First Order Elliptic Systems: A Function Theoretic Approach

295 Pages·1983·3.199 MB·English

by  Robert P. Gilbert and James L. Buchanan (Eds.)

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
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United Kingdom Edition published by
ACADEMIC PRESS, INC. (LONDON) LTD.
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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. II. Title. Ill. 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 9 8 7 6 5 4 3 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.
This page intentionally left blank
Contents
Preface ix
0. Introduction 1
1. Elliptic Systems in the Plane
I. Introduction 6
2. Hyperanalytic Functions II
3. Generalized Derivatives and the Hypercomplex Pompieu Opera tor 24
4. Generalized Hyperanalytic Functions and Liouville's Theorem 32
5. Cauchy Representation for Generalized Hyperanalytic Functions 44
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
5. The Hilbert Problem for Generalized Hyperanalytic Fulzctions 75
6. The Hilbert Problem in the Purely Hypercomplex Case 83
7. The Riemann-Hilbert Problem for Hypercomplex Functions 87
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 CONTENTS
3. Global Similarity Principle 121
4. The Riemann-Hilbert Problem 133
5. Hyperanalytic Functions Having Distributional Boundary Data 136
6. Nonlinear Problems and Reductions to Linear Problems 140
7. 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 269
Index 275
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 functions-
the 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 accu-
mulation 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