Functional Analysis: Applications in Mechanics and Inverse Problems

255 Pages·1996·10.403 MB·English

by  L. P. Lebedev, I. I. Vorovich, G. M. L. Gladwell (auth.)

FUNCTIONAL ANALYSIS
SOLID MECHANICS AND ITS APPLICATIONS
Volume 41
Series Editor: G.M.L. GLADWELL
Solid Mechanics Division, Faculty o/Engineering
University o/Waterloo
Waterloo, Ontario, Canada N2L 3Gl
Aims and Scope of the Series
The fundamental questions arising in mechanics are: Why?, How?, and How much?
The aim of this series is to provide lucid accounts written by authoritative research
ers giving vision and insight in answering these questions on the subject of
mechanics as it relates to solids.
The scope of the series covers the entire spectrum of solid mechanics. Thus it
includes the foundation of mechanics; variational formulations; computational
mechanics; statics, kinematics and dynamics of rigid and elastic bodies; vibrations
of solids and structures; dynamical systems and chaos; the theories of elasticity,
plasticity and viscoelasticity; composite materials; rods, beams, shells and
membranes; structural control and stability; soils, rocks and geomechanics;
fracture; tribology; experimental mechanics; biomechanics and machine design.
The median level of presentation is the first year graduate student. Some texts are
monographs defming the current state of the field; others are accessible to fInal
year undergraduates; but essentially the emphasis is on readability and clarity.
For a list o/related mechanics titles, see final pages.
Functional Analysis
Applications in Mechanics
and Inverse Problems
by
L. P. LEBEDEV
Department ofM athematics and Mechanics,
Rostov State University,
Russia
I. I. VOROVICH
Department ofM athematics and Mechanics,
Rostov State University,
Russia
and
G. M. L. GLADWELL
Department of Civil Engineering,
University ofW aterloo,
Ontario, Canada
KLUWER ACADEMIC PUBLISHERS
DORDRECHT / BOSTON / LONDON
Library of Congress Cataloging-in-Publication Data
Lebedev, L. P.
Functlonal analysis appllcatlons In mechanlcs and lnverse
problems / by L.P. Lebedev, I.I. Vorovlch, and G.M.L. Gladwell.
p. cm. -- (SOlld mechanlcs and ltS appllcations ; v. 41>
Inc I udes 1 ndex.
ISBN 0-7923-3849-9 (HB . alk. paper)
1. Functional analysls. I. Vorovich. IOSlf Izrallevlch. 1920-
II. Gladwell, G. M. L. III. Title. IV. Serles.
CA320.L348 1996
515' ,7'02453--dc20 95-47427
ISBN-13: 978-94-010-6649-5 e-ISBN-13: 978-94-009-0169-8
DOl: 10.1007/978-94-009-0169-8
Published by Kluwer Academic Publishers,
P.O. Box 17, 3300 AA Dordrecht, The Netherlands.
Kluwer Academic Publishers incorporates
the publishing programmes of
D. Reidel, Martinus NiJboff, Dr W. Junk and MTP Press.
Sold and distributed in the U.S.A. and Canada
by Kluwer Academic Publishers,
101 Philip Drive, Norwell, MA 02061, U.S.A.
In all other countries, sold and distributed
by Kluwer Academic Publishers Group,
P.O. Box 322, 3300 AH Dordrecht, The Netherlands.
Printed on acid-free paper
All Rights Reserved
© 1996 Kluwer Academic Publishers
Softcover reprint of the hardcover 1st edition 1996
No part of the material protected by this copyright notice may be reproduced or
utilized in any fonn or by any means, electronic or mechanical,
including photocopying, recording or by any infonnation storage and
retrieval system, without written permission from the copyright owner.
Table of Contents
Preface vii
1 Introduction .......... . 1
1.1 Real and complex numbers. 1
1.2 Theory of functions . . . . . 9
1.3 Weierstrass' polynomial approximation theorem 14
2 Introduction to Metric Spaces 19
2.1 Preliminaries ..... . 19
2.2 Sets in a metric space .. 25
2.3 Some metric spaces of functions 27
2.4 Convergence in a metric space 29
2.5 Complete metric spaces. . . . 30
2.6 The completion theorem ... 32
2.7 An introduction to operators . 35
2.8 Normed linear spaces ..... 40
2.9 An introduction to linear operators 45
2.10 Some inequalities .. 47
2.11 Lebesgue spaces ... 50
2.12 Inner product spaces 57
3 Energy Spaces and Generalized Solutions 63
3.1 The rod ......... . 63
3.2 The Euler-Bernoulli beam 72
3.3 The membrane ... 76
3.4 The plate in bending 81
3.5 Linear elasticity ... 83
3.6 Sobolev spaces 86
3.7 Some imbedding theorems 88
4 Approximation in a Normed Linear Space 97
4.1 Separable spaces ........... 97
4.2 Theory of approximation in a normed linear space 101
4.3 Riesz's representation theorem. . . . . . . . . . . 104
4.4 Existence of energy solutions of some mechanics problems. 108
4.5 Bases and complete systems . . . . . . . . . . . . . . . . . 111
vi Table of Contents
4.6 Weak convergence in a Hilbert space .... 118
4.7 Introduction to the concept of a compact set 124
4.8 Ritz approximation in a Hilbert space. . . . 125
4.9 Generalized solutions of evolution problems. 129
5 Elements of the Theory of Linear Operators 137
5.1 Spaces of linear operators 137
5.2 The Banach-Steinhaus theorem 140
5.3 The inverse operator 143
5.4 Closed operators ., ,-148
5.5 The adjoint operator 153
5.6 Examples of adjoint operators 158
6 Compactness and Its Consequences 163
6.1 Sequentially compact == compact 163
6.2 Criteria for compactness . . . . . 167
6.3 The Arzela-Ascoli theorem .... 170
6.4 Applications of the Arzela-Ascoli theorem .. 174
6.5 Compact linear operators in normed linear spaces 179
6.6 Compact linear operators between Hilbert spaces 185
7 Spectral Theory of Linear Operators ....... 189
7.1 The spectrum of a linear operator . . . . . . 189
7.2 The resolvent set of a closed linear operator 193
7.3 The spectrum of a compact linear operator in a Hilbert space 195
7.4 The analytic nature of the resolvent of a compact linear operator 202
7.5 Self-adjoint operators in a Hilbert space 205
8 Applications to Inverse Problems .. . . 213
8.1 Well-posed and ill-posed problems. 213
8.2 The operator equation . . . . 214
8.3 Singular value decomposition 220
8.4 Regularization......... 223
8.5 Morozov's discrepancy principle 228
Index 233
Preface
This book started its life as a series of lectures given by the second author
from the 1970's onwards to students in their third and fourth years in the
Department of Mathematics at the Rostov State University. For these lectures
there was also an audience of engineers and applied mechanicists who wished
to understand the functional analysis used in contemporary research in their
fields. These people were not so much interested in functional analysis itself as
in its applications; they did not want to be told about functional analysis in
its most abstract form, but wanted a guided tour through those parts of the
analysis needed for their applications.
The lecture notes evolved over the years as the first author started to make
more formal typewritten versions incorporating new material. About 1990 the
first author prepared an English version and submitted it to Kluwer Academic
Publishers for inclusion in the series Solid Mechanics and its Applications. At
that stage the notes were divided into three long chapters covering linear and
nonlinear analysis. As Series Editor, the third author started to edit them.
The requirements of lecture notes and books are vastly different. A book has
to be complete (in some sense), self contained, and able to be read without
the help of an instructor. In the end these new requirements led to the book
being entirely rewritten: an introductory chapter on real analysis was added,
the order of presentation was changed and material was added and deleted. The
last chapter of the original notes, on nonlinear analysis, was omitted altogether,
the original two chapters were reorganized into six chapters, and a new Chapter
8 on applications to Inverse Problems was added. This last step seemed natural:
it covers one of the interests of the third author, and all the functional analysis
needed for an understanding of the theory behind regularization methods for
Inverse Problems had been assembled in the preceding chapters. In preparing
that chapter the third author acknowledges his debt to Charles W. Groetsch and
his beautiful little book Inverse Problems in the Mathematical Sciences. Chapter
8 attempts to fill in (some of) the gaps in the analysis given by Groetsch.
Although the final book bears only a faint resemblance to the original lec
ture notes, it has this in common with the~: it aims to cover only a part of
functional analysis, not all of it in its most abstract form; it presents a ribbon
running through the field. Thus Chapter 2 introduces metric spaces, normed
linear space, inner product spaces and the concepts of open and closed sets and
completeness. The concept of a compact set, which was introduced in Chapter
1 for real numbers, is not introduced until Chapter 4, and not discussed fully
until Chapter 6.
Chapter 3 stands somewhat apart from the others; it illustrates how the
idea of imbedding, appearing in Sobolev's theory, ru:ises in continuum analysis.
From Chapter 2 the reader may pass directly to Chapter 4 which considers
the important problem of approximation, and introduces Riesz's representation
theorem for linear functionals, and the concept of weak convergence, in a Hilbert
viii
space. In keeping with the aim of following a ribbon through the field, the
presentation of the concepts of weak convergence, and of the adjoint operator
in Chapter 5, are limited to inner product spaces.
The theory of linear operators is discussed, but not covered (!), in Chapters
5 and 7. The emphasis here is on the parts of the theory related to compact
linear operators and self-adjoint linear operators.
It is the authors' fervent wish that readers will find the book enjoyable
and instructive, and allow them to use functional analysis methods in their
own research, or to use the book as a jumping board to more advanced and/or
abstract texts.
The authors acknowledge the skill and patience of Xiaoan Lu in the prepa
ration of the text.
L.P. Lebedev
1.1. Vorovich
G.M.L. Gladwell
July, 1995
1. Introduction
Everyone writes as he wants to, and as he can.
Anton Chekhov, The Seagull
1.1 Real and complex numbers
A book must start somewhere. This is a book about a branch of applied math
ematics, and it, like others, must start from some body of assumed knowledge,
otherwise, like Russell and Whitehead's Principia Mathematica, it will have to
start with the definitions of the numbers 1, 2 and 3. This first chapter is in
tended to provide an informal review of some fundamentals, before we begin in
earnest in Chapter 2.
We will start with the positive integers 1,2,3",,; zero, 0, and the negative
integers -1, -2, -3", ·.From these we go to the rational numbers of the form
min, where m,n are integers (we can take m = 0, but n =j:. 0). However we
soon find that having just rational numbers is unsatisfying; there is no rational
number x = min such that x2 = 2. For suppose there were such a number.
If m, n had a common factor (other than ± 1) we could divide that out, and
arrange that m, n had no common factor - we say they are mutually prime.
Our supposition is that m21n2 = 2, or m2 = 2n2• But if m2 is even, so is m.
Thus m = 2p, for some integer p, and 4p2 = 2n2, so that n2 = 2p2. Therefore
n2 is even, and hence n is even. Thus m, n have a common factor, 2, contrary
to hypothesis. This contracliction forces us to conclude that there is no rational
number x such that x2 = 2.
However, we can find a sequence of rational numbers xI, X2, ••• whose
squares get closer and closer to 2 as n increases. Note: Sequence always means
an infinite sequence. For let Xn+! be obtained from Xn using the formula
+
Xn+l = Xn 22 1xn . (1.1.1)
If we start from Xl = 1, we find the sequence
3 17 577
1- - - ... (1.1.2)
, 2' 12' 408'
We note that
2 1. Introduction
+ 2)2 2)2
(X2 (X2 _
X2 _ 2 = n _ 2 = -'---'n.:.....----'-_
n+l 4X2 4X2
n n
so that the Xn are all too big, i.e. x~ > 2 for n = 2, 3, .. " and
X~+1 - 2 < (x~ - 2)2/8. (1.1.3)
= = =
When n 2, x~ - 2 9/4 - 2 1/4. Thus
2 2 )2
1 1 1 ( 1
X3 - 2 < '8 . 42' x4 - 2 < '8 8.42 '
so that, given any small quantity f > 0 we can find an integer N such that
x~ - 2 < f for all n > N. Thus the terms in the sequence x~, x~, ... get closer
and closer to 2; we write this
lim x~ = 2.
n-too
and say the sequence x~, x~" .. converges to 2. We can make a formal definition
of a convergent sequence:
Definition 1.1.1 The sequence Xl, X2, •• " which we write as {xn}, converges
to a if, given f > 0 we can find N. (depending on f) such that, for all n > N.
IX
we have n - al < f.
Notice that at present all the numbers in this definition, i.e. Xl, X2,"'; a, f,
must be interpreted as rational numbers.
Problem 1.1.1 Show that a sequence cannot converge to two different limits,
i.e. that a convergent sequence (one that has one limit) has a unique limit.
In our example the sequence {x~} converges to 2, but there is no (ratio
nal) number to which the sequence {xn} converges. However, we note that the
members of the sequence get closer to each other. For
I I-I
X~+21_IX~-21IX~-21
Xn - Xn+l - Xn - -2-- - -2-- < 2 (1.1.4)
Xn Xn
so that as n -+ 00, IXn - Xn+1l-+ O.
We can also show that we can make the difference of any two members of
the sequence as small as we please merely by taking the indices large enough.
For suppose m > n, then
We use (1.1.3) and (1.1.4). Since x~ - 2 < 1 for n 2:: 2 we can replace (1.1.3)
by the weaker statement