fonseca i., leoni g. - bulk and contact energies nucleation and relaxation(1999)(30).ps

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S I A M J . M A T H . A N A L .

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1 9 9 9 S o c i e t y f o r I n d u s t r i a l a n d A p p l i e d M a t h e m a t i c s

V o l . 3 0 , N o . 1 , p p . 1 9 0 { 2 1 9 , F e b r u a r y 1 9 9 9 0 0 2

B u l k a n d c o n t a c t e n e r g i e s :

N u c l e a t i o n a n d r e l a x a t i o n

I r e n e F o n s e c a

y

a n d G i o v a n n i L e o n i

z

A b s t r a c t . A n i n t e g r a l r e p r e s e n t a t i o n f o r m u l a i n B V ( ; R

p

) f o r t h e r e l a x a t i o n H ( u ; ) w i t h

r e s p e c t t o t h e L

1

t o p o l o g y o f f u n c t i o n a l s o f t h e g e n e r a l f o r m

H ( u ; ) : =

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h ( x ; u ( x ) ; r u ( x ) ) d x +

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) ;

i s o b t a i n e d . H e r e R

N

i s a n o p e n , b o u n d e d s e t o f c l a s s C

2

, T i s t h e t r a c e o p e r a t o r o n @ a n d

H

N 1

i s t h e N 1 d i m e n s i o n a l H a u s d o r m e a s u r e . T h e m a i n h y p o t h e s e s o n t h e f u n c t i o n s h a n d a r e

t h a t h ( x ; u ; ) i s q u a s i c o n v e x a n d h a s l i n e a r g r o w t h , a n d t h a t ( x ; ) i s L i p s c h i t z . T h e u n d e r s t a n d i n g

o f n u c l e a t i o n p h e n o m e n a f o r m a t e r i a l s u n d e r g o i n g p h a s e t r a n s i t i o n s l e a d s t o t h e s t u d y o f c o n s t r a i n e d

m i n i m i z a t i o n p r o b l e m s o f t h e t y p e

i n f

H ( u ; ) +

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;

w h e r e K i s a n o n e m p t y c o m p a c t s u b s e t o f R

p

, a n d : K ! R i s a c o n t i n u o u s f u n c t i o n . I t

i s s h o w n t h a t i f ( x ; ) i s a d o u b l e w e l l p o t e n t i a l v a n i s h i n g o n l y a t a n d , t h e n m i n i m i z e r s u o f

t h e t o t a l e n e r g y a r e g i v e n b y p u r e p h a s e s , t h a t i s , t h e r e e x i s t s

u

s u c h t h a t u ( x ) = f o r L

N

a . e . x 2

u

( l i q u i d ) a n d u ( x ) = f o r L

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a . e . x 2 n

u

( s o l i d ) . T h i s c o n c l u s i o n i s c l o s e l y r e l a t e d

t o r e s u l t s p r e v i o u s l y o b t a i n e d b y V i s i n t i n , a n d w h e r e t h e i n t e r f a c i a l e n e r g y i s a s s u m e d t o s a t i s f y a

g e n e r a l i z e d c o - a r e a f o r m u l a . H e r e t h i s c o n d i t i o n i s r e p l a c e d b y a p r o p e r t y w h i c h m a y b e v e r i e d b y

e n e r g i e s f o r w h i c h t h e c o - a r e a f o r m u l a m i g h t n o t h o l d .

K e y w o r d s . f u n c t i o n s o f b o u n d e d v a r i a t i o n , n u c l e a t i o n , r e l a x a t i o n , b u l k a n d c o n t a c t e n e r g i e s ,

g e n e r a l i z e d c o - a r e a f o r m u l a

A M S s u b j e c t c l a s s i c a t i o n s . 4 9 J 4 5 , 4 9 Q 2 0 , 4 9 N 6 0 , 7 3 T 0 5 , 7 3 V 3 0

1 . I n t r o d u c t i o n .

T h i s p a p e r i s d i v i d e d i n t o t w o p a r t s . I n t h e r s t p a r t w e o b t a i n a n i n t e g r a l

r e p r e s e n t a t i o n f o r m u l a i n B V ( ; R

p

) f o r t h e r e l a x a t i o n H ( u ; ) w i t h r e s p e c t t o t h e

L

1

t o p o l o g y o f f u n c t i o n a l s o f t h e g e n e r a l f o r m

( 1 . 1 )

H ( u ; ) : =

Z

h ( x ; u ( x ) ; r u ( x ) ) d x +

Z

@

( x ; T u ( x ) ) d H

N 1

( x ) ; u 2 W

1 ; 1

( ; R

p

) ;

w h e r e R

N


2

, T i s t h e t r a c e o p e r a t o r o n @

a n d H

N 1

i s t h e N 1 d i m e n s i o n a l H a u s d o r m e a s u r e . T h e m a i n h y p o t h e s e s o n t h e

f u n c t i o n s h a n d a r e t h a t h ( x ; u ; ) i s q u a s i c o n v e x a n d h a s l i n e a r g r o w t h , a n d t h a t

( x ; ) i s L i p s c h i t z .

y

D e p a r t m e n t o f M a t h e m a t i c a l S c i e n c e s , C a r n e g i e M e l l o n U n i v e r s i t y , P i t t s b u r g h , P A 1 5 2 1 3

z

C e n t e r f o r N o n l i n e a r A n a l y s i s , C a r n e g i e M e l l o n U n i v e r s i t y , P i t t s b u r g h , P A 1 5 2 1 3 a n d 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 P e r u g i a , P e r u g i a , I t a l y 0 6 1 2 3 .

1 9 0


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n u c l e a t i o n a n d r e l a x a t i o n 1 9 1

U n d e r a d e g e n e r a t e c o e r c i v i t y a s s u m p t i o n o n h ( x ; u ; ) w e o b t a i n t h e f o l l o w i n g

i n t e g r a l r e p r e s e n t a t i o n f o r u 2 B V ( ; R

p

)

( 1 . 2 )

H ( u ; ) =

Z

h ( x ; u ( x ) ; r u ( x ) ) d x +

Z

h

1

( x ; u ( x ) ; d C ( u ) )

+

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K

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( x ) ; u

+

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u

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( x )

+

Z

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( x ) ;

w h e r e r u i s t h e d e n s i t y o f t h e a b s o l u t e l y c o n t i n u o u s p a r t o f t h e d i s t r i b u t i o n a l d e r i v -

a t i v e D u w i t h r e s p e c t t o t h e N { d i m e n s i o n a l L e b e s g u e m e a s u r e L

N

, ( u

+

u

) i s t h e

j u m p a c r o s s t h e i n t e r f a c e S ( u ) , a n d C ( u ) i s t h e C a n t o r p a r t o f D u . F o r t h e c a n o n i c a l

m o d e l w h e r e h ( x ; u ; r u ) : = j r u j , > 0 , t h e r e l a x e d e n e r g y H ( u ; ) r e d u c e s t o

( 1 . 3 ) H ( u ; ) =

Z

j D u j +

Z

@

( x ; T u ) d H

N 1

; u 2 B V ( ; R

p

) :

I n t h e s c a l a r c a s e w h e r e p = 1 t h e l o w e r s e m i c o n t i n u i t y o f t h e f u n c t i o n a l ( 1 . 3 ) w a s

p r o v e d b y M a s s a r i a n d P e p e M P ] w h e n ( x ; u ) : = ^ j u j , w i t h j j , a n d b y M o d i c a

M o 2 ] u n d e r t h e a s s u m p t i o n t h a t

( 1 . 4 ) j ( x ; u ) ( x ; u

1

) j j u u

1

j

f o r a l l x

2@ a n d a l l u , u

1

2R .

O n e o f t h e m o t i v a t i o n s f o r t h e i n t r o d u c t i o n o f a r e l a x e d e n e r g y i s t h a t n o n c o n -

v e x v a r i a t i o n a l p r o b l e m s m a y n o t h a v e a m i n i m i z e r i n t h e s p a c e o f s m o o t h f u n c t i o n s ,

t h e r e f o r e i n o r d e r t o a p p l y t h e d i r e c t m e t h o d o f C a l c u l u s o f V a r i a t i o n s o n e h a s t o e x -

t e n d t h e o r i g i n a l f u n c t i o n a l . A l t h o u g h S o b o l e v s p a c e s a r e c o n s i d e r e d t o b e t h e n a t u r a l

e x t e n s i o n t o t h e s p a c e o f s m o o t h f u n c t i o n s , i n r e c e n t y e a r s t h e t h e o r y o f p h a s e t r a n s i -

t i o n s , a n d t h e n e e d t o d e t e r m i n e e e c t i v e e n e r g i e s f o r m a t e r i a l s e x h i b i t i n g i n s t a b i l i t i e s

s u c h a s f r a c t u r e s a n d d e f e c t s , h a v e l e d u s t o f u r t h e r e x t e n d t h e d o m a i n o f f u n c t i o n a l s

o f t h e f o r m ( 1 . 1 ) i n o r d e r t o i n c l u d e f u n c t i o n s u w h i c h p r e s e n t d i s c o n t i n u i t i e s a l o n g

s u r f a c e s . M o t i v a t e d s o m e w h a t b y L e b e s g u e ' s d e n i t i o n o f s u r f a c e a r e a , S e r r i n i n S e 1 ,

S e 2 ] p r o p o s e d t h e f o l l o w i n g n o t i o n f o r t h e r e l a x e d e n e r g y o f H ( u ; ) ( i n t h e c a s e w h e r e

0 )

H ( u ; ) : = i n f

f u

n

g

n

l i m i n f

n ! 1

H ( u

n

; ) : u

n

2 W

1 ; 1

( ; R

p

) ; u

n

! u i n L

1

( ; R

p

)

o

:

O n e o f t h e m a i n i s s u e s i n t h e C a l c u l u s o f V a r i a t i o n s c o n c e r n s t h e s e a r c h a n d c h a r a c -

t e r i z a t i o n o f a n i n t e g r a l r e p r e s e n t a t i o n f o r H ( u ; ) i n t h e s p a c e B V ( ; R

p

) .

I n t h e s c a l a r c a s e w h e r e p = 1 a n d h ( x ; u ;

) i s c o n v e x , t h e i n t e g r a l r e p r e s e n t a t i o n

( 1 . 2 ) w a s r s t o b t a i n e d b y G o m a n a n d S e r r i n G S e ] w h e n h = h ( r u ) ( s e e a l s o

R e ] ) , a n d b y G i a q u i n t a , G . M o d i c a a n d S o u c e k G M S ] f o r h = h ( x ;

ru ) . T h e s e

r e s u l t s w e r e t h e n e x t e n d e d b y D a l M a s o D M ] w h o c o n s i d e r e d t h e g e n e r a l c a s e w h e r e

h = h ( x ; u ; r u ) , a n d e m p h a s i z e d t h e i m p o r t a n t r o l e o f t h e c o e r c i v i t y c o n d i t i o n i n

e s t a b l i s h i n g ( 1 . 2 ) . I n d e e d , D a l M a s o s h o w e d t h a t , w h i l e ( 1 . 2 ) h o l d s f o r n o n n e g a t i v e

f u n c t i o n s h = h ( u ; r u ) w i t h o u t a n y l o w e r b o u n d o n h , w h e n h = h ( x ; r u ) , o r , m o r e

g e n e r a l l y , w h e n h = h ( x ; u ; r u ) , t h e r e p r e s e n t a t i o n ( 1 . 2 ) m a y f a i l u n l e s s o n e r e q u i r e s


3/30

1 9 2 i r e n e f o n s e c a a n d g i o v a n n i l e o n i

a w e a k c o e r c i v i t y a s s u m p t i o n o f t h e f o r m

( 1 . 5 ) h ( x ; u ; r u ) g ( x ; u ) j r u j :

I n t h e v e c t o r i a l c a s e w h e r e p > 1 a n d h ( x ; u ; ) i s q u a s i c o n v e x , A m b r o s i o a n d D a l M a s o

A D M 2 ] p r o v e d ( 1 . 2 ) w h e n h = h ( r u ) a n d w i t h o u t ( 1 . 5 ) . I n d e p e n d e n t l y , F o n s e c a a n d

M u l l e r F M 2 ] o b t a i n e d t h i s r e s u l t f o r g e n e r a l f u n c t i o n s h ( x ; u ; r u ) w h i c h v e r i f y ( 1 . 5 ) .

I n a l l t h e w o r k s m e n t i o n e d a b o v e 0 , a n d o n e o f t h e p u r p o s e s o f t h i s p a p e r

i s t o e x t e n d t h e s e r e s u l t s t o t h e n e w c a s e w h e r e p o s s i b l y 6 0 . T h e r e l a x a t i o n

o f f u n c t i o n a l s o f t h e t y p e ( 1 . 1 ) a r i s e s i n t h e v a n d e r W a a l s { C a h n { H i l l i a r d t h e o r y o f

p h a s e t r a n s i t i o n s f o r u i d s ( c f . C H 1 , C H 2 , v d W ] ) . I n t h i s c o n t e x t t h e b o u n d a r y t e r m

R

@

( x ; T u ) d H

N 1

r e p r e s e n t s t h e c o n t a c t e n e r g y b e t w e e n t h e u i d a n d t h e c o n t a i n e r

w a l l s , w h e r e ( x ; u ) i s t h e c o n t a c t e n e r g y p e r u n i t a r e a w h e n t h e d e n s i t y i s u ( s e e C ,

G ] ) .

W e p r e s e n t h e r e t w o r e l a x a t i o n r e s u l t s . I n T h e o r e m 3 . 5 w e s h o w t h a t , w i t h o u t

a n y a p r i o r i c o e r c i v i t y o n t h e f u n c t i o n h , t h e f u n c t i o n a l o n t h e r i g h t h a n d s i d e o f ( 1 . 2 )

a c t u a l l y g i v e s t h e i n t e g r a l r e p r e s e n t a t i o n f o r t h e f o l l o w i n g r e l a x e d e n e r g y

H

b

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j j u

n

j j

W

1 1


4/30


w h i c h i s e s s e n t i a l l y t h e c o n d i t i o n f o u n d b y M a s s a r i a n d P e p e M P ] a n d b y M o d i c a

M o 2 ] . W e n o t e t h a t o n c e ( 1 . 1 ) i s w r i t t e n i n t h e f o r m ( 1 . 6 ) , w e l l k n o w n i n t e g r a l

r e p r e s e n t a t i o n r e s u l t s f o r r e l a x a t i o n o f b u l k e n e r g y f u n c t i o n a l s a p p l y ( s e e F M 2 , F R ] ) .

T h i s i s i l l u s t r a t e d r s t o n S e c t i o n 2 i n t h e s c a l a r c a s e p = 1 a n d f o r a c l a s s o f e n e r g i e s

i n c l u d i n g t h e t y p i c a l o n e s c o n s i d e r e d b y V i s i n t i n ( s e e V 1 , V 2 ] ) ,

h ( x ; u ; ) : = j j ; ( x ; u ) : = ^ u :

M o r e g e n e r a l m o d e l s , r e l e v a n t t o t h e s t u d y o f a n i s o t r o p i c m a t e r i a l s a n d n o n l i n e a r

c o n t a c t f o r c e s , a r e s t u d i e d i n S e c t i o n 3 . I n p a r t i c u l a r , d e g e n e r a t e b o u n d s f o r t h e b u l k

e n e r g y d e n s i t y h , a s i n t r o d u c e d i n F M 2 ] i n t h e v e c t o r i a l c a s e , m a y p r o v e t o b e u s e f u l

i n t h e a n a l y s i s o f p h a s e t r a n s i t i o n p r o b l e m s .

I n t h e s e c o n d p a r t o f t h e p a p e r , S e c t i o n s 4 a n d 5 , w e a r e c o n c e r n e d w i t h c o n -

s t r a i n e d m i n i m i z a t i o n p r o b l e m s o f t h e t y p e

i n f

H ( u ; ) +

Z

( x ; u ( x ) ) d x : u 2 B V ( ; K )

;

w h e r e K i s a n o n e m p t y c o m p a c t s e t o f R

p

, a n d : K ! R i s a c o n t i n u o u s

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

t r a n s f o r m a t i o n s a n d i n n u c l e a t i o n p h e n o m e n a ( c f . V 1 , V 2 ] ) . A c c o r d i n g t o t h e v a n

d e r W a a l s { C a h n { H i l l i a r d t h e o r y o f p h a s e t r a n s i t i o n s ( c f . C H 1 , C H 2 , v d W ] ) , t h e t o t a l

e n e r g y o f a u i d o f t o t a l m a s s m a n d d e n s i t y u ( x ) c o n n e d i n a b o u n d e d c o n t a i n e r

R

N

, i s g i v e n b y

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E

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2

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w h e r e t h e c o a r s e { g r a i n e n e r g y W

1

( u ) i s a d o u b l e w e l l p o t e n t i a l v a n i s h i n g o n l y a t a n d

a n d c o r r e s p o n d i n g t o t h e s t a b l e t w o { p h a s e c o n g u r a t i o n o f t h e u i d , t h e g r a d i e n t

t e r m "

2

j r u j

2

m o d e l s t h e i n t e r f a c i a l e n e r g y a c r o s s a s m o o t h t r a n s i t i o n l a y e r , w i t h "

a s m a l l p a r a m e t e r , a n d W

2

r e p r e s e n t s t h e c o n t a c t e n e r g y b e t w e e n t h e u i d a n d t h e

c o n t a i n e r w a l l s . T h e s t a b l e c o n g u r a t i o n s o f t h e u i d c o r r e s p o n d t o s o l u t i o n s o f t h e

p r o b l e m ( s e e C ] )

i n f

E

"

( u ) : u 2 W

1 ; 1

( ; R ) ;

Z

u d x = m

:

C o n r m i n g a c o n j e c t u r e o f G u r t i n G ] , M o d i c a i n M o 2 ] w a s a b l e t o s h o w t h a t i f a

s e q u e n c e o f m i n i m i z e r s f u

"

g c o n v e r g e s i n L

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0

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s o l v e s t h e

l i q u i d { d r o p p r o b l e m

i n f f H ( u ; ) : u 2 B V ( ; f ; g ) g ;

w h e r e H ( u ; ) i s t h e r e l a x e d e n e r g y o f

H ( u ; ) =

Z

j r u j d x +

Z

@

T u d H

N 1

; u 2 W

1 ; 1

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H e r e ^ d e p e n d s o n l y o n W

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a n d W

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. T h e l i q u i d { d r o p p r o b l e m a d m i t s a s o l u t i o n i f

a n d o n l y i f j j 1 . A n a n a l o g o u s r e s u l t i s d u e t o A l b e r t i , B o u c h i t t e a n d S e p p e c h e r

A B S ] w h o r e c e n t l y s h o w e d t h a t i f t h e p a r a m e t e r " i n f r o n t o f t h e c o n t a c t e n e r g y i n


5/30


( 1 . 7 ) i s r e p l a c e d b y

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i s a d o u b l e w e l l p o t e n t i a l w h i c h v a n i s h e s o n l y a t

1

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, t h e n t h e l i m i t

p r o b l e m i s g i v e n b y a d i e r e n t m o d e l f o r c a p i l l a r i t y w i t h l i n e t e n s i o n . I t i s w o r t h

n o t i n g t h a t i n t h i s c a s e t h e e e c t i v e e n e r g y t a k e s t h e f o r m

H ( u ; ) =

Z

j D ( G ( u ) ) j

+ i n f

Z

@

j G ( T u ) G ( v ) j d H

N 1

+

K

Z

@

j D v j

2

: v 2 B V ( @ ; f

1

;

1

g )

f o r u 2 B V ( ; f ; g ) a n d H ( u ; ) = 1 o t h e r w i s e . H e r e G i s a p r i m i t i v e o f 2

p

W

1

. I t

c a n b e s e e n i m m e d i a t e l y t h a t i n t h i s c a p i l l a r i t y m o d e l t h e c o n t a c t e n e r g y i s n o n l o c a l

a n d s t r o n g l y n o n l i n e a r , a n d a g a i n t h i s l e a d s u s t o c o n s i d e r i n g f u n c t i o n s o t h e r t h a n

( x ; u ) = u ( s e e V 1 , V 2 ] a n d S e c t i o n 3 ) .

I n t h e l a s t s e c t i o n o f t h e p a p e r w e p r o v e s o m e m i n i m i z a t i o n r e s u l t s w h i c h a r e

r e l a t e d t o s o l i d n u c l e a t i o n . F o r a c o m p l e t e d e s c r i p t i o n o f t h i s p h e n o m e n o n w e r e f e r t o

t h e r e c e n t m o n o g r a p h o f V i s i n t i n V 1 ] a n d t o t h e b i b l i o g r a p h y c o n t a i n e d t h e r e i n . B y

s o l i d n u c l e a t i o n w e m e a n t h e f o r m a t i o n o f a n e w s o l i d p h a s e , t h a t i s o f a c o n n e c t e d

c o m p o n e n t o f s o l i d i n a l i q u i d . I f t h e n e w s o l i d p h a s e i s f o r m e d i n t h e i n t e r i o r o f t h e

l i q u i d , t h e n u c l e a t i o n i s c a l l e d h o m o g e n e o u s , w h i l e i f i t i s a l s o i n c o n t a c t w i t h o t h e r

s u b s t a n c e s , s u c h a s t h e c o n t a i n e r , i m p u r i t i e s d i s p e r s e d i n t h e l i q u i d o r n u c l e a n t s , t h e n

w e n a m e i t h e t e r o g e n e o u s n u c l e a t i o n ( c f . V 1 , C h . V I I . 2 ] ) . B y t h i n k i n g o f t h e s e

i m p u r i t i e s o r p a r t i c l e s a s h o l e s i n t h e d o m a i n , w e c a n r e p r e s e n t t h e c o n t a c t e n e r g y

b y a n i n t e g r a l t e r m o v e r t h e b o u n d a r y o f . F u r t h e r m o r e , s i n c e t h e n e w s o l i d p h a s e

i s f o r m e d t h r o u g h c r y s t a l l i z a t i o n , a n d c r y s t a l s a r e a n i s o t r o p i c , t h e c l a s s i c a l i s o t r o p i c

i n t e r f a c i a l e n e r g y

R

j D u j i s n o w r e p l a c e d b y

R

h ( x ; D u ) . I n t h e a p p l i c a t i o n s o n e

s e e s o f t e n h ( x ; D u ) = j A ( x ) D u j , w h e r e A ( x ) i s a n o n n e g a t i v e d e n i t e N N t e n s o r

( c f . V 1 , p . 1 5 7 ] ) .

T h e m a i n r e s u l t s o f t h i s p a r t a r e T h e o r e m s 5 . 1 a n d 5 . 4 , w h e r e w e s h o w t h a t

m i n i m i z e r s u o f t h e t o t a l e n e r g y a r e g i v e n b y p u r e p h a s e s , t h a t i s , t h e r e e x i s t s

u

s u c h t h a t u ( x ) = f o r L

N

a . e . x 2

u

( l i q u i d ) a n d u ( x ) = f o r L

N

a . e . x 2 n

u

( s o l i d ) . T h i s r e s u l t i s c l o s e l y r e l a t e d t o T h e o r e m 2 i n V 2 ] , w h e r e t h e i n t e r f a c i a l e n e r g y

i s a s s u m e d t o s a t i s f y a g e n e r a l i z e d c o - a r e a f o r m u l a . W e r e p l a c e h e r e t h i s c o n d i t i o n b y

s o m e h y p o t h e s e s w h i c h a r e e a s y t o v e r i f y a n d a l l o w u s t o i n c l u d e i n t e r f a c i a l e n e r g i e s

o f t h e f o r m

R

h ( x ; D u ) , w h e r e h ( x ; ) i s c o n v e x a n d p o s i t i v e l y h o m o g e n e o u s o f d e g r e e

o n e , a n d f o r w h i c h t h e c o - a r e a f o r m u l a m i g h t n o t h o l d .

2 . R e l a x a t i o n : A S i m p l e M o d e l P r o b l e m .

I n w h a t f o l l o w s R

N


2

, a n d T s t a n d s f o r

t h e t r a c e o p e r a t o r o n @ .

I n t h i s s e c t i o n w e c h a r a c t e r i z e t h e r e l a x e d e n e r g y H w h e n

( A

1

) h : R

N

! 0 ; + 1 ) i s a c o n t i n u o u s f u n c t i o n ;

( A

2

) h ( x ;

) i s c o n v e x f o r a l l x

2 ;

( A

3

) j j h ( x ; ) C ( 1 + j j ) ; f o r s o m e ; C > 0 , a n d f o r a l l ( x ; ) 2 R

N

;

( A

4

) ( x ; u ) : = ^ u ; f o r s o m e j j a n d f o r a l l ( x ; u ) 2 R .


6/30


T h e m a i n i d e a i s t o r e w r i t e

H ( u ; ) : =

Z

h ( x ; r u ( x ) ) d x +

Z

@

T u ( x ) d H

N 1

( x ) ; u 2 W

1 ; 1

( ; R

p

) ;

a s a b u l k e n e r g y a n d t h e n a p p l y k n o w n r e l a x a t i o n r e s u l t s f r o m W

1 ; 1

t o B V ( s e e , e . g . ,

D M , F M 2 ] ) .

W e s t a r t w i t h s o m e n o t a t i o n . F o r a n y 2 S

N 1

: = f x 2 R

N

: j x j = 1 g l e t

f

1

;

;

N 1

;

gb e a n o r t h o n o r m a l b a s i s o f R

N

v a r y i n g c o n t i n u o u s l y w i t h , a n d

Q

: = f x 2 R

N

: j x

i

j


7/30


a n d

l i m

r ! 0

1

r

N

Z

f y 2 B ( x

0

; r ) : ( y x )

u

( x ) 1 . F i n a l l y , D u m a y b e r e p r e s e n t e d a s

D u = r u L

N

+ ( u

+

u

)

u

H

N 1

b S ( u ) + C ( u ) ;

w h e r e r u i s t h e d e n s i t y o f t h e a b s o l u t e l y c o n t i n u o u s p a r t o f D u w i t h r e s p e c t t o t h e

N { d i m e n s i o n a l L e b e s g u e m e a s u r e L

N

. T h e s e t h r e e m e a s u r e s a r e m u t u a l l y s i n g u l a r .

L e m m a 2 . 1 . T h e r e e x i s t s ' 2 C

1

0

( R

N

; R

N

) w i t h j ' ( x ) j 0 ,

( 2 . 1 )

\U

j

=

fx

2U

j

: (

j

( x ) )

N


8/30


H e n c e

Z

@

T v ( x ) d H

N 1

( x ) =

X

2

Z

@

( x ) T v ( x ) d H

N 1

( x )

=

Z

v ( x ) d i v

0

@

X

2

'

( x )

1

A

d x +

Z

0

@

X

2

'

( x )

1

A

d ( D ( v ( x ) ) ) :

T h e p r o o f o f L e m m a 2 . 1 i s c o m p l e t e i f w e s h o w t h a t ' ( x ) : =

P

2

'

( x ) s a t i s e s

j ' ( x ) j 0 , w e m a y u s e t h e r e l a x a t i o n a r g u m e n t s i n t r o d u c e d i n F M 1 , F M 2 ] , t a k -

i n g i n t o a c c o u n t t h a t t h e r e l a x a t i o n m e t h o d i s l o c a l , s o t h a t o n e a c h b a l l B ( x

0

; " )


9/30


t h e r e i s a n u p p e r b o u n d j ' ( x ) j < 0 s u c h t h a t

( 3 . 1 ) g ( x ; u ) j j h ( x ; u ; ) C g ( x ; u ) ( 1 + j j )

f o r a l l ( x ; u ; ) 2 R

p

M

p N

, w h e r e M

p N

i s t h e v e c t o r s p a c e o f p N m a t r i c e s ;

( H

4

) f o r e v e r y x

0

2 a n d > 0 t h e r e e x i s t s " > 0 s u c h t h a t

( 3 . 2 ) h ( x

0

; u ; ) h ( x ; u ; ) ( 1 + g ( x ; u ) j j )

f o r a l l x 2 w i t h j x x

0

j " a n d f o r a l l ( u ; ) 2 R

p

M

p N

;

( H

5

) t h e r e e x i s t C

0

> 0 a n d m 2 ( 0 ; 1 ) s u c h t h a t

j h

1

( x ; u ; ) h ( x ; u ; ) j C

0

g ( x ; u ) ( 1 + j j

1 m

)

f o r a l l ( x ; u ; ) 2 R

p

M

p N

, w h e r e t h e r e c e s s i o n f u n c t i o n h

1

o f h i s d e n e d a s

h

1

( x ; u ; ) : = l i m s u p

t ! 1

h ( x ; u ; t )

t

;


10/30


( H

6

) a d m i t s a n e x t e n s i o n 2 C ( R

p

; R ) \ C

1

( R

p

; R ) s u c h t h a t

j r

x

( x ; u ) j a

1

( x ) + C

1

( 1 + j u j

q

c

)

f o r L

N

a . e . x 2 a n d a l l u 2 R

p

, w h e r e a

1

2 L

1

( ; R ) , C

1

> 0 a n d q

c

i s t h e S o b o l e v

e x p o n e n t q

c

: = N = ( N 1 ) i f N > 1 a n d q

c

0 t h e r e e x i s t s " > 0 s u c h t h a t

( 3 . 3 ) j r

u

( x

0

; u ) r

u

( x ; u ) j g ( x ; u )


0

j " a n d f o r a l l u 2 R

p

;

( H

7

) g ( x ; u ) j r

u

( x ; u ) j f o r a l l ( x ; u ) 2 R

p

.

R e m a r k 3 . 1 . ( i ) C o n d i t i o n s ( H

1

)

( H

5

) w e r e c o n s i d e r e d b y F o n s e c a a n d M u l l e r

( s e e F M 2 ] ) , w h o t r e a t e d t h e c a s e w h e r e 0 . A c t u a l l y , i n F M 2 ] h y p o t h e s i s ( H 4 )

i n c l u d e d :

( H 4 )

1

f o r e v e r y c o m p a c t s e t K b R

p

t h e r e e x i s t s a c o n t i n u o u s f u n c t i o n

! : 0 ; 1 ) ! 0 ; 1 ) , w i t h

! ( 0 ) = 0 , s u c h t h a t

j h ( x ; u ; ) h ( x

1

; u

1

; ) j ! ( j x x

1

j + j u u

1

j ) ( 1 + j j )

f o r a l l ( x ; u ; ) , ( x

1

; u

1

; ) 2 K M

p N

.

A s i t t u r n s o u t , t h i s p r o p e r t y f o l l o w s f r o m ( H 1 ) , ( H 2 ) , ( H 5 ) . I n d e e d u s i n g t h e f a c t

t h a t t h e r e c e s s i o n f u n c t i o n h

1

o f h i s s t i l l q u a s i c o n v e x a n d i s p o s i t i v e l y h o m o g e n e o u s

o f d e g r e e o n e i n t h e v a r i a b l e ( s e e F M 2 , M ] ) , b y ( H 1 ) , ( H 2 ) , ( H 5 ) , i t i s p o s s i b l e t o

s h o w t h a t h

1

i s a c t u a l l y c o n t i n u o u s i n R

p

M

p N

a n d , i n t u r n , t h a t h s a t i s e s

c o n d i t i o n ( H 4 )

1

o f F M 2 ] . W e o m i t t h e d e t a i l s .

( i i ) B y t h e M e a n V a l u e T h e o r e m a n d c o n d i t i o n s ( H

3

) a n d ( H

7

) w e h a v e

( 3 . 4 ) j ( x ; u ) ( x ; u

1

) j j r

u

( x ; u ) j j u u

1

j j j g j j

L

1

j u u

1

j

f o r a l l x 2 a n d a l l u , u

1

2 R

p

. T a k i n g u

1

= 0 i t f o l l o w s b y ( H

6

) a n d ( 3 . 4 ) t h a t

( 3 . 5 ) j ( x ; u ) j j j g j j

L

1

j u j + j j ( x ; 0 ) j j

L

1

f o r a l l ( x ; u ) 2 R

p

. T h i s g r o w t h c o n d i t i o n , t o g e t h e r w i t h ( 3 . 1 ) , i m p l i e s i n p a r t i c u l a r

t h a t t h e f u n c t i o n a l H ( u ; ) i s w e l l d e n e d a n d n i t e f o r u 2 W

1 ; 1

( ; R

p

) .

( i i i ) A t y p i c a l e x a m p l e o f t h e e n e r g y d e n s i t i e s i s ( s e e V i s i n t i n V 1 , V 2 ] , a n d

S e c t i o n 2 )

( 3 . 6 ) h ( x ; u ; ) : = j j ; ( x ; u ) : = ^ u ;

w h e r e > 0 a n d ^ 2 R . I t i s e a s y t o s e e t h a t c o n d i t i o n s ( H

1

) ( H

6

) h o l d w i t h

g ( x ; u ) : = , w h i l e a s s u m p t i o n ( H

7

) r e d u c e s t o t h e i n e q u a l i t y j j . M o r e g e n e r a l l y ,

( H

6

) i s t r i v i a l l y s a t i s e d i f = ( u ) .

( i v ) I f i n ( 1 . 3 ) w e t a k e ( x ; u ) : = ^ j u j f o r ( x ; u ) 2 @ R

p

( c f . M P ] ) , t h e n i t i s

p o s s i b l e t o e x t e n d t o R

p

a s f o l l o w s

( x ; u ) : = ^

p

j u j

2

+

2

( x ) ;

w h e r e 2 C

1

( ; R ) i s s u c h t h a t ( x ) > 0 f o r x 2 a n d ( x ) = 0 f o r x 2 @ .

C o n d i t i o n s ( H

1

) ( H

7

) a r e t h e n v e r i e d w i t h g ( x ; u ) : = , p r o v i d e d j j . T h e


11/30


p r o b l e m o f n d i n g a n e x t e n s i o n o f : @ R

p

! R t o R

p

w h i c h s a t i s e s ( H

6

) ( H

7

)

f o r t h e f u n c t i o n a l ( 1 . 3 ) , a n d w h e n ( 1 . 4 ) h o l d s , w i l l b e a d d r e s s e d i n a f o r t h c o m i n g p a p e r .

O u r g o a l i n t h i s s e c t i o n i s t o g e n e r a l i z e t h e i n t e g r a l r e p r e s e n t a t i o n o b t a i n e d i n

S e c t i o n 2 t o t h e r e l a x e d e n e r g y o f H ( u ; ) i n B V ( ; R

p

) w i t h r e s p e c t t o t h e L

1

t o p o l o g y , t h a t i s

H ( u ; ) : = i n f

f u

n

g

n

l i m i n f

n ! 1

H ( u

n

; ) : u

n

2 W

1 ; 1

( ; R

p

) ; u

n

! u i n L

1

( ; R

p

)

o

:

F r o m t h e d e n i t i o n o f H ( u ; ) i t f o l l o w s i m m e d i a t e l y t h a t t h e f u n c t i o n a l H ( u ; ) i s

l o w e r s e m i c o n t i n u o u s i n L

1

( ; R

p

) .

B e f o r e s t a t i n g t h e m a i n t h e o r e m s o f t h i s s e c t i o n w e i n t r o d u c e t h e s u r f a c e e n e r g y

a s s o c i a t e d t o t h e f u n c t i o n h . F o r x e d a , b

2R

p

w e d e n e

A( a ; b ; ) a s t h e c l a s s o f a l l

f u n c t i o n s 2 W

1 ; 1

( Q

; R

p

) s u c h t h a t

T ( y ) =

a i f y = 1 = 2

b i f y = 1 = 2

a n d w h i c h a r e p e r i o d i c o f p e r i o d o n e i n t h e r e m a i n i n g d i r e c t i o n s

1

;

;

N 1

. T h e

s u r f a c e e n e r g y a s s o c i a t e d t o t h e f u n c t i o n h , K

h

( x ; a ; b ; ) , i s d e n e d b y

K

h

( x ; a ; b ; ) : = i n f

Z

Q

h

1

( x ; ( y ) ; r ( y ) ) d y : 2 A ( a ; b ; )

:

F o r a d e t a i l e d s t u d y o f t h e p r o p e r t i e s o f t h e f u n c t i o n K

h

( x ; a ; b ; ) w e r e f e r t o F R ] .

F o r u 2 B V ( ; R

p

) w e d e n e t h e f u n c t i o n a l

L ( u ; ) : =

Z

h ( x ; u ( x ) ; r u ( x ) ) d x +

Z

h

1

( x ; u ( x ) ; d C ( u ) )

+

Z

S ( u ) \

K

h

( x ; u

( x ) ; u

+

( x ) ;

u

( x ) ) d H

N 1

( x ) +

Z

@

( x ; T u ( x ) ) d H

N 1

( x ) :

T h e o r e m 3 . 2 . L e t ( H

1

) ( H

7

) h o l d . I f u 2 B V ( ; R

p

) t h e n

H ( u ; ) = L ( u ; ) :

C o r o l l a r y 3 . 3 . I f h = h ( x ; ) t h e n

H ( u ; ) =

Z

h ( x ; r u ( x ) ) d x +

Z

h

1

( x ; d C ( u ) )

+

Z

S ( u ) \

h

1

( x ; ( u

+

( x ) u

( x ) )

u

( x ) ) d H

N 1

( x )

+

Z

@

( x ; T u ( x ) ) d H

N 1

( x ) :

T h e p r o o f o f C o r o l l a r y 3 . 3 f o l l o w s f r o m R e m a r k 2 . 1 7 i n F M 2 ] .

R e m a r k 3 . 4 . ( i ) R a t h e r s u r p r i s i n g l y , i n g e n e r a l t h e f u n c t i o n a l L ( u ; ) i s n o t


1

i f t h e d o m a i n i s o n l y L i p s c h i t z . T h i s f a c t w a s r s t

p o i n t e d o u t b y M o d i c a i n M o 2 ] w h o g a v e t h e f o l l o w i n g s i m p l e e x a m p l e . L e t : =

( 0 ; 1 ) ( 0 ; 1 ) R

2

a n d t a k e h a n d a s i n ( 3 . 6 ) , w i t h


12/30


( H

1

) { ( H

7

) a r e s a t i s e d ( s e e R e m a r k 3 . 1 ( i i i ) ) , a n d

L ( u ; ) =

Z

j D u j +

Z

@

T u d H

1

; u 2 B V ( ; R ) :

C o n s i d e r t h e s e q u e n c e

u

n

( x

1

; x

2

) : =

0 i f x

1

+ x

2

1 = n

n i f x

1

+ x

2


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D M ] t h a t H ( u ; ) = L ( u ; ) w h e n h = h ( u ; ) s a t i s e s o n l y ( 3 . 7 ) , w h i l e p o s s i b l y

H ( u ; ) 1 a n d w h e n 0 , A m b r o s i o a n d D a l M a s o

A D M 2 ] p r o v e d t h a t H ( u ; ) = L ( u ; ) w h e n h = h ( ) s a t i s e s o n l y ( 3 . 7 ) . I n d e -

p e n d e n t l y , F o n s e c a a n d M u l l e r F M 2 ] h a v e o b t a i n e d t h i s r e s u l t f o r g e n e r a l f u n c t i o n s

h ( x ; u ; ) w h i c h v e r i f y ( 3 . 1 ) , s t i l l i n t h e c a s e w h e r e 0 .

W e p r o c e e d w i t h t h e p r o o f s o f T h e o r e m s 3 . 2 a n d 3 . 5 . W e s t a r t w i t h s o m e p r e l i m -

i n a r y r e s u l t s . I n w h a t f o l l o w s , a n d u n l e s s o t h e r w i s e s p e c i e d , w e a l w a y s a s s u m e t h a t

c o n d i t i o n s ( H

1

) ( H

7

) h o l d .

L e m m a 3 . 7 . I f u 2 B V ( ; R

p

) t h e n t h e f u n c t i o n v ( x ) : = ( x ; u ( x ) ) 2 B V ( ; R )

a n d

D v =

r

x

( x ; u ) L

N

+ D u

T

r

u

( x ; u ) o n n S ( u )

( ( x ; u

+

) ( x ; u

) )

u

H

N 1

b S ( u ) o n S ( u ) :

M o r e o v e r

T v ( x ) = ( x ; T u ( x ) ) :

T h e p r o o f o f L e m m a 3 . 7 i s s t r a i g h t f o r w a r d i n l i g h t o f r e l a t e d r e s u l t s o n t h e c h a i n

r u l e f o r B V f u n c t i o n s ( s e e A D M 1 ] a n d t h e r e f e r e n c e s c o n t a i n e d t h e r e i n ) .

B y L e m m a s 2 . 1 a n d 3 . 7 , i f u 2 B V ( ; R

p

) t h e n

( 3 . 8 )

Z

@

( x ; T u ( x ) ) d H

N 1

( x ) =

Z

( x ; u ( x ) ) d i v ' ( x ) d x +

Z

' ( x ) d ( D ( ( x ; u ( x ) ) ) :

I n t u r n , b y ( 3 . 8 ) w e c a n r e w r i t e t h e f u n c t i o n a l H ( u ; ) a s

( 3 . 9 )

H ( u ; ) =

Z

h ( x ; u ( x ) ; r u ( x ) ) + ' ( x ) r u

T

( x ) r

u

( x ; u ( x ) )

d x

+

Z

( x ; u ( x ) ) d i v ' ( x ) d x +

Z

' ( x ) r

x

( x ; u ( x ) ) d x :

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

t w o e n e r g y i n t e g r a l s

R

h ( x ; u ; r u ) d x a n d

R

@

( x ; T u ) d H

N 1

. I n p a r t i c u l a r , i t i s

n o w c l e a r t h a t w i t h o u t a c o n d i t i o n o f t h e t y p e

h ( x ; u ; ) j r

u

( x ; u ) j j j

o n e m a y h a v e H ( u ; ) = 1 , a s i n t h e e x a m p l e i n R e m a r k 3 . 4 ( i i ) .

D e n e f ( x ; u ; ) : = h ( x ; u ; ) + ' ( x )

T

r

u

( x ; u ) f o r ( x ; u ; ) 2 R

p

M

p N

,

s e t

F ( u ; ) : =

Z

f ( x ; u ( x ) ; r u ( x ) ) d x ; u 2 W

1 ; 1

( ; R

p

) ;

a n d l e t

F ( u ; ) : = i n f

f u

n

g

n

l i m i n f

n ! 1

F ( u

n

; ) : u

n

2 W

1 ; 1

( ; R

p

) ; u

n

! u i n L

1

( ; R

p

)

o

:

L e m m a 3 . 8 . I f u 2 B V ( ; R

p

) t h e n

H ( u ; ) = F ( u ; ) +

Z

( x ; u ( x ) ) d i v ' ( x ) d x +

Z

' ( x ) r

x

( x ; u ( x ) ) d x :


14/30


P r o o f . C l e a r l y i t i s e n o u g h t o s h o w t h a t

l i m i n f

n ! 1

H ( u

n

; ) = l i m i n f

n ! 1

F ( u

n

; )

+

Z

( x ; u ( x ) ) d i v ' ( x ) d x +

Z

' ( x ) r

x

( x ; u ( x ) ) d x

f o r a n y s e q u e n c e f u

n

g W

1 ; 1

( ; R

p

) s u c h t h a t u

n

! u i n L

1

( ; R

p

) . W e r s t o b s e r v e

t h a t , s i n c e ' 2 C

1

0

( R

N

; R

N

) , t h e f u n c t i o n s ' a n d d i v ' a r e b o u n d e d i n . M o r e o v e r ,

b y ( 3 . 4 )

j ( x ; u

n

( x ) ) ( x ; u ( x ) ) j j j g j j

L

1

j u

n

( x ) u ( x ) j f o r L

N

a . e . x 2 .

H e n c e

l i m

n ! 1

Z

( x ; u

n

) d i v ' d x =

Z

( x ; u ) d i v ' d x :

B y ( H

6

) , b y v i r t u e o f t h e S o b o l e v i n e q u a l i t y , a n d d u e t o t h e f a c t t h a t ' i s b o u n d e d ,

t h e f u n c t i o n a l u 7!

R

' r

x

( x ; u ( x ) ) d x i s c o n t i n u o u s i n L

1

( ; R

p

) ( s e e K , T h . 2 . 1 ] )

a n d t h u s

l i m

n ! 1

Z

' r

x

( x ; u

n

) d x =

Z

' ( x ) r

x

( x ; u ) d x :

W e a r e n o w r e a d y t o p r o v e T h e o r e m 3 . 2 .

P r o o f o f T h e o r e m 3 . 2 . B y L e m m a 3 . 8 , i n o r d e r t o n d a n i n t e g r a l r e p r e s e n t a t i o n

f o r H ( u ; ) i n B V ( ; R

p

) i t i s s u c i e n t t o d e t e r m i n e o n e f o r F ( u ; ) . T h e i d e a i s t o

a p p l y T h e o r e m 2 . 1 6 o f F M 2 ] . I n o r d e r t o d o s o w e n e e d t o s h o w t h a t t h e f u n c t i o n

f ( x ; u ; ) = h ( x ; u ; ) + ' ( x )

T

r

u

( x ; u )

s a t i s e s c o n d i t i o n s ( H

1

)

( H

5

) w h i c h a r e e s s e n t i a l l y t h e s a m e o f F M 2 ] .

C o n d i t i o n ( H

1

) i s t r i v i a l l y v e r i e d s i n c e t h e f u n c t i o n s a n d ' a r e o f c l a s s C

1

.

A s f i s t h e s u m o f a q u a s i c o n v e x f u n c t i o n a n d a f u n c t i o n l i n e a r i n , i t i s c l e a r t h a t

f ( x ; u ; ) i s s t i l l q u a s i c o n v e x a n d t h a t

f

1

( x ; u ; ) = h

1

( x ; u ; ) + ' ( x )

T

r

u

( x ; u ) ;

w h i c h , i n t u r n , i m p l i e s t h a t

j f

1

( x ; u ; ) f ( x ; u ; ) j = j h

1

( x ; u ; ) h ( x ; u ; ) j C

0

g ( x ; u ) ( 1 + j j

1 m

)

b y ( H

5

) . T h u s f v e r i e s a l s o ( H

2

) a n d ( H

5

) .

W e p r o v e ( 3 . 2 ) . F i x x

0

2 a n d > 0 . T h e r e e x i s t s " > 0 s u c h t h a t f o r x 2

w i t h j x x

0

j " a n d ( u ; ) 2 R

p

M

p N

h ( x

0

; u ; ) h ( x ; u ; )

1

3

( 1 + g ( x ; u ) j j ) ; j ' ( x ) ' ( x

0

) j

1

3

;

j r

u

( x

0

; u ) r

u

( x ; u ) j

1

3

g ( x ; u )


15/30


b y t h e c o n t i n u i t y o f ' , ( 3 . 2 ) a n d ( 3 . 3 ) . H e n c e

f ( x

0

; u ; ) = h ( x

0

; u ; ) + ' ( x

0

)

T

r

u

( x

0

; u )

= f ( x ; u ; ) + h ( x

0

; u ; ) h ( x ; u ; )

+ ' ( x

0

) ' ( x ) ]

T

r

u

( x ; u ) + ' ( x

0

)

T

r

u

( x

0

; u ) r

u

( x ; u ) ]

f ( x ; u ; ) +

1

3

( 1 + g ( x ; u ) j j ) +

2

3

g ( x ; u ) j j

w h i c h i s ( 3 . 2 ) , a n d w h e r e w e h a v e u s e d ( H 7 ) a n d t h e f a c t t h a t j ' ( x

0

) j 1 .

F i n a l l y , c o n d i t i o n ( H

3

) i s r e p l a c e d b y t h e c o n d i t i o n

( 3 . 1 0 ) g ( x ; u ) j j ( 1 j ' ( x ) j ) f ( x ; u ; ) 2 C g ( x ; u ) ( 1 + j j )

w h i c h f o l l o w s f r o m ( 3 . 1 ) a n d ( H

7

) . A l t h o u g h ( 3 . 1 0 ) i s w e a k e r t h a n c o n d i t i o n ( H

3

) i n

F M 2 ] , t h e p r o o f t h e r e c a r r i e s o u t e v e n w i t h ( 3 . 1 0 ) . I n d e e d , c o n d i t i o n ( H

3

) w a s u s e d

i n F M 2 ] o n l y t o s h o w t h a t

F ( u ; )

Z

f ( x ; u ; r u ) d x +

Z

f

1

( x ; u ; d C ( u ) )

+

Z

S ( u ) \

K

f

( x ; u

; u

+

;

u

) d H

N 1

( x ) :

T h e p r o o f o f t h i s i n e q u a l i t y r e l i e s o n t h e b l o w { u p a r g u m e n t i n t r o d u c e d i n F M 1 ]

w h i c h i s a l o c a l a r g u m e n t , i n t h e s e n s e t h a t i n o r d e r t o p r o v e t h e t h r e e m a i n p o i n t w i s e

i n e q u a l i t i e s ( 2 . 1 0 ) { ( 2 . 1 1 ) i n F M 2 ] a t p o i n t s x

0

2 , o n e i s o n l y i n t e r e s t e d i n w h a t

h a p p e n s i n a b a l l B ( x

0

; " ) . S i n c e i n o u r c a s e j ' ( x

0

) j <

0

0 , i f w e

t a k e " s u c i e n t l y s m a l l w e c a n a s s u m e t h a t j ' ( x ) j

0

f o r a l l x 2 B ( x

0

; " ) a n d t h u s

( 3 . 1 0 ) r e d u c e s t o

g ( x ; u ) j j ( 1

0

) f ( x ; u ; ) 2 C g ( x ; u ) ( 1 + j j )

f o r a l l ( x ; u ; ) 2 B ( x

0

; " ) R

p

M

p N

, w h i c h i s t h e l o c a l v e r s i o n o f ( H

3

) i n F M 2 ] .

I n c o n c l u s i o n , w e m a y a p p l y T h e o r e m 2 . 1 6 o f F M 2 ] ( s e e R e m a r k 3 . 1 0 b e l o w ) t o

o b t a i n t h a t f o r u 2 B V ( ; R

p

)

F ( u ; ) =

Z

h ( x ; u ; r u ) + ' r u

T

r

u

( x ; u )

d x +

Z

h

1

( x ; u ; d C ( u ) )

+

Z

( ' r

u

( x ; u ) ) d C

T

( u ) +

Z

S ( u ) \

K

f

( x ; u

; u

+

;

u

) d H

N 1

;

w h e r e

K

f

( x ; a ; b ; ) = i n f

Z

Q

h

1

( x ; ( y ) ; r ( y ) ) + ' ( x ) r

T

( y ) r

u

( x ; ( y ) )

d y :

2 A ( a ; b ; )

:

G i v e n a n y 2 A ( a ; b ; ) w e h a v e

Z

Q

' ( x ) r

T

( y ) r

u

( x ; ( y ) ) d y =

Z

@ Q

' ( x ) n ( y ; Q

) ( x ; T ( y ) ) d H

N 1

( y )

= ' ( x ) ( ( x ; b ) ( x ; a ) ) ;


16/30


a n d s o

K

f

( x ; a ; b ; ) = K

h

( x ; a ; b ; ) + ' ( x ) ( ( x ; b ) ( x ; a ) ) :

I f w e n o w u s e L e m m a s 2 . 1 , 3 . 7 , a n d 3 . 8 , w e n a l l y o b t a i n t h a t H ( u ; ) = L ( u ; ) .

T h i s c o n c l u d e s t h e p r o o f o f T h e o r e m 3 . 2 .

R e m a r k 3 . 9 . T h e c o n t i n u i t y h y p o t h e s e s ( H

1

) , ( H

4

) a n d ( 3 . 3 ) m a y b e r e p l a c e d

b y

( H

1

)

0

h i s C a r a t h e o d o r y ;

( H

4

)

0

f o r a l l ( x

0

; u

0

) 2 R

p

a n d f o r a l l > 0 t h e r e e x i s t s " > 0 s u c h t h a t

j h ( x ; u

1

; ) h ( x ; u

2

; ) j " ( 1 + j j )


0

j " , u

1

, u

2

2 B ( u

0

; " ) a n d 2 M

p N

;

p r o v i d e d

7!f ( x ; u ;

) i s c o e r c i v e ( e . g . i f g ( x ; u )

>

j j r

u

j j

L

1

f o r s o m e > 0 ) .

I n t h i s c a s e , i n T h e o r e m 3 . 2 w e w o u l d u s e t h e i n t e g r a l r e p r e s e n t a t i o n o b t a i n e d b y

B o u c h i t t e , F o n s e c a a n d M a s c a r e n h a s B F M ] i n p l a c e o f t h e c o r r e s p o n d i n g r e s u l t b y

F o n s e c a a n d M u l l e r F M 2 ] .

P r o o f o f T h e o r e m 3 . 5 . B y L e m m a 3 . 8 i t i s e n o u g h t o n d a n i n t e g r a l r e p r e s e n t a -

t i o n f o r t h e c o r r e s p o n d i n g F

b

( u ; ) i n B V ( ; R

p

) . L e t f

"

( x ; u ; ) : = f ( x ; u ; ) + " j j ,

f o r " 2 ( 0 ; 1 ) , w h e r e , a s b e f o r e , f ( x ; u ; ) : = h ( x ; u ; ) + ' ( x )

T

r

u

( x ; u ) , a n d d e n e

F

"

( u ; ) : = i n f

f u

n

g

n

l i m i n f

n ! 1

F

"

( u

n

; ) : u

n

2 W

1 ; 1

( ; R

p

) ; u

n

! u i n L

1

( ; R

p

)

o

;

w h e r e

F

"

( u ; ) : =

Z

f

"

( x ; u ( x ) ; r u ( x ) ) d x ; u 2 W

1 ; 1

( ; R

p

) :

W e c l a i m t h a t

l i m

" ! 0

F

"

( u ; ) = F

b

( u ; ) :

F i x u 2 L

1

( ; R

p

) . F o r a n y g i v e n > 0 t h e r e e x i s t s a s e q u e n c e f u

n

g W

1 ; 1

( ; R

p

) ,

w i t h s u p

n

j j u

n

j j

W

1 1

= M


17/30


C o n v e r s e l y , x u 2 L

1

( ; R

p

) a n d " > 0 . T h e n t h e r e e x i s t s a s e q u e n c e f u

"

n

g

W

1 ; 1

( ; R

p

) s u c h t h a t u

"

n

! u i n L

1

( ; R

p

) a s n ! 1 a n d

( 3 . 1 1 )

F

"

( u ; ) + " l i m

n ! 1

Z

f ( x ; u

"

n

; r u

"

n

) + " j r u

"

n

j ] d x l i m i n f

n ! 1

Z

f ( x ; u

"

n

; r u

"

n

) d x :

W i t h o u t l o s s o f g e n e r a l i t y w e c a n a s s u m e t h a t F

"

( u ; )


18/30


f o r a l l ( x ; u ; ) 2 R

p

M

p N

, w h i c h i s s t r o n g e r t h a n c o n d i t i o n ( H

3

) . T o c i r c u m v e n t

t h i s d i c u l t y c o n s i d e r t h e f u n c t i o n f

"

( x ; u ; ) d e n e d a s i n t h e p r o o f o f T h e o r e m 3 . 5 .

S i n c e i t s a t i s e s c o n d i t i o n s ( H

1

) ( H

5

) a n d ( 3 . 1 5 ) , t h e i n e q u a l i t y ( 3 . 1 4 ) h o l d s f o r f

"

.

A l s o t h e i n e q u a l i t y f f

"

c l e a r l y i m p l i e s t h a t

F ( u ; S ( u ) \ ) F

"

( u ; S ( u ) \ )

Z

S ( u ) \

K

f

"

( x ; u

; u

+

;

u

) d H

N 1

( x ) : >

I f w e n o w l e t " ! 0 a n d u s e ( 3 . 1 3 ) , w e c o n c l u d e t h a t ( 3 . 1 4 ) h o l d s a l s o f o r f .

4 . M e s o s c o p i c s c a l e .

W e a r e i n t e r e s t e d i n t h e f o l l o w i n g c o n s t r a i n e d m i n i m i z a t i o n p r o b l e m

i n f

H ( u ; ) +

Z

( x ; u ( x ) ) d x : u 2 B V ( ; R

p

) ; u ( x ) 2 K f o r L

N

a . e . x 2

;

w h e r e K i s a n o n e m p t y c o m p a c t s e t o f R

p

, a n d : K ! R i s a C a r a t h e o d o r y

f u n c t i o n s u c h t h a t

( 4 . 1 )

j ( x ; u )

j a

0

( x ) f o r

L

N

a . e . x

2 a n d f o r a l l u

2K ,

f o r s o m e f u n c t i o n a

0

2 L

1

( ; R ) . I n a p p l i c a t i o n s i n p h a s e t r a n s i t i o n s , o f t e n K = f a ; b g

o r K i s a c o n v e x s e t .

F o r u 2 L

1

( ; R

p

) w e d e n e t h e f u n c t i o n a l

I ( u ; ) : = H ( u ; ) + I

K

( u ; ) ;

w h e r e

I

K

( u ; ) : =

R

( x ; u ( x ) ) d x i f u ( x ) 2 K f o r L

N

a . e . x 2

+ 1 o t h e r w i s e :

L e m m a 4 . 1 . I f ( H

1

) ( H

7

) h o l d t h e n t h e f u n c t i o n a l I ( u ; ) i s l o w e r s e m i c o n -

t i n u o u s i n L

1

( ; R

p

) .

P r o o f . C o n s i d e r u

n

, u 2 L

1

( ; R

p

) s u c h t h a t u

n

! u i n L

1

( ; R

p

) .

I f l i m i n f

n ! 1

I ( u

n

; ) = 1 t h e r e i s n o t h i n g t o p r o v e . A s s u m e t h a t l i m i n f

n ! 1

I ( u

n

; )


19/30


( F

1

) t h e r e e x i s t a f u n c t i o n 2 C ( R

p

; R

p

) \ C

1

( R

p

; R

p

) a n d a f u n c t i o n b 2

L

1

( ; R ) s u c h t h a t

( 4 . 2 ) j r

x

( x ; u ) j b ( x ) f o r L

N

a . e . x 2 a n d f o r a l l u 2 K ,

a n d

( 4 . 3 ) h

1

( x ; u ; ) j r

u

( x ; u ) j j j

f o r a l l ( x ; u ; ) 2 K M

p N

;

( F

2

) f o r L

N

a . e . x 2 t h e f u n c t i o n ( x ; ) : K b R

p

! ( x ; K ) b R

p

i s i n v e r t i b l e

a n d ( x ; y ) 7! ( ( x ; ) )

1

( y ) i s C a r a t h e o d o r y . I n a d d i t i o n , t h e r e e x i s t s a f u n c t i o n c 2

L

1

( ; R ) s u c h t h a t

( 4 . 4 ) j ( x ; )

1

( v ) j c ( x ) f o r L

N

a . e . x 2 a n d f o r a l l v 2 ( x ; K ) :

L e t

D ( I ) : = f u 2 L

1

( ; R

p

) : I ( u ; )


20/30


( 3 . 5 ) w e h a v e

( 4 . 6 )

Z

h ( x ; u

n

; r u

n

) d x +

Z

h

1

( x ; u

n

; d C ( u

n

) )

+

Z

S ( u

n

) \

K

h

( x ; u

n

; u

+

n

;

u

n

) d H

N 1

M

1

;

f o r s o m e c o n s t a n t M

1

i n d e p e n d e n t o f n . B y ( H

5

) , ( 3 . 1 )

1

a n d ( 4 . 6 )

Z

h

1

( x ; u

n

; r u

n

) d x

Z

( h

1

( x ; u

n

; r u

n

) h ( x ; u

n

; r u

n

) ) d x + M

1

C

0

j j g j j

L

1

+ C

0

j j g j j

m

L

1

Z

h

1 m

( x ; u

n

; r u

n

) d x + M

1

:

U s i n g H o l d e r ' s i n e q u a l i t y a n d ( 4 . 6 ) a g a i n , w e c o n c l u d e t h a t t h e r e e x i s t s M

2

2 ( 0 ; 1 )

s u c h t h a t f o r a l l n

( 4 . 7 )

Z

h

1

( x ; u

n

; r u

n

) d x +

Z

h

1

( x ; u

n

; d C ( u

n

) )

+

Z

S ( u

n

) \

K

h

( x ; u

n

; u

+

n

;

u

n

) d H

N 1

M

2

:

D e n e v

n

: = ( x ; u

n

( x ) ) . A s i n L e m m a 3 . 7 , w e c a n s h o w t h a t v

n

( x ) 2 B V ( ; R

p

)

w i t h

( 4 . 8 ) D v

n

=

r

x

( x ; u

n

) L

N

+ r

u

( x ; u

n

) D u

n

o n n S ( u

n

)

( ( x ; u

+

n

) ( x ; u

n

) )

u

n

H

N 1

b S ( u

n

) o n S ( u

n

) :

F u r t h e r m o r e

( 4 . 9 )

Z

j D v

n

j =

Z

j r v

n

j d x +

Z

j r

u

( x ; u

n

) d C ( u

n

) j

+

Z

S ( v

n

) \

j ( v

+

n

v

n

)

v

n

j d H

N 1

:

B y R e m a r k 2 . 1 7 i n F M 2 ] a n d t h e f a c t t h a t S ( v

n

) = S ( u

n

) a n d

v

n

=

u

n

, w e c a n

r e w r i t e t h e l a s t i n t e g r a l a s

Z

S ( u

n

) \

K

j j

( x ; v

n

; v

+

n

;

u

n

) d H

N 1

w h e r e

K

j j

( x ; v

n

; v

+

n

;

u

n

) : = i n f

(

Z

Q

u

n

j r ( y ) j d y : 2 A ( v

n

( x ) ; v

+

n

( x ) ;

u

n

)

)

:

G i v e n 2 A ( u

n

( x ) ; u

+

n

( x ) ;

u

n

) , i t i s c l e a r t h a t t h e f u n c t i o n ( y ) : = ( x ; ( y ) ) b e l o n g s

t o A ( v

n

( x ) ; v

+

n

( x ) ;

u

n

) a n d t h a t r ( y ) = r

u

( x ; ( y ) ) r ( y ) . B y ( 4 . 3 ) t h i s i m p l i e s


21/30


t h a t

K

j j

( x ; v

n

; v

+

n

;

u

n

)

i n f

(

Z

Q

u

n

j r

u

( x ; ( y ) ) j j r ( y ) j d y : 2 A ( u

n

( x ) ; u

+

n

( x ) ;

u

n

)

)

K

h

( x ; u

n

; u

+

n

;

u

n

) :

T h e r e f o r e , a l s o b y ( 4 . 2 ) , ( 4 . 3 ) , ( 4 . 7 ) , ( 4 . 8 ) a n d ( 4 . 9 )

( 4 . 1 0 )

Z

jD v

n

j

Z

j r

x

( x ; u

n

)

jd x +

Z

j r

u

( x ; u

n

)

j j ru

n

jd x

+

Z

j r

u

( x ; u

n

) j j d C ( u

n

) j

+

Z

S ( u

n

) \

K

h

( x ; u

n

; u

+

n

;

u

n

) d H

N 1

j j b j j

L

1

+ M

2

:

F i n a l l y , s i n c e v

n

( x ) 2 ( x ; K ) f o r L

N

a . e . x 2 , ( x ; K ) i s a c o m p a c t s e t o f R

p

,

a n d b y ( 4 . 1 0 ) , t h e r e e x i s t s a s u b s e q u e n c e , s t i l l d e n o t e d f v

n

g , w h i c h c o n v e r g e s s t r o n g l y

i n L

1

( ; R

p

) a n d p o i n t w i s e a l m o s t e v e r y w h e r e t o a f u n c t i o n v 2 B V ( ; R

p

) ( s e e Z ,

C o r . 5 . 3 . 4 ] ) , w i t h v ( x ) 2 ( x ; K ) f o r L

N

a . e . x 2 . D e n e u ( x ) : = ( ( x ; ) )

1

( v ( x ) ) .

B y ( F

2

) t h e f u n c t i o n u i s m e a s u r a b l e . S i n c e u

n

( x ) = ( ( x ; ) )

1

( v

n

( x ) ) i t f o l l o w s t h a t

u

n

( x ) ! u ( x ) f o r L

N

a . e . x 2 , t h u s u ( x ) 2 K f o r L

N

a . e . x 2 . M o r e o v e r , b y

( 4 . 4 ) w e h a v e t h a t

ju

n

( x )

j c ( x ) f o r

L

N

a . e . x

2 , t h e r e f o r e b y L e b e s g u e D o m i n a t e d

C o n v e r g e n c e T h e o r e m u

n

! u s t r o n g l y i n L

1

( ; R

p

) . B y L e m m a 4 . 1 w e c o n c l u d e t h a t

I ( u ; ) i n f f I ( u ; ) : u 2 D

1

g :

C o r o l l a r y 4 . 3 . A s s u m e t h a t c o n d i t i o n s ( F

1

) a n d ( F

2

) i n T h e o r e m 4 . 2 a r e r e -

p l a c e d b y t h e a s s u m p t i o n

h

1

( x ; u ; ) j j f o r a l l ( x ; u ; ) 2 K M

p N

,

f o r s o m e > 0 . T h e n D

1

= D ( I ) a n d t h e r e e x i s t s a f u n c t i o n u 2 D

1

s u c h t h a t

I ( u ; ) = i n f f I ( w ; ) : w 2 D

1

g :

P r o o f . I t s u c e s t o t a k e ( x ; u ) : = u i n T h e o r e m 4 . 2 . T h e n v

n

= u

n

c o n v e r g e

s t r o n g l y i n L

1

( ; R

p

) t o a f u n c t i o n v 2 B V ( ; R

p

) , a n d t h e r e f o r e u : =

1

v i s t h e

d e s i r e d m i n i m i z e r .

5 . N u c l e a t i o n : t h e s c a l a r c a s e .

I n t h i s s e c t i o n w e s t u d y t h e c o n s t r a i n e d m i n i m i z a t i o n p r o b l e m i n t r o d u c e d i n

S e c t i o n 4 , r e s t r i c t e d t o t h e s c a l a r c a s e p = 1 , w h e n K i s a c l o s e d , c o n n e c t e d s u b s e t o f

R ( n o t n e c e s s a r i l y b o u n d e d ) , a n d w h e n t h e p o t e n t i a l ( x ; u ) i s g i v e n b y

( x ; u ) : =

1

( x ; u ) + ( x )

2

( u ) ;

w h e r e

1

( x ; u ) i s a C a r a t h e o d o r y f u n c t i o n , c o n c a v e i n t h e u v a r i a b l e , i s a n o n n e g a -

t i v e , m e a s u r a b l e f u n c t i o n , a n d

2

i s a c o n t i n u o u s f u n c t i o n s u c h t h a t

( 5 . 1 )

a l l t h e c o n n e c t e d c o m p o n e n t s o f S : = f u 2 i n t K :

2

( u ) <

2

( u ) g a r e b o u n d e d ;


22/30


w h e r e

2

i s t h e c o n v e x e n v e l o p e o f

2

. A s r e m a r k e d i n V 2 ] , ( 5 . 1 ) h o l d s i f

l i m s u p

j u j ! 1

2

( u )

j u j

= 1 :

F u r t h e r m o r e w e a s s u m e t h a t

( 5 . 2 ) ( x ; u ) L

1

L

2

j u j f o r L

N

a . e . x 2 a n d f o r a l l u 2 K ,

f o r s o m e L

1

, L

2

> 0 .

U n d e r a p p r o p r i a t e a s s u m p t i o n s o n t h e f u n c t i o n s h a n d , w e p r o v e t h a t m i n i m i z -

e r s u 2 L

1

( ; R

p

) o f

I : v 2 L

1

( ; K ) 7! H ( v ; ) +

Z

( x ; v ( x ) ) d x

h a v e t h e p h a s e s t r u c t u r e

u ( ) K n S :

I n p a r t i c u l a r , i f K = a ; b ] , i f

2

i s c o n c a v e i n a ; b ] , a n d i f u i s a m i n i m i z e r o f I , t h e n u

m u s t h a v e a 2 - p h a s e s t r u c t u r e , i . e . t h e r e e x i s t s a s e t

0

s u c h t h a t u ( x ) = a f o r L

N

a . e . x 2

0

a n d u ( x ) = b f o r L

N

a . e . x 2 n

0

. T h i s r e s u l t h a s i m p o r t a n t a p p l i c a t i o n s

i n n u c l e a t i o n p h e n o m e n a w h i c h h a v e b e e n s t u d i e d e x t e n s i v e l y b y V i s i n t i n i n V 1 , V 2 ] ,

w h e r e u s u a l l y K i s b o u n d e d ,

1

( x ; u ) : = ( x ) u , 2 L

1

( ; R ) i s p r o p o r t i o n a l t o t h e

r e l a t i v e t e m p e r a t u r e , a n d ( x )

2

( u ) i s t h e d o u b l e w e l l p o t e n t i a l ( x ) ( b u ) ( u a )

( s e e R e m a r k 5 . 2 b e l o w ) . G i v e n a s i m p l e f u n c t i o n u 2 L

1

( ; K ) o f t h e f o r m

( 5 . 3 ) u ( x ) =

k

X

i = 1

c

i

!

i

( x ) ;

w i t h c

i

2 K , L

N

( !

i

) > 0 f o r a l l i = 1 ; ; k , a n d L

N

( n

k

i = 1

!

i

) = 0 , w i t h o u t l o s s o f

g e n e r a l i t y w e m a y a s s u m e t h a t

( 5 . 4 ) i n f K c

1

< c

2


23/30


( i i i ) T h e f u n c t i o n c 7! V ( c ) i s c o n c a v e i n K .

I n a d d i t i o n , s u p p o s e t h a t t h e f u n c t i o n a l u 7!

R

( x ; u ( x ) ) d x i s c o n t i n u o u s i n

D ( V ) \ L

1

( ; K ) . T h e n

i n f f V ( u ) + I

K

( u ; ) : u 2 D ( V ) g = i n f

V ( u ) + I

K

( u ; ) : u 2 D ( V ) ; u ( x ) 2 K n S

f o r

L

N

a . e . x

2

:

R e m a r k 5 . 2 . ( i ) T h e f u n c t i o n a l V ( u ) + I

K

( u ; ) i s w e l l d e n e d b y ( 5 . 2 ) .

( i i ) T h e o r e m 5 . 1 i s c l o s e l y r e l a t e d t o T h e o r e m 2 i n V 2 ] , w h e r e K = R a n d

c o n d i t i o n s ( i ) a n d ( i i ) a r e r e p l a c e d b y t h e a s s u m p t i o n t h a t V s a t i s e s t h e g e n e r a l i z e d

c o - a r e a f o r m u l a

( 5 . 6 ) V ( u ) =

Z

R

V (

f x 2 : u ( x ) t g

) d t :

I t i s e a s y t o s e e t h a t ( 5 . 6 ) r e d u c e s t o ( 5 . 5 ) f o r f u n c t i o n s u o f t h e f o r m ( 5 . 3 ) - ( 5 . 4 ) .

T h e r e f o r e ( 5 . 5 ) i s w e a k e r t h a n ( 5 . 6 ) . O n t h e o t h e r h a n d , c o n d i t i o n s ( i ) a n d ( 5 . 6 ) d o

n o t s e e m t o b e r e l a t e d . I n d e e d , c o n s i d e r t h e f u n c t i o n a l

V ( u ) : =

Z

j D u j +

R

m a x f u ( x ) ; 0 g d x i f H

N 1

( S ( u ) \ ) = 0

0 o t h e r w i s e :

F r o m t h e p r o o f o f T h e o r e m 5 . 4 b e l o w i t f o l l o w s t h a t V s a t i s e s h y p o t h e s e s ( i ) | ( i i i ) o f

T h e o r e m 5 . 1 . T a k e u ( x ) : = 1 i n ( 5 . 6 ) ; t h e n V ( 1 ) = L

N

( ) , w h i l e t h e r i g h t h a n d s i d e

o f ( 5 . 6 ) i s i n n i t e . T h e r e f o r e ( 5 . 6 ) f a i l s . W e n o t e t h a t V i s n o t l o w e r s e m i c o n t i n u o u s

i n L

1

.

W e r e m a r k t h a t T h e o r e m 5 . 1 m a y b e a p p l i e d t o a l a r g e c l a s s o f f u n c t i o n a l s o f t h e

f o r m ( 1 . 1 ) , f o r w h i c h t h e c o - a r e a f o r m u l a m i g h t n o t h o l d .

( i i i ) I f , i n a d d i t i o n t o h y p o t h e s e s ( i ) { ( i i i ) i n T h e o r e m 5 . 1 , w e a s s u m e t h a t V i s


1

( ; R ) , t h a t K = R , a n d t h a t t h e r e e x i s t a s e t ! 2 E w i t h

0 0 , a n d "

n

: =

1

n

o t h e r w i s e . C l e a r l y u

n

! c

i n L

1

( ; R ) , t h e r e f o r e b y t h e l o w e r s e m i c o n t i n u i t y o f V a n d ( 5 . 5 )

0 V ( c ) l i m i n f

n ! 1

V ( u

n

) = l i m

n ! 1

"

n

V (

!

) = 0 ;

w h e r e w e h a v e u s e d t h e f a c t t h a t V (

!

) f r o m

h y p o t h e s e s ( i ) a n d ( i i ) o f T h e o r e m 5 . 1 a n d f r o m t h e l o w e r s e m i c o n t i n u i t y o f V .


24/30


I n o r d e r t o s h o w 4 ) , x u 2 D ( V ) . B y ( i ) t h e r e e x i s t s a s e q u e n c e f u

n

g S

1

c o n v e r g i n g t o u i n L

1

( ; R ) a n d L

N

a . e . x 2 s u c h t h a t

V ( u ) l i m

n ! 1

V ( u

n

) = l i m

n ! 1

Z

R

V (

f x 2 : u

n

( x ) t g

) d t

Z

R

l i m i n f

n ! 1

V (

f x 2 : u

n

( x ) t g

) d t

b y ( 5 . 5 ) a n d F a t o u ' s L e m m a . S i n c e L

N

( f x 2 : u ( x ) = t g ) = 0 f o r a l l t 2 R n M ,

w h e r e L

1

( M ) = 0 , w e x t 2 R n M a n d t a k e a s u b s e q u e n c e f u

n

k

g o f f u

n

g s u c h t h a t

l i m i n f

n ! 1

V (

f x 2 : u

n

( x ) t g

) = l i m

k ! 1

V (

f x 2 : u

n

k

( x ) t g

) :

T h e n f

f x 2 : u

n

k

( x ) t g

g c o n v e r g e s p o i n t w i s e t o

f x 2 : u ( x ) t g

f o r L

N

a . e . x 2 a n d ,

b y L e b e s g u e D o m i n a t e d C o n v e r g e n c e T h e o r e m , a l s o s t r o n g l y i n L

1

( ; R ) . T h e r e f o r e

b y t h e l o w e r s e m i c o n t i n u i t y o f V

l i m i n f

n ! 1

V (

f x 2 : u

n

( x ) t g

) V (

f x 2 : u ( x ) t g

)

f o r L

1

a . e . t 2 R , a n d w e c o n c l u d e t h a t

Z

R

l i m i n f

n ! 1

V (

f x 2 : u

n

( x ) t g

) d t

Z

R

V (

f x 2 : u ( x ) t g

) d t :

W e d o n o t k n o w i f t h e r e v e r s e d i n e q u a l i t y o f 4 ) h o l d s , i . e . i f t h e c o - a r e a f o r m u l a

( 5 . 6 ) i s s a t i s e d .

L e t

: = i n f

V ( u ) + I

K

( u ; ) : u 2 D ( V ) ; u ( x ) 2 K n S f o r L

N

a . e . x 2

:

L e m m a 5 . 3 . I f u 2 S

1

t h e n

V ( u ) + I

K

( u ; ) :

P r o o f . A s I

K

( u ; ) =

1f o r u =

2L

1

( ; K ) i t s u c e s t o p r o v e t h e r e s u l t f o r

u 2 S

1

\ L

1

( ; K ) . B y ( 5 . 1 ) w e c a n d e c o m p o s e t h e o p e n s e t S a s a d i s j o i n t u n i o n o f

b o u n d e d i n t e r v a l s

S =

r 2 R

( a

r

; b

r

) :

F o l l o w i n g V i s i n t i n V 2 ] w e r e p l a c e t h e f u n c t i o n

2

b y

~

2

( u ) : =

8


25/30


c

i

a

r

f o r a l l i < l ( t h e c a s e s w h e r e l = 1 o r l = k c a n b e t r e a t e d a n a l o g o u s l y ) .

D e n e t h e f u n c t i o n

( t ) : =

k 1

X

i = 1 ; i 6= l 1 ; l

i + 1

( c

i + 1

c

i

) +

l

( t c

l 1

) +

l + 1

( c

l + 1

t ) +

k

X

i = 1 ; i 6= l

i

( c

i

) +

l

( t )

f o r t 2 a

r

; d ] , w h e r e d : = c

l + 1

i f c

l + 1

b

r

a n d d : = b

r

i f c

l + 1

> b

r

. S i n c e

2

( u ) ~

2

( u )

b y c o n s t r u c t i o n , t h e n c l e a r l y V ( u ) + I

K

( u ; ) V ( u ) +

~

I

K

( u ; ) = ( c

l

) . O b s e r v e

t h a t s i n c e ~ ( x ; ) =

1

( x ; ) + ( x ) ~

2

( ) i s c o n c a v e i n a

r

; d ] , t h e n t h e f u n c t i o n

l

( )

i s a l s o c o n c a v e i n a

r

; d ] , a n d ( t ) , b e i n g t h e s u m o f a l i n e a r f u n c t i o n a n d a c o n c a v e

f u n c t i o n , a t t a i n s i t s m i n i m u m a t o n e o f t h e e n d p o i n t s Q o f a

r

; d ] . I t f o l l o w s t h a t

V( u ) + I

K

( u ; )

V( u ) +

~

I

K

( u ; ) = ( c

l

)

( Q ) =

V( u ) +

~

I

K

( u ; )

w h e r e

u ( x ) : =

8

>

>

>

:

P

k

i = 1 ; i 6= l

c

i

!

i

( x ) + a

r

!

l

( x ) i f Q = a

r

P

k

i = 1 ; i 6= l

c

i

!

i

( x ) + b

r

!

l

( x ) i f Q = b

r

P

k

i = 1 ; i 6= l ; l + 1

c

i

!

i

( x ) + c

l + 1

!

l

!

l + 1

( x ) i f Q = c

l + 1

:

I f k = 1 , n a m e l y i f u ( x ) c , t h e n

( 5 . 7 ) V ( u ) + I

K

( u ; ) V ( u ) +

~

I

K

( u ; ) = V ( c ) +

Z

~ ( x ; c ) d x :

A s s u m e t h a t c 2 ( a

r

; b

r

) f o r s o m e r 2 R . S i n c e b y ( i i i ) a n d b y t h e c o n s t r u c t i o n o f ~

2

t h e r i g h t h a n d s i d e o f ( 5 . 7 ) i s a c o n c a v e f u n c t i o n o f c 2 a

r

; b

r

] , i t s i n m u m i s a t t a i n e d

a t o n e o f t h e e n d p o i n t s , s a y a t b

r

, a n d t h u s w e c a n r e p l a c e u ( x ) b y u ( x ) : = b

r

2 K n S .

W e c o n c l u d e t h a t i t i s e n e r g e t i c a l l y p o s s i b l e t o r e d u c e a t l e a s t b y o n e t h e n u m b e r

o f v a l u e s c

i

b e t w e e n a

r

a n d b

r

. R e p e a t i n g t h i s p r o c e d u r e f o r t h e n i t e n u m b e r o f

i n t e r v a l s ( a

r

; b

r

) w h i c h c o n t a i n a t l e a s t o n e o f t h e c

i

, b y m e a n s o f a n i t e i n d u c t i o n

a r g u m e n t w e c a n c o n s t r u c t a s i m p l e f u n c t i o n ^ u o f t h e f o r m

u ( x ) =

k

X

i = 1

c

i

!

i

( x ) ;

w h e r e

k k , s u c h t h a t ^ u ( ) K n ( a

r

; b

r

) f o r a n y r 2 R a n d V ( u ) + I

K

( u ; )

V( u ) +

~

I

K

( u ; ) . S i n c e

2

( u ) = ~

2

( u ) f o r u

2K

nS , i t f o l l o w s t h a t I

K

( u ; ) =

~

I

K

( u ; )

a n d t h u s V ( u ) + I

K

( u ; ) V ( u ) + I

K

( u ; ) . T h i s c o n c l u d e s t h e p r o o f o f t h e

l e m m a .

P r o o f o f T h e o r e m 5 . 1 . L e t u 2 D ( V ) \ L

1

( ; K ) . B y ( i ) t h e r e e x i s t s a s e q u e n c e

f u

n

g S

1

\ L

1

( ; K ) c o n v e r g i n g t o u i n L

1

( ; K ) s u c h t h a t

l i m s u p

n ! 1

V( u

n

)

V( u ) :

M o r e o v e r , b y h y p o t h e s i s

l i m

n ! 1

Z

( x ; u

n

( x ) ) d x =

Z

( x ; u ( x ) ) d x ;


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a n d s i n c e b y L e m m a 5 . 3 V ( u

n

) + I

K

( u

n

; ) , i t f o l l o w s t h a t

V ( u ) + I

K

( u ; ) ;

a n d w e c o n c l u d e t h a t

i n f f V ( u ) + I

K

( u ; ) : u 2 D ( V ) g :

T h e r e v e r s e d i n e q u a l i t y i s t r i v i a l l y s a t i s e d .

I n o r d e r t o a p p l y T h e o r e m 5 . 1 t o f u n c t i o n a l s o f t h e f o r m ( 1 . 1 ) , w e c o n s i d e r t h e

s p e c i a l c a s e w h e r e

h = h ( x ; ) i s p o s i t i v e l y h o m o g e n e o u s o f d e g r e e o n e i n ; ( x ; u ) : = ^ u ; 6= 0 ;

a n d h y p o t h e s e s ( A

1

) ( A

7

) i n S e c t i o n 2 a r e s a t i s e d . C l e a r l y h ( x ; ) = h

1

( x ; ) , a n d

b y T h e o r e m 2 . 2 ( s e e a l s o C o r o l l a r y 3 . 3 , ( 3 . 9 ) , a n d L e m m a 3 . 8 ) , f o r u 2 B V ( ; R ) w e

h a v e

H ( u ; ) =

Z

f ( x ; r u ) d x +

Z

f ( x ; d C ( u ) )

+

Z

S ( u ) \

f ( x ; ( u

+

u

)

u

) d H

N 1

+

Z

u d i v ' d x ;

w h e r e , w e r e c a l l ,

f ( x ; ) = h ( x ; ) + ' ( x )

f o r a l l ( x ; )

2

R

N

:

F u r t h e r m o r e , w e a s s u m e t h a t t h e p o t e n t i a l a l s o s a t i s e s t h e g r o w t h c o n d i t i o n

( 5 . 8 ) ( x ; u ) b

1

( x ) + M

1

( 1 + j u j

q

c

) f o r L

N

a . e . x 2 a n d a l l u 2 K ,

w h e r e b

1

2 L

1

( ; R ) , M

1

> 0 a n d , a s b e f o r e , q

c

i s t h e S o b o l e v e x p o n e n t q

c

: =

N = ( N

1 ) i f N > 1 a n d q

c


27/30


T h e m a i n r e s u l t o f t h e s e c t i o n i s t h e f o l l o w i n g t h e o r e m

T h e o r e m 5 . 4 . I f ( H

1

) ( H

7

) a n d ( 5 : 8 ) a r e v e r i e d t h e n

i n f f I ( u ; ) : u 2 B V ( ; K ) g = i n f f I ( u ; ) : u 2 B V ( ; K n S ) g :

R e m a r k 5 . 5 . T h e o r e m 5 . 4 n o l o n g e r h o l d s i n g e n e r a l i f ( x ; ) i s n o n l i n e a r .

I n d e e d , c o n s i d e r t h e s i m p l e c a s e w h e r e : = ( c ; d ) , K : =

1 ; 1 ] ,

H ( u ; ) : =

Z

j D u ( x ) j

Z

@

s i n ( T u ( x ) ) d H

N 1

( x ) ; u 2 B V ( ; R )

a n d ( x ; u ) : = a ( 1 u

2

) . H e r e K n S = f 1 ; 1 g , a n d i f > t h e n a l l c o n d i t i o n s

( H

1

) ( H

7

) a r e s a t i s e d . L e t u 2 B V ( ; R ) , w i t h u ( x ) 2 f 1 ; 1 g f o r L

N

a . e . x 2 .

T h e n I ( u ; ) =

R

j D u ( x ) j 0 . O n t h e o t h e r h a n d , i f w e t a k e u ( x )

1

2

t h e n

I ( u ; ) = 2 +

3

4

a ( d c )


28/30


L e t u

n

( x ) : =

P

k

n

i = 1

d

n ; i

!

n i

( x ) , w h e r e

d

n ; i

=

8

>

:

s u p K i f c

n ; i

s u p K

c

n ; i

i f i n f K < c

n ; i


29/30


a n d f o r i 2 f 2 ; ; k 1 g

( 5 . 1 2 )

i 1

X

j = 1

Z

@ !

i

\ @ !

j

\

f ( x ;

i

) d H

N 1

k

X

l = i + 1

Z

@ !

l

\ @ !

i

\

f ( x ;

l

) d H

N 1

=

Z

@ (

k

l = i

!

l

) \

f ( x ;

i

) d H

N 1

Z

@ (

k

l = i + 1

!

l

) \

f ( x ;

i + 1

) d H

N 1

;

w h e r e ^

i

a n d ^

i + 1

a r e , r e s p e c t i v e l y , t h e o u t w a r d u n i t n o r m a l s t o t h e s e t s

k

l = i

!

l

a n d

k

l = i + 1

!

l

. I t n o w f o l l o w s f r o m ( 5 . 1 1 ) { ( 5 . 1 2 ) t h a t

V ( u ) = c

k

V (

!

k

) c

1

V (

k

l = 2

!

l

) +

k 1

X

i = 2

c

i

V (

k

l = i

!

l

)

k 1

X

i = 2

c

i

V (

k

l = i + 1

!

l

) ;

w h i c h i s ( i i ) .

F i n a l l y , b y ( 5 . 2 ) , ( 5 . 8 ) a n d t h e S o b o l e v i n e q u a l i t y , t h e f u n c t i o n a l u 7!

R

( x ; u ( x ) ) d x

i s c o n t i n u o u s i n B V ( ; K ) ( s e e K , T h . 2 . 1 ] ) . T h e r e f o r e w e c a n n o w a p p l y T h e o r e m

5 . 1 ( w i t h ( x ; u ) r e p l a c e d b y ( x ; u ) + u d i v ' ( x ) ) t o o b t a i n t h a t

i n f f I ( u ; ) : u 2 B V ( ; K ) g = i n f f I ( u ; ) : u 2 B V ( ; K n S ) g :

C o r o l l a r y 5 . 7 . A s s u m e t h a t K = a ; b ] i n T h e o r e m 5 . 4 . T h e n t h e r e e x i s t s a

f u n c t i o n u 2 B V ( ; a ; b ] n S ) s u c h t h a t

I ( u ; ) = i n f f I ( u ; ) : u 2 B V ( ; a ; b ] ) g :

P r o o f . B y T h e o r e m 5 . 4

i n f f I ( u ; ) : u 2 B V ( ; a ; b ] ) g = i n f f I ( u ; ) : u 2 B V ( ; a ; b ] n S ) g = :

T o c o m p l e t e t h e p r o o f i t s u c e s t o a p p l y C o r o l l a r y 4 . 3 , w i t h K : = a ; b ]

nS , t o n d

u 2 B V ( ; a ; b ] n S ) s u c h t h a t I ( u ; ) = .

R e m a r k 5 . 8 . I f w e a s s u m e t h a t

2

i s c o n c a v e i n a ; b ] , t h e n S = ( a ; b ) a n d

c o n s e q u e n t l y t h e m i n i m i z e r u i n C o r o l l a r y 5 . 7 h a s t h e p r o p e r t y t h a t u ( x ) 2 f a ; b g f o r

L

N

a . e . x 2 .

A c k n o w l e d g m e n t s . P a r t o f t h i s r e s e a r c h w a s u n d e r t a k e n d u r i n g G . L e o n i ' s v i s i t t o

t h e C e n t e r f o r N o n l i n e a r A n a l y s i s u n d e r t h e C N A / C M U { C N R p r o j e c t . T h e r e s e a r c h

o f I . F o n s e c a w a s p a r t i a l l y s u p p o r t e d b y t h e N a t i o n a l S c i e n c e F o u n d a t i o n t h r o u g h t h e

C e n t e r f o r N o n l i n e a r A n a l y s i s , a n d b y t h e N a t i o n a l S c i e n c e F o u n d a t i o n u n d e r g r a n t

N o . D M S { 9 5 0 0 5 3 1 . T h e a u t h o r s a r e d e e p l y i n d e b t e d t o L u c T a r t a r f o r m a n y f r u i t f u l

a n d s t i m u l a t i n g d i s c u s s i o n s i n t h e s u b j e c t o f t h i s w o r k .

R E F E R E N C E S

A B S ] A l b e r t i G . , G . B o u c h i t t

e a n d P . S e p p e c h e r , P h a s e t r a n s i t i o n w i t h l i n e t e n s i o n e e c t ,

s u b m i t t e d t o A r c h . R a t . M e c h . A n a l .

A D M 1 ] A m b r o s i o L . a n d G . D a l M a s o , A g e n e r a l c h a i n r u l e f o r d i s t r i b u t i o n a l d e r i v a t e s , P r o c .

A m e r . M a t h . S o c . , 1 0 8 ( 1 9 8 8 ) , p p . 6 9 1 - 7 0 2 .

A D M 2 ] A m b r o s i o L . a n d G . D a l M a s o , O n t h e r e l a x a t i o n i n B V ( ; R

m

) o f q u a s i { c o n v e x i n t e -

g r a l s , J . F u n c t . A n a l . , 1 0 9 ( 1 9 9 2 ) , p p . 7 6 - 9 7 .


30/30


A M T ] A m b r o s i o L . , S . M o r t o l a a n d V . M . T o r t o r e l l i , F u n c t i o n a l w i t h l i n e a r g r o w t h d e n e d

o n v e c t o r { v a l u e d B V f u n c t i o n s , J . M a t h . P u r e s e t A p p l . , 7 0 ( 1 9 9 1 ) , p p . 2 6 9 - 3 2 2 .

B F M ] B o u c h i t t

e , G . I . F o n s e c a a n d L . M a s c a r e n h a s , A g l o b a l m e t h o d f o r r e l a x a t i o n , s u b -

m i t t e d t o A r c h . R a t . M e c h . A n a l .

C ] C a h n J . W . , C r i t i c a l p o i n t w e t t i n g , J . C h e m . P h y s . , 6 6 ( 1 9 7 7 ) , p p . 3 6 6 7 - 3 6 7 2 .

C H 1 ] C a h n J . W . a n d J . E . H i l l i a r d , F r e e e n e r g y o f a n o n { u n i f o r m s y s t e m I : i n t e r f a c i a l

e n e r g y , J . C h e m . P h y s . , 2 8 ( 1 9 5 8 ) , p p . 2 5 8 - 2 6 7 .

C H 2 ] C a h n J . W . a n d J . E . H i l l i a r d , F r e e e n e r g y o f a n o n { u n i f o r m s y s t e m , J . C h e m . P h y s . ,

2 8 ( 1 9 5 9 ) , p p . 6 8 8 - 6 9 9 .

D M ] D a l M a s o G . , I n t e g r a l r e p r e s e n t a t i o n o n B V ( ) o f { l i m i t s o f v a r i a t i o n a l i n t e g r a l s ,

M a n u s c r i p t a M a t h . , 3 0 ( 1 9 8 0 ) , p p . 3 8 7 - 4 1 6 .

E G ] E v a n s L . C . a n d R . F . G a r i e p y , M e a s u r e T h e o r y a n d F i n e P r o p e r t i e s o f F u n c t i o n s , C R C

P r e s s , 1 9 9 2 .

F ] F o n s e c a I . , L o w e r s e m i c o n t i n u i t y o f s u r f a c e e n e r g i e s , P r o c . R . S o c . E d i n . , 1 2 0 A ( 1 9 9 2 ) ,

p p . 9 9 - 1 1 5 .

F M 1 ] F o n s e c a I . a n d S . M

u l l e r , Q u a s i { c o n v e x i n t e g r a n d s a n d l o w e r s e m i c o n t i n u i t y i n L

1

,

S I A M J . M a t h . A n a l . , 2 3 ( 1 9 9 2 ) , p p . 1 0 8 1 - 1 0 9 8 .

F M 2 ] F o n s e c a I . a n d S . M

u l l e r , R e l a x a t i o n o f q u a s i c o n v e x f u n c t i o n a l s i n B V ( ; R

p

) f o r i n -

t e g r a n d s f ( x ; u ; r u ) , A r c h . R a t . M e c h . A n a l . , 1 2 3 ( 1 9 9 3 ) , p p . 1 - 4 9 .

F R ] F o n s e c a I . a n d P . R y b k a , R e l a x a t i o n o f m u l t i p l e i n t e g r a l s i n t h e s p a c e B V ( ; R

p

) , P r o c .

R o y . S o c . E d i n b . S e c t . A , 1 2 1 ( 1 9 9 2 ) , p p . 3 2 1 - 3 4 8 .

G S e ] G o f f m a n C . a n d J . S e r r i n , S u b l i n e a r f u n c t i o n s o f m e a s u r e s a n d v a r i a t i o n a l i n t e g r a l s ,

D u k e M a t h . J . , 3 1 ( 1 9 6 4 ) , p p . 1 5 9 - 1 7 8 .

fonseca i., leoni g. - bulk and contact energies nucleation and relaxation(1999)(30).ps

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