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(z , or by /(<£>). Theorem 1.2.1. A linear functional f on ^ (SI) is a distribution in SI, if and only if for every compact set K C SI there exist constants C and m such that \ = (Rn), we have f * -<*■ b) If A € Wfl CO (nr) and / is a function such that ftp € W^°°(Un) W£°°(IR n ), then we define the product fh by = < « ( • ) , e x p ( - t A M - ) >, veWgiW). .=< h,L' , Vy> € -Vc||2<e, e > 0 . j= 0 as k -> oo, which is impossible. The lemma is proved. T h e o r e m 7.4.7. Suppose that Ra < oo and M = Sup(fibkRk)(J2akRt) ">o W o / U=0 77iera the imbedding (7.4-13) holds.
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),
6
DIFFERENTIAL
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
For a = i we have the fundamental solution of the Cauchy problem for the Laplace equation:
^(*> x ) = h[0ix + «0 - 6(* ~ »<)]■ Another interesting example is in Maslov [5]; this author considers the equation of the vibrations of atoms in a one-dimensional crystal lattice with spacing h on a circle of length 27rrjv = Nh, where N is the number of atoms: ^
= ^K
+ 1
-2«
n
+ U „_ 1 )
(n = l , 2 , . . . , i V ) .
(0.12)
Here un denotes the deviation of the n-th atom from the position of equilibrium, and u0 = UN,UI = njv+i-
We consider a smooth 2nrjv-periodic function u(x,t) points. Then (0.12) can be rewritten as follows: Q2
or in the form
= T J ( U ( X + h,t)-
taking values Uj at the lattice
2u(x, t) + u(x -
h,t)),
,, ,„dd22uu ih dd\\ ,2 , 2. ,o / ih „ h2—+4:C sm [ ~ ^ " s - « = 0 2 dt \k 2 fo;
(0.13)
exp I i ( -ih-Q- J j u(x,t) = u(x + h,t),
(0.14)
,
by using the identity
which can be verified either by the Taylor expansion or by the Fourier transform. In this case Maslov found the family of characteristic equations corrsponding to the problem (0.12)
fds\2
c2 . J \
ds\
sin
=0
{») -^ {rd-x)
n
(°-15)
that are differential equations of infinite order and play a very important role in the asymptotic analysis of the vibrations of atoms in a one-dimensional crystal lattice.
INTRODUCTION
7
From the above mentioned examples, we see that from the viewpoint of either appli cations or of theoretical investigation DOIOs are a subject that should be studied. It is clear that operators like (0.10) or (0.14) and also spaces like (0.11) play an impor tant role in the study of partial differential equations, in mathematical physics, and especially, in the study of ill-posed problems. In 1958, Gelfand and Shilov (Gelfand and Shilov [1], Vol II, p. 167, Vol. Ill, p. 31) already emphasized the importance of DOIOs, and considered them as one of two main methods for studying problems of partial differential equations. In 1975 Dubinskii (Dubinskii [1]) and his pupils have initiated a systematical inves tigation of DOIOs and differential equations of infinite order based on their study of Sobolev spaces of infinite order. In Dubinskii [1-3], Balashova [1-5], Konajev [1], Kobilov [1] boundary value problems for differential equations of infinite order with coefficients of polynomial growth are considered. In his theory of Sobolev spaces of infinite order, Dubinskii proved the nontriviality criteria of Sobolev spaces of infinite order, the extension criteria, embedding theorems . . . ; and based on the theory of nonlinear equations of Minty [1], Visik [1], Browder [1], Dubinskii [7], Lions [1], he proved also several existence and uniquenes theorems for differential equations with coefficients of polynomial growth. Dubinskii was also the first to consider pseudodifferential operators with analytic symbols and to use DOIOs to study the problems of partial differential equations and of mathematical physics from a quite new view point. We note also that there are many others who studied and used DOIOs and differential equations of infinite order, we cite further only the names of Gelfand and Shilov [1], Ovsyannikov [1,2], Boutet de Monvel [1], Leont'ev [1,2], Korobeinik [1, 3]. In this book we outline mainly our own results on the subject, though some related results of other authors have been added. We neither try to give a comprehensive exposition of the results of many other authors nor to give a survey on the subject. The reader can find excellent surveys on differential operators of infinite order and on differential equations of infinite order in Davis [1], Carmichael [1], Leont'ev [1,2], Dubinskii [3-5] and references therein. Furthermore we mention that the theory of differential operators of infinite order in complex domains has also been developed extensively. We quote the names of Leont'ev [1], Korobeinik [2,3], who made valuable contributions to this subject. Finally, we cite the series of papers by Sato, Kawai and Kashiwara [1], in which they developed an algebra of differential operators of infinite order defined on the space of all functions holomorphic in (F1, with applications to differential equations. Our book is organized as follows. In the first chapter some auxilary results needed later are outlined. The second chapter is devoted to the theory of pseudo-differential
8
DIFFERENTIAL
OPERATORS
OF INFINITE
ORDER
operators with real analytic symbols, and the next chapter is devoted to its applica tions to problems of mathematical physics. Some approximation methods for pseudodifferential equations are given in Chapter 4. In Chapter 5 a mollification method for ill-posed problems with some numerical examples is described. Chapters 6-8 are devoted to the theory of Sobolev-Orlicz spaces of infinite order and its applications to nonlinear elliptic equations of infinite order with arbitrary nonlinearities. For the reader's convenience we write to every chapter an introduction and "Bibliographical Notes" As a guide for the reader, we indicate below the principal connection between the chapters of the book.
Chapters 2, 3, 6, 7, 8 and Sections 1.8, 1.16-1.20 of Chapter 1 have been written by Tran Due Van, the other parts have been written by Dinh Nho Hao. This book has been written during several stays of the authors at the Hanoi Institute of Mathematics, the University of Tokyo, Freie Universitat Berlin, Universitat -GH Siegen. They have been supported by the National Basic Research Program in Na tional Science of Vietnam, the Japan Society for the Promotion of Science (JSPS), the Alexander von Humboldt Stiftung, the Deutscher Akademischer Austauschdienst (DAAD) and the Deutsche Forschungsgemeinschaft (DFG). The supports of these institutions and organizations are kindly acknowledged. The authors thank Profes sors Yu. A. Dubinskii (Moscow), H. Komatsu (Tokyo), R. Gorenflo (Berlin), H.-J. Reinhardt (Siegen) and Lee Peng Yee (Singapore) for their interest in writing this book. Concerning the English style, Professors R. Gorenflo, H. J. Reinhardt and Ms. H. M. Ho (World Scientific Publ.) have carefully looked through the text and have given precious and helpful advice. They deserve sincere thanks for this tedious work. Hanoi - Berlin - Siegen, 1993 Tran Due Van and Dinh Nho Hao
Chapter 1 PRELIMINARIES
The purpose of this chapter is to sum up very briefly some results from functional analysis, distribution theory and approximation theory which will be frequently used in the subsequent chapters. Neither proof, nor historical remarks are given. For a more detailed analysis of almost all topics discussed here the reader is referred to Hormander [1], Vladimirov [1], and Nikolskii [1]. 1.1. Spaces Ck(f?) and Lp(fi). Let ft be an open set in the Euclidean space IRn, and let / be a continuous function in ft. By the support of / (in ft), denoted by supp/, we then mean the closure in ft of {x | x G ft,/(x) ^ 0}. The support is thus the smallest relatively closed subset of ft outside which u vanishes. Definition 1.1.1. By C*(ft), 0 < k < oo, we denote the set of all functions / defined in ft, whose partial derivatives of order < k all exist and are continuous. By C'o(ft) we denote the set of all functions in Ck(Q) with compact support in ft. By a we shall denote a multi-index, that is, n—tuples (cti,... ,an) of non-negative integers. Put a\ =
Ql!a2!
■ ■ ■ anl, ** = * - » - x f , (f) = (fj) • • • ( * ) =
\a\ = a-i + ■ ■ ■ + an. With Dj = id/dxj
^ T ^ y ,
and the imaginary unit i, we set
Da = Z)f1 •••££". The set of all functions f(x) from C*(ft), for which all derivatives D°f(x), \a\ < k, can be continuously extended to ft is denoted by C*:(ft). The norm in C*(ft) is defined by \\f\\cHQ) = sup \Daf(x)\. xen, \a\
=
||u||oo =
QfjuO)|'
P
, for
l
ess maxn|u(x)|.
It is known that with this norm i p (ft) is a Banach space.
9
DIFFERENTIAL
10
OPERATORS
OF INFINITE
ORDER
Further, put L p ,i«(ft) = {/|
(SI) are called test
(SI) is dense in Lp(Sl), 1 < p < oo.
Definition 1.2.1. Every linear continuous functional on ^ (SI) is called a distribu tion in ft. The set of all distributions in SI is denoted by © ' ( f t ) . The action of a distribution / on a test function
(1.2.1)
If the integer m in (1.2.1) can be chosen independent of K, the distribution u is said to be of finite order in ft, and the smallest such integer m is called the order of u in ft. The set of all distributions of finite order in ft is denoted by &'F(Sl). Theorem 1.2.2. Suppose that the sequence of distributions / i , / 2 , •.. from . © '(ft) has the property that for every cp € .©"(ft) the number sequence < fk,f> converges as k —* oo. Then the functional f over @ (ft) defined by < f,ip > : = Km < fk,ip
>
is also a distribution in ft. Given a distribution / in ft, we can define its restriction to an open set ft' C ft simply by restricting the domain of definition of the linear functional / to CQ°(SI'). We shall say that two distributions / i and / 2 in © '(ft) are equal in a neighborhood of a point x g ft if the restrictions of / i and f2 to some open neighborhood of x are equal.
11
PRELIMINARIES
A distribution is completely determined by its local behavior; in fact, the following theorem holds. Theorem 1.2.3. Let / j and $?, be two distributions in ft such that every point in ft has a neighborhood where / i = / 2 . Then fi = f2 in ft. Definition 1.2.3. For / e ®"(ft) the support of / is defined as the set of points in ft which have no neighborhood where / is equal to 0. The support of / is denoted by supp/. 1.3. Multiplication of distributions by functions. Definition 1.3.1. For / € ®"'(ft) and a £ C°°(ft) we define the product of / and a by the formula
(-I)'" 1 < f,D*v
>, V<^ € ^ ( f t ) .
(1.4.1)
It is clear that (1.4.1) defines a new distribution Daf and that the mapping / —» Daf in i ^ ' ( f t ) is continuous. 1.5. Distributions with compact support. Theorem 1.5.1. The set of distributions in ft with compact support is identical with the dual space of C°° with the topology defined by the semi-norms
y^
£sup|£>
|c|<*
where K ranges over all compact subsets of ft and k over all non-negative integers. Denote by £ (ft) the space C°°(ft) equipped with this topology. Accordingly, the space of distributions with compact support in ft is defined by £'(il).
DIFFERENTIAL
12
OPERATORS
OF INFINITE
ORDER
Proposition 1.5.1. Every distribution with compact support in ft is of finite order in ft. Theorem 1.5.2. A distribution whose support only contains the point y is a finite linear combination of the Dirac measure at y and its derivatives. 1.6. Convolution of distributions. The convolution f * g ot two functions / and g is defined by ( / * 9)(x) = J f{x-
y)g{y)dy = j f{y)g{x -
y)dy.
This definition is meaningful, if / € Z»i,i0c(IRn), <7 G L\tComv{MP'), or both functions / and g belong to Zi(ET). If
/ez,p(nr), 5 ei,(nr), 1
)|/*5lk<||/IWI$|fv In particular, if p = 1 and q G [0, oo], then
ll/**lk< 11/11* IWU,. Definition 1.6.1. For / G & '(IR") and
(f*g)(x)
=
(f(.)Mx--))-
Theorem 1.6.1. If f e & '(IR") and
jyv.
Theorem 1.6.2. If ip and ip are in @> (IR ) and f € ®'{WLn), (f*tp)*il>
=
then
f*(
If h G IR" then we define the translation operators r^ by (Tk(p)(x) =
PRELIMINARIES
13
sequence ipj —> 0 in ^ ( I R " ) . Then there exists one and only one distribution f such that F
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<3Datp{x)\
for all multi-indices a and j3. The topology in Sf is defined by the semi-norms in the left-hand side of (1.7.3). Proposition 1.7.1. The Fourier transformation y> *-* (p maps £? continuously into
fy. Proposition 1.7.2. The Fourier inversion formula (1.7.2) is valid in Sf Definition 1.7.2. A continuous linear functional u on Sf is called a tempered dis tribution. The set of all tempered distributions is denoted by £/". Definition 1.7.3. If u € . 5 " ' , then the Fourier transform u is defined by < u,ip > = < u,tp >, ip € £r.
14
DIFFERENTIAL
OPERATORS
OF INFINITE
ORDER
Theorem 1.7.1. Fourier's inversion formula is valid for every u € *S", that is, it, = (27r)nu. The Fourier transformation maps £/" into Sf Theorem 1.7.2. If u £ L 2 (IR n ), the Fourier transform u is also in LiiJR?) and Parseval's formula is valid, f \u\2dx = (2w)n J \u\2dx. Theorem 1.7.3. If f £ £', then its Fourier transform f can be represented in the form / ( O = (/(•). «?(*)«*"). (1-7-4) Here, n is any function of &
that is equal to 1 in a neighborhood of supp / .
The right-hand side of (1.7.4) is also defined for every complex vector £ € 071 and is an entire analytic function of £, called the Fourier-Laplace transform of / . 1.8. The space of analytic functionals. Let K be a, compact subset of lR n . We denote by A[K] the space of all real analytic functions in some neighborhood of K. That is, if
(1.8.1)
We say y>j —> 0 in A[K] as j —> oo if there is a constant h > 0 such that sup j £ K
J?\ h\"\a\
-> 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\
+ e\£\),
C = £ + *»? 6
where L = sup x 6 i f |z|. Here \x\ denotes the Euclidean norm of x.
(1.8.3)
PRELIMINARIES
15
Conversely, if F(Q is an entire analytic function satisfying the estimate (1.8.3) with some constant L > 0, then F(Q is the Fourier-Laplace transform of a unique element u in A'[SL], where SL = {x € IR", \x\ < L). We can consider A'[ffi] to be a subset of A'[/f2] if Ki C K2 and set
A'[m.n] =
|J
A'[K].
A'CCR"
Then we have the inclusion £'(IR") C £ W ( I R " ) C £<M»>'(IRn) C A'[JR.n],
(1.8.4)
where £'(IR n ) is the space of tempered distributions, £{M*>'(IRn), (£(M»>'(IR")) is the space of ultradistributions of Roumieu type (of Beurling type) of class Mv (for the theory of ultradistributions the reader is referred to Komatsu [2], Bjock [1]). Remark 1.8.1. In all papers on ultradistributions (see, for example, Komatsu [2], Hormander [2], Bjock [1]), the Denjoy-Carleman class of several variables is defined as the set of functions / in C°° with sup \Daf\ < C ■ fcWj|fW( |a| = 0 , 1 , . . . . But it is a natural way to define the Denjoy-Carleman class of several variables as sup|£>°/|
|or| = 0,1,
(In the case of classes of analytic functions we have the estimate sup \Daf\ < C ■ h^a\.) Then we may consider the space of ultradistibutions £<M°>'(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<">'(IR") and £<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 (<) := sint/t for t =£ 0, sinc(0) := 1. Theorem 1.11.2. If f e 9TU with f{r) € LipL(a; (^(IR1)), 0 < a < 1, then
/w-|:/£)-(?-»)
=
0(AT- r - a )
/or AT suc ft that N + 1 > 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<3*/2. J — TT
PRELIMINARIES
19
We denote by L*(—7r,7r) the space of 27r-periodic functions whose restrictions to (—"■)"■) are in Lp(-w,Tt). For a function / € L*(—TT,TV) we define the mollification operator M^' 1 as follows /*(*) := M?f(x)
=l f
Vk'(x -
Of(Odt
J — 7T
7i
Then fk is a trigonometric polynomial of order 2k and
\\f-f%<*E'k(f)p. Here El(f)v is the best approximation of the function / by trigonometric polynomials of order not higher than k in the Lp(—w,ir) norm. 1.12.2. T h e case of functions given in W1. The function ,
.
V
»W
1 TT COS(VXJ) — cos(2^x, :
= ZZ 11
~2
where v is a positive number, is called the de la Vallee Poussin kernel and has the following properties: i) Vi/(z) is an entire function of exponential type of degree 2v relative to each variable Zj, j — 1 , 2 , . . . , n, bounded and summable on H n ; ii) ( | )
n / 2
K : = ^ / K ( O e - ^ = l o n A „ A , := {\XJ\ < u,j = 1,2,... ,n},
iii) ^nJVl/(x)dx iv) l*j
= 1,
| K ( x ) | dx < (2V5)» := M {v > 1).
Furthermore
K=
n M),
3=1
where i /*(>?) = til) =
12v^JLV
\/i
I 00
n
(M <»),
(" < \V\ <2v), 2v), (2i> (21/ < |,J). M).
Now, for a function / 6 L p (lR n ) we define the mollification operator M* as follows MU := * „ ( / , * ) = Q "
2
K * / = ^ / K ( x - y)/(y)dy.
20
DIFFERENTIAL
OPERATORS
OF INFINITE
ORDER
The function MXJ belongs to an2„,p(Irtn), and (see Nikolskii [1, p. 301 - 308])
\\M1J-f\\,<(l
+ M)E„(f)p,
where
inf
Eu{f)p
||/ --9»h-
1.13. T h e Dirichlet kernels Let v > 0. The function
A,(*)=n^^ j=i
x
i
is called the Dirichlet kernel and has the following properties (see Nikolskii [1, p. 316 -318]): i) It is an entire function of exponential type of degree v relative to each variable Zj, j = 1,2,..., n, belonging to XP(1R"), for all p 6 (1, oo];
i i ) ( i r / 2 a = ( i w = { ;0
on A v = {x\ \XJ\ < v] outside A„
iii) -At I Du(x)dx = 1. iv) The convolution
Su{f, x) = D„*f= J L . J for / € Lp
Du{x
- y)f(y)dy
(1 < p < oo) is belonging to an„iP(]R"). In addition UA,*/||p
where KP depends only on p. v) If w„ G 9n„, p (Irr), then Sv{uJ) = w„. vi) F [ A , * / ] = / o n A „ . vii) 11/ - 5 „ ( / ) | | , < (1 + « , ) £ „ ( / ) , . We recall now two important theorems of Markov type in approximation theory.
21
PRELIMINARIES 1.14. M a r k o v t y p e t h e o r e m s x £ IR1, can be represented in the form
Suppose that a function f(x),
nx) = r
■00 J —CO
G(x-y)h{y)dy - 0 0 < x < 00), G( -y)h(y)dv ((-oo<x
where G(x) is a Krein kernel. A function G(x) is said to be a Krein kernel, if A_ \G{x)\ < ——- 2
l+x
( - 0 0 < x < 00)
and for every v > i>0 > 0 there exist a real number a and an entire function G„ g //(IR1) of exponential type < v such that
Gv(x)) sin (u(x -- o r ) ) (G{x)-G--v{x))sm(y{x-a)) (G(x) does not change its sign on the real axis. T h e o r e m 1.14.1. (Achieser [1, p. 222]) If a function f can be represented as the convolution of a Krein kernel G(x) with a function h € L p (lR 1 ),p G [l,oo], then, for any v > v0,
E„(f)p < n„(a)||ft||„ where Q 2fc+1 »" / fl„(a) = - Y. — i — ^ {e- ( 7T
e '<
2A+1
»" f G(^a
T h e o r e m 1.14.2. (Achieser [1, p. 222]) / / / is a periodic function of the above form, G(x) is a Krein kernel and h is a periodic function in Lp(—ir,ir), p > 1, then, for any v > VQ,
Et(f)„ < n^aJHfclli,,^,. 1.15. T h e second i n t e r p o l a t i o n m e t h o d of B e r n s t e i n (see Natanson [1, p. 410413]). If a function is given only at discrete points, then we should interpolate it by a contin uous function. There are many methods devoted to this problem, but for our purpose we describe here only the second interpolation method of Bernstein. Suppose that the function / 6 CIT,, (the set of all continuous periodic functions on [—7r,7r]) is given only at equidistant points Xk := xkn' = 2kir/(2n + l),k = 0 , 1 , . . . ,2n.
22
DIFFERENTIAL
OPERATORS
OF INFINITE
ORDER
The second interpolation method of Bernstein works as follows: Put
f.[/;x]
=IM*+^)+T-(*-
r»(i)
=
AW
-.
2n + lJi
where AW + £ ) [«4"> 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) = $(<), $(i) > 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 < *(<) + §(s), V t , s e l 1 . For the iV-function $(<) the following inequalities hold:
*(#) < /?*(*), «(#) £e [0,1], 0*(O, /3e[o,i], l. * ( # ) > /?*(*), /?>!■ *(#) /?*(*), /?>
(1.16.1)
23
PRELIMINARIES
Definition 1.16.2. The JV-function $(<) is said to be satisfied the A 2 -condition, if #(2t) < K
*(*)= [\(r)dr, Jo
v
where i) ip(t) = \t\ ~H, 1 > p < oo ( $ and $ both satisfy the A 2 -condition); ii) tp(t) = sign <(exp(|t|) - 1) ($ satisfies the A 2 -condition, but $ does not); iii) y>(t) = sign <(ln(l + \t\)) (<& satisfies the A 2 -condition, but $ does not). Let now G be a domain in IR" and $(<) be an JV-function. Definition 1.16.3. Let £ { $ , G } be the set of all measurable functions / : G —» IR1, such that />(/, * ) := / *(f{x))dx < +oo, (1.16.2) JG
moreover, almost everywhere equivalent functions are identified. £ { $ , G} is called an OTUCZ class.
If / € ■£{$,
t (/ G l/(*)l«k]^(/,*)
1
mes G
\
(1.16.3)
mes G
holds. In general, Orlicz classes need not be linear spaces. £ { $ , G} is a linear space if $ satisfies the A 2 -condition. Definition 1.16.4. Let £ { $ , G } be the Orlicz class introduced in Definition 1.16.3. Then the space L{$,G} of all measurable functions / : G —» IR1, such that af € £ { $ , G} for some a > 0, is called an Orlicz space. Thus, L{$, G} := \ f : G —> IR1 | / i s measurable, / $(af)dx
< +oo for some a > 0 >
Definition 1.16.5. Let ( $ , $ ) be a complementary pair of iV-functions. We define the Orlicz norm in the Orlicz space i { $ , G } as ll/H* := sup y
\fg\dx | p(g, * ) < l } .
(1.16.4)
24
DIFFERENTIAL
OPERATORS
OF INFINITE
ORDER
Definition 1.16.6. The gauge norm in the Orlicz space L { $ , G} is defined as follows 11/11(4, := inf{fc > 0 | [ *(f/k)dz
(1-16.5)
JG
Proposition 1.16.1. L { $ , G } is a Banach space with either Orlicz norm or gauge norm (1.16.5). Moreover, these norms are equivalent
H/IIW < ||/U» < 2|1/H(#).
(1.16.4)
(1-16.6)
R e m a r k 1.16.1. The norm (1.16.5) is often called Luxemburg's norm. We call it the gauge norm following Rao and Ren [1, p. 97]. Definition 1.16.7. The closure in L { $ , G } of the set of all bounded measurable functions / : G -» ET is denoted by £ { $ , G}. It is well known that the space E{<5>,G} is separable, but the Orlicz space £ { $ , G} is separable iff $ satisfies the A2-condition. The space L { $ , G} is reflexive iff both $ and $ satisfy the A 2 -condition. In general, E{$, G}* = £ { * , G } . Remark 1.16.2. We have summarized in this section several properties of Orlicz spaces defined by ./V-functions only. However, these properties still remain valid for Orlicz spaces defined by Young functions. For the general theory of Orlicz spaces we refer the reader to Krasnoselskii and Rutiskii [1], Adams [1] and Rao and Ren [1]. 1.17. Sobolev-Orlicz spaces. Let G be a domain in IRn, and $ a be a family of iV-functions, \a\ < m. Definition 1.17.1. The anisotropic Sobolev-Orlicz space of order m > 0, denoted by W,mL{$a, G}, is the set of equivalence classes of functions from the Orlicz space I { * o , G} given by WmL{$a,G}
:=
{f\Daf € L{$a,G},\a\ < m,Daf being the derivative of / in the sense of distributions } .
Set m
H l » '= E
ll£>NI(*<.)>
(1.17.1)
|o|=0
where || • \\[
/ $a(D"u/k)dx
< 11.
(1.17.2)
25
PRELIMINARIES L e m m a 1.17.1. The norms (1.17.1) and (1.17.2) are equivalent.
Furthermore,
IMI(«> < ||«ll» < M||«||(«), where M = ZT=0S(i),
S(i) = Card {a = (au ... ,an) \ \a\ = i}.
The proof of Lemma 1.17.1 can be found in Tran Due Van, Le Van Hap and R. Gorenflo [1]. o Further, we define the space W mL{$a,G} as the closure of the set C£°(G) in the weak topology a{UL{9a,G},UE{9_a,G}), where TIH**, G}{U E{$a, G}) is the product of the spaces L{$a,G} {E{$a, G}), \a\ < m (see, Gossez [2]). Let W m £ { $ a , G} be the closure of C%°{G) in HTL{$ a , G) by either the norm (1.17.1) or the norm (1.17.2). P r o p o s i t i o n 1.17.1. The space WmL{$a,G} (W mL{$a,G}) is a Banach space with either the norm (1.17.1) or the norm (1.17.2). Furthermore, Wm+1L{$a,G} - » WmL{$a,G} m = 0,l,....
(W
m+1
L{$a,G}
-.iy
m
L{$a,G})
and || ■ || ( m + 1 ) > || ■ || (m) ,
It is well known that the dual space (WmL{$a,G})" of the space WmL{$a,G) consists of the additive set functions unless $„ satisfy the A 2 -condition. However, ( W m L { $ a , G} j consists only of point functions. If $ 0 are the complementary Nfunctions to $ a , then (wmL{^>a,G}J
=
W~mE{$a,G}.
More precisely, we set W""E{K,
G] := | h | h(x) = £ (-l)^Daha(x) I |o|<m
1, J
and take the norm as \\h\\-m - <mp { / \gh\dx | g £WmL{*a,
G}, \\g\\{m) < l } .
Moreover, w-mE{$a,G]
-♦ r
| m t
%{i,G)
and || • ||-m < II ■ ||-(m+i) (see, for example, Rao and Ren [1, p. 389]).
26
DIFFERENTIAL
OPERATORS
OF INFINITE
ORDER
1.18. The Denjoy-Carleman classes and the quasianalyticity of functions of several real variables. Let Ma, \a\ = 0 , 1 , . . . , 0 < Ma < co, be a sequence of real numbers, and G be a domain in IRn. Definition 1.18.1. The Denjoy-Carleman class associated with the sequence denoted by C[Ma], is the set of infinitely differentiable functions given by C[Ma) := {/ 6 C°°(G) | \Daf(x)\
< AKaMa,
{Ma},
\a\ = 0 , 1 , . . . } ,
where the positive constants A and K may depend on / . Definition 1.18.2. The class C[Ma) is called a quasianalytic class if u e C[Ma] n C™(G) implies that u(x) = 0 in G. Otherwise, (i.e. if there exists u g C[M a ] PI CQ'(G) non-quasianalytic class.
such that u(x) ^ 0) it is called a
For the case n = 1 we have the following results. Theorem 1.18.1. (Mandelbrojt [1]) Each of the following conditions is necessary and sufficient in order for the class C[M^ to be quasi-analytic: a) if/3m = mik>m Ml'\
then £ ~ = 1 J - = +oo; m
b) ifT(r)
= s u p m >\jr;
y°° In T(r) then I ^-dr
= +co;
c) either liminf m _ 00 M^/ m < oo, or liminf m _ 00 M\/m
= +oo,
= +00, where M£ is a convex regularization of the sequence Mm by means of logarithms. The multi-dimensional case is reduced to the one-dimensional one by Lelong's theo rem. Theorem 1.18.2. (Lelong [1]) The class C[Ma] is quasi-analytic if and only if the class C[Mm], Mm = inf| 0 | = m {M„} is quasi-analytic. The construction of functions with compact support of one real variable plays an important role in Chapter 6.
27
PRELIMINARIES
L e m m a 1.18.1. (Mandelbrojt [1]) Let fi0 = l,fiN > 0, (JV = 1,2,...) be a sequence of numbers satisfying the condition ^i + fi2 +
\-HN + --- < a / 3 , a > 0.
(1.18.1)
Then there exists a function v € Co'(—a,a), such that 1. 2.
1,(0) = 1, DNv(-a)
= DNv(a) = 0, TV = 0 , 1 , . . . ;
N
\D v(t)\<(liop1...tiN)-\
ie(-a,a),
JV = 0 , 1 , . . . .
(
'
- )
Proof. We choose a continuous function w0 satisfying the conditions: 1.
0 < w0(<) < 1, if <€ ( - a , a);
2.
v0(t) = l, if* € ( - a / 3 , a/3);
3.
« o (0 = 0, iff € ( - a , - 2 a / 3 ) U (2a/3, a).
Besides, let i>o(^) be an even function. Further, we define the sequence of functions vm(t) by the following recursion formula v
m(t) := -— /
vm-i(r)dr,
m = 1,2,....
In view of Condition (1.18.1) these functions are finite on the interval (—a, a). It is also clear that the functions vm(t), m = 1,2,... are even and differentiable at least up to order m. Let us prove that the sequence of functions vm(t) (strictly speaking a subsequence) converges to a function v(t) € C^(—a,a) a s m - t o o . To verify this, first we shall prove the following inequality max \Dmvm{t)\
< (nofn ■ ■ ■ pm)~l,
m = 0,1,...
(1.18.3)
<£(-o,a)
In fact, for m = 0 this inequality is evident. Further, for k < m we have Dkvm(t)
= - ! - [Dk-1vm_1(t
+ /*m) - D^Vm-tit
- fim)} ■
(1.18.4)
In particular (k = m), we obtain max |D m i> m (i)| < — t€(-a,<0'
-
max |£> m_1 t> m _i(t)|.
/im t e ( - a , a ) '
Thus, the inequality (1.18.3) is valid if this inequality is valid for m — 1, too. There fore, the inequality (1.18.3) is valid for all m.
DIFFERENTIAL
28
OPERATORS
OF INFINITE
ORDER
Now, let k < m — 1 be arbitrary. Using the inequality (1.18.4) we obtain that for any point t G (—a, a) there exist points t\,..., tm-k such that Dkvm(t)
= Dkvm^(U)
= ■■■=
Dkvk(tm-k).
Consequently, from this and the inequality (1.18.3) we have max \Dkvm(t)\<
max \Dkvk(t)\
t6(-a,a)
< ( ^ i ■ ■ ■ A**)-1-
(1.18.5)
te{-a,a)
Using Arzela's theorem and a diagonal process, and the last inequality, we see that there exists a subsequence of {vm(t)} (that denoted also by {vm(t)}) and a function v € C£°(—
vm(t) -> v(t),...,
-»
Dkv{t),...
uniformly for t £ (—a,a). It is clear that v(t) is a function to be found. It only remains to remark that the in equality (1.18.2) may be obtained from the inequality (1.18.5) when m —y oo. Lemma 1.18.1 is proved. 1.19. Semifinitary functions. Definition 1.19.1. Let a be a certain number. A function / : IR1 —> d? is said to be semifinitary if f(x) = 0 for x < a. T h e o r e m 1.19.1. (Shilov [1]). Let f(z) half-plane of (f, for which \z\k\f(z)\
< ukea\
be a holomorphic function
k = 0,1,...,v
in the lower
= Imz,
or, equivalently,
l/WI < r*(i + |*|)-V for all r] < 0, where Tk,vk are constants depending only on k. Then f{z) is a holo morphic extension of the Fourier transform of some semifinitary function fix). 1.20. Weakly nonlinear equations. We consider operator equations of the type A{u) = h,
(1.20.1)
where the operator A (which is generally speaking nonlinear) possesses two prop erties: coercivity and weak compactness. We call such equations weakly nonlinear
29
PRELIMINARIES equations.
Let X be a Banach space which we assume to be separable and reflexive. Further, let X* be its dual space. The action of y* € X* on an element x € X is denoted by
A{u),u> rr—
► OO, i f \\u\\x
- > OO.
u
\\ \\x Here || • \\x is the norm in X. II. Weak Compactness. If um —» u (m, —» oo) weakly in X, then for any v E X lim < A(ufc),i; > = < A(u),v >, k—^oo
where {v./,} is a subsequence of the sequence {um}. We have the following result. Theorem 1.20.1. Let the operator A be satisfied Conditions I and II. Then for any element h € X* the equation (1.20.1) has at least one solution u in X. The proof of this theorem can be found in Lions [1] and Dubinskii [7].
Chapter 2 PSEUDO-DIFFERENTIAL OPERATORS W I T H REAL A N A L Y T I C S Y M B O L S
It is well known that the classical pseudo-differential (PD) operators can be repre sented, with the aid of the Fourier transform, in the form
L(D)u(x)
= J L j j L{Z)u{ty*d£.
(2.0.1)
The Fourier transform technique enables us to convert the construction of the theory of PD-operators to the construction of the algebra of symbols in the space of the dual variables. In this chapter we shall develop a theory of constructing PD-operators with real variables in the original variables x € IR" by using the technique of the algebra of differential operators of infinite order. In its turn, this algebra is based on the theory of test and generalized function spaces W^°°(IR n ) and Wr(J°0(lRn), respectively. By definition, the test function space WG°°(IR") contains precisely all functions f(x) defined in IR" that have analytic continuations f(z) to the complex space (F as entire functions which are the Fourier-Laplace transforms of analytic functionals u € A'[G\ with supports in G. Locally WG°°(IR") can be regarded as the inductive limit of the spaces of such functions, when the type of the entire functions tends to an appropri ate limit (see Proposition 2.1.2 below). Thus, due to the Paley-Wiener theorem, our approach to the construction of the space of generalized functions WQ°°(W) is based on an extension over compact sets in the Fourier dual space IR? rather than in the space IR", as it has been usually done. Since the function exp(ia:£) for every fixed £ belongs to the test function space WG°°(1R") one can define the Fourier transform of generalized functions in Wg°°(lR") in the same way as one does for the classical Fourier transform. A remarkable prop erty of the generalized functions in WQ°°(]R") is that their Fourier transforms are analytic in G. Thus, we have the following diagram
30
PD-OPERATORS
WITH REAL ANALYTIC
SYMBOLS
31
where A[G] is the algebra of analytic functions in G, and A'[G] is the tspace of analytic functionals with supports in G. It is clear that the PD-operators introduced in this chapter are identical with usual PD-operators of the form (2.0.1), and therefore one should be able to obtain all the results of this chapter by using this integral representation. However, the technique of differential operators of infinite order has several significant advantages, the principal one being that it does not require the introduction of dual variables. We note that the formalism of differential operators of infinite order has been frequently used in solving many problems of mathematics, mechanics and physics,... Without claiming to give a complete bibliography, we mention here the book of Davis [1] published in 1936 with a bibliography of over 40 pages, and recent works by Hirschman and Widder [1], Widder [1], Lur'e [1,2], Vlasov [1,2], Bondarenko [1], Bondarenko and Filatov [1], Podstrigach [1], Hills and Irwin [1]. Sections 2.2.1-2.2.3 of this chapter are devoted to the test function space H / Q°°(IR") and the space of generalized functions W,J°°(IR") in a neighbourhood of zero and to the properties of differential operators of infinite order. The main result of this section is the structure theorem for generalized functions: every generalized function h € VFg°°(IR") can be represented in the form h(x) = I ( D ) ( 2 A A ) - n e x p ( - z 7 4 ) , where L(D) is a pseudo-differential operator with the symbol L(£) = fe(£)exp(£2). In Section 2.2.4 we construct an algebra of pseudo-differential operators with real analytic symbols associated to an arbitrary domain G C IR? and study its properties. At the conceptual level, this chapter is a continuation of Part III of Dubinskii's book [5], where a theory for the pair of the spaces (H°°(G),H~°°(G)) has been developed using the duality of the Hilbert space L 2 (IR n ) (see Section 2.1). We use here an other duality, namely that of the space of analytic functions and the space of analytic
32
DIFFERENTIAL
OPERATORS
OF INFINITE
ORDER
functionals in G: (A[G],A'[G]). This duality allows us to inherit the properties of analytic functionals in constructing our spaces (W£°°(IR n ), Wj°°(IR")) and to make these spaces more convenient for applications (see Chapters 3, 4, 5). 2.1. The space of test functions in a neighborhood of zero. Let now x € 1 R " , I > 1, and £ € IR£ be real variables, 0 < R < oo be a real number and let BR := {( € IR" | |£| < R}, where |£| = max\£j\,j = 1,2, ...,n. Assume that f(x) : IR" —» fl?, that is, f(x) is a function defined on the whole Euclidean space IR" taking complex values, in general. Definition 2.1.1. The space of test functions WR°°(JRn) is the set of functions f(x) satisfying the following condition : / admits analytic continuation as an entire function to QT and for each e > 0 there exist constants r < R and Cc such that \f{x + iy)\ < C £ exp(r|y| + e|x|), V x + iy = z € (ST .
(2.1.1)
From the Paley-Wiener (Theorem 1.8.1) we arrive at the following proposition. Proposition 2.1.1. A function f(x) belongs to W^°°(IR n ) if and only if its analytic continuation f(z) is the Fourier-Laplace transform of an analytic functional u with support in BR (supp u CC BR), that is
u = f. We introduce a topology in WR°°(]Rn) as follows: Definition 2.1.2. A sequence of functions fn £ W^°°(lR.n) is said to converge to / € Wfl°°(IRn) if and only if: for each e > 0 there exists a positive r < R such that sup |/„(z) - f(z)\ e x p ( - r | y | - e|x|) -> 0, for n -> oo. zed!" Let {rfc} be a sequence of positive numbers, r^ < rk+i,k = 1,2,... and r* —> R. In this case Br C BR. We define a space W^°°(IR n ) as a set of entire functions / such that their analytic continuations on G? admit the estimate: for each e > 0 there exists a constant Cc such that \f{x + iy)\ < C £ exp(r 4 |j/| + e|x|). It is not hard to see that W£-(IR") = lim i n d ^ o , W+°°(1R").
PD-OPERATORS
WITH REAL ANALYTIC
33
SYMBOLS
By virtue of the property of the inductive limit (see, Komatsu[l]) and the PaleyWiener (Theorem 1.8.1) we have Proposition 2.1.2. The sequence {/„} converges to f in VK^°°(IR") if and only if there exists a compact set K C BR such that un -> u in A'[K], where un = / „ , u = f. We list here some examples of test functions in Wjf (]Rn). 2.1.1. T h e Dubinskii space JJ~(]R n ). Let « ( 0 := £
M=o
a
°¥a>
a
° > 0,
(2.1.2)
be any function that is analytic in BR. (AS usual, (,2a = (l"1 ■ ■ -{J""). We note that if R = +oo, then
Il/llla
JBR JBR
Definition 2.1.4. We say that a sequence of functions fm,m to / in HR°(TRn) if
= 1,2,..., converges
||/» - /||a,. -» 0 for any weight function a(£) of the type (2.1.2). Remark 2.1.1. The space i7jf (IRn) was introduced and studied in detail in Dubin skii [4,5]. We note that for n = 1 this space was used by Miranker [1] for studying the heat equation backwards in time. We can alternatively define the space i/]j'(IR.n) as follows:
34
DIFFERENTIAL OPERATORS OF INFINITE ORDER
Let n(t\ a(0 = \J £&■« •"**"' ^ 5 * ' + 0O, £,$BR be such that
(2.1.3)
2*jr(]R») = {/ € i2(IRB) I H/tlt := | R n a(0|/(0| 2 df < +00}. This means that the finiteness of the norm || ■ ||2,a implies the equality /(£) = 0 for (almost all) £ $ BR and consequently, the integration is, in fact, only over BR. We note the special case R = +00 which is of frequent occurence in mathematical physics and in the theory of partial differential equations. In this case the condition i) does not appear in the definition of the space H^n(JRn). More precisely, fl3?.(]R") = {/ € L a (B") I H/llt := /
a(0\f(OM
< +00}.
for any entire function a(() of the type (2.1.3). In other words, the inclusion / 6 H^n(SC) is equivalent to the inclusion / e £2,a(]R"), where Z2,a(IR'1) is the Lebesgue space with arbitrary weight a(£) > 0. Let us now give some simple examples of test functions / £ HR°(JRn): sinrx e"x — 1 sin2(rx/2) where r € (0,72). After simple calculations we see that their Fourier transforms are
/sinrx\ =_ ff 1/2, /sinrxN 1/2, a*s e ((-r,r); -r,r); \ x J \\ 0, x ££( (-r,r), -r,r), (i, ,r); f i, ,xe F(e^-1\ = G ( 0(0,r); V x y £(0,r), 1\ 0, x*g(0,r), F
F
/W(rx/2)\ V
y
[ = | Il
-i, i, 0, 0,
*6(0,r); i6(-r,0)i *x *^((--rr,,rr)),,
all having compact support. Moreover, it is evident that
ll/llt = /MOlfM < +co, that is, / 6 #2?(IRn).
PD-OPERATORS
WITH REAL ANALYTIC
SYMBOLS
35
Proposition 2.1.3. The inclusion f e H%°(HT) holds, if and only if there is a positive number r < R such that supp / C BT C BR. Proof. Since the sufficiency is trivial, we shall prove only the necessity. To do this we assume the contrary: There exists an increasing sequence of domains BR^ —> BR as m —> oo such that for all m = 0 , 1 , 2 , . . .
L
lB
l/(0| J # = «m "> +00.
If we choose a sequence of positive numbers /3m such that Q0/30 + Qi/?i + • • ■ = +oo, then we can construct a function
°°
11/112,. = L «(0l/(0l a « > £ «»A» = +°°•'B*
m=0
In order to do this, we set
-ott) = E i e 2 a |a|=0 -"o
(here, as usual, £2or = £ 2 a i • • ■ £21°"), where fc0 is chosen so that a0(.Ro) > Po- Similarly, we set
«.(0= E -ie 2 a |o|=lo+l -"o
where fci is chosen so that ai(-Ri) > /?i, and so on. Finally, we set
a(0 = «<>(£) +«i (£) + ••-■ It is easily seen that a(f) is analytic in f?H and, simultaneously,
u/iit = L a(oi/(«r« oo
-
= E / m=0 ■ /B «m+l\fiK m
oo
«(0l/(0l2# > E a - ^ = +°°m=
0
This implies that / 0 / ^ ( I R " ) , which contradicts our assumption. The necessity is proved, which completes the proof of Proposition 2.1.3. Using the Paley-Wiener theorem, from Proposition 2.1.3, we can obtain the following corollary.
36
DIFFERENTIAL
OPERATORS
OF INFINITE
ORDER
Corollary 2.1.1. A function f in i 2 (IR n ) belongs to HR°(B.n) if and only if there exists an entire function f(z), z — x + iy € G?, such that f{z)\z-x = f(x) and \f(z)\
in Radyno [2] is continuously imbedded
ExPd/dtLpOR-1) =
an^nt1).
2.1.3. The classes of all polynomials Pm(x) = 1 + oix + . . . - ) - amxm, all quasipolynomials exp(i\x)Pm(x), X 6 BR, the trigonometric functions n " sins,-, fli cosi, .. .belong to our space Wjf(IR"). This fact allows us to define the Fourier transform of generalized functions in the same way as one does for the classical Fourier transform (since exp(iAa;) belongs to the test space WRa(JRn), see Section 2.3 below). Moreover, in the case R = +oo, we can approximate the space of finite smooth functions (like Lp, Ck, ...) by means of functions from W^(IR"). 2.2. Differential o p e r a t o r s of infinite order (DOIO). Let the function L(£) be expanded into the Taylor series oo
HO = E
M=o
a
« r , ( € BR, aa = DaL{0)/a\,
(2.2.1)
and assume that (2.2.1) converges for all £ € BR. We now consider the action of the DOIO with constant coefficients oo
L(D) = £
N=°
aaDa,
(2.2.2)
PD-OPERATORS
WITH REAL ANALYTIC
SYMBOLS
37
in the space W ^ ^ I R " ) . A main result of this section is the following theorem. Theorem 2.2.1. A DOIO with the symbol as above acts invariantly and continuously in W£°°{]Rn). Proof. Let f(x) be an arbitrary function in W£°°(filn). Denote by / the inverse Fourier transform of / . Then f(() is an analytic functional and supp/ C Br for some r £ (0, R). We shall show that in W£°°(IR") there exists the limit Hm Ln(D)f(x),
aaDa.
Ln(D) = £
For this purpose we define the analytic functional g(£) 6 A'[5 r ] by the formula 9(0 = £ ( £ ) / « ) . 8 u p p / c B r ,
r
Since Ln(£) —♦ L(£) in A[BT] we have
\im(U0f(0M0) = ^MfU),U(M0) =
in
A'[Br).
From Proposition 2.1.2. we obtain lim Ln{D)f(x)
= g(x) G VK+TO(IR").
n—*oo
We put L(D)f(x) proved.
= g(x) by definition, and the invariance of L(D) in W£°°{1Rn) is
Now let fn(x) -> f(x) in W£°°(Irt n ). Then there exists a number r < R such that /» -» / in A'[Br]. Consequently L ( 0 / „ ( 0 -» £ ( 0 / ( 0 i n A'[J5rJ. Applying the Fourier transform we obtain the continuity of L(D) l\mL(D)fn(x)
=
L(D)f(x).
Example 2.2.1. We consider the function u{t, x) := exp I at—
)
38
DIFFERENTIAL
OPERATORS
OF INFINITE
ORDER
where t € J R \ a € (C\ \a\ = 1 and
riy/aoo
—a2
/-v/aoo
-s2
- s)ds, t<0,
u(t,x) = ' tp(x)
t = 0,
»
■ s)ds,
t<0,
where the contour (—iy/aoo, iy/aoo) is, for example, the straight line in the plane z = x+iy joining the two points —iy/a and iy/a (similarly for the contour (—y/aoo, y/aoo)). First we suppose t > 0. Since
— —1 e 4ai v>(x -- s)ds 2( 7 rai) 1 / 2 7-^co ^
=
—
fVSoo = £ ~ ^ W ( i ) . —JT / e4a! > ,. -{-s)"ds
= £ fcj 2l*at)wUj^
S ds
? > (2.2.3)
where (^<^(x) = dk
2 ( ™^/-^ e4 ' ( - S) ^ = ji(-*W/
if it = 2m = 1, if fc = 2m(m = 0 , 1 , . ■ • ) .
where T(-) is the Gamma function. Taking into account the fact that
r(m+i) = (m-i)(~
3\ 2/
3 1 (2m-l)!V5F 2 2 ~~ 2 2 " - 1 ( m - l ) ! '
we immediately deduce from the last formula and from (2.2.3) that /■•ysoo -»2
I —
—r- /
2 ( ? r o 0 1 / 2 -/-X/SOO
e4al(-5)*ds
- £ (2-)! (a<) = ex
for £ > 0, as required.
p( a 'jy^)>
PD-OPERATORS
WITH REAL ANALYTIC
SYMBOLS
39
The computations for t < 0 are completely analogous. The fact that this integral takes the value tp{x) for t = 0 can be proved by classical arguments usually applied in the study of the Poisson integral. Finally, we note that u(t,x)
can be written in the form 1
u(t,x) =
—
ry/a sign too 7775/ 2
2(Traty' J
/
-.
-,/a sign too
-£_ e 4at mlX
—
^
s)ds, ;
or, what is the same, after substitution 5 = y/a sign i£, 1 /-oo _ J i / u(i, x) = - \fa sign <M d£. 2.3. The space of generalized functions W-°°(]R n ). 2.3.1. The definition of W^°°(Rn). We denote by WR°°(W) the space of all con tinuous linear functionals defined on WJj°°(IRn). We call the elements of W^°°(lRn) generalized functions. The space W/^°°(lRn) has all the standard properties; for ex ample: a) If h € W^°°(IR n ) then dh/dxt € W^°°(]R") can be defined by
w
(fh,tp) = (h,ftp),
for all tp £
^ff+»(I'),
Let ft € Wfl°°(]Rn) and let L(D) be a DOIO whose symbol L(() is analytic in BR. Then {L(D)h,tp) := (h,L(-D)tp), tp € W+°°(IR"). This expression is well-defined, because L(—D)tp € W£°°(lR n ) for any test function tp € w^°°(nt n ). As a consequence of Theorem 2.2.1 we have T h e o r e m 2.3.1. The space H^°°(IR n ) is invariant under differential operators of infinite order whose symbols are analytic in BR.
40
DIFFERENTIAL
OPERATORS
OF INFINITE
ORDER
From Theorems 2.2.1 and 2.3.1 we obtain Theorem 2.3.2. The set of all DOIOs with symbols analytic in BR constitutes an algebra of operators isomorphic to the algebra A(BR) of functions analytic in BR. This isomorphism is defined by the correspondence L{D) <-> L(() :
aL(D) ± PB(D) «-» aL(O±0£(O, L(D)-B(D) « L(0-B(0In particular, if Z/_1(£) is also analytic in BR, then L~1(D) := I/L(D) L{D).
is inverse to
2.3.2. Examples of generalized functions in W^°°(]R' 1 ). Example 2.3.1. If \h{x)\ < exp(—a|x|),a > 0, then h(x) determines a generalized function by the formula (h,(p):=f
h(x)
\/
(2.3.1)
A generalized function h in W^°°(lRn) is called regular if its action is representable in the form (2.3.1). Example 2.3.2. The delta function 8{x) determines a singular generalized function over W£°°(IR") by the formula <*(■),*(•)> = P(0).
This is well-defined because every ip € W£°°(IEtn) is continuous. Example 2.3.3. Consider the generalized function oo
E(t,x) := exp {at^j
6(x) =
(ak)>
k=0
-««(*), £^6™(x),
where a S Q? and t S IR1. The action of this generalized function on tp € W^ (E.1) is given by the formula
a< (E(t,-)M-)) = (H-Uxp ' e x p ([at— £ij MO) Taking into account Example 2.2.1, we obtain
W^
_ 11 r \/-v/a / o " sign
— s2
= 2UaW Lja^lJ^^ds-
PD-OPERATORS
WITH REAL ANALYTIC
SYMBOLS
41
Consequently, E(t, x) is the analytic functional determined by the function —s«2 1 -e^at 1/2 2(™i) with the contour of integration (—y/a sign t, ^a sign t). Example 2.3.4- Consider the generalized function
E{t,x) =
sinh \at4-) V^(z): a—
^J:Ur^
1
where a 6 <S?,t e ffi. . The action of this generalized function on
=
(*(•),
f(-)
dx
= {^iaiz^)dT)=isy^dT-
^
We claim that for functions tp{x) that decay sufficiently rapidly as x —* +oo (for more details, see later), this functional can be written in the form E(t, x) = —[0{x + at) - 6(x - at)], 2a where 0(x + h) is a shift (in general, complex) of the Heaviside function (6(- + h)M-))
■■= (0{-)M- - * ) > = /
¥>(* -
h)dx.
Jo
Indeed, in accordance with this definition, 1 — (0(- +at)-9(--at), la
1 t°° 1 t°° ?(■)) = — ifi(x - at)dx - — I
We take the following contour of integration
(2.3.3)
42
DIFFERENTIAL
OPERATORS
OF INFINITE
ORDER
and suppose that ip(x) decays sufficiently fast as x —► +00, so that Cauchy's theorem holds for the whole contour L = L\ U Li U L3, that is / ip(z)dz = 0. This formula is clearly equivalent to 0
/
r+00
tp(x — at)dx — / .00
rat
JO
ip(r)dT.
J—at
Comparing this equality with (2.3.2) and (2.3.3), we immediately obtain E(t,x)
= ^-[6{x + at) - 0{x - at)]. 2a
2.3.3. The Fourier transform in W^°°(TR,n). We introduce the Fourier transform h of h in WR°°(TELn) by the formula fefl~)>
= (2*)n(h(-)M-)),
(2-3.4)
where ip e W ^ ^ I R " ) is any test function and
<*(•), «r*> = (2r)-(A(.),*(--0) = (2*rM£), hence
i ( 0 = (2iO»(fc(.), «"*«),
(2.3.5)
and, in particular, 5(f) = 1. It is clear that the formula (2.3.5) defines the Fourier transform h(£) of a generalized function h € W^°°(lR n ) in the same way as the clas sical Fourier transform, and h(£) is analytic in BR. Thus, we have the following theorem. Theorem 2.3.3. The Fourier transform of the generalized function h € W^°°(IR") is an analytic function in BR. We are going to prove a structure theorem for generalized functions in
W^°°(JRn).
Theorem 2.3.4. Every generalized function h(x) e W^°°(SRn) can be represented in the form h{x) = L{D)(2y/*)-n exp(-a; 2 /4), (2.3.6)
PD-OPERATORS
WITH REAL ANALYTIC
43
SYMBOLS
where L(D) is a pseudo-differential operator with the symbol Z(£) = A(£)exp(£ 2 ) an alytic in BR. Proof. Since h € H^°°(IR n ), the function h(£) is, by Theorem 2.3.3, analytic in BR. Therefore
ko = ko • i. Applying the inverse Fourier transform to this equality we obtain h(x) =
B(D)6(x),
where B(D) is a pseudo-differential operator with the symbol B(£) = h(0- Thus, if we can prove the following lemma, then the proof of Theorem 2.3.3. will be accom plished. L e m m a 2.3.1. The delta function 6(x) over WRIJR71)
can be represented in the form
n
6{x) = ( 2 - A ) - e x p ( - A ) e x p ( - x 2 / 4 ) ,
(2.3:7)
where A is the Laplace operator. Proof of Lemma 2.3.1. Consider the Cauchy problem
^1-Ae(f,x)
= 0,
e(r,x)| ( = 0 =
(2.3.8) ao
n
6(x)eWR (m ).
(2.3.9)
The existence and uniqueness of t(t,x) follow from Theorem 3.3.1 below. From that we have e(i,x) = exp(tA)S(x), in particular e(l,x) = exp(A)£(z).
(2.3.10)
By virtue of Theorem 2.3.2. or by Theorem 3.1.1 below, from (2.3.10) we obtain 6(x)=exp(-A)e(l,x).
(2.3.11)
Now we consider the generalized function e(t,x). The action of this generalized func tion on
(2.3.12) 1
We shall calculate the expression exp(—tA)y>(x). Let ip 6 W ^ ^ I R ) (for the sim plicity we consider the case n = 1). It follows from Theorem 2.1.1. that the function u(t, x) = exp ( - < — j
44
DIFFERENTIAL
OPERATORS
OF INFINITE
ORDER
belongs to W£°°(IR n ) for t > 0. Thus, (2.3.12) is well defined. We claim that can be written in the form (Poisson representation)
m ds s)ds. u(t, » ) = * / , e-V<*V(* -- *) -
«>'*=*5sL*-* «-
u(t,x)
(2-3-13)
Indeed, since y is an entire function,
,- / ( 4 0 , ( . - s)ds = - i = / -4=/ /(«V(.-«)«fa ^ ' M^ n\v i ^dx"t - ^' U M ) e 2y/¥tJw 2y/TCt M 2
l
¥
x
K
v
;
0
2 . e-Hm {-sTds = ^£-, nii*v(.) ^ > 4 l= /i e-< /(«)(_ srds. ! da;" 2 A / S ^R =
1
dxn
^Qn\
V
2yfiTtJl&
'
Next, by the substitution s = 2y/it] we find
2^rtL^m^ds
= ^-*&
L"*™ 0,
{
n = 2m + 1, l, otherwise.
l ^ /1P ,. 'v 2(22mm- -! l()m U-/ l5)F! -F=(4*) mS 2m — , 1, ,x,
0F
2
otherwise.
- (m-l)!
Then we derive directly from the last formula and (2.3.14) that I t 2y/FtJ*>
*,m
{x_ K
1 ^d*»V(x) ™\^ dxim
s)ds ' =
e X p
(-^)^
( a ; )
'
as required. In accordance with the formula (2.3.12) we find that -s)ds. 7 e"^'(iM-s)ds.
1 < e(t, * ' *■)M-)>= * > >='2T7W»L 2(v^Fi)T
-» 2 /(4«) v ? ( _
(2.3.15) _e-x2/(«)
It follows that tit, x) is a regular generalized function defined by the function In particular, by setting t = 1, from (2.3.11) we obtain 2
•x /4 X p 6(x)-^ ) == eexp(( -AA) ) ( 2 ^ e _C l 2 / 4 (2^)" The lemma is completely proved.
■=—. 2(v/7rf)n
PD-OPERATORS
WITH REAL ANALYTIC
SYMBOLS
45
2.4. The algebra of pseudo-differential operators with analytic symbols. Let G be an open set of H ? . Definition 2.4.1. The space of test functions Wjt°°(]Rn) is the set of functions / ( x ) satisfying the following conditions: i) f(x) admits analytic continuation as an entire function to (Tf, ii)
/ ( ( ) is the Fourier-Laplace transform of an analytic functional u £ A'[IRn], with supp u C K C G, where K is a compact set.
We now turn to a description of the structure of WQ°°(]Rn). We denote by BR(X) = {£ € IR" : |£ — A| < R) a sphere centred at A completely contained in G and, in accordance with the definition of WQ°°(JR,n), we set WR(Rn)
= {f[x)
| /(C) = u(C),u € A'[JRn], supp u C
BR(X)}.
B
Then we see that f(x) g W£~(1R ) if and only if e x p ( - i A s ) / ( x ) € H^~(ffi. n ). Hence we can write symbolically ^,?(IRn)=exp(iAx)W+°0(IR"), where WR°°(]Rn) is the space of test functions in the neighborhood of zero constructed in Section 2.1. Further, it is not hard to show that any function / € WQ°°(filn) can be represented in the form
/(*) = Z]«A*(a:). where U\k{x) € WRk,xJJRn), BRk{Xk) C G and J is a finite set of indices. Indeed, if / E W j ^ I R " ) , then / £ A'[SRn] and s u p p / = K C G. We can choose the compact subsets Kk such that K = \JKk and A"fc C BRk,xk, k £ I. Then we have / = £ f c 6 / ilxk, where UA4 6 ^'[-^it] ( s e e Bjock [1], H6rmander[l]). Consequently / = T,kelu*k-> where «** e W+^ITBT). Definition 2.4.2. A sequence {/„} is said to converge to / in W^°°(W) exists a compact set L C G such that
if there
/„ -» / in AIL]. Now let L(£) be an arbitrary complex-valued function that is analytic in G. It is possible to choose BRk{Xk) so that in each BRk{Xk) the function L{t) can be expanded as the Taylor series oo
Hi) = £ <•«(**)« - A *) a , * e /, £ e fl*(A*). |c|=0
46
DIFFERENTIAL
OPERATORS
OF INFINITE
ORDER
For any function u G Wg°°{TRn) we have "(z) = I > A t ( s ) , fce/ and we define the action of L(D) on u(x) by oo
L(D)u(x)
:= E
E
o*(*k)(D -
XkI)auXk(x).
kel |a|=0
Here I is the identity operator. By Theorem 2.2.1 L(D)u(x) is again a function in Wg°°(]R n ). Moreover, it can be shown by using the Fourier-Laplace transform that this definition does not depend on the number of functions u\k representing the func tion u(x), that is the action is well-defined. We denote by (W<*°°(nr))* the space of all continuous linear functionals on W ^ I R " ) and we set W£°°(IR n ) = (W+^IR"))*. Let L(() be a function analytic in G. We assign to it a pseudo-differential operator L(D) acting in W G ° ° ( ] R " ) according to the formula < L(D)h(-)M-) >■■ = < h(-),L(-D)9(-) >, where h € Wa°°(JRn), and ip £ W^°°(]R n ). The definition is correct since L(-D)p € H^+^(IR n ) for y e W£°°(lRn). Applying Theorem 2.3.2. we obtain the following re sults. Theorem 2.4.1. A PD-operator L{D) with symbol L{Q analytic in G acts invariantly and continuously in WQ^QR71) (and hence also in WQ°°(JRn)). Theorem 2.4.2. The set of operators L(D) with symbols L(£) analytic in a domain G and defined on WQCO(TRn) forms an operator algebra that is isomorphic to the algebra of functions analytic in G. This isomorphism is defined by the correspondence L(D) <-> L(Q. Here
aL(D)±pB(D)
~
L{D)-B(D)
~
aI({)±/JB(0, L(£)B(t).
In particular, if L~l(£) is also analytic in G, then L~1(D) := I/L(D) inverse to L(D). For any function L(£) analytic in G the maps
is the operator
L{D) : W(£0°(IR.n) -> Wg°°(JRn) are continuous. This statement remains valid if everywhere the upper index +oo is replaced by —oo.
PD-OPERATORS
WITH REAL ANALYTIC
SYMBOLS
47
Bibliographical N o t e s . The important paper on the algebra of real differential op erators of infinite order is that of Dubinskii [4], where the duality (#g>(]R n ), #o°°(IR n )) is studied in connection with the action of differential operators of infinite order. The spaces W(£°°(]R") and ^ " " ( I R " ) are introduced by Tran Due Van [14] based on the duality (A[G], A'[G]) (see also Tran Due Van and Dinh Nho Hao [1]). In the pa pers by Trinh Ngoc Minh and Tran Due Van [1], Trinh Ngoc Minh [2,1], Tran Due Van [13] the duality of the Schwarz space (.5s*, .9**') is used to construct the spaces W+°°{G) and W-°°(G) (the space VK+°°(G) is a subset of the space W ( * 00 (]R n )). The interesting property of WJ°°(IR") is that F(Wj°°(]R n )) = C°°(G), i.e., the Fourier transform of a generalized function from Wj°°(]R n ) is a function infinitely differentiable in G. The theory of analytic PD-operators in infinite-dimensional spaces and Vladimirov-Volovich super-analysis is given in Khrenikov [1,2]. The spaces of vectors of exponential type of Radyno [1,2] are also closed to our spaces Wg w (lR°). The material in this chapter is taken from Tran Due Van [14].
Chapter 3 A P P L I C A T I O N S TO P S E U D O - D I F F E R E N T I A L E Q U A T I O N S
In this chapter we shall give some applications of the results in Chapter 2 to pseudodifferential equations and their Cauchy problems, and boundary value problems, etc. The algebra of differential operators of infinite order constructed in Chapter 2 can be regarded as an operational calculus of the classical Heaviside type ( 0 . Heaviside [1]) for the case of differential, pseudo-differential and functional equations in several variables. Section 3.1 is devoted to the solvability of pseudo-differential equations L(D)u(x)
= h(x),
x £ IR",
(3.1.1)
with the symbol L(£) analytic in some domain G of 1R" Thanks to our algebra of differential operators of infinite order, the solution of this equation can be obtained by purely algebraic means and can be written in the form
u(x) u{x)= -
I
h{x) L(D)L(D) >
h(x),
where I/L(D) is the pseudo-differential operator with the symbol 1/£(£). We also show that, when the symbol L(£) has singularities in G, the Fredholm property of the operator L(D) depends on the set of zeros of the symbol £(£). In Sections 3.2-3.4 we study the Cauchy problems for pseudo-differential equations, whose symbols are analytic in a domain G, and establish that they are well-posed within the framework of the (Wg°°(lRn), W^°°(IR n ))-theory. In particular, the ex istence of a fundamental solution of the Cauchy problem in the space of generalized functions valued in VK(J0°(lRn) is proved. For ordinary pseudo-differential operators, the correctness of the Cauchy problem depends on the number of zeros of the symbols. In Section 3.5 we show that many boundary value problems in a strip (a, 6) x IR" can be reduced to the Cauchy problems for systems of pseudo-differential equations studied in Sections 3.2-3.4. Section 3.6 is devoted to multi-dimensional integral equations of the first kind with entire kernels. Such equations arise, for example, in the study of inverse problems of mathematical physics (see, for example, Anikonnov [l]-[3]). Here we shall apply the technique of PD-operators with real analytic symbols to integral equations of the first kind by reducing them to differential equations of infinite order. The technique of
48
APPLICATIONS
TO PSEUDO-DIFFERENTIAL
49
EQUATIONS
replacing integral equations by differential equations has been used by many authors (see, for example, Anokonov [1], Zabreiko et. al. [1]). The purpose of Section 3.7 is to introduce a method for solving functional and functional-differential equations based on the theory of pseudo-differential equations with real analytic symbols. Finally, in Section 3.8 we study the equation (3.0.1) for the case when the symbol Z(£) may vanish in the domain G by constructing a subalgebra of the algebra of PD-operators with real analytic symbols acting in function spaces with weights. 3.1. Problems in the whole Euclidean space. Let L(D) be a pseudo-differential operator with analytic symbol £(£). We consider the equation L(D)u{x)
= h{x),
xelRJ.
(3.1.1)
The results of Chapter 2 have the following consequence. Theorem 3.1.1. Suppose that L(£) and £ _ 1 (£) are analytic in some domain G C K?. Then for any h € W Q ° ° ( I R " ) Eq. (3.1.1) has a unique solution U{X) =
ljDJh{x)-
(3 L2)
-
Example 3.1.1. We consider the Helmholtz equation (u> is a complex parameter) Au(x)+u2u
= h{x),
xeJrT.
(3.1.3)
Note that the symbol of the Helmholtz operator and the inverse operator / / ( A + u 2 I ) are analytic for £2 ^ w 2 , £ € IRn. Hence, the whole space IRj? is the common domain of analyticity of the symbols of both operators, except when u> is real. The space Wj[E°(IRn) consists of all functions h(x) that admit an analytic continuation as entire functions and are the Fourier-Laplace transform of h € A'[IR"]. Consequently, for any such function we find that the solution of (3.1.3) is given by
u(x) = -^-JjH*)-
(3-1-4)
However, when w is real, then G = IR£ \ 5 , where S is the sphere £2 = u>2- Hence in this case (3.1.4) yields a solution of (3.1.3) for any function h(x), h C A'[lRn], supp h C G. Example 3.1.2. We consider the problem of the existence of a fundamental solution for the operator L(D), that is, the solvability of the equation L(D)S{x)
= 6{x),
x£jRn.
50
DIFFERENTIAL
OPERATORS
OF INFINITE
ORDER
Suppose that the domain of analyticity of the symbol L({) and L'1 (£) is not empty, G yt 0. Then clearly <$(•) € Wo°°(Irl n ); therefore, the fundamental solution €(x) exists as a functional on WQ°°(TR.n) and is given by
eis) = j ^ M . Example 3.1.3. We consider the equation with shifts u(x + l) + u{x-l)
= h(x),
xeIR1.
(3.1.5)
Using Taylor's formula, we can write this as a differential equation of infinite order. We set 2 c o s h [ — \u(x)
= h{x),
zeIR1.
(3.1.6)
Since cosh(i£) ^ 0 for £ ^ TT/2 + kit, k = 0, ± 1 , . . . , we find that (3.1.6) or, what is the same, (3.1.5) for any h G W£°°(1R}), where G = IR1 \{TT/2 + kir\k € ZZ}, has a unique solution u(x) = -sech I — ) h(x). If h € W ^ . ^ I R
1
) (i. e. supp h(0 C (-7r/2,7r/2)), then U
(
3
:
)
=
2 £ - ( 2 ^ ^ ^
where the Eim are the Euler numbers: EQ = 1, -Eg = — 1, E+ = 5, etc, (see, for example, Abromowitz and Stegun [1, p. 810]). In particular, if h(x) = Pm(x), where Pm{x) is a polynomial of degree m > 0 ( it is clear that Pm(-) e W(i~ /2 ^ ( I R 1 ) ) . Then the solution
u(x) = sech
^ (i) Pm(a; )
is also a polynomial of the same degree. Example 3.1.4- We consider Eq. (3.1.1) with h(x) = Pm(x)exp(i\x), where P m (x) is a polynomial and A € G is arbitrary. We know that h € Wo^lR"). Hence 1
°°
U
W = 77mM^)= E 6 «(A)p-A/r[P m (x)ex P (ax)] 00
=
e x p ( a x ) 5 3 ba{\)DaPm(x)
:=
exp{i\x)Qm{x),
APPLICATIONS
TO PSEUDO-DIFFERENTIAL
EQUATIONS
51
where Qm(x) is a polynomial of degree m. Now, let us discuss the equation (3.1.1) when the symbol L(f) has singularities in G, h 6 W ^ ^ I R " ) . Applying the Fourier transform to both sides of (3.1.1), we obtain
mHo = ko,u^h Let u € W£°°(lRn)
(3.1.7)
be an arbitrary solution of (3.1.1). Then supp fl(() C supp fc(f) U {f € G,L(f) = 0}.
Hence, if fc(f) = 0, then supp 6(f) C {f € G | 1(f) = 0}. We denote the set {f e G | 1 ( 0 = 0} by 0 ( 1 , G). We recall that a linear continuous operator $ : X —> Y is called a Fredholm oper ator, if dim Ker $ < +oo, dim Coker $ < +oo and Im $ is closed in Y. Clearly, Im L(D) = H^°°(IR n ) for any 1(D) with symbol analytic in G. Therefore, these operators are Fredholm operators if and only if dim Ker L(D) < +oo. Proposition 3.1.1. If the operator L(D) with symbol analytic in G is a Fredholm operator in Wg°°(IR'1), then the set 0(L,G) is finite. Proof. Suppose that L(f) has an infinite number of different zeros fijfg,... in G. Then the generalized functions 6(£ — fi), 6(£ — f 2 ), • • • are linearly independent. Since L(h) = 0, we have (L(-)S{- -&),
HQSit) = 0, it follows that supp5(f) = {0}. Consequently, 5(f) can be represented in the form
5(f) = E ^ ( , ) ( 0 , «.ea?. •=0
Thus, we have dim Ker L(D) < +00.
52
DIFFERENTIAL
OPERATORS
OF INFINITE
ORDER
For the case n > 2 Proposition 3.1.2. is no longer correct. We consider the counter example constructed in Trinh Ngoc Minh [2, p. 112] . Let n = 2 and L(£) = l/2(£i + £2)- It is clear that 0 is a unique zero of £(£). Hence, for all k > 2 there exist numbers ca ^ 0 such that
S(t) = £ «-*
d"'"^'30
+ Y,Ljll(t,d/dt,D)uk(t,x)
= hj(t,x),
(3.2.1)
k=i
'
dkuj{0,x)/dtk
=
where
m
j
Lik{t,d/dt,D)= £
_ 1
(3.2.2)
ai
iy^-D)^,
the mj > 0 are integers, the Lijk(t, D) are arbitrary pseudo-differential operators, and for each t the symbols £,-,•*(*,£) are analytic functions of £ in some domain G C 1R? that depend continuously on t C IR 1 . Denote by W|°°''(]R n ) the space of vector-valued functions u(x) = (u\(x),..., ui(x)), m G Wg"(K!»),i = 1,...,/, and by C*1 k'(JR},W£°°'t(JRn)) the space of vectorvalued functions u(t,x) = ( u i ( t , i ) , ...,ue(t,x)) which for each i G IR1 are vectorvalued functions in WG°°' (IRn) with Ui(t,x) depending continuously on t together with their derivatives up to order &,- (i = 1, ...,£), respectively. Theorem 3.2.1. Let
Cmi
m
'(lR\W2 00 ''(]R n )).
Proof. For simplicity of exposition we consider the case £ = l,rrij = m. First of all, we observe that by Duhamel's principle it suffices to consider the case h(t,x) = 0. Indeed, if U(t, s, x) is a solution of the problem Omrj
m-l
air;
L(d/dt,D)U := ^ + E W > ) | £ = 0,
APPLICATIONS
TO PSEUDO-DIFFERENTIAL
^ ° - ' k' « ) dt am-1f/(o115ix)
EQUATIONS
53
= 0, * « 0 , l , . . . , m - 2 , h(s,x),
dt™-
where 5 £ (0,<) is arbitrary, then the solution u(2,x) of the problem L(d/dt, D)u(t, x) = h(t, x), dku(0, x)/dtk
= 0, k = 0 , 1 , . . . , m - 1,
is given by the formula u(<,x) = / U(t — s,s,x)ds. Jo Taking into account this remark, instead of Problem (3.2.1)-(3.2.2), we can consider the following Cauchy problem L(d/dt,D)u = ^+m£Lk(t,D)^=0, D)™ dku(0,x) QtH
' dtk
= o,
(3.2.3)
= ¥>*(*), fc = 0 , l , . . . , m - l ,
(3.2.4)
where
= 0, = Sik ( 0 < J f c , j < r a - l ) ,
u\k)
~dk/dthui.
Here 8^ is the Kronecker symbol and £ is a real parameter. Since the Lk(t, £) depend analytically on £ in G, each u,(t, £) is an analytic function of f in G. We assign to each such "basic" solution Uj(£,£) a DOIO Ui(i, £>), whose action is continuous in WQ°°(IR,"), in accordance with Chapter 2. Clearly, the formula m
u(t,x) =
^2ui(t,D)tpi(x) i=0
then determines the required solution. To prove the uniqueness of the solutions of the problem (3.2.1)-(3.2.2) we note that the Fourier-Laplace x-transform «(£,£) of the solution u(t,x) is a solution of the Cauchy problem for the system of ordinary differential equations valued in the space of hyperfunctions and is therefore unique. Hence, so is u(t,x). This completes the proof.
DIFFERENTIAL
54
OPERATORS
OF INFINITE
ORDER
3.3. T h e Cauchy p r o b l e m in t h e space Wj°°(]rt n ) a n d its fundamental so lution Theorem 3.3.1. Let tpk e W£00'<(IR'1) and h{t,x) € C° °(IR\ WG°°''(IR n )). Then there exists a unique solution of the Cauchy problem (3.2.1)-(3.2.2) in the space Cmi m'(IR\WG00''(nr)). Proof. As in the proof of Theorem 3.2.1, for simplicity we consider the case when £ = l,rrij = m. By Duhamel's principle it suffices to consider the case h(t,x) = 0. Namely, we prove that for any
u(t,x) -=j(t,D)^,Uj(t,D)ipj{x) u(t,x)='£u Vi(x)
(3.3.1)
3=1
gives the generalized solution of Problem (3.2.1)-(3.2.2) (recall that Uj(t,D) are the PD-operators whose symbols Uj(t,£) are the solutions of the initial value problem for the ordinary differential equation
L(d/dts()uj(t,{) = o, «f(
is a PD-operator) is a solution of the homogeneous equation fpn
L(d/9t, D)w = ^
m-l
+
g
Qfc
Lk(t, D)£g
= 0,
then the function w'{t,x)
= U{t,-D)<j>{x),
with <j> e W~ G (IR n ),
satisfies the equation
L'{d/dt,D)w'
:
=
f)m..,m
dtm 01
m-l
+ £ M*. k=0
-"£"• /)*,„*
Consequently, for any tpj € WG"°°(lRn) and for any v € Wffc(]RB) we have < L ( a / 0 t > 0 ) u i ( t , 0 ) w , t , ) = {iphL*{B/dt,D)u4{t,-D)v)
= 0.
This means that the function (3.3.1) is a generalized solution of the original equation (3.2.1).
APPLICATIONS
TO PSEUDO-DIFFERENTIAL
55
EQUATIONS
Now, we verify that this solution satisfies the initial conditions (3.2.2) over the space W™G(]Rn). Indeed, in accordance with the construction of the operators Uj(t, D) (see the proof of Theorem 3.2.1) we have dk
dT^j(t,D)\t=0-- = hjl, where / is the identity operator in W(f(]R"). Hence dk -^Uj[t,-D)\t=o
= 6kjI,
where / is the identity operator in W™G(ttln). Consequently, for any v 6 W™G(JR.n), \JftkU3(t,D)w,v\
\t=o = U>j,-^v.j{t,-D)v\
|(=0 =
(
that means ^u(*,x)|(=o =
Vk(x)
in We°°(IR n ).
Thus, the existence of the solution of Problem (3.2.1)-(3.2.2) in C m (lR 1 , ^ " " ( I R " ) ) is proved. In order to prove the uniqueness of the generalized solution we note that the Fourier transform h of any functional h € WG°°(JRn) is a function analytic in G (Theorem 2.3.3). Further, if u(t, x) € C m (lR 1 , W^°°(IR")) is a generalized solution of the Cauchy problem (3.2.1)-(3.2.2), then its Fourier transform u(i,£) satisfies the problem m-l
« W ( « ) + E £ k ( « ) « w ( « ) = M',0, u«(0,O =
MO^
and therefore is unique. The theorem is completely proved. We now consider the problem of the existence of a fundamental solution of the Cauchy problem. The fundamental solution of the Cauchy problem for the opera tor L(d/dt,D) (3.2.3) is a function £(t,x) that solves the problem L{d/dt,D)£(t,x) £(0,z) = 0,...,£< m - 2 >(0,x)
= =
0,r^0, 0,S(m-V{0,x)=6{x).
If we know the fundamental solution of this Cauchy problem we can obtain (at least formally) a solution of the Cauchy problem with arbitrary initial conditions
56
DIFFERENTIAL
OPERATORS
OF INFINITE
ORDER
>Pk{x), 0 < fc < m — 1, by means of the operations of differentiation and convolution. Since S(-) € W^°°(IR") we deduce from Theorem 3.3.1 the following corollary. Corollary 3.3.1. For any operator L(d/dt,D) of the form (3.2.3) with symbols Lk(t,0 analytic in G the Cauchy problem has a unique fundamental solution with values in W£°°(]rt n ). 3.4. The Cauchy problem for ordinary pseudo-differential equations. Let L{D) be a pseudo-differential operator with the symbol L(£) analytic in G C E 1 which has a finite number of different zeros in G. Theorem 3.4.1. Let £1,^2, ■ ■-,£m be the zeros of L(£) in G with their multiplicity ni,ri2,... ,nm, respectively. Then there exists a unique solution u € WQ°°(JRl) of the Cauchy problem heW^iJR1),
L(D)u(x)
=
h(x),
(
=
Ci, a €
u ''(x 0 )
(3.4.1)
i = 0, l , . . . , 7 i - 1, n = n\ + ... + nm.
(3.4.2)
Proof. By virtue of Proposition 3.1.2 it follows that any solution of (3.4.1) can be represented by the formula m
njt-l
u(x) = uo(i) + £
£
Ajye^i*',
fc=0 J=0
where u0(x) is a solution of (3.4.1), and the Xkj are arbitrary numbers. We shall show that if u(x) satisfies (3.4.2) then the numbers Ajy are uniquely defined. In fact, rewrite (3.4.2 ) in the form
"v
Vi(so)
Ci —
A2
.^r%0) '.■
^ ' ( z o ) .
UQ(X0)
c2 - ul(x0)
(3.4.3)
5
/ \ Lc„-uJ,( 1 - 1 )'(rc 0).
A„ .V
where
^(x) = e'^,...,Vni(a;) = e^x"'-1, ¥>«,+!(*) = e 6 * . . . , ipni+n2(x)
=
.,tpn{x) =
t^x«*-\ ixU nm e x -\
Ai = A10, A2 = A n , . •• > A ni = A i > n , _ i , . . . , An = A m n
1.
APPLICATIONS
TO PSEUDO-DIFFERENTIAL
EQUATIONS
57
Clearly, Vn(Xo)
Vi(zo) ■
det L^i
'(x0)
.
^'(xo)
to.
• vtrvfav
Hence (3.4.3) has a unique solution ( A i , . . . , A n ). 3.5. Boundary-value problems. In the strip (a, 6) x IR" we consider the boundaryvalue problem dmu(t,x) dtm ^ , s-Q
v=\
^bjs
d'u(a,x)
+ J2L,(t,D) j=o
„;; ; = o , t e (a,6),sent"
a t
(3.5.1)
=
=
i>i(x),
u l
d-u{b,x)
at'
j =m-l,
...,m-l,
x G IR",
(3.5.3)
where the Lj(t,D) are PD-operators with symbols Lj(t,£) analytic in some domain G 6 IR" and depending continuously on t € IR1, and \\bjs\\ is a non-singular numerical matrix. We shall show that this problem may be reduced to a Cauchy problem for a system of pseudo-differential equations with initial conditions that are, in their turn, solutions of some systems of differential equations of infinite order. Consider the family of Cauchy problems for the systems of ordinary differential equa tions with a real Darameter £ G G:
dmUok(t,0 dtm
(3...5.4) j=o
vJk(t,o -- ^ Vjk{a,Q
----
u l
d>uok(t,0
Bih,
(3.5.5) (3.5.6)
where j , k = 0,1, ...,m — 1. These problems have unique solutions Uok, V}k depending analytically on £ (Kamke [1]). Further, we form the equations J ] Vj,(b,D)
(3.5.7)
DIFFERENTIAL
58
OPERATORS
OF INFINITE
ORDER
where Vjs(b,D) are PD-operators with symbols Vjs(b, £). We rewrite (3.5.7) in the form m—1
£
m—l—1
Vi,(b,D)
£
s=m—/
Via(b,D)
(3.5.8)
3=0
Then (3.5.8) is a system of / pseudo-differential equations with analytic symbols, where the fs(x), s = m — /,..., m — 1 are unknown functions. T h e o r e m 3.5.1. Letipj,j = 0,1, ...,ra — I — 1, ^ , j = m — (, ...,m — 1, belong to the space Wo°°(lR.n). Then the solution of the problem (3.5.1)-(3.5.3) can be represented in the form m—l
u(t, x) = £ 0E-*(*, D)^(x)
(3.5.9)
fe=0
and belongs to the space C m (IR 1 , WQC°(JR.n)), where the ifik, k — m — I, ...,m — 1 are solutions of the system (3.5.8). Conversely, if u(t,x) is a solution of the problem (3.5.1)-(3.5.3), then the functions •E? dku{a,x) Vs=2~,b'k—^Tk—'m
s=
m-l,...,m-l,
k=o
are solutions of the system
(3.5.8).
Proof. The fact that (3.5.9) satisfies (3.5.1) and (3.5.2) is a consequence of (3.5.4) and (3.5.5). We shall show that (3.5.9) satisfies (3.5.3). Indeed, using (3.5.7), we have for j = m — /,..., m — 1 ■E*
^
is
d'u(b,x)
at' Ul
1=0
^ ^ V
d>UQk{b,D)
.
= 2 - 2 - h>—or*—■vw) fc=0
3=0
°l
m-1
=
E^.( f c .^)v.(*)=>i(i).
3=0
To prove the second part of Theorem 3.5.1 we observe that m—1
/m-l
x)\ ,(b,D) £ ftEjUb,D)(^fb =Vi(*), j = m- I,...,m-1, ! sk^^j at*
a=0
\t=0
follows from the fact that
u(t,x) = 2^ u0.(t,D)l This completes the proof.
22b>>< Qtk
1 ■
APPLICATIONS
TO PSEUDO-DIFFERENTIAL
59
EQUATIONS
3.6. Multi-dimensional integral equations of t h e first kind with entire kernels. We shall consider the multi-dimensional integral equations (£ti)(i) = /
k(x, y)u(y)dy = f(x),
(3.6.1)
where u is unknown, / is given, i € IR", and k(x,y) is a complex-valued kernel. First, consider the case when k(x,y) = k{x). Denote by A'(£) the Fourier transform of the kernel k(x). Assuming that the function K(£) is analytic in some domain G C IR£ and the convolution theorem is applicable, we obtain from (3.6.1) K(OHO
= /(«•
(3.6.2)
Applying the inverse Fourier transform to (3.6.2), from the results of Chapter 2, we have K(D)u{x) = f(x), x e AT, (3.6.3) where K(D) is the pseudo-differential operator with the symbol A'(£) analytic in G. Conversely, if A"(£) is analytic in G and u(x) is a solution of Eq. (3.6.3) then u(x) satisfies Eq. (3.6.1). We have shown in this case that Eqs (3.6.1) and (3.6.3) are equivalent. Thus, as an immediate consequence of Theorem 3.1.1 we have Theorem 3.6.1. Let K(£) be analytic in the domain G C Ht|? and K(£) ^ 0 for f € G. Then for any f € W Q ° ° ( I R " ) there exists only one solution u(x) in Wc°°(IR n ) of Eq. (3.6.1). Furthermore, it can be written in the form U{X)
= A^)/(X)'
where I/K(D) is the pseudo-differential operator with the symbol l/K(() analytic in G. This statement remains valid if everywhere the upper index +oo is replaced by — oo. Now, we consider the case when the function K(£) is an entire analytic function, i.e. K(£) can be expanded into the Taylor series Kit) = £
o-f,
«- € <E\ f € R J
(3.6.4)
whose domain of convergence is the whole space IR£ . Suppose that the function K(£) is an entire elliptic symbol, that is K(t) is an entire analytic function of the form (3.6.4) and tf(f) > A' m (f) := E|S|=o a «f a > 0 for f € IR{n. We can rewrite Eq. (3.6.3) in the form x £ aaDau{x) l»|=o
= f(x),
x € IRn,
(3.6.5)
60
DIFFERENTIAL
OPERATORS
OF INFINITE
ORDER
and in this case the equations (3.6.1) and (3.6.5) are equivalent. Invoking Theorem 3.6.1 we deduce that for any / € W^(JRn) there exists only one solution of Eq. n (3.6.1) in W^S°(lR ). Let / now belong to Z/2(IR"). A function u(x) is called an L2—solution of Eq. (3.6.1) if u(x) satisfies Eq. (3.6.5) in the L2—sense, i.e. the series oo
£ aaDau{x) l"l=o
(3.6.6)
converges in the metric of the space ^(IR™). Further, set
W~{K(t),HI") := {«(*) | \\u\\l{i) = I #(0 a |fi(0| 2 # < °o}.
(3.6.7)
Theorem 3.6.2. Let K(£) be an entire elliptic symbol. Then for any f 6 i 2 (!R n ) there exists a unique L2—solution of Eq. (3.6.5) in the space W°°(K(£),]Rn). Proof. We define the solution u(x) of Eq. (3.6.5) by the formula u(x) == F~ * >
—
rfHO)
U(0/' ( $ )
■
_1
where F is the inverse Fourier transform. We have to show that u(x) is an Li—solution of Eq. (3.6.5). In fact, using Parseval's equality, we get
ll/-| o -^IIL = / R „l/(i)l 2 (i-^f) 2 ^. In view of the ellipticity of the entire symbol K(£), we have for any e > 0
" ~ I.-"'* s . L
l/(£)|I +
« h* (' - w)1 l/(f **< '•
if R and m (m = m(R)) are sufficiently large. It follows that the series (3.6.6) converges in the space Z/2(IR") to the function / . Thus, u(x) is a unique L2—solution of Eq. (3.6.5). It is easily seen that
JTR" JlR"
1/(01** < oo. K3(om)M --Jm." Jm."
Thus, u(x) belongs to W°°(K(£),JRn).
The theorem is completely proved.
APPLICATIONS
TO PSEUDO-DIFFERENTIAL
EQUATIONS
61
Now, we consider the multi-dimensional equation ^
k(x, y)u{y)dy = f(x),
xemr.
(3.6.8)
We show that the investigation of the integral equations of first kind (3.6.8) under some assumptions can be reduced to the solution of pseudo-differential equations with analytic symbols. For this purpose we rewrite Eq. (3.6.8) in the form /
K(x,y)e-^u(y)dy
=
f(x),
where K(x,y) = k(x,y)e'xy. Suppose that for every fixed i the function K(x,y) is an entire analytic function with respect to the variable y. Expand K(x,y) into the Taylor series in a neighborhood of the point y = 0 : oo
K(x,y)=
£
aa(x)y".
(3.6.9)
|«|=0
If u(x) (the Fourier transform of the solution u(y)) is infinitely differentiable and the differentiation under the integral sign is legitimate, it is easily seen that the function u(x) satisfies the differential equation of infinite order oo
K(x,D)u(x)
= Y, aa(x)D°'u(x)
= f(x).
(3.6.10)
|c|=0
Suppose further that the coefficients aa(x) of the Taylor expansion (3.6.10) have the form aa(x) = aa(l +ea(x)), where aa are certain numbers and ea(x) are certain functions of the variable x. Assume that the following conditions are satisfied: 1) K(y) = Z)u|=o a a2/ a ' s
an
entire elliptic symbol.
2) £(y) = | sup tt £a(y)\ belongs to ioo(]R"), where £a{y) is the Fourier transform of ea(x), and \\e\\La, < 1. In the case that the kernel k(x, y) has the form k{x,y) =
k(y)e-ix\
i.e. / is the Fourier transform of (uk)(y), and the differential equation (3.6.10) has constant coefficients aa, we invoke Theorem 3.6.1 to deduce that if k(y) is analytic in some domain G C K^ then
62
DIFFERENTIAL i) for any / € W£°°(tiln) A'[IRn], supp uCG;
OPERATORS
OF INFINITE
ORDER
there exists a unique solution u of Eq. (3.6.8), u €
ii) for any / € Wc°°(IR n ) there exists a unique solution u of Eq. (3.6.8), where u(—x) belongs to A[G\, i.e. u(x) is analytic in (—G). We now turn to the general case. Denote by L2,K()0&n) the space of functions u(y) such that / K{yf\u{y)\2dy < +oo. Theorem 3.6.3. Let the conditions 1) and 2) be satisfied. Then for any function f € Z/2(lRn) there exists a unique solution u(y) in the space Z/2,A'(.)(IRn) of Eq. (3.6.8). Proof. We have to show that for any / £ L2{Wl) there exists only one solution u(x) in W°°(V(y),]R")ofEq. (3.6.10) (see Theorem 3.6.2). Since u(x) is in W°°(K(y),Rn), we have
I' K(y)2\Hy)\2dy < +oo.
Jnn Consequently, u(y) belongs to L2,/f()(lR-n)-
Let us consider Eq. (3.6.9). Rewrite it in the form K{x, D)u(x)
=
K(D)u(x)
+
Kt{x,D)u{x)
= fix),
(3.6.11)
where K(D)u(x)
:=
^
aaDau(x),
(3.6.12)
:=
5 3 aaea(x)Dau(x). l«l=o
(3.6.13)
OO
K^x^Mx)
Since K(y) is an entire elliptic symbol (the condition 1)), one sees, by virtue of Theorem 3.6.2, that the operator (3.6.12) K(D)u(x):
W°°(K(y),m.n)
^
L2(Mn)
is an isometric isomorphism. Consequently, making the replacement u(x) = K~\D)h{x),
h(x) e L2(m.n)
in Eq. (3.6.11) we get an equivalent equation in /^(IR") h(x) + Lh(x) = f(x),
(3.6.14)
APPLICATIONS where L :=
TO PSEUDO-DIFFERENTIAL
63
EQUATIONS
K1(x,D)R-1(D).
To study the operator Ki(x,D),
we need the following lemma
Lemma 3.6.1. Let condition 2) be satisfied. Then the operator Ki(x,D) is a con tinuous operator from W°°(K(y),m.n) to L2(Mn) and its norm \\Kt\\ satisfies the inequality \\Ki\\ < 1. Proof. From Parseval's equality and the ellipticity of the symbol K(y) we have OO
WK^D)^
=
a
a£a(y) *Dau
E
L2
<
II °°
ar
\\E «L&(y)\(--yrn--y)\dy m
\\\a\=o
i-2
00
<
! £ JU \a\=o
= \\[
\i(y)K(--yr\u(--y)\dy r.„
£(y)K(--y)\u(.-y)\dy L,
=
\\e(y)*K(y)\u(y)\\\^
<
I^IILOO • ll"lk(v)-
Here we have used the fact that the function £(y) is a convolutor in L2(IR") if and only if £ € Z,oo(IRn) (see Hormander [3]). The lemma is proved. According to Lemma 3.6.1 the operator Ki(x,D) is acting continuously from the space W°°(K{y),mr) to L2{JRn) and \\KiW < 1. Consequently,
m<\\Ki\\-\\K-l\\
£ o.ku{x + hk) = g(x), x e n r , k=0
(3.7.1)
64
DIFFERENTIAL
OPERATORS
OF INFINITE
ORDER
and of functional-differential equations rtmii
m _ 1
19*7/
^ + i ^ ( ^ + W = j ( M ) ,
telR1, x€lRn,
(3.7.2)
and their Cauchy problems. Here ak G d?, hk S IRn, fc = 0 , 1 , . . . , m, u is an unknown complex-valued function of real variables and g is a given function. 3.7.1. First we consider functional equations of the type m
J2 aku(x + hk) = g(x), x e
ffi",
(3.7.3)
k=0
where ak € Q?, hk € IR", fc = 0 , 1 , . . . , m, u(x) is an unknown complex valued func tion of n real variables x = (xi,..., xn), and g(x) is a given function. We set, by definition, oo
La
:= Y' —da/dxa. Q! \°\=o First, let u(x) be an entire function. Then Eq. (3.7.3) can be written in the form exp (hd/dx)
m
J ] aA exp{/ifcd/dz}"(z) = 5(1), fc=o or L(a, A, d/dx)u(x)
= g(x),
where m
L(a, /i, 9/9x) := 5 3 a * exp(ft*d/cfo), k=0
is a pseudo-differential operator with the symbol L(a,h,i()
:=
^akexp(ihk£). k=0
Notice that for any complex numbers a0, ...,am and hk £ IRn, k = 0 , . . . , m the function L(a, h, i£) is a real analytic function of the variable £ in the whole space H? . Denote the set {£ e JR£ | L(a,h,i£) = 0} by 0(£) and the set IR" \ 0 ( I ) by 0 . Theorem 3.7.1. For any g € W^°°(TRn) there exists a unique solution u € W„ °°(]Rn) of Eq. (3.7.3). Furthermore, this solution can be written as U{X)
= L(a,h,d/dx)9{x)-
(3-7-4)
APPLICATIONS
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EQUATIONS
65
Proof. We see that both operators L(a, h, d/dx) and I/L[a, h, djdx) are pseudodifferential operators with symbols L(a,h,i£) and l/£(tt, h,i£) respectively, which are analytic in fi. With this fact in hand, we can apply Theorem 3.1.1 and the proof is straightforward. Remark 3.7.1. It is known that all polynomials belong to VKf|co(IRn) for any domain Cl C IR£, fi ^ 0. Let now g(x) be a polynomial. Since 1/L(a,h,i() is analytic in ft, it can be locally expanded in the Taylor series, and (3.7.4) can be represented by the formula oo
u x
i)
9c{a,h,^o)da/dxag(x),
= Yl \a\=0
where ga{a,h:(o) are certain coefficients, £o is a certain point in ft. It is clear that this series terminates and reduces to a finite sum. We have thus established that the solution of (3.7.3) for any polynomial g(x) is also a polynomial in the variable x. This agrees with the results obtained by Baker [1]. Let us consider some concrete equations of the type (3.7.3). Example 3.7.1. The case m = 0 : L(a, h, i£) = a0 exp(ih0£) ± 0, V£ <E IR£, a0 ^ 0. We observe that ft = IR|? and u(x)
{aoexp{h0dldx)~lg{x)
= 1
— OQ exp( — h0d/dx)g(x) =
a.Q 9{x - h-a)-
It is easily seen that the spaces W^?°(IR") are dense in function spaces of finite smoothness like Ck{JRn), Lp(Mn), Sobolev spaces W£(W) etc. ...Therefore, the right-hand side in the last formula is valid for functions g(x) belonging to function spaces of finite smoothness. Example 3.7.2. The case m = 1, a0 = - a i ^ 0, h0 ^ hi: a0u{x + h0) + aiu(x + hi) = g(x). The operator on the left-hand side of (3.7.5) has the symbol L(a,h,£)
= aoexp(ih0Z)
+
aiexp(ih^).
(3.7.5)
66
DIFFERENTIAL
In this case Q = JRn \ U = . —t I
h 0-
hi
OPERATORS
OF INFINITE
ORDER
| * € 2 z l . Then for any g € W£°°(IR n ) : J
u(x) = (aQexTp(h0d/dx) +
exp(h1d/dx))~1g(x),
and u{x) belongs to W^°°(IR n ). In particular, consider the equation u(x + 1) + u(x - 1) = g(x), x € U1.
(3.7.6)
Using Taylor's formula, we can rewrite this as a differential equation of infinite order. Set 2cosh (■£-] u{x) = g(x), xeJR1.
(3.7.7)
Since cosh(i£) ^ 0 for £ ^ ir/2 + kw, k € S , we see that there exists a unique solution of Eq. (3.7.7) (or, what is the same, of Eq. (3.7.6)) for any / € W ^ I R 1 ) : u(x) = -sech
(^Wz),
where fi = Ht1 \{TT/2 + kir, ke 2Z] (see also Example 3.1.3). Example 3.1.3. Consider the equation n
£
u(x + k)= g(x), x € IR1,
(3.7.8)
where n is an integer positive number. We rewrite it in the form L
The operator Ln(d/dx)
»[fc)u = £ has the symbol nH)
±U
ek*u(x) = g(x).
~
sin((l/2)0 '
(3.7.9)
(3-7-10)
(for the formula (3.7.10), see Stromberg [1]). We define the domain Q. as n = IR1 \{£ = 2fc7r/(2n + 1 ) |
keZZ}.
Then taking account of Theorem 3.7.1, we see that for any g 6 WQ°°(]R}), exists a unique solution of Eq. (3.7.9) (and so, of Eq. (3.7.8)) U(X)
sinh((l/2)d/dx) smh((n + l/2)d/dx)9(X'-
there
APPLICATIONS
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EQUATIONS
67
Example 3.7.4- Let us consider the equation £ f l ~ - T T ) <X + fc) = Six), xem}. (3.7.11) U k=-n \ +i / By an argument analogous to that used in Example 3.7.3 we deduce that for any g € ^ " " ( I R 1 ) , Eq. (3.7.11) possesses a unique solution u{x) in V K ^ I R 1 ) : _ (n + l)[8inh((l/2)d/«fe)]' 9[ [sinh(((n + l)/2)d/dx)f
{>
h
where fi = IR1 \{£ = 2kir/(2n + 1), k g S } . 3.7.2. Functional-differential equations. We investigate functional-differential equations of the type m
Qk
J2ak-K^Dm-ku(t,x k=o
+ hk)=g{t,x),
teJR},
z € DT,
(3.7.12)
"'
where a^ G (C, A^ G IR", fc = 0 , . . . ,m. According to the results of Chapter 2 we can rewrite (3.7.12) in the form L(a, h, d/dt, D)u(t, x) = g(t, x), where the operator L(a, h, d/dt, D) has the symbol m m
L(a,k,i(o,£) : = £fl*(»&)*f"~*ra
Set O(L) := {(&,() £ IR | L{a,h,i(0,0 = 0} and ft = M n + 1 \ 0 ( I ) . By virtue of Theorem 3.7.1 one can prove the following result. Theorem 3.7.2. For any g G Wn°°(]R n+1 ) there exists a unique solution u € W^°°(JR.n+1) of Eq. (3.7.12) and this solution can be written in the form
x)u(t, u{t X)= >
9
9(t x) L(a,h,d/dt,D) L(a ,h,d/dt,D) > - ^'
x).
As an example we consider the equation (Chambers [1]): T \^oo
a,-^-u{x + la) = g{x), x € IR1, dx
where a( € (E1, a € IR 1 . We can rewrite (3.7.13) in the form ^00 - E af e x P dx l=-oo d
((, d \ (' a "7~dx I u(x) \ J
=
9(x)> ^ € IR1 •
(3.7.13)
DIFFERENTIAL
68
OPERATORS
OF INFINITE
ORDER
Consequently, E
aiexplla—\u(x)
= I g(r)dT + c := G{x,c),
where c is some arbitrary constant. We see that the operator OO
L\ [dx) ''
l=-oo
has the symbol oo
mo = E ^ilaiSuppose that L(i£) is analytic and does not vanish in some domain fi C IR^ . Then in view of Theorem 3.7.2, there is a unique solution u € W ^ ^ I R 1 ) °f Eq. (3.7.13) for any G G W ^ f f i 1 ) and u x
() =
L
TrdTG(x,c)-
\dZ)
Assume now that B(i£) = 1/L(i£) has the form oo
^enai, L(iO-B(iO = l-
B(iO= E n=—oo
Then the coefficients bn satisfy the relations oo
E
a,bk-i
= 1, k = 0,
(=-oo oo
E aih-i = 0, k ± 0, l=-oo
and for G € W ^ I R 1 ) we obtain the formula oo
"(z)=
E
oo
bne™d'dxG(x,c)
n=—oo
= E
bnG{x + na,c).
(3.7.14)
n=—oo
The solution (3.7.14) agrees with the particular solution constructed in Chambers [1]. In particular, the solution of the equation —u{x) + —u(x + a) = g(x), a 6 IR\ x G IR1, can be represented by the formula u(x) =
f
(1 + exp(cid/dx)) Jo
g(r)d.T
+ c.
APPLICATIONS
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EQUATIONS
69
For any n e R 1 the operator (1 + exp(ad/dx)) _ 1 has the symbol (1 + exp(iaf)) - 1 , which is analytic in ft = IR1 \{(2k + l)ir, k £ ZZ}. We then have u x
()
=
T,
;—TTTG(X.C)
(l + e x p ( a £ ) ) * OO
Tl(-1)neMand/dx)G(x,c)
= n=0 oo
E ( - l ) n G ( x + an,c),
=
n=0
where G(x,c) = /0* g{r)dT + c € W ^ I R 1 ) . 3.7.3. Cauchy problems for functional-differential equations. In the space Rn+1 of the variables t € IR1 and i 6 E " we study the Cauchy problem of order m > 1 : m_1 dmu dku -g^ + E «*(*) 3 ^ ( * + hk)=g(t,x), (3.7.15) dku(0,x) ^ '
;
, x , = y t ( » ) , fc = 0 , l , . . . , m - l ,
, (3.7.16)
where /ifc € IR", o t (<) G (^(IR1), fc = 0 , 1 , . . . ,m - 1. Theorem 3.7.3. Let tpk G W^„°°(IR;)
and 5 (<,x) € C°(IR1, W|„°°(IR;) ). Tfcen m
there exists a unique solution u(t,x) € C (JR},W^°(lR2)
) o/ the Cauchy problem
(3.7.15)-(3.7.16). Proof. Rewrite Eq. (3.7.15) in the form
^+ELk(t,D)-^>=9(t,x),
= g(t,x),
where the Lk(t, D) = a.k(t) exp(ihkD) are pseudo-differential operators with symbols Lk{t,0 = a-k{i) • exp(z/&itf), which are analytic functions with respect to the variable ( in the whole space R% for every fixed ( 6 E 1 . Then the proof is immediate from Theorem 3.7.1. Example 3.7.5. Consider the Cauchy problem
^ % ^ + £ aku(x + hk) = o,te R1, x e nr, u(0,x) = ip(x), x € IR",
(3.7.17) (3.7.18)
70
DIFFERENTIAL
OPERATORS
OF INFINITE
ORDER
where ak 6 0?, h € Htn, * = 0 , 1 , . . . . m ,
=
52±-LLt(D)
'■
where m
£(£>) = £
akexp{ihkD).
k=0
In the case m = 0 we obtain that the problem
^hEl
+ u (t x
+ h0) = o,tem},xeiRn,
(3.7.19)
at u(0, *) = V{x),
(3-7.20)
has a unique solution given by the formula u(t,x) = exp(-texTp{h0d/dx})(p(x). Consequently, 00
n
00 (—i)n [—t) u(t,x) u(t, x) = = n=o 5252—j—exp{nhod/dx}ip(x) n r~ exp{nh0d/dx}(p(x) n—0
ft*
= E^-f^+nAo). n=0
"■
In particular, the fundamental solution of the Cauchy problem for the equation (3.7.19) is the generalized function
£(t,x)='£^-S(x
+ nh0),
n=0
where 6(x) is the Dirac function which is defined on W^?°(IR") and belongs to
Remarks 3.7.2. It is not difficult to obtain variants of the results presented in this subsection for systems of equations. The theorems in this section remain valid if hk €
APPLICATIONS
TO PSEUDO-DIFFERENTIAL
EQUATIONS
71
in the spaces W<3°°(IRn). The equation (3.8.1) has a unique solution when the sym bols L(£) and L(()~x are analytic in G. In this section we consider Eq. (3.8.1) for the case when L(£) has singularities of a certain degree, i.e., it vanishes on the boundary of G or at a point £0 G G with a definite order. We construct here an algebra of PD-operators acting in function spaces with weights. Let G be a bounded domain in lRn with boundary dGDefinition 3.8.1. A test space Hg^W1) satisfying the following conditions:
is a space of all functions / € L2{Mn)
i) The Fourier transform / of / vanishes almost everywhere exterior to G, i.e., /(£) = 0, for almost all ($ G; ii) For a € W the inequality
\\f\\L~ /l/(0IVKK<+» JG
holds, where
p(0 = mm||^-r||. Examples of test functions in HQ (JRJ1) are elements of the space HQITR") (see Sec tion 2.1). It is clear that HQJJBT') contains functions that are not in HQ{W1); the function F _1 (exp(—l//»(£)), where F~l indicates the inverse Fourier transform, does belong to Hg^W), but not to Hg(WLn). Thus, Hg-(nn) C Hg^W1), but
Hg(lRn) $ H%p(m.n). Definition 3.8.2.
L»(G,if{t)), a e m
We say that / „ -> / in #g> p (IR n ), if / „ -* f in all spaces 1
Now, we construct an algebra of PD-operators acting invariantly and continuously in HQJW). We begin by defining the algebra of symbols A(G,p): A(G,p) := | L : G -> (C | L is an analytic function, 3a € IRSsup \L(()/pa(t)\
< oo I
To every symbol L(£) in A(G,p) we make correspond a PD-operator defined by L(D)f(x) := [
for / e HgJW).
exp(ix0L(0f(0d(,
(3.8.2)
72
DIFFERENTIAL
OPERATORS
OF INFINITE
ORDER
Since supp/ C G, the relation (3.8.2) is equivalent to L(D)f(x)
:= / exp(iXOL(Of(i)dC
(3-8.2')
JG
It is easily seen that the definition (3.8.2) is correct and defines an algebra of PDoperators isomorphic to the algebra A(G, p). We note that if in G we use the expansion
i(0 =
oo |c|=0 |o|=0
wr,
then the corresponding operator acts as a differential operator of infinite order 00
L{D) = £
aaDa.
|o|=0
We say that an operator L on Z.2(IR") has the symbol L(t) if, for each / € /^(Ht"), the relation Lf(£) = £(£)/(£) holds. The following result is valid for such operators. Theorem 8.3.1. The space HQ (TRn) is invariant under the operator L having an analytic symbol L(£) in G, if and only if L(£) € A(G,p). Proof. Sufficiency. We assume that L(i) S A(G, p) and / e Hgp(JELn). We shall prove that L(D)f € H^(JRn). Let a0 e IR1 be such that \L(£)\/pc"'(t) a £TR},
< N < oo for some positive TV. Then for
/m„ |Z7(0IV«K = J^JL(OH()\2pa(Od( < ^ 2 ^l/(OlV +2oo UK
By the condition of the theorem L(D)f(x)
/ „ . \rf(0\2Pa(Od(
belongs to Hg'p(JRn).
= / m „ | L ( 0 / ( O l V ( 0 * < oo-
(3-8.3)
Let OT : ={
■=r\aeB}Ll(G,pa(0),
APPLICATIONS
TO PSEUDO-DIFFERENTIAL
EQUATIONS
73
where Z,i(G,p°(f)) is the space of all functions that are summable with weight pa((.) and take real values. In fact, if ip GOTand ip = | / | 2 with / e Hg^W1), then = / 1/(01 V ( 0 # < oo, Va € R 1
L HO\p"(m
JG
JG
Conversely, if 0 £ Mu and 0(f) > 0,Vf S G, then -F _ 1 ((0(f)) 1 / 2 ) is, as is easily seen,
in flg^nt"). A symmetric function 0 and tp± £ Mi. It is easily verified that a weight function pa(£),a € R , is a multiplier of the space Mi. Then, (3.8.3) defines a continuous linear functional |Z,(f)|2 on 9Jt, i.e., on the set of non-negative functions. Hence, |Z(f)| 2 can be defined as a functional on Mi by the formula
|£(OI2(v) = |£(0lV+ - |£(0I V It follows that |L(f)| 2 £ {Mi)' Now, we prove that (Mi)* = U a e R . i c o ( G , p - " { i ) := M*,
(3.8.4)
where £oo(G, /3 _ a (0) i s * n e space of functions ip for which ess sup |¥>(fV -a (OI < °°To prove (3.8.4) we need the following result. Lemma 3.8.1. The relation (Li(G,/>"(£))* = L^G,
p-"(())
holds.
Proof of Lemma 3.8.1. The imbedding Z , O O ( G , / J " D ' ( 0 ) C (Li(G,/> Q (f))* is obvious. Let (p € {Li(G, pa{0Y-
Then there exists an isometric mapping 0
:
Li(G)-+Li(G,p"(()),
(0(/))(O := /(0p-(0Hence ¥>o0 is a continuous linear functional on Li(G). There is
¥>(/) = V W 7 ) = / ¥>W(0/(£R-
74
DIFFERENTIAL
OPERATORS
OF INFINITE
It follows that ip is a regular functional and ip(t) = ip'{£)f»a(£)- Hence, f{£)p ¥>'(£) € L^G). The lemma is proved.
ORDER "{£) =
We see that the relation (3.8.4) is a trivial consequence of Lemma 3.8.1. It follows from (3.8.4) that |I(£)| 2 € M M , i.e., there exists a 0 € Ht1 such that \L(0\2Pa"(0 € ioo(G). On the other hand, L(£) and f{£) are continuous. Hence ess sup | L ( f l | V » ( 0 = sup \L(0\2P°°(0
< oo,
or
sup|£(OlV°(0<°°This completes the proof of the theorem. Now, we consider the space of generalized functions # e ~ ( n t n ) := (#±£? p (lR n ))\ As in Section 2.1 we define the action of an operator£(£)) with the symbol L(£) on a generalized function as follows H^iW1).
< Lh,
Here L* is the linear PD-operator with the symbol L(—£). As a consequence of The orem 3.8.1, we find that the operator L(D) with the symbol L(£) 6 A(—G,p) acts invariantly on the space (H™G (IRn))*. Finally, we have T h e o r e m 3.8.2. If L{() and I ( £ ) _ 1 are in A(G,p), hand side h € H^(JRn) has a unique solution
u{x)=
then Eq. (3.8.1) with the right
L(D)h{x)
inH^(JRn). Example 3.8.1. Consider the Helmholtz equation Au(x) + u>2u(x) = h(x), where u> is a real number. Let ft = {£ 6 W,(2 < u2} . Then the operators A + u2I and / / ( A + w2I) have symbols in A(Q,p). For h G H^(lRn) this equation has a unique solution of the form u{x) =
AT^ih{x)-
APPLICATIONS
TO PSEUDO-DIFFERENTIAL
EQUATIONS
75
Example 3.8.2. Let Q, = [for, (k + l)w] C IR1, where k is an integer, and consider the equation sin I — J u(x) = h(x), with h € HQP(]R}). The operator sin (d/dx) has the symbol sin(£), which is easily verified to be in A(ft, p) together with its reciprocal 1/ sin(£). It follows from Theorem 3.8.2 that this equation has a unique solution "( x )
=
.
I
sm
h x d\ ( )
(sj
in ^ ( I R 1 ) . Example 3.8.3. Consider the unit ball SI = {£ 6 ]Rn, ||£|| < 1} in IR^ and the equation Aw(x) = h(x),
xeW1
Let the weight function be po(£) := M\\,£ £ &, a n d le* the definition of h € i/^, o (]R") be analogous to the definition of h £ / / ^ ( I R " ) , the only difference being that the weight function po(0 here is not a function of the distance to the boundary but a function of the distance to zero, i.e., the singularity of the symbol £ 2 . Thus, for any
h € #n,X( IRn )' t h e r e
exists
a unique solution
«(*) = -£h(x)
infOm"). Bibliographical N o t e s . Our treatment given in Sections 3.1-3.5 follows Tran Due Van [14]. We have been influenced by the paper by Dubinskii [4]. The results of Sections 3.6-3.7 are due to Tran Due Van and are published here for the first time. The section 3.8 is based on Trinh Ngoc Minh's paper [2]. Various applications of the algebra of PD-operators with real analytic symbols can be found in Dubinskii's book [5], Trinh Ngoc Minh's Thesis [1], Samarov's papers [1, 2] (see, also Medeiros [1], Samarov and Aksenov [1], Umarov [1,2], Khrenikov [1,2], Casanova [1], and many others). We also refer the reader to Davis's book [1] (and the references therein) on the dif ferential operators of infinite order. Differential operators of infinite order have been used by Hirschman and Widder [1] to invert convolution transforms and by Widder [1], Hills and Irwin [1] in their study of the heat equation.
Chapter 4 APPROXIMATION METHODS
It is well known that the "initial function method" has been widely used for numeri cally solving broad classes of equations of elasticity theory and construction mechanics (see, e.g., Agarev [1], Bondarenko [1], Bondarenko and Filatov [1], Lur'e [1, 2], and Vlasov [1,2]). However, why this method does work so well is not known. It is clear that the "initial function method" is nothing but the method of DOIO! In this chapter, based on the technique of pseudo-differential operators with real analytic symbols, we give a very elementary approximation scheme for pseudo-differential equations, and in particular, differential equations of infinite order. We prove also why the "initial function method" works well in many cases. Some numerical experiments are given at the end of this chapter. 4.1. Approximating the symbols by algebraic polynomials 4.1.1. Approximation by the Taylor series Let L(£) be an analytic function in A„ = {| K 6 RMfcl < »i,Vj > 0, j G { l , 2 , . . . , n } } ,
m = E «<»?", E KK < oo. H>o |o|>o We must, for tp € W£™(JRn), approximate the expression g(x) = L(D)
aaD°v{x).
\a\>0
Since £(£) is analytic in A,,, there exists a vector e = (ei,e2, ...,£;v) with all £; > 0 such that £| a |> 0 \aa\{i' + e) a < oo. The function L(£) is analytic in A„ + £ and can be analytically extended to □ H . £ = {Z I ZJ =Xj + iyh\xj\
< vi + ti,3 = l , 2 , . . . , n } .
The following theorem shows that the functions gN = LN(D)ip = T,\a\
76
converges in the
77
APPROXIMATION METHODS
Further, if ip g mu„ then g(x) = T.\a\>oac.Dc'ip(x) converges in the norm of Lp(JRn), and the following inequalities hold for p G [l,oo] and for p = 2 in particular: \\g\\p =
\\L(D)
\\g% =
||£(£)
\\9-9N\\P
< ( £
\\9-9N\\2
< max ^ j ^ y p
< max \L(z)\ (l + - ) |M| P , (4.1.1)
KK)|M|P<
(4.1.2)
max J ^ l L ( l
+
l) N |H| p ,(4.1.3)
1Mb —> 0 as TV -> oo,
where, for abbreviation, L(v) = >J |attJi/", 1 H— = (1 H |a|>0
'
*
'
(4.1.4)
)-.-(l H
«*
)•
«W
The proof of this theorem follows from an estimation of the Taylor coefficients aa of L(£) and the Bernstein-Nikolskii inequalities (Theorem 1.9.2). Now, we consider the Cauchy problem —
-
L(D)u = 0,
(4.1.5)
n
u\t=0 =
(4.1.6)
By Theorem 3.2.1 we have oo
u(t,x) = e
The following theorem shows
uN(t,x) = -£tkLk(D)
exp [«£(„)] | H | , .
(4.1.7)
78
DIFFERENTIAL
OPERATORS
OF INFINITE
ORDER
In particular, if p = 2, then |Ki,0-«N(<,0ll2<^^(tmax|i(Oir+1exp[imax|L(O|]|M
(4.1.8)
The right-hand sides of (4-1-7) and (4-1-8) tend to zero as N —> oo. The proof of this theorem is similar to that of Theorem 4.1.1, therefore we omit it. 4.1.2. Approximations by other algebraic polynomials. Let G := {£ | 0 < £i < 1, i £ [1,2,...,re]}. We consider the Bernstein polynomials BN(L,0 = £ ... E L (!,...,%) PNkMll-PN^n) fc,=0
where PNkj(£j) = (k)(j'(^~ [1, p. 102]) then
kN=0
V
V
£i)N~k'■ KdL/d(j
IV
(4.1.9)
/
gLipj^ 1 for every variable £, (Lorentz
\L(0-BN(L,0\<±Mfi{12~^.
(4.1.10)
Let n = 1, G = [a, 6] and a = £0 < 6 < ••• < £n = b. Set
MO = K - &)...« -to),MO = jr J ^ ^ L
(4.1.11)
We have, with a suitable < € (0,1), the well-known remainder formula
RN(0 = |i(0 - MOI = t y + i y ) l | ^ ( ° | -
(4 L12)
"
If the points of interpolation are taken as the zeroes of the Chebyshev polynomials T N ( 0 (Gelfond [1, p. 14 ff]), then MO = MO and R (?) ^ \L{N+1)(tQ\ w(0
(b-a)N
- IF+i)!—2^^~-
Let n = 1, G = [-1,1]. Denote by PN(0 imation of L(£) in G. Set EN{L)
for ever
y N, t
ne
where VN is the space of polynomials of degree at most N. Then
46G
-
polynomial of best approx
= inf sup | Q ( 0 - 1 ( 0 1 ,
EN(L) = S*p\PN(0 - £ ( 0 1 ,
, , „ „ (4 L13)
(4.1.14)
APPROXIMATION
79
METHODS
and from Lorentz [1, p.66] we have EN(A)
< MmZ%(Ou(L^\ZN(0),
HJV =
N € {m,m + 1,...},
EN(() = max ( — — - , — J , H0(£) = 1.
Here m is an arbitrary natural number. From these estimates and Theorem 4.1.1. we can deduce the following assertion. Theorem 4.1.3. For every
< C-WvWi,
(4.1.15)
P C ^ - M ^ I I * < ™^L+;m \\L{D)
^ T 1Mb, (4-U6)
< EN(L)\\V\\2
(4.1.17)
4.2. Approximating the symbols by trigonometric polynomials 4.2.1. Periodic symbols. Let A„ = {£ | | ^ | < vh j £ {l,...,n}}. If L(() is a periodic function with period 2i/, analytic for all £ 6 ]Rn, then L(() can be expanded in a Fourier series
H0=
I > * e x P »*— , 1*120
v
L
(4-2.1)
\
[£ = l^-,..., £"-)) with Fourier coefficients 2~n ak = ^—-f ak=
f /
1 L(()exp ( 0 exp —ik— - * k — di. #.
V\...VnJ£>„
(4.2.2)
VV
[
It is clear that the series (4.2.1) converges absolutely. Set LN(X) = Eo<|Jt|
LN(D)
J2 <W (* + — ) .
o<|*l<W
V
(4.2.3)
" /
T h e o r e m 4.2.1. / / y SSJt^ tftera LN(D)ip(x) converges to L(D)
80
DIFFERENTIAL
OPERATORS
OF INFINITE
ORDER
number X, and let for any nonnegative integer n-vector I = (/j,..., /„) < p, be 2
-n
.vn J&„
\L(l)(t)?di <M\
Then
CM —J-J-IMU
\\L{D)
forpel^cc).
(4.2.4)
Here C does depend on X but not on M and N. If A„ = [—IT, v] then \\L(D)ip-LN(D)ip\\p<e(N)^^MP,
for N > 2, p e [ l , o o ] . (4.2.5) 9 Here q is any natural number, e(N) < mqui{Lq,j^),mq is a defined constant, ui(-) is the modulus of continuity (see Chapter 1). Proof. We have
\\L{D)v-LN{D)V\\f<
Y.
I^MMIP-
\k\>N+l
On the other hand, from Nikolskii [2, p.281-283], V* I \k\>N+l
1^
CM
N*-2
where C does depend on X but not on M and N. The first inequality of the theorem is proved. The second inequality follows from the inequality
£
M < * ( # ) — ,
\k\>N+l
9
\
1\ /
See Bary [1, p.70-71], Laurent [1] and Nikolskii [2]. Corollary 4.2.1. If n = 1 and G = [-7r,7r], then there exist a number 6 G (0,1) and a constant C such that oN+l
\\L(D)
< 2C^—^Hvllp.
In fact, from Bary [1] it follows that there exist 0 g (0,1) and C such that |I*|
(4.2.6)
APPROXIMATION
METHODS
81
Hence \\L(D)V
- LN(D)
<
£
K||M|p
|fc|>AT+l
< c
^IIIVIIP = 2 C ^ - | | ^ | | P .
£
i
|M>JV+I
"
It is worthwhile to note here that Formula (4.2.3) is convenient for practical use. 4.2.2. Non-periodic symbols. If the function L(£) is not 2i/-periodic, then for any e > 0 we can extend it to a 2(u + e)-periodic function A(£) such that L(£) = A(£) in A„. Indeed, there always exists a function i»(£) € C^°(A v+£ ) such that ^(£) = 1 in Ay (see Nikolskii [1]). Hence, A(£) = il>(£)L{£) can be extended to a 1(v + e)-periodic function in C°°, and furthermore .4(£) = L(£) in A„. Therefore we can simply apply the above procedure to our problem . We can also carry out the above procedure in another way. Writing L(£) as a sum
m-
L(t) -L(- -0 = HO + HO, 2
L{() + L(- -0 + 2
(4.2.7)
we see that Li(£) is an even analytic function and L2(£) is an odd analytic function. Hence L
i(0
Here on =
= £ <*i*exp ik |fc|>o
2_n V\V2
a.
(4.2.8)
V
r
/ Mfl«
exp[ia£}L2{()
= Yl
a
P
ex 2*ex Pp i
V
#, (4.2.9)
V
\k\>0 ■K
and 2
«2* =
7T
-n
— /
L2(0 exp [-i'(far/i/ + a)(\ d(.
.vn n J& JAu V\V2...V
Clearly, there exists a constant M' such that
maxf-^^/
\DlLt(i)\*dC,
2
— / \DlL,(t) + itHt)\2dA < M'. (4.2.10)
82
DIFFERENTIAL OPERATORS OF INFINITE ORDER
Consequently, by Nikolskii [2, p. 281-283], there exists a constant C" such that
{
max!
Y E
|*|> |>AH-1 W+1 1|*
I«I*I. l«i*l.
)
<
CM' CM' <-TS-
E E
wM
YE
a exP\i(k-a2 2*k ex p i(k-
|*|>N+1 |*|>N+1
Jj
^N
Defining L*N(D)= ^(£>)=
Y E
0<|fc|<7V
ex r.,7ri>-]+ a«i* lkexp\ik— P +
0<|fc|<JV
- a)D\ a)D ,
(4.2.11)
we have
L* -- E N{D)y{x) = L'N{D)ip{x)= Y.
«i*¥> (* (* + + **--)) ++ «i«»
EE
<<wp(x W (* ++ fc- -k--a). a) ■
(4.2.12) (4-2-12)
It follows now directly the following theorem. Theorem 4.2.3. L%(D)ip(x) is an approximation to L(D)
(4.2.13)
4.3. Trigonometric interpolation. In cases, where it is difficult to calculate the Fourier coefficients of L{() we can use trigonometric interpolation. Let A„ = [-*,«] (n = 1) and L(£) be 27r-periodic. With (,k = % ( - m < k < m), define Lm(£)
=
sequence of numbers ah, »
i
m-l
I
Js=l-m
^
£ ' ckexp[ik(],
k=-m
where, for any
«* = 55«-m+ «-m+ E aatk + 5«™. -am, E a* Z l Z
fc=-m
cc
,.
= =
1
m
m
k=-m
«3 E i(6)expHi6],j = = - -m,...,m. m....,m. 5T Y!' m)eM-itik]J Z a n
(4.3.1) (4.3.1) (4.3.2)
Since (John [1, p.91-93]) c
|\L(0 £ ( ~ 0 -MMO O c ^ max max 11^(01, 11^(01, I
(4.3.3)
we have \\Lmn(£)Y> ||L (D)
m
max max |Z«(0| | Z « ( 0 | IM| |M|23,
[-ir.jr]
(4.3.4)
APPROXIMATION
83
METHODS
where m
Lm{D)
Y!
ckV(k + x).
(4.3.5)
k=—m
If L(£) is not 27r-periodic we can again write
Ko_att±a=fl + a£LjStfl.Il(0+x1tt) and repeat this procedure for Za(£) and exp[?|£]£ 2 (0- Note that this technique can be generalized to multidimensional cases, too. Remark 4.3.1. In the case when we know that the exact solution belongs to the class 9Jt„p then Theorems 4.2.1, 4.2.3 give us a very effective method for solving the above problem even if the data are inexactly given. In fact, let the function tp be given in Lp with some error 6, \\
(4.3.6)
Then instead of (4.2.3) we shall use L(D)
=
£
aVp,(x
0<|*|
+- ^ - ) . "+
V
£
(4.3.7)
/
From Theorem 4.2.1 we have \\L(D)
<
\\L(Dyp-LN{D)
+
'" M rlMI P + ( £
l«*D*
<
-^r\M\P
<
■^l\\f\\,+
\\LN{D)
+ (2Nn + l)M6 M(2n + l)NS.
N* 2
Now, with N =
( 2(2iV + l)6 [C\MP(2X-1))
\
"+1
+ 1,
(4.3.8)
where [x] is the entire part of x, we get Holder continuous dependence on the error 6 (see, e.g., Baumeister [1]). (It is easy to verify that, with N in (4.3.8), we have an error estimate of the type CS^.)
84
DIFFERENTIAL
OPERATORS
OF INFINITE
ORDER
4.4. Approximating the data by sine functions. Using Theorems 1.11.1 and 1.11.2 we see, if ip € EDt„p(l < p < oo) then M*)~
E
J-)sinc(--fc)
k=~N
\
V
(4.4.1)
V
I
is an approximation to y?(x). Furthermore y?;v(x) belongs to SBl„p. It follows that
L{D)VN(x) = E v ( - ) L(D) sinc (— - k) is an approximation to
(4-4-2)
L(D)tp(x).
In practice it is difficult to calculate L(D)sinc (f - k). One should therefore approx imate L(D) smc{f - k) by Lm(D)smc(—-k):=
V
a a D a sinc(— - fc).
0<|a|<m
It is not hard to prove that <$(x)
:= I m ( 5 ) E =
v(^)«inc(^-fc)
£ > ( - ) (
E
(4.4.3)
a ^ s m e p - f c ) )
*irW V " / \O
vx
v
is an approximation to g(x) = L(D)tp(x) in SJl^p. ' 4.5. Examples. In this section we give some illustrative examples to our above general schemes. In Subsections 4.5.1 -4.5.3 the idea of trigonometric approximations of the symbols is demonstrated. After approximating the symbol by trigonometric polynomials we obtain that the action of the operator with this symbol on a function
; (4.5.1)
85
APPROXIMATION METHODS We have (I/(--£-2+a2))h(x).
u(x) = The symbol of the operator If(—£t
+ a2) is l / ( f + a 2 ) and in [—7r,ir] we have
i ine+a2) == 2^
°° J2 ckexp[ik(], k= — oo
where Ck
I
exp[—ik£\ dt e + a,2
= — [exp[ka]Ei(—ka + ikir) — exp[— ka]Ei(ka + ikn)2a — exp[ka]Ei( — ka — ikir) + exp[—ka]Ei(ka — ikir)] (Prudnikov et al. [1, p. 139]). The special function Ei(z) is also treated in Lebedev [1, Chapter 3]. There we can find that Ei(z) = C + ln(-2) + £ - J - for | a r g ( - 2 ) | < 7 r , k=\
k\k
where C = 0,5772157... is Euler's constant, ln(—z) = In | — z|+iarg (—z), | arg(—z)\ <
Thus, we have the following solution formula for our problem: u x
( ) = 7T 2n k=-£
Ckh x
( + *)•
(4.5.2)
4.5.2. T h e Cauchy problem for the heat equation fill
fj 11
— = a2-^;, at ox'
- o o < x < oo, t> 0, u\t=o =
We have w(i,x) = exp The operator exp(ta2-^)
(**£)
ifi(x).
has the symbol
exp(-ta2£2) = Y
4 ( 0 exP
v
(4.5.3)
86
DIFFERENTIAL
OPERATORS
OF INFINITE
ORDER
with 2vdk(t) = f exp(-ia 2 £ 2 - i-Jfefldf = v ' J-u r v v ' = /
d£ =
exp
J—V
-.w-ii.)-(«*+,£*)■ see Prudnikov ei a/. [1, p. 139], and Lebedev [1]. Note that erf z = /0Z e~slds. Thus we have
«(*,«)= ±
dk(t)V(x + - ) .
Jt=-oo
V
V
(4.5.4)
)
4.5.3. The Cauchy problem for the inverse heat equation ^ = -a2!^, at ox We have
-oo
*>0,
«(f, a;) = exp
« [ « = *> €
OT^IR1).
(4.5.5)
(-a2* J^) v(x).
The operator
has the symbol exp(a2^)= this expansion being valid for -v
£ fc(t) exp ( £ * { ) ,
< x < v, with
2ubk{t) = ]\xp(a2te)exp
{-£%)
« =
-^(o&)-(^-iSi). (see Prudnikov e< a/. [l,p. 139], Lebedev [1]). Here erfi c = £ e'2rfs. Thus we have u(t,*) = 2i/ £ t=-oo
bk(t)
(4.5.6)
"
4.5.4. The Cauchy problem for the wave equation.
a2u(t,x) . . f d2 d2\ A gf2 = Anu{t,x)= I —+ ... + —\u(t,x), u(o,i) = vK>) e w+°°(iR"),
xeTBT, 0 0,(4.5.7) (4.5.8) (4.5.9)
87
APPROXIMATION METHODS where G is a bounded domain in IR". It is easily verified that u(t,x)
-
cosh (ty/£\ oo
smh
^2k
oo
(J^)ijj{x)
(4.5.10)
/2fc+l
- r ( 2 l ) ! ^w + E ( a + 1 ) ,A> w . In the case n = 1 we have . . „(t,_) =
exp[id/dx] + exp[-td/dx] V?(x + .) + y>(x - t) 2
, . „(_) +
1 /*+*
+5r
exp[td/dx] - exp[-td/dx]
_____
0(.)*.
_ ,, W, (4.5.H)
_ JI-(
This is the well-known D'Alembert formula For the case n = 2, by the above formula, it is easy to calculate u(t,x), when
+
d2 1 dxi J
COs(^lXi)cOs(l/2X2)
=
Y_ I
k
COS 2 *(^iXi)cOS 2 A r
2k
(v2X2)
kTo\ J =
( — 1)
(f 2 + V2)
COs(l/!Xi) COs(i/ 2 X2).
Then
~ (-lyftf «(*,*) = =
_
+ v2 f
T^n
. . . . cos(i/ 1 x 1 )cos(i/ 2 x 2 )
cos(t\Jv2 + J/|)COS(I/IX 1 )COS(I/2X2).
(4.5.12)
Now, we treat the case n = 1 as an illustrative example for the application of our different approximation methods. For simplicity, let ip = 0 and put _(i,£) := (exp[i„g+exp[-*t£|)/2. First we approximate _(*,£) by the Taylor series
M U ) := X) ^ ( t f ) 8 *
(4-5.13)
Thus
Iw 6, ^H ¥>(*) = aw(t, *) = _
TJ^J^HX).
(4.5.14)
DIFFERENTIAL
88
OPERATORS
OF INFINITE
ORDER
Now, for ip e S&t^lR1), we have oo
\\u(t,-)-uN(t,-
j.2kv2k
(4.5.15)
0, : £ ptfM-
(2JV + 2)!
COB^JH^OHp.
We see that if i/ is small then u ^ ( i , x ) converges very fast to
u(t,x).
Secondly, we approximate L(t,£) by trigonometric interpolation. Let Cj =
1
JZ*i exp[it£k] + exp[-it£k]
2^ £
2
where £* = kir/m,
•■,""1
j = -m,...,m,
(4.5.16)
—m < k < m. Then
1 He =
,
«PHJ&-1,
> cos
2™ L=o
f . M cos I j t—
— I H— cos(tj/) cos(jTr)
(4.5.17)
V m)
Thus we have u*m(t,x):
(4.5.18)
As a numerical experiment we have calculated % ( * , l),uj),(t, 1) for ¥>(z) = sin(ux)/x with various n,m,v. In Table 1 one should see "error coefficients" u2N+2/(2N + 2)! cos v in (4.5.15) for various values of v and N. Graphics of u(t, 1), u^(t, 1), iijjff, 1) for various JV, m, 1/ are drawn in Figures 1 to 4. Bibliographical N o t e s . The initial function method has been developed in Agarev [1], Bondarenko [1], Bondarenko and Filatov [1], Lur'e [1, 2], Vlasov [1, 2]. These authors give many algorithms and applications to the problems of elasticity theory and construction mechanics. The approximation methods for differential equations of infinite order have received very few consideration. The results of this chapter are taken from Tran Due Van and Dinh Nho Hao [1] and from Tran Due Van, Dinh Nho Hao, Trinh Ngoc Minh and R. Gorenflo [1]. Further applications of the methods in this chapter are given in Dinh Nho Hao and R. Gorenflo [1], Dinh Nho Hao, Tran Due Van and R. Gorenflo [1], Tran Due Van, Dinh Nho Hao and R. Gorenflo [1], and Tran Due Van, Nguyen Duy Thai Son and Dinh Zung [1].
APPROXIMATION METHODS N \ v !! 1 2 3 4 5 6 7 8 9 10 N N
!! ! ! ! ! ! ! ! ! !!
\S '!
1 2 3 4 4 5 6 7 8 8 9 9 10 10
.1 -1
.2
.3 •3
.4
.5
.41E-5 .14E-8 .25E-12 •27E-16 .27E-16 .21E-20 .11E-24 .48E-29 .16E-33 .41E-38 .41E-38 .89E-43
.65E-4 .87E-7 .62E-10 .28E-13 .84E-17 .18E-20 .31E-24 .40E-28 .42E-32 .42E-32 .37E-36
.32E-3 .97E-6 .16E-8 .16E-11 .11E-14 .52E-18 .20E-21 .58E-25 .14E-28 .14E-28 .27E-32
.98E-3 .52E-5 .15E-7 .27E-10 .32E-13 .28E-16 .19E-19 .99E-23 .42E-26 .42E-26 .14E-29
.23E-2 .23E-2 .19E-4 .85E-7 .24E-9 .45E-12 .61E-15 .61E-15 .64E-18 .52E-21 .52E-21 .34E-24 .34E-24 .19E-27
2
3 3
4 4
5 5
-.70E1 -.33E1 -.37E1 -.10E1 -.16E0 -.11E1 -.19E0 -.16E-1 -.11E-2 ' -.23E-1 -.54E-4 -.20E-2 -.20E-5 -.13E-3 -.60E-7 -.70E-5 -.14E-8 -.30E-6 -.28E-10 -.10E-7
.74E1 .74E1 .62E1 .62E1 .27E1 .27E1 .76E0 .14E0 .14E0 .20E-1 .20E-1 .21E-2 .17E-3 .17E-3 .11E-4 .11E-4 .60E-6
1 1
N NN
89
2
!! !• .! L ! ! ! ! ! !
.23E-1 •23E-1 .75E-3 .13E-4 .15E-6 .11E-8 .62E-11 .26E-13 .84E-16 .22E-18 -48E-21 .48E-21
-.28E0 -.37E-1 -.26E-2 -.12E-3 -.36E-5 -.78E-7 -.13E-8 -.17E-10 -.18E-12 -.16E-14
v « ! N\ \v
6
7
1 2 3 4 5 6 7 8 9 10
! ! i ! Ii ! i ! ! i ! !
..52E2 52E2 . .62E2 62E2 .40E2 . 40E2 .16E2 .16E2 .44E1 .44E1 .86E0 .86E0 .13E0 .13E0 .15E-1 .15E-1 .14E-2 .14E-2 -11E-3 .11E-3
N N N \ Vv i !
11 11
11 2 2 3 3 4 4 5 5 6 6 7 7 8 8 9 9 1 0 10
i ! i ! >i i ! I ! i. ' i !
.27E1 .11E2 .24E2 .32E2 .32E2 .29E2 .29E2 .19E2 .19E2 .97E1 .97E1 .38E1 .38E1 .12E1 .12E1 .32E0 .32E0
.75E2 .12E3 .12E3 .11E3 .11E3 .59E2 .22E2 .59E1 .59E1 .12E1 .12E1 .19E0 .19E0 .25E-1 .25E-1 .26E-2 12 .73E3 .35E4 .90E4 .14E5 .14E5 .16E5 .16E5 .12E5 .12E5 .75E4 .35E4 .13E4 .13E4 .41E3 .41E3
Table 1: Values of uw+1/(2N
8
9
10
-.25E2 -.53E2 -.61E2 -.43E2 -.21E2 -.73E1 -.20E1 -.41E0 -.69E-1 -.69E-1 -.96E-2
-.25E3 -.67E3 -.97E3 -.88E3 -.54E3 -.24E3 -.81E2 -.21E2 -.46E1 -.46E1 -.80E0
-.35E3 -.35E3 -.12E4 -.12E4 -.21E4 -.21E4 -.23E4 -.23E4 -.18E4 -.18E4 -.96E3 -.96E3 -.40E3 -.40E3 -.13E3 -.13E3 -.34E2 -.34E2 -.75E1
13
14
15
■ 11E4 .11E4 . .61E4 61E4 . .18E5 18E5 . .34E5 34E5 . .44E5 44E5 . .41E5 41E5 .29E5 . 29E5 .1SE5 .1SE5 . .71E4 71E4 .26E4
. 22E3 .22E3 . .14E4 14E4 . .50E4 50E4 .11E5 .11E5 • .16E5 16E5 -17E5 .17E5 .14E5 . 14E5 . .91E4 91E4 .47E4 .20E4 . 20E4
+ 2)\ for various N and v
-.16E4 -.12E5 -.48E5 -.48E5 -.12E6 -.21E6 -.21E6 -.25E6 -.25E6 -.24E6 -.24E6 -.18E6 -.18E6 -.10E6 -.10E6 -.51E5
90
DIFFERENTIAL OPERATORS OF INFINITE ORDER
Fig. 1: Graphics of u(U),« 8 (M) and uj(t,l) for v = 0.1.
Fig. 2: Graphics of u(t, l),u 5 (i, 1) and «;(*, 1) for u = 0.5.
91
APPROXIMATION METHODS
Fig. 3: Graphics of u ( U ) , u 5 ( U ) and uj(t,l)
for
u=l.
Fig. 4: Graphics of u(f, l),u 7 (t, 1) and uJ(t,l)for v = 2.
Chapter 5 A MOLLIFICATION M E T H O D FOR ILL-POSED P R O B L E M S
5.1. Introduction. A specific class of the methods for solving ill-posed problems, to say it in Payne's terminology (Payne [1]), is the "restriction of data method". This method has been used by Miranker [1] in the early sixties for the heat equation back wards in time. His "restriction of data" space is the space H°° of Dubinskii [4] which is only a subset of the space W°° (Tran Due Van [14], Tran Due Van and Dinh Nho Hao [1]. This method gives us, however, only well-posedness classes of the problems under consideration but does not show us how to use them to solve the problems in a stable way. The aim of this chapter is to suggest a mollification method that can improve upon the above-mentioned shortcomings of the restriction of data method. Namely, we shall use the well-posedness classes to solve a class of ill-posed problems 111 Si stable way. Our idea is very simple and natural: if the data are given inexactly then we try to find a sequence of "mollification operators" which map the improper data into well-posedness classes of the problem (mollify the improper data). Within these mollified data our problem becomes well-posed. And when these facts are in hand we try to obtain error estimates and then by minimizing them with respect to mollification parajjicters W€ shall get optimal or "quasi-optimal" mollification eters A detailed description of the method will be given in Section 5.2. Manselli, Miller [1] and Murio [1,3] have used a mollification method, (they use the Weierstrass kernel (Achieser [1], Natanson [1], Nikolskii [1]) to construct mollifica tion operators) for solving some ill-posed problems for the heat equation, but their method is working only in the case of the Hilbert space L2, and moreover they could not find reasonable mollification parameters that are essential in the study of ill-posed problems. From the stability estimates in these authors' papers one cannot see the fact that the approximate solutions depend on the data in a Holder continuous fash ion. Our method is more general than the method of these authors. It is working for Banach spaces, and an important feature is that we succeed in obtaining explicit mol lification parameters which give us error bounds of optimal order (see, e.g., Louis [1]). In the applications of our method to concrete problems (numerical differentiation, the parabolic equations backwards in time, the Cauchy problem for the Laplace equation, the non-characteristic Cauchy problems for parabolic equations,...) we get very sharp stability estimates of Holder continuous type. In these cases our method is optimal in the sense that it gives the same order of Holder continuous dependence on the data as for the regularized problems. Furthermore the method may be implemented numerically1 using° fast Fourier transforms . It ais worth m s m to IU note uuic that xi i. for IUI the cue firs* just fir™. time aa uniform stability estimate of Holder continuous type of the solution of the heat equation backwards in time in the space IJlR^forallp e (1 ool could be
eZblished 92
A MOLLIFICATION
METHOD FOR ILL-POSED
PROBLEMS
93
by our mollification method. A new simple sharp pointwise estimate of H61der type for the weak solution of a non-characteristic Cauchy problem for parabolic equations in a finite domain is established. T. Seidman and L. Elden in their work (Seidman and Elden [1]) have proposed an "optimal filtering" method for the sideways heat equation. However, their method is working only for the Hilbert space L2. Our method seems to be simpler and it is applicable to more complicated problems. In many cases our method is more flexible than the projection method of Natterer [1,2], since our mollification operators need not be a projection and Natterer's method is for Hilbert spaces only. Maslov [3] was the first to set up a regularizing algorithm whose convergence was equivalent to the existence of a solution to an ill-posed problem. We hope also that our mollification method and deep results of approximation theory can give us qual itative properties of many ill-posed problems. Tran Due Van and his coworkers (Tran Due Van, Nguyen Duy Thai Son and Dinh Zung [1]) have proposed a mollification method (they use a Fejer type kernel to con struct mollification operators) for solving the well-posed Cauchy problem for the wave equation. The concept "regularizable" is given in Vinokurov [1]; for works related to our method see, e.g. Andersson [1], Baumeister [1], Carasso [1], Davies [1], Domanskii [1], Ivanov [1], Ivanov and Mel'nikova [2] -[3], Engl and Manselli [1], Khudak [1,2], Liskovec [1], Louis and Maass [1], Maslov [l]-[4], Vinokurov and Menihes [1], Walter [1]. For application of our method see, e.g., Dinh Nho Hao and R. Gorenflo [1], Dinh Nho Hao, Tran Due Van and R. Gorenflo [1]. This section is a generalized and improved version of our early results (Dinh Nho Hao and R. Gorenflo [1], Dinh Nho Hao, Tran Due Van and R. Gorenflo [1]). In Section 5.2 the method is described, applications of the method to concrete problems, namely numerical differentiation, parabolic equations backwards in time, the Cauchy prob lems for the Laplace equation, the non-characteristic Cauchy problem for parabolic equations, are given in Section 5.3. 5.2. Mollification m e t h o d 5.2.1. Description of the method. Many direct and inverse problems lead us to the following problem: Let A be a linear (possibly unbounded) operator on a Banach space X with the norm || - ||. How to calculate y := Ax
94
DIFFERENTIAL
OPERATORS
OF INFINITE
ORDER
when x is given approximately by z£ (generally, xc are not in the domain of A): < £?
\\x-X<\\
This problem is, generally, ill-posed. We suggest the following mollification method for it: 1) Establish a sequence of subspaces Xa, a > 0, of X such that Xa D X0 if a > P, Xx = X, and in which our problem is well-posed: there exists a function a(-,-) such that \\Ax\\
ifx
€
XanD(A).
2) Find a sequence of operators Ma : X -> Xa with the following properties i) there exists a function m(v) such that \\Max - x\\ < m(a,||x||), with liir^^oo m(a, \\x\\) -► 0 when x is fixed. (Approximation property), ii) MaA
=
AMa,
iii) | | M a | | < c, where c is a positive constant. (In the case Ma is the orthog onal projection of X onto Xa we have ||Ma|| = 1.^ 3) Ifx is given approximately by x* (generally, x< are not in the domain of A):
||x-x'|| we take the following mollification
< e,
ofxe:
xl -» Max'
€
XanD{A).
Succeedingly, instead of calculating y = Ax we shall calculate y'a =
AMaxc.
Remark 5.2.1. (Important Remark) It is important that we must always stay in the chosen stability classes, e.g. Xa, of the problems. Although the mollified data may be perturbed, these new ones must be in a chosen stability class of the problems. This fact must be also taken into account when the problem has been discretized. Remark 5.2.2. The first property in 2) shows how good the space Xa approximates the space X. The second property in 2) is restrictive, but it is valid in many cases, for example, when Ma is a convolution operator and A is a pseudo-differential operator
A MOLLIFICATION
METHOD FOR ILL-POSED
PROBLEMS
95
with constant coefficients. The following simple lemma gives an error estimation. Lemma 5.2.1. With the above ■properties the following inequality is valid liar - Vail < m(o, ||il*||) + a(a, \\Ma{x - x<)\\). Proof. From 2ii), we have y-yca
=
Ax-AMaxl
=
Ax - AMax + AMax -
=
Ax - MaAx + A(Ma(x
AMaxc - x€))
Thus, from 2i) and 1) we obtain ||y-l£||
<
\\Ax~MaAx\\
<
m(a,\\Ax\\)
+ +
\\A(Ma(x-x°))\\ a(a,\\Ma(x~x')\\).
The lemma is proved. When a{a, ■) is an increasing function in the second variable we have l l v - l £ l l < m(a,\\Ax\\)
+ a{a,c£).
Let ||Ax|| < C {a-priori
information).
In practice, we often have m(a, \\Ax\\)
<
C'a-\
where v > 0 is a positive constant. This estimate shows how good the spaces (or in general, the classes) Xa approximate the space X. To answer the question how to choose Xa and how to obtain these estimates is a principal problem of approximation theory (Nikolskii [1]). (The inverse embedding theorems (v shows how "smooth" are the elements of Xa), are also very important in our method.) We are now dealing with the case of unbounded a(a, \\x\\) for a € [0, oo) or with the cases of ill-posed problems. We often meet one of the two following situations: a) a(a, \\x\\) = c(\\x\\)a\
where 7/ is a positive constant: mild ill-posedness, a
b) a(a, \\x\\) = c(||x||)e ', where rj is a positive constant: severe ill-posedness.
96
DIFFERENTIAL
OPERATORS
OF INFINITE
ORDER
The remaining question is how to choose the mollification parameters a depending on e? To answer it we try to obtain 0(e) = inf {m(a, ||Ax||) + a{a, \\Ma(x -
x£)||)}
and its argument a := a(e). Having done this we hope that 0(e) -> 0 as e ~+ 0 in which case the problem has been regularized. Remark 5.2.3. It is clear that if 0(e) -» 0, as e -> 0 for certain a and m, and for an element x, then x is a weak solution of the problem x := A~ly. Thus, our concept is corresponding to that of Maslov [3], Domanskii [1]. 5.2.2. A technical lemma. The following lemma gives us a hint to find mollification parameters: Lemma 5.2.2.
Let c(/?)e). nficc(e) (e) = inf(j8 + c(/?)e).
i) lfc(0) = coexp(s^-"), i / > 0 , tftera i) lfe(0) = coexp(s/?-"), u > 0, then 1 nc(e)=filn nc(e) = ( - In 5Cove(-j,
\^
-
-
os e —» 0. as t - 0.
+ o(l) o(l) )I +
The infimum is attained, if 0 iis the sofotion /3(e) o/ the equation Tfce 4 m u m is attained, l / / 3 is the solution 0(e) of the equation v
3 (e _= - 1L / ^ffV+l e - "c~*0" CQSV CQSU
which can be written in the form
( i l n - , 1 , \ u+iw + 0(1)1 ^m~ = (\ n - ( i l n ^ e ^ .+!)/
1 -v }
ii) Ifc(/3) Ifc(/3) = = co/? Co/?"", tf > 0, then then ", n
r ( , 5 * T ++ , - *r^). ). n,M iip(t)==(coe)"+ MMn^ The infimum The infimum is attained, attained, ifif : 00 := = ( (cone)^. ■=0(e) 0{e)= W ) * .
as e —> 0. a S t ^ °-
A MOLLIFICATION
METHOD FOR ILL-POSED
PROBLEMS
Proof, i) (We follow directly Talenti and Vessella [1]). Let h := 0 +
coexpisp-^e.
Then = 1 - cosWT"- 1 e x p ( * r > -
ftf)
To find the 10 nna me minimum of f\\P) we put fi\P) '— 0. Then 1
/T-'exp^-")
CQSV€
With r := $
, A := sco^e,
we have r" + 1 exp(sr") = i Let r"
:=-ln-r-. s Ag
Then ~ = expfsr"). Ag Thus,
r ^ -\qI = -A'
It follows -T-T
1,
1
andgMrT=^ln^.
r"+1=q, From the last equality it is clear that aqV+\
+\nq
= In-,
and if A ~» 0, then q -> oo.Thus 1 In 9 \ - ( " + 1 ) / " sf/("+i)/
1 +
Since
9
+1
>/"
-+ oo when A -> 0, we obtain -S&fi
Thus,
(i/ l , _ l >l- y< "
-
•e
0, as A -
-(I/+I>/I/
'»!)"
0 (and so q -> oo).
-+ 1 a s A — 0.
97
98
DIFFERENTIAL
OPERATORS
OF INFINITE
ORDER
Hence q= : ( I l n I )
( +1)/
" +
'
0(1)
as A — >
0.
It follows r" = - In — , 3 + o(l) as e -» 0. « AClln^^+D/Rewriting 0 = r'1 we obtain the second assertion. The first assertion follows now directly from the second one. ii) Let M0)
:= P + Cop-"*-
Then f2[fi)
=
l-rjcop-^e.
Putting f2(0) = 0 we obtain fi := 0(e) =
(cor/e)1^1).
The first assertion is now clear. The lemma is proved. Remark 5.2.4. The estimates in Lemma 5.2.2 are exact but 0(e) depends on Co and is therefore not convenient for practical use. The following argument gives us simpler, but not optimal estimates. i) We try to choose 0 := 0(e) such that 0 + coexp(s0-")e->O, as e -
0.
Thus 0 should be chosen such that 0 - 0 and coexp(s0-")e -> 0 simultaneously. We have exp(s0-")e = exp( S 0-" + In e) -» 0 as e -» 0, or 5 0 - " + In e -+ - o o as e - 0. If we take S0~* = -Sine, where S is an arbitrary number in (0,1), then it is clear that coexp(30-")e 0:=0(e)
=
oe^-W'-O
as e - 0,
= ( - l n ^ T ^ - O
as e -» 0.
A MOLLIFICATION
METHOD FOR ILL-POSED
PROBLEMS
99
Therefore (__lne)
/3 + Coexp( 5/ 0-")e
-iu* ii) If we take
0 := 0(e) =
+Coe(i-«)/» -1/*
as e - > 0 .
(Vefn^\
then it is easily seen that
:= £ + cor"
Q0(e)
=
«*[,*+*,- - * I .
In this case 0 does not depend on Co but flp(e) still shows the H61der continuity with respect to the error. 5.2.3. Examples. Now we give some examples that may lead us to the problem of calculating the action of an operator A to an element x in a space X. a) Find a solution of the initial problem for the heat equation 82 u
Ou u(x,0)
= u 0 (x), x e IR",
we have (see Chapters 3, 4) u(x,t) --= exp
' a2" ' ^ .
This formal solution has a meaning when, e.g., u0 € Wg>{m.n). (It is well known from classical results that u 0 can belong to wider classes of functions). b) The solution of the initial problem for the heat equation backwards in time
d2
du
ft = - * * « > » . * * « • u(i,0) = uo(»), x e nr, can be written in the form «(s,t) = exp
-
-•
-H)-a*-«o
100
DIFFERENTIAL
OPERATORS
OF INFINITE
ORDER
This formal solution makes sense, e.g., when u 0 € H ^ ( l R n ) . c) The solution of the Cauchy problem for the Laplace equation d2u
d2u
0 , 0 < x < a, - c o < y < oo,
dx~2 + W2 "(0,y) =
= V(!/), - o o < y < oo
ux(0,y)
can be written in the form u{x,y)
=
sinh [ix dldy] ,. , id/dy ^
, .
cosh
+
^
%
f, (-i)"+1x2"+1 rf2"^) £J (2« + l)! ^ 2 n '
~(-iyx2ncP»ip(y) h (*•)>
d) The solution of the noncharacteristic Cauchy problem for the heat equation 82u
du ZZl
ti(0,t) =
, - o o < f < oo, 0 < x < 1,
ux(0,<)
= 0(4), - o o < t < oo.
can be written in the form
u(x,t)
—
sinh
fi
= cosh
x
V
3
?
—
l
Ut
•2" dnV »(t)
n)\ d *"
[*M
-ih(t\
fi
Idl !n+l
,
^ x dtn + ~ o ( 2 " + D!
We see that in the first case the pseudo-differential operator exp^/dx2] has the sym bol exp[-*£2] that is bounded. Therefore, the initial problem for the heat equation is well-posed. In the last three cases, all operators exp[-ttP/dx2], cosh[ixd/dy],... have unbounded symbols, therefore the three last problems are ill-posed in classical spaces. In the next sections we shall apply our method to various ill-posed problems, like numerical differentiation, parabolic equations backwards in time, the Cauchy prob lem for the Laplace equation, one- and multi-dimensional non-characteristic Cauchy problems for parabolic equations. Here the spaces LP(EC), L;(-TT,W) (1 < p < OO) play the role of the space X; and <ma,p oo rhe epace eo all trigonometric colynomials so degree not higher than k play the roie of subspaces Xa; Ma is a convolution operator generated by a kernel in Chapter 1; and A is a differential or a pseudo-differential operator with constant coefficients. It is well known that the convolution operators
A MOLLIFICATION
METHOD FOR ILL-POSED
PROBLEMS
101
Ma commute with such an operator A (see Hormander [1]). 5.3. Numerical differentiation. The process of numerical differentiation is well known to be an ill-posed problem, it has been discussed by many authors, and a large number of different solution methods has been proposed. For references we refer the reader to Baumeister [2], and the new papers Murio [4], Murio and Guo [1] and refer ences therein. Murio in his papers [4], Murio and Guo [1] has applied the mollification method of Manselli and Miller [1] and of himself (Murio [l]-[3]) with the Weierstrass kernel to the problem of numerical differentiation. He developed, indeed, the results of Vasin [1] further. In this subsection we apply our method to this problem. The method seems to be simpler and clearer than that of Murio. Furthermore, we can use fast Fourier transforms to implement it numerically. 5.3.1. The case of a function defined on JR.1. Suppose that the function / € LP{JR}), 1 < p < oo, is given inexactly: (5-3.1)
\\f-fc\\P<«,
where e is a positive number. We have to reconstruct from fc the function / itself or its derivatives. At first, we mollify f€ by using the analogue of the de la Vallee Poussin kernel (see Chapter 1.12) :— Vvyj-iX) hf —,r>./; atVx-K
—
l1
f^ H
VTcJ-oo
Ccos(i/(x s(t/(a:
°
-- y)) — cos(2;/(a; ~ y)) - co<2<x - '-^-fMdy. »)) f (v) dy
{x - yy
(5 3 2)
The functions / * are now belonging to the space OTj^lR1) and therefore the problem of calculating its derivatives is well-posed (since in this space all high frequencies are cut off!). Following the general scheme in Section 5.2, instead of calculating / W , we calculate (/*)<*>. Theorem 5.3.1. Suppose that the function f belongs to L^JR1) and there exist its k-th and (k + \)-th derivatives that also belong to ^ ( M 1 ) . Suppose further that the inequality (5.3.1) and ll/ ( A + 1 ) L < Eu (5.3.3) where E\ is a positive, finite number, are valid. Then, for every fixed u £ (0,oo), the problem of calculating derivatives of"" defined in (5.3.2) is well-posed. Furthermore, with v' = t - 1 ^ ||/W-(/rTWllp
< c^,
where c is a constant depending only on E\ and k.
(5.3.4)
102
DIFFERENTIAL
OPERATORS
OF INFINITE
ORDER
Proof. From the properties of / f (see Section 1.12), we have
ll/W ~ (K)l%
< ll/W-(/o)(fc)|lP+ll(/o)(i)-(/e")W|lP < <
(1 + 2V3)E„(f{% (l + 2V3)E„(fW)p
+ (2^)fc||(/o) - ( / n i l , + 2V3(2v)ke.
The last inequality follows from the property of the de la Vallee Poussin kernel, and Bernstein-Nikolskii's inequality for functions in mv,p(TR}) (see Theorem 1.9.2). Further, the inequality (5.3.3) yields li/W-(/f)(%
+ 2V3{2v)ke
<
(l + 2VZ)E„(fl%
<
c(l + 2v^)||/<* + 1 >|| p l + 2 v ^ ) * e
<
0 ( 1 + 2 ^ 3 ) ^ 1 + 2^(2*/)^,
with a positive constant c. Since c and E% are in general not known, we choose v* = e ~ x / W . Then \\}(k)-{r:t\
< {41 + 2 ^ 3 ) ^ + 2 * + ^ } ^ ,
implying a Holder continuous dependence on the error e. Thus, the theorem is proved. Our method is interesting in the sense that we get the Holder continuous dependence on the error in all spaces Lp, and since the mollified data # are belonging to SH^IR 1 ) (the Fourier transforms of its elements have a compact support lying in (-v, v)) we can use fast Fourier transforms for numerical implementation. 5.3.2. The case when the function / is given in ( - * , * ) . We consider the case that / is given only in ( - * , * ) . Suppose that the derivative of the first order / W exists, and we have to find it from a "measurement" /«: ll/-/£|lM-^)<£.
(5.3.5)
For simplicity, we assume that / € CW, the space of all continuous periodic functions in (-7r,7r), and so f, € L*p(-K,ir). Now, as in Section 5.3.1, we use the de la Vallee Poussin kernel to mollify the data fc: U
^h-2n*L
If*
cos((n + l){x - y)) - cos((2n + l)(x - y))
ain a ((*-y)/2)
^ ^
(5 3 6)
-"
A MOLLIFICATION
METHOD FOR ILL-POSED PROBLEMS
103
The function ft is a trigonometric polynomial of order In (see 1.12.1), and therefore the problem of calculating its derivatives is stable, and very simple. Using Bernstein's inequality (Nikolskii [1], p.92-93) for trigonometric polynomials instead of BernsteinNikolskii's inequality, we can, as above, prove T h e o r e m 5.3.2. Suppose that the function f belongs to Lp{-w,w) D CW and there exist its first and second derivatives that also belong to £„(-*-,*•). Suppose further that |\f2% < E2, and the inequality (5.3.5) is valid. Then, with n" = [e_1/2], ([a] denotes the entire part of the number a) | | / ( 1 ) - ( / : * ) ( 1 ) l l P < «fc where c is a constant depending only on E2, and ff order 2n* defined by (5.3.6).
(5.3.7)
is a trigonometric polynomial of
5.3.3. The case when the Function is given only at discrete points. In practice, the following situation is often met. A function / is measured at some dis crete points with an error, and one has to find its derivatives from these data. This problem is more complicated than the above. However, our mollification method is still working in this case and gives the very same results as above. Suppose that the function / € CW, the set of all continuous periodic functions on [ - * , * ] , and / is given only at equidistant points xk := xf> = 2A:7r/(2n + 1), k = 0 , 1 , . . . , 2 n , with an error \f(xk) - / t l < e, k = 0 , 1 , . . . , 2n,
(5.3.8)
where e is a given positive number. Now, we mollify / by the second interpolation method of Bernstein. Namely, we put /—««/[/;;*].
(5-3.9)
The function Un [f;x] has been described in Section 1.15, and it is a trigonometric polynomial of order n. Suppose there exist the first and the second derivative of / . From Bernstein's in equality for trigonometric polynomials and from Theorem 1.15.1 we have \Df-DUn[f';x}\
<
\Df-DUn[f;x}\
=
\Df - Un [Df;x]\ + \D(Un [f;x] - Un [f';x})\
<
(l + 2* + + 2U(2?r + 47r2)e,
+
\D(Un[f;x]-Un[r;x])\ 4*2)E:(DfU+u;(Df,^-[)
DIFFERENTIAL OPERATORS OF INFINITE ORDER
104
with D as operator of differentiation. Suppose further that max \D2f\ < E3,
(5.3.10)
where E3 is a positive constant. Then
\Df-DUn[T;x]\
< c1{l+2x + 4x*)E3± + c2E3^-[ <
(ot(l
+ 2TT + 4TT2) + C2TT) E3-
+
M2*+4*2)t
+ 4(TT + 2 T T > €
< ft- + ftne. n Again, since a, c 2 , E3 are in general not known, we choose (with denoting [a] the entire part of a)
n := n Then
1 r 1 = s*.+1.
\Df - DUn. [f; x}\ < ( f t + ft(l + >/e)) V& Thus, we have proved the following theorem. Theorem 5.3.3. Let f be a function in ft*, given inexactly at equidistant points xk := x[n) = 2kn/{2n + 1), k = 0 , 1 , . . . , 2n, iy / * I / O * ) - / S | < « , fc = 0 , 1 , , . . , 2n. Furthermore, suppose f G C 2 [0,2TT] and ll^/lloo < ^ 3 , uAere £3 *s a ^ni*e posiiiwe ni/m6er. Then, ma,x \Df - DUn.[f';x]\
< cyfe.
Here c is a finite positive number depending only on E3, n' = [ 4 ] + 1, and
DUn.[f';x} -' L n>)
9
V
2n + lJ
2n*
<4 (*) ==
o„ , 1 Z)/t c o s ( m : r *).
*£?« =
2n + 1
ifeS
V
2n + l / J J '
A MOLLIFICATION
METHOD FOR ILL-POSED
105
PROBLEMS
5.4. The heat equation backwards in time. The problem of finding solutions of parabolic equation backwards in time is well known to be ill-posed. It is ill-posed in the sense that a small perturbation in the data may cause large errors in the solution. There have been many attempts to solve this problem numerically and stably. The readers are referred to Baumeister [2], Carasso, Sanderson and Hyman [1] and references therein. In this section we shall use our method to solve the following one-dimensional problem (the results for the multi-dimensional problem are analogous, and therefore we do not write them down here): find u(xt)) from ut(x,t)
= uxx{x),),
u{x,T)
=
(5.4.1)
-oo < x < oo, 0
MT'x), - o o < x < o o .
(5.4.2)
This problem is ill-posed. The following simple examples will show this fact. Example 54-1: With Uj(x)
= e*« we have
n
U (x,t)
=
e(T-t)d
2
/dx2
einT
_ e(T-t)n*
einx
_
We see that ||ixJ||£oo(R1) = 1, ^ut ll«"(-.«)lko(H») = e ( r "' ) n 2 -» °° *> n -> cx5, and 0 < t < T. Thus, the problem is ill-posed in the L^-topology. Example 5.4.2: With u"T{x) = 2 ^ , we have
-**N*(fl =
T
=*«■ *) == e( -
f e< T "-*)e
1°
2
iei<"
Thus,
Kt-Miw
= IK(s<)Hfa(R»)
s=
I/
e 2 ' T _ t ^ d£)
—> oo
as v —> * oo, and 0 < t < T. On the other hand K H ^ R 1 ) the Z,2-topology.
= 1- xt means that the problem is also ill-posed in
By our method, to our knowledge, for the first time a uniform stability estimates of Holder type in Lp-norm (1 < p < oo) for the problem (4.ll)-(4-12) is established. In the case p = 2 we have very sharp estimates.
106
DIFFERENTIAL
OPERATORS
OF INFINITE
ORDER
Since it is simpler to deal with the case when the data are in L2, we consider two cases of the data in L2 and in Lp (1 < p < oo) separately. For convenience, we write || • ||„ instead of || • hp(m}) in this section. 5.4.1. The L 3 case. Suppose that u(-,t) and u T (-) are to be in L2(1R}), and instead of the exact uT(-) we have only the measured data u'T(-) E ^ ( H 1 ) such that \K-
(5.4.3)
uT\\2 < e.
We mollify uf, by the Dirichlet kernel (see Section 1.13)
1s Uijt
Since «{. G I 2 (IR 1 ), <
¥ XLrp
-
roo , sinti^fx, —! V\ ) H u w*^ " '^» r W
V^F ■/-«*, x-y e ^ ^ ( I R 1 ) (see Section 1.13).
(5. 1.4)
""
Now, instead of solving the equation (5.4.1) with the data in (5.4.2), we solve it with the new mollified data u/. Denote the solution of this new mollification problem by uc<": we get the following system: =
u\'"{x,t) u"(x,T)
u^(ar, i), - o o < x < oo, 0 < t < T,
= uY{x),
(5.4.5)
- o o < x < oo.
(5.4.6)
The solution of Problem (5.4.5)-(5.4.6) is stable, since the data uf belong to S H ^ M 1 ) (see Chapters 3, 4). We recall here that the Fourier transform of a function in SJt^IR 1 ) is an ordinary function and its support is a compact set lying in [-*, v]. Theorem 5.4.1. Letu be a solution of (54.1)-(5.4.2) anduT be given approximately by (54.3). Then, for every fixed v € (0,00), the solution u'-" of the mollified problem (5.4.5)-(5.4.6) is stable. Suppose that for a nonnegative integer k there exists the derivative dku(x,Q)/8xk, {&>u{x,0)/dx° := ti(i,0) := Ua(x)), and that \\dku(x,0)/dx%<Eh,
(5.4.7)
where Eh is a finite positive number (for convenience, suppose e < Eh). Then, for t small enough,
|K-,0-n,0lla< 1+
TlD
^
J
U ^ + ln(ln^)-i/2j J
£ ? e+Tln ^
- * ^
V e)
(5.4.8) (the first term in the right-hand side is bounded above by 1 + {2Tf'2), 2 1/2
E \-*/ \ :=(\ ln E (ln —^ "*.(!■.&(», ir 7H 7\) 1j i/*
•
and
<5-4-9)
A MOLLIFICATION METHOD FOR ILL-POSED PROBLEMS
107
In the case Eh is not known we have, with -- (i.in}-(i iyk/2y\k/2y\
(5.4.10)
:=(i.in}-(i v --:= n niy
the estimate
((
\\ k*^2\ l%\
/I
1
Ek
+ [l I+EA
/ / we are interested only in u(x,0), /
IK,o)-^"-(-,o)|| | K , 0 ) - ^ " - ( - , 0 ) | | 22 <^ <^
/2 -' ^Ul'0-* *
£
"(lne)"" <*{*»~)~**
then, for k>\,
I1 j In a
(5.4.11) (5.4.11)
,
fc£
^ * . 1 + 1 / t + oo(l) (l)
1 /O I
In
as ease-0, —> 0, (5.4.12)
where Wiere
k/2
( v{:=
I
V2 ^ln
5f*
2eTf 2eT(1 ln|%V
+0(i)
J
as£-0. t —t 0.
(5.4.13)
For a num&er 5 £ (0,1), andfc> 1 we have, with I 8 \1/2 i/** as e -► -» 0, 0, := I1--— —I lnneeJl *? :=
(5.4.14)
the estimate |K,0)-^(.,o)||
2
(
f. \ ~kl2 ~ i ^ -— | > lnej eJ as e -» 0. 35 0.
(5.4.15)
W»tft iAes mollification parameter v{* we still have a Holder estimate of type (5.4.8) for the solution u{-,t) in [0,T]. € OT„, (IR), we have (see Chapters 3, 4) Proof. Since w^" uc/ € a»„l22(IR), Proo/. , T , «r(*). «(*,*) = e< e
From (5.4.16), (5.4.3) and Bernstein's inequality (Theorem 1.9.2), we obtain obtain u (T t)u ||u°'"(-,i) - uuet-'"(-,t)|| (-,t)\\<< ee(T ||u°'"(-,i) --t)u \,\,
(5.4.16) (5-4-16)
108
DIFFERENTIAL
OPERATORS
OF INFINITE
ORDER
On the other hand, from (5.4.7) we get
B«(.,*)-u°^(-,*)lla B«(.,*)-U°^(-,*)lla
=
H-,*)-u°'"(-,*)||a \\u(;t)-^(;t)\\2
(o)w«f - (£H -"[*Ssl]®H«f -
(/:
= =
e-*e
^sin(i/x))
i^ _ 2F
X
j^mmf%m u (0) [L>. [L>»ml||4' || = maxj /
(
<
k
-t? 2
6
-t(
7 2
0
)
:)
2
dt
—tu2
6
\ 1/2 \1/2
di\
h
)
\\u(0k)\\2.
Thus, Thus,
||U(,,i
M -(-t)|| 2 <e(
r
-^ 2 e+ e 7 2 ^.
Substituting v = V (or v = v**) in the last inequality we obtain the estimates (5.4.8) and (5.4.11). The estimates (5.4.12) and (5.4.15) follow directly from Lemma 5.2.2 and Remark 5.2.1. The theorem is proved. 5.4.2. T h e Lp ( i < p < oo) case. To our knowledge, up to now there have been very few considerations for the heat equation backwards in time, when the data are given in Lp, 1 < p < oo. By our method, we can establish a Holder estimate for the case of the data in L^IR 1 ), 1 < p < oo. Unfortunately, we do not succeed in the case p = 1. Suppose that, the solution «(.,<) of Problem (5.4.1)-(5.4.2) and the Cauchy data uT(-) are to be in L ^ H 1 ) , 1 < p < oo, and instead of the exact uT(-) we have only the measured data u'T £ L^1R1) such that
K-ur|U<e.
(5.4.17)
The difficulties here are that if p > 2, then the Fourier transform of u(; t) is, generally, a distribution, therefore the above estimation procedure for the L, case is no longer applicable. We now use our mollification method and the results of pseudodifferential operators with real analytic symbols to regularize our problem. We mollify the data uf, by the analogue of the de la Vallee Poussin kernel (see 1.12)
M - a*(2vy) cos(^)7oe(2^^ 4-^KH.r == —^r^-,)cos r «T(* ■y) „2 "■y u§.- - » Ml(ucT) -- = ut/
wvJ-oo WV J-tx
y2
(5.4.18)
A MOLLIFICATION
METHOD FOR ILL-POSED
109
PROBLEMS
Since 14 e ^ ( I R 1 ) , MJ" € ^ ^ ( M 1 ) (see 3.1.2). Now, instead of solving the equation (5.4.1) with the data in (5.4.2), we solve it with the new mollified data B$". Denoting the solution of this new mollified problem by ul'u, we get the following system = w£(*,*)i - o o < i < o o , 0 < ( < r , = u^{x), - 0 0 < x < 00.
ut'"{x,t) u^(x,T)
(5.4.19) (5.4.20)
The solution of this problem is stable, since the data B J " belong to 9tt2„,p(]R) (see Chapters 3, 4). We recall here that the support of the Fourier transform of a function in Wa.g.flR) is a compact set lying in [-», j/]. Theorem 5.4.2. Letu 6e a so/u«io7i of (5.4.1)-(5.4.2) and uT be given approximately by (5.4-lV- Then, for every fixed v € (0,oo), the solution u<'" of the mollified problem (5.4.19)-(5.4.20) is stable. Assume further that ||u(-,0)|| p < Eh, where Eh is a finite positive number. Then, for e small enough, \\u(-,t) - u^(-,t)\\p
< 2 ^ 3 [(c*Eh)1-t'Tet'T
where c* = 1 / ( 2 ^ ) ( 1 + 2v^)e3/2, ,
,
+ {c'EHf-^^e'l^]
(5.4.21)
and : =
(^
l n
£^)
1 / 2
.
(5.4.22)
In the case Eh is not known we have, with
p* : = f_L l a iV
(5.4.23)
the estimate ||tt(.,*) - u^'(-,t)\\P
< 2V3 [e^ + c'E^IW]
.
(5.4.24)
1
c
Proof. Since u f £ C M I R ) , we have «-(*,*) = e(T-i)ti2/i*V(z)-
(5-4-25)
Thus,
|«n,«)-»-»<-,<)| 11
v
'iiP
|K->«-«;K-»W| iip
11
< £(^|||1M:(,T-^)| (Bernstein-Nikolskii's inequality (Theorem 1.9.2)) °° (T — t)n
11
i < n=0 Ei£TT -(2^rK(«r - « T ) | L "• n=0
"•
(due to the properties of Ml)
< 2>faF-4*#t
{v
> i).
110
DIFFERENTIAL OPERATORS OF INFINITE ORDER
(From this estimate we see that the solution w£'" of the problem (5.4.19)-(5.4.20), for every fixed v € (0,oo), is stable.) On the other hand, (Widder [1]) u(x ,t)-- = /
ifc(x--y,t)u(y,0)dy,
for p > 1,
J—oo
where k(x,t) =
V47Tf
Set u{y,0):=uo(y). We have u{x,t)
= V^k*u0.
The following lemma is essentia L e m m a 5.4.1. The following estimation holds £„(fc(-,*))* < ^~tv\
if " > \/3/(2f)
(5.4.26)
{see 1.12 for the definition of Ev{-)„). Proof We have
k\i,t)-.
1
-ti\
Since k{-,t) is an even function in I ^ I R 1 ) , the function
m,t)=
k(x,t)cos(x£)dx
= e-«
-OO
is also an even function of the variable ( G IR1. Furthermore
m o > 0 ,^il s o ,^M, 0 ,«M, 0 , [ o r £ ,^). Now, it follows from the Sz.-Nagy criterion (Achieser [1, p. 190]) that the function [k{x,t) - kv{x,t)]cos{vx) does not change iis ssgn in IR1, for every v > Js/^Zt), where
Ml,1) = ^±Z*l£ H yK*l^). v
i=-oo
X ~ rZv ~ V
111
A MOLLIFICATION METHOD FOR ILL-POSED PROBLEMS
. y/vm
It follows from Theorem 1.14.1 that, for v I
4 ~ tf((2Z + l ) M ) . t exp[-*(2Z+l) 2 i/ J ] ^ 4 _fl/2 2/+1 - we
4 ~ TfeS The lemma is proved. Note that u°'"(x,t) =
tF-W^Wz)
It follows
= v&
* u0
= y/^tk-^V^k]*^. Thus, on one side, from the lemma, if v > y/s/(2t), then
M.,t)-u°"(.,*)ll, <
k-]Hv-*k
< (l + <
IKIp
2V3)^(ifc)l|Ki
(l+2V3)V-2||wo||P.
On the other hand, if 1 < v < ^3/(2*), then \\u(;t)-U°*(.,t)\\,
<
H;t)-\[lK{-)*k(.,t ) ll«o||p 1
< <
(l|fc(-,<)lli + ^BK(-)llill*(-,*)lli)ll«oll, i + \F2V/3TTJ
|j u0 || P
112
DIFFERENTIAL OPERATORS OF INFINITE ORDER =
(l + 2V6^)e 3 /V 3 / 2 |K|| P
<
(l + 2v^)e 3 /V"- 2 || U o|| P .
Hence, for v > 1, \\u(;t) - u°'"(-,i)llp < (l + 2 v ^ r }
e3'2e-^\\u0\\p.
Consequently, with v > 1 and the condition ||«(-,0)||p < Eh, we obtain \\u(;t)-U<'"(;t)\\p
< 2y/3
e ^ - ^ e + ^ = ( l + V^)e^e- f e 2 |Kfl,
< 2V3 e ^ l ^ +
c-e-^j
The proofs of the estimates (5.4.21) and (5.4.24) are now straightforward. We note here that the method works well also for other parabolic equations back wards in time, and other domains, for example, for the equation ut = a(-A)5'6u (see Carasso, Sanderson and Hyman [1]), for finite domains, or for boundary value problems, etc. However, the scheme remains the same, so we do not repeat the details. 5.5. The Cauchy problem for the Laplace equation. The Cauchy problem for the Laplace equation can also be treated by the mollification method analogously to that for parabolic equations backwards in time. We therefore omit, in this section, all proofs. An extensive list of literature on the Cauchy problem for elliptic equations can be found in Dinh Nho Hao, Tran Due Van and R. Gorenflo [1]. We consider the problem D2u a^ "(0,y) =
d2u + ^ = 0, 0 < x -
a> - ° ° < y < ° ° ,
ux{0,y) = 0, -oo < y < oo.
(5.5.1) (5.5.2)
Suppose that the solution u{x, ■) and the data
\\
V21F y-00 V^F^-°o 1
lv
1
y - l
Since v?< € ^ ( H t ) , V> ' € € l ^ I R ) (see Section 1.1.).
K
' ' '
A MOLLIFICATION
METHOD FOR ILL-POSED
113
PROBLEMS
Now, instead of solving the equation (5.5.1) with the data in (5.5.2), we solve it with the new mollified data tp^'. Denoting the solution of this new mollified problem by u£'", we get the following system «K e
u '"(0,y)
+ =
u
vv = °> 0 < x < a, - o o < t/ < oo, t,u
(5.5.5)
= 0, - o o < y < oo.
(5.5.6)
The solution of this new mollification is stable, and by the method of Section 5.4.1 we can prove the following theorem Theorem 5.5.1. Let u be a solution of (5.5.1)-(5.5.2) and
|K*,-)-^"*(*,-)||2
< 11+
aU
(2£.)'/'£ 1 -/- (In ™lYkX,\
/—nr
(5.5.7)
where ««•" has 6een rfe/iiiaf by (5.5.4), and
^Iln^fln^) a e \ e I a
t
\
(5.5.8)
e /
In the case Ee is not known we have, with U
1 , 2 / . 2> -* *'-=alnl(lnJ
(5.5.9)
the estimate k\
I l + 2£e
H*, •)-«'•""(*.-)lb< \
alnj
2^-kx/a
W + in(mi)-k)
■^"(kf) (5.5.10)
If we are interested only in u{a,y), then, for k > 1 we have, with ( u{: =
1,
(
4*£«*
2£a(iln^|^J
)
m
J
it
as e —> 0,
(5.5.11)
114
DIFFERENTIAL
OPERATOSS
OF INFINITE
ORDER
the estimate f
\\u(a--)-u^'(a,-)\\2<2Eek
4kEek
-In
V* 2 e a « ( l l ^ )
1 T m
f
+ "(1)\
ase^0. (5.5.12)
For a number S e (0,1), k > 0, if we take
V := S lne
" [- a )
ase -» 0.
(5.5.13)
Then
|M,0)-««■•**(•, 0)||2~2J5,t
ase — 0.
»
(5.5.14)
"
With this mollification parameter we still have a Holder estimate of type (5.5.7) for the solution u(x-)) in [0,a]. The case of periodic solutions can be considered in the same manner, however, instead of using the Dirichlet kernel, one should use the de la Vallee Poussin kernel. 5.6. The noncharacteristic Cauchy p r o b l e m for parabolic equations. The noncharacteristic Cauchy problem for parabolic equations is the main subject of in verse heat conduction problems. The problem is well known to be severely ill-posed. An extensive list of literature on noncharacteristic Cauchy problems for parabolic equations can be found in Dinh Nho Hao and R. Gorenflo [1]. In this section we consider three model problems: one-dimensional and multi-dimensional problems in infinite domains, and a one-dimensional problem for a finite domain. We use the mollification method to solve these problems stably and to get stability estimates of Holder type. 5.6.1. The one-dimensional noncharacteristic Cauchy problem for a parabolic equation. We consider the following noncharacteristic Cauchy problem (see Knabner and Vessella [2]): tt
a{x)uxx
b(x)ux-c(x)u
= 0,
x€(0,/),t6lR1,
u(0,t) =
(5.6.1) (5.6.2) (5.6.3)
Here / > 0, a, 6, c are given functions such that for some A, A > 0 A
a € W '°°[0, /],
x€[0,Z],
(5.6.4)
b e W^°°[0, /], c 6 Ioo[0, I].
(5.6.5)
A MOLLIFICATION
115
METHOD FOR ILL-POSED PROBLEMS
Set [*a(s)-Wds, L := A(l). Jo ' As we consider the problem in time in L2{m}), we assume A{x):=
ip € L2(1R1). If v(x, 0 is a solution of the Cauchy problem a(x)v„(x,
0 + b(x)vx(x, 0 + c(x)v(x, f) - i(v(x, f) = 0,
»e[o,q, ^effi.1,
(5.6.6)
1
w(0,O = l,f€lR , e s (0,O = 0,4elR 1 .
(5.6.7) (5.6.8)
Then (see Knabner and Vessella [2]) u(x,t) =
F->[v(X,0m],
and i » ( U ) * 0 , V^elR. Furthermore, there exists a constant C = C(A, A) > 0 such that for x € [0, /], £ € E \ > C [1 + i + | | ^ a | | ; + iiZ?6|U + (1 + / 2 )||6||i + ||c|[, the following inequalities hold \v(x,0\ < Cexp [n\\D*a\\l + \\b\fj] exp (v^2A(x))
.
|„(l,fl| > C-> exp [-Z2 (HB»a||; + ||6||L) ] exp ( v & ) . Since the conditions (5.6.4) and (5.6.5) hold, ||Z>2a||2 + HJD4HL + ||6||«, + ||c||, is bounded. It follows that for all£ € IR, there exist constants Cu C2 (that depend only on A, A, /, ||Z>a«||i, Il^lli. IIMI- and ll*lli) s u c h t h a t \v(x,()\
< C1exp(V^A(x)),
K«)|
> C2exp{y/mL).
(5.6.9) (5.6.10)
Now, suppose that the exact Cauchy data if € IR1 are given inexactly by
y-v%
(5.6.H)
116
DIFFERENTIAL
OPERATORS
OF INFINITE
ORDER
The problem (5.6.1)-(5.6.2) is ill-posed because of high frequencies of the Fourier transforms of the Cauchy data. To avoid this fact, we mollify the measured data
,' - » ^ = Mfr := J = I" ^(t) Sin(( ; (< ~ T)) rfr. \/27r J—a>
(5.6.12)
t—T
From ^ £ L2(]R}), we conclude that if* € SR^IR 1 ) (see Section 1.13). Now, instead of solving the equation (5.6.1) with the Cauchy data <^£, we solve it with the mollified data
= 0,x€ (0,/),t € 1R1,
(5.6.13) (5.6.14)
u^(0,t) = ^(t),teM\ <•"((), i ) = CM S I R 1 .
(5.6.15)
There exists a solution of the problem (5.6.13)-(5.6.15) that is unique and stable, since the high frequencies of ft* are cut off (see Chapters 3, 4). T h e o r e m 5.6.1. For every fixed problem (5.6.13)-(5.6.15) is unique lem (5.6.1)-(5.6.3) such that there integer, and \\duh(l,-)/dt% < Epk,
v G (0,oo), the solution ue'" of the mollified and stable. If u is a solution of the Cauchy prob exists 8uk{l,-)/dtk £ L^IR1), k is a nonnegative where Epk is a finite positive number. Then
\\u(x,.)-u^(x,.)\\2
< [1 + £/^ln^ | {CE " [ In^+m(in^))-2*jV
MW/Ltc y-A(*),L f l n ^ V ' V t ) (5.6.16)
where Cu C2 are constants from (5.6.9)-(5.6.10),
and
K:=2(lln C ^(ln^)- 2 y. IfCi,C3
and Epk are not known, then with
we have \\u(x,-)-
- «'•%, Oils < [ Ci + C2£:pA i / x / ^ l n « _. | e x - ^ / J \ \ lnI+In(in A) /
e)
A MOLLIFICATION
METHOD FOR ILL-POSED
117
PROBLEMS
With
2
*;••=
fv2
L
2V2kC2Epk
)
1
, as e —► 0,
^(fm^r-) "'
we get a better estimate I «(/,-)-u'-"(/,-)||2
<
C2Epk
41 m
2V2kC2Epk
LeC^fln^^)
1+2*
+ "(1)
as t —» 0. For a Tiwmier j G (0,1), fc > 0 i/ tue take v\ ■- f - ^ - l n e ]
u s e -► 0.
77iera, ||u(I, •) - u'* (I, .)|j, ~ C 2 £„, ( - ^ lne)
.
Proo/. It is clear that ||U°-(X,.)-U-(X,-)I|2
=
| K * ( Z , . ) - , ~ * ( z , Oil,
= |Kx,.)FK^-^)]||2 = (jyMF{M?{
1/2
< Since »{U) /
& exp f
1/ i / 2
2
\ A(i) 1 e.
0, for £ € HI1, u(x,t) =
F-*[v(x,Om}=F-*
U)H
The condition ||3u*(7,-)/«**Ha < Epk yields
||«(*,-)-«•"(». OIIJ =
«(*,•) «(».-) »(/,.)
II2 2
\M>»
«(i ,* ) ( ^jfc\*w " v . ;
\ V2
dM
118
DIFFERENTIAL
OPERATORS
OF INFINITE
^ <
sup ,, J r/.s, \t\>* v(l,Q (£)*
< <
c^M^^Lk(^,,u{l ,)ldt%2 C 2 e (. M -,)v^i-||^ ) /^||
ORDER
\\dku(l,-)/dt%
< Thus, H«(», ■) - ««■"(*, -)||2 < (h ex P ( | f f/2 A(x)) e +
c^W)-*)^±
E p k
.
The proofs of the estimates in the theorem axe now straightforward. 5.6.2. M u l t i - d i m e n s i o n a l p r o b l e m s . Let n be an integer > 2. Seta; = {xux2,,.., xn) := (xux). We consider the problem (Dinh Nho Hao and R. Gorenflo [1]): "t
=
«*,«, + ■ ■ • + « I „ I „ ,
(5.6.17)
0 < Xi < 1, —CO < X 2 , X3, . . . , X n , < < OO,
u(0,*,t) Ull(0,x,t)
=
(5.6.18) (5.6.19)
(Note that the problem considered in Guo and Murio [1] can be reformulated to such a problem.) We assume
«(
=
(M- + £ 2 ) « ( X I , £ , T ) ,
= £(£,T), = 0,
-OQ
0<
Xl
< 1, - 0 0 < & , . . . , £ , < 00,
<&,...,tn,T
-QO<6,..,,e.,T<00,
where £2 := ff + . . . + £*. From this system it is not hard to see that 8 ( H , * , T ) = c o s h e r + £')#£, r),
(5.6.20)
where Jir + 0r + Vp= ^ = ( ^/ s= ^~
2 ^ 2++ W ^ 2 )+ +e e + + ti ss ii gg nn rr ^y ^+ T( a^ 2p-Te 2^ ) -.
T
It is easily seen that (V3-v^)/2exp ^ I ^ T ' + K ^ + P ) ^ ) < < I cosh(xiy/iT + ?)\ < v/2exp f * i ^ / ( P ) a + T» + £»)/2) .
(5.6.21)
A MOLLIFICATION
METHOD FOR ILL-POSED
119
PROBLEMS
Since cosh(x1^/?^TT^) + 0, V x € [0,1], = ^ ( ^ ^ . r )
u(xux,t)
= ^M^v^TF)
V
(5 622)
COsh V«T + f2
Suppose that instead of the exact
(5.6.23)
We see from (5.6.20) and (5.6.21) that the problem (5.6.17)-(5.6.19) is severely illposed, a small error in the Cauchy data may cause very large perturbations in the solution u(x1,x,t). To overcome this difficulty, we mollify
—y V '
:=
1
/
/
7o~~\2 /
sinh(x2-j/2)]
, / ,
\2,TT) 7 R " -
1
Jm.
1
- yB)]
sinM*„
X2 — J/2
x
xn — yn
(5.6.24)
xsinM«-T)] t — T
Here v := fa, i/2), i/i and i/2 are positive numbers. We say is positive and finite, if Vy and v2 are positive and finite. a r t ^ n r ) , the class of all entire functions of exponential of a real vector (s,t) € IRn lie in L2{W), and so supp ^
that the vector v = (vuv2) From Section 1.13, tpc>" S type v which as functions C [-fi, ^ i ] " - 1 x [-**, v2}.
Now, instead of solving (5.6.17)-(5.6.19), we solve its mollified problem (we denote the solution of this new mollified problem by «<•"): «r
=
<„+•••+«&». 0 < X i < 1 ,
(5-6-25)
- O O < X2,X3, ••• ,In,<
< 00,
ii"(0,i,J)
=
^"(x,t), -oo<x2,,..,xn,t<+oo,
(5.6.26)
u^(0,*,t)
=
0, - o o < x 2 , . . . , x n , * < o o .
(5.6.27)
Theorem 5.6.2. Let u be a solution of (5.6.17)-(5.6.19) and? be given approximately by
■, •) - « ^ ( * n '. Oil. < 2V2K^(4Emhrel-*\
(5.6.28)
where v*m:=(am,(a'my),
l
a'm : =
l n ^ .
(5.6.29)
*ȣ
\/(y/n + n-l)/2 /n tfie case Emh is not known, with C"(
I, -
*
1W
J(v^ + n-l)/2
1";
«
(5-6.30)
120
DIFFERENTIAL
OPERATORS
OF INFINITE
ORDER
we have (5.6.31)
\\u(xU;-)-u^(Xu;.)\\2<(V2Kn+4Emh)e^.
Proof. Since ■) - S ^ f o . ' . Oils = (/
,
|cosh ( x , y ^ + 7 ) ( # £ , T)-?(t,r))
< v^/c„exp (xJfy^
[ dfdr) '
+ i/?) / 2 J e.
+i ^ l M
On the other hand, (5.6.22), (5.6.21) and (5.6.24) yield ||«(* x , -, ■) - u°'"{xu -, -)|j 2 = ||5(*j, -, •) - &>(xu -, OH2 (r
f
|cosh(x1N/F+?)
- ^ W U I cosh^/FTF
J ' ^ V
U(
7 2
'^rJ| * J
< 4exp ((*, - l ) t / ( ^ ( n - I K + ^+(n-
1)*?) / ^ ||u(l, -, -)|[a.
Thus, ||»(«i, •, •) - • v ( * i , v ) | | 2 < | | t » - « 0 | " | | a + | | « 0 | , > - « ^ | | a < v^c n exp L ^ f n
- I K + ^ + (n - l ) , ? ) / ^ e
+4exp(a; 1 - l)\J(y/(n - l)v\ + v* + {n - l)^)/i||w(l,-,-)|| 2 . Taking {v2f = i/, = a > 0, and putting by P := (a,a2), Emh, we have
then, since ||u(l, -, -)||a <
M**»v)-«t,B(*i,v)l|a< exp
+ n-- l ) / 2 ) e
+ > / 2 ( 3 - V 5 ) e x p ((xx - l ) a ^ / ( ^ + « - l ) / 2 ) £ m h . The proofs of the estimates in the theorem are now straightforward. 5.6.3. T h e case of finite d o m a i n s . The case of finite domains is more difficult than the above cases for infinite domains, since we cannot apply the Fourier trans form. However, our mollification method is still working, and we can get a pointwise
A MOLLIFICATION
121
METHOD FOR ILL-POSED PROBLEMS
stability estimate of Holder continuous type. Our estimate seems to be simpler and sharper than those of Knabner and Vessella [1] but does not include them. We consider the noncharacteristic Cauchy problem in a finite domain (Knabner and Vessella [1]): («(*)«.)« u(0,i) ux(0,t)
= ut, 0<x
(5.6.32) (5.6.33)
=
(5.6.34)
0, 0 < < < r .
Here, i) /, T are given positive numbers, ii) a e Loo[0,1} such that 1 < a(x) < A, A is a positive given number > 1, iii)
Definition 5.6.1. A weak solution of the problem (5.6.32)-(5.6.34) in H^(QT) := (0,/) x (0,T)) is a function « € # u ( < ? : r ) satisfying the integral equality / for all n € H^iQj)
(QT
{a{x)uxnx + utT)} dxdt = 0
that vanish for x = I, and u(0,<) = tp(t).
te[0,T}.
Here H^(QT) is the space of functions u € L2{QT) such that ux, ut G L2{QT). H^(QT) is a Hilbert space with the scalar product (u, v)m,HQr)
= I
JQT
(uv + uxvx +
And
utvt)dxdt.
Lemma 5.6.1. (Knabner and Vessella [1]) Let u solve the problem (5.6.32)-(5.6.34). Then for all n£JN,a,6e (0,1/2) and for x G [0,1(1 - a)], t e [6T, T] the following estimates hold: cn+lQn+ln2n
| | u | | in cn+1Qn+1n2n llullr,
dnnu(x u(x,t)t)
dt«
-
n1
d du(x t) du(x,t) dt » dx
^
w«
jm*' 2(
ccn+2Qn+2n W n
1
n+ > \\u\\ L in N"III.2(<3T)
Pn+l
^
where c is a constant that depends only on A and l~2T, and Q =
(8-1+o-2)2.
•
122
DIFFERENTIAL
OPERATORS
OF INFINITE
ORDER
(From Knabner and Vessella [1] one sees that Q = {6'1 + o - 1 ) 2 However, when the author of Dinh Nho Hao [2] "re-proved" this lemma he observed that, indeed
Q = (6-1 + a""2)2
From Lemma 5.6.1 we see that the function tp has to be a function of a Holmgren class two (Dinh Nho Hao and R. Gorenflo [1], Dinh Nho Hao [2]). Furthermore, we can assume (see Ginsberg [1]) that pW(0) = V<">(7) = 0, Vn = 0 , 1 , . . . .
(5.6.35)
Suppose that, instead of the exact
We now mollify ^ bb yhe de el Vallee Poussin kernel. Namely, we eake esee eection 1.12.1) v*
-*/'":= Ml&1) =
~~\r
K (* - r) *&)*•
(5.6.36)
From Section 1.12.1 we see that
u^ {Q,t)
=
u't'n,
0<x
=
0
t£»(0,t) = 0,
(5.6.37) (5.6.38) (5.6.39)
that is unique and stable. Theorem 5.6.3. Let u be a weak solution of the problem (5.6.32)-(5.6.34), such that conditions i) -Hi) and (5.6.35) are satisfied. Suppose that, instead of the exact Cauchy data
Let further u^n be a solution of the mollified problem (5.6.37)-(5.6.39). every finite n, this solution is unique and stable. If
Hkwri < E,
(5.6.40) Then, for
(5.6.41)
where E is a positive finite number, then, with
1
1
n" := ~y/21 hl-t
2
(5.6.42)
A MOLLIFICATION
123
METHOD FOR ILL-POSED PROBLEMS
(here [a] is the entire part of a), we have for x € [0,x„], t G [t0,T], 0 < x0 < I, 0 < t0 < T, we have the estimate \u(x, •) - u^'(x,
.)| < (6w2/Ty-*<> + (i/V^y/('^<2T/<°+4/
(5.6.43)
VlA.
where c is a constant depending only on A and Proof. We have, see (Knabner and Vessella [1]), OO
[£*(»-•>;
k=0
1
where
M*) = !> [* ds [° ~ L H7) L
MX)
A
»~^)di,k>\.
Hence, \\u°'n{x,-)
w£,n(i,-)IUoo(o,T)
<
°° x2k 53/ 9 1 .»{2n)*l|A^(v
mi>
s =
V£)IUoo(o,T)
(67r / T ) cosh(a;V27i)e.
Furthermore, see Natanson [1, p. 150] and Section 1.12.1, |u(x, ■) - u°' n (x, -)| = \u{x, •) - Ml (u(x, •)) | < iEl(u(x,
•))«,•
On the other hand, from Jackson's theorem (see Natanson [1]), we have ££(«(*»■))„
max
5 fc+1 u(x,-)
<
nA:+ i
.
ll"IU2Wr) /12cQy +1 (/ _
aw
(V* > 0)
] l)2(k+1)
for x g [0, Z(l - a)], t € [M\ r ] , a, S € (0,1/2), Q = (6'1 + a" 2 ) 2 Thus,
^:wx,-)L
2fc
0 fc ,
V *■ > 1 ,
124
DIFFERENTIAL
for x € [0,/(l - a)], t € \ST,T], \H\L2(QT)/VT,
OPERATORS
OF INFINITE
ORDER
a, 6 6 (0,1/2), Q = {S'1 + a " 2 ) 2 . Here
Cl
:=
C2 := 12c.
If we take k such that, n := [cQk4], where [a] is the entire part of a, then we have k < (n/(c2Q)f\ And therefore
^w1
( V /2 \
/ E*n{u{x,.)U
c1[^)\'k
<
(n y(*F =
-l
ex (—— 1
( n r , (n\\ = Cjexp 1 - . / — - r l n l — 1 1 .
\ ^2 ^5 /
\
V * >*
\ * »/ /
Hence, |«(s, . ) - » * * ( * , ")l ^ <
(6»Vr)exp{iV^e + c , e x p f - J ^ l n f ^ J J ( 6 , r 2 / T ) e x p ( V 2 x ^ ) e + Cx exp
Suppose that we are interested in «(x,i), 0 < x < x0 < /, 0 < i 0 < t < T, where x0 and t0 are given. Then, with 6 := to/2,
^
,
4
f%T
W — VI. 20,10,1 — 1
T , . _
\2 , J
•
Thus, K x , - ) - « £ '"(x,-)| < (67r 2 /T)exp (V2xy/K) e
+>y««-p(->w+i-. vs). The estimate (5.6.43) is now clear. However, this estimate becomes worse and worse when x0 ~» / and t0 -t 0, and it blows up for x 0 = I and t0 = T. 5.7. Numerical case study. In this subsection we give an example for numerical differentiation. Let the function / e Ia(JR) be given with a noise fc:
U - fh<e. We want to calculate derivatives Dnf of / from f€.
A MOLLIFICATION
METHOD FOR ILL-POSED
PROBLEMS
125
For the numerical calculation of £>"# we remember the facts that
pirn) =rerw) = ww.)*- (iK) and that F(\VU) has compact support in (~2vt2v). Thus we approximately calculate F- 1 ((id)«F(/ £ ")) using the inverse fast Fourier transform (FFT) and FFT instead of the inverse Fourier Transform F'1 and the Fourier Transform F, respectively. The mollification £ of the disturbed function / , is approximately obtained by means of quadrature formulae for the convolution integral (±VV) * fc. We consider the test function fi(x) = e - 2 , and the noise level e = 0.1 (very large!). The error function is S e (*)
= 4 = ^ = = , V^ v 1 + x2
that is / € = / i + f c .
The numerical results are given in Figures 1-4. Bibliographical N o t e s . The mollification method for ill-posed problems has been published first in Miller and Manselli [1]. Following their idea, Murio and his students published a large circle of papers on applying it to ill-posed problems (see, for example, Murio [1-4], Murio and Guo [1]). A variant of this method is given also in Engl and Manselli [1]. Tran Due Van and his coworkers (Tran Due Van, Nguyen Duy Thai Son and Dinh Zung [1]) have proposed a mollification method (from a quite different point of view) (they use a Fejer type kernel to construct mollification operators) for solving the well-posed Cauchy problem for the wave equation. The method in this chapter has been developed by Dinh Nho Hao [1]. It improved his joint results with Gorenflo (Dinh Nho Hao and R. Gorenflo [1]) and with Tran Due Van and R. Gorenflo (Dinh Nho Hao, Tran Due Van and R. Gorenflo [1]). Several applications of the method in this chapter are given also in Dinh Nho Hao, Reinhardt and Seiffarth [1].
126
DIFFERENTIAL
OPERATORS
OF INFINITE
Figure 1: Function / i and its perturbation (error bound e = 0.1)
Figure 2: Function £ and its mollification (error bound e = 0.1)
ORDER
A MOLLIFICATION METHOD FOR ILL-POSED PROBLEMS
Figure 3: Derivatives of /i and of its perturbation (error bound e = 0.1)
Figure 4: Derivatives of h and of its mollification (error bound e = 0.1)
127
Chapter 6 N O N T R I V I A L I T Y OF SOBOLEV-ORLICZ SPACES OF I N F I N I T E O R D E R
In this chapter we introduce a class of Banach spaces which are monotonic limits of Sobolev-Orlicz spaces of order N, as N -> oo. We call such spaces Sobolev-Orlicz spaces of infinite order (see definitions in Sections 6.1-6.5). The elements of these spaces are infinitely differentiable, moreover, they may satisfy certain boundary con ditions or conditions at infinity. In contrast to finite order Sobolev-Orlicz spaces, the very first question, which arises in the study of Sobolev-Orlicz spaces of infinite order, is that of their nontriviality (or non-emptiness), i.e. the question of the existence of a non-zero element u in these spaces. It turns out that the answer depends not only on jV-functions defining the spaces, but also on the domain of definition of their elements etc... Therefore, in Sections 6.2-6.5, we give necessary and sufficient conditions for nontriviality of Sobolev-Orlicz spaces of infinite order in four of the most commonly encountered cases in analysis: a bounded domain 0, C 1R"; the torus Tn = S1 x • • • x S \ where 51 is the unit circle; the full Euclidean space IRn and angular domains in IR\ But before doing this we present in Section 6.1 some results on monotonic limits of Banach spaces, a concept introduced by Yu. A. Dubinskii (Dubinskii [2]). These results are very useful in the study of function spaces of infinite order. Let X be a Banach space with the norm || ■ \\x. We say that X ii nontrivial, ii fhere exists an element x € X, with \\x\\x / 0. 6.1. Monotonic limits of Banach spaces. The aim of this section is to introduce a concept of the limit of monotonic sequences of Banach spaces. This concept pro vides a natural tool for investigating function spaces of infinite order, in particular, for studying their geometric and embedding properties. We note that the main feature of the construction of monotonic limits is that it is closed in the category of Banach spaces, meaning that the limiting space is also a Banach space. Let X1DX2D---DXmD---
(6.1.1)
be a decreasing sequence of Banach spaces whose norms satisfy the inequalities
lklli<||x||2<---<Wm<...,
128
(6. L 2)
NONTRMALITY
OF SOBOLEV-ORLICZ
SPACES OF INFINITE ORDER
129
where || ■ || m denotes the norm in Xm. Definition 6.1.1. The space Xcv := {x £ n™=1Xm
| l|*U»:=4Jiktt*|l»<«»},
is said to be the limit of the monotonically decreasing sequence of Banach spaces (6.1.1). (Henceforth for brevity we shall call X„, a monotonic limit.) It is quite likely that X„ may be trivial, that is, it may consist only of the zero element. However, if I M is nontrivial, then one can show that it is also a Banach space. We shall suppose that the convergence in Xm is compatible, i.e. if xm -» x in Xm and xn -» x' in Xm+U then x = x'. Theorem 6.1.1. If Xm are Banach spaces and if X^ a Banach space.
is nontrivial, then X^
is also
Proof. Obviously, it is necessary to verify the completeness of X^. Indeed, let { i „ } n E K , £ ! « , , be a fundamental sequence. This means that for any e > 0 there exists a number N such that \\xn+p - x„||oo < t,
n>N,
p>0.
(6.1.3)
p > 0,
(6.1.4)
Consequently, for every m = 1,2,..., a fortiori \\xn+p - xn\\m <e,
n>N,
i.e. the sequence xn is a fundamental sequence in the space Xm. Since the convergence in Xm is compatible, there exists an element
x s n%=1xm, so that xn —»■ as in every Xm, as n -» oo. Letting p - t o o w e get from (6.1.4) that ||i-x„||m<e,
n> N,m =
1,2,,...
Letting m -» oo in the last inequality we have ||* - *n||oo < e, It follows that xn^xmXao,
i.e. the space Xx
n > N. is complete. The theorem is proved.
130
DIFFERENTIAL
OPERATORS
OF INFINITE
ORDER
Example 6.1.1. Let {LPm}mew, with LPm := LPm(G), be a sequence of Lebesgue spaces in a bounded domain GcM'.We can assume without loss of generality that mes G = 1 and hence Pl < P2 < ■ ■ ■ ■ pm < ■ ■ ■ implies that
W„
D---LPL
^Loo,
D
where /,«, is the Banach space of measurable functions in G with the norm |ju||oo := ess max|u(*)|, x € G. This example illustrates clearly the difference between the monotonic limit and, say, the projective limit, since obviously ^
C lim p r o j , , , ^
LVm,
and the inclusion is strong, since the projective limit includes also unbounded func tions. An important example of monotonic limits is Sobolev-Orlicz spaces of infinite order which will be given in the following sections. Now, we describe the concept of the limit of a monotonically increasing sequence of Banach spaces. Let X_i C l - 2 C - C l . „ C (6.1.5) be a sequence of normed spaces with the norms
NI-i>NI-»> — >IWI—>■••• Consider the linear space * _ „ := {x e l C = 1 X . m I | | * | | _ , := lim | | s | | _ m } . i n
ii
m—*00
"
"
■*
The "norm" || • ||_«, may have a nontrivial kernel and may thus be only a semi-norm. In that case, we identify Y_„ with the corresponding factor space related to the ker nel. Hence, we obtain a normed space Y.^ with the norm || ■ !!_«,. Definition 6.1.2. The completion of * ! „ with respect to the norm || • H ^ is called the limit of the monotonically increasing sequence (6.1.5) and denoted by X-oo = lim AL m . m—*oo
NONTRIVIALITY
OF SOBOLEV-ORLICZ
131
SPACES OF INFINITE ORDER
Example 6.1.2. Let, as in Example 6.1.1., LPm be Lebesgue spaces in the domain G C IRn, mes G = 1. We denote by X
-m
= Lpin,
{p'm = pm/{pm
~ 1)),
the space dual to Xm = LPm. It is obvious that Lp[ C ^ C - C V „ C
>£i,
where Lx is the classical Lebesgue space of summable functions. In other words, lim Lp,m = Lx or, equivalently,
lim Lp, = Lx.
Let us now consider the general case. Namely, let (6.1.6)
•■•■XmCC-•CXxcX.xC---CX_mC---
be a sequence of Banach spaces Xm and its dual spaces A"_m = (Xm)* We shall assume that the dualities < x',x >m in the pairs {X-m,Xm) are compatible in the following sense: if x' G A"_m, then < X',i >m=< x',x > m „ holds for any m> m0 and x G Xm. Denote by A ^ and * _ « , the corresponding limits, i.e. X«, = lim X m ,
X.,*, = lim
■m—*-oo
X-m.
m—+oo
Our aim is to point out the cases when these spaces are nontrivial. Theorem 6.1.2. Let the space Xx trivial.
be nontrivial.
Then the space A"_M is also non-
Proof. Let xo € Xx and x0 ^ 0. We fix a number m0 and consider x0 as an element of Xmo. Using a well known corollary of the Hahn-Banach theorem, we find a bounded linear functional x' € A"_mo such that < x\x > m o = 1- Since the dualities in the chain (6.1.6) are compatible, < x',x >m= 1 for all m > m0. It remains to remark that Hi'lLoo = lim \\x'|Lm = lim —*',X°>-
> 0,
since x0 is not equal to zero. The theorem is proved. T h e o r e m 6.1.3. Let one of the following conditions be fulfilled:
132
DIFFERENTIAL
OPERATORS
OF INFINITE
ORDER
i) the spaces Xm, m = 1,2,..., are reflexive, ii) the imbeddings Xm+1 C Xm, m = 1,2,..., are compact. Then the triviality of Xx
implies the triviality o / A L M .
Proof. Let X^ = {0}, i.e. let the space Xx be trivial. We shall show that then for every x' £ U~ = 1 X_ m Hi'll-c := lim | | i ' | | _ m = 0. 171—S-OO
It will follow that the space X^
is trivial.
Indeed, let x> € X_ m o , where m0 is the smallest number among those m for which x' G X_m. Taking into account the compatibility of the dualities, we have for all m > m0 n l„ < X',X >m < x',x >m \\x _ m := sup — — = sup 7^—r . x€Xm
X£Xm
\\x\\m
\\X\\m
Suppose that the norms do not tend to zero. Then we have a sequence of elements xm€Xm, m = 1,2,..., such that <x',xm > m o x€X„
where e is a positive number (strictly speaking, we have a subsequence of the sequence {x m }, but this is immaterial). Equivalent^ <x',ym>mo>e, where ym =
||y m || m = 1,
(6.1.7)
am/|\xm||m.
Since Hfc.ll, < ||ym||m for s < m, then in view of Condition i) or Condition ii) we may consider the sequence {ym} (in general, only a subsequence) as tending to an element ye
n™=1xm,
and converging in any Xs (s = 1,2,...) weakly. Moreover, it is obvious that from (6.1.7) we get <x',y>mo>e, ||s/||oo < l. Since the space Xx is trivial, these relations are impossible. This contradiction proves our theorem. Theorem 6.1.4. Let I M be a reflexive and strictly convex space together with its dual space X^. Then the formula Aloo = X^
NONTRMALITY
OF SOBOLEV-ORLICZ
SPACES OF INFINITE ORDER
133
is valid. Proof. Indeed, from the assumptions of the theorem there exists a dual operator J : Xx -> X^ acting homeomorphically. Since || • \\.m > || • \\XL, for any sequence of elements hn £ U^=1.X_m, which is a fundamental sequence in the norm || ||_„ there exists one and only one element x = lim^oo J~l(K) in *oo- Identifying every fundamental sequence hn € U£=1X_m with the corresponding element i e Jf;, we get the assertion of the theorem. Hence, the complete dual chain in the last case is the topologically dual sequence *oo <
C Xm C • • • C Xi C X-i C • C X'm C
> X%.
Example 6.1.3. Combining Examples 6.1.1 and 6.1.2 we obtain the complete chain (Pi > 2) loo < C - C i f t C ^ - C l f t C ► In. It is well known that Lt £ (£«,)•. This example demonstrates that the assumptions of Theorem 6.1.4 are essential. 6.2. Nontriviality of Sobolev-Orlicz spaces of infinite order in a bounded domain. Let ft be a bounded domain in ttT and with boundary 5 0 . We consider the space of functions u(x): := {u\u G C 0 ° ° ( n ) , H | M < o o } ,
LW°°{$a,n} where
Jfc>0 £/ n $a(^^)^<
(6.2.1)
||u||(0O) := inf Ik > 0
Here {#*}|„(=o,i,... is a sequence of JV-functions (see 1.16). L W°°{$a,n} is called a Sobolev-Orlicz space of infinite order on the bounded domain 0 . We define also Sobolev-Orlicz classes of infinite order as follows: oo
£W^{<S>a,n}:=\u
f
0 «€Cnft),/>°»:= : = £ / *«(i> «)«i <
oo
Remark 6.2.1. The reader should notice that by the form of brackets used in the definition of these spaces we want to indicate, that they do not depend on a particular index a. In fact, by { $ a , n } we mean the whole sequence { { $ a } , 0 } . o L e m m a 6.2.1. The space LJT
o , * . , « } is the linear hull of the class C W - { * 0 , n } .
134
DIFFERENTIAL
OPERATORS
OF INFINITE
ORDER
1^;::;;-;;::
Proof. Let u € C W ° ° { $ a , n } and p°°(u) = K. It is sufficient to consider the case K > 1. Then
Hence, ||u||(<x>) < tf, i.e. u € i W ° ° { * a , n } . This proves fh<- Icmm,, Lemma 6.2.2. For any « e i l T f i , ! ! } lira Hullun = JV—oo"
"l
;
N W
" "l
'
Proo/. It suffices to consider the case lim \\u\\{N) = Mi
>0.
Then, for any non-negative integer N, we have
inf I k > 0 This implies N
r
|>(^H-H (Dau(x)\
Hence,
IMI(oo) < ML
(6.2.2)
Otherwise, if 0 < ||u|| (oo ) = M2, we have, by the definition of the norm || • || (oo) , that for every t > 0 there exists fc0 > 0 such that M2 < fc0 < M2 + e and
s>m-
M
Thus, for every non-negative integer N,
£^m*« l«M"
Hence, IMI(N) < *0 < M 2 + £, and lim \\u\\tm <M2 + e.
NONTRIVIALITY
OF SOBOLEV-ORLICZ
SPACES OF INFINITE ORDER
135
Since e is an arbitrary non-negative number, we have lim \\u\\(N)
< M 2.
From (6.2.2) and the last inequality we obtain Lemma 6.2.2. o Theorem 6.2.1. L VK°°{$ a ,ft} is a Banach space. Furthermore it is the monoN tonic limit of the decreasing sequence of Sobolev-Orlicz spaces W L{$a,n}, N = 0 , 1 , 2 , . . . , and lim |M|(N) = IH(oo). Proof. Obviously, we have WlL{$a,n)
D W2L{$a,fi}
D ■ ■ ■ D WNL{*a,n}
D ■ ■ ■,
and o ., , and each of WNL{$a,n} 6.2.2
Hb<WCT<--<Ww<--, is a Banach space (see Section 1.17). Further, by Lemma
It follows that L w H < & a , f t } is the monotonic limit of the decreasing sequence of the spaces WNL{$a,n} (see, Section 6.1). By Theorem 6.1.1, if L W°°{$a,fi} is nontrivial, then it is a Banach space with the norm || ■ || (oo) . The theorem is proved. We introduce the sequence of numbers Ma: M
Ma
_ f ^»;'(1/B«ifl), ( l / m e s f i ) , *$ aa ^^ 00 "~ \ +00, $a = 0 '
where *;*(*) is the function inverse to $ Q (<), \a\ = 0 , 1 , 2 , . . . . Theorem 6.2.2. The space LW/°°{$ Q ,n} is nontrivial if and only if the sequence of numbers Ma, \a\ = 0 , 1 , 2 , . . . defines a nonquasianalytic class of functions of n real variables. Proof. Sufficiency. The sufficiency relies on a well-known lemma concerning compactly supported functions of a single real variable and on criteria of Mandelbrojt, Bang and Lelong on the quasianalyticity (Section 1.18).
136
DIFFERENTIAL
OPERATORS
OF INFINITE
ORDER
We shall construct a nontrivial function u € L W°°{$a,fl}. It follows from the condition of the theorem, and the Mandelbrojt-Bang criterion (Theorem 1.18.1, Part c), and the Lelong criterion (Theorem 1.18.2) that OO
T MCNIMCN,, < oo, M N = min Ma, t o N' N+1 N="
(6.2.3)
In accordance with (6.2.3) there exists a number N0 such that OO
MCN/MCN+1 < a/3,
£
We can then find positive numbers fi0, m,.,., Ato + H +•• ■ + mo + 4 Set
1 Mc »N = Hljr~> P
M
a >0.
/iNo and /9 £ (0,1) such that £
MCN/MCN+1 < a/3.
N = N0 + l,N0 +
(6.2.4)
2,....
N+1
It is obvious, that by virtue of (6.2.4), the sequence of numbers fiN, N = 0 , 1 , 2 , . , . , satisfies (1.18.1). Thus, by Lemma 1.18.1, there exists a function ^ € C0°°(-a,a), v(t) £ 0, of a single real variable, satisfying (1.18.2). Taking into account the definition of the numbers fiN, we find that \DNu{t)\<^Q...^N)-'
N = 0,1,2,...,
(6.2.5)
where, without loss of generality, we can assume that the constant K, which is inde pendent of TV, is equal to one. Next, let x° - (x°,.,., x°) be an interior point of fi, and let a > 0 be such that the cube U(x°, a) = {x\\xx°\ < a} lies entirely in fi. We put u(x) = u(xx -*?)•■■ v{xn - x°n) and show that u € L W ° ° { * a , f i } . Ma, \a\ = N, we obtain
Indeed, from (6.2.5) and since M° < MN <
\D°u(x)\ < j}NM% < f)NMN < pNMa,
\a\ = N.
Further, taking into account the definition of the numbers M„, we have *a(D°u(x))
< f a{^M a) x " ~
<- ^ r , mes
\a\ = 0 , 1 , 2 , , . . . fi '
(6.2.6)
NONTRMALITY
OF SOBOLEV-ORLICZ
137
SPACES OF INFINITE ORDER
Consequently, oo
£
.
OO
/ $a(D°u(x))dx
OO
< £
|or|=0
oo
/?W = E ( E
|o|=0
N=0
l
)PN = E Ot^")^
\a\=N
< oo,
N=0
0
_
since 0 < 0 < 1. By the same token u G L W~ {*„,«}. Necessity. Let M 0> |a| = 0,1, • ■ ■, generate a quasianalytic class C[Ma] of functions of n real variables, and let u £ L W ° ° { * a , n } . We show that u = 0. Since u € LH/°°{$a,ft}, we can assume p{D°u,9a) < 1, |a| = 0 , 1 , 2 , . . . . From Jensen's integral inequality and the last condition, we have ( L\Dau(x)\dx\ *"1 mesfl I mes 3£ **PP*ymg ^ 0
1 -m^"n' J mes "
l«l = 0,1,2,
(6.2.7)
«" DOtn siues 01 ^o.z.f j we ODtain i \u uyx )\<xx — mes it * a ^i/mes acj, |«|
runner, ior arDiirary a ana ^
u, I , z , . . ..
yv.&.o)
Vsi? * *" is>n) t *n, we nave
where u(£) is the Fourier transform of u{x). Put ( = r)6, where JJ G I R \ 0 = («!,■■• A ) , and the |0j| > 1 are fixed. We then find from (6.2.9) and the fact that u(x) has compact support that for all positive integers N>2 (l+T,3)\fl\N-2\u(Ve)\
\a\ = N.
(6.2.10)
We note that these inequalities are valid for all a, |a| = N. Therefore (l+r,2)\vf-2H¥)\<
KmeSnMN,
TV = 2 , 3 , , . . .
where MN = m i n M a , | a | = N. The last inequalities signify that for the function i/(y), y € JR.1, which is the inverse Fourier transform of the function % 0 ) with respect to the variable rj, the inequalities \DN~2v(y)\
< Kmes fi MN,
N = 2,3,...
138
DIFFERENTIAL
OPERATORS
OF INFINITE
ORDER
hold, i.e. the function v(y) belongs to the class C[MN+2\. Since the Ma define a quasianalytic class of functions of n real variables, according to Lelong's theorem (Theorem 1.18.2), the class C[MN] and, along with it, the class C[MN+2] are quasianalytic. On the other hand, according to the Paley-Wiener theorem, v € C^fffiL1), consequently v(y) — 0. From this we conclude that on an arbitrary line £ = nO, where 8 = (0i, • • - A ) , |0;| > 1, the functions u(f) = 0; hence, u(£) = 0 in lR n , i.e. u(x) = 0 in Q, which is what we wanted to prove. Using Theorem 1.18.2, it is not difficult to formulate Theorem 6.2.2 in term of quasianalytic classes of functions of a single variable. We have, in particular, T h e o r e m 6.2.2'. The space Z,W°°{$.,n} ™ nontrivial if and only if the sequence of numbers MN = min{M 0 | |e*j = N},N = 0,1, • • • generates a nonquasianalytic class of functions of a single real variable. From Theorem 6.2.2', and also Theorem 1.18.1, we obtain the following algebraic necessary and sufficient conditions for the nontriviality of the spaces
LW°°{$a,£l}.
Corollary 6.2.1. Each of the following conditions is necessary and sufficient for the space LW°°{$a,n}
to be nontrivial:
a) Ifpn = \niN>n{mmH=N b) IfT{r)
Ma}llN,
then £ J ° 0? < +oo.
= s u p ( r N / M N ) , then /J» r~2 In T(r)dr < +oo.
c) l i m i n f ^ ^ Ml/N
= oo and ZN=0MCN/MCN+1
< +oo.
As is well known, it is sometimes impossible to calculate $(i) analytically from the JV-function $(<), and conversely. In practice, therefore, the following propositions are useful. Let x(ft) De the characteristic function of the domain fi, and let ( $ , $) be a complementary pair of Ar.functions (see Section 1.16). Corollary 6.2.2. The space LW°°{$a,Q} is nontrivial if and only if the sequence of numbers ||x(n)||#., |*| = 0 , 1 , . . . defines a nonquasianalytic class of functions of n real variables. Here \\ ■ ||«j, denotes the Orlicz norm (1.164). Corollary 6.2.2 is proved by means of a direct calculation of the norm ||x(^)|U • Namely, ||x(n)lk=
sup
f \X(n)u(x)\dx
=
sup
f \u(x)\dx.
(6.2.11)
NONTRIVIALITY
OF SOBOLEV-ORLICZ
SPACES OF INFINITE ORDER
Let u{x) be a function in £ { * a , n } such that p{u,$a) Theorem 6.2.2, we have
139
< 1. Then, as in the proof of
/ \u(x)\dx < mes SI • ^ ( 1 / m e s 0 ) .
(6.2.12)
Taking the function u0a(x) = ^ ( l / m e s ft)x(ft), we have p(tio„,*«) = 1 and j \u0a(x)\dx = mes n • ^ ( l / m e s SI).
(6.2.13)
From (6.2.11)-(6.2.13) it follows that ||x(n>||*. = mes n • ^ ( l / m e s SI), \a\ = 0 , 1 , , . . . The last equalities and Theorem 6.2.2 yield Corollary 6.2.2. In a similar way, we obtain the following result. Corollary 6.2.3. The space LW°°{$a,Sl} is nontrivial if and only if the sequence llxWII^L), M = 0 , 1 , " ■, defines a nonquasianalytic class of functions of n real variables" Here \\ ■ \\m denotes the gauge norm (see 1.16.5). Example 6.2.1. We consider the class W°°{a,p Q } introduced by Dubinskii [I], [2], [6] as toiiows ^notice tnat t>y \cx^paj we mean tne wnoie sequence \Oi^pa j\a\=ofiimmm): VF°°{a,pQ} := {u\ue wnere
C^(Q)7^2aa\\Dau\\ll
< oo}, aa > 0,1 < pa < oo,
• «a is tne i/« (Jcj~norm, \$ is a Douncieci domain m ut\.
we cienne
MN = min Ma, \a\=N
where, for \a\ = 0 , 1 , . . . , M
' a;1/p°, 1
+oo,
aa >0 aa = 0
Then, according to Theorem 6.2.2, we can state the criterion for nontriviality of W°°{a,pa}
in the following form:
The class W°°{a,Pa} is nontrivial if and only if the sequence of numbers MN,N 0 , 1 , . . . defines a nonquasianalytic class of functions of a single real variable. Thus, Theorem 6.2.2 gives a new criterion of nontriviality of W°°{a,pa} us to calculate the numbers Mn explicitly.
=
and enables
140
DIFFERENTIAL
It should be noted that the class W°°{a,pa}
OPERATORS
ORDER
may not become a linear space when pa -»
oo. In such cases, we introduce the space L W°°{a,pa} as a subset as follows: LW°°{a,Pa}
OF INFINITE
which contains
W°°{a,pa}
:= {u <E C0°°(n), \\u\\{oo) < +00},
where \\u\\M := Jnf Ik > 01 £ a^-^H^wH^ < 11 . IIM 12. II* I h I
(6.2.14)
The space Z W ° ° { a a , p a } is a Banach space with the norm (6.2.14) for an arbitrary sequence pa. Example 6.2.2. Let *„(*) = exp[t 2 a a ] - 1, aa > 0. Without loss of generality, we can assume mes 0 = 1 . For each a the function $ a ( t ) is an JV-function. We readily calculate that 2 1
*?(!) = a?' ^) ".
According to Theorem 6.2.2, the space LW°°{a,pa} is nontrivial, for example, for aa = (\a\\)-2u, v > 1, and trivial for aa = (|a|!)- 2 ", v < I. Example 6.2.3. Let *„(*) = (1 + a a | t | ) l n ( l + aa(\t\)) - aa\t\,aa > 0. For each a , $ a is an iV-function. In order to apply Corollary 6.2.3 we need to calculate ||x(0)||(« a ):
milkIk > > 0| jf*„ 0\J^J^-\dx< (^\ dx
- 1).
L W °°{(1 + o ffl |t|)ln(l + 8,(1*1)) - aa\t\,n} (|a|)-"W,^>l.
= aa(e- l)" 1 .
According to Corollary 6.2.3 the space is nontrivial, for example, for aa =
Example 6.2.4. Let ».(*) = exp(a; 1 |t|) - o-*|*| - 1, aa> 0. Obviously, $ a is an iV-function for each a, |a| = 0 , 1 , . . . . From Example 6.2.3 and the equivalence of the Orlicz and the gauge norms, it follows that llx(n)||». > llx(n)||(-*j = a a ( e - 1 ) - 1 . Consequently, according to Corollary 6.2.2, the space IV^{exp(a-|t|)-a-|t|-l,n}
NONTRIVIALITY
OF SOBOLEV-ORLICZ
is nontrivial, for example, for aa = (\a\)"^,
SPACES OF INFINITE
ORDER
141
v>\.
6.3. Nontriviality of Sobolev-Orlicz spaces of infinite order on t h e ndimensional torus. Let us denote by Tn the n-dimensional torus. Consider the space LW°°{$a,T"}
:= {u € C°°(T") | ||«|| (oo) < oo},
with the norm
1 \\u\\{ao):=ini\k>Q/\tjM T^{~)^
(6.3.1)
IcI^O-7"'"
where u(x) is a periodic function of period 2TT in every component of x. As usual, the question of the nontriviality of the space LW°°{$a, Tn) arises. We are interested only in the spaces LW°°{$a,Tn} which have infinite dimension, i.e. which contain an infinite set of linear independent periodic functions. Such spaces are called nontrivial in this case. Theorem 6.3.1. The space LW°°{
£ *•(£) < oo.
(6.3.2)
l«l=o where q^ =
q^---q^.
Proof. Obviously, Condition (6.3.2) is sufficient. In fact, in this case the periodic functions uv(a) = exp( J g„, x) = exp(t ? 1 „i, + ■ • • iqnuxn) belong to LW^i^, Tn}. To prove the necessity, let us assume the contrary: the series (6.3.2) converges only for a finite set of multi-indices q = (qu ..,, qn), namely, for | 9 l | < TV,,... \qn\ < Nn, where Nj are integers. We prove in this case LW°°{a,Tn} C
L{exp{iq,x)),
where L(exp{iq,x)) is the linear hull of the functions exp(iq,x) with | $ | < Nhj = l,...,n. The latter contradicts the condition that LW°°{$a,Tn} is infinite dimen sional.
142
DIFFERENTIAL
OPERATORS
OF INFINITE
ORDER
In fact, if u S £ W 0 0 { * 0 , T n } , then it is an infinitely diiferentiable periodic function and therefore u(x)= where
OO
Y y kho
c„ {2ir)-nnJ j C, == (2n)~
Moreover, for any a,
ccexp(iq,x), qexp(iq,x),
u{x)exp(-iq,x)dx. nu(x)exp{-iq,x)dx.
OO
a a Z)°u(x) D u(x)= = ^ c , ( iYlc e x paex ( iPg(iq,x), ,x), g ) g(iq) ||,|=o ,|=0
and consequently, n a \c ( 2 » ) -\J| / qq \l < (2w)' k,9
a u(x)exV(-iq,x)dx\ nD u(x)exp(-iq,x)dx\
a \D \Dau(x)\dx. u(x)\dx.
< (27r)(2w)-nn f
(6.3.3) (6.3.3)
From Jensen's integral inequality and (6.3.3) we deduce $ a (c,g°) (c,ga)
<
$a((2K)-n $a((2K)J nJJD JDau(x)\dx)
<
(27r)-" / (2*)-"
a $JDa$JD u(x))dx. u(x))dx.
(6.3.4) (6.3.4)
Further, our assumption means, in particular, that the series (6.3.2) diverges in points of type ( 0 , . . . , Nj; + 1,...,0), j = 1 , , . . . n. It is clear that any multi-index complementary to the multi-indices q, where \qj\ < JV,, majorizes an index of this type, that is ( | 9 l | , , , . , 1^1,,.., | Q j [ , , . . , k„|) |g„|) > ( 0 , , . . , A A^T,- ++ 11,,......,,00)).. where j € { 1 , . . . , n } is some integer. Since u € LW°°{$ W ° { $ a„, , T " } , we can assume
|or|=0
Obviously, that for any j OO
P°°(u)>
£ o,=0 a,=0
p(Dau,$a),
where the indices a run through the set of values ( 0 , . . . , ah ..,, 0). Hence, by virtue of (6.3.4), we have OO
/»"(«)
>
E(2tr)»$a(cf^) a
3 OO OO
>
a £(27r) £ *m{c,{Nj *a(c,(iVi + Il ) D ' ) = +oo, t(2ir)nn Y, CTj=0
NONTRIVIALITY
OF SOBOLEV-ORLICZ
SPACES OF INFINITE ORDER
143
(e > 0), if only c, ji 0. Since the latter inequality is impossible, we obtain that for any u 6 I W ° ° { $ a , Tn} the Fourier coefficients c, are equal to zero, provided at least one qj > Nh 1 < j < n. Thus, if the series (6.3.2) converges for only a finite set of the points q = (qu- ■ .,?„), then the space LW°°{$a,Tn} of periodic functions has finite dimension. That con tradicts our condition. The theorem is proved. Remark 6.3.1. The condition (6.3.2) does not rule out the possibility that the space LW°°{$a,Tn} consists only of functions that depend on a fewer number of variables, is nontrivial. To exclude this degenerate case we can require, for example, that LW°°{$a,Tn} is dense in L2(Tn). It is easily seen that this requirement is necessary and sufficient in order that Condition (6.3.2) be satisfied for some sequence qv such that
mm(qlv,...,qnu)^oo as v -* oo. In the future we consider only such spaces. Example 6.3.1. Let *„(<) = exp[(|a|!)- 2A t 2 ] - 1, \a\ = 0 , 1 , . . . , A > 0. Then, for the sequence qv = (if,... ' ,v), 5 '
v = 1,2,... we have
OO
CO
E*„(tf) = E[«p(^ l (H0-") -iK+oo, n
a=0 ~— n
i.e. for every q, the condition (6.3.2) is satisfied. By virtue of Theorem 6.3.2 the space LW°°{$a,Tn} is nontrivial. Note that the corresponding space LW°°{$a,Tn} is trivial for 0 < A < 1 (see Example 6.2.1). Example 6.3.2. (Dubinskii [31,[61) The Banach space oo
H ^ { a a , p , T " } := {u g C°°(T") | ||u||' = £
aa\\Dau\\lp
< oo},
|a|=0
where aa > 0, is nontrivial and dense in L2(Tn) if and only if its characteristic function OO
v(« v(o = |c|=0 £ ««r |»|=0 is an entire function. This follows from Theorem 6.3.2. In particular, the space
-{&"•}
144
DIFFERENTIAL
OPERATORS
OF INFINITE
ORDER
is nontrivial. The latter space is an "energy space" for the equation cos ( j-)
«(x) = h(x),
xeT1
(see Chapter 7 below). 6.4. Nontriviality of Sobolev-Orlicz spaces of infinite order defined on a full Euclidean space. We consider the Sobolev-Orlicz space of infinite order of functions denned on W1: := {u | Dau e L{* a ,Hl B },||ti|| ( o o ) < oo},
LW°°{$a,W} with
||u|| (00 ) := inf | i f c > 0 |
£
/
*a{Dau(x)/k)dx
(6.4.1)
We also define the Sobolev-Orlicz class of infinite order as follows: £W°°{$a,]Rn} P"(»)
:=
{u g C°°(lR n ) | p°°{u) < oo},
:
£
=
/
*«(Dau(x))dx
< -foo.
By the methods of Sections 6.1-6.3 we can prove the following result. Ti nheeoo rr ee m m B 6.4.1. (6.4-1) and . u . LW~{* LW y*a > ]R»} « . ) «is. a« Banach n m space space vnth mm the me norm ™™l»-4JJ ana is is the me hnpn-r nil! linear hull of the class CW {*„, U }. Furthermore LWJ* ffi. }i» thermnotomc limit oj the decreasing sequence of bobolev-Orhcz spaces W L { $ a , IR }, JV = 0 , 1 , . . . and hm^\\u\\(N) = H|(oo). We now lormulSjLc tne m
is nontrivial if and only if there exists a
OO
£
*.(,!-") < oo.
(6.4.2)
|a|=0
Proof. Necessity. We need the following result which is, in our opinion, of independent interest. T h e o r e m 6.4.3. Let the function f{x) be such that its generalized derivatives Daf LPa(TRn), 1 <pa
liminf||7?"/IIJ/H>d/, |a|-*oo
€
NONTRIVIALITY
OF SOBOLEV-ORLICZ
SPACES OF INFINITE
ORDER
145
where df :=sup
{mm
161 | £ = (&, . . . , £«•) € supp
/ 6ein7 tte Fourier transform of f, \\ ■ \\Pa aeing the norm in the Lebesgue space i P „(iR n ). A proof of this theorem will be given later. Using this theorem we show that if LW°°{$a,lRn} is nontrivial, then there exists a number q > 0 such that Condition (6.4.2) is fulfilled. It is obvious that the problem of nontriviality of LVK°°{$a,]Rn} becomes interesting if $ 0 -fr 0, since otherwise the function u{x) = const belongs to LW°°{$a,TRn}. In view of this we always assume that $o ^ 0. Let u € £W°°{$a,Mn}, that ||w||(oo) < 1. Then,
and u(x) ^ 0. Without loss of generality we may assume
/ „ ®a(Dau{x))dx
fT{u) = E
< 1.
(6.4.3)
|a|=0-'1R"
Since u{x) ^ 0 and $ 0 is an A-function, $ 0 ^ 0, it follows that u(x) ^ const. If we show that (6.4.3) yields (6.4.2), then the necessity will be established. To this end we construct a function g(x) by putting g(x) =
i meSjt)(U, ir) meSjB(0, ' ) JB(O,T) JB(0,r )
u(x < +
(M,
where r is a positive number, 5 ( 0 , r) is the ball of radius r, centered at 0. Then, g e C°°(IR n ) and
j ^ r /
D"g(x) =
Dau(x + i)di.
mes.tf(U,rj JB(O,T)
Using Jensen's inequality we obtain, for any i S M n and \a\ = 0 , 1 , , . . , $a(\Dag(x)\)
±-—p(Dau,$a).
<
(6.4.4)
Since $ 0 is an A-function, (6.4.4) yields
II^IIHR") < +°°, t.iWVsKu.vn)
<
p{Dau a) ^ mesB(o,r)
(6-4-5) (6 4 6)
'-
146
DIFFERENTIAL
OPERATORS
OF INFINITE
ORDER
From (6.4.5) we see that the function g(x) satisfies all conditions of Theorem 6.4.3, and it follows that UmmfHD-plli' 1 -' >dg. (6.4.7) On the other hand, since g(x) =£ const, |MU„(]R") < oo, comparing the proof of Theorem 2.1 in Dubinskii [3] we obtain that g is not concentrated on the hyperplans £ = 0, j = 0 , 1 , . . . , n ; hence dg > 0. We now choose a number q > 0, q < dg. By (6.4.7) it follows that >
W9\\w)
(6-4.8)
Without loss of generality we may assume that (6.4.8) is valid for all a, |aj = 0 , 1 , . . . . Then, combining (6.4.3), (6.4.6) and (6.4.8) we get OO
OO
I
*(
E*.
I
mes5(0, r)
1
,»«)< |a|=0
mesi?(0, r)
The necessity is established. Sufficiency. Suppose that Condition (6.4.2) is satisfied. We shall construct a nontrivial function u € LW°°{$a,TRn}. To do this we consider a function u € C£°(G),«(£) ^ 0, where G = { £ € l R n | \Zj\
JG
e^«(0«
is a desired one. In fact, for any a, a direct calculation yields \D"u(x)\ < Kq^\a\n+1
J ] mi l l (l, I**! - 1 - 1 '"), k=i
where K does not depend on a. Then **a.(\D°>u(x}\) (l^-u(«)|) < i n { l l | x\xtk|\-^'»)* - 1 - 1 /a-(Kq[%r*}. )*0(^H|a|-+i}. < nftm min{l,
(6.4.9)
We put A=
[
f[mm{l,\xk\~>-V»}dx.
By (6.4.9) it follows that
l p(D°u,* )
a
(6.4.10)
NONTRIVIALITY
OF SOBOLEV-ORLICZ
SPACES OF INFINITE ORDER
147
On the other hand, for |Q| large enough, A' 9 !<"|ar +1 < ,1-1.
(6.4.11)
Combining (6.4.10), (6.4.11) and (6.4.2) we get
/>"» = This means that u € CW°°{^a, completely proved.
\a\=0
W},
hence u G LW°°{<Sfa,WC}. Theorem 6.4.2 is
Proof of Theorem 6.4.3. We show that liminf||Z>«/||VW> minlrfl, J
|of|—KX>
? = (£,...,#), 3*
(6.4.12) ■
for any & G supp / and <*, > 0. Let £° G supp / , i° / 0,i = l , . . . , n . For definiteness we may assume (° > 0, j = 1 , . . . , n and £? = min£°, j = 1 , . . . ,n. We fix a number e > 0 such that £ ° - 2 e > 0 and £° G G, where G is a domain with smooth boundary r and G C K where A' = {5eR"
| £° - £ < & < $ + e, / = 1 , . . , , n}.
Since £° G s u p p / there exists a function v € C0°°(G) such that £° G supp vf. Taking into account that / G .9 s "' (this follows from / G LK(R*)) we obtain =(2*)n >,
(6.4.13)
for any function p G C0°°(IRn). Putting in (6.4.13)
(6.4.14)
1
with ^(x) = F - ^ M ] = v * io(s). Here F" denotes the inverse Fourier transform. The distribution »(£)/(£), just as every distribution of compact support, can be represented in the form
HOHO = £ Daha{0,
(6-4.15)
|or|<m
where m > 0 is a positive integer and the ha(() are ordinary functions on G (for example, fe0 G Cl{G)). We may assume m > In + 1. Hence it follows that there exists a function z € Cm{G), Daz\r = 0, |a| < m - 1, such that
*(0/(0 = W ( 0 = £ (-l)HD"(Z>»i(£)) |a|<m
148
DIFFERENTIAL
OPERATORS
OF INFINITE
ORDER
(it is obvious that S(£) is a solution of the elliptic problem £
L2mz(0=
D<*ha(OtD°S\r
= 0,
|a|<m-l;
|a|<m
the advantage of the latter representation compared to (6.4.15) is that it is unique). From (6.4.14) we now have
=
=
(2TT)" < / , ¥ > > ,
or, what is the same,
(6.4.16)
where, as we recall, w € C0°°(G) is an arbitrary function. It is obvious that the left side of (6.4.16), hence also the right side, admits closure with respect to an arbitrary function w G C2m{G) n C?(G); in this connection if, = w *v € i„ 0 (IR n ), l/pa + l/qa = 1, |a| = 0,1, Completely analogous arguments for the functions Daf(x) < z(.),L2m{.°w(.))
reduce to the relation
> = (2x)»(-l)M < / , „ > ,
(6.4.17)
where w € C2m(G) n C?(G).
Suppose that u>0 € C2m(G) n C?(G) is a solution of the equation
Since Ci - 2e > 0, {J = min£°, j = 1 , . . . , n, we have 0 £ G, and the function
satisfies the equation
W«°«*«(0) = K?-2e)M*(0.
Putting (6.4.18) into (6.4.17) we obtain « J - 2 « ) H < 5,5 > = ( a r H - l ) M < 7?-/,V- > ,
(6.4.18)
where ^ = v * wa € Lqa(W) (since v € i ^ l R " ) , w« € I,„(IR n ); the last statement will be shown later). Then, V|o| = 0 , 1 , . . . ( $ - 2 e ) l " l < z,z > < (2T)n\\v\\1\\wa\\qa\\Daf\\Pa.
(6.4.19)
NONTRIVIALITY
OF SOBOLEV-ORLICZ
SPACES OF INFINITE ORDER
149
We now prove that there exists a constant C > 0 such that (2ir)"||«||i||w.|[ fc < C,
|a| = 0 , l , . . .
(6.4.20)
Indeed, using the Leibniz formula we obtain that
^a-g^-tf-
C-w^MO-
J[ ak ■ ■ • (a* + 7i - V
On the other hand, since wa(x) = F^w^),
\a\ = 0 , 1 , . . . , we have
(2TT) W , ( x ) = H ) l " l ( t f - 2e)H / e^D^(raMiM. Ja Observing that £? - t < (,-, j = 2 , . . . , n, we conclude for any f G G, |/0| < Iin that
r.n^wi - ^ - 2 e W f S I ,
(Z7T) \X Wa(X)\
x IT a* fc=i
5
px
2L. 1 „]//}
„\|X
(«* + 7* - 1) //G | r 7 ^ - ^ o ( O I ^ } • J
MI-<) ,§«-,»:S-(-+-
(6-4.21) -1),
with iflS=max{j^|r'i^-nr«o(Oi«
1 7
We now estimate the expression n
n«f'K+Ti-i). *=i
Since |7| < \0\ < 2n we have m a x ( a , + 7 i - 1) < (\a\ + [ 7 | - 1) < [a| + 2n. Hence (6.4.22) .. On the other hand, (6.4.23) Combining (6.4.21), (6.4.22) and (6.4.23) we get for any |/3| < 2n
(2*)*|x/lw0(*)| < ff, ( f ? ~ 2 e ] " 2 2 "(| Q | + 2n)2".
(6.4.24)
150
DIFFERENTIAL
OPERATORS
OF INFINITE
ORDER
It is obvious that
lim(|a| + 2n) 2 *(^- 2 £ y a | =0. Then, from (6.4.24) follows the existence of a constant K2 such that for any x € lRn and \P\ < 2n \x0wa{x)\ < K,, |a| = 0,1, (6.4.25) From (6.4.25) we have |a| = 0 , l , . . . .
(l + xi)...(l+xl)\wa(x)\
(6.4.26)
We now show that for any p, 1 < p < oo, we have wa € L p (lR n ), and
I K I | P < / ^ " , H = o,i,....
(6.4.27)
Indeed, with p = oo it is obvious that
IKIU < Ks, and with 1 < p < oo we obtain from (6.4.26)
KM, = (JnJWa(x)\^x)1/P
* *(/a+**d+o)
-*<■•>***■•■
Thus, C = (2flr)B|M|ifr3irB yields (6.4.20). Further, combining (6.4.19) and (6.4.20), we obtain « ? - 2 e ) H < M > < C\\D°f\\Pa. (6.4.28) Taking the |a|-root of both sides of (6.4.28) and letting \a\ ^ oo we get l i m i n g ? - 2 e ) < i , i >VI-I <
ljyMCl/W||Zr/(,VI.|.
Therefore,
^-2 e
NONTRMALITY
OF SOBOLEV-ORLICZ
SPACES OF INFINITE ORDER
151
Let us now consider the weighted spaces of functions u : H " -» (C: LW00^,^}
:= {u e C°°(]Rn) | ||u||(00ir<>) < oo},
with /Rn(l + |x|2)-$ (
ll"ll(oo,ra) := inf I * > 0 | £
| J ?
y ) dx
(6.4.29)
where ra > 0 are real numbers. By arguments analogous to those in Section 6.4.2 we can prove the following lemma. Lemma 6.4.1. LW°°{ra,$a}
is a Banach space with the norm
(64-29).
To formulate the criterion of nontriviality for LW°°{ra, $ a } we need a notion of the order near zero for TV-functions (see Nguyen Nhu Doan [1]). Definition 6.4.1. We define the order near zero of an JV-function as ord * = sup{Jb > 0 | *(At) < A**(t),0 < X,t < 1). Theorem 6.4.4. Let the condition s u p ^ — = M
(6.4.31)
R e m a r k 6.4.1. We see from Theorem 6.4.2 and Theorem 6.4.4 that the spaces LW°°{$a,W} and LW°°{ra,$a} are nontrivial simultaneously. Later on we shall show that Condition (6.4.30) is necessary for the nontriviality of LW°°{ra,$a}. Proof of Theorem 6.4.4. Necessity. If u € LW°°{ra,$a}, then u € LW°°{$a,lRn}. Using Theorem 6.4.2 we immediately obtain the necessity of Condition (6.4.31). Sufficiency. We have to prove that there exists a non-zero function u 6 I W ° ° { r a , $ a } . To this end, we take a function u € C0°°(G), u ( 0 ^ 0, where G={feIRn
\q/4
=
l,..,,n},
152
DIFFERENTIAL OPERATORS OF INFINITE ORDER
and put
u(x) = / ^m<%JG
VVC SJIOW I .licit il t
J~/VV
\'ati
x D u(x)
* „ } . In fact, using the identities i —
j e
(
-
L) (t, MivJJ
?^«(0.
jyt£» and the equalities
,7^-7)! we obtain, for any x € JR.", \0\ < N, \a\ > 0, that
\X^D"U(X)\
< f memm
<^
H ^ ^ H
(6.432)
where
C7N = max{ j£ ^^WM*
< A 101 < ^ }
By (6.4.32) there exists a constant C > 0 such that (1 + |*| a )"+*'|iyu(*)| < C|a| 2 (" +M )( g /2)l°l,
(6.4.33)
where N = 2(n + Af). Since
lim |«r/2'-' = 0, |a|-tc»
we get from (6.4.33) for |a| large enough the estimate (i + \x\9)^M\iyu(x)\
< ,I-I.
Consequently, *.(|D"u(s)|) < *„(,H(1 + |x| 2 )-(" +M ))
< (l + isp)-<»+^-$Q(?M),
(6.4.34)
wherefcais the order near zero of $„, i.e. fc0 = ord $„. Without loss of generality we may assume that (6.4.34) holds for all a, \a\ = 0,1,.... Combining (6.4.34), (6.4.30)
NONTRIVIALITY
OF SOBOLEV-ORLICZ
SPACES OF INFINITE ORDER
153
and (6.4.31) we get T,
/
\x\2Y'^a{\Dau{x)\)dx<
(1 + OO
-
\xn°-^k°a(qM)dx
(i +
< £ **(qH) L (i + \x\2rndx < oo, /K
M=o i.e. u € LW°°{ra,
$ a } . The proof of Theorem 6.4.4 is complete.
We show now that Condition (6.4.30) is necessary for the nontriviality of LW°° {ra, $ Q } by constructing a counter-example. For simplicity we consider the case n = 1 and *fc = ak\t\"k,ak >0,pk> 1. In this case, ord $A = pk. L e m m a 6.4.2. Let 1 < pk < oo, and ak > 0, k = 0 , 1 , . . . be real numbers such that there exists a number q0 > 0 for which OO
X>Ag^
g
(6.4.35)
£ > q * » = oo, g > g 0 .
(6.4.36)
oo
Ar=0
TTjen / € LW°°{ak\t\^,m}} functions. Proof. Let / € LW^^^WL1}, 6.4.3 we have
implies f G B,0, where Bqo is the set of bounded entire
where $fc(<) = ak\t\'-.
Then, by virtue of Theorem
lim ||Z?*/lli{* > dh
(6-4-37)
k—*oo
whereby = sup{|£| |£ € supp /(£)}- On the other hand, since / € we may assume
LW°{afc|i|'",IR1},
oo
ElP a /li;!|<+oo.
(6.4.38)
Combining (6.4.35), (6.4.37) and (6.4.38) we obtain dj < go, and then because supp fa) C [-go, go] from the Paley-Wiener theorem / is an entire function of expo nential type g0. Since / € Lpo it follows that f £ Bqo. This completes the proof of Lemma 6.4.2.
154
DIFFERENTIAL
OPERATORS
OF INFINITE
ORDER
We now begin to construct a counter-example. We have to find sequences of numbers ak >0,pk,l
= oo,
(6.4.39)
k
and the condition (6.4.31) is satisfied, but LW°°{rk, **} is trivial, i.e. LW°°{rk, $*} = {0}. For this purpose we choose two sequences {ak} and {pk} such that there exists a number q0 satisfying Conditions (6.4.35) and (6.4.36). Further, we put rk = k + pkkk + tk,
(6.4.40)
where tk is a sequence of numbers, atf* > 1. Obviously, suprk/pk
= (k + pkkk + tk)/pk = oo,
that means the condition (6.4.39) is satisfied. We will show LW°°{rk,$k} = {0}. Assume the contrary, namely that there exists / G LW°°{rk,$k},f ^ 0. Without loss of generality we may assume ||/||(oo,rt) < 1 whence OO
_
£ / , ( ! + \x\2Y^k(\Dkf(x)\)dx Then / € LW°°{$k,
< oo.
(6.4.41)
JR}} and by Lemma 6.4.2 / € Bqo. Since / e BK we have lim sup \Dkm\%,k
= °l > 0,
(6.4.42)
where as denotes the type of an entire function / of exponential type a. We observe that for any x0 € IR1 the function gx„{x) := f{x + x0) is also an entire function of exponential type <JS. Applying (6.4.42) to gXo{x) we obtain a, = l i m s u p | £ * 5 x o ( 0 ) r = l i m s u p | ^ / ( x 0 ) | 1 / * . Consequently, by putting x0 = 1, a^limsuplD^l)!17*-
(6.4.43)
Let a = CTf/2. Then by (6.4.43) it follows that there exists a subsequence {km} such that 2V~<|£>*"/(l)|,m = 0,l,....
(6.4.44)
NONTRIVIALITY
OF SOBOLEV-ORLICZ
SPACES OF INFINITE ORDER 155
On the other hand, since / is bounded on IR\ it follows by virtue of the BernsteinNikolskii inequality (Section 1.9) that
IPVlloo < 4I1/IUThen, \Dkf(x)
- Dkf(l)\
<\x- 1 | < T ) + 1 | | / | U .
< j ' \Dk+if(t)\dt
Consequently, if \x - 1| < (c2k)~\c
= a / ||/|| 0 0 ,then
\Dkf(x)-Dkf(l)\
(6.4.45)
Combining (6.4.44) and (6.4.45) we get | * - l | < ( c 2 » - r , l m = 0,lf....
*** < \Dk"f(x)\,
(6.4.46)
Now, by (6.4.46), we obtain ^
k
Si f
fc (l+xy^k(\D f(x)\)dx £/J(l+*'T**(IX> /(*)|).fa
k=o k=0
> £ / _ m=0
,„
^(i+x2ykmak^kmPkmdx
- -
m =o
- -
y ^ 2 rr kk m at m
> > y^2
at
CT*mI'fc'"(c2fcm)_1. akmJ""»(c2km)~1.
(6.4.47)
On the other hand, (6.4.40) shows that &
^km"
\ C^
)
— C
y&
Q
)
Observe that for any a > 0 l i m 2 * V = oo. Consequently, for k large enough, T>-akm*k^(c2k~r > c-H^a^r
> 1.
(6-4.48)
From (6.4.47) and (6.4.48) it follows that CO
,
£ /
n+xr^(\Dkf(x)\)dx = oo.
m
k=oJ -
Thus, we arrive at a contradiction to (6.4.42). The proof of that Condition (6.4.30) is necessary for the nontriviality of W ° { r 0 , $ Q } is completed.
156
DIFFERENTIAL
OPERATORS
OF INFINITE
ORDER
Example 6.4.I. Let $„(i) = exp[t2aa] - l,o„ > 0. According to Theorem 6.4.2, the space W » { $ 0 , ] R " } is nontrivial, for example, for aa = (|a|!)-^,i/ > 0. Note that the corresponding space L i y ° ° { $ a , f t } in a bounded domain ft is trivial for v < 1 (see Example 6.2.1). £xamp/e M-2- We consider the Sobolev spaces of infinite order := {u € C°°(IR n ) | ||u||(oo)}
W°°{aa,pa}(JRn)
||u||(oo) := inf L > 0 I £
aJb-»||jD"«|12: ™ , < 1 \
(6.4.49)
where aa > 0,pa > 1 (the sequence {pa} may be unbounded). With the norm (6.4.49) W°°{aa,pa}(lR}) is a Banach space. This space is nontrivial if and only if its "characteristic" function OO OO a apa
V(0 :== £
is analytic in a neighbourhood of zero.
H=o
°s
6.5. Nontriviality of Sobolev-Orlicz spaces of infinite o r d e r in angular d o m a i n s . In this section we study the Sobolev-Orlicz spaces of infinite order in the cone-shaped domain GcEn: G = {xenr
\xn>t{x\
+ --- + xl_1f'2},
£
>0.
We define the Sobolev-Orlicz space of infinite order generated by the collection of AT-functions *„(*) as follows: LW°°{* a ,G}
:=
{u(x) € C~(G) I £>°ti|8G = 0,|a| = 0 , 1 , , . . , }
||u|| (oo ,
:=
inf j k > 0 I ] T J <5>a(D°u/k)dx < 1 1 .
By the method of Section 6.2 we can prove that LW°°{$a, the norm (6.5.1). Furthermore LlTfi,
lim \\u\\m We note also that LW°°{$0,G}
G}, N = 0 , 1 , . . . , and
= \\u\\(oo).
contains the following class as a subset
:= {u(x) £ C°°{G) \ Dau\dG OO
p°»
G) is a Banach space with
G} is the monotonic limit of the decreasing
sequence of Sobolev-Orlicz spaces of order N: WNL{$a,
£VV°°{*„,G}
(6.5.1)
\*\=oJG
= 0, |Q| = 0 , 1 , . . . }
.
■= Y. |c|=0''G
$a(Dau(x))dx
< 00.
NONTRIVIALITY
OF SOBOLEV-ORLICZ
SPACES OF INFINITE
ORDER
157
Let $;» be the function inverse to *„(*). Then we define the sequence of numbers M Ma
. ffs-Hi), <^1(1)' $ * 0^^o, 0 , \\ +00, +00,
$$ aa == 0, 0,|a| |a| == 0,1, 0,1,
The main result of this section is the following theorem. T h e o r e m 6.5.1. The space L W™^, G} is nontrivial if and only if the sequence of numbers Ma, \a\ = 0 , 1 , . . . defines a non-quasianalytic class of functions of n real variables. Remark 6.5.1. In this case, the criterion for nontriviality of the space L W°°{$a, G} has the same form as in the case of bounded domains (see Section 6.2). From The orems 6.5.1 and 1.18.2, analogously to the bounded case (Section 6.2), we can get algebraic criteria for nontriviality of the space LW°°{®«, G). It also follows that in the case of Sobolev classes of infinite order, i.e. *«,(*) = aa\t\»°,aa > 0,pa > 1, the numbers MN are defined by MN=
min la- 1 /""}
|«|=JV L "
>'
Therefore, application of Theorem 6.5.1 to the case *„(<) = a Q |i| p ° yields a new cri terion that is more explicit than that in Kobilov [1,2]. Proof of Theorem 6.5.1. Sufficiency. Assume that the assumptions of the theorem hold, that is, the sequence {Ma} generates a non-quasianalytic class of functions of n real variables. We have constructed in the proof of Theorem 6.2.2 a function u(x), u{x) £ 0 and ||u||(0O) < 00, having support in G C R". This proves that the condition of the theorem is sufficient for the space LW°°{$a,
G} to be nontrivial.
Necessity. Let Ma, \a\ = 0 , 1 , . . . generate a quasianalytic class C[Ma) of functions of n real variables, and let u <E LW°°{SQ, G). We show that u = 0. Without loss of generality we assume u e £ W °°{$ a ,G} and p{Dau,$a) < 1, \a\ = 0 , 1 , . . . . Put GR := G n 5 ( 0 , R), where R > 0 is an arbitrary number and
^-{t'lloU{?"' From Jensen's integral inequality we have 1
mes C
1 ~
mesGR
~
mesGR
^ m e f c l«l = 0,1,...
(6.5.2)
158
DIFFERENTIAL
OPERATORS
OF INFINITE
ORDER
Applying * - J to both sides of (6.5.2) we obtain / JQR
\DauR(x)\dx<mesGR-$:1(l/mesGR). I KV i\ -
(6.5.3)
Now, following Kobilov [2], we consider an auxiliary function tp(x) = uR(x).e-x". arbitrary a and £ G HT we have
For
iriwoi < w ^ 4 k £ 5 & <
K-\\DauR\\l,
jfc=0
n
II
1
Ti—l
\
(6.5.4)
where K is independent of a (see the estimate (2.3) in Kobilov [2]). Set & = «/ ' Trfc + fri * = l , . . . , n - l ; 2(n - 1)
^ . . . , U e E
1
,
From (6.5.4) for all a, [a| = JV we obtain the inequalities (1 +£)\ZnF~2
< /fl||B"Ufl||l,
(6-5-5)
where
^ " ) = ^( 2 7^lj^ + 6 l '---'2(^i)^ + 6 ' 1 - 1 ' ^ ) Now, starting from the definition of Ma and MN and from (6.5.5) and (6.5.3) we obtain (l + en)\UN-2\MZn)\
=
2,3,...
that is, the function v(y) belongs to the Hadamard class C[MN+2], which is quasianalytic (since C[MN] is a quasianalytic class). Therefore, to prove that v(y) = 0 it suffices to show that at some point y0 <= IR1 the function v{y) vanishes, together with its derivatives of all order.
NONTRIVIALITY
OF SOBOLEV-ORLICZ
159
SPACES OF INFINITE ORDER
By Theorem 1.19.1 it suffices to prove that & ( £ , ) allows a holomorphic extension £i(z„) to the lower half-plane of d? and satisfies there the inequalities
kl*"#iWI<7*.,*»GN. The last inequality has already been proved in Kobilov [2, p.49]. Thus, &(£„) satisfies all the assumptions of Theorem 1.19.1 and so u(y) is semifinitary. Hence, v(y) = 0. It now follows that &({„) = 0 for all £n G B \ that is, on any ray 6
=
2(^l) e n
+
6i
'
«=
1
."M»-1.
the function & ( & , - • • ,£») = 0. As a consequence <£(£) = 0 in B " That is uR{x) = 0 in G, or u(x) = 0 in GH. Since # is arbitrary, u(x) = 0 in G. The necessity is proved. Remark 6.5.1.
From theorem 6.5.1 we see that the nontriviality of the Sobolev-
Orlicz spaces L W°°{$a,G} for the conical domain G is precisely the same as that for a bounded domain (Theorem 6.2.2). We stress that the angle of the cone, which is determined by the number e > 0, is arbitrary but less than w. It is interesting to remark that when t = 0 the domain G reduces to a half-space, and the criterion for nontriviality has another character (see Theorem 6.5.2 below). We define the Sobolev-Orlicz space of infinite order in the half-space as follows: iVr°{$mo,[0,oo)xB"}
:=
{U(M)eC~([0,oo)xB")|.D>(0,z) m = 0,l,..,,||u||(oo)<+oo},
= 0,
with ||u|| (oo) := inf Ik > 0 |
f)
J°° f $ma{D?Dau(t,x)/k)dtdx
< l\ .
(6.5.6)
L W°°{*m«, [0, oo) x B " } is a Banach space with the norm (6.5.6). [0,oo) x B " } is nontrivial if and only if the
Theorem 6.5.2. The space LW°°{$ma, following conditions are satisfied: i) There exists a number d>0
such that for all a OO
bm := £ •*(**) tofcO
< +«■
160
DIFFERENTIAL
OPERATORS
OF INFINITE
ORDER
ii) The sequence of numbers M
m
_ f K\
bm>0
~ \ +00, 6m = 0 '
defines a nonquasianalytic class of functions of a single real variable. Further we consider the case of functions defined in the domain G = ft x W, where fl is either a bounded domain in IR" or an angular domain in HT I W " { * ^ C } := {«(«,*) € C°°(G) I D°u(t,x)\dSl ||u||(oo) := inf {* > 0 I £
= 0,|a| = 0 , 1 , . . . } ,
JG^(DaxD?u(t,x)/k)dxdt
£
< 1 j .(6.5.7)
i J T M ^ G } is a Banach space with the norm (6.5.7). Theorem 6.5.3. The space LW°°{$a0, conditions are satisfied: i) There exists a number d>0
G} is nontrivial if and only if the following
such that for all a: OO
ba := £
I0l=o
•+{&)
< +00.
ii) The sequence of numbers
«"{&. til •"*'-■ decrees a nonquasianalytic class of functions of n real variables. The proofs of Theorems 6.5.2 and 6.5.3 are, indeed, a combination of those of Theo rems 6.2.2, 6.4.2 and 6.5.1, and therefore we omit them. Bibiographical Notes: The treatment in Section 6.1 is based on the paper by Dubinskii [2] (see also Dubinskii's book [3], Chapter 4, and Dubinskii's survey [6]). The Section 6.2 follows the paper by Tran Due Van, Le Van Hap and R. Gorenflo [1]. The methods of the proof of Theorem 6.2.2 and Corollaries 6.2.1, 6.2.2 are due to Tran Due Van [4], [6], (see also Tran Due Van's book [12], Chapter 2). Section 6.3 is a simple modification of Section 2.2 in Tran Due Van [12]. Section 6.4 is mainly based on Tran Due Van, Ha Huy Bang and R. Gorenflo's paper [1]. The key of the proof of Theorems 6.4.2 and 6.4.3 is due to Ha Huy Bang [3]. The results of the last section of this chapter are published for the first time, they are due to Tran Due Van. For the criteria of nontriviality of Sobolev spaces of infinite order we refer the reader to Dubinskii's book [3] and Tran Due Van's book [12] (the case of weighted spaces).
Chapter 7 S O M E P R O P E R T I E S OF SOBOLEV-ORLICZ SPACES OF I N F I N I T E O R D E R
In this chapter some problems of the trace theory and the imbedding theory of infiniteorder Sobolev-Orlicz spaces are considered. To study the inhomogeneous boundary value problems one needs to work out an ap propriate theory of traces of functions lying in the space of solutions and a suitable method for solving the corresponding homogeneous problem. With this aim, we give in Sections 7.1, 7.2 some necessary and sufficient conditions guaranteeing the exis tence of traces of functions lying in Sobolev-Orlicz spaces of infinite order. We note that in contrast to the function spaces of finite order, the problem of the extension of a trace for infinite-order function spaces is nontrivial even in the one-dimensional case (see Section 7.2). Section 7.3 is devoted to the structure of spaces dual to Sobolev-Orlicz spaces of infinite order. We prove a structural theorem for these dual spaces and study their separability. In the last section some imbedding theorems for Sobolev-Orlicz spaces of infinite order are established. It begins with a general functional approach of Dubinskii [2,3] with applications to the case of Sobolev-Orlicz spaces of infinite order and ends with some algebraically easily verified imbedding criterions for Sobolev spaces of infinite order on the line IB,1. Since many problems related to this chapter need a serious investigation, we raise in Sections 7.2 - 7.4 some open questions. 7.1. Traces and extensions. Let G C Mn be an arbitrary domain with boundary T. Let us consider in this domain the space lW~{$a,G}:=LeC«>(G)\
||«||(oo):=inf{fc>0|
£
JQ*a(D°ufk)dx
< ill
and let us ask: What properties must the sequence of functions }w{x'), x' £ T, |w| = 0 , 1 , . . . , possess to ensure the existence of a function u e C°°(G) such that Duu\v = fu{x%
M = 0,1,...
and
IMIM < 161
+00?
162
DIFFERENTIAL
OPERATORS
OF INFINITE
ORDER
We then say that the sequence of functions / „ is the trace of the function u € W°°{<Pa, G], and the function u is an extension of the sequence fu(x% |w| = 0 , 1 , . . . , inW°°{$a,G}. Definition 7.1.1. (Donaldson and Trudinger [1]) The domain G C HT is called admissible if the imbedding Wj(G) -» Lp.(G),p* = np/{n
-p)
holds for all p > 1. Before formulating the main results, let us make some observations. Put Wa{G} := {u\Dau € L{$a,G}}. Let N > 0 be a natural number. We suppose that for any TV > 0 the boundary functions fu(x'), |w| = 0 , 1 , . . . , TV - 1, can be extended into the interior of the domain G as a function in the class WN := n H <jvW 0 {G}, i.e., that there exists a function u <E WN such that D"u\T
= fu(x>),
M
Hw < °° (see the definition of the norm || • \\(N) in Section 1.17). Theorem 7.1.1. Suppose that the domain G is admissible. A sequence of boundary functions f„(x'), \u\ = 0 , 1 , . . . , x' € T, is the trace of a function u € LW°°{^a, G} if and only if the following conditions are fulfilled: i) For every N = 1 , 2 , . . . , the function fu{x'),\u\ extension in WN;
< TV - l,x'
e T, admits an
ii) There exists a sequence of extensions uN € WN such that IKII(W) < const. Proof Indeed, if the sequence /„(*'), |w| = 0 , 1 , . . . , x' G T, is the trace of a function uG LW°°{i!>a,G}, then for any TV = 1,2,..., the family fu(x'), |w| < T V - l , x ' e T is also its trace in the space WNL{
PROPERTIES
OF SOBOLEV-ORLICZ
SPACES OF INFINITE
163
ORDER
[1]) and the diagonal process, we get that there exist a subsequence of the sequence uN(x) and a function u G C°°{G) such that this subsequence converges to u uniformly (locally) together with all its derivatives, respectively. It is clear that u(x) satisfies the conditions
D"u\r = fu{x'),
|w| = 0,l,..., s ' e r .
Moreover, from Condition ii) we obtain that ||u||(oo) < K, and therefore u{x) is the desired extension. The theorem is proved. R e m a r k 7.1.1. We see that the functions uN{x) are the solutions of the problems
J2(-l)MD(
= 0,
(7.1.1)
|er|=0
£T«|r 1
=
/„(*'),
|w|<JV-l,
(7.1.2)
1
where
(0,a)} (see Section 6.2)
£W°°{*n,(0,a)}
:=
{«€Co°°(0,a)
| ||«|| ( 0 O )
(7.2.1)
Hl(oo) £ r*n(Dnu/k)dx
J
(7.2.2)
Suppose that LW°°{<S>n,(0,a)} is nontrivial. Then, by Theorem 6.2.2, we have liminf MlJ« = co, £
-j^-
< +co, Mn = { * £
a / a )
'
£ f J' ■
(7.2.3)
164
DIFFERENTIAL
OPERATORS
OF INFINITE
ORDER
It is well known (Borel, 1896) that, if an arbitrary sequence of numbers bn, n = 0 , 1 , . . . , is given, then there is a function u € C°°(0, a) (a is a positive number) such that Dnu(0) = bn, n = 0 , l , . . . which means that any sequence of numbers is the trace of some infinitely differentiable function u{x). However, such a function hardly ever belongs to LW°°{$c G}, where W°i,
a
, G) := {u G C°°(0, a) | \\u||{ao) < oo}.
Let us now formulate the problem. Suppose that two sequences of numbers bm and Cn,m = 0,0,--- ar• given. It .I tisired to find a function u(x) in LW°°{$a, G} sucs that Dmu(0) = bm, Dmu(a) = cm, m = 0 , l , . . . . As it will be seen below, it is sufficient to consider the case c m — 0, m — 0 , 1 , . . . , i.e. to consider such conditions on the sequence {bm} under which there exists a function u € LW°°{$n,G} satisfying the conditions Dmu{Q) = bm,Dmu(a)
= 0, m = 0 , 1 , . . .
(7.2.4)
T h e o r e m 7.2.1. Let bm be a sequence of numbers satisfying the inequalities \bn\ < Mnn\,
(7.2.5)
where M is a positive constant. Then, for a given infinite-order Sobolev-Orlicz space LW°°{$n,G}, there exists a function in it satisfying Conditions (7.2.4). Proof. The condition (7.2.5) means that the numbers bm are the boundary values of an analytic function in a neighbourhood of zero, i.e. the series
f(x) := £ bmxm m=0
converges in some interval [0,6]. We may assume b < a. Further, since the sequence of numbers Mcn satisfies Conditions (7.2.3), due to Lemma 1.18.1, for any b > 0 and 0 < q < 1, there exists a function F £ C,T(0,&) satisfying the following conditions: i) F(0) = 1, DkF(0) = 0, fc = l , 2 , . . . . , ii) DkF(b) = 0, ifc = 0 , l , . . . ; iii) max ie(0 ,6) \DkF(x)\
< qkMl k = 0 , 1 , . . . .
PROPERTIES
OF SOBOLEV-ORLICZ
SPACES OF INFINITE
ORDER
165
Now put , .
f F(x)f{x),
x €[0,6]
(7.2.6)
where the function F(z) is denned by Conditions i)-iii) with q < 1/2. We show that this function is the desired one. In fact, the function u(x) defined by (7.2.6) satisfies Conditions (7.2.4). It remains to prove that ||u||(oo) < oo. To this end, using Leibniz's formula, we get for any x € (0,6) \Dnu(x)\
< (2L)n f >
k)lM£(q/L)k,
-
(7.2.7)
k=0
where L > 0 is a constant. By virtue of the logarithmic convexity of the sequences {M^} and {n!}, we obtain from (7.2.7) the inequalities (7.2.8)
\DnU(x)\<(2Lrn]f2Un)k,
k=0
where ln = {M'/n\y'nq/L. same,
We shall show that /„ -> oo as n -
oo, or what is the (7.2.9)
I i m ( n ! / M ^ = 0. n—«»
Indeed, for any N and n > N the estimates 'K-l
Mn-2
[ M< M*n_x
< (MS)"""
"
< (Aft) l'n (eNn _
M£_t
T
1 M
n-N
1
N)
(7.2.10)
hold, and by virtue of (7.2.3) we have en =
J2
Mt-dMi
^ 0, as iV ^ oo.
k=N+l
Consequently, using Stirling's formula, we find from (7.2.10) that
s
{£T
[«*(»-*) ^™] <
fK V21r n \ 1 / n n [ M% ) n-N\
fne
1/n
n
7 e
(7.2.11)
166
DIFFERENTIAL
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OF INFINITE
ORDER
where K is a positive constant. Obviously, for a fixed N,
(KV2l^V/n n V M» J n - N \
(n-N\N/n —» 1 asn —► oo, eN )
therefore for n large enough we get from (7.2.11) the inequality /n!\1/n<2eN
(7.2.12)
WJ - e ' By virtue of (7.2.11), the inequality (7.2.12) yields 00, n * oo, (n!/M<) 1 / n - t O , as rn —
i.e. the limit relation (7.2.9) is valid. Now, from (7.2.8) and the fact that ln -> oo as n -► oo we get m+l _ i
\D"u{x)\ |JD"«(X)|
n, n' n! ^ l f *^ < < (21) {2LTn\±{l 2 I )n» " ! £nf( ' " ) << ( {2L) <
Mcnn < ??Af£, q"Mcn, 2{2L)nn\lnn < 2{2q)nM'
for n large enough and for all x € (0,6). Here q0 < 1. Hence, oo
ra ra
n n (D u(x))dx T / $•»(iru(*))ix
oo OO
< <
#£*»(<«) n=0
OO
<
/ f i D 9 o * » ( ^ )
Thus, we have proved that ||u||(oo) < tf2, and this completes the proof of Theorem 7.2.1. Consider now a more general case. Set OO
-m = E W m+*/^»+*+lfc=0 fc=0
(7.2.13) (7-2.13)
It is clear that sm —% 0 as m —> oo. Theorem 7.2.2. Let r be a positive number such that °°
(
rm
1
< +oo. (*y*}<+»-
X;j6 I ) | 6mm|max| | m a x 5m | 3 —, (M^j m-,
(7.2.14) (7.2.14)
PROPERTIES
OF SOBOLEV-ORLICZ
SPACES OF INFINITE
Then, there exists a function u € LW°°{$n, (7.2.4).
167
ORDER
(0,o)} satisfying the boundary conditions
Proof. We construct a desired function u € LW°°{$n,
(0,a)} as follows:
i) Indicate the "basic functions" vm € C°°(0,6),(6 < a), satisfying the conditions £>num(0) = «5nm, n = 0 , l , . . . . ii) Showing that Dmv(b) = / C m ! for the function OO
«(*) := ^
6„w m (x),
iii) Using Theorem 7.2.1 to construct a function w € LW°°{$n, (6,a)} such that Dmw(b) = Dmv{b), Dmw{a) = 0, m = 0 , 1 , . . . , and show that the function f"(x), {
;_\
a: € ( 0 , 6 ) ,
w{x), i 6 [6,a)
is the desired one. Step i): Construction of basic functions. Denote by c the sum of the series (7.2.14). Let m be a non-negative integer and 6 < r/2. Choose the sequence of numbers
bkm =
M:/(dkmMl+k),
where dm = 4s m /6 and sm is defined by (7.2.13). It is easily seen that
-e"4A
Mm+k <3'
Consequently, by virtue of Lemma 1.18.1, there exists a sequence of functions such that 1) F m ( 0 ) = 2 e ,
Fm{x),
D n F m (0) = 0, n = 0 , l , . . . ;
n
2) D Fm(b) = 0, n = 0 , l , . . . ; 3)
max\DkFm(x)\
it = 0 , l , . . . .
(7.2.15)
x6(0,6)
Now, for x.£ (0,6), put
vo(x) = Um(x)
Y/o(x),
Vi(x)
= ilF^ydr,,...,
= 2c(m-2)! jf (z " e r " 2 /o^"" F - ^ ) * * • - = 2,3,.,..
168
DIFFERENTIAL
OPERATORS
OF INFINITE
ORDER
These functions vm are the desired basic functions. It is clear that they satisfy the conditions Dnvm{0) = 6nm, n, m = 0 , 1 , 2 , , . . . In addition, starting from the construction of vm{x) and (7.2.15), we get that for any 16(0,4) \D"vm(x)\ 1
< —P^
;l
\Dnvm(x)\
TV, n < m - , ,
-
2c(m-l-n)!
<
£
2c
'",
n>m.
(7.2.16) (7.2.17)
jm
Step ii): Construction of the main extension. For x € [0,6] put CO
v(x) = £
bmvm(x).
(7.2.18)
It is evident that Dmv(b) = 6m, m = 0 , 1 , . . . Consider the values of v(x) and its derivatives at the point x = b. We have CO
Dnv{b) = £2 6mDnvm(6), m=n
since DNvm{b) we find
= 0, m < n. From this, taking into account the inequality (7.2.16), 1
°°
A
h m ~1~n
* ^-M*^*™.
(7.2.19)
where if is a positive constant. Step iii): Removal of the " residual" at x = b. The function w(x) constructed in Step ii) does not satisfy the homogeneous conditions at x = 6, but the boundary values of its derivatives satisfy the inequality (7.2.19). Therefore, by Theorem 7.2.1, there exists a function w e W ° { $ „ , (6, a)} such that Dnw{b) = Dnv(b), Dnw{a) = 0, n = 0 , 1 , . . . . Finally, put ttW=|»W, v
'
*e[0,6) \ w(x), x S [ M ] '
169
PROPERTIES OF SOBOLEV-ORLICZ SPACES OF INFINITE ORDER
We shall show that u(s) is the desired function. In fact, we have only to prove that uelvr°{$n,(0,6)},since supp v PI supp w = 0. By virtue of (7.2.16) and (7.2.17) we obtain the estimates: n
i
oo
lure
i
j
Lm-l-n
Hence,
• w ( ^ . ) ) ^ ( { E ^ }M,) /£
1 +2
+
cW (m_* 1J_ ^! \
v»»
(7.2.20)
n) /
We estimate h and 72 separately. Because of (7.2.14) we have 1
-
2c *-„
J
^.ri
„ if
Mc
— O 2^ ZC
n(
M'
M
m=0
(7.2.21)
a
m
Further, combining (7.2.9) and (7.2.14), we obtain, for n large enough,
h
1
/
!,
6
-
\bm\dm(2bT-i\
- 2*" I * m £ + 1 c ( m -l)! j (7.2.22)
1/rom I i.z.z i) CLUQ 11.z.zz i n ioiiows LIICLL
1 i c a
-
|6m|g—
m=0
m
1
°°
+ 2c*n(6-"n!)mg+i
16 Id (26) m _ 1
(m_ 1 } ,
.
Consequently,
?"(«) = jt [" **(Dnv(x))dx i-i
-
2 c a
n=0
2^
L l+
m=0
^m
2lL2'
2 c n=0
m=n+l
(m - 1)!
170
DIFFERENTIAL
OPERATORS
OF INFINITE
ORDER
By virtue of (7.2.14) we have oo
KHM'J-1
In < £ m=0
oo
f > * < oo (b< r/2). A=0
Further, combining (7.2.14), (7.2.11) and (7.2.3) we obtain OO
La < c Yl *n(6""n!) < oo. n=0
Thus, p°°(v) < oo and, in particular, this means that v € LW°°{$n, quently tt € W ° { * n , (0,a)}. The theorem is proved.
(0,6)}, conse
Problem 7.2.1. Consider the Sobolev-Orlicz space of infinite order in a strip G = [0,a]xIR B LW°°{*ma,G} := {u e C°°(G); \\u\\(oo) < oo} (see Definition of LW°°{$ma,G} in Section 6.5). Further, suppose that there are given two families of functions ¥>m(z), < M i ) , m = 0 , 1 , . . .. The question is to find sufficient conditions imposed upon
D?u(a,,x) = ip(x),
such that
m = 0,l,...
(Some sufficient conditions for the case * m a ( £ ) = ama\tf, ama > 0,p > 1 are given in Dubinskii [3, Chapter 3] and Balashova [2, Sections 3].) Problem 7.2.2. Extend the results of Balashova [2, Sections 1, 2] to the case of LW~{*m,(0,a)} for * m ( | ) = a m * ( 0 , where $ ( 0 is an JV-function, and am > 0. 7.3. Dual spaces. In this section we shall study, based on the constructions in Sec tion 6.1.1, the dual spaces to Sobolev-Orlicz spaces of infinite order. To be definite we limit our considerations to the case of spaces LW°°{$a, fl} and £ W - ~ { * a , 0 } of functions defined in a bounded domain Q, C IRn. We emphasize that all results given below may be obtained also for the cases of a torus, the full Euclidean space 1R", the angular domain etc... Let n be a domain in IRn. Consider the space LW°°{
lim
W»L{*mm,
N—t-oo
where W * I { * B ) f t } is the Sobolev-Orlicz space of order N (see Sections 1.17 and 6.2).
PROPERTIES OF SOBOLEV-ORLICZ SPACES OF INFINITE ORDER
As is well known, the space W-NE{*a,n} is dual to WNL{$a,n} Consider now the increasing sequence of Banach spaces
171
(Section 1.17).
w~xE{*a, n} c w-2E{*a,n} c • • • c w-NE{*a,n} • • • INI-i>NI-2>--->NU>--Denote by EW-°°{$a,n}
the completion of
um: -.oW-mE{$a,n} in the norm IMI(-oo) = lim | | i | | _ m . II
*l V
/
J7J—J.OO
The space W-°°£{#a,fi} is the monotonic limit of the sequence of Banach spaces W-NE{*a,(l} (see Definition 6.1.2). By virtue of Theorem 6.1.2 we have Theorem 7.3.1. Let the space LW°°{
be nontrivial. Then, the space
We assume henceforth that the derivative <pa(t) of the function $„(/) is strictly in creasing. Theorem 7.3.2. The following representation is valid:
Ew-°°{$a,n}
1I
l«l=o
K e E*a(n),
t I I M C J <+00} |o|=0 J
Proof. We shall show that there is a unique series OO
£
(-IJMD-M*),
|c|=0
defined in the statement of the theorem corresponding to an arbitrary element of the
space
EW"»{$a,n}.
Indeed, let hs € U~ = 1 W-mE{$a,n},s = 1,2,-•• be a fundamental sequence in the norm II • (—oo)Denote m0 = m0(s) the smallest among all numbers m for which
172
DIFFERENTIAL
OPERATORS
OF INFINITE
ORDER
h3 € W-mE{$a,Sl}. Then, as is known from the theory of Sobolev-Orlicz spaces, the functions h,(x) can be represented in the form m m
M*)=E(-i)wM-W, |«|=0 |«|=0
where h,a 6 Eia{il)
(see Section 1.17).
Consider now the family of the Dirichlet problems of infinite order OO
L{Us)
=
T£(-i)MlT(<pa(iy>u.(z)))
= h.(x),
(7.3.1)
|c|=0
fl-'u.laiteo = 0, M = 0 , 1 , . . . ,
(7.3.2)
where <pa is the derivative of $ „ . According to the results of Sections 8.1 and 8.2 below, Problem (7.3.1)-(7.3.2) has a unique solution us € L W°°{<S>a,Q}. Moreover, for any m > m0 the inequality
IKII(m) < K < +oo
(7.3.3)
is valid. The estimate (7.3.3) implies that there exists a function ti € Z ( T { $ „ , f 2 } such that «, -» u uniformly together with all its derivatives, respectively. We now prove that u(x) is the unique limit of the sequence of functions Indeed, let {uk(x)} be another subsequence of the sequence {u,(x)} so that u*(z)-►»(x)
{us(x)}.
in C0°°(fi))
where w(x) is a function in LW°°{<S>a,n}- We have to show w{x) = u{x). Obviously,
<
\/veLW~{$a,n}
(see the definition of number m0{r) at the beginning
Hence,
< L(uT) - L(uk),v > < \\hr - M l w — « • mlkllo From this, letting m -> oo, we get < L(uT) - Lluk),v
> <
||fcr-MI(-oo)IM|«»l.
PROPERTIES
OF SOBOLEV-ORLICZ
SPACES OF INFINITE
173
ORDER
Since the original sequence {hk(x)} is fundamental in the norm || • ||(oo), letting k -» oo, r - m o w e obtain < L(u) - L(w),v > < 0, where v e L H/°°{* a ,fi} is an arbitrary function. In view of the uniqueness of the solution of the Dirichlet problem of infinite order (Theorems 8.1.1, 8.1.2, 8.2.1 below) u(z) = w{x). Thus, for every fundamental sequence {/is(x)}s=i,2,..„ there exists one and only one function u € L W°°{$a, ft} such that U , ( x ) = l-^hs)
-
u(x),
uniformly with all its derivatives. It is easily seen that if the fundamental sequences {hs{x)} and {h',(x)} are equivalent, the corresponding functions u(x) and u'(x) are equal, i.e. u(x) = u'{x). It remains to note that for any function u € I l T { i , 0 } , the series L{u) defines an element of the completion of U%=1W-mE{$a,n} in the norm || • ||(_<„). Therefore, the desired map L(u) <-> {hs{x)},
a=
1,1,...
is defined. The theorem is proved. Corollary 7.3.1. Suppose that the functions $ „ and $ a satisfy the Then
^-condition.
EW °°{$a,n} = [Lw°°{$ l a,n}) , V °" V /
0
\ *
0
where ( LW°° {„, ft) ] is the dual space of the space L VF°°{Q,ft}. Proof. Indeed, from the proof of Theorem 7.3.2 we see that the operator L(u) maps W L ^ W T X t i n r v of monotone'onerators in the'reflexile r W h s l c e l aennes a nomeomurpni&iii
L ^
i(«) ; ir{$ a ,fl}^(ir{$ a ,n}) , since L{u) is a strongly monotone operator. Thus, the corollary is proved. T h e o r e m 7.3.3. Suppose that $ a and $ a satisfy the ^-condition 0, !,■■•. Then the spaces LW~{$a,
SI} and EW~°°{$a,n}
for all a, \a\ =
are separable.
174
DIFFERENTIAL
OPERATORS
OF INFINITE
ORDER
Proof. First note that £W_m{$a,n},m = 1,2,... are separable. We shall show that EW-°°{*a,fl} is separable. Indeed, let Bm be a countable set which is dense in EW-m{$a,Sl}. Then B = U™=1Bm is also countable. Moreover, B is dense in
U~ =1 £W- m {i,fi} in the norm || • \|(_x). According to Theorem 7.3.2 any element h £ £ W - ° ° { $ a , n } can be represented in the form
*(«) = £ (-iyalDaha(x) = lim hm(x), where
M*) = £ (-i)wi>-*.(») e w-mE{*a,n}.
Thprpfnrp anv element h of FW^-co-f fll ran be anoroximated bv elements of the « n f with arhitrarv tmall error It follows that eap soace EW~ft M so also
Lrahle
'
The separability of L W°°{$a, 0 } follows from the fact that L(u);XW°°{
PROPERTIES
OF SOBOLEV-ORLICZ
SPACES OF INFINITE
175
ORDER
7.4.1. Imbedding criterion for monotonic limits of Banach spaces. Let
be a sequence of imbedded Banach spaces and let X^, = limbec XT be the monotone limit of this sequence. We recall that
x00 = \ue rc,Jf T | ||tx|too := Hm ||u||r < 00}, where \\u\U < \\u\\2 < •• • are the norms in XuXt,...
(see Section 6.1).
Likewise, we consider the Banach space F«, = l i m ™ ^ Ym, where Yt D Y2 D ■ ■ ■ D Ym D • • ■ is a second sequence of decreasing Banach spaces. We are interested in the question of the existence of the imbedding operator
It is clear that the problem of imbedding X^ C F« is deep only if both of these spaces are nontrivial. Thus, we shall further assume that the spaces X M and Y*, are nontrivial. Theorem 7.4.1. (unconditional criterion). The space X „ can be imbedded in Y^ if and only if the following conditions are satisfied: i) for every m > 0 the imbeddings
hold; ii) the limit lim ||ioo,m|| = M, where ||«oo.™|| W the norm of the operator i ^ ,
(7.4.1) exists and is finite. Moreover,
||ioo,oo||= m li5g b ||i 00 , m ||.
(7.4.2)
176
DIFFERENTIAL
OPERATORS
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ORDER
Proof. The sufficiency of Conditions i), ii) is evident. Indeed, if u g I „ , then in view of Condition i) u € Ym for all m = 1,2,..., too, moreover,
M*. < Hv»IIM*-Letting m -> oo and taking into account Condition ii) we get the inequality
H|K<Jm llwllHkoIt follows that the imbedding X„ C Y^ holds. Necessity. Let the imbedding X^ C Y^ be fulfilled. Then, a fortiori, there exists the imbedding X^ CYm, where m = 1,2,... is arbitrary. Therefore, it remains to establish Condition ii). Let us assume the contrary, i.e. Jirn |Hoo,m|| = + ° o .
(7.4.3)
The contradiction will be, obviously, obtained if we can find an element u € Xx such that linw^oo \\u\\Ym = oo. For this purpose let us choose a family of numbers Nk -> oo, consider the open unit ball
k = 0 , 1 , . , . . and
S(0,l) = { u 6 X o o | \\u\\Xa> < 1 } in the space Xx. The condition (7.4.3) means that there exists a number m0 and an element u0 € 5(0,1) such that ||«0||ymo > N0. In view of the continuity of the imbedding operator ioo,m0 there exists a ball 5(w 0 , t0) C 5(0,1) with center u0 and radius e0 > 0 such that the inequality
Hk, 0 >f, W g5Keo), holds. Since the sequence of bounded linear operators is simultaneously bounded or un bounded on every ball, Condition (7.4.3) implies that there exists a number m t and an element m € 5(ui,e 0 ) such that |K||y m ] > Nt. Now, taking into account the continuity of the operator »«,,„,,, we find a ball S{uu ex) C S(u0, e0) (the notation is clear) such that the inequality JV, \m\Ym% > Y >
u 6
%i'ei)'
PROPERTIES
OF SOBOLEV-ORLICZ
SPACES OF INFINITE
177
ORDER
holds. In the same way we find a family of imbedded balls ^(uo, eo) D S{uu e,) 3 • - • O S(uh tk) D ■ • • such that for any u 6 5(«fc, e*) the inequalities N k . > y . * = 0,l,..,,Jb,
(7-4.4)
where ms -> oo as s -> oo, are valid. In particular, for the centers of the balls uk6S(uk,ek) the inequalities
>f,s-
IKIIJ^
= 0,1,,..,*,
are valid, too. Without loss of generality one can assume that the family of balls S{uk, ek) tends to an element u 6 Xx. On the other hand, if k -> ooo ,i tollows sfom ((.4.11 )hat tor rny 5
ll«llr-. > y , where 7V„ -» oo, ,a s -► to. As was slready yemarkedd ,his sroves she eecessity yo our Conditions i) and ii). The validity of the formula (7.4.2) is evident. The theorem is thus proved. The criterion proved in Theorem 7.4.1 is defined by the behaviour of the norms of imbedding operators ^OO.TTl
•y
—> v
• -^oo
*
1
m = 1,2,---,
mi
i.e. in this criterion the limit space is used. It is also natural to have a criterion for the imbedding X^ C Y^ in which only the spaces XT and Ym are used. For this we suppose that the spaces Xr are reflexive and, in addition, the following condition is fulfilled: For any m > 0 there exists a number r{m) > 0 such that the imbeddings ir,m : Xr -» Ym hold; moreover, these imbeddings are compact.
((.4.5)
178
DIFFERENTIAL
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OF INFINITE
ORDER
Under this condition we have Theorem 7.4.2. (conditional criterion). The space Xx can be imbedded in Y^ if and only if the limit lim^,*, Km,-.*, ||r r , m || exists and is ffnite. Moreover, ||«ooooll= Km lim||r r , m ||.
(7.4.6)
m—*-oo r—*oo
Proof The sufficiency of the condition of our theorem is evident. As for its necessity, the proof of this fact is reduced to the proof of the necessity of Condition ii) of The orem 7.4.1. Indeed, there holds the following Lemma 7.4.1. Let Condition (7.4-5) be fulfilled. formula
Then, for any m there holds the
Hm I I ^ H I I ^ I I , where *oo,m ■ -^*-oo
Proof Let u G Xx. we have
* ^m-
Then, a fortiori, u G Xrr r — 1,2,..., and, therefore, from (7.4.5) Nk<|]ir,m|||HUr,r>r(m).
Letting r -» oo and taking into account that the norms ||i r , m || do not increase, we get the inequality ||ti||ym < Km ||ir, m |||k|U«,. From this, it follows immediately that |lwil
(7.4.7)
Let us show that, in fact Wi^mW =
\im\\ir,m\\.
Indeed, if we assume
Wi^W < \im\\ir,n\\, then for all sufficiently large r there are elements uT G XT such that
I H k < l , |K||y»> 11100^11+e,
(7.4.8)
where e > 0 is a fixed number. Since the spaces XT are reflexive, one can suppose (choosing a subsequence if needed) that in every X3, s = 1 , 2 , . . . , the sequence {u r } r >, converges to an element u G n?=1X, weakly.
PROPERTIES
OF SOBOLEV-ORLICZ
SPACES OF INFINITE
ORDER
179
Further, since in view of (7.4.8) \\"r\\x, < 1 for r > s, we have ||u|U, < 1 for every a = 1 , 2 , . . . , too. It follows that u £ X « and \\u\\Xo ^ 1. On the other hand, taking into account the compactness of the operator tr,m : XT —¥ Ym
we get that there exists a subsequence (we denote it as uT too) and an element u such that uT converges to u in Ym strongly, i.e. in the norm of Ym. Consequently, passing to the limit in the second inequality (7.4.8), as r -> oo, we obtain that
Nk.>llw.ll + e, where e > 0 is a number. The latter inequality is impossible in view of the definition of the norm of the operator i^m. Lemma 7.4.1 and, therefore, Theorem 7.4.2 are proved. Remark 7.4.1. It is evident that, the property of the reflexivity of XT and the compactness of the imbeddings (7.4.5) were used only for the proof of necessity. Suf ficiency is true without these conditions, i.e. the imbedding Xx C Y^ is true if there exist only the imbeddings (7.4.5) and the limit mentioned in Lemma 7.4.1 is finite. Let us turn to the question of compactness of the imbedding operator ^00,00 • -<^oo
-Yc
We recall that the operator i^*, is compact if it maps the unit ball of the space 1 M into a compact set of the space Yx, i.e. if the unit ball in X„, is a compact set in the norm of Yx. Let us suppose that the following condition is fulfilled: For any m > 0 the imbedding X^ C Ym is compact. Under this condition we have the following Theorem 7.4.3.
The imbedding ^00,00
■y
—k v
is compact if and only if the convergence
Hlr~=JimHk
(7.4.9)
180
DIFFERENTIAL
is uniform on the unit ball of the space Proof. For the proof of assumption the unit ball itive integer. Therefore, u € n£ = 1 Y m such that un
OPERATORS
OF INFINITE
ORDER
Xx.
the sufficiency we remark that in compatibility with our 5 C X M is compact in Ym, where m is an arbitrary pos there exist a sequence {u n } n =i, 2i ... € S, and an element -> u in Ym for all m.
Since for any m BU.HK, < | K l k
< | | » o o , o o | | | K | U . . < ll.oo.ooH,
(7-4.10)
we have u € Yoo- It remains to show that un -> u in Y*. Indeed, in view of (7.4.9) and the inequality (7.4.10), for any e > 0 there exists a number m > 0 so that for any n = 1,2, ■ ■ • h-un\\y„<\\u-un\\ym+t. From this, for n large enough, II" - u»||y < 26, since, as it was remarked, un -» u in Ym for any fixed m. The latter inequality means that un - m in y „ . The sufficiency is proved. The necessity will be proved by contradiction. Namely, let us suppose that the imbed ding i«,tao is compact but Condition (7.4.9) is not satisfied. This means that there exist a number e0 > 0, a subsequence of natural numbers (without loss of generality one can assume that this subsequence is the whole series of natural numbers) and a sequence of elements um € S such that [Kiln. - I K I k > *.
(7-4.11)
On the other hand, since the imbedding *«,,«, is compact, the sequence of elements um € 5 may be considered as a sequence in Y^. Let ueY^be the limit element, i.e.
ll«-«»k.->0,
m-»oo.
Consequently, if m -> oo, then
IKIk-IKIk, < H k - N k +ll^-^lk + ll^-^lk < IHk-IHk + zilu-t^lk-o. The latter is, however, impossible in view of (7.4.11). The necessity and, therefore, Theorem 7.4.3 are proved.
PROPERTIES
OF SOBOLEV-ORLICZ
SPACES OF INFINITE
181
ORDER
7.4.2. A p p l i c a t i o n s t o t h e spaces LW~{*a, G}. We apply the previous results to the Sobolev-Orlicz spaces of infinite order. Let LW°°{*a,G}
:= {«€ C°°{G) I ||u||(00,«) < oo},
LW™{qa,G]
:= {u e C°°{G) | ||u||(oo,») < 00}
be the pair of Sobolev-Orlicz spaces of infinite order, G be a domain in K." assume $„ ^ 0 and $„ ^ 0. We are interested in the question of the imbedding of the space LW°°{$a, space LW°°{ya, G). For the study of this question we recall that LW°°{*a,G}
'
LW°°{Va,G}
lim lim
G} in the
WNL{$a,G}, l
JV—00
=
We
'
WNL{Va,G},
N-HX N
where W L{*a,G}
N
and W L{<Sla,G} are Sobolev-Orlicz spaces of order N.
Let
in m
' '' Zl^*%^^w~ui,'Gr{
2oo,<x> : LW
{9a,G)-+LW
(7-412)
{Wa,G}
be the imbedding operators. Then from Theorem 7.4.1 we get the following results. T h e o r e m 7.4.4 (unconditional criterion). The space Z,W°°{$ a ,G} can be imbedded in the space LW°°{Va,G} if and only if the following conditions are satisfied: i) for anym>0
there exists the imbedding *oo,m : LW°°{$a,
G} -* WmL{Va, G};
ii) the limit l i m ^ o o ||i00.m|| exists and is finite. In this case ||*oo,ooII = J i m ||4o,m||-
T h e o r e m 7.4.5. (conditional criterion). Suppose that for any m > 0 there exists a number n{m) such that for all n > n(m) the imbeddings (7.4.12) are valid and there exists the finite limit lim lim I I L J I . m-*-oo n—»-oo
Then, LW™{$a,G}
-+ W ° { t f „ , G } .
182
DIFFERENTIAL
OPERATORS
OF INFINITE
ORDER
Remark 7.4.2. If the functions $ a , $ „ , f „, $ 0 satisfy the A 2 -condition, and |a| = 0,1, • ■ • , then the condition of Theorem 7.4.5 is necessary for the imbedding ioo,oo : LW°°{$a,G}
-.
LW°°{9a,G}
and Hico^H = lim lim ||i n , m ||. 7.4.3. Sufficient algebraic conditions. In this subsection we give some simple algebraic conditions under which the imbedding
LW°{^,G}
^
LW^^l^G] p
is valid for the case G c t f and $„(*) = an|*| ,tfa = bn\t\", an > 0,6 n > 0,p > 1, that is for the case of Sobolev spaces of infinite order defined in G C HI1. I. The case G = M 1 Let an > 0 and bn > 0, n = 0 , 1 , 2 , . . . be arbitrary sequences of numbers, and let 1 < p < oo and 1 < r < oo. We define W»{«.,i«,r}(H1)
:= {/(as) £ C0O(IR1) | H/lljoo,.) < oo},
where
II/IU>:=X>^/II;, JI=0
and || • H, is the norm in LP(JR}). In order that W°°{an,p,r}(JR}) really be a space of infinite order we assume that a0 jL 0 and {a„} contains an infinite subsequence of positive elements. As we know (see Section 6.4), the space W°°{an,p,r}{m}) is nontrivial if and only if Ra > 0, where Ra is the radius of convergence of the series
n=0
Everywhere in this subsection we write W™ in place of W°°{an,p,r}(m}) assume Ra > 0. We introduce similarly another Sobolev space W§° := W°°{bn,p,r}{WLl) order. We are interested in the existence of the imbedding W? Wb°°. w? ^<—» W£°.
and we
of infinite (7.4.13)
It is clear that the inequalities bn
Vn,
(7.4.14)
PROPERTIES
OF SOBOLEV-ORLICZ
SPACES OF INFINITE
ORDER
183
and K > 0, suffice for the imbedding (7.4.13). However, these inequalities are too restrictive, since they require that the coefficients bn vanish for at least those indices n for which an = 0. The theorems below are free from this deficiency. We first prove some auxiliary results. The following result is a consequence of Kolmogorov [1] and Stein [1]. L e m m a 7.4.2. Suppose that 1 < p < oo and the function f £ C ^ I R 1 ) is such that | | £ > " / l | P < o o . n = 0 , l , . . . . Then
\\okf\\;<(*m\f\\rk\\Dnft
o
for n = 2 , 3 , . . . . Lemma 7.4.3. Suppose that the sequence 0 = n 0 < ni < • • • of integers and the function u € C°°(IR1) are such that \\Dn*u\\p < oo, k = 0 , 1 , . . . . Then the limit
du=\im\\D»u\\l'» exists. Proof. If u(x) is a constant, then the statement of the lemma is trivially true. Suppose that u{x) is not a constant. The Gagliardo-Nirenberg theorem (see, for example, Besov, Ilin and Nikolskii [1], §15.1) and the conditions of the lemma give us that ||£>nu||p < oo, n = 0 , 1 , . . . From this and Lemma 7.4.2 we get UU»«t|; < (7r/2)"|| U ||^||i3"u|| p fc ,
0 < k
Consequently, Vrc > 2, 0 < k < n
\\Dku\\i'k < {tml,k\nirmnk)\\l>n
(7-4-15)
It follows from (7.4.15) that there exists an index k0 = k0(u) such that 1 < (Tr/2y'k\\u\\£-kVW
< 2, k0 < k < n.
(7.4.16)
Next, suppose that the sequence {km} is such that d=
lim \\Dk-u\\l'k-
= limsup ||0"ti||l/".
(7.4.17)
We first consider the case d = oo. It then follows from (7.4.15) that there exists an index m 0 such that for all m > m0 ||Z>*»u||;/*» < 2\\Dnu\\l'n,
\/n>km.
(7.4.18)
184
DIFFERENTIAL
OPERATORS
OF INFINITE
ORDER
Combining (7.4.17) and (7.4.18), and taking d = oo, we get that n 00<2HmM\]D u\&\ n—+o6
v
But this means that the limit du = limn-,*, ||£>"Hlp/n exists. Next, assume that d < oo. Then for any t > 0 there exists an index mx = m,(e) such that for all m > m, \\Dk-u\\yk(ir/2)-*/*-[f«[[-f—*-)/fr*-)
(7.4.19)
> d-t, >
l-£,
Vn>fc m .
(7.4.20)
On the other hand, for all m > 1 \\Dk-u\\l'kDirectly from (7.4.19)
< ^l2ylk-\\ut-k^nk^\\D-u\\y\
(7.4.21)
(7.4.21) we obtain limmf||Dnu||J/">(
Hence liminfll^ll^^imsupll^H^ and so the limit du = l i m ^ ||B»tt||V» exists. The lemma is proved.
Lemma 7.4.4. Suppose that the sequence {km} and the function u{x) satisfy the condition of Lemma 7.4.3, and u{x) is not a constant. Then du>0. Proof. It follows from the conditions of the lemma that 0 < \\Du\\p < oo. Using Lemma 7.4.2 we get \\Du\\;<
(x/2)«MJ- l l|JD»u||„
Vn>2.
Hence,
^ ' • w r <»«-<»,
Vn>2.
From the last inequalities and Lemma 7.4.3 we obtain
4>2|1'ft>0. The lemma is proved.
PROPERTIES
OF SOBOLEV-ORLICZ
SPACES OF INFINITE
185
ORDER
L e m m a 7.4.5. [0, R\lT) C R
| u € W?} C [0, RlJr],
(7.4.22)
Proof. We first prove the second inclusion in (7.4.22). Clearly we need treat only the case Ra < oo. We argue by contradiction, assuming that there exists a function u € W™ such that du < R>Jr. Then \\D-u\\^>(Ra
+ e0)^,
Vn>n0
for some e > 0 and no > 1. From this we get immediately that u £ W™. . u t this contradicts the assumption u € W™. Let us prove now the first inclusion. Choose a non-constant function g 6 W™ (such a function exists: see Section 6.4). Then it follows from Lemma 7.4.4 and the proven second inclusion that 0 < dg < RlJT. We next fix an arbitrary number A > 0 such that Xdg < RlJ\ and we let f{x) = g{Xx). Then the equalities \\Dnf\\p = A " - 1 ^ ! ^ ! ! , , ,
Vn > 0
immediately yield df = Xdg and / € W™. Since the number A {Xdg < R}Jr) is arbi trary and 0 € W~, we get what we needed, and the lemma is proved. C o r o l l a r y 7.4.1. i) If Ra > Rb, then W£° ^ Wb°°. ii) IfRa < Rb, then W~ >-» Wh°°. Hi) IfRa = Rb and £ ~ KK < oo, then Wa°° — Wb°°. The following transformation of series will be needed below. L e m m a 7.4.6.
Suppose that 0 < an and 0 < xn < xn+l, oo
oo
] T anxn = x0el + ^(xk+1 n=0
where el =
n = 0 , 1 , . . . . Then
- xk)4+1,
(7.4.23)
k=0
Z™=kam,Vk>0.
Proof. For n = 0 , 1 , . . . n
Sn
= X 0 eS+ £ ( * * + ! - * * K + i fc=o = xn+1e°n+1 + x0(e"0 - el) + x,{e\ - t\) + ■ ■ ■ + * „ « - < + 1 ) = a0x0 + a1x1 H --- + anxrl + xn+1el+1. (7.4.24)
186
DIFFERENTIAL
OPERATORS
OF INFINITE
ORDER
We first consider the case £ anxn = oo. From the equalities (7.4.24) we get Sn > Y%anxn for all n > 0. Consequently, OO oo
xoeg + E ^ + i --xk)el+1=limSn = oo. k=0
Suppose now that E < " A < oo. Then, in view of (7.4.24), in order to get (7.4.23) it remains to prove that limn_ 00 xncan = 0. But this follows immediately from the inequalities OO
0 < xne„ = xn(an + an+i H
a X
) < anxn + a n + 1 x„+i + • • ■ = Yl
kk
k=n
and the convergence of the series £ a n x n . The lemma is proved. T h e o r e m 7.4.6.
Let Ra > T 2 / 4 . Then for the imbedding (7.4.13) it suffices that
sup ^ = M < oo. (7.4.25) n>0 el Proof. Obviously, (7.4.13) holds if and only if there exists a number Mx such that \H\U,b)<M1\\u\\l00ia).
VueWa°°,Mp
(7.4.26)
= l.
Let Wr W2
:= {u(x)eWr\\\u\\p=l,\\Dku\\l'k<^/i,k := {u(x)eWa°°\\\u\\p
= k
=
l,2,...},
1 k
l,3k0:\\D °u\\ J °>w2/4}.
We fix an arbitrary function u € W2 (the existence of such a function follows from Lemmas 7.4.3 and 7.4.4) and we denote by k0 = k0(u) the smallest among all the indices k for which ||D*u||V* > TT 2 /4. Then \\D»u\\p < \\Dn+lu\\p,
(7.4.27)
\/n>k0.
Indeed, by Lemma 7.4.2, \\Dk°u\\; < (T/2)"||77"U||J>,
Vn>k0,
( | l ^ w X " '
< \\Dnu\\^,
\/n > k0.
From this, | < (^||2?*»«||p)1/fc0
< \\D"u\\\l\
Vn >
Further, Lemma 7.4.2 yields
\\Dnu\\;+l < <
(v(nf2)^\\ir^u\\r f2)^\\Dn*lu\\;.
fc0.
(7.4.28)
PROPERTIES
OF SOBOLEV-ORLICZ
SPACES OF INFINITE
ORDER
187
Consequently,
|ir s
,+ * u5sqy< 1J" '"""
From this and (7.4.28) we immediately get (7.4.27).
Further, Lemma 7.4.6 and the inequalities (7.4.25) and (7.4.27) yield oo
oo
DMDNi; = ll^0«ll^0 + E(ll^+1"li;-ll^«li;)4+1 <
M
\\Dk^\\TA0+E(\\Dk+^\\;-\\Dku\\;)el+1 k=k0
= M Y, «»ii0B«n;k=k0
Consequently, u € W{°, and W2 C Wb°°, because the function u € W2 is arbitrary. By the definitions of W% and W2 and by Lemma 7.4.5,
(x 2 /4,^ r ) c R | II e w2} c K I « e w?0}. From this we obtain i? 0 < Rb. Let Mi = M + (l/a 0 )6(7r 2 72 2 r ). Then 7r 2 72 2r < Ra < Rb implies Mi < oo. We now show that Mi is the desired number. Indeed, for u € W2, jfco-l
°°
HU) = EMi>"«li;+EMI^«||; fco-I
/7T2r\
6 M nW «II; " a*) ++JT«-ll^-ie < 2_ EM n=ko
<
n=0
n=0
\
Z
/
n=*o
< * ( £ ) + W U , < M U (" + ^ 6 (£)) = ^B«IIUFinally if u € W,, then
i T M < » ( ; ) s M u i ' ( ^ ) <"*»;»,., The proof of the theorem is now complete. L e m m a 7.4.7. Let u € WB°°. TTien /or oraj/ n = 0 , 1 , . . .
188
i) \\D" ■*n; <
DIFFERENTIAL
(i:
r
i£
OPERATOSS
71
k=0
<
ii) \\B»u\\; < 2 ( f ) r s u p t > 0 [ f a - H O l l h l l U . )
OF INFINITE
ORDER
ifRa < oo; # * • = °°-
The next result follows in an obvious way from Lemma 7.4.7. Corollary 7.4.2. Let Ra < oo and
fwH T iiCTi irit iTTiutmuaty
(7.J.-13)
\_*oroiiary i . 1 . 0 . i/et i t 0
\ —1
(7-4'29)
ituitio.
00 a/ut
£6nSup[ra-(0]
(7.4.30)
Then the imbedding (7.4-13) holds. R e m a r k 7.4.3. Lemma 7.4.7 and Corollaries 7.4.2, 7.4.3 were proved in Dubinskii [2] for the case r = p. These results are also valid for the case r ^ p. In Lemma 7.4.3 we established that l i m * . ^ ||I> B /|| J / n exists. A concrete computation of this limit is needed in what follows. Lemma 7.4.8. Suppose that 0 = n 0 < ni < • ■ ■ is a sequence of integers, and f e C°°(IR) is a function such that D«*f € Lp, k = 0,1 Then
l
ds = af := sup{|£| | ( e supp where /(£) is the Fourier transform of ffx). Proof. It follows from Lemma 7.4.3 that Dnf £ Lp, n = 0 , 1 , . . . and the limit df = lim||.D n /||J/ n exists. Note that from f € Lp follows f e & if / £ ,9". It will be assumed that f[x) is not a constant. We first prove the inequality df < vf.
(7.4.31)
It clearly suffices to consider the case af < oo. By the Paley-Wiener theorem, / ( « ) is an entire function of exponential type a,. Using the Bernstein-Nikolskii inequality
PROPERTIES
OF SOBOLEV-ORLICZ
SPACES OF INFINITE
(see Section 1.9), we get | | D n / | | p <
ORDER
189
This immediately gives us
We now prove the reverse inequality (7.4.32)
df > a,.
Indeed, consider first the case p = oo. We prove (7.4.32) by contradiction. Suppose that dj < erf. Then there exist numbers C < oo and a < 07 such that ||Z?"/|| P < Can,n = 0 , 1 , . . . This and the inverse Bernstein theorem imply that / ( z ) is an entire function of exponential type a < 00. Consequently, by a theorem of Schwartz (see Nikolskii [1], §3.1.5), supp / C {£ I |f I < a}. (7.4.33) From (7.4.33) and the definition of a; we have 07 < a, which is impossible. Suppose now that 1 < p < 00. We construct the new functions ri/k
fk(x)
= k
Jo
f(x + t)dt,k = 1,2,...
(7.4.34)
Then the continuity of / in LP(]R) implies 11/* - /UP — 0,fc— 00.
(7.4.35)
It is easily seen from (7.4.35) that fk -» / weakly in .9*. Consequently, />(£) -♦ / ( f ) weakly in Sf (see Nikolskii [1], §1.5). Further, by (7.4.34) and Jensen's inequality, for all n > 0 \DnMx)\v
\Dnf{x + t)fdt,
Vfc > 1.
This gives us for all n > 0 the inequalities \\Dnfk\\co <
fc1/p||^n/l|P
Vfc>l.
(7.4.36)
From (7.4.36) and (7.4.32) with p = 00 it follows that for all k > 1
ah = Jim \\D»fk\\H» < Jim ||IT/||}/» = ds.
(7.4.37)
It is clear that if we can prove the inequality af
then we obtain the required inequality (7.4.32) immediately from (7.4.37).
(7.4.38)
190
DIFFERENTIAL
OPERATORS
OF INFINITE
ORDER
To get (7.4.38) it suffices to prove that |6)|
(7.4.39)
k—»oo
for any fixed point £0 6 supp / ( £ ) . We prove (7.4.39) by contradiction. Suppose that there exist a point £0 6 supp /(£) (for definiteness assume £0 > 0), a number e > 0, and a subsequence {nk} (for simplicity of notation assume that nk = k, k = 1,2,...) such that
(7.4.41)
/
Proof. For A > 0 let T A (/) = A _1/ »7(A _1 a:). Then the equalities \\DnTx(f)\\p
= X-n\\Dnf\\p,
n = 0,l,...,
imply that T\ is a bijective and isometric mapping from W°°{an,p,r}(JR}) onto W , °°{a n A'"',p,r}(]rl 1 ). Next, let A = R-1^. The imbedding (7.4.13) holds if and only if W°°{anR:,p,r}(m}) - » W~{6 r fl;,p l r}(Hl 1 ). (7.4.42) We now prove (7.4.42). Let / £ Wx{anRZ,p,r}(m}). Then the fact that the radius of convergence of the series J] anR2(,n is equal to 1 and Lemmas 7.4.5 and 7.4.8 give us that d; = a/ < 1. This and the Paley-Wiener and Bernstein-Nikolskii theorems imply | | £ B + 1 / | | „ < °AD*fh < \\D«f\\P, n = 0 , l , . . . (7.4.43) We now fix an arbitrary positive integer N. From the Abel transformation, the condition of the theorem, and the inequalities (7.4.43),
EWISVII;
= \\DNf\\;E^K + i:(\\Dnf\\;-\\D^f\\;)±bkRka
n=0
n=0
n=0
k=0
< M \\\D f\\; £ anR2 + E W / | | ; - ll^+1/li;) £ akRka N
n=Q
= Mj2anR»jD«f\\;, n=0
n=0
lt=o
PROPERTIES
OF SOBOLEV-ORLICZ
SPACES OF INFINITE
ORDER
191
Letting N —> oo, we get oo
oo
n=0
n=0
Y,bnK\\Dnft^MYi"nK\\Dn!\\Tp. This means that the imbedding (7.4.42) holds. The theorem is proved. It is natural to ask whether the imbedding (7.4.13) is compact. T h e o r e m 7.4.8. The imbedding (7.4-13) is not compact. Proof. It suffices to prove that the imbedding W™ >-+ Lp(m})
(7.4.44)
is not compact. We choose an arbitrary entire exponential function f(z) of type o-f < 1 such that / £ W£°,f(x) is not a constant, and f(x) —> 0 as \x\ —* oo (it is not hard to show that such a function exists), and we let £ = {sr„(-) = / ( - + n ) | n = 0 , l , . . . } . Then the family E of these functions is bounded in W^°, because ||fl>n||(oo,ii) — ||/||(oo,a)i n = 0 , 1 , . . . We prove that E does not contain a sequence convergent in Lp. On the other hand, by an inequality of Nikolskii (see Nikolskii[l], §3.3.5.),
||ft.iU<2|M„ n = 0,l,.... Consequently, if E contains a subsequence convergent in Lp, then it converges in L^. To complete the proof of the theorem we show that E cannot contain a subsequence convergent in Loo- Indeed, let x0 be a point such that |/(xo)| = ||/||oo- Then |x 0 | < oo, because f(x) —» 0 as |x| —■> oo. For all n >. 0 and m > 0 \\9n ~ ffn+m|U = ||/(- + n) - / ( • + n + m)||oo > |/(io) - /(xo + m)|. Consequently, for all n > 0 liminf||5 n -ff n + m ||oo > liminf |/(x 0 ) - f(x0 + m)\ = m—^oo
\f{x0)\.
m—^oo
It is easily seen from the last inequalities that E cannot contain a subsequence con vergent in Loo. The theorem is proved. II. The case of a finite interval. Suppose that the numbers an > 0 form a sequence containing an infinite subsequence of positive elements, l < p < o o , l < r < o o , and (c, d) is a finite interval. We call W°°{an,p,r}(c,d)
:= {u{x) € C0°°(c,d) | ||u|| (oo , a) < oo}
192
DIFFERENTIAL
OPERATORS
a Sobolev space of infinite order on the interval (c,d). Wa°° instead of
OF INFINITE
ORDER
Everywhere below we write
W°°{an,p,r}{c,d).
We introduce similarly a Sobolev space Wb°° of infinite order, and we consider the imbedding Wr ^Wb°°. o As we know (see Section 6.4), Wa°° is nontrivial if and only if f ; sup aj/< nr) < oo.
(7.4.45)
(7.4.46)
m=l n > m
We assume henceforth that (7.4.46) holds. Note that (7.4.46) implies Ra = co. Con sequently, repeating the arguments given in the proof of Theorem 7.4.6, we get Corollary 7.4.4. If (7.4-25) holds, then the imbedding (7.4-45) holds. Corollary 7.4.5. / / (7-4-30) holds, then the imbedding (7-4-45) holds. In contrast to the case of an infinite interval, the imbedding (7.4.45) can be compact. Theorem 7.4.9. Suppose that lim -a- = 0.
(7.4.47)
Then the imbedding (7.4-45) is compact. Proof. By virtue of Theorem 7.4.3 for compactness of an imbedding, it follows that (7.4.45) is compact if and only if for all e > 0 there is an N(e) such that °° o £ bn\\D«u\\; < e, Vu €Wa°°, |M|(oo,a) < 1.
(7.4.48)
Obviously, (7.4.48) holds if and only if for all e > 0 there is an N(e) such that °° o E *>n\\Dnu\\; < e, W ew«°°, |MI(=o,a) < 1, IHI, < l.
(7.4.49)
n=N
We now prove (7.4.49). Indeed, repeating the arguments in the proof of Theorem 7.4.6 and using Condition (7.4.25) (which follows from (7.4.47)), we obtain Ra < Rb. Consequently, Rt, = oo.
PROPERTIES
OF SOBOLEV-ORLICZ
SPACES OF INFINITE
ORDER
193
Now fix an arbitrary number t > 0. Then there exists an index n0 such that
(7-4-5°)
t
EMW 2 / *£■ - 2 2r
Next, for each function u £Wa°°,
\\u\\v < 1, let \ \\Dku\\\lk > TT 2 /4}.
kQ = k0(u) := mm{k
Then, as shown in the proof of Theorem 7.4.6, \\Dnu\\p<\\Dn+1u\\p,
n>k0.
(7.4.51)
Let ni be an index such that 4 / < < e/2, n > m .
(7.4.52)
o Let N = max{n 0 ,rai}. Then for all u €VKa°°, ||u||(oo,a) < 1, || u ||p < 1> we have oo
OO oo
B
/ _ 2 r \ "7 t
OO °o
£ MIZTT.II; E 6nii^ «n; < £&n (Vr) + n=N n=N n=JV n=7V \ / n=max{A:o,zV} n=max{A:o,Ar} Z
oo
< 2+
M|£n*C
£ n=max{fco,N}
From this and (7.4.51), (7.4.52), and Lemma 7.4.6 we get oo
oo
n
£ n~ma.x{k0
MP 4;
£
«-IP"«II;<2-
n=ma.x{k0,N)
The above inequalities immediately yield oo
£MPNi;<e. The theorem is proved. Problem 7.4.1. Establish a variant of Theorem 7.4.3 for the case of Sobolev-Orlicz spaces of infinite order. Problem 7.4.2. Extend Theorems 7.4.6 -7.4.9 to the simplest case of Sobolev-Orlicz spaces of infinite order in ffi.1, i.e. to the case $ n M = o„$(i), an > 0, $(£) is an ./V-function.
194
DIFFERENTIAL
OPERATORS
OF INFINITE
ORDER
Problem 7.4.3. Extend Theorems 7.4.6 - 7.4.9 to the case of general Sobolev-Orlicz spaces of infinite order in 1R1. Bibliographical Notes. Most of the analysis of the Sections 7.1 and 7.2 is both a completion and extension of the results given in Tran Due Van [9], [12]. Section 7.3 follows the paper by Tran Due Van, Le Van Hap and R. Gorenflo [1]. The imbedding criteria for monotonic limits of Banach spaces (Theorems 7.4.1 7.4.3) are due to Dubinskii [2], [3], and their applications to Sobolev-Orlicz spaces of infinite order (Theorems 7.4.4, 7.4.5) are due to Tran Due Van. The subsection 7.4.3 is based on the paper by Ha Huy Bang [4]. We note that many geometric properties of the Sobolev-Orlicz spaces of infinite order can be found in Ha Huy Bang's Ph. D. Thesis [3], Dinh Dung's papers [1,2] and Nguyen Nhu Doan's paper [1]. The reader can find many deep results of the imbedding theory and the trace theory of Sobolev spaces of infinite order in Dubinskii's book [3], and in Balashova's papers [l]-[5].
Chapter 8 ELLIPTIC E Q U A T I O N S OF I N F I N I T E O R D E R WITH ARBITRARY NONLINEARITIES
This chapter is concerned with the existence and the uniqueness of boundary value problems for elliptic operators in divergence form : oo
£ (-l)aDaAa(x,u,...,Dau). M=o
(8.0.1)
These problems have been extensively studied since 1975 by Dubinskii, Balashova, Agadzhanov, Kobilov, Tran Due Van, Umarov and others for the case in which the coefficients Aa have polynomial growth (see Dubinskii's book [3] (1986) and his 1991 survey [6]). Our intention here is to study the Dirichlet problem for the operator (8.0.1) in the case when the coefficients Aa have growth of an arbitrary order. As an example, we take oo
Y. (-l)aDa(<pa(Dau(x)) = h(x), (8.0.2) l«l=o where the ipa : ]R} ^ IR1 are continuous, odd, nondecreasing functions, and ipa(+oo) = +oo. For large values of t these functions tpa(t) may behave, e.g, crudely speaking, like polynomials, like exponentials or like logarithms. A boundary value problem of infinite order for the operator (8.0.1) is, in a certain sense, a limit of boundary value problems for equations of finite order. In this connec tion, it should be noted that the theory of boundary value problems for equations of finite order with arbitrary nonlinearities has been intensively developed in papers by Vishik [1], Donaldson [1], Fougeres [1], Gossez [1-3], Benkirane and Gossez [1], Gossez and Mustonen [1], Robert [1] and others in close association with the development of the theory of monotone operators acting in nonseparable nonreflexive Banach spaces. We note that the theory of monotone operators can be applied in its entirety to nonlinear elliptic equations of infinite order. However, in the case of reflexive "energy spaces", we shall see that the monotonicity property of the operator (8.0.1) does not play (in contrast to the theory of equations of order 2m) a crucial role in the proof of the existence of a solution, but represents only one of the possible sufficient conditions for the uniqueness. In this respect nonlinear elliptic equations of infinite order turn out
195
196
DIFFERENTIAL
OPERATORS
OF INFINITE
ORDER
to be related to systems of nonlinear algebraic equations, for the solvability of which, as is well known, a coercivity condition is sufficient and no conditions of monotonicity are required ( see, for example, Lions [1], Dubinskii [7] et al. ). In particular, this means that one can analyze the solvability not only of the infinite-order nonlinear elliptic and parabolic equations, but also of nonlinear hyperbolic equations of the type ^ at
- £ (-VMD°Aa(t, M=o
x, u,..., D"u) = h(t, x).
This chapter deals only with elliptic equations of infinite order with arbitrary nonlinearities. In Sections 8.1 8.3 we consider the homogeneous and inhomogeneous Dirichlet problems for the operator (8.0.1) in a bounded domain in IR™ and on the torus Tn. Several characteristic examples illustrating the theory are given. The re sults are new, even for the case of a polynomial growth of the coefficients Aa (see, Example 8.2.2). The last section is devoted to the Dirichlet problems for nonlinear degenerate equations of infinite order with arbitrary nonlinearities. We also note that in contrast to the theory of nondegenerate equations (Sections 8.1-8.3), certain degenerate equations of infinite order are solvable only if the number of boundary conditions is finite. 8.1. Elliptic boundary value problems. Let flbea bounded domain in IR",n > 1, with boundary 3f2. We consider the Dirichlet problem oo
L{x,D)u
:=
52(-lpDaAa(x,u,...,Dau)
= h(x).
^ « U = 0,|w| = 0 , l , . . . .
(8.1.1) (8.1.2)
Here the Aa(x,(,) are, generally speaking, nonlinear functions of £ = (£0, •••,&*), ((a £ IR1). We assume that for each a the function Aa(x, £) satisfies the Caratheodory condition and also the following conditions. I. There exist an TV-function $„(*), a function aa € L{$a,Q}, a continuous bounded function c\(\t\),t € IR a (l < c*(|<|) < const) and a constant 6 > 0 such that
\Aa(x,0\ < aa(x) + tf;I»«(<£<|k|6,)l where oo
J2 lla«ll(*<») < °°II. There exist functions bm € i i ( f i ) , ga € E{$a,Sl}, a continuous bounded function c£(|i|)( C i(|i|) < c£(|rf|),< g IR1 and a constant d > 0 such that for
ELLIPTIC
EQUATIONS
OF INFINITE
197
ORDER
every m
£ (A,(z,0 -a,(x))fc > <* 5] *«(4(lfi*l)e.) - M*), |d|=m
|a)=m
where S |o|=0
HPc||(*a) < +00,
J^
/ |6m(a;)|dx < +00.
m=0J{1
III. The N- functions $ a ( 0 are such that the space Z,iy°°{$ a ,n} is nontrivial (see Section 6.2). IV. The strict monotonicity condition: For arbitrary £ = (£0, ■•-,£<*),£' = (£0, •••)£») and x € fi we have the inequality m
-a>o,
£ ( 4 , ( x , 0 - ^ ( z , £'))(&, ■
|c|=0 |c|=0
and the equality is valid if and only if ( = £', m = 0,1, We assume in (8.1.1) that h G EW-°°{$a,n} spaces LW°°{
(see Section 7.3). The duality of the
is determined by ha(x)Dav(x)dx,
^2 /
where 00
h(x) = £
(-I)WIJ'MI).
I<*M> The scalar product is obviously a meaningful expression (see Section 7.3). 0
Definition 8.1.1. A function u G LW°°{$a,Q}
is said to be a generalized solution 0
of the problem (8.1.1)-(8.1.2) if for an arbitrary function v € LW°°{$a,Q} < L(x,D)u,v
we have
> = < h,v >
Let us start with the case when the functions $ a , $ a satisfy the A2-conditions,
M = o,i,.,.
T h e o r e m 8.1.1. Let the conditions I-IH be satisfied. hand side h G EW~°°{$aiQ} Problem (8.1.1)-(8.1.2)
Then for an arbitrary right-
there exists at least one generalized solution u(x) of
in the space I l T { $ 0 , f i } .
198
DIFFERENTIAL
OPERATORS
OF INFINITE
ORDER
Proof. The idea of the proof is as follows. First, the "partial" equation of order 2N (the partial sum of the series (8.1.1)) is perturbed by a sufficiently "small" linear equation of order 2N + 2. The corresponding boundary value problem is always solv able. Then the passage to the limit N —> oo is performed. Thus, let us consider the sequence of the following Dirichlet problems of order 2N + 2(N = 0,1,...): (-l)N+1caD2auN(x)
£
+ L2N{uN{x))
= hN(x),
DuuN(x)\9n
= 0,
j
(g
|a|=JV+l
3)
(8.1.4) N
|w|<JV,
where N
W«)
:=
£ |a|=0
hN{x)
:= l"l=o
(-l)^DaAa(x,u,...,Dau), Z(-l)MDaha(x),
and ca > 0 are suitable small constants (see below). Further, we define : = ^ ( 0 ) n # * ! { • , . , n>,
fr
|| ■ | | ^ : - || ■ || o ^
+ || • || „ ^
n }
,
where W^ + 1 (fi) is the Sobolev space of order N + l and WNL{$a, fi} is the SobolevOrlicz space of order N (see Section 1.17). The space I f " is a reflexive separable Banach space. It is not difficult to see that the problem (8.1.3)w-(8.1.4)jv is weakly nonlinear ( the principal part of (8.1.3)AT is linear). By virtue of Theorem 1.20.1, there exists a function uN € 1 ^ which is a solution of the problem (8.1.3)AT - (8.1.4)w in the following sense <
caDauN,Dav>
£
+
VveWN.
(8.1.5)
\a\=N+i
Our aim is to get estimates for the solutions UJV. TO this end, putting v = u^ in (8.1.5) we have the identities <
5Z
caDauN,DauN>
+ < L2N(uN),uN
>=< hN,uN
> .
\a\=N+l
Further, using the Young inequality we get the estimate \
< e £ |a|=0
p(c^(\DauN\)DauN),^a)+
£) |a|=0
p(ha/e,$a)
(8.1.6)
ELLIPTIC
EQUATIONS
OF INFINITE
199
ORDER
tY,P(<£)(\DauN\)DauN),
<
+ K(e)\\h\\{^h
(8.1.7)
|c|=0
where 0 < t < 1 is a constant, and K(e) is a constant independent of N. Because of the condition II we obtain N ! > | > - J2 /
< L2N{uN),uN
\ga{x)DauN{x)\d.
|a|=0"'" AT
AT
+d £ p(c™(\D*uN\)DauN),$a)|a|=0
£
/
\bM(x)\d
|a|=0""
> -E*«M ! ) (i^i)),i) |c|=0
+(d - 1 ) £
^ ( c i 2 ) ( | Z ) Q U w | ) D a ^ ) , «„) - K.
(8.1.8)
|o|=0
Here, and in the sequel, we use K,Ki,..., to denote various positive constants inde pendent of N. Choosing e sufficiently small and fixing it, we find from (8.1.6), (8.1.7) and (8.1.8) that £
ca / \D°uN\2dx Q
|a|=N+l
+ £ p(c™(\D"uN\)DauN),$a)
< Ki.
(8.1.9)
\a\=0
Since all the functions $<»(£) —> oo as t —» oo, from (8.1.9) and Jensen's inequality (1.16.3) we get the estimate \DauN(x)\
< K2,
N = 0,1,..,
\a\
From this we obtain that the sequence of functions unr(x) (and all of its derivatives) (a priori, a subsequence) converges uniformly in fi to a function u(x) £ C{f(Q) (and to its derivatives, respectively). Of course, every function UAT(X) has an increasing, but always, finite smoothess; therefore, the convergence u^f(x) —> u(x) in Co°(f2) means that Dauw{x) —* Dau(x) uniformly in fi, beginning with an index N = N(a) large enough. From (8.1.9) we also obtain the estimate £
p{cW(\DauN\)DauN),
which shows that u e L
o
W°°{$a,ti}-
*ft) < # 3 ,
WV = 0,1,...,
(8.1.10)
200
DIFFERENTIAL
OPERATORS
OF INFINITE
ORDER
We now prove that the limit u(x) is a generalized solution of the problem (8.1.1)(8.1.2). Let us denote by L W°°{ca,il} LW°°{ca,Sl}
the Sobolev-Orlicz space of infinite order
:= {«(*) | u e C0°°(ft), £
cJ,/ 2 ||D°u|| M 0 ) < +oo}.
Then the following imbedding is valid. o L e m m a 8.1.1. For any nontrivial space LW°°{$a, ft} there exists a nontrivial space o £W°°{c„,ft}, so that
ir{* 0 ,n}^irK,n}.
Proof of Lemma 8.1.1. Indeed, we choose Cy
2
< (Ma+1)-\
if M a + 1 ± 0,
ca = 0,
if M 0 + 1 = 0,
(8.1.11)
where the numbers Ma are defined in Section 6.2 and a + 1 = (ai + 1,..., an + 1). o o Let u e I W ° ° { $ a , f t } . We shall show that u € LW°°{ca,£l}, where the numbers ca are defined by (8.1.11). Without loss of generality we assume ||u||(oo) < 1- Then oo
£ l|0"«ll<«.) < +°°-
(8-1-12)
|c|=0
Further, since u € Co°(ft) the inequality \Dau{x)\ < J \Da+lu(x)\dx
(8.1.13)
is valid. By virtue of Holder's inequality and (8.1.11) we have <%* Ja\D°+1u(z)\dx
<
4 / 2 |P a + l «ll(*,, + 1 ) llx(fi)ll(* o M )
= 4 / a *ii(l/m« <
ft|pa+1«||(4a+l)
mesnU/y+^l^,,.
(8.1.14)
From (8.1.13) and (8.1.14) it follows that
4/2||£au|U(n) < ^ll^+1«ll(».+I), H = 0,1,■ • ■ Then the last inequalities and (8.1.12) yield oo
£
oo
c)l*\\D"u\\Lm
\\D"+1u\\(*a+1)
< +00.
ELLIPTIC
EQUATIONS
OF INFINITE
o This means that u G LW°°{ca,fl}.
201
ORDER
Lemma 8.1.1 is proved.
Using Lemma 8.1.1 we show that the limit u(x) of the sequence {ujv(x)} is the desired solution of the problem (8.1.1)-(8.1.2). C WN,VN and by (8.1.5) we get the identity
From the fact that LW°°{ca,tt} ca
J2
+
VveL
W/°°{c a ,n}.
\a\=N+l
(8.1.15) Consider the expression
J2 ca
(8.1.16)
.
\a\=N+1
Using Young's inequality and (8.1.9) we have for any e > 0 the estimate
££ \a\=N+l |o|=JV+l
a ccaa
Y: £
+£-+2A Ca ca\\D°u \\D°uNN \\l\\i m {n)
* |o|=N+l |a|=N+l
Y: (Q) Y, ca\\D"v\\i ca\\D"v\\l (Q)
\a\=N+l |a|=JV+l
< Z
^£
||«|=AT+1 a |=AT+l
By choosing e sufficiently small and TV sufficiently large we can make (8.1.16) arbi trarily small. Then from the identity (8.1.15) it follows that lim
V]
ca < D°'UN,D°'V
= lim < L2N(uN),v>= N—KXJ
for any function u € L
> + < L2N{UN),V
>
lim < hN,v > = < h,v > N—*00
0
W^i^cSl).
On the other hand, lim < L2N(uN),v
>=< L(x,D)u,v
>, Vt> € L
W°°{$a,to}-
N—*oo
Indeed, let N0 be a fixed number and let N > N0- We have < L(x,D)u,v
[ {Aa(x,u,...,Dau)-
£
= +
> — < L2N(UN),V
oo
£
>=
Aa(x,uN,...,DauN)}Dav{x)dx
/ A c ,(x,u,...,Z) 0, u)D a t ! (a;)da;
i i ^ +l . ■. Jo |o|=JV 0
(8.1.17)
202
DIFFERENTIAL
+
E
OPERATORS
OF INFINITE
ORDER
Aa(x,uN,...,DauN)Dav(x)dx
/
(8.1.18)
= h+I2 + I3.
We now consider each term on the right-hand side of (8.1.18) separately. By virtue of the Caratheodory condition and the fact that u^ -+ u in C$°(fl), we see that Ii tends to zero as N — ► oo. Next, using the condition I and Young's inequality, we estimate J2 : \h\ 141
oo oo
<
E £
/
aaaa{x)D"v(x)dx {x)Dav(x)dx
\a]=No+l | a |=iVo+l ■'""
+b £E
1K K1^a(c^\Dau[)DaU{x))Dav(x)dx
I Jti
||=No+l CO
<
CO oo
£E |a|=No+l oo
+6
p(a />(««>**) + a,$a)+
P(0°«>*.) pPa«^c)
E |a|=W \a\=N00+l +l
oo
p(D°v,$ />(I>at>,$ a)+ a)+
£ E |a|=N 0 +l
pCci^di^uDB-u,*,,). p{^\\Dau\)Dau^a). (8.1.19)
E £ |a|=No+l
From (8.1.9) it follows that OO
/>(
£
< +oo.
(8.1.20)
|O|=0
Combining (8.1.19),(8.1.20) and the condition I we obtain that I2 can be made arbi trarily small if No is taken sufficiently large. Finally, we estimate I3. Taking the condition I and Young's inequality into account we see that \h\ 1^1:<<
/
a aaaa(x)\D (i)|
+b +6
£ S
rr -
uNfl\)\D v(x)\dx UND I^'MX)!^ •; 1 *$a0(cy(\D ( c (, ,, (|/? / $; |a|=N +l 0 1
a
0
JV
< < «i E £
AT
p(aa°a,* + (i (l ++ 6) b) E £ ,**) a) + P(
a | = N00+l +l 1|o|=N
|o|=iVo+l |o|=iV 0 +l AT
+eb +e& £E
|a|=JV 0 +l !a|=JV„+l
< <
a
aa p(D v/e,$ p{D v/e. , * a a))
a
p(c^(\D uN\)D ' « NuN , $,^aa)) P( ctftlD-uwl)//
|a|=N 0 +l oo oo
< « e £E
a
JQ
OO 00
p(aa°a,$ + (l £ ,*a)a ) + (i + b) E P(
a|=N 00+l l|o|=N +l
a a p{D p(D v/e,v/t^ , * c a.))
ELLIPTIC
EQUATIONS
OF INFINITE
203
ORDER
N
+«&
p(c{a){\DauN\)DauN,^Q),
E
0<e
(8.1.21)
|a|=W q +l
By virtue of the condition I, the inequalities (8.1.10), (8.1.21) and v € LW°°{$a,ft}, we can make I3 arbitrarily small by choosing t sufficiently small and N0 sufficiently large. Thus, as a result, we find lim < L{x,D)u-
LN(uN),v
> = 0, Vu € L VF°°{$ a ,fi}.
Consequently it follows from (8.1.17) that u(x) is a generalized solution of the prob lem (8.1.1)-(8.1.2). This completes the proof of the theorem. Theorem 8.1.2. Let the strict monotonicity condition (the condition IV) be satis fied. Then the generalized solution of the problem (8.1.1)-(8.1.2) is unique. Proof. Suppose that Ui and u2 are two generalized solutions of the problem (8.1.1)(8.1.2). We have ux - u2 € LW°°{$a,ti) < L(x,D)ui
and
— L{x,D)u2,ui
— u 2 > = 0.
Then, from the condition IV it follows that u\ = u2. The theorem is proved. We now consider the general case. Assume that the domain fi satisfies the segment property, i.e., there exists a locally finite open cover {(/,•} of the boundary of H and a corresponding sequence {yj} of nonzero vectors such that if x € H n Uj for some j , then x -f tyt■ € fi for 0 < t < 1. Theorem 8.1.3. Let Q be a domain in IR™ with the segment property and let the conditions I-IV be satisfied. Then for any h € .EVK -0o {$ a ,fi} there exists a unique o generalized solution u of the problem (8.1.1)-(8.1.2) in the space L W°°{$a,Q}. Proof. We consider the sequence of "partial" problems of the problem (8.1.1)-(8.1.2),
L2N(uN(x))
:=
52(-l)WDaAa(x,uN,...,DauN(x))
= hN(x),
(8.1.22W
\a\
ITuN(x)\m
= 0,\u>\
(8.1.23)^
204
DIFFERENTIAL
where
hN(x) = J2
OPERATORS
OF INFINITE
ORDER
(-i)MDah.(x).
\a\
From the conditions I, II and IV it follows that the problem (8.1.22)jv - (8.1.23)JV is solvable in the Sobolev-Orlicz space WNL{<5>a, ft} (see Remark 8.1.1below), i.e. there exists a function uN GWN L{$a,Sl}
such that
VveWNL{$a,n}.
(8.1.24)
Since L W °°{$a,ft} <-*W N Z { $ 0 , f t } , it follows that (8.1.24) is true for all v €
LW°°{$a,n}. Further, as in the proof of Theorem 8.1.1, we can show that UN converges to u in C£°(IR") and u is a generalized solution of the problem (8.1.1)-(8.1.2). The unique ness follows from the strict monotonicity condition IV. The theorem is proved. Remark 8.1.1. The solvability of (8.1.22)JV-(8.1.23)JV was established by Gossez in [1],[2] for the case $ a = $ (isotropic case) only. However, his method can readily be applied for the anisotropic case. 8.2. A model equation. Examples. We consider the following problem oo
L{D)u
:=
Y, (-l)MDa(
= h{x),
= 0, |w| = 0 , l , . . .
xen,
(8.2.1) (8.2.2)
For each a the function (pa : IR1 —* 1R1 is continuous, odd, nondecreasing and ipa(+oo) = +oo. We put $a(t) = j
<pa{r)dT, \a\ = 0 , 1 , . . .
(8.2.3)
The functions $„(<) are N- functions. As a consequence of Theorems 8.1.1 and 8.1.2 we have the following result. Theorem 8.2.1. Suppose that the functions $„(<) in (8.2.3) o space L jy°°{$cr,^} and ft satisfies the segment property. Then o (8.2.2) has at least one generalized solution in L W°°{$a, ft} for / / the functions <pa are strictly increasing, then the solution of (8.2.2) is unique.
define the nontrivial the problem (8.2.1)h € EW~°°{$a, ft}. the problem (8.2.1)-
ELLIPTIC
EQUATIONS
OF INFINITE
205
ORDER
Proof. We have only to show that the conditions I and II in Section 8.1 are satisfied for Eq. (8.2.1). This is a consequence of the following lemma. L e m m a 8.2.1. Suppose that the N-function $(<) is continuously differentiable, and its derivative is ifi(t). Then there exists a continuous function c(t), t € 1R1, 1 < c(|i|) < 2, such that
i) |„(i)| < «-»*(c(|i|)*)> ii) V(t)t = 1>(c(\t\)t). Proof of Lemma 8.2.1. We have $(*) <
$(c(t)t) =
M*)l < i- 1 *^)*)-
In an analogous fashion we consider the case ( < 0. This completes the proof of Lemma. Remark 8.2.1. If $ a and $ a satisfy the A 2 -condition for |a| = 0 , 1 , . . . , then Theorem 8.2.1 holds without the segment property of fi. Example 8.2.1. Case of coefficients with rapid growth. We consider the problem •£l(-l)HDa(aaDauexp[aa(Dau)2])
= D"u\dii
h(x),
= 0, |Q| = 0 , 1 , . . . .
(8.2.4) (8.2.5)
For aa = (|a|!) - 2 ", v > 1, the space L W°°{exp(aat2) — 1,0} is nontrivial (see, Example 6.2.2). Consequently, because of Theorem 8.2.1, Problem (8.2.4)-(8.2.5) has o a unique generalized solution u 6 L W°°{exp(aat2) — 1,0} for an arbitrary function h in the corresponding space £ , i y _ 0 o { $ c t , f i } . For aa = (|a|!)~ 2,/ , v < 1 the problem (8.2.4)-(8.2.5) reduces to the triviality 0 s 0 .
206
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ORDER
Example 8.2.2. Case of coefficients with power-law growth. Let oo
J2(-l)HDa{aa}Dau\'*-2Dau) Duu\da
=
h(x),
(8.2.6)
=
0, |a| = 0 , l , . . .
(8.2.7)
This problem was considered in Dubinskii [1,3] for the case of boundedness of the sequence {pa}. Here {pa} is an arbitrary sequence (it may happen that pa —> oo as a —+ oo). In accordance with Theorems 8.1.1 and 8.1.2, the problem (8.2.6)-(8.2.7) o o has a unique generalized solution in LW°°{aa\t\'">,Sl} if the space LW0O{aa\t\p''^} is nontrivial (see, Example 6.2.1). Example 8.2.3. Case of coefficients with slow growth. We consider the problem oo
Y,(-l)MDa(aasign(Dau)lR(l l«t=o
+ aa\Dau\))
=
h(x),
(8.2.8)
D"u\d(l
=
0, | a | = 0 , l , . . .
(8.2.9)
If aa = (|a|!) - ", f > 1, this problem has, by virtue of Theorem 8.2.1, a unique generalized solution in the nontrivial space LW°°{(1 + aa\t\) ln(l + a a |t|) - a a |i|,fi} (see Example 6.2.3) for h e L ^ - ° ° { e x p ( a ; 1 | i | - a ^ | t | - 1,0}.
8.3. Periodic problems. Inhomogeneous boundary problems 8.3.1. Periodic problems. As is well known from the results of Sections 6.2 and 8.1, the "energy space" corresponding to the Dirichlet problem (8.1.1)-(8.1.2) is nontrivial if the coefficients Aa are rapidly decreasing to zero as | Q | —► oo. For example, the space LW°°{exp((\a\])-2H2)
- 1,(1}
is nontrivial for A > 1 and trivial for A < 1 (see Example 6.2.2). Consequently, if A < 1 the Dirichlet problem for the equation oo
^ ( - l ) l a l J D a ( | a | ! ) - w J D ° U e x p ( ( H ! ) - 2 A p ^ ) 2 ) = Mx), x € Q,
(8.3.1)
ELLIPTIC
EQUATIONS
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207
ORDER
has no meaning. At this time, the space LW°°{exp((|a|!)-2V)-l,r"} is nontrivial for 0 < A < 1 (see Example 6.3.1), then the periodic problem for Eq. (8.3.1) is correct. Probably, the distinction of these results is not random but is connected with the nature of the operators of infinite order which are determined by entire or analytic functions. Let us note that almost all operators which occur in physics (for example, the shift operators) act on periodic functions or on functions that are defined on the full space IRn. Let us now consider on the torus Tn,
n>\
the equation
oo
L(x,D)u
:= J2 (-l)MDaAa{x,u,...,D"u) M=o
in which the functions Aa(x,£) following condition
satisfy the conditions I, II, IV in Section 8.1 and the
V. The TV-functions <&a(t) are such that the space LW°°{$a,Tn} trivial (see Section 6.3.) The space EW^^^T")
(8.3.2)
= h{x),
is non-
is defined in the same way as £ W - ° ° { $ a , n } .
Theorem 8.3.1 Let the conditions I, II, IV, V be satisfied. Then for an arbitrary h G jEW - °°{$ a , T"1} there exists a unique generalized solution u of the problem (8.3.2) in the space W 0 ^ , ! " } . Proof. Let us consider the following sequence of nonlinear equations of order 2N(N = 0 , 1 , . . . which correspond to the partial sums of the series (8.3.2) L2N(uN):=
J2(-l)MDaA«(x,UN,---,DauN) H=o
= hN(x),
xeT",
(8.3.3)
where
hN(x) := £
{-lpD"ha{x).
It is well known from the theory of operator equations acting in nonreflexive Banach spaces (see, e.g., Gossez [2]) that Eq. (8.3.3) has a unique solution ujv € WNL{$a,Tn}, moreover, the inequality N
J2p(D"un,$a)
(8.3.4)
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ORDER
is valid, where K(h) > 0 is a constant independent of N. From this, using the compact imbedding theorems of Sobolev-Orlicz spaces (see, e.g., Adams [1], Rao and Ren [1]) and a diagonal process, we obtain that the se quence of functions UN(X) (a priori, a subsequence) converges in C°°(Tn) to a func tion u(x). By speaking of the convergence uN(x) -* u(x) in C ° ° ( r n ) we understand that DauN(x) -» Dau(x) uniformly in Tn beginning with a sufficiently large index N depending on a. We have this in mind in the sequel. By virtue of (8.3.4) we have that u e I V K ° ° { $ a , r n } . Further, using the conditions I, II, IV and the inequality (8.3.4) we obtain (in the same way, as in the proof of Theorem 8.1.1, 8.1.3) the identity veLW°°{$a,Tn},
which means that u(x) is the unique generalized solution of the problem (8.3.2). The theorem is proved. Example 8.3.1. We consider the equation oo
Y. (-l)MD"(<pa(Dau(x)))
= h(x), x e T",
(8.3.5)
|c|=0
where <pa : 1R1 —> IR1 is a continuous, odd and nondecreasing function and ipa(+oo) = +oo. This equation satisfies the conditions I, II by virtue of Lemma 8.2.1. If the space LW°°{$a,Tn} is nontrivial, where the function $ a is defined by the formula (8.2.3), then Eq. (8.3.5) has at least one generalized solution in LW°°{$a,Tn} for any h € EW~°°{^aiTn}. In particular, Eq. (8.3.1) has a unique generalized solution u i n W ° ° { e x p ( ( | a | ! ) - w t 2 ) - l , T " } , A > 0. Example 8.3.2. Let oo
£
(-l)^D"(aaPa\Dau\^-iDau)
= h(x),
x € Tn,
(8.3.6)
|c|=0
where aa > 0, pa > 1 are such that the space ,LVK 0o {a a |i|'' a , T*n} is nontrivial (see Sec tion 6.3). The equation (8.3.6) has a unique generalized solution u in LW°°{aa\t\Va,Tn} for any h € EW-°°{aa\t\"",Tn}, where 1/Pa + l/qa = 1. In particular, the equation cos ( — ] u(x) = h(x),
x 6 T1,
(8.3.7)
ELLIPTIC
EQUATIONS
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209
ORDER
possesses a unique generalized solution in the space Z,W°°{(2n)!t2, T1}. We note that in this case the Dirichlet problem reduces to the triviality 0 = 0. 8.3.2. Inhomogeneous boundary value problems. In a domain fi C IR" with boundary dil the following inhomogeneous boundary value problem is considered: oo
L(x,D)u
:=
£
( - l ) l a | D < M a ( x , u , . . . , D a u ) = h{x),
Dwu\dQ = fu,(x'),x'£dn,
(8.3.8)
(w|=0,l,-...
(8.3.9)
Let us suppose that the conditions I-III of Section 8.1 are fulfilled. Moreover, we shall assume that the boundary values (8.3.9) admit the extension / € LW°°{$a,ft} into the domain fi, i.e. there exists a function / € LW°°{$a,fi} such that
D"f{x)\da = /„(*'),x' e dn, \u\ = o, 1, ■ ■ ■ (see Sections 7.1 and 7.2). Definition 8.3.1. The function u £ LW°°{^a,n} is called a generalized solution of the problem (8.3.8)-(8.3.9) if there exists a function / € LW°°{$a,n} such that u - f £ LW°°{$a,tt}
and the identity < L(x,D)u,v
is valid for any function v €
>=< h,v >
o LW°°{$a,il}.
Theorem 8.3.2. Suppose that the conditions I-III are satisfied. Then, for any h € £ W - 0 0 { $ a , f i } , there exists at least one generalized solution u € iVK°°{$„,fl} of the problem (8.3.8)-(8.3.9). Proof. The proof of this theorem is carried out in the same way as in the case of the homogeneous boundary value problems (Theorem 8.1.1). We, therefore, confine ourselves to a brief outline for the case $ a , $ a satisfy the A 2 -condition. An approximate solution can be found in the form UN{X) = f{x) +
zN(x),
where ZN(X) is a solution of the boundary value problem of order 2./V+2 (JV = 0,1,...) £ \c\=N+l
(-l)N+1aaD2azN+
Yl(-^)MDaAa(x,uN,---,DauN)
= hN, (8.3.10)
\a\=0
D"zN{x)\da
= 0, |u| = 0 , 1 , . . . , AT(8.3.11)
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DIFFERENTIAL
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ORDER
Here aa > 0 is a sequence of numbers tending to zero sufficiently fast as TV —» oo and
hN(x) = J2
l«l=o
i-l)WaaDaha{x)
is the partial sum of the series oo
H*) = E
|c|=0
(-i)wDaha(x).
It follows from the conditions of our theorem that the problem (8.3.10)-(8.3.11) is solvable for any N = 0 , 1 , . . . , i.e. there exists a family of solutions z^(x). Moreover, the following estimates E
*a\\DazN\\l2lil)
+ £
|cj=JV+l
P(cW(\D°zN\)D»zN,
$„) < K,
(8.3.12)
|a|=0
are valid. The constant K > 0 in (8.3.12) depends on f(x) zN(x), TV = 0 , 1 , . . . .
and h(x) but not on
From the last inequalities it follows that there exists (as usual) a subsequence of functions zpf(x),k = 1,2, ■•■, of the sequence {ZN(X)} which together with their o derivatives converge uniformly to some function z(x) in L VF°°{$ a ,fi} and to its derivatives, respectively. The function u(x) = f(x) + z(x) is a generalized solution of the original problem (8.3.8)-(8.3.9). The theorem is proved. 8.4. Degenerate nonlinear elliptic equations of infinite order. In this section we study the Dirichlet problem for nonlinear elliptic equations of infinite order, de generate on the boundary of a cube in the Euclidean space IRn. In G := {x | 0 < Xi < T,-, i = 1, ■ ■ ■, n) C )Rn we consider the equation oo
L(x, D)u := £ {-lpDaAa{x, M=o
u, ■ ■ ■, Dau) = h{x),
(8.4.1)
where the functions Aa(x, £) are continuous with respect to the variable £ = (£0, • • •, £ a )i and measurable with respect to x € G. We assume that the following conditions are satisfied: a) There exist an JV-function * a ( i ) , a function aa G £ { $ a , G } , a continuous bounded function c^(|t|),< € HI1, a weight function pa(x) = xj" 1 • ■ • x£»", (qa > 0, i = 1, ■ • •, n) and a constant b > 0 such that
K M ) | < aa(x) + 6*;V(*)<McL(l6,|)W),
ELLIPTIC
EQUATIONS
OF INFINITE
211
ORDER
where
Y
|a|=0
lla«ll(5=) < +°°-
b) There exist functions bm € £i(G), ga € £ { $ a , f i } , a continuous bounded function c£(|i|) > cj,(|tj) and a constant d > 0 such that for every m
Y {Mx,t)-9«{xma>d
|a|=m
where
oo
Y *(*)*«(4(I&l)kl)-M*). |a|=m oo
-
Y Itff4(«„) < +°°> ]C / \bm{x)\dx < +C
l«|=o
Let i be fixed, 1 < j < n. It is known, (see, L. D. Kudryavsev [1], S. M. Nikolskii [1], H. Triebel [1]), that the number of boundary conditions for i , = 0 depends on the behaviour of the quantity Li := a, - [(qa, + l)/pa] - 1
(8.4.2)
in the case $<*(') = \t\p°, Pa > 1- For arbitrary Af-functions $Q(£) the number of boundary conditions for x,- = 0 depends also on the behaviour of the quantity (8.4.2) where pa are the order near zero of the iV-function $ t t (see Definition 6.4.1). If limsupL, = +oo, a,—*oo
then the number of boundary conditions for X{ = 0 must be infinite; if l i m i n f i ; = s;, s, < +oo, cti—»oo
then for x; = 0 we must have the boundary conditions D°;u\x-o=0,a,:
= 0,1, • " , * , * > 0,
and no boundary conditions are required if s,- < 0. It easily follows that there are the following distinct boundary conditions: £ " > U = o = 0, ^»;«i«.=T. = ° i and
■Dx'«U=o = 0, D°'M*i=Ti=0,
a, = 0 , 1 , . . . , a; = 0 , 1 , . . . ,
Vt = l , . . . , n , V« = l , . - . , n ,
«j = 0 , 1 , . . . , s h Vi = l , . . . , n , a, = 0 , 1 , . . . , Vt = l , . . . , n .
(8.4.3)
(8.4.4)
212
DIFFERENTIAL
OPERATORS
OF INFINITE
ORDER
(Relations (8.4.4) include the case in which s,- < 0, i.e., no boundary conditions are needed on the part of the boundary dG where i , = 0.) All other situations are clearly described by various combinations of Conditions (8.4.3) and (8.4.4). We shall con sider in detail Problem (8.4.1),(8.4.3) and Problem (8.4.1),(8.4.4). The investigation of other boundary value problems for Eq. (8.4.1) can be carried out similarly. 8.4.1. Solvability of Problem (8.4.1), (8.4.3). First, we study the weighted Sobolev-Orlicz spaces of infinite order, which are the "energy spaces" corresponding to Problem (8.4.1), (8.4.3). Let us consider the space := {u e C0°°(G) | Hl(oo,„a) < +oo},
LW°°{*a,pa} where
IMI
/ pa(x)*a(u/k)dx
M=
I
< 1! ,
(8.4.5)
J
and $ a are iV-functions, pa{x) = x\ai ■■■x%°», qai > 0, i = \,... ,n. The space o LW°°{$a,Pa} is a Banach space with the norm (8.4.5); moreover, it is a monotonic limit of the decreasing sequence of Sobolev-Orlicz weighted spaces of order m (see Section 6.1 for the definition of a monotonic limit of Banach spaces). As in Section 6.2 we consider the sequence of numbers Ma: j * 0 ( l / m e s G), $ a ^ 0 ° ~ \ +oo, *a = 0 •
M
We assume that qaJpa i i
^ i
< const, Vi = 1 , . . . ,n and m = 0 , 1 , . . . , and n
^
+
\a\=m+Ni+—+Nm+n
where Ni = sup qaJpa Definition 6.4.1).
Ma
=
i11?"1 M ° + ^ + i '
™=
0,1,...,
\a\=m
and pa is the order near zero of the ./V-function $ a (2) (see
o Theorem 8.4.1. The space L VK°°{$a,/>a} is nontrivial if and only if the sequence of numbers Ma, |a[ = 0 , 1 , . . . , determines a non-quasianalytic class of functions ofn real variables. This theorem can be proved by combining the technique of Section 6.2 and that of Tran Due Van [7], [12], where the weighted Sobolev spaces of infinite order (i.e. $„(<) = aa\t\Va,aa > 0,pa > 1) are studied in detail. Therefore we omit the proof.
ELLIPTIC
EQUATIONS
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213
ORDER
We define the space in which the right-hand side h(x) in (8.4.1) should lie as follows: := \h(x)
EW"»{*a,pa}
| h(x) := £ ) ( - 1 ) H D » M X ) 1 ,
where ||fc||(-oo,„) := inf ik | 5 3 / pa(x)$a(hjk)dx l |«|=o-/G
< 1} .
The duality of LW°°{* a ,Pa} and £ W - ° ° { $ a ) p 0 } is defined by the relation Pa(x)Dav(x)ha(x)dx
:= ^3 / 0
Definition 8.4.1. A function u € L VK°°{$a,/>a} is called a generalized solution of Problem (8.4.1), (8.4.3) if, for each v €
LW°°{$a,pa}
< L(x,D)u,v
> = < h,v >
At first, let us consider the case when the functions <& a ,$ a satisfy the A 2 -condition. o Theorem 8.4.2. If the space L VK°°{$ a ,p a } is nontrivial and Conditions a), b) are satisfied, then Problem (8.4-1), (8.4-3) has at least one generalized solution in for any h €
LUritoPa}
EW-°°{*a,pa}.
Proof. The proof of this theorem is similar to that of Theorem 8.1.1. Namely, let us consider the sequence of the following Dirichlet problems of order 2m + 2 (m = 0 , l , . . . , ) : 53
caD2oum
+ L2m(um)
= hm,
(8.4.6)
= 0, \u\ < m,
(8.4.7)
|a|=m+l
LTu\dG where m
M*)=
£ {-l)MDaK,(x), H=o
"*
-£(-l)MDaAa(x,um,---,D"um),
L2m(um)= M=°
and the numbers ca > 0 are tending sufficiently fast to zero when \a\ —> oo. The problem (8.4.6),(8.4.7) is weakly nonlinear (see Section 1.20) and, therefore, has a
214
DIFFERENTIAL
OPERATORS
OF INFINITE
solution in the space W?+1(G)nWmL{$a,Pa}, where WmL{$a,pa Sobolev-Orlicz weighted space with the norm H W c ) := inf ( k\
I
£
/ pa(x)$a(D°u/k)dx
\°>\=oJG
ORDER
} is an anisotropic
< 11 .
)
By the method similar to that of Section 8.1 we obtain the inequality £
/
Pa(x)$a(D*um)dx
(8.4.8)
< K,
where K > 0 is a constant independent of m. By virtue of (8.4.8) we can select a subsequence of the sequence {um(x)} (which we denote also by {um(x)}) such that Daum —> Dau uniformly when m - t o o . Thus, as in Section 8.1, the inequality (8.4.8) o implies u G LW°°{$a,pa}Select now ca and £W°°{c a |<| 2 ,1} such that LW^^^p*}
^>
LW°°{ca\t\2,l}
(this can be done by a direct construction as in Lemma 8.1.1). Then we can show that < L(x,D)u,v>=<
h,v>,
Vu6
LW°°{$c,,pa},
and the proof is completed. We next give a uniqueness result. We assume that Conditions a) and b) are satisfied and that Problem (8.4.1), (8.4.3) satisfies the strict monotonicity condition, i.e.,
c)
£(A,(*,0->t«(*,O)(6.-&)><> |o|=m
for £ = (for • •, 6»), (' = (Co,- ■ ■ i £,), a n c l t n e equality holds only when & » = £ , , m = 0,1,.... Corollary 8.4.1.
Suppose that $ „ may not satisfy the A2-condition.
tions a), b), c) are satisfied and the space L W°°{$a,pa}
If Condi
is nontrivial, then Prob
lem (8.4-1), (8-4-3) has a unique generalized solution u € L W°°{^a,Pa} heEW-~{$a,Pa}.
for any
The proof of this corollary is similar to that of Theorem 8.1.4 and we omit it.
215
ELLIPTIC EQUATIONS OF INFINITE ORDER
8.4.2. Solvability of Problem (8.4.1), (8.4.4). Let L Wa°°{$a,pa} be the space of functions u(x) that are infinitely differentiable in G and satisfy Condition (8.4.4) on dG, and are such that ||u||(oo,P<1) < +co (see the definition of the norm || • ||(oo,p„) in Formula (8.4.5)). o For n = 1 Theorem 8.4.1 remains true for the space L Wf{$a,pa}This means o that L Wf{$m, pm] is nontrivial if the sequence Mm,m = 0 , 1 , . . . , determines a nonquasianalytic class of functions of one real variable. For n > 1 we need the notion of pointwise quasi-analyticity of functions of several real variables (see Matsaev and Ronkin [1], Tran Due Van [12]). Definition 8.4.2. The class C[M a ] is called pointwise quasianalytic if the following condition holds: u € C[Ma] and 3x0 € G : Dau{xQ) = 0, V|Q| = 0 , 1 , . . . , implies u = 0 in G. Otherwise, the class C[M 0 ] is called pointwise nonquasianalytic. o By the same method as in Section 6.2 we can obtain that the space L Wf{$a,Pa} is nontrivial if and only if the sequence of numbers Ma determines a pointwise non quasianalytic class of functions of n real variables. o Our basic assumption is that the space LWf{$a,Pa} LWf{$a,pa}
is called a generalized solution of Problem (8.4.1), (8.4.4) if < L(x,D)u,v
for each v €
is nontrivial. A function u €
>=< h, v >
LW?{$a,pa}.
Theorem 8.4.3. / / Conditions a), b), c) are satisfied and the space
o
o is nontrivial, then Problem (8.4-1), (8.4-4) has a unique solution u € LWf corresponding to each h € EW~°°{$a,Pa}-
LWf{^a,pa) {$Q,pa}
Proof. Consider the following Dirichlet problem for the equation of order 2m obtained from (8.4.1) by replacing the infinite series by a partial sum: m
L2m(um)
:= £
(-l)^DaAa(x,
u m , - • ■, Daum) = hm(x),
(8.4.9)
L>a'um\Xi=0 = 0, a, = 0 , 1 , . . . , m - [ ( < ? „ . + l ) / p a ] - l ,
(8.4.10)
Da'um\x-T,
(8.4.11)
=0,
a, = 0 , 1 , . . . .
216
DIFFERENTIAL
OPERATORS
OF INFINITE
ORDER
In view of Conditions a), b), c), Problem (8.4.9)-(8.4.10) has a unique solution um(x) in the Sobolev-Orlicz space W™L{$a,pa) of functions satisfying Conditions (8.4.10),(8.4.11) on the boundary of G with the norm ||u||(m,p.):=inf|fc>0| k
£
/ pa(x)$a(Dau/k)dx
\c\=0JG
{
< 11. )
We obtain the inequality
/ pa(x)<^a(Daum)dx < K,
£
JG
\a\=o
where K = K(h) > 0 is a constant independent of m. Hence the sequence {um} (a priori subsequence) converges locally and uniformly together with all derivatives to a function u G C0X(G) and its derivatives, respectively. The function u satisfies the o boundary conditions (8.4.4) and belongs to LWf{$a,pa}. It is easily proved, as in the proof of Theorem 8.1.1, that u(x) is a generalized so lution of Problem (8.4.1), (8.4.4), and its uniqueness is a consequence of the strict monotonicity condition. This completes the proof of Theorem 8.4.3. Example 8.4-1- In G consider the equation oo
Y, (-l)MDa(pa(x)ipa(Dau(x)))
= *(*),
(8-4.12)
where the function ipa : IR1 —> IR1 is continuous, odd, increasing and ¥>a(+oo) = +oo. We put *.(*)=
(\a{T)dr. Jo For simplicity we consider the case <pa(t) = aa\t\Pa~'it, i.e., the case of coefficients with power-law growth, aa > 0, pa > 1. i) If aa = (|a|!)-" p ° with v < 1, then the space L WT{$c,,pa} Eq. (8.4.12) reduces to the identity 0 = 0. ii) Suppose that aa = (|a|!) _ " P a ,
is trivial; hence
v > 1, and
lim sup Li = +oo, i =
l,2,...,n,
(see (8.4.2)), and let Eq. (8.4.12) be supplemented by the boundary conditions ZT| a G = 0, M| = 0 , 1 , . . . Then there exists a unique generalized solution u G LW°°{$0,/9a} (8.4.12), (8.4.13) if h G £ W - 0 O { $ c r , pa}.
(8.4.13) of Problem
ELLIPTIC
EQUATIONS
OF INFINITE
ORDER
217
iii) If qai = 2Q, — 3,p a = 2 and aa = (|a|!) _ " p<> , v > 1, then no boundary conditions are needed on the part of dG where s; = 0, i = 1 , . . . , n. A generalized solution of Eq. (8.4.12) satisfying the conditions £ £ > L = r , = 0,oj = 0 , 1 , . . . , t == 1,2,... ,n, o is in LWf{$a,pa}
with s = - 2 .
Bibliographical N o t e s . The theory of infinite-oder nonlinear differential equa tions with coefficients of polynomial growth was developed by Dubinskii, Balashova, Kobilov, Konjaev and many others (see Dubinskii's survey [6]). The solvability of nonlinear differential equations of infinite order in spaces generated by spectral oper ators was considered in Umarov [1], [2] and Klenina [1]. Infinite-oder degenerate nonlinear differential equations with coefficients of polyno mial growth were considered in great detail in Tran Due Van's papers [l]-[3], [7], Some degenerate elliptic equations were considered in Nguyen Nhu Doan [1], Nguyen Minh Chuong and Le Quang Trung [1]. The Cauchy problem for certain linear par tial differential equations of infinite order in Banach spaces was studied by Ricceri [1]. The analysis of this chapter is both a completion and extension of the results given in the papers by Tran Due Van [4]-[6], [8], [11] and [12] and by Tran Due Van, R. Gorenflo and Le Van Hap [1]. Using the methods of this chapter one can obtain the existence and uniqueness of solutions of infinite-order nonlinear parabolic and hyperbolic equations with arbitrary nonlinearities. For the case of polynomial growth of coefficients see, e.g, Dubinskii's book [3] and Tran Due Van's book [12].
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