Linear and quasilinear complex equations of hyperbolic and mixed type

267 Pages·2002·1.474 MB·English

by  Guo Chun Wen

Linear and Quasilinear Complex
Equations of Hyperbolic
and Mixed Type
Asian Mathematics Series
A Series edited by Chung-Chun Yang
Department of Mathematics, The Hong Kong University of
Science and Technology, Hong Kong
Volume 1
Dynamics of transcendental functions
Xin-Hou Hua and Chung-Chun Yang
Volume 2
Approximate methods and numerical analysis for elliptic complex equations
Guo Chun Wen
Volume 3
Introduction to statistical methods in modern genetics
Mark C.K. Yang
Volume 4
Mathematical theory in periodic plane elasticity
Hai-Tao Cai and Jian-ke Lu
Volume 5
Gamma lines: On the geometry of real and complex functions
Grigor A. Barsegian
Volume 6
Linear and quasilinear complex equations of hyperbolic and mixed type
Guo Chun Wen
Linear and Quasilinear
Complex Equations of
Hyperbolic and Mixed Type
Guo Chun Wen
School of Mathematical Sciences,
Peking University, Beijing, China
London and New York
First published 2002
by Taylor & Francis
11 New Fetter Lane, London EC4P 4EE
Simultaneously published in the USA and Canada
by Taylor & Francis Inc,
29 West 35th Street, New York, NY 10001
Taylor & Francis is an imprint of the Taylor & Francis Group
© 2002 Guo Chun Wen
This edition published in the Taylor & Francis e-Library, 2006.
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ISBN 0–415–26971–7
Contents
Introduction to the series viii
Preface ix
Chapter I
Hyperbolic complex equations of first order 1
1 Hyperbolic complex functions and hyperbolic
pseudoregular functions 1
2 Complex forms of linear and nonlinear hyperbolic
systems of first order equations 10
3 Boundary value problems of linear hyperbolic complex
equations of first order 18
4 Boundary value problems of quasilinear hyperbolic
complex equations of first order 25
5 Hyperbolic mappings and quasi-hyperbolic mappings 35
Chapter II
Hyperbolic complex equations of second order 39
1 Complex form of hyperbolic equations of second order 39
2 Oblique derivative problems for quasilinear hyperbolic
equations of second order 43
3 Oblique derivative problems for general quasilinear
hyperbolic equations of second order 50
4 Other oblique derivative problems for quasilinear
hyperbolic equations of second order 59
5 Oblique derivative problems for degenerate hyperbolic
equations of second order 66
vi Contents
Chapter III
Nonlinear elliptic complex equations of first and second order 79
1 Generalizations of Keldych–Sedov formula for
analytic functions 79
2 Representation and existence of solutions for
elliptic complex equations of first order 90
3 Discontinuous oblique derivative problems for
quasilinear elliptic equations of second order 95
4 Boundary value problems for degenerate elliptic
equations of second order in a simply
connected domain 108
Chapter IV
First order complex equations of mixed type 119
1 The Riemann–Hilbert problem for simplest first
order complex equation of mixed type 119
2 The Riemann–Hilbert problem for first order
linear complex equations of mixed type 126
3 The Riemann–Hilbert problem for first order
quasilinear complex equations of mixed type 134
4 The Riemann–Hilbert problem for first order quasilinear
equations of mixed type in general domains 138
5 The discontinuous Riemann–Hilbert problem for
quasilinear mixed equations of first order 143
Chapter V
Second order linear equations of mixed type 157
1 Oblique derivative problems for simplest second
order equation of mixed type 157
2 Oblique derivative problems for second order
linear equations of mixed type 162
3 Discontinuous oblique derivative problems for second
order linear equations of mixed type 171
4 The Frankl boundary value problem for second
order linear equations of mixed type 177
5 Oblique derivative problems for second order
degenerate equations of mixed type 194
Contents vii
Chapter VI
Second order quasilinear equations of mixed type 200
1 Oblique derivative problems for second order
quasilinear equations of mixed type 200
2 Oblique derivative problems for second order
equations of mixed type in general domains 209
3 Discontinuous oblique derivative problems for second
order quasilinear equations of mixed type 218
4 Oblique derivative problems for quasilinear equations
of mixed type in multiply connected domains 227
References 240
Index 250
Introduction to the Series
The Asian Mathematics Series provides a forum to promote and reflect timely
mathematical research and development from the Asian region, and to provide
suitable and pertinent reference on text books for researchers, academics and
graduate students in Asian universities and research institutes, as well as in the
West. With the growing strength of Asian economic, scientific and technological
development, there is a need more than ever before for teaching and research
materials written by leading Asian researchers, or those who have worked in or
visited the Asian region, particularly tailored to meet the growing demands of
students and researchers in that region. Many leading mathematicians in Asia
were themselves trained in the West, and their experience with Western methods
will make these books suitable not only for an Asian audience but also for the
international mathematics community.
The Asian Mathematics Series is founded with the aim to present significant
contributions from mathematicians, written with an Asian audience in mind, to
the mathematics community. The series will cover all mathematical fields and
their applications, with volumes contributed to by international experts who have
taught or performed research in Asia. The material will be at graduate level or
above. The book series will consist mainly of monographs and lecture notes but
conference proceedings of meetings and workshops held in the Asian region will
also be considered.
Preface
In this book, we mainly introduce first and second order complex equations of
hyperbolic and mixed (elliptic-hyperbolic) type, in which various boundary
value problems for first and second order linear and quasilinear complex equations
of hyperbolic and mixed type are considered. In order to obtain the results on
complex equations of mixed type, we need to first discuss some boundary value
problems for elliptic and hyperbolic complex equations.
In Chapters I and II, the hyperbolic pseudoregular functions and quasi-hyperbolic
mappings are introduced, which are corresponding to pseudoanalytic functions
and quasiconformal mappings in the theory of elliptic complex equations. On the
basis of hyperbolic notations, the hyperbolic systems of first order equations and
hyperbolic equations of second order with some conditions can be reduced to
complex forms. In addition, several boundary value problems, mainly the Riemann–
Hilbert problem, oblique derivative problems for some hyperbolic complex equations
of first and second order are discussed in detail.
In Chapter III, firstly the generalizations of the Keldych–Sedov formula for
analytic functions are given. Moreover, discontinuous boundary value problems
for nonlinear elliptic complex equations of first and second order are discussed.
Besides some oblique derivative problems for degenerate elliptic equations of
second order are also introduced.
In Chapter IV, we mainly consider the discontinuous boundary value problems
for first order linear and quasilinear complex equations of mixed type, which
include the discontinuous Dirichlet problem and discontinuous Riemann–Hilbert
problem. In the meantime we give some a priori estimates of solutions for the
above boundary value problems.
For the classical dynamical equation of mixed type due to S. A. Chaplygin
[17], the first really deep results were published by F. Tricomi [77] 1). In Chapters
V and VI, we consider oblique derivative boundary value problems for second
order linear and quasilinear complex equations of mixed type by using a complex
analytic method in a special domain and in general domains, which include the
Dirichlet problem (Tricomi problem) as a special case. We mention that in the
books [12] 1), 3), the author investigated the Dirichlet problem (Tricomi problem)
for the simplest second order equation of mixed type, i.e. u +sgnyu =0 in
xx yy