A Descomposition Theorem for Solving Initial Value Problems in Associated Spaces (Tutschke)

8/12/2019 A Descomposition Theorem for Solving Initial Value Problems in Associated Spaces (Tutschke)

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R E N D I C O N T I D E L C l R C O L O M A T E M A T l CO D i P A L E R M OS e r i e I I , To m o X L n l 1 9 9 4 ) . p p. 4 1 9 -4 3 4

A D E C O M P O S I T I O N T H E O R E MF O R S O L V I N G I N I T I A L

V A L U E P R O B L E M S I N A S S O C I A T E D S P A C E SR . H E E R S I N K - W . T U T S C H K E * )

I n i ti a l v a lu e p r o b l e m s o f Ca u c h y - K o v a l e v s k a y a t y p e c a n b e s o lv e dn o t o n ly i n s p a c e s o f h o lo m o r p h i c o r g e n e r a l i z e d a n a ly t i c f u n c ti o n s b u ta l s o i n m o r e g e n e r e a l a s s o c i a t e d s p a c e s p r o v id e d t h e a s s o c i a t e d s p a c eis de f ined by a pa r t ia l d i f fe ren t ia l equa t ion whose so lu t ions pe rmi t anin t e r i o r e s t im a t e . Th e p r e s e n t p a p e r i s a im e d a t s o lv in g i n i t i a l v a lu ep r o b l e m s w i th i n i t i a l f u n c t i o n s w h ic h c a n b e d e c o m p o s e d i n c o m p o n e n t sb e lo n g in g t o a f a m i ly o f a s s o c i a t e d s p a c e s .

Th e m a in r e s u l t i s a d e c o m p o s i t i o n t h e o r e m o b t a in e d i n tw ov e r s io n s: T h e f ir s t v e r s i o n u s e s f a m i l i e s o f s c a l e s o f B a n a c h s p a c e sw i th r e s p e c t t o t h e s p a c e l i k e v a r i a b l e s ) , w h i l e t h e s e c o n d o n e a p p l i e s

th e c o n t r a c t i o n - m a p p in g p r i n c ip l e t o a Ba n a c h s p a c e o f f u n c t i o n s d e f i n e din a c o n i c a l d o m a in a n d d e p e n d in g o n b o th t h e t im e a n d t h e s p a c e l i k evar iab les .

1 S t a t e m e n t o f t h e p r o b l e mI n i t i al v a l u e p r o b l e m s o f t y p e

u1 ) dt - ~u

* ) Th e a u th o r s t h a n k P r o f. G . F I C H E RA f o r s o m e h in t s im p r o v in g t h e fi n al v e r s i o nof the paper .


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4 20 R rmERSI~ . w. r~rsc H =( 2 ) u ( 0 ) = u 0c a n b e s o l v e d i n a s c a l e o f B a n a c h s p a c e s B , , 0 < s < S o, p r o v i d e dt h e ( li n e a r ) o p e r a t o r ~ h a s s u i t a b le p r o p e r t i e s i n t h e g i v e n s c a l e( c f . , f o r i n s t a n c e , [ 1 3 ], [ 1 2 ] o r [ 1 4 ]) . I f t h e a b s t r a c t o p e r a t o r e q u a t i o n( 1 ) o r i g i n a t e s f r o m a p a r t i a l d i f f e r e n t i a l e q u a t i o n , i n m a n y c a s e st h e o p e r a t o r ,9 a c t s i n a s o - c a l l e d a s s o c i a t e d s p a c e d e f i n e d b y ar e l a t e d d i f f e n t i a l e q u a t i o n ~ ' u - - - - 0 , i .e . t h e o p e r a t o r ~ t r a n s f o r m st h e a s s o c i a t e d s p a c e i n t o i t s e l f w h e r e t h e s p a c e s B s c a n b e d e f i n e da s t h e s e t o f a ll s o l u t i o n s o f fC u --- 0 i n a f a m i l y o f s u b d o m a i n s D , ,e x h a u s t i n g t h e d o m a i n D i n w h i c h th e i n i ti a l f u n c t i o n u 0 is g i v e n .

I n m a n y c a s e s t h e a s s o c i a t e d s p a c e i s n o t u n i q u e l y d e t e r m i n e db y t h e d i f fe r e n t i a l o p e r a t o r ~ , i .e . t h e r e e x i st v a r io u s s o - c a l l e dc o a s s o c i a t e d s p a c e s d e f i n e d b y(3) (qvu = 0w i t h d i f fe r e n t a s s o c i a t e d d i f f en t ia l o p e r a t o r s ~ v . T h e p r e s e n t p a p e ri n v e s t i g a t e s i n i ti a l v a l u e p r o b l e m s w i t h i n i ti a l f u n c t i o n s n o t b e l o n g i n gt o a s i n g l e a s s o c i a t e d s p a c e b u t p e r m i t t i n g d e c o m p o s i t i o n s i n t oc o m p o n e n t s b e i n g e l e m e n t s o f a fa m i l y o f c o - a s s o c i a t e d sp a c e s .

2 . A gener l pproxim t ion t h e o r e m .I n o r d e r t o i n t r o d u c e n e c e s s a r y s y m b o l s r e g a r d , f i r s t , t h e u s u a l

c a s e o f a l i n e a r o p e r a t o r ~ a c t i n g in a f i x e d l y c h o s e n s c a l e o fB a n a c h s p a c e s B s , 0 < s < So e q u i p p e d w i t h t h e n o r m I1 IIs. T h eb a s ic a s s u m p t i o n o n , 9 is t h e c o n d i t i o n

C( 4 ) I I ~ u - ~ ' v l l , ' ~ - - I l u - o 11 ,S S I

t h a t h a s t o b e s a t i s f i e d f o r a n y s , s ' w i t h 0 < s ' < s < So . S u p p o s e ,f u r t h e r , t h a t t h e i n i t i a l f u n c t i o n u 0 c a n b e e s t i m a t e d b y

K( 5 ) l i e u 0 1 1 , ~ - - .S 0 S


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D E C O M P O S r r l O N T H E O R E M F O R S O L V I N G I N IT I A L 42

T h e n t h e s u c c e s s i v e a p p r o x i m a t i o n s u k) a s u s u a l l y d e f i n e d b y

/u (k +l) = u o + d r 9 ~ u ( k ) ( r )s a t i s f y t h e i n e q u a l i t y

6 ) I l u k + l ) t ) - u k ) t ) l l , _ < ~ C e t ) k + l .0 - - S

P r o v i d e d t i s r e s t r i c t e d t o t h e in t e r v a l(7 ) 0 < t < q - -So -- SC ew h e r e q i s a n y n u m b e r w i t h 0 < q < 1 , t h e n i t c a n b e e a s i l y s e e nf r o m ( 6 ) t h a t t h e l im i t f u n c t i o n u , e x i st s in B , a n d c a n b e e s t i m a t e db y

K q k + l8 ) I l u k ) t ) - - u , t ) l l s _ C e 1 - qR e p l a c e n o w t h e s c a l e B ~ b y a f a m i l y o f s c a l e s B } v ), v = 1 , 2 . . . . .

w h e r e s r u n s a g a i n i n 0 < s < s o. S u p p o s e t h e c o n d i t i o n s ( 4 ) a n d( 5 ) a re s a ti s f i e d i n e a c h o f t h e g i v e n s c a l e s w h e r e t h e c o n s t a n t s Ca n d K a r e t o b e r e p l a c e d b y C , a n d K ~ r e sp . P r o v i d e d t h e s c a l e sB } ") a r e i n j e c t e d i n t o a s c a l e B , a n d f i n it e l i n e a r c o m b i n a t i o n s o fe l e m e n t s b e l o n g i n g t o s p a c e s B } v) w i t h t h e s a m e s a r e d e n s e i n B , ,t h e n t h e f o l l o w i n g t h e o r e m h o l d s :

T H E O R E M 1 . Suppose t he g i ven opera tor Jr ac t s i n t hesubsca l es B} v) bu t no t i n t he who le s ca l e Bs i t s e l f T hen f o r everycho i ce o f an i n it ia l e l em en t Uo in Bs a nd f o r every 6 > 0 t hereexis t s an e lement r io in B, wi th

I l u o - ~ 7 o l l~ < esuch tha t the in it ia l va lue pro blem 1) wi th the in i t ia l fun ct io n fi0ins tead o f uo i s so lvable b y a so lu t ion f i , t ) ex i s t ing in an in tervalo f t ype 7).


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A D E C O M P O S I T I O N T H E O R E M F O R S O L V I N G I N I T I A L 4 3

is n o t c o n v e r g e n t f o r al l c a s e s. A n a l o g o u s l y , t h e c o n s t a n t s C v a r en o t b o u n d e d f r o m a b o v e ( w i t h o u t a n y lo ss o f g e n e r a li t y w e m a ya s s u m e , h o w e v e r , t h at th e C v h a v e a p o s it i v e i n f i m u m b e c a u s e e v e r yC ~ c a n b e r e p l a c e d b y a l a r g e r o n e ) . I n o r d e r t o b e i n a p o s i t io n t od e c o m p o s e t h e i n i t i a l f u n c t i o n i n t o a i n f i n i t e s e r i e s , w e a s s u m e t h a tt 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 :

9 T h e s e r i e s ( 1 1 ) i s c o n v e r g e n t9 T h e c o n s t a n t s C ~ c a n b e i n c l u d e d i n p o s i t i v e b o u n d s , i . e . t h e r e

e x i s t c o n s t a n t s C , a n d C ** s u c h t h a t 0 < C , < C ~ < C* * fo re v e r y v .N o w w e r e s t r i c t t t o t h e i n t e r v a l

SO - - 50 < t < q C * * ew h e r e q i s a g a i n b e t w e e n 0 u n d 1. T h e n t h e e s ti m a t e ( 6 ) a p p l i ed t ot he v t h c o m p o n e n t y ie l d s

[[U~k+l) t) k) K~ qk+lI f _ C , eT a k i n g i n t o c o n s i d e r a t i o n t hi s e s t i m a t e a n d t h e c o n v e r g e n c e o f

t h e se r ie s ( 1 1 ) as w e l l , t h e m a j o r r e a r r a n g e m e n t t h e o r e m s h o w s t h a tt h e n o r m s o f t h e e le m e n t s o f t h e f o l l o w i n g i n f in i te m a t r i x c o n v e r g ef o r a n y a r r a n g e m e n t o f i t s e l e m e n t s :

(U01 U(1 ) __ U01 U(12)__ U~ ) . . .1 ) 2 1 )

u 0 2 u 2 - - u 0 2 u 2 ) - - u 2 . . .~ ~ ~

I n v i e w o f th e c o n v e r g e n c e o f t h e n o r m s t h e e l e m e n t s t h e m s e l v e sc o n v e r g e , to o . T h a t w a y t h e f o l lo w i n g t h e o r e m h a s b e e n p r o v e d :

T H E O R E M 2 . The init ial value proble m 1) w ith the ini tialf unc t i on

UO ~ Z UOu


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4 2 4 R. ~m~s~r~K . w. ~a.~ 'scar~c a n b e s o l v e d b y s p l i t t in g u p t h e i n it i a l f u n c t i o n i n to t h e g i v e nf a m i l y o f s u b s c a l e s p r o v i d e d t h e s e r i e s (1 1 ) c o n v e r g e s , t h e C ~ a r eb o u n d e d a n d t h e in f i m u m o f th e C v i s p o s i ti v e .

4 A s e c o n d v e r s io n o f t h e d e c o m p o s i t i o n t h e o r e mL e t B d e n o t e s o m e k i n d o f ( n o r m e d ) f u n c t io n s p a c e s u c h a s

a n L e - s p a c e o r a s p a c e o f H / 5 1 d e r- co n ti n u ou s f u n c t io n s . R e g a r d a( b o u n d e d ) d o m a i n D in R n a n d a n e x h a u s t i o n o f D b y s u b d o m a i n sD s , 0 < s < S o , h a v i n g t h e f o l l o w i n g p r o p e r t i e s :

9 F o r an y p a i r s , s ' w i t h 0 < s ' < s < s 0 t h e d i s ta n c e o f D s ,,f r o m t h e b o u n d a r y o f t h e la r ge r d o m a i n D s c a n b e e s ti m a t e d b y

( 1 2 ) d i s t ( D s , , O D ~ ) > C o (S - s )w i t h a c o n s t a n t Co n o t d e p e n d i n g o n s a n d s ' .

9 T o e v e r y p o i n t x e D ( w i t h X ~ X o w h e r e Xo i s f i x e d l y c h o s e n i nD ) t h e r e e x i s t s a u n i q u e l y d e t e r m i n e d s ( x ) s u c h t h a t x b e l o n g st o t h e b o u n d a r y o f D ~ t ~ ) .D e n o t e t h e f u n c t i o n s p a c e B w i t h re s p e c t to t h e d o m a i n D ~ b y

B s . D e n o t e , f u r t h e r , t h e n o r m i n B s b y I 1 I 1 ~ . F i n a l l y , d e f i n e t h ef o l l o w i n g c o n i c a l s e t M i n t h e ( t , x ) - s p a c e :( 1 3 ) M = { ( t , x ) 9 x E D , 0 < t < O/(So - s ( x ) ) }w h e r e S ( X o ) = 0 a n d 0 w i l l b e f i x e d l a te r . O b v i o u s l y , t h e i n t e r s e c t i o no f t h e p l a n e t = i w i t h M i s D z , i f g i s d e f i n e d b y(1 4) i" = O(So - g) .

N e x t c o n s i d e r ( r e a l - , c o m p l e x - o r v e c t o r - v a l u e d ) f u n c t i o n su = u ( t , x ) d e f i n e d i n M s u c h t h a t u ( t ' , x ) b e l o n g s t o /~ i w h e r e i"a n d ~ ar e c o n n e c t e d b y ( 1 4 ). I n t r o d u c e t h e p s e u d o - d i s t a n c e

td ( t , x ) = s o - s ( x ) - 11


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DECOMPOSITION THEOREM FOR SOLVING INITI L 4 2 5m e a s u r i n g t h e d i s t a n c e o f ( t , x ) f r o m t h e l a t e r a l s u r f a c e o f M .D e f i n e t h e f u n c t i o n a l

I lu( t , ~)11. = su p I lu( t , x ) l l , x ) d ~ x )M

w h e r e c r i s a f i x e d l y c h o s e n p o s i t i v e n u m b e r . D e f i n e , f i n a l l y , t h es p a c e B . M ) c o n s t i s t i n g o f a l l f u n c t i o n s u = u t , x ) f o r w h i c hI l u t , x ) l l . i s f i n i t e . I n c o m p a c t s u b s e t s o f M i n w h i c h

d t , x ) > 6 > 0t h e s - n o r m o f u t , x ) ( f o r f i x e d t ) c a n b e e s t i m a t e d b y t h e * - n o r m .T a k i n g i n t o c o n s i d e ra t io n t h e c o m p l e t e n e s s o f B s o n e c a n s h o w ,t he r e f o r e , t ha t B . i s c om pl e t e , t oo .

N o w w e a r e g o i n g t o r e t u r n t o a n a s s o c i a t e d p a i r ~ ' , f ro f d i f f e r e n t i a l o p e r a t o r s w h e r e t h e c o e f f i c i e n t s o f t h e ( f i r s t o r d e r )o p e r a t o r ~ m a y d e p e n d o n t. L e t B ~ b e t h e s e t o f a ll e l e m e n t su = u t , x ) o f B . b e i n g a s o l u t i o n o f t h e d i f f e r e n t i a l e q u a t i o nf qu = 0 f o r f i xe d t .

S u p p o s e a n i n t e r i o r e s t i m a t e i s t r u e f o r t h e d e r i v a t i v e s o f t h es o l u t i o n s o f t h e d i f f e re n t i a l e q u a t i o n f q u = 0 , i. e. i n a s u b d o m a i nh a v i n g t h e p o s i t i v e d is t an c e O f r o m t h e b o u n d a r y t h e n o r m o f t h ef i r s t o r d e r d e r i v a t i v e s a n d , c o n s e q u e n t l y , t h e n o r m o f , ~ ' u , t o o , c a nb e e s t i m a t e d b y

C(15 ) I I u l ID _< - - I lu l ID0w h e r e C d o e s n o t d e p e n d o n u a n d O ( cf . a l s o [ 7] ; in t e r i o r e s t i m a t e sf o r g e n e r a l i z e d a n a l y t i c f u n c t i o n s o n e c a n f i n d i n [ 1 4 ] , w h i l e t h ec a s e o f g e n e r a l i z e d a n a l y ti c v e c t o r s i s i n v e s t i g a t e d in [ 4] ; th e p a p e r[ 5 ] , f i na l l y , c on t a i n s i n t e r i o r e s t i m a t e s f o r ge ne r a l e l l i p t i c e qua t i onsu n d s y s t e m s a s w e l l ) . U n d e r t h e a b o v e a s s u m p t i o n w e a r e g o i n g t oi n v e s ti g a te th e b e h a v i o u r o f t h e o p e r a t o r ~ i n th e c o n i c a l d o m a i nM . T a k e a n y p o i n t ( t , x ) e M , i .e .

d t , x ) = s o - s x ) t > O.17D e f i n e

r - - - d t , x )~ r + l


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4 2 6 R . ~ S I ~ - W. n .r rs cH~a n d

S i n c e o n e h a s

o n e h a s

= s x ) + r .

d t , x ) < s o - s x )

o r


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DECOMPOSITION THEOREM FOR SOLVING INITI L 4 7

be c a use the no r m on the l e f t - ha nd s ide i s de f ine d a s

~0 ~:)dru p # ' u ( r , d a t , x )M s x )

a n d b e c a u s e

o d r 0 1d ~+ l ( r , x ) < a d ~ ( t , x )( i n t h e s p e c i a l c a s e o f t h e E u c l i d i a n d i s t a n c e i n t h e c o m p l e x p l a n eth i s e s t ima te i s u se d in the pa pe r [ 16 ] , wh i l e f o r a s c a l e d i s t a nc e inthe c omple x p l a ne suc h a n e s t ima te c a n he f ound in [ 14 ] ) .N o w s u p p o s e t h e e x p r e s s i o n

I I u 0 I I s < x ) s o - s x ) ) ~i s bounde d f o r a g ive n in i t i a l f unc t ion u0 . The n u0 c a n be in t e r p r e t e da s e l e m e n t o f B e , M ) n o t d e p e n d i n g o n t . T h e i n e q u a l i t y ( 1 7 )impl ies , f i r s t ly , tha t the fo l lowing s ta tement i s t rue :

LEMMA 1. T h e o p e r a t o r f18) U t , x ) = U o X ) + ~ ' u ( r , x ) d rt r a n s f o r m s B ~, M ) i n to i t s e l f

S e c o n d l y , a p p l y i n g t h e e s t i m a t e ( 1 7) t o ~ u - # ' v ( w h e r e ~ i ssuppose d to be l i ne a r ) , one se e s tha t t he ope r a to r ( 18 ) i s c on t r a c t ivein c a se 17 i s sma l l e nou gh . Th a t w a y the f o l low ing s t a t e m e n t ha sb e e n p r o v e d :

THEOREM 3. I n co n i ca l d o m a i n s M w i t h w i t h a s u f f i c i en t l ys m a l l h e i g h t 0 th e in i t ia l va l u e p r o b l em 1 ), 2 ) ca n b e s o l ve d b yt h e c o n t r a c t i o n m a p p i n g p r i n c i p l e p r o v i d e d a n i n t e r i o r e s t i m a t e i st r ue f o r ~ r .

N o w a s s u m e t h at # i s a s s o c i a te d t o a f a m i l y o f (c o - a s s o c i a te d )o p e r a to r s fr T h e n i n t ro d u c e t h e s p a c e s B , ( M ) . D e n o t e t h e c o n s t a n t


10/16

428 R. rm~S NK - w. rtrrscH~C i n ( 1 5 ) b y C ~ i n th e c a s e o f so l u t i o n s o f th e d i f f e r e n ti a l e q u a t i o nfC~u = 0

S u p p o s e , m o r e o v e r , t h a t a g i v e n i n i t i a l f u n c t i o n c a n b ed e c o m p o s e d i n to t h e s e ri es ( 10 ) w h e r e f~ v u = 0 . I n v i e w o f T h e o r e m3 t he i n i t i a l va l ue p r ob l e m w i t h t he i n i t i a l f unc t i on u~ i s s o l va b l e i na c o n i c a l d o m a i n w h o s e h e i g h t d e p e n d s o n C ~ (c f . ( 1 7 )) . O b v i o u s l y ,T h e o r e m 3 i m p l i e s t h e f o l l o w i n g s t a t e m e n t :

THEOREM 4. Suppose the initial function uo can be representedby a series in solutions of co-associated differential equations~,u = O. Suppose the constants Cv occuring in the interior estimate15) of the solutions of those differential equations are uniformlybounded. Then the initial value problem is solvable in a conicaldomain of type 13) for which the quantity 0 characterizing itsheight can be estimated by

1coo u)O< s pC~e o + 1A g a i n , i f th e C v a r e n o t b o u n d e d o n e c a n f i n d a n a r b i tr a r il y

g o o d a p p r o x i m a t i o n u 0 , o f a g i v e n i n i t i a l f u n c t i o n u 0 s u c h t h a t t h ei n i t i a l va l ue p r ob l e m w i t h t he i n i t i a l f unc t i on u0 , i s s o l va b l e i n ac o n i c a l d o m a i n o f t y p e ( 1 3 ) .

5 ExamplesE X A M P L E 1 . F i r s t w e r e g a r d a n e x a m p l e i n w h i c h a n ( i l l - p o s e d )

i n i t i a l v a l u e p r o b l e m w i t h a n a r b i t r a r y continuous i n i t i a l f u n c t i o n c a nb e s o l v e d a p p r o x i m a t e l y . R e g a r d a n o p e r a t o r o f t y p e( 1 9) ~ u = Cluz + C2u~ + C3(uz) + C4 u~) + Au + B~w h e r e t h e d e s i r e d f u n c t i o n u d e p e n d s o nB a u e r - P e s c h l o p e r a t o r s

v v + 1)ffvu = Uz~ + u1 + z ~ ) 2

t a n d z . T h e n a l l


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DECOMPOSITION THEOREM FOR SOLVING INITI L 4 9

wi th a r b i tr a r y v = 1 , 2 . .. a r e a s soc ia t e d to ~ p r ov ide d the Cj h a v et h e f o r m

Cl = ao + a lz + a2z 2, C2 = a2 - a l z + aoz 2,C3 = bo + bl?: + b2z2, Ca = b2 - b l z + b0z2

whe r e A , B , a o , a l , a 2 , bo , b l , b2 a r e a r b i t r a r y ( c on t inuous )f unc t ions de pe nd ing on t ( c f . [ 6 ] ) .

W e a r e go ing to so lve a n in i t i a l va lue p r ob le m o f t ype ( 1 ) ,( 2 ) w he r e ~ i s g ive n by ( 19 ) a nd the in i t i a l f unc t io n Uo i sc o n t i n u o u s . S i n c e a n a r b i t r a r y c o m p l e x - v a l u e d f u n c t i o n u o c o n t i n u o u si n t h e c l o s u r e o f a b o u n d e d d o m a i n i n t h e c o m p l e x p l a n e c a nb e a p p r o x i m a t e d u n i f o r m l y b y l i n e a r c o m b i n a t i o n s o f s o l u t i o n so f B a ue r - P e sc h l d i f f e r e n t i a l e qua t ions o f t he a bove type ( se e , f o rins tance , [2] ) , i . e . , one can f ind a l inea r combina t ion r io of so lu t ionso f f in i t e ly ma ny B a ue r - P e sc h l e qua t ions suc h tha t

IlUo - if011 < ew i t h r e s p e c t t o t h e s u p r e m u m n o r m . T h e f a c t o r v v + 1) of u int h e B a u e r - P e s c h l e q u a t i o n i m p l i e s t h a t t h e c o r r e s p o n d i n g c o n s t a n t sC ~ a r e no t b ou nd e d f o r a l l v = 1 , 2 . . . . . I n t he c a se o f f i n i te ly ma n yB a u e r - P e s c h l e q u a t i o n s , h o w e v e r , t h e c o r r e s p o n d i n g f i n i t e l y m a n y C ~a r e bounde d a nd , t he r e f o r e , t he c o r r e spond ing so lu t ions e x i s t i n ac o m m o n c o n i c a l d o m a i n o f t y p e ( 1 3 ) , i . e . , t h e i n i t i a l v a l u e p r o b l e mwi th the in i t i a l f unc ion ~0 i s so lva b le in t ha t c on ic a l doma in .

N e x t w e e s t i m a t e t h e d i s t a n c e o f t w o a p p r o x i m a t e s o l u t i o n s .N o t e t h a t t h e f o l l o w i n g e s t i m a t e i s a l w a y s a p p l i c a b l e i f o n l y t h ec ons t r uc t e d a pp r ox ima te so lu t ion i s a f i xe d - po in t o f a n ope r a to r o ftype ( 18 ) ( o r i t i s t he sum o f f in i t e ly ma ny f ixe d po in t s ; i n c a sethe ope r a to r ( 18 ) c a n be u se d f o r c ons t r uc t ing the e xa c t so lu t ion o fa n i n i t i a l v a l u e p r o b l e m ( a n d n o t a n a p p r o x i m a t e o n e ) t h e f o l l o w i n ge s t i m a t e s h o w s t h e w e l l - p o s e d n e s s o f t h e i n i t i a l v a l u e p r o b l e m u n d e rc ons ide r a t ion ) . R e ga r d two f ixe d po in t s t~ ( t , x ) a nd ~ t , x ) w h e r etT ( t , x ) be lon gs to the in i ti a l va lue s tT0 ( x ) , wh e r e a s t~ ( t , x ) i s t hea pp r ox ima te so lu t ion w i th the in i t i a l va lue s t~0 ( x ) . The r e f o r e , oneha s

f ~ 3 , x ) d ~t i ( t , x ) = uo(x ) + 0


12/16

4 3 0 R H E E R S I N K W. T U T S C H K Ea n d a n a n a l o g o u s e q u a t i o n f o r ~ ( t , x ) . T a k e i n t o c o n s i d e r a t i o n t h a t a ni n it ia l f u n c t i o n w i t h a f in i te n o r m h a s a f in i te , - n o r m , t o o . P r o v i d e dt h e o p e r a t o r ( 1 8 ) i s c o n t r a c t i v e w i t h t h e f a c t o r q , 0 < q < 1 , o n eo b t a i n s

I I~ i( t, x ) - ~ ( t , x ) l l , < I I~ i0 (x ) - ~ 0 ( x ) l l , ++ q . I I~ ( t, x ) - ~ ( t , x ) l l , ,

i .e . t h e d i s t a n c e o f th e t w o a p p r o x i m a t e s o l u t i o n s ~ i( t, x ) a n d ~ ( t , x )c a n b e e s t i m a t e d b y

I1~( /, x ) - t~ ( t , x ) l l , < I I~0(x ) - ~0 (x ) l l , .1 - q

( 2 0 )E X A M P L E 2 . S u p p o s e ~ i s t h e ( l i n ea r ) C a u c h y - R i e m a n n o p e r a t o r

O wC ( t , z ) - - z - - + A ( t , z ) wo zw h e r e t h e c o e f f i c i e n t s A ( t , z ) a n d C ( t , z ) a r ea n d h o l o m o r p h i c i n z . T h e n f o r e v e r y c o n s t a n tn o n - n e g a t i v e i n t e g e r k t h e d i f f e r e n t i a l o p e r a t o r sO wf f w - - - O ~ .* wO~

c o n t i n u o u s i n t0 a n d f o r e v e r y

a r e a s s o c i a t e d t o ~ ( c f. [9 ] ). S u p p o s e n o w t h a t t h e c l o s u r e o f ag i v e n d o m a i n D i st c o n t a i n e d i n t h e o p e n d i s k w i t h r a d iu s I n 2c e n t r e d a t z = 0 . T h e n e v e r y c o m p l e x - v a l u e d f u n c t io n c o n t i n u o u si n b c a n b e a p p r o x i m a t e d u n i f o r m l y b y li n e ar c o m b i n a t i o n s o ff u n c t i o n s d e f i n e d b y

z l e x p ( Z ' n ~\ n m /w i t h 0 < / z < m ( s e e [ 8 ]) . M o r e o v e r , t h e s e f u n c t i o n s a r e s o l u t i o n so f th e d i f fe r e n t ia l e q u a t i o n ~ w = 0 w i t h

/za = n - - - a n d k = n - 1 ,n m r ni .e . th e c o e f f ic i e n t s ar e u n i f o r m l y b o u n d e d . C o n s e q u e n t l y , t h ec o r r e s p o n d i n g c o n s t a n ts C a re u n i f o r m l y b o u n d e d , t o o (s i n c e t h ed i f f e r e n t i a l e q u a t i o n ~ ' w = 0 d o e s n o t c o n t a i n ff~ e v e r y s o l u t i o n c a n


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DECOMPOSITION THEOREM FOR SOLVING INITI L 43

b e r e p r e s e n t e d a s p r o d u c t o f a h o l o m o r p h i c f u n c t i o n a n d a f a c t o rn o w h e r e v a n i s h i n g a n d n o t d e p e n d i n g o n t h e s p e ci a l c h o i c e o f t h eso lu t ion ; t hus the un i f o r m bou nde dne s s o f t he c ons t a n t s C i s t r uea l s o b y u s i n g t h e s u p r e m u m n o r m ) . S u m m a r i z i n g t h e s e a r g u m e n t s ,i t ha s be e n se e n tha t t he a bove in i t i a l va lue p r ob le m i s so lva b le f o rin i t i a l f unc t ions be ing c on t inuous , on ly .

No t i c e tha t t he so lu t ion c ons t r uc t e d a bove i s no t a so lu t ion o fone o f t he d i f f e r e n t i a l e qua t ions f ~w = 0 ( f o r fi xe d t ) , bu t i t c a nb e a p p r o x i m a t e d u n i f o r m l y b y s o l u t i o n s o n f i n i t e l y m a n y d i f f e r e n t i a le qua t ions o f t ha t t ype ( whe r e the f i r s t o r de r de r iva t ive s c onve r geloc a l ly un i f o r m ly ) . I f t he ope r a to r ~ de f ine d by ( 20 ) c on ta in sthe a dd i t iona l t e r m D t , z ) ( s u c h o p e r a t o r s o c c u r i n t h e H . L e w ye xa mple , s e e [ 11 ] o r [ 10 ] ) , t he n a ( ~ o f t he a bove type i s a s soc ia t e do n l y i f D t , z) sa t i s f ie s the d i f fe ren t ia l equa t ion

OD = o~kD.a~I n i t ia l va lue p r ob le m s w i th ope r a to r s , .~ o f t ype ( 20 ) c a n a l so

be so lve d in the spa c e s H~ o f po lya na ly t i c f unc t ions ha v ing thef o r m

2 1 ) r ~w h e r e ~ i s an a r b i tr a ry h o l o m o r p h i c f u n c t io n ( c o n c e r n i n g p o l y a n a l y t i cf unc t ions se e [1 ] ). S inc e ~ t r a ns f o r ms H~ in to i t s e l f one c a n so lvein i t i a l va lue p r ob le ms w i th in i t i a l f unc t ions o f t ype ( 21 ) . On theo t h e r h a n d , th e c o n s ta n t C d e s c r i b i n g t h e a c t io n o f ~ d o e s n o td e p e n d o n v b e c a u s e ~ d o e s n o t c h a n g e th e f o r m ( 2 1 ) o f af u n c t i o n u n d e r c o n s i d e r a t i o n . U n i f o r m l y a p p r o x i m a t i n g a n a r b i t r a r yc o n t i n u o u s c o m p l e x - v a l u e d f u n c t i o n b y f u n c t i o n s o f t y p e ( 2 1 ) ( t h i si s p o s s ib l e i n v i e w o f t h e W e i e r s tr a s s- S t o n e a p p r o x i m a t i o n t h e o r e m ) ,i t f o l lows a ga in tha t c on t inuous in i t i a l f unc t ions a r e pe r mis s ib l e i nthis case .

A n o t h e r a p p r o a c h t o s o l v i n g i n i t i a l v a l u e p r o b l e m s f o r t h ec l a s s i c a l C a u c h y - R i e m a n n o p e r a t o r i n t h e c a s e o f c o n t i n u o u s i n i t i a lf u n c t i o n s s t a r t s f r o m t h e u n i f o r m a p p r o x i m a t i o n o f t h e g i v e n i n i t i a lf u n c t i o n b y p o l y n o m i a l s i n x a n d y . S u c h p o l y n o m i a l s c a n b ee x t e n d e d t o p o l y n o m i a l s i n t w o c o m p l e x v a r i a b l e s z l a n d z z t o


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43 2 R. HEERSINK W. TUTS CHKE

w h i c h t h e c l a s s i c a l C a u c h y - K o v a l e v s k a y a t h e o r e m i s a p p l i c a b l e .H o w e v e r , th e s o l u t io n s d o e s n o t e x i st in a c o m m o n d o m a i n , i ng e n e r a l , b e c a u s e o n e h a s n o t a u n i f o r m e s t i m a t e f o r t h e c o m p l e xe x t e n s i o n s . T h e a b o v e a p p r o a c h e s ( u s i n g c o - a s s o c i a t e d d i f f e r e n t i a lo p e r a t o r s o r s p a c e s o f p o l y a n a l y t i c f u n c t i o n s ) e n s u r e t h e e x i s t e n c eo f t h e s o l u t i o n s i n a c o m m o n c o n i c a l d o m a i n .

E X AM P LE 3 . S u p p o s e ~ i s a n o p e r a t o r o f t y p eO w~ w = + A w + B ~ oO z

w i t h p i e c e w i s e c o n s t a n t c o e f f i c i e n t s . T h e n a l l o p e r a t o r s ~ ' w i t hO wf ~ w - - - - ( ~ + i 9 ~ A ) w - B ( o

w h e r e Z i s a n a r b i t r a r y r e a l n u m b e r a r e a s s o c i a t e d t o , ~ ' . A l l t h e s eo p e r a t o r s fr a r e a s s o c i a t e d to ~ ( cf . [ 1 4 ]) . N o w t a k e a n y b o u n d e ds e q u e n c e o f re a l n u m b e r s ~ .~ , v = 1 , 2 . . . . . e . g ., r e g a r d a n y s e q u e n c ec o n t a i n i n g a ll r a ti o n a l n u m b e r s b e t w e e n - 1 a n d + 1 . D e n o t e th eo p e r a t o r ~ ' w i t h ~ . = ~.~ w i t h f r T h e n t h e c o r r e s p o n d i n g c o n s t a n t sC ~ a re b o u n d e d . S u p p o s e a g i v e n c o m p l e x - v a l u e d f u n c t i o n w o c a nb e r e p r e s e n t e d b y t h e s e r i e s

I13 ~== ~ 1 30vo

w h e r e t h e W o ~ a r e s o l u t i o n s o f t h e a s s o c i a t e d d i f f e r e n t i a l e q u a t i o n( ~ w = 0 a n d t h e s e r i e s c o n v e r g e s w i t h r e s p e c t t o th e H t~ l de r n o r m( w i t h a f i x e d H 6 1 d e r e x p o n e n t ) o r w i t h r e p e c t t o t h e L p - n o r m ( w i t hp > 2 ) . T h e n t h e d e r i v a t i v e s c o n v e r g e , t o o ( b e c a u s e t h e H - o p e r a t o ris a b o u n d e d o p e r a t o r , c f. [ 1 5 ]) , a n d , c o n s e q u e n t l y , t h e a b o v ed e c o m p o s i t i o n t h e o r e m y i e ld s t h e s o l v a b il it y o f th e i n i t ia l v a l u ep r o b l e m w i t h t h e i n i t i a l f u n c t i o n w 0 .

A n a n a l o g o u s r e s u l t i s t r u e f o r o p e r a t o r s a c t i n g i n m o r eg e n e r a l s p a c e s o f g e n e r a l iz e d a n a l y ti c f u n c t i o n s b e c a u s e a s s o c i a t e dd i f f e r e n t i a l o p e r a t o r s a r e u n i q u e l y d e t e r m i n e d u p t o t h e r e a l p a r t o ft h e c o e f f i c i e n t o f w o n l y ( f o r d e t a il s s e e [ 1 4 ] ). T h e s a m e s i t u a t i o no c c u r s f o r i n i t i a l v a l u e p r o b l e m s w i t h g e n e r a l i z e d a n a l y t i c v e c t o r sa s i n it ia l e l e m e n t s ( c o n c e r n i n g t h e f o u n d a t i o n s o f th e t h e o r y o f


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DECOMPOSITION THEOREM FOR SOLVING INITI L 4

g e n e r a l i z e d a n a l y t i c v e c t o r s cf . [ 3 ], w h i l e in i t i a l v a l u e p r o b l e m s w i t hg e n e r a l i z e d a n a l y t i c i n it ia l v e c t o r s a r e i n v e s t i g a t e d i n [4 ] ).

R E F E R E N C E S[1] Balk M.B . , Po lyana ly t i c func t ions . Ber l in 1991 .[2 ] Bauer K. W. , Uber e ine der Di f f e ren t i a lg le i chung (14-z~ , )2 wz~ +n n + 1)w =

0 z u g e o r d n e t e F u n k t i o n e n t h e o r i e . B o n n e r M a t h e m a t i s c h e S c h r i f t e n , v o l .23, 1965.

[3 ] Bojar sk i B . , Theory o f generalized analytic vecto rs. A n n . P o l o n . M a t h . ,vo l . 17 , 281-320 , 1966 ( in Russ ian) .

[4 ] Crod e l A. , Si~tze vom Cauchy-Kowalewskaja-Typ f i ir pa rt iel le komplexeDifferentialgleichungssysteme in Klassen verallgemeinerter a nalytischerVektoren. Thes i s (Dis ser t a t ion A) , Hal l e Univer s i ty , 1986 .

[5 ] Dougl i s A. , Ni renberg L . , Interior estimates for el liptic systems o f part ialdifferential equations. C o m m . p u r e a p p l . M a t h . , v o l . V I I I , 5 0 3 - 5 3 8 , 1 9 5 5 .

[ 6] H e e r s i n k R . , T u t s c h k e W . , Solution o f init ial value ploblem s o fCauch y-Kowalew skaja type satisfying a par tial secon d order differentialequation of prescr ibed type. G r a z e r M a t h e m a t i s c h e B e r i c h t e N r . 3 1 2 ,1991.

[ 7] H e e r s i n k R . , T u t s c h k e W . , Solution of initial value problems inassociated spaces. L e c t u re N o t e s 6 8 1 / 1 8 o f t h e S e c o n d W o r k s h o p o nF u n c t i o n a l A n a l y t i c M e t h o d s i n C o m p l e x A n a l y s i s a n d A p p l i c a t i o n s t oP a r ti a l D i f fe r e n t i a l E q u a t i o n s , I n t e rn a t i o n a l C e n t r e f o r T h e o r e t i c a l P h y s i c sTr ies t e , 25-29 J anuary 1993 .

[ 8] H e e r s i n k R . , T u t s c h k e W . , On the approximation o f continuouscomp led-valued fun ctio ns using generalized an alytic fun ction s, F o u r n M a t h .S c i e n c e s ( U . N . S i n g l M e m o r i a l V o l u m e , P a r t . I ) , 2 8 ( 1 9 9 4 ) , 4 7 - 6 3 .[ 9] H e e r s i n k R . , T u t s c h k e W . , On associated an d co-associated comp lexdifferential operators, Zei t s chr . Anal . Anwend ( in p r in t ) .

[ 1 0 ] J o h n E , Partial differential equations. 4 t h e d . N e w Y o r k / H e i d e l b e r g / B e r l i n1982.

[ 1 1 ] L e w y H . , An examp le o f a smooth linear par tial differential equ ationwithout solution. A n n . o f M a t h . , v o l . 6 6 ( 1 9 5 7 ), 1 5 5 - 1 5 8 .

[12] Ni renberg L . , Top ics in nonlinear Functional A nalysis. N e w Y o r k 1 9 7 4 .R u s s i a n tr an sl .: M o s c o w t 9 7 7 .

[13] T reves E , Basic linear differential operators. N e w Y o r k / S a n F r a n z i s c o/ London , 1975 .


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[14] Tutschke W., Solution o f in itial valu e problems in classes of generalizedanalytic functions. Teubner Leipzig and Springer-Verlag 1989.[15] Vekua I.N., Generalized analytic functions. 2nd edit. M osc ow 1988 inRussian; E ng l. transl. Reading 19 62 , Germ an transl. Berlin 1963).[16] W alter W., An elementary proof of the Cauchy-Kowalevsky theorem.

Am er. Math. M onthly, vol.92 1985), 115-125.P e r v e n u t o i l 3 d i c e m b r e 1 9 9 3 . Rudolf HeersinkTechnical University GrazDepartment of MathematicsSteyrergasse 30/3

A-8010 Graz Austria)Wolfgang Tutschkeat present: Tec hn ical University GrazDepartment of MathematicsSteyrergasse 30/3A-8010 Graz Austria)

A Descomposition Theorem for Solving Initial Value Problems in Associated Spaces (Tutschke)

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