81) implies that Eq. (2.77) can be solved for c and c?. We can do so, provided that

for all choices of a and v for which

J(*Re(Aw', v) ds

=

0,

I

= 1,

..., N ,

(2.83)

and

From Theorem (2.5), (2.83) implies that

-

N l -a/w*I(z) /=I 2

(2.85)

is a piecewise generalized hyperanalytic function. Let us show that (2.83) and (2.84) imply that w*[a, v] is a solution to (RH*). Note that since we may take A and B to be identically zero in @-, D*w*[a, v] = 0 in W. Moreover, from the Plemlj formulas, 1 w*_[a,v] = --i['-'A' 2 -

-

21r &

( W L dA(5)' + W 5 , dA(5)r)v(5)

1 cN -Q/W*/, 2

/=I

ds

(2.86)

i.e., from (2.75), Re{i['w*_[a, v]} = 0.

(2.87)

99


0= =

l6 lee

Re(AOek, u) ds

+ I&(,, I2= I uIt’w*’)d i N

Re(ek, - 2t’wi[a, ul) ds.

Thus we are left with the boundary value problem D*w*[a, VI = 0,

(2.88)

Re{it’wi[a, u ] } = 0,

(2.89)

J6 Re(ek, t ’ w i [ a , u ] ) ds = 0,

k = 1,

..., r

+ 1.

(2.90)

Recall that D’(w*/ri) = 0. Hence the first two conditions of (2.88) and (2.89) are equivalent to the Riemann-Hilbert problem D’(w*[a,vl/ti)

=

0,

Re{if’r:w?[a, u ] } = 0

(2.91)

a;,

on each domain k = 0, ..., m. The complex part of it’r: is i(i - qoF),where qo is the diagonal element of the quasi-diagonal matrix Q. On each ak,k = 1 , ..., m , we have

As we saw previously, this problem has no solutions other than w*[a, u] = 0 in @ ik, = 1 , ..., m. For problem (2.91) in the index is

a;

1

21r Acj,

arg( - i ) ( i - G) = 1

100


The function w*[a,v] has the representation w*[a,vl(z)

Thus

from (2.89) and (2.90). Consequently, w*[a,v](z)has a zero of order two at infinity. Upon reflection across the unit circle, this corresponds to a zero of order two at z = 0. From the representation (2.63) of solutions to the Riemann-Hilbert problem, we conclude, successively, that the constants co and c are zero. Hence w*[u,v] = 0 in a;. Thus we have established that the boundary value problem (2.88-2.90) admits only the trivial solution in @-. From the Plemlj formulas, w*,[u,V] - w*_[u, V] = i t ’ - ’ A ’ ~ ,

i.e., w*,[a,v] = i [ ’ - ’ A ’ v .

Hence iv = A’-’(’w*,[a, v], and thus Re{A’-’(’w*,[a, v]} = 0. This establishes the desired result that w*[a,v] is a solution to (RH*) when conditions (2.83) and (2.84) are satisfied. Let us now return to condition (2.82), which we must show holds under hypothesis (2.81) of the theorem. We have just established that Y = -iA’-’[’w*,[a, v]. Also, we have

101


Thus

from hypothesis (2.81) of the theorem. This concludes the proof of the theorem. Finally we work out a formula for the Riemann-Hilbert problem that is analogous to (2.50) for the Hilbert problem.

Theorem 2.13 (Bojarski, 1966) Let ko denote the number of solutions to the homogeneous problem (RH) and 6 denote the number of solutions to (RH*). Then ko - k,* =

K

-

(m - l)(r + 1).

Recall that m is the connectivity of the domain

(2.92)

a+.

PROOF Let us first find the rank of the matrix M with rows of the form

/& Re(Aw', v) ds,

...,

/&

Re(AwN,v) ds.

(2.93)

The matrix M is generated by letting [a, v] range over all independent admissible pairs [a, v]. The number of rows is N - k t + J*, where k f is the rank of the matrix (Re{i('w*'}, p'),={,j=Y.

102


From the considerations of the proof of Theorem (2.12), it follows that for any choice of [a, v] for which (2.93) vanishes, w*[a, v] is a solution to (RH*), and conversely. Thus the rank of A4 is k , := N - K f + J* - k,*. From Lemma (2.11) and Eq. (2.77), the number of distinct choices of the set F := {el,..., c , + , ,c I , ..., cN}that correspond to solutions to (RH) is (r + 1 + N ) - k l . For a fixed F and any solution po of (2.74) corresponding to that choice, all solutions to (2.74) are given by po + bjpj. However, in view of the indeterminacy in p , stated in Theorem (2.10), (r + 1)m of the bjpJ do not give independent solutions to (RH) when added to po.Thus from (2.78) there are J - kf - ( r + l)m choices of the bj corresponding to linearly independent solutions to the homogeneous problem (RH). Combining all of the above results, we obtain

,.

+ 1 + N ) - ( N - k f - J* - k* + J - k f - ( r + 1)m = k,* + J - J,* - ( r + 1)m - 1 = kg + K - ( r + 1)m - 1 ,

ko = (r

0)

which was to be shown.

10. THE RIEMANN-HILBERT PROBLEM IN THE PURELY HYPERCOMPLEX CASE

We consider in this section the boundary value problem Dw = aw

+ bts

in 8,

Re(yw) = 4 on

&,

(2.94)

where the coefficients a, b are hypercomplex Holder continuous functions in a bounded simply connected region 8 with piecewise Holder continuous boundary. In := 8 U a, a and b are to be continuous. Moreover, y has one Holder continuous derivative, and is to be a real Holder continuous function on 8.It is natural for us to define the index of this problem as (cf. Section 7) K

:= Ind 7 = Ind To,

(2.95)

where yo is the complex part of y . We discuss first the case of index zero, since then we may reduce problem (2.94) to Dw = 50

+ 6G

where ii := y-I Dy

+

a,

in 8 ,

-

Rew

b := y b y - ' ,

=

4 on

and

6,

w := yw.

(2.96)

103

10. PURELY HYPERCOMPLEX PROBLEM (RH)

To see that such a transformation is valid it is necessary to demonstrate that the inverse y-I exists in a. This follows directly from the fact that lyol # 0 on a, and hence it is possible to continue harmonically the components of y := R exp( -ie) into the interior of C9 such that Ro := (yol nowhere vanishes. Our problem may now be written as the system DWk =

Re W k

=

+ boGk + fk

z@k

4k

in on

8,

6,

k = 0 , 1,

..., r,

(2.97)

where

This is equivalent to the integral equation formulation o k

=

fik

+ P(zOwk + 6OGk + fk),

where (see Section 7) 4k({)[dnG1(Z75) - idG"(Z, {)I

nk(z) :=

Ck

and Ck is an arbitrary constant that is fixed by imposing the additional conditions Im W k ( 6 ) dnG"(Z,{),

Ck := i

k

= 0,

..., r .

The operator P is defined in terms of Green's functions as

-

G:'(z, 01) d5 A dz.

(2.98)

Taking advantage of the fact that q is nilpotent yields a concise representation for o as 0

where

=

R

+ P [/:oC ( - q r I ) ' ( G w + 6w - qR')

1

,

(2.99)

104


Moreover, by introducing R := P(l

-

4II-I

r

=

P C(-qII)&, k=O

(2. loo)

we obtain the hypercomplex integral equation w =

n

-

R(qf2') + R(iiw

+ 673).

(2.101)

From the imbedding inequalities given in Begehr and Gilbert (1977), it may be seen that (2.101) has a unique solution; moreover, it may be constructed by an iterative procedure. For the case of positive index we introduce a transformation Y

$(z) := n ( t ( z ) - t(zd),

u := $-Iw,

v=

I

(2.102)

where the points z, lie in a. The differential equation becomes Du = iiu + h in a, and the boundary condition takes on the reduced form Re@) = 4 on a, with

ii := a ,

6 := $b/$,

and

p := T$.

(2.103)

What is significant is that Ind p = 0, and we have modified the problem to the case previously discussed. The homogeneous boundary value problem for u(z), which is normalized such that

is known to have nontrivial solutions u h such that the complex part of uho of u h does not vanish in @. Hence the homogeneous solutions w h of (2.94) may be written in the form wh(z) =

A$(z)Uh(z),

(2.104)

where r

From this it is easily seen that each solution of the homogeneous (2.94) having ( K + 1) zeros must be identically zero. The general solution to (2.94) may be written as w = $(z)(u, + huh), where uo is a particular solution of the reduced equation and A is a hyperreal constant. With regard to the number of linearly independent solutions, Begehr and Gilbert (1977) have proved the following theorem.

105

10. PURELY HYPERCOMPLEX PROBLEM (RH)

Theorem 2.14

Dw

The homogeneous boundary value problem =

aw

+ biV

in

a,

Re(yw) = 0 on

&

(2.105)

has exactly ( 2 +~ 1 ) linearly independent solutions with nonidentically vanishing complex parts over the algebra of real hypercomplex numbers dr+land exactly ( 2 + ~ l)(r + 1 ) independent solutions over [w.

That there are at least this many independent solutions is clear from the preceding discussion. To show there are no more than ( 2 ~+ 1 ) solutions, with nonvanishing complex part over dr+I , we show that each such solution can be written as a linear combination over dr+lof ( 2 + ~ 1) hypercomplex solutions. To this end, let

be a nontrivial solution of the system

2 A;)Ep(Zu) = 0, 2K

p=O

1

S

v

S K,

where the E,(z) (0 S p S 2 K ) are ( 2 K + 1) linearly independent SOlUtiOnS that were constructed relative to the n different points z, (v = 1 , ..., K ) . For each p we define the functions

c 2K

W,(Z)

:=

k= I

hp)Ek(Z),

where the ALP) are hyperreals that are uniquely determined as the solutions of the hypercomplex system WZ,(Z,)

=

apU,

w2p-~(zu) = iapv.

In complex form this becomes

2x

2 rn

k = l /=O

wk,m- / ( z u ) = i6omapU 1

0 S m S r , 1 s plv s K . To show that these inhomogeneous equations have a unique solution it is sufficient to demonstrate that the system

106


possesses only the trivial solution. This follows directly from the fact that if

is a solution of (2.105) that is, moreover, independent of wo(z),it must be the trivial solution. From this we conclude, since the f i k ( Z ) are linearly independent, that the X k are all zero. In this way we may construct a system of functions w,, p = 0, ..., 2 ~ satisfying , the conditions wo(z,) = 0,

w2,(zu)=

a,

is,,,

~ Z , - ~ ( Z , )=

1 6 p , Y s K . It is easily seen that each solution of (2.105) may be represented as a linear combination of the w,. For instance, if we set X 2 , := Re w(z,,) and A*,-, = Im w(z,), then the function w - EF=lh,w, is a hyperreal multiple of w o ,i.e.,

2 A,~,(z). 2K

howo(z) =

W(Z)

-

= ,

I

We can, moreover, show that there are at most (2K + l ) ( r + 1 ) linearly independent solutions over R. To this end, we introduce the functions w ~ , and , ~ w ~ , - , as , ~ linear combinations of the Ek:

which, moreover, satisfy the conditions WZp.k(Zu)

=

e

k

apu

9

w2p- I , k ( Z u )

= ieka,u

-

These two conditions for determining the coefficients AP*”’ may be reformulated in terms of their components as

0 S Y , m S r , 1 S p , 1 S K . If we now fix wo,u:= e’wO (0 s Y s r ) , then each solution of (2.105) may be written as a linear combination of the w,Jz).

107

10. PURELY HYPERCOMPLEX PROBLEM ( R H )

+

We must still show that there exist exactly 2~ 1 solutions over d r + lFollowing . Begehr and Gilbert (1977), let wo(z) be a nontrivial solution of (2.105) that vanishes at each of the given points zk (1 s k s K). If we have 2~ additional solutions that moreover satisfy the conditions

W Z ~ Z =J & I ,

W X -~ Z I= )

1 6 k, I

i&,

S K,

then these solutions also have nonvanishing complex parts and form with wo(z) a linearly independent system over dr+ I. Let us define v=l,u#r

then if w is a solution of (2.105), u = w/fk is a solution of the homogeneous boundary value problem of index 1, Du

= au

+ bfku/fk

in 8 ,

Re(yfku) = 0 on

6.

(2.106)

If u,, ub are two nontrivial solutions.of (2.105) that satisfy the boundary conditions Re u, = Im u, = 0 on a, then it is known (see Haack and Wendland (1972)) that without loss of generality we may assume that Im uaO> 0, Re ubo > 0 on 8.Furthermore, let u,(z), u,(z) be two solutions of (2.105) that satisfy u,(zk) = up(zk)= 0 and the inhomogeneous boundary conditions Re(rfkv,) = -Re(yfkv,) Re(yfkup) = -Re(yfkub) Then the two functions w 1 := fk(u, of (2.105) that satisfy wl(zJ

=

+ u,),

wz(zJ = 0,

W d Z k ) = fk(Zk)U,(Zk)

# 0,

6, 6. w2 := fk(ub + u,) on on

I # k,

1sIs

are solutions

K,

wz(z2) = f k ( Z k ) U b ( Z k )

# 0.

Consequently, one has Im(wIO(zk)wZO(zK)) # 0, and the linear equations A$Ywi(Zk) + A $ ? w ~ ( z ~ )= 1, Z-IW(Z~)

+ A % - I ~ z ~ =) i

having nonvanishing may be solved for hyperreal A:;, A$, A-;: I , real parts. In this way we may construct two solutions wP = A Z ’ W ~

+ Ag’~2,

/A

=

2k - 1,2k,

108


with the properties w X ( z I )= tikI, W ~ ~ - , ( Z=~ itikl. ) By doing this for each k , we obtain 2~ + 1 independent solutions over .dr+I. The problem with negative index may be handled in a similar manner; see in this regard Begehr and Gilbert (1977) and Haack and Wendland (1972).

Reductions to Hyperanalyticity

1. INTRODUCTION

An approach that was used very successfully by I. N. Vekua (1962) was to reduce the boundary value problems associated with the complex equation aw

- + aw

az

+ bw

=

0

to a related boundary value problem for analytic functions. This technique made use of the so-called Carleman similarity representation of a generalized analytic function w . This representation states that in its domain of definition w(z) may be written as the product of an analytic function and a function that does not vanish in a. All attempts to date to find an adequate generalization of this global result for solutions to the system

+ k2 2 ek (akjwj + B ~ ~ =w o~ ) =O r

DW

k

I

j=O

(3.1)

have failed. There are, however, several special cases of (3.1), for example where Bjk = 0, where we can find global similarity representations. Whenever this is possible the full strength of the Gakhov-Vekua approach to boundary value problems is also available for the hypercomplex case. This possibility merits further investigation, and this is one purpose of the present chapter. Another topic that will be treated is the exploitation of this representation for solving the Riemann-Hilbert boundary value problem. 109

110

3. REDUCTIONS TO HYPERANALYTICITY

2. SIMILARITY PRINCIPLES

Two functions will be called “similar” if they have zeros of the same orders at the same points. For example, Theorem (1.37) shows that any solution of (3.1) which is continuous and bounded on C is similar to a constant. A second example is the similarity principle for pseudoanalytic functions. By a pseudoanalytic function we mean a solution of (3.1) with r = 0; i.e., a solution of

The following theorem is proved in Bers (1953). Theorem 3.1 Let aooand boobe Holder continuous in a domain 8.For any solution wo E C’7a(@)of (3.2), there is a function f analytic in @ such that and a function so that is Holder continuous in

wo(z) = esO(z)f(z). See also Haack (1952) and I. N. Vekua (1952, 1954) who also proved this result at about the same time. We have an analogous, although local, result for solutions of (3.1). Theorem 3.2 (Kuhn, 1974) Suppose the coefjicients akl and bkl are Holder continuous. Let w be a solution of (3.1) with Holder continuous jirst partial derivatives such that wo is not identically zero. I f z o is a point such that W k ( z 0 ) = 0 for k = 0, ..., r , then in some neighborhood of zo there is a lower triangular, nonsingular matrix E(z) = {Ekj}and integers no, ..., n, such that

Moreover, each Ekj has the representation &j

=

Ukj

eXp(Sk),

where sk is Holder continuous and gkj

=

ykj -k

0 Ukk

k

/=o

Gj

k

G

r,

analytic. For j < k ,

ffkjl(-)

z

- zo

I 7

z - zo

with ykj Holder continuous and the ffkjl’scomplex constants.

111


PROOFWithout loss of generality zo = 0. The result will be proved by induction. Our induction hypothesis is that there exists a neighborhood S of 0 and a complex-valued function s, E C0.Ys)such that m

2 cmj(Z)zfl’,

w m ( z ) = exp(sm(z))

j=O

where

(1) s, has a generalized derivative with respect to Z such that

a

-sm

az

E LQ(S)

for every q 2 1. ( 2 ) nj is a positive integer. (3) umm is analytic in S with umm(0)# 0. (4) F o r j < rn

where amjlE C and

with Rmj E Lp(s) for some p > 2. (Note that this implies that ymj E Co,a(S)for a = ( p - 2)/2.) First we prove the result for wo. In this case

i.e., wo is pseudoanalytic. Thus we can apply Theorem (3.1) to obtain the result. The function so is obtained as follows: let

lo

otherwise;

we define

Since wo E C’(S),Po is bounded in S and hence in Lq(s).Also so E

112


Co,u(s),0 < CY < 1, and possesses a generalized derivative with respect to z, aso/az = Po (weak). If the function f in the statement of Theorem (3.1) has a zero of order no at 0, then we define U ~ ( Z:= )

f( z ) -. Zno

This completes the first step of our induction hypothesis. Now assume that our induction hypothesis holds for w, such that 0 s m S k - 1 . The proof for wk will depend upon several lemmas which will be demonstrated at the conclusion of the proof. From (3.1)

Define

As in the case m = 0 we have sk E Co*a(s),0 < (Y < 1 , and dsJaZ = Pk (weak) and Pk E Lq(S),q 3 1 . If we define J, := wk exp(-sk), then

By the Cauchy integral representation there is an analytic function such that

in S. Using the induction hypothesis, it can be shown that

C#J

113


where hkilE @, Rki E Lp(s), p > 2. (See Lemma ( 3 3 . ) So

Since

- - 1_

5-z

z + - 1+-

52

5

... + -Zni 5ni-I

Zni

+

5Y5 - z)’

it follows that

(3.3) where N := maxosisk-I(ni - 1) and where the complex constants kp are sums of convergent integrals. Employing a forthcoming result (Lemma (3.3)), we have k- I I=O

&(I) ((5 - z)

c akil(j),

k- I

/

5 d,$dq

=

-

/

/=0

z # 0.

Define

T(z) := +(z)

+ 2 kpzp. N

p=O

Then from (3.3) and the definition of $,

wdz)

=

$(z) exp(sdz)) (3.4)

Suppose T(z)

+ 0. From (3.4) r(0)

=

0, and since

r

is analytic in S,

114


there is a nk

3

I such that Ukk(Z)

:=

UZ) Znk

is analytic in S and ukk(0)# 0. Hence (3.4) becomes

as desired. Suppose T(z) = 0. Let nk = no, ukk= 1 , and

Then from (3.4),

Hence by induction we have all the results of the theorem except ukk(z).Since the nonsingularity of E(z). The determinant of E(z) is ukk(0) # 0 for all k and each ukkis continuous in a neighborhood of zero, we conclude that E(z) is nonsingular in some neighborhood of zero. Finally we prove the following three lemmas used above.

n;=,

Lemma 3.3 For m = 0, 1 , 2,

..., and z

E S, z f 0

PROOFLet S := (5 : 151 < R}, and note that the integral above is convergent over the region G,(z) :=

If 5

(5

= pi',then

:E

< 151

IzI

- E} U

(5 : I z I +

E

< 151 < R } .

115


From the Cauchy formula

Thus

as

E

+ 0.

Hence the result.

Lemma 3.4 Let f E Co@(S),0 < a < 1 , and let g(z) be bounded in S with g E L'(3). Then

(This lemma will be used in the proof of Lemma ( 3 . 9 . )

PROOFIn 3 we have If(z)

where /3 = ( 1

-

-

f(0)l

S

KlzJ" and Jg(z)l S M . Hence

a)(2 + a). Since /3 < 2 ,

IZI-~

E

L'(S), and thus

Lemma 3.5 For ck as defined in the proof of the Theorem (3.2),

where Akil E C and Rki E LP(S) for some p > 2 .

116


PROOFRecall that

c

k- I

ck =

UkjWj

j=O

+

bkjvj

From our induction hypothesis,

where

:=

yji

+

pjj

for i < j, and ujj is analytic and ujj(0) = 0. Thus akjwj

bkjwj

=

Ukj

eXp(Sj)

c

i

i

i=O

Z"'Uji

+ b k j eXp(Sj) i2 =O

Zni?Fji

Since akj, bkj, exp(sj), ujj, and yji are continuous and Z / z is bounded, we conclude that Fkijis in L4(s) for every q 3 1. The functions ujiand sj have generalized derivatives with respect to Z and hence z. Thus so does w j . Differentiating we have

+ ic =O j- I

zn;-l

exp(sj)

+ niyji + nipji + z az apji

117


Note that zujj(asj)/(az) + njujj + in Lp(S), p > 2, by the induction

a + nj-yji(0) + nipji + z -pji. az Since yji E CoVa(S), we have from Lemma (3.4)

for some p > 2. Also we have az

/-I

/=I

Hence there exist complex constants ajil such that

Hence we conclude that there exist functions Gij E Lp(s) for some > 2 such that

p

Using this and (3.5) we find

By Lemma (3.4),

118

If

hij/


:= Lkij(o)(Y,$,then we may write

So finally

where v

=

max(j, I). Taking

C (Hkij

k- 1

Rki :=

J-1

k- I

Mkij),

Aki/

:=

2

j=v

Akij/

yields the result. In keeping with the general notion of similarity principles we would like to interpret the integers nj as the orders of the zeros of the wk. However, in the proof of the theorem we encountered a case, namely r = 0, where the choice of nj was not unique. Nevertheless, there is a quantity that can be constructed from these exponents that can be interpreted as the “order” of the zero. Define Nk(Z0) :=

{(Y

: (Y

3

0, W k ( Z )

= o ( l Z - Zol?

for z

zo}.

Since zo is fixed, we will use the abbreviated notation N k . Clearly all N k are nonempty. The quantity that will be taken to be the order of the zero at zo is Pk(Z0)

:= sup{a E Nk(zO)}*

The possibility p k = + m is not excluded. We see from Theorem (3.2) that when all Wk’S vanish at zo, then p k 3 1 for every k. Moreover Corollary 3.6 (Kiihn, 1974) Let zo be a zero of w. Then under the assumptions of the Theorem (3.2), p := min

Osksr

nk =

min p k .

Osksr

PROOF From the proof of the theorem no = po, and so mink p k < m.

119


Now k

wk(z) =

j=O

Ekj(z)(z

-

zO)n’,

and since the Ekj are bounded in a neighborhood of zo, we have mino,,, nj E Nk.Hence p s minosjsk nj s Pk for k = 0, ..., r, i.e., p minoSksrPk. If I := min{i : p = ni},then

(1) for 1 = 0, minOSkSrP k s p o = no = p . ( 2 ) for 1 a 1, we have nj > nl f o r j = 0, ..., I W/(Z) = E//(Z)(Z- ZOY

+ O(l)(z - ZOY+l

-

1, hence

as z +, zo.

(3.6)

Since E,(zo) # 0, p I = p , and thus minIskSrPk s p , which proves the result. Note further that from (3.6)

wdz) = E//(ZO)(Z - ZOY + (E//(Z) + E//(ZO))(Z - zoY + O(l)(z =

A (z

-

z0Y + o(l)(z

where A := E,,(zo)and o(1)

-

+0

zoY

-

zoY+l

as z + zo,

as z + zo.Moreover, A # 0. Hence

zo)p + o(lz

-

zolp)

as z + zo.

At this point mention should also be made that the zeros of the components wk of w will not in general be isolated, even though those of w must be isolated, as just shown. For example, the system

awl

222o

az z2 + 2’

=

has solution wo = z, w I = = 2(x2 - y‘). We now extend the idea of generating pairs. The original idea is due to Bers (1953) for pseudoanalytic functions and was developed further by Gilbert and Hile (1976)for pseudohyperanalytic functions. Bers de-

120


fined the generating pair { F , G } to be a pair of solutions of pseudoanalytic functions, i.e., solutions of aw

-=

az

a(z)w(z) + b(z)w(z),

a , b Holder continuous,

(3.7)

such that neither function has zeros and Im{FG} > 0. To extend this notion, let F and G be two solutions of Dw = A(z)w(z) + B(z)w(z),

A , B Holder continuous,

(3.8)

where A, B, and w are hypercomplex. We will call such w pseudohyperanalytic functions. Suppose further that Fo = complex part {F} and Go = complex part {G} have no zeros and that Im{~oGo}> 0. Equating the complex parts of (3.8) we get

Thus wo is pseudoanalytic, and by the Bers theory there exists a generating pair {Fo,Go}.Using these functions in the first step of the induction proof in Theorem (3.2), we easily see the existence of generating pairs for pseudohyperanalytic functions. Note that any function w can be written w(z) = F(z)4(z) + G(z)J,(z),

where and J, are real hypercomplex. The representation is unique, for suppose 0 = F ( z ) ~ ( z+) G(z)J,(z), i.e.,

Then or

0=

40

FOG0

+ -J,o

IFOr

Equating real and imaginary parts gives +o = and so forth.

3r0

=

0. Next,

121

3. GLOBAL SIMILARITY PRINCIPLE

Theorem 3.7 (Kuhn) Let A and B be Holder continuous in a domain a. Let w be a solution of Dw = AW

+ BW.

If zo E 8 , then either W(Z)

F(z)$(zo)

in 8 or there is a constant A

+ G(z)$(zo)

# 0 and an integer p such that

(z -

as z

ZOY

+ o(lz

- zolp)

(3.9)

+-zo.

PROOFDefine 4 ~ :=) W(Z) - F(z)$(zo) - G(z)$(zo).

Since w , F, and G are all solutions of (3.8), so is u. If uo is not identically zero in a neighborhood of z o , then we apply Theorem (3.2) to obtain (3.9). If uo is identically zero in a neighborhood of zero, but uI is not, then we can still apply the theorem to obtain (3.9). If all u k vanish in some neighborhood of zo, then they all vanish in a. For vo is pseudoanalytic in (3 and so must vanish in @ by Theorem (3.1). But then u1 is pseudoanalytic, and so forth. 3. GLOBAL SIMILARITY PRINCIPLE

Let @ be a domain. For more generality we momentarily drop our hypercomplex notation and adopt matrix notation. We consider the system aw

aw = AW az

-+ Q-

az

+ BW

(3.10)

where Q, A, and B are lower triangular matrices and w is a vector. We assume that the components of Q, A, and B are bounded and measurable and that Iqkk(z)I 6 qo < 1 for some constant qo and all z E a,

122


k = 0, ..., r. We seek a representation wfz) = E(z)u(z),where E(z) is a nonsingular matrix. Define

and C := A + H. If w = Eu, then from (3.10)

+ E-uaza + QE-.au az This differential equation for u will be simplified if E satisfies

a

a + Q-E az az

-E

-

(3.11)

CE = 0.

Hence we require that dE/aZ, aE/az E Lp@ for some p > 2, and that E satisfy (3.11) almost everywhere. Because Q and E are both lower triangular, (3.11) is automatically satisfied above the main diagonal. For k L j , (3.11) is equivalent to the system

or

a a - c..E.. = 0 , -E.. az JJ + q..-E.. JJ az JJ JJ JJ

j = 0,

, ..., r

(3.12a)

k > j. (3.12b)

All solutions of (3.12a) are of the form is a homeomorphic solution of

a a -azx . + q . .az- x .

aj(xj(z)) exp +j(z)r =

where

xj(z)

0,

JJ

ajis an arbitrary analytic function defined on xj(S),w E Lp(@)for some p

> 2 is a solution of

123


and 4j(z) = Jawj (see I. N. Vekua, 1962, pp. 207ff.). Since we can certainly choose aj # 0 in 8,it follows that there exists a nonvanishing solution Ejj of ( a ) which has generalized derivatives in LP(@ for some p > 2. The other terms in a column can be computed successively from (3.12b). (The proofs of the above statements will be discussed later.) Using the matrix E , we define v := E-'w. By our choice of E we have (3.13) Recapitulating, we have the following result: Theorem 3.8 (Kuhn) Let w be a function with generalized derivatives w z , w, E Lp(@), p > 2. Then there exists a function v(z) which satisfies (3.13) and a nonsingular matrix E(z) so that in 8 , w = Ev.

When Q and E commute, we have au av -+Q-=O,

az

az

i.e., u is a solution of the principal part of the equation. An example of when this happens is given in the next corollary. Corollary 3.9 (Kuhn) If w(z) is a solution of

aw

aw

-+ Q-

az

az

= Aw,

where Q and A are quasi-diagonal, then there exists a solution u of av av -+Q-=O

az

az

and a nonsingular quasi-diagonal matrix E(z) such that w = Ev. PROOF Here C solution of E

=

A , and so, proceeding as above, we construct a

aE

al? az

-+ Q-

az

=

AE.

124


Define

Then E := { E k j } is quasi-diagonal and nonsingular, since &,&) (otherwise?!, is singular). We now show that aE

aE

-+ Q-az az

AE

=

0.

# 0 (3.14)

For k <j this follows from the fact that E, Q, and A are lower triangular. Writing out the system (3.14) we have for k 2 j, as in (3.12),

or, since

qk+m,j+m

-

q k j , ak+m.j+m

-

akj,

A

aE00

+

az

a$- j.0 az

which is true by (3.12), since

is true by (3.12). As noted earlier, quasi-diagonal matrices commute, hence we have E-'QE = Q

125


and thus av av -+Q-=O.

az

Corollary 3.10 (Kuhn, 1974) Let

az

zo E (3 and let w(z) be a solution of

where Q and A are quasi-diagonal. Suppose further that w $ 0 in a. Then i f z o is a Zero of w(z), there exists a constant A # 0 and a positive integer p such that

( z - zo)P + o(lz - z0r)

w(z) =

as

z

+

zo.

REMARK A similar representation theorem was also obtained earlier

by Gilbert (1973). Let us now consider

W?

+ QwZ + AW + B E = 0,

(3.15)

where w := ( w , , ..., wr)‘ and Q , A , B are (r + 1) x (r + 1) matrices. The matrices A and B are not assumed to be lower triangular, but we assume Q to have the form

Q :=

[l: 920

PZ

-..

lr,r-l

.**

*

9

(3.16)

0 P1

where all coefficients are taken to be Holder continuous. From the Bers-Vekua theory we know that for the case r = 0, the local behavior about a zero is given by w(z) = a(z - zo)” (1 + O(lz - Z O ~ ) )as z + Z O , which means w behaves as a holomorphic function. For the case r 2 1 , Habetha (1976) proved the following result:

Theorem 3.11 Let w be a solution of (3.15) with Q given by (3.16) and Q , A , B locally Holder continuous zo a zero of w of order N. If wjo is

126


the first component of w having a zero exactly of order N , we have

z

+

zo,

Wj(Z)

=

O(lz -

Z,lN+'),

j = 1,

...,j ,

- 1,

(3.17)

wj(d = pjN(z - zo, m) + O(lz

-

z,I~+~),

j

=j

,

+ I , ..., r.

Here ajo # 0, pi, is a homogeneous polynomial of degree N and is a quasi-conformal transformation of a neighborhood of zor (j(zo) = z o , which is a solution of

REMARK Theorem (3.5) only gives information about the first component having a zero of order exactly equal to N. For the other components no further information is provided. Let us note the following local similarity principle, due to Pascali (1965, 1967) and Habetha (1976), for systems of the form WZ

+ AW + BW

=

0.

(3.18)

An elliptic system having this normal form is often referred to as a Pascali system after D. Pascali (1965, 1967) who first studied them. We shall now prove a local similarity for such systems. A global result for bounded domains will be proved in Section 7. Note that the (r + 1) x (r + 1 ) coefficient matrices A and B are not assumed to have lower triangular form. Theorem 3.12 (Pascali, 1965, 1967; Habetha, 1976) Let (3 be a domain and suppose that A , B E Lroc(@).Let zo E (3 and suppose that w is a solution to (3.18) that is continuous in a neighborhood of zo. Then there 1) x (r + 1) matrix S that is continuous and nonsingular in is an (r C and a vector JI which is analytic in a neighborhood of zo such that w has the local representation w = SJI.

+

PROOFEquation (3.18) may be rewritten wz

+ c w = 0,

(3.19)

where C := (ajk + bjkFk/wk)J,k=O. For the given point zo let

127


D ( z ~E,) := { z : Iz

-

zol s

E}.

Let

For E sufficiently small the operator JcC,w is a contraction. Thus the equation w + JcC,w = e J , where e’ := (0, ..., 0,1,0, ..., 0)’ (prime denotes transpose), has a unique solution S J which is continuous in @. See Theorems (1.25) and (1.42). It is easily verified in the same manner as in the theory of ordinary differential equations that (det S)z

+ tr C , det S = 0

(3.20)

in @ and moreover that det S(m) = 1. Here “det” means determinant and “tr” trace. From Theorem (3.1) it follows that det S # 0 in C. Since S-’is a solution to (s-’)?- S - ’ C = o in D ( Z ~E ,) , it follows that ( S - ’ W ) ~= 0 in D(zo, E ) , which gives the result. We return to the equation (3.11) to obtain a representation for the matrix E . To normalize our solution we ask that it become the identity at infinity. It is convenient to associate the kth column E‘k’of the matrix E with the hypercomplex function (3.21) which satisfies the integral equation

The matrix E is seen to have the representation (3.23)

n:=,

where det E = E‘:’ # 0 in @. From this it is clear that E - ’ exists, and we may use the operator D* of (3.13). It was remarked that when E is quasi-diagonal then Q and E commute so that E-’QE = Q . In general, E-’QE is not quasi-diagonal; however, it is nilpotent.

128


In vector form we write Equation (3.1) as

I

Dw + AW + BW = 0. (3.1)’ We consider the special case where A and B are quasi-diagonal and 4 may be represented as the matrix Q, 0

0

0

...

0

0

41

...

0

41

If the vector w := (wo, ..., w,)’ is now identified with the hypercomplex function w := EL=owv(z)ev,with r

Y

then w is seen to have a Carleman representation

+

+kek which satisfy D*+ = 0. for hypercomplex functions := We investigate next the local behavior of 4 about its zeros. If the first component of 4 which does not vanish identically is + k , then D*+ = 0 may be written as a4k _ - 0,

az

a

az

-+&,

+

c qy*,a

u-

I

p=k

$+p

=

0

(k < u < u

-

I).

The zeros of +(z) must be isolated since +k is analytic. We denote by D,,p the space of functions in Lp having first order Sobolev derivatives that are in Lp also. Lemma 3.13 Let z = 0 be a zero of $(z), then in a sufficiently small neighborhood of 0, the components have the representations

where nu E N, and the qJpv E C” for some a , 0 < a < 1. Moreover,

129


JllLu E Dl,p (2 < p ) , and $pp(z)is analytic, with $,,(O) is a polynomial of degree amin{p - v - 1 , nu - 1).

# 0. Here p p v

PROOFThe proof is analogous to the Theorem (3.2), and is also given in Begehr and Gilbert (1982). For expository purposes we reproduce the main points of the argument. The result of Lemma (3.2) may also be presented in the form m+ 1 1 (3.26) m + l z = x(z) + JZ( m +E l )

-(z)

9

where J is the complex Pompieu operator and x(z) is analytic. To establish (3.25), we assume it is true for p = 0, 1 , ..., k - 1 ; then

k- I

(3.27) It is not difficult to show that (a/az)+,,, E Lp, which implies that E Lp. Furthermore, &,, is a polynomial whose degree is less than or equal to min{k - 1 - v , nu}. $ku

Applying the Cauchy-Pompieu operator to (3.27) yields

where the bracketed terms may be expressed in the form

= analytic function

+ JlkU(z)zn"

130


and

From this it follows that

where $ k ( Z ) is analytic, and $do) = 0. If $ k ( z ) f. 0, then there exist positive integers nk such that $ k ( Z ) = Znt$kk(Z). If $k = 0, then we renormalize our representation by setting $&) = 1 and represent +&) as

+

($kO(z)

+

PkO(Z)

- l)Zno*

The nk are not, of course, uniquely determined by the above procedure. In Begehr and Gilbert (1982) an improved version of the above result is established. which we list below.

Lemma 3.14 In a neighborhood of a zero, which we take for convenience to be at the origin, +,(z) may be represented as

where a, is either 0 or 1 , nu E N, and $, E C" n D I , ~0, < a < 1 , p > 2. Moreover, ,$ , is analytic, qW,(O) # 0 , and pPu is a polynomial whose degree smin{p - v - I , nu - I}.

Theorem 3.15 Let z = 0 be a zero of a solution to (3.1)'. Then in a sufficiently small neighborhood of the origin a vector solution w(z) has the representation w(z) = ESZ

(3.29)

where E is the lower triangular matrix (3.23); the entries of S ( z ) := (sJ

131


are defned by

with s,,(z) := +,,(z), and Z is the column vector

z := ( f f O Z " 0 ,

ff[Zn',

..., a,z n)r .

(3.31)

It is possible for us to demonstrate that the representation (3.28) has uniquely determined coefficients {a,, nu} if we require that for the pth component the restriction n, < nu holds whenever a, = a, = 1 , 0 s v s p s r - 1. To this end, we notice that s, has the form

4,

G ffuS,uZnw u=o P

=

(0 s p s v - 1).

To show uniqueness let us assume the existence of a second normalized representation, namely,

4,

c putpuzm= P

=

u=o

(0 6 p s r ) ,

where & and mu fulfill the conditions m, < mu whenever p, = p,, = 1 for 0 s u < p s r, where the tPu are bounded functions and t,, is analytic with tpp(0) # 0. We show that the a, and n, must be uniquely determined, by considering all the various possibilities that may occur. with analytic soo(z)and too(z). If p = 0 this implies that aosooz)Io= pOtoozmo This implies a. = Po, and if a. = Po = 1, then no = mo.On the other hand, if a. = po = 0, we proceed to p = 1 and consider alsllz;' = P l f l l z mand ' obtain the analogous result. Now we consider the generic case and assume, at first, that nu = m, when a, = p, = 1 for all v < p. The two possible representations for 4, imply that

If n, = m , we have

which implies a,s,,(O) - &t,,(O) = 0, and as s,,(O) and t,,(O) are not zero, it follows that a, = p,. Supposing, on the other hand, that

132

n,

#


m,

, or in particular that n,

C m F , leads

to

m~ which implies From this we conclude that lim,,o a p ~ , , ~ n w=-P,t,,(O), that a, = p, = 0. So a, = p, in both cases, and if a, = p, = 1 then n, = m,. Since the pairs {a,,, n,) are uniquely determined by the above procedure, the integer N defined by

N

:= min{n, : 0

sv

=S

r, a, = 1)

is a characteristic number that we refer to as the order of the zero. This is clearly also the order of w = E$J. Having discussed the local behavior around zeros of hyperanalytic functions it seems natural to investigate the nature of poles. It appears conceivable, for example, for just one component of a hypercomplex function to be singular. If this occurs should such a point qualify as a pole? It turns out to be more appropriate to demand that the reciprocal function have a zero in order that the function have a pole. Repeating much of the argument used in connection with the discussion of zeros, it may be seen (Begehr and Gilbert, 1982) that the pth component may be represented in the form

( p = 0, 1, ..., r), where as before apE (0, 1) and the p,,, are polynomials. To show this is indeed the case we provide a proof by induction. Suppose (3.32) holds true for all v such that 0 =S v s p - 1, then as before

which leads to an expression of the form

where $,u is Holder continuous and ppu is another polynomial. The function &(z) may be represented as a sum of functions p,,,(z)z-”” where

133

4. RIEMANN-HILBERT PROBLEM

each p,,(z) is a solution of the equations

More precisely,

with $,,(z) analytic and with $,,(O) = 0. From where $,(z) := $,,(z)z-~~, this we may argue to obtain (3.32) in an analogous manner to the case of zeros. As before it is possible to obtain a normal form by demanding n, < n, whenever a, = a, = 1 for 0 S Y < p S r - 1. Under these conditions the pairs {ap,n,} are unique. We choose the order of a pole to be the characteristic number defined by P := max{n, : 0

6

v

6

r , a,

=

I}.

We note, however, that this definition chooses the order of the most singular component. There is something lacking in this definition, as it is the orders of the zeros and poles of the complex coefficient which play the significant role in hyperanalytic function theory. Since in the case of poles we have that n, > n, when p > v, the function o(z) := [t(z) - t(ZO)lnoW(Z)

may be singular when w(z) has a pole at zo and no is the order of the pole of wo.Likewise if w(z) has a zero at zo,the function V(Z) := [t(z)- t(Zo)l-nOW(Z)

may also be singular at zo.

4. THE RIEMANN-HILBERT PROBLEM

In this section we investigate the special Riemann-Hilbert problem in a', Dw+Aw+BW=O

(A) :=

Re(Xw) = 0

on

r:=a,

(3.33)

where a' is rn-fold connected and bounded by the smooth closed Jordan curves ri(i = 0, ..., rn). Tois designated as the curve containing all the others in its interior. We consider the special case where A = 1 in order

134


to develop a reflection principle that helps to discuss the nature of zeros and poles of w on r. By performing a conformal mapping, a' is transformed into a domain lying within the unit disk and having a component y of its boundary lying on the unit circle. It may be shown that it is possible to reflect w(z) across the segment y into the reflection a, of a. The reflection is performed by the identification zE@, (3.34) The function w,(z) is then seen to satisfy the differential equation D,w,

+ A,w, + B,W,

=

0

in

U a, U y ,

(3.35)

and the boundary condition w i = w;

on y.

(3.36)

The coefficients are given by

The operator D , is defined as

a

a

D , := - + Q , - , az az

1:. (1) Q 0.

It may also be seen that strong ellipticity is invariant under diffeomorphisms of class C2.Because of the above result, Bojarski showed it is sufficient to investigate the equation

i(zw

-

zd

=

44zw + zw, z, w).

Furthermore, he showed that this system can be resolved into the form zw =

N z , w ,zw),

(3.60)

where h now is seen to satisfy the conditions (1) (2)

N z , w ,0) = 0, INz, w , 51) - h(z, w9 5211

4o(z,w)l& -

521,

where qo(z,w ) < 1 . Such systems (3.60) are referred to as A-systems. Since any solution of (3.60) is also a solution of a Beltrami system ZT =

14(w>l< 40

4(w)zw,

for w E @,

it can be shown that if z = z(w)is a homeomorphism, then the inverse function w = w(z)is also a homeomorphism. Iwaniec (1976) continues the investigation of equations of the form w, = qdz, w ,w,)w, + 42(z,w ,w,)% =

m z , w ,w,) = 4(z,w ,w,)w,.

(3.61)

For @ and 9, the unit disks in the z, w planes, Iwaniec (1976) proved that if the coefficients qi(z,w , 5) satisfy the strong ellipticity condition in @ x 59 x @ and, moreover, the qi are measurable in z and uniformly equicontinuous in w E 9 with respect to (z, 5 ) E @ x @, then there exists a generalized homeomorphic solution of the system (2.61) mapping @ onto 6 with the normalization w(0) = 0, w(1) = 1. A theorem concerning multiply connected domains was also proved (Iwaniec, 1976). Begehr and Hsiao (1980, 1981, 1982), in a series of papers, treat various nonlinear boundary value problems in the plane using the imbedding method of Wacker (1970), as was done in the work of Wendland (1974). Begehr and Hsiao (1980) investigated the boundary value problem of negative index wT = H(z,w ) Re{e"w}

=

in @,

$(z,w)

on

& = r,

(continues)

144


X-'

I r

Im{e'*w}crds = I

w(zk) = q k ( w ) ,

The system

K(w),

n = -1nd

k s n,

S

w7 = ql(z)w, + q2(z)wz+ h(z, w ) Re{e"w)

=

+(z, W )

w ( z ~= ) u~(w),

k

=

on

r,

1,

..., - n

T

E N.

in 8 , (3.62)

was investigated. Here it was assumed that 1q1(z)1 + (q2(z)(s qo < 1 . The approach follows the usual imbedding arguments developed in earlier papers and makes use of a homeomorphism of a Beltrami system to reduce a linearized form of the differential equation to one of the form wT = A(z)w + B(z)tii

+ C(z).

(3.63)

7. LIOUVILLE'S THEOREM AND THE SIMILARITY PRINCIPLE FOR PASCAL1 SYSTEMS

The results of Sections 1-5 assumed and made frequent use of the lower triangularity of the matrices A and B in equation (3.10). In this section we show that in the case of Pascali systems W,

+ AW + BW

=

0

(3.64)

a similarity principle and a generalized Liouville theorem can be established in the absence of such an assumption. As in Theorem (3.12), we assume A , B E Lp(@), p > 2. Moreover, we assume that the domain @ is bounded. Outside of @ set A, B = 0. Since the form of (3.64) does not change under the transformation 2 = kz, k real, we assume for convenience @ C Co, the closed unit disk. As in Theorem (3.12) it suffices to demonstrate the similarity principle for solutions to w7

+ cw

In general the equation

M W := w

= 0.

(3.65)

+ J ~ C W= 4

(3.66)

145

7. SIMILARITY PRINCIPLE FOR PASCAL1 SYSTEMS

may have nontrivial solutions for 4 = 0. This is illustrated by Habetha’s example in Section 4 of Chapter 1. If (3.66) is regarded as defining w in all of @, then solutions to Mw = 0 correspond to continuous solutions to (3.65) which vanish at infinity. Let Xdenote the space of such functions. By Corollary (1.27), the Hilbert space Fredholm theorem is applicable with the inner product [ w , vl := //c:w, $ du,.

Here, as in Chapters I and 2, (., .) denotes Euclidean inner products. The adjoint operator under the inner product [., is M*v := v

-

a]

--

C’J,,V,

(3.67)

where z E Co and the prime denotes matrix transpose. As in Chapter 1 we establish a correspondence between the null space of M* and continuous solutions to the adjoint differential equation w;

-

C’W* = 0,

(3.68)

which vanish at infinity by

(3.69)

W* = J Q .

Let X* denote the space of solutions to (3.68) that vanish at infinity. By the Fredholm theorem, N a n d X* have the same dimension N over @. Preliminary to proving the two theorems mentioned above, let us establish the unique continuation property. Theorem 3.25 If a solution to (3.65) is continuous in a domain 8 and vanishes on an open subset of 8 , then w = 0 in 8 .

0) is closed relative to 8. By the local similarity principle, Theorem (3.12), and the unique continuation property for analytic functions, the same set is open; hence it must be 8.

PROOF By continuity, { z E 8 : w(z, Z)

=

Any element w of X i s analytic in C\Co and hence has the expansion m

(3.70) where a 3 I and the vector a. # 0. The possibility of w 3 0 in C\Co is precluded by the unique continuation property. The number a is no greater than N, for if not, { Z ~ W } would ~ = ~ be an independent set of solutions in X of dimension N + 1. We refer to - a as the degree at infinity of w and a. as the initial vector.

146

3. REDUCTIONS TO HYPERANALMICITY

a*

Lemma 3.26 There are sets of functions 83 := {Cj}?, C X and := {C*'}zI E X * , with No + NO* 6 r + 1 (the dimension of the system (3.64)), having degrees at infinity -aj and -Pj and initial vectors {a{}y2 and { b { } y j l , respectively, such that { ~ ~ @ j }~ N= ol ,* k~ =j -O1 and {z 1'Nto9BJ-1 I , & = O are bases for X and X* over 62. Thus Xj"=",aj = Xyil Pi = d.=The vectors {ah, ..., a?, bh, ..., b:."}form a linearly independent subset of Cr+l with ( a { , bgk) = 0 , j = 1, ..., N o ; k = 1, ..., No*.

'-*J

PROOF Let - y 3 - N be the minimal degree at infinity possessed by any element of X. The set of all elements having degree - y forms a subspace X-, of X. Let X f y be a complementary subspace of J Y - ~ The . elements of X?, possess minimal degree - y + 1 and the elements of degree - y + 1 form a subspace X-,+, of XYy which then has a comin , NYy.Proceeding in this manner we obtain the deplement K Y + composition X = K Y@ -.. @ .IY-~. If w E X is of degree -a, then w Ex -0 ~ --.0 x - ~Let . := {w-j"}Y:,k=I be a basis over 62 for x where { W - ' . ~ } ; J = , is a basis for JY-~.Proceeding in the order (-7, l), ..., (-7, k y ) , ( - y + I ) , ..., ( - 1, k , ) we eliminate from W dl elements w-'sm that can be written as (3.71)

..

where pj,k is a polynomial of degreej - 1. We denote the set that remains by W := {Cj}?:,. Consider the set { a O - ' v k } ~ ~of, kinitial =l vectors of elements in 9. Suppose that Xjy_/ X.km=l cjk = 0 for some set of complex constants {cjk} with clm # 0. Then

is in X-, @ .--@ J C - ~ Consequently . w - ' . ~has a representation of the form (3.71) and is thus not in hence the initial vectors {a(}y:l of the elements of form a linearly independent set. We can carry out a corresponding construction for X * . We denote the := {C*j}yl1 and the set of initial vectors by {12'o)j=I. set obtained by Let { - aj}y2 and { - Pj}yl I designate the degrees at infinity of the elements of and respectively. From (3.65) and (3.68) we have for w E X and w* E X*

a;

a*

a

a*,

(w,w*)z =

(W?, w*)

+ (w,wt)

= =

(-cw,w*) + (w,w?) (w,w: - C'w*)

=

0.

147


Since both vectors vanish at infinity, Liouville’s theorem implies ( w , w*) = 0 in C. In C\C0 both w and w* have expansions of the form (3.70), and thus ( a & ,bh) = 0 for j = 1, ..., N o ; 1 = 1, ..., No*. Hence No + NO* 6 r + 1. Since neither No nor No* is zero unless both are by the Fredholm theorem, N o , No* < r + 1. Moreover {a;, ..., u p , b;, ..., b:’} must be a linearly independent set. Finally, let us show that L := { ~ ‘ i ? j } y 2 ~is a basis for X. Every w E X has the representation w = Z;=l cjkw-Jgk for some cjk E C. By virtue of (3.71) we can write w = Zy.l pj@j for some set of polynomials {pj}yiI with the degree of pi less than a j . Upon rearrangement we conclude that w = Xy2, ZP2d djkzkBjfor some set of complex numbers {djk}.Thus L spans X. To establish linear independence suppose that ZyiI XP5, djkzki?’ = 0 in C for some set of constants {djk}.In C\Co each @ j has an expansion of the form (3.70). Thus in C\Co

From the linear independence of the set {a&}yiIwe conclude that for j = 1, ..., No the constants dj,aj-k-I = 0 for k = 0, ..., aj - 1, successively. From Lemma (3.26) the following generalization of Liouville’s theorem follows at once. Theorem 3.27 (Buchanan, 1982) Every w E X has the representation

w

2 p,@’, No

=

,=I

where i?j E

&I, pj is a polynomial of degree less than aj, and No < r

+ 1.

Note that No < r + 1 implies that No = 0 if r = 0. Hence w = 0, which is Liouville’s theorem for the Bers-Vekua equation (3.2). We are now in a position to prove the similarity principle for Pascali systems. The principal difference between the general case and the lower triangular cases is that the similarity matrix will no longer necessarily be bounded at infinity. Its determinant however does remain bounded. Theorem 3.28 (Buchanan, 1983) Let w be a continuous solution to (3.64) in some domain (3 C Co. There is a ( r 1) X (r + 1) matrix S which is continuous and nonsingular in C and a vector t,b which is analytic in (3 such that w has the representation w = St,b in (3.

+

PROOFLet s j = @ j E &I for j = 1, ..., N o . Let (ai}yi1and {&}?it, pj be as in Lemma (3.26). A set of vectors may be

aj and

148


chosen to satisfy the following conditions: No (1) { p jO r}+j =l - N f i forms +l a basis for {u:, (2) for j = 1, ..., No*,

p’o E {UA,

..., uoNo , b:, ..., b;’}’,

...)up, b:, ..., b’o-l, b y , ..., boNg , po& + I , ...,p;+’-No}’.

Let pj:=

2 Pij-kZk,

j

k=O

=

1,

..., r

+ 1 - N,,

where the vectors p i are just indicated and mj = pi for j = 1, ..., No*; mj = 0 for j = No* + 1, ..., r + 1 - N o . The remaining coefficients of the polynomial vector will be chosen so that M w = p J has solutions. By the Fredholm theorem, the necessary and sufficient condition for this is [ p j , u] = 0 for each solution to M*u = 0. By Green’s theorem, Lemma (3.26), and (3.69) this is equivalent to f b ( p j , z‘W*‘) dz = 0,

I

= 1,

..., N,*,

k

=

0,

...,PI

-

1.

(3.72)

For (3.72) to hold we must have (3.73)

whenever k - PI + mi - s = -1. Since mj = 0 for j = No* + 1, ..., r + 1 - N o , this is possible only if k = PI - 1 and s = 0. The requirement (3.73) is then (do,bk) = 0, which is satisfied by the choice (1) of p i made above. For j = 1, ..., N,*,mj = Pj. When s = 0 and 1 # j, (3.73) becomes ( p i , b;) = 0, which is in keeping with (2) of the criteria for choosing p i . Finally, when s = 0 and j = I, then k PI + mi - s = k 3 0, and hence (3.73) is met vacuously. Since the bk are linearly independent we can solve

successively for p’; , ..., p i j . Thus (3.73) can be satisfied. Let sj”O be a solution to Mw = p-’ in Co. Since p j is entire, each = sj”O can be extended to C continuously by (3.66). Moreover,


149

+

p' O(2-I) in C\Co. Let S := (sl, ..., s'+'). Then S is a matrix solution to (3.65). Moreover in C\Co (recall Xy21aj = XyJl Pi),

det S

=

I det(aA, ..., a.No,PO, ..., POr + l - N o ) + O(z-'),

and thus by the choice of p i , j = 1 , ..., r + 1 - N o , det S(m) is a nonzero constant. By the same argument as in the proof of Theorem (3.12), det S # 0 in @. Thus if w is a solution to (3.65) in a,then (S-lw), o in a.

Function Theory over Clifford Algebras

1. INTRODUCTION

Let d,be a Clifford algebra over a quadratic n-dimensional real vector space V, with orthogonal basis e := { e l , ..., en}.Then d,has as its basis e l , ..., e n ; e1e2, ..., e,-le,; ...; ele2.-.e n . Hence an arbitrary element of the basis may be written as f?A = e,, e,, where A := { a 1 ,..., a h } C (1, 2, ..., n ) and 1 s aI < < a h s n. In general, one has ef = E i , e i ( E ; E R) and eiej + ejei = 0 (i # j), where at least one ei # 0. For simplicity in exposition we shall impose the restriction that ef = - e l , i = 2, ..., n , and = 1. We notice that the real vector space V, consists of the elements z := xlel

+ x2e2 +

***

+ x,e,;

(4.1)

V, is not closed under multiplication for n > 2. We may identify V, with R" by the obvious correspondence z = xlel

+

+ x,e,

t-)

(xl, x2, ..., x,,).

We shall refer to xI as the real part of z. If for z E V, we define the conjugate as

-

z := xlel - x2e2 -

- Xnen,

then ZZ = el(x: + + xi); hence, we normalize our system by taking el = 1 to be our unit element. If the absolute value for an element in n - 0

150

151

1. INTRODUCTION

d,, is taken to be Ic(' = (sum of the squares of the 2"-' basis vectors), then

1zI'

= 121' = Zz = zZ.

z(;)

(4.2)

We note that if z # 0, then we have

);(

=

z

= l;

hence, all nonzero elements of V, have multiplicative inverses. We now introduce two definitions due to Hile (1972). Definition For c E d", ( 1 ) c is conjugable if there is a d E d,, such that cd = dc = lcI2 = /dlz. This element d is called the conjugate of c and is denoted by T . ( 2 ) c is invertible if there is a b E d,, such that cb = bc = 1. This b is said to be the inverse of c and is written b = c-' = l / c .

Note that if c is conjugable, then @ = c. The proofs of the following two propositions are left as exercises. Claim For c E d,,, c is conjugable if and only i f c = 0 or c has an inverse satisfying

Moreover, i f c # 0 is conjugable, then

Claim For z,

5E

V, , lZ5) = lZ115l9

-

z5 =

5 z 9

and z-'5-' = (5z)-'

( z , 5 # 0).

Let D be an open connected set in R". If U := X;=,,@ Apw is the exterior algebra with basis {dx,, ..., dx,}, we consider the differential forms $(x) := XA,H$A,H(X)eAdxH where x E D c R". Furthermore, the functions are assumed to be of class C'(D) ( r 3 1). Integration

152

4. FUNCTION THEORY OVER CLIFFORD ALGEBRAS

of +(x) over the p-chain

r C D is defined as

The set of C'-functions in D with values in d is denoted by FX) := {f I f : D --* dn,f(x) := EAfA(x)eA}. We introduce also the operator

-

a :=

such that

a 2" e a ax, -

a = ~

af

-

The operator

:=

a is defined as

a,A

: FI;'

+FX-I)

(4.3)

afA eaeA-. ax,

(4.4)

Since dnis not a commutative algebra, the following expressions will, in general, be distinct:

au au au := - + e 2 ax,

+ ... + e n -au 8x2 ax,

au au u a := - + -e2 ax, ax2

+

+ -au en. aXN

Definition Let D C Rn be a domain and let u E F g ) . Then (1) u is left regular in D if au = 0 in D; (2) u is right regular in D if us = 0 in D.

For n = 2, eu = 2 (a/aT)u, and thus in this left regular is equivalent to analytic. Note that the formal product of the operators a and 3 is

the Laplacian operator. Thus the following is clear Theorem 4.1 Let D be a domain in R", with u E Fa). I f u satisfies Au = 0 in D , then

au is left regular in D ; ( 2 ) ua is right regular in D . (1)

153

1. INTRODUCTION

We now establish Stokes-Green theorems for functions in Fg). We follow here the notation and scheme developed by Delanghe (1970).t Theorem 4.2 r f M C D is an n-dimensional, differentiable, oriented manifold, and f E F$) ( r 5 1 ) and r is an arbitrary n-chain on M , then

where

c n

d u :=

a= I

( - l)a-'eadZa,

dj?, := dx' A as

* - a

A dxa"A &"+'A

A dx".

PROOF The proof is an immediate application of the Stokes theorem,

REMARK Iff is left regular in D , then for any arbitrary n-chain I' on M C D,

lr

duf = 0.

(4 * 7)

Another variant of our Stokes theorem is for two functions f, g E FZ). If r is any n-chain on M ,then

where the parentheses indicate whether 3 acts to the right or the left. The proof is standard and the reader is referred to either Delanghe (1970) or Hile (1972). It is clear from (4.8) that iff, g E F$) are, respectively, right and left

t

In this regard one should also see the doctoral dissertation of Hile (1972).

154


regular in D then for any n-chain

r on M we have

f d u g = 0.

2. REGULAR FUNCTIONS

Delanghe (1970) and Hile also give a Cauchy formula for functions f E Fg). As Hile's representation is actually a Cauchy-Pompieu formula, we present this result. Theorem 4.3 (Cauchy-Pompieu) I f S C D is a compact n-dimensional, differentiable, oriented manifold with boundary then for each z E 5. (the interior of S)

s

and

where w, := 2 d 2 / r ( n / 2 ) .

PROOFWe permit the abuse of notation and use z to stand for both a point in D and for the hypercomplex element z = x1 + xze2 + + x,en. For n 3 3 we note that

whereas for n

=

2,

,

In both instances

as both

(5

-

zIZ-,(n

2

3 ) and ln(5 - zI are harmonic. If D , :=

155


D\{5 : 15 - zI s c } where z E b is fixed and the &-ballabout z lies completely within b , one has by the Stokes-Green theorem

About the fixed point z , f(5) may be approximated as =

Hence as

E

-+

C f A ( z ) e A+

where lim o A ( E ) + 0.

oA(E)

A

E+O

0 one obtains for the integral over the &-sphere

-from which we obtain the Cauchy-Pompieu representation (4.9). If

af = 0, then (4.9) yields the Cauchy representation. A calculation analogous to the above verifies the representation (4.10). Delanghe (1970) has introduced the idea of a totally regular hypercomplex variable. These are hypercomplex variables of the form

z :=

u=l

x,e;,

where e; E d n ,

such that all powers z p are regular, i.e.,

-

a(zp) = 0,

v p E N.

(4.11)

To show that there are indeed such variables we exhibit the example zk := xk - Xlek,

-

azk

=

ek - ek

=

0.

Let the symbol (e;, , ..., eLP-J:= ,...,u p - l ) eAI ... eLp-l for all possible permutations with repetition of the sequence ( a l , ..., ap-J. Then the necessary and sufficient conditions for z to be totally regular are given by Theorem 4.4 (Delanghe) A hypercomplex variable z = Zt=lx,eL is totally regular if and only if the coefJicients e; satisfy the relations

f:

u=1

eU(e;,,..., e ~ ~ - , )=e b0

f o r all (a,,..., a P - ~E) { I , ..., n}”-’.

(4.12)

156


PROOFThe only ifpart follows by induction. It is clear that one wants to show

For p = 1 this is

-

a(z) =

C e,eb G-

=

0.

Since z p - ’ = &,....,a p - l ) xaI xap-l(ekl, ..., ekP-J,and the induction step requires that 3(zp-I) = 0, the necessary condition for total regularity is (4.12).

To see that (4.12) is also a sufficient condition we calculate directly

By expanding the coefficients, we obtain

C e,eb

=

U

0,

which shows that a(zP) = O. As Delanghe (1970) has remarked, if one has the commutivity relation eAei

=

eie;

( a , P = 1, ..., n),

(4.13)

then z = EE=l x,eb is totally regular if it is regular. In addition to the variables zk, we also introduce -

zk - zk:= (xk - a&

+ (xI - a l ) e k .

(4.14)

The homogeneous polynomials of degree p defined as

where the sum is taken over all permutations with repetitions of the

157


sequence (kl, ..., kp), are useful in obtaining an alternative form for the Taylor representation. For example, if p',"' is a homogeneous regular polynomialt of degree p , then it may be represented as

where the sum is taken over all possible combinations with repetitions of the elements (2, ..., n} in sets of p elements [Delanghe (1969, 1970a)l. Indeed, if [Pr'] is the set of all homogeneous polynomials of degree p in xl, ...,x, ,then [Pi'] is a right module over d,. The set of homogeneous polynomials

@':= {vi:!..kp : ( k j , ..., k p ) E (2, ..., n}P}

(4.16)

is a set of generators for the right module [P',"']. Delanghe (1970) refers to functions f: R" + ~4, ,f = &&A, as regular analytic in D when the f are real analytic and either (af)(x) = 0 or (fS)(x) = 0 in D. Expanding f ( x ) as a real Taylor series about x = 0. we obtain

1 ax,,

axap

+

...

(4.17)

where (ax,,

apf

ax,)(o)

= ~(ax,~aX,)(O)eA.

The homogeneous part of pth degree we designate as Qp

(

2

:= 1 x,~ * * * x , ~ ax,, P ! (at.....U p )

- -.axap)(Oh

apf

When f is regular and analytic in an open neighborhood of the origin, Qp is regular in D for all p . The polynomial Qp is an element of the right module [Pi'],i.e.,

Q&>

=

(ki

...

.,kp)

t All homogeneous polynomials in x , , x 2 , borhood of the origin are regular in R".

vk~~..kp(x)ckl~..kp

..., X,

which are regular in an open neigh-

158


where

ckl...kp

E

dnfor all ( k , , ..., kp) E (2,

..., n)”. As

we conclude ckl...kp = (apf)/(axk, ... a x k p ) ( o ) . From this one obtains an alternative form of the Taylor expansion for an analytic regular function about the origin, namely,

fh)= f(0)

Hence, a function f that is analytic and regular in an open neighborhood of the origin can always be expressed in terms of the n - 1 hypercomplex variables zz, ..., z,. On the other hand, as we have

c

d k i , ....kp)

zkl

*”

zkp

E @’,

v(kl, ..., kp) E (2,

..., n}’,

Hence if in a certain open neighborhood of the origin

, then f is analytic and regular. where the c k , ...kp E d,, If we consider regular analytic functions in a neighborhood of a E D , then we are led to consider series of the form (Delanghe, 1969, 1970a) f ( x ) = f(d

(4.19) This representation leads to the

159


Definition A function f: R" + d,,is regular-analytic in D C R" if for each a E D there exists a neighborhood N u ) C d such that f has a Taylor development about a, i.e., /

m

\

where c:!,.~, E d,,. As our normalization is somewhat different from that of Delanghe, we rephrase his main theorem of (1970) as Theorem 4.5t Iff E F g ) is regular in D C R", then for each point a E D there exists a ball B(a, r) C D such that in j ( a , r)

The proof makes use of the Cauchy formula (4.9) with Jf = 0, namely,

from which we may conclude that

..., cup) E { 1 , n}Pwhen z E b . Without loss of generality for each (a,, we take the origin to lie in'd and seek an expansion of (5 - 73/15 - zl" about the origin in a ball B(0, r ) C D ,i.e.,

+

* a * .

Putting this into the Cauchy representation for f(z) and identifying the derivatives o f f by means of (4.21) produces our result by the usual arguments concerning uniform convergence. t This result can be strengthened: the function f has a Taylor expansion in the largest open ball contained in D which converges absolutely and uniformly on compact subsets of this ball. This was shown independently in 1980 by Ryan (1983). Sommen (1983), and Goldschmidt (1983).

160


On the other hand, iffis left regular in the shell @ := 8 (0, R)\B(O, r ) , R > r, it is possible to develop f i n a Laurent-type series. To this end, ) we first mention that using the definition of the functions v ~ ! . . k , ( x given by (4.15) in the Taylor expansion (4.19) yields an expression of the form (4.22)

In order to obtain a Laurent expansion, we assume that there exists a shell @, := &O, R,)\B(O, r , ) C (3 such that (t/z + l)rl < (t/z - l ) R I . Further, we introduce the notation BI := B(0, R , ) , Bz := B(0, rl) as well as B ; := B(0, R ; ) and B; := B(0, r ; ) , with r; = (fi + l)rl and R ; = (lh- l)R1. From Cauchy's formula for all Z E l! 1\B2 we have

=: f d z )

+ f2(z).

Since fl(z) corresponds to the first integral, it is regular in admits in 8; an expansion

fi is w\B; : As

regular outside

8, and

B, we have the following expansion valid in

m

(4.23)

where (4.24)

161


and the

w k l...k,

are expansion coefficients for the right-regular function (4.25)

where (4.26)

and the conjugate coordinates Zk

:=

zk

are given by

xk - Xlek,

k

=

1, ..., n.

Using the Stokes theorem one may replace b, by B in the integral representation for the a k ]...kp and B2 by B in the integral representation for the b k l . . . k p . Hence we obtain the Laurent expansion

(4.27)

with (4.28)

and (4.29)

where convergence is uniform in 8; n (R"\&). Moreover, the representation (4.27-4.29) is unique (Delanghe, 1972). We refer to the series involving the v k l...kp and the w k , ...k, as the $rst and second series, respectively. As might be expected, a point x E R" is considered a regular point off if there exists an open neighborhood in which f is regular. A point that is not a regular point is a singular point. A point x is said to be a pole of order m if (1) f is singular at x, and (2) f has a Laurent expansion whose second series truncates at p = m.

If the second series of the Laurent expansion does not truncate, then we say f has an isolated'essential singularity at x.

.

162


For convenience, we assume in what follows that our singular point is situated at the origin. If in the second series of the Laurent expansion the term (X/lxln)bo appears, then bo is called the residue off at x = 0; hence, (4.30) where the ball B has been suitably chosen. Iff has a pole of first order at the origin, another useful method of determining the residue is given by (Delanghe, 1972): (4.31) Finally, we have an analog of the Cauchy residue theorem for an n-dimensional compact, differential, orientable manifold S, such that, within 3, f has k singular points a , , ..., ak. Then (4.32) Delanghe (1972) goes on to develop a Mittag-Leffler-type theorem. Next we turn to the inhomogeneous equation -

(4.33)

au = U.

In order to obtain a more general formulation we introduce the concept of a weak or Sobolev derivative for functions with range in a Clifford algebra. Definition If (3 C R" is a domain, u , u E L:oc(@), then u if for every 4 E Ct((3),

= eu

(weak)

(4.34) where CF(@) is the space of infinitely differentiable functions with compact support contained in (3. By restricting attention to real-valued 4, we see immediately that au is uniquely defined up to a set of measure zero. We now prove a regularity theorem for weak solutions of 8u = u. The theorem will be based upon the following lemma of Hormander (1958). Lemma 4.6 Let Mat(2) be a matrix with M rows and N columns whose entries are partial differential operators with constant coeflcients. Let a system of differential equations with constant coeflcients be given in

163


the f o r m Mat(3)u = f,

(4.35)

where u := ( u , , ..., uN),f := (f l , ...,f M ) are vectors whose components are distributions defined on a domain CY C R". Let 5 := (5,. ..., 5") be a complex n-tuple and define Mat(6) to be the matrix that results from replacing (a/axj) by itj in Mat(a). Assume also that the distance from the real point toto the set V := {t:rank of Mat(6) < N } tends to infinity with to.Then i f the distribution u satisfies (4.35) in @ and f E Cm(@),u is also in Cm(@). Theorem 4.7 (Hile, 1972) If @ C R" is a domain and u E L;J@), u E C=(@),and au = u (weak) in @, then u E C"(@).

PROOFThe function u can be written as From (4.34)

u = uI

+ e2u2+

+ e,u,.

In particular this must be true when 4 is real-valued, and so, since 4 u can have nonzero coefficients only for those basis elements appearing on the right hand side, we conclude that u is of the form u = ul

+ e2u2+ ... + e,v, +

x

where the summation in the last term is over 2 4 real-valued we have

ejekujk

S j

sk

d

n. Now for

where once again the summation in the last term is over 2 < j d k S n. This gives us the following system of 1 + (n(n - 1))/2 distributional equations

164


which we write as Mat(&)

=

f.

We now show that the set V defined in Lemma (4.6) contains only the = (0, ..., 0). First assume [I # 0. Among the point 5 := (51, ..., rows in Mat(& are the following (disregarding the factor i):

en)

(51

-52

51

-53

-5n)

..*

(53

0

41 ..-

0) 0)

(&

0

0

51).

(52

0

***

***

It is easy to show that these n vectors are linearly independent. Hence # 0 we have 5 $L V. for all 4 such that = 0, & # 0. In this case Mat(() has among its rows Next assume the following n row vectors:

el

(0 (&

(0 (0 (0

-52

0

-5n)

54

-53

-53

52

0 0

-54

0

52

***

0) 0) 0)

-en

0

0

-**

52).

0

*-*

**.

These are linearly independent, hence 5 $L V for this case also. Finally, assume that & = & = ... = & - I = 0, but & # 0. Then the following n rows of Mat(5) are linearly independent. (0

-*.

0

(5k

(O

-5k

(0

"'

(0 (0

...

(0

*..

..-

-5k

...

-[&+I

'**

-6n)

0) 0)

...

-C$k

0

0

...

0

-5k+I

5k

"'

0) 0) 0)

0

"*

(k).

-5k+2

0

-5n

0

...

Thus V contains only 5 = 0 and hence clearly satisfies the hypotheses of Lemma (4.6). Applying this lemma we conclude that u E C"(8).

165

3. HILBERT MODULES

Corollary 4.8 (Hile, 1972) If (3 is a domain in R", u E L:m(@), and 0 = au (weak) in (3, then u is (left) regular. 3. HILBERT MODULES

It is convenient for the development of our function theory having values over a Clifford algebra to introduce the idea of a Hilbert module. By this we mean a linear space X whose scalars are elements from a finite dimensional X* algebra d.The space X is endowed with addition of vectors and right-multiplication by scalars. These operations are governed by the following rules (wheref, g , h E X,a , b E d): (a) f + g = g + f , (b) (f + g ) + h = f + (8 (4 f ( a + b ) = fa + f b , (dl (f + g)a = fa + ga, (el ( f a ) b = f ( a b ) , ( f ) f . 0 = 0 , f . 1 = f.

+ h),

The algebra d may be over either the real or complex field, and as a finite X* algebra it has the following properties: (1) d is a Banach algebra with unity 1 (or I ) , whose underlying Banach space is a finite dimensional Hilbert space with inner product (,) and norm given by llall = ( ( a , a))'". ( 2 ) d has an involution a + a* satisfying

(a*)*

=

a

(ab)* = b*a* (ha)* = xa*(X real or complex).

( 3 ) (ab, c ) = ( b , a*c) = ( a , cb*). A n element a in d is said to be symmetric if a = a*, and positive if both a is symmetric and (ab, b ) 2 0 for all b in d. We call an element e in d a projection if e = ez = e*. Although not part of the definition, another property of a finite dimensional X* algebra is: (4) $ a is positive, then a has the representation n

a =

i= 1

h:ei,

where the ei's are projections in d,eiej = 0 if i f j, and each X i is a positive real number. Moreover a has a unique positive square root given

166


n

all2 =

i= I

hiei.

As further postulates on our Hilbert module, we assume that X itself has an inner product (f,g ) , defined for f,g in X and with values in d, and governed by the rules (g) (f + g , h) = (f,h) + ( g , h), (h) (f,g)* = (g,f), (i) (f, f ) is positive, and (f,f ) = 0 if and only iff = 0, (j) (f,gal = (f,g)a for a in d. We shall assume that the norm in d is normalized so that the norm of the identity is 1 . For any element a E d we define a truce by tr a := (a, I ) ; hence, tr Z = 1, tr(6a) = ( a , b*) = tr(ab), and tr b*a = (a, b). The modulus of an element a E d is defined as la I := t r [ ( ~ a * ) ” ~where ] (aa*)’/*denotes the unique positive square root of au*. We remark that our two norms are equivalent in that la1

6 llall G

ka

for some k > 1 and a E A. Our norm on the function space X is defined by

IIfIIn

:=

I(f,f)l’”

=

tr(f,f).

(4.36)

Finally, our last postulate regarding X is ( k ) 2f is complete with respect to

Il.ll~.

Definition A family X := X ( 8 ) of functions defined on a domain with values in the algebra d is said to be a Hilbert function module X if (1) it is closed under addition and right multiplication by elements of d,(2) it is endowed with an inner product with values in d,and (3) the postulates (a)-(k) hold.

A linear functional L on a Hilbert module X having values in d is said to be bounded if lLfl s

Ilfllx,

V f E ,3f,

where M is independent off. Goldstine and Horwitz (1966) have proven the next two theorems. Theorem 4.9 Iff, g E X,then

167

3. HILBERT MODULES

Equality holds, except for f or g zero, if and only i f g = f a , where a*a is a real multiple of the unity element in d.

Theorem 4.10 If L is a bounded linear functional on X , then there exists a unique g in X such that, f o r f in X , L f = ( 8 ,f ) . Definition A Hilbert function module X(@)is said to have a reproducing kernel K = K ( x , y ) if ( I ) K is defined pointwise in @ x @ with values in d, (2) for each fixed x in @ the function K , , given by K,(y) = K ( x , y ) , is in %'(a), (3) f ( x ) = K,f ) for x E @ , f E X(@). Definition A Hilbert function module X ( @ )has property P if for each

x in @ there exists a nonnegative constant M ( x ) such that

Iml

M(x)

Ilf

1 1 1

for all f in X. X ( 8 ) has property P' if moreover M ( x ) is bounded on compact subsets of (as a function of x). Theorem 4.11 #(a)has a reproducing kernel i f and only i f property P. The reproducing kernel, i f it exists, is unique.

%(a)has

PROOF The proof for the case of Hilbert modules is similar to that of Hilbert spaces; see, for example, Fichera (1954) or Meschkowski (1962). We list now several other results which are proved in the work of Gilbert and Hile (1977). Theorem 4.12 satisfies

The reproducing kernel f o r a Hilbert function module a x , Y ) = K(Y, X I * .

(4.37)

Theorem 4.13 If X(@)has a reproducing kernel, then a sequence that has property converges in the norm of X(@)converges pointwise. If %(a) PI, convergence is uniform on relatively compact subsets of a.

Iff and g are elements of %(a),then we say that they are orthogonal if ( f , g ) = 0. A collection {an} C X ( @ )is orthonormal when the elements are pairwise orthogonal and for each anthere exists a projection en E d such that ( a n , a n ) = en, anen =

168


Theorem 4.14 (Goldstine and Horwitz, 1966) If X is separable, then for X,in the sense that each there exists an orthonormal basis g in X has the representation

where the series converges in the norm of X,and moreover,

We notice that the Fourier coefficients of K, are ( a n ,

K,)

=

hence,

w,, an)*= an(x)*;

2 @,,(y) an(x)*. n

Kx

=

K(x,Y ) =

(4.38)

n

As in the usual case, the reproducing kernel can be characterized as a solution to certain extremal problems when it is possible to invert K ( x , x) in the algebra d.We clarify the invertibility problem with the following two lemmas: Lemma 4.15 (Gilbert and Hile, 1977) Necessary and suficient f o r K ( x , x) to be invertible for x E @ is that f(x) is invertible for some f in %(a).Furthermore, if a E d,then aK(x, x) = 0 if and only if af(x) = Ofor a l l f i n X(@).

PROOF If uf(x) = 0 for all f i n %(a), we choose in particular f = K, and obtain aK,(x) = aK(x, x) = 0. Conversely, assume aK(x, x) = 0. We then have (K,a*, K,a*) = a(K,, K,)a* = aK(x, x)a* and we conclude K,a* = 0. But then, fo rfi n =

a(K,,f)

=

=

0

%(a),

(Kxa*,f) =.(O,f)

=

0.

The first part of the lemma follows from the second part and from the fact that f(x) is invertible if and only if uf(x) # 0 for all nonzero a in d. Lemma 4.16 (Gilbert and Hile, 1977) For each x in @ a n d f i n X(@) there exists an element a in d such that f ( x ) = K ( x , x)a.

PROOFIf K(x, x) is invertible, then a = K ( x , x)-' f(x). For the more general case, we observe from K(x, x) = (K,, K,) that K(x, x) is positive

169

3. HILBERT MODULES

definite and has the representation K ( x , x) =

c hiei, i

where each hi is positive and the ei's are projections mentioned earlier. Defining e := Z

-c i

and

ei

d := e

+ xi ( h i ) - ' e i

yields the following:

+ e)d = I , %'(a),

( K ( x , x) Forfin

with ed

de

=

=

e,

and

K ( x , x)

=

0.

( K ( x , 4 + e)df(x) = K(x, x)df(x) + efW. But by Lemma (4.15) we have ef(x) = 0, which implies the result with a = df(x).

f(x)

=

We introduce at this time two extremal problems. Problem Z is to minimize llfllR for f E %'(a) subject to the side condition f ( x ) = b for x E @ a fixed point. Problem ZZ is to minimize Ilfllxforf€ %'(a) subject to the side condition If(x)l = 1 for x E @ a fixed point. Theorem 4.17 (Gilbert and Hile, 1977) If b has the representation b = K ( x , x)a, then Problem Z has the unique solution g = Kxa.

PROOF We write g in the form g = Kxa + h , where h E X(@).The condition g(x) = b gives b = K(x, x ) a + h(x), and hence h(x) = 0. Moreover, lIg1l2 = t r k , g ) = tr[M,a, Kxa) + a * ( K x ,h ) + ( h , Kx)a + ( h , h)l. Since ( K x ,h ) = h(x) = 0 , we have llgll& = llKxall&+ llhll&. This quantity is minimized when llhllBe= 0, h = 0 , and thus g = Kxa. We notice that when K(x, x) is invertible Problem I has the unique solution g = K,K(x, x ) - ' b . Furthermore, for the special case where b is the unity of d we have ( g , g ) = K(x, x)-'(K,., Kx)K(x,x)-' = K(x, x)-', and therefore Kx = g K ( x , x) = g ( g , g ) - ' . Theorem 4.18 (Gilbert and Hile, 1977) ZfK(x, x) # 0 , then a necessary and sufficient condition for Problem ZZ to have a solution g is that it have the representation g = Kxa, where a E d and a*a = IK(x, x)I-'Z. PROOF Iff E X(@), If(x)l = 1, then by Theorem (4.9) we have 1 = If(x)l

and

llfllx

5

=

W

x

9

f)l

ll~xll~llfllYt

~ l K x ~ ~Moreover, &'. equality holds above if and only iff =

170


Kxa, where a E d , a*a = pZ, p > 0 ( p is positive since a*a is positive definite). Thus Problem I1 is solved by those functions g of the form g = Kxa, where a*a = pZ, and where (g(x)l = 1. The last condition gives I

=

IK(~,

x)al = tr[(K(x, x)aa*K(x, x))"~]

tr ~ ( xx), = p ' / ' I ~ ( xx)l ,

= p'/'

and thus p = IK(x, x)I-'. The paper by Gilbert and Hile (1977) lists numerous other results concerning Hilbert modules; however, since we are primarily interested in the applications to function theory we pass over these to an application to solutions of 3.f = 0. It is well known that the finite-dimensional Clifford algebras have matrix representations, and since it will be advantageous from time to time for us to use the matrix formulation we introduce this here. To this end let A', A 3 , ...,A , be rn x rn matrices with complex constant entries. Furthermore, assume that the matrices satisfy

where Z is the identity matrix. We consider complex matrix solutions in R" of the partial differential equation

-af:= Let x := (xI,x 2 , define

z-af

ax,

..., x,)

+

ax2

af = 0. + A,, -

(4.40)

E @ C R" where @ is a domain, and let us

+ + xi)'/', := xIZ + ~ 2 A 2+ + x,A,,

1x1 := (xi X*

af + -

*.*

X := x ~ Z- ~ 2 A 2-

.**

-

x,A,.

As we saw earlier the C'(@) solutions of (4.40) have a Cauchy integral representation, namely, for x E @ (4.41)

where v is the outward-pointing unit normal vector on surface area of the unit sphere in R".

and

o,,

the

171

3. HILBERT MODULES

As is the case for analytic functions, we can use the representation

(4.40) to show that the uniform limit of solutions to (4.40) is also a solution to (4.40). Now, let @ C R" be a fixed domain, and let d,, be some subalgebra with unity of the algebra of m x m constant matrices such that dnis

closed under conjugate transposition. We define (a, b) := tr ab*,

a , b E dn,

where as usual the trace function is normalized so that tr I = 1. For our we define inner product with values in d,,,

here f and g are defined in @ and take values in d,,. We ask now that Ye(@) be the subfamily of C'(C5) solutions of (4.40), having values in d,,, such that

It is readily verified that axioms (a)-(j) given earlier for a Hilbert module hold for the subfamily Ye(@). The completeness axiom (k) requires some justification. For a fixed x E @, we choose B(x, p) C @ to be a sphere with center x and radius p. Using the Cauchy representation (4.41) for the surface B(x, p), we have

Using (4.38) and the conjugacy definition, we obtain ( Y ) ( y - x)* = Iy -

and hence f(x) = which yields

0,'

p'-"

I I

f ( x ) = np-"w,'

IY --XI = P

XI2

z

=

p2 I ,

f(Y)ddY),

IY - - X l S P

f(Y)dY*

We recall that for a E d, llallap = (tr(aa*))'I2, la1 = tr(aa*)"2,

172

and 1.1


S

1 ) ~ ) ) ~Then . using the Cauchy-Schwarz

S

inequality one obtains

np np-"wi I (wnpn/n)'"IIflILZ.(~.

Hence we have an inequality of the form

If(x)l c C K " / *IlfllLZ.W

9

(4.42)

where R, is the distance from x to a. From (4.40) it follows that norm convergence in X(@)implies uniform convergence on compact subsets of @. Axiom (k) then follows directly. Finally, (4.42) implies that X(@) has a reproducing kernel, and axiom (k) then implies it has an orthogonal expansion (4.38).

REMARK It is clear from our development that if we had considered, instead of matrix solutions (4.40), solutions of jf = 0 having values in d,,, the Cauchy formula for hyperanalytic functions would permit us to establish for this case the validity of (4.42). See in this regard also Delanghe and Brackx (1980). 4. LIOUVILLE'S THEOREM

We prove next a version of the Liouville theorem. To this end we introduce the Definition (Delanghe, 1970c) The point f(x) if

~0

is called a regular point for

(1) there exists a real number E > 0 such that for all r E (0, E ) , f ( x / r ) is regular in the open domain &O, b)\B(O, a) for suitably chosen a and b with 0 < a < b < + m. (2) limr+of(x/r) = f(m)exists and does not depend on x.

Theorem 4.19 (Delanghe, 1970b) Zffis left regular in R" U f is a constant.

{cQ},

then

PROOF We take x to be an arbitrary but fixed point of R" and choose

173

4. LIOUVILLE’S THEOREM

E > 0 sufficiently small so that llxll < 1 / ~ Then . from Cauchy’s formula for x E &O, l/r), 0 < r < E , we have

As the value of the integral is independent of r, if

exists, it must equal f(z). To show that the limit does indeed exist we first introduce the hyperspherical coordinates

cos o1 cos o2

=

5,-

a . 0

52 = T cos

cos e2

= T

cos

sin e2,

=

sin e l ,

I

5,

T

cos 8,-

I,

sin 8,- I ,

s 2 ~ and , - ~ / 2 s Oi s ~ / for 2 i = 1, ..., n - 2. where 0 s Introducing a change of integration parameter 5 = (‘/r, we have after a short calculation

\a= I

As limr,of(t’/r) = f(w) exists, we have after passing to the limit under the integral sign

= f(4.

Since f ( x ) equals f(w) everywhere, f(x) is a constant.

174


5. (u-HOLOMORPHICFUNCTIONS

A function theory in the large, i.e., independent of pointwise partial differentiation, can be constructed on the basis of the areolar derivative (Pompieu, 1912; Theodorescu, 1931). A function f E Co(@) is a-holomorphic in the domain @ if there is a 4 E Co(@)such that

for every a' C 6'. We write

which is bounded by a simple, closed, rectifiable curve

\JP

I f f € C'(@), then (Df/Dw) = (af/aZ).The Pompieu representation (1.28) remains valid for these generalized Cauchy-Riemann operator. When a function f and its successive areolar derivatives (D,f/Dw,) are uniformly bounded in n, we have the expansion

where the &[f] are a-holomorphic. This notion extends to m dimensions where the definition of areolar derivative is (Moisil and Theodorescu, 1931)

/;(n,

*

y)4(M) du,

+

1.

@(Q> d w = ~ 0

for every w C @ and bounded by a regular surface u. Here y = {y,, ..., ym} is a given, fixed, collection of constant matrices, n, is the interior normal to ua + M , and (n, y ) = Ey=l nj,yj. The function = D 4 , when it exists, is called the left spatial y-derivative. For this derivative we have the Pompieu representation

Various aspects of a function theory for a-holomorphic functions have been developed by Coroi-Nedelcu (1959, 1960, 1965a,b, 1967a,b).

175

6. GENERALIZED REGULAR FUNCTIONS

6. GENERALIZED REGULAR FUNCTIONS IN R"

Goldschmidt (1980) considered generalizations of regular functions by investigating solutions to the equations

-

C, cA(x)HAw(x) A

aw -

= F(x),

(4.43)

where the CA(x)are hypercomplex-valued functions and HAis a mapping that we define as follows. Let H i , 2 s i s n, be the linear mapping of A onto itself that maps ei +-- e,, but for j # i leaves ej fixed. Then we define HA:= Ha, ... H a p , A := {Crl, ..., aP}, where 2 s al < < ap s n. Furthermore, we shall assume that the cA(x) E Lp(@). We say, as usual, that w is a weak solution of (4.43) when we have

For w and u

E C ' ( 8 ) the

Green's theorem (Hile, 1972)

+

~ p ~ dx) w

1

u JW dx = . u d m

which implies Jcv[ ( ( u a ) ~+ u

C, CAHAW)+ A

( JW u

-

u

A

u duw.

=

11

C, CAHAW

dx (4.44)

The notation Re(a) means the coefficient of e , (or 1). We have then from the definition of H i , Re

~(CAHAW) dx

Re

w

+ U ( ~ W- C CAHAw)

Hence from (4.44) we have Re

I[

(US + C, HA(uCA) A

=

Re

u duw.

I,

=

HA(uCA)Wdx.

1

dx (4.45)

If we introduce an inner product for functions having values in the

176


(4.46)

then the formal adjoint to the differential equation (4.43) is seen to be

~8 + 2 HA(ucA) A

= F.

(4.47)

Furthermore, it is clear that w is a weak solution of (4.43) if and only if

[

Re/ w (($3)

+ 2 HA(4CA))w+ 4 F ] dx A

=

0,

V+ E Ct(C4). (4.48)

By applying the operator

to the left-hand side of equation (4.43) we construct the operator

which we may consider in R" if it is assumed that CA = 0 in the exterior of (3. Consequently, in what follows we abbreviate Ja by J . Using the definition of the inner product ( ,) given above, a direct calculation shows that the formal adjoint to M is given by (Goldschmidt, 1980) M*U := u -

where

2H~(C~J*U)

(4.50)

A

Whereas to the differential equation (4.43) there corresponds an integral equation

'I

Mw

= F,,

F , ( x ) := JF(x) + - . -du,w(r) = JF(x) W" (~ Iti - xXI''

+ @(x),

(4.51)

to the adjoint equation (4.47) there is the corresponding adjoint integral equation M*U = u HA(CAJ*U) =F ~ . (4.52)

2 A

t In what follows we use an extended notion of conjugate (Goldschmidt, 1980). For the hypercomplex number u = Xa,, . oke,l ... e,,, defining ii = Xu,, . (, - 1)'ee,, ... eel.

177


Before proceeding further with our investigation of these operator equations we present some results concerning the operator Js . Our approach follows that of Hile (1972); further information may be found in the works of Iftimie (1965, 1966). Theorem 4.20 (Hile, 1972)

JW E Lf,(g), and

If F C R" is a domain and u E L'(@), then

u = ;~(J@U)

(weak).

PROOF Let a0C @ be a bounded domain and denote its characteristic function by XB.. Then

M ( n , @d/Jv(t)l dt = M ( n , W l u ,

@I*.

This justifies the use of Fubini's theorem in the following:

Hence Jau E L:w(@). Let 4 E C:(@). To complete the proof we make use of the Cauchy-Pompieu representation

178


The above interchange in the order of integration is permissible, since

Corollary 4.21 (Hile, 1972) Let G C R" be a domain, u E Ll(@),and Then w E L,!,J@) with u = 8w (weak) in

a.

w = J,

+J

~ u

in @ where J, is left-regular. Conversely, i f J, is left-regular in @ and u E L'(@), then J, is in L,',J@) and

-

+ JCYU)=

+ JCru

u.

PROOF If u E L'(@), then by the above theorem Jmu E L,',,J@). Also, from the above theorem it follows that for w E L:,(@) and u = 8w (weak) that -

a(w - JCUU)

=

u -u

0.

=

By Corollary (4.8), J, := w - JCru is left-regular in (3. The converse follows immediately from the above theorem. Definition For each x , t E W, n

3

2 , and v

2

0, we define

2 Ix(Y-klt(k Y

P,(x, t ) :=

k=O

(Observe that P,(x, t ) = 1.) Lemma 4.22 (Hile) For nonzero t , x E R", n

3

2 , and v

3

0

179


PROOFBy squaring and expanding the left-hand side we obtain

Hence, if we could show that P,(x, r)lx

lxlxly - tltlYl

-

rl,

(4.53)

we would be finished. To this end, we note that if t or x is zero, or if It1 = 1x1, the result is trivial, so assume t , x # 0 and It1 # 1x1. Then (4.53)is equivalent to lxlxly

- t(tl'(2s Ptlx -

?I 2

But 0 s Ix

- t12 - (1x1

-

Itl)2 =

2lxlIrl -

Hence (4.53)is equivalent to

s P?(x, t )

IX(YJflY

= lXIYltlY

tx - xt.

+ ...,

which is clearly true and, hence, our result follows. The next lemma may be found in Hellwig (1964)or Miranda (1970); for example, Lemma 4.23 (Hadamard) Let @ C R" be a bounded domain, n 3 2 , and let a and p satisfy 0 < a, p < n, a + p > n. Then for all xI, x2 E R" such that x1 # x 2 , we have -

xII-alt - x21-@dt s M ( a , p)IXI - x21n - a - p

*

Theorem 4.24 (Hile, 1972) Let @ C R" be a bounded domain and let u E Lp(@), n < p < 03. Then w = Jcru is in Bovu(R")twhere a = t Bo."(w")is the space of bounded, Holder continuous functions in w".

180


where (l/p) + (l/p') = 1. As n < p < m, we have 1 < q < n / ( n and n - 1 < (n - 1)q < n. Hence, -

-

I),

~ l ( ' - "dt) ~s M ( n , q , @)

for all x E R". This proves (1). To prove (2) we first note that

An application of Lemma (4.23) yields (W(XI) -

w(xz>l

For 1 s k s n - 1 we have 0 < kq < kn/(n (n - k)q = nq > n. Lemma (4.23) implies that JJt

+

1) < n, and kq

+

- x ~ J --~x J~ J~ - ( ~dt- s ~ )M(q)Jx, ~ - x~\~('-~),

from which our result follows when this is applied to (4.54). From the regularity theorem (4.27) that will be proven later it is clear that the operator J@ is compact even on L2(@)not just on Lp(@), n C p < 03; hence, we may apply the Fredholm theory of compact operators

181


to conclude that the homogeneous equations Mw = 0 and M*w = 0 have at most a finite number of linearly independent solutions. If {wl, ..., wN} is a basis for the null space of M and {ul, ..., uN.} a basis for the null space of M*,then by the Fredholm alternative we have that N = N ’ . Furthermore, if the basis functions are orthonormalized with respect to (,), i.e., ( W i , Wj) = (u.1 ’ u.) J = 6.. lJ 9

then (4.51) is solvable if and only if ( k = 1, ... N ) ?

(Fly uk) = 0

9

and (4.52) is solvable if and only if

(k

(Fz, wk) = 0

1,

=

..-)N ) .

This suggests that, as usual, one constructs the new integral equation N

Mlw := Mw

+ 2I (w,w k ) U k k=

=

Fl.

Taking the scalar product with ui yields

=

(w,Wi) =

(Uj,

FI).

Hence, it follows that the solutions to the homogeneous equation Mlw

=

0

are unique, since in this case we have for each solution w that (w,wj) =

(uk,

0)

=

0.

Goldschmidt (1980) then goes on to show that from the Fredholm theory we know that a resolvent kernel rA(x, t) exists such that whenever there exists a regular function @ (i.e., = 0) with @ E C0(@ n C’(@) such that (Fl, uk) = 0 ( k = 1, ..., N) and FI := JF + @, then the solution to (4.51) may be represented in t.he form w(x) = R(F1) +

here the dk are all real constants.

c N

k= I

dkwk,

182


As is usual in the Fredholm theory the kernels r A satisfy an integral equation; in this case we obtain (Goldschmidt, 1980)

N

Changing orders of summation and using the notation

u

A A B := (A\B)

(B\A)

= (A

u

BNA

n B),

one obtains after renaming summation indices (Goldschmidt, 1980)

(4.56)

Since Fl(t) is arbitrary, as @ is arbitrary, we consider F l ( t ) of the form ecC#I(t),where C#I is a real-valued function. As our equation is linear, it will be seen that this can be done with no loss of generality. The inner product terms may then be expanded as (uk,

FI)u~(x) = Re

vk(t)eA(t) dfuk(X)

=

Re(=h(x)W

dt.

Replacing these terms in (4.56) and taking into account the arbitrariness of +(t) leads to the conclusion that the coefficient of +(t) must vanish identically, namely,

2 B

[

rB(X,

x

t)

+ K(t,X)CB(t) +

rAt, 2) d5

]

HBe, +

2 A

K ( t , X)CAAB(t)HAM

2 Re(~,uk(t))vk(x)= 0. N

k=l

183


From Goldschmidt (1980, p. 17, “Remark” (1.3)), it can be seen that the above system may be solved as an algebraic system for the square bracket term, namely, t) =

+ K(t, x)c~(t) + -2-“

c N

k= I

c A

K ( 5 , ~ ) C A A B ( S ) H A A B ~ A ( S0, d5

vk(X)HB(Uk(t))

(4.57)

The operator adjoint to MI can be seen (Goldschmidt, 1980, p. 40) to have the form N

MTV =

M*V

c(V,vk)wk

k= I

As before, it is possible to obtain a representation for the resolvent R* to MT, namely (Goldschmidt, 1980), R*F2(x) = F2b)

+

c A

H A r A ( f r x ) H A F 2 ( t ) dt.

The identity Mf(R*F2)= F2 then leads to the expression

and proceeding as before we are able to change orders of summation and solve for the bracketed terms

184


or

(4.59)

Goldschmidt (1980) is also able to obtain a generalized Cauchy representation in terms of Vekua-type kernels for solutions of the differential equation (4.43) which are of class Co(@), that is, he obtains representations of the form w(X) =

c A

c N

l b RA(X, f ) H A (dgtw(t)) +

k=l

dkwk(X),

(4.60)

where the dk := ( w , wk) (cf. equation (1.56)). The generalized Cauchy (Vekua-type) kernels are defined as

In order to obtain an integral equation for the kernels RA we multiply (4.57) on the right by HBK(5,t ) and integrate with respect to t. We obtain

Replacing the indices AAB by C and summing over C , we can replace

185


If the symbol 6(B) is introduced by 6(B) =

1

for B = $I

0

for B # $4,

then we obtain the following integral equation for the fl,(x, t):

(4.61)

In an analogous manner we may multiply on the right by HBK(q, t) and integrate with respect to t to obtain another integral equation (Goldt ) , namely, Schmidt, 1980) for the t ) - a(B)K(t,x ) +

c A

flA(x, q)HACAAB(7))HBK(f,

7) d q

From the Hadamard estimate (4.25) and (4.61) above it is now obvious that for (x, t) E (3 x (3 one has an estimate of the form

Ifldx, t ) - 6(B)K(t,x)l

Ix

C

-

f("+-l'

(4.62)

where a = ( n / p ) - 1 . In order to show that the generalized Cauchy kernels have the appropriate residue we apply the Green's theorem to the region @& := (3\ { t : It - XI s E } with u := - H B f l B ( x , t ) and w a continuous solution

186


of our differential equation, i.e.,

Applying HB to this equation and summing over B yields

By replacing AAB = C in the second integral and summing instead over C we notice that this cancels the first integral, and by using the identity X B HBu = 2"-' Re(u) we finally reduce our equation to the form

The local behavior of the fi&, Schmidt, 1980)

t ) then yields the representation (Gold-

As noted by Goldschmidt (1980), the construction of the generalized Cauchy kernels from the integral equation (4.62) is quite difficult because of the coupled nature of the equations. To decompose the system, GoldSchmidt multiplies the system by the element eD, operates on the product with H B , and sums over B. After some manipulation, an integral equation

187


In order to prove some regularity theorems about solutions to (4.43) we first establish some results concerning singular integrals. To this end we make the Definition We say w E Lp*"(R")if IwI, Iw(")I E Lp(A,,) where A,, is the unit sphere in R" and w(")(x):= 1x1- "w(1 /x)

for x E AN.

This space is normed by

&Ip

Iwlp.u:= Iw,

+ Iw"),&I p .

The following result has been established independently by several authors (see, for example, Hile, 1978; Iftimie, 1966). Theorem 4.25 r f u E Lp"(Rfl), n < p < m, then the function w = Ju(:= JRnu) is in Bo.a(Rn),where a = ( p - n ) / p . Moreover, w satisfies

lw(.dl s W n , P>I.l p.n * (2) Iw(xl) - w(x2)l M(n, p)Iulp,nlxI - x21a* (3) For any a > 1 , there is a constant M(n, p, a ) such that for

(1)

1x1 3 a

Iw(x)l s M(n, p ,

aw (weak) in R". PROOF We write (4)

u =

~)(U(~,,,~X~~'~-(~-').

188


For @ ( x ) we have

(4.65)

where -(n-l)p'

dt.

To treat Z(x) we consider two cases: (a) 1x1 2 1/2. We have (I-n)p'

dt.

Since

implies (n - 1)p' < n , p' be applied to obtain

+

(n

-

1)p' = np' > n, Lemma (4.23) can

Z(x) s IxI(I-")p'M(p,n)

1;

ln-p'-p'(n-I)

(4.66)

2(n-l)p' This proves (1). For (2), it follows from Theorem (4.24) that I W l )

- W Z ) l

s M ( n , P ) l V , AnlplXI - x*Ia.

189


By Lemma (4.24) n- I

since kp' Now

6

(n - 1)p' < n.

and

6

2a21-alx1 - x z y .

Hence Z(xl, x 2 , k ) s M(n, p)(xl -

XZ(("-I)~'.

(c) If lxll 6 1/2, (xzl 3 1/2, we proceed as in (b). (d) If lxl(, Jx2(3 1/2, then

We have kp' s (n - 1)p' < n,

(n - k)p' kp'

+ (n - k)p'

6

(n - l)p' < n,

= np'

> n.

190


By Lemma (4.23) n(l - p ' )

Z(Xl, x2,

k) 6 2 " P ' M ( n , p ) 1

IXI

=

2np'4n(1- P')

-

XZln(l-P')

2np'4n(I -p')IXI

-

XzI(-

I)P'

Now returning to (4.67) we have (a) For IxIJ,lx2l s 1 / 2 , IxI - x2[ 1, and hence 1xI - x2[s [xI- x2IP. (b-d) In all these cases we have (XI

- X2[(Z(XI,

X2,

k))'/"' s M ( n , p)Jx1- X21JXI - X 2 1 ( e - I ) p ' / P ' =

M n , P)lXl - &Ia.

Hence (2) is proved. For (3), we note that for 1x1 > 1 , (tl s 1, It -

XI

2

lltl - 1x11 2 I1 - 1x11,

and so

For a, p > 0, p - a < 0, the function p/(s - 1)* is decreasing - for > 1 . Hence for a = n - 1, p = n - 1 - ( n / p ) we have

s

for 1x1

3 a.

Thus

191


Since n / ( n - 1) s p , we have p' lG(x)l

-

n s (n/p) - (n

-

1) and so

s M ( n , p)lu("),A n ~ p [ x [ ( " ~ " ) - ( n - ' ) .

This gives (3). To prove (4) we proceed as in Theorem (4.20). From the above theorem and Corollary (4.21) we have Corollary 4.26 I f w E L,',,(R"), u E Lp"(Rn),n < p < (weak) in R", then w - J u is left regular in R".

03,

and u = aw

Theorem 4.27 I f u E Lp3"(g), 1 S p s n , then the function w := JBU belongs to Ly(@) where @ is a bounded domain in R" and y is an arbitrary number satisfying the inequality l n. We conclude, using Theorem (4.25), that w E CO."(@J. Goldschmidt investigated uniqueness of the solutions to the integral equation (4.51) w - J~

2 0 CAHAW

=J

~ + F

E cosy@);

which leads to the question of whether the homogeneous equation

possesses merely the trivial solution. He finds that if the region @ is sufficiently small then only the trivial solution exists; in particular this is true if the CA E Lp(@), and (4.70) where p > n and (l/p) + (l/q) = 1 . It is also possible to prove a global result, i.e., for = [w", if we require the coefficients C A to be sufficiently small in R". We recall that by reflecting through the unit sphere we may transform the LP-integral over the exterior of the unit-ball into a weighted integral over the unit ball, namely,

195


Recalling the definition of the spaces Lp*'(R"), this provides the proof of the identity, LP([w") =

LPAWP)

Wn).

Furthermore, from the inequality

dx

we may conclude that for p s ( 2 n / p ) s v the inclusion LP.*([W")-J LP(R") = LP.(*"/P)([W") 3 LP*"([W")

(4.71)

holds. We are now in a position to prove Theorem 4.28 Let A(x) E Lp."(R"), p > n. Then Pf := J(Af) is compact f o r q > p n / ( p - n). Furthermore, Pf E Ca(R")where in Lq~o(R") (4.72) and IIPfllC.

Furthermore, as 1x1

--* m

M P , ~)lAlp,~lflq*o.

(4.73)

the upper estimate

l(Pf)(x)lS M(P9 ~)IAlp.nVlq,OIXI- p holds with /3 := n(l - ( l / p ) - ( l / q ) ) + 1 .

(4.74)

PROOF If l / r := ( l / q + l / p ) < l / n , then r > n , and for A the unit ball we have that Holder's inequality implies IAf,

and

AIr

s IA, Alplf,

Alq

9

196


Consequently, IAflr,, s IAJ,,,lflq,owhich implies P f E Ca(Rn)with 0 < Q < 1 - n ( l / p ) + (1/9). Theorem (4.25) then implies IPflc-

M(P, ~ ~ l ~ l p , f l l f l q . o ~

and for any a > 0 there exists a constant M ( p , q , a) such that

Theorem 4.29 Let C,(X) E Lp*"(Rn),p > n. Then the operator (4.75) is compact in the space C(R"), and maps this space into Ca(W) with (Y := ( p - n ) / n . Furthermore, we have

IIWIIc- s M ( P ) where llfllI

:= supRnlf(x)l.A s 1x1 +

I

lpfl s M ( p )

holds.

IC lp9n

CA

CA

IVIImr

(4.76)

the estimate

~~f~~~~~~"""p"("q'"'-'

(4.77)

1P.n

PROOF The inequalities (4.76) and (4.77) follow from conclusions (1) and (2) of Theorem (4.25) when we notice that XCJE Lpq"(R"),p > n for f E C(R") and that IXCAfJp,ns X I C A I ~* ,llfll=. ~ Let LpLp'(@)be the intersection of the spaces Lp(@),and Lp'(@)with p > n and 1 < p r < n. LpLp'(@) becomes a Banach space with the norm

Ilf, @llp,p,

:=

I f 7

@Ip +

I f 7

alp,.

(4.78)

Theorem 4.30 L e t f E LpLp'(@),p > n , 1 < p r < n; then the function v := .If satisfies

lv(x)J M ( p , p')llf,

W p , p ~ for

x E R",

(4.79)

197


PROOFThe proof of this theorem is similar to the two-dimensional version in the book by I. N. Vekua (1962). The inequality (4.80) follows from inequality (2) of Theorem (4.24) as the constant M(n, p ) of (2) does not depend on the domain @. We notice, however, that M(n, p, a) of inequality (1) does. Hence to establish (4.79) we proceed by assuming f - 0 in Fin\@ and decompose Jf as

From Holder's inequality we obtain

with M ( P , P ' ) :=

I [(1 - (q - l)(n - 1) w, wn

+

((.

-

a n

l)(q' - 1) - 1

)'"'I.

(4.81)

From the above we are able to establish the uniqueness of the solutions to the integral equation (4.51). Theorem 4.31 Let the coeficients CA(x) E LpLp'(Rn) and satisfy the

198


inequality

(4.82)

then the only solution in C(R") of w = T ( ~ C A H A Wis) the trivial solution.

PROOF As J := LpLp'(R")-+ C(Rn)we have lJfl =s ~~Jll~,p~llfllp,p~ where means the norm in the dual space LpLp'(Rn).From w = J(ZCAHAw) we obtain the estimate

1[.11f, .

with M ( p , p') given by (4.78). Taking the supremum norm of the left hand side yields llwllm s M ( p , p')II~ll~llXC~ll~,~, from which we may conclude that if (4.82) holds llwllrn = 0. Using Theorem (4.29) we may prove analogously the next result. Theorem 4.32

Let the coeficients C,(X) E Lps"(Rn) satisJL the inequality (4.83)

where M ( p ) is the constant in inequality (1) of Theorem (4.25); then w = J(ECAHAw) possesses only the trivial solution in the space C(R"). Under the conclusions of Theorems (4.3 1) or (4.32) the representation formula (4.55) reduces to w(x) = R(Fl) := F,(x) +

x A

rA(x, t)HAFl(t)dt,

(4.84)

and the Cauchy integral representation (4.63) becomes w(X) =

C Jb B

t)HB(datw(t))*

(4.85)

As the null spaces of the operators M and M* are now empty, N = N' = 0, and all expressions involving the bases { w I , ..., wN} and { u l , ..., uN} disappear from the various representations that were heretofore derived. Goldschmidt has referred to conditions such as (4.70) or those we have given, which ensure uniqueness of solutions to w = Pw, as "Liouville properties" (1980, p. 67), because in the case of generalized analytic

7. OVERDETERMINED ELLIPTIC SYSTEMS

199

functions, the existence of a Liouville theorem was sufficient to prove uniqueness. We shall refer, in the present work, to a Liouville property as the following: Definition The differential equation aw - ~ C A H A W= 0 is said to have a Liouville property if every bounded, continuous solution in R" can be expressed as a linear combination of m s n given solutions {wl, ..., wm}, which are themselves bounded and continuous in R".


We turn next to the study of overdetermined first order elliptic systems, which contain as a subcase the generalized regular functions of the previous section. In the present investigation it is more convenient for us to use matrix notation. Let x := (xI,..., x k ) ' E R ~ u, := (uI, ..., u,)' E R" or @", and consider the first order system (4.86) where the Pi ( i = 1, ..., k ) are m x n matrices with m > n. We note that for m = n, it has been shown by Adams, Lax, and Phillips (1965) that there exists a maximum value of k for which a system can be elliptic. As has been observed by Hile and Protter (1977b), it is useful to know when it is possible to differentiate the system (4.86) and obtain a second order system. It is assumed, of course, that the coefficients and solution are sufficiently smooth. To perform this.reduction we seek n x m matrices Ri (i = 1, ..., k) such that

Ripj + RjPi = '26,l,, ,

(4.87)

where I,, is the unit n x n matrix. As has been shown by Hile and Protter (1977a,b), when m = n one may use a Clifford algebra formulation. Definition The overdetermined system (4.86) is called elliptic at the point x if the matrix

(4.88) has rank n for all 5 := (5,. t 2 ,..., 5 k ) E Rk\{O}.

200


The condition (4.87) is clearly sufficient to ensure ellipticity, as multiplying by titjand summing over the indices yields

which shows that P(x, 5) has a left inverse and hence has rank n. Definition (Hile and Protter, 1977b) The system (4.86) will be said to be non-degenerate if the m x kn matrix

P := (PI, P2,

..a,

Pk)

is of rank m < nk. Furthermore, the system will be referred to as highly overdetermined if m 3 tn(k + I). By introducing the matrices R and T as R := ( R l , R 2 , ..., R k ) and

...

Ph 0 ... 0 0 P2 P, ... Ph 0

... 0

... 0

...

...

0

0

... ...

0

P& I Pk

0

0

T =

. ...

.

PI 0

... P2 0

...

P,

(4.89)

Hile and Protter write the conditions (4.87) as (4.90)

RT = I , where I is the matrix of n rows and tnk(k =

[ I n Onxn(k-1) I n

+

1) columns defined by

Onxn(k-2) I n Onxn(k-3)

***

I n onxn In],

(4.91)

and O p x g is a zero matrix with p rows and 4 columns. Clearly T has mk rows and tnk(k + 1) columns. Theorem 4.33 (Hile-Protter, 1977b) Let the system (4.86) be highly overdetermined, and let r := fnk(k 1). Then if T has a left inverse U satisbing UT = I , ,

+

the system (4.90) is solvable for R . Indeed, R (4.86) is elliptic.

=

IU, and the system

20 1


If the system has nk equations, then it can be seen that the system is always reducible to the form (4.87). If each term (aui/axj)is considered to be an unknown, it follows from the nondegeneracy condition that these terms may always be solved for. The next result permits the introduction of a so-called canonical form from which it is easier to analyze the systems. Theorem 4.34 (Hile and Protter, 1977b)

(1) Zf in the system (4.86), the matrix P I has maximum rank, it may be assumed to be of the form

[?..I

pi=

(4.92)

(2) Zf there exist matrices Ri (i = 1, ..., k ) satisfying (4.87), then they may be assumed to have the form R I = [ Z n , On,,], R j = [ - A i , bi] the matrices bj (i = 2,

..., k ) are

(i = 2,

..., k ) ;

(4.93)

solutions of the system

biaj + bjai = 2tiijZn

+ AiAj + AjAi

(4.94)

(i, j = 2, 3, ..., k ) , where the A i , the ai, and the bi are n x n , ( m - n ) x n , and n x ( m - n ) matrices, respectively.

The proof of this theorem, while somewhat technical, is not difficult; we omit it here and refer the reader to the original work. We notice, however, that for the special case m = n , the system (4.86) is no longer overdetermined; the matrices ai and bi in (4.94) therefore are not present, and (4.94) becomes AiAj + AjAi = 26,Z

( i , j = 2,

...,k).

(4.95)

In this case the matrices {A, := I n ,A 2 , ..., Ak} form a basis for the Clifford algebra associated with the quadratic form --(xi + + xi). Hile and Protter (1977b) prove the following: Theorem 4.35 Let u E C'(R) be a solution of L[ul :=

au 2 P i-axj = f ( x , u),

i=l

x ER

c Rk,

(4.96)

202


where If(x, u)l S Mlul, and let the matrices Pi satisfy the relationst P,*Pj + PTP, = 26,I. I f u = 0 in some open subset of R, then u

(4.97)

=

0 in R.

This result is based on a sequence of lemmas and theorems. To this end we list: Lemma 4.36 (Hile and Protter, 1977b) Let R C Rk be bounded and have a C1-boundary, and let M be an n x n , skew-Hermitian matrix + @" is in C ( a ) n C 1 ( R )with bounded with C 1 ( a )entries. I f v := first derivatives in 51, and u = 0 on h, then

a

2 Re

J uxi- Muxjdx R

=

I,(u*Mxjuxi- u ~ M x i v xdx j)

( i , j = 1, 2 ,

..., k). (4.98)

The proof is obtained by integration by parts (for details, see Hile and Protter, 1977b).The next theorem is somewhat more complicated; hence, we sketch the proof. Theorem 4.37 (Hile and Protter, 1977b) Let R be as in Lemma (4.36). In addition, suppose x1 # O for all x := (xI, ..., xk) E Suppose u E C ( a ) fl C 1 ( R )with boundedfirst order derivatives in R. Ifu = 0 on h, and i f L has coeflcients Pi as in the hypothesis of Theorem (4.35) and j3 is a nonnegative integer the following holds:

a.

(4.99)

PROOF Setting u

=

x f v we have

c k

Lu

= xf

and

3

PiUXi

i= 1

+ pxf-'p1u

2 Re ]RPIux, * ( p x ; ' P l u

+

k i=2

P p , ) dx.

From (4.90) the first integral on the right-hand side becomes, upon int P: is the conjugate transpose of P,.

203

7 . OVERDETERMINED ELLIPTIC SYSTEMS

tegrating by parts,

2p Re J,(x;lP1u,, - P l u )dx

=

p Jn x;2p-21u12dx.

The remaining integrals vanish since M := P:Pi is skew-Hermitian, and by (4.98) this is seen to vanish. The goal of the above described sequence of results is the following: Theorem 4.38 (Hile and Protter, 1977b) Let R C Rk have a smooth boundary and let xo be a boundary point with the property that there exists a closed ball intersecting only at xo. If u E C(a)f l C'(R), with bounded first derivatives, the conditions (4.96) and (4.97) hold, and u = 0 on h f l B for some ball B with center at xo, then u = 0 in R fl B 1for some ball B' concentric with B .

a

PROOF By using a translation, rotation, and change of scale it may be assumed that the ball B is centered at the origin, B has a radius 1, and xo = (1, 0, 0, ..., 0). It can be shown that the conditions (4.96) and (4.97) are preserved by the above transformation. If necessary, truncate R so that it is bounded; by reflecting through the boundary of B , the image of R is now entirely within the sphere. The differential equation takes on the form

where g satisfies a Lipschitz condition and the matrices Qi satisfy the relations (4.97). The point xo remains fixed under the reflection and the image of R is strictly convex at .yo, i.e., the hyperplane of support, {x, = l}, intersects solely at {x'}. Now, as Hile and Protter (1977b) show, the inequality (4.99) can be sharpened to hold for all p 3 Po > 0, where Po is seen to depend only on the dimension k. A standard argument now shows u = 0 in R f l B 1 ,for some ball B' with center at xo.

a

REMARKIf u is a solution of the equation (4.96) where the coeficients satisfy (4.97) and i f u = 0 in some open set of R, then u = 0 in R. These results may be applied to several special, higher order cases by using a reduction of order which leads to first order overdetermined systems (Hile and Protter, 1972). Example 1

A4 :=

c k

i= I

=

g ( x , 4, V4).

(4.100)

204


Introducing the new unknowns UO

:=

4,

uj :=

di

(i

=

1, 2,

..., k),

we obtain the overdetermined system duo _

( j = I , 2,

axj - uj

aui- -auj = ax, axj

(i,j

=

..., k)

1 , 2 , ..., k; i>j)

(4.101)

Writing this in the form (4.2.1 l), the condition (4.97) is seen to be satisfied; hence, as is well known, the unique continuation property holds for (4.100). Example 2 (4.102)

A*+ = g(x, 4, V 4 , A+, V(A4)). We perform a reduction of order by setting uo := uk+l :=

4, A49

uj :=

+xi

uk+l+j :=

(i

= 1,

(A4),

2, ... k ) , )

( j = 1, 27

... k). 9

Then one obtains the first order overdetermined system ( i , j = 1, 2,

..., k; i > j ) ,

( i , j = 1 , 2,..., k ; i>j),

If we designate the vector u by u := (uo, u l , ..., U Z k + l ) , then the system (4.103) may be written in matrix form as

205


where

pj:=

[

Pi 0 0

1,

Pj

and the Pj are the same as in the previous example. Consequently, one may show that (4.102) also obeys a unique continuation principle. Hile and Protter (1977b) also investigated the case of non-highly overdetermined systems and found conditions under which (4.97) is satisfied for certain matrices Ri. For the non-highly overdetermined systems the m x n coefficient matrices Pi (i = 1, ..., k ) satisfy the restriction n s m < (1/2)n(k + 1). The reader is referred to the above-cited work for further details. Hile and Protter (1977a) prove the following maximum principle which has an application to the type of first order systems discussed in this chapter. Theorem 4.39 (Hile and Protter, 1977b) Let u := (ul, u2, ..., u,) be a solution in R of the uniform second-order elliptic system,

L :=

a2 2 aij(x)-.axiaxj rn

(4.104)

i,j=l

The coeficients, moreover, are assumed to be bounded, complex-valued functions. Then there exists a constant k 2 0, depending only on the coeficients of (4.104), such that i f a is a positive C2-functionin R, then alu12 := a C ~ l I u i 1cannot 2 achieve a positive maximum at any point x satisfying a-'La - 2a-'(AVa) Va > K ; here A := (aij)is the coeficient matrix of the principal part of L.

-

As an illustration of this theorem let us consider a first order system of the form (4.105) where u := (ul, u 2 , ..., u,) is a column vector and the m x m matrices Pi (i = 1, 2, ..., k ) and Q have complex-valued C'-components, defined in a domain R C Rk. Furthermore, let us assume that the solution is a complex-valued C2-function in R. Let us assume it is possible to choose

206


m x m matrices Ri (i

=

1 , 2,

..., k) in such a way that (4.106)

where I is the m x m identity, L is a second order elliptic operator as in the previous theorem, and S is an m x m matrix operator of order less than two. Then it follows from the theorem of Hile and Protter that a maximum principle for (4.105) holds. Let us investigate the consequences of (4.106) by equating the principal parts of (4.106). It follows that the matrices Rj must satisfy the equations Ripi

+ Ripj = 2aijI

for i # j ,

R i p i = aiiI.

(4.107)

As before, the matrix A := (aij)is to be symmetric. Furthermore, A is to satisfy the condition of uniform ellipticity, namely, for some constant co > 0, we have

2

i ,j = I

k

aij(x)yiyj

CO

2 y;,

r=l

v y := ( y l , ...)Y k ) E [w.

(4.108)

Hile and Protter (1977b) consider several cases, distinguished by whether k or m is greater than or equal to 2. When we have real principal terms, the case where m = 2, and k > 2 is not elliptic. When both m and k are 2, (4.105) can be put into the form of a Cauchy-Riemann system. The case where m 3 2 and k = 2 leads to a Douglis system. Consequently, we limit further discussion to the case m > 2, k > 2. First, it can be shown that if the Ri exist, then the system (4.105) is elliptic. Multiplying (4.107) by the real scalars hiAj and summing, we obtain

As A is positive definite, it follows that det(Ei hipi) # 0 unless the hi are all zero; we conclude therefore that (4.105) is elliptic. Adams, Lax, and Phillips (1965) have shown that for a given k there are severe restrictions on the value of m for which elliptic systems exist. Let R(m) (C(m))be the maximum number of real (complex) m x m matrices { P I ,...,Pk} such that det(E hipi) # 0 for all real hi not identically zero. If we set m := (2a + 1)2', b := c + 4d where a, b, c, d are nonnegative integers with 0 6 c < 4, then it has been shown (Adams et al., 1965) that

R(m) = 2" + 8d,

207

8. HIGHER ORDER ELLIPTIC SYSTEMS

and C(m) = 2'

+ 2.

As an illustration of this result they note that there are no real first order elliptic systems of 6, 10, 14, 18, ... equations in more than two independent variables. If on the other hand, it turns out that (4.105) is elliptic, then it may be possible to find matrices R j (i = 1, 2, ..., k ) such that (4.107) holds. The determination of these matrices is equivalent to the existence of a set of m x m matrices Q z , ..., Qk satisfying QiQj

=

(i

-QjQi

#j )

(i, 2, 3,

Q : = -1

...1,

where each Pi is a linear combination of the Q ; . The matrices I , Qz,..., Qk form the basis of a Clifford algebra associated with the quadratic form -(x: + ... + x:). Clearly, for these Clifford algebras it is possible to determine first order elliptic systems for which a maximum principle holds.

8. FUNCTION THEORY FOR HIGHER ORDER ELLIPTIC SYSTEMS WITH ANALYTIC COEFFICIENTS

Douglis and Nirenberg (1953) have defined a notion of ellipticity for the general higher order system N

2Ljj(uj)= A,,

j= I

i = 1,

..., N .

(4.109)

where each Lij is a differential operator of the form Ljj :=

7

la sm,

a;(x)D".

Here we employ the usual conventions

and 6" := are

J&,

(j"'. The characteristic polynomials of the operators L,

eij(x,5) := C

la1= m,,

a;(x)ta,

I s i, j s N .

208


The coefficients uz are assumed to be real-valued. The system (4.109) is said to be elliptic if there are integers sI, ..., s,, t l , ..., I , such that si + tj = mij and if det(Qij(x, 4)) # 0 for 4 E l%"\{O}. The system is elliptic in a domain @ if it is elliptic at each point of @. Under the assumption that the coefficients uz are real analytic functions of x, Habetha (1973) has generalized the proof of John (1950, 1955) concerning the local existence of a fundamental solution for the higher order elliptic equation to systems satisfying the Dough-Nirenberg definition of ellipticity. That is, for an operator XLij elliptic in @ we can find in the neighborhood of each x E 8 an N x N matrix (K,p(x, y)) that is analytic in @ x @ - {(y, y)} such that

where 6ie is the Kronecker delta. The adjoint system to (4.109) is

c L$(Uj) N

j= I

where L$

=

c

Ial-,,

0,

(4.110)

( - 1)I"ID"u4(x).

(4.111)

=

Suppose Q$ = Qij, and let t := maxi ti, s := minj sj. Then s,+ = ti + t, and t* = t. From Green's theorem we have

-

t,

ti* = sj

where Mij is a bilinear form in u and u of order at most mij - 1 . The fundamental solution for the adjoint system (4.11l), (K,e(x, y)), is given by (K%(x,Y ) ) = (Kej(Y, A vector 6 is said to be characteristic at the point x with respect to the system (4.109), if det(Qij(x, 4)) = 0.

A surface is characteristic at x if its normal vector is characteristic. Habetha proves the following generalization of the Cauchy-Kowalewski theorem:

209

8. HIGHER ORDER ELLIPTIC SYSTEMS

Theorem 4.40 (Habetha, 1973) Let the coefficients of (4.109) be analytic in a domain let the order of the Lij be si ti, and let S be an analytic, regular, ( n - 1)-dimensional manifold in a.At each noncharacteristic point xo on S there exists a neighborhood U(xo)such that one canjind numbers rj, depending upon xo and the normal direction to S, such that max (0, ti s) s rj s tj. In such a neighborhood, the Cauchy problem

+

a,

+

N j= I

L,(u~)= J;.

in

u(x,)

dkuj Gjk for 0 s k 6 r j - 1 ; j = 1 , ..., N in S n U(xo) dtk is uniquely solvable. Here 5 is the normal to S, and the Gjkare arbitrary analytic functions. When all rj = 0 no data can be spec$ed. The rj vary between the prescribed limits and a more precise determination is not possible. However, x r j = m , where m is the order of the system. --

As a corollary to the Cauchy-Kowalewski theorem Habetha shows that if the coefficients and the right-hand side of the system (4.109) are analytic in a domain a,then each solution satisfying uj E C'j(@) is also analytic. A function theory can be constructed upon these results. Habetha derives the Cauchy integral representation 4x1

=

-1

U(Xd

C Mij(uj(y),K d x , Y ) ) d s ij

for solutions to (4.109) in a neighborhood U(xo)of some given point xo and also a Cauchy's theorem. Habetha (1973) also establishes a Taylor and Laurent series.

xy=,

Theorem 4.41 Suppose the elliptic system Lij(uj) = 0 (i = 1 , 2, ..., N ) has analytic coefficients. Then in some neighborhood U of xo, for each multi-index a there exists a matrix solution V"(x, xo) := ($&, xo))such that each solution of the system may be represented by means of a uniformly convergent, termwise indejinitely differentiable series

in U . This theorem follows from the fact that the formal powers V:&, xo)

210


satisfy the Cauchy data DPUj(X0) p ! 6,,6,,

( j = 1,

..., N ) .

The above theorem may be seen to generalize the Taylor theorem of complex analysis, as in that case we may take x I = x , x2 = y , and L I 1 := alax, L12:= -a/ay, LZl := a/ay, and L22 := a/ax. The index condition is then satisfied by taking sI = s2 = 0, tl = t2 = 1. Theorem 4.42 If the elliptic system ZYzILij(uj)= 0 has analytic coefficients, then in some punctured neighborhood of xo, U'(xo),every solution analytic in U'(xo)may be developed as a uniformly convergent, termwise indefinitely differentiable series N

N

The v:k are the solutions described in the previous theorem, whereas the Vjf' correspond to the adjoint system. The Laurent coefficients are given by the formulae

where p is chosen such that {Ix - xol 6 p } C V'(x0).

The Taylor and Laurent expansions described above may be shown to be unique representations. About a zero of a solution to the homogeneous system (4.109), Habetha provides the following result: Theorem 4.43 Let u be a solution of (4.109) in U(xo)having a zero at xo, that is, all components of u have zeros of order tj + pi > 0 at xo. Let p := minj pi and uojbe the homogeneous polynomial of degree ti + p beginning the expansion of u j. Then uo := ( u o l ,..., uON)is a solution of the differential equation with constant coefficients obtained by taking the principal part of (4.109) and substituting in x = xo. 9. COMMUTATIVE ALTERNATIVES FOR HIGHER DIMENSIONAL FUNCTION THEORY

For an algebraic theory of vectors in R", n > 2, it has long been known that at least one of the following must be surrendered: (1) product com-

21 1

9. COMMUTATIVE ALTERNATIVES

mutativity ; (2) product associativity; or (3) invertibility of all nonzero elements (T. E. Phipps, Jr., private communication; Hamilton, 1853). The Clifford algebra formulation described previously relinquishes commutatity ; however some work has been done with algebras lacking one of the latter properties. We briefly consider these approaches. The rudiments of a function theory based upon a nonassociative algebra have been developed by T. E. Phipps, Jr. (private communication) for three dimensions. The independent variable is z = zo + ilzl + i2z2where il and i2 obey the (nonassociative) multiplicative laws i:”i;m

=

i p + l i 22m + l =

(2n)!(2m)!( - l),+, 22(n+m)n!m!(n + m)!’

0.

The kth term w := dk)zk,dk):= ahk) + il&’ + i2uik’,of the formal power series CF=odk)zkcan be shown to have components each of which are harmonic polynomials and satisfy the “Cauchy-Riemann equations”

These equations are not however derivable from the assumption that a derivative exists at a point independent of the path of approach. Functions having this property exist only for associative algebras in the finite dimensional case (Ketchum, 1928). The components of w = dk)zkhave explicit, but complicated, formulae (T. E. Phipps, Jr., private communication). If we define aw

aw2 - aw0 + i l -aw, +i2-, az az, azo azo

21 2


then the desired formula

is obtained. Any vector z = zo

with 7. := zo

-

+

ilzl

i l z l - i2z2, and

+

i2z2 has an inverse

+ 1/2(z: + 22); 2

z2 := zz = zz = z;

however, in the absence of associativity the relation az = b t* z = a-'b fails. The solution to az = b in fact is not necessarily unique. Polynomials C;= dk)zk have the fundamental factorization U ~ ( Z-

z,)"'(z - z2Ia2 ..*( Z -

Zm)Pm,

but the "zeros" zj are not uniquely determined. A function theory for R" can be based upon commutative, associative algebras in which the equation az = b may have zero divisors. One such class of algebras has been studied by Ketchum (1928) and Edenhoffer (1976a). In this case the algebra d n ( y ) is spanned by the elements {eo = 1, e l , ..., en- I}, which are assumed to obey a multiplicative law of the form n-

where the

yijk

I

are arbitrary other than the two requirements n-

I

n-1

which make the algebra commutative and associative. The product of two elements a = XC;S~akek, b = ZC,!:; bkek in d,,(y) can be computed as the product T(a) b

-

where b = (bo, ..., bn-l)' and T(a) is the matrix

An element a E &(y)

is invertible if and only if det r ( a ) # 0.

213


We associate with a the norm llall = (maximum eigenvalue of

r(a))1’2.

The Cauchy-Riemann equations

af = e i - a, !

850

at‘i

i = l , ..., n - 1 ,

can be obtained from the assumption of path independence as in R2. Cauchy’s theorem, Morera’s theorem, Cauchy’s integral representation, power series, Liouville’s theorem, and so forth can now be proved by means similar to those employed in analytic function theory. As an application of such a function theory we note that it is possible to prove a Riemann mapping theorem for a certain algebra of the type d 4 ( y ) (Edenhoffer, 1976b). The theorem states that given any simply connected planar region we may find a hypernanalytic function f(z) defined on the four-dimensional cylinder containing @ (projection is parallel to the zero divisors) such that ( f ’ ( 7 j - I exists and f maps @ onto a two-dimensional manifold whose projection is the unit circle. This theorem can be used to solve the biharmonic Dirichlet problem. For another application of an algebra of type d 4 ( y ) to the four-dimensional Laplacian, see Belbruno (1981). Finally, we note that it is sometimes useful to employ infinite dimensional commutative algebras

Ketchum (1929) introduced such an algebra in order to produce a function theory that generates all solutions to Laplace’s equation in three dimensions. For a description of this algebra and further function theoretic results on it, see the monograph of Snyder (1968). Roaculet (1955, 1975) has constructed infinite-dimensional algebras generalizing the complex algebra x + iy for Laplace’s equation of n + 2 dimensions, which we write as

a2u= 0, 2 a2u -+ ax; ay2

j=o

and even more general equations with constant coefficients. For Laplace’s equation we seek a representation of solutions in the form + xn& + y e . By analogy with U = f ( w ) , where w := xo + x& + the two-dimensional case we require the algebraic elements 8, ..., +, to satisfy the characteristic equation 1

+ 6;+ ... + &e2

=

0.

214


Let &,

47' ... 4a"

:=

9

&;

:=

$71

... 4*"e.

Here the ajcan be positive or negative integers. With the multiplication l-ules (a = p = ( P I , .**, P n ) ) b

l

9

. * a 9

%I),

& a ' Ep

&,

-

&:*&;

& ;

=

Ea+p,

= = -&:+p

-

5

p= I

Ea+p+2ep9

where ep is the n-tuple (aip),we obtain a commutative, associative algebra d,, having elements of the form w = x

+

ey,

where x and y are the formal infinite sums X =

2

laKX

XaV,

Y =

C

I+=

Y~V.

The n-tuples a once again may have positive or negative integers as their entries. Rosculet introduces two subspaces

(2') (A)

Ea(lxal + lyal) < m IxJ, lyal s k f for all a.

The space 2' is closed under multiplication; the product 7 E ( A )is in (A).For w = x + Oy E (2')define ~ ( x := ) xlxal a

+ ~a l

w7, w E

(2'),

~ a l .

The power series (4.113)

is said to converge absolutely if

.

It can be shown that the series (4.113) converges absolutely if

215


From the condition of rnonogeneity, i.e., the condition that the derivative txists independent of direction of approach, we can derive the Cauchy-Riemann equations. A function f E C'(@) of the variable w = xo + x 1 4 + ... + xn@' + Oy is monogenic in @ if the exterior product df A dw is zero in @. Here we take f to be of the form where

and X, U, and Xa V , converge. The functionf(w) is monogenic in equations

a aY

=

if and only if the Cauchy-Riemann

a i:-Va-*ejr axi

j=o

a

a

a

a

-va-ei = -Va-ej, ax, axj -va-ei = -Ua aY axi are satisfied for all n-tuples a. The monogeneity o f f implies that its components are harmonic. Cauchy's theorem, Cauchy's integral representation, and so forth may be established from Green's theorem. For further details see the book by Rogculet (1975).

5 ~

~

~~

Partial Differential Equations of Several Complex Variables

1. INHOMOGENEOUS CAUCHY-RIEMANN EQUATIONS

IN POLYCYLINDERS

The prerequisite for reading this chapter is some knowledge of functions of several complex variables, such as might be obtained from Cartan (1963), the first chapter of Fuks (1963), of Gunning and Rossi (1965), or Gilbert (1969). We consider in this section inhomogeneous Cauchy-Riemann differential equations for functions of several complex variables defined in polycylinders @ := a, x ... x a,,. That is, we consider systems of the form aw

-=

a zj

h j ( z I ,..., z,,),

j

=

1, 2,

..., n,

z,E ajC @,

(5.1)

where w(z) := w ( z I , ..., z,,). If w is a solution of (5.1) having Sobolev derivatives of second order, then certain consistency relations must hold, namely,

(i, j = 1, 2, ..., n). The equations (5.2) are necessary integrability conditions. If higher order derivatives (in the sense of Sobolev) exist, which 21 6

217

1. EQUATIONS IN POLYCYLINDERS

are independent of the order of differentiation, then h. . = hJ,..'JA . . , A n,

(5 * 3)

where

and (i,, ..., i,) is a permutation of ( j , , ...,j , ) . For the polycylinder := a1x ... x a,,, we designate the Pompieu operator with respect to the z,-coordinate as (J,g)(

..., z,, ...I

1

:= 7r

II

g( ..., 5,,

...) -.d t , dr), 5, - z,

(5.4)

In order to integrate the system (5.1), we first need to recall some information concerning multiple Pompieu integrals. The first is that we have a Fubini-type theorem holding for integrals of the form

aj1

ajA,

x ... x the integral That is, when h is integrable over the region has the same value when the orders of integration are changed. Hence for k {jl,..., j , } we may write in operator notation the elementary result

e

Jj,

J j l - ,J k J j l

Jj,

h = JkJj,

JjI-1 Jjl

***

Jj,h,

which implies, furthermore, that

a az k

-J j ,

* * a

Jj,-, J k J j l

*.*

Jj,

= Jj,

a * *

Jj,h.

e

We may also differentiate with respect to z k when k { j l ... jA}, since for every #I E c@$)we have, by integrating by parts, that

which in the operator form is expressed as

218

5. SEVERAL COMPLEX VARIABLES

for

k

{ji,

..-,jA}-

Tutschke (1973, 1977) gives the following representation as an integral for the system (5.1): n

w =

C (-

A= 1

Z*

l)A+lJj, *..JjAhjl..gA.

(5.5)

Here Z* is meant as a sum over all indices j , j , that are different from one another. The hj,...jAare to be bounded functions that satisfy (5.3). Actually, this condition is automatically satisfied when the condition (5.2) is met. Let us rewrite the right-hand side of (5.5) as

where the notation z*kmeans that sums are over all indices j , , different from k. Clearly the second sum may be expanded as

..., j , (5.7)

Differentiating (5.7) now with respect to Zk yields n- 1

from this it is clear that by differentiating (5.6) the first sum will cancel the one above. Hence, aw/(aZk) = hk, and the representation (5.5) yields a solution of (5.1) in the polycylinder 8 := a1X --.x 8,. We turn next to the construction of nonsingular solutions to the partial differential system aw.

az

= &(z, w1,

..., w,)

( j = 1,

..., rn).

(5.8)

Applying the Pompieu operator we may write this as a system of integral equations wj(z) = ( J @ , f ) ( w:= ) @j(Z)

where the Q j ( z ) are holomorphic in 8.

219

1. EQUATIONS IN POLYCYLINDERS

We shall make the following assumptions concerning the A:

(HI) For z E @ and w E BR := { w : lwjl < R } theJ;are continuous and therefore bounded by some KR. (H,) For z E @ thefi satisfy a uniform Lipschitz condition

I ~ ( Z , W I , ..*, W m ) - f i ( z ,

@I,

*..,

m

@,)I

s LR C l w j j= I

-

@jl,

(5.10)

when w , itt E BR.

The linear space of continuous, complex vectors w := ( w , , ..., w,) defined over @ forms a Banach space when equipped with the norm llwll := max sup Ilwjll. j

@

Let us call this space M. If the holomorphic vector Q, := (a1,..., am) E BRr C BR, and w E BR, then the function Wj(z)defined as the right-hand side of (5.9) is bounded above by

(5.11)

where m(@)is the Euclidean measure of @. Consequently,

Now if m(@)is sufficiently small, for example, if m(@)s

T(R - R')* 9

4KR

(5.12)

then W lies in the ball BR C M. Theorem 5.1 (Tutschke) Let Q, := (Q1, ..., am) be holomorphic in @ and continuous in @. Furthermore, let Q, E BR- C BR and the inequality,

m(@)6 min

{T(R

'A}

- R')2

4KR

(5.13)

hold. Then there exists a unique solution to the equation (5.9). This solution is at the same time a solution of the differential system (5.8).

220


PROOFThat a solution of (5.8) is also a solution of (5.9), and vice versa, is evident. In order to show that a unique solution of (5.9) exists, we demonstrate that Ja,f is a contraction mapping. If w , Kj are two elements from BR then WJZ) -

Wj(Z)

and from the Lipschitz condition (5.10) one obtains

s

and

I

6 7T

max suplwj i

@

Kjjl dz dv

5

-

z'

(5.14).

We turn next to investigating solutions of (5.8) with isolated singularities. In addition to assuming that t h e 6 satisfy Lipschitz conditions (5.10) we ask that for all z E @ and arbitrary ( w I , ..., w m ) , theA be continuous and bounded, i.e.,

~ A ( zW, I , .*.,

wrn)l s K .

(5.15)

From this inequality we conclude that

(5.16) If prescribed, isolated singularities are given at the points {zo,,}, which may have limit points on the boundary of 8,then from (5.16) the integrals

where @ := @\{zo,,, z ~ ,...}, ~ , exist and define continuous functions in the entire z-plane. From the properties of the Pompieu operator (see, for example, I. N. Vekua, 1962), it is clear that the functions Q j ( z ) defined by @j(Z)

satisfy

aQj/az

=

:= wj(Z) - (Jwj)(Z)

(5.17)

0, which implies they are holomorphic. As the func-

22 1

2. SYSTEMS IN SEVERAL UNKNOWNS

tions Qj(z)about an isolated singular point may have expansions of the form

C

4-m

Q,j(z) =

y=

-OE

a j 3 z - zoUy,

the solutions must have the form wj(Z) =

=

Qj(Z) +

(JWj)

2 ag)(z

v= -OE

-

zou)v

+ continuous function.

From Theorem (5.1) it then follows that there exists a unique solution vector w := ( w I , ..., w,) to (5.17) that corresponds to a prescribed holomorphic vector @ := (al,..., @,), providing m(@)< 7r/(4m2Li). The same result may be shown to hold even in the case where Q, has isolated singularities. Since the condition (5.17) ensures that (Jwj)(z)is continuous, the estimate (5.14) also holds in the present case. We summarize this discussion with: Theorem 5.2 (Tutschke) Let the right-hand sides of the differential system awj/az = &(z, w I , ..., w,) ( j = 1, 2, ..., m ) satisJL uniform Lipschitz conditions in the components of w with Lipschitz constant L. Furthermore, let the 1h1 be uniformly bounded for z E % and all wj, and the measure of @ be bounded by m(@)< .rr/(4m2Li).Then to each holomorphic vector Q, := (al,..., a,) having prescribed isolated singularities at the points {zap} C @ there exists one and only one solution w := ( w l , ..., w,) to this differential system. This solution has the same singular behavior at the {zorr}as @.

2. INHOMOGENEOUS CAUCHY-RIEMANN SYSTEMS IN SEVERAL UNKNOWNS

In this section we will investigate differential systems of the form (5.18)

(i = 1, ..., m ; j = 1, ..., p 6 n) in the polycylinder @ := X x an. To this end, we first consider systems of one complex variable and m

222


unknowns, awi

az =A(Z, W I , ..., W m ) ,

-

i = 1,

..., m ,

z E @,,

(5.19)

which we write in abbreviated form as aw

az = f ( z , w ) ,

z E

@I.

We are interested in the circumstances under which there exists some sort of similarity principle for (5.19). More precisely, we seek holomorphic functions of one complex variable Q l , ..., Qm such that in the domain of definition wj = Qjwoj, where the woj are continuous, nonvanishing functions. Tutschke (1974) has offered a variation on a theorem due to Bers (1953), Haack (1952), and I. N. Vekua (1952, 1954) that answers this question. The main idea is to recognize that the functions wj need only to satisfy a differential inequality of the form (5.20)

or in the terminology of Bers (1953), be approximate analytic, in order for a similarity principle to hold. Indeed, Bers (1953) has shown that a function w(z) continuous in a bounded domain (8 in the plane, which possesses a Sobolev derivative aw/aZ, satisfying the inequality (5.21)

can have only isolated zeros. We shall prove this result for (5.20), however, following an idea of I. N. Vekua. Let N be the set of points in @, where w = 0, and let us define the function in @\N,

g ( z ) :=

in N.

(5.22)

Furthermore, let us define (5.23)

It is easy to show that o is continuous and a solution of the inhomogeneous Cauchy-Riemann equation (I. N. Vekua, 1962) am _ - g.

az

(5.24)

223


Moreover, in the usual manner we can show that the function

@(z) := w(z) exp( - o)

(5.25)

is holomorphic in a; see, for example, I. N. Vekua (1962) or Tutschke (1977). If in the differential equation (5.26)

f(z, w ) satisfies a uniform (in z) Lipschitz condition If(z, w) - f ( z ,

where w ,

ii,

3 1

Llw - GI,

are two continuous solutions of (5.26), then w = i6

+ @(z)expw,

where @(z) is holomorphic. We consider next differential equations of the form

i = 1,..., m ;

j = 1,..., n ; z j E a j .

.

,

It is possible to show the existence of a similarity principle, wj = ajexp oj, where the aj are now holomorphic functions of the complex variables Zl,

..., Zn.

We recall that for n complex variables the integral representation (5.5) for nonhomogeneous Cauchy-Riemann equations contains totally mixed derivatives of no more than order n. Hence, if we attempt to repeat the arguments of the last section, we should require that the mixed derivatives satisfy bounds of the form

I

aAw

azi,

az,,

1

(5.28)

SKlwl,

for the various i,, ..., i A , 1 s A s n. Let N be the set of all points of where w vanishes and let the functions gil...ih be defined as g l.] " ' l ,. =

{+ ... L(LZ) az. az,, 0

a?, w

in a\N, in N.

(5.29)

As w is continuous, N is relatively closed in a, and thus @N \ is open. In @N \ the terms in the expression for gi,...,* consist of sums of products

224


of generic terms having the form 1

akw

These terms are bounded by the constant K, hence it follows that the Ig;,...;,lare also bounded by some constant. It is therefore not difficult to demonstrate (see, for example, Tutschke, 1977, p. 91) that n

o =

C (-

A= I

l)A-lC'J;,* * * Ji,

(5.30)

g;,...;,

is almost everywhere a solution of the nonhomogeneous Cauchy-Riemann equation in

a;,

aw _ - gi

( i = 1 , ..., p ) .

az;

(5.31)

From this representation we can prove: Theorem 5.3 (Tutschke) Zf w is a solution of the differential inequalities (5.20), it may be represented as

w = @ exp a, where @(z,,

..., zn) is a

(5.32)

holomorphic function.

PROOF To realize this result we argue using induction. The case = 1 has already been shown to be true in the last section; hence, we suppose that the result is true for p - 1 variables. Let w be a solution of (5.27) in p complex variables, and let us define w , whose derivatives are given by (5.31), by means of the gjl..+*and the Pompieu representation. It is necessary for us to exhibit some continuity properties of w. To this end, we consider a generic term from (5.5). Because of the properties of Pompieu operators

p

Jil

9..

Ji, g i l . . . i h

(5.33)

is a continuous function of each of the variables z;,, ..., ziA separately, (v = i l , ..., i A ) , but also in the entire not only in each domain z,-plane. We consider the dependency next on a variable zj, wherej 4 { i l , ..., i A } . Let zoj be a fixed point of aj and consider w fixed at this zoj as a function of the variables (Cil, ..., ti,) E x ... x Again by the induction hypothesis, since A S p - 1 , the set of zeros of w must be an analytic set. This set can be either a proper subset of x ... X Bi, or identical to the entire product. In the first case there exists a continuum K C ailx - - * x where tv is dift'erent from zero and such

a,,

a;,

a;,.

225


that the measure of (ailx x aiA)\K is arbitrarily small. This means because of (5.28) an arbitrarily small error is made when the domain of integration in (5.33) is replaced by K. Because of the continuity of w there exists a 61 > 0 such that w # 0 for all lzj - zojl 6 61 and all (ti,,..., &,) E K, and hence for these values

(' azil

gi,,,,i* = - ... - - -

azi, a

(5.34)

az,a w

Because of the continuity of w and the derivatives appearing in (5.34),

gil+, is uniformly continuous, and Igil...ih (..., z j ,

...) - gil...i* (..., z o j , ...)I s &

(5.35)

for lzj - zojl s 6 sufficiently small. Hence, by replacing the integration domain in (5.33) by K and making use of the estimate ( 5 . 3 9 , it is easy to see that the terms in (5.33) must be continuous in the variable z j . In the second case we have for zj = zoj that w vanishes identically for all (&, ..., &,) E ailx ..- x BiA.Therefore there must be at least one zj in each neighborhood of zoj where w vanishes identically in ai,x x ai,(if we exclude isolated points zoj). In this case if we hold (&, ..., ti,) fixed, w as a function of zj does not have isolated zeros. However, by the similarity principle for one variable, since w satisfies the inequality (5.18), w must vanish identically as a function of z j . Hence (5.33) is zero for all values of zj, and consequently, a continuous function of zj. The remaining possibility, that w vanishes for zoj, but not identically for points in a neighborhood at zoj, occurs only for in ailx ... x aiA isolated points in aj. If we define Q, := w exp( -o),and consider @ as a function of zj alone, then

If w # 0, then wgi

=

aw/azj,

and aQ,/azj

=

0. When w = 0, then

g j = 0, and by the inequality (5.28) we also have a w / E j = 0. So in

this case aQ,/azj = 0 also, and we see that aQ,/aZj = 0 almost everywhere. All the poinp in ajwhere o is not continuous are isoJated; let us call this point set aj.The function Q, is also continuous on aj,and from the Weyl lemma it is holomorphic there. The Riemann theorem on removable singularities allows us to show that Q, must be holomorphic in the entire aj,as Q, is bounded in ajbecause of (5.25). The Hartogs' theorem [see Gilbert (1969, p. Il)] shows, then, that Q, is a holomorphic function in the variables z I , ..., z,.

226


From the above theorem, several consequences follow, which we list below. We refer the reader to Tutschke (1977) for further details. Corollary 5.4 If the J j and its derivatives with respect to Zi,w,, and i7, to the ( n - 1)st order in (5.27) satisfy a Lipschitz condition, and wi = wi ( z , , ...,z,) is a classical solution of (5.27), then wi may be represented in the form wi = Qiexp(oi), (5.36)

where the

Qi

are holomorphic functions of the variables

z1,

..., z,.

Corollary 5.5 Z f J j and its derivatives with respect to Zj and w, to the ( n - 1)first order satisfy a Lipschitz condition and t h e J j are holomorphic in w,, then the factorization (5.36) above holds for every continuous solution in the Sobolev sense.

3. EXISTENCE THEOREMS FOR SOLUTIONS OF PARTIAL DIFFERENTIAL EQUATIONS IN SEVERAL COMPLEX VARIABLES

We follow, in our discussion below, the ideas of Tutschke (1977). Tutschke investigates completely integrable systems of the form (5.37) ~ , z, are i = 1, ..., m ; j = 1, ..., p (Gn), where the variables z ~ + ..., parameters rather than being essential to the differential system. We shall assume henceforth that p = n. It is assumed, furthermore, that the domains @k in the zk-plane are bounded, and that theJj are defined and continuous in the variables ( z l , ..., z,, w l , ..., w,) for zk E @k and (w,( d R and are holomorphic in w,. We ask that the J j have totally mixed derivatives (i.e., no repeated differentiation) with respect to the Zk up to order n - 1 , and the w, have arbitrary derivatives up to order n - 1 . We ask, furthermore, that when the zj E aj and 1wI, d R, J j and the above cited derivatives be bounded in modulus by a constant K R . Finally, the J j and its derivatives should satisfy a uniform Lipschitz

227

3. EXISTENCE THEOREMS

condition Ig(Zl9

-

-..)Z n , ~

1

..., , Ern)

-

g(zl,

..., ~

n ~, 1

,

Wm)l

m

S LR

ClE; - w;I

(5.38)

i= I

for z k E & and Iw,I 6 R. If (wl, ..., w,) is a continuous solution of the differential system in @ := a;, then it is clear that the continuous functions @; which are defined by

m=,

are holomorphic in @. As before, in order to solve the differential system (5.37), we replace it by an operator equation in a suitable Banach space and examine whether a fixed point exists. In the case at hand, the operator equation is the integro-differential system (5.39), which may be rewritten as w = Jqfw,

(5.40)

where w := ( w I , ..., w,) and @ := (al,..., a,). The Banach space B is the space of rn-tuples (wl, ..., w,) of complex-valued functions that have continuous, totally mixed derivatives up to order n - 1. The norm of B is defined to be (5.41)

From the condition Ifujl s K R , since aw,/aZj = fuj, we have that (aw,/azjl 6 K R . Similar inequalities hold for higher order derivatives; for example, from --A a2wm - a f

a t j ati

m ' +

ati

,=I

afuj

aw,

aw,

azi

(5.42)

we obtain

Clearly, the totally mixed derivatives of up to order n - 1 must be bounded. In order to have R as a bound not only for IwI but also the absolute value of its derivatives, we introduce the variable transformation

228


KZ, ( j = 1 , ..., n). The constant K 2 K~ > 1 is to be chosen sufficiently large in order to provide that all solutions of (5.37) and also their totally mixed derivatives are bounded in absolute value by R. Then if M R is the ball of radius R in B, i.e.,

2, :=

MR

:=

{W

E B : llwll S

R},

then each such solution must be an element of M R . With the above coordinate transformation the system (5.37) becomes aw;

1

-

= = -f;,(Z,,

az,

If w E

MR,

K

..., z,,

WI,

..., wm),

(5.43)

then its image under the mapping

is an element of B; we wish to show that W E M R . By applying successive differentiations with respect to the different coordinates 2; it is not difficult to show that the totally mixed derivatives of W;are of the form ( I / K ) ( S ~+ S 2 ) . Here S1 is a product-sum of derivatives of the L, and totally mixed derivatives of w,, and S2 is made up of multiple, double integrals of such product-sums. We list below several lemmas due to Tutschke (1977) which are useful in demonstrating that the mapping has a fixed point. Lemma 5.6 (Tutschke) If the holomorphic functions Qj are bounded in modulus by R ' , and w E M R , then

where C , = C l ( R , KR) and C2 = C2(R, KR, M(&,)). Furthermore, by choosing the M(&,) sufficiently small, C2 may be made as small as desired. Lemma 5.7 (Tutschke) Let w, B be two elements of exists the following estimate

MR,

then there

Here C3 = C3(R,KR, L R ) , and C4 = C4(R,KR, LR, M(&,)), where L R is the Lipschitz constant of (5.38). The constant C4 may be made arbitrarily small by choosing the m(&,) arbitrarily small.

229

3. EXISTENCE THEOREMS

We define next a useful class of auxilliary functions associated with the differential system (5.37). Let the functions wi = wi(Zl, ..., Z,,) be given and let us define the Fijby

(5.45)

Then a simple computation yields the formal identity (Tutschke, 1977)

where

vanishes identically for completely integrable systems. The function class AX)...jfiwill be defined as the family of functions that may be represented as a homogeneous polynomial in the Fijand their - totally - mixed-derivatives up to rth order with respect to the ..., Zj"-,,Z,,,, ..., Z,. For our purposes it will be useful to show that certain functions are of class A~;!..jfi. We notice, moreover, in the special case where w := (wl, ..., w,) is a solution to (5.37), that Fij vanish identically. In this instance, any function of class AX!, is also identically zero. From (5.44) we obtain

zjl,

+ -K1f i k- + -K1 hz=(2 4 ) A - I

z .. 'k

ji

...A- I

-

J j , ...JjAV

a ...- a ...azj,-, aZk aZj2fijt a

*

(5.48)

Then if Wi is a solution that we denote as wi, this becomes

If just after performing each indicated differentiation in (5.48) (aw,)/(aZj)

230


is replaced by ( l / ~ ) (+yFuj ~ ,~then ) it is clear that the bracketed expression in the summation of expression (5.48) must be of class AF.I]:k. Since this term depends on the indices jl, ...,j , , k and also i, we shall designate the bracketed term by L;l...jAk.In this new notation, (5.49) becomes (5.50)

Since the wi have totally mixed (continuous Sobolev) derivatives with respect to the variables z,, ..., z,, up to order n - 1, the Fikhave totally mixed derivatives up to order n - 2 with respect to the variables z,, ..., z k - 1, zk+1,..., z,,. Hence, we consider the space R” of all m x n matrices F whose elements are defined in the polycylinder @ and have totally mixed derivatives of order up to n - 2. We choose a norm on R”, as before, to be of the type (5.51)

Let M” denote the ball of radius R” in R”. If the right-hand side of (5.50) is designated by JZ F , then this equation is written as F = JZ F. In order to see whether this solution actually exists we consider the mapping H = JZF where F E M”. It is possible to obtain an upper bound the norm of JZF in R” (Tutschke, 1977), namely, 1 ~ ( C+ S c6)9

)IJZFll”

(5.52)

where cS= C, ( K ~RJ , and c6 = c 6 ( K ~R”, , m(&,)). Likewise, an estimate similar to that of Lemmas (5.6-5.7) may be obtained for F , E M”, namely, IIJZF

-

JZFll

1

-(C7 K

+ c8)(lF - PIIS,

(5.53)

~(6)).

where C, = c7( K ~R”), , and c8 = c8 ( K ~R”, , From the above discussion it is clear that J@,Jmaps M R into itself and is contracting, and furthermore, JZ maps M”into itself and is contracting. Hence, there exists a unique fixed point w E MR. Using this fixed element w , the Fik and the operator JZ are then both determined. Now as JZ is also contracting, there exists a unique fixed point to (5.50), that is, an element H = JZH E M”. On the other hand, as (5.50) is homogeneous, it must have the trivial solution since L:.l..jAk vanishes identically when the Fij vanish identically. From the identity (5.45) we realize that w is

231

4. REAL-LINEAR EQUATIONS

a solution of the differential system

awi - 1 _ - - - - f i j ( Z I , ..., z,,

az,

K

WI,

..., w,).

Hence we conclude that if m := min, m(gJ is sufficiently small, then the system (5.37) possesses a unique solution in @ := x X a,. 4. REAL-LINEAR EQUATIONS IN TWO COMPLEX VARIABLES

In the preceding sections, solutions to the system of equations (5.37):

were considered. It was assumed that f depended upon the wj’s but not the WjJs. I f f does depend upon V j , then bounds on the totally mixed derivatives are no longer obtainable from (5.42). In this section we indicate that introducing V into the right hand side of (5.37) may substantially change the nature of solutions by considering two simple examples: aw

- -- Ajw,

a Zj

j = 1,2.

(5.54)

(5.55) Inasmuch as our purpose is expository, we make the following assumption in order to abbreviate topological arguments: we take Aj and Bjto be analytic functions of their real arguments (xl, yl, x2, y2) (zj := xj + i j j ) and seek solutions that are analytic in a neighborhood of a fixed given point (xy, yy, x!, y!). The assumption of analyticity permits us to solve, instead of (5.54) and ( 5 . 5 3 , the systems aw -(z,

a5j

5)

aw -(z,

a5j

5) = Bj(Z,5)w*(5,z),

= Aj(Z, 5)w(z,51,

j = 172, j = 1 , 2,

(5.56)

(5.57)

where 5 := ({1, 52), zj = xj + iyj, ti = xj - iyj; xi, y j E C, the functions Aj, Bj,and w are now analytic in the four complex variables ZI,z2, 51,

232 52,


and

f * ( L z ) :=

x*,

t2,

ZI ,221.

It is easily seen by power series arguments that

and w * ( ( , z ) = W(z, Z) when xj and yj are restricted to real values [see I. N. Vekua (1967)l. Let us consider the system (5.56) first. Cross-differentiating yields

and hence, because of our assumption of analyticity, (5.56) can have nontrivial solutions only if (5.58)

Equation (5.56) also implies (5.59)

Let us impose the nonsingularity assumption A2(Z0,to) # 0

where (z',

5')

(5.60)

abbreviates (xy, yy, x!, y!). Also we let -AI A := -. A2

Equation (5.59) can be solved by solving the ordinary differential equation (5.61) to obtain a general solution

52

=

t 2 ( z 1 z2, , tl, a ) , where

in a neighborhood of (z', 5'). We may solve for a as a function of ( z , 5) in a neighborhood of (z', 5'). The function a ( z , 5) satisfies the


233

differential equation

Equation (5.59) may be written as

i.e., w = w ( z I ,z 2 , a). The system (5.56) reduces to the single equation

and hence (5.62)

Thus, solutions depend upon an arbitrary analytic function of two complex variables. In the one complex variable case solutions to

(5.63) have the resolvent representation

where the resolvent kernals rj depend upon A and B. Hence the solution depends upon an arbitrary function of a single complex variable irrespective of whether A or B is identically zero. We shall now see that this is not the situation in the two-variable case, i.e., as opposed to (5.54), solutions to (5.55) are never as arbitrary as an analytic function of two variables. Instead, they depend upon either an arbitrary analytic function of one complex variable or a constant.

234


To see this, we follow the approach of Buchanan (1982) which is based upon the methods of Koohara (1971), (1976). As with (5.54), we seek a change of coordinates which will reduce the problem to a single equation. We assume henceforth that B2(Z0,to) # 0.

(5.65)

We look for a change of coordinates in the form

61

=

ZI,

C2

=

52(~,51,

71 =

51, 71

=

72(~,5).

(5.66)

The Jacobian of this transformation is

which we require not to vanish at (zo, .)'5

In

Define

(t,7)-coordinates (5.57) becomes

Subtracting the second equation from the first, we obtain

The equation ah

ah

851

a52

-+B-=O

(5.68)

may once again [cf. (5.61)] be solved by the method of characteristics. The differentia1 equation

has general solution 52 =

t2 (zl, z2,

a),where

235


in a neighborhood of (zo, ( z , 5) with

Thus we may solve for a as a function of

.)'5

If we introduce the coordinates (91,

22, g l , a21

:= (Zl,Zzr

5r,&

r2).

then all solutions to (5.68) have the form A = A(?, ,Z2, We henceforth require that the functions g2and q2 in our as-yet-undeterminedcoordinate change (5.66) both be solutions to (5.68) and satisfy J [ t , q](zO, )'5 # 0. With this added condition, equation (5.67) becomes

and thus the system (5.57) reduces to the single equation (5.69)

where

and we have assumed without loss of generality that

It is easily shown that

az2 - 0.

and

a 5 2- - B -

ar, I

a711

(5.70)

Consequently, aY _ - 0.

(5.71)

871

From (5.69) and w = w(&, t2,q2) we have

a

-(Cw*) a711

=

0,

236


and hence from (5.70),

ac

(5.72)

By the involution of a functionf(z, 5) we shall meanf*({, z). Taking the involution of (5.72) and then changing to (c, 77) coordinates yields the auxillary equation

(5.73) The involution is to be regarded as taken with respect to the original variables unless otherwise indicated. We shall now use the auxillary equation (5.73) to find sets of compatibility conditions on the coefficients BI and B2 of (5.57) that permit there to be solutions. For our purposes a regular case will be one in which none of the differential equations which w must satisfy has (z", 5') as a singular point. Only regular cases will be considered. If, in equation (5.73), we choose the coordinates (5,q) to be the system (2, C) defined in connection with (5.68), then we have

aw + B*

aZI

%+ az2

Since w = w(Z1, z2, to i 1 ,

(2

+ B*&)

i2), we

a22

d" = -

(5

log C)*W. (5.74)

a52

have, upon differentiation with respect

We take as the defining condition of the first case CASEI

Since

i2satisfies (5.68) it is not difficult to see that

&(& + B*z)at2 a51

a21

=

a ~ * a t ,-

--

at1

a22

(x) *at2 -. aB*

a52

Hence, if the condition of Case I is satisfied, the left-hand side of (5.75) vanishes. Consequently, we have the necessary condition

237


(5.76) Now suppose that

In keeping with the regularity criterion stated above, we impose the condition

(5.77) for some function DI regular in a neighborhood of (2', t o ):= (2(zo, 5'). For the defining condition of the second case we take

t(z',

5'1,

CASEI1

Equation (5.75) can now be written as

(5.78) where we assume that there is a function E l , regular in a neighborhood of (2', to), such that

(5.79) Finally, let us suppose that

We assume that there is a regular function F1 in a neighborhood of (2', to) such that

-= F l y . a51

a51

We take the defining condition of the third case to be

CASEI11

(5.80)

238


If the defining condition of Case I11 is met, equation (5.78) implies upon differentiation by 5, that (5.81)

It is necessary that

for there to be nontrivial solutions to (5.81). Thus we have exhausted the regular possibilities for (5.74) to have solutions. All other cases in which there are solutions will be instances when (2', l o )lies on the singular manifolds of B, D 1 , E l , or F I .Note that since

the defining conditions of all three cases can be written in terms of the original variables. Hence the cases are independent of the choice of the function f 2 . Cases I and I1 each have several subcases, and we shall not go into the details of the solution of each. Let us discuss one subcase, the one considered by Koohara (1971), which is distinguished from the rest by having a greater degree of arbitrariness in its solutions. It may be shown that if the defining condition of Case I holds, then the coordinates t2 and q2 may be chosen so that both are solutions to the system

ax

ax

-+B-=O,

ac2

ax + B * -ax az I

822

=

0

(5.82)

and moreover, 62 (5, z ) = q&, 5) (see Buchanan, 1982). Because of the latter property, involutions with respect to the original variables may be written as involutions with respect to the (5, $-variables. Equation (5.73) can be written as

aw = 861

(2

log c*)w.

(5.83)

(5.84)

239


is a necessary condition for there to be nontrivial solutions. From (5.83) w(t-1, 5 2 7 7 2 ) = w1(52,172)G(519

52, v2)

(5.85)

where and

WI(52,

q 2 ) := W ( t Y , 5 2 , 7 2 )

That G is independent of q1is implied by (5.84). Substituting (5.85) into (5.69) yields awl

awl

- + yarl2

where

a52

=

u w ,+ Vwf,

(5.86)

The defining conditions of the particular subcase we wish to consider are (5.87)

An easy computation shows that

where J is the Jacobian of the transformation. Hence (5.87) implies that aB* - - 0. 852

In view of the defining condition of Case I, we conclude that B := -B1/B2 must be a function of and (2 alone. Since this is the case, we may take t2= f ( z l ,z2) and qz = f * ( ( , , for some analytic5 For this choice y = 0 and equation (5.86) becomes an equation of the form (5.63) and hence has solution of the form (5.64).

c2)

Theorem 5.8 If in the system of equations (5.55) the quotient of BI and Bz depends (perhaps) upon Z but not z , then solutions to (5.55) exist in a neighborhood of (zo, 6') and depend upon an arbitrary analytic fwzction

240


of a single complex variable. In all other cases the solution depends upon an arbitrary constant. For further details concerning the last statement see Buchanan (1982). Hence solutions to (5.55) are never as arbitrary as solutions to (5.54), all of which depend upon arbitrary analytic functions of two complex variables. 5. NONHOMOGENEOUS CAUCHY-RIEMANN EQUATIONS IN ANALYTIC POLYHEDRA

In this section we shall generalize some of the previous results to more complicated domains. To this end, we must first develop integral representations analogous to the Cauchy-Pompieu formula for polycylinders. C @" which may be decomposed We begin by considering regions into a sum of p-dimensional simplexes all..., ashaving (p - 1) dimensional simplexes rl, ..., r, as their boundaries. Furthermore we assume that the rk are smooth. Next we introduce the concept of differential forms. A differential form f is said to be of type (p , q ) when it may be written in the form (5.88)

where dz, A dzJ := dzi, A ..-A dzipA dzjl A ... A dZjq. Furthermore, one has the basic rules for exterior multiplication, -dzjAdzi,

i #j,

dZiAdZj = -dzjAdZi,

i #j,

(5.89)

af + 8f,

(5.90)

dziAdzj

and

=

dzi A dzi = dZj A dZj = 0.

If we define for scalar functions

then df = where for (p, q) forms

df,,, A dz, A dZJ =

af

=

2 a& IJ

AdzIAdzJ,

(5.91)

241

5. EQUATIONS IN ANALYTIC POLYHEDRA

We notice that d2f = a2f + (~38+ %)f + a'f = 0, which implies that = g has a solution the nonhomogeneous Cauchy-Riemann system only if 3g = 0. We restrict ourselves temporarily to the case of C2 and recall the Green's formula in R4 = C2, where we identify zl := x1 + ix3, z2 :=

zf

x2

+ ix4.

If 9 is a domain with a smooth boundary and f and g are in C2(a), then

Here dx := dxl A dx2 A dx3 A dx4, ua:= ( - I),-' uafi:= ( -

dxl A

0 . .

A [dx,]

l)a+fi-ldx, A [dx,] A

A dx,, **.

A [dx,] A

A dx,.

(5.92)

The square bracket is meant to indicate that a term was deleted from the exterior product. We notice that uaB= -uBaand uau= 0. In terms of the complex variables zI,z2 this becomes

Iff is taken to be a holomorphic function of zl, z2 and g is the special kernel

then we obtain the Bochner-Martinelli representation (Fuks, 1963; Vla-

242


dimirov, 1966) for holomorphic functions in domains bounded by piecewise smooth surfaces, namely,

(5.93)

This result, which generalizes easily to domains B C C", takes the form (n - 1)

2

f ( 4 ) k = I (&

- &) (161

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