P R O B L E M S O F C O N V E R G E N C E

W E A K A P P R O X I M A T I O N

O F T H E M E T H O D O F

V. A. Novikov UDC 518:517.944

In th is p a p e r the m e t h o d of w e a k a p p r o x i m a t i o n (MWA) is u s e d fo r so lv ing c e r t a i n s y s t e m s of l i n e a r and q u a s i l i n e a r e q u a t i o n s . P r o b l e m s of c o n v e r g e n c e of the MWA in r e f l e x i v e Banach s p a c e s in the c a s e of a s p e c i f i c s p l i t t i n g w e r e c o n s i d e r e d in [1]. In the p r e s e n t p a p e r the r e s u l t s of [1] a r e e x t e n d e d to the c a s e of any s p l i t t i n g , and we a l s o p r o v e c e r t a i n c o n v e r g e n c e t h e o r e m s fo r s p a c e s tha t a r e con juga te to a Banach s p a c e . M o r e o v e r , we c o n s i d e r the u s e of the MWA in the so lu t ion of a b o u n d a r y - v a l u e p r o b l e m f o r a q u a s i - l i n e a r s y s t e m .

1 . C o n v e r g e n c e o f M W A i n C e r t a i n C l a s s e s o f B a n a c h S p a c e s

1. A u x i l i a r y D e f i n i t i o n s and T h e o r e m s . L e t T and Tt be l i n e a r n o r m e d s p a c e s . By T ' and T~ we s h a l l !

deno te the con juga t e s p a c e s of T and T 1. Sup'pose tha t the l i n e a r o p e r a t o r A a c t s f r o m T ' to T 1. (In th is s e c -

t ion we s h a l l c o n s i d e r on ly l i n e a r o p e r a t o r s . )

L e t ~ ( A ) b e the d o m a i n of de f in i t i on of the o p e r a t o r A.

D e f i n i t i o n 1. An o p e r a t o r A i s s a i d to b e s e q u e n t i a l l y * - w e a k l y c l o s e d if i t s g r a p h , i . e . , a s e t of the f o r m {(x, A x ) I x ~ (A)}, f o r m s a s e q u e n t i a l l y * ' w e a k l y c l o s e d s u b s p a c e of the s p a c e T ' x T~. In o t h e r w o r d s ,

an o p e r a t o r A is s e q u e n t i a l l y * - w e a k l y c l o s e d if and on ly if: ~-wk ,.wk

if x ~ ) ( A ) , x,, , x and Ax,, , y, t h e n x ~ . ~ ( A ) and Ax----y.

T H E O R E M 1. If an o p e r a t o r is s e q u e n t i a l l y * - w e a k l y c l o s e d , then i t is c l o s e d [2]; the c o n v e r s e is in

g e n e r a l not t r u e .

T h e p r o o f of the f i r s t a s s e r t i o n is o b v i o u s . F o r p r o v i n g the s econd a s s e r t i o n l e t us c o n s t r u c t the f o l -

l owing e x a m p l e .

We c o n s i d e r a s e t of s e q u e n c e s ~ = {~n} and de f ine the fo l lowing n o r m e d s p a c e s :

m i s a s p a c e of bounded s e q u e n c e s wi th a n o r m

c is a s p a c e of s e q u e n c e s fo r which t h e r e e x i s t s lira ~n and which has the s a m e n o r m ; i t is e v iden t that

c c m ;

l i i s a s p a c e of s e q u e n c e s such tha t oo

I t i s w e l l known [3] tha t ( l l ) ' = m and tha t any l i n e a r con t inuous func t iona l on l 1 can be e x p r e s s e d in the f o r m

('1, ~) = ~ 'h~,, '1 ~ l l , ~ ~ m. 4 = !

L e t us c o n s i d e r an o p e r a t o r A : m --" R 1, w h e r e Rt is the r e a l a x i s , de f ined b y the f o r m u l a

A~----- l im~, ~ ) ( A ) - - : c .

I t is e a s y to s e e tha t the o p e r a t o r A i s l i n e a r and c l o s e d . L e t us t ake the s e q u e n c e ~ ( A ) in the f o r m

6 : , , )

T r a n s l a t e d f r o m S i b i r s k i i M a t e m a t i e h e s k i i Z h u r n a l , Vol. 18, No. 5, pp. 1125-1139, S e p t e m b e r - O c t o b e r ,

1977. O r i g i n a l a r t i c l e s u b m i t t e d June 21, 1976.

798 0 0 3 7 - 4 4 6 6 / 7 7 / 1 8 0 5 - 0 7 9 8 5 0 7 . 5 0 �9 1978 P l e n u m Pub l i sh ing C o r p o r a t i o n

*-wk k*-wk Then it is evident that ~ ~ 0,and A~ , - ~ t for k ~ ~o, Thus we have constructed an example of a closed ope r - a tor that is not sequentially *-weakly closed.

Definition 2. We shall say that an opera to r A : T ' --* T~ admits a sequentially *-weakly closed extension if there exists a sequentially , -weakly closed opera tor S : T ' -* T~ such that Sx = Ax for any x~D(A).

We have the following c r i t e r ion of sequentially *-weakly c losure of an opera tor .

LEMMA 1. An opera tor A : T ' -~ T~ admits a sequentially *-weakly closed extension if and only if the following condition is satisfied:

*-wk *-wk if x~ .q) (A) , xn---~O and Ax,,---+y, then y----0.

This lemma can be proved in exactly the same way as the c r i te r ion of c losure of a l inear opera to r [2!. Let us p resen t an example of an opera to r that admits a sequentially *-weakly closed extension. Let us consider a l inear differential express ion

= cj (x) # IJl-.~h

with coefficients cj(x)~Ck(Q),where gt is an open set of the space R n. Let us define a l inear opera tor A that. maps the region ~ (A) - -C~(Q)~L=(Q) in to the space Leo(S-t) by the formula (Af)(x) = Dxf(x). It is easy to show that the opera tor A admits a sequentially *-weakly closed extension.

Definition 3. An opera tor A is said to be sequentially . -weakly continuous on a set M ~ ( A ) u n d e r the following condition:

*-wk . * -Wk_ if x~, x~Jr and x~ ~ x, then Axn ~ A:c.

We have the following theorem.

THEOREM 2. Let X be a Banach space, and let X' be its conjugate space. Let us assume that the opera to r A:Fq)(A) -+X', 5D(A)~X' is sequentially *-weakly closed, and let M~-~{x t x~ ) (A ) , I]Axll~c<oo}.

Then the set M c will be sequentially *-weakly closed and the opera tor A will be sequentially . -weakly continuous on it.

The proof of this theorem is an exact repetition of the proof of Theorem 2 of [I] if we take into account

that the space which is conjugate to a Banach space is *-weakly complete, and that any bounded set ~n it is . -weak ly compact [2].

In the Banach space E let us consider Cauchy ' s problem

l du

(I) - ~ - = A ( t ) u , 0~<t~<T, (1.1)

[u (0 )=u0 , 0 < 0 < T , u ~ U 0 ~ E , U 0 ~ E , (t.2)

where A(t) is in general an unbounded opera tor with a var iable domain of definition ~)(A (t))that is dense in E, and A(t) :E -* E for any fixed t.

Definition 4 [1]. A continuous function u(t) is said to be a solution of P rob lem (D if:

1) The function u(t) has a general ized derivative du/dt ~ L ~ ( 0 , T; E);

2) Eq. (1.1) holds for a lmost all t ~[0 , T];

3) the conditions (1.2) are sat isf ied.

The definition of the spaces Lp(0, T; E), 1 _< p _< r canbefound in [4].

If for a u 0 ~U0 the P rob lem (I) has a unique solution u(t), we shall formal ly introduce a resolving opera - tor S(t, e),

~(t) =S(t, O)uo.

Let us present some definitions that are s imi la r to those used in [5].

Definition 5. P rob lem (I) is said to be co r rec t ly formulated in E if:

1) For any u 0 ~ U 0 there exists a unique solution of P rob lem (I);

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2) the o p e r a t o r S(t, 0) sa t i s f i e s the fo l lowing condi t ions :

a) S(t, tl)S(tl, t 2) =S( t , t2) , 0 -< t 2 -< t 1 -< t - T ;

b) llS(t, t~)ll - M(T), 0 -< 0 - t - T ;

c) S(t, t) = I, I be ing the ident i ty o p e r a t o r ;

d) S(t + At, t)u 0 -* U 0 fo r At --" 0, At _> 0.

The p r o p e r t i e s c) and d) fol low f r o m the def ini t ion of the solut ion and f r o m condi t ion b).

Def ini t ion 6. P r o b l e m (I) is sa id to be p iecewise c o r r e c t if t he re ex is t s a finite pa r t i t ion of the in te rva l [0, T ] : [T 0, T1] . . . . . [Tn-&, Tn] , To = 0, Tn = T such that this p r o b l e m is c o r r e c t l y f o r m u l a t e d on any in te rva l [Tk, Tk+ l ] , k = 0, 1 . . . . . u - 1. Suppose tha t the o p e r a t o r A(t) can be r e p r e s e n t e d in the f o r m

h A ( t ) = ~ A i(t).

i=l

Let us c o n s i d e r the o p e r a t o r h

A, (t) = ~ a~ (~, t) Ai ( t )

With r e g a r d to the coef f ic ien t s a i0 - , t) we shal l a s s u m e that they a r e funct ions that take the i r va lues in R1, be ing p iecewise cont inuous in t, u n i f o r m l y bounded in ~- and t, and sa t i s fy ing the condi t ion

1 ~ (z, 0) dO---~l for z - ~ 0 (1.3) t

u n i f o r m l y in t, i = 1, 2 . . . . . k .

It is e a s y to see that f o r 7 ~ 0 the funct ions ai(~-, t) weakly a p p r o x i m a t e uni ty, whe rea s on the c l a s s of

funct ions h

V ~ - { u ( t ) ] a ) u ( t ) ~ N ~ (At (t)); b) u (t), A(t) u ( t ) ~ L ~ c ( 0 , T;E)} i=t

the o p e r a t o r s AT(t) weakly a p p r o x i m a t e the o p e r a t o r A(t) (the c o r r e s p o n d i n g defini t ions can be found in [51).

T o P r o b l e m (I) le t us a s s i g n the p r o b l e m

( du.~ (I,)i--~-=A~(t)ux' O<~t<~T,

[ ~ ( 0 ) = u0, 0 ~ 0 < r

Defini t ion 7. P r o b l e m (I T) is sa id to be u n i f o r m l y p iecewise c o r r e c t in E if it is p iecewise c o r r e c t in E fo r any pos i t ive ~', and if the bound (b) in Def in i t ion 5 holds u n i f o r m l y in T.

Le t us note that in c o n t r a s t to [5] we do not r e q u i r e in the Defini t ions 5-7 the cont inui ty of the o p e r a t o r S(t, 0) in the tota l i ty of v a r i a b l e s t and 0. In fact , unde r the condi t ions used in the fo rmu la t i on of the t h e o r e m s ,

this p r o p e r t y is a u t o m a t i c a l l y sa t i s f i ed .

Le t us a s s u m e that E has a subspace E I with the fol lowing p r o p e r t i e s :

1) E 1 is dense in E, i . e . , E1 -~E;

2) E i ~ n ~ (A(t)); t

h (1.4) 3) ~ iIA~ (t) uHE 411uIi~<~cllu[tE, vu~E~,

q=l

w h e r e c = cons t and does not depend on u. In al l the subsequent t h e o r e m s we sha l l denote by E 1 a subspace

tha t s a t i s f i e s 1 -3 .

Le t us wr i t e II. ll~--~ il II, It" IJE, = il tli.

T H E O R E M 3. Suppose that P r o b l e m (I~-) is u n i f o r m l y p iecewise c o r r e c t in E and E 1. Then the o p e r a t o r ST(t , 0) wil l be s t rong ly cont inuous in E in the to ta l i ty of v a r i a b l e s t, 0, 0 -< 0 -< t - T , un i fo rmly in T.

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P r o o f . It evident ly suf f ices to p rove the s t r o n g cont inui ty of the o p e r a t o r St(t , 8) fo r any % ~ E t, s ince E1 is dense in E. Le t us c o n s i d e r the inequal i ty

ii[S,(td-At, Od-AO)--S,(t,O)]uoll~ll[S~(td-ht, 0d-AO)--S~(t, 9d-AO)]uo[]q-}![S~(t, 0d-A0)--S~(t, 0)]ao[I. (i .5)

The second t e r m in the r i g h t - h a n d s ide of (1.5) can be wr i t t en as fo l lows:

a) If A0 > 0, then

N [S,(t, 0~-, AO)--S,(t,

b) if A0 < 0, then

o) ]uol l= l l&( t , O+AO) [ I - -& (o+Ae, e)]uoli<~M(r)li[I--S,(O+Ae, o)]udl; (1.6)

II [&(t, Od-AO) --S, (t, e) lush ~M(T)ll&(0, Od-A0) - - I ] u0L (1.7)

F r o m (1..5)-(1.7) we can see that f o r p rov ing the t h e o r e m it su f f i ces to show that the f i r s t t e r m in the r i gh t - hand s ide of (1.5) and the r i g h t - h a n d s ides of (1.6)-(1.7) tend to z e r o when At, Ae - - 0 u n i f o r m l y in r . S imi - l a r l y , a s in the l e m m a of [1] i t is e a s y to show that P r o b l e m (It) is so lvable in E fo r any u 0 ~ E ~, Then ST(t, 0 + AO)u 0 wil l be a so lu t ion of the p r o b l e m

and

{ ~ = A~ (t) uT,

ut (0 § 50) = u0

[St (t + ,At, 0 + AO)-- St (t, O + AO)]u o =

t - - A t t ~ A ~ ' du.~

"'dTdt = [ A t(t) utdt , 7

whence we can eas i ly obtain by v i r tue of the u n i f o r m boundedness of c~i(r, t) and with the use of (1.4) the fo l - lowing bound:

i? i [ $ , ( t + A t , Od-AO)--S~(t ,O+AOl]uoi l<q I]S~(t,O-~AO) uo!!Idt . t

T a k i n g into accoun t the u n i f o r m p iecewise c o r r e c t n e s s of P r o b l e m (I r) in E 1, we can r ewr i t e this bound as fo l lows:

ll[&(t+A~, 0d-A0)- -&i t . O+AO)]uol!~c21iuoLl,IAtl,

whence it e a s i i y fol lows that the f i r s t t e r m in the r i g h t - h a n d s ide of (1.5) tends to z e r o fo r &t - - 0 u n i f o r m l y in t, O, and r . Le t us note tha t h e r e and in the fo l lowing we sha l l denote by ci va r i ous cons tan t s . It is e a s y to s ee that tile fac t tha t the r i g h t - h a n d s ides of (1.6)-(1.7) tend to z e r o u n i f o r m l y in t, 0, and r when A0 -*0 can be p roved in the s a m e way; this c o m p l e t e s the p roo f of the t h e o r e m . Le t us note tha t a s i m i l a r a s s e r - t ion has been p roved in [1], but h e r e the cont inui ty is p roved u n i f o r m l y in ~.

In a Banach space E let us c o n s i d e r the nonhomogeneous Cauchy p rob l em

(I') -~7 = A (t) u + f (t),

[u (0) -- u0, f (t) ~ L• (0, T; E).

The solut ion of P r o b l e m (I') can be defined in the s a m e way as the so lu t ion of P r o b l e m ([).

T H E O R E M 4. Le t

a) P r o b l e m (I) be p iecewise c o r r e c t in E and El;

b) uo~E~, ](t)~L~(O, ?; E).

If u(t) is a solut ion of P r o b l e m (I'), it can be r e p r e s e n t e d in the f o r m

u (0 = s (t, to) U(to) + t" s (t, o) / (~) dO, t > to > O,

w h e r e S(t, 0) is a r e s o l v i n g o p e r a t o r of P r o b l e m (I).

The p roof of this t h e o r e m is an exac t r epe t i t ion of the p roof of T h e o r e m 3 of [1] if we use T h e o r e m 3 of the p r e s e n t paper as appl ied to P r o b l e m (I).

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R e m a r k 1. T h e o r e m 4 evident ly holds a l so fo r the p r o b l e m

, [dl~ = A~ (t) u~ (t) -+- ]~ (t),

2. C o n v e r g e n c e T h e o r e m s . T H E O R E M 5. Le t P r o b l e m (IT) be u n i f o r m l y p tecewise c o r r e c t in E and E l , le t u 0 ~ E l , and le t u(t) be a so lu t ion o f P r o b l e m (I) such that u : [0, T] - - E 1. T h e n u~-(t) which is a solut ion of P r o b l e m (I v) will be c o n v e r g e n t to u(t) in C(O, T; E), and hence u(t) will be a unique solut ion of P r o b l e m (I). The r e s o l v i n g o p e r a t o r of P r o b l e m (I) is s t rong ly cont inuous in E in the to ta l i ty of v a r i a b l e s t and 0, 0 -< 8 - t _ < T .

w h e r e

P roo f . Le t us wr i t e vT(t) = UT(t ) -- u(t). Then vT( t ) will be a solut ion of the fol lowing Cauchy p r o b l e m :

I dv,~ ] E = A, (t) v, + I, (t),

(I{) ! [ v ~ (o) = o ,

h

f~ (t) ~ A t (t) u - - A ( t ) u= ~ (a i ('~, t) - - t ) A t (t) u. 4=t

F r o m the condi t ion of u n i f o r m p iecewise c o r r e c t n e s s of P r o b l e m (I~-) in E 1 it fol lows that fT(t} ~ L~(0, T; E), and hence we can apply T h e o r e m 4 to P r o b l e m (Iv).

Thus , t

v~ (t) = J" s~ (t, ~) I~ (n) d~]. 0

The m e a n va lue s of the funct ion ~0 : [0, T] ~ E wil l be def ined in the s a m e way as in [1], i . e . ,

% ( t ) = - ~ i ~ ( O + t ) d O , t ~ [ 5 , T - - S l - ~ A S , 6 > 0 , e < 6 .

It is e a s y to p rove the fol lowing a s s e r t i o n : If q~(t) ~ L p ( 0 , T; E), 1 _< p < m, then

[I ~ (t) - w (t)[Icp(.,~;~) --,- o fo, ~.- ,- o.

Le t

v~ (t) = .t' S, (t, ~1) [~ in) d~l, 0

where 1{ (t) = ~ ( ~ (-c, t) - t ) ( & (t) ~)~

i = !

and (At(t)u) e is the m e a n value of the funct ion Ai(t)u.

It is e a s y to see that vve(t) ~ VT(t) f o r e ~ 0 u n i f o r m l y in t and T. Th i s a s s e r t i o n can be ver i f i ed d i r ec t l y by tak ing into accoun t the u n i f o r m p iecewise c o r r e c t n e s s of the P r o b l e m (IT) in E and El . Le t us note that

VTe(t) is a solut ion of the p r o b l e m

[,~ (0) = 0

By apply ing to the P r o b l e m (I e) the s c h e m e of p roof of the c o n v e r g e n c e t h e o r e m p r e s e n t e d in [6] [the funct ions (Ai(t)u) e a r e cont inuous in t! ], we find a t once that f o r any f ixed e the funct ions VTe(t) tend to z e r o u n i f o r m l y in t when T ~ 0. But in this c a s e it fol lows d i r e c t l y f r o m the inequal i ty

II v, (01I <~ II,,, (t) - v{ (011 + 11 ~,{ (t)11

tha t vT(t} -*0 u n i f o r m l y in t when T --* 0. The s t r o n g cont inui ty of the r e s o l v i n g o p e r a t o r of the P r o b l e m (I) can be d i r e c t l y ve r i f i ed . This c o m p l e t e s the p roo f of the t h e o r e m .

Le t E be a space that is con juga te to a Banach space X, i . e . , E = X ' . Le t us f o r m u l a t e a n e c e s s a r y con-

di t ion to be sa t i s f i ed by E.

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Proof. m a t e s :

Condi t ion t . A space E conjuga te to a Baaach space X sa t i s f i e s condi t ion 1 if (Ll(0, T ; X)) ' = L~(0 , T~ E) f o r any /9 < T, and fo r any (p ~ L~(0 , T ; E) and ~b ~ L 1(0, T ; X) we have

T

<% ~) = ; <(p (t), ~ ( t ) )dr , (1o8) 0

w h e r e ( , } is the value of the funct ional on an e l emen t . E d w a r d s [7] has p r e s e n t e d a number of suff ic ient c o n - di t ions fo r a space to sa t i s fy the Condi t ion 1. One such condi t ion is the s epa rab i l i t y of the space X.

T t t E O R E M 6. Le t E and E 1 be spaces tha t a r e conjugate to Banach s p a c e s X and X1, and let E sa t i s fy the Condit ion 1. Next, le t P r o b l e m (I~-) be u n i f o r m l y p iecewise c o r r e c t in E and El , u 0 ~ E t , and let the o p e r - a t o r s Ai(t), i = 1, 2 . . . . . k, be sequent ia 'dy * -weak ly c losed for any fixed t. Hence if u~.(t) is a solut ion of P r o b l e m (I~.), then:

1) u~-(t) fo r "r ~ 0 c o n v e r g e s in C(0, T; E) to a funct ion u(t);

2) u(t) is a unique solut ion of P r o b l e m (I);

3) the r e so lv ing o p e r a t o r of P r o b l e m (I) is s t rong ly cont inuous in E in the to ta l i ty of v a r i a b l e s t and 0, 0 _ < 0 _ < t _ ~ T .

F r o m the u n i f o r m c o r r e c t n e s s of P r o b l e m (I~) in E and Et we obtain the fol lowing a p r io r i e s t i -

1I u~ k~(0 ~;~) = il & (t, o) uo [t~(o,~;~)~ M (T)II ~0!i, (1.9)

,i dt ,iL~(O,T;E) =I1A~u~[IL:de,r;E)~m. ax[ c~ (~, t) l ~ l iU~IiL~(O,T;Z) ~C31IU~IIL~(~,r,E,)<_ ~ 'I c~, u0 IL (1.!0) ~,r,t i - I

il u~lll <~ c5 II uoll. (1.11)

i]A~u~l[ ~ c6H u0[]. (1.12)

I t fol lows f r o m Condi t ion 1 that L~(0 , T; E) is con juga te to a Banach space LI(0 , T ; X). But s ince any bounded se t in a space that is conjugate to a Banach space is , - w e a k l y c o m p a c t [2], it fol lows f r o m (1.9) arid (1.10) that t he re ex i s t s e q u e n c e s ~U~-n} , ~dUl.n/dt } such that

~t * - w k ~t, dUxn*-Wk �9 ~ ~ 7/-----~ v (i .13)

for ~-n -*0 in L~o(0, T; E). From (1.13) it follows directly that v(t) is a generalized derivative of u(t), and since (see [2])

du i if v t[Loo(o,~;~) ~ lira ' r i il

it follows that u(t) is a continuous function to within a set of measure zero [4]. Since E I is conjugate to a Banaeh space XI, it evidently follows from (I.II) that

u: [0, T] ---~E~. (1.14)

F r o m the bounds (1.9)-(1.10) and f r o m Ascoli~s t h e o r e m [7] it fo l lows, in par t icular~ that Urn(t ) * - ~ u ( t ) fo r

T n --~0 in E u n i f o r m l y in t. But in this ease it fol lows f r o m the sequent ia l ly * -weak c l o s u r e of the o p e r a t o r s Ai(t), i = 1, 2 . . . . . k, the bound (1.12), and T h e o r e m 2 that fo r any f ixed t the funct ion u ( t ) ~ ) ( A ~ ( t ) ) a n d

* - w k A, (t) u ~ (t) - ~ A, (t) ~ (t) (1 .15)

in E for T n ~ 0 , and a l so that

Le t us c o n s i d e r the d i f f e rence dU ~n

dt

and show that fo r any funct ion r ~ L t ( 0 , T ; X),

dt

A d t ) u ( t ) ~ L ~ ( O , T; E).

- - - - A , ~ - - ( ~ i - ~ -- A~) -**~(t)

(t .16)

dt ' @ -- (A~nur Au, ~ ) - ~ 0 ( i . I7 )

~03

for T n - - 0 . The f i r s t t e rm tends to zero by virtue of the above proof. Since (1.10) and (1.16) easi ly yield [IAT~lUTn - - Aul I I~ (O,T ;E ) -< c 7, it suffices for proving that the second t e rm in (1.17) tends to zero, to show

that it tends to zero on any set [2] that is s t rongly dense in Ll(O, T; X). Such a set c a n b e taken in the form of the set G of simple functions that take their values in X. The construct ion of such a set is descr ibed in [2]. With the use of (1.8) we can write the second t e r m in (1.17) as follows:

T T

<A~ u~,~-- Au, F> = [ <Ax u~,-- A~,u, F> dt ~ . ! <A~ u - - Au, F> dt, (1.18) 0 O

where F ~ G . By using the form of the function F, we can rewri te the f i rs t t e rm in the r ight-hand side of (1.18) in the form

T p

.! (A~nu~n -- ATnu, F) dt = j~i !j (A~nu~n -- ATnu, Fi) dt, e

where Bj is a sys tem of pairwise disjoint sets in [0, T] with #Bj < ~ (#Bj being the measure of Bi), and F j ~ X. F r o m the definition of the opera tor AT(t) it follows direct ly that for any j = 1, 2 . . . . . p the r ight-hand side of this equation can be rewri t ten as follows:

h

.[ <A~nU~n-- A~ u, Fj> dt = ~_~ [ a i ('~,, t) <A,ux n -- A~u, s dt. (1.19) Bj ~=i 'Bj

F r o m the uniform boundedness of oq0"n, t), the formulas (1.10), {1.15), (1.16), and Lebesgue ' s theorem [8] it follows direct ly that the r ight-hand side of (1.19) tends to zero when T n ~ 0. The convergence to zero of the second t e r m in the r ight-hand side of (1.18) can be obtained in the standard manner (see, e .g . , [8]) f rom (1.3) and Theorem 1' of [8]. Thus we find that du/dt = A(t)u a lmost everywhere in E, du/dt E L~o(0, T; E), and evi- dently u(0) = u 0. Thus, u(t) will be a solution of P rob lem (I), and it follows f rom (1.14) that it sa t isf ies all the conditions of Theorem 5. This completes the proof of the theorem.

Remark 2. By using the scheme of proof of Theorem 6 and also Theorem 2 of [1], we easi ly obtain the

following theorem.

THEOREM 7. Let E and Et be reflexive Banach spaces , let Problem (IT) be uniformly piecewise co r r ec t in E and El, u 0 ~ E 1, and let the opera to r s Ai(t), i = 1, 2 . . . . . k, bec losed in E for any fixed t. IfUT(t) is a

solution of P rob lem (IT), then:

1) u~-(t) for T --*0 converges in C(0, T; E) to a function u(t);

2) u(t) is a unique solution of Problem (I);

3) the resolving opera tor of P rob lem (I) is s t rongly continuous in the totality of var iables t and 0, 0 -<

0 - - t - T .

Remark 3. The appropria te ly reformula ted Theorems 5-7 hold also in the case of a nonhomogeneous

problem.

2 . C o n v e r g e n c e T h e o r e m s o f MWA f o r a B o u n d a r y - V a l u e P r o b l e m

In this sect ion we shah cons ider a regular ized equation of variable type. This equation has been p ro - posed by N. N. Yanenko in connection with the possibil i ty of simulation of se l f -osc i l la tory and turbulent flows. P r i o r to formulat ing the problem, let us make some r emarks concerning a priori es t imates for solutions of equations of var iable type. Let us cons ider the equation

au.~_ au a ( a u ) at + u ~ - = T x - x ~0 ~u , x ~ [ 0 , l], (2.1)

and the initial and boundary conditions

ult=o=uo(x), u(O, t)--~u(l, t)---~O. (2.2)

Integral a pr ior i es t imates of the solution of the problem (2.1)-(2.2) were obtained in [9] for a special type of function r176 However, the form of the function w(Su/Sx) is in fact not important; only the constraints on the growth of the function ~o(Su/0x) are significant. Thus we can a s se r t the following. Let

a) w(p) be a smooth function such that

804

w h e r e N is a n u m b e r , and r fo r I pl < N can t ake n e g a t i v e v a l u e s ;

b) t h e r e e x i s t s a c o n s t a n t k such tha t

w h e r e ~ and ~3 a r e p o s i t i v e c o n s t a n t s .

P

h

T h e n the s o l u t i o n of the p r o b l e m (2.1)-(2 .2) w i l l have the s a m e i n t e g r a l a p r i o r i e s t i m a t e s a s in [9]. E s t i m a t e s in C~(0, 1) when co(Ou/Ox) s a t i s f i e s only the cond i t i on (a) w e r e o b t a i n e d in [10, 11].

Suppose tha t w(Ou/Ox) s a t i s f i e s the c o n d i t i o n (a) . L e t us m u l t i p l y Eq. (2.1) by 2u and i n t e g r a t e f r o m 0 to h

d_ t u~dx -- 2 ~o -g-~. =- dx = O, d t , , ~ , a x

0 0

whence by v i r t u e of the p r o p e r t y (a) we e a s i l y ob t a in

o q f -

dt "o

w h e r e k(N) is a c o n s t a n t . By i n t e g r a t i n g th is i n e q u a l i t y f r o m 0 to t and d r o p p i n g p o s i t i v e t e r m s in the l e f t - hand s i d e , we ob ta in

o r

t l

f t ~ u~dxdt ~ (c + k (N) t), ~o

2 Cl + ,! l[ Ux HL,(0j) dt ~ -7- + kl (N), t ~ t o > O. 0

(2.3)

t

L e t us w r i t e u (x, t) = l _ 1' udt. We a l s o w r i t e the obvious i n e q u a l i t i e s t .

0

I- i-~ t" u~dx == u S t dz ~< i t f x ' , L 2 0 , l ) :

" I ~iiux.~(o,~ ) . cA L .~ t ~12( u~dt d x = T " % t ~ k l ( N ) ' o 0

t ~ > t 0 > O ,

and f r o m F r i e d r i e h s i nequa l i t y

we f ind tha t II6llL2(0, l ) and i!~xltL2(0, l ) a r e u n i f o r m l y bounded in a t t t .

Now l e t us f o r m u l a t e the p r i n c i p a l p r o b l e m .

We sha l l c o n s i d e r a s y s t e m 3

0 - ~ - + 8 - W - = ~= -~77~ L ~a~,] 0z--Tj (2.4)

with initial and b o u n d a r y cond i t ions

O u 0~2 u, - ~ = 0, u (x, 0) = u 0 (x), t ~ [0, T]0 (2.5)

w h e r e e is a s m a l l p o s i t i v e p a r a m e t e r , x ~ R ~ , .0. is a bounded r e g i o n with a s u f f i c i e n t l y s m o o t h b o u n d a r y

On, u = ( u l , . , u~), \ o z , ] = t - - ~ 1 ~ . % --~x~l ,,:~, ~'~>0, and the cond i t ion tha t ,(~ s ign v a r y i n g is

is s a t i s f i e d : ul 2 - 4v 2 = a > 0. F o r e = 0 we o b t a i n an equa t ion of v a r i a b l e t ype . To the p r o b l e m (2.4)-(2.5) we a s s i g n the s p l i t p r o b l e m

805

(II~) ~at T --~=,Z ~' ( % t ) ~ L t -~zi) OziJ'

t Ou~ ] u~, ~ o e = 0 , u~(x, 0 ) = u 0(x), t ~ [ 0 , T],

where c~i('r , t)= 35ij for t ~(m'~ -4-S ~__._ii % ml: -1- i z], 5ij being Kronecker's delta, i, j = 1, 2, 3, re'r< T, m= 0, I, 2, . . . .

R e m a r k . One of the d i f f icul t ies of u s ing the MWA in so lv ing b o u n d a r y - v a l u e p r o b l e m s for nonl inear equat ions c o n s i s t s in p rov ing that at each subsequen t s t ep we p r e s e r v e the s m o o t h n e s s with r e s p e c t to the v a r i a b l e s that have b e e n p a r a m e t e r s . This d i f f icul ty can be c i r c u m v e n t e d h e r e as fo l lows. We spl i t only the non l inea r t e r m s , w h e r e a s the l i nea r t e r m (of h igher o r d e r ! ) is not spl i t ; as wil l be shown below, this m a k e s it poss ib le to p r e s e r v e the s m o o t h n e s s with r e s p e c t to al l the v a r i a b l e s . How this dif f icul ty can be c i r c u m - vented in so lv ing C a u c h y ' s p r o b l e m c a n be seen in [12, 13].

Thus let us c o n s i d e r the f i r s t pa r t i a l s tep (j = 1) of the null s t ep (m = 0). At this pa r t i a l s tep we solve the fol lowing p r o b l e m :

0 - / + ~ - ~ = 3 - ~ k o ~ ] o~, j'

0 ~ - 0 u (x, 0) = u0 (x) t ~ [0, ~,'3]J �9 11,, On O~-

Le t u 0 (x) ~ I~V~ (~2). va l [0, v /3] , and that the solut ion of this p r o b l e m has the fol lowing p r o p e r t i e s :

o~ Lo(0, 1'; ~ (.o.)) u ~ L ~ ( 0 , T; W2(Q)), u t ~ .

But in this c a s e (see [4])

~ c ( o , T; ~ (a ) ) At the second pa r t i a l s tep (j = 2) of the null s tep we solve the p rob l em

0~ = 0 u, an oe u lt:=~13 = uli3 (x) , t ~ [zt3, 2~/31,

(2.6)

In the s a m e way as in [10] it is e a s y to show that the p r o b l e m (2.6) is so lvable in the i n t e r -

(2.7)

(2.8)

(2.9)

w h e r e ul/3(x) = u(T/3, x), and u(t, x) is a so lu t ion of the p r o b l e m (2.6). As it fol lows f r o m (2.8), u~/3 (x) ~ [?V 2 (~) and we can aga in r e f e r to the so lvabi l i ty of the p r o b l e m (2.9), with the obta ined solut ion having the p r o p e r t y (2.7)-(2.8) . In exac t ly the s a m e way we go o v e r to the th i rd pa r t i a l s tep, and then the p r o c e s s is r epea ted . Thus we find that P r o b l e m (I v) is so lvable in the i n t e rva l [0, T] , and that the solut ion aT(t, X) has the p r o p e r - t ies (2.7)-(2.8), with the fo l lowing equat ion be ing sa t i s f i ed [10]:

3

where the symbol ( , ) denotes the s c a l a r p roduc t in L2(~2). Now let us obta in bounds fo r the solut ion of P r o b - l em (I T) that do not depend on v. Le t us c o n s i d e r the f i r s t pa r t i a l s tep, i . e . , the p rob l em (2.6). Le t us mul t ip ly the equat ion by u and in t eg ra t e o v e r ft. Af t e r apply ing the fo rmu la of in tegra t ion by pa r t s , we o b -

tain

i d f u 2 d x s d _ 3 y v [ a u k / a u , 2 , - - - .

In the same way as in [10], we hence eas i ly obtain

1/2 (l[ ul? + e I[AuL[ 2) ~ I/2 (ilu01!~-4- el[Au0J[ 2) -]-kz.

( as ~ and [I-[I is the n o r m in L2(fD. whe re k = max k ( N l i , N~i); Nli, N2~ a r e the roo t s of the po lynomia l v ~ az i ]

By a s i m i l a r p r o c e d u r e fo r the second and the th i rd pa r t i a l s tep, and then going ove r f r o m s tep to s tep ,

we eas i ly obtain the bound

806

% (IIu, IP-Fu IIAu, ll ~) ~ ~I~ (l!uoll~+s IlAudl :) + 3 ~ r ,

whence it follows d i rec t ly that

Ifu~l , oo 4c. (2.11) L.~(,0,T; WT) (fl))

F r o m (2.11), by us ing the inclusion of W~([Z) in W~(9) [14], we d i rec t ly obtain

il u, II ~ ~ (2.12)

In (2.10) le t us wr i te v = uTt, which is poss ib le by v i r tue of the p rope r t i e s of the solution of P rob l em (IIT), and

then in tegra te f r o m 0 to T . Hence

The r ight -hand side of (2.13) can be est imated as fo l lows:

zs i~ o~ ~ a~ o a~ d~dt << - aT, / ~ x

r QT T \ =

• # ( ~ i f exet~2~o ['to~! ~ ! , / ~ = i T "QT

where % is a posi t ive constant to be de te rmined below. We use H6Ide r ' s inequali ty and Jung ' s inequality. By using the inclusion of Lp(ft) in Lq(ft) for p >q, and the f o r m of the function v(0u/Sxi), it is easy to obtain f r o m (2.14) the bound

3 T

T O~i r, 8o fi U.-t 112 Z ( x ' t ) v / a u * ~ Our a au , d x d t ~ c ) l l u T x t t ~ , t ~ ) d t 4 _ ~ _ i t t~/IL'(Qr)' (2.15) 0

By using the es t ima te of ilUiIw~(s in t e r m s of IIAul] [15], F r i e d r i e h s ' inequality, and the bound (2.15), we can

wri te (2.13) in the f o r m T

0

By set t ing % = 5 and by using (2.12), we obtain

. T I [ L : ( Q T ) T 2 ii dt L~-(QT) ~'~C; (2.16)

where c does not depend on ~-. But in this case (2.13) yie lds d i rec t ly the bound

L~(QT ) A- S 'l ~ ! ' <.~ c. (2.17) I, d t []L~-(QT)

F r o m (2.11) and (2.17) it follows that we can se lec t a sequence {U~k } that has the following p rope r t i e s :

8U'rh 8u ~ [0, " at ~--~-weaklyin L., T; W~,(~)) for x h-*-0. (2.19)

From (2,,18)-(2.19) it follows that in the linear terms of (2.10) we can go over to the limit for r k ~0. ^.e

us show that for any function q~(t)~L2(0, T) and any function v ~ II.'[ (.Q) for ~'k -" 0 we have

F o r proving (2.20) it evidently suff ices to show that for T k ~ 0,

o f ( ~ ( ' r k , t ) - - 4 ) I 0 , ~ a,, a~, . - QT

Now let

(2.20)

(2.21)

807

S r [o~. ,, o%_v(o..~ o~]-oo QT

(2.22)

F r o m (2.18) and T h e o r e m 1' of [8] it fo l lows that (2.21) ho lds . By us ing the f o r m of v(0u/Oxi) , we can r e w r i t e the l e f t -hand side of (2.22) as fo l lows:

T

o

= a~(%, t ) T(t) , f ~ ~x~ ~dxdt--

0 T

[. \-~xi ) Ox---7~ k Ox~ ) Ox i J ~ d x d t . 0

It evident ly suf f ices to prove the c o n v e r g e n c e of the l a s t t e r m in the r i g h t - h a n d side of (2.23), s ince the con- v e r g e n c e of the o t h e r t e r m s can be p roved in the s a m e way. Indeed,

T [(.Ou~h~2Ourk T I ~ a x a t = ~ ai ( r~, t) •

a n 0

T . ! Q ,.~ \\t/~ f i. [ O,QI ' 8/3 0~,t,/3

F r o m the embedding of Vet(l-t) in W~(~-t) a s fo i lows.

t r,'o,~ ,~ o~,.,. _ f o,~ ,~-o~.~] ('(r, o%

a

L e t us note that

[14] and f r o m (2.17) it follows that the l a s t inequal i ty can be r e w r i t t e n

�9 , 1/2/ T - I]2 ot~ ,; dt rf, 2 (t) dt ~ :L,~a> . \'6

(2.24)

" %2 ' Ozz~: ~ ~ L: (0, T; ~ (~)), ~ ~ L,_ (0, T; L~ (9.)).

Since the embedd ing of W~(fl) in Wl(fl) is comple t e ly cont inuous t14], it fol lows f r o m T h e o r e m 5.1 of [4] that

, % - ~ strongly ~, L~(0, T; W~(~)). But in this e a s e the r i g h t - h a n d s ide of (2.24) wi l l tend to z e r o when 1" k -* 0. Thus we have shown that the

s equences UTk c o n v e r g e to a funct ion u that has the fol lowing p r o p e r t i e s :

V v ~ I~V2 (~2), u ]t=0 = u0 (x). (2.26)

By us ing the bounds (2.11), (2.16), and (2.17), which evident ly hold a l so for a l imi t funct ion, it is easy to

show that the so lu t ion (2.25)-(2.26) is unique.

Thus we have p roved the fo l lowing t h e o r e m .

T H E O R E M 8. Le t Uo(X ) ~TV~(9.). Then the re ex i s t s a unique solut ion of the p r o b l e m (2.4)-(2.5) [in the sense of (2.25)-(2.26)] , and for T ~ 0 the solut ion of P r o b l e m (II T) c o n v e r g e s to it . - w e a k l y in L~ (0, T; VV~ (fl)).

808

The author e x p r e s s e s his grat i tude to N. N. Yanenko and G. V. Demidov for useful d i scuss ions .

L I T E R A T U R E C I T E D

1. G . V . Demidov and V. A. Novikov, "On convergence of the method of weak approximat ion in a re f lex ive Bausch space , " Funkts . Anal. P r i lozhen . , 9, No. 1, 25-30 (1975)o

2. K. Yoshida, Funct ional Analys is , Sp r inge r -Ver l ag , Ber l in (1965). 3. A . N . Kolmogorov and S. V. Fomin, E lements of the Theo ry of Functions and Functional Analysis [in

Russian] , Nauka, Moscow (1968). 4. J . L . Lions, Ce r t a in Methods of Solution of Nonlinear Boundary-Value P r o b l e m s , Dunod and Gauth ie r -

Vi l lars , P a r i s (1969). 5. Z . G . Gegechkori and G. V. Demidov, "On convergence of the method of weak approximat ion ," Dokl.

Akad. Nauk SSSR, 213, No. 2, 264-266 (1973). 6. Z . G . Gegechkorf and G. V. Demidov, "On convergence of the method of weak approximat ion ," Chislen.

Metody Mekh. Sploshnoi Sredy, 4, No. 2, 43-50 (1973). 7. R. Edwards , Funct ional Analys is , Holt, Rinehar t , and Winston, New York (1965). 8. G . V . Demidov and S. A. K a n t o r , "Convergence t heo rems of the method of weak approximat ion ,"

Chislen. Melody Mekh. Sploshnoi Sredy, 3, No. 1, 35-45 (1972). 9. N . N . Yanenko and V. A. Novikov, "On a fluid model with a s ign -va ry ing v i scos i ty coeff ic ient ,"

Chislen. Melody Mekh. Sploshnoi Sredy, 4, No. 2, 142--147 (1973). 10. T . I . Zelenyak, \7 A. Novikov, and N. N. Yanenko, "On the p r o p e r t i e s of solutions of equations of

va r iab le type," Chislen. Metody Mekh. Sploshnoi Sredy, 5, No. 4, 35-47 (1974). 11. T . I . Zelenyak, "An equation with a s ign-vary ing diffusion coeff ic ient ," in: Mat. Prob[ . Khim. , No. 1,

111-115, Computat ional Cen te r of the Siber ian Branch of the Academy of Sciences, Novos ib i r sk (1975). 12. Yu, Ya. Belov and G. V. Demidov, "Solution of Cauchy 's p rob lem for s y s t e m s of equations of Hopf type

by the method of weak approximat ion ," Chislen. Metody Mekh. Sploshnoi Sredy, 1, No. 1, 3-16 (1970)~ 13. \7 A. Novikov, "Use of method of weak approximat ion in solving Cauchy ' s p rob lem for a quas i l inear

s y s t e m of parabol ic type, ~ Chislen. Metody Mekh. Sploshnoi Sredy, 6, No. 3, 58-75 (1975). 14. S . L . Sobolev, Applicat ions of Functional Analys is in Mathemat ica l Phys i c s [in Russfan~, Sib. Old.

Akad. Nauk SSSR, Novos ib i r sk (1962). 15. O . A . Ladyzhenskaya , Mathemat ica l Theo ry of Viscous Incompres s ib l e Flow, Gordon (1969)o

T H R E E P R O B L E M S ON M I N I M A L I S O T O P I E S A N D

A P P R O X I M A T I O N O F C U R V E S A N D S U R F A C E S

IN H O M E O M O R P H I S M S

I . Y a . O l e k s i v a n d I . N. P e s i n UDC (513.832:513.835)+517.51

The purpose of the cons idera t ions which follow below is to d i r ec t the at tention of the r eade r to a c i r c l e of p rob l ems on the boundary of topology and function theory (mixed p rob lems) . We hope that the formula t ion of the p rob l ems which we p ropose will a rouse in te res t .

The solution of the min imal isotopy p rob lem is due to the second of the authors (it has not been pub- l ished previous ly) . The solution of the second and third p rob l ems , due to the f i r s t author, was announced in [1, 2]. A r epo r t on mixed p rob lems was made in 1974 at a conference on analytic functions in Krakov .

The formulation of the problem which we propose is one of several possible ones concerning minimal isotopies (homotopies). We therefore will not strive in our exposition to exhaust the subject completely, but instead intentionally leave some of the questions unanswered and invite the reader to formulate naturally arising generalizations. For example, since the location of sacks (cf. below) is completely characterized by a certain graph, the problem considered can be formulated in terms of gTaph theory.

T r a n s l a t e d f r o m Sibirskii Matemat icheski i Zhurnal , Vol. 18, No. 5, pp. 1140-1158, Sep t embe r -O c to b e r , 1977. Original a r t i c l e submit ted March 31, 1976.

0037-4466/77/1805-08 09 $07.50 �9 1978 Plenum ]~Jblishing Corporation 8 09