NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS
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NORTH-HOLLAND MATHEMATICS STUDIES
44
Notas de Matematica (73) Editor: Leopoldo Nachbin Universidade Federal do Rio de Janeiro and University of Rochester
Nonlinear Partial Differential Equations Sequential and Weak Solutions
ELEMER E. ROSINGER National Research Institute for Mathematical Sciences Pretoria, South Africa
1980
NORTH-HOLLAND PUBLISHING COMPANY - AMSTERDAM
NEW YORK
OXFORD
North-Holland Publishing Company, I980
All rights rrscwerl. No purt of rhispublicurioti mu,v he, rrprvducecl. siori>clin N retrievtrlsy.stett~. or trunsmirtrd, in uny form or by uny nwuns, c+c~trotiic, mc~chanicd. photocopying. rcw)rtling or othrrwisc.. without the prior pertnissioti of the, copvrighr owtwr.
ISBN: 0 4 4 4 8 6 0 5 5 ~
Pu hlishm: NORTH-HOLLAND PUBLISHING COMPANY AMSTERDAM NEW YORK O X F O R D Sokc disirihurors for ihr U.S.A. und Cmnudri: ELSEVIER NORTH-HOLLAND. INC. 5 2 VANDERBILT AVENUE, NEW YORK, N.Y. 10017
PRINTED IN T H E NETHERLANDS
Dedicated t o my parents i n law, Renee a n d Max Neufeld
FOREWORD The p r e s e n t volume i s a sequel t o t h a t p u b l i s h e d i n t h e s e r i e s S p r i n g e r L e c t u r e Notes i n Mathematics, as v o l . 684, i n 1978. D u r i n g t h e l a s t two years, a number o f new r e s u l t s o b t a i n e d by t h e a u t h o r have c o n t r i b u t e d t o t h e c l a r i f i c a t i o n o f t h e general framework on which t h e p r e v i o u s volume was based, and have come t o e n r i c h and l i f t t o a new l e v e l o f g e n e r a l i t y t h e r e s u l t s p u b l i s h e d e a r l i e r . An a t t e m p t i s made here t o p r e s e n t these developments i n a s e l f - c o n t a i n e d manner. I n developing t h e p r e s e n t method, t h e a u t h o r ' s main concern has been t o t r y t o f i n d a u n i f i e d way o f d e a l i n g w i t h weak s o l u t i o n s o f general non=
l i n e a r PDEs, a-ich would e x h i b i t a c e r t a i n n a t u r a l , o b j e c t i v e t r a i t , thus going beyond t h e somewhat adhoc appearance t h e customary f u n c t i o n a l a n a l y t i c methods, w i t h t h e i r w e a l t h o f l i n e a r t o p o l o g i c a l s t r u c t u r e s and f u n c t i o n spaces used, m i g h t now and t h e n suggest. I n t h i s respect, a way presented i t s e l f n a t u r a l l y by n o t i c i n g t h a t t h e s e q u e n t i a l , i n p a r t i c u l a r t h e weak s o l u t i o n s o f a n o n l i n e a r PDE T(D)u(x) = f ( x ) ,
x
E
R, Q domain i n Rn,
can be seen as elements i n t h e a l g e b r a o f continuous f u n c t i o n s C(N x Q ) . Consequently, t h e known r i i d i t y between t h e t o p o l o g i c a l p r o p e r t i e s o f a completely r e g u l a r space+-ti-an t e a l g e b r a i c p r o p e r t i e s o f i t s continuous f u n c t i o n s , a p p l i e d t o N x R, r e s p e c t i v e l y C(N x Q), might make i t p o s s i b l e t o t r a n s l a t e t h e t o p o l o g i c a l p r o p e r t i e s i n t o t h e a l g e b r a i c ones and v i c e versa, a procedure which c o u l d prove t o be an advantage i n t h e s t u d y o f n o n l i n e a r PDEs. F o r i n s t a n c e , one c o u l d expect t h a t a good deal o f t h e d i s c u s s i o n o f s e q u e n t i a l s o l u t i o n s m i g h t be k e p t on an a l g e b r a i c l e v e l , s i n c e these s o l u t i o n s a r e elements i n C ( N x Q), which p a r t i c i p a t e s m a i n l y through i t s a l g e b r a i c s t r u c t u r e . Moreover, t h e n a t u r a l t o p o l o g i c a l pro= p e r t i e s o f N x R would p r o v i d e t h e o b j e c t i v e c h a r a c t e r i s t i c o f t h e method aimed a t . A d d i t i o n a l , i m p o r t a n t aspects might obvious1 be a i n e d from t h e presenceof t h e chains o f d i f f e r e n t i a l subalgebras ( C X (a)) 8 i n C(N x Q), w i t h II E N, which appear as t h e n a t u r a l domains o f d e f i n i t i o n f o r t h e n o n l i n e a r PDOs i n v o l v e d , when t h e i r a c t i o n i s extended t o sequences o f f u n c t i o n s . F o r t u n a t e l y , these e x p e c t a t i o n s c o u l d be f u l f i l l e d t o a c e r t a i n e x t e n t . However, t h e method presented here o f s t u d y i n g s e q u e n t i a l and weak s o l u = t i o n s o f n o n l i n e a r PDEs can o n l y be considered as a f i r s t s t e p towards f u l l use o f t h e power o f f e r e d b y t h e t h e o r y o f a l g e b r a s o f continuous,functions a p p l i e d t o t h e p a r t i c u l a r case o f C(N x a ) , w i t h R a domain i n R
.
I am happy t o acknowledge here my s p e c i a l g r a t i t u d e t o P r o f . L. Nachbin, E d i t o r o f Notas de Mathematica i n N o r t h - H o l l a n d Mathematical S t u d i e s , f o r suggesting t h a t t h i s volume should be w r i t t e n . F o r d i s c u s s i o n s on v a r i o u s aspects o f t h i s work, g r a t e f u l thanks a r e a l s o due t o M.C. Reed and E. Schechter of Duke U n i v e r s i t y , C. M. Dafermos and W.A. Strauss o f Brown U n i v e r s i t y , H.C. Kranzer o f Adelphi U n i v e r s i t y and J. Horvath o f C o l l e g e Park. vi
vi i
FOREWORD
Work on the book was begun in Israel and continued during short-term v i s i t s to several universities in Switzerland and the U S A . The second part was completed in Pretoria, South Africa. The author i s most grateful t o a l l those who helped him with t h e i r coments a t that stage, special thanks being due t o Professor D.H. Jacobson, Director of the National Research I n s t i t u t e for Mathematical Sciences of the CSIR, Pretoria, who was kind enough t o extend a longer invitation t o his Institute and place a t the author’s disposal a l l the necessary f a c i l i t i e s f o r research. A l s o sincere t h a n k s are due t o Dr. E.
Fredriksson, Publisher a t NorthHolland, f o r his courteous and e f f i c i e n t collaboration.
The task of editing and typing a manuscript i s one f o r which an author cannot be thankful enough: I can only express my special admiration t o Mr. F.R. Baudert and Mrs. M. Russouw and acknowledge my heavy indebtedness t o them, f o r t h e i r patience and efficiency.
E.E.R.
Pretoria
PRELIMINARIES The method o f s o l v i n g n o n l i n e a r PDEs b y c o n s t r u c t i n g 'weak s o l u t i o n s ' i s w i d e l y used. One reason f o r t h i s i s t h a t r a t h e r b a s i c and s i m p l e n o n l i n e a r PDEs w i t h r e g u l a r i n i t i a l o r boundary c o n d i t i o n s may l a c k r e g u l a r s o l u t i o n s . A well-known example i s t h e case o f shock wave s o l u t i o n s o f n o n l i n e a r hy= p e r b o l i c c o n s e r v a t i o n laws. I n such cases, owing t o t h e presence o f singu= l a r i t i e s such as d i s c o n t i n u i t i e s o r l a c k o f d e r i v a t i v e s o f s u f f i c i e n t l y h i g h o r d e r , t h e s o l u t i o n s o b t a i n e d w i l l s a t i s f y t h e n o n l i n e a r PDEs i n a weak sense o n l y . U s u a l l y , these weak s o l u t i o n s can be i n t e r p r e t e d as d i s = t r i b u t i o n s . However, i n t h e case o f n o n l i n e a r PDEs, d i s t r i b u t i o n s o l u t i o n s may f a i l t o s a t i s f y t h e equations i n a d i s t r i b u t i o n a l sense, s i n c e even elementary n o n l i n e a r o p e r a t i o n s on d i s t r i b u t i o n s , f o r i n s t a n c e products, may l e a d o u t s i d e t h e d i s t r i b u t i o n s . I n t h a t way, t h e d i s t r i b u t i o n a l frame= work proves t o be r a t h e r narrow when s o l v i n g n o n l i n e a r PDEs. Moreover, as t h e well-known example o f H. Lewy shows, even l i n e a r PDEs may f a i l t o have distribution solutions.
A s u f f i c i e n t l y wide framework f o r s o l v i n g n o n l i n e a r PDEs w i l l be presented i n t h i s work b y c o n s i d e r i n g t h e weak s o l u t i o n s , i n p a r t i c u l a r t h e d i s t r i = b u t i o n s o l u t i o n s as ' s e q u e n t i a l s o l u t i o n s ' , i.e. b y c o n s i d e r i n g them as g i v e n b y sequences o f continuous, o r more r e g u l a r f u n c t i o n s . A b a s i c ad= vantage o f t h i s approach i s t h a t t h e ' s e q u e n t i a l s o l u t i o n s ' a r e i n a natu= r a l way elements o f c e r t a i n a s s o c i a t i v e and commutative algebras o f 'gene= ralized functions' containing the distributions D'cAmC
... c A P c ... c A o , p € N n
previously introduced by the author. These a l g e b r a s possess p a r t i a l d e r i v a t i v e o p e r a t o r s D ~ : A P - + A P ' ~ ,P E N " ,
EN^,
q < p ( i i = ~ u
{m))
which s a t i s f y t h e L e i b n i t z r u l e o f p r o d u c t d e r i v a t i v e s . I n t h a t way, as shown i n t h e p r e s e n t work, t h e p o l y n o m i a l n o n l i n e a r PDEs Pij Z ci(x) D U(X) = f ( x ) , x R c Rn , l G i Q h l<JGki w i t h g i v e n c., f E C"(f2) and unknown u , can under q u i t e general c o n d i t i o n s f i n d d i s t r i b h i o n s o l u t i o n s , weak s o l u t i o n s o r ' s e q u e n t i a l s o l u t i o n s ' w i t h = i n t h e algebras Ap, w i t h p E , p " p i j , f o r 1 G i Q h , 1 Q j Q ki.
mn
The need f o r d e a l i n g with t h e whole c h a i n o f a l g e b r a s Ap, w i t h p E In , a r i s e s because c e r t a i n p r o d u c t s c o n t X i T 5 i g s i n g u l a r f a c t o r s , such as t h e D i r a c 6 d i s t r i b u t i o n and i t s d e r i v a t i v e s , may l e a d t o d i f f e r e n t r e s u l t s depending on t h e a l g e b r a s t h e y a r e considered i n . T h e r e f o r e i n o r d e r t o o b t a i n s u i t a b l e p r o p e r t i e s o f t h e m u l t i p l i c a t i o n one has t o choose t h e p r o p e r a l g e b r a s . F o r i n s t a n c e , as can be shown, based on a well-known counterexample g i v e n b y L. Schwartz i n t h e one-dimensional case n = 1, under n a t u r a l c o n d i t i o n s concerning pKoducts o f c" f u n c t i o n s and d e r i v a = t i v e s o f Lo1 f u n c t i o n s , t h e a l g e b r a A cannot accommodate t h e r e a l t i o n x
6(x) = 0 viii
PRELIMINARIES
ix
which gives an gpper bound on the order of singularity exhibited in x = 0 E R' by the Dirac 6 distribution, and which will hold in the algebra A" The counterexample mentioned was occasionally interpreted as i l l u s = trating t h a t a suitable distribution multiplication i s impossible, i .e. that i t i s impossible t o embed the distributions into differential alge= bras. However, as mentioned above, such embeddings are possible and use= ful i f chains of algebras are considered.
.
I t i s particularly important t o notice t h a t from the p o i n t of view of solving polynomial nonlinear PDEs, the essential features of the above algebras of 'generalized functions' are the following. F i r s t , the algebras are l a r e enough t o contain the usual weak or distribution solutions. Actua Tp y , as seen in Chapter 2 they will accommodate the existence of 'sequential solutions' for wide classes of nonlinear PDEs.
Secondly, olynomial nonlinear operations can freely be performed under the best a ge raic circumstances granted by associativity and comutativity.
eb--
Thirdly, the partial derivative operators s a t i s f y the Leibnitz rule of product derivative. Finally, the algebras are obtained in a simple, constructive way, a s quotient algebras of sequences of continuous or more regular functions. Therefore t h e i r elements, the 'generalized functions', in particular the 'sequential solutions' weak solihions or distributions, are classes of sequences of continuous or more regular functions, much in the s p i r i t of the usual approximation methods, a circumstance that will be specially helpful when constructing solutions for nonlinear PDEs and establishing t h e i r properties. The fact t h a t with the exception of the algebra Am, the elements of the other algebras Ap are not indefinitely derivable, i s no particular dis= advantage as long as the polynomial nonlinear PDEs considered are of f i n i t e order and can therefore be dealt w i t h i n the algebras Ap, with p f i n i t e and greater than the order of the equations involved. I n t h i s connection i t i s worth noticing t h a t the indefinite derivability of the distributions in D' does n o t compensate f o r the lack of suitable nonlinear operations within the distributions. I n conclusion i t may be remarked t h a t as long as one deals with PDEs of f i n i t e order, the essential fea= tures of the spaces of 'generalized functions' are t h e i r large size as well as algebra structure, and n o t necessarily the indefinite derivability of t h e i r elements. The study of general linear PDEs i n i t i a t e d two major departures from the classical framework. The f i r s t , due t o L. Schwartz, brought an increase in the 'reservoir' of b o t h data and possible solutions f o r linear PDEs, by introducing the distributions. T h a t departure proved t o be particu= l a r l y f r u i t f u l in the study of linear PDEs with constant coefficients. The second departure, due t o J . J . Kohn and L. Nirenberg, one of the aims of which was t o deal with linear PDEs with variable coefficients, extended the class of equations considered , by introducing pseudodifferential equations, an extension s t i l l within the linear framework. The approach t o nonlinear PDEs suggested in t h i s work represents a further departure along the direction initiated by L . Schwartz in that the equations considered are classical , while the possible solutions can be=
PRELIMINARIES
X
l o n g t o algebras which include the distributions. The method presented for polynomial nonlinear PDEs can easily be extended t o much wider classes of classical nonlinear PDEs.
The present work deals with the following problems related t o 'sequential solutions' of polynomial nonlinear PDEs : existence, s t a b i l i t y , resolu= tion of singularities and regularity. The characteristic idea of the approach i s f i r s t t o develop the consequences resulting from the basic algebraic properties of particular differential algebras of classes of sequences of regular functions on domains in Euclidean spaces. Here an important role will be played by the basic algebraic, topological and cardinality properties of algebras of sequences of continuous functions on domains in Euclidean spaces. Analytic methods are employed during l a t e r stages only. Methods based on the properties of linear topological structures on various function spaces will not be used in the present work. More precisely, the 'sequential solutions' being elements of dif= ferential algebras (Cm ( Q ) ) N contained in the algebra of continuous functions C ( N x R), the idea i s t o pursue the study of polynomial non= linear PDEs as f a r as possible, using algebraic properties of C(N x Q) and t o ological properties of N x R, which - owing t o the known ri i d i t be= tween t e topological properties of a completely regular space an the algebraic properties of C ( X ) - determine each other uniquely. The consequ= ent development will therefore have a certain objective character result= ing from the natural topological properties of N x n, in contrast t o the somewhat arbitrary or ad hoc nature of the developments based on the use of various 1 inear topological structures on various function and 'genera= lized function' spaces.
+
-%-$
Lately there has been an awareness that important results on PDEs can be deduced from basic, rather simple algebraic and possibly analytic or topo= logical properties of several classical function spaces , w i t h o u t recourse to the customary wealth of linear topological structures. In the case of linear PDEs with constant coefficients for instance, R. Struble could prove the existence of fundamental solutions, using basic algebraic properties of a special module structure on the space of smooth functions. I n view of the algebraic c l a r i t y of t h i s result and several other similar ones, i t may perhaps be considered t h a t a multivariable operational cal= culus has been developed. The role of purely algebraic structures proves t o be substantial also when important properties o f nonlinear PDEs are being investigated. A recent survey of Yu. Manin demonstrated in detail the interplay between differential a1 gebra and the classical or geometri= cal methods in the study of a wide class of nonlinear PDEs related t o the Korteweg-de Vries equation, with the resulting important role played by the basic algebraic structures involved. When studying PDEs, there are essentially three reasons f o r considering topologies on the associated function or 'generalized function' spaces: f i r s t , in order t o define partial derivatives; secondly, in order t o approximate existing exact solutions; and f i n a l l y , in order t o define 'weak solutions' as elements in completions of certain function spaces. In the study of exact solutions of polynomial nonlinear PDEs w i t h i n the framework of quotient algebras of sequences of functions, the considera= tion of topologies on function spaces can t o a certain extent be avoided. Indeed, the partial derivatives being definednterm by term on sequences of functions, only the standard topology on R will be involved. Further,
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t h e a l g e b r a i c s t u d y o f e x a c t s o l u t i o n s does n o t n e c e s s a r i l y r e q u i r e appro= x i m a t i o n methods. F i n a l l y , t h e ' r e s e r v o i r ' o f p o s s i b l e ' g e n e r a l i z e d s o l u = t i o n s ' can always be k e p t l a r g e enough, w i t h o u t recourse t o t o p o l o g y , s i m p l y b y c o n s i d e r i n g s u i t a b l e q u o t i e n t a l g e b r a s o f sequences o f f u n c t i o n s . I n t h i s connection, t h e method i n t h i s work aims t o reassess t h e balance between a l g e b r a and t o p o l o g y i n t h e s t u d y o f n o n l i n e a r PDEs: f i r s t , b y a ' b l o w u p ' o f t h e e s s e n t i a l a l g e b r a i c phenomena i n v o l v e d , f o l l o w e d b y t h e use o f t h e necessary l e v e l o f t o p o l o g y , which up t o a s i g n i f i c a n t ex= t e n t w i l l n o t exceed t h a t customary i n t h e s t u d y o f t h e a l g e b r a o f con= t i n u o u s f u n c t i o n s C(N x a). Other t o p o l o g i c a l s t r u c t u r e s m i g h t become useful i n c o n n e c t i o n w i t h t h e r e g u l a r i t y p r o p e r t i e s o f ' s e q u e n t i a l s o l u = tions'. As i s w e l l known, t h e s t u d y o f n o n l i n e a r PDEs has r e v e a l e d t h r e e fundamen= t a l phenomena, v i z t u r b u l e n c e , s o l i t a r y waves and shock waves. The p r e s e n t work d e a l s w i t h aspects o f t h e l a t t e r phenomenon, considered under i t s general form o f d i s t r i b u t i o n , weak o r ' s e q u e n t i a l s o l u t i o n s ' o f n o n l i n e a r PDEs. The f a c t t h a t these s o l u t i o n s a r e e l e m e n t s ' o f spaces o f ' g e n e r a l i z e d f u n c t i o n s ' possessing a s t r u c t u r e o f a s s o c i a t i v e and commutative algebra, makes i t p a r t i c u l a r l y easy t o deal w i t h polynomial n o n l i n e a r PDEs. How= ever, i t i s i m p o r t a n t t o n o t i c e t h a t t h e presence o f a s t r u c t u r e o f a l g e = b r a on t h e ' g e n e r a l i z e d f u n c t i o n s ' - u n l i k e t h e v e c t o r space s t r u c t u r e o f d i s t r i b u t i o n s , w i t h m u l t i p l i c a t i o n defined p a r t i a l l y , f o r special p a i r s o f f a c t o r s o n l y - o f f e r s a n o t h e r advantage, i n t h a t , i t y i e l d s a b a s i c and simple algebraic t o o l f o r modelling s i n g u l a r i t i e s i n the s o l u t i o n o f polynomial n o n l i n e a r PDEs. Indeed, i4TI-e i n groups, o r v e c t o r spaces as t h e case may be, a l l t h e o p e r a t i o n s - e x c e p t m u l t i p l i c a t i o n w i t h t h e scalar zero - are i n v e r t i b l e , t h e rings, o r e l s e algebras, a r e the next s i m p l e s t a l g e b r a i c s t r u c t u r e s which owing t o t h e presence o f z e r o d i v i s o r s , possess a n o n t r i v i a l n o n - i n v e r t i b l e , t h u s ' s i n g u l a r ' o p e r a t i o n , v i z m u l t i = p l ic a t i on. I n s h o r t , one may say t h a t t h e method presented i n t h i s work e s t a b l i s h e s a n a t u r a l and s t r o n g c o n n e c t i o n between t h e s t u d y o f general n o n l i n e a r PDEs and t h e s t u d y o f a l g e b r a s o f continuous f u n c t i o n s C ( N x a ) , where R C Rn i s a domain. Chapter 1 s t a r t s w i t h an i m p o r t a n t m o t i v a t i o n f o r t h e i n t r o d u c t i o n and s t r u c t u r e o f t h e a l g e b r a s o f ' g e n e r a l i z e d f u n c t i o n s ' , prompted b y s t a b i l i = t y c o n s i d e r a t i o n s c o n c e r n i n g weak o r ' s e q u e n t i a l s o l u t i o n s ' . As p o i n t e d o u t i n t h e case o f n o n l i n e a r PDEs, t h e s t a b i l i t y o f weak o r ' s e q u e n t i a l s o l u t i o n s ' - u n l i k e t h e s t a b i l i t y o f s o l u t i o n s o f l i n e a r PDEs, which i s t r i v i a l and assured a u t o m a t i c a l l y - becomes an e s s e n t i a l and n o n t r i v i a l problem, even i f q u i t e o f t e n overlooked. A f t e r d e f i n i n g t h e n o t i o n o f ' s e q u e n t i a l s o l u t i o n ' , a method f o r t h e r e s o l u t i o n o f s i n g u l a r i t i e s o f ' s e q u e n t i a l s o l u t i o n s ' i s presented, and i t i s shown t h a t i n s p i t e o f t h e g e n e r a l i t y o f t h e framework, t h e r e g u l a r i t y o f s o l u t i o n s o f e l l i p t i c o r h y p o e l l i p t i c l i n e a r PDEs i s preserved.
I n Chapter 2 necessary and/or s u f f i c i e n t c o n d i t i o n s f o r t h e e x i s t e n c e o f ' s e q u e n t i a l s o l u t i o n s ' a r e g i v e n f o r l a r g e c l a s s e s o f n o n l i n e a r PDEs. By i n t r o d u c i n g c e r t a i n s i m p l e c o n d i t i o n s o f t o p o l o g i c a l t y p e , we a r r i v e a t r e s u l t s which o f f e r an i n s i g h t i n t o t h e s t r u c t u r e o f t h e a l g e b r a s o f 'generalized functions'. Chapter 3 p r e s e n t s t h e c o n s t r u c t i o n o f t h e a l g e b r a s o f ' g e n e r a l i z e d func=
PREL IMI N ARI ES
xi i
tions' containing the distributions used in the sequel. In Chapter 4 , a general result i s given concerning the resolution of singularities of distribution solutions for polynomial nonlinear PDEs. This result i s then applied t o the case of known weak or distribution solutions of f i r s t and second-order nonlinear wave equations. Further, a general necessary and sufficient junction condition across hypersurfaces of discontinuities i s established for weak solutions of a large class o f systems of PDEs, containing among others the equations of magnetohydro= dynamics and general relativity. An extension of that result gives an insight into the structure of nonlinearities of a particularly large class o f systems of PDEs, containing many o f the equations encountered in phys i cs
.
Chapter 5 deals with the s t a b i l i t y of weak or distribution solutions given in sequential form. Results are presented on maximal s t a b i l i t y , as well as the interplay between s t a b i l i t y and exactness. The study o f maximal s t a b i l i t y leads in a natural way t o a characteriza= tion of the necessary structure of the algebras of 'generalized functions' containing the distributions. Based on the properties of 'zero f i l t e r s ' , results are presented in Chapter 6 that specialize and strengthen those arrived a t in the theory of algebras of continuous functions. I n Chapter 7 , within the a1 gebras of 'generalized functions ' , a scattering problem i s solved with potentials arbitrary positive powers of the Dirac 6 distributions, i .e. exhibiting the strongest local singularities studied in the literature. The wave function solutions obtained in the one and three-dimensional cases are regular, except on the support of the poten= t i a l s , where the discontinuities accompanying the transmission and reflec= tion of waves are determined. I n view of the special role played in the construction of the algebras of 'generalized functions' by certain products involving the Dirac 6 d i s t r i = bution and i t s derivatives, Chapters 8 and 9 are devoted t o the study of such products, also o f certain unusual properties of the Dirac 6 d i s t r i = bution within the mentioned algebras.
An essential and characteristic property of multiplication within the a l = gebras of 'generalized function' which i s extensively used throughout the present work, i s i t s local character, i.e. the property that the product of two 'generalized functions' on an open subset depends only on the restriction of the two factors t o t h a t subset. In order t o clarify t h a t property as well as several related ones, including certain aspects o f division within the algebras of 'generalized functions' , Chapter 10 i s devoted t o the study of support and local properties of 'generalized functions'. The work ends w i t h an appendix which presents in some detail various con= nections between the algebras of 'generalized functions' and related notions and constructions in the Calculus.
TABLE OF CONTENTS
CHAPTER 1 : S e q u e n t i a l S o l u t i o n s o f N o n l i n e a r PDEs
1
0
Introduction
1
1
The Problem o f S t a b i l i t y o f S e q u e n t i a l S o l u t i o n s
2
2
D e f i n i n g t h e Q u o t i e n t Spaces and Algebras
5
3
The N o t i o n o f S e q u e n t i a l S o l u t i o n
10
4
Exactness o f S e q u e n t i a l S o l u t i o n s
12
5
I n t e r p l a y between S t a b i l i t y , G e n e r a l i t y and Exactness
13
The D e f i c i e n c y o f t h e D i s t r i b u t i o n a l Approach Concerning Exactness
17
7
R e s o l u t i o n o f Nowhere Dense S i n g u l a r i t i e s
19
8
P r e s e r v a t i o n o f E l 1 i p t i c i ty an3 Hypoel 1 i p t i c i t y
24
9
General N o n l i n e a r PDEs
25
10
V a r i a b l e Transforms
27
11
Stronger Conditions f o r P a r t i a l Derivative Operators
30
12
Systems o f N o n l i n e a r PDEs
32
13
A Cesaro C h a r a c t e r i z a t i o n o f t h e Nowhere Dense
6
Ideal CHAPTER 2 : Necessary and/or S u f f i c i e n t C o n d i t i o n s f o r t h e Existence o f Sequential Solutions
34 37
Introduction
37
Subsequence Q u a s i I n v a r i a n t S e q u e n t i a l S o l u t i o n s
38
A p p l i c a t i o n s t o L i n e a r and N o n l i n e a r PDEs
44
Subsequence I n v a r i a n t S e q u e n t i a l S o l u t i o n s
46
Resolvent Sets
51
Domains o f Sol v a b i 1 it y
55
Remarks on Lemma 2
62 xiii
xiv
TABLE
OF CONTENTS
CHAPTER 3 : Algebras C o n t a i n i n g t h e D i s t r i b u t i o n s
65
0
Introduction
65
1
Embedding t h e D i s t r i b u t i o n s i n t o Q u o t i e n t A1 gebras
65
2
S i m p l e r I n c l u s i o n Diagrams
68
3
R e g u l a r i z a t i o n s and Chains o f Q u o t i e n t Algebras Containing the D i s t r i b u t i o n s
69
4
Existence, C h a r a c t e r i z a t i o n and C o n s t r u c t i o n o f Regularizations
75
A d d i t i o n a l Chains o f Q u o t i e n t Algebras C o n t a i n i n g Distributions
88
Q u o t i e n t Algebras C o n t a i n i n g Large Subspaces o f Distributions
97
5 6
7
Q u o t i e n t Algebras Possessing S p e c i a l P r o p e r t i e s
104
8
Definition o f Positive Generalized Functions
107
9
Powers f o r C e r t a i n
L i m i t a t i o n s on Embedding Smooth Functions i n t o Chains o f Q u o t i e n t Algebras
110
10
Special Classes o f Regular I d e a l s
113
11
The P r o o f o f Lemma 1
115
12
Example o f S h r i n k i n g Non-6 Sequence
117
13
I n e x i s t e n c e o f t h e L a r g e s t Regular I d e a l s
118
CHAPTER 4 : R e s o l u t i o n o f S i n a u l a r i t i e s o f Weak S o l u t i o n s f o r Polynomi a1 N o i l inear PDEs
121
Introduction
121
The Case o f Simple Polynomial N o n l i n e a r PDEs
122
R e s o l u t i o n o f S i n g u l a r i t i e s o f N o n l i n e a r Shock Waves
131
R e s o l u t i o n o f S i n g u l a r i t i e s o f Klein-Gordon Type Nonl in e a r Waves
133
R e s o l u t i o n o f S i n g u l a r i t i e s o f Weak S o l u t i o n s f o r General Polynomial Nonl inear PDEs
134
J u n c t i o n C o n d i t i o n s and R e s o l u t i o n o f S i n g u l a r i = t i e s o f Weak S o l u t i o n s f o r t h e Equations o f Magneto-hydrodynamics and General R e l a t i v i t y
139
TABLE OF CONTENTS
xv
6
Resoluble Systems o f Polynomial N o n l i n e a r PDEs
152
7
Computation o f t h e J u n c t i o n C o n d i t i o n s
155
8
Examples o f Resoluble Systems o f PDEs
156
CHAPTER 5: S t a b i l i t y and Exactness o f S e q u e n t i a l and Weak S o l u t i o n s f o r Polynomial N o n l i n e a r PDEs
0
Introduction
163 163
1 B a s i c Facts and Remarks Concerning S t a b i l i t y
163
2
Maximal S t a b i l i t y
165
3
I n t e r p l a y between S t a b i l i t y and Exactness
167
4
Remarks i n t h e F i n i t e Smoothness Case
16 7
5
S t a b i l i t y o f Regular Weak S o l u t i o n s
168
6
S t a b i l i t y o f S o l u t i o n s w i t h Jump D i s c o n t i n u i t i e s f o r Resoluble Systems o f PDEs
170
CHAPTER 6: C h a r a c t e r i z a t i o n o f t h e Necessary S t r u c t u r e o f t h e Algebras C o n t a i n i n g t h e D i s c r i b u t i o n s Introduction
173
Zero F i l t e r s
174
D e n s i t y C h a r a c t e r i z a t i o n o f Z-Fi 1t e r s
176
Large C"-Regular I d e a l s
178
F u r t h e r P r o p e r t i e s o f Z-Fi 1t e r s
182
C o n s t r u c t i o n and C h a r a c t e r i z a t i o n o f a Class o f Vanishing I d e a l s
185
Countable Reduced Products o f V e c t o r Spaces
194
CHAPTER 7: Quantum S c a t t e r i n g i n P o t e n t i a l s P o s i t i v e Powers o f the Dirac 6 D i s t r i b u t i o n
0
173
Introduction
199 199
1 Wave F u n c t i o n S o l u t i o n s and J u n c t i o n R e l a t i o n s
199
2
Weak Wave F u n c t i o n S o l u t i o n s
20 1
3
Smooth Representations f o r t h e D i r a c 6 D i s t r i b u = t i o n and Smooth Weak Wave F u n c t i o n S o l u t i o n s
2 10
Wave F u n c t i o n S o l u t i o n s i n t h e Algebras Contain= ing the Distributions
214
4
TABLE OF CONTENTS
xvi
5
S t a b i l i t y and Exactness o f t h e Wave F u n c t i o n Sol u t i ons
CHAPTER 8 : Products w i t h D i r a c 6 D i s t r i b u t i o n s
220 22 3
0
Introduction
223
1
A Class o f R e g u l a r i z a t i o n s
223
2
Products w i t h D i r a c 6 D i s t r i b u t i o n s
229
3
Application t o a R i c c a t i D i f f e r e n t i a l Equation
234
4
Val id i t y o f Formulas i n Quantum Mechanics
2 35
5
The E x i s t e n c e o f t h e Sequences o f F u n c t i o n s i n Z6
242
Remark on N i l p o t e n t Elements i n t h e Q u o t i e n t A1 gebras
249
L i n e a r Independent F a m i l i e s o f D i r a c 6 D i s t r i = butions a t a Point
251
0
Introduction
251
1
Compatible Q u o t i e n t Algebras and Independent V a r i a b l e Transforms
251
L i n e a r Independent F a m i l i e s o f D i r a c 6 D i s t r i b u = tions a t a Point
255
Generalized D i r a c 6 D i s t r i b u t i o n s
257
6 CHAPTER 9
2 3
CHAPTER 10: Support and Local P r o p e r t i e s
265
0
Introduction
26 5
1
Support o f Elements i n Q u o t i e n t Algebras
265
2
L o c a l i z a t i o n w i t h i n t h e Q u o t i e n t Algebras
270
3
Equivalence between S=O and supp
4
Domains o f Sol vabi 1it y f o r Polynomi a1 Nonl i n e a r PDEs
S=0
278 2 79
FINAL REMARKS
281
APPENDIX 1: N e u t r i x C a l c u l u s and N e g l i g i b l e Sequences o f Functions
285
APPENDIX 2: The Embedding I m p o s s i b i l i t y R e s u l t o f L.Schwartz
2 89
APPENDIX 3: A N o n l i n e a r E x t e n s i o n of t h e Lax-Richtmyer Equivalence between S t a b i l i t y and Convergence o f
xvi i
TABLE OF CONTENTS
APPENDIX 4: REFERENCES
D i f f e r e n c e Schemes
29 3
The Cauchy-Bolzano Q u o t i e n t Algebra Construe= t i o n o f t h e Real Numbers
30 1 30 5
NOTE TO THE READER A t a f i r s t l e c t u r e , t h e Reader may c o n c e n t r a t e on t h e f o l l o w i n g s e c t i o n s : Foreword Prel i m i naries Chapter 1, Sections 0-5,
7 , 9 , 10, 12
Chapter 3, Sections 0-4, 7, 8, 10 Chapter 4, Sections 0-6 Chapter 5, Sections 0-6 Chapter 7, S e c t i o n s 0-5 However, f o r a deeper understanding o f t h e method presented i n t h i s work, t h e f o l l o w i n g s e c t i o n s a r e important: Chapter 1, Sections 6, 11 Chapter 2, Sections 0, 1, 3-5 Chapter 3, Sections 5, 6, 9, 13 Chapter 6, Sections 0-5 F i n a l Remarks Appendices 1, 2, 4
xviii
NOTAT I ONS
N = I0,1,2,
R
=
[PI
N
u
= p1 +
...... } {m}
...
P G q * Pi 4 q i
+ pnY where p = (pl, Y 1
iG n
...,p,)
where P
xix
E
N- n
= (pl
,... y p n ) y q
=
( 4 ly...q,)E -n N
This Page Intentionally Left Blank
CHAPTER 1 SEQUENTIAL SOLUTIONS OF NONLINEAR PDEs
0.
Introduction
The a i m of t h e p r e s e n t c h a p t e r i s twofold, v i z t o s e t u and m o t i v a t e t h e general framework used i n t h e sequel i n t h e s t u d y & u e n t i m a r t i = c u l a r weak o r d i s t r i b u t i o n s o l u t i o n s o f polynomial n o n l i n e a r PDEs. I n t h e f o l l o w i n g chapters, t h a t framework w i l l u s u a l l y be r e s t r i c t e d t o chains o f algebras containing the d i s t r i b u t i o n s
D' c Amc
... c
Ap c
... c A" ,
p E
8;
however, t o understand f u l l y t h e phenomena i n v o l v e d i t i s a d v i s a b l e t o c o n s i d e r t h e framework i n i t s f u l l g e n e r a l i t y . The s e q u e n t i a l s o l u t i o n s w i l l be elements o f v e c t o r spaces o r algebras o f ' g e n e r a l i z e d f u n c t i o n s ' which a r e c l a s s e s o f sequences o f f u n c t i o n s on domains i n Euclidean spaces. There a r e a t l e a s t two e q u a l l y i m p o r t a n t reasons f o r c o n s t r u c t i n g v a r i o u s spaces o f ' g e n e r a l i z e d f u n c t i o n s ' . The f i r s t , as mentioned i n t h e P r e l i = m i n a r i e s , i s connected w i t h t h e way p a r t i a l d e r i v a t i v e o p e r a t o r s can be d e f i n e d f o r ' g e n e r a l i z e d f u n c t i o n s ' . The second reason, which w i l l be t h e f i r s t t o become atmarent as t h i s c h a p t e r i s read, i s t h e i n t e r p l a y between the s t a b i l i t e n e r a l i t and exactness p r o p e r t i e s o f s e q u e n t i a l i o l u t i o n s of p ~ ' n E s . Given t h e n e c e s s i t y o f d e a l i n g w i t h v a r i o u s spaces o f ' g e n e r a l i z e d func= t i o n s ' , t h i s c h a p t e r p r e s e n t s t h e b a s i c a l g e b r a i c ideas i n v o l v e d i n t h e c o n s t r u c t i o n s encountered w i t h i n t h e n e x t chapters. General a l g e b r a i c r e s u l t s on t h e s t a b i l i t y , r e s o l u t i o n o f s i n g u l a r i t i e s and r e g u l a r i t y o f s e q u e n t i a l s o l u t i o n s f o r polynomial n o n l i n e a r PDEs w i l l be presented. I n l a t e r c h a p t e r s some o f t h e s e r e s u l t s w i l l be strengthened w i t h t h e h e l p o f a d d i t i o n a l a n a l y t i c o r t o p o l o g i c a l methods a p p l i e d t o v a r i o u s p a r t i c u = l a r cases o f PDEs. T h i s c h a p t e r a c t u a l l y p r e s e n t s a 'blow u p ' o f t h e e s s e n t i a l a l g e b r a i c phenomena i n v o l v e d i n t h e s t u d y o f s e q u e n t i a l s o l u t i o n s o f polynomial non= l i n e a r PDEs, p o i n t i n g o u t
-
t h e wa p a r t i a l d e r i v a t i v e s can be d e f i n e d on a l g e b r a s o f ' g e n e r a l i z e d i ions' func?
as w e l l as
1
2
E.E.
-
Rosinger
+
t h e i n t e r l a between t h e s t a b i l i t y , g e n e r a l i t y and exactness proper= t i e s o sequential solutions.
The d i s c u s s i o n amounts t o a reassessment o f t h e balance between a l g e b r a and t o p o l o g y i n t h e s t u d y o f PDEs, s t r e s s b e i n g placed on t h e r i g h t stage a t which t h e t o p o l o g y s h o u l d be i n t r o d u c e d , and i t s c o r r e c t l e v e l . Indeed, i t becomes obvious t h a t t o o e a r l y a c o n s i d e r a t i o n o f t o p o l o g y can p r e c l u d e e s s e n t i a l a l g e b r a i c o p t i o n s r e l a t e d t o t h e above-mentioned i n t e r p l a y . Moreover, t h e c h o i c e o f t h e r i g h t t o p o l o g i c a l s t r u c t u r e s m i g h t prove t o be p a r t i c u l a r l y i m p o r t a n t .
A somewhat l e s s r e l a t e d example i s o f f e r e d i n t h e Appendix 3, where t h e Lax-Richtmyer equivalence between t h e s t a b i l i t y and convergence of d i f f e r = ence schemes f o r l i n e a r e v o l u t i o n equations i s extended t o t h e correspon= d i n g n o n l i n e a r case b y n o t i c i n g t h a t s t a b i l i t y as w e l l as convergence a r e p r o p e r t i e s which do n o t n e c e s s a r i l y r e q u i r e t h e completeness o f t h e u n d e r l y i n g t o p o l o g i c a l s t r u c t u r e , and t h a t t h e r e f o r e t h e i n i t i a l p r o o f i n t h e l i n e a r case, based on t h e u n i f o r m boundedness p r i n c i p l e o f l i n e a r o p e r a t o r s i n Banach spaces, can be r e p l a c e d b y a s i m p l e r , r a t h e r t r i v i a l one, o n l y i n v o l v i n g c o n t i n u i t y , compactness and boundedness i n normed v e c t o r spaces , a p r o o f t h a t w i l l be v a l i d i n t h e n o n l i n e a r case as w e l l . 1.
The Problem o f S t a b i l i t y o f Sequential S o l u t i o n s
The p r e s e n t work w i l l deal m a i n l y w i t h t h e c l a s s o f p o l y n o m i a l n o n l i n e a r PDEs :
(1)
z:
T-r-
Ci(X)
1 Q i G h
where u : R + R ' i s t h e unknown f E C o ( Q ) a r e given; and ci, partial derivatives.
(2)
Dpij
U(X) = f ( x )
,
x E
R
l G j G k i u n c t i o n , w h i l e R c Rn i s non-void, open , w i t h pij E Nn , denote t h e usdal
,b,j
-
The o r d e r o f t h e PDE i n (1) i s
m=max{lpij]
1 1 G i G h
,
l Q j < k i l
p r o v i d e d t h a t none o f t h e c o e f f i c i e n t s ci vanishes everywhere on R. I t w i l l be convenient t o a s s o c i a t e w i t h t h e PDE i n ( l ) , t h e corresponding PDO
(3)
T(D) =
Z
ci(x)
I G i G h
Dpij
1 G J Gki
which o b v i o u s l y generates a mapping
(4)
T ( D ) : Cm(R) * c" (a).
A c l a s s i c a l s o l u t i o n o f (1) i s any f u n c t i o n
(5)
JI
E
cm(Q)
f o r which t h e mapping (4) s a t i s f i e s
,
X E R
SEQUENTIAL SOLUTIONS
(6 1
3
T(D)$ = f on R.
I n case one i s i n t e r e s t e d i n s o l u t i o n s o f (1) which a r e n o t s u f f i c i e n t l y smooth a l l o v e r R i n o r d e r t o be c l a s s i c a l s o l u t i o n s , a usual procedure i s t o c o n s t r u c t weak o r i n general, s e q u e n t i a l s o l u t i o n s g i v e n b y s u i t a b l e sequences o f f u n Z Z n s , f o r i n s t a n c e (7)
s =
... , $,
($0,+1,
...), w i t h $,
E
Cm(Q), f o r v
E
N.
The meaning a s s o c i a t e d w i t h a s e q u e n t i a l s o l u t i o n ( 7 ) i s t h a t i t 'conver= ges' t o an ' i d e a l ' s o l u t i o n S o f ( l ) , supposed t o b e l o n g t o a c e r t a i n ' c o m p l e t i o n ' o f a s u i t a b l e f u n c t i o n space on n. F o r i n s t a n c e , i n case s i s a usual weak s o l u t i o n , t h e n S i s a d i s t r i b u t i o n . When c o n s t r u c t i n g s e q u e n t i a l s o l u t i o n s ( 7 ) o f ( l ) , t h e c o r r e c t and f u l l i n f o r m a t i o n on t h e ' i d e a l ' s o l u t i o n S o b v i o u s l y r e q u i r e s a knowledge o f a l l t h e sequences o f f u n c t i o n s t = s + v =
(8)
+
XI,
( X O Y
= ($0
+
($0,
$1,
..., $,,
... x,, . . . I
xo,
Y
$1
-t x 1 ,
...) t
=
. . . I
$,
+
x,,
..
with (8.1)
v = (XO,
x1,
... , X Z , .. . ) , where
xvE
which sequences t a l s o 'converge' t o t h e same ' i d e a l ' s o l u t i o n S . I n o t h e r words, t h e s i g n i f i c a n c e o f a weak o r s e q u e n t i a l s o l u t i o n o f (1) g i v e n b y a s i n l e sequence of f u n c t i o n s ( 7 ) can p r o p e r l y be a p p r e c i a t e d o n l y i f i t s s t a i l i t y p r o p e r t y (8) i s known as w e l l .
-9b
I n t h e case o f a l i n e a r PDE w i t h Cm-smooth c o e f f i c i e n t s and t h e sequen= t i a l s o l u t i o n (7)?EFGiny a d i s t r i b u t i o n S E U'(R), t h e s t a b i l i t y p r o p e r = t y (8) can e a s i l y be e s t a b l i s h e d . Indeed, owing t o t h e l i n e a r i t y and c o n t i n u i t y o f t h e corresponding PDO, one can t a k e i n (8.1) any sequence o f f u n c t i o n s weakly convergent t o z e r o i n U'(R). However, i n t h e case o f a n o n l i n e a r PDE, t h e problem o f t h e s t a b i l i t y o f a s e q u e n t i a l s o l u t i o n i s no l o n g e r t r i v i a l and i t s s t u d y i s e s s e n t i a l , even i f i t i s q u i t e o f t e n overlooked. Indeed, t h e p r o p e r t i e s o f a sequen= t i a l s o l u t i o n o b t a i n e d b y a p a r t i c u l a r c o n s t r u c t i o n o f a s i n l e sequence o f f u n c t i o n s ( 7 ) , c o u l d happen t o be a r a t h e r i r r e l e v a n t a c c i e n t , a r e s u l t o f t h e d i s c o n t i n u i t o f t h e c o r r e s p o n d i n g n o n l i n e a r PDO on a c e r = t a i n space o f i s r i u i o n s .
+
d
The f o l l o w i n g v e r y s i m p l e example i l l u s t r a t e s t h e above-mentioned p o i n t . Consider i n t h e one-dimensional case, ( n = l ) , w i t h R = R', t h e p o l y n o m i a l n o n l i n e a r POE o f o r d e r m=O:
(9)
( ~ ( x ) ) '= 1
,
x E R'.
A d i r e c t computation w i l l show t h a t t h e sequence of f u n c t i o n s (10)
s =
($0,
$1,
...,
$"Y
...I
4
E.E.
Rosinger
with (10.1)
$,(x)
=
&?-' cos vx,
for
v
E N,
x
E
R',
g i v e s a usual weak s o l u t i o n o f ( 9 ) - i . e . t h e f u n c t i o n s ( $ ) 2 converge i n However, t h e sequence o f f u n c t i a n s (10) i s a l s o P ' ( R ' ) t o 1, when v m. a usual weak s o l u t i o n o f t h e e q u a t i o n -f
(11)
U ( X ) = 0,
x
E
R'.
As seen i n t h e l i t e r a t u r e , [ 1 3 5 , 160, 96, 126, 1251 weak s o l u t i o n s f o r
n o n l i n e a r PDEs a r e o f t e n o b t a i n e d as l i m i t s - i n s u i t a b l e d i s t r i b u t i o n spaces - o f sequences o f f u n c t i o n s ( 7 ) , whose terms qv s a t i s f y l i n e a r i z e d o r r e g u l a r i z e d v e r s i o n s o f t h e i n i t i a l n o n l i n e a r PDEs. U s u a l l y , o n l y p a r t i c u l a r sequences o f f u n c t i o n s ( 7 ) a r e e x h i b i t e d , and t h e y r e s u l t from m o n o t o n i c i t y , f i x e d p o i n t o r compactness arguments. The l a s t - m e n t i o n e d case appears t o be most i n need o f subsequent s t u d y i n o r d e r t o determine t h e s t a b i l i t y p r o p e r t i e s o f t h e s e q u e n t i a l s o l u t i o n s presented. The non-uniqueness o f weak s o l u t i o n s s a t i s f y i n g n o n l i n e a r PDEs t o g e t h e r w i t h t h e usual i n i t i a l o r boundary c o n d i t i o n s and o b t a i n e d as d i s t r i b u t i o n a l l i m i t s o f p a r t i c u l a r l y c o n s t r u c t e d sequences o f f u n c t i o n s ( 7 ) may a l s o r e q u i r e a s t a b i l i t y a n a l y s i s i n t h e above-mentioned sense. I t follows t h a t specifying a sequential s o l u t i o n (7) together w i t h i t s s t a b i l i t y p r o p e r t y ( 8 ) , means t o g i v e a c l a s s o f sequences o f f u n c t i o n s i . e . an element
S = s t U = { t ( t = s t v , v E U} i n a q u o t i e n t space (12)
E
(then s
S/U
E
S, S E E )
where U c S a r e s u i t a b l e v e c t o r spaces o f sequences o f f u n c t i o n s on Q . v a r i o u s i n s t a n c e s , i t w i l l be c o n v e n i c n t t o deal w i t h p a r t i c u l a r cases o f t h e q u o t i e n t spaces (12), g i v e n by q u o t i e n t a l g e b r a s (13)
A
In
= A/l
where A i s an a l g e b r a o f sequences o f f u n c t i o n s on R, and 1 i s an i d e a l i n A. The most general space o f f u n c t i o n s considered i n t h e sequel i s t h e alge= b r a M(R) o f a l l r e a l - v a l u e d , mefisurable and a.e. f i n i t e f u n c t i o n s on R. T f o l l o w s t h a t t h e space (M(R)) o f a l l sequences o f f u n c t i o n s i n M(R), i s i n a n a t u r a l way an a s s o c i a t i v e and commutative algebra, when consider= e d w i t h t h e term-by-term o p e r a t i o n s on sequences of f u n c t i o n s . The most general cases o f q u o t i e n t spaces ( 1 2 ) and q u o t i e n t a l g e b r a s (13) However, i t w i l l be u s e f u l t o deal a l s o w i t h t h e subalgebras
w i l l be o b t a i n e d when V , S , I , A c(M(R))~.
R
E
w
of (M(n))N, and r e s t r i c t U ,
S,
I,
(C"Q))N
,
A i n (12) and (13) a c c o r d i n g l y .
5
SEQUENTIAL SOLUTIONS
One o f t h e b a s i c advantages o f d e f i n i n g t h e ' g e n e r a l i z e d f u n c t i o n s ' as e l e = ments o f t h e q u o t i e n t spaces (12) o r q u o t i e n t algebras ( 1 3 ) i s t h a t i t o f f e r s t h e p o s s i b i l i t y o f a d i r e c t and e x p l i c i t s t a b i l i t y a n a l y s i s o f t h e s e q u e n t i a l s o l u t i o n s o f polyE%%T n o n l i n e a r PDEs. 2.
D e f i n i t i o n o f t h e Q u o t i e n t Spaces and Algebras
As a l r e a d y mentioned, and w i l l now become e v i d e n t , t h e r e a r e several rea= sons f o r c o n s t r u c t i n g v a r i o u s q u o t i e n t spaces ( 1 2 ) and q u o t i e n t a l g e b r a s (13) when d e a l i n g w i t h s e q u e n t i a l s o l u t i o n s o f PDEs.
+
One o f t h e main reasons i s t h e s p e c i f i c i n t e r l a , i n t h e case o f each p a r t i c u l a r problem, between t h e c o n f l i c t i n g s t a i l i t y , g e n e r a l i t y and exactness p r o p e r t i e s o f t h e s e q u e n t i a l s o l u t i o n s i n v o l v e d . Indeed, t h e s t a b i l i t y p r o p e r t y o f s e q u e n t i a l s o l u t i o n s r e s u l t s i n an i n = t e r e s t i n q u o t i e n t spaces ( 1 2 ) and q u o t i e n t a l g e b r a s (13), h a v i n g v e c t o r spaces V , o r e l s e i d e a l s 7 , t h a t a r e l a r g e , p o s s i b l y maximal w i t h i n t h e a d d i t i o n a l c o n d i t i o n s t h e y w i l l have t o s a t i s f y . On t h e o t h e r hand, i n o r d e r t o i n c l u d e among t h e s e q u e n t i a l s o l u t i o n s a l l t h e c l a s s i c a l , weak o r d i s t r i b u t i o n s o l u t i o n s , t h u s s e c u r i n g t h e g e n e r a l i t y o f t h e n o t i o n o f s e q u e n t i a l s o l u t i o n , one i s i n t e r e s t e d i n l a r e q u o t i e n t spaces (12) and q u o t i e n t algebras (13). C l a s s i c a l s o l u t i o n s w i 1 ob= v i o u s l y be i n c l u d e d among t h e s e q u e n t i a l s o l u t i o n s , i n case embedding c o n d i t i o n s o f t h e f o l l o w i n g form a r e r e q u i r e d :
+
F c E
(14)
= S/V
or else
(15)
G
C
A = A/Z
where F and G a r e s u i t a b l e v e c t o r subspaces, o r e l s e subalgebras i n M ( R ) . I n o r d e r t o i n c l u d e t h e weak o r d i s t r i b u t i o n s o l u t i o n s among t h e sequen= t i a l s o l u t i o n s , t h e f o l l o w i n g embedding c o n d i t i o n s w i l l be r e q u i r e d :
(16)
D'(Q) c
(17)
U'(n) c A
E = S/V = A/I.
Obviously, t h e above requirements t h a t b o t h V and E i n (12), o r e l s e b o t h Z and A i n ( 1 3 ) s h o u l d be l a r g e , a r e t o a c e r t a i n e x t e n t c o n f l i c t i n g . The impact o f t h e exactness p r o p e r t y o f s e q u e n t i a l s o l u t i o n s w i l l be presented i n S e c t i o n 4. Meanwhile, i t i s u s e f u l t o r e w r i t e t h e above embeddings (14) and (15) under t h e form o f t h e i n j e c t i v e mappings F 3 +
(18)
-> u($)
+
V
E
E = S/V
o r else
(19)
G
3
JI >-
u($) t
rE
A =
AIL
6
Rosinger
E.E.
where for each $ u($)
E
M(R) we denote
...
= ($,$,
y
$,
.
.
.
by .
I
.
.
1
the sequence of functions with a l l the terms equal t o the function $. Obviously
u
=
I
tu(Ji)
E
M(R)I
i s a subalgebra in ( M ( Q ) ) N , which through the mapping $ u($) i s isomor= phic with M ( i 2 ) . Moreover, Q = { u ( o ) } i s the null ideal, while u(1) i s the unit element in (M(R))N. +
Now, i t i s easy t o see t h a t (18) i s equivalent t o the existence of the inclusion diagram:
satisfying (20.1)
V n UF =
Q
and (19) to:
2 -' ' G satisfying (21.1)
I n
uG = Q
where we used the notation:
'F = {u($)
I
$
F 1
and similarly for UG.
I t i s worth mentioning t h a t , even i f n o t in an explicit way, conditions of type (20.1), (21.1) on inclusion diagrams (20), (21) were used in [2161 connected with the 'neutrix calculus' developed i n i t i a l l y in order to simp1 ify methods i n asymptotical expansions. With the terminology in 12161, the sequences of functions i n V o r I are 'negligible'. In t h i s sense the meaning of (20.1) for instance, i s that the difference JI -x, w i t h $,x E F, i s not 'negligible' unless $ = x. Therefore the quotient space E = S/V 'distinguishes' between different functions in F. Similar= ly for the condition ( 2 1 . 1 ) . More details concerning the 'neutrix calcu= lus' can be found in Appendix 1. I t i s specially important t o point out that in the case of inclusion dia=
SEQUENTIAL SOLUTIONS
7
grams ( 2 1 ) , t h e r a t h e r simple condition ( 2 1 . 1 ) possesses a p a r t i c u l a r l y g r e a t power. Indeed, as seen l a t e r in Chapter 6 , f o r l a r g e and important c l a s s e s o f inclusion diagrams ( 2 1 ) , the condition (21.1) means t h a t the ' n e g l i g i b l e ' sequences of functions w E 7 have t o vanish densely on R , i.e. the sets
R
W
=
{x
E
R
I
3v
N : w v ( x ) = 03
E
a r e dense in R , where wv, with w E N , denotes t h e functions t h a t a r e t h e terms o f the sequence o f functions w. In the sequel, i t will be convenient s l i g h t l y t o r e s t r i c t t h e inclusion diagrams ( 2 0 ) and ( 2 1 ) , i . e . the q u o t i e n t spaces ( 1 2 ) and q u o t i e n t alge= bras ( 1 3 ) , which give t h e spaces of 'generalized f u n c t i o n s ' . Suppose given a vector subspace F C M ( R ) and a subalgebra G C M ( R ) . We s h a l l denote by
rs
the e t o f a l l q u o t i e n t spaces E in F and
v->
= S/V,
where V and S a r e vector subspaces
s
i
with
V n
UF
= O_.
W e s h a l l denote by ALG
the s e t o f a l l q u o t i e n t algebras A I i s an ideal i n A and I->
=
A / 7 , where A i s a subalgebra in G
N
,
A
i
with
7 n UG =
!!
Obviously F = UF/Q
E
VSF
G
= UG/Q E
ALG
The space of d i s t r i b u t i o n s D'(n) w i l l i n t h e sequel be t h e u o t i e n t space o f b a s i c i n t e r e s t . I t w i l l be convenient t o use several equiva ent con= s t r u c t i o n s giving D'(n). To i l l u s t r a t e , given a vector subspace F c M ( Q ) , denote by SF the s e t of a l l sequences s E FN of measurable functions weak= l y convergent in D'(n), and by V F t h e s e t of a l l sequences V E S F weakly convergent t o zero i n D ' ( n ) , i . e . c o n s t i t u t i n g the kernel of the l i n e a r mapping
s7____
(24.1)
SF 3 5
> G , .>E
D'(Q)
E.E. Rosinger
8
where (24.2)
,..., +", ..... ) .
assuming t h a t s = ($o, J1l
-
Obviously, i n case F i s l a r e enough i.e. s e q u e n t i a l l y dense i n V(Q)f o r instance, c o n t a i n s a e polynomials, o r c o n t a i n s qQ),e t c . , t h e mapping (24) i s a l i n e a r s u r j e c t i o n , t h e r e f o r e t h e mapping
rrft
(25)
SF/WF 3
+
(5
< S,'
W F ) >-
w i l l be a v e c t o r space isomorphism. spaces (26)
D'(Q2) =
>E
P'(Q)
I n t h a t way, we o b t a i n t h e q u o t i e n t
S F / V F E VSF
p r o v i d e d t h a t F c M(R) i s s u f f i c i e n t l y l a r g e , as mentioned above. The cases o f s p e c i a l i n t e r e s t w i l l be those where F = CR (n), w i t h R E fl given. I n t h e n e x t .chapters, we f u n c t i o n s ' from Al. , i . e . c o n s t r u c t them i n t u c h a r e s u l t i n g from (23) t h e we s h a l l have G = $(a),
s h a l l deal o n l y with spaces o f ' g e n e r a l i z e d w i t h q u o t i e n t algebras A = MI, and we s h a l l way t h a t a p a r t from t h e embedding c o n d i t i o n (15) embedding c o n d i t i o n ( 1 7 ) w i l l a l s o h o l d . U s u a l l y , witha. E s u i t a b l y chosen.
w
A second main, reason f o r c o n s t r u c t i n g v a r i o u s q u o t i e n t spaces and algebras when d e a l i n g w i t h s e q u e n t i a l s o l u t i o n s o f PDEs i s t h e way a r t i a l d e r i v a = t i v e o p e r a t o r s can be d e f i n e d on algebras c o n t a i n i n g t h e E,i t i s i m p o r t a n t t h a t i t s h o u l d n o t be assumed t h a t each o f t h e alge= b r a s c o n t a i n i n g t h e d i s t r i b u t i o n s i s i n v a r i a n t under a r b i t r a r y p a r t i a l d e r i v a t i o n , even i f some o f them w i l l be. I n o t h e r words, t h e p a r t i a l d e r i v a t i v e o p e r a t o r s s h o u l d be a1 lowed t o a c t between d i f f e r e n t algebras. Moreover, t h e e x i s t e n c e o f p a r t i a l d e r i v a t i v e s s h o u l d b e assumed o n l y 9 t o a c e r t a i n order.
I n t h i s connection, we s h a l l d e f i n e t h e p a r t i a l d e r i v a t i v e o p e r a t o r s w i t h = i n t h e s l i g h t l y more general framework. (27)
Dp :
E
-+
A
,
p E N',
IpI Q R
where E = S/W E VSF, A = A / l E AL a r e s u i t a b l y chosen and R E fl may depend on E and A. As seen i n S e c t i g n 11, t h e assumption t h a t E = A and II 2 1 i n ( 2 7 ) , which a u t o m a t i c a l l y i m p l i e s t h a t R = m, may under r a t h e r general c o n d i t i o n s l e a d t o a somewhat p a r t i c u l a r m u l t i p l i c a t i o n i n t h e a l g e b r a A, g i v i n g f o r i n s t a n c e
62 = 6.D6 = 0 where 6 i s t h e D i r a c d i s t r i b u t i o n . I f i t i s r e q u i r e d t h a t t h e p a r t i a l d e r i v a t i v e o p e r a t o r s i n ( 2 7 ) should t a k e values i n q u o t i e n t algebras, t h i s requirement w i l l p r o v i d e t h e oppor= t u n i t y o f e x t e n d i n g t h e mappings ( 4 ) generated b y t h e {olynomial n o n l i n e a r PDO i n ( 3 ) t o mappings T(D) : E + A between spaces o f g e n e r a l i z e d func= t i o n s ' , as w i l l be seen i n S e c t i o n 3. I n t h a t way a p r o p e r framework i s
SEQUENTIAL SOLUTIONS
9
o b t a i n e d f o r f i n d i n g s e q u e n t i a l s o l u t i o n s f o r polynomial n o n l i n e a r PDEs. The q u o t i e n t s t r u c t u r e o f E and A i n ( 2 7 ) suggests d e f i n i n g t h e p a r t i a l d e r i v a t i v e o p e r a t o r s as t h e term-by-term p a r t i a l d e r i v a t i v e s of sequences o f f u n c t i o n s . F o r instance, i n t h e case
sc
(C"(n))N
one can d e f i n e
Dp : E
-f
A,
p
Nn,
E
Ip
by
provided t h a t DPV
C
I,
DPS C A ,
Vp E Nn,
I p I < R.
However, i t i s p o s s i b l e t o proceed i n a l e s s r e s t r i c t i v e manner. assume f i r s t t h a t F, G E M ( Q ) s a t i s f y (28)
DP(F n
cR(n))c
Obviously, (28) h o l d s whenever G Given now E = S / V
E
Vp E Nn,
G,
Indeed,
IpI G R .
3 C"((n).
VSF and A = A/I E ALG denote R E < A
vs; t h e subset o f a l l q u o t i e n t spaces E = S / V (30)
E
s c V + (C"(Q))N.
Obviously, (30) h o l d s whenever F c CR(n). see ( 2 6 ) - t h a t (31)
VSF which s a t i s f y
D'(Q)
=
sR/ V R
E VSRR
c
(Q)
, VR
It f o l l o w s i n p a r t i c u l a r
-
E Fj
where f o r t h e sake o f s i m p l i c i t y , we used t h e n o t a t i o n : (31.1)
f
NOW, t h e p a r t i a l d e r i v a t i v e o p e r a t o r s ( 2 7 ) can be de i n e d as f o l l o w s . Given E = S / V E VSf and A = A/IEALG, such t h a t E Q A, d e f i n e f o r P E NnY
IPI G9.
E . E . Rosinger
10
(32)
D P : E + A
by means of the r e l a t i o n (32.1)
Dp(s
DPt
t V) =
t
I,
V s
s-t
E
E
S
where
t
(32.2)
E
s
n (CR(Q))N,
V.
k
I t i s easy t o see t h a t t h e p a r t i a l d e r i v a t i v e operators ( 3 2 ) a r e we 1 defined, extend the usual p a r t i a l d e r i v a t i v e s o f functions i n F n C (Q), and a r e l i n e a r . Furthermore, in case F i s a subalgebra i n M(n) and E i s a quotient algebra from ALF, t h e p a r t i a l d e r i v a t i v e operators ( 3 2 ) s a t i s f y the Leibnitz r u l e f o r product d e r i v a t i v e s . 3.
The Notion of Sequential Solution
I t follows e a s i l y from t h e previous section t h a t t h e mapping ( 4 ) generated by the polynomial nonlinear PDO in ( 3 ) can be extended t o mappings between spaces of 'generalized f u n c t i o n s '
T(D) : E
(33)
-+
A
m
whenever E = S.:V E VSF , A = A/ZEAL and E G - see ( 2 ) - of the PDi) in ( 3 ) , while % a n d G a r e subspace and a subalgebra i n M(R) such t h a t ( 2 8 ) c o e f f i c i e n t s c i a s well a s t h e right-hand term f l o n g to G.
where m i s t h e order respectively a vector is s a t i s f i e d and t h e i n the PDE i n ( l )be= y
A,
Indeed, under the above assumptions, t h e mapping (33) can be defined by (33.1)
T(D)(s
t V) =
T(D)t
t I,
WS E S
where
t
(33.2)
E
s
n (C"'(R))~,
s-t
E
v
while T ( D ) t in (33.1) i s t h e term-by-term application of the mapping ( 4 ) t o the sequence o f Cm-smooth functions t . The f a c t t h a t t h e mapping (33) i s well-defined r e s u l t s from the following obvious property of t h e polynomial nonlinear PDO i n ( 3 ) :
T(D)(ttv)
(34)
=
T ( D ) t + E t,-Dpav,
Vt,v E ( C ~ ( ~ ) ) ~
a
P where t, are products o f c i , D j t and possibly D p i j v y while p, are some of pi j. Indeed, owing t o (34) , one obtains t h e implications
s n ( c m ( n ) ) N ,v
t
E
-
T(D)t
E
I
E
v
n
(ttv) -
SEQUENTIAL SOLUTIONS
11
m
since E
<
A.
We can now proceed t o d e f i n e t h e n o t i o n o f a s e q u e n t i a l s o l u t i o n o f a polynomi a1 nonl in e a r PDE .
+
Suppose g i v e n t h e m-th o r d e r polynomial n o n l i n e a r PDE i n ( 1 ) and a sequence s E (M(R))N o f measurable f u n c t i o n s on R. Then s i s c a l l e d a se u e n t i a l s o l u t i o n o f ( l ) , o n l y i f f o r a c e r t a i n v e c t o r subspace F and su a ge r a G(Q) s a t i s f y i n g ( 2 8 ) and w i t h G c o n t a i n i n g t h e c o e f f i c i e n t s c . and t h e r i g h t - h a n d t e r m f i n ( l ) , t h e r e e x i s t a quotAent space E = s / v kVSF and a q u o t i e n t a l g e b r a A = A / l E ALG , w i t h E < A, such t h a t (35.1)
+
(then S = s
S E S
V
E
E =
s/V)
and t h e mapping ( 3 3 ) maps S E E i n t o f E A, i . e . (35.2)
T(D)(s + V ) = u ( f )
+
1
I n view o f ( 3 4 ) and (30), t h e c o n d i t i o n ( 3 5 ) i s o b v i o u s l y e q u i v a l e n t t o
(36) s-t
T(D)t-u(f)
V,
E
E
I
which i s t h e same as t h e f o l l o w i n g two: (37.1)
S E S
and
v
t E
s
n ( C ~ ( R ) :) ~
V
* T(D)t-u(f)
(37.2) s-t
E
E
I.
When i t i s u s e f u l t o emphasize t h e q u o t i e n t space E and q u o t i e n t a l g e b r a
A i n t h e above e q u i v a l e n t c o n d i t i o n s ( 3 5 ) , ( 3 6 ) o r ( 3 7 ) , we s h a l l c a l l s a s e q u e n t i a l s o l u t i o n i n E + A o f t h e polynomial n o n l i n e a r PDE i n (1). Obviously, t h e above d e f i n i t i o n o f a s e q u e n t i a l s o l u t i o n can be extended t o t h e case when t h e r i g h t - h a n d t e r m i n t h e PDE under (1) i s no l o n g e r a continuous f u n c t i o n b u t a ' g e n e r a l i z e d f u n c t i o n ' b e l o n g i n g t o a c e r t a i n q u o t i e n t a l g e b r a . Such cases w i l l , however, n o t be discussed i n t h e p r e s e n t work. Remark 1 ( a ) Suppose s (1) and
E
(M(C?))N i s a s e q u e n t i a l s o l u t i o n i n E
E = S/V then t h e element S = s
+
,A
=
-+
A o f t h e PDE i n
A/I;
V o f t h e q u o t i e n t space E = S / V w i l l s a t i s f y t h e
12
E.E.
Rosinger
polynomial n o n l i n e a r PDE i n ( l ) , i n t h e usual a l g e b r a i c sense, w i t h t h e m u l t i p l i c a t i o n i n t h e q u o t i e n t a l g e b r a A1, assuming t h e p a r t i a l d e r i = v a t i v e o p e r a t o r s were d e f i n e d a c c o r d i n g t o ( 3 2 ) . I n o t h e r words, t h e s o l u t i o n S i s an element o f t h e q u o t i e n t space E, w h i l e t h e PDE i s s a t i s = f i e d i n t h e q u o t i e n t algebra A. ( b ) Obviously, t h e g i v e n s e q u e n t i a l s o l u t i o n s i n E A, can a l s o be a s e q u e n t i a l s o l u t i o n i n v a r i o u s o t h e r E l + A ' , p r o v i d e d t h e q u o t i e n t space E ' and q u o t i e n t a l g e b r a A ' a r e s u i t a b l y chosen. That f a c t , as seen i n S e c t i o n 5, may p r e s e n t an e s s e n t i a l advantage when c o n s i d e r a t i o n i s g i v e n t o i m p r o v i n g t h e s t a b i l i t y , g e n e r a l i t y and exactness (see S e c t i o n 4 ) o f t h e given s e q u e n t i a l s o l u t i o n . -f
4.
Exactness o f S e q u e n t i a l S o l u t i o n s
One o f t h e main advantages o f t h e f a c t t h a t t h e mappings (33) a s s o c i a t e d w i t h t h e PDO i n ( 3 ) t a k e values - even i n t h e p a r t i c u l a r case o f a l i n e a r PDO - i n q u o t i e n t algebras, i . e . t h a t t h e PDE i n (1) i s s o l v e d i n q u o t i e n t algebras, can be seen i n connection w i t h t h e exactness p r o p e r t y o f sequential solutions, defined i n t h i s section. I n t h i s c o n n e c t i o n we f i r s t n o t i c e t h a t t e PDE i n (1) has a c l a s s i c a l s o l u t i o n o n l y i f t h e r e e x i s t s t E (Cm(n))R, such t h a t (38)
wt = T ( D ) t - u ( f )
E
Q.
I n t h e sense o f t h e s e q u e n t i a l s o l u t i o n s d e f i n e d i n S e c t i o n 3, t h e exten= s i o n o f (38) takes t h e f o l l o w i n g form. The sequence of f u n c t i o n s s E ( M ( R ) ) N i s a s e q u e n t i a l s o l u t i o n o f t h e PDE i n ( l ) , o n l y i f (39)
wt = T ( D ) t
-
u(f)
E
I,V t
E
S n (Cm(n))N, s - t
E
V
where n o t a t i o n s corresponding t o t h o s e i n ( 3 ) have been used. Now, t h e e s s e n t i a l f e a t u r e o f t h e c o n d i t i o n 39) i s t h a t - as I i s an i d e a l i n A - n o t o n l y does w t e n t e r t h e z e r o c l a s s i n t h e q u o t i e n t a l g e b r a A/I, b u t so does each p r o d u c t w t . 2 , w i t h z E A, i . e .
(40)
w t . ~c I,
v
t
E
s n(cm(n))N,
s-t
E
v.
The c o n d i t i o n (40) i s an e x p l i c i t a l g e b r a i c t e s t on t h e e r r o r w caused b y t h e n o n - c l a s s i c a l , s e q u e n t i a l s o l u t i o n t when substit-info equation (1). That t e s t , besides demanding t h e ' I - n e g l i g i b i l i t y ' - see S e c t i o n 2 o f t h e e r r o r w t , demands a l s o t h e ' I - n e g l i g i b i l i t y ' o f each o f i t s ' p r o = j e c t i o n s ' w t . 2 , w i t h z E A. We n o t i c e t h a t (40) implie-g), i n case u ( 1 ) E A . Obviously, t h e c o n d i t i o n (40) i s a good e x t e n s i o n o f (38), p r o v i d e d t h a t 7 i s small and A i s l a r g e . Thus, c o n d i t i o n ( 3 8 ) corresponds t o t h e b e s t case, w m = Q and A.We s h a l l c a l l (40) t h e exactness p r o p e r t y o f t h e s e q u e n t i a l s o l u t i o n s o f t h e polynomial n o n l i n e a r P D t i n (1).
As mentioned i n S e c t i o n 2 and w i l l become e v i d e n t l a t e r i n Chapter 6, under r a t h e r general c o n d i t i o n s t h e ' I - n e g l ig i b i 1it y ' w i 11 mean a t 1e a s t t h e v a n i s h i n g o f t h e sequences o f f u n c t i o n s w E 7 on dense subsets i n n.
13
SEQUENTIAL SOLUTIONS
Any f u r t h e r a l g e b r a i c o r t o p o l o g i c a l c o n s i d e r a t i o n s taken i n t o account when c o n s t r u c t i n g t h e q u o t i e n t a l g e b r a s A = A / I i n which t h e PDE i n (1) i s s o l = ved, can be seen as a t t e m p t s t o i m rove t h e exactness p r o p e r t i e s o f t h e s e q u e n t i a l s o l u t i o n s b y sharpening t e ' I - n e g l i g i b i l i t y ' through t h e enlargement o f t h e dense subsets i n R on which t h e sequences o f f u n c t i o n s w E I vanish.
+
5.
I n t e r p l ay between S t a b i 1 it y , General it y and Exactness
R e c a l l i n a now t h e corresoondinq D a r t o f S e c t i o n 2 , we a r e i n a b e t t e r t)osi= t i o n t o g p e c i f y t h e i n t e k p l a y i e t w e e n t h e s t a b i l i t y , g e n e r a l i t y and e x a c t = ness p r o p e r t i e s o f s e q u e n t i a l s o l u t i o n s .
-
N . Suppose t h a t s E ( M ( R ) ) i s a s e q u e n t i a l s o l u t i o n o f t h e polynomial n o n l i = n e a r PDE i n (1) which s a t i s f i e s any o f t h e e q u i v a l e n t c o n d i t i o n s ( 3 5 ) , ( 3 6 ) o r ( 3 7 ) , f o r a c e r t a i n v e c t o r subspace F and subalgebra G i n M ( R ) , as w e l l as f o r a s u i t a b l e q u o t i e n t space E = I / V E V S p and q u o t i e n t a l g e b r a A = A/IEALG, w i t h E A.
2
As seen i n S e c t i o n 2 , t h e s t a b i l i t y i n t e r e s t r e q u i r e s
(41)
V
maximal
w h i l e t h e general it y i n t e r e s t demands
(42)
E = S/V
maximal.
NOW, t h e exactness i n t e r e s t w i l l demand
(43.1)
I
minimal
A
maximal.
and
(43.2)
As mentioned i n S e c t i o n 2, t h e c o n d i t i o n s ( 4 1 ) and ( 4 2 ) a r e o b v i o u s l y i n A - see ( 2 9 ) the conditions c o n f l i c t . I n view o f t h e f a c t t h a t E ( 4 1 ) and ( 4 3 . 1 ) w i l l a l s o be c o n f l i c t i n g .
-
2
The manner i n which these c o n f l i c t s a r e r e s o l v e d depends on t h e p a r t i c u l a r problem considered. Examples w i l l be g i v e n i n Chapters 4 , 5 and 7 . How= ever, a general approach t o t h a t i n t e r p l a y , as p r e s e n t e d i n t h e sequel , may a l s o be u s e f u l .
-
I n c o n n e c t i o n w i t h t h e s t a b i l i t y and g e n e r a l i t y p r o p e r t i e s , i t i s conven= see ( 4 1 ) and ( 4 2 ) t o define a partial order ient
-
-
f o r q u o t i e n t spaces Ei = Si/Vi
(44.1)
F1 c F2
E VS
w i t h i E {1,2) , by t h e conditions Fi
v e c t o r subspaces i n
M(R)
E.E. Rosinger
14
t
(44.2)
-> s2
t
, w i t h V2 n s1 = V1.
5 -> s1
-
I n connection w i t h t h e exactness p r o p e r t y , i t i s convenient t o define a p a r t i a l order A1
f o r q u o t i e n t algebras Ai = Ai/Ti
see (43)
-
A2
E
, with i
ALG
E
(1,2)
i
conditions (45.1)
G1 c G2 subalgebras i n M(Q)
(45.2)
i2 c
r1
-
, by the
c A~ c A ~ .
Indeed, i t i s obvious t h a t whenever i t i s p o s s i b l e t o r e p l a c e m E < A, i n the conditions d e f i n i n g the sequential s o l u t i o n s i n E m E' =G A ' , w i t h E -I E' , A I I A',
-+
A, b y
general it y and exactness p r o p e r t i e s o f t h e sequenti a1 sol u= A ' , w i l l be improved.
t h e s t a b i 1it y tion s i n EI-1
The f o l 1owing two p r o p o s i t i o n s w i l l y i e l d general c o n d i t i o n s under which maximal s t a b i i t y and g e n e r a l i t y , or exactness can be achieved. Proposition 1 I f A E ALG, i n t h e c o n d i t i o n s d e f i n i n g t h e s e q u e n t i a l s o l u t i o n s o f t h e
PDE i n ( l ) , i s considered f i x e d ,
spaces
E
E
E
VS:
can be r e p l a c e d b y q u o t i e n t
VST which s a t i s f y t h e c o n d i t i o n s mentioned and y i e l d maximal
s t a b i l i t y and g e n e r a l i t y f o r t h e s e q u e n t i a l s o l u t i o n s . Proof I t i s e a s i l y seen t h q t t h e s e t o f q u o t i e n t spaces E ' E V S F , ,
such t h a t
E -I E ' and E l =G A, i s chain-complete w i t h t h e p a r t i a l o r d e r -I i n ( 4 4 ) . T h i s , i n view of Z o r n ' s lemma, completes t h e p r o o f .
Proposition 2 IfE E V S T
,
i n the conditions d e f i n i n g the sequential s o l u t i o n s o f the
PDE i n ( l ) , i s considered f i x e d , algebras
A
E
A E ALG can be r e p l a c e d b y q u o t i e n t
Ai-T; which s a t i s f y t h e c o n d i t i o n s mentioned and y i e l d maximal
exactness for t h e s e q u e n t i a l s o l u t i o n s .
U
15
SEQUENTIAL SOLUTIONS
Proof I n view o f Z o r n ' s lemma, i t s u f f i c e s t o show t h a t t h e s e t o f q u o t i e n t a l g e = b r a s A ' E ALGl, w i t h A -H A ' and E A ' , i s chain-complete w i t h t h e
2
Assume t h a t Ah = A X / l h
p a r t i a l order 4 i n (45).
, with
AL
h
E
A ,
Gh
i s a 4 c h a i n and m A 4Ah,
(46)
E
<
E
v
Ah,
h E
r\.
Using t h e n o t a t i o n
5;
(47)
T = h 2 A 1 A Ay=
= h e AG A Y
i t i s obvious t h a t 5; i s a subalgebra i n M ( R ) and
We show t h a t h E A. Then
7
i s an i d e a l i n w.2
(48)
7
E
1-1' Indeed, i n case Ah --cI A
7
i-I
V li li
A. E
Assume w E
7
A
u X E A
Ah
i s a subalgebra i n
cN.
and z E Ah, f o r a c e r t a i n
A.
then
c l h c Ah c A
li
hence i . A h c I .A
W.Z E
Otherwise, A
1-1
l i p
c1
li
.
4 Ah, hence
l x c I c A cAh 1-11-1
t h e r e fo r e w.z E i . A h
C
l h Ah c l Ac I
lJ
and t h e p r o o f o f (48) i s complete Now t h e f a c t t h a t t i o n (23).
A
=
A/T
E ALE f o l l o w s e a s i l y b y d i r e c t c h e c k i n g o f c o n d i =
F i n a l l y , (46) and (47) o b v i o u s l y i m p l y t h a t A 4 A and E
m
< A.
0
Remark 2 Given a s e q u e n t i a l s o l u t i o n s i n E + A, i t i s i m p o r t a n t t o n o t i c e t h a t i t s s t a b i l i t y and g e n e r a l i t p r o p e r t i e s depend on t h e q u o t i e n t space E, w h i l e i t s exactness p r o p e r t y epends on t h e q u o t i e n t a l g e b r a A. The r e l a t i o n between s t a b i l i t y and generality,on t h e one hand and exactness on t h e other, A (see F i g . 1 ) . i s implied by the f a c t t h a t E
4
16
maximal stabi 1 i ty
=
E
s
m
A
maximal
/
v
A
=
A
~
/
h
A
r E.E.
Fig. 1
1
= =
DP(V n
(cm(n)lN)c I ,
s c v + (cm(n))N
DP(S n
maximal exactness
(cm(n)lN) c
A,
v
p E N",
1p1
=
m
I
maximal minimal-
Rosinger
maximal generality
I maximal maximalminimal
SEQUENTIAL SOLUTIONS
17
6. The D e f i c i e n c y o f t h e D i s t r i b u t i o n a l Approach t o Exactness We s h a l l show t h a t from t h e p o i n t o f view o f t h e exactness p r o p e r t y o f s e q u e n t i a l s o l u t i o n s , i t i s n o t convenient t o deal w i t h s e q u e n t i a l s o l u t i o n s i n E -t P ' ( R ) , o r i n p a r t i c u l a r , w i t h s e q u e n t i a l s o l u t i o n s i n P ' ( n ) -t P ' ( n ) - see ( 2 6 ) and (31) - which a c t u a l l y r e p r e s e n t t h e usual d i s t r i b u t i o n a l ap= proach. We s h a l l f i r s t i n t r o d u c e s e v e r a l n o t a t i o n s . G c M(R) o f measurable f u n c t i o n s . ForJ c quences o f measurable f u n c t i o n s , denote b y A&
du,
t h e l a r g e s t subalgebra 8 c o n l y i f J i s a subalgebra i n Denote b y
J
uppose g i v e n a subalgebra a v e c t o r subspace o f se=
J)
such t h a t J . 8 c J . Then o b v i o u s l y J c A G ( J ) i n which case J w i l l be an i d e a l i n A G ( J ) .
4,
IG
N
t h e s e t o f a l l v e c t o r subspaces 7 c G such t h a t
7 n UG = Q, I u UG c AG ( 7 ) . IG i s a subalgebra i n GN, s i n c e 7 c A G ( 7 ) .
Obviously, each 7 E for 7
E
Finally,
IG we use t h e n o t a t i o n : AG(1) = AG ( 1 ) / 1 -
It f o l l o w s e a s i l y t h a t AG(7)
(49)
E
ALG
,
V 7
E
IG
and c o n v e r s e l y
(50) i.e.
IG , V A = A / l E ALG , N IG contains e x a c t l y a l l 7 E G f o r which t h e r e e x i s t A = A / l E
7
E
ALG.
Moreover
V A = A/7
E
ALG :
(51) A
AG(7)
i n o t h e r words, each q u o t i e n t a l g e b r a A = A / ? AGW.
E
ALG i s a subalgebra i n
E.E.
18
Rosinger
I t f o l l o w s f r o m (51) and (45) t h a t
(52)
V A = A/IEALG :
(52.1)
A
(52.2)
V E
+A & I ) VS:
E
:
m E
m A
d
E
d
AG ( I )
N i s a sequential s o l u t i o n i n E + A , then s w i l l also Therefore i f s E ( M ( R ) ) be a s e q u e n t i a l s o l u t i o n i n E + A ( I ) and, i n view o f (52.1) and (43.2), t h e l a t t e r s e q u e n t i a l s o l u t i o n wiF1 have t h e best exactness p r o p e r t y f o r t h e given I.
Suppose now g i v e n a s e q u e n t i a l s o l u t i o n s i n E + D ' ( Q ) , w i t h
D'(Q)
(53)
=
SH / V H
E
VSH
- see ( 2 6 ) - H i s a v e c t o r subspace i n M(n), s e q u e n t i a l l y dense i n D'(n). Then, i n view o f (39), (40) and (43.2), t h e l a r g e r
where
AH( V H ) i s , t h e b e t t e r t h e exactness p r o p e r t y o f s. However, t h e above s e t i s u s u a l l y r a t h e r narrow. Indeed, s i n c e a p r o d u c t o f a d i s t r i b u t i o n i n D'(n) and a f u n c t i o n which i s n o t Cm-smooth, i s n o t n e c e s s a r i l y again a d i s t r i b u t i o n i n D ' ( n ) , i t f o l l o w s t h a t r e l a t i o n
UH g AH (V,)
(54)
may h o l d , i n case H
(55) i n case H (56)
Cm(n).
Moreover
g AH ( V H )
VH 3
6
Cm(n).
Indeed, w i t h t h e n o t a t i o n i n (31.1), one o b t a i n s
( vm. vm)n
smg vm,
as t h e f o l l o w i n g one-dimensional example (n=1) w i l l i l l u s t r a t e . Take s E (Cm(n))N g i v e n i n m ( l O ) . Then, as seen i n S e c t i o n 1, SE Vm and S.SE Sm; however, s . s $C V , s i n c e w i t h t h e n o t a t i o n i n ( 2 4 ) , < s . s , .> Now any o f t h e r e l a t i o n s (54) o r ( 5 5 ) w i l l i m p l y
SH t h e r e f o r e i t may happen t h a t t h e exactness c o n d i t i o n (40) w i l l no l o n g e r be s a t i s f i e d when t h e q u o t i e n t a l g e b r a A = A / I i s t h e r e r e p l a c e d b y t h e q u o t i e n t space o f d i s t r i b u t i o n s (53).
= 1.
SEQUENTIAL SOLUTIONS
7.
19
R e s o l u t i o n o f Nowhere-Dense S i n g u l a r i t i e s
Another advantage o f s o l v i n g polynomial n o n l i n e a r PDEs i n q u o t i e n t a l g e b r a s o f classes o f sequences of f u n c t i o n s i s t h e p o s s i b i l i t y o f f e r e d f o r t h e r e s o l u t i o n o f nowhere-dense s i n g u l a r i t i e s e x h i b i t e d by q u i t e a l a r g e c l a s s o f s o l u t i o n s o f t h e P D t s considered, as s p e c i f i e d i n t h e f o l l o w i n g d e f i n i = tion. Given t h e m-th o r d e r polynomial n o n l i n e a r PDE i n ( l ) , c a l l a sequence of f u n c t i o n s S E (Cm(n))N a s o l u t i o n w i t h nowhere-dense s i n g u l a r i t y o f t h e PDE i n ( l ) , o n l y i f
3
r
c R, c l o s e d , nowhere dense :
V X E R \ 3 F E N :
(57)
V V E
r
:
N , v > p :
T(D)sV(x) = f ( x ) i . e . , i f t h e terms s o f t h e sequence of f u n c t i o n s s s a t i s f y t h e PDE i n (1) a t each p o i n t of'the open, dense s e t R \ r i n R, p r o v i d e d t h a t v E N i s s u f f i c i e n t l y large. A p a r t i c u l a r case o f s o l u t i o n s w i t h nowhere-dense s i n g u l a r i t i e s , i m p o r t a n t i n a p p l i c a t i o n s , w i l l now be discussed.
A subset
r
c
R i s c a l l e d CR-smooth, f o r g i v e n
r
(58)
=
f o r a c e r t a i n y: R
I X E R I ~ ( x )= o -f
E
il E
R,
only i f
R ~ I
R R ~ , ~ E C
.
Suppose $ E Cm(n\ r), i s a s o l u t i o n on R \ where r c R i s Cm-smooth.
r
o f t h e m-th o r d e r PDE i n (l),
Given CY E Cm(Rg) such t h a t CY vgnishes on a neighbourhood o f o E Rg and takes t h e v a l u e 1 o u t s i d e a bounded neighbourhood o f o E R9; d e f i n e t h e sequence o f f u n c t i o n s s E ( F ( R b)y) ~ ICc((V+l)Y(X))
(59)
.q(x)
i f x E R\
s (x) = V
10
if x
then obviously V X E R \
(60)
V v
E
sv(x) T h e r e f o r e i f $ E L!oc(Q
r:
N :
3 p E
N , v 2 p: =
+(XI. T(D)sv(x)
\r),
one o b t a i n s
= f(x).
E
r;
r
E . E . Rosinger
20 s E
(61
sm;
and i f i n a d d i t i o n t h e Lebesque measure o f
(62)
r
On t h e o t h e r hand, i f
i s zero, then
.
= J,
<S,'Z
r
i s nowhere dense, i t f o l l o w s t h a t
s i s a s o l u t i o n w i t h nowhere-dense s i n g u l a r i t y o f t h e PDE i n (1).
(63)
The q u o t i e n t algebra o f classes o f sequences o f f u n c t i o n s used i n t h e re= g u l a r i z a t i o n o f nowhere-dense s i n g u l a r i t i e s , i s g i v e n b y And = ( c o (R)l N j r n d
(64)
N where Ind i s t h e s e t o f a l l t h e sequences w E ( C o ( n ) ) o f continuous func= t i o n s on R which s a t i s f y t h e v a n i s h i n g c o n d i t i o n
3 r c R, closed, nowhere dense : V X E R \ r : 3 EN: V v E N, v
>u
:
wv(x) = 0 ; i n o t h e r words, t h e terms w o f t h e sequence w v a n i s h a t each p o i n t o f t h e provided t h a t v E N i s s u f f i c i e n t l y large. open, dense subset R \ r in'R, I t i s easy t o see t h a t
(66)
Ind i s an i d e a l i n
(Co(n))N
and
We s h a l l c a l l Ind t h e nowhere-dense i d e a l and And t h e nowhere-dense q u o t i e n t a1 gebra. Examples o f n o n t r i v i a l sequences o f continuous f u n c t i o n s on R, b e l o n g i n g t o Indy can be o b t a i n e d as f o l l o w s . Suppose z E (c(n))N, y : R -+ RS, YE^ and
c1 E
Co(Rg), w i t h supp a c Rg compact. w,(x)
Using t h e n o t a t i o n
= a ( v y ( x ) ) . z V ( x ) , Vw
r
= Cx E
R
I
y (x)
Define then w E
N, x
= 0 E Rg},
E
E
( C o ( n ) ) N by
R.
i t i s e a s i l y seen t h a t
21
SEQUENTIAL SOLUTIONS
int as
r
r
B
=
=w
w i l l s a t i s f y (65).
cl(o)
r
s i n c e (65) h o l d s f o r
lnd
Also
o
=
E
= W
=
E
rnd
0.
The b a s i c p r o p e r t i e s o f t h e i d e a l Ind are given i n : P r o p o s i t i o n 3: Each sequence w E 'Ind o f continuous f u n c t i o n s on R s a t i s f i e s t h e f o l l o w i n g stronger version o f t h e vanishing c o n d i t i o n (65):
3
r
c R, closed, nowhere dense :
V X E R \
(69)
3
r :
p E N, V c
R \
r
neighbourhood o f x:
V V E N , V > ~ , Y E V :
WJY) F o r any R
E
N,
= 0.
the f o l l o w i n g r e l a t i o n holds:
N DP(lnd n ( c ' l ( f l ) ) ) c lnds
(70)
v
p E
Nny
IpI < R .
Proof The r e l a t i o n (70) f o l l o w s e a s i l y from (69). R e l a t i o n (69) i s an obvious consequence o f Lemma 1 below. Indeed, assume w E lnd and r g i v e n i n ( 6 5 ) . Then R' = R \
r
i s open and dense i n R and V x
E
R': 3
E
N :V v
E
N, v
> p : w,(x)
= 0.
T h e r e f o r e Lemma 1 a p p l i e d t o each non-void, c l o s e d s u b s e t F c n ' , w i l l y i e l d a non-void, r e l a t i v e l y open subset GF c F, such t h a t wv vanishes on G , f o r v E N s u f f i c i e n t l y l a r g e . Denote b y a" t h e u n i o n o f a l l non-void, oben subsets G c R' on which w vanishes f o r v E N s u f f i c i e n t l y l a r g e . I t f o l l o w s t h a t R" i s dense i n s1', t h e r e f o r e R" i s a l s o dense i n R, as R' i s dense i n n. NOW, one can t a k e r = n \ n" i n (69). 0
Lemma 1 Suppose E i s a complete m e t r i c space and E ' i s a t o p o l o g i c a l space. Sup= pose g i v e n t h e continuous f u n c t i o n s f : E -+ E l and fv : E -+ E l , f o r v E N, such t h a t
V x
E
E :I p
E N : V v E N, v
>u
:fV(x) = f(x).
Then f o r each non-void, c l o s e d subset F c E t h e r e e x i s t s a non-void, r e l a = t i v e l y open subset G c F and I.! E N, such t h a t f v ( x ) = f ( x ) , V x E G, v E N, v a p .
22
E . E . Rosinger
Proof For F
C
E non-void, c l o s e d and l~ F =
{X E
lJ
E
N, denote
FI fV(x) = f ( x ) , V u
E
N, u 2
u}.
Then t h e h y p o t h e s i s o b v i o u s l y imp1 i e s t h a t F =
u P E N
But FU, w i t h
~ J E N,
F . lJ
a r e c l o s e d i n F , i n view o f t h e c o n t i n u i t y o f f and fv.
NOW, as F i s i n i t s e l f a complete m e t r i c space, t h e B a i r e c a t e g o r y argument i m p l i e s t h a t f o r a t l e a s t one 1-1 E N, t h e r e l a t i v e i n t e r i o r o f F i n F i s lJ not void. 0
We s h a l l a l s o need t h e f o l l o w i n g q u o t i e n t algebras d e r i v e d f o r g i v e n R E from And
A,
(71) Obviously (72)
(1
And
E
AL~I C
(n)
And
t h e l a s t r e l a t i o n r e s u l t i n g from ( 7 0 ) and ( 2 9 ) . Before p r e s e n t i n g t h e r e s u l t on t h e r e s o l u t i o n o f s i n g u l a r i t i e s , i t i s worth n o t i c i n g t h a t t h e s o l u t i o n s w i t h nowhere-dense s i n g u l a r i t i e s d e f i n e d i n ( 5 7 ) a r e i n f a c t l o c a l s o l u t i o n s o f t h e PDE i n ( l ) , on open, dense sub= s e t s i n n, as shown i n t h e f o l l o w i n g p r o p o s i t i o n . Proposition 4
N Suppose s E ( p ( n ) ) i s a s o l u t i o n w i t h nowhere-dense s i n g u l a r i t y o f t h e PDE i n (1). Then t h e r e e x i s t s an open, dense subset n' c n, such t h a t VxEn' : 3 1-1 E N, V neighb. o f x : V v E sV
N,v>u:
i s a s o l u t i o n o f t h e PDE i n ( l ) , on V
Proof A p p l y i n g Lemna 1 t o dense subset R' c R \ V x
E
Q\r i n ( 5 7 ) , r , such t h a t R ' :3 p
E
i t f o l l o w s t h a t t h e r e e x i s t s an open,
N, V neighb. o f x : V V E
N, v
>p
:
T(D)sv = f on V. But R' i s a l s o open and dense i n R, s i n c e
a.
r
i s c l o s e d and nowhere dense i n 0
SEQUENTIAL SOLUTIONS
23
We a r e now i n a p o s i t i o n t o s t a t e t h e general r e s u l t on t h e r e s o l u t i o n o f nowhere-dense s i n g u l a r i t i e s o f s o l u t i o n s f o r p o l y n o m i a l n o n l i n e a r PDts. Theorem 1 N Suppose s E ( C ? ( Q ) ) i s a s o l u t i o n w i t h a nowhere-dense s i n g u l a r i t y of t h e m-th o r d e r polynomial n o n l i n e a r PDE i n (1). Then s i s a s e q u e n t i a l solu= t i o n i n AZd And o f t h e PDE i n (l), i . e . s d e f i n e s an element S = s + -f
( l n d n ( @ ( n ) ) N ) i n t h e q u o t i e n t a l g e b r a dA:
t h a t s a t i s f i e s t h e PDE i n ( l ) ,
i n t h e usual a l g e b r a i c sense, w i t h t h e m u l t i p l i c a t i o n i n t h e q u o t i e n t algeb r a And (see ( a ) i n Remark 1, S e c t i o n 3 ) . Proof I n view o f ( 5 7 ) , i t f o l l o w s e a s i l y t h a t T(D)S - u ( f ) E
rnd;
t h e r e s t f o l l o w s from ( 7 2 ) , w i t h R = m.
n
The general r e s u l t o f Theorem 1 above w i l l be a p p l i e d t o s o l u t i o n s o f t h e p a r t i c u l a r form ( 5 9 ) . F i r s t , f o r given R E
8 , we d e f i n e t h e q u o t i e n t space (see ( 3 1 ) , ( 2 6 ) )
u ; ~ ( Q =)
(73)
sR/(vR n rnd).
Then, i n view o f ( 3 0 ) , (29), (64) and (70), we have t h a t (74) and t h e l i n e a r mapping (75)
u;ld(~) 3 ( s
+ vR
n I~~)->
(s+vR)E ~ ' ( n ) v, s
E
sR,
i s surjective Corollary 1 Suppose $ E Cm(C2\ r ) i s a s o l u t i o n onmR \ r o f t h e m-th o r d e r polynomial i s C -smooth and nowhere dense. Then n o n l i n e a r PDE i n ( l ) , where r C $, b y means o f ( 5 9 ) , d e f i n e s a s e q u e n t i a l s o l u t i o n i n A l d And o f t h e PDE -f
) and r has z e r o Lebeseque measure,$ , b y means o f ( 5 9 ) , d e f i n e s a s e q u e n t i a l s o l u t i o n i n U,id + And o f t h e PDE i n ( l ) , i n (1).
I f $ E Lioc(Q \
r
w i t h U;ld g i v e n b y ( 7 3 ) , f o r R = m. Proof T h i s f o l l o w s f r o m Theorem 1 and ( 6 3 ) , ( 6 1 ) , (62) and ( 7 4 ) .
0
24
E.E. Rosinger
Remark 3 ( a ) F o r t h e method f o r t h e r e g u l a r i z a t i o n o f s i n g u l a r i t i e s presented i n t h i s s e c t i o n , t h e assumption t h a t t h e s i n g u l a r i t i e s a r e concentrated on nowhere-dense subsets i s e s s e n t i a l a t two stages. F i r s t , i n o r d e r t o e s t a b l i s h t h a t lnd i s an i d e a l i n (Co(n))N(see ( 6 6 ) ) , use i s made o f t h e f a c t t h a t t h e union o f two nowhere-dense subsets i s again nowhere-dense. Secondly, i n o r d e r t o e s t a b l i s h (67), use i s made o f t h e f a c t t h a t a con= t i n u o u s f u n c t i o n v a n i s h i n g o u t s i d e a nowhere-dense subset i s i d e n t i c a l l y zero. Obviously, b o t h r e l a t i o n s (66) and ( 6 7 ) a r e e s s e n t i a l i n o r d e r t o obtain (68).
(b) The power o f Theorem 1 l i e s i n t h e f a c t t h a t o n l y nowhere-dense s i n g u l a r i t i e s a r e r e q u i r e d , o f which i t i s known t h a t they may have arbi= t r a r y p o s i t i v e Lebesque measures. ( c ) The method f o r t h e r e g u l a r i z a t i o n o f s i n g u l a r i t i e s presented i n t h i s s e c t i o n i s a c t u a l l y t h e framework f o r more s p e c i a l i s e d cases o f a p p l i c a = t i v e i n t e r e s t , presented i n Chapter 4. 8. P r e s e r v a t i o n o f E l l i p t i c i t y and H y p o e l l i p t i c i t y I t i s shown i n t h e p r e s e n t s e c t i o n t h a t i n s p i t e o f notion o f sequential s o l u t i o n introduced i n Section s o l u t i o n s f o r e l l i p t i c and h y p o e l l i p t i c l i n e a r PDEs f o r i n s t a n c e i n t h e case o f s e q u e n t i a l s o l u t i o n s i n
the generality o f the 3, t h e r e g u l a r i t y o f w i l l be preserved, DAd + And.
Suppose g i v e n t h e 1i n e a r PDO (76)
L(D) =
C
1
ci(x)
DP i
,x
E
R.
I n case t h e c o e f f i c i e n t s ci i n ( 7 6 ) a r e a n a l y t i c f u n c t i o n s on R, we s h a l l say t h a t L ( D ) i s e l l i p t i c on R s a t i s f i e s the equation (77)
L(D) U ( X ) = f ( x )
,
o n l y i f any d i s t r i b u t i o n S E D'(R) which
x E R'
on an open, dense subset R ' c R , w i l l be an a n a l y t i c f u n c t i o n on v i d e d t h a t f i s a n a l y t i c on R!
a', pro=
F u r t h e r , i n case t h e c o e f f i c i e n t s c . i n ( 7 6 ) a r e Cm-smooth f u n c t i o n s on
R, we s h a l l say t h a t L(D) i s h y p o e l l i p t i c on R o n l y i f any d i s t r i b u t i o n S E D ' ( S 2 ) which s a t i s f i e s t h e e q u a t i o n ( I / ) on an open, dense s l b s e t R ' C R, w i l l be a C -smooth f u n c t i o n on R ' , p r o v i d e d t h a t f E c (R').
I t i s easy t o see t h a t i n t h e case o f a l i n e a r PDO i n (76) w i t h c o n s t a n t c o e f f i c i e n t s , t h e usual n o t i o n s o f e l l i p t i c i t y , o r e l s e h y p o e l l i p t i c i t y c o i n c i d e w i t h those mentioned above. I n t h e case o f non-constant c o e f f i = c i e n t s , t h e above n o t i o n s of e l l i p t i c i t y , o r e l s e h y p o e l l i p t i c i t y , appear t o be somewhat s t r i c t e r t h a n t h e usual ones. However, i f t h e usual e l l i p t i c i t y , or e l s e h y p o e l l i p t i c i t y , on a g i v e n open s e t R C Rn i s as= sumed t o be h e r e d i t a r y f o r open, dense subsets Q' C R, i t w i l l be i d e n t i = c a l w i t h t h e e l l i p t i c i t y , o r e l s e h y p o e l l i p t i c i t y , i n t r o d u c e d above.
25
SEQUENTIAL SOLUTIONS
The n e x t two theorems w i l l p r e s e n t r e s u l t s on t h e p r e s e r v a t i o n o f e l l i p t i = c i t y , o r e l s e h y p o e l l i p t i c i t y , w i t h i n t h e framework o f s e q u e n t i a l s o l n s . Theorem 2 Suppose t h e m-th o r d e r l i n e a r PDO i n ( 7 6 ) i s e l l i p t i c . Then any s e q u e n t i a l s o l u t i o n i n DInd(R)+ And of t h e PDE i n ( 7 7 ) i s an a n a l y t i c f u n c t i o n on a c e r t a i n open, dense subset o f R, p r o v i d e d t h a t f i s a n a l y t i c on R and D;ld(R) i s g i v e n b y ( 7 3 ) , w i t h 2 a m . Proof Assume
R
SE
i s a sequential s o l u t i o n i n
D'nd (R) * A n d
o f t h e PDE i n ( 7 7 ) .
- u ( f ) E Ind; t h e r e f o r e ( 6 9 ) i m p l i e s t h e e x i s t e n c e o f an open, dense subset R' c R, such t h a t
Then L ( D ) s
V x E R' :
=j D
EN,
V neighb. o f x : Vv E
N, v
>u:
L ( D ) s v = f on V . I t f o l l o w s t h a t t h e d i s t r i b u t i o n S = <s, . > E D ' ( R ) s a t i s f i e s t h e PDE i n (77) on t h e open, dense subset R' c 0. Thus i n view o f t h e e l l i p t i c i t y o f t h e PDO i n ( 7 6 ) , S w i l l be a n a l y t i c on R' .
n
Theorem 3 Suppose t h e m-th o r d e r l i n e a r PDO i n ( 7 6 ) i s h y p o e l l i p t i c . Thgn any And o f t h e PDE i n ( 7 7 ) i s a C -smooth s e q u e n t i a l s o l u t i o n i n UAd(fi) +
f u n c t i o n on a c e r t a i n open, dense subset o f R, p r o v i d e d t h a t f and UAd(fi) i s g i v e n by ( 7 3 ) , w i t h R m.
E
c"(Q)
Proof T h i s i s s i m i l a r t o t h a t f o r Theorem 2.
9.
General N o n l i n e a r PDEs
The method f o r s o l v i n g PDEs i n q u o t i e n t a l g e b r a s o f c l a s s e s o f sequences o f f u n c t i o n s can be used n o t o n l y i n t h e case o f polynomial n o n l i n e a r PDEs, b u t a l s o f o r a much w i d e r c l a s s o f n o n l i n e a r PDEs, as shown i n t h i s section. An m-th o r d e r continuous n o n l i n e a r PDE i s b y d e f i n i t i o n o f t h e form
(78)
F(x,u(x),
...,DPU ( X ) ,...)
where p E N n y I p I Gm, f E
(78.1)
R= carCp E Nnl
=
f(x), x
C"(n) and F
E
E
R,
C"(nxR'),
with
I p I Gm}.
Obviously, an m-th o r d e r p o l y n o m i a l n o n l i n e a r PDE o f t y p e (1) i s a p a r t i = c u l a r case o f t h e PDE i n ( 7 8 ) . I n o r d e r t o s o l v e PDEs o f t y p e (78) w i t h i n q u o t i e n t a l g e b r a s , we s h a l l
26
Rosinger
E.E.
d e f i n e mappings o f t y p e ( 3 3 ) f o r t h e a s s o c i a t e d PDOs (79)
. ,Dpu(x) ,... ),
F ( D ) u ( x ) = F(x,u(x),..
x E
a.
I n t h i s connection, suppose g i v e n a q u o t i e n t space E = S / V E VS; quotient algebra A = A / l
E
and a
ALG, where F and G a r e r e s p e c t i v e l y a v e c t o r
subspace and a subalgebra i n M(a). We then use t h e n o t a t i o n : F(D) (80 1 E 6 A only i f (80.1)
F(D) ( F n cm(n)) c G
(80.2)
F(D)(S n (cm(n)lN) c A
(80.3)
F(D)(t+v)
-
F(D)t
E
I, V t
E
S n (Cm(R))N, v E V n (Cm(fi))N
where F(D) in, t h e above r e l a t i o n s means t h e term-by-term a p p l i c a t i o n o f t h e mapping F(D) : Cm(n)
-f
(n)
C"
generated b y (79) , t o sequences o f f u n c t i o n s . Now i f t h e r e l a t i o n s (80) h o l d , one can d e f i n e t h e mapping (81)
F(D) : E
-+
A
by (81.1)
F(D)(s+V) = F ( D ) t + l ,
Y s E S
where (81.2)
t
E
sn(cm(n))N, s-t
E
v.
With t h e h e l p o f t h e mapping ( 8 1 ) , we can d e f i n e t h e n o t i o n o f s e q u e n t i a l s o l u t i o n o f a continuous n o n l i n e a r PDE. Suppose g i v e n t h e m-th o r d e r continuous n o n l i n e a r PDE i n (78) and a se= quence s E ( M ( R ) ) N o f measurable f u n c t i o n s on R. Then s i s c a l l e d a s e q u e n t i a l s o l u t i o n o f (78), o n l y i f f o r a c e r t a i n v e c t o r subspace F and subalgebra G i n M(R), t h e r e e x i s t a q u o t i e n t space E = S / V E VSF and a q u o t i e n t a l g e b r a A = A / l EALGy w i t h E F&D) A , such t h a t (82.1)
s E
S
and t h e mapping (81) maps S (82.2)
+
(then S = s
F(D)(s+v) = u ( f )
E
V E E = S/v)
E into f
E
A;
.e.
+ r.
I n view o f ( 8 1 ) , (80) and ( 3 0 ) , t h e c o n d i t i o n ( 8 2 ) i s o b v i o u s l y e q u i v a l e n t to
27
SEQUENTIAL SOLUTIONS
t E
sn
(cm(n)lN :
(83) s-t E V , F(D)t
-
u(f) E 7
which i s t h e same as t h e f o l l o w i n g two: (84.1)
s E S
and
vt
E
s
n (cm(n)lN :
(84.2) s - t E V * F(D)t
-
u ( f ) E 7.
The c r u c i a l p o i n t i n t h e above d e f i n i t i o n s i s t h e r e l a t i o n ( 8 0 ) , more e x a c t = l y the c o n d i t i o n (80.3). I n t h i s connection, t h e n o n - t r i v i a l examples given i n the next proposition are important. Proposition 5 Suppose g i v e n t h e m-th o r d e r continuous n o n l i n e a r PDO i n ( 7 9 ) . following r e l a t i o n s hold: F(D) F(D) And And, D',d(R) And.
Then t h e
Proof I n view o f ( 6 4 ) , t h e r e l a t i o n s (80.1) and (80.2) o b v i o u s l # h o l d i n b o t h cases. Regarding t h e r e l a t i o n (80.3), assume t E (Cm(n)) and v
E
Ind n (Cm(R))N.
where-dense subset V X E R \
1 p
E
Then, a c c o r d i n g t o ( 6 9 ) , t h e r e e x i s t s a closed, no=
r
c R, such t h a t
r :
N, V neighb. o f x :
V v ~ N , v ) > p : v
V
= 0 on V .
Therefore F(D)(tv+vV)
-
F ( D ) t v = 0 on V, V v
E
N, v > p
which means t h a t F(D)(t+v)
-
F(D)t
E
Ind;
and t h i s completes t h e p r o o f o f (80.3).
10.
0
V a r i a b l e Transforms
As seen i n t h e p r e v i o u s s e c t i o n , t h e q u o t i e n t spaces and a l g e b r a s o f c l a s s e s
E .E.
28
Rosinger
of sequences o f f u n c t i o n s can i n a n a t u r a l way accommodate n o t o n l y alge= b r a i c o p e r a t i o n s w i t h p a r t i a l d e r i v a t i v e s b u t a l s o a wide range o f such continuous o p e r a t i o n s . I n t h i s s e c t i o n , a n a t u r a l procedure i s presented o f e f f e c t i n g dependent and independent v a r i a b l e t r a n s f o r m s w i t h i n t h e framework o f t h e q u o t i e n t spaces and algebras mentioned. These v a r i a b l e t r a n s f o r m s w i l l be used i n Chapters 3,7 and 8. Suppose g i v e n a q u o t i e n t space E = S / V c V S F and a q u o t i e n t a l g e b r a A =A/IEALG, where F and G a r e r e s p e c t i v e l y a v e c t o r subspace and a subal= gebra i n M( n).
Dependent v a r i a b l e t r a n s f o r m s from E t o A can be d e f i n e d as f o l l o w s . pose g i v e n a mapping (85)
Sup=
T : R’ + R’
such t h a t
F
(86.1)
T o $ E G,
V $
(86.2)
T o s E A,
V s E S
where (-ros),(x) (86.3)
=
E
T ( S ~ ( X ) ) V, v E N , x E R, and f i n a l l y ,
T o ( s + v ) - T O s E 7 , V s E S, v E V .
Then o b v i o u s l y i t i s p o s s i b l e t o d e f i n e a mapping (87)
E
T :
+ A
by (87.1)
T(s+V) = T o s
+
7, V s
E S.
A mapping (85) s a t i s f y i n g t h e c o n d i t i o n s ( 8 6 ) , w i l l be c a l l e d an pendent v a r i a b l e t r a n s f o r m .
E
-t
A-de=
A simple example i s g i v e n by t h e mapping T
: R’
-+
R’, w i t h ~ ( y =) ay
+ by
f o r y E R’,
where a,b E R’ a r e given, which i s o b v i o u s l y an E + A-dependent v a r i a b l e transform, provided t h a t G c o n t a i n s F , as w e l l as t h e c o n s t a n t f u n c t i o n s on Q, and E ---I A (see ( 4 4 ) ) .
I f t h e mapping (85) i s n o t defined everywhere on R ’ , f o r i n s t a n c e , t h e p o s i ti ve power mapping (88)
T~
:
[O,m)
+
R’,
w i t h ~ ~ ( =yyay ) for y E [Op),
where c1 E ( O , m ) b e i n g g i v e n (see i t s a p p l i c a t i o n s i n Chapters 3 and 7 ) , i t i s p o s s i b l e t o proceed i n a more general way as f o l l o w s . Suppose g i v e n a mapping
SEQUENTIAL SOLUTIONS (89)
A+
T :
R',
29
A C R 1
and t h e r e e x i s t s ST c S , s a t i s f y i n g t h e c o n d i t i o n s (90.1)
u
(90.2)
-r 0 s E A , V s
-
v
s , S ' E ST : s-s' E
s = s'
€ST.
Using t h e n o t a t i o n (91)
E
T
I
= {(stv) E E
s
E
ST}
i t i s o b v i o u s l y p o s s i b l e t o d e f i n e a mapping
(92)
T : E
+ A
T
by (92.1)
o s + I, V s E S ~ .
T ( s + V ) = -r
A mapping ( 8 9 ) s a t i s f y i n g t h e c o n d i t i o n s ( 9 0 ) w i l l be c a l l e d an E dent v a r i a b l e p a r t i a l t r a n s f o r m .
-t
A-depen=
To i l l u s t r a t e , we show how a r b i t r a r y p o s i t i v e powers can be d e f i n e d w i t h i n t h e above framework. We use t h e n o t a t i o n (93)
F
+
I
F
= {$ E
It f o l l o w s t h a t i f G 3 {$ E F n
-rCL o $ E G , V
C'(n),
e (a)
I
CL
E
(0,m)I (see ( 8 8 ) )
then
2 0, w x E n ~ F+. c
IJe a l s o use t h e n o t a t i o n
(94)
s,
=
Is
E
s
n (F+)
Nl
T~ o s E A , V
CL
E
(0,m)I.
Suppose g i v e n a s p l i t t i n g
(95)
s
=V@S'
s a t i s f y i n g the condition (95.1)
UF
c S'
which i s p o s s i b l e i n view o f (22). O b v i o u s l y S+ n S ' s a t i s f i e s ( g o ) , t h e r e f o r e u s i n g t h e n o t a t i o n (96)
E,
=
{(StV)E
E
one can, f o r each C L E(O,m), (97) by
yCL
: E,
+
A
sES+ n S ' } d e f i n e t h e mapping
30
E.E.
(97.1)
= Ta0 S
Ta(S+V)
+ I, v
S
Rosinger ES+n S ' .
In view of ( 9 5 . 1 ) , we have
and the p o s i t i v e power mappings ( 9 7 ) coincide with t h e usual p o s i t i v e powers of f u n c t i o n s , when applied t o E F+.
+
We now turn t o independent v a r i a b l e transforms. Suppose given a quotient space E = s/V E VSF and a quotient algebra A = A / I E A L ~ , where F i s a vector subspace i n M(R), while G i s a subalgebra i n M ( R ' ) , w i t h Q cR", R' c R n ' , non-void, open. Suppose given a mapping (99)
w : R +R'
such t h a t (100.1)
+o
(100.2)
s o w
w E G, V E
where ( s o w),(x) (100.3)
v o w
+
A, V s
E
F
E
S
= sv(w(x)), E
r, v
v
E
V v EN, x E
n, a n d f i n a l l y
V.
Then obviously i t i s possible t o define a mapping (101)
w :
E
+
A
by
(101.1)
W(S+V)
=
s o
w
+
r, v
s E
s.
A mapping ( 9 9 ) s a t i s f y i n g t h e conditions ( l o o ) , will be c a l l e d an E dependent variable transform. 11.
+
A-in=
Stronger Conditions f o r P a r t i a l Derivative Operators
I t i s shown in t h i s s e c t i o n t h a t t h e r e a r e very simple a l g e b r a i c considera= t i o n s which govern the way p a r t i a l operators can be defined on the quotient spaces or algebras. Namely i t i s necessary t o assume p a r t i a l d i f f e r e n t i a l operators t h a t a c t between d i f f e r e n t quotient spaces and a1 gebras, other= wise one automatically ends up with a p a r t i c u l a r m u l t i p l i c a t i o n f o r the Dirac d i s t r i b u t i o n and i t s d e r i v a t i v e s . This i s t h e main reason f o r deal= ing w i t h chains of quotient algebras when studying polynomial nonlinear
PDEs. In order t o simplify the presentation we s h a l l consider t h e one-dimensional case ( n = 1 ) only.
Suppose A is an a s s o c i a t i v e and commutative a l g e b r a , containing
-
t h e real-valued polynomials on R'
-
t h e d i s t r i b u t i o n s i n U'(R')
with support a f i n i t e number of
SEQUENTIAL SOLUTIONS
31
p o i n t s , i .e. l i n e a r combinations o f D i r a c d i s t r i b u t i o n s and t h e i r d e r i v a t i ves
.
Suppose a l s o t h a t
( 102 )
t h e m u l t i p l i c a t i o n on A induces on t h e p o l y n o m i a l s t h e usual m u l t i p l i c a t i o n and t h a t polynomial $ ( x ) = 1, V x E R ’ , i s t h e u n i t element i n t h e a l g e b r a A;
(103)
t h e r e e x i s t s a l i n e a r mapping D : A ( 2 7 ) ) such t h a t
(103.1)
D i s i d e n t i c a l w i t h t h e usual d e r i v a t i v e , when a p p l i e d t o p o l y = nomials o r d i s t r i b u t i o n s w i t h s u p p o r t a f i n i t e number o f p o i n t s
-f
A, c a l l e d d e r i v a t i v e (see
i n R’; D s a t i s f i e s the L e i b n i t z r u l e f o r product d e r i v a t i v e s
(103.2)
D(a.b) = (Da).b t a.(Db), W a, b
A
E
and f i n a l l y (see P r e l i m i n a r i e s ) (x-x,).
(104)
6x
=
0 E A , Y xOE R ’ ,
0
denotes t h e D i r a c d i s t r i b u t i o n w i t h s u p p o r t i n xo E R ’ .
where 6x 0
Proposition 6 W i t h i n t h e a l g e b r a A, t h e f o l l o w i n g r e l a t i o n s h o l d : (105)
( x - x ~ ) ~ . D= ~0~E A,
I x0
E
R’,
p, q E N, p > q
xO
(106)
+
(p+l).DP6
( X - X ~ ) . D ~= ~0~E&A,~ tf xo
(107)
6
R’,
p E N
0
xO
( X - X ~ ) ~ . ( )‘D ~= ~0 E A, tf x 0 E R’,
p,q E
N, q 2 2
xO
(108)
= 0 E A, Y xo E R ’ .
( 6 x ) z = dX .Ddx 0
0
0
Proof A p p l y i n g D t o (104) and t a k i n g i n t o account ( 1 0 3 ) , we have (109)
+
6
(X
-xo).D6
xO
= 0 E A , Y x0 E xO
R’
-
which, m u l t i p l i e d b y ( x - x ) , g i v e s i n view o f (104) - t h e r e l a t i o n (x-x,)’ D6, = 0 E A, V xo E R ’ . ‘Applying D t o t h e l a t t e r r e l a t i o n and t h e n mul= t i p ? y i n g b y ( x - x o ) , we have i n t h e same way t h e r e l a t i o n ( X - X ~ ) ~ . D =’ ~ ~ 0
0
E
A , V xo
E
R’.
Repeating t h e procedure, we have (105).
The r e l a t i o n (106) i s t h e r e s u l t o f repeated a p p l i c a t i o n o f D t o (109). Now, m u l t i p l y i n g (106) b y ( X - X ~ ) we ~ , have (p+l)(X-Xo) P .D P
.
+ (x-x,) p+l DP+lBI(.Q
= 0 E A, V xo E R’,
p
E
N.
32
E.E.
Rosinger
M u l t i p l y i n g t h i s r e l a t i o n b y (DP6x )q-l
and t a k i n g i n t o account (105), we
0
have (107). F i n a l l y , t a k i n g p=O and q=2 i n (107), we have ( 6
X
)' = 0
E
A , V xo E R ' .
A p p l i c a t i o n o f D t o t h i s l a s t r e l a t i o n completes ?he p r o o f o f (108). 12.
0
Systems o f Polynomial N o n l i n e a r PDEs
Suppose, i n s t e a d o f t h e polynomial n o n l i n e a r PDE i n ( I ) , we a r e given a svstem o f such PDEs (110)
1 G i
c@~(x) 1Q j
Q
kBi
DpBiju
a
(x) = fB(x), x ~i j
Q
1< @ < b y where u = ( u ,...,u ) : Ra a r e t h e unknown f u n c t i o n s , w h i l e f B E C"(Q1 a3.e givefl. The o r d e r o f t h e system i n (110) i s +
(111)
m,= max { I p 8ij(
1
l G @ < b , 1 G i < ha, 1 G j
p r o v i d e d t h a t none o f t h e c o e f f i c i e n t s c denote the corresponding PDOs as followsBi (112)
TB(D)u(x) = 1 G1i Q h 8
Q
C B ~ ,
ksi]
vanishes everywhere on
a
n. L e t
B iS
lGBGb,
o r , i n an even more compact form
(113)
T(D) = ( T i ( D ) 9
- - ,Tb(D))
which o b v i o u s l y generates a mapping
(114)
T(D) : (Cm(n))a
+
(C"(R))b
I t f o l l o w s t h a t a c l a s s i c a l s o l u t i o n o f (110) i s any f a m i l y o f f u n c t i o n s
(115)
0 = (Qly...YQa)qCm(wa
f o r which t h e mapping (114) s a t i s f i e s (116)
T ( D ) @ = f on R
where
(116.1)
f = (fly...,
fb)
I n analogy t o (7), the s e q u e n t i a l s o l u t i o n s o f (110) w i l l be g i v e n by se= quences o f f a m i l i e s o f f u n c t i o n s
(117) with
s = ( ~o,~ly...y~vy.....)
us
SEQUENTIAL SOLUTIONS
33
i n t h a t way
The p r e c i s e d e f i n i t i o n can be o b t a i n e d i n analogy t o t h e one i n S e c t i o n 3 as f o l l o w s . F i r s t , we r e a r r a n g e t h e f u n c t i o n s i n s, t r a n s f o r m i n g t h e sequence o f f a m i = l i e s o f f u n c t i o n s i n (118) i n t o a f a m i l y o f sequences o f f u n c t i o n s
(119)
s = (S1'
.... Sa)E((Cm(n)) N a
where (119.1)
sclv = +va
' for 1 G a Q a ,
V E
N
Suppose now g i v e n F and G r e s p e c t i v e l y a v e c t o r subspace and a subalgebra i n M(Q), such t h a t (28) i s s a t i s f i e d and t h e c o e f f i c i e n t s cBi as w e l l as t h e r i g h t - h a n d terms fB i n t h e s stem o f PDEs i n (110), b e l o n g t 8 G. pose f u r t h e r g i v e n E = S/W E V S i and A = A / I EALG such t h a t E ,
Sup= Then,
t h e mapping (114) can be extended t o a mapping (120)
T(D) : Ea
+
Ab
d e f i n e d by (120.1)
T(D)(sl+W
,... ,Sa+W)
=
(T1(D)t+l
,... , T b ( D ) t + l ) ,
V sl,...,sa
E
S
where (120.2)
t = ( t
( 120.3)
ta E
(120.4)
T B ( D ) t = ( T B ( D ) t l,...,Ta(D)ta),
s
ta)
n ( c m ( f i ) l N , sa-ta
E
V , Y 1G a G a
Y 1 -(B G b
and T ( D ) t , i n (120.4) i s t h e t e r m by t e r m a p p l i c a t i o n o f t h e mapping genergted by (112) t o t h e sequence o f f u n c t i o n s tUE (Cm(fi))
.
A f a m i l y o f sequences o f f u n c t i o n s s =
( S l,...'Sa)
E
sa
i s c a l l e d a sequential solution i n E o n l y if t h e mapping (120) maps S = (Sa,.
into
where
.. ,Sa)
E
Ea
-f
A o f t h e system o f PDEs i n (110),
,
34
E . E . Rosinger
S = s t a a
V E
U 1 <&aa.
E = S/V,
I t i s easy t o see t h a t t h e above c o n d i t i o n i s e q u i v a l e n t t o
'7(cm(n))N) a ..
t = ( t l,...,ta)qS
(121)
V l G a G a , 1GB G b :
B
-
E
V , TB(D)t
-
u(fB) E 7
which i s t h e same as t h e f o l l o w i n g c o n d i t i o n
v
t E ( S "(cm(n))N)a
:
/
( 122 1
\
Y V1
lGRGb :
* T (D)t-u(fB) E
\B
With t h e above d e f i n i t i o n o f a s e q u e n t i a l s o l u t i o n f o r a system o f polyno= rnial n o n l i n e a r PDEs, i t i s obvious t h a t t h e s t u d y o f such s o l u t i o n s can a c t u a l l y be reduced t o t h e simultaneous s t u d y o f a f i n i t e number - i n t h i s case g i v e n by a - o f s e q u e n t i a l s o l u t i o n s i n t h e sense o f S e c t i o n 3, which i n t h i s case given by b o f polynomial n o n l i n e a r s a t i s f y a f i n i t e number PDEs.
-
-
The way t h a t r e d u c t i o n a c t u a l l y works can be seen i n Sections 5,6, Chapter 4, connected w i t h t h e r e s o l u t i o n o f s i n g u l a r i t i e s o f weak s o l u t i o n s f o r general systems o f polynomial n o n l i n e a r PDEs. 13.
A Cesaro Type C h a r a c t e r i z a t i o n o f t h e Nowhere Dense I d e a l
The nowhere dense i d e a l 7 d i n t r o d u c e d and used i n Sections 7 and 8, w i l l a l s o p l a y an i m p o r t a n t r o r e i n Chapters 3,4,5 and 7. Here, an a l t e r n a t i v e c h a r a c t e r i z a t i o n o f t h e i d e a l lnd, based on a Cesaro t y p e c o n d i t i o n w i l l be presented. Given a sequence o f continuous f u n c t i o n s w t h e sequence o f continuous f u n c t i o n s I;jP E (123)
GpV
= w
P
+
... + w
~ + ~U , v
E
E
( P (Q))N and 1~.E N, we d e f i n e
( e (n))N
by
N
We s h a l l denote by
E" t h e s e t o f a l l t h e sequences o f continuous f u n c t i o n s e E ( P satisfy the condition ev(x)
(124) Obviously l / e
E
E"
C
# 0, tl v
E
N, x
E
R
(c" (Q))N, whenever e N
(a))N which
E
E"
.
An i d e a l 7 i n ( C (a)) i s c a l l e d Cesaro i d e a l , o n l y i f each w E I s a t i s f i e s the c o n d i t i o n
SEQUENTIAL SOLUTIONS
Q' c Q non-void, open :
Y
1 0' c Q' non-void, open, 1-1
(125)
35
x
Y
C O ( Q ) , supp
E
-'( l / e )
u(x)w
E
x
c
E
N, e E E " :
Q":
'I
I n view o f ( 6 9 ) , i t i s easy t o see t h a t 7 i s a Cesaro i d e a l . The con= verse o f t h a t p r o p e r t y i s a l s o t r u e , i n tfig f o l l o w i n g sense. Proposition 7 I f a c o f i n a l i n v a r i a n t Cesaro i d e a l 'I s a t i s f i e s t h e c o n d i t i o n (see ( 2 3 ) )
7 n u
(126)
C"
(a)= Q
then (127)
'nd
'I
Proof Assume w
7 and denote by
E
r the s e t of a l l the p o i n t s
X E
Q which s a t i s f y t h e c o n d i t i o n
Y V neighbourhood o f x,
(128)
1
YE
WJY) Obviously show t h a t
r
V,vE f;
N, v
3
E
Y :
u:
0
i s closed, t h e r e f o r e i n o r d e r t o p r o v e (127), i t s u f f i c e s t o
(129)
int
r
=
0.
Assume t h e n
Q' = i n t r # 0 But = w.w
E
r
hence, (125) a p p l i e d t o z w i l l y i e d f i " e E P, such t h a t
Yx
E
P O ) ,suppx c R "
(130) u(x)
Z* ( l / e )
E
L
c Q ' non-void,
open,
E N and
E.E.
36
Rosinger
Assume now x E R". B u t R" C R' c r , t h e r e f o r e (128) w i l l y i e l d y and w E N, w 2 u, such t h a t w,(y) # 0. Then d e n o t i n g
R"'
= Cy' E
R"
R " I w w ( y ' ) # 01
Q"
i t follows that
E
i s non-void and open.
I n view o f (123) a p p l i e d t o z, i t i s obvious t h a t (131)
# 0 on R"', U A
2:
We d e f i n e (132)
w'E
W'
P
=
N, A
E
>w
(c" ( R ) ) N by x.?' wtp'ev+p'
where x E c" (R) , w i t h supp o f (130) w'
E
p
x
R"' i s a r b i t r a r i l y given.
C
Then i n view
1
since 1 i s cofinal invariant. We can d e f i n e t E(c" ( R ) ) N by 0 (133) t,(x)
i f x(x)
=
0
i fx ( x )
P
=
ewtp(x)/z;+,(x) s i n c e supp
x
0
C 0 ' ' ' and t h e r e l a t i o n (131) h o l d s
Now (132) and (133) w i l l y i e l d u(x) = w l t
E
N
1.(C0 (0))
c1
which i n view o f (126) w i l l i m p l y t h a t x=OonR c o n t r a d i c t i n g t h e a r b i t r a r y c h o i c e o f x. T h e r e f o r e t h e p r o o f o f (129) i s 0 completed and t o g e t h e r w i t h it, a l s o t h e p r o o f o f (127).
CHAPTER 2 NECESSARY A N D / O R SUFFICIENT CONDITIONS FOR THE
EXISTENCE OF SEQUENTIAL SOLUTIONS
0 . Introduction
The problem of t h e existence of sequential s o l u t i o n s of a PDE can be divided i n t o two parts. Given the m - t h order polynomial, nonlinear PDE in ( l ) , Chapter 1 , and a sequence of functions s E (C'm(n))N,t h e problem i s whether t h e r e e x i s t quotient spaces E = S/VEVST and quotient algebras A = A / l E A L G , m with E < A, F a vector subspace in h l ( R ) , G a subalgebra in M(R) , and G 3 C'(n), such t h a t s i s a sequential solution in E considered.
-+
A of t h e PDE
I n terms of the sequence of functions s , t h e problem - see Section 3 , Chapter 1 - i s whether t h e following two r e l a t i o n s can be s a t i s f i e d : (1)
s
(2)
wS
s
E
=
T(D)s - u ( f )
E
I.
A t f i r s t glance, t h e second r e l a t i o n seems t o be more d i f f i c u l t t o f u l f i l , since I has t o be an ideal in A , a n d moreover has t o s a t i s f y t h e r e l a t i o n (see ( 2 3 ) , C h a p t e r 1)
I n t h i s Chapter we s h a l l deal with t h a t p a r t of t h e existence problem r e l a = ted t o sequential s o l u t i o n s of polynomial nonlinear PDEs. More s p e c i f i c a l = ly, we s h a l l e s t a b l i s h necessary and/or s u f f i c i e n t conditions f o r the existence of quotient algebras A E AL such t h a t sequential s o l u t i o n s C"
(a)
i n E -+ A of t h e PDE in ( l ) , Chapter 1, e x i s t f o r c e r t a i n quotient spaces t . In o t h e r words - see Remark 2 , Section 5 , Chapter 1 - we s h a l l be in= t e r e s t e d only in t h e exactness p r o p e r t i e s of sequential s o l u t i o n s . More p r e c i s e l y , given the m - t h order polynomial nonlinear PDE in ( l ) , Chap= ter 1, and a sequence of functions s € ( C m ( n ) ) N , we s h a l l e s t a b l i s h neces= s a r y and/or s u f f i c i e n t conditions f o r t h e existence of quotient algebras such t h a t the ' e r r o r ' sequence ( s e e Section 4 , Chapter 1) A = A/lEAL c" (n) wS = T(D)s - u ( f ) s a t i s f i e s the condition
ws
E
7
37
38
E.E.Rosinger
as w e l l as p o s s i b l e a d d i t i o n a l ones. I t w i l l be convenient t o deal here w i t h a more r e s t r i c t e d n o t i o n o f sequen= t i a l s o l u t i o n , which besides t h e a l g e b r a i c c o n d i t i o n s i n S e c t i o n 3, Chapter 1, w i l l a l s o s a t i s f y c e r t a i n c o n d i t i o n s o f t o o l o g i c a l n a t u r e , w i t h o u t how= e v e r t h e necessary presence o f a t o p o l o g y on t e spaces o f f u n c t i o n s i n = volved. An example o f such a t o p o l o g i c a l - t y p e c o n d i t i o n i s t h a t r e q u i r i n g a l l t h e subsequences o f a s e q u e n t i a l s o l u t i o n o f a c e r t a i n PDE t o be a l s o s e q u e n t i a l s o l u t i o n s o f t h a t PDE.
+
1.
Subsequence Q u a s i - I n v a r i a n t Sequential S o l u t i o n s
F i r s t , we s h a l l c o n s i d e r s e q u e n t i a l s o l u t i o n s s a t i s f y i n g a r a t h e r weak t o p o l o g i c a l - t y p e c o n d i t i o n , c a l l e d subsequence q u a s i - i n v a r i a n c e and d e f i n e d next. Given t h e m-th o r d e r polynomial n o n l i n e a r PDE i n ( l ) , Chapter 1, a sequence n))N i s c a l l e d a subsequence q u a s i - i n v a r i a n t s e q u e n t i a l o f f u n c t i o n s SE (P( s o l u t i o n o f t h e PDE considered, o n l y i f f o r any subsequence s ' i n s, t h e r e e x i s t s a q u o t i e n t a l g e b r a A ' = A ' / I ' E AL co,n), such t h a t
wS1
E
ri.
A u s e f u l , simple c h a r a c t e r i z a t i o n o f t h e above t y p e o f s e q u e n t i a l s o l u t i o n s i s now presented. Given a sequence w E ( C o ( Q ) ) N o f continuous f u n c t i o n s , define = %/Iw
where
$
i s t h e subalgebra i n ( C " ( f 2 ) )
N generated b y {wlu
Uco(n),
while
generated b y w.
Iw i s the ideal i n Proposition 1
N
A sequence o f f u n c t i o n s s E (Cm(Q)) i s a subsequence q u a s i - i n v a r i a n t s e q u e n t i a l s o l u t i o n o f t h e PDE i n (l), Chapter 1, o n l y i f A
E
ws '
AL
C" ( f2)
f o r each subsequence s ' i n s . Proof T h i s f o l l o w s from Lemma 1 below. Lemma 1 Ifw E
( c " ( Q )N) t h e n
Aw E ALCe(n), o n l y i f w
E
I for a certain
A = A/IEALC0(,). Proof Assume A = A / I E A L P ( Q ) and w E I .
Then i n view o f ( 2 3 ) , Chapter 1, we
39
CONDITIONS FOR SOLUTIONS have UC" ( R )
{w) hence
c A. w
c 7 vAcA,
Now
1 * w.A c 1
E
".Aw
c 1
=)
1,
c 7;
hence, again i n view o f ( 2 3 ) , Chapter 1, we have lW n uC O ( R )
t h e r e f o r e A,
AL
E
C"
(a)
=
2;
C" ( n) *
The converse i s obvious, s i n c e w E lw, Denote now b y
R t h e s e t o f a l l sequences w the condition
E
(C"(n))N o f continuous f u n c t i o n s t h a t s a t i s f y
V R' c R non-void, open, w ' subsequence i n w,
$'
E
Co(n') :
(3) w ' = u ( + ' ) on R' * $ ' Obviously, w E
R, o n l y i f
w'E
=
0 on
R'.
R , f o r each subsequence w ' i n w.
The b a s i c c h a r a c t e r i z a t i o n o f a subsequence q u a s i - i n v a r i a n t s e q u e n t i a l s o l u t i o n i s g i v e n i n t h e f o l l o w i n g theorem. Theorem 1 N A sequence o f f u n c t i o n s s E ( C m ( R ) ) i s a subsequence q u a s i - i n v a r i a n t se= q u e n t i a l s o l u t i o n o f t h e PDE i n ( l ) , Chapter 1, o n l y i f
(4)
ws
E
R.
Proof T h i s f o l l o w s from P r o p o s i t i o n 1 above and P r o p o s i t i o n 2 below.
U
Prooosition 2 N Given a sequence w E(c(Q)) o f continuous f u n c t i o n s , t h e n w E R , o n l y i f ,A, E ALcD(Q), f o r each subsequence w ' i n w. Proof
Assume t h a t w
E
R.
Then i t s u f f i c e s t o
show t h a t A,
= Aw/Zw E AL c"(n), i . e .
E . E . Rosinger
40
I t i s easy t o see t h a t an element o f t h e i n t e r s e c t i o n i n ( 5 ) has t h e form
... + w R+ 1. U ( Q ..~ .,, qJk E C o ( Q ) .
u ( + ) = w.u(qJo) +
(6)
where R
E
N and $, q ~
+
i t s u f f i c e s t o show t h a t = 0 on R. R'C n i s non-void, open, such t h a t
(7)
# 0, v x
$(X)
E
Therefore i n o r d e r t o prove ( 5 ) ,
Assume t h a t t h i s i s n o t t h e case and
R'.
Denoting by w , w i t h v E N, t h e continuous f u n c t i o n s on R t h a t a r e t h e terms i n t h e Yequence w, t h e r e l a t i o n ( 6 ) w r i t t e n term-by-term, y i e l d s (8)
(w,(x))
kt 1
.qJg(x) t
... t
+ (-$(x))
w,(x).+,(x)
= 0,V v E N ,
XE
R.
Therefore ( 7 ) w i l l i m p l y t h a t t h e i n f i n i t e m a t r i x
....... wo(x)
(wo j x ) ) R+l I I I I I I
/
I I
I I I I I
I I I
I I I
I
I
(wv(x))
\
Rt 1
....... w v p
I I
\
I I
;
\ ,
I I I
I I I
I I I I I I
1
I
I I
I I
I I
I I
I I
I
I
I I I
I I
I
//
I
has rank a t most
"
I I
I
I
-.I
1
/
I 1
R t l , f o r any g i v e n x E R'.
NOW, a well-known p r o p e r t y o f Vandermonde determinants i m p l i e s t h a t t h e i n f i n i t e sequence o f numbers wo(x),.
.. ,wv(x),. .....
c o n t a i n s a t most k l d i f f e r e n t terms, f o r any g i v e n x E 0,'. T h e r e f o r e Lemma 2 below w i l l g r a n t t h e e x i s t e n c e o f a closed, nowhere-dense subset rl C R', such t h a t each x E R' \ r ' possesses an open neighbourhood R" C R' \ r ' , w i t h t h e p r o p e r t y t h a t t h e i n f i n i t e sequence o f f u n c t i o n s
wo
.... ,w v y ..........
when r e s t r i c t e d t o R", c o n t a i n s o n l y a f i n i t e number o f d i f f e r e n t func= t i o n s . I n o t h e r words, t h e r e e x i s t s a subsequence w " i n w and $" E co(nlo) such t h a t
(9)
w" = u ( q ~ " ) on
R".
41
CONDITIONS FOR SOLUTIONS Now, w E R w i l l i m p l y t h a t w i l l contradict (7).
J," = 0 on
6 R.
Conversely, assume t h a t w
a".
And then ( 9 ) t o g e t h e r w i t h (8)
Then
non-void, open, w ' subsequence i n w; J,' E Co(n') :
(10)
3 R' c R
(10.1)
w ' = u(J,') on
(10.2)
J,'(x) # 0,
v
a'; x
E
R'.
We n o t i c e t h a t i n view o f (10.1) i t may, b y t a k i n g J,' = w ' , f o r any v E N, be assumed t h a t $' E C" (R). Then f o r any $ E C" (R), withvzupp J, c R' , we have W'.U($) = u ( $ ' . $ )
E
Now as )I i s a r b i t r a r y , t h e r e l a t i o n (10.2) w i l l i m p l y lWl n Uco(R) hence Awl F ALCo(n), c o n t r a d i c t i n g t h e h y p o t h e s i s .
# Q , 0
Lemma 2
....
Suppose t h e sequence w = ( w ,w i s such t h a t f o r any g i v e n 9 E R, he&! W,(X)
Y . .
......)
o f continuous f u n c t i o n s on R sequence o f numbers
. YWV( x ) f . . ...
c o n t a i n s o n l y a f i n i t e number o f d i f f e r e n t terms. Then t h e r e e x i s t s a closed, nowhere-dense subset r c R, such t h a t t h e sequence o f f u n c t i o n s W o,..
. ,wv,. ......
r e s t r i c t e d t o a s u i t a b l e neighbourhood o f any g i v e n x E R \ o n l y a f i n i t e number o f d i f f e r e n t terms.
r,
contains
Proof Denote by
r
wo
t h e s e t o f a l l p o i n t s x E R such t h a t t h e sequence o f f u n c t i o n s
.... ,wv
9 . .
......
when r e s t r i c t e d t o any neighbourhood o f x, c o n t a i n s i n f i n i t e l y many d i f = f e r e n t terms. I t i s easy t o see t h a t r i s c l o s e d . T h e r e f o r e i t o n l y remains t o prove t h a t r has no i n t e r i o r . I t s u f f i c e s t o show t h a t
Indeed, denote R' = i n t r and assume R ' # 4 . Then I-' c o r r e s p o n d i n g as above t o a',w i l l s a t i s f y r 1 = R ' , hence c o n t r a d i c t i n g (11). I n o r d e r t o o b t a i n ( l l ) , t h e B a i r e c a t e g o r y argument w i l l be used i n two successive steps. F i r s t , f o r p E N , define the closed s e t
42
E.E.
Rosinger
tlvEN, v >p+ 1 : A
P
(13)
= {X E R
R'
jhEN, X Q p :
u
=
A;.
PEN Indeed, denote for x E R' Mx = {(A,v) E N x N and t a k e
P E
I
X < u Q p , w,(x)
N, such t h a t
I/(P+~)dmin then o b v i o u s l y x E
~ l w ~ ( x ) - w , ( x ) l /(x,V)
A;.
Now i n a d d i t i o n t o ( 1 3 ) , we show t h a t (14)
A; closed, tl P
E
N.
Indeed, d e n o t i n g
M = {(X,V) we have
# wv(x)}
E
NxN
X < u
E
M~};
43
CONDITIONS FOR SOLUTIONS
Now ( 1 3 ) and ( 1 4 ) t o g e t h e r w i t h t h e B a i r e c a t e g o r y argument i m p l y t h a t i n t A; f 0 Denote then 62'' = i n t A[',. for a certain E N. o b v i o u s l y be complete i f we show t h a t ,)
(15)
R" c R
The p r o o f o f ( 1 2 ) w i l l
r.
\
Assume t h e r e f o r e x E R" and V c R" an open, connected neighbourhood o f x . We s h a l l prove t h a t t h e sequence o f f u n c t i o n s
. . ,w
wo,.
. . . ..
y . .
when r e s t r i c t e d t o V, c o n t a i n s a t most 1-1 + 1 d i f f e r e n t terms. Indeed, i f v 6 N, w 2 LI + 1, t h e n w V ( x ) = w,(x), f o r a c e r t a i n X E N , X < u , s i n c e
But then (16)
wv
wh
=
V.
on
Assume indeed t h a t ( 1 6 ) i s f a l s e . Denote
V'
=
ix' E
v
V"
=
Ix" E
v
1
V " = {x" E
1
V
/
1
for a certain y
V
E
W"(X') = w,(x')l W"(X") # w x ( x " ) 1 ;
then x E V ' , y E V", V = V ' u But V " i s a l s o c l o s e d , s i n c e
(17)
Then w ( y ) # w,(y),
V",
V' n
V"
and
= @
V'
i s obviously closed.
w I( X I ' ) - wx ( X ' ' ) l > 1 / ( 1 ' + 1 ) I
\
t h e i n c l u s i o n 3 b e i n g obvious, w h i l e t h e converse r e s u l t s as f o l l o w s . Take x" E V", t h e n t h e r e e x i s t s n E N, o < , such t h a t w ( x " ) = w ( x " ) , 0 s i n c e v > p t 1 and
Hence w,(x")
,
# w,(x")
x" E
v"
C
V
C
t h e r e f o r e n, A
R"
C
<
and
A'
P
w i l l imply t h a t
I wv ( x
'1
) -w A ( x " )
I
=
I wo ( x
"
) -wx ( x '1 ) I
> 1/ ( D t 1) ;
and t h i s completes t h e p r o o f o f ( 1 7 ) .
As t h e decomposition V = V ' U V " t h a t has been o b t a i n e d c o n t r a d i c t s t h e connectedness o f V, i t f o l 1ows t h a t ( 16) h o l ds . Now, ( 1 6 ) i m p l i e s ( 1 5 ) , which i m p l i e s ( 1 2 ) . proved.
Thus f i n a l l y ( 1 1 ) has been U
V.
44
2.
E.E.Rosinger
A p p l i c a t i o n s t o L i n e a r and N o n l i n e a r PDEs
We s h a l l p r e s e n t several a p p l i c a t i o n s o f t h e c h a r a c t e r i z a t i o n o f subse= quence q u a s i - i n v a r i a n t s e q u e n t i a l s o l u t i o n s , o b t a i n e d i n Theorem 1 above.
I t f o l l o w s e a s i l y from t h a t theorem t h a t T(D)-'(u(f)
t
R) c (cm(n)lN
i s t h e s e t of a l l subsequence q u a s i - i n v a r i a n t s e q u e n t i a l s o l u t i o n s o f t h e PDE i n ( l ) , Chapter 1. Therefore t h e e x i s t e n c e o f such s o l u t i o n s i s equi= valent t o the condition
(18)
V R'C
n non-void,
open, s'subsequence i n s, $ ' = 0 on
T s ' = u ( f t $ ' ) on R'
+'
E
f(n)
:
a'.
The PDO i n ( 3 ) Chapter 1, w i l l t h e r e f o r e be c a l l e d expansive o n l y i f
3 SE ( P ( n ) ) N :
(19)
tl s ' subsequence i n s: n
int
VYPE
N
Z(T(D)sb
where z ( g ) = { x E Ig(x) = f u n c t i o n g E C"(n).
-
T ( D ) s ' ) = I$ P
0) denotes t h e z e r o - s e t o f t h e continuous
Theorem 2 I f t h e PDO i n ( 3 ) , Chapter 1, i s expansive, then t h e corresponding PDE
T(D)u(x) = f ( x ) , x
E
R,
possesses a subsequence q u a s i - i n v a r i a n t s e q u e n t i a l s o l u t i o n s E ( cm(n)) N f o r any g i v e n f E Co(n). Proof T h i s f o l l o w s e a s i l y from ( 1 9 ) and ( 1 8 ) .
n
We s h a l l now show t h a t several well-known l i n e a r o r n o n l i n e a r PDOs a r e expansive, and t h a t t h e r e f o r e t h e corresponding PDEs possess subsequence q u a s i - i n v a r i a n t s e q u e n t i a l s o l u t i o n s f o r any g i v e n continuous r i g h t - h a n d term. F i r s t , l e t us c o n s i d e r t h e l i n e a r PDO o f (20)
L(D) =
a a t iaxl ax2
H. Lewy, h 2 0 1:
a, x =
2i(x t i x ) 1 2 ax3
which f o r c e r t a i n f E Cm(R3) g i v e s l i n e a r PDEs
( x x ,x ) 1' 2 3
E
R3,
CONDITIONS FOR SOLUTIONS
(21)
L(D) + ( x ) = f ( x ) , x
E
45
R3
w i t h n o t even l o c a l d i s t r i b u t i o n s o l u t i o n s . We s h a l l show now, t h a t t h e o p e r a t o r L(D) : C 1 ( R 3 )
C "(R3)
-f
corresponding t o (20), i s expansive and t h a t t h e r e f o r e t h e e q u a t i o n ( 2 1 ) has subsequence q u a s i - i n v a r i a n t s e q u e n t i a l s o l u t i o n s SE ( cl( R 3 ) )N, f o r any g i v e n f E c"(R3). Indeed, d e f i n e s
E
( C 1 ( R 3 ) ) N by
s V ( x ) = v(x1+x2+x3), V v E N, x = (x1,x2,x3)
E
R3;
t h e n a s i m p l e computation y i e l d s
-
Z(L(D)sv
L(D)s
1-I
1
= {($,
- by
x 3 ) I x3 E R 1 I y V v Y u E
N,
v
#
t h e r e f o r e t h e r e l a t i o n (19) h o l d s . As a second example, l e t us c o n s i d e r t h e n o n l i n e a r c o n s e r v a t i o n l a w (22)
ut(x,t)
+ a(u(x,t)).u,(x,t)
=
0, x E R1, t 2 0
and assume t h a t (22.1)
a
non-constant on any i n t e r v a l i n R1.
C'(R'),
E
We s h a l l show t h a t t h e n o n l i n e a r PDO T ( 0 ) : C1(R)
+
C"(n)
w i t h R = R' x(O,m), d e f i n e d b y t h e l e f t - h a n d t e r m i n (221, i s expansive. Indeed, d e f i n e S E (C'(R))N b y sv where h v y kv
(23)
+ kvt, V
(x,t) = h x V
E
R1\
h .k # hu.kvy v u We s h a l l show t h a t i n t Z(T(D)sv
E
R,
w i t h v E N, s a t i s f y t h e c o n d i t i o n
{O},
V
(23.1)
(24)
v E N, ( x , t )
-
V,
1-1 E N, v f
u.
T(D)sP) = 4 , V v, p
E
N, v
# u.
Indeed , assume t h a t T(D)sv = T ( D ) s
u
on R '
f o r a c e r t a i n R ' c R non-void, open and v, P E N , v f 1-1. computation y i e l d s kv
+
hva(hvx+kvt)
= k
!J
+
h a ( h x+k t ) , V ( x , t ) l
J
u
u
Then a d i r e c t E
R';
p;
46
Rosinger
E.E.
hence, a p p l y i n g t h e p a r t i a l d e r i v a t i v e s a/%, o f t h e above r e l a t i o n , we have = h k a'(h,,xtk
h,k,a'(h,xtk,t)
uu
P
o r e l s e a/ax, t o b o t h terms
t)
v h:
(x,t)
E Q'
,
a'(h,xtk,t)
= h i a l ( h xtk,,t)
which w i l l o b v i o u s l y c o n t r a d i c t ( 2 3 . 1 ) and ( 2 2 . 1 ) , t h u s c o m p l e t i n g t h e p r o o f o f ( 2 4 ) . Now ( 2 4 ) o b v i o u s l y i m p l i e s t h a t t h e sequence o f f u n c t i o n s i n (23) s a t i s f i e s (19).
A f i n a l example i s t h e second-order n o n l i n e a r wave e q u a t i o n Utt(x,t)
(25)
-
u x X ( x , t ) + f(u(x,t),u,(~,t),u,(x,t))
x
6
R',
{dl
C
R3.
= 0,
t>O
where one assumes t h a t (25.1)
f
C'(R3) non-constant on any subset ( a , b ) x
E
Ccl
x
We s h a l l prove t h a t t h e n o n l i n e a r PDO (26)
T(D) : c 2 ( Q )
+
P(Q)
= R ' X(O,cu), d e f i n e d b y t h e l e f t - h a n d t e r m i n ( 2 5 ) , i s expansive. with Indeed, a d i r e c t computation w i l l show t h a t t h e o p e r a t o r (26) and t h e sequence o f f u n c t i o n s ( 2 3 ) s a t i s f y ( 1 9 ) .
As p a r t i c u l a r cases o f t h e PDE i n ( 2 5 ) , t h e n o n l i n e a r Klein-Gordon equa= tion
-
Utt
w i t h a, m
E
utt
uxx
t
R'\{O),
- uxx
au
m
= 0, x E
R ' , t >O,
as w e l l as t h e sine-Gordon e q u a t i o n t
a s i n u = 0, x
E
R',
t 20,
w i t h a E R ' \ l o ) , o b v i o u s l y s a t i s f y t h e c o n d i t i o n (25.1); hence t h e corresponding n o n l i n e a r PDOs, d e f i n e d b y t h e i r l e f t - h a n d terms, a r e a l s o expansive. 3.
Subsequence I n v a r i a n t Sequential S o l u t i o n s
As can be seen from t h e examples i n S e c t i o n 2, t h e n o t i o n o f subsequence q u a s i - i n v a r i a n t s e q u e n t i a l s o l u t i o n i s r a t h e r t o o general. We s h a l l t h e r e f o r e i n t r o d u c e a more r e s t r i c t e d v e r s i o n o f t h e above n o t i o n b y demanding a c e r t a i n u n i f o r m i t c o n d i t i o n on t h e subsequence i n v a r i a n c e . The r e s u l t i n g n o t i o n o so u t i o n w i l l t u r n o u t t o have a t l e a s t p a r t l y customary p r o p e r t i e s , such as t h a t i t s a t i s f i e s t h e PDE i n ( l ) , Chapter 1, on c e r t a i n subsets i n a. A b a s i c c h a r a c t e r i z a t i o n of these s o l u t i o n s w i l l a l s o be presented.
+
cm(Q))
N A sequence of f u n c t i o n s s E i s c a l l e d a subsequence i n v a r i a n t s e q u e n t i a l s o l u t i o n o f t h e PDE i n ( l ) , Chapter 1, o n l y i f t h e r e e x i s t s a q u o t i e n t a l g e b r a A = A / I E A L c o ( a ) , such t h a t
CONDITIONS FOR SOLUTIONS (27)
wSi
I,
E
47
V s ' a subsequence in s .
A useful, simple characterization o f the above type of sequential solu= tions i s now presented. Given a sequence w E ( c " ( ~ )of) ~continuous functions, define AW = Aw/Iw
where Aw i s the subalgebra i n (cD(n))N generated by U c" a l l the subsequences w' i n w , while 1' i s the ideal in the subsequences w' in w.
(a)together AW
with generated by a l l
ProDosition 3 A sequence of functions s E ( E " ( ~ )i)s ~a subsequence invariant sequential solution of the PDE in ( l ) , Chapter 1, only i f
AWS
E
ALP(,,.
Proof First we notice that for any subsequence w' in ws, there exists a subse= quence s ' in s such t h a t w' = w s l . W
Assume now t h a t (27) holds. Then obviously A ' c Therefore in view of ( 2 3 ) , Chapter 1, wS
I
('
(n) c ~ nc" u(Q)
A, hence
TS
c I.
=a
which means t h a t
The converse i s immediate.
0
An a1 ternative, simple characterization can be obtained as follows. N Call a subset H c ( M ( R ) ) subsequence invariant, only i f 11 w E
H , w ' subsequence in w :
(28)
W'E H. subsequence invariant, only i f Call a quotient-algebra A = A / l E A L C" (a) T and A are subsequence invariant, and denote by
the set of a l l such quotient algebras.
w
E(CD(R))N
Obviously, for any given
E.E. Rosinger
48
1' and Aw a r e subsequence i n v a r i a n t .
(29)
Proposition 4
A sequence o f f u n c t i o n s s i s a subsequence i n v a r i a n t s e q u e n t i a l s o l u t i o n o f t h e PDE i n ( l ) , Chapter 1, n l y i f t h e r e e x i s t s a subsequence i n v a r i a n t q u o t i e n t a l g e b r a A = A j l E A L $(il) , such t h a t
ws
E 1.
Proof Assume A = A / I E A L
sB C"
(a)and
YE
W
Then o b v i o u s l y A
1.
S
W
c A, hence 1 c 1.
Therefore i n view o f (23), Chapter 1, we have
u ~ ~ c (I ~n u)c"(n)
7"s n i . e . Al's
E
ALCo(n).
= Q
Now P r o p o s i t i o n 3 i m p l i e s t h a t s i s t h e r e q u i r e d t y p e
o f solution.
The converse f o l l o w s f r o m (29) and P r o p o s i t i o n 3.
0
I n o r d e r t o o b t a i n n e x t i n Theorem 3 t h e b a s i c necessary c o n d i t i o n on sub= sequence i n v a r i a n t s e q u e n t i a l s o l u t i o n s , s e v e r a l d e f i n i t i o n s a r e needed.
A quotient algebra A = A / l EAL
n'c n non-void open (30)
A' = A'/l'
E
C"
(n)
i s c a l l e d h e r e d i t a r y , o n l y i f f o r any
ALCo(Rl)
where 7 ' and A ' a r e o b t a i n e d from 1 , o r e l s e A, b y r e s t r i c t i n g t o f u n c t i o n s i n t h e corresponding sequences o f f u n c t i o n s , i.e. 'I' = { w '
I
=
( w o,. '
. . yw;,.
. .)
1
w = (wo,.
. . ,w " , . . . . ) E
0' t h e
13
where w ' = w I ; s i m i l a r l y f o r A ' . When needed f o r t h e sake o f c l a r i = ty, we ? h a l l " e ! ? p l i c i t e l y s p e c i f y n' , u s i n g t h e n o t a t i o n s :
t h e s e t of a l l t h e h e r e d i t a r y q u o t i e n t algebras A = A / I E A L ~ ~ ( ~ ) . F u r t h e r , we s h a l l say t h a t t h e q u o t i e n t a l g e b r a A = A / l E A L C O ( n ) only i f
is full,
CONDITIONS FOR SOLUTIONS
a' c
V z E A,
49
R non-void, open:
(31)
where l / ( z t h e terms
) denotes t h e sequence o f continuous f u n c t i o n s on
1 R' l/zv,
f o r v E N,
Q',
with
We s h a l l denote by
t h e s e t o f a l l t h e f u l l q u o t i e n t algebras A = pJ1 E A L ~ ~ ( ~ ) . Obviously
A
=
A/l EAL
C" ( Q)
(32)
-
A = A/I
n) *
E A C" L (~
A = (C"(Q))N F i n a l l y , we denote b y
-
R
t h e s e t o f a l l t h e sequences o f f u n c t i o n s w E ( C " ( Q ) ) vanishing condition
N
s a t i s f y i n g the
V Rl c R, non-void, open: ~ L J E N :
(33)
V v E
3 x
N,
E
>u:
v
R':
wv(x) = 0. Proposition 5 Indc
2
c R
(see ( 6 5 ) , Chapter 1 )
and a l l t h r e e s e t s of sequences o f f u n c t i o n s a r e subsequence i n v a r i a n t . Proof The i n c l u s i o n 'Indc
2
i s obvious.
Assume now w E R \ R. Then, t h e r e e x i s t S2' c S2 non-void, subsequence i n w and $ ' E c O ( R ' ) , such t h a t
(34.1)
w ' = u ( $ ' ) on
S2'
open, w ' a
E.E. Rosinger
50 (34.2)
+ ' ( x ) # 0, V x E
n'.
But i n view o f (33) i t f o l l o w s t h a t N , v 2 ~ :'3 xu E R ' : W ' ( X ) = 0. v v which c o n t r a d i c t s Now (34.1) w i l l i m p l y t h a t + ' ( x ) = 0, V v E N , v
3
N : V v
PIE
E
(34.2).
V
I t i m n e d i a t e l y f o l l o w s t h a t lnd, 2 and
+
R s a t i s f y (28).
0
We now discuss t h e b a s i c necessar c o n d i t i o n on subsequence i n v a r i a n t s e q u e n t i a l s o l u t i o n s , w h i c is an analog o f t h e corresponding p a r t o f Theorem 1, S e c t i o n 1. Theorem 3 i s a subsequence i n v a r i a n t Suppose t h e sequence o f f u n c t i o n s s E s e q u e n t i a l s o l u t i o n o f t h e PDE i n (l), Chapter 1, such t h a t
(35) f o r a c e r t a i n subsequence i n v a r i a n t , h e r e d i t a r y and f u l l q u o t i e n t a l g e b r a A = A/1 E ALSB H F.
c" I
d
Then ws s a t i s f i e s t h e v a n i s h i n g c o n d i t i o n
(36)
ws
E
2.
Proof Assume t h a t (36) i s f a l s e . Then i n view o f (33) t h e r e e x i s t v o i d , open and a subsequence s ' i n s , such t h a t
(37)
# 0, V v
w;(x)
E
N, x
E
R' c
n
non-
a'
where, f o r t h e sake o f s i m p l i c i t y , we have used t h e n o t a t i o n
w'
wsl.
As w ' i s o b v i o u s l y a subsequence i n ws, t h e c o n d i t i o n (35) w i l l i m p l y ( 38)
W'E
7
s i n c e A=M i s subsequence i n v a r i a n t . (38), (37) and (31) w i l l y i e l d u(1)
I?'
).(1/(w' = (w'iRl
But A = A / 1 i s a l s o f u l l , t h e r e f o r e ,
))
E
1
In. Aln' I;.
which o b v i o u s l y i m p l i e s t h a t
thus c o n t r a d i c t i n g t h e f a c t t h a t A = A/I i s h e r e d i t a r y .
c 7
CONDITIONS FOR SOLUTIONS
51
Next we have a simple sufficient condition on subsequence invariant sequen= t i a l solutions, which i s also an analog of the respective implication in Theorem 1 , Section 1. Theorem 4 Suppose the sequence o f functions s
E
(6"(Q ) ) ~s a t i s f i e s
the condition
Then s i s a subsequence invariant sequential solution of the PDE in ( l ) , Chapter 1. Moreover, there exists u sequence invariant, hereditary and , such t h a t full quotient algebra A = A/l E AL sBy'yp
r(n)
ws
E
1
Proof
Taking A 4.
=
A n d , t h i s follows from (64), Chapter 1.
0
Resolvent Sets
The results i n the previous section establish an interest in subse uence invariant sequential solutions, and in view of Proposition 4 +e s ow t a t subsequence invariant quotient a1 gebras offer the natural framework for f i n d i n g such solutions. Let us use the notation R~~ = u
r
where the union i s taken over a l l the subsequence invariant quotient a l = SB gebras A = A j l E ALC,(n). Obviously (39)
T(D)-'(u(f)
+
RsB)
c
(C?(n))N
will be the s e t o f all subsequence invariant sequential solutions of the PDE in ( l ) , C h a p t e r T I t follows that the solution in the above sense of the PDE mentioned in= volves two steps. The i r s t , independent of the PDE, consists in suitable c h a r a c t z z a t i o n s of R SB The second, dependent on the PDE considered, consists in suitable characterizations of the inverse image in (39), primarily answering the question as t o whether t h a t inverse image i s nonvoid.
.
I t i s easy to see t h a t Theorem 1 yields the following:
Corollary 1 RsB
C
R.
I n view of the above, we shall next introduce a general definition.
52
E.E. Rosinger
Given a p r o p e r t y P, v a l i d f o r c e r t a i n q u o t i e n t algebras A = A / I E A L C O ( n ) , den0 t e by
t h e s e t o f a l l those q u o t i e n t a l g e b r a s h a v i n g t h e p r o p e r t y P. A sequence o f f u n c t i o n s S E w i l l be c a l l e d a P-sequential s o l u t i o n o f t h e PDE i n ( l ) , Chapter 1, o n l y i f
f o r a c e r t a i n A = A / l EAL
P C o p ) '
The s e t o f sequences o f continuous f u n c t i o n s on R
P
= U
a, given
by
I
where t h e u n i o n i s taken o v e r t h e q u o t i e n t a l g e b r a s A = A / r E A L P
,
is
c a l l e d the P-resolvent set. NOW, t h e necessary, o r e l s e s u f f i c i e n t c o n d i t i o n s on subsequence i n v a r i a n t s e q u e n t i a l s o l u t i o n s o b t a i n e d i n Theorems 3 and 4, w i l l r e s u l t i n t h e f o l 1owing c o r o l 1a r y .
Corol l a r y 2
I t f o l l o w s t h a t a b e t t e r c h a r a c t e r i z a t i o n o f t h e r e s o l v e n t s e t R SB ,H ,F
pre-supposes a n a r r o w i n g o f t h e gap between Ind and R . w i l l be presented i n Chapter 6.
Related r e s u l t s
However, i t should be n o t e d t h a t t h e simultaneous demand f o r t h e t h r e e p r o p e r t i e s 'subsequence i n i a r i a t ' , ' h e r e d i t a r y ' and ' f u l l ' c o n c e E i K j s e q u e n t i a l s o l u t i o n s s E(C ( Q ) ) [ m i g h t l e a d t o r a t h e r p a r t i c u l a r - s o l u = t i o n s . Indeed, i n view o f t h e v a n i s h i n g c o n d i t i o n ( 3 3 ) d e f i n i n g R , t h e inclusion
i n C o r o l l a r y 2 means t h a t t h e e r r o r sequence ws E R S B y H * F corresponding t o s , w i l l vanish r a t h e r o f t e n on R , v i z V fi' c R non-void, open :
3
(40)
P E N :
tt v
E
N, v 2 ~ :
R ' n Z(T(D)sV-f)
# 9.
I n o t h e r words, t h e zero-sets Z(T(D)sv-f) w i t h v
E
N, a r e ' a s y m p t o t i c a l l y
CONDITIONS
FOR SOLUTIONS
53
dense' i n R, a s i t u a t i o n which i n a way i s more p a r t i c u l a r t h a n t h e u n i = form convergence o f T(D)sv t o f on R, when v -+ m, as seen n e x t i n t h e r e l a = t i o n (48). Now among t h e above-mentioned t h r e e p r o p e r t i e s , t h e f i r s t two a r e o f topo= l o g i c a l n a t u r e , w h i l e t h e l a s t i s a l g e b r a i c . The demand f o r t h e f i r s t 3 2 t h e b s e q u e n c e i n v a r i a n t ' p r o p e r t y , seems t o be j u s t i f i e d , n o t l e a s t because t h e n o t i o n o f s o l u t i o n used i n S e c t i o n s 1 and 2 proves t o be r a t h e r general. T h e r e f o r e i f c o n d i t i o n (40) appears t o be t o o s t r o n g , t h e 'here= d i t a r y ' o r ' f u l l ' p r o p e r t i e s c o u l d be r e l i n q u i s h e d . I n t h i s case, i n view o f (31) and (32), t h e q u o t i e n t a l g e b r a s A = A / l i n v o l v e d , m i g h t have A n o t l a r g e enough, a s i t u a t i o n which i n view o f (43.2), Chapter 1, w i l l n o t be most f a v o u r a b l e t o t h e exactness o f t h e corresponding s e q u e n t i a l so= 1u t i o n s . Remark 1 The d i f f i c u l t y o f t h e problem concerning t h e r e l a t i o n between t h e r e s o l = vent s e t s encountered above i s i l l u s t r a t e d i n t h e r e l a t i o n s
F
(41)
RsB
I t A
proved b y t h e f o l l o w i n g examples. I t i s easy t o see t h a t i t i s p o s s i b l e t o con= Assume n = l and R = ( 0 , l ) . s t r u c t a sequence o f continuous f u n c t i o n s w €(C'(n))N, such t h a t
o
(43)
z(wV) = { ( 2 i t 1 ) / 2 ~ l /
(44)
W ~ ~ ( tXW) ~ * ~ ( X=) 1, tl v E N, x
v
< i<2"-1), E
v E N
n.
Then (45)
w
E
R
\ RsB.
f o l l o w s e a s i l y from (43) and ( 3 3 ) , w h i l e t h e Indeed, t h e r e t i o n w E relation w g R can be deduced as f o l l o w s . Assume i t i s n o t t r u e and t h a t w E Z, f o r a c e r t a i n A = A / ~ E R S B . Take w ' , w" subsequences i n w , given by
48
w\], = w ~ W:~ =, wZvtl
,V
v E N.
Then w ' , w" E 7 , s i n c e 7 i s subsequence i n v a r i a n t . w i l l yield
w'
t
w"
= u(1) E 7
NOW, t h e r e l a t i o n (44)
n Up (n),
c o n t r a d i c t i n g (23), Chapter 1, and c o m p l e t i n g t h e p r o o f o f ( 4 5 ) . Define
,A, where ,A,
= Auc/7uc
N i s t h e a l g e b r a o f a l l t h e sequences z E (c" (n)) o f continuous
54
E.E.
Rosinger
a
i s the ideal f u n c t i o n s on R, u n i f o r m l y bounded on c mpacts i n R, w h i l e iyc o f a l l t h e sequences WE ( c " ( R ) ) o f continuous f u n c t i o n s on R, u n i = i n ,A, f o r m l y convergent t o zero, on compacts i n R.
Then o b v i o u s l y
SB ,H
(46)
Auc = Auc/Iu,
E ALCo
(n);
however (47)
,A,
D e f i n e now
wEiUC
e
= A,,/Z,~
by
= l/(v+l),
w,(x)
AL:~(~).
V v EN, x
E
R;
then obviously (48)
W
E
RSByH \
t.
The r e l a t i o n s (45) and ( 4 8 ) w i l l o b v i o u s l y y i e l d ( 4 1 ) and (42). C a r e f u l c o n s i d e r a t i o n o f t h e sequence o f continuous f u n c t i o n s g i v e n i n (43-45), leads t o t h e f o l l o w i n g r e s u l t . Denote b y R~~ t h e s e t o f a l l t h e N sequences w E (c0(n)) o f continuous f u n c t i o n s on R, s a t i s f y i n g
R'
V
(49)
3x
E
lim v+m
C
R non-void, open :
R' : w,(x)
= 0.
Theorem 5 The f o l l o w i n g r e l a t i o n h o l d s Rdc C RSByH.
Proof Assume w
E Rdc.
We s h a l l show t h a t
I n view o f (23), Chapter 1, i t s u f f i c e s t o prove t h e r e l a t i o n
Now an element o f t h e above i n t e r s e c t i o n o b v i o u s l y has t h e f o r m
CONDITIONS FOR SOLUTIONS where a, ba E N \ l o } , follows that
J I , $a E "(R)
and w , ' ~ a r e subsequences i n w.
55 It
t h e r e f o r e , i n view o f ( 4 9 ) , we have non-void, open : 3 x E R' : $ ( x ) = 0
V 0' c
which r e s u l t s i n
v
$(X) = 0,
x
i s continuous on R.
E
R
T h i s completes t h e p r o o f o f ( 5 1 ) , and hence o f
t i s easy t o see t h a t ( 5 0 ) and ( 4 9 ) w i l l y i e l d
t h e r e f o r e w E RSByH. 5.
0
Domains o f S o l v a b i l i t y
The problem o f t h e r e g u l a r i t y o f s e q u e n t i a l s o l u t i o n s w i l l be approached i n t h i s s e c t i o n . The method i s r a t h e r d i r e c t as i t aims t o e s t a b l i s h p r o = p e r t i e s o f t h e s e t o f p o i n t s x E R which have neighbourhoods on which t h e s e q u e n t i a l s o l u t i o n s a r e c l a s s i c a l s o l u t i o n s . The main t o o l s used i n t h i s c o n n e c t i o n w i l l be Lemma2,Section 7, Chapter 1 and Lemma 2 , S e c t i o n 1. Suppose, t h e sequence o f f u n c t i o n s t E (?(a)) N i s a s e q u e n t i a l s o l u t i o n i n E + A o f t h e m-th o r d e r PDE i n ( l ) , Chapter 1, where m E = S / V E VS:m(n), A = A / 7 E AL e(n)' A Obviously, t i s a classical solution, only i f (52)
T(D)tv(x) = f ( x ) , V x
E
R, v
E
N
The aim o f t h i s s e c t i o n i s t o e s t a b l i s h a ' b e s t a p p r o x i m a t i o n ' o f t h e con= d i t i o n (52) i n t h e case o f s e q u e n t i a l s o l u t i o n s . We s h a l l denote by
n E + A the set o f a l l the points x E R s a t i s f y i n g the condition
S
,V
(53)
1 t
E
(53.1)
wt
T(D)t
-
neighbourhood o f x:
u(f) E 2
E.E.
56
Rosinger
(53.2)
VvEN,v>p:
T(D)tV(Y) = T ( W p ( Y ) The s e t RE
~
A w i l l be c a l l e d t h e domain o f l o c a l s o l v a b i l i t y i n E -+ A o f
t h e PDE i n ( l ) , Chapter 1, w h i l e t h e s e t rE
-+
A
=n\
QE+A
w i l l be c a l l e d t h e l o c a l s i n g u l a r i t y i n E
(55)
t e
''+A=
Wt
n
s n(cm(n))N
cl
-+
A o f t h e mentioned PDE.
u u cNve N
A (T(D)tv
-
T(D)tp)
v>u
E I
where we denoted A(g) = R \ Z ( g ) ,
f o r g E C" ( Q )
Proof It s u f f i c e s t o show (54), t h e r e s t o f P r o p o s i t i o n 6 r e s u l t i n g e a s i l y .
Assume t h a t x belongs t o t h e l e f t hand t e r m i n (54). t E S , w i t h wt E 1 , such t h a t x
E
int u p E
n Z(T(D)t" N vEN v> p
-
Then t h e r e e x i s t s
T(D)tp)
Then
v > p
w i l l be a neighbourhood o f x.
Moreover, i f y
3 p E N ; V V E
N ,v>u :
T ( D ) t V ( y ) = T(D)t,,(Y)
E
V then o b v i o u s l y
CONDITIONS FOR SOLUTIONS
therefore x
E
RE
-+
57
A
Conversely, assume t h a t x obtain that
E
RE
+
A.
Then, w i t h t h e n o t a t i o n s i n (53), we
v2l-l
therefore
v2l.l The r e l e v a n c e o f t h e domain o f l o c a l s o l v a b i l i t y and o f t h e l o c a l s i n g u l a r < = ty w i l l be p r e s e n t e d i n Theorems 6 and 7.
F i r s t , we denote by
'E
-+
A
the set o f a l l the points x E
V neighbourhood o f x, p
(56)
3 t
E
(56.1)
wt
1
(56.2)
V v E N, v 2 p:
E
S,
T(D)tv = T(D)tp
-+
N :
E
on V
Qi
A w i l l be c a l l e d t h e domain o f s t r o n g l o c a l s o l v a b i l i t y i n A o f t h e PDE i n ( l ) , Chapter 1, w h i l e t h e s e t
The s e t E
satisfying the condition
~
w i l l be c a l l e d t h e s t r o n g l o c a l s i n g u l a r i t y i n
E
+
A o f t h e mentioned PDE.
Proposition 7
Ri
-+
A i s open,
i-;
+
A i s c l o s e d and
Proof Again i t s u f f i c e s t o show t h a t (57) i s v a l i d .
Assume t h a t x belongs t o t h e r i g h t - h a n d t e r m i n (57). Then t h e r e e x i s t s t E S n(Cm(Q))N, w i t h wt E 1 , as w e l l as p E N, such t h a t
58
E.E. x Eint
- T(D)tp)
Z(T(D)t\,
n
Rosinger
v E N v > p
Then
n
V = int
V E
- T(D)tp)
Z(T(D)tv N
v > p
w i l l be a neighbourhood o f x, w i t h t h e p r o p e r t y t h a t T(D)tv = T ( D ) t u therefore x
S
RE
E
-+
on V, V v
N, v > p
A
Conversely, assume t h a t x E RE obtain that
n
V C
E
Z(T(D)tv
A.
+
Then, w i t h t h e n o t a t i o n s i n (56), we
- T(D)tu)
v E N
v>u theref o r e x
int
E
n v E N v
Z(T(D)tv
-
T(D)tu)
>!J
0
Theorem 6
rE+A
$-+AcRE+A’ (59)
E‘
-+
c
r:
+
A and
A “ ~ - + A = ‘ ~ + A ‘
r E + A i s nowhere dense i n R ,
i n o t h e r words
(60)
$i
~
A i s dense i n R E
-+
A
Proof The i n c l u s i o n s as w e l l as t h e e q u a l i t y i n (59) a r e obvious. Therefore, i t o n l y remains t o show t h a t (60) i s v a l i d . B u t (60) f o l l o w s e a s i l y f r o m Lemma 2, S e c t i o n 1. 0 An example sented now.
o f r e g u l a r i t y p r o p e r t y o f s e q u e n t i a l s o l u t i o n s w i l l be p r e =
Call a subset H C (M(n))N c o f i n a l i n v a r i a n t , only i f V w E(M(Q))N :
(61)
3 w ’ E H , p E N (V;r:i~ap:
*
W E
H
CONDITIONS FOR SOLUTIONS
59
Theorem 7 Suppose t h e q u o t i e n t a l g e b r a A = A / 7 has I c o f i n a l i n v a r i a n t . I f x E Q;
A, t
~
E
c o n d i t i o n s (56.1-2)
S, an open neighbourhood
LI E
N s a t i s f y the
, then
T ( D ) t v = f on V, U v
(62)
V o f x and
E
N, v 2~
i n o t h e r words, t, E Cm(Q), w i t h v E N, v 2 p, a r e c l a s s i c a l s o l u t i o n s o f t h e PDE i n ( l ) , Chapter 1. Proof We d e f i n e $
E
C" (Q) b y $ = T(D)t
(63)
LI
Then, i n view of (56.2),
wtv = T(D)tv - f = $ - f on V, V v
(64) Assume now
i t follows t h a t
x
(65 1
E
E
N, v 2 L.
c" (a), such t h a t SUPP
x
v
c
Then (66)
wt.u(x)
E
7 . uco
C I.A C
7
B u t (64) and (65) w i l l i m p l y t h a t
(67)
W ~ . X= ( $ - f ) x
,V v
E
N, v 2
LI
Now, i n view o f t h e f a c t t h a t 7 i s c o f i n a l i n v a r i a n t , (66) and (67) w i l l yield u((dJ-f)x)
E
7
Then, a c c o r d i n g t o (23), Chapter 1, i t f o l l o w s t h a t
(VJ-f)
x
= 0
t h e r e f o r e , i n view o f t h e f a c t t h a t
x i s a r b i t r a r y , we can conclude t h a t
$ = f which t o g e t h e r w i t h (63) w i l l y i e l d ( 6 2 ) .
0
In view o f (60) i n Theorem 6, i t s u f f i c e s t o know t h e s i z e o f t h e domain o f l o c a l s o l v a b i l i t y nE -f A. I t i s obvious t h a t t h e s i z e o f RE
I f we denote
f
A i n c r e a s e s t o g e t h e r w i t h S and I.
E .E. Rosi nger
60
then ( 5 4 ) can o b v i o u s l y be w r i t t e n i n t h e form
therefore
- wu )
n
u
P E N
vEN v >u
11 w
lTYf, R' c R non-void, open:
A (wv
i s dense i n R
or, e q u i v a l e n t l y E
3 X E
R' :
Moreover, i n view o f t h e i n c l u s i o n
U
'T,f
l i m Z(wv) c RE int v -+m
-+
A
Concerning t h e p o s s i b l e n a t u r e o f t h e f a m i l y ( Z ( w ) I v E N ) o f subsets i n , where w E I i s given, t h e f o l l o w i n g two exampyes p r e s e n t i n t e r e s t i n g cases.
Q
(e( Q ) ) Nwhich s a t i s f i e s m Z(WV) = b
Example 1 : w E
(73)
v-+m
the conditions
CONDITIONS FOR SOLUTIONS
U
(74)
61
R' c R non-void, open:
3 P E N :
UvGN,v>u: Z(WV) n R' # 0 We s h a l l c o n s i d e r t h e c a w R = R', s i n c e t h e c o n s t r u c t i o n can e a s i l y be ex= tended t o a r b i t r a r y R C Rn n o n - v o i d and open. Suppose ( x Iv E N) i s dense i n R and c r e a s i n g t8 zero, so t h a t
(75)
{xV +
Nln{xv V +
~ E~
E
E
E~
u IV E
> 0, w i t h v E N, a r e s t r i c t l y de=
N} = flyV A,p
f o r i n s t a n c e , ( x v l v E N) a r e t h e r a t i o n a l numbers and v E N. We d e f i n e w
= (x-x
w,(x)
o
-E
v
)...(X-X~-E~),
U x E R
,v
E
Then o b v i o u s l y
..
{ X ~ + E ~ , . , x ~ + E ~V ~v , E
Z(wW) =
N,
t h e r e f o r e , ( 7 3 ) r e s u l t s e a s i l y from ( 7 5 ) . Suppose now g i v e n R'
x
u'
f o r a certain X
u'
f o r a suitable
u' E +
C
R non-void and open.
Then
R'
E
N.
Ev
u"
~.r = max
E
E
N.
Therefore
R', U v
E N,
w >u''
Taking now
{u', ~ " 1
i t obviously follows t h a t X
1J-I
=
E~
(c" (R))N by
E
+
E Z(wv)
n a', Y
v E
N,
v 2
u
and t h e p r o o f o f (74) i s completed. Example 2 : w €(C?
(n))N which s a t i s f i e s t h e c o n d i t i o n s
l i m mes A (w,) = 0 v-+int l i m Z(wv) = D (78) v +m where ( x v l v E N) i s dense i n n.
N, X # 1-1
E
N
J2/(v+l), with
62
Rosinger
E.E.
We s h a l l c o n s i d e r t h e case R = ( 0 , l ) c R', C Rn non-void and open b e i n g obvious.
the extension t o a r b i t r a r y
R
Denote f o r v
E
N
t (XI, - xol
6\,= min and t a k e w
E
0
G v } /(v+l)
( C ' ( S ~ ) )such ~ that
u
(O,l)\
Z(wV)
(79)
0 Gp <
OQ,,,
(x,,
-
6 v , xx
+
6 v ) y tl v E N
Then, o b v i o u s l y mes Z(wv)
1
-
2(v+1)6,
hence
l i m mes Z(wv) = 1
v
+ m
and t h e p r o o f o f (77) i s completed. Now, we s h a l l e s t a b l i s h t h e p r o p e r t y i n ( 7 6 ) . (79) w i l l y i e l d
If p
E
N i s given
, then
(xx - G v J x
+6J
therefore
v
l i m Z(wv)
= (O,l)\
+ m
u
0 E N
U v E N
"
OGXGv
v > p
which w i l l o b v i o u s l y imply (76), s i n c e
lim
v
6,,
= 0
+ m
F i n a l l y , (78) f o l l o w s f r o m (76), i n view o f t h e f a c t t h a t dense i n a .
{xVlv E N} i s
Connected w i t h Example 1 above, an i n t e r e s t i n g problem i s whether t h e r e e x i s t sequences o f c o n t i n u o u s f u n c t i o n s w E (Co ( R ) ) N which s a t i s f y (73) and (74), as w e l l as (see ( 2 3 ) , Chapter 1)
(80)
I(w) n
where
uco
(a)
=
!!
7 ( w ) i s t h e subsequence i n v a r i a n t i d e a l i n (C'
N (a)) generated
by
W.
6.
Remarks on Lemma 2
I t i s easy t o see t h a t Lemna 2, S e c t i o n 1, i s a c t u a l l y v a l i d f o r sequences o f continuous f u n c t i o n s d e f i n e d on l o c a l l y connected t o p o l o g i c a l spaces o f
CONDITIONS FOR SOLUTIONS
63
second B a i r e category and t a k i n g values i n m e t r i c spaces. W i t h i n the men= t i o n e d framework, Lemma 2, Section 1, i s an extension o f Lemma 1, Section 7, Chapter 1, which can be formulated as f o l l o w s :
I f E i s a 1ocall.y connected t o p o l o g i c a l space, o f second B a i r e category, E ' i s a m e t r i c space and f : E -+ E ' , fw: E + E l , w i t h v E N, are continuous functions w i t h t h e Pro0ert.v V
X E
E :3pE
then, there e x i s t s
Y x f
E
E
N : Y v E N, v > p
rc E \ r: 3
: fV(x) = f ( x )
closed and nowhere dense, such t h a t p E N, V neighbourhood o f x:
=fonV,VvEN,v>p
V
The n o n - t r i v i a l i t y o f the r e s u l t i n Lemma 2 w i l l be i l l u s t r a t e d i n two examples. Example 3 : w E (C" ( ~ 2 ) which )~ s a t i s f i e s the c o n d i t i o n s
NI
< 2, U
(81)
car{wv(x)lv
(821
U R' c $2, R' non-void, open,
E
lQ1
c a r twv
=
r = 0
(85)
v E > o :
carEwv
R'
3
r
:
C" (R) such t h a t supp w
=
I€ u
V E
and d e f i n e w E (C"
,
Define
E
lw
NI
E
< 2,
NI
U x
R1, which s a t i s f i e s the c o n d i t i o n s E
R
= car N
Indeed, take
R
[O,ll
E N
R n (0,~).
=
=
{ O l and (81) as w e l l as (82) are v a l i d .
car{wv(x) Iw
(84)
where RE
E
w E (C' (n))N, w i t h R C
Example 4:
(83)
r
R
w((~tl)((~+2)~-l)),V x E R, w
=
Then obviously
E
I v E N 1 = car N
Indeed, take R = R ' and w w E (C' (Q))N by w,(x)
x
(1/(2w+2),1/ 2vtl) ) N by
1if x
E
(1/ 2vt2),1/ (2w+l))
wv(x) = 0 if x E n \ then i t i s easy t o see t h a t (83-85
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CHAPTER 3 ALGEBRAS CONTAINING THE DISTRIBUTIONS
0.
Introduction
As pointed out in Section 5, Chapter 1, a proper study of the sequential, in particular the weak or distribution solutions of PDEs, requires a s u i t = able determination of the interplay between the s t a b i l i t y , generality and exactness properties of the solutions involved. In view of the already traditional role played by distributions in solving linear or nonlinear PDEs, we shall from now on aim a t a level of generality n o t below the distributional one. Having defined t h i s aim, certain n a t u r a l methods will follow of dealing with the interplay between s t a b i l i t and exactness, leading t o chains of quotient algebras containing d s t r i bu = tions, o f the form
(1)
D'(n) c Am c
..... c A'c ..... c A O ,
II
E
4,
with the PDO in ( 3 ) , Chapter 1, acting between quotient algebras
(2)
T ( D ) : A'
with R , k E 1.
d,
R
+
Ak
- k > m y where m i s the order of T ( D ) .
Embedding the Distributions into Quotient Algebras
We shall start from the quotient space representation of the distributions (3)
o'(n)
=
s2/ v' ,
R E
fi
introduced in ( 3 1 ) , ( 2 4 ) , (25) and (26), Chapter 1.
A simple way of embedding the distributions into quotient algebras would be to construct inclusion diagrams of the form
r
9
- A
(C"(n))
65
N
E.E. Rosinger
66
R
w i t h A a subalgebra i n ( C ( Q ) )
r
(4.1)
nsR=
N
and I an i d e a l i n A, such t h a t
fi
g e n e r a t i n g the l i n e a r embedding of U'(R) i n t o t h e q u o t i e n t algebra A = A/7 AL R d e f i n e d by
c
(81,
P'(Q) =
sR/fi3( s
$1
t
(s
-f
t
r)
E A =
MI,
However, i n c l u s i o n diagrams o f t y p e ( 4 ) cannot be constructed, s i n c e
(fi.VR)
(5)
ve,
nsR F
w R€W
as can be seen u s i n g t h e argument i n t h e p r o o f o f (56), Chapter 1. Another simple way o f embedding t h e d i s t r i b u t i o n s i n t o q u o t i e n t algebras would be t o c o n s t r u c t i n c l u s i o n diagrams o f t h e form > A
7
w i t h A a subalgebra i n
>
(CR(WN
and I an i d e a l i n A, such t h a t
f i n A = 7 and R 9. V t A = S
(6.1) (6.2)
g e n e r a t i n g t h e l i n e a r i n j e c t i o n o f t h e algebra A = 4 / I o n t o U ' ( n ) , defined by A = wi 3
(S t
r)
+
(s
t
v')
E
D'(Q)
=
sR/vR.
Here, t h e problem a r i s e s connected w i t h t h e c o n d i t i o n (6.2) , which prevents A from c o n t a i n i n g some o f t h e f r e q u e n t l y used ' 6 sequences', as i s e v i d e n t from t h e lemma given below (see p r o o f i n S e c t i o n 11). Lemma 1 Suppose g i v e n s
E
(C"(S2))
N
such t h a t supp sv s h r i n k s t o xo
E
R, when v
-+
m.
I f s i s a sequence o f non-negative f u n c t i o n s , t h e f o l l o w i n g two p r o p e r t i e s are equivalent:
(7) lim
(8)
I
s,(x)dx
= 1;
v + m Q
If s
E
and
, then
<s,*> = 6 xO
s2
6 9.
ALGEBRAS CONTAINING THE DISTRIBUTIONS
67
However, the f a i l u r e of inclusion diagrams of types ( 4 ) and ( 6 ) can be overcome. Indeed, i n [169 - 1761 i t has been proved ( s e e a l s o Theorem 1 , Section 3, as well as t h e r e s u l t s i n Section 4 ) t h a t the following, more complicated inclusion diagrams can be constructed: 7
' (c-7 n) ) N
> A
I
w i t h A a sukalgekra i n ( C " ( R ) j N ; 1 an ideal in A ; spaces in V , S r e s p e c t i v e l y , such t h a t
(9.1)
I ~ S = V ;
(9.2)
V
(9.3)
sm =
m
n S = V;
and v , s vector sub=
and
Vrn t S;
and t h e r e f o r e generating t h e l i n e a r embedding of D'(n) i n t o t h e q u o t i e n t algebra A = A/I E AL c=(n), defined by U'(0) =
sm/vw
(10)
S/
W
s
+
v
A
111
vw-s
t
i som
=
S/7 W
v
s tl.
1 in , i n j
The intermediate q u o t i e n t space S / V plays t h e r o l e of a r e g u l a r i z a t i o n of the q u o t i e n t space representation of d i s t r i b u t i o n s in (3). The construe= t i o n of q u o t i e n t algebras containing t h e d i s t r i b u t i o n s e s s e n t i a l l y depends on the choice o f the r e g u l a r i z a t i o n s S/V , which a l s o determine t h e s t a b i = lity and exactness o f t h e sequential s o l u t i o n s in t h e q u o t i e n t algebras mentioned.
The next f i v e s e c t i o n s w i l l t h e r e f o r e deal with t h e way r e g u l a r i z a t i o n s and the corresponding q u o t i e n t algebras containing t h e d i s t r i b u t i o n s can be constructed. Remark 1
The form of t h e inclusion diagrams ( 9 ) i s necessary i n the following sense. the I f we assume t h a t f o r a c e r t a i n quotient a m = A/IsALy (n) ' following r e l a t i o n holds:
E. E
68 D'(Q
C A
. Rosi nger
;
1
i . e . i f we assume t h e e x i s t e n c e o f a commutative diagram sur < s,*> E D'(Q)
S3s\
s
inj
7 A~
t
f o r a s u i t a b l e v e c t o r subspace S c A n Sm, then t a k i n g
u = I n s we o b t a i n an i n c l u s i o n diagram ( 9 ) s a t i s f y i n g a l l t h e p r o p e r t i e s , except perhaps f o r the i n c l u s i o n U
c"(n)
2.
S*
Simpler I n c l u s i o n Diagrams
The choice o f t h e r e g u l a r i z a t i o n S/U i n t h e i n c l u s i o n diagram ( 9 ) c o u l d be reduced t o the choice o f S only, as U would r e s u l t from ( 9 . 2 ) . However, i t w i l l be more convenient f i r s t t o split S according t o (11)
S
=U@S'
where S ' i s a v e c t o r subspace i n S, and then t o r e p l a c e t h e problem o f t h e choice o f S b y t h e problem o f t h e choice o f t h e p a i r (U,S'),which under c e r t a i n c o n d i t i o n s (see ( 2 1 ) ) w i l l from now on be termed r e g u l a r i z a = tions. Indeed, assuming t h a t t h e s p l i t t i n g (11) holds, t h e r e l a t i o n s (9.2) and (9.3) become
Now the problem o f f u l f i l l i n g (9.1) remains. I t i s obvious t h a t i f t h e r e e x i s t s an i d e a l 7 i n A which s a t i s f i e s (9.1) then the s m a l l e s t i d e a l i n A c o n t a i n i n g U , i . e .
(13)
l ( U , A ) = t h e i d e a l i n A generated b y V
w i l l a l s o s a t i s f y (9.1).
Since u ( 1 ) has a simple s t r u c t u r e , v i z
E
U
C"(Q)
c A i n (9), t h e ideal l ( U , A )
.
7 ( U , A ) = t h e v e c t o r subspace i n A generated by U . A N I n t h e p a r t i c u l a r case when A = (C"(n)) , we s h a l l f o r t h e sake o f s i m p l i = c i t y use t h e n o t a t i o n :
(14)
(14.1)
l(U) = 7(V,
(C"(n))N.
ALGEBRCSCONTAINING THE DISTRIBUTIONS
69
Owing t o t h e s i m p l e s t r u c t u r e o f t h e i d e a l 7 ( V , A ) , we g a i n a good i n s i g h t i n t o the s t r u c t u r e o f the q u o t i e n t algebra A = A / 7 ( V , A ) c o n t a i n i n g the d i s t r i b u t i o n s . T h e r e f o r e t h e i n c l u s i o n diagrams ( 9 ) w i l l o n l y be c o n s i = dered under t h e p a r t i c u l a r f o r m
with (15.1)
7(V,A) n (V
(15.2)
S" =
0s')
= V
V" 0 s ' .
I t i s u s e f u l t o n o t i c e t h a t (15.1) can be w r i t t e n i n t h e s i m p l e r , equiva= l e n t form
(15.3)
7 ( V , A ) n S' =
2.
-
I n t h e case o f t h e i n c l u s i o n diagrams ( 1 5 ) , t h e l i n e a r embedding o f D'(n) i n t o q u o t i e n t a l g e b r a s i n ( l o ) , w i l l assume t h e f o l l o w i n g p a r t i c u l a r form:
D'(n)
= S"/V"
s
+
A = A/7(V,A)
+(V@S')/V
Urn - s isom
t V
-
s t I(V,A).
lin,inj
I n c o n s t r u c t i n g t h e i n c l u s i o n diagrams ( 1 5 ) , we have n o t o n l y t h e problem o f choosing t h e p a i r s ( V , S I ) , b u t a l s o t h a t o f choosing t h e subalgebras A A s i m p l e s o l u t i o n of t h i s l a t t e r problem w i l l however, be g i v e n i n t h e n e x t s e c t i o n . Therefore t h e problem o f c o n s t r u c t i n g c h a i n s o f quo= t i e n t a l g e b r a s ( I ) , c o n t a i n i n g t h e d i s t r i b u t i o n s , w i l l be reduced t o t h e problem o f c o n s t r u c t i n g s u i t a b l e r e g u l a r i z a t i o n s ( V , S ' ) .
.
3, R e g u l a r i z a t i o n s and Chains o f Q u o t i e n t Algebras C o n t a i n i n g t h e Distributions I n t h i s section, D e f i n i t i o n 1 presents t h e b a s i c n o t i o n o f r e g u l a r i z a t i o n , which l e a d s t o t h e b a s i c r e s u l t o f Theorem 1 on t h e c o n s t r u c t i o n o f c h a i n s
70
E.E.
Rosinger
of q u o t i e n t algebras c o n t a i n i n g t h e d i s t r i b u t i o n s . T h i s r e s u l t w i l l be used i n t h e n e x t two chapters i n t h e s t u d y o f polynomial n o n l i n e a r PDEs. A second way o f c o n s t r u c t i n g chains o f q u o t i e n t algebras c o n t a i n i n g l a r g e v e m u b s p a c e s o f d i s t r i b u t i o n s i s presented i n Sections 5 and 6. We f i r s t i n t r g d u c e s e v e r a l u s e f u l a u x i l i a r y n o t a t i o n s and d e f i n i t i o n s . subset HC (C (n))N i s c a l l e d d e r i v a t i v e i n v a r i a n t , o n l y i f
(17)
i?H
Obviously, Given V
cH,W
PE
A
Nn.
H = (C"(n))N i s d e r i v a t i v e i n v a r i a n t .
,S
C
( 18)
(Cm('d))
{v
VL=
N and L E V
I
E
8 , we use t h e n o t a t i o n s :
P C v E V , V
PE
Nn,
(PI
and (19)
A' (V S , ) = t h e s m a l l e s t d e r i v a t i v e i n v a r i a n t subal gebra i n
t h a t contains V L
(C"(Q))N
t
S.
F i n a l l y , w i t h t h e h e l p o f (13), we i n t r o d u c e t h e n o t a t i o n :
!"(VS) = I ( V , A % S ) ) .
(20)
The b a s i c n o t i o n f o r t h e sequel i s i n t r o d u c e d i n t h e f o l l o w i n g d e f i n i t i o n . Definition 1
A p a i r ( V S )of v e c t o r subspaces i n V", regularization only i f (21.1)
I(V) n s = Q
(21.2)
S"
(21.3)
Uc"(n)
S"
respectively i s called a
( s e e (14.1));
c v"@s C
S
The e x i s t e n c e o f r e g u l a r i z a t i o n s w i l l be proved i n S e c t i o n 4. Meanwhile i n Theorem 1, i t w i l l be shown how r e g u l a r i z a t i o n s l e a d t o t h e construe= t i o n o f chains o f q u o t i e n t a l g e b r a s c o n t a i n i n g t h e d i s t r i b u t i o n s . F i r s t , however, s e v e r a l comments w i 11 be made. The above c o n d i t i o n (21.1) i s e q u i v a l e n t t o (15.3),
A
= (C"(Q))N
assuming t h a t
i n (15).
C o n d i t i o n (21.2) i s i d e n t i c a l t o (15.2) o r (12). C o n d i t i o n (21.3) p l a y s a s p e c i a l r o l e : i t ensures t h a t t h e m u l t i p l i c a t i o n s and p a r t i a l d e r i v a t i v e o p e r a t o r s Jn t h e q u o t i e n t a l g e b r a s c o n t a i n i n g t h e d i s t r i b u t i o n s , when a p p l i e d t o c -smooth f u n c t i o n s , c o i n c i d e w i t h t h e usual corresponding o p e r a t i o n s on f u n c t i o n s .
ALGEBRAS CONTAINING THE DISTRIBUTIONS
71
A simple and important property of regularizations (V,s) is given by (22)
V
c V" #
which i s arrived a t a s follows. Assume V = V", then V" c V c I ( V ) , hence (21.1) and (21.2) imply 7 ( V ) n Sm c Vm, and ( 5 ) i s contradicted. The significance of (22) i s t h a t i t shows t h a t the problem o f the exis= tence o f regularizations ( U S ) i s n o t t g i v i a l , a t l e a s t as f a r as the existence of the vector subspaces YC V are concerned ( f o r further de= t a i l s , see Sections 4-6, as well as Chapter 6 ) . W e now s t a t e the basic result on the construction of chains of quotient algebras containing the distributions. Theorem 1 If ( V , S ) i s a regularization, then f o r each II s t r u c t the following inclusion diagram:
E
fl i t i s possible t o con=
which will s a t i s f y the condition (23.1)
rR(v,s)n (vPl@
S) =
vII
o r , equivalently , the simpler condition (23.2)
r'(v,s)
n
s
=
Q
as well as the condition (23.3)
S" = V"
@
S.
Proof The inclusions in (23) and the relation (23.3) follow e a s i l y , and we shall prove only (23.2).
72
E.E.
Rosinger
11 i s easy t o n o t i c e t h a t V c V o = V , and u(1) E A ' ( V , S ) , t h u s 7 ( V , S i c I( V ) , s i n c e l'(\',S) i s t h e v e c t o r subspace i n dl V , S ) generated b y Vk . A (V,S), w h i l e I ( V ) i s t h e v e c t o r subspace i n (C"(n)) generated b y
k
V.(c"(11))~. Now (23.2) w i l l f o l l o w from (21.1).
0
The f o l l o w i n g d e f i n i t i o n now g i v e s t h e c o n s t r u c t i o n of t h e b a s i c types o f chains o f q u o t i e n t algebras c o n t a i n i n g t h e d i s t r i b u t i o n s . Definition 2 Supposing (24)
(V,s) i s A'(V,S)
a r e g u l a r i z a t i o n , we i n t r o d u c e t h e n o t a t i o n : =
(V,S), w i t h R E 8 .
A'(V,S)/lR
It f o l l o w s from Theorem 1 t h a t (see ( 3 0 ) , Chapter 1)
A'
(25)
( v , s )=
(v,s)i s
whenever
( v , s )E A L ' ~ ( ~ )V Q,
A'(v,s)/I'
E
8
a regularization.
The b a s i c a l g e b r a i c p r o p e r t i e s o f t h e q u o t i e n t algebras i n ( 2 4 ) a r e pre= sented i n t h e f o l l o w i n g theorem. Theorem 2 Suppose 1)
( v , ~ )i s
a regularization.
Then we have t h e f o l l o w i n g
A ' ( v , ~ ) , w i t h II E 8 , i s an a s s o c i a t i v e and c o m u t a t i v e a l g e b r a o f classes o f sequenc s o f c"-smooth f u n c t i o n s on n, h a v i n g t h e u n i t element u ( 1 ) t 1
fi ( v , ~ ) .
2)
The f o l l o w i n g diagrams a r e commutative:
Yt,k
k ( s t q-(V,S)) = s t 7 ( V , S ) i s an a l g e b r a homomorphism,
ALGEBRAS CONTAINING THE DISTRIBUTIONS
73
w h i l e t h e l i n e a r i n j e c t i v e mapping E~ i s d e f i n e d by
D’(n) (26.1)
=
e;
s”/v”
W
s
W
V
-
v
s
t
W
vR-
The m u l t i p l i c a t i o n i n A‘(V,S), w i t h R usual m u l t i p l i c a t i o n o f f u n c t i o n s .
3)
“(V,S)
S)/VII
>s
lin,inj E
h,
induces on
d( V,S)
C”(n) t h e
Proof 1 ) and 2 ) f o l l o w e a s i l y from Theorem 1. 3 ) f o l l o w s from 2 ) and ( 2 1 . 3 ) .
0
Remark 2 I t i s i m p o r t a n t t o n o t i c e t h a t t h e a l g e b r a homomorphisms
yII,k
: A‘(V,S)
A k ( V , S ) : 1, k
-+
E
&,
k
< .t,
a r e n o t n e c e s s a r i l y i n j e c t i v e ; however, i n view o f 2 ) i n Theorem 2, t h e i r r e s t f i t i o n s t o D ’ ( R ) are injective. Now t h e b a s i c r e s u l t concerning t h e way p a r t i a l d e r i v a t i v e o p e r a t o r s can be d e f i n e d on t h e q u o t i e n t a l g e b r a s i n ( 2 4 ) , i s presented i n t h e f o l l o w i n g theorem. Theorem 3 Suppose (V,S)
i s a regularization.
Then we have t h e f o l l o w i n g
a-k AR(V,S)
2)
The p a r t i a l d e r i v a t i v e o p e r a t o r s a r e d e f i n e d b y t h e l i n e a r mappings
(27)
Dp : AR(V,S)
f
Y il, k
h,
1)
Ak(V,S):
E
K R ( s e e ( 2 9 ) , Chapter 1)
Ak(V,S)
-+
with (27.1)
Dp ( s + l ‘ ( V , S ) ) f o r II, k
3)
*)
E
Dps t I k ( V , S ) ,
=
N, k
G II
,p
E
Nny
V s E A‘l(V,S),
l p l < II - k,*(see
The f o l l o w i n g diagrams a r e commutative:
i t i s assumed t h a t
m
-
~0
=
m
( 3 2 ) , Chapter 1).
E . E . Rosinger
74
for any R , k , R ' , k ' E 8 , k G R , k ' G IpI GR-k, IpI G R ' - k ' . 4)
I.', k'<
k , R' GR, p
E
Nn
The partial derivative operators in ( 2 7 ) satisfy the Leibnitz rule for product derivatives:
_c
q GP A k ( V , S ) and R , k with Dp, Dq, Dp-q : AR(V,S) IpI G R-k as well as S,T E A ( V , S ) .
5)
E
8 , k GR, p
E
Nn,
The partial derivative operators in (28), when applied t o C -smooth functions, coincide with the usual partial deriva= tives of functions.
Proof 1)
In view of (18), i t i s easy t o see t h a t
(29)
Dp VR c V k ,
V p E Nn,
IpI
GR-k.
Moreover (30)
Dp AR(U,S) c Ak(V,S), V p
E
Nny
since VR C Vk implies AR(V,S) c A k ( V , S ) , while in view of (19), A k ( V , S ) i s deri vati ve invariant. The ref0 re k
N n y 1p) < Il-k. Il Indeed, (23) implies that u(1) E U a c A ( V , S ) ; hence, in view of ( 2 0 ) , R C,(W R 7 ( V , S ) i s the vector sub&pace in A ( V , S ) generated by V .A ( V , S ) . B u t Dp, applied to sequences of C -smooth functions, s a t i s f i e s the Leibnitz rule for product derivative. Thus, (29) and (30) will imply (31), which in view of (29), Chapter 1, completes the proof of 1). (31)
Dp I R ( U , S ) c I ( V , S ) , \f p
E
75
ALGEBRAS CONTAINING THE DISTRIBUTIONS
2)
f o l l o w s from ( 3 2 ) , Chapter i and 1) above.
3)
f o l l o w s from ( 2 6 ) and ( 2 7 ) ,
4)
f o l l o w s from ( 2 6 ) and ( 2 7 ) .
5)
f o l l o w s from ( 2 7 ) and (21.3).
Remark 3 a)
Given a r e g u l a r i z a t i o n ( V , S ) , t h e q u o t i e n t a l g e b r a s AR(V,S),
.t E
fly
a r e n a t u r a l l y o r g a n i z e d i n a c h a i n b y t h e commutative diagrams o f a l g e b r a homoporphisms and p a r t i a l d e r i v a t i v e o p e r a t o r s i n ( 2 8 ) . Whenever polynomial n o n l i n e a r PDEs as i n ( l ) , Chapter 1 a r e b e i n g b) d e a l t w i t h , t h e c o r r e s p o n d i n g PDOs i n ( 3 ) , Chapter 1, can be c o n s i d e r e d t o a c t as f o l l o w s (see ( 3 3 ) , Chapter 1) : T(D) : AR(V,S)
(32)
-t
Ak(VyS)
p r o v i d e d t h a t R , k E fly k G R , R-k >m!: m b e i n g t h e o r d e r o f t h e PDO con= s i d e r e d an t h e c o e f f i c i e n t s ciy as w e l l as f i n t h e PDE i n ( l ) , Chapter 1, b e i n g Cf-smooth f u n c t i o n s . The f a c t t h a t t h e PDOs i n (32) m i g h t a c t between d i f f e r e n t q u o t i e n t c) a l g e b r a s f o l l o w s from t h e way p a r t i a l d e r i v a t i v e o p e r a t o r s can b e d e f i n e d on t h e q u o t i e n t a l g e b r a s (see S e c t i o n 2, Chapter 1). d) I f t h e PDO i n ( 3 ) , Chapter 1, a c t s between q u o t i e n t a l g e b r a s as i n (32) , t h e s t a b i l i t o f t h e s e q u t n t i a l s o l u t i o n s o f t h e c o r r e s p o n d i n g PDE i n ( l ) , C h d d e p e n d s on 7 i V , S ) ; t h e e n e r a l i t y depends on A ( V , S ) ; w h i l e t h e exactness depends on I ( V , S ) and A j r W i t h g e n e r a l i t chosen t o be a t l e a s t d i s t r i b u t i o n a l l e v e l - see S e c t i o n 0 - i . e . w en D'(i2)cAR(VyS), f o r R E fl, t h e main c o n f l i c t w i l l be between s t a b i l i t y and exactness. Maximal s t a b i l i t y w i l l r e q u i r e maximal ideals-w h i l e maximal exactness w i l l r e q u i r e among o t h e r s , minimal i d e a l s 1 ( V , S ) .
+
4.
Existence, C h a r a c t e r i z a t i o n and C o n s t r u c t i o n o f R e g u l a r i z a t i o n s
We f i r s t , i n Theorem 4, g i v e a u s e f u l c h a r a c t e r i z a t i o n o f r e g u l a r i z a t i o n s . We n e x t p r e s e n t , i n Theorems 5-7, r e s u l t s on t h e e x i s t e n c e and construe= t i o n o f regularizations. Theorem 4
I f 7 i s an i d e a l i n ( C " ( i 2 ) ) such t h a t (33)
in^
=
* ) i t i s assumed t h a t
N
and t h e r e e x i s t v e c t o r subspaces S , T c Sm
v"n~=q;
m
-
=
m
76
E.E.
@
( 3 4)
I n s w c urn
(35)
Sm c Urn
q) S \-
( 36)
Uc"(n)c
s
Rosinger
T;
(3T;
c;j .-
and
7;
then f o r any v e c t o r subspace V c I
( 3 7)
(w,s
ri
V"
5 T) i s a r e g u l a r i z a t i o n .
Conversely, any r e g u l a r i z a t i o n i s o f t h e form i n ( 3 7 ) . Proof The f a c t t h a t (V,s ( 3 5) and ( 3 6 ) .
@ T)
s a t i s f i e s (21.2) and (21.3) f o l l o w s e a s i l y f r o m
We now prove t h a t (2 1 . 1 ) i s a l s o v a l i d . Indeed, v c 7 i m p l i e s t h a t ~ ( v c) I , s i n c e 7 i s an i d e a l i n (Cm(n))N. T h e r e f o r e
I(V) n
(v"
(SOT) c 7 n (s(t' -,
T ) c ( 7 nSw) n
(sC+:T) c
ET) n (SET)
with the l a t t e r inclusion implied by (34).
I ( V ) n (s@T)
Now i n view o f ( 3 5 ) , we have
c T.
Then t h e i n l c u s i o n I ( V ) c 7 and ( 3 3 ) i m p l y t h a t
I(v) n
(GT)= !!
which completes t h e p r o o f o f ( 2 2 . 1 ) . regularization.
Therefore
(V,s ( 3 7 ) i s indeed a
Conversely, assume g i v e n a r e g u l a r i z a t i o n ( V , s l ) . Take t h e n 7 = I S = Q and T S'. I n view o f ( 1 4 . 1 ) , I w i l l be an i d e a l i n ( ~ " ( 6 2 ) ) ( 33) i s o b t a i n e d as f o l l o w s . F i r s t , we n o t i c e t h a t
A v. ) ,Now
vm n T = v m n s l = q i n view o f ( 2 1 . 2 ) .
Then I n T = I n S' =
p owing t o (21.1).
The r e l a t i o n (34 ) i s o b t a i n e d f r o m (21.2) as f o l l o w s :
I n S" c
sm c v" @ sl
=
vm @T.
F i n a l l y , t h e r e l a t i o n s (3 5 ) and ( 3 6 ) f o l l o w from (21.2) and (21.3) respec= t i vely. 0 The c h a r a c t e r i z a t i o n o f r e g u l a r i z a t i o n s o b t a i n e d i n Theorem 4, reduces t h e problem o f t h e i r e x i s t e n c e t o t h a t o f t h e e x i s t e n c e o f t h e i d e a l s 7 i n (Cm(n))N, s a t i s f y i n g t h e c o n d i t i o n s (33-36). T h i s c o n s i d e r a t i o n j u s t i f i e s the following d e f i n i t i o n .
ALGEBRAS CONTAINING THE D I STR IBUT1ONS
77
Definition 3 An i d e a l 7 i n LCm(Q))N i s c a l l e d r e g u l a r , o n l y i f f o r s u i t a b l e v e c t o r sub= t h e c o n d i t i o n s i-6) are s a t i s f i e d . spaces S,T C S I t t u r n s o u t t h a t a simple, p r a c t i c a l l y u s e f u l c h a r a c t e r i z a t i o n o f t h e s e i d e a l s can be achieved. T h i s c h a r a c t e r i z a t i o n w i l l be suggested b y t h e necessary c o n d i t i o n o b t a i n e d i n t h e f o l l o w i n g c o r o l l a r y (see F i g 2 ) .
Corollary 1 I f t h e i d e a l 1 i n (Cm(Q))
N
i s regular, then
Proof The r e l a t i o n s (34) and (36) i m p l y t h a t
Therefore (35) w i l l y i e l d
which, i n view o f ( 3 3 ) , w i l l i m p l y ( 3 8 ) .
0
+
Our n e x t aim, achieved i n Theorem 5 , i s t o prove t h a t t h e necessar t i o n (38) on r e g u l a r i d e a l s i n (Cm(n))N w i l l a l s o be 2 s u f i c i e n t c ocondi= ndition, w i t h i n t h e framework o f a l a r g e c l a s s o f i d e a l s i n ( C (n))N,ned r a t h e r weak t o p o l o g i c a l - t y p e c o n d i t i o n i n t r o d u c e d i n ( 4 9 ) . C a l l an i d e a l 7 i n (Cm(Q)) V (39)
W
E
~
N
,E N ~
by a
vanishing, o n l y i f :
j v ~ N , v > ~x , € 5 2 : WJX)
= 0.
Obviously, lnd d e f i n e d i n ( 6 5 ) , Chapter 1, when r e s t r i c t e d t o
(Cm(n)) N , i s
a v a n i s h i n g i d e a l and s a t i s f i e s c o n d i t i o n ( 3 8 ) . Proposition 1 I f 7 i s a v a n i s h i n g i d e a l , t h e n t h e r e e x i s t v e c t o r subspaces T c Sm s a t i s = fying the conditions
(40)
i n ~ = v ~ Qn ~ =
(41)
vm t ( I
n sm) =
vm @
T.
(see ( 3 3 ) ) and (see ( 3 4 ) ) .
78
E.E.
Rosinger
The 'Ashqelon Rose'
Fig 2
ALGEBRAS CONTAINING THE DISTRIBUTIONS
79
Proof Taking E = Sm, A = Urn and B = 1 n Sm in Lemma 2 below, i t s u f f i c e s t o show that codi! 1 n Vm Gcodirn I ns Urn We f i r s t n o t i c e t h a t
7 n U?
(42)
codimm I n Urn < dim 7 n Sm < c a r 1 n Sm. I n s N B u t 1 n SY(C'(n)) and c a r Co(Q) = car R ' , t h e r e f o r e codimm 1 n Vw < ( c a r R ' ) c a r = car R ' . 7 ns Now in order t o prove ( 4 2 ) i t s u f f i c e s t o show t h a t 1 n U" 2 car R ' .
cod@
(43)
U
We define vCY.E Urn, with
a',
( V ~ ) ~ ( X =)
(O,l),
by
V v E N,
x
CY. E
E
a.
If i s easy t o see t h a t (v,~cY. E ( 0 , l ) ) is a l i n e a r independent family in U . Denote by U the vector subspace i t generates. Then (44)
7nu=q.
Indeed, assume w
(46)
I n U.
c
w -
(45) w i t h hi
E
Then w
R ' and ai
E
l ai
(0,l);
hence V
c
=
WJX)
V implies
A. v
l d i
E
Xi(ai)
,
V v E N, x
E
R.
l < i < h B u t w E I , t h e r e f o r e (46) a n d (39) w i l l y i e l d (47)
c
~
~
(= 0, a f~o r i)n f i~n i t e l y many v E N .
l < i < h
I t then follows t h a t A
=
... = x
= 0.
(48) 1 h Indeed, i f h = l , then s i n c e a1 # 0 and i n view of ( 4 7 ) , ( 4 8 ) i s obvious. I f h > 1, i t can be assumed t h a t 0 < a1 < ... < ah < 1 and x i # 0 , w i t h 1 G i Q h . Then dividing in (47) by a h , we have Al("l/ah)" +
... +
~ ~ - ~ ( n ~+ -ih ~ = / 0,a f~o r) i ~ n f i n i t e l y many v
E
N,
0, s i n c e 0 < a - / a < 1, w i t h 1 Q i G h - 1 . Relation (48) w h i c h implies A having been probe=d, we o b t a i n (441 fkom ( 4 5 ) . Now (44) w i l l obviously
E.E.
80
Rosinger
imply (43). Lemma 2 I f A and B a r e v e c t o r subspaces i n E and
codim B
A n B Gcodim A n B A
then t h e r e e x i s t v e c t o r subspaces C i n E, such t h a t
A n C = B (1 C = 0 ( t h e n u l l space) and A t B = A @ C , Proof
I i E I ) and ( bJ. I j E J ) a r e a l g e b r a i c bases i n A and B respec= 1 t i v e l y , such t h a t ( c k ! k E K ) , w i t h K = I n J and c = a = b , f o r k E K , i s an a l g e b r a i c base i n A r-1 B. I n view o f t h e hypbthesks, t k e r e e x i s t s an i n j e c t i v e . m a p p i n g c1 : ( J \ K ) + ( I \ K ) . Then C can be chosen as t h e v e c t o r subspace h a v i n g t h e a l g e b r a i c base ( a t b. 1 j E J \ K). Indeed, i f x E A n C, then a(j) J
Assume ( a .
hence
which i m p l i e s p
j
= 0, V j E J\K, t h u s x = 0.
Now i f x
E
B n C, then
hence
i m p l y i n g t h a t p . = 0, V j E J\K, s i n c e F i n a l l y , i t sufSices t o show t h a t BcA+C. Assume x E B.
hence
Obviously
c1
i s injective.
Thus again x = 0.
ALGEBRAS CONTAINING THE DISTRIBUTIONS
therefore, indeed x
E
A
t
81
C.
0
I t can be seen t h a t i f E i s finite-dimensional, the condition of the above inequality of codimensions i s essential. Indeed, i f E = A t B and C = E implies B n C # 0 , as otherwise we have dim A < dim 6, then A the contradiction
(3
dim E >dim B
t
dim C
=
dim B
t
dim E
-
dim A > dim E.
The vanishin ideals used in Proposition 1 above can be simply and usefully charagterize within the framework of thefollowing ideals. An ideal I in ( C ( R ) ) N i s called cofinal invariant, only i f (see (61), Chapter 2 )
--%
v
(
w
E (Cm(n))N:
V v E N Y v > ~ :
)
-
W E l .
Obviously, Ind defined in ( 6 5 ) , Chapter 1 , when restricted to ( C m ( n ) ) N , i s a cofinal invariant ideal. Proposition 2 A cofinal invariant ideal i s vanishing, only i f i t i s proper.
Proof Assume 7 i s a cofinal invariant ideal. First assume that ( 3 9 ) i s n o t valid. Then there e x i s t w E I and p E N , such t h a t w ( x ) # 0 , V v E N , v > L I ,x E R. Define w ' E ( C ( Q ) ) N by w'(x) = 1, for 8 E N,v < p , x E R, while w' = w , in case 2 2 1-1. Then W'E'I, since I i s cofinal invariant B u t obvYously l/w' E ( C (g))N. Therefore u(1) = w ' . ( l / w ' ) E 7 . (C"(n))fl c 7 , in which case I = ( C ( R ) ) N , hence 7 i s n o t proper. The converse i s obvious. 0 We now present the characterization of regular ideals within the framework of cofinal invariant ideals. Theorem 5 A cofinal invariant ideal 7 in (Cm(R)) N i s regular, only i f
in which case 7 will also be a vanishing ideal. Proof Assume t h a t (50) holds. Then I f ( C " ( S ~ ) ) ~ ; therefore in view of Proposi= tion 2 , I i s a vanishing igeal. Thus Proposition 1 will grant the existence of vector subspaces T C S satisfying (40) and ( 4 1 ) .
82
E.E.
Rosinger
Assume f o r t h e t i m e b e i n g t h a t T can be chosen so as a l s o t o s a t i s f y (51)
(V
m
(9 T ) n P
= T
Take a v e c t o r subspace U ' (52)
U
C
m
(n) s a t i s f y i n g
uc"(n)
%"( n) = (7-
Ucm( n).
(?
)
(:I
U'.
It follows t h a t
(53)
(Vm@
T ) n U' = Q
since
(v"G~
) Unl c
( T n Ucm(n))
((v"
GT)n U c y n ) )
n U' =
Q
n U' =
t h e l a t t e r two e q u a l i t i e s b e i n g consequences o f (51) and ( 5 2 ) . I t f o l l o w s t h a t one can choose v e c t o r subspaces S C Sm s a t i s f y i n g ( 3 5 ) , as w e l l as t h e c o n d i t i o n
(54)
0
Uc"(n)
(see ( 3 6 ) ) .
Thus i t f o l l o w s indeed t h a t 7 i s a r e g u l a r i d e a l , p r o v i d e d ( 5 1 ) can be proved. To do so, we s h a l l use Lemma 3 below, as w e l l as Lemma 2 , a l r e a d y used i n t h e p r o o f o f P r o p o s i t i o n 1. Using t h e n o t a t i o n (55)
E =
s",
A =
v",
B = Ucm(n) and C = 7 n S m
we have (56)
A nB = B nC = 0
( t h e n u l l space)
t h e l a t t e r e q u a l i t y b e i n g i m p l i e d by ( 5 0 ) . Lemma 3 below, we have
zi
=
vrn @
B
=
( I n s")
(ucm
(fi)
n
(v"
t
Now, w i t h t h e n o t a t i o n o f
( I n s")))
(57)
@ (uCm(n)
Our aim i s t o a p p l y Lemma 2 t o the r e l a t i o n (58)
codim
B
1 7
B
Q
A, B
codim
n (V" t ( I n S"))). C E.
To do so, we have f i r s t t o prove
A n B.
a
Indeed, an argument s i m i l a r t o t h a t used i n t h e p r o o f o f P r o p o s i t i o n 1 i n order t o e s t a b l i s h (42), y i e l d s
ALGEBRAS CONTAINING THE DISTRIBUTIONS
codim
A nB
83
B S c a r B < c a r R'.
B Now i n o d e r t o o b t a i n (58) i t s u f f i c e s t o show t h a t codim A
(59) To do so
A nB
> c a r R'.
we s h a l l a g a i n use t h e sequences o f f u n c t i o n s va E Vm, w i t h
(0,l , d e f i n e d i n t h e p r o o f o f P r o p o s i t i o n 1, as w e l l as t h e v e c t o r
a E
subs pace V generated b y them, and prove t h e r e l a t i o n
Bnv=Q.
(60)
Indeed, assume
c;
E
B
n V.
Then
c
w =
w i t h Ai E R' and ai €(O,l WJX)
But
ii
E
B
ii E
implies that
; hence
c
=
1
(62) with w
E V
"ai
1 < i< h
(61)
w
= w
i U($)
I n Smand J,
(63)
E
$ ( x ) = 0,
C"(n).
u
The p o i n t i s t h a t
x E
n
.
Indeed, assume (63) i s f a l s e and t h a t open, a r e such t h a t $(x)<
-2E,
u
x
E
>
0 and
n' c n, w i t h n' non-void
E 51'.
Then i n view o f (61) and (62) we have
(64)
3 u E N : wv(x) =
U V E N , V > ~ , X E Q ' : WV(X)
- $(x)>
E
.
Now d e f i n e w ' E (Cm(n))N b y
Then o b v i o u s l y
r
(65)
W' E
since w E 1
, and I
i s cofinal invariant.
Take t h e n any
E. E . Rosi nger
84
x
(66)
E
?(a), w i t h
supp
x
c R'
and i n view o f (64), d e f i n e t E (C"(R))N by
I x
tV
i f v < p
=I
Then (65) gives
Therefore
i n view o f (50) x ( x ) = 0,
u
, we
have
x E R
which i s absurd, s i n c e x can be chosen a r b i t r a r i l y w i t h i n t h e c o n d i t i o n (66). T h i s completes t h e p r o o f of ( 6 3 ) . Now (62) and (63) i m p l y t h a t
i
=
w E
B
n
v
n i n s"
c
r
n
v
=
Q
t h e l a t t e r e q u a l i t y r e s u l t i n g from (44) i n t h e p r o o f o f P r o p o s i t i o n 1, as w e l l as t h e f a c t t h a t - as we have n o t i c e d - I i s vanishing. T h i s com= p l e t e s t h e p r o o f of (60). We can t h e r e f o r e conclude t h a t (59) h o l d s , s i n c e V c V" c i s an a l g e b r a i c base i n V .
fi and (vale.
(0,l))
Since (59) i m p l i e s ( 5 8 ) , we can f i n a l l y a p p l y Lemma 2 t o fi agd B given i n (57) and o b t a i n t h e e x i s t e n c e o f v e c t o r subspaces i n E = S , such t h a t
c
a n c = B n c = 0 and a + B = A @ E. Then (56)mand Lemma 3 below i m p l y t h e e x i s t e n c e of v e c t o r subspaces D c E = S , such t h a t
A nD (67)
C nD = 0;
A t C = A @ D ;
and
Bn(AtC) =BnD. Taking, f i n a l l y , T = D, t h e l a t t e r two r e l a t i o n s i n (67) w i l l y i e l d (51). The converse f o l l o w s from C o r o l l a r y 1. Lemna 3 I f A, B and C a r e v e c t o r subspaces i n E and
0
ALGEBRAS CONTAINING THE DISTRIBUTIONS
A nB
85
B n C = 0 ( t h e null space),
=
then trle following two properties are equivalent: 3 D c E vector subspace: A n D = C n D = O
(68) A + C = A @ D
+
B n (A
C) = B n D
and
E
3 (69)
c E vector subspace:
Ant=@ n t = o A + B = A @ C
where
A
=
B
=
@ c @ A
(B n (A t (B n (A
c))
+ c)).
Proof Assume (69) holds. Then (68) results by direct verification, i f one takes 0 = ( B n ( A + C ) ) @ C. Assuming (68) and taking any vector subspace C C E such t h a t D = ( B n D) @ C, direct verification will yield ( 6 9 ) . 0 The cofinal invariant regular ideals actually s a t i s f y stronger conditions than those in (33-36), as can be seen in the following corollary. Corollary 2 If 7 i s p cofinal invariant regular ideal, there exist vector subspaces c S such t h a t
S, J
(70)
S"=
(71)
7 n T = V" n J = p ;
(72)
V"
V"@S@T;
+ (7
n.!?) = V"
@
J;
and
(73) in which case, for any vector subspace V c 7 n Vm (74)
(V, S
@ J)
will be a regularization.
86
E.E.
Rosinger
Proof See (41) i n P r o p o s i t i o n 1 and (54) i n t h e p r o o f o f Theorem 5. Several f u r t h e r general r e s u l t s on r e g u l a r i d e a l s a r e now presented. Theorem 6 N An i d e a l I i n (C"(n)) i s r e g u l a r , o n l y i f t h e r e e x i s t v e c t o r spaces T c Sm s a t i s f y i n g t h e c o n d i t i o n s ( 3 3 ) and (M),as w e l l as
wm n
(75)
t T) =
(Ucm(,)
Q
.
Proof Assume t h a t I i s r e g u l a r ; that
then i n view o f ( 3 3 ) , (35) and ( 3 6 ) , i t f o l l o w s
Now an element o f t h e l e f t - h a n d term i n (75) has t h e form v = u(IJJ) t t w i t h v E Urn, IJJ E
C"(n) and
Thus i n view o f (76) we have
t E T.
u(q)
= v
-
t E
u(IJJ)
= v
-
t = t'
~
~
m
n(
(~w)" @T )
CT,
i.e. (77) with t '
E
T.
Now Wm n T =
Q
i n ( 3 4 ) a p p l i e d t o (77) w i l l y i e l d
v = u ( o ) , t ' = -t and t h i s completes t h e p r o o f o f ( 7 5 ) . Conversely, assume t h a t (75) h o l d s .
We then show t h a t (76) i s v a l i d .
Indeed, an element o f t h e l e f t - h a n d t e r m i n (76) has t h e form (78) w i t h IJJ E
U($)
C"(n) , v
= v t t E
v = U(IJJ)
Vm and t
-
t E
E T.
vm n
Therefore t
( U ~ W ( ~ )
T).
Hence, i n view o f ( 7 5 ) , we have v = u ( o ) . Now (78) w i l l y i e l d ( 7 6 ) . I t o n b remains t o show t h a t (36) h o l d s f o r s u i t a b l e v e c t o r subspaces To do so, we t a k e a v e c t o r subspace
S c S t h a t a l s o s a t i s f y (35). U' c Ucm(n) such t h a t
ALGEBRAS CONTAINING THE DISTRIBUTIONS
(79)
UC"(
n)
= ( %"(
n)
nT)
@
87
U I .
the l a t t e r inclusion resulting from ( 7 6 ) . T h u s , ( 7 9 ) will yield ( 8 0 ) . NOW, in view of (80), there e x i s t vector subspaces S c Sm satisfying (35) and such t h a t U ' c S. I n t h a t case, ( 7 9 ) will obviously imply (36). 0 Theorem 7
N If a vanishing ideal 7 in (Cm(n)) s a t i s f i e s the condition (81)
(v"
+ 1) n
~
~
m
=(
Q~ , )
then 'I i s a regular ideal and there e x i s t regularizations (I n
v",~
@ T)
satisfying the conditions (70-72) as well as (82)
(see (73) and ( 3 6 ) ) .
UC"(S-2)
Proof Since 7 i s vanighing, Proposition 1 will grant the existence of vector subspaces T c S t h a t s a t i s f y (71) and ( 7 2 ) . B u t condition (81) i s ob= viously equivalent t o
%"( n) n
(v" +
( 7 n s"))
=
Q
;
or, in view of ( 7 2 ) , equivalent t o the condition uCm(n)n (v"
@ T) =
Q
.
Now the existence o f vector subspaces S c Tosatisfying (70) and (82) easily follows. Proposition 3 The ideal (see (65) , Chapter 1) (83)
I;d
= 7,d
(Cm(n))N = Ind
0
88
E.E. Rosinger
i s v a n i s h i n g , c o f i n a l i n v a r i a n t and r e g u l a r , and s a t i s f i e s t h e c o n d i t i o n (81). Proof It s u f f i c e s t o show t h a t
w = v
(84) with w
E
zd;,
v
E
s a t i s f i e s (81).
Assume
t U($)
V m and $ E C”(n).
Then o b v i o u s l y w
E
Sm and < w,.>
=$
.
Now ( 6 9 ) i n Chapter 1 w i l l i m p l y t h a t $(x) = 0, V x E Q. Then i n view o f 0 ( 8 4 ) , we have w = v E Vm, which completes t h e p r o o f o f (81). Remark 4 a) The c h a r a c t e r i z a t i o n o b t a i n e d i n Theorem 4 e L t a b l i s h e s an e q u i v a l e n c e between r e g u l a r i z a t i o n s and r e g u l a r i d e a l s i n ( C ( Q ) ) N , The r e g u l a r i d e a l s 1 - i n t h e c o f i n a l i n v a r i a n t case - have t h e s i m p l e c h a r a c t e r i z a t i o n pre= sented i n Theorem 5, v i z
I t i s i m p o r t a n t t o n o t i c e t h a t i n view o f c o n d i t i o n (23) i n Chapter 1 de= f i n i n g q u o t i e n t a l g e b r a s A = A / l E ALCm(n), t h e above c o n d i t i o n o b t a i n e d
i n Theorem 5 i s t h e b e s t o s s i b l e . T h i s i n d i c a t e s t h a t as we advanced from i n c l u s i o n diagrams (g+lusion diagrams (23) and q u o t i e n t a l g e = b r a s ( 2 4 ) c o n t a i n i n g t h e d i s t r i b u t i o n s , no undue r e s t r i c t i o n was imposed. I t a l s o i n d i c a t e s t h e abundance o f r e g u l a r i d e a l s and c o r r e s p o n d i n g l y o f r e g u l a r i z a t i o n s , which can t h e r e f o r e be r e s t r i c t e d i n a meaningful way i n accordance w i t h t h e requirements o f t h e p a r t i c u l a r problems b e i n g c o n s i = dered (see Chapters 4,5 and 7 ) . b ) Various r e g u l a r i z a t i o n s can be a s s o c i a t e d w i t h a r e g u l a r i d e a l i n a c o n s t r u c t i v e way. The c o n s t r u c t i o n presented i n t h e p r o o f s o f P r o k o s i t i o n 1 and Theorem 5, o n l y i n v o l v e s t h e c h o i c e o f v e c t o r subspaces i n S , supposed t o s a t i s f y s i m p l e decomposition p r o p e r t i e s . S p e c i f i c i n s t a n c e s o f t h e c o n s t r u c t i o n o f r e g u l a r i z a t i o n s w i l l be presented i n t h e n e x t chap= t e r s , i n connection w i t h t h e s t u d y o f p a r t i c u l a r t y p e s o f p o l y n o m i a l non= l i n e a r PDEs, as w e l l as a s c a t t e r i n g problem i n p o t e n t i a l s p o s i t i v e powers o f the Dirac d i s t r i b u t i o n . 5.
A d d i t i o n a l Chains o f Q u o t i e n t Algebras c o n t a i n i n g D i s t r i b u t i o n s
As seen i n 3), Theorem 2, and 4 ) , Theorem 3, t h e q u o t i e n t a l g e b r a s con= t a i n i n g t h e d i s t r i b u t i o n s d e f i n e d i n ( 2 4 ) i n d u c e on C ( Q ) t h e usual m u l t i = p l i c a t i o n and p a r t i a l d e r i v a t i v e s f o r f u n c t i o n s . As seen i n t h e p r e s e n t s e c t i o n , i t i s p o s s i b l e t o c o n s t r u c t c h a i n s - i n t h i s case, f i n i t e ones t o o - o f q u o t i e n t a l g e b r a s i n such a way t h a t t h e y w i l l i n d u c e - € E - G s u a l - m u l t i p l i c a t i o n and p a r t i a l d e r i v a t i v e s on t h e l a r e r spaces o f f u n c t i o n s CK(n), w i t h 9. E These q u o t i e n t algebras, under t e c o n d i t i o n s presented i n Theorem 16, S e c t i o n 6, w i l l a l s o c o n t a i n l a r g e v e c t o r subspaces o f t h e d i s t r i b u t i o n s i n U’(n).
n.
-+
ALGEBRAS CONTAINING THE DISTRIBUTIONS Suppose g i v e n II
E
R.
A subset H c ( C R ( n ) ) (85)
89
N
i s c a l l e d C'-derivative
invariant, only i f
c H , V p E Nn.
Dp(H n ( C . e + l P I : f i ) ) N )
Obviously, H = (CR( Q ) ) ~i s C' - d e r i v a t i v e i n v a r i a n t . Given V ,S c ( C o ( n ) ) N , we use t h e n o t a t i o n s (86.1)
~1%
=
s
n(~'(n))~
and (86.2)
VII
= {v E V I I I
1
Dpv
E
V
,V
p €Nn,
[PI
<&I.
We a l s o use t h e n o t a t i o n (87)
A'( V,S) = t h e s m a l l e s t C'-derivative N i n (CR((n)) which c o n t a i n s VR + S
i n v a r i a n t subalgebra
I R'
F i n a l l y , we use t h e n o t a t i o n (88)
7 (' V,S) = Z ( VR,AL( V , S ) )
where, f o r a subalgebra A c ( Co( n)) and a v e c t o r subspace V c A , we now use t h e n o t a t i o n (see ( 1 4 ) )
I ( V , A ) = t h e i d e a l i n A generated b y V N and i n p a r t i c u l a r , when A = ( C o ( n ) ) (see 1 4 . 1 ) ) (89)
(89.1)
l ( V ) = l ( V , (co(n))N).
I t i s c l e a r t h a t d e f i n i t i o n ( 1 7 ) i s a p a r t i c u l a r case o f (85), correspon= d i n g t o R = m S i m i l a r l y , d e f i n i t i o n s (18) and (14) a r e p a r t i c u l a r cases o f (86.2) and (89) r e s p e c t i v e l y .
.
I n t h e d e f i n i t i o n s ( 8 7 ) , (88) and (89.1) we use t h e same n o t a t i o n s as i n ( 1 9 ) , (20) and (14.1) r e s p e c t i v e l y , u n l e s s t h e r e i s a need f o r s p e c i f y i n g t h e d i s t i n c t i o n . T h i s a l s o a p p l i e s t o t h e n e x t d e f i n i t i o n , where t h e t e r m r e g u l a r i z a t i o n i s used, i n a sense d i f f e r e n t t o D e f i n i t i o n 1. However, i f e k a c t s p e c i f i c a t i o n i s needed, t h e n o t i o n i n D e f i n i t i o n 1 w i l l be c a l l e d C - r e g u l a r i z a t i o n , w h i l e t h a t d e f i n e d n e x t w i l l be c a l l e d C " - r e g u l a r i z a t i o n . Definition 4
A p a i r ( V , S ) o f v e c t o r subspaces i n V",S" l a r i z a t i o n , o n l y if (90.1)
Z(V) n S =
(90.2)
So c V"
(90.3)
U
p
CO(n)= S.
(see (89.1))
;
@ S;
r e s p e c t i v e l y , i s c a l l e d a regu=
and
90
E. E
. Rosi nger
An analog of the property (22) will be valid f o r C"-regularizations ( V,S) viz
as can e a s i l y be seen. The basic property of r e g u l a r i z a t i o n s i s now presented. Theorem 8 I f ( V , S ) is a r e g u l a r i z a t i o n , then f o r each II E fl i t i s possible t o con= s t r u c t t h e following inclusion diagram:
which will s a t i s f y t h e condition (92.1)
la(V,S) n (Va
@ s / & )= vII
o r e q u i v a l e n t l y , the simpler condition (92.2)
IR(v,.s)ns
III
=
2.
Proof This i s s i m i l a r t o the proof of Theorem 1.
0
The r e s u l t of Theorem 8 j u s t i f i e s t h e following d e f i n i t i o n , i n which, un= l e s s t h e r e is a need f o r specifying the d i s t i n c ion, t h e same notations will be used as i n Definition 2. Definition 5 Suppose (V,S) i s a r e g u l a r i z a t i o n , we s h a l l then denote R R (93) A ( V , S ) = A ( V , S ) / l ' ( V , S ) , w i t h R E fl I t follows ( s e e (30), Chapter 1) t h a t (94)
A?V,S) = AR(V,S)/IR(V,S) E AI!ck(n),
V II E
R,
ALGEBRAS CONTAINING THE DISTRIBUTIONS
f o r each r e g u l a r i z a t i o n ( V , S )
91
I
I t i s easy t o see t h a t t h e quotient algebras defined in ( 9 3 ) have a l g e b r a i c p r o p e r t i e s , a s well as properties r e l a t e d t o t h e p a r t i a l d e r i v a t i v e opera= t o r s , s i m i l a r t o those in Theorems 2 a n d 3 , except f o r the following s t r o n g e r ones. Theorem 9 I f ( V , S ) i s a r e g u l a r i z a t i o n , then k
1 ) the m u l t i p l i c a t i o n in A ( V , S ) , f o r il m u l t i p l i c a t i o n of functions;
E
l,induces on C k ( n ) t h e usual
2 ) t h e following r e l a t i o n s hold ( s e e ( 2 9 ) , Chapter 1 ) : k-k
Ak(V,S)
<
A ( V , S ) , V il, k
E
fly k < R ;
3 ) the p a r t i a l d e r i v a t i v e operators
(95)
D p : A'(V,S)
Ak(V,S)
-+
defined by (95.1)
Dp(s t I q V , S ) )
=
Dps
+
Ik(V,S),
V s E Ak(V,S),
f o r any k , k E 1, k < R , p E N n y IpI < k - k , c o i n c i d t w i t h t h e usual p a r t i a l d e r i v a t i v e s of f u n c t i o n s , when applied t o C -smooth f u n c t i o n s . Proof
1) The proof of t h i s follows from ( 9 2 ) . 2 ) We f i r s
n o t i c e t h a t in view of ( 8 6 ) we have
(96)
i n particu (96.1)
V'
c Vk.
B u t (87) y i e l d s (97)
A'( V , S ) c A k ( V , S )
which will imply
(98)
D P A ~ ( V , S )c A ~ ( v , s ) v, p E N ~ ,
G a-k,
k k s i n c e A ( V , S ) i s by d e f i n i t i o n C - d e r i v a t i v e i n v a r i a n t , hence D P ( A ~ ( V , S )n ( ~ ~ + l p l ( nc) A) ~~( v , . s ) ,v p
E
N~
while (97) obviously y i e l d s Ak(V,S)
c Ak(V,.S)
n
(Ck+lpl(a))N, V p
E
N n y IpI G Il-k.
92
E . E . Rosinger
Final l y (99)
DPIR(V,S) c l k ( V , S ) , Y p
E
Nn,
IpI
Q R-k
s i n c e , i n view of (92),lR(V,S) i s ihe vector subspace indliV,S), generated by VR.A ( V , S ) , and Dp applied t o C -smooth functions s a t i s f i e s t h e Leibnitz rule f o r product d e r i v a t i v e ; thus (96) and (98) will indeed r e s u l t i n (99). 3) The proof of t h i s follows from ( 9 2 ) .
0
Remark 5 When dealing w i t h the polynomial nonlinear PDEs given in ( l ) , Chapter 1, the corresponding PDOs in (3), Chapter 1, can be considered t o a c t a s f o l = lows (see (33), Chapter 1 ) : (100)
T ( D ) : A'( V,S)
-+
k
A (V,S)
provided t h a t ( V , S ) i s a r e g u l a r i z a t i o n , R, k E 8 , k < R, R-k > m y m being the order of t h e PDO considered, apd the c o e f f i c i e n t s c i , a s well as f i n the PDE i n ( l ) , Chapter 1, being C -smooth functions. The existence, characterization and construction of regularization can be d e a l t w i t h i n a manner s i m i l a r t o t h a t used f o r r e g u l a r i z a t i o n s , presented in Section 4. Next, we b r i e f l y mention the necessary modifications t o both the statement and proof of t h e r e s u l t s in Section 4. Theorem 10 N I f 1 is an ideal i n (Co(n)) and t h e r e e x i s t vector subspaces S, T c Sosuch that (101)
I n T = V " c T = Q
( 102 1
' I ~ S O C V "
(103)
S o = V " @ S @
( 104 1
Uco(n)c S
@
T
T
@ T;
then f o r any vector subspace V c 1 n (105)
(V,S
@ T)
v"
i s a regularization.
Conversely, any regularization is of t h e form i n (105). Proof This is s i m i l a r t o t h a t of Theorem 4.
0
In the characterization of r e g u l a r i z a t i o n s obtained i n Theorem 10, t h e i r existence i s equivalent t o the existence of i d e a l s 7 i n (Co(n))Ns a t i s f y i n g the conditions i n (101-104). This f a c t justifies t h e following d e f i n i t i o n
ALGEBRAS CONTAINING THE DISTRIBUTIONS
93
in which, unless there i s a need f o r specifying the d i s t i n c t i o n , the same term will be used as in Definitign 3. When such a need a r i s e s , the ideals in Definition 3 will be called C -regular, while those defined next will be called C"-regular. Definition 6
An ideal I in (C"(n))Ni s called regular, only i f , f o r s u i t a b l e vector subspaces S, 7 c S the c o n d i t i o n s m l - 1 0 4 ) are s a t i s f i e d . Before obtaining in Theorem 11 the characterization of regular ideals i n the above sense, i t will be useful t o make some remarks. We notice t h a t the definition (39) of vanishing ideals in (C"(Q)) N can be extended t o ideals in ( C o ( R ) ) N . I n t h a t case, Ind will be a vanishing N ideal in (C"(Q)) . Proposition 4 If I i s a vanishing ideal in ( C " ( Q ) ) N ,then there e x i s t vector subspaces T C So satisfying the conditions (106)
I nT
( 107 1
v0
t
=
v"
Q and
(see (101))
v0 @ T.
(see (102))
n T =
(I n so) =
Proof This i s similar t o the proof of Proposition 1, where now we take E = S o , A = v" and B = 1 n S " .
a.
I t should al&o be noticed t h a t the definition (49) of c final invariant ideals in ( C ( f i ) ) N can be extended t o ideals in ( C o ( Q ) ) In t h a t case, I n d will also be a cofinal invariant ideal. Moreover, w i t h i n t h a t extended framework of ideals in ( C o ( Q ) ) N , Proposition 2 remains valid. Theorem 11 A cofinal invariant ideal 1 in ( C ( Q ) ) Ni s regular, only i f (108)
I n U q n ) = !! ;
in which case I will also be a vanishing ideal. Proof This i s similar t o t h a t of Theorem 5 , with the following modifications.
First, Proposition 4 will grant the existence of vector subspaces T c 9 satisfying (106) and (107). On the other hand, T c
s"
can be assumed (see (51-53)) t o s a t i s f y
94
E . E . Rosinger = (T
(110)
U
(111)
( vJo 'G
C" (a)
T)
n uCo(s2))@
n U'
u'
; and
= Q
where U ' i s a s u i t a b l e v e c t o r subspace i n U c o ( n ) .
Indeed, (109) can be
o b t a i n e d s i m i l a r l y t o ( 5 1 ) , b y a p p l y i n g Lemmas 2 and 3 t o E = S o , A = V " , B = Uco (n) and C = 7 n So as w e l l as t o
1=
E
V"
0(Uco(n)
= (7
nso)
n (V" t ( 7
n s ' ) ) ) and
( u ~ ~n ( ~ t) ( 7 (VO
"so))).
Now (111) w i l l g r a n t t h e e x i s t e n c e o f v e c t o r subspaces S c So s a t i s f y i n g (103) and (112)
U' c s.
Then S, T and any vect.or subspace V c 7 n V" w i l l s a t i s f y (101-104). In= deed (101) and (102) f o l l o w from (106) and (107). F i n a l l y , (104) w i l l be 0 o b t a i n e d i n view o f (109) and (112). Corollary 3 I f 7 i s a c o f i n a l i n v a r i a n t , r e g u l a r i d e a l i n ( C " ( S ~ ) ) ~then , there e x i s t v e c t o r subspaces S , T c So, such t h a t t h e f o l l o w i n g s t r o n g e r v e r s i o n of t h e c o n d i t i o n s (101-104) i s s a t i s f i e d :
(113)
l n ~ = v " n ~ = q
(114)
V"
(116)
UC"(Q)
t
( r n s a ) = vn @
J
0
Proof T h i s f o l l o w s from t h e p r o o f o f Theorem 11.
0
The r e s u l t corresponding t o Theorem 6, i s g i v e n in t h e f o l l o w i n g theorem. Theorem 12 N An i d e a l T i n ( C O ( 0 ) ) i s r e g u l a r , o n l y i f t h e r e e x i s t v e c t o r subspaces T c So s a t i s f y i n g t h e c o n d i t i o n s (101), (192) as w e l l as (117)
V' n (Uco(n) + T ) = Q.
Proof Assume t h a t 7 i s r e g u l a r ; lows t h a t
t h e n i n view o f (101), (103) and (104), i t f o l =
ALGEBRAS CONTAINING THE DISTRIBUTIONS
95
Now an element i n t h e l e f t - h a n d t e r m o f (117) has t h e form v =
t t
U($)
w i t h v E V o , $ E Co(n) and t E T . Thus i n view o f (118) we have u(+) = v - t E T . Hence, a c c o r d i n g t o (101), v E V" n 7 = 2, and (117) has been proved. Conversely, assume t h a t (117) h o l d s . An element o f t h e l e f t - h a n d t e r m o f (118) has t h e form U($) = v
+
t
w i t h $ E Co(n), v E v" and t E T . T h e r e f o r e v = u ( $ ) - t and, i n view o f (117) we have t h a t v E 2, and t h i s completes t h e p r o o f o f (118). I t o n l y remains t o show t h a t (104) h o l d s f o r s u i t a b l e v e c t o r subspaces S c So s a t i s f y i n g a l s o (103). To do so, we t a k e a v e c t o r subspace U'C
uco(n) such
(119)
UC"
that
(n)
=
(0) n T )
0u1.
Then i n view o f (119) we have (120)
ul
n
(v" @ T ) = 2
ul
n
(v" @TI
since =
u l
n
( u ~ . ( ~n) (v" @
TI) c
ul
nT
t h e l a t t e r i n c l u s i o n r e s u l t i n g from (118). Now i n view o f (120) t h e r e e x i s t v e c t o r subspaces S c So s a t i s f y i n g (103) and such t h a t U ' c S . I n t h a t case, (120) w i l l o b v i o u s l y i m p l y (104). 0 The corresponding v e r s i o n o f Theorem 7 i s now presented. Theorem 13 I f a v a n i s h i n g i d e a l 7 s a t i s f i e s t h e cond t i o n
(121)
(v"
+ 7) n
=
2
t h e n 7 i s a r e g u l a r i d e a l and t h e r e e x i s t r e g u l a r i z a t i o n s
(I n V,S
@ T)
s a t i s f y i n g t h e c o n d i t i o n s (113-115) as w e l l as
E.E.
96
Rosinger
Uc"(n) =
(122)
(see (104) and ( 1 1 6 ) ) .
Proof Since 7 i s v a n i s h i n g , P r o p o s i t i o n 4 w i l l g r a n t t h e e x i s t e n c e o f v e c t o r subspaces T c So s a t i s f y i n g (113) and (114). B u t c o n d i t i o n (121) i s ob= viously equivalent t o U
C"
n (Vo
(a)
t
(7
ns'))
=
2
o r , i n view of (114), e q u i v a l e n t t o t h e c o n d i t i o n
Now t h e e x i s t e n c e o f v e c t o r subspaces S c So s a t i s f y i n g (115) and (122) follows e a s i l y .
0
The adapted v e r s i o n o f P r o p o s i t i o n 3 i s g i v e n i n t h e f o l l o w i n g p r o p o s i t i o n . Proposition 5 The nowhere-dense i d e a l Ind (see ( 6 5 ) , Chapter 1) i s v a n i s h i n g , c o f i n a l i n v a r i a n t and r e g u l a r , and s a t i s f i e s t h e c o n d i t i o n (121). Proof I t s u f f i c e s t o show t h a t Ind s a t i s f i e s (121).
w
(123) with w
E
Assume indeed
= v t u(+)
Ind, v E v" and I$E Co(n).
Then o b v i o u s l y w E So and < w,.>
= JI.
Now, (69) i n Chapter 1 w i l l i m p l y t h a t $(x) = 0, V x E n. Then, i n view 0 o f (123), we have w = v E v", which completes t h e p r o o f o f (121). I n view o f t h e r e s p e c t i v e c h a r a c t e r i z a t i g n s i n Theorems 5 and 11, t h e con= n e c t i o n i s e a s i l y e s t a b l i s h e d between C - r e g u l a r and C " - r e g u l a r i d e a l s ; v i z i f t h e c o f i n a l i n v a r i a n t i d e a l I i n (Co(n))N i s C o - r e g u l a r , t h e n (see (86.2)) (124)
1
=
r
n
(cm(n)lN
a
w i l l be a ?-regular An i d e a l J i n (?(a))
ideal i n
N
(C"(n)) N .
i s c a l l e d Cm-strongly r e g u l a r , o n l y i f
=
f o r a c e r t a i n C"-regular ideal I i n O b v i o u s l y (see ( 8 3 ) )
(P(n)).
ALGEBRAS CONTAINING THE DISTRIBUTIONS
97
I ; di s a Cm-strongly r e g u l a r i d e a l .
(125)
IL seems t o be r a t h e r l e s s s i m p l e t o e s t a b l i s h t h e c o n n e c t i o n between C - r e g u l a r i z a t i o n s and C " - r e g u l a r i z a t i o n s . T h i s again i l l u s t r a t e s t h e v a l u e o f t h e c h a r a c t e r i z a t i o n s i n Theorems 4 and 10, which reduce t h e regu= l a r i z a t i o n s t o r e g u l a r i d e a l s , and t h e v a l u e o f t h e c h a r a c t e r i z a t i o n s o f these r e g u l a r i d e a l s , i n Theorems 5 and 11.
6.
Q u o t i e n t Algebras c o n t a i n i n g Large Subspaces of D i s t r i b u t i o n s
I t i s shown i n t h i s s e c t i o n t h a t i t i s p o s s i b l e t o c o n s t r u c t c h a i n s o f q u o t i e n t a l g e b r a s (93) (1
A (V,S), with
v. E fl,
each o f which c o n t a i n s t h e d i s t r i b u t i o n s i n D ' ( R ) , w i t h t h e p o s s i b l e ex= ception o f the functions i n
cR(n).
CO(R) \ Definition 7
An i d e a l I i n (Lo(R)) if
N
I nso c
t 126 )
il
i s c a l l e d C -smooth, f o r a c e r t a i n g i v e n il E
V"
t
fi, o n l y
( I n?).
Proposition 6 In i, s ,a C'-smooth
i d e a l i n ( C ' ( R ) ) ~ , f o r any g i v e n
(1 E
fl.
Proof Assume (Kw that
Iw
E N ) i s an ' n c r e a s i n g f a m i l y o f compact subsets i n
R, such
U K c R, compact :
(127)
3 v e N : K c Kv
w h i l e JI,
E
C'(R),
with w
E
N, s a t i s f y the conditions
(128.1)
supp J I ~c R compact
(128.2)
$w = 1 on K
W
Assume now w
E
.
lnd n 9 and c o n s t r u c t t h e sequence o f f u n c t i o n s w '
E
lnd
de f ined by w$(x) = $v(x)wv(x), Since w
E
V w
E
N, x
E
R.
9,i t i s easy t o see t h a t t h e r e l a t i o n s (127) and (128) i m p l y
98
(129)
E.E.
w' - w
Rosinger
v".
E
Now, i n view o f L g m a 4 below, i t i s p o s s i b l e t o c o n s t r u c t a sequence of f u n c t i o n s w" E ( C (R))N such t h a t f o r any u E N , t h e r e l a t i o n s h o l d : (130.1)
supp w;
(130.2)
Iw;(x)-w;(x)
C
B(O,l/(v+l)) t supp w \
I
G 1/( v t l ) ,
v
x E R
where, f o r x E Rn and p z 0, we used t h e n o t a t i o n (131)
B(x,p) =
Ily-xll < p }
E Rn
w i t h 1I II t h e Euclidean norm on Rn. Then (132)
Ind.
W"E
Indeed, w s a t i s f i e s ( 6 9 ) , Chapter 1, i . e . V X E R \
r:
3 LI E N, V C R \ VvEN,v>p:
r
supp wv c R \
v.
neighbourhood o f x :
Therefore i n view o f (130.1) i t f o l l o w s t h a t V X E R \
3 p' V
E
r:
N , V'c R v>u':
\r
neighbourhood o f x:
vEN,
supp w;
c R \ V'
which o b v i o u s l y i m p l i e s (132) if ( 6 5 ) , Chapter 1 i s taken i n t o account. F u r t h e r , we show t h a t (133)
W"
- w'
E
v.
Indeed, assume $ E D(R). inequalities
Then i n view o f (130.2) we have f o r v E N, t h e
f lwp)-";(") I * I + ( x ) I d x G ( f I J l ( x ) I d x ) / ( v + l ) . R R Now t h e r e l a t i o n s (129), (133) and (132) w i l l y i e l d w E
v"
t
( I n d n sm)
which completes t h e p r o o f o f (126) f o r a r b i t r a r y II
E
17.
ALGEBRAS CONTAINING THE DISTRIBUTIONS
99
Lemma 4
.
Given JI E C" (R w i t h supp $ c R compact, and c o n s t r u c t E d"(n) such t h a t (134.1)
supp
(134.2)
1
x
+ supp
c B(O,E)
-
x(x)
$(XI1
Gn
Y
E.
q > 0, i t i s p o s s i b l e t o
$ and
tl x
R.
E
Proof Obvious y , i t can be assumed w i t h o u t l o s s o f genera i t y t h a t 0 supp $ c R. We assume f u r t h e r p > 0, such t h a t B(O,E)
I
s u p { ~ $ ( x ) - ~I ) x and t a k e w
E
E
R, y
E
Q n B(X,P))
E
R and
rl ;
satisfying the conditions
C"(R)
2 0 , tf x E R;
.(X)
supp w c B(0,p);
and
w(x)dx = 1.
R
x
We n e x t d e f i n e
x(x) =
E
Cm(R) b y
I NY)dX-Y)dY.
R
Now (134.1) w i l l o b v i o u s l y f o l l o w . Ix(x)-+(x) =
R
I
I
R
Moreover l4X-Y)dY =
I$(X)-$(Y)
I$(x)-$(y) I w ( x - y ) d y G r i
, tf
x E
Q
n B(x,P)
and t h i s completes t h e p r o o f o f (134.2).
0
We now prove a r e s u l t i m p l y i n g t h a t t h e c h a i n s o f q u o t i e n t a l g e b r a s (93) can a l s o c o n t a i n t h e d i s t r i b u t i o n s i n U ' ( n ) , w i t h t h e p o s s i b l e e x c e p t i o n o f f u n c t i o n s n o t s u f f i c i e n t l y smooth. Theorem 14
N R Suppose t h g i d e a l I i n ( C o ( R ) ) i s v a n i s h i n g , i s C -smooth f o r a c e r t a i n g i v e n R E N, and s a t i s f i e s t h e c o n d i t i o n (135)
( v" + I ) n Uco(n) =
Q.
Then t h e r e e x i s t v e c t o r subspaces R,T c SR
(136)
1n T = V nT=Q
(137)
V t t n T ) = v @ ~
, satisfying
the conditions
100
E.E.
Rosinger
Therefore using the n o t a t i o n
we o b t a i n a r e g u l a r i z a t i o n
Moreover
f o r e v e r y v e c t o r subspace V c I n V " .
Proof Taking E =
s',
and B = 7 n S R
A = y'
and u s i n g t h e same argument as i n t h e p o o f o f P r o p o s i t i o n 1, we o b t a i n t h e e x i s t e n c e o f v e c t o r subspaces T c S s a t i s f y i n g
E
II
2 and ( I n sR)= v'@ T .
(142)
I n T = V nT
(143)
V'
+
=
Now (142) w i l l o b v i o u s l y i m p l y (136), s i n c e T c SR and Vo n S' Also, t h e r e l a t i o n (143) o b v i o u s l y i m p l i e s t h a t (144)
vo
t
( r n sR)=
i f we t a k e i n t o account v" n T =
(145)
v
+ ( I ns') c
vo
@
2in vo
=
T
(136).
But
+ ( I n S O ) c V" + ( I n's')
t h e l a t t e r i n c l u s i o n r e s u l t i n g from (126). (145) w i l l i m p l y (137).
Now t h e r e l a t i o n s (144) and
I n view o f (135) and (137), we have
I t f o l l o w s t h a t t h e r e e x i s t v e c t o r subspaces
R'c So such t h a t
Then, t a k i n g
i n Lemna 5 below, i t f o l l o w s t h a t t h e r e e x i s t v e c t o r subspaces
C = R c F = SR
vR .
ALGEBRAS CONTAINING THE DISTRIBUTIONS
101
T h i s completes t h e p r o o f o f (138). Now i n view o f Theorem 10, t h e r e l a t i o n s (136-139) w i l l o b v i o u s l y i m p l y (140).
R
Finally, noticing that i n view o f (86.1).
@T
, the
c 'S
r e l a t i o n (141) f o l l o w s e a s i l y , 0
Lemma 5 I f F, A and B a r e v e c t o r subspaces i n E and
(146)
E = A t F
(147)
A n B = 0 ( t h e n u l l space)
then t h e r e e x i s t v e c t o r subspaces C c F, such t h a t (148)
A n C = O
(149)
A
@B
= A
@
C.
Proof Assume, i n view o f (146), t h a t (150)
E=A@F'
f o r a s u i t a b l e v e c t o r subspace F ' C F. F u r t h e r , assume t h a t ( a . l i E I ) , ( b j [ j E J ) and ( f k l k E K) a r e a l g e b r a i c bases i n A, B and F ' r k p e c t i v e l y . Then i n view o f (146)
c
b. = J
Xji
c
ai +
p j k fk,
Y
j
E
J,
k E K
~
E
I
hji
# 0 and p j k # 0
where
f o r o n l y a f i n i t e number o f i E I , k v e c t o r subspace i n F ' generated by c
=
j
c
pjk
fk,
E
K respectively.
with j
E
Now t a k e C as t h e
J.
k E K
Then, i n view o f (150), t h e r e l a t i o n (148) f o l l o w s e a s i l y . i s obvious t h a t
Moreover, i t
102
E.E. A t B C A
+
Rosinger
C.
F i n a l l y , we demonstrate t h e i n c l u s i o n C c A t
Indeed, i f x
E
B.
C, then
where p. # 0 f o r o n l y a f i n i t e number o f j E J .
I t now f o l l o w s e a s i l y t h a t
J
x =
1 Pjbj1 oj j € J j E J
z
i € I
hji ai E A
+ B;
and i n view o f (147) and ( 1 4 8 ) , t h i s completes t h e p r o o f o f (149).
0
The analog o f Theorems 1 and 8 i s presented i n t h e f o l l o w i n g theorem. Theorem 15
Q
If ( V , S + T ) i s a r e g u l a r i z a t i o n g i v e n as i n (139-140), t h e n f o r each k E fly k . .R , i t i s p o s s i b l e t o c o n s t r u c t t h e i n c l u s i o n diagrams
which w i l l s a t i s f y t h e c o n d i t i o n
R
@ T)
= Vk
o r equivalently, the simpler condition
(151.2)
Ik(V,S
@
T)
n (U k
c
(0)
@ R @ T)
=
2
.
Proof T h i s i s s i m i l a r t o t h e p r o o f o f Theorem 1.
0
We now g i v e t h e main r e s u l t i n t h i s s e c t i o n , i n d i c a t i n g t h e v e c t o r subspaces
ALGEBRAS CONTAINING THE DISTRIBUTIONS
103
o f d i s t r i b u t i o n s which can be embedded i n t o t h e q u o t i e n t a l g e b r a s ( 9 3 ) . Given a v e c t o r subspace H c S o y we use t h e n o t a t i o n
DpL)
(152)
=
{S = < s , * >
s E HI.
Theorem 16
Lt'
I f (V,S T ) i s a r e g u l a r i z a t i o n g i v e n as i n (139-140), t h e n t h e f o l = l o w i n g embeddings h o l d :
Ck(n)
(153)
9
c Ak(V,
UQT(R)
S
@ T),
U k
E
w,
k
as d e f i n e d b y t h e l i n e a r mappings
isom
lin,inj
I
s
t
Ik(V,s
@ T ) E Ak(V,s @ T ) .
Proof T h i s f o l l o w s e a s i l y from t h e i n c l u s i o n diagram i n ( 1 5 1 ) .
0
Remark 6
N
i n (Co(n)) s a t i s f i e s a ) I n view o f P r o p o s i t i o n s 5 and 6, t h e i d e a l Ind t h e c o n d i t i o n s i n Theorems 14, 15 and 16, f o r any g i v e n R E
w.
b)
The s i z e o f t h e v e c t o r subspaces o f d i s t r i b u t i o n s i n ( 1 5 3 ) :
(154)
Ck(n)
@ Dk
T(R)
,k
E
a,
k
c o n t a i n e d i n t h e r e s p e c t i v e q u o t i e n t a l g e b r a s ( 9 3 ) , can e a s i l y be apprecia= ted, i f i t i s n o t e d t h a t i n view o f (138), we have (155)
D'(Q) =
CO(i-2)
@ Dk
0
$2).
T h e r e f o r e t h e o n l y d i s t r i b u t i o n s t h a t m i g h t n o t be c o n t a i n e d i n t h e quo= t i e n t algebras k
A (V,S
@ T),
with k E
8, k
E.E.
104
Rosinger
obtained i n Theorem 16, a r e those i n k (? (Q) \ C (a),w i t h k E
R,
k GR.
c ) I t i s i m p o r t a n t t o p o i n t o u t t h a t t h e r e s u l t i n (153), Theorem 16, y i e l d s what i s i n a sense t h e l a r g e s t v e c t o r subspaces o f d i s t r i b u t i o n s t h a t can be embedded i n t o t h e corresponding q u o t i e n t a l g e b r a s ( 9 3 ) . In= deed, as shown i n S e c t i o n 9, i t i s nat p o s s i b l e , as i n Theorem 16, t o con= s t r u c t q u o t i e n t algebras t h a t would possess embeddings Ck-2(Q)
c Ak(V,S
@ T).
d) I t i s obvious t h a t each o f t h e types o f chains o f q u o t i e n t algebras (24) o r (93) presents c e r t a i n p r o p e r t i e s t h a t a r e s t r o n g e r t h a n those o f t h e o t h e r . F o r instance, each q u o t i e n t a l g e b r a i n chains of t y p e (24) c o n t a i n s a l l t h e d i s t r i b u t i o n s i n D'(Q), w h i l e t h e embeddings o f d i s t r i b u = t i o n s i n t o t h e q u o t i e n t algebras i n chains o f t y p e (93) a r e l i m i t e d by t h e c o n d i t i o n (178), as seen i n S e c t i o n 9. On t h e o t h e r hand, m u l t i p l i c a t i o n and p a r t i a l d e r i v a t i v e o p e r a t o r s w i t h i n t h e c h a i n s o f q u o t i e n t aigebras (24) c o i n c i d e w i t h t h e corresponding usual o p e r a t i o n s o n l y f o r C -smooth f u n c t i o n s , w h i l e w i t h i n t h e q u o t i e n t algebras AR(V,.S) o f c h a i n s o f t y p e (93), t h e y c o i n c i d e w i t h t h e corresponding o p e r a t i o n s f o r C k i m o o t h func= t i o n s , f o r any R E fl (see Theorem 9, S e c t i o n 5 ) .
7.
Q u o t i e n t Algebras h a v i n g Special P r o p e r t i e s
I t w i l l be convenient i n t h e sequel t o deal w i t h cases o f q u o t i e n t bras c o n s t r u c t e d i n Sections 3 and 6, t h a t have s p e c i a l a d d i t i o n a l t i e s needed f o r c e r t a i n a p p l i c a t i o n s . Several o f t h e s e a d d i t i o n a l t i e s w i l l depend on corresponding p r o p e r t i e s o f t h e subalgebras o f ces of f u n c t i o n s A , which d e f i n e t h e r e s p e c t i v e q u o t i e n t a l g e b r a s (15) and i n p a r t i c u l a r ( 2 4 ) and ( 9 3 ) )
alge= proper= proper= sequen= (see
A = A/7 (V,A). I t i s o u r aim i n t h e p r e s e n t s e c t i o n t o develop a u n i f i e d approach t o t h e s t u d y o f t h e above-mentioned p r o p e r t i e s o f t h e q u o t i e n t algebras (24) and ( 9 3 ) . Since t h e subalgebras A can be seen w i t h i n e i t h e r (Cm(n))N o r (CR(Q))N, w i t h R E N, corresponding t o t h e q u o t i e n t algebras ( 2 4 ) and ( 9 3 ) r e s p e c t i v e l y , two more o r l e s s para1 l e l approaches would be needed. How= ever, i t i s p o s s i b l e t o p r e s e n t them simultaneously, w i t h t h e h e l p o f a n o t i o n i n t r o d u c e d now. I n o r d e r t o deal w i t h t h e r e q u i r e d p r o p e r t i e s o f t h e subalgebras t h a t have been mentioned, we s h a l l c o n s i d e r p r o p e r t i e s P valid f o r certain inclusions
(156)
H c (CR(Q))N
where R E For t h e sake of s i m p l i c i t y , we s h a l l say t h a t H has t h e r o = p e r t y P i n C whenever t h e p r o p e r t y P i s v a l i d for t h e i n c l u s i d Definition 8
N A p r o p e r t y P v a l i d f o r c e r t a i n i n c l u s i o n s ff c ( C a ) ) w i t h a r b i t r a r y R E N, i s c a l l e d a d m i s s i b l e , o n l y i f f o r any R E (157.1)
(CR(Q))N has t h e p r o p e r t y P i n CR;
ALGEBRAS CONTAINING THE DISTRIBUTIONS
105
n Hi has t h e p r o p e r t y P i n C Q , whenever each Hi c ( C R ( n ) ) N , i E 1 w i t h i E I, has t h e p r o p e r t y P i n C:
(157.2)
Suppose P and Q a r e a d m i s s i b l e p r o p e r t i e s , thenQP i s c a l l e d s t r o n g e r t h a n Q ( d e n t t e d by P >Q) o n l y i f each subset H C ( C ( ~ 2 ) )which ~ has property P i n C , w i l l a l s o have p r o p e r t y Q i n CR.
.. ,P n
Obviously, i f P1,.
are admissible properties, then t h e i r conjunction
P = P1 and
...
and P2
i s a l s o an a d m i s s i b l e p r o p e r t y and P = maxt P1,...,
i n t h e sense o f t h e above p a r t i a l o r d e r 2 .
N D e f i n i g t h e p r o p e r t y P b y t h e c o n d i t i o n t h a t H C ( C Q ( R ) ) has t h e p r o p e r t y 1 P i n C o n l y i f H = (CR(Q))N, one o b v i o u s l y o b t a i n s t h e s t r o n g e s t a d m i s s i b l e property. We now p r e s e n t some a d m i s s i b l e p r o p e r t i e s a l r e a d y encountered o r t o be met w i t h i n t h e sequel. The d e r i v a t i v e i n v a r i a n t p r o p e r t y (see ( 1 7 ) , ( 8 5 ) ) i s d e f i n e d b y
H
(158)
C
(CQ(R))N i s d e r i v a t i v e i n v a r i a n t i n C II,
only i f DP(H n ( c ' + I ~ l ( s ; ) ) ~ )c
(158.1)
H,
u
p
E
N ~ .
The p o s i t i v e power i n v a r i a n t p r o p e r t y i s d e f i n e d by H c ( c ' ( R ) ) ~ i s p o s i t i v e power i n v a r i a n t i n
(159)
cQ,
only i f
tls€H:
(159.1)
i ) sv(x) 2 0 , U v
i i ) sa
E
E
N, x E R'
(~'(fi))~,tl a E ( o , ~ )
\
where sa
E
(CQ(R))N i s d e f i n e d b y (sa),(x)
= ( S ~ ( X ) ) ~U, v E N,
x E R.
The subsequence i n v a r i a n t p r o p e r t y (see (28), Chapter 2 ) i s d e f i n e d b y (160)
H
C (C
& (n))N i s subsequence i n v a r i a n t i n C R Y
106
E.E.
Rosinger
only i f
tl s E (cR(n))N :
(160.1)
f
lItEHZ
*
S E
ti.
s subsequence i n t F i n a l l y , t h e c o f i n a l i n v a r i a n t p r o p e r t y (see ( 4 9 ) ) i s defined b y (161)
H c
( ~ ' ( n ) )N
i s c o f i n a l i n v a r i a n t i n C!
only i f (161.1)
v
s E (CR(n))N :
Now, i n o r d e r t o o b t a i n t h e q u o t i e n t 8lgebras (24) and (93) possessing s p e c i a l p r o p e r t i e s , we have t o r e d e f i n e (19-20) , o r e l s e (87-88), i n a way which i n c l u d e s these p r o p e r t i e s . Suppose g i v e n a f i x e d a d m i s s i b l e p r o p e r t y P. F i r s t , wegresent the modified version o f (19). V , S c ( C (n))N and R 6 N, denote
( 162 1
F o r given v e c t o r subspaces
APyR(V,S) = t h e s m a l l e s t s u k a l g e b r a s i n (Cm(n))N which has t h e p r o p e r t y P i n C and c o n t a i n s VR t S
.
The m o d i f i e d v e r s i o n o f 87 i s o b t a i n e d as f o l l o w s . F o r g i v e n v e c t o r subspaces V, S c ( C ( n ) ) kat!d R E N, unless t h e r e i s a need t o s p e c i f y t h e d i s t i n c t i o n from (162) , denote a g a i n
s.
APYR(V,S) = t h e s m a l l e s t su a l g e b r a i n ( C R ( n ) ) N which has t h e p r o p e r t y P i n C and c o n t a i n s VR + S R' Now t h e m o d i f i e d v e r s i o n s o f (20) and (88) can w i t h t h e h e l p o f (89) be g i v e n u s i n g t h e same n o t a t i o n , w i t h no r i s k o f c o n f u s i o n : (163)
( 164 1
rPsR(v,s)
I
= 7(vR, A ~ ' ' ( V , S ) ) .
It i s easy t o see t h a t i f ( V , S ) i s a C m - r e g u l a r i z a t i o n o r a C " - r e g u l a r i z a = t i o n , a r e s u l t w i l l a p p l y s i m i l a r t o t h a t o f Theorem 1, o r e l s e Theorem 8. Therefore i n b o t h cases we can d e f i n e t h e c h a i n o f q u o t i e n t a l g e b r a s
(165)
APyR(V,S) = APyR(VyS)/7P'R(V,S), w i t h R E
4,
ALGEBRAS CONTAINING THE DISTRIBUTIONS
107
that will have algebraic properties corresponding t o those in Theorem 2. I n order t o obtain partial derivative operators on the chains of quotient algebras (165) , similar t o those in Theorem 3 , or e l s e Theorem 9 , i t has to be assumed t h a t the admissible property P i s stron e r than the deriva= tive invariant admissible property in (158). Actua l+ y, i f the a d m w property P i s identical t o the one in (158), definition (165) will give the same chains of quotient algebras as those in ( 2 4 ) , o r e l s e (93), i . e .
Ap"(V,S)
(166)
V II g
= AR(V,S),
N,
according as ( V , S ) i s a C"-regularization, or e l s e a C"-regularization. We shall in the sequel always assume t h a t any admissible property P being considered, i s stronger-the derivative invariant property in (158). When in addition t o t h a t , the admissible property P i s stronger t h a n the admissible properties in (159), (160) o r (161) , the corresponding chains of quotient algebras (165) will be called positive power algebras, subse= uence a1 gebras, or cofinal a1 gebras respectively, and the superscript P g ?e dropped, for simplicity. 8.
Definition o f Positive Powers f o r Certain Generalized Functions
I n connection with the scattering problem solved in Chapter 7 , we shall need t o define a r b i t r a r y positive powers f o r the Dirac distribution: powers t h a t will yield the potentials studied. Since the framework for the dis= cussion will be given by the quotient algebras containing the distributions presented in Sections 3 and 6, i t i s useful t o specify for t h a t particular case the general method of defining positive powers presented in Section 10, Chapter 1. Suppose therefore t h a t we are given a Cm- or a C"-regularization ( V , S ) . Our task i s t o d fine arbitrary positive powers within each of the quo= t i e n t algebras A % ( V , S ) , with J? E N . Given II
E
fi, we use the notation: r ) + ( x ) > O , V x E Rn
c;(n)
(167)
= )I{
1.
E CR(Q)
**)
($1"
E
cR(n),
R C e r t a i n s i m p l e f a c t s about C+(R) s h o u l d be noted. $ E C R ( R ) and $ ( x ) > 0, V x E R
exist
)I E
C (R), w i t h
R, then $
$(x) 2 0 , V x
example o f such a f u n c t i o n i s
E
E
2 $ ( x ) = x1
v
a
E
For instance, i f
Cf(n). But i f
R, such t h a t
.....xn,2
(0,m)
$ $
R 2 1, t h e r e
Ct(R).
V x = (xl,
An
...,xn)
E
R,
i f 0 E R. However, t h e r e a r e non-negativemCm-smooth f u n c t i o n s on n t h a t v a n i s h on subsets o f 8 and y e t belong t o C + ( R ) . F o r example, i f 0 E R and
I
=
1
exp(-l/(xl +
0
... t
x,)) f o r xi > 0 , w i t h 1 < i < n
otherwise
108
E.E.
then $
E C,"(
Rosinger
Q).
F o r a v e c t o r subspace P c So we use t h e n o t a t i o n :
D ; , ~ , ~ ( Q )=
(168)
c
< s , - x P(Q)
and c a l l such d i s t r i b u t i o n s P-
I
s E P n (C;(Q))~I R C -non-negative.
Now, u s i n g t h e i d e a of t h e general method f o r dependent v a r i a b l e t r a n s f o r m , q i v e n i n ( 8 9 - 9 8 ) , ChaDter 1, i t i s D o s s i b l e t o o b t a i n t h e f o l l o w i n a r e s u l t on p o s i t i v e powers w i t h i n t h e q u o t i e n t algebras c o n t a i n i n g t h e d i s t r i b u = tions. F i r s t , we c o n s i d e r t h e case of chains o f q u o t i e n t algebras c o n t a i n i n g t h e distributions, introduced i n (24). Theorem 1 7 Suppose g i v e n a C m - r e g u l a r i z a t i o n ( V J ) satisfying the condition
(169)
U($)
P,
E
v
s
and a v e c t o r subspace p c
$ E CY(n).
Then
Cy(n) c
1)
,(Q) c A ' ( V , s ) ,
V R
E
w.
Moreover, i n t h e case o f p o s i t i v e power a l g e b r a s i t f o l l o w s t h a t :
2)
For R
(170)
E
1 and
c1 E ( O , m ) ,
we can d e f i n e t h e mapping c a l l e d t h e a - t h power
D;),m,
,(a)
-+
3 S
Scl E AR(V,.S)
by
(170.1)
Sa = s a t
Z R ( V , S ) , w i t h S = < s,.> , s
E
P n (C,"(n))N
.
The mapping ( 1 7 0 ) w i l l have t h e p r o p e r t i e s
1 = S;
(171.1)
S
(171.2)
SatB
(171.3)
(Sa)m = SaWm
If p E Nn and I p I
= Sa.SBy V a,@ E ( 0 , m ) ;
,V
a
(Op), m
E
N \ {O}.
= 1, t h e p a r t i a l d e r i v a t i v e o p e r a t o r (see
Dp : AR(V,S)
+.
Ak(V,S), k E
R,
(27))
k <~-l,
s a t i s f i e s the r e l a t i o n
(171.4) 3)
DPSa = a.S"-l.DpS,
V a E(1,m).
The mapping ( 1 7 0 ) , a p p l i e d t o f u n c t i o n s i n usual a - t h power o f f u n c t i o n s .
Cy(n), i s i d e n t i c a l t o t h e
109
ALGEBRAS CONTAINING THE DISTRIBUTIONS
Proof 1)
f o l l o w s from (169) and ( 2 6 ) .
2)
f o l l o w s f r o m (169) and ( 2 6 ) , as w e l l as (27).
3)
f o l l o w s from 1 ) and 2 ) .
0
Remark 7 a ) The c o n d i t i o n (169) can e a s i l y be f u l f i l l e d , s i n c e t h e Cm-regulariza= t i o n ( V , S ) s a t i s f i e s (21.3). b ) I n o r d e r t o d e f i n e a r b i t r a r y p o s i t i v e powers f o r t h e D i r a c & d i s t r i b u = t i o n w i t h i n the chains o f quotient algebras containing d i s t r i b u t i o n Lntro= duced i n ( 2 4 ) , we can use Theorem 17 as f o l l o w s . Suppose g i v e n $ E C+( R) such t h a t $(x)dx = 1.
(172)
s1
N
I f 0 E R, d e f i n e s E S"n(C:(R))
by
s U ( x ) = ( w l ) $ ( ( V t l ) x ) , Y u E N , x E R;
(173) then
< s>;
(174)
=
6.
Now we can d e f i n e a r b i t r a r y p o s A t i v e powers o f 6 w i t h t h e h e l p o f t h e map= p i n g (170), p r o v i d e d t h a t t h e C - r e g u l a r i z a t i o n ( V , S ) s a t i s f i e s t h e condi= tion
s
(175)
E
s.
T h i s procedure, i n r a t h e r more general form, w i l l be used i n Chapter 7. We now p r e s e n t t h e analog o f Theorem 17, i n t h e case o f c h a i n s of q u o t i e n t a1 gebras (93). Theorem 18 Suppose g i v e n a C " - r e g u l a r i z a t i o n ( V , S ) and a v e c t o r subspace P c S , satisfying
( 176 1
u($) E
P,
$
E
c p .
Then
Moreover, i n t h e case o f p o s i t i v e power a l g e b r a s , i t f o l l o w s t h a t :
2) F o r R
E
R and
C ~ (EO , m ) ,
we can d e f i n e t h e mapping c a l l e d t h e
power
(177)
Sa
E
AR( V ,s)
&
E . E . Rosinger
110
by (177.1)
SO1 = s'
t
IR(V,.S), w i t h S = < s , * >
, sE
p n ( c (' sl))N.
The mapping (177) w i l l a l s o have t h e p r o p e r t i e s g i v e n i n (171.1-4) 3)
The mapping (177) a p p l i e d t o f u n c t i o n s i n Cf(sl) i s i d e n t i c a l t o t h e usual a - t h power o f f u n c t i o n s .
Proof 1)
f o l l o w s from (176) and ( 9 2 ) .
2)
f o l l o w s from (176),
3)
f o l l o w s from 1 ) and 2).
(92) and ( 9 5 ) . 0
Remark 8 a)
The c o n d i t i o n (176) can e a s i l y be f u l f i l l e d , s i n c e t h e C"-regulariza= t i o n ( v , ~ ) s a t i s f i e s (90.3).
b)
I n view o f Theorem 16, we can d e f i n e a r b i t r a r y p o s i t i v e powers o f t h e D i r a c 6 d i s t r i b u t i o n a l s o w i t h i n t h e chains o f q u o t i e n t a l g e b r a s i n t r o = duced i n ( 9 3 ) , by u s i n g t h e procedure g i v e n i! (172-175). More pre= cisely, i f E $(a) f o r a c e r t a i n g i v e n RE N , t h e a r b i t r a r y p o s i t i v e powers o f 6 w i l l be d e f i n e d i n each o f t h e q u o t i e n t a l g e b r a s
+
k
A
(us),
with k €
8,
kGR.
D e t a i l s o f t h i s procedure a r e presented i n Chapter 7. 9.
L i m i t a t i o n s on t h e Embedding o f Smooth Functions i n t o Chains o f Q u o t i e n t A1 gebras
I t i s shown i n t h i s s e c t i o n t h a t , i n c o n t r a s t t o t h e case o f t h e chains o f q u o t i e n t algebras i n (24) which c o n t a i n a l l t h e d i s t r i b u t i o n s i n D ' ( n ) , smooth f u n c t i o n embeddings o f t h e form
( 178)
L!-~(R)
c
A (us) , R
E
N, R
2,
w i t h i n t h e chains o f q u o t i e n t algebras ( 9 3 ) a r e not possible. T h i s r e s u l t f o l l o w s from an a d a p t a t i o n o f L . Schwartz's well-known c o u n t e r example - see Appendix 2 - and f o r t h e sake o f s i m p l i c i t y i s presented o n l y f o r t h e one-dimensional case n = l , w i t h R = R
'.
Suppose g i v e n t h e a1 gebras (179)
A
2
, A1
and A'
t o g e t h e r w i t h t h e 1 i n e a r mappings c a l l e d d e r i v a t i v e s D D A2 + A1 -+ Ao (180) which s a t i s f y t h e L e i b n i t z r u l e f o r p r o d u c t d e r i v a t i v e s .
ALGEBRAS CONTAINING THE DISTRIBUTIONS
111
Suppose f u r t h e r t h a t t h e f o l l o w i n g c o n d i t i o n s a r e s a t i s f i e d : (181)
A
2
contains the functions x, x ( L n l x 1 - 1 )
and x 2 ( a n 1x1 -1)
.
and t h e m u l t i p l i c a t i o n
in A
2
9
i s such t h a t
2 ( x ( a n l XI -1)l.x = x (an1 XI - 1 ) ;
A
(182)
1 contains the functions 1, x and x(Enlx1-1)
1 and t h e c o n s t a n t f u n c t i o n 1 i s t h e u n i t element i n A ; A'
(183)
i s a s s o c i a t i v e and c o n t a i n s t h e f u n c t i o n s 1 and x
and t h e c o n s t a n t f u n c t i o n 1 i s t h e u n i t element i n A';
( 184 1
t h e mapping D : A' A 2 x and x (an 1x1-1)
1
+
applied t o the functions
i s t h e usual d e r i v a t i v e on C'(R ') ; t h e mapping D : A1
(185)
+
A'
applied t o the function
X
i s t h e usual d e r i v a t i v e on C ' ( R ' ) . Theorem 19 F o r any (186) where (187 1
6 E A',
the relation
x.6 = 0,
. signifies
m u l t i p l i c a t i o n i n A',
implies
6 = 0.
Proof The i d e a i s t o a p p l y t h e method used i n t h e p r o o f o f Theorem 1, Appendix 2, w i t h i n t h e above, more general framework. To do so, we f i r s t p r o v e t h a t t h e f o l l o w i n g r e l a t i o n h o l d s i n A': (188)
(D2(x(hlxl-l))).x
= 1.
Indeed, i n view o f t h e L e i b n i t z r u l e s a t i s f i e d b y t h e mapping D : A* we have
*) f o r x = 0, b o t h o f t h e f u n c t i o n s have b y d e f i n i t i o n t h e v a l u e z e r o .
+
A
1
,
E.E.
112
Rosinger
D((x(9.,n(x(-l)).x) = ( D ( x ( a n \ x \ - l ) ) ) . x t x (E.nlx(-l) i f (184) i s a l s o t a k n i n t o o a c c o u n t . Then (185) and t h e f a c t t h a t t h e + A also satisfies the Leibnitz rule, w i l l y i e l d l i n e a r mapping D : A
D2( (x(9.n
x l - l ) ) . x ) = (D2 (x(Rnlx
-l))).x
+
2 D(x(anlx1-1)).
It follows that
(189)
2 2 ( D2( x ( Rn x l - l ) ) ) . x = D ( x ( a n l x - 1 ) )
- 2
D(x(Rnlx1-1))
B u t (184) w i l l y i e l d
if (181) i s taken i n t o account.
2 D(x ( a n l x l - 1 ) ) = 2 x(Rnlx1-1)
+
x;
hence, (182), (185) and t h e l i n e a r i t y o f t h e mapping D : A’ (190)
D 2 ( x2 ( a n l x l - 1 ) ) = 2 D ( x ( a n l x 1 - 1 ) )
E
A’
and assuming (186) f o r a c e r t a i n 6
E
0 = x
-1
.(x.6)
= (x
-1
yield
Thus u s i n g t h e n o t a t i o n
(183) and (188) w i l l i m p l y
A’,
.x).6
A’
+ 1.
Now (189) and (190) w i l l o b v i o u s l y y i e l d (188). x - l = D2(x(Rnlxl-1))
-f
=
6.
0
I t f o l l o w s from t h e r e s u l t i n Theorem 19 t h a t t h e a l g e b r a A’ cannot c o n t a i n t h e D i r a c 6 d i s t r i b u t i o n , known t o s a t i s f y ( 1 8 6 ) , a r e l a t i o n g i v i n g an upper bound on t h e s i n g u l a r i t y e x h i b i t e d a t x = 0 b y t h a t d i s t r i b u t i o n . The e s s e n t i a l rea on why t h e c h a i n o f algebras (180) have t h a t i n c o n v e n i e n t f e a t u r e i s t h a t A i s supposed t o c o n t a i n t h e f u n c t i o n
3
(191)
x(Rnlx1-1)
E
C(R1) \
C1(R’).
I t i s i n t h i s sense t h a t Theorem 19 i m p l i e s t h e i m p o s s i b i l i t y o f t h e embed= dings (178). T h i s i m p o s s i b i l i t y r e s u l t i s i m p o r t a n t because none o f t h e algebras i n (180) was supposed commutative and o n l y t h e a l g e b r a Ao was r e = q u i r e d t o be a s s o c i a t i v e . Moreover, t h e f u n c t i o n i n (191) has a s i n g u l a r i = t y o n l y a t x = 0, i . e .
x(Rnlx1-1)
E
C ~ ( R \COI); ’
i n o t h e r words (192)
s i n g supp x(Rnlx1-1) =
{ol.
The i n t e r e s t i n g b u t undecided q u e s t i o n remains as t o whether smooth func= t i o n embeddings o f t h e form (193)
CR-’(Q)
are possible o r not.
C AR(V,S),
R E N, R 2 1
ALGEBRAS CONTAINING THE DISTRIBUTIONS 10.
113
Special Classes of Regular Ideals
In order t o obtain l a t e r in Section 4, Chapter 4, the general r e s u l t s on the resolution of s i n g u l a r i t i e s of weak s o l u t i o n s f o r polynomial nonlinear PDEs, i t i s useful t o consider special c l a s s e s of regular i d e a l s and e s = t a b l i s h those of their p r o p e r t i e s t h a t will be needed. These special c l a s s e s of regular i d e a l s w i l l be obtained by p a r t i c u l a r i z i n g t h e notions introduced i n Definition 3, Section 4 , o r e l s e Definition 6 , Section 5 . Definition 9 Suppose given a vector subspace R c Sm such t h a t (194)
R ) n V"
(Ucm(Q) t
=
2.
An ideal I in (C"(Q)) N ismcalled C", R-regular, only i f f o r s u i t a b l e vector subspaces S , T c S , the following r e l a t i o n s hold:
v"
(195)
I nT
(196)
rnsrnc V"@T
(197)
S m C V"
( 198)
UC"(Q)
=
nT =
0T
@ S t
2
R C S B T .
Obviously, the above notion i s a p a r t i c u l a r case of t$at introduged i n Definition 3, Section 3. Indeed, i f an ideal 1 i n ( C (g))Ni s C , R-regu= l a r , thgn i t i s a l s o C -regular. Moreover, I will be C -regular only i f i t i s C , R-regular, f o r R = 2. I t i s a l s o easy t o see t h a t (194) follows from (197) and (198). In a s i m i l a r way, by p a r t i c u l a r i z i n g Definition 6 , Section 5 , we can define
C , R-regular i d e a l s .
In order t o shorten the presentation, we s h a l l deal only w i t h C", R-regu= l a r i d e a l s and notice t h a t a l l t h e r e s u l t s i n this s e c t i o n obtained f o r them are valid a l s o f o r C? , R-regular i d e a l s . A necessary condition on C", R-regular i d e a l s i s obtained in the following extension of Corollary 1, Section 4.
Proposition 7 I f the ideal 1 in (C"(Q))N ( 199 1
I n
i s C", t
R-regular, then
R ) = 9.
Proof The r e l a t i o n s (196) and (198) imply t h a t I
(UC"(S-2)
n (S
t
@ T).
R) c (I n
s")
n ( u ~ " ( ~t) R )
c (v"
0T ) n
114
E.E. Rosinger
Therefore (197) w i l l y i e l d
which, i n view o f (195), w i l l i m p l y (199). We now p r e s e n t a s u f f i c i e n t c o n d i t i o n on C", f o l l o w i n g e x t e n s i o n o f Theorem /, S e c t i o n 4.
R-regular i d e a l s i n the
Theorem 20
I f a v a n i s h i n g i d e a l T i n (Cm(n)) ( 200 1
N
1 ) n (Ucm(n)
(Vm t
s a t i s f i e s the condition
+ R)
=
2,
then I i s Cm, R-regular, and t h e r e e x i s t C m - r e g u l a r i z a t i o n s
( r n v", s @
T)
satisfying the conditions (201)
I ~ T = V " ~ T =
(202 1
v"
(203)
Sm=Vm@S@
(204)
Ucm(n)
t
(r
n
s")
i- R C
=
2
vm @
T
T S.
Proof Since 'I i s vanishing, E r o p o s i t i o n 1, S e c t i o n 4 w i l l g r a n t t h e e x i s t e n c e o f v e c t o r subspaces T C S t h a t s a t i s f y (201) and (202). B u t c o n d i t i o n (200) i s equivalent t o ( ~ ~ m t ( ~ R) )
n (vm t
( r n S" 1)
=
2;
t h e r e f o r e , i n view o f (202) i t i s e q u i v e l e n t t o t h e c o n d i t i o n
From these c o n s i d e r a t i o n s t h e e x i s t e n c e o f v e c t o r subspaces S c S m s a t i s f y = i n g (2g3) and (204) f o l l o w s e a s i l y . Then i n view o f Theorem 4, S e c t i o n 4, 0 (I n V , S T ) w i l l indeed be a C m - r e g u l a r i z a t i o n .
0
Remark 9
a
The between t h e necessarx c o n d i t i o n i n P r o p o s i t i o n 7 and t h e s u f f i c i e n t c o n d i t i o n i n Theorem 20 on C , R - r e g u l a r i d e a l s , can e a s i l y be seen by comparing t h e r e l a t i o n s (199) and (200). As seen l a t e r i n Sections 1-5, Chapter 4, i n c o n n e c t i o n w i t h t h e r e s o l u = t i o n o f s i n g u l a r i t i e s o f i m p o r t a n t c l a s s e s o f weak s o l u t i o n s f o r p o l y n o m i a l n o n l i n e a r PDEs, t h i s gap t u r n s o u t t o be i r r e l e v a n t , s i n c e c o n d i t i o n (200) w i l l be a u t o m a t i c a l l y s a t i s f i e d .
ALGEBRAS CONTAINING THE DISTRIBUTIONS
11.
115
The Proof o f Lemma 1
The p r o o f 0s Lemma 1 i n S e c t i o n 1 i s g i v e n h e r e . The e q u i v a l e n c e between ( 7 ) and (8) f o l l o w s e a s i l y . The second p a r t i s proved as f o l l o w s . = Rn, xo = 0 E Rn.
For s i m p l i c i t y we s h a l l assume t h a t
R
For a E R ' and v
E
N denote
=
I x E Rnl s v ( x ) > a 1
E(a,v)
F i r s t , we prove t h e r e l a t i o n
K
(205)
v
-, ~0
i
s v ( x ) d x 2 1, U a
E
R'
E(a,v)
Assume i t i s f a l s e
.
Then
3 a € R'
,E >O,
U V E N
, v >
I
N :
p'€
p':
Sv(x)dx
<1-
E
E(a,v)
.
But, s E S o , i s , - > = 6 and supp s ( v ) s h r i n k s t o 0 E Rn, when w -t Therefore, assuming $ E lI(Rn) and $ = 1 on a neighbourhood o f 0 E Rn, one obtains ~0
1
=
$(O)
= lim
In sv(x)$(x)dx = l i m I
v+-tRR"
v+mR
s (x)dx
It follows t h a t
1
-
E/Z
r
sv(x)dx
Rn
Now, f o r v E N, t h e r e l a t i o n s h o l d
< 1
sv(x)dx
+
E(a,v)
a
I
dx
SUPP SV
Therefore, one o b t a i n s f o r v E N, v 2 max {IJ',~
1 - ~ / 2 < 1 - ~ + a J SUPP sv
dx
" 3the
inequality
E.E. Rosinger
116
which i s absurd since supp sv s h r i n k s t o 0 E R’, completed.
and t h e p r o o f o f (205) i s
We prove now t h a t t h e r e e x i s t av E [ O p ) , w i t h v E N, such t h a t (206)
l i m av v+cQ
and Tiiii
m
v +
sv x)dx 2 1
E(av,v)
m
Indeed, a c c o r d i n g t o (205), t h e r e e x i s t v E N, w i t h (207)
vo < v1 <
...
< v
<
P
u
...
u
E
N such t h a t
and
= i n f Ip E N Y v E N and
Define now av Then aV Q avtly
I
v
Q v u l , with v
E
N.
av , = u , Y p E N 1-1 due t o (207), hence, t h e f i r s t r e l a t i o n i n (206) i s proved. Taking i n t o account (208) the second r e l a t i o n i n (206) f o l l o w s from (209). (209)
F i n a l l y , we prove
l-iiii-
(210) v
-+
J E(av,v)
(sv(x))’dx
= t
Indeed, (sV)’ 2 avsv on E(av,v), Y v E N. Therefore
J
(s,(x))’dx
Y v E N, s i n c e sv
>av
E (av YV 1
- E(av,v) J
sv(x)dx
The r e l a t i o n (210) w i l l r e s u l t now f r o m (206).
I , (Sv(x))’dx = t m vfi + w R Then s’4 S” s i n c e supp s t = supp sv
2 a 2 0 on E(av,v) V
,
Y v E N
Obviously,
(206) i m p l i e s
s h r i n k s t o 0 E Rn when v
-+
m
Remark 10 The c o n d i t i o n o f n o n n e g a t i v i t y o f t h e sequence s i n t h e f i r s t p a n t o f Lemma 1, S e c t i o n 1 can be removed i n s p e s i a l cases. For i n s t a n c e , assume s given by s v ( x ) = av
$(bvx)
, bv
,
V v
E
N, x
E
Rn
,
R1 and l i m l b v l = t m , Then, i t i s easy t o see v -+m t h a t t h e equivalence between ( 7 ) and ( 8 ) i n t h e mentioned lemna, w i l l s t i l l be v a l i d .
where $ E U(Rn), av
E
However, a s seen n e x t i n S e c t i o n 12, t h e s h r i n k i n g o f the s u p p o r t o f s together w i t h the condition (8) are not s u f f i c i e n t i n order t o obtain y7). _ .
ALGEBRAS CONTAINING THE DISTRIBUTIONS
12.
117
Example o f S h r i n k i n g Non-6 Sequence
We s h a l l c o n s t r u c t SE
( c (R’))N
and $
E
U (R’)
such t h a t supp sv s h r i n k s t o 0
(211)
E
R’,
when v
-+
m
s v ( x ) d x = 1 , V V N~ R’ n e v e r t h e l e s s , t h e r e l a t i o n does not hold
I
(212)
l i m Il sv(x)$(x)dx = $(O)
(213)
v+mR
t h e reason f o r t h e f a i l u r e o f (213) b e i n g t h a t t h e f o l l o w i n g c o n d i t i o n i s not satisfied (214)
sV(x) > O ,
F i r s t we d e f i n e
x
: R’
-+
V v E N, x
< l/v
l / ( v + l ) < 1x1
if
and v
E
Mi01
otherwise
0 E
R’
R’ by
(-l)vlxl
and we d e f i n e t
E
by
(R’-+R’)N
otherwise
0
Then
while
1/ (2v+2)
1/ (2v+2) (216)
I
tv(x)dx =
R’
v + l Qu<m
6u2 + 6 A =
We d e f i n e now t ’ tb(X)
E
I
- 1/2v+2)
X(x)dx = 2
0
+1 c (R’
v -+
> R’)N
= tv(x)/cv,
O by Y v
I
E
N, x E R’
X(x)dx =
E.E. Rosinger
118 Then (216) w i l l i m p l y j l t$
(217)
( x ) d x = 1, V v
E
N
R
w h i l e (215) w i l l i m p l y t $ ( x ) x ( x ) d x = l / l 2 ( ~ + l ) ~,cV~ v
I
(218)
E
N
R'
B u t an easy computation w i l l y i e l d t h e i n e q u a l i t y
the r e f o r e
Now, i n view o f (218), i t f o l l o w s t h a t
1 t $ ( x ) x ( x ) d x > ~ ~ / 4 ( v + 1,) V~ v
E
N\{O)
R'
hence
(219)
fi
J l t \ ; ( x ) x ( x ) d x 2 1/4 > 0
= x(0)
v + m R A p p l y i n g a simple smoothing t o t ' and x and t a k i n g i n t o account (217 and (219), i t i s o b v i o u s l y p o s s i b l e t o o b t a i n s E (C "(R'))N and IJJ E D(R ) so t h a t (211-213) w i l l be v a l i d .
1
13.
I n e x i s t e n c e o f L a r g e s t Regular I d e a l s
Connected w i t h t h e c o n d i t i o n
(220)
I
n Uc"(R)
Q
N on i d e a l s 7 i n (C"(Q)) encountered i n C o r o l l a r y 1, Theorem 5 and Remark 4, S e c t i o n 4, i t can be n o t i c e d t h a t t h e r e a r e no l a r g e s t b u t o n l y maximal such i d e a l s . Indeed, i n case R = R'
, define
w ' , w"
wV' ( x ) = 1 + s i n v x , w;(x)
=
E
(C"(R)) N by
1
t cosvx,
V v E N, x E R
Then 7 ' = w'.(C"(R))N,
I"
= w".(C"(R))N
a r e i d e a l s i n (C"(R))N which s a t i s f y (220).
I
=
7'
+ I"
However
ALGEBRAS CONTAINING THE DISTRIBUTIONS
119
i s an i d e a l i n (Cm(Q))N which does n o t s a t i s f y (220). Indeed w = w ' t w"
1
E
and wV ( x ) = 2 t s i n v x
t
cosvx > 0, V v
E
N, x E
n
therefore
u ( i ) = w.(i/w)
E
r.(cm(n))N
c
r
I t f o l l o w s t h a t no i d e a l i n (Cm(n))N which c o n t a i n s fy (220).
1' and I", w i l l s a t i s =
Obviously, a s i m p l e m o d i f i c a t i o n o f t h e above example w i l l i m p l y t h e -= istence o f ideals 7 i n (C(n))N satisfying the condition
(221)
I n Uc o ( n )
=
Q
encountered i n Theorem 11, S e c t i o n 5 . The above p r o p e r t y can be seen as a reason f o r t h e i n e x i s t e n c e o f a unique, ' c a n o n i c a l ' c h a i n o f q u o t i e n t a l g e b r a s (24), r e s p e c t i v e l y (93), and there= f o r e can be seen as an a d i i t i o n a l reason - besides t h e one o f f e r e d by t h e i n t e r p l a y between c o n f l i c t i n g s t a b i l i t y , g e n e r a l i t y and exactness i n t e r e s t s concerning s e q u e n t i a l s o l u t i o n s - f o r d e a l i n g w i t h v a r i o u s chains o f quo= t i e n t a l g e b r a s (24) o r (93).
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CHAPTER 4 RESOLUTION OF SINGULARITIES OF WEAK SOLUTIONS FOR POLYNOMIAL NONLINEAR PDEs
0.
Introduction
The aim o f t h e p r e s e n t c h a p t e r i s t o a p p l y t h e general method f o r t h e reso= l u t i o n o f nowhere-dense s i n g u l a r i t i e s o f s e q u e n t i a l s o l u t i o n s f o r p o l y n o m i a l n o n l i n e a r PDEs p r e s e n t e d i n S e c t i o n 7, Chapter 1, t o t h e i m p o r t a n t p a r t i c u = l a r case o f weak s o l u t i o n s o f t h e PDEs mentioned. I n o r d e r t o deal w i t h weak s o l u t i o n s , t h e s i n g u l a r i t i e s w i l l have t o be r e s t r i c t e d t o nowhere-dense subsets w i t h z e r o Lebesque measure. However, t h e r e s u l t s presented i n t h i s c h a p t e r remain s u f f i c i e n t l y general t o i n c l u d e as r a t h e r s i m p l e p a r t i c u l a r cases most o f t h e known types o f s i n g u l a r i t i e s e x h i b i t e d by weak s o l u t i o n s o f f i r s t - and second-order n o n l i n e a r PDEs, [ 3, 14-17, 67, 87-90, 115,117, 149, 161-163, 1781, f o r i n s t a n c e those o f shock wave s o l u t i o n s f o r n o n l i n e a r h y p e r b o l i c c o n s e r v a t i o n laws, K l e i n Gordon wave equations, as w e l l as t h e e q u a t i o n s o f magnetohydrodynamics o r general r e l a t i v i t y . S t a t e d s i m p l y , t h e r e s o l u t i o n o f s i n g u l a r i t i e s o f weak s o l u t i o n s o f t h e PDE i n (1) means t h a t t h e weak s o l u t i o n s considered w i l l s a t i s f y t h e PDE i n ( l ) , i n t h e usual a l g e b r a i c sense, w i t h m u l t i p l i c a t i o n and p a r t i a l d e r i = v a t i v e o p e r a t o r s d e f i n e d w i t h i n t h e c h a i n s o f q u o t i e n t algebras (24) o r Tv3) , Chapter 3. Thus t h e r e s u l t s p r e s e n t e d i n t h i s c h a p t e r can be seen as b e l o n g i n g t o a ' p o l y n o m i a l n o n l i n e a r o p e r a t i o n a l c a l c u l u s ' f o r PDEs. T h i s ' o p e r a t i o n a l c a l c u l u s ' o f f e r s t h e p o s s i b i l i t y , among o t h e r s , o f i d e n t i = f y i n g t h e c l a s s o f r e s o l u b l e systems o f p o l y n o m i a l n o n l i n e a r PDEs, charac= t e e d b y an upper bound on t h e c o m p l e x i t y o f t h e i r n o n l i n e a r i t i e s , a c l a s s which i n c l u d e s many - i f n o t most - o f t h e e q u a t i o n s of p h y s i c s . I n S e c t i o n 1, a p a r t i c u l a r case w i l l be c o n s i d e r e d o f t h e general polyno= m i a l n o n l i n e a r PDE (see ( l ) , Chapter 1)
and a c o r r e s p o n d i n g r e s u l t on t h e r e s o l u t i o n o f s i n g u l a r i t i e s w i l l be e s t a b l i s h e d i n Theorem 1. T h i s r e s u l t w i l l be f u r t h e r p a r t i c u l a r i z e d , i n S e c t i o n s 2 and 3, t o t h e case of shock wave s o l u t i o n s o f n o n l i n e a r h y p e r b o l i c c o n s e r v a t i o n laws, o r K l e i n-Gordon t y p e n o n l i n e a r waves.
121
E.E. Rosinger
122
A much more general r e s u l t on t h e r e s o l u t i o n o f s i n g u l a r i t i e s o f weak s o l u = t i o n s i s presented i n S e c t i o n 4. I n S e c t i o n 5 a c h a r a c t e r i z a t i o n i s presented o f weak s o l u t i o n s by j u n c t i o n c o n d i t i o n s across the hypersurfaces o f t h e i r d i s c o n t i n u i t i e s , f o r a l a r g e c l a s s o f systems o f polynomial n o n l i n e a r PDEs i n c l u d i n g among o t h e r s t h e equations o f magnetohydrodynamics and general r e l a t i v i t y . T h i s r e s u l t i s then extended i n S e c t i o n 6 t o t h e c l a s s o f r e s o l u b l e systems o f polynomial n o n l i n e a r PDEs, a c l a s s which i n c l u d e s many o f t h e equations o f physics. The PDE i n (1) can be considered w i t h i n t h e framework o f t h e chains o f q u o t i e n t algebras i n (24) o r (93), Chapter 3, corresponding t o t h e smooth= ness o f t h e c o e f f i c i e n c t s c. and r i g h t - h a n d t e r m f. I n t h e case ci, f€C"(n), f o r i n s t a n c e both types Ehains o f q u o t i e n t a l g e b r a s may be used. Other= wise, i n case c . , f E C ( Q ) , w i t h R E N, t h e most obvious procedure would be t o use t h e c i a i n s o f q u o t i e n t algebras (93) , Chapter 3.
01
The choice o f t h e framework i s a l s o i n f l u e n c e d by t h e smoothness e x h i b i t e d a p a r t from t h e i r s i n g u l a r i t i e s by t h e weak s o l u t i o n s b e i n g considered.
To a v o i d a l e n g t h y p r e s e n t a t i o n , t h e p r o o f s w i l l be g i v e n w i t h i n one o f t h e two frameworks o n l y , t h e necessary m o d i f i c a t i o n s b e i n g mentioned when= e v e r t h e o t h e r framework c o u l d a l s o be used. As mentioned i n d), Remark 6, S e c t i o n 6, Chapter 3, i t i s u s e f u l t o make a p a r a l l e l s t u d y o f t h e problems considered, w i t h i n b o t h t y p e s o f chains o f q u o t i e n t algebras (24) and (93), Chapter 3, b e c a u s e c h t y p e has c e r t a i n advantages compared with the other. We now g i v e a simple and u s e f u l c l a s s i f i c a t i o n o f t h e polynomial n o n l i n e a r PDEs i n (1). Given k E
1.
fly t h e PDE i n ( 1 ) i s c a l l e d C k -smooth, o n l y i f ci,
f E C
k
(n).
The Case o f Simple Polynomial N o n l i n e a r PDEs
An m-th o r d e r polynomial n o n l i n e a r PDO i n ( 3 ) , Chapter 1, i s c a l l e d simple, o n l y i f i t can be w r i t t e n i n t h e form
where L. (D) a r e m-th o r d e r l i n e a r PDOs w i t h continuous c o e f f i c i e n t s , w h i l e Ti a r e b o l y n o m i a l s o f t h e form Tiu(x)
c. . ( x ) ( u ( x ) ) j ,
C
=
1 G j
x
E
Q,
lJ
w i t h cij E Cm(Q).
I n t h e p r e s e n t s e c t i o n and t h e n e x t two, we s h a l l deal w i t h PDEs c o r r e s = ponding t o PDOs i n (2), i . e . those h a v i n g t h e f o r m
(3)
T(D)u(x) = f ( x ) , x
E
a,
where f E Co(Q) i s given. Obviously, i f cij
E
Cmtk(Q) and f E C k (Q) t h e PDE i n ( 3 ) i s C k -smooth.
RESOLUTION OF SINGULARITIES
123
The nonlinear hyperbolic conservation laws, as well as the nonlinear secondorder wave equations studied in Sections 2 and 3, are obviously o f the above form ( 3 ) . I n general, the following large class of quasilinear PDOs are also of the above form ( 3 ) :
T(D)u(x) =
(4)
c ( x ) D u( x ) + T ' ( D ) u ( x ) , x
C
E
R,
oENn
IPI = m
where c P PDO.
E
c" ( Q ) and T'(D) i s an ( m - 1 ) t h order simple polynomial nonlinear
Suppose given k
8.
E
A function
u : Q
+
R'
i s called a piecewise Ck-smooth weak solution of the simple polynomial nonlinear PDF in ( 3 ) only i f the following five conditions are satisfied. There exists a family G of C k -smooth mappings set (5)
r
=
{ x E Q
y :
n
-f
RSy,
such that the
l y E G : y ( x ) = O E R gY 1
i s closed, has zero Lebesque measure - and i s therefore nowhere-dense and
(61
u
E
ck(n
\
-
r).
Further (7)
u
b
qoc(Q) I
where b = max { b . 1 4 i a1 , with the notation in (2.1), the weak solutioA property holds: (8)
1
R
(
C
l
Tiu(x)Lf(D)+(x)
-
and moreover,
f(X)+(X))dx = 0 W JI
E
U(n),
where L f ( D ) i s the formal adjoint o f L i ( D ) . Finally, forgeach y E G, there exists a bounded and balanced neighbourhood o f 0 E R Y, such that BY { Y - ' ( B ~ ) ( y E GI i s locally f i n i t e in n (91
.
Solutions of important classes of nonlinear hyperbolic conservation laws , as well as nonlinear second-order wave equations are known t o be piece-wise smooth weak solutions i n the above sense, [ 161-163, 1781
.
A f i r s t result in the case o f
Cm-smOothness, in next presented on the resolution of singularities of weak solutions for nonlinear PDEs.
124
E.E.
Rosinger
Theorem 1 Sgppose u : n R ' i s a piecewise C"-smooth weak solution of the m - t h order C -smooth simple polynomial nonlinear P D t i n ( 3 ) . Then i t i s possible t o construct C -regularizations ( V , S ) , such t h a t -f
u
1)
= s i-7 ' ( V , . S )
where s
E
E
S does
AR ( V , S ) , V k
not depend
E
8,
on R ;
u s a t i s f i e s the PDE in ( 3 ) , in the usual algebraic sense, with multiplication in Ak(V,S) and the partiaT derivative operators
2)
Dp : AR(V,.S)-+ Ak(V,S), p R-k am.
E
N n y IpI Gm with R , k
E
8,
Proof Obviously, i t may be assumed that
c"(n)
u 4
(10)
otherwise 1) a n d 2) are t r i v i a l l y s a t i s f i e d , in view of Theorems 2 and 3, Section 3, Chapter 3.
For each y
E
G, take
(11) such that 0 on a certain neighbourhood V
(11.1)
a
=
(11.2)
a
= l o n RgY 'BY
Y Y
Then for each u given by
(12)
SJX)
E
Y
of 0
E
RSy
and
(see (9)).
N, define a regularizing function sv : R
-f
R'
for u ,
=
I
0
if
X E
r.
We shall show t h a t each of these functions s . i s C" -smooth, and that therefore the resulting regularizing sequence'of functions s = (so,.. .,sU,...) s a t i s f i e s the condition (13)
s
E
(C"(Q))N.
Indeed, in order to see t h i s , assume v Now i f x
E
R
\r,
then
E
N given, fixed.
RESOLUTION OF SINGULARITIES
Iy
(14)
i s finite
G l(v+l)y ( x ) E By)
E
125
as
Y * x
E B
(V+l)Y(X)
E
Y
-1 1
(x By)
and B b e i n g balanced, we have Y
1
v+l
By
= By
;
t h e r e f o r e (14) f o l l o w s from ( 9 ) .
B u t (14) i m p l i e s t h a t t h e p r o d u c t
is i n (12) c o n t a i n s o n l y a f i n i t e number o f f a c t o r s # 1. T h e r e f o r e s w e l l - d e f i n e d on R \ I'. B u t R \ r i s open, t h e r e f o r e we can t a k e a"compact neighbourhood V of x such t h a t V C R \ r. Then, as i n (14), we have
(15)
(y
E G
I
(v+l)y(V) n By #
8) i s f i n i t e ,
which means t h a t i n t h e p r o d u c t
-l-ray((v+1)y(" Y E G
o n l y a f i n i t e number o f f a c t o r s a r e n o t i d e n t i c a l l y 1 on V. Thus ( 1 2 ) w i l l i m p l y t h a t sv
i s Cm-smooth a t x E R \
Assume now x E r. Then y ( x ) = 0 E Rgy, f o r a c e r t a i n t a k i n g a neighbourhood V ' o f - x such t h a t (see (11.1))
Y(V)
=
1
G.
Therefore
(1213
sv = 0 on V.
Thus o b v i o u s l y sv i s Cm-smooth a t x
E
D e f i n e now t h e sequence o f f u n c t i o n s w
(17)
y E
vy
we have i n view o f (11.1) and
(16)
r.
r. E
(Cm(n))N by
w = T(D)s - u ( f ) .
Then w i s t h e e r r o r i n t h e PDE i n ( 3 ) , as a r e s u l t o f t h e replacement o f t h e p i e c e w i s e smooth weak s o l u t i o n u by i t s r e g u l a r i z i n g sequence s d e f i n e d i n (12). The main p o i n t i n t h e p r o o f o f Theorem 1 i s t o c o n s t r u c t C a - r e g u l a r i z a t i o n s (V, S ) such t h a t R w € I ( V , S ) , tl R E fl (see (20), Chapter 3). (18)
126
E.E. Rosinger
i . e . such t h a t t h e e r r o r sequence w b&comes ' n e g l i g i b l e ' (see S e c t i o n 2, Chapter 1) i n t h e q 3 G n t a l g e b r a s A ( V , S ) , with R E N. For t h i s purpose, several a d d i t i o n a l p r o p e r t i e s w i l l be needed o f t h e r e g u l a r i z i n g sequence s , o r e r r o r sequence w. F i r s t we prove
r,
V K c R \
(19)
compact :
3pEN :
VvEN,v>u
:
sv = u, wv = 0 on K. Indeed, i n view o f ( 9 )
# 0) i s f i n i t e
GK = Ey E G ly(K) n By
(20)
since
# !J
y(K) n By
# 8.
K n ?(By)
We use t h e n o t a t i o n : a = i n f (Ily(x)ll Iy E GK, x E K I (21) Y where, f o r each y E G, II II i s a norm on Rgy and i t i s assumed t h a t
Y
SUP
SUP
y E G
xyEB
Y
IIx II y y
<
m
.
which i s p o s s i b l e , i n view o f ( 9 ) , g r a n t i n g t h e boundedness o f each B w i t h y E G. Since K i s compact, K n r = 8 , and i n view o f (20), GK f i n i t e , i t follows that
ix
a > O t h e r e f o r e , i n view o f (22), t h e r e e x i s t s p E N, such t h a t sup x E B
Y
Gu.a,
IIx II
Y
V y E G.
y y
Then (v+l)y(K) c Rgy \ B
(24)
Indeed, i f y
E
Y Y
V y
E
G, v
GK and x E K, t h e n (21) i m p l i e s
a G Ily(x)ll
Y
hence p.a
< A(vtl)y(x)ll
Y
E
N, u
p
.
,
RESOLUTION OF SINGULARITIES
127
which, i n view of ( 2 3 ) , Y i e l d s
B By.
(V+l)Y(X)
On t h e o t h e r hand, i f y we have
E
G \ GK, then from t h e d e f i n i t i o n o f GK i n ( 2 0 ) ,
By
=
0
Y(K)
B i s balanced, y i e l d s
which, i n view o f t h e f a c t t h a t
Y
( ( v + l ) y ( K ) ) n B = 0, which completes t h e p r o o f o f ( 2 4 ) . Now, ( 1 2 ) , ( 2 4 ) , ( 1 3 ) , ( 6 ) , (8) and (1 4 ) w i l l e a s i l y i m p l y ( 1 9 ) .
A f u r t h e r p r o p e r t y o f t h e r e g u l a r i z i n g sequence s i s g i v e n i n
(25 1
<s,.>
SECrn,
u
=
which f o l l o w s e a s i l y f r o m ( 1 3 ) , ( 1 9 ) , ( 5 ) and ( 7 ) , p r o v i d e d t h a t we assume b 2 1 i n ( 7 ) , s i n c e o t h e r w i s e T(D) i n ( 2 ) and t h e r e f o r e ( 3 ) , become t r i v i a l . F i n a l l y , t h e e s s e n t i a l p r o p e r t y o f t h e e r r o r sequence w i n o r d e r t o o b t a i n (18), i s g i v e n i n (26 1
WET.
Indeed, assume $ r e 1a t i ons
E
U(n) and v
E
N, t h e n ( 1 7 ) , ( 5 ) , ( 6 ) and (8) i m p l y t h e
T h e r e f o r e i n o r d e r t o prove (26) i t s u f f i c e s t o show t h a t
(27)
lim v+m
I
) T i ~ L ( ~ ) - T i u ( x ) I d x = 0, tl 1 G i G h, K C R , compact.
K
To do so, we f i r s t n o t i c e t h a t f o r x E R \ relation TiSV(X)
(28)
-
Tiu(x)
=
r
( 2 . 1 ) and ( 1 2 ) y i e l d t h e
128
E.E.
Rosinger
B u t i n view o f (11) we have ( a Y ( ( v + l ) y ( x ) ) ) j - l l G 1, V j E N, v
(29) Y E
E
N, x E R.
G
Furthermore t h e argument used t o p r o v e (19) w i l l e s t a b l i s h t h a t (30)
lim v+m
(m ( a Y ( ( v + l ) y ( x ) ) ) j - l )
= 0, V j E N,
x
E
R .'l\
y E G
Indeed, t a k i n g K = {XI, r e l a t i o n (24) y i e l d s 9 ( v + l ) y ( x ) E R Y\BY, tf Y E G, v E N, v 2 P for a certain
E
N. Hence, i n view o f (11.2) we have
( a Y ( ( V + i ) Y ( x ) ) ) J = 1, tt j
Y
E
N, x
E
n \ r,
v E N, v 2 p
EG
and thus ( 3 0 ) . Now (28) , (29) and (30) , t o g e t h e r w i t h ( 7 ) and t h e f a c t - see ( 5 ) - t h a t has zero Lebesque measure, w i l l i m p l y ( 2 7 ) , which completes t h e p r o o f o f (26).
r
The above r e s u l t s on t h e r e u l a r i z i n g sequence s and e r r o r sequence w a f = f o r d t h e p o s s i b i l i t y o f cons r u c i n g t h e r e q u i r e d C - r e g u l a r i z a t i o n s ( V , S ) .
47€-
t h e i d e a l i n (C"(n))N generated by CDPwlp E NnI. Indeed, denote by Iw Then i n view o f (69), Chapter 1 and (19) we have
(31)
'w
"nd'
Assume now g i v e n any i d e a l 7 i n (Cm(Q)) (32)
N
such t h a t
Ind.
1
Then, as seen i n t h e pgoof o f Theorem 7 , S e c t i o n 4, Chapter 3, t h e r e e x i s t v e c t o r subspaces T c S s a t i s f y i n g t h e c o n d i t i o n s (33)
r
n
(34)
V"
(35)
(V"
+
= v~m n T
= q ; v" @
( 7 n Sm) =
@
T ) n Uc"(n)
=
2.
But i n view o f ( 1 0 ) : (36)
s
9
Vm@
@
Ucm(n)
Indeed, assuming t h i s i s f a l s e and
s = v + u(*)
+
t
T;
T.
and
RESOLUTION OF SINGULARITIES
129
f o r c e r t a i n v E V", IJJ E C " ( Q ) and t E J, then i n view o f (25), (34) and (32) above, as w e l l as (69), Chapter 1, we have i n V'(Q) the r e l a t i o n
,
u = IJ t < t , * > But u
E
C"(Q \
r)
(37)
r
nowhere-dense
i n view o f (6), t h e r e f o r e
SUPP
since Q \
supp
r
c
i s open, as according t o ( 5 ) ,
r
i s closed.
It follows t h a t
u = $ o n ~ r. \
But i n view o f ( 5 ) ,
r
has zero Lebesque measure, w h i l e i n view o f ( 7 )
u E qoc(Q) since i t may be assumed t h a t b >l i n ( 7 ) otherwise the PDE i n ( 4 ) becomes trivial.
D'(Q) the r e l a t i o n
Thus (37) w i l l y i e l d i n u =
IJJ E
C"(Q)
c o n t r a d i c t i n g (10) and thus completing t h e p r o o f o f (36). (36) there e x i s t vector subspaces S c S , such t h a t
Now i n view o f
s;
(38)
SE
(39)
UCm(Q)
(40)
Sm =
c
and
S;
Vm@
S
@ J.
Now (33), (34), (40) and (39) w i l l , according t o Theorem 4, Section 4, Chapter 3, i m p l y t h a t
SO
(v, J) i s a Cm-reguIarization (41) f o r any vector subspace V n 1 C V" Since by d e f i n i t i o n {Dpwlp E N n I C lw C I, i t i s obviously possible, i n view o f (26), t o choose V so t h a t t h e f o l l o w = i n g condition i s satisfied:
(42)
(Dpwlp
E
NnI
t a k i n g f o r instance V = 7
C V;
r~ V".
F i n a l l y , i n view o f Theorems 2 and 3, Section 3, Chapter 3, the r e l a t i o n s 0 (38), (17) and (42) complete the p r o o f o f Theorem 1. The above r e s u l t o f Theorem 1 on t h e r e s o l u t i o n o f s i n g u l a r i z a t i o n of weak s o l u t i o n s f o r n o n l i n e a r PDEs, i n t h e case o f i n f i n i t e smoothness, has i t s counterpart f o r f i n i t e smoothness, presented i n the f o l l o w i n g theorem.
130
E.E.
Rosinger
Theorem 2 k, R 1 i s a piecewise C-imooth weak s o l u t i o n o f t h e m-th o r d e r If smooth s i m p l e polynomial n o n l i n e a r PDE i n ( 3 ) , and kl 2..
;Sqpose u
: R
-f
(43) then
C"
i t i s possible t o construct (?-regularizations
1) where s does
2)
(n)
u = s t IR(V,S)
not depend
AR(V,S),
E
'b R
E
N, il
( V , S ) such t h a t Q
kl
on 2;
u s a t i s f i e s t h e PDE i n ( 3 ) , i n t h e usual a l g e b r a i c sense, w i t h m u l t i p l i c a t i o n i n Ak(V,S) and t h e p a r t i a l d e r i v a t i v e opera t o r s DP : A ~ ( v , s ) with R, k
-f
A~(w,s), p
E
N",
\PI d m
N, R-k 2 m y R G kl and k d minCkl-m,k2).
E
Proof We use the n o t a t i o n h = min{k -m k I . can c o n s t r u c t a r e g u l a r i z i n g l e q i e i c e (44)
s
u =
E $1,
As i n t h e p r o o f o f Theorem 1, we
<S,.>
and t h e corresponding e r r o r sequence (45 1
b o t h s a t i s f y i n g (19). (46)
-
w = T(D)s
{DPwIp
u ( f ) E Vh
Therefore E
Nn,
h l c Ind n V'.
IpI
Assume now g i v e n any i d e a l I i n ( C
(47)
IDpwlp
E
such t h a t h) c I c I n d
Nn Y
and I i s Ck'-smooth (see (126), Chapter 3). Obviously, such a c h o i c e o f I i s p o s s i b l e , i n view o f (46) and P r o p o s i t i o n 6, S e c t i o n 6, Chapter 3. Using now t h e same argument as t h a t i n t h e p r o o f o f Theorem 14, S e c t i o n 6, Cheyter 3, f o r R=kl,it f o l l o w s t h a t t h e r e e x i s t v e c t o r subspaces T C S satisfying
v"
(48)
I nT
(49 1
V'
(50)
(v" @ T )
t
=
nT =
2;
( 7 nso) = V @ 7; n UC" ( a ) =
and
Q.
B u t i n view o f ( 4 3 ) , and u s i n g an argument s i m i l a r t o t h a t which e s t a b l i s h e d (36), we have
(51)
sB
v"
@uco
(a)
0T .
RESOLUTION OF SINGULARITIES NOW, i n view o f (51), t h e r e e x i s t v e c t o r subspaces R '
u"
S" =
@
@
UC" (n)
0R '
R's
131 C
So , such t h a t
@ 7.
Thus t a k i n g i n Lemma 5, S e c t i o n 6, Chapter 3
v" @ Up(n) 0 + R's @ J, B
E = S o , F =Ski A =
= R'
i t f o l l o w s t h a t t h e r e e x i s t v e c t o r subspaces
C=R
F = Sk l
C
satisfying
@
(V'
Up(no@ R's @ 7) n R
=
9
and
s"
(52)
=
u"
0 Uco
@
R's
0R
@ 7.
Introducing the notation (53)
S
=
Up (n) @ R's @ R
t h e r e l a t i o n s (48), (49) and (52) w i l l i m p l y t h a t (54)
(V, S
@
T ) i s a c" - r e g u l a r i z a t i o n
f o r any v e c t o r subspace V S e c t i o n 5, Chapter 3.
C
1 n 'V
, ifwe
t a k e i n t o account Theorem 10,
B u t i n view o f (46), i t i s p o s s i b l e t s choose V so as t o s a t i s f y
w E Vh.
(55)
Thus i n v i e w o f t h e v e r s i o n s o f Theorems 2,3 and 9, Chapter 3, correspon= d i n g t o C " - r e g u l a r i z a t i o n s , t h e r e l a t i o n s (53), (45) and (55) complete t h e p r o o f o f Theorem 2. 0 2.
R e s o l u t i o n o f S i n g u l a r i t i e s o f N o n l i n e a r Shock Waves
Suppose g i v e n t h e n o n l i n e a r h y p e r b o l i c c o n s e r v a t i o n law (56)
ut(x,t)
t
a(u(x,t)).ux(x,t)
t 0,
x
E
R',
t > 0,
w i t h the i n i t i a l condition (56.1)
u(x,O) = u o ( x ) , x
E
R'.
We s h a l l suppose t h a t t h e f u n c t i o n (57)
a : R' -+ R'
i n (56) i s an a r b i t r a r y p o l y n o m i a l . Then i t i s obvious t h a t (56) i s a f i r s t - o r d e r C--smooth s i m p l e p o l y n o m i a l n o n l i n e a r PDE on n = R'x(0,m) c R2. Indeed, t h e l e f t - h a n d t e r m i n ( 5 6 ) can be w r i t t e n i n t h e fonn i n (21, p r o =
E.E. Rosinger
132
v i d e d t h a t we t a k e a=2, L1(D) = Dty L2(D) = D x y T1u = u and T2u = b ( u ) where (58)
b : R'
R'
-+
i s a p r i m i t i v e o f t h e f u n c t i o n i n ( 5 7 ) , and thus a g a i n a polynomial. I t i s known t h a t under r a t h e r general c o n d i t i o n s [ 67,1781 f o r Cm-smooth o r piecewise smooth i n i t i a l d a t a u o y t h e e q u a t i o n (56) has shock wave so u=
tions u :
w i t h the following properties.
R',
+
Thgre e x i s t s a f i n i t e s e t G o f -smooth c u r v e s
Cm-smooth f u n c t i o n s
y:
R
+
R',
defin
C
r
Y
=
IX E n
I
Y ( X ) = 01
which d e s c r i b e t h e p r o p a g a t i o n o f t h e shocks, and such t h a t
r),
u E C
(60)
u i s l o c a l l y bounded on R ;
I
(61)
n
\
where
r
(59)
(u(x,t)$(x,t) t
t
=
y g G r y ;
b(u(x,t))$(x,t))dx
d t = 0, tf $ E D ( f i ) .
X
Obviously, such a s o l u t i o n u w i l l be a i e c e w i s e Cm-smooth weak s o l u t i o n o f t h e PDE i n (56), i n t h e sense o f t h e S e f i n i t i o n i n S e c t i o n 1. There= f o r e Theorems 1 and 2 w i l l y i e l d t h e f o l l o w i n g r e s u l t . Theorem 3 Suppose u : R' x(0,m) -+ R1 i s a shock wave s o l u t i o n o f t h e n o n l i n e a r h y p e r b o l i c c o n s e r v a t i o n law i n (56)mti s a t i s f i e s the conditions (59-61). Then i t i s p o s s i b l e t o c o n s t r u c t C - r e g u l a r i z a t i o n s (V,s), such that
1) 2)
tt R
u = s t lR(V,S) E AR(V,S),
E
8,
n o t depend o n R ; where s E S does u s a t i s f i e s t h e PDE i n (56), i n t h e usual a l g e b r a i c sense, w i t h m u l t i p l i c a i i o n i n Ak(V,S) and t h e p a r t i a l d e r i v a t i v e o p e r a t o r s Dt, Dx : A (V,S) -+ Ak(V,S) w i t h R , k E \, R-k > 1.
If
(62)
u .$
c" (n)
then one can c o n s t r u c t
3)
u = s t lR(V,.S) depend on R ;
Y
C " - r e g u l a r i z a t i o n s ( V , S ) , such t h a t 6
AR(V,s),
tt
RE
w,
where s
E
s
does not
RESOLUTION OF SINGULARITIES u s a t i s f i e s t h e PDE i n ( 5 6 ) , i n t h e usual a l g e b r a i c sense, w i t h m u l t i p l i c t t i o n i n Ak(V,S) and t h e p a r t i a l d e r i v a t i v e o p e r a t o r s Dt,Dx : A (V,S) + Ak(V,S) with1,kE N, e-k > 1.
4)
3.
133
R e s o l u t i o n o f S i n q u l a r i t i e s o f Klein-Gordon Type N o n l i n e a r Waves
Suppose g i v e n t h e Klein-Gordon type n o n l i n e a r wave e q u a t i o n (63)
u t t ( x y t ) - u x x ( ~ , t ) = T(D)u(x,t),
x
E
R',
t > 0,
w i t h t h e in i ti a1 condi ti ons (63.1)
u(x,O)
(63.2)
ut(x,O)
= f(x),
x
= q(x),
E
x
E
R'
and
R',
where T(D) i s a f i r s t - o r d e r Cm-smooth s i m p l g p o l y n o m i a l n o n l i n e a r PDO. Then i t i s obvious t h a t (63) i s a second-order C -smooth s i m p l e p o l y n o m i a l non= l i n e a r PDE on R = R' x(0,m) C R', s i n c e i t has t h e form i n (4). I t i s known t h a t under general c o n d i t i o n s [ 161-1631 t h e e q u a t i o n (63) has l o c a l o r g l o b a l s o l u t i o n s u : R' + R', w i t h R' c R open, which have t h e following properties.
There e x i s t a f i n i t e number o f p o i n t s xl, cones w i t h the-ari es g i v e n b y
r-c1
= {(x,t)
rta =
{(x,t)
E E
R'I R'I
...,xo E
R',
originating light-
X-xa t t = 01 , x-xa
-
t = 01
, with
1<
B GO,
such t h a t
(65)
u i s l o c a l l y bounded on R';
where i t has been assumed t h a t T(D) i n (63) has t h e form ( 2 ) and L f ( D ) i s t h e formal a d j o i n t o f Li(D). Obviously, such a s o l u t i o n u w i l l be a p i e c e w i s e Cm-smooth weak s o l u t i o n o f t h e PDE i n (63), i n t h e sense o f t h e d e f i n i t i o n i n S e c t i o n 1. There= f o r e Theorems 1 and 2 w i l l y i e l d t h e f o l l o w i n g r e s u l t , s i m i l a r t o t h a t o f Theorem 3. Theorem 4 Suppose u : R -+ R', w i t h fi C R' x(O,m), i s a s o l u t i o n o f t h e Klein-Gordon t y p e n o n l i n e a r wave e q u a t i o n (63) and t h a t i t s a t i s f i e s t h e c o n d i t i o n s
134
E.E.
(64-66). that
1) 2)
Rosinger
Then i t i s p o s s i b l e t o c o n s t r u c t C m - r e g u l a r i z a t i o n s ( V , S ) such
R u = s t I ( V , S ) E AR(V,S), tl R E 8, where 5 E S does not depend on R ;
u s a t i s f i e s t h e PDE i n (63), i n t h e usual a l g e b r a i c sense,with m u l t i p l i c a t i o n i n Ak(I/,S) and t h e p a r t i a l d e r i v a t i v e o p e r a t o r s Dp : A'(V,S)
+
Ak(V,S),
Y p E Nny I p I
< 2,
w i t h R , k E 8, R-k 2 2.
If
(67)
u
4 co (R)
then i t i s p o s s i b l e t o c o n s t r u c t
c"
- r e g u l a r i z a t i o n s ( V , S ) , such t h a t
V R E fly
3)
u = s t lR(VyS)E AR(V,S), depend on R ;
4)
u s a t i s f i e s t h e PDE i n (63), i n t h e usual a l g e b r a i c sense, w i t h m u l t i p l i c a t i o n i n Ak(V,S) and t h e p a r t i a l d e r i v a t i v e o p e r a t o r s Dp : A'(V,S)
4.
-+
Ak(V,S),
not where s E S does -
Y p E Nny I p I G 2, w i t h % k E fly R-k
> 2.
R e s o l u t i o n o f S i n g u l a r i t i e s o f Weak S o l u t i o n s f o r General Polynomial N o n l i n e a r PDEs
I n t h i s s e c t i o n a general method i s presented f o r t h e r e s o l u t i o n o f s i n g u = l a r i t i e s of weak s o l u t i o n s f o r polynomial n o n l i n e a r PDEs, under t h e i r general form (see ( l ) , Chapter 1) (68)
1 l < i < h
Ci(X)
'T op'ju(x) l<j
= f ( x j , x E R.
This method i n c l u d e s , as a p a r t i c u a r case, t h a t given i n S e c t i o n 1 and a p p l i e d i n Sections 2 and 3. Denote by (69)
T(D) =
c
l < i Q h
'i
l Q j < k i
t h e PDO corresponding t o t h e PDE i n ( 6 8 ) . (70)
m
= max I l p .
.I
1.J
We r e c a l l t h a t
1 G i < h, 1 < j
< ki}
i s c a l l e d the o r d e r o f t h e PDE i n (68), o r e l s e PDO i n (69). The general method f o r t h e r e s o l u t i o n o f s i n g u l a r i t i e s w i l l be a p p l i a b l t o t h e s o - c a l l e d r e g u l a r weak s o l u t i o n s , a c l a s s e x t e n d i n g t h e piecewise smooth weak s o l u t i o n s i n t r o d u c e d i n S e c t i o n 1. Since t h e r e e x i s t c e r t a i n d i f f e r e n c e s between t h e i n f i n i t e - s m o o t h and f i n i t e - s m o o t h cases, we s h a l l have t o deal w i t h them s e p a r a t e l y . F i r s t , we c o n s i d e r t h e i n f i n i t e - s m o o t h case.
!
RESOLUTION OF SINGULARITIES
Amdistribution S
E
D'(n) \ C m ( n ) i s called a (?-regular
135
2
weak solution of
C -smooth PDE i n (68), only i f there exists a sequence of functions s
E
S
satisfying the following three conditions: (71)
s
(72)
w = T(D)s - u ( f )
(73)
Iw i s a Cay R-regular ideal
=
<S,.>
Y
V";
E
(see Chapter 3, Section 1 0 ) ;
where (73.1)
R
=
(73.2)
Iw
=
( i . e . the vector subspace i n S" generated by s ) ; the ideal in ( C m ( Q ) ) N generated by CDPwIp E NnI.
R!s
The following theorem shows t h a t the above notion i s m r e general than t h a t which applies t o the infinite-smooth case of piecewise smooth weak solu= tions, defined i n Section 1. Theorem 5 If a piecewise Cm-smooth weak solution u : n -t R'of a polynomial nonlinear P D t s a t i s f i e s the condition
u B
(74)
Cm-smooth simple
C"(Q)
then, u i s a C :regular
weak solution of t h a t PDE.
Proof This follows e a s i l y from (25), ( 1 7 ) , (26), (36) and (34) as used in the proof of Theorem 1, Section 1.
Let
US
kl > m
now consider the finite-smooth case. (see (70)).
0
Suppose given k l , k 2 E N , w i t h
A distribution
E U'(Q)\ c" (Q) i s called a Ckl-regular weak solution of an m - t h order C"-smooth PDE i n (68), only i f there e x i s t s a sequence o f functions s E Skl satisfying the following three conditions:
(75)
s
=
< s,
.>
(76)
w
=
T(D)s
-
(77)
TW i s a C?
u(f)
E
Vhy w i t h h = min
Ikl-m,k21;
R-regular ideal
where
(77.1)
R
(77.2)
W I
=
R!s;
= the ideal i n
(e
generated by IDPwIp
E
Nn,lpl
hl.
E . E . Rosinger
136
The f o l l o w i n c ] theorem shows t h a t t h e above n o t i o n extends t h a t o f t h e f i n i t e - s m o o t h case o f piecewise smooth weak s o l u t i o n s , d e f i n e d i n S e c t i o n 1. Theorem 6
Ifa {iecewise Ck'-smooth weak s o l u t i o n u : R + R' o f an m-th o r d e r C k z smoot s i m p l e polynomial n o n l i n e a r P D t s a t i s f i e s t h e c o n d i t i o n
u 4 C"
(78)
(a),
then u i s a Ckl-regular
weak s o l u t i o n o f t h a t PDE.
Proof T h i s f o l l o w s e a s i l y f r o m (44), ( 4 5 ) , (46), (51) and (49), as used i n the p r o o f o f Theorem 2, S e c t i o n 1.
0
We now g i v e t h e general r e s u l t on t h e r e s o l u t i o n o f s i n g u l a r i t i e s o f weak s o l u t i o n s f o r PO-1 n o n l i n e a r PDEs. We d e a l f i r s t w i t h t h e i n f i n i t e smooth case. Theorem 7 Suppose S E D'(Q) i s a Cm-regular weak s o l u t i o n o f t h e m-th o r d e r Cm-2mooth polynomial n o n l i n e a r PDE i n ( 6 8 ) . Then i t i s p o s s i b l e t o c o n s t r u c t C -regu= l a r i z a t i o n s ( V , S ) , such t h a t
I)
s
= s t
rR(v,s) E
A ~ ( v , s ) , 4' R E
R,
where s E S does n o t depend on R;
2)
S s a t i s f i e s t h e PDE i n (68), i n t h e usual a l g e b r a i c sense, w i t h
m u l t i p l i c a t i o n i n Ak(V,S)
Dp : AR(V,S)
+
Ak(V,S),
and t h e p a r t i a l d e r i v a t i v e o p e r a t o r s p E Nn,
I p I Gm, w i t h R, k
E
m,
R-k > m .
Proof T h i s f o l l o w s e a s i l y f r o m (71-73) above, and (195-198), S e c t i o n 10, a l s o Theorem 4, S e c t i o n 4, Chapter 3, i f account i s a l s o taken o f Theorems 2 and 3, S e c t i o n 3, Chapter 3.
0
I n a s i m i l a r way, t h e f o l l o w i n g general r e s u l t can be o b t a i n e d on t h e r e = s o l u t i o n o f s i n g u l a r i t i e s o f weak s o l u t i o n s f o r polynomial n o n l i n e a r PDEs, i n t h e case o f f i n i t e smoothness. Theorem 8 Suppose S E D ' ( R ) i s a Ckl-regular weak s o l u t i o n o f t h e m-th o r d e r ck2smooth polynomial n o n l i n e a r PDE i n (68). Then i t i s p o s s i b l e t o c o n s t r u c t c" - r e g u l a r i z a t i o n s ( V , S ) such t h a t
1)
R R S = s + 1 ( V , S ) E A ( V , S ) , 4' R E N, R d kl where s
e s does not
depend on R ; 2)
S s a t i s f i e s t h e PDE i n (68), i n t h e usual a l g e b r a i c sense, w i t h
RESOLUTION OF SINGULARITIES
137
m u l t i p l i c a t i o n i n A k ( V , S ) and t h e p a r t i a l d e r i v a t i v e o p e r a t o r s Dp : AR(V,S)
II G kl,
* Ak(V,S),
p E Nn,
I p I G m , w i t h a, k
E
N, a-k > m y
k G min Ikl-m,k22.
The s u f f i c i e n t c o n d i t i o n on r e g u l a r i d e a l s g i v e n i n Theorem 2 0 , S e c t i o n 10, Chapter 3, can be t r a n s l a t e d i n t o a s u f f i c i e n t c o n d i t i o n on r e g u l a r weak s o l u t i o n s , a s p r e s e n t e d now, i n t h e case o f i n f i n i t e smoothness. Theorem 9 Suppose themPDE i n (68) i s Cm-smooth, and t h a t we a r e g i v e n a d i s t r i b u t i o n S E t ? ' ( Q ) \ C (a). I f t h e r e e x i s t s a sequence o f f u n c t i o n s s E S , such t h a t (79)
s
(80)
w = T(D)s
(81)
lw i s a vanishing ideal
= < S,.
then S i s a (?-regular
>
;
- u(f)
E V";
(see (39), Chapter 3);
and
weak s o l u t i o n f o r t h e Cm-smooth PDE i n (68).
Proof I n view o f (81) and (82) above, as w e l l as t h e p r e v i o u s l y mentioned Theo= 0 rem 20, S e c t i o n 10, Chapter 3, i t f o l l o w s t h a t (73) h o l d s . Remark 1 The r e s u l t o f Theorem 9 y i e l d s i m p o r t a n t i n s i g h t i n t o t h e e s s e n t i a l p r o = p e r t i e s o f r e g u l a r weak s o l u t i o n s , and t h e r e f o r e i n p a r t i c u l a r , p i e c e w i s e smooth weak s o l u t i o n s . Indeed, t h e c o n d i t i o n s (79) and (80) a r e t h e n a t u r a l requirements t h a t t h e r e q u l a r weak s o l u t i o n S s h o u l d be generated b y a r e u l a r i z i n g sequence s t h a t y i e l d s an e r r o r sequence w weakly convergent +TT- t o zero. However, these two c o n d i t i o n s a r e t h e consequence o f t h e p a r t i c u l a r way t h e e n e r a l i t y o f s e q u e n t i a l s o l u t i o n s (see S e c t i o n 5, Chapter 1) was s o l v %-Tie w en c o n s t r u c t i n g c h a i n s o f q u o t i e n t a l g e b r a s ( 2 4 ) and (93), Chap= t e r 3, (see S e c t i o n 0 , Chapter 3 ) . T h i s means t h a t i t i s p o s s i b l e t o dispense w i t h these c o n d i t i o n s when general s e q u e n t i a l s o l u t i o n s a r e b e i n g studied. On t h e o t h e r hand, t h e c o n d i t i o n s (81) and ( 8 2 ) a w e s s e n t i a l w i t h i n t h e framework of any k i n d o f q u o t i e n t a l g e b r a , as d e f i n e d i n Chapter 1. I n = deed, i n view o f P r o p o s i t i o n 1 below, t h e c o n d i t i o n (81) means t h a t t h e e r r o r sequence w, t o g e t h e r w i t h i t s p a r t i a l d e r i v a t i v e s o f a r b i t r a r y f i x e d order
Dpw, p
E Nn,
IpI G
Q.
have t o v a n i s h s i m u l t a n e o u s l y . F i n a l l y , c o n d i t i o n (82) demands a separa= , generatemr= t h e r e g u l a r i z i n g sequence s and t h e i d e a l 1
t i o n between
b i t r a r y p a r t i a l d e r i v a t i v e s o f t h e e r r o r sequence w .
138
E.E.
Rosinger
I n s h o r t , the two c o n d i t i o n s (81) and (82) mean r e s p e c t i v e l y t h e v a n i s h i n g o f t h e e r r o r sequence and t h e s e p a r a t i o n between t h e ' n e g l i g i b l e ' e r r o r sequence (see S e c t i o n 2, Chapter 1) and t h e r e g u l a r i z i n g sequence.
Proposition 1 The i d e a l I,
(see (73.2)) i s vanishing, o n l y i f
Vu,
RE
N :
3 vEN,v>p,xEQ: V p
(83)
E
Nn,
IpI GR:
DPwv(x) = 0. Proof Assume Iw i s v a n i s h i n g .
Obviously, f o r any g i v e n R E N
c
i=
( D P w ) ~E
rw.
p E Nn IPI G!z Therefore a p p l y i n g (39), Chapter 3, t o
i E Iw,
we e a s i l y o b t a i n ( 8 3 ) .
The converse i s obvious.
11
I n t h e case o f f i n i t e smoothness, i t i s easy t o o b t a i n t h e c o r r e s p o n d i n g v e r s i o n s o f Theorem 9 and P r o p o s i t i o n 1, p r e s e n t e d next, as w e l l as t h e corresponding remarks on t h e s u f f i c i e n t c o n d i t i o n s on r e g u l a r weak solu= ti ons. Theorem 10 Suppose t h e m-th o r d e r PDE i n (68) i s Ck2-smooth, and a d i s t r i b u t i o n S E P'(R)\ C(Q) i s given. I f t h e r e e x i s t s a sequence o f f u n c t i o n s s w i t h kl 2 m, such t h a t (84)
s
(85)
w = T(D)s
(86)
Iw i s a v a n i s h i n g i d e a l ;
(87)
(V
E
ski
<s, *>;
=
t
-
u(f)
rw)n(uc"
E Vh,
t
R!S)
w i t h h = min {kl-m,k2}; and =
p;
then S i s a C k ' - r e g u l a r weak s o l u t i o n of t h e m-th o r d e r Ck2-smooth PDE i n (68)
-
Proposition 2 The i d e a l Iw (see (77.2)) i s vanishing, o n l y i f
RESOLUTION OF SINGULARITIES
5.
13Y
J u n c t i o n C o n d i t i o n s and R e s o l u t i o n o f S i n q u l a r i t i e s o f Weak S o l u t i o n s
f l The problem o f f i n d i n g j u n c t i o n c o n d i t i o n s across h y p e r s u r f a c e s f o r s o l u = t i o n s o f t h e e q u a t i o n s o f magnetohydrodynamics o r qeneral r e l a t i v i t y i s u s u a l l y approached e i t h e r by a p p l y i n g i n t e g r a l c o n d i t i o n s o r b y i n t r o d u c i n g ce r t a in s impl ify ing ass ump t ions
.
Both methods p r e s e n t obvious d e f i c i e n c e s when compared w i t h t h e d i r e c t method suggested i n [161 and based on t h e i d e a o f o b t a i n i n g t h e j u n c t i o n c o n d i t i o n s from weak s o l u t i o n c o n d i t i o n s f o r m u l a t e d across t h e h y p e r s u r f a = As t h e n o n l i n e a r i t y o f t h e e q u a t i o n s i n v o l v e d ces o f d i s c o n t i n u i t i e s . w i l l i m p l y t h e presence of p r o d u c t s o f t h e H e a v i s i d e f u n c t i o n w i t h t h e D i r a c 6 d i s t r i b u t i o n and i t s p a r t i a l d e r i v a t i v e s , t h e l a t t e r method cannot be implemented w i t h i n t h e d i s t r i b u t i o n a l framework. For t h i s reason, s p e c i a l r e g u l a r i z a t i o n procedures were suggested i n [ 10,14-171 , amounting t o the c o n s t r u c t i o n o f subalgebras c o n t a i n e d i n t h e d i s t r i b u t i o n s and c o n t a i n i n g s i n g u l a r d i s t r i b u t i o n s , such as t h e H e a v i s i d e f u n c t i o n , t h e D i r a c 6 d i s t r i b u t i o n , as w e l l as i t s d e r i v a t i v e s . I n t h e p r e s e n t s e c t i o n , t h e polynomial n o n l i n e a r o p e r a t i o n s on t h e singu= l a r d i s t r i b u t i o n s mentioned w i l l be performed w i t h i n t h e q u o t i e n t a l g e b r a s c o n t a i n i n g t h e d i s t r i b u t i o n s . The r e s u l t i n g a l t e r n a t i v e o f t h e method suggested i n [161 has t h e major advantage, among o t h e r s , t h a t a c l e a r and a l g e b r a i c a l l y simple i n s i g h t i s obtained i n t o the s t r u c t u r e o f the n o n l i = n e a r i t i e s o f a p a r t i c u l a r l y l a r g e c l a s s o f systems o f polynomial n o n l i n e a r PDts encountered i n t h e s t u d y o f p h y s i c s , a r e s u l t p r e s e n t e d under i t s general f o r m i n S e c t i o n 6. The method o f e s t a b l i s h i n g t h e j u n c t i o n c o n d i = t i o n s p r e s e n t e d w i l l a t t h e same t i m e y i e l d t h e r e s o l u t i o n o f s i n g u l a r i t i e s across t h e h y p e r s u r f a c e s i n v o l v e d i n these j u n c t i o n c o n d i t i o n s . I n o r d e r t o a v o i d r a t h e r t r i v i a l t e c h n i c a l c o m p l i c a t i o n s (see S e c t i o n 9, Chapter 1 ) and a l s o t o make i t p o s s i b l e a t t h e same t i m e t o p o i n t o u t t h e e s s e n t i a l u n d e r l y i n g a l g e b r a i c phenomena, o n l y t h e case o f p o l y n o m i a l non= l i n e a r i t i e s w i l l be d e a l t w i t h , a case which o b v i o u s l y covers t h e s i t u a t i o n i n magnetohydrodynamics, as w e l l as general r e l a t i v i t y . Suppose we a r e g i v e n a system o f polynomial n o n l i n e a r PDEs (see (110), Chapter 1):
l
(u ,..., u ) c ( Q ) a i e give$.
where u =
fBE
:R
-f
Ra a r e t h e unknown f u n c t i o n s , w h i l e cBi,
E . E . Rosinger
140
The system o f PDEs i n (89) i s c a l l e d , [ 161, o f -t y p e (MH), o n l y i f each o f the a s s o c i a t e d PDOs Tg(D)u(x) =
l < i < h
% ,
C B ~ ( XT )
C
B
1 < j ,
j
u,
Bi
Y
Bij
XER,
can be w r i t t e n i n the f o r m
where LBp(D) a r e l i n e a r PDOs w i t h c p t i n u o u s c o e f f i c i e n t s and o r d e r mBp, w h i l e PBp(D) a r e l i n e a r PDOs w i t h C BP-smooth c o e f f i c i e n t s and o r d e r at Sometimes i t w i l l be c o n v e n i e n t t o w r i t e t h e c o n d i t i o n (91)
most one.
under the form
X E R,
where P
Bpacl'
1 < B
(D) a r e PDOs o f t h e same t y p e w i t h PBp(D).
As can e a s i l y be seen, [ 161 , t h e equations o f magnetohydrodynamics as w e l l as those o f general r e l a t i v i t y a r e o f t y p e ( M H ) . The Navier-Stokes equa= t i o n s are a l s o of t y p e (MH). I f t h e o r d e r o f t h e system (89) i s
m=max{/pBijl
11<@
l < i < h B Y 1 G j g kB i 1
,
then i f (89) i s o f t y p e (MH), i t may be assumed t h a t
m = 1 t max {m
11 G B < b , 1 < p < r B ) . BP Suppose now given a hypersurface r c R d e f i n e d b y (92)
r
=
{X E
R
I
where y : R -+ R ' , y E s h a l l a l s o suppose t h a t (92.1)
r
Y(X) = 0) Cm.
Obviously t h e subset
r
i n R i s closed.
We
has zero Lebesque measure
:R We s h a l l be i n t e r e s t e d i n f i n d i n g weak s o l u t i o n s u = (ul,...,ua) f o r an (MH) type system of PDEs (89) such t h a t u a r e Cm-smoth on R \ and d i s c o n t i n u o u s across t h e hypersurface r. I t f o l l o w s immediately t h a t u w i l l have t h e form
-+
r
Ra
RESOLUTION OF SINGULARITIES
(93)
U ( X ) = u-(x)
-
+ (u,(x)
u-( x ) ) . H ( x ) ,
x
E
141
Q,
where u , u : Q Ra, u , ut E Cm a r e c l a s s i c a l s o l u t i o n s f o r t h e system o f PDEs-in t89), w h i l e H 7 R + R' d e f i n e d b y -f
(94)
H x)
0
i f y(x) GO
1
i f y(x) > 0
=
i s t h e Heav s i d e f u n c t i o n a s s o c i a t e d w i t h t h e r e p r e s e n t a t i o n o f t h e hyper= s u r f a c e r g i v e n i n (92). The problem can now be f o r m u l a t e d as f o l l o w s : f i n d t h e necessary and/or s u f f i c i e n t j u n c t i o n c o n d i t i o n s on t h e system (89), t h e h y p e r s u r f a c e r and i t s r e p r e s e n t a t i o n t h r o u g h y i n (92), as w e l l as t h e c l a s s i c a l s o l u t i o n s u- and ut o f (89), such t h a t u g i v e n i n (93) i s a weak s o l u t i o n f o r (89). I f w i t h t h e h e l p o f t h e method i n S e c t i o n 12, Chapter 1, t h e above problem i s c o n s i d e r e d w i t h i n t h e framework o f t h e q u o t i e n t a l g e b r a s c o n t a i n i n g t h e d i s t r i b u t i o n s , i t i s easy t o o b t a i n necessary and s u f f i c i e n t j u n c t i o n c o n d i t i o n s . To do so, we s h a l l f i r s t c o n s t r u c t Clll-smoth r e g u l a r i z a t i o n s o f t h e H e a v i s i d e f u n c t i o n (94) a c c o r d i n g t o t h e method p r e s e n t e d i n t h e f o l l o w i n g lemma.
Lemma 1 Suppose g i v e n any f u n c t i o n rl : R'
n
= 0 on
(-m,
n
= 1 on
11,
-1
-+
[O,ll,
r) E
such t h a t
Cm,
1
(95) t a).
We d e f i n e s,. E(Cm(Q))N by
(96)
s (x) rlv
= q((v+l)y(x)),
V v
E
N, x
E
R.
Then t h e f o l l o w i n g two r e l a t i o n s h o l d :
(97)
(s,.) R = S ( ~ ) R
(98)
(sr))'.
,
V II
E
N \ { O } , and
Sm, <(sr))',*> = H, V 2,! E N \ { O } .
Proof The case ~ = il s okvious. I f R > 1, i t i s easy t o see t h a t (I-,)' : R1-+[O,l], (q)'E Cm and ( q ) a l s o s a t i s f i e s (95), t h e r e f o r e t h e problem i s reduced t o t h e case R=l. 0 The Cm-smooth r e g u l a r i z a t i o n s of regularizations
(99)
H o b t a i n e d above w i l l generate p-smooth
s = (sl,...YSa~((Cm(~))N)a
142
E.E. Rosinger
o f t h e p o t e n t i a l weak s o l u t i o n u i n (93) given by t h e r e l a t i o n (see (119), Chapter 1): (99.1)
-
sav(x) = (u-),(x)
+ ( ( U ~ ) ~ ( X )(u-)a(xj)snv(x)y
V 1 Q a
VE
N,
X E R
which w i l l o b v i o u s l y i m p l y s
(100)
a
E
Sm
y
< Say'>
= Ua'
Y 1Qa
A t t h i s stage the problem i s t o i d e n t i f y t h e c o n d i t i o n s which w i l l make s given i n (99) a weak s o l u t i o n f o r t h e system (89). I n t h i s connection, t h e f o l l o w i n g r e s u l t w i l l be u s e f u l . Suppose given on R a f i r s t - o r d e r , l i n e a r and homogeneous PDO P(D) w i t h continuous c o e f f i c i e n t s . For g i v e n f u n c t i o n s E C'(Q) d e f i n e t h e r e g u l a r i z a t i o n s t,t' E (C1(R))N as f o l l o w s :
+-,$ty+',$i
with v
E
- $_(X))S,,(X)
t v ( x ) = $-(XI
+
t$(x) =
+ ($i(X)
N, x
E
t, t '
$(:'XI
($,(x)
-
$:(x))snv(x)
R, assuming t h a t n S ' , < t, .> = $-
E
+
C'(Q) i n ( 9 5 ) . (+t -+-)Hy
Then o b v i o u s l y
< t', *> = $ ' t
($i-$L)H.
Proposition 3 The f o l l o w i n q r e l a t i o n s h o l d :
(101)
t P(D)t' E
(m21
< tP(D)t',
So ->
P ( ~ ) + -i
= $- P(D)+L t
+-
+
bt
+
mJt +J+;- +L)P(D)H.
P(D)+L)H +
+
Proof -
It i s easy t o see t h a t f o r each v
E
N, t h e f o l l o w i n 9 r e l a t i o n h o l d s :
RESOLUTION OF SINGULARITIES
143
= stv
(SJZ
where 5 = (n)' also s a t i s f i e s ( 9 5 ) . Lemma 1 i s taken into account.
This completes the proof i f (98) in 0
Corollary 1 If P ( D ) i s a linear PDO on R w i t h continuous coefficients and order a t most one, then the following relations hold: (103)
t P(D)t'E
(104)
< t P(D)t', - > = 9- P(D)$'-
+
-
-t
So
f
($+ p(D)$;
+
3
$-
P(D)$'-) H
- $y?(D)H;
($+ + +-)($;
where Q ( D ) i s the first-order homogeneous p a r t of P ( D ) . Proof Assume P ( D ) has the form P ( D ) $ ( X I = Q(D)$(x)+ d(x)$(x) + e ( x ) , x E Q ,
where Q ( D ) i s the first-order homogeneous part of P ( D ) , while d,e Then relation (103) follows easily from (101). Further, for given v
t, d t; = $- d :$ +
($+
-
+
Co(n).
N, we have
E
= ($-
E
+
(qt - $_)Snw)d($l (+; f
-
($- d (+;
$y($; -
($+
+
-
$-Id
- $'-bnw) = $1)Srl,
+
$LXSqwY;
therefore, in view of Lemma 1,
< t d t ' , .>
d :$
= $-
+
($+ d $;
-
$- d $'-)H.
Finally, i t i s easy t o see that
< te , * >
= $-
e
+
(qt e
- $-
e)H*
The l a s t two relations together with (102) will obviously yield (104). The result on junction conditions for discontinuous solutions of systems of PDEs of type (MH) will be presented next in Theorems 11 and 1 2 . F i r s t we consider the i n f i n i t e l y smooth case, i . e . the case when the co= efficients c . and right-hand terms f inoothesystem (89) as well as y defining in 742) the hypersurface r age c -smooth. Moreover, the func= (95) used in the regularizations (96) and ( 9 9 ) will also be tions assumed C -smooth.
0
144
E. E. Rosi nger
Proposition 4
n
Ra a r e two Cm-smooth s o l u t i o n s o f t h e m-the o r d e r (MH) i n (89). Then f o r any c -smooth r e g u l a r i z a t i o n s given i n (99), t h e f o l l o w i n g r e l a t i o n s h o l d f o r e v e r y 1 < B < b: Sjppose u-,
:
u+
+
C -smoothmpolynomial n o n l i n e a r system o f PDEs o f t y p e
( 105 1
TB(D)s
( 106 1
< TB(D)s,
where
Q
BWcl
, (D)
sm
E
*
1)
(uJa1
>
=
(D)H)
QBpaal
i s t h e f i r s t - o r d e r homogeneous p a r t of P
B paa
I
(0).
Proof I n view o f (91.1) and (99), i t f o l l o w s t h a t
B u t (103) i n C o r o l l a r y 1 i m p l i e s t h e r e l a t i o n s m
SapBpaal
(D)SaI
t h e r e f o r e the l i n e a r i t y o f L TB(D)s
E
E
BP
s ; (D) w i l l g i v e
sm
as w e l l as
Now (104) i n C o r o l l a r y 1 w i l l g i v e
where Q
8P a
I
(D) i s t h e f i r s t - o r d e r homogeneous p a r t o f P
BPW
I t f o l l o w s t h a t f o r 1 Q g Q b, t h e f o l l o w i n g r e l a t i o n h o l d s :
, (D).
RESOLUTION OF SINGULARITIES
145
Since u was supposed t o be a c l a s s i c a l s o l u t i o n s o f t h e system (89), we have T (D)u
e
-
= 0 on
R,
which completes t h e p r o o f .
U
B e f o r e p r e s e n t i n g t h e r e s u l t i n Theorem 11, we need t h e f o l l o w i n g d e f i n i = tion. Given two f u n c t i o n s u-, u+ : R
+
Ra,
u-,
U+ E
U ( X ) = u-( x ) + ( ~ + ( x )- u-(x))H(x),
c
m
we d e f i n e u : R X E
+
Ra by
R,
and use t h e n o t a t i o n :
< a,
u @ c"(n)>. a Then t h e f u n c t i o n s u , u+ w i l l be c a l l e d Cm-independent on 11, o n l y i f f o r any ha E R1, w i t h C I , t h e f o l l o w i n g i m p l i c a t i o n i s v a l i d :
I =
{all < a
I f a = l , then u-, u+ a r e t r i v i a l l y C"-independent on r, which i s why t h e above c o n d i t i o n was n o t demanded i n Theorem 1 i n S e c t i o n 1. I n terms o f t h e system o f PDEs i n (89), t h e case a = l corresponds t o t h e s i t u a t i o n when one unknown f u n c t i o n u : R R ' has t o s a t i s f y b PDEs. -f
Obviously, ifu Moreover, u
E
C" t h e n u-, u+ a r e
c" o n l y i f D~(u-),(x) = D~(u,),(x),
C"-independent
on
r.
E
tl 1 G a
< a,
x
E
r,
p
E
Nn
.
Theorem 11 Suppose u , ut : R -f Ra a r e two C"-smooth s o l u t i o n s o f t h e m-th o r d e r C"'-smoothpolynom~al n o n l i n e a r system o f PDEs o f t y p e (MH) i n (89) and suppose g i v e n a C -smooth h y p e r s u r f a c e (92). Then t h e f u n c t i o n
(107)
~ ( x )= u-(x)
+ (u+(x)
-
~ - ( x ) ) H ( x ) , x E R,
146
E . E. Rosi nger
where H i s t h e Heaviside f u n c t i o n (94) a s s o c i a t e d w i t h t h e hypersurface (92), i s a weak s o l u t i o n o f t h e system (89), o n l y i f t h e f o l l o w i n g j u n c t i o n c o n d i t i o n s a r e s a t i s f i e d f o r each 1 Q B b:
where Q
BPclcl
I
( D ) i s t h e f i r s t - o r d e r homogeneous p a r t o f P
B Pas ' (D).
I n t h a t case, i f t h e f u i c t i o n s u , u+ a r e Cm-independent on p o s s i b l e t o c o n s t r u c t C - r e g u l a r T z a t i o n s ( V , S ) , such t h a t
( 109 1
ua
= s
a
where scl (110)
t
rk(v,s)E A'(v,s), E S,
with 1 Q a
Q
v
1
R E
r,
it i s
i,
a, do n o t depend on R ;
u s a t i s f i e s each o f t h e PDEs o f t h e system (89), i n t h e usual k a l g e b r a i c sense, w i t h t h e m u l t i p l i c a t i o n i n A ( v , S ) and t h e p a r t i a l d e r i v a t i ve o p e r a t o r s DP : A ~ ( v , s )-+ A ~ ( v , s ) , p w i t h k, k E R, k-k > m .
E
N",
/PI
my
Proof I f t h e j u n c t i o n c o n d i t i o n s (108) hold, then i n view o f P r o p o s i t i o n 4, t h e f u n c t i o n (107) w i l l be a weak s o l u t i o n o f t h e system ( 8 9 ) .
Conversely, assume u i n (107) i s a weak s o l u t i o n o f t h e system ( 8 9 ) . Since we a r e i n t h e Po-smooth case, t h e o p e r a t i o n s i n t h e d e f i n i t i o n o f u can be performed w i t h i n D'(n), however t h e same does n o t h o l d f o r t h e o p e r a t i o n s on u performed by t h e PDOs T (D) , w i t h 1 Q B Q b y as these o p e r a t i o n s w i l l I (D)ff. Nevertheless, a c c o r d i n g t o (105) i n Propoi n v o l v e p r o d u c t s H.P s i t i o n 4, t h e (?-smo&%egularization s o f u, c o n s t r u c t e d i n (99) has t h e p r o p e r t y t h a t T (D)s i s weakly convergent f o r e v e r y 1 Q f 3 Q b . Therefore t h e assumption !hat u i s a weak s o l u t i o n o f t h e system (89) i m p l i e s t h a t < TB(D)s,* >
= fB , Y 1
Q B Qb.
This, i n view o f t h e r e l a t i o n s (106), completes t h e p r o o f o f t h e converse. Assuming now t h a t t h e j u n c t i o n c o n d i t i o n s (108) a r e s a t i s f i e d and t h e s o l u t i o n s u , u a r e (?-independent on r, we proceed w i t h t h e c o n s t r u c t i o n o f f ' - r e g u T a r i t a t i o n s ( V , S ) such t h a t (109) and (110) w i l l h o l d . Obviously, we can assume t h a t
(111)
u q
c?.
We use the n o t a t i o n :
RESOLUTION OF SINGULARITIES
(112 1
w
B = TB(D)s - u ( f B ) , f o r 1 G 5
Q
147
b
and p r o v e t h e r e l a t i o n s
Indeed, assume V C 0 \ r such t h a t V i s an open b a l l and i t s c l o s u r e does n o t i n t e r s e c t r. Then, i n view o f t h e c o n t i n u i t y o f Y, t h e r e e x i s t s E > 0 such t h a t one and o n l y one o f t h e f o l l o w i n g two c o n d i t i o n s i s s a t i s f i e d :
(114)
~ ( xQ )
(115)
Y(X)
E,
-
> E, v
tt
X E
x
E
V ; or
v.
Assume t h a t (114) h o l d s and t a k e u E N such t h a t (Pt1)E G -1. view o f ( 9 5 ) , ( 9 6 ) and (99.1) i t f o l l o w s t h a t
(116)
U 1
sav(x) = (u&(x),
Q
a, v
E
N, v 2 1-1
,x
E
Then i n
V.
I n t h e same way i t can be shown t h a t (115) w i l l i m p l y
(117)
sav(x) = ( U + ) ~ ( X ) ,V 1 Q a
Q
a, v
E
N, v 2
u,
x
E
V.
T h e r e f o r e t h e f a c t t h a t u and ut a r e c l a s s i c a l s o l u t i o n s o f t h e system (89) w i l l r e s u l t i n t h e r e l a t i o n s
(118)
wBV(x) = 0, U 1 G 6 G b y v
E
N,
V
2 U , x E V,
c o m p l e t i n g t h e p r o o f o f (112), as r i s nowhege de se i n a, a c c o r d i n g t o (92.1). L e t us denote by 1 , t h e i d e a l i n ( C (a)) generated by n {DPwB 11 G B b, p E N 1 . Then t h e r e l a t i o n (118) w i l l o b v i o u s l y i m p l y that
a
(119)
'w
'nd.
Assume now g i v e n any i d e a l 1 i n (Cm(Q))N
( 120 1
such t h a t
1 c 1 c lnd. W
Then, as seen i n t h e p r o o f 02 Theorem 7 i n S e c t i o n 4, Chapter 3, t h e r e e x i s t v e c t o r subspaces T c S s a t i s f y i n g t h e c o n d i t i o n s
(121)
i n T = v"n T = Q;
( I22 1
V"
(123)
(Ip"
(1 n S")
t
a
= V"
@
T)n U
P ( Q =)
T;
and
Q-
B u t assumption (111) means t h a t
31 G
a : u
a
7 c"(n).
Thus r e a s o n i n g s i m i l a r t o t h a t used i n t h e p r o o f o f (36) w i l l y i e l d
b' l G a Q a :
148
E . E . Rosinger
However, we s h a l l need and prome t h e f o l l o w i n g s t r o n g e r r e s u l t . by S t h e v e c t o r subspace i n S generated by Esa 1 < a d a, ua then'
I
su)n(f @
(124)
( ~ ~ m t( ~ )
7
Denote C"(R)};
= O_
Indeed assuming t h a t o u r s u p p o s i t i o n i s f a l s e and t h a t
where J, E Cm(n), I = {a 11 < IY. < a, uCI $! Cm(n } , X, E R' , v E Vm and t E T , t h e n i n view o f ( l o o ) , (120) and (120 , as w e l l as (69), Chapter 1, t h e f o l l o w i n g r e l a t i o n i n D'(n) i s obtained:
c
J, t
X, ua
= < t, - >
, supp
a>
nowhere dense.
a €1
But u a €
Cm(n \ r ) , V 1 < supp
since R
\r
< t, - > c
i s open, as
r
a, as a consequence o f (107).
Therefore
r i s closed.
Then
X,ua = O o n R \ r €1 NOW, s i n c e r has zero Lebesque measure (see ( 9 2 . 1 ) ) , t h e r e l a t i o n (126) w i l l give i n U ' ( Q ) the e q u a l i t y (126)
C
J,t
CI
t h e r e f o r e (125) w i l l i m p l y v + t
E
v",
which i n view o f (121) w i l l r e s u l t i n (128)
t = u(0) E Q,
But t h e r e l a t i o n (127) which i s v a l i d i n U'(Q) o b v i o u s l y i m p l i e s XCIuCI=-J,onR
Z CIE
s i n c e J, EC"(R),
I
Xa
C CIE
uCIE Cm(R \
I
r)
and, i n view o f (92.1),
r is
nowhere dense i n R . Thus i t f o lows t h a t
s i n c e u-, g i ve
J, = 0 on R and
Xa
u+ were assumed
C
v = u(0) E Q,
= 0, V C I E m
-
I
ndependent on
r.
Now
125) and (128) w i l l
RESOLUTION OF SINGULARITIES
149
which completes the proof of ( 1 2 4 ) . In view of (124) there e x i s t vector subspaces s c s" such t h a t (129)
UC"(S2)
(130)
S"
=
+
c s;
v" @ S @
T.
Then (121), ( 1 2 2 ) , (130) and (129) will , according t o Theorem 4, Section 4, Chapter 3, imply t h a t (131)
0 T)
s
(v,
i s a c"-regularization
for any vector subspace v c I nV'. B u t by definition {DPwB [ I d6 d b y P E Nn 1 c I, c I ; and moreover, the junction condition (108) means t h a t
w
E P ,V l d B 6 b ;
B therefore i t i s obviously possible to choose
(132)
I
{DPwB
<
l<
taking f o r instance v
=
by
I n
PE
Nn} c
v satisfying the condition
v,
v".
NOW, i n view of Theorems 2 and 3, Sectilon 3, Chapter 3, the relations (129), (112) and (132) complete the proof o f Theorem 11. 0
The necessary and s u f f i c i e n t junction condition, as well as the resolu= tion of s i n g u l a r i t i e s presented above i n Theorem 11 f o r the case of i n f i = n i t e smoothness, have t h e i r f i n i t e l y smooth counterpart i n Theorem 12. R a , u-, u+ E c", we define u : R + Ra Given two functions u-, u+ : R -f
bY U(X) =
u-(x)
+ (u,(~)-u_(x))H(x)~ X E Q,
and use the notation J = {a\ 1 d a < a , u d P(Q)>. a Then the functions u-, u+ will be called (?-independent on r , only i f f o r any Xa E R *, w i t h a E J , the following implication i s valid: (
Xaua€ P Q ) ) -(Acl=
C
0, +'a E J ) .
clEJ
Obviously, u-, u+ are (?-independent on Tin case a = l or u over u E C "only i f u-(x)
=
u+(x),
v
x
Er.
E P.
More=
150
E . E . Rosinger
Theorem 12 ose u , u : R -+ Ra a r e two ckl-smooth s o l u t i o n s o f t h e m-th o r d e r $ @ % n o o t l i pofynomial n o n l i n e a r k s y s t e m o f PDEs o f t y p e (MH) i n (89) and kl m. Suppose a l s o q i v e n a c '-smooth hypersurface (92). Then t h e function (133)
+ (u,(x)-u-(x))H(x),
U ( X ) = u-(x)
x
E Q,
where H i s t h e H e a v i s i d e f u n c t i o n (94) a s s o c i a t e d w i t h t h e hypersurface (92), i s a weak s o l u t i o n o f t h e system (89), o n l y i f t h e j u n c t i o n condi= t i o n s (108) a r e s a t i s f i e d f o r each 1 Q B G b . I n t h a t case, i f t h e f u n c t i o n s u , u+ a r e C"-independent on p o s s i b l e t o c o n s t r u c t C " - r e q u l a r i z a t i o n s ( V , S ) such t h a t :
r,
it is
(134)
IR(V,S) EAR(U,S), tl 1 G a G a y R E N, & Q k l where s a E S, w i t h 1 Q a G a y do n o t depend on R;
(135)
u s a t i s f i e s each o f t h e PDEs o f t h e system (8i3) i n t h e usual a l g e b r a i c sense, w i t h t h e m u l t i p l i c a t i o n i n A ( V , S ) a n d W p a r t i a l d e r i v a t i v e operators
uQ
=
sat
DP:AR(UyS) -Ak(V,S), p E Nny Ip I Q m , w i t h R L-k > m y R Qkl and k G m i n I kl-rn,k21.
,k
E
N,
Proof The p r o o f t h a t u i s a weak s o l u t i o n o n l y i f (108) holds, i s s i m i l a r t o t h e p r o o f o f Theorem 11, To prove t h e second p a r t o f t h e theorem we use t h e n o t a t i o n h = m i n ( kl-my
k2].
Obviously, we can assume t h a t (136)
u q
c
I t can e a s i l y be seen t h a t t h e sequence s c o n s t r u c t e d i n (99) s a t i s f i e s
(137)
s
€skl,<s Q
w h i l e t h e sequences w (138)
W
B
B
= TB(D)S
ay
. > = u
a
,
V
I
Q
~
Q
~
c o n s t r u c t e d i n (112) s a t i s f y
-
U(fS)
E
Ind nvh
v
1 "6
Qb.
Therefore (139)
{DPwB 11 Q B Qb, p
ENn,
Assume t h e n g i v e n any i d e a l I i n (2'
( 140)
{DPwB 11 Q B Qb, p
ENny
I p I Q h ) C Ind n Lf'. N such t h a t
(n))
IpI Qh}c I
c Ind
and I i s 81-smooth (see (126), Chapter 3) which i s p o s s i b l e i n view o f
RESOLUTION OF SINGULARITIES
151
(139) and P r o p o s i t i o n 6, S e c t i o n 6, Chapter 3 . I n t h i s case, t h e argument i n t h e p r o o f o f Theorem 14, S e c t i o n 6, ChapteL 3, a p p l i e d f o r L = kl w i l l y i e l d t h e e x i s t e n c e o f v e c t o r subspaces T c s which s a t i s f y (141)
l n T = V " n T = q ;
(142)
v"
(143)
(f @
( r n s o ) = v"
t
T) n
@ T; and
(n)
UCo
=
0.
I f we denote by Su t h e v e c t o r subspace i n s" generated by Isc, 1 1 c, G a, % 9 C" (Q)}, t h e n s i n c e u , us were assumed C" -independent on r, t h e argument used t o o b t a i n (124) in-the p r o o f o f Theorem 11, w i l l now y i e l d
(144)
Wc"(q
t SUP
(f'
0 T ) = 0..
Now, i n view o f (144), t h e r e e x i s t v e c t o r subspaces R' c So such t h a t
f'
So =
0 (UC" (Q) + Su) 0 R '
@
T.
Therefore, t a k i n g i n Lemma 5, S e c t i o n 6, Chapter 3
F=Skl
E=S',
9
+ Su) @T
A = f' @ ( U c o ( n )
,B
=
R',
i t f o l l o w s t h a t t h e r e e x i s t v e c t o r subspaces
C=
CF=Skl
R
which s a t i s f y
W c . (a)+ S u )
(Lp
@ T) n R
=
0,
and (145)
So =
v"
a(Ue(Q) + Su)
0R0
T.
Introducing the notation
s = (UC (Q1 + SU) @ R (146) t h e r e l a t i o n s (141), (142) and (145) w i l l i m p l y t h a t (147)
(v,
s
@T)
i s a e-regularization
f o r any v e c t o r subspace V Chapter 3.
C
1 n V'
, according
t o Theorem 10, S e c t i o n 5,
B u t i n view o f (139), i t i s p o s s i b l e t o choose V such t h a t (148)
we
E
Vh,
V 1 G 6 G b.
Now t h e r e l a t i o n s (146), (138) and (148) w i l l complete t h e p r o o f , i f we t a k e i n t o account t h e v e r s i o n s o f Theorems 2,3 and 9, Chapter 3, correspon=
E . E . Rosinger
152
d i n g t o C" - r e g u l a r i z a t i o n s . 6.
Resoluble Systems o f Polynomial N o n l i n e a r PDEs
The necessary and s u f f i c i e n t j u n c t i o n c o n d i t i o n s across hypersurfaces o f d i s c o n t i n u i t i e s o f weak s o l u t i o n s f o r systems o f PDEs o f type (MH) and the r e s o l u t i o n o f t h e corresponding s i n g u l a r i t i e s presented i n t h e pre= vious s e c t i o n a r e extended here t o a l a r g e c l a s s o f systems o f PDEs which c o n t a i n s many - i f n o t most - o f t h e equations m o d e l l i n g p h y s i c a l pheno= mena. The r e s u l t o b t a i n e d can be i n t e r p r e t e d as an upper bound on t h e degree o f n o n l i n e a r i t y t h a t may be expected i n most o f t h e equations o f physics. I t i s , p a r t i c u l a r l y i n t e r e s t i n g t h a t t h e r e s u l t i n g c l a s s o f systems o f PDEs - which we s h a l l h e r e a f t e r r e f e r t o as r e s o l u b l e - i s c h a r a c t e r i z e d by i t s s p e c i a l behaviour i n r e g a r d t o weak s o l u t i o n s w i t h d i s c o n t i n u i t i e s across hypersurfaces. More p r e c i s e l y , t h e upper bound on t h e degree o r c o m p l e x i t y o f n o n l i n e a r i t y , p r e v i o u s l y mentioned, i s i n f a c t a s u f f i c i e n t c o n d i t i o n f o r o b t a i n i n g simple, a l g e b r a i c j u n c t i o n c o n d i t i o n s c h a r a c t e r i z i n g weak s o l u t i o n s w i t h d i s c o n t i n u i t i e s across hypersurfaces. The s i g n i f i c a n t advantage o f d e a l i n g w i t h t h i s problem w i t h i n t h e frame= work o f t h e q u o t i e n t algebras c o n t a i n i n g t h e d i s t r i b u t i o n s soon becomes apparent. I n f a c t , one o f t h e m a j o r advantages o f o u r method o f d e a l i n g w i t h n o n l i n e a r PDEs w i t h i n t h e framework o f t h e q u o t i e n t algebras i s t h e i d e n t i f i c a t i o n o f t h e c l a s s o f r e s o l u b l e systems o f PDEs. Definition 1 The system o f polynomial n o n l i n e a r PDEs i n (89) i s c a l l e d r e s o l u b l e , o n l y i f each o f t h e a s s o c i a t e d PDOs i n 90) can be w r i t t e n i n t h e form TB(D)u(x) =
(149)
C l < p < r
B
whenever (149.1)
u(x) = $(x) t x(x).o(x),
where I ), x : Q and T a r e m' BP
x E a,
R a y w : n + R', $, X, w E Cm E Nn, R , P P - t h o r d e r polynomial n o n l i n e a r PD8s i n $ an!'X.
+
BP
The p a i r ( m u , ml'), where
m' = m a x {m'
BP
(1989b, lGp
(149.2)
m"
max
I ]pep I
l G B B b , lGpGrB)
i s c a l l e d t h e s p l i t o r d e r o f t h e r e s o l u b l e system (89). Proposition 5
A system o f PDEs o f t y p e (MH
i s resoluble.
E
N
RESOLUTION OF SINGULARITIES
153
Proof Assume t h e PDOs i n d90) c o r r e s p o n d i n g t o t h e system (89) a r e o f f o r m (91) and t a k e u : R R g i v e n i n (149.1). Then (91), o r e q u i v a l e n t l y (91,1), yields -f
But
u-QBpaaI
(D)u =
1
Q8paal
(D)(a)’-
Now t h e r e l a t i o n (149) f o l l o w s e a s i l y .
0
The c h a r a c t e r i s t i c b e h a v i o u r o f t h e r e s o l u b l e systems o f p o l y n o m i a l non= l i n e a r PDEs towards weak s o l u t i o n s w i t h d i s c o n t i n u i t i e s a c r o s s hypersur= faces i s now p r e s e n t e d i n t h e case o f i n f i n i t e smoothness. Theorem 13 Suppose t h e m-th o r d e r Cm-smooth p o l y n o m i a l n o n l i n e a r system o f PDEs i n (89) i s r e s o l u b l e . Given u , u+ : R Ray two Cm-smooth s o l u t i o n s o f (89), and a Cm-smooth Ra by h y p e r s u r f a c e (92), d e f i n e t h e f u n c t i o n u : fi -+
-+
(150)
~ ( x )= u_(x)+(u+(x)-u_(x))H(x)
,x
E
fiy
where H i s t h e H e a v i s i d e f u n c t i o n (94) a s s o c i a t e d w i t h t h e h y p e r s u r f a c e (92). Then u i s a weak s o l u t i o n o f (89), o n l y i f t h e f o l l o w i n g j u n c t i o n c o n d i = t i o n s a r e s a t i s f i e d f o r each 1 G B G b :
(151)
c Gr l G p
(TB,(D)(u-,~+-~-)).~BpH
= f8 ’
8
r,
I n t h a t case, i f t h e f u n c t i o n s u , u+ a r e Cm-independent on p o s s i b l e t o c o n s t r u c t C - r e g u l a r i z a t i o n s ( V , S ) such t h a t
( 152 1
ua = sa where s,
( 153 1
+ E
w,
R R ‘I (V,S)EA ( V , S ) , S
it is
tt 1 Gc1 < a , R E , w i t h 1 G c1 G a, do n o t depend on R;
u s a t i s f i e s each o f t h e PDEs o f t h e system (89) i n t h e usual a l g e b r a i c sense, w i t h t h e m u l t i p l i c a t i o n i n Ak(U,S) a n d T l F p a r t i a1 d e r i v a t i v e o p e r a t o r s Dp : AR(V,S)
-+
Ak(V,S),
p
E
Nn,
I p ( Gm, w i t h
R, k
E
fl,R-k
am.
154
E . E . Rosinger
Proof F o r u g i v e n i n (150) (see a l s o ( 9 3 ) ) l e t m u s c o n s i d e r t h e CEsrnooth r e g u l a r i = z a t i o n (99) o b t a i n e d when rl i n (95) i s C -smooth. Then, f o r each 1 G G b, the f o l l o w i n g r e l a t i o n s hold:
Indeed, i n view o f (149) we have f o r 1 G D G b and v (156)
TB(D)sv =
(TBp(D) (U-,U,-U-)).
C
l G p G r
E
N, t h e r e l a t i o n s
DPDP (sqv)
5%
*
D
But, i n view o f (97) i n Lemma 1, S e c t i o n 5, R
( SllV)
(1
Bp = s SV
, where 5
= ( q ) Bp.
Moreover TDP(D)(u_,ut-u-) t h e r e f o r e the products U' ( Q ) .
E
c"(Q);
T ( D ) ( u ,u,-u).D PP -
P 'pH
i n (156) a r e w e l l - d e f i n e d i n
Reasoning thus, (154) and (155) w i l l f o l l o w e a s i l y from (98) i n Lemma 1, S e c t i o n 5. Assume now t h a t t h e j u n c t i o n c o n d i t i o n s (151) h o l d . Then i n view o f (154) and (155), u w i l l o b v i o u s l y be a weak s o l u t i o n o f ( 8 9 ) . Conversely, i f u i s a weak s o l u t i o n o f (89) t h e n (154) and (155) w i l l ob= v i o u s l y i m p l y t h e j u n c t i o n c o n d i t i o n s (151). The second p a r t o f t h e p r o o f i s s i m i l a r t o t h a t g i v e n f o r Theorem 11,Sec= 0 t i o n 5. The corresponding theorem f o r f i n i tely-smooth systems i s now given, Theorem 14 Suppose t h e m-th o r d e r and (m' ,m")-th s p l i t o r d e r Ckl-smooth n o n l i n e a r system o f PDEs i n (89) i s r e s o l u b l e and k l > m " - l .
polynomial
k Suppose g i v e n u , u : R -+ R a y two C 2-sm00th s o l u t i o n s o f (89), and a Ck3-smooth hypeFsurPace (92), w i t h k2-m' m"-1, k 3 > m". Then t h e f u n c t i o n u : fl (157)
+
Ra d e f i n e d by
U ( X ) = u_(x)+(u+(x)-u_(x))H(x),
x
E
where H i s t h e H e a v i s i d e f u n c t i o n (94) a s s o c i a t e d w i t h t h e h y p e r s u r f a c e (92), i s a weak s o l u t i o n o f (89), o n l y i f t h e f o l l o w i n g j u n c t i o n c o n d i t i o n s a r e s a t i s f i e d f o r each 1 G L? G b :
RESOLUTION OF SINGULARITIES
155
I n t h a t case, i f t h e f u n c t i o n s u , u+ a r e C"-independent on c o n s t r u c t C" - r e g u l a r i z a t i o n s (V,S) such t h a t u
(159)
c1
R
+ I (V,S)
= scl
where scl
E S,
E
R A (V,S), Y 1 < a
RE
r,
one can
N, R <min{k2,k3},
w i t h 1 < a d a, do n o t depend on R ;
u s a t i s f i e s each o f t h e PDEs o f t h e system (89) i n t h e usual a l g e b r a i c sense, w i t h t h e m u l t i p l i c a t i o n i n Ak(V,S) and-fXF p a r t i a l derivative operators k Dp : AR(V,S) + A ( V , S ) , p E Nny I p I < m w i t h R , k E N,
(160)
R-k
> m,
R
< min{k2,k31
and k ,< min Ck2-m' ,k3-m",kll.
Proof T h i s f o l l o w s i f t h e f i r s t p a r t o f t h e p r o o f o f Theorem 13 and t h e second p a r t o f t h e p r o o f o f Theorem 12, S e c t i o n 5, a r e adapted and combined i n an obvious way.
7
0
Computation o f t h e J u n c t i o n C o n d i t i o n s
The j u n c t i o n c o n d i t i o n s (151) c o n t a i n as a p a r t i c u l a r case t h e j u n c t i o n c o n d i t i o n s (108), and a t t h e same t i m e have a more compact form, t h e r e f o r e we s h a l l p r e s e n t h e r e o n l y t h e way t h e y can be computed e x p l i c i t l y . We s h a l l assume t h e n o t a t i o n s and c o n d i t i o n s i n Theorem 14, S e c t i o n 6. The b a s i c r e l a t i o n s used i n t h e sequel a r e Dp H = (DPy).6
(161)
,V
p E Nny \ P I = 1
and DP ( DR6) = ( D P V ) . ( D ~ + ~ ~Y) , p E N",
(162)
I P I = 1, R E N,
as w e l l as
y.6 = 0
(163.1)
+
(163.2)
(R+l)DR6 = 0, V L E N
where 6 i s t h e D i r a c d i s t r i b u t i o n c o n c e n t r a t e d on t h e hypers r f a c e r i n (92) which i s r e p r e s e n t e d by t h e mapping y : R + R', y E C 1 3 It i s w e l l known t h a t r e l a t i o n s (161) and (162) a r e v a l i d i f f o r i n s t a n c e
.
grad
y(x)
# 0, V x
E
r.
I t i s easy t o see t h a t r e l a t i o n s (161) and (162) w i l l y i e l d
where K
PR
(D) a r e polynomial n o n l i n e a r PDOs o f o r d e r R + l which can be ob=
156
E.E.
Rosinger
t a i n e d from t h e recurrent r e l a t i o n s (164.1)
Kpo(D)y = Dpy, V p E Nny I p I = 1
(164.2)
Kptqll RE
(D)Y = D q ( K p R 0 Y ) N, R Q I p I .
+
(KpR-l(D)Y
S u b s t i t u t i n g now t h e r e l a t i o n s (164) i n t o t h e j u n c t i o n c o n d i t i o n s (151) and r e a r r a n g i n g t h e terms a c c o r d i n g t o t h e i n c r e a s i n g o r d e r o f d e r i v a t i v e s o f t h e D i r a c 6 d i s t r i b u t i o n , we have (GB0 (D) (U-Y u,-U-YY)
) .H
+
(165 1
where G
BR
(D) a r e polynomial n o n l i n e a r PDOs i n u-, u+-u-
and y.
With t h e
h e l p o f t h e r e l a t i o n s (163), i t i s p o s s i b l e t o e l i m i n a t e i n (165) t h e D i r a c 6 d i s t r i b u t i o n and i t s l o w e r d e r i v a t i v e s and f i n a l l y o b t a i n r e l a t i o n s o f t h e form . H t ( T i ( D) ( u-,ut-u-,y)
(166) ( T i ( D) ( u-,ut-u-,y))
) .Dm"-l6=fBYV 1 Q
G by
w i t h Th(D), T"(D) polynomial n o n l i n e a r PDOs i n u , u -u and y; r e l a t i o n s which can be Been as t h e e x p l i c i t f o r m o f t h e j u x c t i h c o n d i t i o n s i n (151). Indeed, as u- and u were assumed t o be known s o l u t i o n s o f t h e r e s o l u b l e system (89), t h e r e l a t i o n s (166) w i l l g i v e c o n d i t i o n s on y i n t h e form o f polynomial n o n l i n e a r PDEs, i.e. c o n d i t i o n s on t h e p o s s i b l e hypersurfaces o f d i s c o n t i n u i t i e s o f weak s o l u t i o n s o f t h e r e s o l u b l e system (89). 8.
Examples o f Resoluble Systems o f PDEs
The aim o f t h i s s e c t i o n i s t o e x p l i c i t a t e i n two p a r t i c u l a r cases o f low o r d e r t h e c o n d i t i o n (149) d e f i n i n g t h e r e s o l u b l e systems o f PDEs. Given t h e system o f polynomial n o n l i n e a r PDEs i n (89), i t s o r d e r o f n o n l i = n e a r i t y i s by d e f i n i t i o n
I
IT = maxlk 1G B G by 1Q i Q hsl Bi (167) Obviously, t h e system o f PDEs i n (89) i s n o n l i n e a r , o n l y i f n
> 2.
I n view o f (2), Chapter 1, t h e o r d e r o f t h e system o f PDEs i n (89) i s c a l l e d the i n t e g e r (168)
m = max{)pBij
11
1 Q B Q b , 1 G i Q h B , 1 G j QkBil
F i r s t , we c o n s i d e r t h e s i m p l e s t systems o f polynomial n o n l i n e a r PDEs i n (89) which correspond t o (169)
1 ~ = 2 , m = l
and which c o n t a i n t h e n o n l i n e a r i t y e x h i b i t e d by t h e Navier-Stokes equations.
RESOLUTION OF SINGULARITIES
-
where
Pi Di = D , w i t h pi = (O,..
(170.1)
157
i
. ,O,l,O,..
., O ) ,
f o r 1 =G i
Q
n
Now, a d i r e c t and easy computation w i l l show t h a t t h e PDOs i n (170) s a t i s = fy t h e c o n d i t i o n (149), o n l y i f t h e a s s o c i a t e d PDOs (171) T;(D)u
c
x) =
1 Q a , a ' a.< 1 6 i , j Q n
Bm'i j (x)Diua(x)D J.ua'
CV
also s a t i s f y that condition. S u b s t i t u t i n g (149.1) i n t o (171), i t f o l l o w s t h a t a l l t h e r e s u l t i n g terms i n (171) can be w r i t t e n i n t h e form r e q u i r e d i n (149), w i t h t h e e x c e p t i o n o f t h e terms i n
Therefore, (173)
t h e system o f PDEs i n (89) w i l l be r e s o l u b l e , o n l y i f
V
cBaaIij(x)
V
= -coaaIji(x),
V x E R, 1 Q B < b y 1 Qa,cr' Q a ,
1
IT =
2, m=2
The i n t e r e s t i n g f e a t u r e o f t h i s case i f t h a t owing t o t h e presence o f second-order terms i n (89) , t h e r e s o l u b i 1 it y c o n d i t i o n (173) w i 11 appear under a weaker form. I t i s easy t o see t h a t the a s s o c i a t e d PDOs i n (83) w i l l have t h e form
( 175 1
1G B Q b T&D) = T;3(D) + T i ( D ) , where T;((D) has t h e f o r m g i v e n i n (170), w h i l e
E . E . Rosinger
158
where D . D , f o r 1 G i , j Q n (see (170.1)) J i S u b s t i t u t i n g (149.1) i n t o (176), an easy d i r e c t computation shows t h a t t h e r e s u l t i n g terms i n (176) can be w r i t t e n i n t h e form r e q u i r e d i n (149), w i t h t h e p o s s i b l e e x c e p t i o n o f t h e f o l l o w i n g ones, which f o r t h e sake o f s i m p l i c i t y , a r e presented w i t h o u t m e n t i o n i n g 1GB G b and x E R (176.1)
Dij
=
c
(177)
1 Qcx,c1' Q a 1Q i , j Q n
dAAli j
+
DjX,X,i
-t
xcxDk xcxl DhwD i j~+x,x,ID.
xcx x ~ l ,w
Dijwt
DiWDhkW+XaDhkX,rWD. 1J.w+X,DhX,IDkWDi *wD
jw t
U)
hk Rearranging t h e above terms a c c o r d i n g t o t h e p r o d u c t s wD. .w,
1J
DiwD.w,
J
DiwD.
IJ
w, 0. .wD
Jh
IJ
hk
w
t h e e x p r e s s i o n o b t a i n e d w i l l be t h e sum o f t h e f o l l o w i n g f o u r terms
RESOLUTION OF S I N G U L A R I T I E S
159
But, t h e f o l l o w i n g i d e n t i t y o b v i o u s l y h o l d s = 2wD. .w+2D.uD.o , Y 1 i,j G n 1 J 1J 1J I n view o f t h e symmetry i n i,j o f (179), i t i s obvious t h a t t h e terms i n (172), (178.1) and (178.2) can be w r i t t e n i n t h e form r e q u i r e d i n (149), provided t h a t
(179)
D. .w2
( 180 )
c
1 G a,aI G a
+
c
1 G a,a' G a l G h Q n
V
V
(Cclalij+caalji)Xaxal (d& I ij h + d z I ihj+d,
II I I
+
j ih+daa I l l I j h j )&Dh&
I
+
E.E. Rosinger
160
I n view o f t h e symmetry i n j , h o f t h e f a c t o r D i w D . t p , the terms i n (178.3) can o b v i o u s l y be w r i t t e n i n the form r e q u i r e d i n f i 4 9 ) , p r o v i d e d t h a t
c
t
1 Q ~ 1 . ~ 1 'Q a
1v 1v +dlV 1v ( ( dm j h k i tdaa ' ih k j aa' h j k i+daa' ij k h DkXaXa'
V l Q i Q n ,
t
1 Q j Q h Q n
F i n a l l y , i n view o f t h e symmetry i n i,j, r e s p e c t i v e l y h,k, as w e l l as i n ( i , j ) , (h,k) o f t h e f a c t o r D . . w D w, t h e terms i n (178.4) can a l s o be w r i t t e n i n t h e form r e q u i r e d ' j n (lhb), p r o v i d e d t h a t
z
( 182 1
1 Qa,a'
Q
a
( d 1v a a ' i j h k + d al Va ' j i h k +d"a a ' i j k h + dal Va ' j i k h +d"clcl'hkij
t
1Q h G k Q n . As the f u n c t i o n x = (xl, ...,x ) : R -+ Ra i n (149.1) i s a r b i t r a r y , i t i s obvious t h a t t h e above condit?ons (180-182) can be w r i t t e n i n an equiva= l e n t and s i m p l e r form i n which a l l t h e sumations
c 1 Q a , a' G a a r e c a n c e l l e d , t h e r e s u l t i n g r e l a t i o n s b e i n g t h i s t i m e v a l i d f o r any 1 G a,a' G a. I n view o f t h i s f a c t , we can o m i t these i n d i c e s c1 and a ' , and r e p l a c e &,xaI by 8, r e s p e c t i v e l y 8 ' . Then, t h e c o n d i t i o n s (180-182) w i l l o b t a i n t h e f o l l o w i n g form: (183)
(cyjtcyi
)eel
+
c
(d;'jh+d"'
1
+dikhj
+ dhil.!)
t d " ' +d"' ) e D, jih jhi
8' t
+d" +diV +diV +d" jkhi kjih ikjh kijh
t
Dke Dh 8'
= (dijtdji)8
t
kihj
ihj
8'
+
'
+d!'! ) D 13 9 ' l G h Q n ( d ? j h J1h h lV td"
c k < n (dijhk 1G h G
jihk
td'" td'.: ) ijkh jikh
t
e Dhk e l , V 1 G
i G j G n
RESOLUTION OF SINGULARITIES
161
and (184)
+
(d;'jh+d;';j)Bf3'
( (djhki+di&j+dhjki+di!kh)Dk 1v 1v
C
+
00'
l < k < n lV td"
+ d l v id1' ) O D k @ ' ) = 0, Y 1 hijk kihj jihk
(dkijh
< i
1<j < h
as w e l l as 1v 1v 1v 1v + d l v . . + d l v 1v ( d i jhk+djihk+d:Ykhtdjikh+dhki j hkJ, k h i j + d k h j i
(185)
= 3'
l < h < k < n
V l < i <j
The c o n d i t i o n s (183-185) can f u r t h e r be s i m p l i f i e d s i n c e t h e f u n c t ons 8,B' : R -+ R' a r e a r b i t r a r y . Indeed, assuming 8 ' = 1 i n (183), i t f o l l o w s t h a t (186)
(cyj+cyi)O
=
(d".+d'!.)O
+
J1
iJ
C
l < h < n
(d;'.h+dj';h)Dh
Y 1<
8,
< j < n
t h e r e f o r e , t a k i n g 0 such t h a t supp O c R ' and B = 1 on Q", where R' C R and V,R' a r e a r b i t r a r y n o n - v o i d open s e t s , t h e r e l a t i o n (186) yields
SL'' C
(187)
'ijtCji
=d!'.+d'! ij
ji'
Y l < i < j < n
S u b s t i t u t i n g (187) i n (186) and t a k i n g B ( x l,...,x,) i t follows that (188)
hd:;
+
c
l < h < n -
+ Bn(xn),
z
'<
( d li j V h k +d!! ~ , h +dl! k 1 ~ k+d" hjihkl0hk
. .,xn)
t h e r e f o r e , t a k i n g 8 ' (xl,. (189) y i e l d s d/'jh+d;'~j+dj';h+d3'hi
= Bi(xl)
+
= 0, Y 1 < i
... + B;l(xn),
< j < n,
1< h
S u b s t i t u t i n g now (190) i n (189) and t a k i n g O'(xl,
B= 1, i t f o l =
(d;'jh+d!'' +d"' +d"' i h j j i h jhilDh " =
l < h d k < n
(190)
...
d!'! = 0, Y 1 < i < j G n , 1 < h < n J1h
F u r t h e r , s u b s t i t u t i n g (187) and (188) i n (183) and t a k i n g lows t h a t (189)
+
= B1(xl)
...,x
) = l
the f o l l o w i n g r e l a t i o n i s obtained
eh(xh) e i ( x k )
j
<
the r e l a t i o n
162
('"1
E.E.
d:Yhk
Rosinger
= 0, +d" + d l V +d!! jihk i j k h Jlkh
v
1< i
<j
1
F i n a l l y , s u b s t i t u t i n g (187), (188), (190) and (191) i n (183), i t f o l l o w s that (Ig2)
diihi
+d" + d l V +d" +d?' +d" +d" = 0, ikhj kihj jkhi kjih ikjh kijh
1< i < j
I n t h a t way, t h e r e l a t i o n (183) was r e p l a c e d by t h e e q u i v a l e n t s i m p l e r r e l a t i o n s (187), (188), (190-192). I n a s i m i l a r way, t h e r e l a t i o n (184) can be r e p l a c e d by t h e f o l l o w i n g three r e l a t i o b s
F i n a l l y , t h e r e l a t i o n (185) i s o b v i o u s l y e q u i v a l e n t t o
Now we can conclude t h a t a system o f polynomial n o n l i n e a r PDEs i n (89) with IT =
2, m = 2
w i l l be r e s o l u b l e , p r o v i d e d t h a t t h e c o e f f i c i e n t s i n (170) and (176) s a t i s f y t h e c o n d i t i o n s (187), (188) , (190-196). Obviously, (187) i s a weaker, g e n e r a l i z e d form o f (173). I t can be n o t i c e d t h a t t h e c o n d i t i o n o f r e s o l u b i l i t y (149) w i l l a c t h a r d e s t on t h e h i g h e s t o r d e r terms, f o r i n s t a n c e , c o n d i t i o n (173) when IT = 2 and m = 1, or c o n d i t i o n s (191), (192) and (194-196) when IT = 2 and m = 2.
The systems o f polynomial n o n l i n e a r l y f a r from having a l l the possible ponding t o a c e r t a i n g i v e n IT and m. i n g t h e i r r e s o l u b i l i t y may prove t o
PDEs encountered i n p r a c t i c e a r e usual= combinations o f n o n l i n e a r terms c o r r e s = Therefore a d i r e c t check f o r e s t a b l i s h = be much more e f f i c i e n t .
CHAPTER 5 S T A B I L I T Y AND EXACTNESS
OF SEQUENTIAL AND WEAK SOLUTIONS FOR POLYNOMIAL NONLINEAR POEs
0. I n t r o d u c t i o n The g e n e r a l i t y p r o p e r t i e s o f s e q u e n t i a l , i n p a r t i c u l a r weak o r d i s t r i b u t i o n s o l u t i o n f o r polynomial n o n l i n e a r PDEs b e i n g s e t t l e d i n t h e way s p e c i f i e d i n S e c t i o n 0, Chapter 3, t h e i n t e r p l a y between t h e s t a b i l i t y and exactness p r o p e r t i e s o f t h e mentioned s o l u t i o n s w i l l be d e a l t w i t h i n t h e p r e s e n t chapter. As s p e c i a l l y p o i n t e d o u t i n S e c t i o n 1, Chapter 1, one o f t h e b a s i c reasons f o r s t u d y i n g s e q u e n t i a l , weak o r d i s t r i b u t i o n s o l u t i o n s o f polynomial non= l i n e a r PDEs w i t h i n t h e framework o f q u o t i e n t algebras i s t h e n e c e s s i t y o f d e a l i n g w i t h t h e problem o f s t a b i l i t y o f these s o l u t i o n s . T h e r e f o r e , a p r i m a r y a t t e n t i o n w i t h i n t h e p r e s e n t c h a p t e r w i l l be g i v e n t o t h e s t a b i l i t y p r o p e r t i e s o f t h e s o l u t i o n s f o r n o n l i n e a r PDEs. We s h a l l o n l y deal w i t h t h e case o f chains o f q u o t i e n t a l g e b r a s ( 2 4 ) , Chap= t e r 3. The r e s u l t s p r e s e n t e d f o r t h a t case, can e a s i l y be r e f o r m u l a t e d f o r t h e chains o f q u o t i e n t a l g e b r a s ( 9 3 ) , Chapter 3, as i n d i c a t e d i n S e c t i o n 4. The r e s u l t s c o n c e r n i n g s t a b i 1 it y and exactness w i 11 be p r e s e n t e d on two l e v e l s . F i r s t , i n Sections 1-4, t h e problems w i l l be d e a l t w i t h on a gene= r a l l e v e l , i . e . r e s u l t s concerning t h e s t a b i l i t y and exactness o f any se= q u e n t i a l s o l u t i o n f o r any polynomial n o n l i n e a r PDE c o n s i d e r e d w i t h i n t h e chains o f q u o t i e n t a l g z a s ( 2 3 ) , Chapter 3, w i l l be presented. Then, i n S e c t i o n s 5 and 6, r e s u l t s concerning t h e s t a b i l i t y o f t h e weak s o l u t i o n s f o r t h e classes o f polynomial n o n l i n e a r PDEs s t u d i e d i n Chapter 4 w i l l be pre= sented.
1. B a s i c Facts and Remarks Concerning S t a b i l i t y Suppose g i v e n a C m - r e g u l a r i z a t i o n ( V , S ) . Then, t h e s t a b i l i t y p r o p e r t y o f s e q u e n t i a l weak o r d i s t r i b u t i o n s o l u t i o n s o f polynomial nonTinear PDEs, c o n s i d e r e d w i t h i n t h e framework o f t h e c h a i n s o f q u o t i e n t a l g e b r a s c o n t a i n = i n g t h e d i s t r i b u t i o n s (see ( 2 4 ) , Chapter 3)
(1)
A R ( V Y s ) = AR(V,S)/IR(V,S),
i s d i r e c t l y expressed b y t h e
(2)
IR(V,S), R
E
size o f
il E
8,
the ideals
R
t h a t i s , b e t t e r s t a b i l i t y means l a r g e r i d e a l s ( 2 ) , and i n e x t r e m i s , maximal s t a b i l i t y means maximal i d e a l s ( 2 ) .
163
164
E.E.
Rosinger
R e c a l l i n g now t h e way g i v e n i n (17-20), Chapter 3, t h e i d e a l s ( 2 ) were c o n s t r u c t e d , i t i s easy t o see t h a t these i d e a l s i n c r e a s e t h e i r s i z e , when= ever V o r S do so. However, g i v e n a C m - r e g u l a r i z a t i o n ( V , S ) , t h e s i z e o f S cannot be increased, i n view o f t h e f o l l o w i n g : Lemma 1 -~ If (V1,S1) and (V2,S2) a r e Cm-regularizations,
t h e n S1
C
S2 i m p l i e s S1 = S2.
Proof I t f o l l o w s e a s i l y from (21.2 , S e c t i o n 3, Chapter 3, which can a c t u a l y be w r i t t e n under t h e e q u i v a l e n t form m
S
=
vmQ S
0
Nevertheless, given a C m - r e g u l a r i z a t i o n ( V , S ) , t h e f o l l o w i n g changes i n S are possible. Lemma 2 I f ( V , S ) i s a C m - r e g u l a r i z a t i o n and S 1 i s a v e c t o r subspace i n Smsuch t h a t
then
(V,Sl)
i s also a
Cm-regularization.
Proof Obviously ( V , S l ) w i l l s a t i s f y (21.2) and (21.3) i n D e f i n i t i o n 1 , S e c t i o n 3, Chapter 3. Concerning t h e r e l a t i o n (21.1) i n t h e mentioned d e f i n i t i o n , we n o t i c e t h a t T ( V ) n SI c I ( V ) n = Z(V) n ( V
@ S)
(v
@ sl)
c Z(V)
ns
=
= Q
the l a s t i n c l u s i o n r e s u l t i n g from t h e f a c t t h a t V
c Z(V)
0
On t h e o t h e r hand, i n c r e a s i n g V i s a l s o n o t a t r i v i a l t a s k , i n view o f t h e r e l a t i o n (see ( 2 2 ) , Chapter 3 ) :
(3)
V C
V"
#
which i m p l i e s t h a t t h e maximum s i z e f o r V , g i v e n b y V", i s i n a c c e s s i b l e and we s h a l l have t o s e t t l e f o r V h a v i n g maximal s i z e s o n l y .
165
STABILITY AND EXACTNESS
Therefore, once again the necessit of dealing with various chains of quo= t i e n t algebras (24) or ( 9 3 ] d e d from the inexistence of canonical chains of the mentioned types, 2.
Maximal Stability
I n view of Lemma 1, we shall suppose given a fixed vector subspace S c Sm which s a t i s f i e s (see ( 2 1 . 2 ) and (21.3), Chapter 3) the conditions:
(4)
S" = V"
(5)
%"(n)
0
S
=
and study thg s t a b i l i t y of golutions as only spaces V c V which give C -regularizations such vector subspaces V , for instance V = p. Section 1, we shall be interested in maximal
depending on the vector sub= ( V , S ) . Obviously, there exist However, as mentioned in V.
Suppose now given a polynomial nonlinear PDE (6)
C ci(x) l ~ i Q h l < j < k i
and the corresponding PDO (7)
T(D) =
ci(x)
C
l < i < h with the order (8)
m
=
max{ l p . . ] 1J
1
u(x)
DpiJ
=
f(x), x
E
il
.n Dpij
l < j < k i
1 < i < h, 1
Q j Q
kil
We shall suppose that the PDE in ( 6 ) i s 0-smooth, that i s , c i , f (see Section 0 , Chapter 4 ) .
E
C"(n)
I t will be useful t o deal with the s t a b i l i t y of solutions for the PDE in ( 6 ) within the framework of the following rather general notion of solution.
Definition 1 A sequence of functions s E ( C m ( n ) )N i s caLled an S-sequential solution of the PDE i n (6), only i f there exists a C -regularization ( V , S ) , such that (9)
s E AR(V,S),
V R
E
W
-
u(f)
E
IR(V,S), 'd 1 E \
(10) w
=
T(D)s
The meaning of the above notion of solution results from:
-Proposition 1 N If s E ( C " ( 0 ) ) i s an 2-sequential solution o f the m-th order PDE in (6), then, for a certain C -regularization ( V , S ) , the following properties hold: 1) S
=
s
t l'l(V,S)
E A'(V,S),
Y 1E
fl
E . E . Rosinger
166
2)
S satisfies th plication i n A
i n t h e usual a l g e b r a i c sense, w i t h R (PDE U S )i nand(6), the p a r t i a l d e r i v a t i v e operators
Dp: A'(U,S)
Ak(VS)
+
,
p
ENn,
IpI
my with R , k g
m,
multi=
R-k> m
Proof I t f o l l o w s e a s i l y from ( 9 ) , ( 1 0 ) above and Theorems 2 and 3, S e c t i o n 3, Chapter 3.
n
The g e n e r a l i t y o f t h e above n o t i o n o f s o l u t i o n r e s u l t s from: Proposition 2 Ifa d i s t r i b u t i o n S E P ' ( R ) \ C m ( n ) i s a Cm-regular weak s o l u t i o n o f t h e PDE i n ( 6 ) , the! t h e r e e x i s t s a sequence o f f u n c t i o n s s E S and a v e c t o r sub= space S C S s a t i s f y i n g ( 4 ) and (5), such t h a t
(11)
s
(12)
s i s an S - s e q u e n t i a l s o l u t i o n o f t h e PDE i n ( 6 )
=
<s,
*>
Proof I t f o l l o w s e a s i l y from (71-73), Chapter 4.
NOW, i n o r d e r t o p r e s e n t t h e r e s u l t on t h e maximal s t a b i l i t y o f S-sequential s o l u t i o n s o f t h e PDE i n ( 6 ) , i t i s convenient t o i n t r o d u c e t h e f o l l o w i n g d e f in i t i on. Definition 2
A sequence o f f u n c t i o n s s E ( C ~ ( R ) )i ~s c a l l e d a maximally s t a b l e S-sequen= t i a l s o l u t i o n o f t h e PDE i n ( 6 ) , o n l y i f t h e r e e x i s t s a c m - r e g u l a r i z a t i o n (V,s),w i t h V maximal, and s a t i s f y i n g ( 9 ) and ( 1 0 ) . Theorem 1 Any s - s e q u e n t i a l s o l u t i o n o f t h e PDE i n ( 6 ) i s maximally s t a b l e . Proof t h e s e t o f a l l t h e v e c t o r subspaces V C Vm f o r which (V,s) Denotemby RV i s a C - r e g u f a r i z a t i o n . We s h a l l show t h a t R Vs i s c b a i n complete. Indeed, i n view o f (21.1), Chapter 3, a v e c t o r subspace U CmV belongs t o RVS , o n l y i f I( V ) n S = Q , where I( V ) i s t h e i d e a l i n ( C (f2))N g e n y a t e d b y V. But, i t i s easy t o see t h a t I( V ) i s t h e v e c t o r subspace i n ( C (f2))N generated b y V. (Cm(n))N. Therefore, RVS w i l l indeed be c h a i n complete. NOW, i n view o f ( 9 ) and ( l o ) , as w e l l as Z o r n ' s Lemma, t h e p r o o f i s com= pleted.
0
Remark 1 R e s u l t s on t h e s t r u c t u r e o f t h e maximal v e c t o r subspaces V w h i c h d e f i n e t h e m a x i m a l l y s t a b l e s e q u e n t i a l s o l u T E 6 - 3 n d whose e x i s t e n c e i s g r a n t e d by Theo= rem 1, w i l l be presented i n t h e n e x t chapter.
STABILITY AND EXACTNESS
3.
167
I n t e r p l a y between S t a b i l i t y and Exactness
As seen i n Sections 4 and 5, Chapter 1, t h e exactness p r o p e r t y o f sequen= t i a l , weak o r d i s t r i b u t i o n s o l u t i o n s f o r polynomial n o n l i n e a r PDEs c o n s i = dered w i t h i n t h e framework o f t h e c h a i n s o f q u o t i e n t a l g e b r a s ( 2 4 ) , Chapter 3, o r ( 1 ) above, i s d i r e c t l y expressed by t h e size o f the ideals
(13)
ZR(V,S) , R
E
R
E
h'
and algebras (14)
A'(Y,S)
,R
t h a t i s , b e t t e r exactness means s m a l l e r i d e a l s ( 1 3 ) and l a r g e r a l g e b r a s ( 1 4 ) . I n e x t r e m i s , maximal exactness means minimal i d e a l s ( 1 3 ) and maximal a l g e = bras (14). NOW, i n view o f (17-20), Chapter 3, where t h e i d e a l s ( 1 3 ) and a l g e b r a s ( 1 4 ) were d e f i n e d , i t i s easy t o see t h a t t h e i d e a l s (13) decrease i n t h e i r s i z e t o g e t h e r w i t h V , w h i l e i n a c o n f l i c t i n manner, t h e a l g e b r a s ( 1 4 ) i n c r e a s e t h e i r size together w i t h V . a t concerns S , i t was seen i n Lemma 1, Sec= t i o n 1, t h a t n e i t h e r i n c r e a s e n o r decrease i s p o s s i b l e .
+
More p r e c i s e l y , suppose t h e PDO i n ( 7 ) i s considered t o a c t as f o l l o w s (see p c t . 2 and 4, Remark 3, S e c t i o n 3, Chapter 3): T(D) : AR(V,S)
+
Ak(V,S)
w i t h 2 , k E 4 , k < R , R-k > m. Then, maximal s t a b i l i t y f o r t h e s e q u e n t i a l , w&ak o r d i s t r i b u t i o n s o l u t i o n s o f t h e m)E i n (6)maximal ideals 7 ( V , S ) , t h a t i s , maximal V . On t h e p t h e r s i d e , maxima7 exactness f t h e same s o l u t i o n s m e a m m a 1 i d e a l s Z ( V , S ) , and maximal a l g e b r a s A ( V , S ) , t h a t i s , i n t h e same t i m e minimal and maximal V .
R
The way t h e above c o n f l i c t i s s e t t l e d r e q u i r e s i n each p a r t i c u l a r case o f s e q u e n t i a l , weak o r d i s t r i b u t i o n s o l u t i o n considered f o r t h e PDE i n ( 6 ) , a s p e c i f i c a n a l y s i s o f t h e s t a b i l i t y and exactness i n t e r e s t s i n v o l v e d . 4.
Remarks i n t h e F i n i t e l y Smooth Case k2 Suppose, i h e PDE i n ( 6 ) i n C -smooth, f o r a c e r t a i n g i v e n k2 E N, t h a t i s , c., f E C 2(Q). We s h a l l be i n t e r e s t e d i n t h e s t a b i l i t y o f s e q u e n t i a l s a l u t i o n o f t h e in-order PDE i n ( 6 ) , g i v e n b y sequences o f f u n c t i o n s s €(Ck1(Q))N, w i t h k l E N, kl > m. The framework w i l l be g i v e n t h i s t i m e by t h e chains o f q u o t i e n t algebras ( 9 3 ) , Chapter 3. More p r e c i s e l y , we s h a l l c o n s i d e r t h a t t h e PDO i n ( 7 ) a c t s as f o l l o w s (see Remark 5, S e c t i o n 5, Chapter 3): T(D) : A'( V,S) * Ak( V , S )
where ( V , S ) i s a C " - r e g u l a r i z a t i o n , k Q min {kl-m,k23.
w h i l e R, k
E
N, R-k > m y "kl,
I t i s easy t o see t h a t t h e remarks i n S e c t i o n 1, remain v a l i d , under t h e i r corresponding form, f o r C" - r e g u l a r i z a t i o n s and t h e c h a i n s o f q u o t i e n t a l = gebras ( 9 3 ) , Chapter 3, w h i c h t h e y generate.
168
E.E.
Rosinger
Therefore, given a v e c t o r subspace S c So which s a t i s f i e s t h e c o n d i t i o n s
so
(15)
v"
=
@S
Uco(n)
y
cs
kl N we c a l l a sequence o f f u n c t i o n s s E ( C (a)) an S - s e q u e n t i a l s o l u t i o n o f t h e PDE i n ( 6 ) , o n l y i f t h e r e e x i s t s a C " - r e g u l a r i z a t i o n ( V , S ) such that
, tf
(16)
s
(17)
w = T(D)s-u(f)
E
A'(v,s)
R E N, R G kl E
k I (V,S), V k
E
a-k
N,
> m,k < min{kl-m,k2}
I n t h a t case, one o b v i o u s l y o b t a i n s t h a t S = s
+ I
R
( V , S ) E A ( V , S ) , U R E N, R
< kl
and S s a t i s f i e s t h e PDE i n ( 6 ) , i n t h e usual a l g e b r a i c sense, w i t h t h e m u l t i p l i c a t i o n i n Ak( V , S ) and t h e p a r t i a l d e r i v a t i v e o p e r a t o r s
DP : A ~ ( v , s ) f o r R, k
E
N, R-k
+
> my a
A ~ ( v , s ) , ~E Nn , I P I
Q
kl,
Q
my
k G min{kl-m,k21
k Moreover, ifa d i s t r i b u t i o n S E P ' ( R ) \ C " ( R ) i s a C ' - r e g u l a r weak s a l u t i o n o f t h e PDE i n ( 6 ) , then, t h e r e e x i s t s a sequence o f f u n c t i o n s s E S and and a v e c t o r subspace S c So s a t i s f y i n g (15), such t h a t S = < s, s i s an S - s e q u e n t i a l s o l u t i o n o f t h e PDE i n ( 6 ) . kl N Now, c a l l i n g a sequence o f f u n c t i o n s s E ( C (n)) a maximally s t a b l e S - s e q u e n t i a l s o l u t i o n o f t h e PDE i n ( 6 ) , o n l y i f t h e r e l a t i o n s (16) and ( 1 7 ) h o l d f o r a c e r t a i n C " - r e g u l a r i z a t i o n ( V , S ) , w i t h V maximal, one can e a s i l y o b t a i n t h e corresponding v e r s i o n o f Theorem 1. F i n a l l y , i t i s easy t o see t h a t t h e remarks i n S e c t i o n 3 can a l s o be adap= t e d t o t h e case o f f i n i t e smoothness. 5.
S t a b i l i t y o f Regular Weak S o l u t i o n s
Two general classes o f weak s o l u t i o n s f o r polynomial n o n l i n e a r PDEs were t i r s t , i n S e c t i o n 4, Chapter 4, t h e r e g u l a r weak s t u d i e d i n Chapter 4 s o l u t i o n s f o r a r b i t r a r y polynomial n o n l i n e a r PDEs were i n t r o d u c e d as an e x t e n s i o n o f t h e p i e c e w i s e smooth weak s o l u t i o n s o f s i m p l e polynomial n o n l i n e a r PDEs s t u d i e d i n S e c t i o n s 1-3, Chapter 4. Then, i n S e c t i o n 6, Chapter 4, t h e weak s o l u t i o n s w i t h d i s c o n t i n u i t i e s across hypersurfaces were s t u d i e d i n t h e case of r e s o l u b l e systems o f polynomial n o n l i n e a r PDEs, systems e x t e n d i n g t h e t y p e o f e q u a t i o n s o f magnetohydrodynamics and general r e l a t i v i t y , s t u d i e d i n S e c t i o n 5, Chapter 4.
.
The r e s u l t s on t h e mentioned weak s o l u t i o n s presented i n Chapter 4, con= cerned t h e r e s o l u t i o n o f t h e i r s i n g u l a r i t i e s and were o b t a i n e d b y construe= t i n g a r t i c u l a r i n s t a n c e s o f t h e chains o f q u o t i e n t algebras (24) o r (93), C h a p t b 3 T 5 T u c h a way t h a t t h e weak s o l u t i o n s s a t i s f i e d t h e polynomial n o n l i n e a r PDEs i n t h e usual a l g e b r a i c sense, w i t h t h e m u l t i p l i c a t i o n and p a r t i a l d e r i v a t i v e o p e r a t o r s i n t h e c h a i n s o f q u o t i e n t algebras.
STABILITY AND EXACTNESS
169
The aim o f t h i s s e c t i o n i s t o p r e s e n t a b a s i c s t a b i l i t y r e s u l t c o n c e r n i n g t h e r e g u l a r weak s o l u t i o n s f o r a r b i t r a r y p o l y n o m i a l n o n l i n e a r PDEs by con= s t r u c t i n g l a r g e classes o f c h a i n s o f q u o t i e n t a l g e b r a s ( 2 4 ) o r ( 9 3 ) , Chap= t e r 3, e x t e n d i n g those p a r t i c u l a r ones used i n Chapter 4. I n t h i s way, t h e r e s u l t s c o n c e r n i n g t h e r e s o l u t i o n o f s i n g u l a r i t i e s p r e s e n t e d i n Chapter 4 w i l l a l s o be improved. We s h a l l o n l y deal w i t h t h e case o f i n f i n i t e smoothness. The r e s u l t s o b t a i n e d can e a s i l y be r e f o r m u l a t e d i n t h e f i n i t e l y smooth case.
A s i m p l e and u s e f u l t o o l f o r e s t a b l i s h i n g t h e mentioned s t a b i l i t y p r o p e r t y o f t h e r e g u l a r weak s o l u t i o n s i s t h e s u f f i c i e n t c o n d i t i o n on t h e s e s o l u = t i o n s p r e s e n t e d i n Tbeorem 9, S e c t i o n 4, Chapter 4, as w e l l as t h e s u f f i = c i e n t c o n d i t i o n on C , R - r e g u l a r i d e a l s presented i n Theorem 20, S e c t i o n 10, Chapter 3. Supposg t h e PDE i n ( 6 ) i s Cm-smooth and t h e d i s t r i b u t i o n S E U'(Q)\C"(Q) i s a C - r e g u l a r weak s o l u t i o n f o r t h e PDE i n ( 6 ) which admits a sequence S" s a t i s f y i n g t h e c o n d i t i o n s o f functions s
(18)
s
=
(19)
w
=
(20)
lW i s a v a n i s h i n g i d e a l i n (C"(Q))
(21)
(v"
< 5, ' >
T(D)s
-
u(f)
E
t zw)n(ucm(n)
V"
+ R.S) 1
=
N
o_
Chapter 4 )
where (see ( 7 3 . 2 ) ,
lw = t h e i d e a l i n (C"(Q))N generated by IDP,
(22)
Ip E Nnl
Then, we denote by iDs t h e s e t o f a l l v a n i s h i n g i d e a l s 7 i n (Cm(Q))N, such t h a t
(23)
lw c 7
(24)
(vm +
i)n(ucm(n) + R.S) 1 =
2
Theorem 2 Under t h e above c o n d i t i o n s , t h e f o l l o w i n g p r o p e r t i e s h o l d :
1)
The s e t of i d e a l s i D s i s c h a i n complete.
2)
Each i d e a l 1 E i D s i s Cm, R - r e g u l a r , where R =
3)
F o r each i d e a l I E i D s t h e r e e x i s t C m - r e g u l a r i z a t i o n s
(25)
(Invm>@)
satisfying the conditions
1
R s.
170
E . E . Rosinger
4)
Each Cm-regula r i z a t i o n (V,S')win (25) y i e l d s t h e following resolu= t i o n of s i n g u l a r i t i e s f o r t h e C -regular weak solution S of t h e PDE in ( 6 ) :
4.1)
S = s + l R ( V , S ' ) E AR(V,S'), \d R E 1, where s i s given i n (18), s E S ' and does
4.2)
S satisfies t h
not depend
on R
PDE in ( 6 ) in t h e usual a l g e b r a i c sense, with multi= p l i c a t i o n in A ( V , S ' ) and the p a r t i a l d e r i v a t i v e operators
Dp : A"(V,S')
R
+
Ak(V,S'), p
E
N n y IpI Q m , w i t h II, k E
R,
R-k am
Proof 1)
I t follows e a s i l y from (23) and ( 2 4 )
2 ) and 3 ) follow from the s u f f i c i e n t condition on Coo, R-regular i d e a l s presented in Theorem 20, Section 10, Chapter 3.
4)
I t follows from 2 ) and 3 ) i f account i s taken of Theorem 7 , Section 4, Chapter 4.
Remark 2
+
From the point of view of t h e s t a b i l i t of the (?-regular weak solution S , Theorem 2 above presents a specia i n t e r e s t . Indeed, the s e t of ideals iD, being chain complete, t h e application of Zorn's lemma w i l l y i e l d t h e fol 1 owing property \d I E
iDs :
-
1 E iD, maximal:
ic 1 Then, t h e cha ns of quotient algebras
see ( 2 4 ) , Chapter 3 )
AR
associated t o l a r e gr maximal i d e a l s E iDS will o f f e r b e t t e r s t a b i l i t y s o l u t i o n S. properties f o r t e C -rG@TFweak
-8
6.
S t a b i l i t y of Solutions with Jump Discontinuities f o r Resoluble Systems of PDE S
Suppose t h e m-th order Coo-smooth polynomial nonlinear system of PDEs in (89) Chapter 4 , i s resoluble ( s e e (149),Chapter 4 ) .
STABILITY AND EXACTNESS Suppose u ,ut : R system of-PDts.
Ra a r e two
+
cm-smooth s o l u t i o n s o f t h e mentioned
F i n a l l y , suppose t h a t t h e f u n c t i o n u : R
+
U ( X ) = u-(x)
(31)
171
-
(u+(x)
+
Ra d e f i n e d b y
u-( x ) ) H ( x ) , x E 0,
where H i s t h e H e a v i s i d e f u n c t i o n ( 9 4 ) , Chapter 4, a s s o c i a t e d w i t h t h e C -smooth h y p e r s u r f a c e r i n ( 9 2 ) , Chapter 4, i s a weak s o l u t i o n o f t h e mentioned system o f PDEs, t h e r e f o r e s a t i s f y i n g on r t h e necessary and s u f = f i c i e n t j u n c t i o n c o n d i t i o n s (108) o r (151), Chapter 4. We s h a l l a l s o assume t h a t u- and u+ a r e em-independent on 5, Chapter 4 ) .
r
(see S e c t i o n
Then s = ( S ~ , . . . , S , ) E ( ~ " ) ~ c o n s t r u c t e d i n ( 9 9 ) , Chapter 4, has t h e proper= tY
<sa
(32)
ua =
(33)
wB = TB(D)s
,a>
-
, V 1< a < u(fB)
a
v",
E
V 1< B
<
b
We s h a l l denote 1 , = the ideal i n
(34)
(c"(R)) N generated b y I D P w B l 1 G B
Q
b,p
E
F i n a l l y , we denote b y JDS
t h e s e t o f a l l v a n i s h i n g i d e a l s I i n ( c " ( R ) ) ~ , such t h a t (35)
7,
(36)
(v" + r ) n
c 1
+
(uCm((,)
1
R.S) =
2
I n view o f t h e p r o o f o f Theorem 13, S e c t i o n 6, Chapter 4, i t i s easy t o see t h a t
JD, # 4 Theorem 3 Under t h e above c o n d i t i o n s , t h e f o l l o w i n g p r o p e r t i e s h o l d :
1)
The s e t o f i d e a l s JD, i s c h a i n complete.
2)
Each i d e a l 7
3)
F o r each i d e a l 7
(37)
(rn
E
JD, E
i s Cm JD,
, R-regular,
1
R = R s.
there e x i s t Cm-regularizations
7 , SOT)
satisfying the conditions
where
N n}
E.E.
172
Rosinger
(38) (39)
(40) (41) 4)
Each C m - r e g u l a r i z a t i o n ( V , S ' ) i n (37) y i e l d s t h e f o l l o w i n g r e s o l u = t i o n o f s i n g u l a r i t i e s f o r t h e weak s o l u t i o n u o f t h e system o f r e = s o l u b l e PDEs i n ( 8 9 ) , Chapter 4:
4.1)
R R uU = sa + I ( L S ' ) E A ( V , S l ) , 11 1 S a w i t h 1 S a Q a, do __ n o t depend on R .
4.2)
u s a t i s f i e s each o f t h e PDEs o f t h e system ( 8 9 ) , Chapter 4, i n t h e usual a l g e b r a i c sense, w i t h t h e m u l t i p l i c a t i o n i n Ak(V,Sl) and t h e p a r t i a l d e r i v a t i v e operators
Dp : AR(V,Sl)
+
Ak(V,Sl),
p
E
Q
a,
R E
fl, where sa
Nny I p I S m y w i t h R , k
E
E S',
fly a-k
2 m
Proof S i m i l a r t o t h e p r o o f o f Theorem 3, S e c t i o n 5. Obviously, a corresponding v e r s i o n o f Remark 2, S e c t i o n 5, w i l l a p p l y t o Theorem 3 above.
0
CHAPTER 6 CHARACTERIZATION OF THE NECESSARY STRUCTURE OF THE ALGEBRAS CONTAINING THE DISTRIBUTIONS
0.
Introduction
The previous chapters presented the way, a s well a s several aspects con= cerning the u t i l i t y of constructing chains o f q u o t i e n t algebras (24) and ( 9 3 ) , Chapter 3: (1)
A'(V,S)
= AR(V,S)/lR(V,S),R E
R,
corresponding t o Coo-regularizations, respectively C' - r e g u l a r i z a t i o n s ( V , S ) . The aim of this chapter i s t o present several basic r e s u l t s concerning the e x t e n t t o which the way the chains o f q u o t i e n t algebras (1) were construe= ted corresponds t o a necessity.
Actually, the r e s u l t s obtained i n t h i s r e s p e c t w i l l amount t o a c h a r a c t e r i = zation of t h e necessary s t r u c t u r e t h e algebras containing the d i s t r i b u t i o n s h a v e 0 possess. In s h o r t , t h a t c h a r a c t e r i z a t i o n says t h a t the i d e a l s (2)
IR(V,S), R
E
w,
corresponding t o the q u o t i e n t algebras ( l ) , have t o be c o n s t i t u t e d from sequences of functions w E (C'( R ) ) N which vanish densely on R , t h a t i s , the s e t s
RW
= {X E
R 3 v
E N :
w,(x) = 03
a r e dense i n R . A f i r s t , very simple f a c t i n t h i s r e s p e c t was m e n t i n d i n Remark 1, Sec= t i o n 1, Chapter 2, concerning the necessary s t r u c t u r e of the inclusion , Chapter 3, upon which diagrams ( 9 ) and therefore ( 1 5 ) , ( 2 3 ) a s well a s the chains of q u o t i e n t algebras (1) a r e constructed.
(v
However, a s known i n the theory o f algebras of continuous functions, the s t r u c t u r e of q u o t i e n t algebras of type ( l ) , depends c r i t i c a l l y on the s t r u c t u r e of the corresponding i d e a l s ( 2 ) . In t h i s connection, r e s u l t s giving a c h a r a c t e r i z a t i o n o f the necessary s t r u c t u r e of the i d e a l s ( 2 ) will be presented i n this Chapter. As a l s o known, a good i n s i g h t i n t o the s t r u c t u r e o f the i d e a l s ( 2 ) can be obtained from the s t r u c t u r e of the maximal i d e a l s in t h e algebras (3)
AR(V,S) c
C' ( N
x
R), RE
173
174
E.E.
Rosinger
That l a t t e r problem i s n o t a simple one, s i n c e t h e i n c l u s i o n s i n ( 3 ) a r e s t r i c t , t h e r e f o r e t h e Stone-Cech c o m p a c t i f i c a t i o n t h e o r y c h a r a c t e r i z i n g the maximal i d e a l s by v a n i s h i n g c o n d i t i o n s a t c e r t a i n p o i n t s , cannot be used i n a d i r e c t way. Nevertheless, i t w i l l be shown i n t h e p r e s e n t c h a p t e r t h a t f o r a l a r g e c l a s s o f chains of q u o t i e n t a l g e b r a s ( l ) , the c o r r e s p o n d i n g i d e a l s ( 2 ) a r e c h a r a c t e r i z e d by c e r t a i n v a n i s h i n g condi t i o n s . Such v a n i s h i n g c o n d i t i o n s have a l r e a d y been encountered i n t h e p r e v i o u s chapters, f o r instance, i n Sec i o n s 7 and 8, Chapter 1, connected w i t h t h e q u o t i e n t a l g e b r a And = (c" (Q))$/Ind, where t h e i d e a l Ind s a t i s f i e s the v a n i s h i n g c o n d i t i o n (65), Chapter 1. Other t y p e s o f v a n i s h i n g c o n d i t i o n s appeared connected w i t h t h e necessary and/or s u f f i c i e n t c o n d i t i o n s on t h e e x i s t e n c e o f s e q u e n t i a l s o l u t i o n s f o r polynomial n o n l i n e a r PDEs, p r e s e n t e d i n Chapter 2 , where t h e s o c a l l e d r e s o l v e n t s e t s (unions o f i d e a l s , see S e c t i o n 4 ) were r e l a t e d t o t h e v a n i s h i n g c o n d i t i o n s (33) and (49) (see C o r o l l a r y 2, Remark 1 and Theorem 5 ) . The v a n i s h i n g c o n d i t i g n (39), Chap= t e r 3, p l a y e d an i m p o r t a n t r o l e i n t h e c h a r a c t e r i z a t i o n o f C and C'-re= g u l a r i z a t i o n s (see Theorems 5 and 11, Chapter 3 ) . I t i s i m p o r t a n t t o n o t i c e t h a t from t h e p o i n t o f view o f t h e exactness p r o = p e r t i e s o f t h e s e q u e n t i a l , weak o r d i s t r i b u t i o n s o l u t i o n s o f polynomial n o n l i n e a r PDEs (see ( 1 ) , ( 3 ) , Chapter 1)
(4 1
T(D)u(x) = f ( x ) , x E R,
t h e v a n i s h i n g c o n d i t i o n s s a t i s f i e d by t h e i d e a l s ( 2 ) mean corresponding v a n i s h i n g c o n d i t i o n s s a t i s f i e d by t h e e r r o r sequences w = T(D)s
- U(f)
where s a r e t h e r e s p e c t i v e s e q u e n t i a l s o l u t i o n s . Obviously, whenever t h e e r r o r sequence w vanishes, t h e corresponding se= q u e n t i a l s o l u t i o n s s a t i s f i e s t h e PDE i n ( 4 ) , i n an e x a c t sense, t h a t b e i n g t h e s p e c i f i c way t h e method p r e s e n t e d i n t h i s work d e a l s w i t h t h e problem o f t h e r e g u l a r i t y o f s e q u e n t i a l s o l u t i o n s . Therefore, s t r o n g e r v a n i s h i n g c o n d i t i o n s mean b e t t e r exactness p r o p e r t i e s . I n t h i s r e s p e c t , t h e f a c t t h a t the i d e a l s ( 2 ) w i l l necessarily s a t i s f y c e r t a i n vanishing conditions means t h e necessary presence o f a c e r t a i n minimal l e v e l o f exactness p r o p e r = t i e s f o r t h e s e q u e n t i a l , weak o r d i s t r i b u t m u t i o n s o f p o l y n o m i a l non= E a r PDEs (see S e c t i o n s 4 and 5 , Chapter 1, as w e l l as Chapter 5 ) F i n a l l y , i t i s i m p o r t a n t t o mention t h a t t h e maximal i d e a l s o f t y p e ( 2 ) which h e l p i n o b t a i n i n g t h e c h a r a c t e r i z a t i o n o f t h e necessary s t r u c t u r e o f t h e i d e a l s ( 2 ) , a r e d i r e c t l y connected w i t h t h e maximal s t a b i l i t y o f se= q u e n t i a l , weak o r d i s t r i b u t i o n s o l u t i o n s o f polynomial n o n l i n e a r PDEs (see Sections 1,2 and 5 , Chapter 1, as w e l l as Chapter 5 ) .
1. Zero F i l t e r s As known i n t h e t h e o r y o f algebras o f continuous f u n c t i o n s , an e s s e n t i a l f e a t u r e o f t h e i d e a l s I i n C ( X ) , where X i s a c o m p l e t e l y r e g u l a r t o p o l o g i = c a l space, i s t h e i r c o n n e c t i o n w i t h z e r o - f i 1t e r s , c a l l e d s h o r t l y z - f i 1t e r s , generated by subsets i n X, on which the f u n c t i o n s i n I vanish.
STRUCTURE OF THE ALGEBRAS
175
In s p i t e of the more general framework ( 3 ) which corresponds t o the chains of q u o t i e n t algebras ( l ) , i t w i l l s t i l l be possible t o e s t a b l i s h a connec= t i o n between the i d e a l s ( 2 ) a n d c e r t a i n z - f i l t e r s on N X R, o r N . These z - f i l t e r s will a c t u a l l y express the vanishing conditions mentioned i n Section 0 . For a sequence of functions w
Z(W)
E(C
= {(v,x) E
"()) N denote
N X R 1 w,(x)
=
01
and c a l l i t the zero s e t of w .
For a s e t of sequences of functions H Z(H) =
{Z(W)
I
w
E
C
(C" ( R ) ) N denote
HI
and c a l l i t the zero familv of H .
A usual argument in the theory of algebras of continuous functions w i l l give t h e following basic r e s u l t concerning the vanishing conditions s a t i s = f i e d by a l l the c" -regular i d e a l s I and t h e r e f o r e , in view of the equiva= lence e s t a b l i s h e d in Theorem 10, Section 5 , Chapter 3, by a l l vector sub= spaces V which a r e components of c" - r e g u l a r i z a t i o n s ( V , S ) . Recalling the ' way given in (85-89), Chapter 3, the i d e a l s ( 2 ) were constructed i n the case of chains of q u o t i e n t algebras ( 9 3 ) , Chapter 3, one f i n a l l y obtains a vanishing condition s a t i s f i e d by a l l these i d e a l s . As customary w i t h i n t h e framework of the theory of algebras of continuous functions, we s h a l l use a s l i g h t l y modified notion of f i l t e r , defined in t h i s case as follows. Given a topological space X, a s e t C of closed subsets 0 C X i s c a l l e d a z - f i l t e r on X, only i f (5)
0 4 C , X E C
(6)
uo n
... n uh E
C , V h E N , 50 ,...,a h E C
In t h a t case, any non-void subset C'
C
C i s c a l l e d a z - f i l t e r generator on
X.
Theorem 1 Suppose 7 i s a C o - r e g u l a r i d e a l . Moreover, i f (V,S @ 7) t o Theorem 10, Section 5, N X n. F i n a l l y , the zero 19,(V,S @ T ) ,
with R E
Then, Z ( 1 ) i s a z - f i l t e r on N X R . t o I according
R,
Proof
First, we prove the r e l a t i o n (7)
Z(w) # 0
Y
tf W E 1
Indeed, assume i t i s f a l s e . w,(x)
# 0, tl v
Then E
N, x E Q
176
therefore,
E.E. Rosinger
one o b t a i n s a sequence of f u n c t i o n s w ' Y v E N, x
w ~ ( x )= l/wv(x),
E
(C" (R))
E
N by d e f i n i n g
R.
I t f o l l o w s t h a t u ( 1 ) = w.w' E I.(C" ( R ) ) N C 7 , t h e r e f o r e 7 = (C" (R))N, con= t r a d i c t i n g t h e v e r s i o n o f C o r o l l a r y 1, S e c t i o n 4, Chapter 3, corresponding t o C"-regular ideals.
Now we s h a l l prove t h e r e l a t i o n (8)
z(wh) #
z(w1)
0, Y W1y..'yWh
7
Indeed, d e f i n i n g
w = w 1.w 1 t one o b t a i n s w E 7 and (9) Therefore,
... t Wh.Wh ,.. n z(w,)
Z ( W ) = z(w,)n
(8) f o l l o w s from ( 7 ) and ( 9 ) .
The r e l a t i o n s (7-9) o b v i o u s l y mean t h a t Z(7) i s a z - f i l t e r on N X R.
OT)
Assume now t h a t ( V , S i s a C" - r e g u l a r i z a t i o n correspond ng t o 7 according t o (101-105), Chapter 3. Then, V C 7 , t h e r e f o r e Z V ) c Z ( 7 i which means t h a t Z(V) i s a z - f i l t e r generator on N X R. F i n a l l y , n o t i c i n g t h a t i n view o f (85-89), l'(V,S
@T)
c
I, Y R E
Chapter 3
fly
the r e l a t i o n s w i l l f o l l o w Z(l'(V,S
BT))
C
Z ( I ) , V R E fl
and the p r o o f i s completed.
2. D e n s i t y C h a r a c t e r i z a t i o n o f Z - F i l t e r s As mentioned i n S e c t i o n 0, i n t h e remark concerning t h e exactness proper= t i e s o f s e q u e n t i a l s o l u t i o n s f o r polynomial n o n l i n e a r PDks, t h e p r o p e r t y m e i d e a l s ( 2 ) corresponding t o C ' - r e g u l a r i z a t i o n s ( V , S') presented i n Theorem 1 i n t h e p r e v i o u s Section, namely t h a t z(I'(v,s~))
i s a z - f i l t e r on N
x R,
for
R E
8,
can become an i n t e r e s t i n g v a n i s h i n g c o n d i t i o n , o n l y if t h e s i z e o f t h e sub= s e t s i n N X R which b e l o n g t o these z - f i l t e r s proves t o be s u f f i c i e n t l y I n t h i s connection, a f i r s t improvement o f t h e r e s u l t s i n Theorem s i t i o n 1, i s presented now by e s t a b l i s h i n g i n Theorem 3 t h e necessar and s u f f i c i e n t s i z e o f t h e p r o j e c t i o n on R o f t h e s e t s from t h e d 7 7 7 ) corresponding t o a (?-regular i d e a l 7
.
A f i l t e r g e n e r a t o r C on N X R i s c a l l e d dense i n R, o n l y i f f o r each u E C : the s e t
177
STRUCTURE OF THE ALGEBRAS
CX
Ru =
R
E
I
3 v E N : (v,x)
E
01
i s dense i n R. I n other words, the f i l t e r generator Z on N X R i s dense i n R, only i f i t s projection on R p r c = Cpr u I u E C) R i s i s f i l t e r generator of dense subsets i n R .
n
Theorem 2 Under the conditions i n Theorem 1, Section 1, the z - f i l t e r s Z ( 1 ) and w i t h II E \, as well a s the z - f i l t e r generator Z ( V ) a r e dense
Z(IR(V,S')),
in
R.
Proof I n view of the inclusions
obtained in the proof of Theorem 1, Section 1, obviously i t s u f f i c e s t o show t h a t Z ( 7 ) i s dense i n R . In this respect, we notice t h a t according to the version of Corollary 1, Section 4, Chapter 3 , corresponding t o c " regular i d e a l s , the relation holds
Assume now t h a t Z(I) i s not dense i n R . set
I
Rw = C X E R
3~
E N :
Then, for a certain w
w,(x)
E
I , the
= 01
i s not dense i n R . Assume therefore R' C R , non-void, open, such t h a t R' n Rw = 0 and take IJJ E C o ( R ) , with supp IJJ C R'. I t follows t h a t one obtains a sequence of functions w'E ( C o ( R ) ) N by defining
wV' ( x )
0
i f x
E n \
$(x)/wv(x)
if x
E
R'
=
a'
Then, obviously u($)
=
w.w'
E
l.(CO(R))N
c I
which i n view of ( l o ) , will give + ( x ) = 0, tl x
contradicting the f a c t t h a t
)I
E
R
was chosen a r b i t r a r i l y .
And now, the basic characterization of the ideals (2), characterization which constitutes an essential feature of the method f o r dealing w i t h se=
0
178
E.E.
Rosinger
q u e n t i a l , weak o r d i s t r i b u t i o n s o l u t i o n s o f polynomial n o n l i n e a r PDEs p r e = sented i n t h i s work, and which s t a t e s t h a t t h e densely v a n i s h i n g c o n d i t i o n obtained i n Theorem 2 above, i s n o t o n l y necessary b u t i t i s a l s o s u f f i = c i e n t . As b e f o r e , i n t h e c h a r a c t e r i z a t i o n o f C"-regular and C" - r e Z T E i d e a l s i n Theorems 5 and 11, Chapter 3, i t i s convenient t o s l i g h t l y r e = s t r i c t t h e framework w i t h t h e h e l p o f t h e t o p o l o g i c a l type c o n d i t i o n o f c o f i n a l in v a r i ance
.
Theorem 3 N Suppose 7 i s a c o f i n a l i n v a r i a n t i d e a l i n (C" (Q)) i d e a l , o n l y i f t h e z - f i l t e r Z ( 7 ) i s dense i n R.
.
Then 7 i s a C" - r e g u l a r
Proof -
.
Then,
and w
E
7.
3 v
E
Assume Z ( 7 ) i s dense i n R
we
s h a l l prove t h a t t h e r e l a t i o n holds
Indeed, assume (12) for a c e r t a i n (13)
u($) = w
+
E
P (R)
Rw =
IX E R
I
But
N : w,(x)
i s dense i n R, i n view o f t h e hypothesis. (13) w i l l g i v e + ( x ) = 0, Y x
E
v
Then, t h e r e l a t i o n s (12) and
Rw
which i n view o f t h e c o n t i n u i t y o f $ ( x ) = 0,
= 0)
+ obviously
implies
x E R
thus, ending t h e p r o o f o f ( 1 1 ) . Now, a c c o r d i n g t o Theorem 11, S e c t i o n 5, Chapter 3, i t f o l l o w s t h a t 7 i s indeed a P - r e g u l a r i d e a l . The converse r e s u l t s f r o m Theorem 2, above.
0
3. Large C"-Regular I d e a l s
From t h e p o i n t o f view o f t h e s t a b i l i t y o f s e q u e n t i a l s o l u t i o n s t h e f o l l o w = i n g q u e s t i o n a r i s e s . Given t h e f a c t i n Theorem 3 o f t h e p r e v i o u s s e c t i o n t h a t t h e P - r e g u l a r i d e a l s I generate z - f i l t e r s Z ( 2 ) which a r e dense i n R, a p r o p e r t y p a r t i c u l a r l y convenient from exactness p o i n t o f view, what a r e n e v e r t h e l e s s t h e p o s s i b l e s i z e s o f t h e maximal C - r e g u l a r i d e a l s ? That q u e s t i o n i s p a r t i c u l a r l y a c t u a l , s i n c e t h e c" - r e g u l a r i d e a l encoun= t e r e d a l m o s t e x c l u s i v e l y so f a r was t h e nowhere dense i d e a l Ind which sa= t i s f i e s the following strong density condition
STRUCTURE OF THE ALGEBRAS
179
Z(lnd) :
(14)
Y
(14.1)
R, dense in R
(14.2)
R \ R, nowhere dense in R
0 E
a condition which strongly t i p s the conflict between exactness and s t a b i l i = ty in the favour of the former one. The aim of the present section i s t o construct families of P - r e g u l a r ideals 1 which in view of Theorem 2 , Section 2. will automaticallv satisfv the corresponding version of (1411) b u t wili as much as possibie f a i l io s a t i s = fy (14.2). A f i r s t family of such CLregular ideals will be constructed as follows. Suppose given a dense sequence of points in R
5
=
t1,..., 5,,
(Coy
.....
) y
with 5,
E
R, for v
E
N,
and denote 8
5
=
{B
n
{Cpy
.-... I
5,+1>
'7
B c . n , R\ B nowhere dense in R
Then obviously 8, i s a f i l t e r base of dense subsets in R . We denote now the set-of'all sequences o f funcfions w E (P(n))N which s a t i s f y by 3 F
E
BE:
V x E F : 3
P E N :
YvEN,
v>p:
wv(x) = 0 1 will be a cofinal and subsequence invariant
I t i s easy to see that ideal in (c" (Q))N.
5
Let us also denote by J the s e t of a l l sequences of functions w satisfying the condi t i oh
E
N (c" (n))
3 p E N :
(16)
YvEN,v>u:
wv(EJ
=
... = W V ( C V )
=
0
I t i s obvious t h a t J will also be a cofinal invariant ideal in (C' (n))N 5 moreover (17)
'nd
U J
5
cr
5
E.E. Rosinger
180 Proposition 1 The i d e a l s J
5
and I a r e c" - r e g u l a r and
5
Proof Indeed, i n view o f Theorem 11, Sec= We s h a l l show t h a t 7 i s ? - r e g u l a r . t i o n 5, Chapter 3, i! s u f f i c e s t o prove t h a t
uc" (n) = Q
'5
(18)
since I i s c o f i n a l invariant.
5
Assume t h e r e f o r e t h a t u ( $ ) that
E
I5 f o r a c e r t a i n $ E ? (n). Then (15) i m p l i e s
$=OonF for a certain F yield
E
B5.
B u t F i s dense i n R, hence t h e c o n t i n u i t y o f $ w i l l
$=OonQ and t h e p r o o f o f (18) i s completed. The f a c t t h a t J i s a l s o ? - r e g u l a r 5 i n (17).
r e s u l t now f r o m t h e i n c l u s i o n J c 7
5
5
I n o r d e r t o end t h e p r o o f o f P r o p o s i t i o n 1, i t s u f f i c e s t o show t h a t J5 \ Znd
# @
and r e c a l l ( 1 7 ) . Assume t h e r e f o r e $v E c"
and d e f i n e w
E
u x
En:
$,(X)
= 0
(C'
(a),w i t h *x
=
v
E
N, such t h a t
5V
by
$v ( x ) , Y v E N, x E OGU Gv t h e n i t i s easy t o see t h a t w E J \ 7 5 nd' wV(x) =
n 0
The c " - r e g u l a r i d e a l s I a r e p a r t i c u l a r case o f t h e C " - r e g u l a r i d e a l s i n = 5 troduced now. the Suppose F and U a r e f i l t e r s on Q , r e s p e c t i v e l y N and denote by IFYu s e t o f a l l sequences o f f u n c t i o n s w
E
(c" (n))Nwhich s a t i s f y t h e c o n d i t i o n
STRUCTURE OF THE ALGEBRAS
181
3 F EF: V x E F :
3 M
A!:
E
V v E M :
w,(x)
= 0
Then o b v i o u s l y I F is an i d e a l i n ( C " ( i 2 ) ) . Moreover, i n case F ' and b4' a r e f i l t e r s on R, r e s p e c t i v e l y N, t h e f o l l o w i n g i m p l i c a t i o n r e s u l t s e a s i l y
F
C
F ' , hl
C
M ' * 'F,hl
'F:M'
Prooosi t i o n 2
lFYM i s a cofinal i n v a r i a n t ideal, provided t h a t
(21)
hif c M
where Mf i s t h e F r e c h e t f i l t e r on N o f a l l subsets i n N w i t h f i n i t e comple= men t a r y
.
I f f F Y b i, s a c o f i n a l i n v a r i a n t
i d e a l t h e n i t w i l l a l s o be a C " - r e g u l a r
i d e a l , o n l y i f F i s a f i l t e r o f dense subsets i n R. Proof Assume w
E
and w ' E(Co (R))N such t h a t I F ,M W1 ~- - W , , ~ / V E N , V > ~ ~
for a c e r t a i n
E
N.
But
1 F E F: Y
F : 3
X E
ME
M: Y v E M: w,(x)
= 0
therefore
V x
E
F : 3 M
O b v i o u s l y M n { p , p t l ,...,I t h a t W ' E lFYM. I n case I
E
E
M: V v
E
M, s i n c e
M n {u,p+1 ,....I: Mf
C
M.
w;(x)
= 0
I n t h a t way i t f o l l o w s
i s c o f i n a l i n v a r i a n t , t h e n i n view o f Theorem 11, S e c t i o n 5,
F ,M Chapter 3, IF,M w i l l be C'-regular,
only i f
I n case F i s a f i l t e r o f dense subsets i n Q, t h e r e l a t i o n (22) f o l l o w s e a s i l y , i n a way s i m i l a r t o t h e p r o o f o f (18). Conversely, assume (22) h o l d s and e x i s t 9 E C' (Q) such t h a t
F
E
F i s n o t dense i n Q.
Then t h e r e
182
E.E.
Rosinger
(23)
$=OonF
(24)
3 x E R \ F:$(x)#O
But, i n view o f (19), t h e c o n d i t i o n (23) i m p l i e s t h a t w = ~ ( $ 1 ~ F , Mn
uco
(n) n
t h e r e f o r e (24) w i l l c o n t r a d i c t (22). ProDosi ti on 3 i s a subsequence i n v a r i a n t i d e a l .
'F,Mf 'FYMf
i s a C " - r e g u l a r i d e a l , o n l y i f F i s a f i l t e r o f dense subsets i n R.
Proof Assume w Then
E
7
F,Mf
and w ' E (c" ( R ) ) N such t h a t w ' i s a subsequence i n w.
w ' = (wuoy
WulY..
.
Y W
UV
,* . . .)
But 3 F E F : V x E F : j X E
N : V V E N , ~ ~ X : W ~ ( X ) = O
and V A
E
N :3 p E
N : U v E N, v 2 p : 1-1, 2
X
therefore w ' E The second p a r t f o l l o w s from P r o p o s i t i o n 2. The f a c t t h a t the i d e a l s 7
F, M
0
generalize the i d e a l s 7
by n o t i c i n g t h a t
5
can be seen e a s i l y
where F denotes t h e f i l t e r on R generated by 8
5
5'
The monotonici ty p r o p e r t y (20) o f f e r s t h e p o s s i b i l i t y o f c o n s t r u c t i n g l a r g e (?-regular i d e a l s 7 by choosing l a r g e f i l t e r s F and My p r o v i d e d they F, M s a t i s f y t h e c o n d i t i o n s i n P r o p o s i t i o n s 2 o r 3. 4.
F u r t h e r P r o p e r t i e s o f Z-Fi 1t e r s
Another d i r e c t i o n t h e r e s u l t s i n Theorem 1, S e c t i o n 1, can be improved i s presented i n t h i s s e c t i o n .
STRUCTURE OF THE ALGEBRAS A (?-regularization (26)
V
183
( V , S ) i s c a l l e d o f l o c a l type, o n l y i f
n'
c R non-void,
3 s
open :
V@S:
E
# 0
(26.1)
<
(26.2)
3 p E N :
S,*>
D'(n)
E
Vv€N,v>u:
supp sv c 0' Theorem 4 Suppose t h e ( ? - r e g u l a r i z a t i o n i n v a r i a n t . Then
( V , S ) i s o f l o c a l t y p e and V @S
......3 X
Z ( v ) n (Cp,p+l, f o r any v
E
U , 1-1
i s cofinal
n') i s i n f i n i t e
N and R' c R non-void, open.
E
Proof Assume v
V, p
E
E
N and R' c R n o n - v o i d open, such t h a t
# 0, Y v
v,(x)
E
N,
v 2 p, x
E
R'
Obviously, we can a l s o assume t h a t (26.2) h o l d s f o r t h e above p and R ' . D e f i n e t h e n w E (C'(R))N by if x
0
9 R'
wV ( x ) = sv(x)/vv(x) whenever v
E
vV.wv =
(27)
>u . sV , V v
N, v
if x E
R'
It follows t h a t E
N, v 2 p
therefore (28)
v.w
s i n c e s E V @S But v (29)
E
E V
@
and V
@
s S i s cofinal invariant.
V w i l l imply v.w
E
l(V)
and (28), (29) above t o g e t h e r w i t h (21.1) i n Chapter 3, w i l l y i e l d v.w which i n view o f (27) w i l l c o n t r a d i c t (26.1).
E
V, 0
184
E.E.
Rosinger
Remark 1 The c o n d i t i o n s i n Theorem 4 a r e s a t i s f i e d i n t h e case o f i m p o r t a n t classes o f C"-regularizations. Indeed: a ) I t i s easy t o see t h a t U @ S w i l l be c o f i n a l i n v a r i a n t whenever U has t h a t p r o p e r t y . What concerns s e c u r i n q t h a t l a t t e r s i t u a t i o n , i t i s o b v i o u s l y s u f f i c i e n t t o c o n s i d e r c o f i n a l i n v a r i a n t C" - r e q u l a r i d e a l s I and use t h e procedure i n Theorem 10, S e c t i o n 5, Chapter 3, f o r construe= t i n q V which i n t h i s case can be assumed c o f i n a l i n v a r i a n t . b ) I n view of the c o n d i t i o n (90.2) i n t h e d e f i n i t i o n o f C " - r e q u l a r i z a t i o n s , i t fol lows t h a t
v x En: 3 s
ua s:
E
= 6,
<s,*>
(the Dirac 6 d i s t r i b u t i o n concentrated a t x)
t h e r e f o r e , t h e c o n d i t i o n ( 2 6 ) w i l l be s a t i s f i e d i n case
tfx
En:
I s E V @ S : * ) < s,*>
-
- &x
supp sv s h r i n k s t o {XI, when v
**)
-+ m
However, t h e c o n d i t i o n s i n Theorem 4 can be relaxed, as seen i n t h e n e x t two theorems. Theorem 5 Suppose g i v e n a C " - r e g u l a r i z a t i o n (U,S) and U Then Z(V) n f o r any v
E
u
u
-
V, p
E
N and s
E
p E
N
and s
V@S
(Cvl X
V@
S,
+
S U ~ Dsv)
<s,.>
S i s cofinal invariant.
i s infinite
# 0
E
D'(n).
Proof Assume v
E
V,
E
, w i t h a,.> #
tlv€N,v>p: vv # 0 on supp sv D e f i n e then w E (C" ( Q ) ) N by
w (x) =
/o
sv(x)/vv(x)
ifx $ if x
E
supp sv supp sv
0
E
D ' ( n ) , such t h a t
STRUCTURE OF THE ALGEBRAS
185
whenever v E N, v > p . Now a r e a s o n i n g s i m i l a r t o t h e one i n t h e second 0 p a r t o f t h e p r o o f o f Theorem 4, w i l l l e a d t o a c o n t r a d i c t i o n . I n t h e same way one can a l s o prove: Theorem 6 Then
Suppose ( V , S ) i s a C " - r e q u l a r i z a t i o n .
u
Z(v) n
({VI
vEN f o r any v
5.
E
V and s E V @
x
suop s v ) P 0
D'(n).
S,< s,*> # 0 E
C o n s t r u c t i o n and C h a r a c t e r i z a t i o n o f a Class o f V a n i s h i n q I d e a l s
The s t a n d a r d r e s u l t on z - f i l t e r s a s s o c i a t e d t o C" - r e g u l a r i z a t i o n s p r e s e n t e d i n Theorem 1, S e c t i o n 1, has been improved i n two d i r e c t i o n s : f i r s t , i n S e c t i o n 2 , by t h e d e n s i t y c h a r a c t e r i z a t i o n o f t h e p r o j e c t i o n on R o f t h e z - f i l t e r s , and then i n S e c t i o n 4, where l o w e r bounds on t h e s i z e o f t h e elements i n t h e z - f i l t e r s have been g i v e n i n Theorems 4,5 and 6. The aim o f t h i s s e c t i o n i s t o p r e s e n t a c o n s t r u c t i o n and c h a r a c t e r i z a t i o n o f t h e c l a s s o f E-vanishing i d e a l s i n ( C " ( I 2 ) ) N d e f i n e d below. The t o o l used i n t h i s c o n n e c t i o n i s a p r o p e r t y o f reduced p r o d u c t s o f c o u n t a b l e f a = m i l i e s o f v e c t o r spaces, p r e s e n t e d i n t h e n e x t s e c t i o n . The i n t e r e s t o f t h e men i o n e d c o n s t r u c t i o n i s t h a t i t s t a r t s f r o m v e c t o r subs aces i n s a t i s f y i n g the E-vanishing condition ( s e e T 3 e w i t h (C" (39) ,Chapter 3 ) . Given a sequence o f f u n c t i o n s w E ( C " ( R ) ) ZX(w) = Cv E
N Iw,(x)
N and x
E
R, denote
= 01
and c a l l i t t h e zero s e t a t x o f w. Obviously Zx(w) = p r (Z(w) "(N N
x(x1))
and p r Z(w) =
n
Cx
E
R IZx(w) # 81
Given a s e t o f sequences o f f u n c t i o n s H Zx(H) =
CZ,(w)I
and c a l l i t t h e zero f a m i l y a t x
w
E
HI
o f H.
I f E i s a s e t o f subsets i n R, denote
C
( e ( a ) )N
denote
E.E.
186
Rosinger
V E E
WE =
{w E (CO (R))
E:
3 x E E : I
Examples o f h e r e d i t a r y s e t s o f subsets i n R are t h e s e t Ef o f non-void and f i n i t e subsets i n R , as w e l l as t h e s e t E, o f non-void and c o u n t a b l e subsets i n fi
.
The main n o t i o n i n t h i s s e c t i o n i s presented i n t h e f o l l o w i n g d e f i n i t i o n . A v e c t o r subspace V i n ( C D ( R ) ) N i s c a l l e d E-vanishing, o n l y i f
The s e t E o f subsets i n R i s c a l l e d dense i n R
only i f
It i s easy t o see t h a t Ef and Ec a r e dense i n R, and i n general, E w i l l be dense i n R, o n l y i f
V R ' c R non-void, open : (32)
3 E E
E:
E c Rl The i m p o r t a n t p r o p e r t y connected w i t h E-vanishing, w i t h E dense i n R, i s presented now. Proposition 4
I f 7 i s a c o f i n a l i n v a r i a n t E-vanishing i d e a l and E i s dense i n n, then 7 i s a (?-regular i d e a l . Proof I t i s a d i r e c t consequence o f (30), Chapter 3.
(31) above and Theorem 11, S e c t i o n 5,
The c o n s t r u c t i o n o f E-vanishing i d e a l s i s based on t h e p r o p e r t y o f i s h i n g v e c t o r subspaces presented n e x t .
0
E-van=
Proposition 5 N Suppose V i s an E - v a n i s h i n g v e c t o r subspace i n ( e ( Q ) )f o r a c e r t a i n E C Ec. Then, f o r any vl,.. . ,vh E V , t h e r e l a t i o n h o l d s
STRUCTURE OF THE ALGEBRAS
V E
(33)
187
E E :
3 x E E :
...
zx(vl)
zx(vh) # 0
I f E i s h e r e d i t a r y then t h e s t r o n g e r r e l a t i o n h o l d s
(3
1
V E
EE:
3 F
CE:
*) car F **) V
X E
=
car E
F
:
Proof The r e l a t i o n (33) f o l l o w s e a s i l y from C o r o l l a r y 1 i n S e c t i o n 6 . Assume now t h a t E i s h e r e d i a t r y and t a k e E
Then ( 3 3 ) i m p l i e s t h a t
E
E.
F
= E = ixl)
3 x 1 E E :
.
Denote then El = E \ I x l )
I f El = 0 then
completed. Otherwise El E E , s i n c e E i s h e r e d i t a r y . be a p p l i e d t o El and i t f o l l o w s t h a t 3 x 2 E El
Denoting E2 = E l \Cx21 t a k e F = E = Cx1,x21 repeated.
.
T h e r e f o r e (33) can
:
,in
case E2 = 0 t h e p r o o f i s completed as one can Otherwise a g a i n E 2 E E and t h e procedure can be
I n case E i s f i n i t e , one w i l l end up w i t h F = E .
F
and t h e p r o o f i s
Otherwise one o b t a i n s
= { x ~ ~ x ~ 1~ C, . E .
which w i l l a l s o s a t i s f y * ) i n (34), s i n c e E i s countable.
0
The r e s u l t i n P r o p o s i t i o n 5 l e a d s t o t h e f o l l o w i n g method f o r c o n s t r u c t i n g E - v a n i s h i n g , c" - r e g u l a r i d e a l s . Theorem 7
I f U i s an E - v a n i s h i n g vectorsrbspace i n (c" (Q))N, w i t h E (see (89.1),
C
Chapter 3) i s an E - v a n i s h i n g i d e a l i n ( c " ( Q ) ) N .
Ec, then I ( V )
188
E.E. Rosinger
I f i n ' a d d i t i o n V i s c o f i n a l i n v a r i a n t and E i s dense i n C" - r e g u l a r i d e a l .
n, t h e n
7(V) i s a
Proof
N N Obviously l ( V ) i s t h e v e c t o r subspace i n ( C " ( R ) ) generated b y V.(C"(n)) , t h e r e f o r e any w E I ( V ) can be w r i t t e n under t h e f o r m (35)
c
w =
1 G i
v. .z i i
(Co ( 0 ) )N .
where vi E V and zi€ But, i n view o f (33)
E:
Y E €
(36)
3 x E E : zx(vl)
zx(vh) #
.*'
0
w h i l e (35) w i l l obviousJy i m p l y t h a t (37)
Zy(w)
3
Zy(vl)
n
... n ZY ( vh ),
Y y E
n
Now, i t i s easy t o see t h a t (36) and (37) w i l l y i e l d
( The second p a r t o f t h e p r o o f f o l l o w s from P r o p o s i t i o n 4, n o t i c i n g t h a t I(V) i s c o f i n a l i n v a r i a n t whenever V has t h a t p r o p e r t y .
0
I n view o f P r o p o s i t i o n 4 and Theorem 7, t h e r e e x i s t s an i n t e r e s t i n c o f i n a l i n v a r i a n t and E-vanishing i d e a l s , w i t h E dense i n n, s i n c e they w i l l auto= m a t i c a l l y be c" - r e g u l a r i d e a l s . I n t h i s connection, a u s e f u l c h a r a c t e r i z a = t i o n w i l l be o b t a i n e d i n Theorems 11, 12 and 13 below, f o r a p a r t i c u l a r case o f t h e mentioned i d e a l s , d e f i n e d as f o l l o w s . Suppose g i v e n a subset E i n n. An i d e a l 7 i n (Co ( n ) ) N w i l l be c a l l e d E-vanishing, o n l y i f I i s EE-vanishing, where
IX
EE = {{XI E E l I n o t h e r words, Z i s E-vanishing, o n l y i f
v w
E ~ , X E E :
(38)
ZX(N # B Obviously EE i s dense i n
n, o n l y i f E i s dense i n n.
Theorem 8 Suppose 1 i s an E-vanishing i d e a l .
Then, f o r any x
E
E, Zx(I) i s a z - f i l t e r
STRUCTURE OF THE ALGEBRAS
189
on N. Proof I t f o l l o w s e a s i l y f r o m (38) and P r o p o s i t i o n 5.
Theorem 9 Suppose 1 i s a c o f i n a l i n v a r i a n t E - v a n i s h i n g i d e a l . Z x ( l ) i s a z - f i l t e r o f i n f i n i t e subsets i n N.
Then, f o r any x
E
E,
Proof Assume x
E
E, w1 ,...,w, n
Zx(w1)
(39)
.,w;
D e f i n e then wi,..
E
u
1 and
E
.. . n ZX(wh) N
( P(n))
E
N, such t h a t c {o,l,..
.ul
by t h e r e l a t i o n s
if v Gp
(40)
W;,(Y)
=
if v 2 p t 1
wiv(y) Obviously wi,,..,w;
E 1,
since 1 i s c o f i n a l i n v a r i a n t .
Therefore, i n view
o f Theorem 8, i t f o l l o w s t h a t Zx(wi) n
(41)
-.. n
ZX(w;)
# 0
B u t (40) i m p l i e s t h a t
z x ( w1! )
(42)
c zX(wi)
n {~ti,ut2 ,... . . . I , V 1 f i
h
Now, i t i s easy t o see t h a t (39) and (42) w i l l c o n t r a d i c t (41). We s h a l l c a l l a subset M i n N c o f i n a l , o n l y i f
M
3
f o r a c e r t a i n 1-1 E
{u,p+l,
......
N.
Theorem 10 Suppose 1 i s a subsequence i n v a r i a n t E - v a n i s h i n g i d e a l . x E E, Z x ( I ) i s a z - f i l t e r o f c o f i n a l subsets i n N.
Then, f o r any
Proof
I t s u f f i c e s t o show t h a t f o r any w f i n a l i n N. Assume w
E
I, x
E
E and
E
T and x
E
E, t h e s u b s e t Zx(w) i s co=
M an i n f i n i t e subset i n N, such t h a t
E .E. Rosi nger
190
Assume M =
(44)
{uo,ul,..
. , p v , . .. . . . I ,
W;=W
N and d e f i n e w 1 E (c"(n)) by t h e r e l a t i o n
V V E N
uv
Then w' E 7 , s i n c e 7 i s subsequence i n v a r i a n t . Theorem 8, i t f o l l o w s t h a t (45)
ZX(W') P
Therefore, i n view o f
0
But ( 4 4 ) imp1 es o b v i o u s l y t h e r e l a t i o n (46)
Z x ( w ' ) c Zx(w) n M
Now (45) and ( 4 6 ) w i l l c o n t r a d i c t ( 4 3 ) .
0
The p r o p e r t i e s o f E - v a n i s h i n g i d e a l s e s t a b l i s h e d i n Theorems 8,9 and 10 above a c t u a l l y c h a r a c t e r i z e those i d e a l s , as seen below i n Theorems 11,12 and 13. I n o r d e r t o o b t a i n t h e mentioned c h a r a c t e r i z a t i o n s several p r e l i m i n a r y con= s t r u c t i o n s are needed. Suppose g i v e n a f a m i l y M = (Mx
o f z - f i l t e r s on N.
IX E
E)
Denote by
'M t h e s e t o f a l l sequences o f f u n c t i o n s w E(C'(n))
N
such t h a t
Y x E E : (47)
ZX(W) E Mx I t i s easy t o see t h a t lM i s an E - v a n i s h i n g i d e a l which has t h e f o l l o w i n g properties
(48)
'M
'C"
(n)
=
Q i f E i s dense i n n
and V x E E :
(49) z~(7M)
Mx
Under t h e above c o n d i t i o n s , we s h a l l i n t r o d u c e t h e mapping i M-
'M
I n view o f Theorem 8, t h e converse mapping from E - v a n i s h i n g i d e a l s t o f a = m i l i e s o f z - f i l t e r s on N can a l s o be defined, as f o l l o w s . Suppose 7 i s an E -
STRUCTURE OF THE ALGEBRAS
vanishing ideal.
19 1
Then, denote b y
Zr = ( Z x ( T ) I x E E )
t h e f a m i l y o f r e s u l t i n g z - f i l t e r s on N. c o n v e n i e n t t o i n t r o d u c e t h e mapping 5
r
Under these c o n d i t i o n s i t w i l l be
>
The f o l l o w i n g i t e r a t i o n p r o p e r t i e s o f t h e mappings i and 5 w i l l be u s e f u l ProDosi t i on 6 Suppose 7 i s an E - v a n i s h i n g i d e a l .
Then
= ~(c(z))
(50)
1
(51)
Z I = ~(i(Z1))
Proof The r e l a t i o n (50) f o l l o w s e a s i l y . Connected w i t h (51) we n o t i c e t h a t by d e f i n i t i o n
ZI = ( Z x ( z )
IxE
E)
t h e r e f o r e , a l s o by d e f i n i t i o n , we have i(2,)
= {WE
N
(C"(n))
IV
x E E : Zx(w) E Z x ( 7 ) l
which w i l l o b v i o u s l y i m p l y ( 5 1 ) .
0
A t t h i s p o i n t , we can p r e s e n t a c h a r a c t e r i z a t i o n
o f the E-vanishing ideals.
Theorem 11 Suppose 'I i s an i d e a l i n (C" if
r
c
(n))N .
Then I i s an E - v a n i s h i n g i d e a l , o n l y
rM
f o r a c e r t a i n f a m i l y M = (Mx I x
E
E ) o f z - f i l t e r s on N.
Proof Assume 'I i s E-vanishing. Then, i n view o f Theorem 8 and (50) i n Proposi= t i o n 6, i t i s p o s s i b l e t o t a k e
M = The converse f o l l o w s f r o m (49).
0
A c h a r a c t e r i z a t i o n o f E - v a n i s h i n g c o f i n a l i n v a r i a n t i d e a l s i s now presen= ted.
E.E. Rosinger
192
Theorem 12 Suppose 7 i s an E-vanishing i d e a l . If
I i s cofinal i n v a r i a n t then M = (Zx(7) u M f \ x
E
E)
i s a f a m i l y o f f i l t e r generators on N, g e n e r a t i n g f i l t e r s o f i n f i n i t e sub= sets. Conversely, i f M has t h e above p r o p e r t y t h e n
-
I and
i(4)
i(M)
i s a c o f i n a l i n v a r i a n t E-vanishing i d e a l , where
fi and
C
=
(nXl x E
E)
fix i s t h e f i l t e r on N generated by Zx(7) u Mf.
Proof The f i r s t p a r t f o l l o w s from Theorem 9. We prove now t h e converse.
J
= i(fi)
Indeed, assume w E J and
i s cofinal invariant. W;=W
f o r a c e r t a i n 1~
But w
E
VY
E
w'E
(C' (Q))N,
such t h a t
YvEN,v>p,
Then o b v i o u s l y
N.
Zx(wl) n { p , p + l
(52)
F i r s t we show t h a t
,....I
= ZX(w) n ~p,p+l,,...I,
V x E E
J implies that Zx(w) E Z x ( I ) ,
V x
E
E,
t h e r e f o r e , i n view o f (52), we have Zx(w') i n other
words, w '
E
E
fix, 4 x
E
E
J.
Since t h e i n c l u s i o n 7 C J
f o l l o w s e a s i l y from (50) i n P r o p o s i t i o n 6, t h e p r o o f i s completed.
0
F i n a l l y , a c h a r a c t e r i z a t i o n o f E-vanishing subsequence i n v a r i a n t i d e a l s can be o b t a i n e d as f o l l o w s ,
STRUCTURE OF THE ALGEBRAS
19 3
Theorem 13 Suppose I i s an E - v a n i s h i n g i d e a l . I f I i s a subsequence i n v a r i a n t i d e a l , then
(53)
Zx(7)
Hf,
C
V x E E
Conversely, i f ( 5 3 ) holds t h e n
(54)
I c i(nf)
and i(h4) i s a subsequence i n v a r i a n t E - v a n i s h i n g i d e a l , where hi = ( h i x = h i f ] x E E)
Proof I f Z i s subsequence i n v a r i a n t then, i n view o f Theorem 10,ZI w i l l have t h e required property.
N F o r p r o v i n g t h e converse, assume w E i(hl) and w ' E (C" (Q)) i s a subsequence i n w, such t h a t W;=W
, V v E N liV
Then, i n view o f t h e h y p o t h e s i s V
X E
E :
3 v E N : { p , ptl,..
.. .
1 c
ix(w)
But, a c c o r d i n g t o t h e d e f j n i t i o n o f a subsequence, i t f o l l o w s t h a t Y u C N :
llv 2
The r e f o r e V
X E
E :
3 X E N : {A, X t l ,
....1
c Z,(Wl)
which means t h a t w ' E i ( M ) . F i n a l l y , t h e i n c l u s i o n (54) w i l l f o l l o w e a s i l y f r o m ( 5 3 ) .
194
E.E.
6.
Rosinger
Countable Reduced Products o f Vector Spaces
Suppose g i v e n a non-void f a m i l y (Xi
I
i E I)
o f n o n - t r i v i a l v e c t o r spaces on R’ t o g e t h e r w i t h a s e t 8 o f subsets i n t h e i n d e x s e t I . We denote then by YB t h e s e t o f a l l elements
which s a t i s f y t h e c o n d i t i o n 3 8 EB: (55)
B c B(x)
where
(55.1)
B(x) = Ii E I
xi = 0
E
Xi}
I t i s easy t o see t h a t YB
i s a p r o p e r v e c t o r subspace i n X, i . e . Y C X, B f o n l y i f B i s a f i l t e r base on I . r o d u c t of t h e f a m i l y ( X . 1 i E I ) o f v e c t o r spaces B i s t h e q u o t l e n t v e c t o r space.
As known, t h e reduced on R’ accordin-ase
’ ’ 8
xi
= X/Y,
We s h a l l c a l l a p r o p e r v e c t o r subspace Y i n X reduced subspace, o n l y i f Y C Y f o r a c e r t a i n f i l t e r base B on I . Then, t h e f o l l o w i n g r e s u l t can be prove1 e a s i l y . Lemma 1
A p r o p e r v e c t o r subspace Y i n X i s a reduced subspace, o n l y i f By = I B ( x )
IX E
Yl
i s a f i l t e r generator on I .
+
As known, t h e p r o e r i d e a l s i n a r b i t r a r y powers o f R ’ o r C’ a r e reduced subspaces. That a c t among others, e x p l a i n s t h e necessary r o l e p l a y e d i n Non-standard A n a l y s i s by reduced powers, i n p a r t i c u l a r u l t r a powers. I n t h i s connection, t h e q u e s t i o n a r i s e s whether t h e p r o p e r v e c t o r subspaces i n C a r t e s i a n products o f v e c t o r spaces on R’ a r e also reduced subspaces. An a f f i r m a t i v e answer w i l l be o b t a i n e d i n t h i s s e c t i o n under r a t h e r general c o n d i t i o n s i n t h e case o f countable C a r t e s i a n p r o d u c t s o f v e c t o r spaces on The c o u n t a b i l i t y c o n d i t i o n proves t o be e s s e n t i a l , as can be seen R’ i n a s i m p l e c o u n t e r example presented i n Remark 2.
.
F i r s t , we need t h e f o l l o w i n g d e f i n i t i o n .
A v e c t o r subspace Y i n X i s c a l l e d
STRUCTURE OF THE ALGEBRAS
195
vanishing, o n l y i f B(x) # 0
(57)
,U
X E
Y
t h e r e f o r e , a v a n i s h i n g v e c t o r subspace i s a p r o p e r v e c t o r subspace. Obviously, a reduced v e c t o r subspace i s a v a n i s h i n g v e c t o r subspace. I n t h e case o f c o u n t a b l e C a r t e s i a n p r o d u c t s , t h e converse o f t h e above p r o p e r = ty i s e s t a b l i s h e d i n : Theorem 14 Any v a n i s h i n g v e c t o r subspace i n a c o u n t a b l e C a r t e s i a n p r o d u c t o f v e c t o r spaces on R’ i s a reduced v e c t o r subspace. Proof Assume Y i s suffices t o e a s i l y from X + Xi
7:
a v a n i s h i n g v e c t o r subspace i n X. According t o Lemma 1, i t show t h a t B i s a f i l t e r g e n e r a t o r on I. T h a t p r o p e r t y f o l l o w s Lemma 3 belxw, by t a k i n g t h e r e T~ as t h e p r o j e c t i o n mapping and I = I11
Remark 2 I n case o f an uncountable C a r t e s i a n p r o d u c t o f v e c t o r spaces on R’ , t h e r e s u l t i n Theorem above does n o t n e c e s s a r i l y h o l d . Indeed, suppose I = R1 and X . = R ’, w i t h i E I. D e f i n e x = (xi[ i E I ) , y = (yiI i E I ) E X = 1
xi
= i, yi
= 1t i
( ~ / 2- a r c t g i),U i E I
Then o b v i o u s l y (58)
B(x)nB(y) = 0
Denote now by Y t h e v e c t o r subspace i n X generated by {x,y). Then i t can e a s i l y be seen t h a t Y i s a v a n i s h i n g v e c t o r subspace i n X. However, i n view o f (58), Y cannot be a reduced v e c t o r subspace. An a d d i t i o n a l c h a r a c t e r i z a t i o n connected w i t h reduced v e c t o r subspaces i s given in: Lemma 2 Suppose Y i s a p r o p e r v e c t o r subspace i n X. t e r base B on I,o n l y i f
Then Y = YB f o r a c e r t a i n f i l =
E . E . Rosinger
196 Proof
Assume Y s a t i s f i e s (59), then o b v i o u s l y Y = Y 1. Therefore By i s a f i l t e r base on I , s i n c e space i n X .
w i t h t h e n o t a t i o n s i n Lemma
BY
Y i s a p r o p e r v e c t o r sub=
The converse i s obvious.
0
The p r o o f o f Theorem 14 was based on t h e f o l l o w i n g lemma o f a r a t h e r gene= r a l interest. Suppose g i v e n a f a m i l y o f l i n e a r mappings : X
Ti
+
Xi,
for i E I
.
between a r b i t r a r y v e c t o r spaces on R '
Define then t h e mapping
I T ~ X=
B ( x ) = {iE I
X 3 x >-
0
E
Xi}
C
I
Suppose f u r t h e r t h a t I i s a s e t o f n o n - v o i d and countable subsets i n I and denote n B(x) #
= { x E X IJ
XI
0,
'J J E 11
3 -Lemma Suppose Y i s a v e c t o r subspace i n X and Y set
IX E
B Y y J = CJ n B ( x )
XI.
C
Then, f o r each J E I,t h e
Yl
i s a f i l t e r generator on J. Proof Assume t h e statement i n Lemma 3 i s f a l s e . xl,...,xh E Y such t h a t
(60)
J n B(xl)
n
... n B(xh)
=
Then t h e r e e x i s t J E I and
0
D e f i n e t h e mapping
R
h
3
X = (X l,..., Ah) + y y x =
XIXl
+ ...
t
Xh xh
E
Y
and t h e n t h e mapping
J Obviously hi,
(61)
3
i
+
hi = { X E R h
lTi
yX =
o E xi)
w i t h i E J , a r e v e c t o r subspaces i n R
c R~
, 'J
#
Indeed, (60) i m p l i e s t h a t
i
E
J
c R~
h . Moreover
STRUCTURE OF THE ALGEBRAS
197
b r i € J :
3 ki
E {ly...,h}
:
#
‘i ‘ki t h e r e f o r e by t a k i n g
xi
= (0,
...,0,1,0 ,... ,O) E
where t h e o n l y c o o r d i n a t e 1 i n
xi @
Ai,
xi
R
h
, for
i E 3,
i s i n t h e ki-th
position, i t follows t h a t
br i E J
Now t h e r e l a t i o n (61) and t h e B a i r e c a t e g o r y argument a p p l i e d t o R imp 1y t h a t
h
will
s i n c e J i s countable.
Assume then
It follows obviously t h a t 7i
yx
# 0, Y i E J
therefore
B u t yA
E
Y
c X I and thus ( 6 2 ) c o n t r a d i c t s the property defining Xl
n
As seen i n t h e previous s e c t i o n , a p a r t i c u l a r i n t e r e s t presents t h e follow= ing consequence of Lemma 3 above ( s e e [ 1761 , Lemma 1, Section 3 , Chapter 8) *
Given a non-void s e t X , denote by W the s e t of a l l the functions + R'. For w E W and x E X denote
w : NxX
ZX(w) = {v
N J W(U,X) = 01
E
F i n a l l y , f o r a s e t Y of non-void an countable subsets in X denote b r Y E Y t
wy
=
{ W E
w
3yEY: ZY(W) f
1 0
E . E . Rosinger
198 the r e l a t i o n holds V Y E Y :
(63)
3 yEY: '~('1)
' * *
zy(vh) #
0
Proof Take i n Lemma 3 above
x=w I = N x X
(64) Xi=
I =
R' { N x Y I Y E Y ]
where t h e n o t a t i o n s i n t h e r i g h t - h a n d t e r n o f t h e above e q u a l i t i e s a r e those i n C o r o l l a r y 1. F u r t h e r , d e f i n e t h e mappings T
i
= X
+
Xi,
for iE I,
i n Lemma 3, by TiX
= w(v,x)
where a c c o r d i n g t o t h e n o t a t i o n s i n (64) x E X corresponds t o w i E I corresponds t o (v,x) E N x X . Then t h e r e l a t i o n (63) w i l l f o l l o w e a s i l y .
E
W and 0
CHAPTER 7 QUANTUM SCATTERING IN POTENTIALS POSITIVE POWERS OF THE DIRAC 6 DISTRIBUTION
0.
Introduction
Recently, t h e r e has been an i n t e r e s t in quantum s c a t t e r i n g i n p o t e n t i a l s with strong local s i n g u l a r i t i e s [4,29,36,37,151,152,180,188, see a l s o 174, 1761 . The s t r o n g e s t local s i n g u l a r i t i e s of t h e p o t e n t i a l s considered a r e those o f measures which may f a i l t o be absolutely continuous with respect t o t h e Lebesque measure [ 361.
The p o t e n t i a l s t r e a t e d in t h i s chapter, given by a r b i t r a r y p o s i t i v e powers with CL E R' a n d 0 < m < m, of t h e Dirac 6 d i s t r i b u t i o n , present the s t r o n g e s t local s i n g u l a r i t i e s d e a l t with in l i t e r a t u r e . The wave function s o l u t i o n s W obtainsd have t h e s c a t t e r i n g property of con= s i s t i n g from p a i r s Y , Y+ of usual c -smooth s o l u t i o n s of the p o t e n t i a l f r e e motions, each v a l i d on the respective s i d e of the p o t e n t i a l s and con= nected by special junction r e l a t i o n s , on t h e support of t h e p o t e n t i a l s . In case of t h e p a r t i c u l a r p o t e n t i a l rr. 6 , corresponding t o m = l , which i s t h e only one of t h a t type t r e a t e d in l i t e r a t u r e [ 571, t h e junction r e l a t i o n obtained here i s i d e n t i c a l w i t h t h e known one.
1. Wave Function Solutions and Junction Relations The one dimensional wave function Y i s t h e s o l u t i o n of t h e equation (1)
Y" (x) t
(k - U ( X ) ) Y ( X )
=
0, x
E
R',
k E R' ,
where t h e p o t e n t i a l i s defined by (2)
U(X) =
a(s(x))'", x
E
R',
E
R', m
E
(O,m),
In view of t h e f a c t t h a t t h e p o t e n t i a l s ( 2 ) a r e concentrated in x = 0 E R ' (see Chapters 8 and l o ) , t h e problem i s t o find wave function solutionsYof (1,2) having t h e form
(3)
Y(X)
where Y-, (4)
'+"I
Y+E
(x)
Cm(R') t
Y-(x)
if x < 0
Y,(x)
if x > 0
=
a r e s o l u t i o n s of t h e p o t e n t i a l f r e e equation
k "(x) = 0,
x ER' , 199
200
E.E. Rosinger
R', which w i l l a c t u a l l y and s a t i s f y c e r t a i n j u n c t i o n r e l a t i o n s i n x = 0 express t h e s c a t t e r i n g p r o p e r t i e s o f t h e p o t e n t i a l s ( 2 ) . As known [ 5 7 ] , t h a t i s t h e s i t u a t i o n i n t h e p a r t i c u l a r case o f m = l , when t h e j u n c t i o n r e l a t i o n i n x = 0 E R1 between Y- and y, i s g i v e n by
I n t h e general case o f an a r b i t r a r y p o s i t i v e power m t h r e e problems a r i s e :
E
R', o f t h e O i r a c
1)
t o d e f i n e t h e power ( & ( x ) ) ~ ,x
2)
t o prove t h a t t h e h y p o t h e s i s ( 3 ) i s c o r r e c t , and
3)
t o o b a t i n a j u n c t i o n r e l a t i o n extending (5)
E
(0,m),
the following
6 distribution,
The t h i r d problem w i l l be s o l v e d f i r s t , by a usual 'weak s o l u t i o n ' approach presented i n S e c t i o n 2 . T h a t approach w i l l a l s o suggest t h e way t h e f i r s t two problems can be s o l v e d w i t h i n t h e q u o t i e n t algebras c o n t a i n i n g t h e d i s t r i b u t i o n s . A f t e r p r e l i m i n a r y c o n s t r u c t i o n s presented i n S e c t i o n 3, t h e s o l u t i o n o f t h e f i r s t two problems w i l l be g i v e n i n S e c t i o n 4. The j u n c t i o n r e l a t i o n s i n x = 0 E R 1 between o f t h e form
y- and Y+ i n ( 3 ) , w i l l be
1
(
J(IY = ~ ) O) a 1
for a E R' (see [57] and ( 5 ) )
20 1
QUANTUM SCATTERING
with u
=
f
1 and
-
m
G K G t
m
arbitrary.
The interpretation o f the wave function solutions (3,6) corresponding t o the one dimensional q u a n t u m scattering (1) in potentials ( 2 ) i s as follows the potential ( 2 ) has no effect on the motion.
1)
For m
2)
For m = l , the known [ 5 7 ] motion i s obtained.
3)
If m=2, there are two cases: f i r s t , for the discrete levels of the potential we1 1
E
(O,l),
,x
U ( x ) = -(vlT)2 ( 6 ( x ) ) 2
(7.1)
E
.....
R’, v = 1,3,5,7,
the wave function solutions suffer a sign change when scattering through the potential , while the discrete levels o f the potential we1 1 (7.2)
U(X) =
-(VV)~(~(X))’
,x
E
R’,
2,4,6,8,
v =
......
have no effect on the motion. For m
4) (8)
E(2,m)
U(X) =
the scattering t h r o u g h the potential well
c 1 ( 6 ( ~ ) ) x~ , E
R’,
ci € ( - m y
O),
exhibits an indeterminary involving the two arbitrary parameters and K in (6.4).
u
As known [ 5 7 ] , the problem of the three dimensional spherically symmetric quantum scattering with no angular momentum, has the radial wave function solution R given by
( r z R ’ ( r ) ) ’t r2(k-U(r))R(r) = 0 , r
(9)
k
E(O,m),
E
R’,
I n case the potential i s concentrated on the sphere of radius a i s given by a positive power of the Dirac 6 distribution
(10)
U(r)
=
a(s(r-a))m, r
E (~,m),
a
E
R’, m E
E
(0,m)
and
( ~ , m ) ,
the scattering problem (9,lO) can be reduced t o the i n i t i a l one (1,2), with the consequent interpretation for the radial wave function solution R. 2.
Weak Wave Function Solutions
There are two important advanta es in solving the scattering problem (1,2) within the quotient alge ras containing the distributions. F i r s t , the arbitrary positive powers of‘ the Dirac 6 distribution in the potentials ( 2 ) can be defined in a convenient way, as seen in Section 8, Chapter 3. Secondly, owing t o the automatically granted s t a b i l i t y property o f the sequential solutions obtained within the quotient algebras, i t i s possible t o construct the wave function solutions (3,6) using the most convenient particular weak solutions which correspond to the most convenient particular weak representations of the Dirac 6 distribution. Indeed, the s t a b i l i t y property will imply that the wave function solutions obtained do not depend on the particular way they were constructed.
b-9-
202
E.E.
Rosinger
The weak r e p r e s e n t a t i o n o f t h e D i r a c 6 d i s t r i b u t i o n , employed f o r t h e s i m p l e r computation o f b o t h t h e weak s o l u t i o n s ( 3 ) and j u n c t i o n r e l a t i o n s ( 6 ) , i s given i n 6(x) = l i m v+m
(11)
V(wV, l / w v ,x),
x E R’
,
where l i m wV
(12)
=
0 and wv
E
(0,m) f o r v
E
N,
v + m
while
V(o,K,x)
(13)
where w
E
(0,~) and K
E
K if x
E
(0,~)
0 if x
E
R’\(O,o)
=
R’.
The advantage o f t h e weak r e p r e s e n t a t i o n (11) i s t h a t i t i s g i v e n by p i e c e wise c o n s t a n t f u n c t i o n s which make i t easy t o compute t h e corresponding weak s o l u t i o n s . Moreover, t h e weak r e p r e s e n t a t i o n (11) i n v o l v e s o n l y one m o b i l e p o i n t on t h e x - a x i s , namely x = ,w, E (0,~). I t i s i m p o r t a n t t o mention t h a t t h e nons mmetric c h a r a c t e r o f t h e weak r e = p r e s e n t a t i o n (11) besides t h e a d v a n t T j d h G T v i n g o n l y one m o b i l e p o i n t x = w , a l s o corresponds t o a c e r t a i n n e c e s s i t y i m p l i e d by t h e way t h e Dirac’6 d i s t r i b u t i o n can be represented i n s e v e r a l classes o f q u o t i e n t a l = gebras c o n t a i n i n g t h e d i s t r i b u t i o n s ( f o r d e t a i l s , see Chapters 8 and 9 ) .
We s h a l l now proceed t o t h e c o n s t r u c t i o n o f t h e weak wave f u n c t i o n s o l u t i o n s o f (1,2) corresponding t o t h e weak r e p r e s e n t a t i o n (11) o f t h e D i r a c 6 d i s = tri b u t i on. Suppose g i v e n m E ( O p ) , a (1) w i t h t h e p o t e n t i a l (14)
E
R’ and v E
m UV(x) = V(wvs a / ( u v ) sx), x
N. Then t h e usual s o l u t i o n yV o f
E
R’,
w i l l obviously s a t i s f y the condition
(15)
yV
E
c”(R’\
to,Wv)
j n c’(R’)
I t f o l l o w s t h a t Yv r e s t r i c t e d t o ( - m y 0 ) i s t h e s o l u t i o n o f t h e p o t e n t i a l f r e e e q u a t i o n (4), t h e r e f o r e i t w i l l be convenient t o s t a t e t h e i n i t i a l c o n d i t i o n s on Yv a t x = 0 E R’, i . e . ,
Now t h e problem i s t o f i n d t h e c o n d i t i o n s on m E ( 0 , ~ ) and a ER’ which w i l l g r a n t t h e weak convergence on R ’ o f t h e sequence o f f u n c t i o n s
QUANTUM SCATTERING
(17)
Y o y Y p...y Y v
203
,.....
p r o v i d e d t h a t wv, w i t h v
E
N, s a t i s f y i n g ( 1 2 ) have been s u i t a b l y chosen,
Denote t h e r e f o r e b y M(k) t h e s e t o f a l l (m,a)E(O,m) x t i s f y i n g ( 1 2 ) and such t h a t
(“wv))
1i m v + m
( 18)
R ’ f o r which t h e r e e x i s t wv , w i t h V E N , sa=
(::)
=
e x i s t s and i t i s f i n i t e , f o r
y;(””)
any yo,yl
E
C’
i n (16)
The i n t e r e s t i n t h e s e t M(k) i s due t o t h e f o l l o w i n g reason. The f u n c t i o n Y r e s t r i c t e d t o ( w ,a) is a l s o t h e s o l u t i o n o f t h e p o t e n t i a l f r e e e q u a t i o n T h e r e f o r e , as’seen i n Theorem 2 below, t h e sequence o f f u n c t i o n s ( 1 7 ) w i l l converge weakly on R ’ , o n l y if t h e r e l a t i o n (18) h o l d s , i n which case t h e wave f u n c t i o n s o l u t i o n ( 3 ) w i l l be determined by t h e c o n d i t i o n s
(t).
w h i l e t h e j u n c t i o n m a t r i x J(m,a) i n ( 6 ) w i l l have t o s a t i s f y
where yp,yA r e su l t r o
yl;
a r e a r b i t r a r y b o t h i n (19) and (20), w h i l e z o y z l
E
cl
We s h a l l now e s t a b l i s h t h e s t r u c t u r e o f t h e s e t M(k). Theorem 1 The s e t M(k) does n o t depend on k E R 1 and M = ((O,l] x R’)u(121 x {-n2, - 4 ~ ’-%’, ~
.... ~ ) U ( ( ~ y ~ ) x ( - ~ , ~ ) ) U ( ( ~ y ~ ) X I o ) )
Proof Ifx
E
Cm(R’)
x”
i s t h e unique s o l u t i o n o f
( x ) t h X ( X ) = 0, x
with the i n i t i a l conditions
E
R’, h
E
R’,
E.E.
204
( I :) [ =
,a
Rosinger
E
R1, b,c
E
C1
x'(a)
then
where = e x p ( x Ah)
W(h,x) and
Therefore (my&
M(k), o n l y i f
lim
(22)
A p p l y i n g ( 2 1 ) t o t h e f u n c t i o n s yV we o b t a i n
x R1.
Assume now (m,a)E(O,m)
W(k-a/(wV)
m
,wV) = J(k,m,a)
e x i s t s and i t i s f i n i t e ,
\ ) + a
f o r s u i t a b l e uv
, with
v E N, s a t i s f y i n g (12).
I t f o l l o w s t h a t t h e o n l y problem l e f t i s t o make t h e c o n d i t i o n i n ( 2 2 ) ex= p l i c i t i n terms o f k , m and CI .
F i r s t assume
CI
> 0.
k
-
a/(wv)
Then i t can a l s o be assumed t h a t
m
< 0
s i n c e wV > 0 and uV + 0, a c c o r d i n g t o ( 1 2 ) . exPLv
(23)
W(k
-
c1/(wV)
m
Therefore
exp(-Lv)
-
exp(-Lv))
V
#uv 1 = $
Hv( expLv -exp( -Lv)) where
1
+XPLV
expLv
t exp( -Lv)
205
QUANTUM SCATTERING
=
(-k
t
a/(wv) m ) t
lim v+m
H
=
+
H
V
,
Lv
= wv HV
But obviously (24)
V
m
and
I L
lim v+m
(25)
0
if m
E
(0,2)
=
V
Therefore M(k) n([2,m)
(25)
Indeed, i f m
t
(27)
E [
x (0,m)) =
0
2,m) t h e n (24) and (25) i m p l y t h a t t h e t e r m
-
Hv(expLv
i n (23) tends t o t
m
exp(-Lv))
t o g e t h e r w i t h v.
L e t us now assume t h a t m E (0,2). Then t h e t h r e e terms i n (23) e x c e p t t h e one i n (27) have a f i n i t e l i m i t when v + m. Concerning t h e t e r m i n (27), i t i s easy t o see t h a t 0
$ Hv(expLv
lim v + m
-
if m ~ ( 0 ~ 1 )
i f m = l
exp(-Lv)) =
t m i f m E
Therefore M(k)n((l,2)
(28)
x (0,m)) =
0
and (29)
(0,ll x ( 0 , m ) c M(k)
Assume now
~1
< 0.
k
-
Then i t can a l s o be assumed t h a t
d(~" > 0) ~
s i n c e wv > 0 and wv
(30)
W(k-a/(w")
where t h is t i me
m
-f
0, a c c o r d i n g t o ( 1 2 ) .
,av) =
Therefore
(1,2)
.
E E . Rosi nger
206
lim
(33)
L~
if m~(0,2)
(-a)$
if m = 2
+
if m
=
x + m
lim
0
(-HV s i n L v )
m
-
(2,m)
i f m = l
cx
v * m
E
a,
if
m
E
(1,2)
Therefore M(k)n((1,2) x
(35)
=
(-m,O))
B
and (36)
( 0 , l I x (-m,O)
C
M(k)
Then again t h e t h r e e terms i n ( 3 0 ) e x c e p t t h e one Assume now t h a t m = 2. i n ( 3 4 ) have a f i n i t e l i m i t when v + m , w h i l e t h e t e r m i n ( 3 4 ) has t h e f o l l o w i n g behaviour
10
lim
i f a =
IT)^, w i t h p
1,2,3,4,
.....
(-Hv sinLv) =
v-tm +m
otherwise
Therefore (37)
M(k)n({2} x
(-m,O))
I 2 1 x {-(pn)’Ip
1,2,3,4,
..... 1
F i n a l l y , assume t h a t m E ( 2 , ~ ) . Then ( 3 2 ) and ( 3 3 ) i m p l y t h a t HV = lim L = + m V v+m t h e r e f o r e , i n view o f t h e t e r m ( 3 4 ) i n (30), a necessary c o n d i t i o n f o r ob= t a i n i n g J(k,m,a) f i n i t e i n ( 2 2 ) , i s t h a t (38)
lim
v+m
20 7
QUANTUM SCATTERING
lim s i n Lv = 0 v+m T h i s time, owing t o t h e r e l a t i o n s ( 3 1 ) , ( 3 8 ) and ( 3 9 ) , i t f o l l o w s t h a t be= s i d e s t h e c o n d i t i o n (12), wv,.with v E N, w i l l have t o s a t i s f y some a d d i = t i o n a l c o n d i t i o n which i s e a s i l y suggested b y ( 3 9 ) . Indeed, i n view o f ( 3 8 ) , t h e necessary and s u f f i c i e n t c o n d i t i o n f o r (39) i s t h a t (39)
lim (Lv v'm where nv E N, w i t h v
(40)
lim
n
-
nvn) = 0
E
N, and
+
=
v-tm
m
Denote t h e n ev = L v
-
nVn , f o r v
E
N.
I t f o l 1ows t h a t
(41)
lim
ev = 0
V'W
Now t h e problem i s t o o b t a i n wvy w i t h v (42)
+ eV , w i t h v
L v = nv IT
where L, i s given i n (31). such t h a t
k
-
E
E
N, as s o l u t i o n s o f t h e e q u a t i o n s
N.
That can be done as f o l l o w s .
a/wm > 0, V w E (0,A)
and assume B > 0 such t h a t t h e f u n c t i o n 0: (0,A)
-+
(B,m)
d e f i n e d b y (see ( 3 1 ) ) O ( w ) = w(k-a/wm)'
1
,
w
E
(0,A)
has t h e p r o p e r t i e s
0 i s s t r i c t l y d e c r e a s i n g on (0,A) lim
O(w) =
m y
w-+o
lim
0(w) = B
o + A
Then, t h e i n v e r s e f u n c t i o n
e x i s t s , i s s t r i c t l y d e c r e a s i n g on (B,m) and
Assume A > 0
208
E.E. Rosinger
NOW, in view of ( 4 3 ) , t h e unique solution of t h e equation (42) w i l l be given by -1 w = o (nvn t e v ) , with v E N, (45) V and ( 4 0 ) , ( 4 1 ) together with (43) w i l l imply t h a t wV, with v E N , s a t i s f y ( 1 2 ) . In t h a t way, the necessary condition (39) f o r the f i n i t e n e s s of J ( k,m,cl) i s s a t i s f i e d .
I n order t o secure the f i n i t e n e s s of J(k,m,a) i t s u f f i c e s t o choose ov, i n other words nv and ev i n ( 4 5 ) , i n such a way t h a t (46)
lim
( - H v sinLV) e x i s t s and i t i s f i n i t e
v-tm
and (47)
lim #+a2
cosL
v
exists
The condition (47) is easy t o f u l f i l l , s i n c e i n view of (45) and ( 3 1 ) , i t fol 1 ows t h a t Lv = O ( W V ) = nvr (48) therefore, i t suffices that
n V' with v (49) B u t (48) a l s o y i e l d s
E N,
sinLv = s i n e
V
t
ev
have constant p a r i t y
, for v
E
N,
hence, i n view of (38) and ( 4 1 ) , t h e condition ( 4 6 ) i s f u l f i l l e d , only i f (50)
lim ev HV
exists and i t i s f i n i t e
v + m
Since i t i s e a s i e r , we s h a l l compute t h e l i m i t o f the square i n (50), which in view of ( 4 0 ) , (41) and (44) can be obtained as follows
as sv E ( O , l ) , w i t h v E N , and a l l the above operations a r e v a l i d , provided that ev # 0 , f o r v E N (5') B u t , i n view o f (43),(40) and ( 4 1 ) , t h e limit
209
QUANTUM SCATTERING
can assume any v a l u e i n [ O + a ] , depending on a s u i t a b l e c h o i c e o f nv and T h e r e f o r e , i n view o f ( 4 6 ) , ( 4 7 ) , (49) and (22), t h e r e l a t i o n (6.4) e., w i l l f o l l o w e a s i l y , p r o v i d e d t h a t ( 5 1 ) h o l d s . I n o t h e r words (52)
(2,m)
x
M(k)
(-a ,O)C
Now, t h e r e l a t i o n s ( 2 5 ) , (28), ( 2 9 ) , ( 3 5 ) , ( 3 6 ) , (37) and (52) complete the proof.
0
The necessary and s u f f i c i e n t c o n d i t i o n f o r t h e e x i s t e n c e o f weak s o l u t i o n s o b t a i n e d w i t h t h e h e l p o f t h e s e t M c h a r a c t e r i z e d i n Theorem 1, can be pre= sented now. I t i s w g r t h n o t i c i n g t h a t t h e weak s o l u t i o n s o b t a i n e d a r e u n i f o r m l i m i t s o f C -smooth f u n c t i o n s on compacts i n R ' \ I O ) . Theorem 2 Suppose g i v e n m (53)
( 0 , ~ ) and a
E
E
Then t h e sequence o f f u n c t i o n s
R'.
.....
Yo,Y p..., Yv,..
c o n s t r u c t e d i n ( 1 5 ) converges weakly on R' f o r s u i t a b l y chosen s a t i s f y i n g ( 1 2 ) , o n l y i f m,a)E M.
w
V"""
woywl,...,
* *
I n t h a t case, t h e sequence o f f u n c t i o n s ( 5 3 ) has t h e f o l l o w i n g two proper= ties (54) (55)
vv
=
lim v+m
v- on I,
(-m
,O], V v
E
N
= I, u n i f o r m l y on i n t e r v a l s [a,-),
w i t h a z 0.
where Y- and Y+ a r e g i v e n i n ( 3 ) , ( 4 ) and ( 6 ) . Proof I n view o f (12) we can assume t h a t wV < 1 , Y v
EN,
i n which case ( 1 4 ) and (24) w i l l y i e l d t h e r e l a t i o n
(yl: 1;)
= W( k
(
) W ( k ,-wV)
):L
Y V E N
Then i t i s obvious t h a t t h e r e l a t i o n (18) w i l l h o l d , o n l y i f (56)
v
~
e x i s~t s and ~i t i s ~ f i n i t e , ~ for any ) yo,yl
E
(16)
C' i n
210
E.E.
Rosinger
But a p p l y i n g again ( 1 4 ) and ( 2 4 ) , i t f o l l o w s t h a t
t h e r e f o r e , t h e sequence o f f u n c t i o n s Y0,Y1'...'Y"
Y
. .......
w i l l converge weakly on R 1 , o n l y i f ( 5 6 ) h o l d s , which as seen above, i s e q u i v a l e n t t o t h e c o n d i t i o n (m,a)€ M. The r e l a t i o n ( 5 4 ) i s an obvious consequence o f t h e form o f t h e p o t e n t i a l i n (14). F i n a l l y , (55) f o l l o w s e a s i l y f r o m ( 5 7 ) . 3.
U
Smooth Representations f o r t h e D i r a c 6 D i s t r i b u t i o n and Smooth Weak Wave F u n c t i o n Sol u t i o n s
As mentioned a t t h e b e g i n n i n g o f S e c t i o n 2 , t h e r e a r e i m p o r t a n t advantages i n s o l v i n g t h e s c a t t e r i n g problem (1,2) w i t h i n t h e q u o t i e n t a l g e b r a s con= t a i n i n g t h e d i s t r i b u t i o n s . I n view o f t h e way t h e c h a i n s o f q u o t i e n t alge= bras ( 2 4 ) o r ( 9 3 ) i n Chapter 3 were c o n s t r u c t e d and t h e p a r t i a l d e r i v a t i v e o p e r a t o r s a r e a c t i n g between t h e q u o t i e n t algebras o f these chains, i n o r = der t o s o l v e t h e s c a t t e r i n g problem (1,2) w i t h i n t h e mentioned chains o f q u o t i e n t algebras i t s u f f i c e s t o smooth t h e weak s o l u t i o n s i n Theorem 2, S e c t i o n 2 , by ' r o u n d i n g o f f t h e c o r n e r s ' o f Yv a t t h e p o i n t s x = 0 and x = wY , w i t h w E N. I n &he case o f t h e chains o f q u o t i e n t a l g e b r a s ( 2 4 ) i n Chapter 3, we need a C -smoothing w h i l e i n t h e case o f t h e chains o f q u o t i e n t algebras (93) i n Chapter 3, a C2-smoothing w i l l be s u f f i c i e n t . I n view o f ( 1 4 ) and ( 1 5 ) , i t i s obvious t h a t a Cm-smoothing o f t h e represen= t a t i o n o f t h e D i r a c 6 d i s t r i b u t i o n i n ( 1 1 ) w i l l i m p l y a Cm-smoothing o f t h e weak wave f u n c t i o n s o l u t i o n s Y,, , w i t h v E N . T h i s smoothing o f t h e re= p r e s e n t a t i o n ( 1 1 ) i s now c o n s t r u c t e d by ' r o u n d i n g o f f t h e c o r n e r s ' o f V(wv,l/wvy*) a t x = 0 and x = wV, w i t h v E N , w i t h t h e h e l p o f two a r b i = t r a r y f u n c t i o n s 6, y E Cr(R')(see (167), Chapter 3) which s a t i s f y t h e following conditions
* ) B = 0 on
(-my
-11
* * ) 0 G B G M on ( - 1 , l ) 1581 , *
* * * ) B = 1 on [ 1,m) * * * * ) Dp ~ ( 0 #) 0, Y p E N
respectively
* ) y = 1 on (59)
(-m,-ll
* * ) 0 G y G 1 on ( - 1 , l )
* * * ) y = 0 on [ 1,m)
211
QUANTUM SCATTERING
The existence of such functions B and y results from Lemma 1, a t the end of this section. We shall suppose now given
any sequence
(m,a) E M and
which s a t i s f i e s the condition ( 1 2 ) and (18) and generates the weak wave function solution (Yv
I
v
N)
E
through (1) and (14), for a certain given i n i t i a l condition
(16).
We shall take any two sequences (w;
I
I
v E N ) and (w;
v
E
N)
which s a t i s f y the conditions
I
I
WoYW1,..
(62)
lim
6
(w;
YW'
VY"""
+
are pair wise different
=
W ;
0 , where
m=
Then we can define the sequence of functions s6 (63)
max{ l , m }
v+m
s b V ( x ) = B(~/o~)Y((x-w,,)/w;
E (Cm(R'))N
)/uVy4 v
E
given by
N , x E R'
I t i s easy t o see that the following properties result
(65)
11 - L1Sbv(x)dx
2((M+1)5 +
W ;
)/wV ,4 u
E
N
therefore, in view of ( 6 2 ) , i t follows t h a t (66)
s6 E
sm n(C;(R')) N ,
<s6,.>
= 6
Now, the Cm-smooth re presentation o f the Dirac 6 distribution obtained i n (63) and1 replace the representation in (11) and correspgndingly, the weak wave function solutions in (15) will be replaced by the C -smooth weak wave function solutions xv of ( 1 ) generated by the potentials ~ ( s , ~ ( x ) )x~E, R', v E N , conditions given by (see ( 1 6 ) )
E.E. Rosinger
212
where (68.1)
xo < i n f
{-LO;
1
v
E
N 1
Indeed, t h e f o l l o w i n g r e s u l t holds. Theorem 3 I t i s p o s s i b l e t o choose t h e sequences
(u;
I
v E
N) and (u; 1 v
E
N)
s a t i s f y i n g t h e c o n d i t i o n s (60-62) and such t h a t t h e sequence o f Cm-smooth functions
xo
(69)
Y
XI,.
. . ,xv , . . . . . . .
converges i n D ' ( R ' ) t o Y g i v e n i n ( 3 ) , ( 4 ) and ( 6 ) . Proof We s h a l l use a Gronwa 1 i n e q u a l t y argument. and x E R', we denote
BVb) =
/
I n t h i s connection, f o r v
0
- k t 4s,v(x))m
and f i n a l l y = F"(X)
N
1
0
\
HJx)
E
-
Gvb)
Then, o b v i o u s l y t h e f o l l o w i n g r e l a t i o n s w i l l h o l d
QUANTUM SCATTERING
213
and = B,(x)G,(x),
G:(x)
x E
R'
The above r e l a t i o n s w i l l y i e l d X
HV(x) =
X
f
(A,(E)-B,(E))F,(<)dC
xO
1
+
x E R',
B,(E)H,(E)d<,
xO
and a p p l y i n g t h e L" v e c t o r , r e s p e c t i v e l y m a t r i x norms, denoted f o r s i m p l i = c i t y by il ii , i t f o l l o w s t h a t X
IIH,(x)II
1 II A,(C) -
Q
Bv(<)II
II Fv(<)II
d<
t
xO X
111 B,(<)II
+
-11 H,(<)ll
,x
d<
E
R'
xO
Now, t h e Gronwall i n e q u a l i t y i m p l i e s t h e r e l a t i o n X
(70)
IIH,(x)ll
G
7
X
UA,(<)-Bv(~)II~II F,(E)U.(exp
But, o b v i o u s l y IYv(x) whi 1e
and
- x,
I I B v ( ~ ) I 1 dn)dg, x E
5
xO
(x)I
<
1I H,(x)II
,x
E
R'
R'
214
E.E. Rosinger
moreo ve r
where
But, f o r g i v e n v E N , F depends on U, o n l y and does n o t depend on U; o r Thus, i n view o f y71.1), K, i s decreasing i n b o t h U; and It f o l l o w s t h e n from ( 7 1 ) and ( 6 2 ) t h a t t h e f u n c t i o n Y,, f o r eazh g i v e n v E N, can be a r b i t r a r i l y and u n i f o r m l y on R ’ approximated by t h e c -smooth func= t i o n xV, p r o v i d e d t h a t U; and 4 a r e chosen s u f f i c i e n t l y s m a l l . Then, i n 0 view o f Theorem 2 i n S e c t i o n 2, t h e p r o o f i s completed.
U ;
.
.
Lemma 1 There e x i s t f u n c t i o n s B , y E p e c t i v e l y (59).
cy(R’) which s a t i s f y t h e c o n d i t i o n s (58), r e s =
4. Wave F u n c t i o n S o l u t i o n s i n t h e Algebras C o n t a i n i n g t h e D i s t r i b u t i o n s The aim o f t h i s s e c t i o n i s t o show t h a t t h e s c a t t e r i n g problem (1,2) can b e
2 15
QUANTUM SCATTERING
s o l v e d w i t h i n t h e chains o f q u o t i e n t a l g e b r a s (24) o r (93) i n Chapter 3. Therefore, t h e wave f u n c t i o n s s o l u t i o n s o b t a i n e d - see (3,4 and 6 ) - do n o t depend on t h e p a r t i c u l a r way t h e weak wave f u n c t i o n s o l u t i o n s were const== t e d i n S e c t i o n s 2 and 3, t h a t f a c t b e i n g a consequence o f t h e s t a b i l i t p r o p e r t y of t h e s e q u e n t i a l s o l u t i o n s o f PDEs w i t h i n t h e framewor o f t e mentioned a1 gebras
4
.
I n o r d e r t o reach t h e above aim, t h e smooth weak wave f u n c t i o n s o l u t i o n s c o n s t r u c t e d i n Theorem 3, S e c t i o n 3 w m used, t o g e t h e r w i t h t h e smooth r e p r e s e n t a t i o n s o f t h e D i r a c 6 d i s t r i b u t i o n c o n s t r u c t e d i n (63-66). I n t h e case o f i n f i n i t e smoothness, i . e . o f t h e c h a i n s o f q u o t i e n t a l g e b r a s (24) i n Chapter 3, t h e r e s u l t o b t a i n e d i s p r e s e n t e d now. Theorem 4 Suppose g i v e n t h e s c a t t e r i n g problem y'' ( x )
(72)
+
(k
-
x
a ( s ( ~ ) ) ~ ) Y =( x0,)
E
R',
where k E R' and (m,a)E M I n case t h e wave f u n c t i o n s o l u t i o n \y i s n o t smooth, i.e&, \y o r I' i s n o t continuous a t x = 0 E R ' , i t i s possibleconstruct C -regularizations ( v , ~ ) ,such t h a t t h e a s s o c i a t e d c h a i n o f p o s i t i v e ower q u o t i e n t a l g e b r a s c o n t a i n i n g t h e d i s t r i b u t i o n s (see S e c t i o n d p t e r 3) w i l l have the following properties: = s
+
I ~ v,s) (
V,s), V
A'(
w,
where s E
s
does
not depend
on &.
1)
Y
2)
I s a t i s f i e s t h e e q u a t i o n (72) i n t e usual a l g e b r a i c sense, w i t h m u l t i =
E
R E
p l i c a t i o n and p o s i t i v e power i n A
Op : A R ( ~ , ~ +Ak(v,s), )
PE
N,
k ( v , s ) and t h e d e r i v a t i v e o p e r a t o r s
p G 2, & , k E 1,
&-k > 2 ,
Proof _ I _
Since (m,a) E M
, one
(uVlv E N),
can a c c o r d i n g t o Theorems 2 and 3 choose
( ~ 4 E1 N)~ and
(wclv
E
N)
which w i l l s a t i s f y ( 1 2 ) , (18) and (60-62) and w i l l generate t h r o u g h (63-68) t h e sequence o f f u n c t i o n s i n (69) (73)
s = (~,X1'...'XV
y
.......1
with the properties (74)
s E
s",
<
S,'>
=
I
and (75) with
X; ( x )
-t
(K
-
~(s,,(x))~)x,(x)
= 0, Y v
E
N, x E
R',
E.E. Rosinger
216
f o r any g i v e n (76.1)
xo< i n f { - w ' l v € N ) V
where y, y- and Y+ a r e g i v e n i n ( 3 ) , ( 4 ) and ( 6 ) . Now, t h e i d e a o f t h e p r o o f i s t o c o n s t r u c t C z r e g u l a r i z a t i o n s ( v , S ) such that (77)
S6 E
s
(see ( 6 3 ) )
and (78)
s ES
I n t h i s case (66) w i l l a c c o r d i n g t o Theorem 17, S e c t i o n 8, Chapter 3, y i e l d w i t h i n p o s i t i v e power a l g e b r a s t h e r e l a t i o n (79)
6 = s6 + + ( ! I S )E AR(V,S), Y
E
fl
as w e l l as (80)
( s 6 ) " t I'(v,s)
6"'
E
A ~ ( v , s ) ,v
mE
(o,m),
R
E
R
Therefore, (74), (75), ( 7 8 ) and ( 8 0 ) w i l l y i e l d w i t h i n t h e same p o s i t i v e power a1 gebras t h e r e 1a t i o n (81)
Y"
+
(k
-
~ ( 6 ) ~ ) =" 0
thus c o m p l e t i n g t h e p r o o f o f 1) and 2 ) . The c o n s t r u c t i o n o f t h e C m - r e g u l a r i z a t i o n s (V,S) f o r which (77) and (78) hold, w i l l proceed a c c o r d i n g t o t h e method presented i n Theorem 4, S e c t i o n 4, Chapter 3. We denote b y
t h e s e t o f a l l t h e sequences o f f u n c t i o n s w conditions
Y x ER' : (82)
3 P E N :
Y VEN,
v > p :
w (x) = 0 V
and
E
(Cw(R'))N
which s a t i s f y the
QUANTUM SCATTERING 3F
C
217
F finite :
R',
G open, G 3 F :
Y G c R',
3 p 1 E N :
(83)
Y v E N, v
>)I':
supp wy c G i n o t h e r words, wv vanishes a t each x E R ' , f o r v s u f f i c i e n t l y l a r g e and furthermore, supp E~ s h r i n k s t o a f i n i t e subset o f R', when v + m. I t i s N o n t r i v i a l examples o f easy t o see t h a t l6 i s an icJeal i n ( C (R'))N. sequences o f f u n c t i o n s w E l 6 w i l l be g i v e n i n ( 6 ) and ( 9 ) , Chapter 8. F u r t h e r , we denote by T t h e g e c t o r subspace i n sequences o f f u n c t i o n s t E S o f t h e form tv(x)
(84)
D P ~ 6 v ( ~ - ~ o V) , v E N, x
=
where xo E R ' and p
E
E
F g e n e r a t e d by a l l t h e R',
N are a r b i t r a r y .
We s h a l l prove t h a t 1; and T s a t i s f y t h e c o n d i t i o n s ( 3 3 ) and (34) i n Theo= rem 4, S e c t i o n 4, Chapter 3 . I n view o f (84) and ( 6 6 ) , t h e f o l l o w i n g r e l a = t i o n i s obvious
vmn T =
(85)
F u r t h e r , t h e re1 a t i on
f o l l o w s from (83),(84) and ( 6 6 ) as w e l l as t h e w e l l known p r o p e r t y t h a t any d i s t r i b u t i o n w i t h s u p p o r t a f i n i t e subset i s a l i n e a r combination o f t h e D i r a c 6 d i s t r i b u t i o n and i t s d e r i v a t i v e s . We prove now t h e r e l a t i o n
, then
Assume indeed t h a t w E 1: n T
where F
C
E
N and X E R'. Assume now g i v e n xO x q Then, i n view o f ( 6 3 ) , (58) a8d ( 5 9 ) , t h e r e e x i s t s
R ' i s a f i n i t e subset, p E
xo E F and f i x e d . )I
obviously
N, such t h a t t h e r e l a t i o n (88) s i m p l i f i e s a t x = xo, as f o l l o w s
xO
(89)
wv(xo) =
c q E N q QPxo
X
Dq S ~ ~ ( OV ) v, 'oq
E
N, v >pxo
2 18
E.E. Rosinger
u,
But, owing t o (82), we can assume t h a t
was chosen i n such a way t h a t 0
wv(x0) = 0, V u
(90)
N, v
E
>
.
11, 0
F u r t h e r , (63) and (58-60) w i l l o b v i o u s l y y i e l d
,
D q ~ 6 u ( 0 ) = DqS(0)/uu(u:)q
(91)
V u
E
N
Therefore, (89-91) w i l l i m p l y t h e r e l a t i o n s
Considering t h e f i r s t p, ding t o v with q E
N,
%px
0
, as
q G p,
1 o f t h e above r e l a t i o n s , i . e . those correspon=
t
N, uxo< v
E
t p,
unknowns ,
, and
considering
5
= h Dqf3(0), q xoq t h e d e t e r m i n a n t o f t h e r e s u l t i n g homo=
0
0
geneous system o f l i n e a r equations w i l l be t h e f o l l o w i n g Vandermonde d e t e r = m i nant det((l/u(Jq
I
G u Gp,
p ,O
which i n view o f (61), w i l l h
DqS(0) = 0, Y q
t
0
vanish. E
,O
N, q Gp, 0
****) i n (58) w i l l i m p l y t h e r e l a t i o n s
which i n view o f
=O, YqEN,
h
)
Gq G p
0
T h e r e f o r e , (92) w i l l y i e l d
,Oq
(93)
,0
p,
qGp,
xOq
0
Since xo E F was chosen a r b i t r a r i l y , t h e r e l a t i o n s (93) and (88) w i l l y i e l d W E
2
thus c o m p l e t i n g t h e p r o o f o f (87). NOW, t h e r e l a t i o n s (85-87) o b v i o u s l y i m p l y t h a t 1; and T s a t i s f y t h e con= d i t i o n s (33) and (34) i n Theorem 4, S e c t i o n 4, Chapter 3. T h e r e f o r e , i t o n l y remains t o c o n s t r u c t v e c t o r subspaces S' c S" which s a t i s f y t h e condi= t i o n s (35) and (36) i n t h e mentioned theorem, and such t h a t
w i l l a l s o h o l d (see (77) and ( 7 8 ) ) . I n t h i s c o n n e c t i o n we n o t i c e t h a t i n view o f (84), t h e r e l a t i o n f o l l o w s easily
QUANTUM SCATTERING
219
(95) Moreover (96)
S
6
vm(+-\ucm(Rl)
J
s i n c e (74) h o l d s and Y was supposed non smooth. Now, t h e r e l a t i o n s (95)mand ( 9 6 ) w i l l o b v i o u s l y g r a n t t h e e x i s t e n c e o f v e c t o r subspaces S' c S s a t i s f y i n g t h e c o n d i t i o n s
v" i;)
(97)
Sm C
(98)
c s 3 u UC"(R')
S'
'
T
c S'
I n t h a t case ( V , S ) , w i t h S g i v e n i n (94) and V a v e c t o r subspace i n 1; n w i l l be t h e needed C - r e g u l a r i z a t i o n .
v", 0
I n t h e case o f f i n i t e smoothness, i . e . of t h e chains o f q u o t i e n t algebras (93) i n Chapter 3, t h e r e s u l t i n Theorem 4 takes t h e f o l l o w i n g form. Theorem 5 Suppose g i v e n t h e s c a t t e r i n g problem
+
(k
-
a(6(~))~)'? = (0, ~)
where k E R ' and ( m y % )
E
M.
'4' '
(99)
(x)
x
E
R',
I n case t h e wave f u n c t i o n s o l u t i o n Y i s n o t continuous a t x = 0
E
R',
it
i s p o s s i b l e t o c o n s t r u c t C " - r e g u l a r i z a t i o n s ( V , S ) , such t h a t t h e a s s o c i a t e d chain o f
o s i t i v e ower q u o t i e n t algebras 5i-n-Fo owing p r o p e r t i e s :
(see S e c t i o n s 7 and 8, Chapter 3)
w i l l have t e 1)
Y
= s
+ lR(V,S) E
where s E S does
2)
AR(V,S),
not
V R E
depend on R
4,
.
I s a t i s f i e s t h e e q u a t i o n (99) i n t h e usual a l g e b r a i c sense, w i t h mul= t i p 1 i c a t i o n and p o s i t i v e power i n Ak( V , S ) and t h e d e r i v a t r v e o p e r a t o r s k 1-k 2 2 . + A ( V , S ) , p E N, p < 2 , R, k E
Dp: A'(V,S)
m,
Proof -
I t i s s i m i l a r t o t h e p r o o f o f Theorem 4, w i t h t h e f o l l o w i n g m o d i f i c a t i o n s . F i r s t , i n s t e a d o f 7; we c o n s i d e r t h e s e t 7: o f a l l t h e sequences o f func= t i o n s w €(C'(R'))N which s a t i s f y t h e c o n d i t i o n s (82) a n d (83). Then, t h e relations
220
E.E. Rosinger
w i l l r e s u l t i n a way s i m i l a r t o (85-87) and ( 9 5 ) . F u r t h e r , s i n c e Y i s n o t continuous, i t w i l l f o l l o w t h a t
Therefore, t h e r e e x i s t v e c t o r subspaces S' cS" which s a t i s f y t h e c o n d i t i o n s
Then, t a k i n g
and V a v e c t o r subspace i n I" n V " , one obtaines i n view o f Theorem 10, S e c t i o n 5, Chapter 3, t h e neided C " - r e g u l a r i z a t i o n ( V , S ) .
0
Remark 1 The smooth r e p r e s e n t a t i o n o f t h e D i r a c 6 d i s t r i b u t on g i v e n b y s 6 ( 6 3 ) has t h e f o l l o w i n g p r o p e r t y
( 100)
Dp s s v ( 0 ) # 0,
E
S" i n
Y VEN, p E N
c a l l e d t h e s t r o n g presence on t h e s u p p o r t I01 t i a l r o l e i n e s t a b l i s h i n g Theorems 4 and 5 .
C
R'
which p l a y e d an essen=
Obviously, t h i s p r o p e r t y imp1 i e s among o t h e r s t h a t symmetric represents= t i o n s f o r t h e D i r a c 6 d i s t r i b u t i o n cannot be used, s i n c e D ssV(O) # 0, YvEN.
The general n-dimensional v e r s i o n o f t h e p r o p e r t y o f s t r o n g presence on t h e support, as w e l l as s e v e r a l o f i t s consequences w i l l be presented i n Chap= t e r s 8 and 9. 5.
S t a b i l i t y and Exactness o f t h e Wave F u n c t i o n S o l u t i o n s
The aim o f t h i s s e c t i o n i s t o e s t a b l i s h t h e s t a b i l i t y and exactness proper= t i e s o f t h e wave f u n c t i o n s o l u t i o n s o b t a i n e d w i t h i n t h e p a r t i c u l a r i n s t a n = ces o f chains o f q u o t i e n t algebras c o n s t r u c t e d i n Theorems 4 and 5, S e c t i o n 4. T h i s time, i n t h e case o f t h e wave f u n c t i o n s o l u t i o n s I o f t h e s c a t t e r i n g problem (1,2), t h e c o n d i t i o n s f o r s t a b i l i t y and exactness a r e b e t t e r and s i m p l e r t h a n those f o r t h e weak s o l u t i o n s o f polynomial n o n l i n e a r PDEs pre= sented i n Sections 5 and 6, Chapter 5. Indeed, t h e r e t h e problem was t o c o n s t r u c t r e g u l a r i z a t i o n s ( V , S ) which would se a r a t e (see Remark 1, S e c t i o n 4, Chapter 4) t h e ' r e g u l a r i z i n g sequence' s rom t e ' e r r o r sequence' w, a c c o r d i n g t o t h e r e l a t i o n s
-I;---h (101)
w
E V , S
E S
Here, t h e problem i s s i m p l e r s i n c e t h e ' e r r o r sequence' w i s i d e n t i c a l l y zero. Indeed, w i t h t h e ' r e g u l a r i z i n g sequence'
221
QUANTUM SCATTERING
s = (&yX 1 ' " . ' X "
(102)
Y
.....)
E S",
<S,*>
=
Y
f o r t h e wave f u n c t i o n s o l u t i o n Y g i v e n i n Theorem 3, S e c t i o n 3, and t h e smooth r e p r e s e n t a t i o n o f t h e O i r a c 6 d i s t r i b u t i o n . (103)
S6
E
S",
<S6,*,
=
6
g i v e n i n (63), t h e r e g u l a r i z e d v e r s i o n o f t h e e q u a t i o n (1,Z) f i e d e x a c t l y (see ( 7 5 ) )
(104)
D2s + ( k
-
w i l l be s a t i s =
a ( ~ , ) ~ ) s= ~ ( 0 ) E Q
T h i s somewhat p a r a d o x i c a l advantage o v e r t h e case o f t h e weak s o l u t i o n s o f polynomial n o n l i n e a r PDEs s t u d i e d i n Chapters 4 and 5, i s due t o t h e f a c t t h a t i n t h e case o f t h e s c a t t e r i n g problem (1,2) b o t h t h e e q u a t i o n and t h e wave f u n c t i o n s o l u t i o n s have t o be r e g u l a r i z e d a n m a t can be done, as seen i n (104), i n an e r r o r l e s s manner. However, t h e c o n d i t i o n (101) w i l l n o t s i m p l i f y i n a t r i v i a l way, s i n c e t h i s t i m e t h e ' r e g u l a r i z i n g sequence' s 6 o f t h e D i r a c 6 d i s t r i b u t i o n w i l l a l s o have t o be taken i n t o account. Nevertheless, as seen i n t h e p r o o f o f Theorems 4 and 5, S e c t i o n 4, t h e new c o n d i t i o n on t h e r e g u l a r i z a t i o n s ( V , S ) w i l l be (see ( 7 7 ) and ( 7 8 ) ) (105)
s,
S6
E S
which i s not o f s e p a r a t i o n type, l e a v i n g t h e r e f o r e ample space f o r t h e c h o i c e o f ( V,S) as seen n e x t . The method used w i l l be s i m i l a r t o t h e one i n S e c t i o n s 2 and 3, Chapter 5. Suppose t h e r e f o r e t h a t , under t h e c o n d i t i o n s i n Theorem 4, S e c t i o n 4, t h e sequences (102) and (103) have been c o n s t r u c t e d . Then, t h e wave f u n c t i o n s o l u t i o n Y i s c a l l e d an S-sequential s o l u t i o n o f t h e s c a t t e r i n g problem (1,2) o n l y i f t h e r e e x i s t s a (?-regularization ( V , S ) , such t h a t t h e corresponding c h a i n o f p o s i t i v e power q u o t i e n t algebras A'( V,S) = A'( V,S)/IR(V,S)
, with
R
E
R,
s a t i s f y the condition (106 1
s, s 6 E AR(V,S),
4 9.
E
a
Obviously, t h e c o n d i t i o n (106) w i l l h o l d , whenever t h e C m - r e g u l a r i z a t i o n ( V , S ) s a t i s f i e s (105). Now, t h e r e s u l t i n Theorem 4, S e c t i o n 4, can be extended as f o l l o w s . Theorem 6 Under t h e above c o n d i t i o n s , i f t h e wave f u n c t i o n s o l u t i o n Y i s an S;sequen= t i a l s o l u t i o n o f t h e s c a t t e r i n g problem (1,2), then, f o r a c e r t a i n C -regu= l a r i z a t i o n ( V , S ) , t h e f o l l o w i n g p r o p e r t i e s h o l d w i t h i n t h e corresponding c h a i n o f p o s i t i v e power q u o t i e n t algebras: 1)
I = s
+ IR(V,S) E AR(V,S), 4 R
E
a,
where s E AR(V,S) does n o t depend on R.
E.E.
222 2)
Rosinger
Y s a t i s f i e s t h e e q u a t i o n (1,2) i n t h e usual a l g e b r a i c sense, w i t h m u l t i p l i c a t i o n and p o s i t i v e power i n Ak'(V,S) and t h e d e r i v a t i v e opera= tors k Dp : AR(V,S) -+ A ( V , S ) , p E N, p G 2 , i l , k E R , k-k 2 2
Proof I t f o l l o w s from t h e p r o o f o f Theorem 4, S e c t i o n 4.
U
I f t h e wave f u n c t i o n s o l u t i o n Y i s an S - s e q u e n t i a l s o l u t i o n o f t h e s c a t t e r = i n g problem (1,21, then we say t h a t Y i s maximally s t a b l e , o n l y i n case t h e r e e x i s t s a C - r e g u l a r i z a t i o n ( V , S ) , w i t h V maximal, and such t h a t (106) holds
.
Theorem 7 Any wave f u n c t i o n s o l u t i o n Y which i s an S-sequential s o l u t i o n o f t h e s c a t = t e r i n g problem (1,2) i s maximally s t a b l e . Proof S i m i l a r t o t h e p r o o f o f Theorem 1, S e c t i o n 2, Chapter 5.
0
Remark 2
1)
Concerning t h e exactness p r o p e r t i e s o f t h e wave f u n c t i o n s o l u t i o n s o f t h e s c a t t e r i n g p r o b 1 , 2 ) , t h e s i t u a t i o n i s s i m i l a r t o t h e one d e s c r i b e d i n S e c t i o n 3, Chapter 5.
2)
The v e r s i o n o f t h e above r e s u l t s corresponding t o t h e case o f f i n i t e smoothness, i . e . , t o chains o f q u o t i e n t a l g e b r a s ( 9 3 ) i n Chapter 3, can e a s i l y be e s t a b l i s h e d (see a l s o S e c t i o n 4, Chapter 5 ) .
CHAPTER 8 PRODUCTS WITH DIRAC 6 DISTRIBUTIONS
0.
Introduction
The following r e l a t i o n s a r e well known in the theory of d i s t r i b u t i o n s ( x - x ~ ) ~ . D6x~ ( x ) = 0, V x0
E
Rn,
p, q
E
Nn, q $ p
0
where 6 , i s t h e Dirac 6 d i s t r i b u t i o n concentrated in xo. Their importance, among o t l e r s , i s i n the upper bounds they give on the order of s i n g u l a r i t i e s exhibited by t h e Dirac 6 d i s t r i b u t i o n and i t s p a r t i a l d e r i v a t i v e s . As seen in Sections 2 and 11, Chapter 1, Section 9 , Chapter 3 and Appendix 2 , the r e l a t i o n s (1) play an e s s e n t i a l r o l e i n determining the way p a r t i a l d e r i v a t i v e operators can be defined on q u o t i e n t algebras containing t h e distributions.
The aim of t h i s chapter i s t o construct chains of q u o t i e n t algebras ( 2 4 ) and (93) i n Chapter 3, i n which r e l a t i o n s of type (1) a r e v a l i d . Several f u r t h e r p r o p e r t i e s of these chains of q u o t i e n t algebras, f o r i n s t a n c e , the Val i d i t y of known formulas in Quantum Mechanics i nvol vi n g s i ngul a r products with Dirac and Heisenberg d i s t r i b u t i o n s , w i l l be presented. The construction of t h e s e chains of q u o t i e n t algebras i s based on a non= t r i vial a1 gebrai c argument i nvol vi ng t h e computation o f a general i zed Van= dermonde determinant. T h a t computation was offered by R . C . K i n g , who was s o kind t o promptly answer an inquiry of the author and c o r r e c t and prove t h e a u t h o r ' s i n i t i a l conjecture. The r e s u l t s w i l l be presented only i n t h e case o f t h e chains of q u o t i e n t algebras (24) i n Chapter 3, s i n c e they can e a s i l y be reformulated f o r t h e chains of q u o t i e n t algebras ( 9 3 ) i n Chapter 3. 1.
A Class of Regularizations
The construction of t h e chains of q u o t i e n t algebras mentioned i n Section 0 i s a c t u a l l y a generalization t o t h e n-dimensional case of t h e construction o f the chains o f q u o t i e n t algebras encountered i n Theorems 4 and 5 , Section 4 , Chapter 7. For given R
R , we denote by
E
6 223
E.E. Rosinger
224
t h e s e t o f a l l t h e sequences o f f u n c t i o n s w f o l l o w i n g two c o n d i t i o n s
E
N satisfy the (C R( R ) ) ~ which
Y X E R ~ :
(2)
3
W E N :
Y v
>u:
v
EN,
wv(x) = 0 and
3 F c R ~ ,F f i n i t e : Y G c R",
(3)
G open, G
3
F :
N :
3
Y v E N, v >p':
supp wv c G a n N Obviously 1: i s an i d e a l i n (C (R ) ) and
v
s
E
I: n sR:
(4) supp <s,
> i s a f i n i t e subset i n Rn
Examples o f n o n t r i v i a l sequences o f f u n c t i o n s w E I: can be o b t a i n e d as f o l l o w s . Suppose Y E D(Rn) and
(5)
0
+
SUPP y
t h e n we d e f i n e w w,(x)
E
(C"(Rn))N by
= Y((vtl)x),
V v E N, x
E
Rn
It follows easily that
(6)
w E l 6R
y
v
LEN
I f we suppose i n a d d i t i o n t h a t
In Y(x)dx = 1 R and d e f i n e now w E (C"(Rn))N b y (7)
(8)
wv(x) = ( v + l ) n Y ( ( v t l ) x ) ,
then obviously (9) and
w
E
I;n&
v
RE
N
Y w
E
N,
x E Rny
225
PRODUCTS
(10)
<
w,
6
=
*>
I t can be noticed t h a t t h e condition ( 5 ) i s the one which implies thafi w i n ( 6 ) o r ( 9 ) s a t i s f y the vanishing condition ( 2 ) in the d e f i n i t i o n of 16. In order t o c o c s t r u c t t h e corresponding n-dimensional version of the vector subspace T c S used in the proof of Theorems 4 a n d 5 , Section 4 , Chapter 7 , several preliminary constructions a r e needed. For given k
E
N , we denote
1
P(n,a)
=
n(a)
car P ( n y a )
{p
E
\ P I
and =
I t i s easy t o s e e t h a t t h e r e e x i s t s a l i n e a r order 4 on N n such t h a t
Nn
{p(l),p(2),
=
p(1) 4
.....1
p(2) 4
.......
and P ( n , k ) = I p ( 1 ) ,..., p ( n ( k ) ) l , V
N
Given now a sequence of functions s E ( C m ( R n ) ) N , we s h a l l a s s o c i a t e t o i t the following Wronskian type i n f i n i t e matrix of functions, using t h e above l i n e a r order 4 on Nn :
Dp(’)so(x).
;)..
/. x)
..
,
xERn
Further, l e t us denote by
L t h e s e t of a l l t h e i n f i n i t e vectors N A = (Ap
I
pEN)ER
which have only a f i n i t e number of non zero components A
An i n f i n i t e matrix
!J
.
E . E . Rosinger
226
A = (a
W
1
v,p E N )
of real numbers w i l l be c a l l e d column wise non s i n g u l a r , only i f
V A
L:
E
A A
E L * A = O
Finally, we denote by L
6
the s e t of a l l the sequences of functions s i n g three conditions
E
Sm which s a t i s f y t h e follow=
(10)
< s,
(11)
supp s v shrinks t o to1 c R n , when v
*>
=
6 + m
and (12)
W,(O)
i s column wise non s i n g u l a r
The existence of sequences of functions s
E
Z6 w i l l be proved i n Section
5.
In view of the conditions (10) and ( l l ) , the sequences s quences.
E
Z6 a r e &se=
The condition (12) s a t i s f i e d by t h e &sequences S E Z ~i s c a l l e d strong pre= sence on t h e support I01 c Rn in view of i t s meaning i n the following p a r t i = c u l a r case ( s e e a l s o ( l o o ) , Chapter 7 ) . Suppose, we a r e i n t h e one dimen= sional case n = l and given Y E D ( R ' ) such t h a t
Then, we define s
E Sm by
I t i s easy t o see t h a t in t h i s case s will s a t i s f y the condition ( 1 2 ) , only if (13)
Dp Y(0) # 0,
V p E N,
Connected with the meaning of the property o f strong presence on support i t i s relevant t o compare the above condition (13) w i t h i t s opposite i n ( 5 ) . As seen l a t e r i n (101), the n-dimensional version o f t h e condition (13) w i l l be a d i r e c t g e n e r a l i z a t i o n , replacing the d e r i v a t i v e s by p a r t i a l deri vati ves
.
NOW, with t h e help o f t h e &sequences s E Z g we s h a l l define vector sub= spaces T E S as follows. Suppose given s E Z6. Then we denote by TS
227
PRODUCTS
t h e v e c t o r subspace i n ? generated b y a l l t h e sequences o f f u n c t i o n s t E o f t h e form t u ( x ) = DPsV(x-xo), V u
(14) where x
0
E
Rn and p
E
N,
x
E
sm
R",
Nn a r e a r b i t r a r y .
E
Prooosition 1 l;, and Ts s a t i s f y t h e f o l l o w i n g c o n d i t i o n s : (15)
I
m
6
n T = T nv"=p s
(16)
vm t
(17)
( V"@T;)
s
(1; n
sm)= v"@
n ucm(Rn) =
T~
p
Proof The r e l a t i o n n
v"
=
p
f o l l o w s e a s i l y from ( 1 4 ) . The r e l a t i o n ( 1 7 ) f o l l o w s from (16) and ( 1 4 ) . The i n c l u s i o n _C i n (16) f o l l o w s e a s i l y from ( 4 ) and ( 1 4 ) . 3 i n (16) i t s u f f i c e s t o show t h a t I n o r d e r t o prove t h e i n c l u s i o n -
(18)
D p s E f +
1;,Vp€Nn
I t i s easy t o see t h a t thc sequence o f f u n c t i o n s i n (8-10) has t h e p r o p e r t y
(19)
DPw E 1; n Sm, = 0'6,
V p E Nn
Therefore, (18) w i l l r e s u l t from ( 1 9 ) . Now, i t remains t o prove t h e r e l a t i o n (20)
f ' n ~= q 6
Assume t h a t w
(21)
s
E
1: n T S . Then
wv(x) =
c xo E F
C A Dpsv(x-xo), p E Nn 'oP I P I Gmx
V w
E
N, x
E
R',
0
where F i s a f i n i t e s u b s e t i n Rn, m
E xO
Assume g i v e n xo
E
F fixed.
N and
h
E
R'.
xOP
Then, i n view o f ( 1 1 ) , t h e r e e x i s t s px E
N,
0
such t h a t t h e r e l a t i o n (21) w r i t t e n f o r x = x0 w i l l t a k e t h e f o l l o w i n g sim=
228
E.E.
Rosinger
p l e r form
was chosen i n such a way
But, i n view o f ( 2 ) , i t can be assumed t h a t p x 0
that
Then ( 2 2 ) and ( 2 3 ) w i l l y i e l d
Now, we d e f i n e t h e i n f i n i t e v e c t o r A = ( A '
u
(25)
A;
I
E
N)
E
R
N
by
=
otherwise Then, o b v i o u s l y A
E
L
and ( 2 4 ) i s e q u i v a l e n t t o t h e r e l a t i o n
ws(o)
A
E
L
Therefore, i n view o f ( 1 2 ) , i t f o l l o w s t h a t A = 0 which a c c o r d i n g t o (25) w i l l y i e l d (26)
x xOP
= 0, Y p E N ~ , 1p1 <mx 0
As x E F was chosen a r b i t r a r i l y , t h e r e l a t i o n s ( 2 6 ) and ( 2 1 ) w i l l complete 0 t h e p r o o f of ( 2 0 ) . The p r o p e r t i e s o f t h e i d e a l 1 : and o f t h e v e c t o r subspaces T e s t a b l i s h e d i n P r o p o s i t i o n 1, o f f e r t h e p o s s i b i l i t y f o r c o n s t r u c t i n g the'needed r e g u l a r i = zations, Indeed, i n view of (15-17) above, as w e l l as Themorem 4, S e c t i o n 4, Chapter 3, o b v i o u s l y t h e r e e x i s t v e c t o r subspaces s c S , such t h a t
229
PRODUCTS
and
(VSC t)Ts )
(28)
i s a C"-regularization
f o r any v e c t o r subspace V C l i n V"
+
However, from t h e p o i n t of view of t h e s t a b i l i t o f t h e r e l a t i o n s (1) w i t h i n t h e chains of q u o t i e n t algebras c o n t a i n i n g t e i s t r i b u t i o n s , i t i s u s e f u l t o c o n s i d e r a c l a s s o f C " - r e g u l a r i z a t i o n s which i s l a r g e r t h a n t h e one i n (28) and which can be o b t a i n e d as f o l l o w s . For given s
E
Z
6
we denote by
i vs t h e s e t o f a l l t h e p a i r s ( 7 , T ) where I i s an i d e a l i n (C"(Rn))N and T i s a v e c t o r subspace i n s", such t h a t t h e f o l l o w i n g c o n d i t i o n s a r e s a t i s f i e d :
(29)
1 3 1;
(30)
I
(31)
znsrncvm@~
(32)
V"
I
TITs
n T = T n V"
=
2
n ( U c m ( ~ n )+ T ) =
2
I n view o f P r o p o s i t i o n 1, o b v i o u s l y
Given now ( 1 , ~ )E !V , i t f o l l o w s from Theorem 6, S e c t i o n 4, Chapter 3, t h a t t h e r e e x i s t v e c t o r sabspaces s c Smsuch t h a t
(v, s
(33)
@ T)
i s a C"-regularization
f o r any v e c t o r subspace V c 1 n
v" .
We s h a l l denote b y RGS
t h e s e t o f a l l t h e C " - r e g u l a r i z a t i o n s o f t y p e ( 3 3 ) , which o b v i o u s l y c o n t a i n as p a r t i c u l a r cases those g i v e n i n ( 2 8 ) . 2.
Products w i t h D i r a c 6 D i s t r i b u t i o n s
P r o p e r t i e s o f t y p e (1) concerning p r o d u c t s w t h D i r a c 6 d i s t r i b u t i o n s and t h e i r p a r t i a1 d e r i v a t i ves a r e p r e s e n t e d now. Theorem 1 Suppose g i v e n a C " - r e g u l a r i z a t i o n
( V,
s @T
E
RGS and
E.E.
2 30 I; n v" c
(34)
Rosinger
v
Then t h e p a r t i a l d e r i v a t i v e s o f t h e D i r a c 6 d i s t r i b u t i o n have t h e f o l l o w i n g properties
# 0
DPBx
1)
E
-
AR(V, Sc)T ) , Y xo E R n y p E Nn,
R E
R
0
Y ( x - x ~ ) . D (~x~) ~= 0
2)
f o r every Y
0 E
E
AR(V,S
Y x,xOE
Rn,
p E Nn,R
E
1,
Cm(Rn) which s a t i s f i e s t h e c o n d i t i o n
v
q
E
( X - X ~ ) ~ . D( 'x ~) ~= 0
E
AR(V,S
D ~ Y ( O =) 0,
(35)
0T ) ,
N ~ q, ~p
or lql G R
I n particular
3)
(;> T ) , V
x,x0
E
Rn,p,q
E
N
n
REN,q $ p
0
and I q l > R Proof 1)
I n view o f (10) and (29) above, as w e l l as Theorem 3, S e c t i o n 3, Chap= t e r 3, i t f o l l o w s t h a t Dp6 = DPs t IR(V,S (.) T ) , V p
t h e r e f o r e , f o r g i v e n p E Nn and R DP6 = 0
E
AR(V,S
E
E
R,
Nn,
R E
1
the r e l a t i o n holds
@ T)
only i f Dps
(36)
E
IR(V,S
@ T)
But, o b v i o u s l y
Dps
E
Ts
C
T
hence, (36) w i l l y i e l d (37)
DPS E
I('V,S
@ T)
nT c I nT =
Q
i f (30) i s taken i n t o account.
Now, t h e r e l a t i o n s (37) and (10) o b v i o u s l y c o n t r a d i c t each o t h e r , thus the, p r o o f o f 1) i s completed f o r xo = 0 E Rn. I n t h e case o f a r b i t r a r y xo E R the proof i s s i m i l a r .
2) (38)
As above i n l ) , t h e r e l a t i o n f o l l o w s e a s i l y
YY.DP6= u(Y).DPs t IR(V,S @ T ) ,
Suppose g i v e n p (39)
E
Nn and R
w = u(Y).DPs
E
1.
V p
E
NnyR E
4
We d e f i n e w E (Cm(Rn))N by
231
PRODUCTS
Then
w
(40)
E
zJ
rR(v,s
T)
Indeed, f i r s t we prove t h a t D ~ W E
(41)
ri, Y
q E N ~ , lg
I a.
I n t h i s c o n n e c t i o n we n o t i c e t h a t (39) and t h e L e i b n i t z r u l e f o r p r o d u c t derivative yield
r
B u t DPfqers s a t i s f i e s ( 1 1 ) s i n c e s s a t i s f i e s t h i s c o n d i t i o n . T h e r e f o r e , (42) w i l l i m p l y thatnD9w s a t i s f i e s ( 3 ) . F u r t h e r , ( 4 2 ) and t h e f a c t t h a t (35) h o l d s f o r r E N , r < p , w i l l i m p l y t h a t Dqw s a t i s f i e s ( 2 ) , s i n c e as seen above, DPtq-rwsatisfies ( 1 1 ) . I n t h i s way, t h e p r o o f o f ( 4 1 ) i s com= pleted. NOW, we n o t i c e t h a t
w
(43)
E
v-
s i n c e (35) i s v a l i d f o r q E N n y q G p . B u t ( 4 2 ) and (43) t o g e t h e r w i t h (34) w i l l y i e l d
D
~
~
E
I
which i n view o f (18-20),
w E VL
~ Y~
EN^, v ~ Irl ~
,
Chapter 3, i m p l i e s t h a t
c lR(V,S @ T )
and t h e p r o o f o f (40A i s completed. Now ( 3 8 ) and ( 4 0 ) complete t h e p r o o f The case o f a r b i t r a r y xo E Rn f o l l o w s i n a s i m i l a r o f 2) f o r xo = 0 E R way.
.
3)
I t f o l l o w s from 2) by choosing Y(x)
, V x
= xq
E Rn
0
The f o l l o w i n g t h r e e theorems w i l l p r e s e n t p r o p e r t i e s o f p r o d u c t s i n v o l v i n g s e v e r a l f a c t o r s which a r e D i r a c 6 d i s t r i b u t i o n o r t h e i r p a r t i a l d e r i v a t i v e s . F i r s t , we p r e s e n t a r e s u l t on t h e n o n t r i v i a l i t y o f t h e powers o f t h e D i r a c 6 d i s t r i b u t i o n and i t s p a r t i a l d e r i v a t i v e s , which extends 1 ) i n Theorem 1. Theorem 2 Suppose g i v e n a C m - r e g u l a r i z a t i o n (V,S
(44)
vc
1; n
vm
Then, t h e r e l a t i o n s h o l d
@ T)
E
RG,
and
E .E .
2 32
)k #
(DP6,
0
E
AR(V,S
@
Rosi nger
T), Y
X ~ ER
n
,p
€Nn,
R
E N , k E N , k >1
0
Proof As i n t h e p r o o f o f l ) , Theorem 1, we o b t a i n t h e r e l a t i o n Dp6 = Dps t IR(V,S
@
T), Y p E Nn,R
E
fl
therefore (DP6)k = (DPs)k t ZR(V,S Then, f o r g i v e n p E Nn,R (DP6)k = 0
E
AR(V,S
E
R
6)T ) , V p
E
Nn,R
E
R,
k E N, k > I
and k E N, k 21, t h e r e l a t i o n holds
@ T)
only i f (45)
(DPs)k E I'l(V,S
But, i n view o f (18-20),
@
I R (v,s
T)
@ T)
Chapter 3, t h e h y p o t h e s i s (44) y i e l d s
c 1;
t h e r e f o r e (45) w i l l i m p l y t h a t ( D P S ) ~E
11
Now, t h e c o n d i t i o n ( 2 ) a p p l i e d t o (DPs)k w i l l y i e l d (46)
DpsV(O) = 0,
f o r a c e r t a i n 1-1
E
>u
Y v E N, v
N.
But (46) and (12) w i l l o b v i o u s l y c o n t r a d i c t each o t h e r .
0
An expected p r o p e r t y o f t h e p r o d u c t o f two p a r t i a l d e r i v a t i v e s o f t h e D i r a c d i s t r i b u t i o n concentrated i n d i f f e r e n t p o i n t s i s presented now. Theorem 3 Suppose t h e C m - r e g u l a r i z a t i o n (V,S ( 3 4 ) . Then
Dp 6, .Dq6 0
f o r any xo,yo
E
Rn,
@ T)
= 0 E AR(V,S yo xo # yo, and p,q
E
RG,
s a t i s f i e s the condition
@ T) E
Nn,R
E
fl.
Proof As i n t h e p r o o f o f l ) , Theorem 1, we o b t a i n t h a t
DP6
= DPs
xO
Dq6 YO
t lR(V,S
@ T),
Y x0
E
Rn,
t IR(V,S
@ T),
V yo
E
Rn, q
p E Nn,&
E
R
xO
= Dqs YO
E
Nn,& E fl
233
PRODUCTS
,s
where s x
E
S m a r e defined by
yo
0
(47) Therefore
-
DPBX .Dp6 = DPs .Dqs + lR(V,S(5) T ) , V xo,y0 0 yo yo yo R EN
(48)
E
Rn,p,q
€Nn,
L e t us denote w = DPs
.Dqs xo
yo
Since xo # yo, t h e r e l a t i o n s ( 4 7 ) and (11) w i l l y i e l d = 0, Y v
w,(x)
(49)
f o r a c e r t a i n li
w
E
EN.
EN,
v 2 1-1
,x
E
Rn,
B u t ( 4 7 ) , (11) and (49) o b v i o u s l y i m p l y t h a t
11n vm
t h e r e f o r e , i n view of t h e above h y p o t h e s i s ( 3 4 ) , as w e l l as t h e n o t a t i o n i n (18-20), Chapter 3, i t f o l l o w s t h a t
w
E
v
c lR(V,S
@ 7)
Now, i n view o f ( 4 8 ) , t h e p r o o f i s completed.
0
An e x t e n s i o n o f t h e p r o p e r t y i n (1) , r e s p e c t i v e l y i n 3) , Theorem 1, concern= i n g products i n v o l v i n g p o l y n o m i a l s and t h e D i r a c 6 d i s t r i b u t i o n o r i t s par= t i a l d e r i v a t i v e s i s presented n e x t . T h a t p r o p e r t y i s ty i c a l f o r t h e m u l t i = p l i c a t i o n i n t h e c h a i n s o f q u o t i e n t a l g e b r a s c o n t a i n i n-%a g t e istributions and cannot be o b t a i n e d w i t h i n t h e framework o f d i s t r i b u t i o n s . F o r t h e sake o f s i m p l i c i t y we s h a l l deal o n l y w i t h t h e one-dimensional case n = l . Theorem 4 Suppose t h e C m - r e g u l a r i z a t i o n ( V , S Then
@ T)
( X - X ~ ) ~ (( xD) ~) ~=~ 0~ E AR(V,S
E RG,
@ T),
s a t i s f i e s t h e c o n d i t i o n (34).
Y x0 E R',
p, k
E
N , i E N,
0
p > R +
1, k > 2
Proof I n view o f t h e L e i b n i t z r u l e f o r p r o d u c t d e r i v a t i v e s (see 4) i n Theorem 3, S e c t i o n 3, Chapter 3) s a t i s f i e d b y t h e d e r i v a t i v e o p e r a t o r s a c t i n g between t h e q u o t i e n t algebras
E . E . Rosinger
234
i t follows e a s i l y t h a t the r e l a t i o n
D ( ( X - X ~ ) ~ ~ (~x( ) D) ~~ )=~ (,p + l ) ( x - x o ) p ( D q 6 x ( x ) ) ~+ 0
0
(50) t
k(x-xo)Ptl(Dqsx
(x))
k- l D q t l
6,
0
k>
R
holds i n AR(V,S T ) , f o r any p,q,k, i n Theorem 1, i t f o l l o w s t h a t
0
(x)
'Dq6,
(XI
0
E
E
AR(V,S
N, k 2 2 .
But, i n view o f 3)
(.I> T )
0
whenever p r e 1a t i on
t
1 > q and p
t
1
L.
Therefore, ( 5 0 ) and ( 5 1 ) i m p l y t h a t t h e
il
i s v a l i d i n A ( V , S It) -. T ) , f o r any p,q,k,
il E N,
k > 2 and p >max
{q,il}.
But t h e p r o d u c t (x-xo)pt1(Dq6x
( x) ) ~ 0
i n t h e l e f t hand term o f ( 5 2 ) i s computed i n A'+'(V,S i n view o f 3 ) i n Theorem 1, i t f o l l o w s t h a t (x-xo)pt1(Dq6x
(53)
whenever p t 1
T ) , therefore,
G;
( x ) ) ~= 0 E ALtl(V,S
0 > q and p t 1 > il t
t
T)
1.
NOW, t a k i n g q = p i n ( 5 2 ) and ( 5 3 ) , t h e p r o o f w i l l be completed.
3.
0
Application t o a R i c c a t i D i f f e r e n t i a l Equation
An a p p l i c a t i o n o f t h e r e l a t i o n s ( l ) , r e s p e c t i v e l y 3) i n Theorem 1, S e c t i o n 2, t o t h e s o l u t i o n o f a R i c c a t i d i f f e r e n t i a l e q u a t i o n i s presented now. Theorem 5 The R i c c a t i d i f f e r e n t i a l e q u a t i o n y ' = xptqy(ytl)
t Dqt16(x)
,x
E
R',
w i t h p,q E N, p 2 1, has t h e general s o l u t i o n
1 xP+q+l
Y(X) = ce-
t D%(x),
x
E
R', c
E
( - ~ , o I,
p+q+l -1
i n each o f t h e q u o t i e n t algebras AR(V,S
@T),
R
E
N, R G q ,
provided t h a t t h e Cm-regularization (V,S (34).
&) T )
E
RG,
s a t i s f i e s the condition
PRODUCTS
235
Proof Assume c E R ’ and d e f i n e Y
E
Cm(R’) by
xP+q+l ~ ( x =) ce-
p+q+l -1 , x
6
R’
Then
l/Y
E
Cm(R’),
Y
c E (-m,O]
therefore, denoting T = 1 / Y t Dq6 we o b t a i n t h a t
G>
But A (V,S T i i s an a s s o c i a t i v e and commutative a l g e b r a w i t h t h e u n i t element 1 E C ( R ), t h e r e f o r e t h e f o l l o w i n g r e l a t i o n s a r e v a l i d w i t h i n i t
~P+~T(T+I)
= xP+q(o%s)*
+
2 x p + q o q q 1 / ~ )+
(54)
+
X ~ + ~ ( ~ +/ Y xp+qDq6 ) ~
+
X ~ + ~ ( ~ / Y )
p r o v i d e d t h a t c E (-m,O1, NOW, i n view o f 3) i n Theorem 1, S e c t i o n 2, i t f o l l o w s t h a t (55)
I? P xp+qDq6 = 0 E A ( V , S V)
whenever p 2 1 and R
T)
< q.
The r e l a t i o n s (54) and (55) w i l l y i e l d w i t h i n t h e q u o t i e n t a l g e b r a AR(V,S @ T) t h e r e l a t i o n (56)
x P + ~ T ( T + ~=) x P + q ( i / y ) ( i
+ i/q, Y
c E ( - ~ , 0 1,
p r o v i d e d t h a t p 2 1 and R G q . B u t (56) o b v i o u s l y i m p l i e s t h a t T i s t h e s o l u t i o n o f t h e R i c c a t i d i f f e r e n = 11 t i a l e q u a t i o n considered. 4.
V a l i d i t y o f Formulas i n Quantum Mechanics
I n t h e one dimensional case n = l , t h e D i r a c 6 d i s t r i b u t i o n and t h e Heisen= berg d i s t r i b u t i o n s
6+
=
6- =
f (6
1 1 + -(-)) 711 x
11 4 (6 - -(-)) 711 x
a r e u s u a l l y assumed i n Quantum Mechanics t o s a t i s f y t h e f o l l o w i n g r e l a t i o n s :
E.E.
2 36
-
1
(57)
A2
(58)
: 6 = -
( 5 9)
6- -
(60)
(-)A
1
=
X
1 1 1 (x)2 = - ; ; I (F) 1
1
2 -
Rosinger
D6
-&
1
D6 1
1
(yT)
1
- v;;T (3)
D 6
These r e l a t i o n s were proved i n [ 68,140,71 ,using s p e c i a l , p a r t i c u l a r regu= l a r i z a t i o n s f o r those o f t h e o p e r a t i o n s i n v o l v e d which cannot be d e f i n e d i n t h e framework o f t h e d i s t r i b u t i o n s . F o r i n s t a n c e , t h e expression ti2
- y71
1 2
($
i n t h e l e f t hand t e r m o f (57), was r e g u l a r i z e d ' e n b l o c k ' i n [140,1 o u t r e g u l a r i z i n g any o f i t s two terms
, with=
1
62 o r (y)z
The aim o f t h i s s e c t i o n i s t o prove t h a t t h e above r e l a t i o n s (57-60) a r e v a l i d w i t h i n t h e framework o f a l a r g e c l a s s o f chains o f q u o t i e n t algebras c o n t a i n i n g t h e d i s t r i b u t i o n s . The i n t e r e s t i n t h i s r e s u l t i s i n t h e f a c t t h a t - as usual - t h e l a r g e s i z e o f t h e mentioned c l a s s w i l l i m p l y a con= v e n i e n t s t a b i l i t y p r o p e r t y o f t h e r e l a t i o n s (57-60), thus making them i n = de endent o f t h e p a r t i c u l a r r e g u l a r i z a t i o n s i n v o l v e d . I n a d d i t i o n , w i z i n t e ramework o f t h e mentioned chains o f q u o t i e n t algebras, t h e r e l a t i o n s (57-60) w i l l be v a l i d i n t h e usual a l g e b r a i c sense, w i t h a l l t h e o p e r a t i o n s involved - addition, substraction, m u l t i p l i c a t i o n , derivative - effectuated w i t h i n these a1 gebras
T-5-
.
The r e s u l t i n t h i s c o n n e c t i o n i s presented i n Theorem 6, a f t e r s e v e r a l pre= 1im i n a r y d e f i n i t i o n s .
z6.
Then we d e f i n e t h e sequence Suppose g i v e n a sequgnce o f f u n c t i o n s s E o f f u n c t i o n s t, E ( C (R1))N by t h e c o n v o l u t i o n s i n D ' ( R ' ) tsv= s V
1 Y * (1)
v
E
N,
which a r e o b v i o u s l y w e l l d e f i n e d , s i n c e , i n view o f (11)
V v
s y E P(R'),
E
N,
while
(5)
E D'(R1)
Moreover (61)
ts E Srn,
We denote b y
.
= ($1
PRODUCTS
237
t h e v e c t o r subspace i n Sm generated by Dpts, with p E N .
I n view o f Lemmas 1 and 2 below, i t f o l l o w s t h a t t h e r e l a t i o n h o l d s (see (16) and ( 1 7 ) )
(v"
(62)
'+
~
~
~T ~~ n)) pS =
m ( + j(
Q
Now, i n view o f ( 6 2 ) i t f o l l o w s t h a t t h e r e e x i s t C m - r e g u l a r i z a t i o n ( V , S , + T ) E RG, such t h a t (63)
Ps'
sic.,\
T
We s h a l l denote by RG;
the s e t o f a l l the Cm-regularizations ( V , S
c+
T ) E RG,
which s a t i s f y ( 6 3 ) .
Theorem 6
Z6 such t h a t t h e r e l a t i o n s (57-60)
There e x i s t sequences o f f u n c t i o n s s E a r e v a l i d w i t h i n t h e q u o t i e n t algebras
(?
AR(V,S
T ) ,R
_I
Cm-regu a r i z a t i ons
f o r any
u,
E
V,S
i+>
T)
E
RG;.
Proof Assume
\y
E P(R'
such t h a t dx = 1
and (13) h o l d s .
We d e f i n e t h e n s
E
s"
Sv(X) = ( v + l ) Y ( ( v + l ) x ) , U v
by E
N, x E R '
As seen i n S e c t i o n 1, i t f o l l o w s t h a t
s E Z6 F o r p E N, l e t us denote
I
Mp = sup(lDPY(x)l
x E R'I
and l e t us assume t h a t
supp Y c for a suitable L
E
[ -L,L
I
( 0 , ~ ) . Then, i t f o l l o w s e a s i l y t h a t
I x ~ + ~ D ~ s ~ (< x )M I .Lp,Y v t h e r e f o r e s i s a '&sequence'
E N , x E R', p E N P i n t h e sense of [ 138,140,7].
2 38
E.E.
Rosinger
(VS @ T )
Assume now g i v e n a C " - r e g u l a r i z a t i o n (29) and ( 6 3 ) , we o b t a i n t h a t
Then, i n view o f
E RG;.
E)
s, t s E S
T
t h e r e f o r e (10) and (61) w i l l y i e l d t h e r e l a t i o n s (64)
6 , = $(s
1 + ;;r t s ) + IR(V,S
GT),V R
(65)
6- = $(s
- ;;1r t s ) +
G-)
(66)
a),( 1
= s ts +
IR(V,S
E
8,
T), Y R E
8,
T), V R E
IR(V,S
We d e f i n e now t h e sequences o f f u n c t i o n s t ' , t " (67)
tl = ( s
+
1
t" = (s
tJ,
711
-
1
711
8
,t", E(Cm(R'))N by
t S ) 2 , t"' = s ts
and r e c a l l t h a t t h e f o l l o w i n g r e l a t i o n s were proved i n [ 140,71 (68)
t ' ,t" , t " I
(69)
=
=
(70) (71)
=
E
-
1
711
-i-
TI1
-
sm D6-
?1~ (F) f1
D6--;1 ; 2 ( ~1 )
t o 6
B u t t h e r e l a t i o n s (64-67) and (69-71) w i l l o b v i o u s l y y i e l d (58-60). There= f o r e , i t o n l y remains t o prove t h e r e l a t i o n ( 5 7 ) . I n t h i s r e s p e c t we no= t i c e t h a t from t h e d e f i n i t i o n o f 6+ i t f o l l o w s t h a t i n each o f t h e associa= t i ve and commulati ve a1 gebras AR(V,S
@ T),
with R
E
1,
t h e r e l a t i o n holds 6: = i ( 6
+
1 1
-TI1 ( - ) ) 2x
=
as2 -
1
1 ($2
+
1
-2-3
(1d 6
which i n view of (58) and (60) w i l l o b v i o u s l y i m p l y t h e r e l a t i o n ( 5 7 ) . And now, t h e two lemmas - t h e f i r s t one o f a r a t h e r independent i n t e r e s t concerning p r o p e r t i e s o f t h e d i s t r i b u t i o n s i n D ' ( R ' ) , needed i n t h e p r o J f of t h e r e l a t i o n (62). We denote b y
D p
l)
t h e s e t o f a l l t h e d i s t r i b u t i o n s i n D'(R') which have t h e s u p p o r t a f i n i t e subset i n R'. F u r t h e r , we denote b y
Db( R l ) t h e s e t o f a l l t h e d i s t r i b u t i o n s T E D'(R') w i t h t h e p r o p e r t y t h a t
2 39
PRODUCTS
9 GP
~ ~ , . . . , h pR',~
for suitable p EN,
with
hp
# 0.
Obviously D i ( R 1 ) i s a v e c t o r subspace i n P ' ( R 1 ) and
c P&(R ')
Dk(R') moreover (73)
C ~ ( R ' ) n D;(R')
=
COI
The d i s t r i b u t i o n s
(T)1
with m E
P'(R'),
E
N, m 2 1,
have t h e f o l l o w i n g p r o p e r t y . Lemma 1
1
6 c " ( R ~ ) t D'(R'), Y m
(-)
EN,m 21
Proof Assume t h a t t h e above statement i s f a l s e . (74)
1
(3)= y
f o r suitable Y
E
(75) = where we denoted
T
t
C"(R')
x
E
Then
and T
E
P&(R').
It follows that
C"(@
:= R ' \ { O } .
B u t i n view o f ( 7 2 ) we have
q GP for certain p E
N, Ao,...,X
(77)
s
S' =
(i
E
P
E
R',
w i t h Xp
or,(:)
Now, t h e r e l a t i o n s (75-77) w i l l y i e l d
(78)
S ' = P(D)x
E
C"(Q)
where we denoted (79)
P(D) =
I: q E N q G P
Xq Dq
# 0.
Then o b v i o u s l y
2 40
E.E. Rosinger
Then, i n view o f ( 7 3 ) , t h e r e l a t i o n s ( 7 7 ) and (78) w i l l i m p l y
S'
= 0 E z l p
which t o g e t h e r w i t h (74-76) w i l l y i e l d (80)
P(D)
1 (siiii) = P ( D ) Y on R
I f we compute t h e d e r i v a t i v e s i n t h e l e f t hand t e r m o f (80) a c c o r d i n g t o ( 7 9 ) , then t h e r e l a t i o n (80) w i l l y i e l d
9 GP Therefore, t a k i n g t h e l i m i t x
xp
(-1)P
+
0 i n t h e above r e l a t i o n , we o b t a i n
= 0
which c o n t r a d i c t s t h e assumption i n ( 7 2 ) t h a t X
P B e f o r e p r e s e n t i n g t h e second lemma, l e t us denote S :
IS
E
Sm
<s,*> E
Db ( R ' ) }
i s a v e c t o r subspace i n Sm and
: Obviously S
(81)
=
# 0.
'Crn(Rl)
'S:
Q
=
F i n a l l y , f o r a g i v e n sequence o f f u n c t i o n s t E Sm, we denote by
pi t h e v e c t o r subspace i n Smgenerated b y
DPt, with p
E
N.
Lemma 2 I f t E Sm and t $ Q, t h e n
('Cm(
R1)
t
$1
n pt =
Proof F i r s t we prove t h e i m p l i c a t i o n I. Assume t h a t i t i s f a l s e and l e t
2
0
4
+ D;(R~)
~ " ( ~ 1 1
241
PRODUCTS
Then (82) y i e l d s s = u(Y)
(84)
+
c
=
S'
hq D q t
qE N 9GP
f o r s u i t a b l e Y ECm(R1), s 1
E
p
,S ;
E
N , Ao y . .
.,Ap
E R'.
But (83) and (84) w i l l imply t h a t hp # 0
(85)
We denote now P(D) =
(86)
hq Dq
C
aeN 9
Let
x
€Cm(R') be such t h a t
(87)
P(D)x = 'Yon R '
Then (88) since,
t ' = t - U(X)
E
84) and (87) y i e l d P(D)t' = P(D)t
But (88
EUA(R')
S",
- u(P(D)x)
= S' E ! i ! I
implies
=
x+
E
C"(R')
+
Ud(R')
which c o n t r a d i c t s the hypothesis. Conversely, we prove t h e i m p l i c a t i o n *
.
Assume again t h a t i t i s f a l s e and Cm(R')
+
UA(R')
d,*>
E
= y t
then
f o r certain Y (89)
E
Cm(R') and t l E S m such t h a t
s = P ( D ) t ' E Si
where P(D) i s a s u i t a b l e l i n e a r d i f f e r e n t i a l operator o f type (86) which s a t i s f i e s the c o n d i t i o n ( 8 5 ) . I t f o l l o w s t h a t hence
= P(D)Y
+
<s,*>
E.E.
2 42
P ( D ) t = u(P(D)") for a certain v
+
Rosinger
s + v
Vm. Then, i n view o f (89) we o b t a i n t h a t
E
since obviously
(91)
urn c s;
Now, t h e f o l l o w i n g two cases a r e p o s s i b l e . Case 1 : P ( D ) t $ Q. Then (90) w i l l y i e l d
which c o n t r a d i c t s t h e h y p o t h e s i s . Case 2 : P ( D ) t
E
Q
Then P(D)
=
= 0
E
D'(R1)
t h e r e fo r e
E
C"(R')
hence, i n view of (91)
i n which case
and t h e h y p o t h e s i s i s a g a i n c o n t r a d i c t e d .
0
5. The E x i s t e n c e o f t h e Sequences o f Functions i n Z When c o n s t r u c t i n g sequences o f f u n c t i o n s s E Zg i t i s obvious t h a t t h e con= d i t i o n s (10) and (11) can e a s i l y be s a t i s f i e d . T h e r e f o r e , t h e o n l y d i f f i = c u l t y i n t h e general n 2 1 dimensional case i s t o s a t i s f y t h e c o n d i t i o n (12) i n t h e d e f i n i t i o n o f Zg . We s h a l l now c o n s t r u c t sequences o f f u n c t i o n s s E Zg which a r e a g e n e r a l i = z a t i o n t o t h e n > 1 dimensional case o f t h e one dimensional example g i v e n i n S e c t i o n 1, where t h e e s s e n t i a l p r o p e r t y was expressed i n ( 1 3 ) which i s t h e one dimensional e q u i v a l e n t o f t h e c o n d i t i o n ( 1 2 ) i n t h e d e f i n i t i o n o f zg. Suppose t h e r e f o r e g i v e n Y
E
D(Rn), such t h a t
243
PRODUCTS
(92)
l n y(x)dx = and d e f i n e t h e sequence o f f u n c t i o n s s E ( c " ( R ' ) ) ~ by
...
)X~ sv(X) = p 1 ( ~ ) P ( v ) Y ( ~ ~ ( v ,...,p(v)xn),Y n n where t h e mapping (93)
N
(94)
3
v
+
,...,
v E N,x=(xl
~ ( v ) E) Nn
p(v) = ( P ~ ( v )
n i s c o n s t r u c t e d a c c o r d i n g t o t h e f o l 1owing r e 1 a t i o n s (95-98) F i r s t , we d e f i n e a mapping N (95)
v
3
-+
-+
.
k ( v ) E N g i v e n b y (see S e c t i o n 1)
k ( 0 ) = 0 and k ( v t 1 ) = k ( v )
Then we d e f i n e a mapping N 3 v
,..., xn)E
t
n(vtl),V v E N
h ( v ) E N by
h(O)=O and h ( v t 1 ) = h ( v ) t v t 1, Y v E N
(96)
F u r t h e r , we d e f i n e a mapping N3 v e(v) = (h(v),
(97)
...,h ( v ) ) ,
-+
e ( v ) E Nn by
V v E N
F i n a l l y , we d e f i n e t h e mapping (94) by t h e r e l a t i o n s (see S e c t i o n 1 ) .
...,1)
(98.1)
p ( 0 ) = (1,
(98.2)
{p(k(v)tl),...,p(k(vtl))l
E
Nn = P(n,v+l)
t
e(vtl), Y v E N
A f i r s t p r o p e r t y o f t h e sequence o f f u n c t i o n s s d e f i n e d i n ( 9 3 ) i s presen= t e d now. Lemma 3
s
E
S" and s s a t i s f i e s t h e c o n d i t i o n s (10) and (11).
Proof I t f o l l o w s from (92) and t h e f a c t t h a t t h e mapping i n ( 9 4 ) has t h e p r o p e r t y
l i m p (v) = v+m i
my
Y i E
{l, ...,n }
which f o l l o w s e a s i l y from (95-98).
0
The e s s e n t i a l p r o p e r t y o f t h e sequence o f f u n c t i o n s s d e f i n e d i n (93) can be p r e s e n t e d now. Proposition 2 The f o l l o w i n g t h r e e c o n d i t i o n s a r e e q u i v a l e n t : (99)
s
(100)
Ws(0) i s column w i s e n o n s i n g u l a r
E
Z6
R
n I
2 44
E.E.
DPy(O) # 0,
(101)
V p
Rosinger
EN^
Proof I n view o f t h e d e f i n i t i o n o f z6 i n S e c t i o n 1, as w e l l as o f Lemma 3, t h e c o n d i t i o n s (99) and (100) a r e o b v i o u s l y e q u i v a l e n t . We s h a l l now e s t a b l i s h t h e equivalence between t h e c o n d i t i o n s (100) and (101). F i r s t , we s h a l l compute t h e Wronskian t y p e m a t r i x W,(O). i t follows easily t h a t
I n view o f (93),
Dps "( 0 = ( ~ ( V ) ) ~ ~ ~ D V~ ~u (E O N , ) ,p € N n Y where we denoted e = (1 . . . , l )
E
N
n
Therefore (102)
w,
(0
where (see Sect A = w h i l e B i s a diagonal m a t r i x , w i t h t h e diagonal elements
(103)
DP(')
~ ( 0 v) E~ N
Now, according t o Theorem 8 below, A i s column w i s e n o n s i n g u l a r . Indeed, i n view o f Lemma 4 below, i t s u f f i c e ? t o show t h a t A s a t i s f i e s t h e condi= t i o n (104). Assume t h e r e f o r e g i v e n u, u E N . Then we choose m E N such that
u = n(mt1) - 1
>a
and k(m)
>;
Now we choose vo = k(m)
t
l , , . . , vu = k(mt1)
Then t h e c o n d i t i o n s *) and **) i n (104) a r e o b v i o u s l y s a t i s f i e d . Moreover, i n view o f (98.2) as w e l l as Theorem 8 below, i t f o l l o w s t h a t t h e c o n d i t i o n **? i n (104) i s a l s o s a t i s f i e d . T h e r e f o r e A i s indeed column w i s e nonsingu= lar. I n t h i s case, t h e r e l a t i o n s (102) and (103) t o g e t h e r w i t h Lemma 4 below im= p l y t h a t W,(O) i s column wise n o n s i n g u l a r o n l y i f t h e c o n d i t i o n (101) i s satisfied. 0 The main r e s u l t i n t h i s s e c t i o n i s presented now. Theorem 7 The s e t Z6 o f sequences o f f u n c t i o n s i s n o t v o i d .
245
PRODUCTS
Proof I n view of P r o p o s i t i o n 2 , i t s u f f i c e s t o show t h e e x i s t e n c e o f Y which s a t i s f y t h e c o n d i t i o n s ( 9 2 ) and (101). I n t h i s connection, we d e f i n e a E Cm(Rn) by xl+. . .+xn , V x = (x1,...,xn) a(x) = e and assume
qRn)
E
E
E
aRn)
Rn
such t h a t
f3 2 0 on Rn and f o r a c e r t a i n neighbourhood V o f 0
E
Rn
B = l o n V Then o b v i o u s l y
I
K =
I f we d e f i n e now \y
a(x)B(x)dx > 0
Rn
=
\y
clf3
E
D(Rn) b y
/K
t h e n t h e c o n d i t i o n ( 9 2 ) w i l l o b v i o u s l y be s a t i s f i e d , w h i l e t h e r e l a t i o n s Dp\y(0) = 1/K > 0, Y p EN’,
w i l l imply (101).
0
I n case a r b i t r a r y p o s i t i v e powers o f t h e D i r a c 6 d i s t r i b u t i o n a r e t o be d e f i n e d w i t h i n t h e chains o f q u o t i e n t a l g e b r a s c o n t a i n i n g t h e d i s t r i b u t i o n s , the following r e s u l t i s useful. Corollary 1
Proof I f we choose i n t h e p r o o f o f Theorem 7,
B
E
CY(Rn)
then i t follows t h a t \y E
CY(Rn)
therefore
s
E
zg “(CY(R
n N
1)
Lemma 4 The i n f i n i t e m a t r i x o f complex numbers A = (aVu
I
v,u
E
N ) i s column wise
E.E.
246
Rosinger
nonsingular, o n l y i f Y;,a€N:
..............a vO‘
I :
Ii
# O
Proof I t f o l l o w s e a s i l y from t h e d e f i n i t i o n i n S e c t i o n 1.
0
And now, t h e theorem on general ized Vandermonde determinants ( f o r n o t a t i o n s see S e c t i o n 1) whose p r e s e n t form, as w e l l as p r o o f was o f f e r e d by R.C.King. Theorem 8 Suppose g i v e n n E N
a
n 2 1.
Then f o r each a E Nn, a 2 e = (1, holds
I
:
...*1) E Nn
and II E N,R 21, t h e r e l a t i o n
I
where p ( j ) = (pl(j) ¶...,pn( j ) ) , f o r 1 G j Q R. Remark 1 The value o f t h e d e t e r m i n a n t depends o n l y on n,a,p( 1) depend on a. Proof
L e t s consider the determinant
.... ,p(R)
and does not
PRODUCTS
Al = d e t ( ( a + p ( o ) ) P ( T ) ) ,
247
where 1 Q o r T Q II
F o r 1 < T < R y t h e -c-th column i n Al i s
... x
(altpl(l))P1(T)x
C,(d
=
x i f a = ( a ly...,
( a n t p n ( l ) ) Pn
an)
...
'
x (antpn(a)) Pn
.
c o n s i d e r t h e column
p(
R)P(T) where 0' = 1 whenever i t occurs. We o b t a i n t h e n (105)
c
C2(T) = C i ( T )
A
where t h e sum C i s taken f o r a l l 1 Q A Q R s u c h t h a t I p ( X ) I A
Introducing t h e determinant A2 = d e t ( p ( ~ ) ' ( ~ , where 1 < u y T < 1,
i t f o l l o w s from (105) t h a t A2 = Al, s i n c e C ~ ( T )i s t h e r - t h column i n A 2 . We s h a l l now s i m p l i f y A2 w i t h t h e h e l p o f t h e f u n c t i o n F : N x N -+ N d e f i n e d by
1
i f k = O
F(h,k) = h(h-1) ...(h - k t l )
if k 2 1
which o b v i o u s l y s a t i s f i e s t h e c o n d i t i o n s
( 106 1
F(h,k)
= 0 * h
- k + 1<0 * h
< k
E.E.
248
(107)
F(h,k) = h !
Now, f o r 1
< T < a, we
Rosinger
d e f i n e t h e column
I. Then i t f o l l o w s t h a t
where
(108.1)
t h e sum w i t h Ji
(108.2)
c
i s taken f o r a l l J = J 1 u
... u J n
# 0
c ~1,2,...,pi(~)-1}, f o r 1 < iQ n ,
p ( ~ ) - ( ~ J 1 ~ , . . . , ~ J n ~ )w, i t h IJi o f elements i n Ji.
q(T) =
I
denoting t h e number
The r e l a t i o n (108) can o b v i o u s l y be w r i t t e n under t h e form
( 109)
C,(T)
=
C,(T) +
c
(~(-j))C2(i(~,J1,...,Jn)), Y 1 Q T G R , jeJ
where (108.1) and (108.2) a r e s t i l l v a l i d and X(T,J~,...,J~) E N i s uniquely d e f i n e d by
p(i(r,J1,...,Jn))= therefore 1
P(T)
-
(lJ1ly...ylJn~)
.
X(-r,J1,... ,Jn) G R and Ii(.r,J1,.. ,Jn) I < Ip(-c)I .
Denoting by A t h e determinant w i t h t h e columns C3( 1), t i o n (109) i m 8 l i e s A3 = A2 and thus
(110)
. ..,C3(a) , t h e
A3 = A1
Now, t h e r e l a t i o n (106) g i v e s f o r any 1 Q
U,T
< 11
t h e equivalences
rela=
2 49
PRODUCTS
T h e r e f o r e , t a k i n g i n t o account (107) and t h e form o f t h e columns C 3 ( ~ ) , w i t h 1 < T G R , t h e r e l a t i o n (110) w i l l end t h e proof.
6.
0
Remark on N i l p o t e n t Elements i n t h e Q u o t i e n t Algebras
We s h a l l g i v e n an example i l l u s t r a t i n g t h a t (111)
Sm = 0 EA'(V,S
/dT) PS
T)
= 0 EAR(V,S@
~ I , m E N , m > 2 , II E N , R 2 1 and ( v , s @ for s E A ' ( V , S ~ T w i t h s E Z6. I n o t h e r words, t h e chains o f q u o t i e n t a l g e b r a s
Q.
T)E
RG,,
T ) , with R EN,
A (V,S
0 -\
c o r r e s p o n d i n g t o C"-regularizations ( V , S T) g i v e n i n ( 3 3 ) , admit non= t r i v i a l n i l p o t e n t elements. F o r t h e sake o f L i m p l i c i t y we s h a l l d e a l w i t h TfG-GG dimensional case n = l and choose t h e C - r e g u l a r i z a t i o n (v,s T) so t h a t (see (29-32))
e,
v = Irn6 n v" s ' L e t us d e f i n e a sequence o f f u n c t i o n s t E ( c m ( R 1 ) )N b y (112)
7 =
(113)
t,(x)
03
7&,T=T
=
xSJX),
V w
x
EN,
E
R'
then obviously t = U ( i d R i ) . S E UCm(Rl).T
@ T),
AR(V,S
C
V R E
t h e r e fo r e
(114)
s
= t t
r'l(v,s @ T )
E A~(V,S
@ TI, v
R
E N
We s h a l l show t h a t
(115)
S # 0
E AR(V,S
@ T),
Y R E
8, R 2 1
Indeed, o t h e r w i s e (114) w i l l y i e l d
t h e r e f o r e , i n view o f ( 2 0 ) , Chapter 3, we o b t a i n t h a t
@ T).
f o r c e r t a i n v . E VQ. and w . E A'l(V,S (see (18) , Chapter 3) i t )allows t h a t DPvi E
But by the d e f i n i t i o n o f VL
V , Y 0 Q i G h , p E N , p
Then, t h e e x p r e s s i o n o f V g i v e n i n (112) y i e l d s DPvi
E
7,;
Y 0 < iG h , p
E
N , p GII
250
Rosinger
E.E.
hence, i n view o f c o n d i t i o n ( 2 ) , we o b t a i n (117)
Dpvi ( 0 ) = 0, U 0 G i
E
N, p G R , V E
N, v 2 p
V
,,
for a suitable E N. B u t , i n view o f t h e L e i b n i t z r u l e f o r p r o d u c t d e r i = v a t i v e , t h e r e l a t i o n s (116) and (117) w i l l o b v i o u s l y i m p l y
< R,v
D P t (0) = 0, Y p E N, p (118) V On t h e o t h e r hand (113) y i e l d s
E
N,
2p
D P t ( x ) = p D p - l s v ( x ) t xDpsv(x), V p V
E
N,p > l , v E
N, x E R'
hence (119)
DPtV(0) = p Dp-lsv(0),
Therefore,
i n case R 21, t h e r e l a t i o n s (118) and (119) w i l l i m p l y
V p
Dqs (0) = 0, Y g E N , q V
E
N, p 2 1, v E N
4 R -1, v
EN,
v 2p
which o b v i o u s l y c o n t r a d i c t s (12) and thus ends t h e p r o o f o f (115).
Now, we s h a l l show t h a t (120)
sm = o
EA~(v,s@
TI, Y m,RE
N, m > R
Indeed, i n view o f (114) we o b t a i n
sm=
tmt
I~(v,s@ T)
v
E A ~ ( V , S @ T), m
EN,
m
RE II
hence, according t o (113) and (10) i t f o l l o w s t h a t
sm=
xm(61m E A ~ ( V , S @ T),V
But t h e m u l t i p l i c a t i o n i n A'(
sm =
V,S@
m E N , m 2 1 , R~ ~
T) i s a s s o c i a t i v e , thus
xm 6 ( 6 p - 1
and i n view o f 3) i n Theorem 1, S e c t i o n 2 xm6 =
o
E A ~ v , s @
T),U m
EN,
m
>I,
m >R
which means t h a t t h e p r o o f o f (120) i s completed. F i n a l l y , t h e r e l a t i o n s (115) and (120) w i l l o b v i o u s l y y i e l d (111).
CHAPTER 9 LINEAR INDEPENDENT FAMILIES OF D I R A C 6 DISTRIBUTIONS AT A POINT
0.
Introduction
The r e p r e s e n t a t i o n s o f t h e D i r a c 6 d i s t r i b u t i o n c o n s t r u c t e d andmused i n Chapter 7 and 8, were g i v e n b y weakly convergent sequences o f C -smooth f u n c t i o n s which s a t i s f y t h e c o n d i t i o n o f s t r o n g presence on t h e s u p p o r t (see (100) i n Chapter 7 and ( 1 2 ) i n Chapter 8 ) . An immediate consequence o f t h i s c o n d i t i o n was t h e nonsymmetr o f t h e mentioned r e p r e s e n t a t i o n s , which means t h a t t h e D i r a c 6 i s t r i u t i o n and i t s p a r t i a l d e r i v a t i v e s a r e n o t i n v a r i a n t under t h e independent v a r i a b l e t r a n s f o r m (see S e c t i o n 10, Chapter 1).
+
(1)
Rn3 x
+
a x E Rn, w i t h a = -1,
w i t h i n the chains o f q u o t i e n t algebras containing t h e d i s t r i b u t i o n s
.
The aim o f t h i s c h a p t e r i s t o p r e s e n t t h e f o l l o w i n g s t r o n g e r r e s u l t : Apply= i n g t o any g i v e n p a r t i a l d e r i v a t i v e DP6, w i t h p E N", o f t h e D i r a c 6 d i s = t r i b u t i o n t h e f o l l o w i n g independent v a r i a b l e t r a n s f o r m s w
(2)
Rn3 x
-+a a x E Rn
, with
a E R1 \ { O )
we o b t a i n l i n e a r independent elements (3)
Dp6( aox) ,. .. ,Dp6( a,x)
w i t h i n t h e chains o f q u o t i e n t a l g e b r a s c o n t a i n i n g t h e d i s t r i b u t i o n s , pro= v i ded t h a t (4)
ao,.
. . ,a m E R'
\
{Ol a r e p a i r w i s e d i f f e r e n t
I n S e c t i o n 3, t h i s r e s u l t i s extended i n o r d e r t o i n c l u d e a l s o g e n e r a l i z e d D i r a c 6 d i s t r i b u t i o n s which have t h e f o r m (5)
lim a+m
an
s(ax)
1. Compatible Q u o t i e n t Algebras and Independent V a r i a b l e Transforms W i t h i n t h i s Chapter, i t w i l l be c o n v e n i e n t t o c o n s i d e r t h e independent v a r i a b l e t r a n s f o r m d e f i n e d i n S e c t i o n 10, Chapter 1, under an a l t e r n a t i v e form p r e s e n t e d now. Suppose g i v e n an independent v a r i a b l e t r a n s f o r m
251
E.E. Rosinger
252
(6)
0:
* Rn,
Rn
w
E
cm
then we can o b v i o u s l y d e f i n e t h e a l g e b r a homomorphism w : ( CTRn)IN *
(7)
( C"pn))N
by ( W ( S ) ) ~ ( X=) s V ( w ( x ) ) , V
(7.1)
S
E ( C TRn))N, v EN,
XE
Rn
We s h a l l say t h a t t h e independent v a r i a b l e t r a n s f o r m w i s i n v e r t i b l e , o n l y if (8)
: Rn
w
-+
Rn e x i s t s and w - l E Cm
Suppose now g i v e n a q u o t i e n t a l g e b r a (see S e c t i o n 2 , Chapter 1)
A = A / I E AL cTRn)
(9)
We s h a l l say t h a t t h e q u o t i e n t a l g e b r a A and t h e independent v a r i a b l e t r a n s = form w a r e compatible, o n l y i f (see (100.2) and (100.3), Chapter 1) w(A) c A and w ( I )
(10) where
c 7
i s t h e mapping d e f i n e d i n ( 7 ) .
Proposition 1 I f t h e q u o t i e n t algebra A i n ( 9 ) and t h e independent t r a n s f o r m w i n ( 6 ) a r e compatible, t h e n t h e mapping
d e f i n e d by
i n an a l g e b r a homomorphism. Proof U
I t i s obvious.
W i t h i n t h i s chapter, we s h a l l o n l y deal w i t h chains o f q u o t i e n t algebras o f t y p e (24), Chapter 1, g i v e n i n
(11)
where (see (33)
(11.1)
(3r),a E N ,
A'(v,s
(v,S
, (34)
0
T ) E RGs and
while
(11.2)
s
E
and (44) , Chapter 8)
zg
v
=
11n
v"
D I RAC
253
D ISTRI BUT I ONS
A u s e f u l c h a r a c t e r i z a t i o n o f c o m p a t i b i l i t y between t h e q u o t i e n t algebras (11) and independent v a r i a b l e t r a n s f o r m s i s presented now. Proposition 2
A q u o t i e n t a l g e b r a i n (11) A'( V,S
@ T)
= A'( V,S
e)
G)
T ) / I R ( V,S
T)
and an i n v e r t i b l e independent v a r i a b l e t r a n s f o r m w i n ( 6 ) a r e compatible, only i f (12)
(9T ) i s an i n v a r i a n t o f
A'(V,S
w i n (7)
Proof The c o n d i t i o n (12) i s by d e f i n i t i o n necessary. We s h a l l show now t h a t i t i s a l s o s u f f i c i e n t . I n t h i s r e s p e c t we o n l y need t o prove t h a t
rR(v,s(5T ) \
(13)
i s an i n v a r i a n t o f w i n ( 7 )
B u t , i n view o f (20), i n Chapter 3, IR(V,S generated by "R
I
= {V E V
DPv
c)
T ) i s t h e i d e a l i n AR(V@T)
V , Y p E Nn,
E
IpI < a } 0
Therefore, i n view o f Lemma 1 below, (13) i s v a l i d . Lemma 1
I f w i s an i n v e r t i b l e independent v a r i a b l e t r a n s f o r m and V = I: then, f o r each R E N , VQ =
V
(V E
I
D'v
E
V
,Y
p
E
Nn,
IpI
n V"
El
i s an i n v a r i a n t o f t h e mapping w i n ( 7 ) . Proof Since w i s i n v e r t i b l e , t h e r e l a t i o n f o l l o w s e a s i l y (14)
w(vm )c
V"
Moreover, t h e r e l a t i o n i s a l s o v a l i d
(15)
w( q c
1;
Indeed, assume t h a t w E I: and denote w ' = w(w). F i r s t we show t h a t w ' s a t i s f i e s t h e c o n d i t i o n ( 2 ) i n Chapter 8. Assume t h e r e f o r e t h a t x E Rn and denote x ' = w(x ) . Then w and x; s a t i s f y t h e Hence mentioned conditpon, s i n c e w E 1; aRd x; €ORn.
(16)
w,(x;)
= 0,
f o r a certain u E N.
4 v
E
N, v 2 p
E.E.
254
Rosinger
so s a t i s f i e s t h e c o n d i t i o n ( 3 ) i n Chapter 8. L e t us denote But w F ' = w . ( F ) , then o b v i o u s l y F ' c Rn i s a f i n i t e subset. Assume now g i v e n an open subset G ' c Rn, such t h a t F ' c G ' , then o b v i o u s l y G = w ( G ' ) c Rn i s open and F c G. T h e r e f o r e (17)
supp wv c G, V v
for a certain plies (18)
E N.
N,v
E
> P I ,
B u t , i t i s easy t o see t h a t t h e r e l a t i o n (17) i m =
supp w; c G I , V v
N, v
E
Now, t h e r e l a t i o n s (16) and (18) w i l l i m p l y ( 1 5 ) . F i n a l l y , t h e r e l a t i o n s (14) and (15) y i e l d (19)
w ( v ) = ~ ( 1 :n
v") c
Assume now g i v e n R E \ and v
~ ( 1 : )n w ( v " ) c
E
11 n v"
=
v
VR and denote v ' = w ( v ) .
I n view o f (19) i t f o l l o w s t h a t E
v
c
DpvI
E
V",
v'
v"
therefore (20)
V p E Nn
B u t (19) w i l l a l s o i m p l y t h a t v'
E
v c
1;
t h e r e f o r e , i t i s easy t o n o t i c e t h a t (21)
Dpv' s a t i s f i e s t h e c o n d i t on ( 3 ) i n Chapter 8, Y p E Nn
F u r t h e r , by d e f i n i t i o n , t h e r e l a t i o n v Dpv E V c li, V p
E
Nn,
E
VR y i e l d s
PI G R
therefore (22)
Dpv s a t i s f i e s t h e c o n d i t i o n ( 2 ) i n Chapter 8, V p
E
N, I p I
E
Nn,lpl
But, b y d e f i n t i i o n (23)
((x)
= vV(w(x)),
V v
E
N,
X E
Rn
Now, t h e r e l a t i o n s (22) and (23) w i l l i m p l y t h a t (24)
DpvI s a t i s f i e s t h e c o n d i t i o n ( 2 ) i n Chapter 8,V p
F i n a l l y , (21) and (24) y i e l d m
(25)
Dpvl
E
16, Y
pE
Nn,
IpI
D I R A C DISTRIBUTIONS
255
The r e l a t i o n s ( 2 0 ) and ( 2 5 ) o b v i o u s l y i m p l y t h a t Dpvl
E
IJ, Y P E Nn,
IpI G g
which means t h a t
d v , ) c VR,
Y
i? E
m
CI
The r e s u l t i n P r o p o s i t i o n 2 o f f e r s a c o n v e n i e n t way f o r c o n s t r u c t i n g chains o f q u o t i e n t a l g e b r a s (11) c o m p a t i b l e w i t h i n v e r t i b l e independent v a r i a b l e t r a n s f o r m s and which w i l l be needed i n t h e sequel. Indeed, t h e mentioned r e s u l t makes i t p o s s i b l e t o use t h e method presented i n S e c t i o n 7, Chapter 3. Suppose g i v e n a s e t M o f i n v e r t i b l e independent v a r i a b l e t r a n s f o r m s . we s h a l l say t h a t , f o r a c e r t a i n g i v e n R E N, t h e i n c l u s i o n
Then
R n N H c(c (R 1)
(26)
has t h e p r o p e r t y PM, o n l y i f f o r e v e r y w E M we have (26.1)
H i s an i n v a r i a n t o f w i n ( 7 )
I t i s easy t o see t h a t PM i s an a d m i s s i b l e p r o p e r t y .
A c h a i n of q u o t i e n t algebras (11) w i l l be c a l l e d M t r a n s f o r m a l g e b r a s , o n l y i f t h e y a r e o f t y p e (165) i n Chapter 3, where P i s an a d m i s s i b l e p r o p e r t y s t r o n g e r t h a n PM. Corollary 1 Any q u o t i e n t a l g e b r a
A'~(v,s
@Tj
= A'(V,S
@T)/I'(IJ,s
T)
o f a c h a i n o f M t r a n s f o r m a l g e b r a s and any i n v e r t i b l e t r a n s f o r m w E M a r e compatible.
with R E
R,
ndependen t va r i ab 1e
Proof
@
By t h e d e f i n i t i o n o f M t r a n s f o r m a l g e b r a s A R ( V , s T . Therefore, t h e p r o o f f o l l o w s from P r o p o s i t i o n 2.
i s an i n v a r i a n t o f
w
2.
L i n e a r Independent F a m i l i e s o f D i r a c 6 D i s t r i b u t i o n s a t a P o i n t
We denote b y M t h e s e t o f a l l i n v e r t i b l e independent v a r i a b l e t r a n s f o r m s wa : Rn -+ Rn? w i t h a E R ' \ { O I , d e f i n e d by a a ( x ) = ax
Y x E R~
I n view o f P r o p o s i t i o n 1 and C o r o l l a r y 1 i n S e c t i o n 1, if
AR(v,s are
Mo
@T),
with R E
transform algebras, then
w,
0
E.E.
256
Dp6(ax) E A'( V,S
@
Rosinger
8, a
T ) ,V II E
E R'\
(01, p E Nn
since DP6 E U'(Rn) c A'(
V,S
& T), Y
II E 8 , p E Nn,
and wa :
A'(v,s(I) T) .+
A'(V,S
0T I , v R
E
8, a
E R]\
toi
The main r e s u l t i n t h i s c h a p t e r i s presented now. Theorem 1 The transforms o f t h e p a r t i a l d e r i v a t i v e s o f any g i v e n o r d e r p E Nn o f t h e Dirac 6 d i s t r i b u t i o n
. . . ,Dp6( amx) ,
Dp6( aox) ,
(+I
a r e l i n e a r independent w i t h i n t h e Mo t r a n s f o r m algebras AR(V,S T ) , with L E N , p r o v i d e d t h a t m G R and ao, ..., am E R ' \ I O ) a r e p a i r w i s e x f f e r e n t . Proof
...,?,,, E R '
Assume t h a t i t i s f a l s e and l o , (27)
Xo DP6(aox) +
(28)
3 ioE {O,
. .. t
...,m l
a r e such t h a t
Am DP6(amx) = 0 E AR(V,S
6) T ) b
# 0
: Ai 0
But, i n view o f (11) above, as w e l l as Theorem 3, S e c t i o n 3, Chapter 3, i t follows t h a t D P =~
DPS
+ rR(v,sQ T )
E
A ~ ( V , S@ T )
t h e r e f o r e , according t o (7), we o b t a i n t h e r e l a t i o n (29)
AiDP6(aix)
L e t us d e f i n e v (30)
E
Dps t lR(V,S i (Cm(Rn))N by
v =
= hiwa
1
<m
O < i
0T )
E AR(V,S
Xi wa (D'S)
i
then (27) and ( 2 9 ) w i l l i m p l y
(31)
v
E
rR(v,s@ T )
t h e r e f o r e , i n view of ( 2 0 ) i n Chapter 3 , we o b t a i n (32)
v =
c
O<j
v w j ' j
f o r s u i t a b l e v . E VR and w . E AR(V,S J J
@T).
0
T),V i d.0,
...,m)
DIRAC DISTRIBUTIONS
25 7
Now, in view of (11.1) above, as well as ( 2 ) in Chapter 8, the relation (32) will obviously imply V q E Nn,
(33)
191 G L :
3uEN: V v c N , v > p :
DqvV(O)
0
=
B u t m d II by the hypothesis, hence, the relations (30) and (33) will y i e l d
v >p
Further, in view o f (11.2) we obtain t h a t
(35)
4 q
E Nn,
3 v
E
U E
N :
N, v > a :
D4Sv(O)
# 0
since the matrix Ws(0) i s column wise nonsingular (see (12) in Chapter 8). Then the relations (34) and (35) will y i e l d
c
Xi ( a i ) k = 0 , V k
E
N, k
0 < i
< R and a ,..., a, E R ' \ I O ) are p a i r wise d i f f e r = e n t , together with a we1 1 known propgrty of the Vandermonde determinants will imply t h a t Now, the hypothesis t h a t rn
A
0
=
... =
Am = 0
which contradicting (28) completes the proof.
0
Corollary 2 The family (DP6(ax)
(36)
1
a
E
R ' \IOl)
of transforms of the p a r t i a l derivatives of any given order p E Nn o f the Dirac 6 distribution i s l i n e a r independent within the Mo transform algebras Am(V,S T)
0
3.
Generalized Dirac 6 Distributions
The following simple interpretation can be attached t o the p a r t i a l deriva= tives of the Dirac 6 distribution (37)
DP ( a x )
,p
E
N", a
E
R ' \(OI
258
E.E. Rosinger
i n p a r t i c u l a r t o the Dirac 6 d i s t r i b u t i o n s (38)
, a € R'
&(ax)
\
Suppose g i v e n a d i s t r i b u t i o n fine (39)
SE
D ' ( R n ) and a € R'
i f S E Sm and S =
w S = <w s ,
a
COI
a
s i n c e t h e d e f i n i t i o n w i l l n o t depend on s .
\ { d , then
we can de=
< s , .>
I n t h i s case, we o b t a i n
Then, i t i s easy t o see t h a t t h e a f o l l o w i n g analog o f t h e r e s u l t s i n Theorem 1 and C o r o l l a r y 2, S e c t i o n 2 i s v a l i d :
If (41)
i n t supp S #
0
then
.,, w
S a r e l i n e a r independent am R' \{Ol, i n o t h e r words, t h e f a = f o r any p a i r wise d i f f e r e n t ao,...,amE mily o f d i s t r i b u t i o n s
(42)
(43)
w
S,.
(ua s
1
a € R' \ I o l )
i s l i n e a r independent. However, i n o r d e r t o secure t h e r e l a t i o n s (42) and (43) w i t h i n t h e d i s t r i = b u t i o n a l framework, t h e c o n d i t i o n 4 1 i s e s s e n t i a l . Indeed, suppoFFEFZt - DP6, f o r a c e r t a i n g i v e n p E NA, !.hen t h e r e l a t i o n (40) w i l l i m p l y
t h e r e f o r e (42) and thus (43) a r e n o t v a l i d . I n t h i s connection, t h e r e s u l t s i n Theorem 1 and C o r o l l a r y 2, S e c t i o n 2, show t h a t w i t h i n t h e framework o f t h e Mo t r a n s f o r m algebras p r o p e r t i e s o f t y p e (42) and (43) can be v a l i d even i f t h e c o n d i t i o n (41) f a i l s t o be sa= t i s f i ed. An a l t e r n a t i v e i n t e r p r e t a t i o n o f t h e r e s u l t s i n Theorem 1 and C o r o l l a r y 2, S e c t i o n 2 can be o b t a i n e d as f o l l o w s . Suppose, t h e D i r a c 6 d i s t r i b u t i o n i s g i v e n by t h e Cm-smooth r e p r e s e n t a t i o n (44)
6 = < s
Y'
'>
where s E Sm i s d e f i n e d i n (93), Chapter 8, w i t h t h e h e l p o f a f u n c t i o n Y E D ( h ) which s a t i s f i e s t h e c o n d i t i o n s (92) and (101) i n Chapter 8. I n t h i s case i t f o l l o w s t h a t (45)
DIRAC DISTRIBUTIONS
259
w h i l e obviously (46)
supp y
SUPP wa
,v
a
E
R ' \ (0)
. I t f o l l o w s t h a t whenever a, b E R ' \ to} and la1 < I b l , t h e d i s t r i b u t i o n 6(bx) = i s 'more c o n c e n t r a t e d ' a t x = 0 E Rn t h a n t h e d i s t r i b u t i o n & ( a x ) = wa6.
I n t h i s connection, t h e r e s u l t s i n Theorem 1 and C o r o l l a r y 2, S e c t i o n 2, show t h a t , n o t u n l i k e i n Nonstandard A n a l y s i s , t h e Mo transfor! a l g e b r a s can d i s t i n g u i s h between elements c o n c e n t r a t e d a t a p o i n t x E R . I n t h e above c o n t e x t , t h e r e s u l t i n C o r o l l a r y 2, S e c t i o n 2, leads t o t h e f o l lowing question: Since t h e d i s t r i b u t i o n s 6 ( a x ) a r e a l l d i s t i n c t and more and more concentra= t e d a t x = 0 E Rn, as a R'\ {OI and la1 i n c r e a s e s , what happens a t t h e l i m i t , when l a ] + m? B e f o r e f o r m u l a t i n g t h i s q u e s t i o n i n a more e x a c t manler, we n o t i c e t h e f o l = l o w i n g f a c t . Suppose g ven a d i s t r i b u t i o n S E D ' ( R ) d e f i n e d by f(x)Y(x)dx, V Y where f E L'(Rn).
E
D(Rn),
i n view o f ( 4 0 ) , we o b t a i n
TKen
n < l a 1 wa S, Y >=
I
1 f(x)Y(;i.x)dx,
V a
E
R'\
w
{O),
Y E D(Rn)
Rn
therefore (48)
lain wa s = K
lim lal+
6
where (48.1)
K
f(x)dx
=
Rn I n view o f (48), t h e p r o p e r way t o f o r m u l a t e t h e above q u e s t i o n i s
lim l a i n 6(ax) = lal+ The aim o f t h i s s e c t i o n i s t o which t h e g e n e r a l i z e d D i r a c 6 o f t h e i r b a s i c p r o p e r t i e s can t r a n s f o r m a l g e b r a s have t o be (49)
?
Oi)
Suppose g i v e n b = (bv phism (50)
v
E
c o n s t r u c t c h a i n s o f q u o t i e n t a l g e b r a s (11) i n d i s t r i b u t i o n s i n (49) can be d e f i n e d and some be e s t a b l i s h e d . I n t h i s connection, t h e Mo r e s t r i c t e d as f o l l o w s .
N)E(R'\t03)N,
t h e n we have t h e a l g e b r a hornomor=
wb : (Cm(Rn
by
(50*1)
(wbs)v(x) =
u
E
N, x E Rn
260
E.E.
Rosinger
i n (50) i s d e f i n e d d i = I t should be noted t h a t t h e a l g e b r a homomorphism r e c t l y , w i t h o u t any recourse t o an independent v a r i a b l e t r a n s f o r m , u n l i k e t h e a l g e b r a homomorphisms ( 7 ) , which correspond t o t h e independent v a r i a b l e t r a n s f o r m s ( 6 ) . Nevertheless, t h e d e f i n i t i o n o f con a t i b i l i t between q u o t i e n t algebras ( 9 ) and independent v a r i a b l e t r m a n naturally be extended t o compactabihity betweeR q i o t i e n t a l g e b r a s ( 9 ) and a l g e b r a -, (Cm(R ) ) , s i n c e i n (10) o n l y t h e a l g e b r a homomorphism w : (Cm(R ) ) homomorphisms ( 7 ) were used. I n t h i s case, a r e s u l t s i m i l a r t o t h e one i n P r o p o s i t i o n 1, S e c t i o n 1, can be o b t a i n e d . F o r t h e a l g e b r a homomorphisms o f t y p e ( 5 0 ) , t h e corresponding v e r s i o n o f P r o p o s i t i o n 2 , S e c t i o n 1 i s o b t a i n e d as f o l l o w s , p r o v i d e d t h a t (51)
v l -+i mm b V =
?
m
a c o n d i t i o n which w i l l be assumed i n t h e sequel. Proposition 3 A q u o t i e n t a l g e b r a i n (11) AR(V,S
@ T)
= AR(V,S
and an a l g e b r a homomorphism (52)
@ T)
A'(V,S
%
@ T)/lR(V,S 4 T )
i n (50) a r e compatible, o n l y i f
i s an i n v a r i a n t o f
%
Proof See t h e p r o o f o f P r o p o s i t i o n 2 , S e c t i o n 1 and Lemma 2 below.
0
Lemma 2 Given t h e a l g e b r a homomorphism Ilea VI1 =
{V E
V
I
DPv
E
%
i n (50) and V =
V , Y p E Nn,
1: n V",
then, f o r each
IpI G k }
i s an i n v a r i a n t o f ub. Proof Assume f i r s t t h a t
We s h a l l prove t h a t (54)
g1:)
= 1;
Indeed, assume t h a t
w E 1; and denote w ' = wb(w).
F i r s t , we prove t h a t w ' s a t i s f i e s t h e c o n d i t i o n ( 2 ) i n Chapter 8.
Assume
D I R A C DISTRIBUTIONS
t h e r e f o r e t h a t xo (55)
Rn\{OI, t h e n i n view o f (50.1),
E
wL(x0) = Ib,l
n
26 1
we o b t a i n
N
wu(buXo), V u E
Now, i n view o f ( 3 ) i n Chapter 8 and ( 5 3 ) , t h e r e l a t i o n (55) i m p l i e s t h a t (56)
w;(x~)
= 0, V u E N, u > p '
f o r a c e r t a i n u' E N, s i n c e V T+i m" II bu x o 1 I = 03, where I1 1I i s any g i v e n norm on Rn. I n case x = 0 E Rn, t h e r e l a t i o n (56) f o l l o w s d i r e c t l y from t h e f a c t t h a t w s a t i s ? i e s t h e c o n d i t i o n ( 2 ) i n Chapter 8.
w,
We prove now t h a t w ' s a t i s f i e s t h e c o n d i t i o n ( 3 ) i n Chapter 8. Since s a t i s f i e s t h e mentioned c o n d i t i o n , t h e r e e x i s t s a bounded s u b s e t B c R a neighbourhood V c B o f x = 0 E Rn, and p " E N, such t h a t (57)
SUPP
(58)
v
wU c 6, Y u
E
n SUPP w V c (01
N, v > p "
c R ~ Y,
EN,
Now, t h e r e l a t i o n s ( 5 3 ) , ( 5 5 ) , (57) and (58) w i l l i m p l y t h a t (59)
SUPP W;
c {O}, Y u
f o r a s u i t a b l e p " ' E N. i s completed.
E N,u > p " '
And i n view o f ( 5 6 ) and ( 5 9 ) , t h e p r o o f o f (54)
F u r t h e r , we s h a l l prove t h e r e l a t i o n (60)
(v)
ab
Assume t h a t v E V . f i c e s t o show t h a t
Then v
(61)
v"
qJV) E
B u t i n view o f (50.1), (62)
E
:I n V",
t h e r e f o r e i n view o f ( 5 4 ) , i t s u f =
t h e r e l a t i o n (61) i s e q u i v a l e n t t o
1 l i m In v v ( x ) Y (6 x ) d x = 0, V Y v+mR V
E
U(Rn)
Now, i n o r d e r t o p r o v e ( 6 2 ) , f i r s t we n o t i c e t h a t v E 1; and t h e condi t i on ( 3 ) i n Chapter 8, i m p l y t h e e x i s t e n c e o f a bounded subset B c Rn and U ' E N, such t h a t (63)
supp
vV c
6, Y v E
N,
v >p'
L e t us t a k e x E D(Rn) such t h a t X = 1 on B . r e l a t i o n (62) i s e q u i v a l e n t t o
(64)
lim
1
V + m $
For given Y E
D
1 v ( X ) X ( X ) Y ( ~x ) d x = 0, Y Y
Then i n view o f (63), t h e E
D(Rn)
V
(Rn).we d e f i n e t h e sequence o f f u n c t i o n s s
E
n N (D(R ) ) by
1 s V ( x ) = X ( X ) \ ~ ( x ) , Y v E N, x E Rn LV Then i t i s easy t o see t h a t t h e f o l l o w i n g r e l a t i o n h o l d s i n U(Rn)
E.E.
262
(65)
l i m sv
Rosinger
Y(0)x
=
v'm
NOW, i n view o f a w e l l known p r o p e r t y o f b i l i n e a r forms on V ( R n ) x D(Rn)), t h e r e l a t i o n VE v" and (65) w i l l i m p l y ( 6 4 ) . I n t h i s way, t h e p r o o f o f (60) i s completed. F i n a l l y , we can prove t h e r e l a t i o n (66 1
%(VJ
c VR
y
Y R E
a
Indeed, assume g i v e n R E and v E V and denote v ' = % ( v ) . and t h e obvious i n c l u s i o n VQ c V w i l e y i e l d
Then (60)
v"
v' E V c therefore (67)
Dpv' E
V",
Y p E Nn
But s i m i l a r l y t o (59) we o b t a i n t h a t (68)
c{O}, Y v E N, v 2 p '
SUPP V;
therefore (69)
DpvI s a t i s f i e s t h e c o n d i t i o n ( 3 ) i n Chapter 8, Y p E Nn
Further, by the d e f i n i t i o n o f (70)
DPV E
v
Suppose now g i v e n x
c 0
r: , v E
Rn.
$ we
obtain
p E N ~ ,
GR
I f xo # 0 E Rn t h e n (68) w i l l y i e l d
DpvI s a t i s f i e s t h e c o n d i t i o n ( 2 ) i n Chapter 8, a t xo (71)
E
Rn\\O},
Yp€Nn
If xo = 0 E Rn then (50.1) and t h e f a c t t h a t i n view o f ( 7 0 ) , Dpv w i t h I p I G R , s a t i s f i e s t h e c o n d i t i o n ( 2 ) i n Chapter 8, a t xo = 0 E Rn, w i l l imply t h a t P + E NnY
Dpv' s a t i s f i e s t h e c o n d i t i o n ( 2 ) i n Chapter 8 a t xo = 0 E Rn, (72)
V p
E
Nn,
IpI < R
But t h e r e l a t i o n s ( 6 9 ) , (71) and (72) o b v i o u s l y i m p l y (66). I n casevlimm bv
=
-
m,
t h e p r o o f o f (66) i s s i m i l a r .
I n view o f P r o p o s i t i o n 3, i t f o l l o w s t h a t t h e d e f i n i t i o n o f M t r a n s f o r m algebras i n S e c t i o n 1, can n a t u r a l l y be extended t o t h e case when M con= t a i n s n o t o n l y independent v a r i a b l e transforms b u t a l s o a l g e b r a hornomor= phism o f t y p e ( 5 0 ) . I n o r d e r t o d e f i n e t h e g e n e r a l i z e d D i r a c 6 d i s t r i b u t i o n s i n (49), l e t us denote
0
D I R A C DISTRIBUTIONS
Mb = Mo
263
U {Wb)
I t f o l l o w s t h a t i n case A'( V 9 - S (J + I T ) , with LEB, a r e Mb t r a n s f o r m a l g e b r a s , t h e n cob DP6 E AR(V,S
( + T) , Y R E
N,
C+, T),V
L
p E Nn,
since DP6
and
E
D'(Rn) C A'(V,S
E
W,
p E Nn,
W b : AL(V,St.+: J) -+ AE(V,S (tj T),V II E 1 The e x t e n s i o n o f Theorem 1 i n S e c t i o n 2 i s p r e s e n t e d now.
Theorem 2 The t r a n s f o r m s o f t h e p a r t i a l d e r i v a t i v e s o f any g i v e n o r d e r p E Nn o f t h e Dirac 6 d i s t r i b u t i o n DP6(aox) ,.. . ,DP6(amx) and t h e g e n e r a l i z e d D i r a c 6 d i s t r i b u t i o n p a r t i a l d e r i v a t i v e
a r e l i n e a r independent w i t h i n t h e Mb t r a n s f o r m a l g e b r a s A L E N , p r o v i d e d t h a t m G J ? and ao,...,am E R1\{O) are Proof Assume t h a t i t i s f a l s e and X o , . . . , ~ , X (73) (74)
A,
D P6(aox) t
X= 0
=r
.. . +
( 3 io E {O,
A,
E R ' a r e such t h a t
DP6(amx) + hob DP6 = 0 E A'( V,S
...,m l
: Xi
@ T)
# 0) 0
B u t i n view o f (11) above, as w e l l as Theorem 3, S e c t i o n 3, Chapter 3, i t follows that
t h e r e f o r e , a c c o r d i n g t o ( 7 ) and (50) we o b t a i n t h e r e l a t i o n s (75)
Xi
DP6(aix) = A1. w ai DPs
Y i
E
-+ IR(V,S @
T ) E AR(V,S
...,m l
{O,
and (76)
Awb DP6 = A w ~DpS
+
IR(VyS
@ T) E
We d e f i n e now v E (Cm(Rn))N by (77)
c
v = O < i
A1 . w ai DPs Gm
t h e n (73), (75) and (76) w i l l y i e l d
+
Xub D's
A'(VyS
@ T)
@
T),
264
E.E.
Rosinger
t h e r e f o r e , as i n t h e p o o f o f Theorem 1, S e c t i o n 2 , i t f o l l o w s t h a t v sa= t i s f i e s t h e c o n d i t i o n 33), which t o g e t h e r w i t h (77) w i l l y i e l d (79) (
c
Qm
OGi
Ai(ai)
ql+
hlbvln+lql)Dp+qs V (O)=O,V
q
E
Nn, 191 G R,v
E
N,
v>p
But (35) and (79) w i l l y i e l d
where f o r any g i v e n u E N i t i s p o s s i b l e t o choose vo,...,vk (81)
vO,...,vR
E N, such t h a t
>a
...,
As ho, h , A , a o , ..., am a r e g i v e n f i x e d constants, t h e r e l a t i o n s (80) and (81) t o g e t h b w i t h ( 5 1 ) w i l l i m p l y t h a t
x =o
(82) since u
E
N i n (81) i s a r b i t r a r y .
Now (80) and (82) w i l l y i e l d C OGi
+(ai)
Qm
k
= 0, V k E N, k Q t
which as i n t h e p r o o f o f Theorem 1, S e c t i o n 2, w i l l f i n a l l y i m p l y (83)
x0
=
... =
Am
0
Since (82) and (83) c o n t r a d i c t ( 7 4 ) , t h e p r o o f i s completed.
0
Corollary 3 The f a m i l y (DP6(ax)
I
a E R'\(01)
o f t r a n s f o r m s o f t h e p a r t i a l d e r i v a t i v e s o f any g i v e n o r d e r p E Nn o f t h e Dirac 6 d i s t r i b u t i o n , together w i t h the generalized Dirac 6 d i s t r i b u t i o n p a r t i a1 d e r i v a t i v e w
DP6
a r e l i n e a r independent w i t h i n t h e M,, t r a n s f o r m a l g e b r a s Am(V,S
@ T).
CHAPTER 10 SUPPORT AND LOCAL PROPERTIES
0.
Introduction
An e s s e n t i a l p r o p e r t y o f t h e p r o d u c t w i t h i n t h e q u o t i e n t a l g e b r a s encoun= t e r e d so f a r , i s i t s l o c a l c h a r a c t e r . T h i s p r o p e r t y extends t h e known p r o p e r t 4 o f those p r o m which can be d e f i n e d w i t h i n t h e d i s t r i b u t i o n s i n U'(R ) , f o r i n s t a n c e
Y S
(1)
E
U'(Rn)), Y E Cm(Rn)
:
supp (yc.S) c supp Y n supp S
o r more general (see [ 17,76,82-85,101])
Y S,S',T,T'
E
U'(Rn)), G c Rn non-void, open : =*
S.T = SIT on G
I n order t o present the l o c a l properties o f the product w i t h i n the quotient algebras i t i s useful t o e x t e n d t h e n o t i o n o f s u p p o r t o f a d i s t r i b u t i o n t o t h e case o f elements o f these q u o t i e n t algebras. W i t h t h e h e l p o f t h e mentioned extended n o t i o n o f s u p p o r t s e v e r a l o t h e r l o c a l p r o p e r t i e s o f elements i n q u o t i e n t a l g e b r a s w i l l be presented.
1.
Support o f Elements i n Q u o t i e n t Algebras
Suppose f o r a c e r t a i n f i x e d 11 S e c t i o n 2, Chapter 1)
A = A/Z
(3) If S
E
E
AL R
c
E
8,
we a r e g i v e n a q u o t i e n t a l g e b r a (see
(0)
A and E c R , we s h a l l say t h a t S vanishes on E, o n l y i f
3 s €A:
(4)
* ) S = s + Z
**) s v = 0 on E, Y v
E
N, v 2 p
f o r a c e r t a i n ~.rE N. The s u p p o r t o f S w i l l be c a l l e d t h e c l o s e d subset
265
266
Rosinger
E.E.
supp S = R
(5)
\ {x E
n
I
S vanishes on a neighbourhood o f x }
Proposition 1
R (n) t h e above n o t i o n o f s u p p o r t i s i d e n t i c a l w i t h t h e
For functions i n C usual one. Proof
I t f o l l o w s from (23) i n Chapter 1.
0
The l o c a l c h a r a c t e r o f t h e m u l t i p l i c a t i o n and a d d i t i o n i n t h e q u o t i e n t a l = g e b r n presented i n t h e n e x t two theorems , w h i c h e extensions o f t h e p r o p e r t i e s i n (1) and ( 2 ) Theorem 1 I f S,T E A t h e n T ) c supp S u supp T
1)
supp ( S
2)
supp (S.T) c supp S n supp T
t
Proof I t f o l l o w s from a d i r e c t v e r i f i c a t i o n .
0
Before p r e s e n t i n g Theorem 2, we s h a l l i n t r o d u c e t h e f o l l o w i n g d e f i n i t i o n . Given S,S' E A and E c R, we say t h a t S = S ' on E
(6) only i f S
-
S ' vanishes on E.
Theorem 2 Suppose S,S',T,T'
E
A and E c
1)
S t T = S ' t T ' o n E
2)
S.T = S ' . T ' on E
n. I f
S = S ' and T = T ' on
E, then
Proof I t f o l l o w s from a d i r e c t v e r i f i c a t i o n .
0
Corollary 1 The e q u a l i t y on E o f elements i n A i s an equivalence r e l a t i o n on A which i s compatible w i t h i t s a l g e b r a s t r u c t u r e .
A u s e f u l e x p l i c i t e x p r e s s i o n o f t h e s u p p o r t i s presented now. Theorem 3 If S E A =
(7)
A / l , then
-
supp S = w €n7 c l v l ~ m msupp (sv t wv)
26 7
SUPPORT
f o r any
SE
A , such t h a t S = s
t
I
E
A.
Proof F i r s t , we prove t h e i n c l u s i o n t o t h e r i g h t hand t e r m o f ( 7 ) .
v
(8)
n supp ( s , + w),
C
=
.
Assume g i v e n X E Q which does n o t b e l o n g Then, t h e r e e x i s t s w E 1 , such t h a t
8, YV
E
N, v > i ~ ,
where V i s a c e r t a i n neighbourhood o f x and
i~
E N i s s u i t a b l y chosen.
B u t t h e r e l a t i o n ( 8 ) o b v i o u s l y i m p l i e s t h a t S vanishes on V , t h e r e f o r e x g supp s. supp S . Then The converse i n c l u s i o n 3 r e s u l t s as f o l l o w s . Assume t h a t x by d e f i n i t i o n t h e r e e x i s t s a neighbourhood V o f x and a sequence o f func= t i o n s s E A, such t h a t (see ( 4 ) ) S = s + I and s v = O o n V , Y v ~ N ,v>p,
(9)
for a certain p
E
N.
But the r e l a t i o n ( 9 ) obviously implies t h a t x
+
-
cl lim v+m
supp s v
t h e r e f o r e x cannot b e l o n g t o t h e r i g h t hand t e r m o f ( 7 ) .
0
I n t h e case o f t h e chains o f q u o t i e n t a l g e b r a s used i n Chapter 7, 8 and 9, some a d d i t i o n a l p r o p e r t i e s o f t h e s u p p o r t a r e presented now. F i r s t , an i m p o r t a n t f e a t u r e o f t h e s u p p o r t o f t h e D i r a c 6 d i s t r i b u t i o n and i t s p a r t i a i d e r i v a t i v e s w i l l r e s u l t i n case t h i s d i s t r i b u t i o n i s represen= t e d b y a c -smooth 6 -sequence which s a t i s f i e s t h e c o n d i t i o n o f s t r o n g sence on t h e s u p p o r t (see ( 1 2 ) i n Chapter 8 ) . T h i s f e a t u r e - p=*r e s e n t e n e x t i n Theorem 4 - g i v e s an a l t e r n a t i v e i n s i g h t i n t o t h e meaning o f t h e mentioned c o n d i t i o n of s t r o n g presence on t h e s u p p o r t . Theorem 4 Within t h e chains o f q u o t i e n t algebras
A'(
v,s
@ T),
witha
E
R,
@
c o r r e s p o n d i n g t o C m - r e g u l a r i z a t i o n ( v,s T ) d e f i n e d i n ( 3 3 ) , Chapter 8, t h e p a r t i a l d e r i v a t i v e s Dp6, o f any g i v e n o r d e r p E Nn, of t h e D i r a c 6 d i s = t r i b u t i o n have t h e f o l l o w i n g p r o p e r t i e s :
1)
supp DP6 = ( O I E Rn
2) 3)
DP6 vanishes on any Ec Rny such t h a t 0
(10)
v c 1; n vm DP6 does n o t v a n i s h on f O I C Rn, i n case t h a t t h e f o l l o w i n g c o n d i t i o n i s s a t i s f i e d (see (44) i n Chapter 8):
4)
4
cl E
Dp6 does n o t v a n i s h on Rn\ EO3, i n case t h a t t h e f o l l o w i n g c o n d i t i o n i s s a t i s f i e d (see (34) i n Chapter 8):
268
Rosinger
E.E.
vcI;n!r
(11) Proof
Assume t h a t (see S e c t i o n 1, Chapter 8) (V,S
6;J ) E
RG,
. Then, i n view o f (10) and (29) i n Chapter 8, f o r a c e r t a i n given s E as w e l l as Theorem 3, S e c t i o n 3, Chapter 3, we o b t a i n t h e r e l a t i o n
@ T) E
DPg = Dps t IR(V,s
(12)
AR(V,s
(5J ) , V R
E
Now, 1) and 2) w i l l r e s u l t e a s i l y from (12) above, as w e l l as (11) i n Chapter 8. 3)
Assume t h a t i t i s f a l s e and DPg vanishes on Rn tion, there e x i s t s a representation DPg = t t
(13)
IR(v,s @ T ) ,
with t
E
\@I. Then, by d e f i n i =
AR(V,s
&?J )
such t h a t t
(14)
=
o
on R~ \ l o } ,
v
R E N, v 2
u
f o r a c e r t a i n P E N. B u t t i s a sequence o f continuous f u n c t i o n s on Rn, t h e r e f o r e (14) y i e l d s = 0 on R
t,
n
Then o b v i o u s l y t E 11 n
(15)
t
vc
E
, V R E N, v > u v",
I%,S
hence i n view o f ( l o ) , we o b t a i n t h a t
@T)
But t h e r e l a t i o n s (13) and ( 1 5 ) w i l l c o n t r a d i c t 1 ) i n Theorem 1, S e c t i o n 2, Chapter 8. 4) (16)
Assume t h a t i t i s f a l s e and t h e r e e x i s t s a r e p r e s e n t a t i o n (13) such that t,(O)
f o r a suitable
(17)
u
w = t
= 0,
V R E N, v > 1.I
E N.
L e t us denote
-
,
DPs
then (12) and (13) w i l l y i e l d
w
E
IR(V.S
@ J)
t h e r e f o r e , i n view o f (20) i n Chapter 3, we o b t a i n v . .w i i R f o r c e r t a i n vi E VL and wi E A (V,S
(18)
W =
C
O < i < h
B u t i n view o f ( l l ) , i t f o l l o w s t h a t
@ J).
SUPPORT
269
m
vi
E VR
c v c 76
therefore, the condition ( 2 ) in Chapter 8, applied t o vO,...,vh viV(O) = 0, Y
(19)
V E
N,
V
E
N.
f o r a certain suitably chosen U'
will yield
>, 1-I'
NOW, the relations (17-19) will imply t h a t
DpsV(O)= 0 , V v
N, v
E
> u' 0
which will obviously contradict (12) in Chapter 8.
We present now in the framework of the chains of quotient algebras in Theorem 4 , an additional information on the general e x p l i c i t expression of the support obtained in Theorem 3. Theorem 5 Suppose the chains of quotient algebras in Theorem 4 s a t i s f y the condition (11). Suppose given for a certain I? E R , an element s = s + 7 R (V,S (
(q
Th
supp sv and
v + w
l%
supp ( s v
t
wv)
v - t m
d i f f e r by a t most a fi n i t e number of points. Proof
I n view of (20) in Chapter 3, we have t h a t
c
w =
OGiGh for certain vi 1 ows t h a t
E
Vil and
wi
v i, .w i R
E A (V,S
GI T ) .
B u t , in view of ( l l ) , i t f o l =
W
v i E VR c v c Z6 the ref ore
w
E
7s
m
since l 6 i s an ideal in ( C " ( R n ) ) N .
I t follows t h a t w s a t i s f i e s the condition ( 3 ) in Chapter 8 and t h u s the proof i s completed.
0
An interesting property of nilpotent elements (see Section 6 , Chapter 8) within the chains of quotient algebras used in Theorem 5 , i s presented now.
270
E.E. Rosinger
Theorem 6 Suppose g i v e n under t h e c o n d i t i o n s i n Theorem 5 , an element S If for a certain m
R A ( V , S (JT).
N \*Qlwe have
E
Sm = 0
E
E
A'( V,S (t)
T)
then supp S i s a f i n i t e subset i n R". Proof Assume g i v e n a r e p r e s e n t a t i o n S = s t lR(V,S
with s
E
AR(V,S
6)T ) .
@ J) E AR(V,S
(,gT )
Then by t h e hypothesis, i t f o l l o w s t h a t
I
Sm = sm t l ' ( V , S
(;j
T ) = 0 E AR(V,S (t, T )
t h e r e f o r e d e n o t i n g w = sm, we o b t a i n
w E lR(V,S (t) T ) Then an argument s i m i l a r t o t h e one i n t h e p r o o f o f Theorem 5 w i l l y i e l d
w E 1; t h e r e f o r e v v m supp wv i s a f i n i t e subset i n Rn. SUPP
wV = supp sV , V v
E
But obviously
N.
thus i n view o f ( 7 ) t h e p r o o f i s completed. 2.
0
L o c a l i z a t i o n w i t h i n t h e Q u o t i e n t Algebras
Given an element S E A = A/l
we s h a l l denote by
t h e s e t o f a l l t h e open subsets G t o see t h a t
G
C
i? on which S vanishes.
Then i t i s easy
= i? \ supp S
GS
I n t h e case o f a d i s t r i b u t i o n S E U ' ( 0 ) and w i t h t h e usual n o t i o n s o f va= n i s h i n g and s u p p o r t f o r d i s t r i b u t i o n s , t h e corresponding s e t GS has t h e w e l l known p r o p e r t y t h a t any u n i o n o f s e t s i n GS i s a g a i n a s e t i n GS. I n particular
SUPPORT
\ supp S =
(21)
U
271
G E Gs
G E Gs
The aim o f t h i s s e c t i o n i s t o p r e s e n t s e v e r a l p a r t i a l e x t e n s i o n s o f ( 2 1 ) t o t h e case o f q u o t i e n t a l g e b r a s ( 3 ) . Theorem 7 Suppose g i v e n S
E
A = A/1 and G1,G2
E Gs.
If
where d denotes t h e E u c l i d i a n d i s t a n c e on Rn,
then
Proof
In view o f t h e h y p o t h e s i s , we can assume s l , s 2 E A s u c h t h a t
(24)
S = S ~ + I = +S I E A = A / I 2 s l v = 0 on G1 , V v E N , v 2 p ,
(25)
s2v = 0 on G2
(23)
f o r a c e r t a i n LI
E
,Y v
EN,
v
>u,
N.
L e t us denote w = s 1 - s2 then (23) y i e l d s
(26)
W E 7
But, i n view of (24) and ( 2 5 ) we o b t a i n f o r v E N, v 2 p
slv(x) - sZv(x)
if x
E
SlV(X)
if x
E
Now, i f we t a k e E = G1 \ G 2 and F = tain d(E,
a \F)
= d(G1\G2,
a \(G2\G1)
\(G1
, the U
relation
G2)
G2\ G1
then i n view o f ( 2 2 ) we ob=
G2\G1) > 0
t h e r e f o r e a c c o r d i n g t o Lemma 1 below, t h e r e e x i s t s Y E Cm(Q) such t h a t
E.E.
272
-4
if x
E
Rosinger
G2\ GI
Y(x) =
(28)
i f x E G1 \ G2
.$ Then we denote
w ' = U(Y).W
(29)
and i n view o f (26), as w e l l as (23) i n Chapter 1, we o b t a i n 1
W'E
Therefore s = (sl t s 2 ) / 2 t
(30)
w'E
A
and i n view o f (24) and (25) s = s t 7 E A = A / 1
(31)
B u t t h e r e l a t i o n s (27-30) y i e l d
(32)
s,,
= 0 on G1 U
G2, V v
E
N,
>
v
and (31) t o g e t h e r w i t h (32) w i l l i m p l y t h a t
G1
U
G2
Gs
E
Lemma 1 Suppose g i v e n t h e subsets E C F
C
R such t h a t
d(E, Q \ F) > 0 then there e x i s t s Y
E
C"(R) w i t h t h e f o l l o w i n g p r o p e r t i e s
1)
O Q Y G l o n a
2)
Y = l o n E
3) 4)
Y = O o n R \ F Y E D(Q) i f E i s bounded
Proof L e t us d e f i n e
x
: Rn
x
(0,m)
-+
R ' by
KE exp(E2/(llxl12
X(X,E)
E'))
i f IIxII <
E
=
0
where
-
if IIxII
2
E
SUPPORT
We s h a l l choose E E
(O,m),
273
such t h a t
d(E, R \ F ) / 2
E
and d e f i n e Y :
Rn
+
X(x-Y,E)dy
=
'+'(XI
R 1 by
E
where E E =
E R n / d(y,E)
GE}
Then i t i s easy t o see t h a t t h e r e s t r i c t i o n o f Y t o R i s t h e r e q u i r e d function.
n
Corollary 2 Suppose g i v e n S E A = A / 7 and G1,G2
(33)
c i G~ n c i G~ =
(34)
c l G1 i s compact
E GS
which s a t i s f y t h e c o n d i t i o n s
0
then
Proof I t i s easy t o see t h a t G1 and G2 s a t i s f y t h e c o n d i t i o n ( 2 2 )
0
Theorem 8 Suppose g i v e n S E A = such t h a t
(35)
A/I
and G1,G2
E
GS.
I f g i v e n a subset G i c
d ( G i , R \ G1) > 0
then
G i u G2
E
GS
Proof We s h a l l use t h e n o t a t i o n s i n t h e p r o o f o f Theorem 7 I n view o f Lemma 1, t h e r e e x i s t s
-j
if x
E
\y E
G2 \ G1
Y(x) =
4
C"(Q)
if x E Gi
such t h a t
GI
2 74
E.E.
Rosinger
I n t h i s case, t h e r e l a t i o n (32) i n t h e p r o o f o f Theorem 7 w i l l y i e l d sV
= 0 on G i
U
G2, Y v E N , v 2~ a
which completes t h e proof. Theorem 9 Suppose g i v e n S E A = f ies t h e c o n d i t i o n s
and an open subset G c R \ supp S which s a t i s =
A/7
(36)
cl G n supp S = fl
(37)
c l G i s compact
then
G
E
GS
Proof I n view o f (36) i t f o l l o w s t h a t Y x E c l G : (38)
]EX
:
E(0,m)
B ( x , E ~ )n R
E
Gs
where we denote B ( x , p ) = {y E Rn
for x
E
Rn,
I
Ily-xll < p }
p E ( O , m ) and II II t h e E u c l i d e a n norm on Rn.
But i n view o f (37) we o b t a i n
f o r c e r t a i n xO,
...,xh
E c l G, which can be assumed p a i r wise d i f f e r e n t .
Now, i f h=O t h e n t h e p r o o f i s completed, s i n c e (38) and (39) w i l l y i e l d G
C
cl G
C
B ( x ~ , E /2) ~ nR 0
Assume then h = l .
E
Gs
We s h a l l denote
G I = B ( x ~ , E ~n~R, ) Gi = B ( x ~ , E ~ ~n/ R~ )
) nR
G2 = B(xo,cx 0
I n view o f (38) t h e s e t s G1,Gi and G2 s a t i s f y t h e c o n d i t i o n s i n Theorem 8, t h e r e fo r e
275
SUPPORT
i f (39) i s taken i n t o account, thus t h e p r o o f i s a g a i n completed.
1 denote
Assume f u r t h e r t h a t h=2.
= B ( x ~ , E /~2 ) 2
G2 = ( B ( x
fli?
/2)) n n 1 follows that
,E
EX
xo Then, i n view o f (38) and O
Thus, t h e s e t s G , G j and G2 s a t i s f y t h e c o n d i t i o n s i n Theorem 8. T h e r e f o r e t h e r e l a t i o n ( 4 0 j w i l l f o l l o w again, i f (39) i s t a k e n i n t o account. I n t h i s way, t h e p r o o f i s a g a i n completed. I t i s easy t o see t h a t t h e above procedure can b e used f o r any h E N, h >D3 i n (39).
Corollarv 3 Suppose g i v e n S E A = A / l perties are equivalent:
(41)
and a s u b s e t H c R, t h e n t h e f o l l o w i n g two p r o =
supp S c H
and
Y
(42)
K
R \H, K compact:
c
3GEGS: K C G
Proof Assume t h a t ( 4 2 ) i s v a l i d and x E R \ H. Then o b v i o u s l y x E G, f o r a c e r = supp S, s i n c e G i s open b y d e f i n i t i o n . I t f a l = tain G E G Therefore x lows i n th?s way t h a t ( 4 1 ) h o l d s .
.
The converse r e s u l t s from Theorem 9, n o t i c i n g t h a t R
\ supp S i s open.
Corollary 4 Suppose g i v e n S E A = lent:
(43)
supp
s
=
A/l
, then
t h e f o l l o w i n g two c o n d i t i o n s a r e equiva=
g
and
Y G c R , G open, bounded :
(44)
G
E
GS
E.E. Rosinger
2 76 Proof I t f o l l o w s from C o r o l l a r y 3.
0
Theorem 10 Suppose g i v e n S E A =
v
Y E
4 / 1 and a c l o s e d subset F c R.
Then, t h e c o n d i t i o n
D(R) :
(45) C R \F * Y . S
supp Y
= 0 € A
implies the condition (46)
supp S C F
Moreover, i n case I i s c o f i n a l i n v a r i a n t (see (61) i n Chapter 2 ) , t h e con= d i t i o n s (45) and (46) a r e e q u i v a l e n t . Proof Assume t h a t (45) i s v a l i d and x E R \F. Since F i s closed, i t f o l l o w s t h a t t h e r e e x i s t s Y E D(Q), w i t h supp Y c R \ F and a neighbourhood V o f x, such t h a t (47)
Y = l o n V
But, i n view o f t h e h y p o t h e s i s \y.S = 0
E
A
t h e r e f o r e , g i v e n any r e p r e s e n t a t i o n (48)
S = s t l
E
A =
A/I,withsEA
we o b t a i n (49)
u(Y).s
E
A
Now, t h e r e l a t i o n s (48) and (49) y i e l d
(50)
s
= u(i-i)s t
r
E
A
L e t us denote t = u(1-Y)s
t h e n (47) y i e l d s tv=OonV,VvEN which t o g e t h e r w i t h (50) i m p l i e s t h a t x p a r t i s completed.
$ supp
S and t h e p r o o f o f t h e f i r s t
Assume now t h a t (46) i s v a l i d and 7 i s c o f i n a l i n v a r i a n t . Then l e t us t a k e Y E D(Q) such t h a t supp Y C il \F. B u t supp Y i s compact, t h e r e f o r e C o r o l =
SUPPORT
277
l a r y 3 i m p l i e s t h e e x i s t e n c e o f G E Gs such t h a t supp Y c G
(51)
S i n c e b y d e f i n i t i o n S vanishes on G t i s f i e s the condition
sv
(52)
= 0 on G, V
for a certain p sume t h a t
(53)
E
N.
v
% we
E
can assume t h a t s i n (48) sa=
N, v > p
E
Moreover, s i n c e 7 i s c o f i n a l i n v a r i a n t , we can as=
p = o
Then, t h e r e l a t i o n s (51-53) w i l l y i e l d = u(0) E
U(Y).S
Q
hence (48) w i l l imply Y . S = 0
E
A.
Corollary 5 Suppose g i v e n S
v
Y
E
E
A =
u(n)
supp Y n
I f 7 i s c o f i n a l i n v a r i a n t then
A/7. : SUPP
S =
0
=*
\y.S =
0 E A
Proof 0
I t f o l l o w s from t h e second p a r t o f Theorem 10.
An i m p o r t a n t decomposition p r o p e r t y f o r t h e elements o f q u o t i e n t a l g e b r a s i s presented now. Theorem 11 Suppose g i v e n S
(54)
E
A = A/7
and I i s c o f i n a l i n v a r i a n t .
If
supp S = F U K w i t h F closed, K compact and F n K =
0
then t h e f o l l o w i n g decomposition h o l d s
s
=
SF
SK
where SF, SK
E
A =
A/I
(55)
and
(55.1)
supp SF n supp SK =
(55.2)
K nsupp
sF
0, w i t h
= F n supp
supp SK compact
sK= 0
Proof Assume t h e open subsets G1,G2,G3,G4
(56)
K c Gl,cl pact
G1
C
C
R such t h a t
G2, c l G2 C G3, c l G3
C
G4, F n c l G4 = 0, c l G4 conp
278
E.E.
Rosinger
L e t us denote K1 = ( c l G4)\G1 then obviously K1 n supp S =
0 and K1 i s compact
t h e r e f o r e , i n view o f C o r o l l a r y 3, t h e r e e x i s t s G Then b y d e f i n i t i o n , t h e r e e x i s t s a r e p r e s e n t a t i o n
+ I€ A
S = s
E
Gs such t h a t
K1
c G.
W l , with s E A
=
such t h a t = 0 on G,
sv
for a certain p t h a t p = 0.
N.
E
V
V E
N, v
>p,
But 7 i s c o f i n a l i n v a r i a n t , t h e r e f o r e we can assume
Now i n view o f ( 5 6 ) , Lemma 1 g r a n t s t h e e x i s t e n c e o f YF YK E D(n) such t h a t
YF = 1 on n \G4,
YF = 0 on c l G3
YK = 1 on c l G1,
YK = 0 on
E
C"(n)
and
R \G2
Then o b v i o u s l y
s = U(YF).S +
U(YK).S
t h e r e f o r e , i f we d e f i n e SF = u(YF)s
+ 1
SK
+
= u(YK)s
E A
7 E A
t h e r e l a t i o n s ( 5 5 ) , (55.1) and (55.2) w i l l obvious h o l d . 3.
Equivalence between S=O and supp S=0
Given S E A = A l l , t h e i m p l i c a t i o n S = 0 E
A =. supp
S = 0
i s obvious. The converse i m p l i c a t i o n i s proved t o h o l d under t h e f o l l o w i n g c o n d i t i o n s . Theorem 12 Suppose g i v e n S E (57)
A
= A l l and 7 i s c o f i n a l i n v a r i a n t . Then t h e c o n d i t i o n
S = O E A =
A/I
i s e q u i v a l e n t t o t h e f o l l o w i n g two ones (58)
supp
s
=
0
SUPPORT
(59)
2 79
S vanishes o u t s i d e o f a compact subset o f
R
Proof Assume t h a t (58) and (59) a r e v a l i d .
Then t h e r e e x i s t s a r e p r e s e n t a t i o n
I E A = A / l , w i t h s ~A ,
S = s +
such t h a t
su
R \K, Y u
= 0 on
E
N, v > p ,
f o r c e r t a i n p E N and compact subset K c R. since I i s cofinal invariant.
B u t one can assume t h a t 1-1 = 0,
L e t us now t a k e Y E o(R), such t h a t Y = l o n K then obviously
s =
U(Y).S
therefore '4.S = S E
(60)
A
0
B u t supp Y n supp S = 2, w i l l i m p l y t h a t
lJ.S = 0
E
= A/I
i n view o f ( 5 8 ) .
Therefore Corollary 5 i n Section
A
which t o g e t h e r w i t h (60) completes t h e p r o o f o f ( 5 7 ) . The converse i s obvious.
4.
0
Domains o f S o l v a b i l i t y f o r Polynomial N o n l i n e a r PDEs
The n o t i o n o f s u p p o r t d e f i n e d f o r t h e elements o f q u o t i e n t a l g e b r a s ( 3 ) o f = f e r s a n a t u r a l way f o r d e f i n i v g domains of s o l v a b i l i t y f o r p o l y n o m i a l non= l i n e a r PDEs. Suppose g i v e n t h e m-th o r d e r polynomial n o n l i n e a r PDE i n (1) , Chapter 1, which we s h a l l c o n s i d e r w i t h i n t h e f o l l o w i n g framework (see S e c t i o n 3, Chap= t e r 1 and S e c t i o n s 0 and 5, Chapter 2 ) : (61)
T(D) :
E * A
where
,A
E
= S/V E VSF
(61.2)
E
m QA
(61.3)
F
v e c t o r subspace i n M ( R ) , G subalgebra i n M(R) and G 3
(61.1)
Then, t h e c l o s e d s u b s e t i n
= A / l E ALG
R given by
C"(n)
zao
E.E. Rosinger
rE+ A
(62)
supp (T(D)S-f)
n
=
S E E
will be c a l l e d the s i n g u l a r i t y i n E
+
A of t h e mentioned PDE.
Obviously, in case t h e mentioned PDE has a sequential s o l u t i o n i n E + A , then rE A = 0. +
The open subset in (63)
given by
n E + A = n \
rE
+ A
w i l l be c a l l e d the domain of s o l v a b i l i t y i n E
-+
A of t h e mentioned PDE.
The r e s u l t s on support obtained i n Sections 1-3 lead t o the following ex= l i c i t expressions f o r t h e subsets o f s i n g u l a r i t y and domain of s o l v a b n i t y h o l y n o m i a l nonlinear PDE, expressions which a r e p a r t i c u l a r l y conve= n i e n t in order t o study t h e v a r i a t i o n of t h e mentioned subsets when t h e i r dependence on E and A i s considered. Theorem 13 The following r e l a t i o n s hold (64)
rE
+
A
E + A
(65)
n
= s Q s w Q 7 cl v - +SUPP ~ (T(D)sV
=
-
f
-
wV)
u u int lim (n \ s u p p ( T ( D ) s V - f SESWE'I v - f m
- wv)
Proof The r e l a t i o n (64) follows e a s i l y from ( 6 2 ) , as well as Theorem 3 in Section 1. The r e l a t i o n (65) follows e a s i l y from (63) and ( 6 4 ) .
0
In view of Theorem 13, i t i s obvious t h a t the domain of s o v a b i l i t y w i l l increase and correspondingly the s i n g u l a r i t y w i l l decrease , whenever S and l i n ( 6 1 . 1 ) increase.
F I N A L REMARKS
The quotient algebras of the chains ( 2 4 ) or (93) in Chapter 3, used within the Chapters 3-9, are particular cases o f the quotient algebras (1)
A = A/I
defined in Section 2 , Chapter 1, by the inclusion diagrams
(2)
i-iP-
+
> G
N
.rnuG=q UG
Indeed, besides the f a c t t h a t in the case of the quotient algebras of the mentioned chains we have (3)
G =
c"(n) or G
c'(n)
the essential particularity i s t h a t the ideals I in A s a t i s f y also the con= di t i on (4)
I i s an ideal in G
N
As seen in Sections 4 and 5 , Chapter 3, as well as in Chapter 6 , t h i s par= t i c u l a r feature of the ideals 1 in (1) and ( 2 ) proves t o be specially con= venient in establishing basic properties of the quotient algebras of the mentioned chains, properties which depend c r i t i c a l ly on the structure of the ideals I . However, as seen in Proposition 2 , Appendix 4 , the rather simple case of the Cauchy-Bolzano quotient algebra (1,2) defining the real numbers, leads t o an ideal 1 in A which does not s a t i s f y the above condition ( 4 ) , f o r G = Q. In other words, the sequential completion of Q i n the usual metric topology requires the f u l l generality of the quotient algebras ( 1 . 2 ) . I t i s therefore natural t o assume that a deeper study of the sequentia1,in particular weak solutions of polynomial nonlinear PDEs will also require quotient algebras (1-3) o f a general form, not necessarily satisfying the condition ( 4 ) . The d i f f i c u l t problem arising here i s the lack of s u f f i c i e n t knowledge concerning the structure of ideals in subalgebras of the algebra of continuous functions on a completely regular topological space.
Even in the case of general quotient algebras (1-3) a further objection 283
E.E. Rosinger
282
m i g h t be r a i s e d . o f functions
(5)
Indeed, as seen i n S e c t i o n 4, Chapter 1, t h e sequences
W E 1
a r e ' e r r o r sequences o r ' n e g l i g i b l e ' sequences o f f u n c t i o n s see S e c t i o n 2, Chapter 1 , as we 1 as Appendix l ) , w h i l e t h e sequences o f f u n c t i o n s (6)
Z G A
a r e t h e ' a d m i s s i b l e ' sequences o f f u n c t i o n s i n t h e q u o t i e n t a gebra ( 1 , 2 ) . Therefore, t h e c o n d i t i o n r e q u i r e d i n (1,2) t h a t (7)
7
i s an i d e a l i n A
m i g h t i n c e r t a i n cases prove t o be t o o s t r o n g , s i n c e i t means t h a t t h e pro= d u c t between a ' n e g i g i b l e ' sequence o f f u n c t i o n s ( 5 ) and an ' a d m i s s i b l e ' sequence o f f u n c t i o n s ( 6 ) i s always a ' n e g l i g i b l e ' sequence o f f u n c t i o n s . However, t h e l i k e l i h o o d o f t h e above o b j e c t i o n seems t o be r a t h e r s m a l l , s i n c e n a t u r a l and minimal assumptions on Iw i l l i m p l y t h a t I s a t i s f i e s t h e c o n d i t i o n ( 7 ) . Indeed, l e t us suppose t h a t (8)
G i s a subalgebra i n
M(n)
and (9)
N 7 i s a subalgebra i n G
which s a t i s f i e s t h e c o n d i t i o n s
(10)
inuG=2
(11)
7 . UG c 7
Then, w i t h t h e n o t a t i o n s i n S e c t i o n 6, Chapter 1, i t f o l l o w s t h a t
(12)
1 i s an i d e a l i n AG(T)
and t h e i n c l u s i o n diagram i s v a l i d
t h e r e f o r e , we o b t a i n t h e q u o t i e n t a l g e b r a
Moreover, i n t h e sense s p e c i f i e d i n (50-52), Chapter 1, each q u o t i e n t a l = gebra (1,2) i s o f t h e f o r m (14). I t i s w o r t h n o t i c i n g t h a t t h e v a r i a n t i n (12) o f t h e c o n d i t i o n ( 7 ) was i m = p l i e d b y t h e assumptions on 7 g i v e n i n (9-11), assumptions w h i c h a r e natu=
FINAL REMARKS
283
r a l and m i n i m a l . Indeed, (10) i s t h e ' n e u t r i x ' c o n d i t i o n (see Appendix 4) which i s anyhow assumed i n ( 2 ) . F u r t h e r , t h e c o n d i t i o n (11) means t h a t t h e p r o d u c t between a ' n e g l i g i b l e ' sequence o f f u n c t i o n s and an ' a d m i s s i b l e ' f u n c t i o n i s always a ' n e g l i g i b l e ' sequence o f f u n c t i o n s . F i n a l l y , t h e con= d i t i o n ( 9 ) which m i g h t seem t o be t h e s t r o n g e s t and thus t h e most q u e s t i o n = a b l e , means t h a t t h e p r o d u c t o f two ' n e g l i g i b l e ' sequences o f f u n c t i o n s i s always a ' n e g l i g i b l e ' sequence o f f u n c t i o n s . I n case t h e c o n d i t i o n s (9-11) a r e a c c e p t a b l e , an o b j e c t i o n m i g h t y e t a r i s e of t h e subalgebra AG ( I ) . Indeed, i n s p i t e o f t h e connected w i t h t h e
size
f a c t t h a t & ( ? ) i s t h e l a r g e s t subalgebra i n GN f o r which t h e i n c l u s i o n diagram (13) i s v a l i d , i t s s i z e m i g h t prove t o be t o o small from some p o i n t s of view, such as f o r i n s t a n c e t h e g e n e r a l i t y p r o p e m s e c t i o n 5, Chap= t e r 1) o f t h e s e q u e n t i a l , i n p a r t i c u l a r weak s o l u t i o n s f o r polynomial non= l i n e a r PDEs
.
This Page Intentionally Left Blank
APPENDIX 1 NEUTRIX CALCULUS AND NEGLIGIBLE SEQUENCES OF FUNCTIONS
Connected w i t h t h e s t u d y o f v a r i o u s a s y m p t o t i c expansions, J.G. Van d e r Corput, 12161, developed a ' n e u t r i x c a l c u l u s ' meant t o deal i n a general and u n i f i e d way w i t h ' n e g l i g i b l e ' q u a n t i t i e s . The b a s i c i d e a o f h i s method presented i n t h e sequel proved t o be o f a w i d e r i n t e r e s t , b e i n g f o r i n = stance u s e f u l i n t h e t h e o r y o f d i s t r i b u t i o n s [53,541.
-
I n t h e c o n d i t i o n s (20.1) and (21.1) i n S e c t i o n 2, Chapter 1, d e f i n i n g t h e q u o t i e n t spaces r e s p e c t i v e l y algebras o f c l a s s e s o f sequences o f f u n c t i o n s on domains i n t u c l i d i a n spaces, we have a l s o made use o f t h e n o t i o n o f ' n e g l i g i b l e ' sequences o f f u n c t i o n s . An o t h e r example can be seen i n Ap= pendix 4, where R ' i s d e f i n e d as a q u o t i e n t a l g e b r a o f c l a s s e s o f sequences o f r a t i o n a l numbers, a c c o r d i n g t o t h e Cauchy-Bolzano method. I n t h a t case t h e ' n e g l i g i b l e ' sequences o f r a t i o n a l numbers w i l l c o i n c i d e w i t h t h e se= quences o f r a t i o n a l numbers convergent t o zero. And now, t h e d e f i n i t i o n o f n e u t r i x . Suppose g i v e n an a r b i t r a r y non-void s e t X and an A b e l i a n group G. j e c t of o u r s t u d y w i l l be t h e f u n c t i o n s
f : X
+
The ob=
G
i n o t h e r words, t h e elements o f t h e C a r t e s i a n p r o d u c t
which i s i n a n a t u r a l way a l s o an A b e l i a n group. The problem i s t o d e f i n e i n a s u i t a b l e and general way t h e n o t i o n o f ' n e g l i = g i b l e ' f u n c t i o n f E Gx.
A g i v e n subgroup (2)
N C Gx
w i l l be c a l l e d a n e u t r i x , only i f YfEN,yEG:
(3)
i'
;(I)x=:J
=+
Y
=
0
i n which case t h e f u n c t i o n s f E N w i l l be c a l l e d N - n e g l i g i b l e .
285
-
2 86
E.E.
Rosinger
The i n t e r e s t i n t h e above n o t i o n comes from t h e f a c t t h a t i n case X has a d i r e c t e d p a r t i a 1 o r d e r < , one can d e f i n e a n e u t r i x l i m i t f o r f u n c t i o n s i n GX as f o l l o w s . Suppose g i v e n a n e u t r i x N c GX, a functi'on f E Gx and y E G. Then we d e f i n e N-
l i m f(x) = y v + w X o n l y i f t h e f u n c t i o n g E G d e f i n e d by
(4)
-
g(x) = f ( x )
yt x E
x,
i s knegligible. I n view o f ( 3 ) i t i s easy t o see t h a t t h e l i m i t ( 4 ) i s unique, whenever i t exists. The c o n d i t i o n ( 3 ) d e f i n i n g a n e u t r i x can be g i v e n t h e f o l l o w i n g a l g e b r a i c c h a r a c t e r i z a t i on. L e t us d e f i n e t h e group monomorphi sm: u : G
+
Gx
and denote UG = u(G) Then UG i s t h e subgroup o f -_constant functions i n G
X
.
L e t us denote by
2 t h e n u l l subgroup i n G
X
Proposition 1
A subgroup N c GX i s a n e u t r i x o n l y i f t h e i n c l u s i o n diagram NL
G
T
x
s a t i s f i e s the condition
o r e q u i v a l e n t l y t h e mapping
d e f i n e d by
NEUTRIX CALCULUS
287
i s a group monomorphism, where 8 i s t h e canon c a l q u o t i e n t epimorph sm. Proof It follows easily.
0
I t i s w o r t h n o t i c i n g t h a t t h e c o n d i t i o n ( 7 ) i s t h e o p p o s i t e o f t h e condi= t i o n t h a t t h e c h a i n o f group homomorphisms U
---+G
X
e
x /N-0
(8)
0 +G
-+G
i s exact.
Indeed, ( 7 ) i s e q u i v a l e n t t o ( 6 ) , w h i l e ( 8 ) i s e q u i v a l e n t t o
(9)
UG = N
X I n view o f ( 7 ) and ( 4 ) , we can i n t e r p r e t G / N as t h e s e q u e n t i a l c o m p l e t i o n o f G o b t a i n e d by u s i n g sequences i n G w i t h i n d i c e s i n t h e d i r e c t e d s e t X . I f we t a k e now
X = N and G = M(R)
then t h e i n c l u s i o n diagrams (20) and (21) i n Chapter 1 a r e p a r t i c u l a r cases o f ( 5 ) above, w h i l e t h e c o n d i t i o n s (20.1) and (21.1) i n Chapter 1 a r e iden= t i c a l w i t h ( 6 ) above. Moreover t h e s e q u e n t i a l s o l u t i o n s of polynomial n o n l i n e a r PDEs d e f i n e d i n S e c t i o n 3, Chapter 1, can be seen as n e u t r i x l i m i t s i n t h e sense o f ( 4 ) above. Indeed, suppose g i v e n t h e m-th o r d e r polynomial n o n l i n e a r PDE (see (1) i n Chapter 1) (10)
T(D)u(x) = f ( x ) , x E R,
w i t h continuous c o e f f i c i e n t s and r i g h t hand t e r m and l e t us c o n s i d e r T(D) : E
-+
A
where
m E
S/V
E
VSmm
c (Q)
,A
= A / 7 E AL e(n) and
E
<
A
Obviously, we can a l s o c o n s i d e r t h e mapping (11)
T(D) : Cm(R)
-f
C"(R)
i n which case f o r each g i v e n sequence o f f u n c t i o n s s E (Cm(R))N i t makes sense t o ask whether o r n o t t h e n e u t r i x l i m i t e x i s t s
And i n case t h e n e u t r i x l i m i t i n (12) e x i s t s , i t w i l l o b v i o u s l y be a func= t i o n i n c" (a). Proposition 2 Suppose g i v e n a sequence o f funct'ons s E S.
Then s i s a s e q u e n t i a l s o l u =
E.E.
288
tion i n E
-+
(13)
I -
Rosinger
A o f t h e PDE i n ( l o ) , o n l y i f
lim
T(D)sV = f
v + m
Proof By d e f i n i t i o n , t h e r e l a t i o n (13) i s e q u i v a l e n t t o T(D)s
-
u(f) E I
which i n view o f (36) i n Chapter 1, completes t h e p r o o f
APPENDIX 2 THE EMBEDDING IMPOSSIBILITY RESULT OF L.SCHWARTZ
Two i m p o s s i b i l i t y r e s u l t s d i s c o v e r e d e a r l y i n t h e development o f t h e t h e o r y o f d i s t r i b u t i o n s came t o have a s i g n i f i c a n t r o l e i n shaping t h a t t h e o r y . H i s t o r i c a l l y t h e second one, i n 1957, due t o H.Lewy, [ 1201 , showed t h a t t h e d i s t r i b u t i o n a l framework i s n o t s u f f i c i e n t f o r t h e s t u d y o f l i n e a r PDEs w i t h v a r i a b l e c o e f f i c i e n t s . Indeed, H.Lewy proved t h a t t h e r a t h e r simple, f i r s t o r d e r l i n e a r PDE:
a u(x) axl
+
a u(x) iax2
-
2i(xl+ix2)
a u(x) -
= f(x),x
3x3
= (x
1' x 2 ,x 3 )
E
R3,
dges n o t possess even l o c a l d i s t r i b u t i o n s o l u t i o n s u, f o r a l a r g e c l a s s o f C -smooth r i g h t hand terms f. The c a r e f u l s t u d y o f t h a t i m p o s s i b i l i t y r e s u l t l e a d t o i n t e r e s t i n g necessary and/or s u f f i c i e n t c o n d i t i o n s f o r t h e e x i s t e n c e o f d i s t r i b u t i o n s o l u t i o n s f o r l a r g e classes o f l i n e a r PDEs w i t h v a r i a b l e c o e f f i c i e n t s , 80,2141. However, t h e message o f H.Lewy's impos= s i b i l i t y r e s u l t remained t h e same: t h e inadequacy o f t h e d i s t r i b u t i o n a l framework, even f o r t h e s t u d y o f l i n e a r PDEs. On t h e o t h e r hand, t h e f i r s t i m p o s s i b i l i t y r e s u l t i n 1954, due t o L.Schwartz, [ 1811 , p o i n t e d o u t t h e d i f f i c u l t i e s i n t r y i n g t o e x t e n d i n s i m p l e ways t h e d i s t r i b u t i o n a l framework, by showing t h a t t h e v e c t o r space o f d i s t r i b u = t i o n s D'(Q) cannot be embedded i n t o a s i n g l e d i f f e r e n t i a l a l g e b r a w i t h convenient properties
.
The d e t a i l s o f t h a t r e s u l t a r e presented now i n t h e one dimensional case n = l and R = R ' , as considered a l s o i n t h e o r i g i n a l v e r s i o n . Theorem 1 Suppose g i v e n an a s s o c i a t i v e a l g e b r a A and a l i n e a r mapping c a l l e d d e r i v a = + A s a t i s f y i n g the L e i b n i t z r u l e f o r product d e r i v a t i v e .
tive D : A
Suppose f u r t h e r t h a t - l ) * ) and x 2 ( L n l x l -1) * I
(1)
the functions l,x,x(LnIxI belong t o A
(2)
t h e c o n s t a n t f u n c t i o n 1 i s t h e u n i t element i n t h e a l g e b r a A
(3)
the multiplication i n A i s such t h a t (X(LMlXl - l ) ) . x = x 2 ( L n l x l -1)
.
* ) f o r x=O,both o f t h e f u n c t i o n s have b y d e f i n i t i o n t h e v a l u e z e r o .
289
E . E . Rosinger
290
(4)
t h e mapping 0 : A -t A a p p l i e d t o t h e f u n c t i o n s 1 , X , X 2 ( l ~ I X J -1) E C ' ( R ' ) i s t h e usual d e r i v a t i v e on C'(R*)
Then, t h e r e i s no 6 E A,
(5)
6
# 0. such t h a t
X.6 = 0
Remark 1 The meaning o f t h e i m p o s s i b i l i t y r e s u l t i n Theorem 1 i s t h a t a d i f f e r e n t i a l a l g e b r a which s a t i s f i e s t h e c o n d i t i o n s (1-4), cannot c o n t a i n t h e D i r a c 6 d i s t r i b u t i o n , known t o possess t h e i m p o r t a n t p r o p e r t y ( 5 ) , g i v i n g an upper bound on t h e s i n g u l a r i t y e x h i b i t e d a t x=O by t h a t d i s t r i b u t i o n . I t i s i n t e r e s t i n g t o mention t h a t t h e above i m p o s s i b i l i t y r e s u l t has q u i t e o f t e n been misunderstood, b e i n g i n t e r p r e t e d as t h e " i m p o s s i b i l i t y o f mul= ti p 1y in g d is t r ib ut ion s " .
As shown i n chapters 1 and 3, t h e way o u t o f t h i s impasse i s t o embed 0' i n t o a c h a i n o f algebras, w i t h t h e d e r i v a t i v e a c t i n g between p o s s i b l y dif= f e r e n t W r a s o f t h e chain. P r o o f o f Theorem 1 The i d e a i s very simple, namely t o c o n s t r u c t i n A a " l e f t i n v e r s e " x-' the function x
(6)
x
for
-1 . x = l
and then, assuming ( 5 ) v a l i d f o r c e r t a i n 6 ~ A , 6 # 0, t o use t h e a s s o c i a t i = v i t y o f t h e product, i n o r d e r t o o b t a i n t h e c o n t r a d i c t i o n
(7)
0 = x-!(x.6)
= (x-l.x).S
= 6
Now ( 6 ) can o b v i o u s l y b e o b t a i n e d b y t a k i n g
However, i n o r d e r n o t t o d i m i n i s h t h e power o f Theorem 1, one should a v o i d demanding t h a t A c o n t a i n s s i n g u l a r f u n c t i o n s o f t y p e ( 8 ) . Here, t h e pre= sence o f t h e d e r i v a t i v e mapping D : A -t A w i l l h e l p a v o i d i n g such a de= mand, by n o t i c i n g t h a t i n t h e sense o f t h e usual d e r i v a t i v e D*(x(LPI~xJ
-1)) = l / x , Y x E R ' ,
x # 0,
and X ( ~ P Z / -1) X ] E f ( R ' ) , assuming t h e f u n c t i o n vanishes f o r x=O. I n t h i s way, i t w i l l be s u f f i c i e n t t o demand t h a t A c o n t a i n s s e v e r a l continuous f u n c t i o n s , as seen i n ( 1 ) . And now, back t o t h e r i g o r o u s c o n s t r u c t i o n o f x - l i n ( 6 ) . we s h a l l show t h a t one can t a k e
(9) and o b t a i n
x
-1
= D 2 ( x ( L ~ l x l- 1 ) ) E A
I n t h i s respect,
29 1
EMBEDDING IMPOSSIBILITY
(10)
( D 2 ( ~ ( L l t I x -l l ) ) . ~= 1
Indeed, i n view o f t h e L e i b n i t z r u l e and t h e l i n e a r i t y o f D, one o b t a i n s
( D 2 ( ~ ( L ~- Il ~) )I ) . ~= D ’ ( ( X ( ~ M ~- X l ) ~) . ~ ) - ~ ( D ( x ( L ? M ~-1)).DX x~
- ( x ( L M ~ x -I 1 ) ) . D 2 x Hence, i n view o f ( 3 ) and ( 4 ) , i t f o l l o w s t h a t
(11)
(D2(X(&kLlXI - l ) ) ) . ~= D 2 ( ~ 2 ( & I -~1 /) ) - 2 D ( x ( & l x l
-1))
But, ( 4 ) w i l l a l s o g i v e
D ( x 2 ( L ~ l x l- 1 ) ) = ~ X ( L P I ~- 1X) I t x therefore
D * ( x ‘ ( L M I x I - 1 ) ) = ZD(X(LII~X~ - 1 ) )
+
1
I n t h a t way, (11) w i l l g i v e ( 1 0 ) .
0
Corollarv 1
I f , under t h e c o n d i t i o n s i n Theorem 1, t h e f o l l o w i n g a d d i t i o n a l p r o p e r t i e s o f A a r e assumed: (12)
the functions 1x1 and x l x l a l s o belong t o A
(13)
the m u l t i p l i c a t i o n
.
i n A i s such t h a t
x.IxI = x l x l (14)
A
+
A applied t o the function
xlxl
E
C’(R’)
t h e mapping D :
i s t h e usual d e r i v a t i v e on C ’ ( R ’ ) then
(15)
D21x1 = 0
Proof I n view o f t h e L e i b n i t z r u l e and t h e l i n e a r i t y o f D, o b v i o u s l y x.D21xl = D 2 ( ~ . I ~ -I ) 2Dx.Dlxl
- D’x.IxI
Thus, i n view o f (13) and ( 1 4 ) , one o b t a i n s
(16)
x . D ~ ~ x =I D 2 ( x l x l )
But, (14) w i l l g i v e D(xlxl) =
21x1
-
2Dlxl
-
E.E.
292
Rosinger
therefore D 2 ( x l x l ) = 201x1 Now, ( 1 6 ) w i l l i m p l y x.D21xl = 0 and i n view o f Theorem 1, t h e p r o o f o f ( 1 5 ) i s completed.
0
Remark 2
1)
(17)
I f D i s t h e d i s t r i b u t i o n a l d e r i v a t i v e on P ’ ( R ‘ ) then D ’ I X I = 26
where 6 i s t h e D i r a c d i s t r i b u t i o n . T h e r e f o r e , t h e r e l a t i o n ( 1 5 ) i n Corol= l a r y 1, shows once more t h e t r i v i a l i t y o f t h e d i f f e r e n t i a l a l g e b r a A con= s idered .
2)
I t i s i m p o r t a n t t o n o t i c e t h a t t h e above embedding i m p o s s i b i l i t y r e s u l t s do n o t suppose t h e c o m m u t a t i v i t y o f t h e d i f f e r e n t i a l a l g e b r a A , b u t o n l y t s a s s o c i a t i v i t y . I n t h i s r e s p e c t , t h e embeddings o f t h e d i s = t r i b u t i o n s i n t o chains o f commutative and a s s o c i a t i v e algebras pre= sented i n chapter 3, p o i n t o u t t h e u t i l i t y o f d e a l i n g w i t h chains instead o f s i n g l e d i f f e r e n t i a l algebras.
APPENDIX 3 A NONLINEAR EXTENSION OF THE LAX-RICHTMYER EQUIVALENCE BETWEEN STABILITY AND CONVERGENCE OF DIFFERENCE SCHEMES
+
The converegence o f a d i f f e r e n c e scheme t o an e x i s t i n g s o l u t i o n o f a PDE i s o b v i o u s l y an ap r o x i m a t i o n p r o p e r t y which does not n e c e s s a r i l y r e q u i r e t h e completeness o t e t o p 0 o g i c a l s t r u c t u r e on t h e space o f n u m e r i c a l and e x a c t s o l u t i o n s . T h e r e f o r e , i t makes sense t o l o o k f o r a c h a r a c t e r i z a t i o n f o r i n s t a n c e , o f s t a b i l i t y t y p e - of t h e convergence o f a d i f f e r e n c e scheme, c h a r a c t e r i z a t i o n w h i c h does n o t i n v o l v e completeness , u n l i k e i t happens i n t h e Lax-Richtmyer p r o o f o f t h e i m p l i c a t i o n ' c o n v e r g e n t * s t a b l e ' i n t h e p a r t i c u l a r l i n e a r case, where e s s e n t i a l use i s made o f t h e u n i f o r m bounded= ness p r i n c i p l e o f l i n e a r o p e r a t o r s i n Banach spaces. I t t u r n s o u t t h a t such a s t a b i l i t y t y p e c h a r a c t e r i z a t i o n o f t h e convergence o f a d i f f e r e n c e scheme can be o b t a i n e d i n a simple, r a t h e r d i r e c t way w h i c h o n l y i n v o l v e s p r o p e r t i e s r e l a t e d t o c o n t i n u i t y , compactness and bounded= ness i n normed v e c t o r spaces. Moreover, due t o i t s s i m p l i c i t y t h e p r o o f o f t h e e q u i v a l e n c e between s t a b i l i t y and convergence w i l l be v a l i d i n t h e general n o n l i n e a r case and under assumptions which a r e weaker t h a n t h o s e i n t h e Lax-Richtmyer l i n e a r v e r s i o n .
Suppose g i v e n a normed v e c t o r space (X,II I I ) and t h e e v o l u t i o n e q u a t i o n
(1.1) (1.2)
U(t) = A(U(t)), t
U(0)
[O,T]
= u
where A: X1 + X,X1 C X , U ( t ) , u E XI. U s u a l l y , t v a r i a b l e , w h i l e t h e elements o f X a r e f u n c t i o n s We s h a l l assume v a r i a b l e x E Rm, t h a t i s , u = f ( x ) , U ( t ) = F ( t , x ) . a c e r t a i n non-void subset XoC X1 d e f i n e d b y i n i t i a l and p o s s i b l y boundary c o n d i t i o n s , t h e f o l l o w i n g "unique s o l u t i o n " p r o p e r t y h o l d s (2)
VuEX0: 31 U : [O,T]
Xo :
1 (u(t+At)-u(t))-A(u(t))ll
(2.1)
lim A*
(2.2)
U(0) = u.
llE
-+
= 0, f o r t
I n t h a t case, one can d e f i n e t h e e v o l u t i o n o p e r a t o r (3)
E: [ 0,T ]
-+
(Xo
+
Xo)
given by 293
[O,Tl
29 4
(3.1)
E.E. Rosinger
E(t)u
=
for t
U(t),
E
[O,Tl, u
E
Xo
which will have t h e semigroup property
(3.2)
E(0) = i d X 0
(3.3)
E(t+s) = E ( t ) E ( s ) , for t,s
E
[O,T], t + s GT.
The problem ( 1 ) with t h e s o l u t i o n ( 2 ) , respectively ( 3 ) , i s c a l l e d -____ properly
posed, only i f the mapping (4)
[ O , T l x Xo
3
(t,u) + E(t)u
E
Xo
i s continuous. As known [116,165], i n the l i n e a r case, the problem (1) i s properly posed, only i f t h e family of l i n e a r operators (5)
E ( t ) , with t
i s uniformly bounded. a1 so hold.
E
[O,Tl
Then, in view of ( 2 ) , i t i s easy t o see t h a t ( 4 ) w i l l
A d i f f e r e n c e scheme i s c a l l e d any family of mappings : Xo + X o y with
t
having t h e fol lowing two properties (6.1)
[ O , T ] x Xo
3
(At,u)
-+
(0,TI
E
.
CAtu
The mapping E
Xo
, f o r At = 0. i s continuous, where Cat = i d xO B c X o y t h e r e e x i s t s M ( B ) > 0 , such t h a t Vu,v
E
B, A t
E
Further, f o r each bounded
(0,TI:
An important c l a s s of e x p l i c i t d i f f e r e n c e schemes s a t i s f y i n g t h e conditions ( 6 . 1 ) and ( 6 . 2 ) a r e those of the form (7)
CAtu = u
+ Atf(At,u), f o r
U E
x,
where f : [ O , T l x X + X i s continuous and s a t i s f i e s the local Lipschitz con= dition
VB c X bounded :
IIf( & , u )
-
f(At,v)ll
< M(B)llu-vll.
Usually, C u represents a function of d i s c r e t i z e d space v a r i a b l e , associa= ted t o t h e A h n c t i o n u E Xo of continuous space v a r i a b l e . In t h a t case,Cat
295
NONLINEAR STABILITY AND CONVERGENCE
w i l l a l s o depend on t h e f i n i t e space increments
and t h e aim o f t h e s t a b i l i t y a n a l y s i s i s t o e s t a b l i s h a r e l a t i o n between Ax and A t , which w i l l g r a n t t h e convergence o f t h e d i f f e r e n c e scheme ( 6 ) t o t h e s o l u t i o n o f ( l ) , p r o v i d e d t h a t A t -+ 0. The d i f f e r e n c e scheme ( 6 ) i s c a l l e d c o n s i s t e n t w i t h t h e problem ( l ) , o n l y i f f o r any compact K c Xo and E > 0, t h e r e e x i s t s ~ ( K , E ) > 0, such t h a t Yu E K, t E [O,T],
A t E (O,T]
:
(8) At
<6(K,E)
* IICAtE(t)u
- E(At)E(t)UY <
€.At
The above c o n d i t i o n o f c o n s i s t e n c y appears t o be somewhat more r e s t r i c t i v e t h a n t h e usual one i n t h e l i n e a r case [165, p.44 c o n d i t i o n ( 3 . 6 ~ ) , i n t h a t i t asks f o r u n i f o r m i t y a l s o i n r e s p e c t t o u E K, f o r each g i v e n corn= p a c t K C Xo. I t i s easy t o see t h a t t h e c o n s i s t e n c y c o n d i t i o n (8) w i l l be s a t i s f i e d , whenever ( 2 ) and t h e f o l l o w i n g c o n d i t i o n l i m 11A t 4
1
At
(CAtE(t)u
h o l d u n i f o r m l y i n t E[O,T]
-
E(t)u)
-
A(E(t)ull = 0
and u E K, f o r each g i v e n compact K
C
Xo.
We s h a l l say t h a t t h e d i f f e r e n c e scheme ( 6 ) i s convergent t o t h e s o l u t i o n of ( l ) , o n l y i f f o r any compact K C Xo and E > D, t h e r e e x i s t s ~ ( K , E ) > 0, such t h a t Yu E K, t (10)
n where Cat
At,
[O,T],
N,
A t E (O,T],
n
E
1 t - n . A t l < q ( k , ~ ) * IIC:t~
-
E ( t ) u l l <E
E
m o a t
denotes t h e n - t h successive i t e r a t i o n o f CAt:
GT :
Xo
-+
Xo.
The above d e f i n i t i o n o f convergence c o i n c i d e s i n t h e l i n e a r case w i t h t h e usual one [ 116,1651. Indeed, i n t h a t case, a convergent d i f f e r e n c e scheme w i l l under t h e c o n d i t i o n s o f Lax-Richtmyer's theorem b e s t a b l e and, as seen i n t h e p r o o f of t h a t theorem, s t a b i l i t y w i l l g r a n t u n i f o r m convergence on [ 0,Tl and bounded B C Xo. The p r a c t i c a l importance o f t h e e q u i v a l e n c e between t h e convergence and s t a b i l i t y of a d i f f e r e n c e scheme i s i n t h e f a c t t h a t t h e s t a b i l i t y o f a d i f f e r e n c e scheme o n l y depends on t h e d i f f e r e n c e scheme i t s e l f and, u n l i k e convergence. i t i s n o t r e l a t e d to t h e p n s s i h l y unknown s o l u t i o n ( 2 ) o r ( 3 ) o f problem (1). I n t h e l i n e a r case 116,165 , t h e s t a b i l i t y o f t h e d i f f e r = ence scheme (6) means t h e uniform boundedness o f t h e family o f l i n e a r opera= tors C,t: w i t h A t E (O,T], n E N, n.At G T . (11) The way t h e e x t e n s i o n i s done i n t h e n o n l i n e a r case i s g i v e n i n t h e f o l l o w = i n g d e f i n i t i o n . The d i f f e r e n c e scheme ( 6 ) i s c a l l e d s t a b l e , o n l y i f f o r
296
E.E.
any compact K c Xo,
Rosinger
0, such t h a t
there e x i s t s L(K)
The e x t e n s i o n o f t h e Lax-Richtmyer equivalence [ 116,1651 between convergence and s t a b i l i t y i s o b t a i n e d as f o l l o w s : Theorem 1 Suppose t h e problem (1) i s p r o p e r l y posed and t h e d i f f e r e n c e scheme ( 6 ) i s c o n s i s t e n t w i t h i t . Then, t h e d i f f e r e n c e scheme ( 6 ) i s convergent t o t h e s o l u t i o n ( 2 ) o r ( 3 ) o f problem ( l ) , o n l y i f i t i s s t a b l e . Proof The i m p l i c a t i o n " c o n v e r g e n t * s t a b l e " . Assume K C Xo compact and denote
B = titu
I
UE
K,At
E ( O , T l , n E N, n A t < T I .
We s h a l l prove t h a t
(13)
BC
Xo bounded.
Indeed, o t h e r w i s e one o b t a i n s u . E K, J such t h a t n A . t g T, j J
(14)
with
j E
A. t E ( O , T l ,
J
N,
n . E N, f o r j, J
N
Since K i s compact and i n view o f ( 1 4 ) , one can e v e n t u a l l y by passing t o subsequences, assume t h a t t h e r e l a t i o n s h o l d
lim
(16)
j+
u ;u E K j
l i m n.A.t = t J J
(17)
j + m
Assume now, t h a t a l s o
then ( 1 5 ) g i v e s n. IICA;,U~ Since u j
E
K, w i t h j
E
-
E ( t ) u . l l t IIE(t)u.lI, w i t h j J J
E
N.
N, t h e r e l a t i o n s ( 1 7 ) and ( 1 8 ) , t o g e t h e r w i t h ( l o ) ,
NONLINEAR STABILITY AND CONVERGENCE
29 7
w i l l give n, j
Jim
+ rn
-
IIC J u . A j t J
E(t)u.ll = J
o
F u r t h e r , (16) and ( 4 ) r e s u l t i n (21)
l i m eE(t)ujll
=
IIE(t)ull
j,,
B u t , ( 2 0 ) and ( 2 1 ) c o n t r a d i c t ( 1 9 ) , thus, t h e assumption ( 1 8 ) i s n o t c o r = r e c t . Then, e v e n t u a l l y b y p a s s i n g t o a subsequence, one can assume (22)
ajt
' 6 > 0,
w i t h j E N.
NOW, ( 1 4 ) and (22) i m p l y nj
' T/6,
N
with j E
hence, b y p a s s i n g e v e n t u a l l y t o a subsequence, one can assume t h a t = n = c o n s t a n t , w i t h j E N. j I n t h a t case, ( 1 5 ) i m p l i e s
(23)
n
(24)
j < IIC,.~U~II n G J n
llC:.tuj - CA.t~II n , w i t h j E N. J J hence (24) i s absurd, s i n c e t h e compact s e t IIC,.tn
UII
I f n=O, t h e n Ca.t = i d X , J 0 w i l l a l s o be bounded. Otherwise, n
w i l l give
t l i m A . t = T~ @,TI,
+
J
> 1 together
K
w i t h ( 1 7 ) , ( 2 2 ) , and (23)
hence, i n view o f (16) and ( 6 . 1 ) one o b t a i n s
j + m J
Iim
IIC,.~UII n
=
IICtn uII -
J
j + c u
n
l i m I I Cn A . ~ U -~ C:.~UII J J
= 0.
j + m
NOW, (25) and (26) w i l l c o n t r a d i c t (24) and t h e p r o o f o f (13) i s completed. I n o r d e r t o p r o v e t h a t t h e a r b i t r a r y compact K c Xo s a t i s f i e s ( 1 2 ) , we s h a l l a p p l y (6.2) t o B g i v e n i n ( 1 3 ) . Assume t h e r e f o r e , u,v E K, A t E (O,T] , n E N, n.At < T. Then, (6.2) a p p l i e d s u c c e s s i v e l y , w i l l g i v e
-
IICitu
<
C!tvII
( l t M ( B ) A t ) l I C~~lu-Cn~*vll~...
u-vII
one o b t a i n s (12) by t a k i n g L ( K ) = e M(B)T* The imp1 i c a t i o n " s t a b l e
=)
Assume K c Xo compact and then (27)
C:t~
-
convergent". E
> 0.
E ( t ) u = (Citu
If u E
-
K,
t E [ O,T]
E(n A t ) u )
t
,A t
(E(n A t ) ,
(O,T] and n E N,
E
-
E(t)u).
298
E.E.
Rosinger
But, o b v i o u s l y
and i n view o f ( 4 ) , t h e s e t
I
K ' = {E(s)v
i n compact, hence, i n view o f (6.1),
I
K" = {CAsE(s)v
AS,S
i s a l s o compact. NOW, i n case n A t scheme ( 6 ) w i l l g i v e (29)
llCiiPCAtE( ( p - 1 ) A t ) u Q
,v
s E [O,T]
K } c Xo
E
the set E
Q
Q
E
K}
Xo
C
T, t h e s t a b i l i t y o f t h e d i f f e r e n c e
- C!EPE(p
At)ull
-
L ( K ' u K") llCAtE((p-1)At)u
But, i n view o f ( 8 ) , i f A t
v
[O,T],
Q
E(p At)Ull
, with
1Q p
Q
n
~ ( K , E ) , then
lICAtE((P-l)At)U
-
E(p At)ull
Q
&.At
hence, (28) and ( 2 9 ) w i l l g i v e
(30)
-
IIC:t~
E ( n At)ull
Q
nL(K'
U K")E.At
Q
L(K'
U
K").T.E.
F u r t h e r , ( 4 ) i m p l i e s t h e e x i s t e n c e o f ~ ' ( K , E ) z 0, such t h a t It
-
n.Atl
<
~ ' ( K , E ) * llE(n A t ) u
-
E(t)ull
Q E
t h e r e f o r e , (27) and (30) r e s u l t f i n a l l y i n llCitu
-
E(t)ull
Q
( 1 t L ( K ' u K").T)E
and t h e convergence o f t h e d i f f e r e n c e scheme ( 6 ) i s proved. Remark 1 The convergence o f a d i f f e r e n c e scheme b e i n g an a p p r o x i m a t i o n p r o p e r t y , i t i s n a t u r a l t h a t t h e above equivalence r e s u l t between convergence and s t a = b i l i t y does n o t r e q u i r e t h a t t h e normed v e c t o r space (X,II 1 1 ) i s complete, t h a t i s a Banach space, n e i t h e r r e q u i r e s t o p o l o g i c a l p r o p e r t i e s f o r t h e sub= s e t Xo C X o f " i n i t i a l c o n d i t i o n s " . S i m i l a r l y t o t h e l i n e a r case, [ 116,1651 , t h e convergence and c o n s i s t e n c y o f t h e d i f f e r e n c e scheme ( 6 ) i s i n f a c t o n l y r e l a t e d t o t h e e v o l u t i o n o p e r a t o r ( 3 ) , t h e r e f o r e , t h e r e s u l t i n t h e above theorem i s v a l i d f o r any semigroup o f o p e r a t o r s ( 3 ) s a t i s f y i n g t h e c o n d i t i o n ( 4 ) . I n t h i s r e s p e c t , t h e connec= t i o n w i t h problem (1) and i t s s o l u t i o n ( 2 ) can be weakened, a s k i n g f o r i n = stance, t h a t ( 2 ) and thus (3.1), o n l y h o l d f o r u E X i , where X; i s a con= v e n i e n t subset i n Xo.
299
NONLINEAR STABILITY AND CONVERGENCE
As shown i n t h e f o l l o w i n g s i m p l e example, t h e p r i n c i p l e u n i f o r m boundedness o f l i n e a r o p e r a t o r s considered w i t h i n t h e framework o f normed v e c t o r spaces i s e s s e n t i a l l y connected w i t h completeness. Example 1 Suppose t h e normed v e c t o r space ( X , II II ) i s d e f i n e d as f o l l o w s
and l e t us d e f i n e t h e f a m i l y o f l i n e a r o p e r a t o r s T : X x by
x
xA
if v =
-+
X, w i t h A
E
N,
x
yv = 0
i f v # x
Then o b v i o u s l y
V
IIT II = A ,
x
E
N,
therefore, the family o f l i n e a r operators T : X u n i f o r m l y bounded. A
-+
X, w i t h
xE
N, i s
not
However , suppose g i v e n x = (x0,x1, then, f o r a c e r t a i n
x
...,xV ......) E x y
E
N , t h e r e l a t i o n holds
= 0, V v E N, v
V
> 11
therefore
T
x = O , V A E N , A > ~
x
I t f o l l o w s i n t h i s way t h a t f o r any g i v e n x
IIT xII
A
,with A
E
X, t h e f a m i l y o f numbers
E N,
i s bounded b y max {IIT xi11 A where
,,
E
x
E
N, 1
,,I
N depends o n l y on x E X.
As t h e normed v e c t o r space (X,II II) i s o b v i o u s l y n o t complete, we can see t h a t t h e r e s u l t i n g framework i s not s u f f i c i e n t f o r g r Z X i n g t h e v a l i d i t y o f t h e
300
E.E.
Rosinger
principle o f uniform boundedness o f linear operators.
APPENDIX 4 THE CAUCHY-BOLZANO QUOTIENT ALGEBRA CONSTRUCTION OF THE REAL NUMBERS
A c l a s s i c a l example o f q u o t i e n t a l g e b r a c o n s t r u c t i o n i s g i v e n b y t h e w e l l known Cauchy-Bolzano method, [218] , f o r c o n s t r u c t i n g t h e s e t o f r e a l num= b e r s R 1 f r o m t h e s e t o f r a t i o n a l numbers Q. L e t us denote A N t h e subalgebra i n Q o f a l l t h e Cauch sequences r = (ro,rl,. r a t i o n a l numbers and l e t us denote b y
. . ,rv,.. . ..)
r t h e i d e a l i n A o f a l l t h e sequences z = numbers which converge t o zero.
(Z~,Z~,...,Z~,
....)
o f rational
We s h a l l denote b y UQ
t h e subalgebra i n A o
Q t h e null i d e a l i n Q
if .a l l
t h e c o n s t a n t sequences.
F i n a l l y , we denote b y
Then t h e f o l l o w i n g i n c l u s i o n diagram i s v a l i d
(1)
+-I Q
i
> QN
’ 0‘
and i t s a t i s f i e s t h e c o n d i t i o n
(2)
rnuQ=Q
Moreover, a c c o r d i n g t o Cauchy-Bolzano
(3)
R’ and A = A / l a r e i s o m o r p h i c f i e l d s .
As s en i n Appendix 1, t h e c o n d i t i o n ( 2 ) above means t h a t 1 i s a n e u t r i x i n Qg and t h e c o r r e s p o n d i n g n e u t r i x l i m i t i s i d e n t i c a l w i t h t h e u s u a l l i m i t f o r r a t i o n a l numbers, i . e . , t h e r e l a t i o n h o l d s
301
of
302
Rosinger
E.E.
I
(4)
-
lim
r
W
V
= lim
r
W
.
whenever r = ( royrl ,.. ,rv , . . . . )
E
v N Q and one o f t h e l i m i t s i n ( 4 ) e x i s t s .
The i d e a l I has s e v e r a l i m p o r t a n t p r o p e r t i e s p r e s e n t e d now. F i r s t we n o t i c e t h a t
(5)
7 i s a maximal i d e a l i n A
since A = A / 7 i s a f i e l d . L e t us now denote by
B t h e subalgebra i n Q obviously
N
o f a l l t h e bounded sequences o f r a t i o n a l numbers. Then
A i s a subalgebra i n B
(6)
The s p e c i a l r e l a t i o n between 7 and B i s presented i n t h e n e x t two p r o p o s i = tions. Proposition 1 7 i s a maximal subsequent i n v a r i a n t i d e a l i n B (see (28) i n Chapter 2 )
Proof I _
Assume t h a t i t i s f a l s e and J i s a subsequence i n v a r i a n t i d e a l i n 8, such that
(71
# L e t us t a k e
(8)
B
7C.3 C
#
then
..., V Y "
z =(zoyzl,
...)E J \ I
Since J c 8, t h e r e l a t i o n (8) g i v e s a p o i n t 5 ) i n t = (zoyzl, zv VY""'
(9)
lim w
z'
v
E
R'\COI
..., ,...... ),
2' = ( t ~ , 2 ~ , . . . , Z '
=
and a subsequence such t h a t
5
But
s i n c e 3 i s subsequence assume t h a t
nvariant.
Moreover, i n view o f ( 9 ) we can o b v i o u s l y
I z ' l 2 I 5 1'2 > 0, V V E N . ....) Therefore d e f i n i n g z " = ( z ~ , ~ ~ ,..J;,.
E
QN b y
CAUCHY-BALZANO QUOTIENT ALGEBRA
30 3
Now, t h e r e l a t i o n s (10-12) w i l l y i e l d
1
z'.z" E J . B
=
C
J
hence (13)
J = B
s i n c e J i s an i d e a l B . Since ( 7 ) and (13) c o n t r a d i c t each o t h e r , t h e p r o o f i s completed. Proposition 2
B i s a maximal subalgebra i n Q
N
i n which 7 i s an i d e a l .
Proof Assume t h a t i t i s f a l s e and C i s a subalgebra i n Q
(14)
B
C
N
such t h a t
C
# (15)
I i s an i d e a l i n C
L e t us t a k e t h e n
z
,..., zV,......)
= (z0,z1
E
C \ B
I t f o l l o w s t h a t t h e r e e x i s t s a subsequence z ' (zo,zl ,..., zv ,.... ) , such t h a t
z
(16)
lim lJ-
I
Izv
=
= (zv 0
,..., zv
,z vl
,.....) i n lJ
m
lJ
Obviously, we can assume t h a t (17)
v 0 < v1 <
...
< lJv <
.....
and (18)
zv
# O , V l J € N
lJ
Then, i n view of (17) and ( 1 8 ) , we can d e f i n e r = (ro,rl ,..., rv,.... )E Q N by (19)
rv = l / z v
if v P
lJ
G
v
i
But, t h e r e l a t i o n s ( 1 9 ) and (16) y i e l d
v
P +1
E . E . Rosinger
304
(20)
r = (ro,rl,
...,r v , . . . . . . )
E
I
therefore, i n view of (15) we obtain (21)
r.z
1.C c I
E
However, in view of (19) we obtain
and the relations ( 2 1 ) and ( 2 2 ) obviously contradict each other.
There=
fore (14) cannot hold.
0
Corollary 1 B i s the largest subalgebra in QN in which 2 i s an ideal.
Proof With the notations in Section 6 , Chapter 1, i t follows from Proposition 2 that B = A (I)
Q
0
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