Differential Operators of Infinite Order with Real Arguments and Their Applications
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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
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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 orbyany means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.
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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
n
2.3. The space of generalized functions W^°°(IR )
39
2.4. The algebra of pseudo-differential operators with analytic symbols
45
Bibliographical Notes
47
v
CONTENTS
vi
Chapter 3. Applications t o 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 Chapter 4. Approximation m e t h o d s
75 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 Chapter 5. A mollification m e t h o d for ill-posed problems
88 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 Chapter 7. Some properties of Sobolev-Orlicz spaces of infinite order
160
161
7.1. Traces and extensions
161
7.2. Traces in LW^^n,
163
(0, a)}
7.3. Dual spaces
170
7.4. Imbedding theorems
174
Bibliographical Notes Chapter 8. Elliptic equations of infinite order with arbitrary nonlinearities
194
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 Bibliography
217 218
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 = du + dv: (du + dv)n(uv)
= vcTu + (n')dvdn-1u
h ( " I d " - 1 ™ + dTvu
+
(0.2)
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),
A xu
= AAA_1K,
and SAu = l / (
e
^
+
^"
+
^-l)
A
(0.4)
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 A
.
A u
dxu
= d ^
.
x+1 u,x A,d
h + A
x+2 u A„d
d ^ 1
h + A
d ^
,. h +
- >
(
°- 5 >
DIFFERENTIAL
2
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 / ' , / " , . . . a r e 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(DX, Dy, Dz,..., A*, Ay, A 2 , . . . ) . The history of the operational calcu 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, hnf{x + an) = f(x), 71=1
introduced the following differential and integral equations of infinite order a0(x)u(x) + ai(x)u'{x)
+ a2(x)u"(x)
b0(x)u(x) + 6 1 (^) M (- 1 >(s) + b2{x)u(-2\x)
-\
=
+ ••■ =
f(x), F(x),
(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(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 °°
Tu=
dnu
Y,an{x)—,
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 pseudodifferential 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
DIFFERENTIAL OPERATORS OF INFINITE ORDER
4
contact pressure q from the equation cos(sjA\/A) + cos(sihvK)
cos(s2h\/A.)
L-*+
— cos(s2h-vA)
vs
L-1\q
=
f(x)-h.
(0.6)
Here A is the Laplace operator, L 1 is the right inverse operator (in the sense of Dubinskii [4]) to the operator 1 [s2 sm(sih\/A) ± Si sin(s2fcvA)], i± = 2SIS2T/A
and si,s2,h are some defined constants. To solve (0.6) one should expand the operators on the left hand side into the Taylor series of v A . If we take only those members of the series containing the powers of h up to degree 1, then we have the equations of the Kirchhoff theory. If we take those members containing the powers of h up to degree 2, then we have the equations of the applied theory. However, both of these theories do not give desired results. The authors of the above mentioned work took, therefore, those members of the series containing the powers of h up to degree 4, and received very satisfactory results. Consider now the well known Cauchy problem
82u dt
2
a
,d2u dx2
u(0, x)
o, ae<E\ H = i, . .
du(Q,x)
(0.7) (0.8)
We note that, (0.7) is the Laplace equation if a = i, is the wave equation if a — 1, The qualitative properties of the corresponding operators are, as is well known, quite different. We would, however, like to consider them here in a unified scheme. To find a formal solution of Problem (0.7) -(0.8), we put £ = —id/dx in (0.7), and regarding | as a complex parameter, we consider the following two Cauchy problems
dt2
0,
WO = tM(U) dt
1, 0,
and (x).
(0.9)
fl.T
Has (0.9) a meaning? And if it makes sense, is it then a solution of the problem (0.7)-(0.8)? To clarify this, it is obviously necessary to give a precise definition of the operator in the right hand side of (0.9) and to specify exactly under which conditions it is applicable. A natural way is to represent (0.9) as the sum of two series OO
2n)
u(t, x) =
= E (2n)! ^
1 °° (x) + i
E
n=0
-4,W(X) (2n + 1)'
(0.10)
and to define the Sobolev space of infinite order
W~:
^^lE^II^IKoo},
(0.11)
where || • ||p is the norm in LP{]B}). Clearly, if
belong to W£°, then u(i, ■) G i p (IR 1 ) for every fixed i, and the formula (0.10) (or (0.9)) determines a classical Z,p-solution of the problem (0.7)-(0.8). In the case a = 1 we can extend W£°, the domain of definition of
(z
1 r*+
V(0#-
2
Finally, we note that the fundamental solution of the Cauchy problem 2 82u 2d u dt2 -- a" — dx2
0, u(0,x) = 0, du/dt(Q,x)
is a hyperdistribution of the form sinh £(t,
(at£) 6(x).
x) l
3x
= S(x),
DIFFERENTIAL
6
OPERATORS
OF INFINITE
ORDER
In the space W°° (see Chapter 2) this functional has the form &{t,x)
= w-[0(x + at) - 0{x - at)], 2a
where the functionals 0(x ± at) are defined by < 0{- + h),ip(-) > : = < 6(-),tp(- -h)>
=
f°° Jo
x ) = h[0ix + «0 - 6(* ~ ».
VS ^(IRn).
T h e o r e m 1.6.4. The convolution is commutative, that is, f\ * / 2 = f2*fi, the distributions f\ and f2 has compact support. We have SU
if one of
PP (A * A) C supp / ! + supp / 2 .
1.7. Fourier transforms of distributions. The Fourier transform / of a function / € Li(IR n ) is defined by /(0 = /e-^f(x)dx,
(1.7.1)
where x£ = a)i£i + • • • + xn(n. If / is also integrable, then we can express / in term of / by means of the Fourier inversion formula /(x) = ( 2 7 r ) - " | e ^ / ( 0 ^ -
(1-7-2)
Definition 1.7.1. By 5 ^ o r &*(Eln) we denote the set of all functions tp G C°°(IR n ) such that sup\x 0 as j ^ o o .
(1.8.2)
Definition 1.8.1. (cf. Martineau [1], Hormander [1], Matsuzawa [1]) We denote by A'[K] the strong dual space of A[K] and call its elements analytic functionals carried byK. Theorem 1.8.1. (Paley-Wiener theorem, cf. Hormander [1], Matsuzawa [1]). If u G A'[K] then the Fourier-Laplace transform u(()=u(exp(-i(.,())) is an entire analytic function such that for every t > 0 \u(0\°/| '(IR") of Roumieu type and £(M (IR") of Beurling type, respectively. Nevertheless, the main results of Komatsu [2], Hormander [2], Bjock [1] on ultradistributions remain valid for £<M'(]Rn) by virtue of Lelong's theorem (see Theorem 1.18.2 below). T h e o r e m 1.8.2. (Schapira [1], Hormander [1]) If u € A'[IR"], then there is a small est compact set K C IRn such that u € A'[A']; it is called the support of u. If K\,..., KT are compact subsets ofW1 Uj € A'(Kj),j = 1,2,..., r so that
and u S A'[K\ U • • • U KT], then one can find
U = Ui + - ■ • -f UT.
For the proofs of Theorems 1.8.1 and 1.8.2 we refer the readers to Hormander [1].
DIFFERENTIAL
16
OPERATORS
OF INFINITE
ORDER
1.9. Entire functions of exponential type that are bounded on R". Definition 1.9.1. (Nikolskii [1, pp. 98-102]) The function 9 '■= 9»{z) ~ Sn,va
i" n ( z i;---> z n)
is called an entire function of exponential type v = ( i / 1 ; . . . , i/ n ), if it satisfies the fol lowing properties: i) it is an entire function in all of its variables, i.e. it decomposes into a power series g(z) = ] T akzk = £ a*, knzkl ...**• A:>0
A|>0, J=l,...,»
with constant coefficients ak = a^,...,&„, and converges absolutely for all complex z = (zu...,zn). ii) For every e > 0 there exists a positive number Ac such that for all complex Zj = XJ + iyj(j = 1 , . . . , n) the inequality \g{z)\ < Aeexp r£(»d
+ e)\zA
.
is satisfied. Definition 1.9.2. OT^IR") := 9Jt„,p (1 < p < oo) is the collection of all entire functions of exponential type v which as functions of a real x 6 IRn lie in i p (IR n ). Theorem 1.9.1. (Nikolskii [1, p. 110]) / / / G a n ^ I R " ) , then its Fourier f has support on A„ := {\XJ\ < v, j = 1,2,... ,n}.
transform
Theorem 1.9.2. (Bernstein-Nikolskii's inequality) (Nikolskii [1, p. 115]) If f artViP(IRn), where v = (y\, u2, ■ • ■, vn), then \\df/dXj\\Lp
< ui\\f\\LP,
£
J = l,2,...,n.
1.10. Difference quotients and modulus of continuity. Suppose f(x) is a func tion and h = (hi,h%,..., hn) is any vector on IR". Let A f c / := A f c /(i) = f(x + h)~
f(x).
PRELIMINARIES
17
Then A * / := A j / ( x ) = A f t A*-V(x)
( A ° / := / , A\ := Afc, * = 1,2,....).
Let H e a unit vector. Put u,*(/,tf)
:=
sup||Af k /(-)|| P ,
Uk(f,6)
:= ul(f,S).
u>h(f,S) is called the modulus of continuity of order k of the function / in the metric of L p (IR n ) along the direction h. It is well known that if / € Lp(lRn) and 0 < p < oo, then lirnw A (/,*) = 0. Let * & » ( / , £ ) » = sup w*(/, £) p . h€IR"
|M=i
Then we have Theorem 1.10.1. (Nikolskii [1], §5.2 ) Suppose f has derivatives of order s, \s\ < p, lying in Lp(TRn). Then for every positive v there exists an entire function fu of spherical type v such that
||07, - Jyf\\p < -£$ T. «k- ( ^ 7 , i ) , •* > l, \s\=p
where D" =
D^....DaNN',
Dj = -id/dxh
j € {1,2,...,N},
a =
(ai,a2,...,aN).
From this theorem we can prove the following Theorem 1.10.2. Let f £ C m (IR B ). Then for any compact set K C 1R" there exists a sequence of entire functions gUk of exponential type Vk (gVk € S9Jli/k<x>0R-n)) such that \\f - 9uk\\c"(K) -* 0 as
k-*co.
18
DIFFERENTIAL
OPERATORS
OF INFINITE
ORDER
1.11. The sampling theorems Theorem 1.11.1. (The sampling theorem (Butzer, Splettstofier and Stens [1]).) Any f e OJt^IR 1 ), l < p < o o , i / > 0 is representable on the whole real line IR1 by
*>-£.'£)-(?-»)• the series being absolutely and uniformly convergent. When p = oo this representa tion does hold if f € ntt^IR 1 ), u < v. Here sine ( i>|*|/(»r£) > 0, 0 < | < 1 and N > r > 0. 1.12. The de la Vallee Poussin kernel 1.12.1. The case of periodic functions in (—7r,7r)n. The function (see Nikolskii [1, pp. 301-308]) V$(x);=V£(x)V£{X)--.VCn(x),
N = (k„k2,...,kn),
kj e { 0 , 1 , 2 , . . . } ,
where Vk*(x) =
cos(fc + l)x — cos(2fc + l)x 4ifcsin2(x/2)
is called the de la Vallee Poussin kernel. We recall here the important properties of this kernel: i) V£ is an even trigonometric polynomial of order 2k; ii) the Fourier coefficients of Vt* with indices I = 0 , 1 , . . . , k are equal to unity;
»i) lfvk*{x)dx = \; iv)
UT\Vk*(x)\dx cos(mx) + &£> sin(mx)] , ■t
2n 2n
'- 2n + l£ / ( l 0 ' = o
2
2n
, i 51f(xk)cos(mxk), 2n + l t=0 2n 2 n) = , „ , , ^ /(xt)sin(mxA). &L = m
2n
+ 1 Jfc=0
T h e o r e m 1.15.1. If f e C2„, then i) I C U / , x ] | < ( 2 x + 4 r 2 ) m a x | / | , ") \Un[f, x] - / ( x ) | < (1 + 2TT + 4* a ) W ) < » + u (/, 2^)^
•
Here ui(f, 5)oo is £/ie modulus of continuity of f in L^. 1.16. Orlicz classes a n d Orlicz spaces. Definition 1.16.1. A function $ : IR1 —* IR1 is called an iV-function if it is continu ous, convex and satisfies the conditions: $(—t) = $( 0 as t > 0, ®{t)/t —> 0 as t —» 0 and $(t)/t —» oo as / —► oo. With such a function $, one can associate another JV-function $ defined by $(t) := sup {ts - * ( « ) } • The function $ is called complementary to $. satisfies Young's inequality:
It is evident that the pair ( $ , $ )
ts < *( /?*(*), /?> /?>!■ *(#)
(1.16.1)
23
PRELIMINARIES
Definition 1.16.2. The JV-function $( 0, for a certain positive constant K. Example 1.16.1. Consider the N-function
*(*)= [\(r)dr, Jo
v
where i) ip(t) = \t\ ~H, 1 > p < oo ( $ and $ both satisfy the A 2 -condition); ii) tp(t) = sign (t) = sign
