Differential Operations of Infinite Order With Real Arguments & Their Applications

247 Pages·1994·81.61 MB·English

by  Tran Duc Van

Differential Operators of
Infinite Order with Real Arguments
and Their Applications
This page is intentionally left blank
Differential Operators
of Infinite Order
with Real Arguments
and Their Applications
Tran Due Van
Dinh Nho Hao
Hanoi Institute of Mathematics
World Scientific
Singapore • New Jersey • London • Hong Kong
Published by
World Scientific Publishing Co. Pte. Ltd.
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DIFFERENTIAL OPERATORS OF INFINITE ORDER WITH
REAL ARGUMENTS AND THEIR APPLICATIONS
Copyright © 1994 by World Scientific Publishing Co. Pte. Ltd.
All rights reserved. This book, or parts thereof, may not be reproduced in any form
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ISBN 981-02-1611-4
Printed in Singapore by Utopia Press.
CONTENTS
Introduction 1
Chapter 1. Preliminaries 9
1.1. Spaces C*(fi) and L„(ft) 9
1.2. Definitions and basic properties of distributions 10
1.3. Multiplication of distributions by functions 11
1.4. Differentiation of distributions 11
1.5. Distributions with compact support 11
1.6. Convolution of distributions 12
1.7. Fourier transforms of distributions 13
1.8. The space of analytic functionals 14
1.9. Entire functions of exponential type that are bounded on IR" 16
1.10. Difference quotients and modulus of continuity 16
1.11. The sampling theorems 18
1.12. The de la Vallee Poussin kernel 18
1.13. The Dirichlet kernels 20
1.14. Markov type theorems 21
1.15. The second interpolation method of Bernstein 21
1.16. Orlicz classes and Orlicz spaces 22
1.17. Sobolev-Orlicz spaces 24
1.18. The Denjoy-Carleman classes and the quasianalyticity of
functions of several real variables 26
1.19. Semifinitary functions 28
1.20. Weakly nonlinear equations 28
Chapter 2. Pseudo-differential operators with real analytic symbols ... 30
2.1. The space of test functions in a neighborhood of zero 32
2.2. Differential operators of infinite order (DOIO) 36
2.3. The space of generalized functions W^°°(IRn) 39
2.4. The algebra of pseudo-differential operators with analytic symbols 45
Bibliographical Notes 47
v
CONTENTS
vi
Chapter 3. Applications to pseudo-differential equations 48
3.1. Problems in the whole Euclidean space 49
3.2. The Cauchy problem in the space
of functions valued in Wg°°(JRn) 52
3.3. The Cauchy problem in the space VKg°°(IR")
and its fundamental solution 54
3.4. The Cauchy problem for ordinary pseudo-differential equations 56
3.5. Boundary-value problems 57
3.6. Multi-dimensional integral equations of the first kind with
entire kernels 59
3.7. Functional equations 63
3.8. A class of real analytic symbols with weights 70
Bibliographical Notes 75
Chapter 4. Approximation methods 76
4.1. Approximating the symbols by algebraic polynomials 76
4.2. Approximating the symbols by trigonometric polynomials 79
4.3. Trigonometric interpolation 82
4.4. Approximating the data by sine functions 84
4.5. Examples 84
Bibliographical Notes 88
Chapter 5. A mollification method for ill-posed problems 92
5.1. Introduction 92
5.2. Mollification method 93
5.3. Numerical differentiation 101
5.4. The heat equation backwards in time 105
5.5. The Cauchy problem for the Laplace equation 112
5.6. The non-characteristic Cauchy problem for parabolic equations 114
5.7. Numerical case study 124
Bibliographical Notes 125
DIFFERENTIAL OPERATORS OF INFINITE ORDER vii
Chapter 6. Nontriviality of Sobolev-Orlicz spaces of infinite order .... 128
6.1. Monotonic limits of Banach spaces 128
6.2. Nontriviality of Sobolev-Orlicz spaces of infinite order in
a bounded domain 133
6.3. Nontriviality of Sobolev-Orlicz spaces of infinite order on
the n-dimensional torus 141
6.4. Nontriviality of Sobolev-Orlicz spaces of infinite order
defined on a full Euclidean space 144
6.5. Nontriviality of Sobolev-Orlicz spaces of infinite order in
angular domains 156
Bibliographical Notes 160
Chapter 7. Some properties of Sobolev-Orlicz spaces
of infinite order 161
7.1. Traces and extensions 161
7.2. Traces in LW^^n, (0, a)} 163
7.3. Dual spaces 170
7.4. Imbedding theorems 174
Bibliographical Notes 194
Chapter 8. Elliptic equations of infinite order
with arbitrary nonlinearities 195
8.1. Elliptic boundary value problems 196
8.2. A model equation. Examples 204
8.3. Periodic problems. Inhomogeneous boundary problems 206
8.4. Degenerate nonlinear elliptic equations of infinite order 210
Bibliographical Notes 217
Bibliography 218
INTRODUCTION
In his infinitesimal calculus, Gottfried Wilhelm Leibniz noted that there are certain
striking analogies between algebraic laws and the behavior of differential and integral
operators. One of these analogies he formulated in what is now known in mathemat­
ical literature as the rule of Leibniz, which states that the form of the nth differential
of a product of two functions resemble that of an nth degree binomial. In fact,
dn{uv) = vdnu + (U jdvd^^u + ■■■+ r jd^^vdu + dnvu. (0.1)
This analogy is immediate from the binomial theorem, if the operator d is expressed
in the form of the sum d = d + d:
u v
(d + d)n(uv) = vcTu + (n')dvdn-1u + h ( " Id"-1™ + dTvu (0.2)
u v
Leibniz wrote about this analogy in a letter to J. Bernoulli (Leibniz [1]) and also in
one of his memoirs (Leibniz [2]).
The observations of Leibniz on the binomial analogy led Joseph Louis Lagrange (1772,
Lagrange [1]) to regarding the differentiation symbols as fictitious quantities, admit­
ting the usual algebraic rules. Lagrange obtained his well-known symbolic formulae
AAu = L^+^v^t - l) , (0.3)
where
Aw = u(x + t,y + 7),z + 0 - u{x,y,z), Axu = AAA_1K,
and
SA = l / ( ^ + ^ " + ^ - l ) A (0.4)
u e
Under the hands of his successors, these equivalents were to form the fundamental
structure for the calculus of finite differences. Nevertheless, Lagrange observed that
the analogy forming the basis of his calculations was obscure even if it did not affect
the exactness of the results obtained. According to him, an analytic proof of this
principle was extremely tedious. Somewhat later, in 1776, Pierre-Simon de Laplace
obtained similar results. He proceeded from the series expansion
. dxu . ,dx+1u, „dx+2u ,.
A A x A
A u = d ^ h + A d ^ h + A d ^ h + -> (°-5>
1
2 DIFFERENTIAL OPERATORS OF INFINITE ORDER
remarking that the coefficients A', A",.. ■ depend only on A but are independent of
the function u.
These works led to a racing development of symbolic calculus. A remarkable event in
this history is the derivation of the formula
f(D)(uv) = uf(D)(v) + Duf(D){v) + ±D2uf'(D)(v) + ■ ■ ■
for a rational function / of the symbol D and /',/", ...are derivatives of / (see
D'Alembert [1] and Hargreave [1]). B. Brisson and Augustin-Louis Cauchy (Cauchy
[1]) studied more general functions f(D, A) of distributive symbols and applied them
to the integration of linear differential equations as well as to linear difference equa­
tions. Cauchy was the first to remark that the series expansion of /(£>, A) in symbolic
calculus can lead to erroneous results. He determined the limiting values for the con­
vergence of this series as well as the technique by which the conclusions realized by
the symbolic method can be verified to a significant level. He extended also his study
to functions F(D , D, D,..., A*, A, A ,...). The history of the operational calcu­
X y z y 2
lus is very interesting and has been presented in many monographs. The interested
reader is referred to the very nice books by I. Z. Shtokalo (Shtokalo [1]) and H. T.
Davis (Davis [1]).
Thus, differential operators of infinite order appeared in the very first days of the
infinitesimal calculus. S. Pincherle in 1886 (Pincherle [1]), in discussing the solution
of the difference equation
771
Y, h f{x + a ) = f(x),
n n
71=1
introduced the following differential and integral equations of infinite order
a(x)u(x) + ai(x)u'{x) + a(x)u"(x) -\ = f(x),
0 2
b(x)u(x) + 6 (^) (-1>(s) + b {x)u(-2\x) + • • ■ = F(x),
0 1 M 2
(see also Amaldi and Pincherle [1]).
Independent of the result obtained by Pincherle, in 1897 C. Bourlet published a paper
entitled: Sur les operations en general et les equations differentielles lineares d'ordre
infini. Bourlet used the term transmutation to denote an operator T which makes
a given function (j>(x), the object of the operation, correspond to another function
T(/>(x), the result of the operation. The transmutation is said to be distributive if for
arbitrary functions cj>(x) and ij>{x) and for an arbitrary constant c, one has
T[tfx) + 4>{x)] = Tcj>{x) + Ti>{x), T[c<t>(x)} = cT</>(x).
INTRODUCTION 3
Pincherle simply called T a distributive operation. Both Bourlet and Pincherle insist
on the proposition that every additive, uniform, continuous, and "regular" transmu­
tation can be represented by a series of the form
°° dnu
Tu= Y,a {x)—,
n
Si d*n
thus bringing the theory of transmutations into intimate association with the theory
of differential equations of infinite order (see Amaldi and Pincherle [1], Pincherle [1],
Bourlet [1-3]).
We give here only a history of the very first days of differential equations of infinite
order. We add also the names of R. D. Carmichael (Carmichael [1]), T. H. Davis
(Davis [1]), E. Hilb (Hilb [1]), O. Perron (Perron [1-3]), G. Polya (Polya [1-2]), J. F.
Ritt (Ritt [1]), G. Valiron (Valiron [1]), ..., who made valuable contributions to the
theory of differential equations of infinite order. For further discussions and refer­
ences on this subject the interested reader is referred to Davis [1] and Carmichael [1].
To these sources of literature on differential equations of infinite order we add some
references on the related theory of difference equations. The most adequate bibliog­
raphy of difference equations is found in N. E. Norlund's Differenzenrechnung. Berlin
(1924), where in 68 pages 1427 references, the work of 540 authors, can be found. The
bibliography of Norlund has been supplimented by more than 300 additional titles
listed at the end of an important summary: Linear q-Difference Equations, by C. R.
Adams, Bulletin of the American Math. Soc. 37(1931), 361-400. We cite also the
books by Boole (Boole [1]), Jordan (Jordan [1]) and Gelfond (Gelfond [1]).
In recent years considerable attention has been paid to the theory of differential
operators of infinite order (DOIOs). A reason for this is, probably, that it has been
successfully applied to various areas of mechanics, elasticity theory, theoretical physics
and mathematical physics, etc. Furthermore, together with the theory of pseudo-
differential operators and the theory of Fourier integral operators (Hormander [1-2],
Treves [1-2], Maslov [4], Egorov [1-3] and C. FefFerman [1], etc), the technique of
DOIOs has been, and continues to be, considered as a powerful method for studying
problems of partial differential equations and of mathematical physics.
As we have seen, DOIOs appear in the calculus of finite differences; rather often they
appear in many problems of mechanics, physics and technology, Without claiming
to give a complete bibliography, we mention the work by Agarev [1], Bondarenko [1],
Bondarenko and Filatov [1], Khlebnikov and Parasak [1], Lur'e [1-2], Podstrigach [1],
Vlasov [1-2], where DOIOs and the technique of DOIOs have been extensively studied.
In Khlebnikov and Parasak [1] (1980), for example, in studying the contact problem
of the winding of isotropic plates by smooth stamps, it is desired to determine the