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Transcript of MRES2
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Y A N G O N U N I V E R S I T Y
D E P A R T M E N T O F M A T H E M A T I C S
S T A T I O N A R Y N A V I E R - S T O K E S E Q U A T I O N S
b y
K H I N S H W E T I N T
T h e s i s s u b m i t t e d i n f u l l m e n t o f r e q u i r e m e n t
f o r t h e d e g r e e o f M a s t e r o f R e s e a r c h i n M a t h e m a t i c s .
M . R e s . T h e s i s M a y , 2 0 0 3
i
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S T A T I O N A R Y N A V I E R - S T O K E S E Q U A T I O N S
b y
K H I N S H W E T I N T
T H E S I S
s u b m i t t e d i n f u l l m e n t o f t h e r e q u i r e m e n t
f o r t h e d e g r e e o f
M A S T E R O F R E S E A R C H
i n
M a t h e m a t i c s o f t h e Y a n g o n U n i v e r s i t y
A p p r o v e d
C h a i r p e r s o n E x t e r n a l E x a m i n e r S u p e r v i s o r
D r . K y i K y i A u n g D r . J . N a s h D r . P y i A y e
H e a d & P r o f e s s o r P r o f e s s o r P r o f e s s o r
D e p t . o f M a t h e m a t i c s D e p t . o f M a t h e m a t i c s D e p t . o f M a t h e m a t i c s
Y a n g o n U n i v e r s i t y P r i n s t o n U n i v e r s i t y Y a n g o n U n i v e r s i t y
I n t e r n a l E x a m i n e r S e c r e t a r y
D r . K . Y o s i d a D r . M y i n t Z a w
P r o f e s s o r P r o f e s s o r
D e p t . o f M a t h e m a t i c s D e p t . o f M a t h e m a t i c s
T o k y o U n i v e r s i t y Y a n g o n U n i v e r s i t y
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A c k n o w l e d g e m e n t s
I w i s h t o e x p r e s s m y t h a n k s t o P r o f . D r . K y i K y i A u n g , H e a d o f M a t h e m a t i c s
D e p t . a n d P r o f . T h a u n g S h e i n , D e p a r t m e n t o f M a t h e m a t i c s , U n i v e r s i t y o f
Y o n g o n , w h o p e r m i t t e d m e t o w r i t e t h i s M . R e s ( M a t h e m a t i c s ) t h e s i s .
A t t h e s a m e t i m e , m y d e e p e s t g r a t i t u i d e a l s o g o s e t o D r . P y i A y e , A s s o c i a t e
P r o f e s s o r , d e p a r t m e n t o f M a t h e m a t i c s , U n i v e r s i t y o f Y o n g o n , f o r h i s v a l u a b l e
s u p e r v i s i o n , c o l l e c t i n g i n l i t e r a t u r e a n d i n p r e p a r a t i o n o f t h i s t h e s i s .
F i n a l l y , I r e m e m b e r t o s a y m y t h a n k s o n m y a d o r i n g p a r e n t s , f o r t h e i r w a r m -
h e a r t e d g u a r d a n d g u i d e l i n e t h r o u g h m y s t u d e n t l i f e . A g a i n , I t h a n k m y p a r -
e n t s , w h o g i v e m e e n o r m o u s f u l m e n t w i t h e t e r n a l k i n d i n p e r p a r i n g t h i s t h e s i s .
M a y , 2 0 0 3 K H I N S H W E T I N T
i i i
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A b s t r a c t
T h e e x i s t e n c e a n d r e g u l a r i t y o f t h e s o l u t i o n o f t h e s t a t i o n a r y N a v i e r - S t o k e s
e q u a t i o n s a r e c o n s i d e r e d i n I R
3
i v
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I n t r o d u c t i o n
v
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C o n t e n t s
A c k n o w l e d g e m e n t s i i i
A b s t r a c t i v
I n t r o d u c t i o n v
1 1
1 . 1 P r e l i m i n a r i e s a n d D e n i t i o n o f G e n e r a l i z e d S o l u t i o n s N o t a t i o n s 1
1 . 1 . 1 L e m m a . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1 . 1 . 2 L e m m a . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1 . 1 . 3 R e m a r k . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1 . 1 . 4 R e m a r k . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2 7
2 . 1 E x i s t e n c e o f G e n e r a l i z e d S o l u t i o n s . . . . . . . . . . . . . . . . 7
2 . 1 . 1 T h e o r e m . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2 . 1 . 2 L e m m a ( 1 ) . . . . . . . . . . . . . . . . . . . . . . . . . 8
2 . 1 . 3 L e m m a . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 4
2 . 2 U n i q u e n e s s o f G e n e r a l i z a t i o n S o l u t i o n o f t h e I n t e r i o r P r o b l e m . 2 0
3 2 2
3 . 1 R e g u l a r i t y o f t h e S t a t i o n a r y N a v i e r - S t o k e s E q u a t i o n s . . . . . . 2 2
3 . 1 . 1 L e m m a ( 1 ) . . . . . . . . . . . . . . . . . . . . . . . . . 2 2
v i
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B i b l i o g r a p h y 4 4
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C h a p t e r 1
1 . 1 P r e l i m i n a r i e s a n d D e n i t i o n o f G e n e r a l i z e d S o l u t i o n s
N o t a t i o n s
F o r t w o p o i n t s x = f x
1
; x
2
; x
3
g a n d y = f y
1
; y
2
; y
3
g i n E
3
, j x y j m e a n s t h d
d i s t a n c e b e t w e e n x a n d y . F o r a n y p o i n t s e t A i n E
3
, @ A i s t h e b o u n d a r y
o f A b e i n g a p o s i t i v e c o n s t a n t , t h e p o i n t s e t ! ( ; A ) = ! ( ) = f x j x 2
A a n d d i s t . ( x ; @ A ) < g i s t h e b o u n d a r y s t r i p o f A w i t h w i d t h
L
p
=
U : R ! R j
Z
U
p
d x
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C
1
( v ) i s t h e c l a s s o f s o l e n o i d a l v e c t o r f u n c t i o n s u 2 C
1
( v )
C
n + h
( v ) = C
n + h
( v ) \ C
1
( v ) ; ( n = 1 ; ; 1 ; 0 h
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1 . 1 . 2 L e m m a
T h e r e e x i s t s p o s i t i v e c o n s t a n t s
0
a n d c
3
= c
3
(
0
) s u c h t h a t
u
2 !
( )
c
3
k r u k
2 !
( )
h o l d s f o r a n y u 2
c
H
1
0
( R ) a n d a n y i n 0 <
0
P r o o f :
D e f t i n t i o n o f G e n e r a l i z e d s o l u t i o n s
S u p p o s e t h a t a v e c t o r f u n c t i o n u a n d a s c a l a r f u n c t i o n p a r e s u c i e n t l y s m o o t h
a n d o b e y t h e s t a t i o n a r y N a v i e r - S t o k e s e q u a t i o n s
u + ( u r ) u + r p = f i n v ;
d i v u = 0 i n v ;
u = o n @ v ;
u ( x ) ! u ( 1 ) j x j ! 1
i n a d o m a i n V R
3
, w h e r e f i s t h e e x t e r n a l f o r c e , i s t h e v i s c o s i t y .
M u l t i p l y i n g ( 1 ) b y a v e c t o r f u n c t i o n ' i n C
1
0
( v ) a n d t h e i n t e g r a t i n g o v e r v , w e
o b t a i n
( r ' ; r u ) + ( ' ; ( u r ) u ) ( d i v ' ; p ) = ( ' ; f )
I f ' 2 C
1
0
( v ) , t h e n ( 5 ) t a k e s a f o r m n o t i n v o l v i n g t h e p r e s s u r e p ,
! ( ' ; u ) = ( ' ; f )
w i t h ! ( ' ; u ) = ( r ' ; r u ) + ( ' ; ( u r ) u )
N e x t w e c o n s i d e r t h e t r i - l i n e a r i n t e g r a l f o r m
A ( u ; v ; ! ) = ( u ; ( v r ) ! ) =
Z
v
u
i
v
k
@
k
!
i
d x ;
w h e r e t h e a r g u m e n t s u ; v ; ! a r e r e q u i r e d t o p o s s e s s t h e f o l l o w i n g p r o p e r t i e s :
1 . e a c h o f t h e m i s e x p r e s s i b l e a s t h e s u m o f a f u n c t i o n i n
c
H
1
0
( v ) a n d a
f u n c t i o n i n C
1
( v ) . H e n c e a n y o n e o f t h e m , i n p a r t i c u l a r , v i s s u b j e c t e d t o
t h e c o n d i t i o n d i v v = 0
2 . E i t h e r u o r ! b e l o n g s t o
c
H
1
0
( k ) , k b e l o n g a b o u n d e d s u b d o m a i n o f V
3
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F o r t h e s e u ; v ; ! t h e d e n i t i o n o f A ( u ; v ; ! ) i n ( 7 ) i s s i g n i c a n t . I n d e e d , w e h a v e
k u k
4 k
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A ( u ; v ; u ) = 0
w h i c h a r e i m m e d i a t e l y v e r i e d b y p a r t i a l i n t e g r a t i o n t a k i n g a c c o u n t o f
d i v v = 0
D e n i t i o n ( 1 )
L e t R b e b o u n d e d . T h e n a v e c t o r f u n c t i o n u i s c a l l e d a g e n e r a l i z e d s o l u t i o n o f
t h e i n t e r i o r p r o b l e m , i f f o l l o w i n g c o n d i t i o n s ( 1 ) a n d ( 2 ) a r e b o t h s a t i s e d :
1 u b b e l o n g s t o
c
H
1
0
( R ) f o r s o m e b s u c h t h a t b 2 C
1
( R ) ; b = o n @ R
2 u s a t i s e s ( 1 ) w e a k l y i n R
1 . 1 . 3 R e m a r k
W i t h t h e a i d o f ( 9 ) , w e c a n s h o w t h a t t h e g e n e r a l i z e d s o l u t i o n u o f t h e i n t e r i o r
p r o b l e m s a t i s e s t h e w e a k e q u a t i o n n o t o n l y f o r e v e r y ' i n C
1
0
( R ) b u t a l s o
f o r e v e r y ' i n
c
H
1
0
( R ) i f f 2 ( L
2
( R ) )
n
D e n i t i o n ( 2 )
L e t R b e u n b o u n d e d . T h e n a v e c t o r f u n c t i o n u i s c a l l e d a g e n e r a l i z e d s o l u t i o n
o f t h e e x t e r i o r p r o b l e m , i f f o l l o w i n g c o n d i t i o n s ( 1 ) a n d ( 2 ) a r e b o t h s a t i s e d :
1 u b b e l o n g s t o
c
H
1
0
( R ) f o r s o m e b s u c h t h a t b 2 C
1
( R ) ; b = o n @ R
b ( x ) u
1
= O
j x j
1
;
r b ( x ) = O
j x j
2
; j x j ! 1
2 u s a t i s e s ( 1 ) w e a k l y i n R
D e n i t i o n ( 3 )
L e t u b e a g e n e r a l i z e d s o l u t i o n o f t h e N a v i e r - S t o k e s b o u n d a r y v a l u e p r o b l e m ,
i n t e r i o r o r e x t e r i o r . T h e n a s c a l a r f u n c t i o n 2 l
l o c
2
( R ) i s c a l l e d t h e p r e s s u r e
a s s o c i a t e d w i t h u , i f u a n d s a t i s f y ( 5 ) f o r a n y ' 2 C
1
0
( R ) . ( 5 ) i s c a l l e d t h e
d e n i n g e q u a t i o n o f
5
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1 . 1 . 4 R e m a r k
W h e n a g e n e r a l i z e d s o l u t i o n u i s g i v e n , t h e a s s o c i a t e d p r e s s u r e i s u n i q u e e x -
c e p t a n a d d i t i v e c o n s t a n t . I n f a c t , s u p p o s e t h a t
1
a n d
2
a r e t h e a s s o c i a t e d
p r e s s u r e . T h e n w e n d ( d i v ' ;
1
2
) = 0 f o r a n y ' 2 C
1
0
( R ) . S u b s t i t u t i n g
' = r h , h 2 C
1
0
( R ) , w e o b t a i n ( h ;
1
2
) = 0 a n d n o t e t h a t
1
2
s a t i s e s t h e L a p l a c e e q u a t i o n w e a k l y . A c c o r d i n g t o a t h e o r e m o f H . W e y l , t h i s
i m p l i e s t h a t
1
2
i s h a r m o n i c i n V . I n p a r t i c u l a r , f r o m ( d i v ' ;
1
2
) = 0
f o l l o w s ( ' ; r (
1
2
) ) = 0 . T h u s w e h a v e r (
1
2
) = 0 i n R a n d h e n c e
1
2
= c o n s t a n t i n R
D e n i t i o n ( 4 )
C o n c e r n i n g t h e i n t e r i o r ( e x t e r i o r ) p r o b l e m a p a i r o f a v e c t o r f u n c t i o n u a n d a
s c a l a r f u n c t i o n i s c a l l e d t h e s t r i c t s o l u t i o n i f u 2 C
2
( R ) \ C
0
( R ) , 2 C
1
( R )
a n d ( 1 ) , ( 2 ) , ( 3 ) , ( 4 ) a r e s a t i s e d . H o w e v e r t h e v e c t o r f u n c t i o n u a l o n e i s a l s o
s o m e t i m e s c a l l e d t h e s t r i c t s o l u t i o n .
6
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C h a p t e r 2
2 . 1 E x i s t e n c e o f G e n e r a l i z e d S o l u t i o n s
2 . 1 . 1 T h e o r e m
A s s u m e t h a t R i s b o u n d e d a n d f 2 f L
2
( R ) g
n
. T h e n t h e r e e x i s t s t h e g e n e r a l -
i z e d s o l u t i o n o f t h e i n t e r i o r p r o b l e m , i f o n e o f t h e f o l l o w i n g c o n d i t i o n s ( 1 ) a n d
( 2 ) f u l l l e d :
1 i s t h e b o u n d a r y v a l u e o f a f u n c t i o n b
2 C
1
( R ) w i t h s m a l l j b
j o r j r b
j
i n t h e s e n s e o f > M
0
C
1
t o b e g i v e n b e l o w .
2 i s t h e b o u n d a r y v a l u e o f a f u n c r i o n b
2 C
1
( R ) e x p r e s s i b l e i n t h e f o r m
b
= r o t a , a 2 C
1
( R ) a n d @ R i s o f c l a s s C
2
P r o o f :
F i r s t l y , w e i n t r o d u c e t h e n o t a t i o n o f C o n d i t i o n ( B ) : n a m e l y , a v e c t o r f u n c t i o n
b i s s a i d t o s a t i s f y C o n d i t i o n ( B ) i f
b 2 C
1
( R ) ; b = o n @ R ( 1 )
a n d i f
j A ( b ; ! ; ! ) j k r ! k
2
( 2 )
i s v a l i d f o r a n y ! 2 C
1
0
( R ) a n d w i t h s o m e c o n s t a n t i n 0 <
N o w , s u p p o s e t h a t w e a r e g i v e n a b s a t i s f y i n g C o n d i t i o n ( B ) a n d t h e n w e s e e k
a g e n e r a l i z e d s o l u t i o n o f t h e f o r m
u = v + b ;
v 2
c
H
1
0
( R )
7
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W e i n t r o d u c e a n d x a s e q u e n c e f
n
g
1
n = 1
o f f u n c t i o n s i n C
1
0
( R ) s u c h t h a t i t s
l i n e a r h u l l i s d e n s e i n
c
H
1
0
( R ) b u t f o r c o n c e n i e n c e w e n o r m a l i z e i t a s
i
;
j
=
i j
=
8
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W (
i
; b + v ) = (
i
; f )
F r o m t h i s w e c a n s o l v e ,
( r
i
; r v ) =
r
i
; r (
1
1
+
2
2
+ +
N
N
)
=
r
i
; r
1
1
+ r
2
2
+ + r
N
N
=
r
i
; r
1
1
+
r
i
; r
2
2
+ +
r
i
; r
N
N
=
r
i
; r
1
1
+
r
i
; r
2
2
+ +
r
i
; r
N
N
=
N
X
j = 1
r
i
; r
j
j
A (
i
; v ; v ) =
i
; v ;
1
1
+
2
2
+ +
N
N
= (
i
; v ;
1
1
) + (
i
; v ;
2
2
) + + (
i
; v ;
N
N
)
= (
i
; v ;
1
)
1
+ (
i
; v ;
2
)
2
+ + (
i
; v ;
N
)
N
=
N
X
j = 1
(
i
; v ;
j
)
j
S i m i l a r l y ,
A (
i
; b ; v ) =
N
X
j = 1
A (
i
; b ;
j
)
j
A (
i
; v ; b ) =
N
X
j = 1
A (
i
;
j
; b )
j
T h e n w e h a v e
N
X
j = 1
n
( r
i
; r
j
) + A (
i
; b ;
j
) + A (
i
;
j
; b ) + A (
i
; v ;
j
)
o
j
= (
i
; f ) ( r
i
; r b ) A (
i
; b ; b )
I n o r d e r t o r e w r i t e ( 8 ) a s a n e q u a t i o n w i t h t h e u n k n o w n N - v e c t o r , w e i n -
t r o d u c e a n N N - m a t r i x T ( ) = f T
i j
( ) g , d e p e n d i n g o n , a n d a n N - v e c t o r
= f
i
g b y
N
X
j = 1
f T
i j
( ) g
j
=
i
; ( i = 1 ; 2 ; ; N )
T
i j
( ) = ( r
i
; r
j
) + A (
i
; b ;
j
) + A (
i
;
j
; b ) + A (
i
; v ;
j
)
i
= (
i
; f ) ( r
i
; r b ) A (
i
; b ; b )
T h e n ( 8 ) i s r e d u c e d t o t h e e q u a t i o n
T ( ) = ;
9
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w h e r e
T ( ) =
0
B
B
B
B
B
B
@
T
1 1
( ) T
1 2
( ) T
1 N
( )
T
2 1
( ) T
2 2
( ) T
2 N
( )
T
N 1
( ) T
N 2
( ) T
N N
( )
1
C
C
C
C
C
C
A
=
0
B
B
B
B
B
B
@
1
2
N
1
C
C
C
C
C
C
A
; =
0
B
B
B
B
B
B
@
1
2
N
1
C
C
C
C
C
C
A
f o r t h e u n k n o w n . F r o m n o w o n w e r e g a r d a s a n e l e m e n t o f N - d i m e n s i o n a l
E u c l i d e a n s p a c e E = E
N
a n d u s e t h e n o t a t i o n
j j =
0
@
N
X
i = 1
2
i
1
A
1 = 2
a n d
=
N
X
i = 1
(
i
i
)
t o d e n o t e t h e n o r m o f 2 E a n d t h e s c a l a r p r o d u c t o f ; 2 E . W e s h a l l s h o w
t h a t T ( )
1
e x i s t s s o t h a t ( 1 0 ) i s r e d u c e d t o = F ( ) , w h e r e F ( ) = T ( )
1
T o t h i s e n d w e e s t i m a t e T ( ) f o r a r b i t r a r y a n d i n E . A s s o c i a t i n g t h e
v e c t o r f u n c t i o n s v ; ! 2
N
C
1
0
( R ) w i t h ; 2 E b y v =
1
1
+
2
2
+
+
N
N
a n d ! =
1
1
+
2
2
+ +
N
N
, w e o b s e r v e t h a t
T ( ) =
1
2
N
0
B
B
B
B
B
B
@
T
1 1
( ) T
1 2
( ) T
1 N
( )
T
2 1
( ) T
2 2
( ) T
2 N
( )
T
N 1
( ) T
N 2
( ) T
N N
( )
1
C
C
C
C
C
C
A
0
B
B
B
B
B
B
@
1
2
N
1
C
C
C
C
C
C
A
T ( ) =
1
2
N
0
B
B
B
B
B
B
@
T
1 1
( )
1
T
1 2
( )
2
T
1 N
( )
N
T
2 1
( )
1
T
2 2
( )
2
T
2 N
( )
N
T
N 1
( )
1
T
N 2
( )
2
T
N N
( )
N
1
C
C
C
C
C
C
A
N 1
T ( ) =
1
T
1 1
( )
1
+
2
T
1 2
( )
2
+ +
N
T
1 N
( )
N
+
1
T
2 1
( )
1
+
2
T
2 2
( )
2
+ +
N
T
2 N
( )
N
+ +
1
T
N 1
( )
1
+
2
T
N 2
( )
2
+ +
N
T
N N
( )
N
1 0
-
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S i n c e
T
i j
( ) =
r
i
; r
j
+ A
i
; b ;
j
+ A
i
;
j
; b
+ A
i
; v ;
j
;
w e c a n s o l v e
1
T
1 1
( )
1
+
2
T
1 2
( )
2
+ +
N
T
1 N
( )
N
=
r
1
1
; r
1
1
+ A
1
1
; b ;
1
1
+ A
1
1
;
1
1
; b
+ A
1
1
; v ;
1
1
+
r
1
1
; r
2
2
+ A
1
1
; b ;
2
2
+ A
1
1
;
2
2
; b
+ A
1
1
; v ;
2
2
+ +
r
1
1
; r
N
N
+ A
1
1
; b ;
N
N
+ A
1
1
;
N
N
; b
+ A
1
1
; v ;
N
N
=
r
1
1
; r
1
1
; r
2
2
; ; r
N
N
+ A
1
1
; b ;
1
1
+
2
2
+ +
N
N
+ A
1
1
;
1
1
+
2
2
+ +
N
N
; b
+ A
1
1
; v ;
1
1
+
2
2
+ +
N
N
=
r
1
1
; r
1
1
;
2
2
; ;
N
N
+ A
1
1
; b ; !
+ A
1
1
; ! ; b
+ A
1
1
; v ; !
=
r
1
1
; r !
+ A
1
1
; b ; !
+ A
1
1
; ! ; b
+ A
1
1
; v ; !
S i m i l a r l y , w e h a v e
1
T
2 1
( )
1
+
2
T
2 2
( )
2
+ +
N
T
2 N
( )
N
=
r
2
2
; r !
+ A
2
2
; b ; !
+ A
2
2
; ! ; b
+ A
2
2
; v ; !
F o r t h e l a s t t e r m ,
N
T
N 1
( )
1
+
N
T
N 2
( )
2
+ +
N
T
N N
( )
N
=
r
N
N
; r !
+ A
N
N
; b ; !
+ A
N
N
; ! ; b
+ A
N
N
; v ; !
C o m b i n i n g ( I 1 ) t o ( I N )
T ( ) =
r
1
1
; r !
+ A
1
1
; b ; !
+ A
1
1
; ! ; b
+ A
1
1
; v ; !
r
N
N
; r !
+ A
N
N
; b ; !
+ A
N
N
; ! ; b
+ A
N
N
; v ; !
r
2
2
; r !
+ A
2
2
; b ; !
+ A
2
2
; ! ; b
+ A
2
2
; v ; !
+
+
r
N
N
; r !
+ A
N
N
; b ; !
+ A
N
N
; ! ; b
+ A
N
N
; v ; !
=
r
1
1
+
2
2
+ +
N
N
; r !
+ A
1
1
+
2
2
+ +
N
N
; b ; !
+ A
1
1
+
2
2
+ +
N
N
; ! ; b
+ A
1
1
+
2
2
+ +
N
N
; v ; !
= ( r ! ; r ! ) + A ( ! ; b ; ! ) + A ( ! ; ! ; b ) + A ( ! ; v ; ! )
1 1
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T ( ) = k r ! k
2
A ( b ; ! ; ! )
b y t h e v a r t u e o f ( 2 . 1 9 ) a n d ( 2 . 1 9 ) .
H e n c e w e h a v e b y ( 3 . 4 ) ,
j T ( ) j =
k r ! k
2
A ( b ; ! ; ! )
k r ! k
2
j A ( b ; ! ; ! ) j
k r ! k
2
k r ! k
2
j T ( ) j
( ) k r ! k
2
j j j T ( ) j ( 2 . 1 . 1 )
O n t h e o t h e r h a n d , b y m e a n s o f ( 3 . 5 ) a n d L e m m a ( 2 . 1 ) , w e o b t a i n
k ! k
2
= ( ! ; ! )
=
1
1
+
2
2
+ +
N
N
;
1
1
+
2
2
+ +
N
N
=
1
1
;
1
1
+
2
2
+
N
N
+
2
2
;
1
1
+
2
2
+
N
N
+ +
N
N
;
1
1
+
2
2
+
N
N
=
1
1
;
1
1
+
1
1
;
2
2
+ +
1
1
;
N
N
+
2
2
;
1
1
+
2
2
;
2
2
+ +
2
2
;
N
N
+ +
1
1
;
1
1
+
1
1
;
2
2
+ +
1
1
;
N
N
+
N
N
;
1
1
+
N
N
;
2
2
+ +
N
N
;
N
N
=
1
1
;
1
1
+
1
1
;
2
2
+ +
1
1
;
N
N
+
2
2
;
1
1
+
2
2
;
2
2
+ +
2
2
;
N
N
+ +
N
N
;
1
1
+
N
N
;
2
2
+ +
N
N
;
N
N
=
2
1
+
2
2
+ +
2
N
=
N
X
i = 1
2
i
k ! k =
0
@
N
X
i = 1
2
i
1
A
1 = 2
= j j
j j = k ! k C
1
k r ! k
C o m b i n i n g t h i s w i t h ( 3 . 1 2 ) w e a r e l e d t o
1 2
-
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( ) j j
2
C
2
1
j j j T ( ) j
j j
C
2
1
( )
j T ( ) j
= j T ( ) j ; w h e r e =
C
2
1
( )
1
j j j T ( ) j
j T ( ) j j j
1
j T ( ) j
j T ( ) j
1
B u t
T ( ) T ( )
1
= j T ( ) j
T ( )
1
= 1
j T ( ) j
T ( )
1
= 1
T ( )
1
= j T ( ) j
1
T h u s w e h a v e T ( )
1
e x i s t s a n d
T ( )
1
( 2 . 1 . 2 )
h o l d s , w h e r e t h e l e f t h a n d s i d e m e a n s t h e n o r m o f t h e l i n e a r t r a n s f o r m a t i o n
T ( )
1
i n E . S i n c e i s t h e c o n s t a a n t v e c t o r i n d e p e n d e n t o n , w e t h u s s e e
t h a t t h e i n e q u a l i t i e s
j F ( ) j = j T ( )
1
j
T ( )
1
j j = d
h o l d f o r a n y 2 E . I n p a r t i c u l a r , t h e c l o s e d s p h e r e o f E w i t h c e n t e r a t t h e
o r i g i n a n d r a d i u s d i s m a p p e d b y F i n t o i t s e l f .
F : S ( d ) ! S ( d )
S i n c e , t h e c o n t i n u i t y o f F ( ) i n i s o b v i o u s ,
j j = j T ( ) j
j T ( ) j j j
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-
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j F ( ) j =
T ( )
1
T ( )
1
j j
T ( )
1
j T ( ) j j j
j F ( ) j j j
w e a r e a b l e t o a p p l y B r o u w e r ' s t h e o r e m a n d c o n c l u d e t h e e x i s t e n c e o f a s o l u t i o n
o f t h e e q u a t i o n F ( ) = . T h i s p r o v e s t h e l e m m a .
C o n c e r n i n g t h e a p p r o x i m a t i n g s o l u t i o n s o b t a i n e d a b o v e w e h a v e
2 . 1 . 3 L e m m a
T h e r e e x i s t s a c o n s t a n t K s u c h t h a t
k r V
N
k K
P r o o f :
M a k i n g u s e o f ( 2 . 1 9 ) a n d ( 2 . 1 9 ) ' w e h a v e
W ( v ; v + b ) = ( r v ; r ( v + b ) ) + A ( v ; v + b ; v + b )
= ( r v ; r v + ; r b ) + A ( v ; v ; v + b ) + A ( v ; b ; v + b )
= ( r v ; r v ) + ( r v ; r b ) + A ( v ; v ; v ) + A ( v ; v ; b ) + A ( v ; b ; v )
+ A ( v ; b ; b )
= k r v k
2
+ ( r v ; r b ) A ( v ; b ; b ) A ( b ; b ; v )
F o r a n y v 2 C
1
0
( R ) . H e n c e s e t t i n g ' = v
N
a n d u
N
= b + v
N
i n ( 3 . 7 ) w e
i m m e d i a t e l y o b t a i n W ( v
N
; b + v
N
) = ( v
N
; f )
k r v
N
k
2
+ ( r v
N
; r b ) A ( b ; v
N
; v
N
) A ( b ; b ; v
N
) = ( v
N
; f )
k r v
N
k
2
A ( b ; v
N
; v
N
) = ( v
N
; f ) ( r v
N
; r b ) + A ( b ; b ; v
N
)
W h e n c e f o l l o w s b y v i r t u e o f ( 3 . 4 ) a n d S c h w a r z ' s i n e q u a l i t y t h a t
k r v
N
k
2
A ( b ; v
N
; v
N
)
= j ( v
N
; f ) ( r v
N
; r b ) + A ( b ; b ; v
N
) j
j ( v
N
; f ) j + j ( r v
N
; r b ) j + j A ( b ; b ; v
N
) j
k r v
N
k
2
j A ( b ; v
N
; v
N
) j k v
N
k k f k + j j j ( r v
N
; r b ) j + j A ( b ; v
N
; v
N
) j
k r v
N
k
2
k r v
N
k
2
k v
N
k k f k + j j k r v
N
k k r b k + j A ( b ; v
N
; v
N
) j
1 4
-
8/3/2019 MRES2
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w h e r e a s w e c a n e s t i m a t e a s
A ( b ; v
N
; v
N
) =
Z
R
f b
i
b
j
g @
j
v
N
i
d x ; x 2 R
= ( b
i
b
j
; r v
N
)
j A ( b ; v
N
; v
N
) j = j ( b
i
b
j
; r v
N
) j
k b
i
b
j
k k r v
N
k
( ) k r v
N
k
2
k v
N
k k f k + j j k r b k k r v
N
k + k b
i
b
j
k k r v
N
k
c k f k k r v
N
k + ( j n u j j r b j + k b
i
b
j
k ) k r v
N
k
= ( c
0
+ c
0 0
) k r v
N
k
( ) k r v
N
k
2
c k r v
N
k
k r v
N
k
c
( )
= k
w i t h a p p r o p r i a t e c o n s t a n t s c
0
; c
0 0
a n d c . C o n s e q u e n t l y w e h a v e k r v
N
k = k
S i n c e k r v
N
k a n d t h e r e f o r e , k v
N
k a r e a b l e t o c h o o s e a s u b s e q u e n c e f v
N
g o f
f v
N
g t e n d i n g t o a v 2
c
H
1
0
( R ) i n t h e s e n s e t h a t v
N
! v
s t r o n g i n L
2
( R )
a n d r v
N
! r v
w e a k l y i n L
2
( R ) . W e s h a l l s h o w t h a t u
= b + v
i s t h e
d e s i r e d g e n e r a l i z e d s o l u t i o n .
I f ' i s a x e d f u n c t i o n i n C
1
0
( R ) , w e h a v e
( r ' ; r v
N
) ! ( r ' ; r v
) ; A ( ' ; v
N
; v
N
) ! A ( ' ; v
; v
) a s N ! 1
B u t ,
A ( ' ; v
N
; v
N
) A ( ' ; v
; v
) = A ( ' ; v
N
; v
N
) A ( ' ; v
; v
N
) + A ( ' ; v
; v
N
)
A ( ' ; v
; v
)
= A ( ' ; v
N
v
; v
N
) + A ( ' ; v
; v
N
v
)
j A ( ' ; v
N
v
; v
N
) j = k ' k
4 R
k v
N
v
k
4 R
k r v
N
k
2 R
k ' k
4 R
C k r ( v
N
v
) k
2 R
k r v
N
k
2 R
A ( ' ; v
; v
N
v
) =
Z
R
( '
i
v
j
) @
j
( v
N
v
)
i
d x ; 8 x 2 R
=
'
i
v
j
; @
j
( v
N
v
)
= ( '
i
v
; r ( v
N
v
) )
j A ( ' ; v
; v
N
v
) j = j ( '
i
v
; r ( v
N
v
) ) j
k '
i
v
k
1 5
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F i x a n a r b i t r a r y p o s i t i v e i n t e g e r n . T h e n f o r a n y N n w e h a v e W (
n
; b + v
N
) =
(
n
; f )
( r
n
; r b ) + ( r
n
; r v
N
) + A (
n
; b ; b ) + A (
n
; b ; v
N
) +
A (
n
; v
N
; b ) + A (
n
; v
N
; v
N
) = (
n
; f )
w h e n c e f o l l o w s b y m a k i n g N ! 1 ,
( r
n
; r v
N
) ! ( r
n
; r v
)
A (
n
; v
N
; v
N
) ! (
n
; v
; v
)
( r
n
; r b ) + ( r
n
; r v
) + A (
n
; b ; b ) + A (
n
; b ; v
) +
A (
n
; v
; b ) + A (
n
; v
; v
) = (
n
; f )
( r
n
; r b + r v
) + A (
n
; b ; b + v
) + A (
n
; v
; b + v
) = (
n
; f )
( r
n
; r ( b + v
) ) + A (
n
; b + v
; b + v
) = (
n
; f )
W ( r
n
; b + b
) = (
n
; f )
F u r t h e r w e n o t i c e t h a t W ( ' ; u
) = ( ' ; f ) i s v a l i d f o r a n y ' i n
F r o m ( 3 . 6 ) ,
v
N
= u
N
b 2
N
u
N
= v
N
b 2
N
u
N
= v
N
b 2
N
F r o m ( 3 . 7 ) ,
W ( ' ; u
N
) = ( ' ; f )
W ( ' ; u
N
) = W ( ' ; u
N
+ b )
= ( ' ; f )
( r ' ; r ( v
N
+ b ) ) + A ( ' ; v
N
+ b ; v
N
+ b = ( ' ; f )
( r ' ; r v
N
+ r b ) ) + A ( ' ; v
N
; v
N
+ b ) + A ( ' ; b ; v
N
+ b ) = ( ' ; f )
( r ' ; r v
N
) + ( r ' ; r b ) + A ( ' ; v
N
; v
N
) + A ( ' ; v
N
; b ) + A ( ' ; b ; v
N
)
+ A ( ' ; b ; b ) = ( ' ; f )
1 6
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W h e n c e f o l l o w s b y M a k i n g N ! 1
( r ' ; r v
) + ( r ' ; r b ) + A ( ' ; v
; v
) + A ( ' ; v
; b ) + A ( ' ; b ; v
)
+ A ( ' ; b ; b ) = ( ' ; f )
( r ' ; r v
+ r b ) + A ( ' ; v
; v
+ b ) + A ( ' ; b ; v
b ) = ( ' ; f )
( r ' ; r ( v
+ b ) ) + ( ' ; v
+ b ; v
+ b ) = ( ' ; f )
( r ' ; r u
) + A ( ' ; u
; u
) = ( ' ; f )
W ( ' ; u
) = ( ' ; f )
T h e n t a k e a n a r b i t r a r y ' i n C
1
0
( R ) . S i n c e i s d e n s e i n
c
H
1
0 s m
( R ) , w e c a n n d
a s e q u e n c e f '
n
g w u c h t h a t '
n
2 a n d '
n
c o n v e r g e s t o ' s t r o n g l y i n
c
H
1
0 s m
( R )
T a k i n g t h e l i m i t o f W ( '
n
; u
) = ( '
n
; f ) w e a r r i v e a t W ( ' ; u
) = ( ' ; f ) b y
v a r t u e o f ( 2 . 1 8 ) .
W ( '
n
; u
) = ( r '
n
; r u
) + A ( '
n
; u
; u
) = ( '
n
; f )
l i m
n ! 1
A ( r '
n
; u
; u
) = A ( ' ; u
; u
)
l i m
n ! 1
( r '
n
; r u
) = ( r ' ; r u
)
( r ' ; r u
) + A ( ' ; u
; u
) = ( ' ; f )
W ( ' ; u
) = ( ' ; f )
A t t h i s s t a g e i t h a s b e e n p r o v e d t h a t a g e n e r a l i z e d s o l u t i o n e x i s t s i f t h e r e e x i s t s
a f u n c t i o n b s a t i s f y i n g C o n d i t i o n ( B ) . W e s h a l l s h o w t h a t t h e c o n d i t i o n ( 1 ) o r
( 2 ) i n t h e t h e o r e m i s s u c i e n t f o r t h e e x i s t e n c e o f s u c h a b . F i r s t l y w e d e a l w i t h
( 1 ) . L e t M
0
a n d M
1
b e c o n s t a n t s s u c h t h a t j b
( x ) j M
0
a n d j r b
( x ) j M
1
h o l d f o r a n y x 2 R . T h e n b y m e a n o f L e m m a ( 2 . 1 ) a n d ( 2 . 1 9 ) w e h a v e
j A ( b
; ! ; ! ) j j b
j k ! k k r ! k
M
0
C
1
k r ! k
2
j A ( b
; ! ; ! ) j = j A ( ! ; ! ; b
) j
= k ! k
2
j r b
j
M
1
C
2
1
k r ! k
2
F o r a n y ! 2 C
1
0
( R ) , w h e r e C
1
i s t h e d o m a i n c o n s t a n t i n ( 2 . 1 ) . T h e r e f o r e b
i t s e l f s a t i s e s C o n d i t i o n ( B ) i f e i t h e r > M
0
C
1
o r > M
1
C
2
1
W e t u r n t o ( 2 ) . W e n o t e t h a t L e m m a ( 2 . 5 ) i s a p p l i c a b l e t o e a c h c o m p o n e n t
o f b o u n d a r y R a n d ( 2 . 1 1 ) r e m a i n s v a l i d w i t h a n a p p r o p i a t e c o n s t a n t C
3
i f
1 7
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w e r e p l a c e !
( ) b y ! ( ) = ! ( ; R ) a n d
( x ) b y ( x ) = d i s t . ( x ; @ R ) W e
c o n s t r u c t t h e d e s i r e d b i n t h e f o r m o f b
= r o t ( h ( ) a ) = r ( h ( ) a ) w i t h a
s c a l a r f u n c t i o n h o f a s i n g l e v a r i a b l e d e n e d b y
h ( t ) = 1
R
t
0
j ( s ) d s
R
1
0
j ( s ) d s
; ( t > 0 )
w h e r e j ( s ) i s a f u n c t i o n w i t h t h e f o l l o w i n g p r o p e r t i e s : j ( s ) i n v o l v e s t w o p a -
r a m e t e r s r a n d w h o s e v a l u e s a r e c o n t a i n e d i n 0 < <
0
, 0 < 0 ) ,
( i i i ) j ( s ) = 0 ; ( 0 s ; ( 1 ) s ) ,
( i v ) j ( s ) =
1
s
; ( 2 s ( 1 2 ) )
T h e r e f o r e , h 2 C
1
0 ; 1 ) a n d
I f 0 t ,
h ( t ) = 1
R
0
j ( s ) d s
R
1
0
j ( s ) d s
= 1
0
R
0
d s
R
1
0
j ( s ) d s
= 1
I f ( 1 ) < t , t h e n t ! 1
h ( t ) = 1
R
1
0
j ( s ) d s
R
1
0
j ( s ) d s
= 1 1
= 0
h ( t ) =
8
0 )
h
0
( t ) =
1
R
1
0
j ( s ) d s
d
d t
Z
t
0
j ( s ) d s
!
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I f t s t + t ,
t m i n j ( s )
t
R
t + t
t
j ( s ) d s
t
t m a x j ( s )
t
T a k i n g t h e l i m i t ! 0 , s = t
T h u s
j ( t )
R
t + t
t
j ( s ) d s
t
j ( t )
h
0
( t ) =
j ( t )
R
t
0
j ( s ) d s
t h
0
( t ) =
t j ( t )
R
0
j ( s ) d s +
R
( 1 2 )
j ( s ) d s +
R
1
( 1 2 )
j ( s ) d s
=
t j ( t )
R
( 1 2 )
1 = s d s
=
t j ( t )
l n ( 1 2 ) l n ( )
M o r e o v e r , a s ! 0 ,
t h
0
( t ) =
t j ( t )
l n ( ) l n ( 0 )
=
t j ( t )
l n ( ) ( 1 )
=
t j ( t )
1
= 0
t h
0
( t ) t e n d s t o 0 u n i f o r m l y w . r . t . a n d t
T h e r e f o r e b y m e a n s o f t h e w e l l k n o w n f o r m u l a
r o t ( h a ) = r ( h a )
= h ( ) ( r a ) + ( r h ( ) a )
= h ( ) ( r a ) ( a r h ( ) )
= h ( ) r o t a a h
0
( ) r
1 9
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A s ! 0 , h
0
( ) t e n d s t o 0
r o t ( h ( ) a ) = h ( ) r o t a
I f 0 , h ( ) = 1
b
= r o t ( h a ) = r o t a = b
= o n @ R
! ( ) = ! ( ; R ) = f x 2 R j d i s t . ( x ; @ R ) < r g
x =2 ! ( ) ) ( x ) = d i s t . ( x ; @ R ) > r
T h u s ( x ) > > ( 1 ) . W e h a v e h ( ) = 0 , b
= r o t ( h a ) = h ( ) r o t a =
0 o u t s i d e ! ( ) a n d b
= h ( ) r o t a h
0
( ) r b e l o n g t o C
1
( R ) , b e c a u s e
i s n o w o f c l a s s C
2
. F u r t h e r m o r e , f o r a n y " > 0 , c h o o s e j ( x ) j
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c h o o s e ' = u
1
u
2
t h e n A ( u
1
u
2
; u
1
u
2
; u
1
u
2
) = 0 b y v i r t u e o f ( ) .
( r ' ; r ( u
1
u
2
) ) = ( r ( u
1
u
2
) ; r ( u
1
u
2
) )
= 0
r ( u
1
u
2
) = 0
u
1
u
2
= 0
u
1
= u
2
T h e r e f o r e t h e g e n e r a l i z e d s o l u t i o n o f t h e i n t e r i o r p r o b l e m i s u n i q u e .
2 1
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C h a p t e r 3
3 . 1 R e g u l a r i t y o f t h e S t a t i o n a r y N a v i e r - S t o k e s E q u a t i o n s
W e d e n e d t h e d i e r e n c e o p e r a t o r s
i
h
b y
i
h
u
( x ) =
1
2
( u ( x + h e
i
) u ( x ) ) ; n 6=
0
H e r e e
i
= (
i 1
;
i 2
; : : : ;
i n
) i s t h e c o n o n i c a l b a s i s o f R
n
3 . 1 . 1 L e m m a ( 1 )
S u p p o s e u 2 H
m
( ) ; m 1 ;
0
A s s u m e d i s t (
0
; @ ) > h > 0 T h e n
i
h
u
m 1
k u k
m
( 1 )
P r o o f :
F o r a n y f u n c t i o n f 2 C
0
( a ; b + h ) w e h a v e
f ( x + h ) f ( x ) =
Z
x + h
x
f
0
( t ) d t
j f ( x + h ) f ( x ) j
2
=
Z
x + h
x
f
0
( t ) d t
!
2
=
Z
x + h
x
Z
x + h
x
f
0
( t ) f
0
( t ) d t d z
= h
Z
x + h
x
f
0
( t ) f
0
( z ) d t
h
2
Z
x + h
x
n
( f
0
( t ) )
2
+ ( f
0
( z ) )
2
o
d t = h
Z
x + h
x
j f
0
( t ) j
2
d t
T h u s
j f ( x + h ) f ( x ) j
2
h
Z
x + h
x
j f
0
( t ) j
2
d t
I n t e g r a t i n g w e g e t
Z
b
a
j f ( x + h ) f ( x ) j
2
d x h
Z
b
a
d x
Z
x + h
x
j f
0
( t ) j
2
d t d x
2 2
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= h
Z
b
a
Z
x + h
x
j f
0
( t ) j
2
d t d x
h
Z
b
a
Z
x + h
x
j f
0
( t ) j
2
d t d x
=
h
Z
b + h
a
j f
0
( t ) j
2
Z
t h
t
d x
!
d t
=
h
2
Z
b + h
a
j f
0
( t ) j
2
d t
= h
2
Z
b + h
a
j f
0
( t ) j
2
d t
h
2
Z
b + h
a
j f
0
( t ) j
2
d t
T h e r e f o r e ,
Z
b
a
f ( x + h ) f ( x )
h
2
d x
Z
b + h
a
j f
0
( t )
2
d t
I f u 2 C
m
( ) ; u
(
m 1 ) 2 C
1
( ) T h e n w e h a v e
Z
u
( m 1 )
( x + h ) u
( m 1 )
( x )
h
2
d x
Z
j u
( m )
( x ) j
2
d x
Z
( u ( x + h ) u ( x ) )
( m 1 )
h
2
d x
Z
j u
( m )
( x ) j
2
d x
Z
i
+ h
u
( m 1 )
2
d x
Z
u
( m )
2
d x
B u t
Z
@
m j
x
j
@
j 1
y
j
i
+ h
u
2
d x =
Z
@
m j
x
j
i
+ h
@
j 1
y
j
u
2
d x ;
f o r e a c h j .
T h e n
Z
@
m j
x
j
i
y
j
@
j 1
y
j
u
2
d x
Z
@
m j
x
j
@
j 1
y
j
u
2
d x ;
f o r e a c h j .
@
m j
x
j
i
y
j
@
j 1
y
j
u
2
l
2
@
m j
x
j
@
j 1
y
j
u
2
l
2
;
f o r e a c h j .
I f j = 1 , t h e n w e h a v e
k
i
+ h
u k
2
m 1
k u k
2
m
S i m i l a r l y , w e h a v e
k
i
h
u k
2
m 1
k u k
2
m
2 3
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T h u s w e o b t a i n
k
i
h
u k
2
m 1
k u k
2
m
L e m m a ( 2 )
S u p p o s e h a s t h e s e g m e n t p r o p e r t y . A s s u m e u 2 H
m
( ) a n d t h a t t h e r e
e x i s t s a c o n s t a n t c > o s u c h t h a t , f o r e v e r y
0
C ; k
i
h
u k
m
c f o r a l l h
s u c i e n t l y s m a l l . T h e n
k D
i
u k m ;
m
c ; w h e r e
m
=
X
m
1
P r o o f :
A s s u m e t h a t m = 0 . F i x
0
C C . B y t h e w e a k c o m p a c t n e s s p r o p e r t y o f L
2
w e
n d a s e q u e n c e h
k
o f r e a l s h
k
! 0 a n d a f u n c t i o n u
i
2 L
2
s u c h t h a t
i
h
k
u ! u
i
w e a k l y i n L
2
(
0
) S i n c e l i m
k
i
h
k
u
0
=
l i m
k
i
h
k
u
0
c ;
k u
i
k
0
c F o r a n y ' 2 C
1
0
( ) w e h a v e
Z
u
i
' d x = l i m
k
Z
i
h
k
u
' d x = l i m
k
Z
1
h
k
n
u ( x + h e
i
) u ( x )
o
' ( x ) d x
L e t y = x + h
k
e
i
; x = y h
k
e
i
; d x = d y T h e n
Z
u
i
' d x = l i m
k
Z
1
h
k
u ( y ) ' ( y h
k
e
i
) d y l i m
k
Z
1
h
k
u ( y ) ' ( y ) d y
= l i m
k
Z
1
h
k
u ( y )
n
' ( y h
k
e
i
) ' ( y )
o
d y
= l i m
k
Z
u
i
h
k
' d y =
Z
u l i m
k
i
h
k
' d x
=
Z
u D
i
' d x :
T h e n w e h a v e u
i
= D
i
u i n t h e w e a k s e r s e i n
0
Z
u
i
' d x =
Z
u D
i
' d x =
j u ' j
2
Z
' D
i
u d x
=
Z
( D
i
u ) ' d x :
T h e r e f o r e k D
i
u k
0
c A l l o w i n g
0
t o v a r y w e o b t a i n t h e c o n c l u s i o n o f t h e
l e m m a f o r m = 0 .
l i m
k
i
h
k
u
i
=
l i m
k
j
h
k
u
i
= k u
i j
k c ;
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f o r s o m e i , j .
B y g i v e n
i
h
u
m
c w e h a v e
i
h
k
u
i
0
j
h
k
u
1
c ; a n d k D
i
u k
1
c
S i m i l a r l y w e h a v e f o r g e n e n a l m ,
k D
i
u k
m
0
B
@
X
m
1
1
C
Ac
k D
i
u k
m
m
c ; w h e r e
m
=
X
m
c
L e m m a ( 3 )
L e t R > 0 a n d l e t G
R
= f j x 2 R
n
j j x j < R ; x
n
> 0 g . S u p p o s e u 2 L
2
( G ) a n d
a s s u m e t h a t t h e r e e x i s t s a n u m b e r c s u c h t h a t , f o r e v e r y R
0
< R ;
i
h
u
0
c
f o r s o m e i 2 1 ; 2 ; : : : ; n 1 . T h e n t h e w e a k d e r i v a t i v e D
i
u b e l o n g s t o L
2
( G
R
)
a n d k D
i
u k
0 G
R
c
T h e d i e r e n c e o p e r a t o r s a c t o n e a c h c o m p o n e n t . T h e o p e r a t o r s
i
h
a c t a l m o s t
l i k e d e r i v a t i v e s :
i
h
( a v ) = a
i
h
( v ) +
i
h
( a )
i
h
( v ) =
i
h
( a ) v +
i
h
( a )
i
h
( v )
H e r e
i
h
( a ) ( x ) = a ( x + h e
i
) i s a t r a n s l a t i o n o p e r a t o r a n d , o f c o u r s e f o r s m a l l
h , i t i s c l o s e t o i d e n t i t y .
T h e o p e r a t o r
n
X
i j = 1
@
@ x
i
0
@
a
i j
( x )
@
@ x
j
1
A
i s s a i d t o b e u n i f o r m l y e l l i p t i c i n a d o m a i n G i f t h e r e e x i s t s M > 0 s u c h t h a t
1
M
j j
2
X
i j
a
i j
( x )
i
j
M j j
2
;
f o r a l l x 2 G ; 2 R
n
w e s h a l l c o n s i d e r t w o k i n d s o f d o m a i n G , b a l l s a n d h a l f b a l l s :
G
R
= f x 2 R
n
j j x j < R g
G
R
= f x 2 R
n
j j x j < R ; x
n
> 0 g
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W e s h a l l d e n o t e b y
g
G
R
t h e s e t
g
G
R
= f x 2 R
n
j j x j < R ; x
n
0 g
A f u n c t i o n w h o s e s u p p o r t i s c o m p a c t a n d i n c l u d e d i n
~
G
R
m a y n o t v a n i s h f o r
p o i n t s o n x
n
= 0
L e m m a
L e t 0 < R ' < R . C o n s i d e r a w e a k s o l u t i o n v , p o f t h e s y s t e m
@
@ x
i
0
@
a
i j
( x )
@ v
m
2 x
j
1
A
+ b
j
( x )
@ v
m
@ x
j
+ g
m j
( x )
@ P
@ x
j
= f
m
m = 1 ; 2 ; ; n
( x ) = g
m k
( x )
@ v
m
@ x
k
w h e r e
a
i j
2 C
1
( G
R
) ; g
m j
2 C
2
( G
R
) ; b
j
2 C
0
( G
R
) ; a n d
f = f
m
2
L
2
( G
R
)
n
; 2 H
2
( G
R
)
T h e p r i n c i p a l p a r t o f ( 7 ) i s a s s u m e t o b e u n i f o r m l y e l l i p t i c , i . e , ( 3 ) h o l d s . T h e
d o m a i n G
R
i s e i t h e r a b a l l ( 4 ) o r a h a l f b a l l ( 5 ) . S u p p o s e v 2
H
1
0
( G
R
)
n
; P 2
L
2
( G
R
) A s s u m e t h a t t h e s u p p o r t s o f v ; p a r e c o m p a c t i n ( G
R
) ( G
R
i n c a s e
o f t h e h a l f b a l l s ) . T h e n t h e r e e x i s t s a c o n s t a n t c d e p e n d i n g o n R ; R
0
a n d t h e
c o e c i e n t o f ( 7 ) , ( 8 ) s u c h t h a t
k D
i
v k
H ( G
R
)
C
h
k f k
L
2
( G
R
)
+ k v k
H ( G
R
)
+ k k
H
2
( G
R
)
+ k P k
L
2
( G
R
)
i
w h e r e i = 1 ; 2 ; ; n i f G
R
; G
R
a r e b a l l s a n d
i = 1 ; 2 ; ; n i f G
R
; G
R
a r e h a l f b a l l s .
P r o o f :
A c c o r d i n g t o l e m m a s ( 2 ) a n d ( 3 ) a l l w e n e e d t o s h o w i s t h a t t h e r i g h t h a n d s i d e
o f ( 9 ) i s a n u p p e r b o u n d f o r
i
h
v
H ( G
R
)
f o r a l l R
0
< R
0 0
< R a n d j h j
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Z
G
R
a
k
j
@ v
m
@ x
j
@ '
m
@ x
k
d x +
Z
G
R
b
j
@ v
m
@ x
j
'
m
d x
Z
G
R
p
@
@ x
j
( g
m j
@
m
) d x =
Z
G
R
f
m
'
m
d x
C l e a r l y ( 1 0 ) i s t r u e , b y c o n t i n u i t y , f o r e v e r y ' 2
H
1
0
( G
R
)
n
S i n c e C
1
0
( G
R
) i s
d e n s e i n H
1
( G
R
) ; w e c a n n d a s e q u e n c e @
m
s u c h t h a t ' 2 C
1
0
( G
R
) a n d '
m
c o n v e r g e s t o ' 2
G
1
0
( G
R
)
n
W e o b t a i n
Z
G
R
a
k
j
@ v
m
@ x
j
@ '
@ x
k
d x +
Z
G
R
b
j
@ v
m
@ x
j
' d x
Z
G
R
p
@
@ x
j
( g
m j
' ) d x =
Z
G
R
f
m
' d x :
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T h e r e f o r e , f r o m ( 1 1 ) w e h a v e
Z
i
h
0
@
a
k
j
@ v
m
@ x
j
1
A
@ '
m
@ x
k
d x
Z
b
j
@ v
m
@ x
j
i
h
'
m
d x +
Z
p
@
@ x
j
g
m j
i
h
@
m
d x
=
Z
f
m
i
h
@
m
d x
w e t r e a t t h e t h r e e t e r m s o n t h e l e f t - h a n d s i d e o f ( 1 2 ) s e p a r a t e l y . T h e r s t o n e
u s i n g ( 2 ) c a n b e c o m p u t e d
I =
Z
i
h
0
@
a
k
j
@ v
m
@ x
j
1
A
@ '
m
@ x
k
d x
=
Z
a
k
j
i
h
@ v
m
@ x
i
! !
@ '
m
@ x
k
d x +
Z
i
h
( a
k
j
)
i
h
@ v
m
@ x
i
!
@ '
m
@ x
k
d x
= a
i
h
v ; '
+
Z
i
h
a
k
j
i
h
@ v
m
@ x
i
!
@ '
m
@ x
k
d x
I a
i
h
v ; '
=
Z
i
h
( a
k
j
)
i
h
@ v
m
@ x
i
!
@ '
m
@ x
k
d x
I n v i e w o f t h e f a c t t h a t a
k
j
a r e u n i f o r m l y L i p s c h i t z w e o b t a i n .
I a
i
h
v ; '
Z
i
h
( a
k
j
)
i
h
0
@
@ v
m
@ x
j
1
A
@ '
m
@ x
k
d x
i
h
( a
k
j
)
Z
i
h
0
@
@ v
m
@ x
j
1
A
@ '
m
@ x
k
d x
1
h
n
a
k
j
( a + h e
i
) a
k
j
( x )
o
i
h
0
@
@ v
m
@ x
j
1
A
L
2
( G
R
)
@ '
m
@ x
k
L
2
( G
R
)
C
1
h
j x + h e
i
x j
@ v
m
@ x
j
L
2
( G
R
)
@ '
m
@ x
k
L
2
( G
R
)
C j e
i
j k V k
H
1
( G
R
)
k r ' k
L
2
( G
R
)
C k V k
H
1
( G
R
)
k ' k
H
1
( G
R
)
T h e c o n s t a n t
C i s i n d e p e n d e n t o f h a n d w i l l c h a n g e d u r i n g t h e p r o o f .
a ( v ; ! ) =
Z
a
k
j
@ v
m
@ x
j
@ !
m
@ x
k
d x
W e e s t i m a t e t h e s e c o n d t e r m u s i n g L e m m a ( 1 )
j I I j =
Z
b
j
@ v
m
@ x
j
i
h
'
m
d x
Z
b
j
@ v
m
@ x
j
i
h
'
m
d x
2 8
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Z
b
j
@ v
m
@ x
j
i
h
'
m
d x
j b
j
j
Z
@ v
m
@ x
j
i
h
'
m
d x
j b
j
j
@ v
m
@ x
j
i
h
L
2
( G
R
)
i
h
'
m
L
2
( G
R
)
k
i
k v k
H
1
( G
R
)
k ' k
H
1
(
G
R
)
;
f o r s o m e c o n s t a n t k
1
; w h e r e R
0 0 0
=
R + R
2
I n o r d e r t o e s t i m a t e t h e t h i r d t e r m w e w r i t e r s t , u s i n g ( 2 )
@
@ x
j
g
m
j
i
h
'
m
= g
m
j
@
@ x
j
i
h
'
m
+
@ g
m
j
@ x
j
i
h
'
m
B u t w e h a v e , g
m
j
@
@ x
j
i
h
'
m
= g
m
j
i
h
@ '
m
@ x
j
a n d
g
m
j
i
h
0
@
@ '
m
@ x
j
1
A
=
i
h
0
@
g
m
j
@ '
m
@ x
j
1
A
i
h
( g
m
j
)
i
h
0
@
@ '
m
@ x
j
1
A
T h u s ,
@
@ x
j
g
m
j
i
h
'
m
=
i
h
0
@
g
m
j
@ '
m
@ x
j
1
A
i
h
( g
m
j
)
i
h
0
@
@ '
m
@ x
j
1
A
+
@ g
m
j
@ x
j
i
h
'
m
N o w s i n c e s u p p i s c o m p a c t i n G
R
(
f
G
R
) w e c a n n d 2 C
2
0
( G
R
) s u c h t h a t
p =
p
T h u s
( x )
@
@ x
j
g
m
j
i
h
'
m
= ( x )
i
h
0
@
g
m
j
@ '
m
@ x
j
1
A
( x )
i
h
( g
m
j
)
i
h
0
@
@ '
m
@ x
j
1
A
+ ( x )
@
@ x
j
( g
m
j
)
i
h
'
m
B u t
i
h
0
@
g
m
j
@ '
m
@ x
j
1
A
=
i
h
0
@
g
m
j
@ '
m
@ x
j
1
A
i
h
( )
i
h
0
@
g
m
j
@ '
m
@ x
j
1
A
( x )
@
@ x
j
g
m
j
i
h
'
m
=
i
h
0
@
g
m
j
@ '
m
@ x
j
1
A
i
h
( )
i
h
0
@
g
m
j
@ '
m
@ x
j
1
A
( x )
i
h
( g
m
j
)
i
h
0
@
@ '
m
@ x
j
1
A
+ ( x )
@ g
m
j
@ x
j
i
h
'
m
2 9
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M u l t i p l y i n g ( 1 4 ) b y
p 1
p
@
@ x
j
g
m
j
i
h
'
m
=
p 1
i
h
0
@
g
m
j
@ '
m
@ x
j
1
A
p 1
i
h
( )
i
h
0
@
g
m
j
@ '
m
@ x
j
1
A
p
i
h
g
m
j
i
h
0
@
@ '
m
@ x
j
1
A
+
p
@ g
m
j
@ x
j
i
h
'
m
j I I I j =
Z
p
@
@ x
j
g
m
j
i
h
'
m
d x
Z
p 1
i
h
0
@
g
m
j
@ '
m
@ x
j
1
A
d x
+
Z
p 1
i
h
( )
i
h
0
@
g
m
j
@ '
m
@ x
j
1
A
d x
+
Z
p
i
h
g
m
j
i
h
0
@
@ '
m
@ x
j
1
A
d x
+
Z
p
@ g
m
j
@ x
j
i
h
'
m
d x
B u t , w e h a v e
Z
p 1
i
h
0
@
g
m
j
@ '
m
@ x
j
1
A
d x
Z
p 1
i
h
0
@
g
m
j
@ '
m
@ x
j
1
A
d x
p 1
L
2
( G
R
)
i
h
0
@
g
m
j
@ '
m
@ x
j
1
A
L
2
( G
R
)
j
p
j
m i n j j
L
2
( G
R
)
i
h
0
@
g
m
j
@ '
m
@ x
j
1
A
L
2
( G
R
)
1
m i n j j
k
p
k
L
2
( G
R
)
i
h
0
@
g
m
j
@ '
m
@ x
j
1
A
L
2
( G
R
)
k
1
k p k
L
2
( G
R
)
k
p
k
L
2
( G
R
)
i
h
0
@
g
m
j
@ '
m
@ x
j
1
A
L
2
( G
R
)
f o r s o m e c o n s t a n t k
1
3 0
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Z
p 1
i
h
( )
i
h
0
@
g
m
j
@ '
m
@ x
j
1
A
d x
Z
p 1
i
h
( )
i
h
0
@
g
m
j
@ '
m
@ x
j
1
A
d x
Z
p 1
1
h
!
n
( x h e
i
) ( x )
o
i
h
0
@
g
m
j
@ '
@ x
j
1
A
d x
2
h
Z
p
i
h
0
@
g
m
j
@ '
m
@ x
j
1
A
d x
2
h
i
h
0
@
g
m
j
@ '
@ x
j
1
A
L
2
(
G
R
)
k
p
k
L
2
( G
R
)
2
h
g
m
j
@ '
m
@ x
j
L
2
( G
R
)
k p k
L
2
( G
R
)
k
2
k p k
L
2
( G
R
)
k ' k
1
H
( G
R
) ;
f o r s o m e c o n s t a n t k
2
Z
p
i
h
( g
m
j
)
i
h
0
@
@ '
m
@ x
j
1
A
d x
Z
p
i
h
( g
m
j
)
i
h
0
@
@ '
m
@ x
j
1
A
d x
i
h
( g
m
j
)
Z
p
i
h
0
@
@ '
m
@ x
j
1
A
d x
C k p k
L
2
( G
R
)
i
h
0
@
@ '
m
@ x
j
1
A
L
2
( G
R
)
C k p k
L
2
( G
R
)
k r ' k
L
2
( G
R
)
C k p k
L
2
( G
R
)
k ' k
H
1
( G
R
)
;
t h e c o n s t a n t C i s i n d e p e n d e n t o f h
Z
p
@ g
m
j
@ x
j
i
h
'
m
d x
Z
p
@ g
m
j
@ x
j
i
h
'
m
d x
@ g
m
j
@ x
j
Z
p
i
h
'
m
d x
k
3
k p k
L
2
( G
R
)
i
h
'
m
L
2
( G
R
)
3 1
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k
3
k p k
L
2
( G
R
)
k ' k
H
1
( G
R
)
;
f o r s o m e c o n s t a n t k
3
s i n c e g
m
j
2 C
2
( G
R
)
j I I I j k
1
k p k
L
2
( G
R
)
i
h
0
@
g
m
j
@ '
m
@ x
j
1
A
L
2
( G
R
)
+ k
2
k p k
L
2
( G
R
)
k ' k
H ( G
R
)
+ C k p k
L
2
( G
R
)
k ' k
H ( G
R
)
+ k
3
k p k
L
2
( G
R
)
k ' k
H ( G
R
)
k
4
k p k
L
2
( G
R
)
2
6
4
i
h
0
@
g
m
j
@ '
m
@ x
j
1
A
L
2
( G
R
)
+ k ' k
H ( G
R
)
3
7
5;
f o r s o m e c o n s t a n t k
4
Z
f
m
i
h
'
m
d x
Z
f
m
i
h
'
m
d x
k f
m
k
L
2
( G
R
)
i
h
'
m
L
2
( G
R
)
k f k
L
2
( G
R
)
k ' k
H ( G
R
)
S u m m i n g u p , w e o b t a i n
I = I I I I I
Z
f
m
i
h
'
m
d x
a + ( I a ) = I I I I I
Z
f
m
i
h
'
m
d x ; w h e r e a = a
i
h
v ; '
a = I I I I I
Z
f
m
i
h
'
m
d x ( I a )
j a j j I I j + j I I I j +
Z
f
m
i
h
'
m
d x
+ j ( I a ) j
a
i
h
v ; '
C k v k
H ( G
R
)
k ' k
H ( G
R
)
+ k
1
k v k
H ( G
R
)
k ' k
H ( G
R
)
+ k f k
L
2
( G
R
)
k ' k
H ( G
R
)
+ k
4
k p k
L
2
( G
R
)
k ' k
H ( G
R
)
+ k
4
k p k
L
2
( G
R
)
i
h
0
@
g
m
j
)
@ '
m
@ x
j
1
A
L
2
( G
R
)
a
i
h
v ; '
k
5
k ' k
H ( G
R
)
h
k v k
H ( G
R
)
+ k p k
L
2
( G
R
)
+ k f k
L
2
( G
R
)
i
+ k
4
k p k
L
2
( G
R
)
i
h
0
@
g
m
j
@ '
m
@ x
j
1
A
L
2
( G
R
)
; f o r c o n s t a n t k
5
a
i
h
v ; '
k
6
h
k ' k
H ( G
R
)
n
k v k
H ( G
R
)
+ k f k
L
2
( G
R
)
+ k p k L
2
( G
R
)
o
+ k p k
L
2
( G
R
)
i
h
0
@
g
m
j
@ '
m
@ x
j
1
A
L
2
( G
R
)
3
7
5
; ; f o r c o n s t a n t k
6
3 2
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V a l i d f o r e v e r y ' 2 ( C
1
0
( G
R
) )
n
. N o w f o r e a c h x e d h , ( s m a l l )
i
h
v 2
H
1
0
( G
R
)
n
T h e r e e x i s t s a s e q u e n c e '
( )
2 ( C
1
0
( G
R
) )
n
c o n v e r g i n g t o
i
h
v i n
H
1
( G
R
n
. B u t
t h e n g
m
j
@ '
( )
m
@ x
j
c o n v e r g e s i n L
2
( G
R
) t o t h e c o r r e s p o n d i n g e x p r e s s i o n g
m
j
i
h
@ v
m
@ x
j
I t f o l l o w s t h a t
i
h
g
m
j
i
h
g
m
j
@ '
( )
m
@ x
j
c o n v e r g e s w e a k l y i n L
2
( G
R
) t o
i
h
g
m
j
i
h
@ v
m
@ x
j
T h e r e e x i s t ! 2 L
2
( G
R
) s u c h t h a t
! ;
i
h
g
m
j
@ '
( )
m
@ x
j
!
! ;
i
h
g
m
j
i
h
@ '
( )
m
@ x
j
0
@
! ;
i
h
g
m
j
@
@ x
j
'
( )
m
i
h
v
m
1
A
=
Z
G
R
!
i
h
0
@
g
m
j
@
@ x
j
'
( )
m
i
h
v
m
1
A
d x
B u t w e h a v e ,
!
i
h
0
@
g
m
j
@
@ x
j
'
( )
m
i
h
v
m
1
A
= ! ( x )
1
h
8
>
:
0
@
g
m
j
@
@ x
j
'
( )
m
i
h
v
m
1
A
( x h e
i
)
0
@
g
m
j
@
@ x
j
'
( )
m
i
h
v
m
1
A
( x )
9
>
=
>
;
=
1
h
8
>
:
! ( x )
0
@
g
m
j
@
@ x
j
'
( )
m
i
h
v
m
1
A
( x h e
i
)
! ( x )
0
@
g
m
j
@
@ x
j
'
( )
m
i
h
v
m
1
A
( x )
9
>
=
>
;
=
1
h
8
>
:
! ( y + h e
i
)
0
@
g
m
j
@
@ x
j
'
( )
m
i
h
v
m
1
A
( y )
! ( y )
0
@
g
m
j
@
@ x
j
'
( )
m
i
h
v
m
1
A
( y )
9
>
=
>
;
=
1
h
n
! ( y + h e
i
) ! ( y )
o
g
m
j
@
@ x
j
'
( )
m
i
h
v
m
=
i
h
( ! ) g
m
j
@
@ x
j
'
( )
m
i
h
v
m
Z
! ;
i
h
0
@
g
m
j
@
@ x
j
'
( )
m
i
h
v
m
1
A
d x =
Z
i
h
( ! ) g
m
j
@
@ x
j
'
( )
m
i
h
v
m
d x
T h e n w e o b t a i n
0
@
! ;
i
h
g
m
j
@
@ x
j
'
( )
m
i
h
v
m
1
A
=
0
@
i
h
! ; g
m
j
@
@ x
j
'
( )
m
i
h
v
m
1
A
3 3
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N o w t h i s l a s t e x p r e s s i o n c a n b e c o m p u t e d u s i n g ( 2 ) .
i
h
0
@
g
m
j
i
h
@ v
m
@ x
j
1
A
=
i
h
0
@
i
h
g
m
j
f @ v
m
@ x
j
i
h
( g
m
j
)
i
h
0
@
@ v
m
@ x
j
1
A
1
A
=
i
h
i
h
( )
i
h
0
@
i
h
( g
m
j
)
i
h
0
@
@ v
m
@ x
j
1
A
1
A
=
i
h
i
h
( )
i
h
g
m
j
i
h
i
h
0
@
@ v
m
@ x
j
1
A
i
h
i
h
( g
m
j
)
i
h
i
h
@ v
m
@ x
j
B u t ,
i
h
i
h
@ v
m
@ x
j
=
@ v
m
@ x
j
a n d
i
h
i
h
@ v
m
@ x
j
=
i
h
@ v
m
@ x
j
i
h
0
@
g
m
j
i
h
@ v
m
@ x
j
1
A
=
i
h
i
h
( )
i
h
i
h
g
m
j
@ v
m
@ x
j
i
h
g
m
j
i
h
0
@
@ v
m
@ x
j
1
A
i
h
0
@
g
m
j
i
h
0
@
@ v
m
@ x
j
1
A
1
A
L
2
( G
R
)
i
h
i
h
( )
L
2
( G
R
)
+
i
h
i
h
g
m
j
@ v
m
@ x
j
L
2
( G
R
)
+
i
h
g
m
j
i
h
0
@
@ v
m
@ x
j
1
A
L
2
( G
R
)
i
h
( )
H
1
( G
R
)
+
i
h
i
h
( g
m
j
)
k v k
H
1
( G
R
)
+
i
h
( g
m
j
)
i
h
v
H
1
( G
R
)
k
7
i
h
( )
H
1
( G
R
)
+ k v k
H
1
( G
R
)
+ k
i
h
v k
H
1
( G
R
)
i
;
f o r s o m e c o n s t a n t k
7
H e r e w e u s e d t h e f a c t t h a t g
m
j
i s C
2
, L e m m a ( 1 ) a n d t h e f a c t t h a t s u p p o r t s
o f v a n d a r e a c t u a l l y i n G
R
. S i n c e
i
h
g
m
j
@ '
( )
@ x
j
c o n v e r g e s w e a k l y i n
L
2
( G
R
) a s ! 1 t o
i
h
g
m
j
i
h
@ v
m
@ x
j
w e m a y a s s u m e , b y p a s s i n g t o a
s u b s e q u e n c e , i f n e c e s s a r y , t h a t t h e i r n o r m i n L
2
( G
R
) a r e u n i f o r m l y b o u n d e d
b y t h e r i g h t - h a n d s i d e o f ( 1 6 ) ( w i t h a l a r g e r c o n s t a n t ) .
i
h
0
@
g
m
j
@ '
( )
m
@ x
j
1
A
L
2
( G
R
)
=
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h
0
@
g
m
j
@ '
( )
m
@ x
j
1
A
i
h
0
@
g
m
j
i
h
@ v
m
@ x
j
1
A
+
i
h
0
@
g
m
j
i
h
@ v
m
@ x
j
1
A
L
2
( G
R
)
3 4
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i
h
0
@
g
m
j
@
@ x
j
'
( l )
m
i
h
v
m
1
A
L
2
( G
R
)
+
i
h
0
@
g
m
j
i
h
@ v
m
@ x
j
1
A
L
2
( G
R
)
i
h
0
@
a p g
m
j
@ '
( )
m
@ x
j
1
A
L
2
( G
R
)
i
h
0
@
g
m
j
i
h
@ v
m
@ x
j
1
A
L
2
( G
R
)
i
h
0
@
a p g
m
j
@ '
( )
m
@ x
j
1
A
L
2
( G
R
)
k
7
i
n
( )
H
1
( G
R
)
+ k v k
H
1
( G
R
)
+ k
i
h
v k
H
1
( G
R
)
R e a d i n g ( 1 5 ) f o r '
( )
a
i
h
v ; '
( )
k
6
'
( )
H
1
( G
R
)
n
k v k
H
1
( G
R
)
+ k p k
L
2
( G
R
)
+ k f k
L
2
( G
R
)
o
+ k p k
l
2
( G
R
)
i
h
0
@
g
m
j
@ '
( )
m
@ x
j
1
A
l
2
( G
R
)
3
7
5
a n d p a s s i n g t o l i m
! 1
s u p w e o b t a i n
a
i
h
v ;
i
h
v
k
6
i
h
v
H
1
( G
R
)
n
k v k
H
1
( G
R
)
+ k p k
L
2
( G
R
)
+ k f k
L
2
( G
R
)
+ k p k
L
2
( G
R
)
i
h
( )
H
1
( G
R
)
+ k v k
H
1
( G
R
)
+
i
h
v
H
1
( G
R
)
a
i
h
v ;
i
h
v
k
8
i
h
v
H
1
( G
R
)
n
k f k
L
2
( G
R
)
+ k v k
H
1
( G
R
)
+ k p k
L
2
( G
R
)
+ k p k
L
2
( G
R
)
k v k
H
1
( G
R
)
+
i
h
( )
H
1
( G
R
)
;
f o r s o m e c o n s t a n t k
8
. N o w f r o m t h e u n i f o r m e l l i p t i c i t y
a
i
h
v ;
i
h
v
1
M
r
i
h
v
2
L
2
( G
R
)
=
1
M
i
h
v
2
H
1
( G
R
)
1
M
i
h
v
2
L
2
( G
R
)
a n d t h u s w i t h L e m m a ( 1 )
a
i
h
v ;
i
h
v
1
M
i
h
v
2
H
1
( G
R
)
k
9
k v k
2
H
1
( G
R
)
; k
9
=
1
M
( 1 8 )
N o w w e u s e ( 1 7 ) , ( 1 8 ) a n d Y o u n g ' s i n e q u a l i t y t o d e d u c e
i
h
v
2
H
1
( G
R
)
M k
9
k v k
2
H
1
( G
R
)
+ M k
8
i
h
v
H
1
( G
R
)
n
k f k
L
2
( G
R
)
+ k p k
L
2
( G
R
)
+ k v k
H
1
( G
R
)
o
+ k p k
L
2
( G
R
)
k v k
H
1
( G
R
)
+
i
h
( )
H
1
( G
R
)
B u t ,
M k
8
i
h
v
H
1
k v k
H
1
=
2 M k
8
h
k v k
2
H
1
3 5
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M k
8
h
i
h
v
h
1
k p k
L
2
+ k v k
H
1
k p k
L
2
i
= M k
8
k p k
L
2
h
i
h
v
H
1
+ k v k
H
1
i
= M k
8
3
h
k p k
L
2
k v k
H
1
3
2 h
M k
8
k v k
2
H
1
+ k p k
2
L
2
M k
8
i
h
v
H
1
k f k
L
2
=
2
h
M k
8
k v k
H
1
k f k
L
2
1
h
M k
8
k v k
2
H
1
+ k f k
2
L
2
i
h
v
2
H ( G
R
)
k v k
2
H ( G
R
)
+
2 M k
8
h
k v k
2
H ( G
R
)
+
3
2 h
M k
8
k v k
2
H
1
( G
R
)
+ k p k
2
L
2
( G
R
)
+
M k
8
h
k v k
2
H
1
( G
R
)
+ k f k
2
L
2
( G
R
)
+ M k
8
k p k
2
L
2
( G
R
)
i
h
( )
H
1
( G
R
)
i
h
v
2
H ( G
R
)
k
1 0
k v k
2
H
1
( G
R
)
+ k p k
2
L
2
( G
R
)
+ k f k
2
L
2
( G
R
)
+ k p k
2
L
2
( G
R
)
i
h
( )
H
1
( G
