Generalized Cauchy-Riemann Systems with a Singular Point

233 Pages·1997·4.032 MB·\233

by  Zafar D Usmanov (Author)

Pitman Monographs and
Surveys in Pure and Applied Mathematics 85
t
Generalized Cauchy-Riemann systems
with a singular point
Main Editors
H. Brezis, Université de Paris
R.G. Douglas, Texas A&M University
A. Jeffrey, University of Newcastle upon Tyne (Founding Editor)
Editorial Board
H. Amann, University of Zurich
R. Aris, University of Minnesota
G.I. Barenblatt, University of Cambridge
H. Begehr, Freie Universitat Berlin
P. Bullen, University of British Columbia
R.J. Elliott, University of Alberta
RP. Gilbert, University of Delaware
R. Glowinski, University of Houston
D. Jerison, Massachusetts Institute of Technology
K. Kirchgassner, Universitat Stuttgart
B. Lawson, State University of New York at Stony Brook
B. Moodie, University of Alberta
S. Mori, Kyoto University
L.E. Payne, Cornell University
D.B. Pearson, University of Hull
G.F. Roach, University of Strathclyde
I. Stakgold, University of Delaware
W.A. Strauss, Brown University
Pitman Monographs and
Surveys in Pure and Applied Mathematics 85
Generalized
Cauchy-Riemann
systems with a
singular point
ZDUsmanov
Institute ofM athematics,
Tajik Academy ofS ciences
Boca Raton London New York
CRC Press is an imprint of the
Taylor & Francis Group, an informa business
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First published 1997 by Addison Wesley Longman Limited
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ISBN 13: 978-0-582-29280-2 (hbk)
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AMS Subject Classifica1ions: (Main) 35J, 35C, 45P
(Subsidiary) 308, 45E, 53C
ISSN 0269-3666
British Library Cataloguin1 in Publication Data
A catalogue record for this book is
available from the British Library
Library of Congress Cataloging-in-Publication Data
Usmanov, Zafar Dzuraevich.
Generalized Cauchy-Riemann systems with a singular point/ Zafar
Dzuracvich Usmanov.
p. cm. - (Pitman monographs and surveys in pure and applied
mathematics, ISSN 0269-3666 ; ???)
ISBN 0-582-29280-8 (alk. paper)
1. CR submanifolds. 2. Singularities (Mathematics) I. Title.
II. Series.
QA649.U86 1997
516.3'62-DC20 96-31177
CIP
Contents
Introduction 1
Chapter 1
Interrelation between sets of general and model
equation solutions 6
1 The method of constructing a general integral operator 6
2 Properties of the functions fii and ÎÎ2 9
3 Properties of a general operator 14
4 The general integral equation 19
5 Additions to Chapter 1 21
6 Unsolved problems 21
Chapter 2
The model equation 23
1 Basic kernels and elementary solutions of the conjugate equation 23
2 Cauchy generalized formula 25
3 Sequences of continuous solutions for the model equation 27
4 ${z) representation by some series. Analogy of the
Liouville theorem. Uniqueness theorem 27
5 Regularity of solutions at a singular point 31
6 Analogy of Laurent series 34
7 Generalized integral of Cauchy type 35
8 Cases of a one-to-one correspondence between the sets
{*(*)} and MC)} 37
9 Riemann-Hilbert problem for solutions of the model equation 39
10 Conjugation problem for solutions of the model equation 53
11 Additions to Chapter 2 57
Chapter 3
The general equation 60
1 Behaviour of solutions at a singular point 60
2 Solutions bounded on the plane 65
3 The canonical form of the Riemann-Hilbert problem 70
4 The Riemann-Hilbert problem with a zero index 72
5 The Riemann-Hilbert problem with a positive index 78
CONTENTS
6 The Riemann-Hilbert problem with a negative index 80
7 The conjugation problem 84
8 The method of constructing additional operators 87
9 Unsolved problems 90
Chapter 4
Modified generalized Cauchy—Rie man n systems
with a singular point 91
1 The general integral equation 91
2 Elements of the theory of the model equation 95
3 Boundary value problems for solutions of the model equation 100
4 The general equation 103
Chapter 5
Generalized Cauchy-Riemann system with the order
of the singularity at a point strictly greater than 1 104
1 Constructing general solutions of an inhomogeneous model equation 104
2 Properties of the functions fii and i^2 109
3 Properties of an operator 118
4 Relation between solutions of general and model equations 121
5 Investigation of the model equation 123
6 Boundary value problems for solutions of the model equation 134
7 Properties of the solutions of the general equation at a
singular point 144
8 The Riemann-Hilbert problem for solutions of the general equation 145
9 The conjugation problem 155
10 Unsolved problems 159
Chapter 6
Infinitesimal bendings of surfaces of positive curvature
with a flat point 161
1 Equations of infinitesimal bendings of a surface with
positive curvature 162
2 Conjugate isometric system of coordinates on the model surface 163
3 Conjugate isometric system of coordinates on the general surface 166
4 Equations of infinitesimal bendings of the general surface 171
5 Geometric significance of boundary conditions 175
6 A general expression of infinitesimal bendings for the model
surface 176
7 Surfaces of types So and S\ 178
8 Properties of infinitesimal bendings of class Cp for
surfaces of type S\ 180
9 Properties of smallness in deformations of class Cp for
the general surface 181
10 Representation for 6L,8M,8N variations through
component ((x,y) of a bending field 183
11 Determination of deformation classes for the general
surface in conjugate isometric coordinates 185
CONTENTS
12 General results for the model surface 187
13 Study of an analytic problem for the general surface 190
14 The influence of a flat point on infinitesimal
bendings of the surface with boundary conditions 196
15 Surfaces with a more complicated structure in
a neighbourhood of a flat point 198
16 Additions to Chapter 6 204
17 Unsolved problems 206
Supplement
Generalized Cauchy-Riemann systems with a singular line 208
References 214

Preface
In this monograph a theory of generalized Cauchy-Riemann systems with polar
singularities of order not less than 1 is presented and its application to the study
of infinitesimal bendings of positive curvature surfaces with an isolated flat point
is given here. It contains results of investigations obtained recently by the author
and his collaborators. In this monograph special attention is paid to the description
of formal methods of constructing general integral operators which are a natural
extension of the classical apparatus of generalized analytic function theory.
The monograph is written not only for specialists in complex analysis, geometry
and mechanics but also for student-mathematicians who could use it as a manual.
Professor Zafar D. Usmanov
Institute of Mathematics
of the Tajik Academy of Sciences
ul. Aini 299, Academgorodok
Dushanbe 734 063
TAJIKISTAN