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or Sufficient Conditions for the Existence of Sequential Solutions
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
hl(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
ws '
E
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 j x ) ) R+l
/
I I
I
I I I I
I I I I
I
\
....... wo(x)
I I I I I I
(wv(x))
I
Rt 1
I I
....... w v p
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
1 I I
"
I I I
I I
I I I I
1
I
I I
/
I I
I
I I I
I 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: int
n
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)
h:
uu
P
o r e l s e a/ax, t o b o t h terms
t)
v
(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,
49
CONDITIONS FOR SOLUTIONS
a' c
V z E A,
R non-void, open:
(31)
where l / ( z t h e terms
1 R' l/zv,
) denotes t h e sequence o f continuous f u n c t i o n s on 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
r(n)
sBy'yp
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
and R . pre-supposes a n a r r o w i n g o f t h e gap between Ind 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
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 i s the ideal i n ,A, 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 = f o r m l y convergent t o zero, on compacts i n R. (46)
Auc = Auc/Iu,
Then o b v i o u s l y
SB ,H 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 V
(49)
R'
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
E I
-+
A o f t h e mentioned PDE.
u u cNve N
A (T(D)tv
-
T(D)tp)
v>u
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
Then
n Z(T(D)t" N vEN v> p
-
Then t h e r e e x i s t s
T(D)tp)
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
-+
Qi
N :
E
on V
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
n
v E N v > p
Rosinger
- T(D)tp)
Z(T(D)t\,
Then
n
V = int
V E
N
- T(D)tp)
Z(T(D)tv
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
N, v > p
A
Conversely, assume t h a t x E RE obtain that V C
E
v E N
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>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
r E + A i s nowhere dense i n R ,
A “ ~ - + A = ‘ ~ + A ‘
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 0 Lemma 2, S e c t i o n 1. 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
u
n
A (wv
- wu )
i s dense i n R
P E N
vEN v >u
11 w
lTYf, R' c R non-void, open:
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,,,
Then, o b v i o u s l y mes Z(wv)
1
-
(x,,
-
6 v , xx
+
6 v ) y tl v E N
2(v+1)6,
hence
v
l i m mes Z(wv) = 1 + 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) + m
= (O,l)\
u
0
E N
U v E N
v > p
"
OGXGv
which w i l l o b v i o u s l y imply (76), s i n c e
v
lim
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"
E
lw
NI
E
< 2,
NI
N
U x
E
R
= car N
(1/(2w+2),1/ 2vtl) ) by
1if x wv(x) =
,
Define
R1, which s a t i s f i e s the c o n d i t i o n s
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
E
(1/ 2vt2),1/ (2w+l))
0 if x E n \ then i t i s easy t o see t h a t (83-85
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
hl(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
ws '
E
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 j x ) ) R+l
/
I I
I
I I I I
I I I I
I
\
....... wo(x)
I I I I I I
(wv(x))
I
Rt 1
I I
....... w v p
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
1 I I
"
I I I
I I
I I I I
1
I
I I
/
I I
I
I I I
I 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: int
n
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)
h:
uu
P
o r e l s e a/ax, t o b o t h terms
t)
v
(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,
49
CONDITIONS FOR SOLUTIONS
a' c
V z E A,
R non-void, open:
(31)
where l / ( z t h e terms
1 R' l/zv,
) denotes t h e sequence o f continuous f u n c t i o n s on 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
r(n)
sBy'yp
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
and R . pre-supposes a n a r r o w i n g o f t h e gap between Ind 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
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 i s the ideal i n ,A, 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 = f o r m l y convergent t o zero, on compacts i n R. (46)
Auc = Auc/Iu,
Then o b v i o u s l y
SB ,H 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 V
(49)
R'
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
E I
-+
A o f t h e mentioned PDE.
u u cNve N
A (T(D)tv
-
T(D)tp)
v>u
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
Then
n Z(T(D)t" N vEN v> p
-
Then t h e r e e x i s t s
T(D)tp)
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
-+
Qi
N :
E
on V
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
n
v E N v > p
Rosinger
- T(D)tp)
Z(T(D)t\,
Then
n
V = int
V E
N
- T(D)tp)
Z(T(D)tv
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
N, v > p
A
Conversely, assume t h a t x E RE obtain that V C
E
v E N
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>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
r E + A i s nowhere dense i n R ,
A “ ~ - + A = ‘ ~ + A ‘
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 0 Lemma 2, S e c t i o n 1. 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
u
n
A (wv
- wu )
i s dense i n R
P E N
vEN v >u
11 w
lTYf, R' c R non-void, open:
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,,,
Then, o b v i o u s l y mes Z(wv)
1
-
(x,,
-
6 v , xx
+
6 v ) y tl v E N
2(v+1)6,
hence
v
l i m mes Z(wv) = 1 + 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) + m
= (O,l)\
u
0
E N
U v E N
v > p
"
OGXGv
which w i l l o b v i o u s l y imply (76), s i n c e
v
lim
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"
E
lw
NI
E
< 2,
NI
N
U x
E
R
= car N
(1/(2w+2),1/ 2vtl) ) by
1if x wv(x) =
,
Define
R1, which s a t i s f i e s the c o n d i t i o n s
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
E
(1/ 2vt2),1/ (2w+l))
0 if x E n \ then i t i s easy t o see t h a t (83-85