Elliptic Functional Differential Equations and Applications

297 Pages·1996·6.386 MB·English

by  Alexander L. Skubachevskii (auth.)

Operator Theory
Advances and Applications
Vol. 91
Editor
I. Gohberg
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Elliptic Functional
Differential Equations
and Applications
Alexander L. Skubachevskii
Birkhauser Verlag
Basel . Boston . Berlin
Author's address:
Alexander L. Skubachevskii
Moscow State Aviation Institute
Volokolamskoe shosse 4
Moscow 125 871
Russia
1991 Mathematics Subject Classification 39A05, 35J99
A CIP catalogue record for this book is available from the Library of Congress, Washington D.C., USA
Deutsche Bibliothek Cataloging-in-Publication Data
Skubachevskij, Aleksandr L.:
Elliptic functional differential equations and applications /
Alexander L. Skubachevskii. -Basel; Boston; Berlin:
Birkhauser, 1997
(Operator theory; Vol. 91)
ISBN-13:978-3-0348-9877-5 e-ISBN-13:978-3-0348-9033-5
DOl: 10.1007/978-3-0348-9033-5
NE:GT
This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the
rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in other ways, and
storage in data banks. For any kind of use the permission of the copyright holder must be obtained.
© 1997 Birkhauser Verlag, P.O. Box 133, CH-401O Basel, Switzerland
Softcover reprint of the hardcover 1st edition 1997
Printed on acid-free paper produced from chlorine-free pulp. TCF
00
Cover design: Heinz Hiltbrunner, Basel
ISBN -13: 978-3-0348-9877-5
To my mother Sof'ya M. Skubachevskaya
and to the memory of my father Leonid S. Skubachevskiz
Contents
Acknowledgments ........................................................ IX
Notation................................................................. X
Introduction ............................................................. 1
I Boundary Value Problems for Functional Differential Equations
in One Dimension
1 Ordinary Differential Equations with Nonlocal
Boundary Conditions .......................................... 19
2 Difference Operators in One Dimension ......................... 26
3 The Boundary Value Problem for the
Differential-Difference Equation ................................ 37
4 Generalized and Classical Solutions ............................ 53
5 Applications to Control Systems with Delay .................... 71
6 The Boundary Value Problem for the Differential-Difference
Equation with Degeneration ..... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
Notes.......................................................... 88
II The First Boundary Value Problem for Strongly Elliptic
Differential-Difference Equations
7 Some Geometrical Constructions 92
8 Difference Operators in the Multidimensional Case ............. 96
9 Necessary and Sufficient Conditions for Strong Ellipticity ....... 109
10 Solvability and Spectrum ....................................... 122
11 Smoothness of Generalized Solutions in Sub domains ............ 125
12 Smoothness of Solutions on a Boundary of
Neighboring Sub domains ....................................... 138
13 Elliptic Differential Equations with Nonlocal Conditions
on Shifts of Boundary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 147
Notes 159
VII
VIII Contents
III Applications to the Mechanics of a Deformable Body
14 The Elastic Model ............................................. 161
15 Variational and Boundary Value Problems...................... 166
16 Smoothness of Solutions ....................................... 177
17 The One-Dimensional Case. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 180
Notes.......................................................... 185
IV Semi-Bounded Differential-Difference Operators with Degeneration
18 Self-Adjoint Extension of a Semi-Bounded
Differential-Difference Operator ................................ 187
19 The Spectrum of Semi-Bounded Differential-Difference
Operators. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 193
20 Smoothness of Solutions of Equations with Degeneration ....... 198
Notes.......................................................... 209
V Nonlocal Elliptic Boundary Value Problems
21 Nonlocal Elliptic Problems with a Parameter .................. . 211
22 Elliptic Equations with Nonlocal Boundary Conditions
in a Cylinder ................................................. . 218
23 Elliptic Differential-Difference Equations in a Cylinder ......... . 225
24 Applications to the Multidimensional Diffusion Processes ...... . 230
25 Elliptic Problems with Nonlocal Conditions near the
Boundary and Feller Semigroups .............................. . 240
Notes 247
Appendix
A Linear Operators ............................................... 249
B Functional spaces .............................................. 256
C Elliptic Problems .............................................. 267
Bibliography ............................................................. 275
List of Symbols .......................................................... 285
Index.................................................................... 292
Acknowledgments
In preparing this book, I am indebted to many people. I would like to express my
hearty thanks to Professors M. S. Agranovich, A. V. Bitsadze, V. A. Il'in, A. G. Ka
menskil, G. A. KamenskiT, A. N. Kozhevnikov, M. A. Krasnosel'skiT, S. G. Krein,
A. D. Myshkis, G. G. Onanov, and V. G. Veretennikov for their constant interest
in my work. I am deeply indebted to Professors A. K. Gushchin. V. P. Mikhailov,
V. A. Kondrat'ev, E. M. Landis, and O. A. Olelnik for the discussions of results at
the seminars of the Steklov Institute of Mathematics and Moscow State Univer
sity. I am very thankful to Professors I. C. Gohberg and S. Verduyn Lunel. Their
advice has helped me to make many improvements to this book.
I would like to express my gratitude to Professor J. Kato and the Kawai
Foundation for the Promotion of l\hthematical Science for their financial sup
port and hospitality during my visit to Japan in 1992. I am very thankful to
Mr. V. V. Pronin for his financial and moral support.
I am grateful to the editorial staff of Birkhiiuser for their highly qualified
assistance in the preparation of the manuscript.
The research described in this book was also made possible in part by Grant
JH6100 from the International Science Foundation and Russian Government, and
by Grant 94-2187 from INTAS.
Notation
lR real numbers
e
complex numbers
lRn n-dimensional real space
en
n-dimensional complex space
[a,b] closed interval {x E lR: a:S x:S b}
(a, b) open interval {x E lR : a < x < b}
Q closure of Q
EB orthogonal sum
o end of a proof
Other notation introduced in the text is listed in the List of Symbols.
Introduction
1. This book is devoted to the theory of boundary value problems for elliptic
functional differential equations. This new field of differential equations has grown
out of the theory of functional differential equations and modern partial differential
equations theory.
Boundary value problems for elliptic functional differential equations have
some astonishing properties. For example, unlike elliptic differential equations, the
smoothness of the generalized solutions can be violated in a bounded domain and
is preserved only in some subdomains. A symbol of a self-adjoint semi bounded
functional differential operator can change its sign. This theory has important
applications to elasticity theory, control theory, and diffusion processes. Elliptic
functional differential equations are closely associated with differential equations
with nonlocal boundary conditions, which arise in plasma theory.
In the one-dimensional case, functional differential equations describe pro
cesses depending on the history of a system. Some results for such equations were
obtained more than 200 years ago. The new classes of functional differential equa
tions arising in mechanics and biology were studied by V. Volterra [1, 2], and a gen
eral theory of functional differential equations was put forward by A. D. Myshkis
[1], R. Bellman and K. Cooke [1], J. Hale [1], and others. Much research in this
field is connected with applications to control systems with delay (see R. Bellman
and J. M. Danskin [1], N. N. Krasovskil [1], Yu. S. Osipov [1]).
Elliptic functional differential equations containing transformations of argu
ments were studied by A. B. Antonevich [1], D. Przeworska-Rolewicz [1], and
V. S. Rabinovich [1]. These authors assume that the transformations of arguments
map a domain onto itself and generate a finite group. Therefore, their results
are similar to well-known results for elliptic differential equations. The situation
changes if the equation has these shifts in the highest derivatives, and the shifts
map the points of the boundary into the domain. The influence of such shifts on
the solvability and smoothness of generalized solutions was studied only in one
dimension in the papers of G. A. Kamenskil and A. D. Myshkis [1] and A. G. Ka
menskil [1].
The theory of elliptic differential-difference equations was constructed by
A. 1. Skubachevskil [1-3, 5, 8-10]. He considered necessary and sufficient condi
tions for ellipticity, solvability, spectrum and smoothness of generalized solutions.
1