AppI Math Optim 27:i05-123 (1993) pplied Mathematics and Optimization 9 1993 Springer-Verlag New York Inc. Asymptotic Development by F Convergence Gabriele Anzellotti and Sisto Baldo Dipartimento di Matematica, Universitfi Degli Studi di Trento, 38050 Povo (Trento), Italy Communicated by D. Kinderlehner Abstract. A description of the asym ptotic development of a family of mini- mum problems is proposed by a suitable iteration of F-limit procedures. An example of asymptotic development for a family of functionals related to phase transformations is also given. Key Words. F-convergence, Asymptotic developments, V functions. AMS Classification. Primary 49J45, Secondary 41A60. O. Introduction It is very common, both in pure and applied mathematics, to have to deal with a family of problems depending on a parameter e > 0, being interested in the asymptotic behavior of the problems as e-, 0. Typically, in the calculus of variations we are given a family of minimum problems min{~(u): u ~ X} (0.1) and we are interested in finding a limit problem min{~-(u): u e X}. (0.2) Of course, from the point of view of the calculus of variations, the limit problem must be such that its minimizers are closely related to the possible limit points of A notion of convergence for functionals, which is very well suited to the variational setting, is the well-known F-convergence, introduced by De Giorgi [DF]. In fact, if a functional ~ is the F-limit of the ~ and if u s are minimizers of ~ and u s ~ u, then u is a minim izer of ~ (see Section 1 for a precise statement).
Transcript
8/11/2019 Anzellotti - Baldo - Asymptotic Development by Gamma Convergence
AppI Math Opt im 27: i05-123 (1993) pplied M athem aticsand Optimization9 1993 Springer-VerlagNew York Inc.
Asympto t ic Deve lopment by F Convergence
G a b r i e l e A n z e l l o t t i a n d S i st oBaldo
Dipart imento di Matematica, Universi t f i Degli Studi di Trento,38050 Povo (Trento), Italy
Communica ted by D. Kinder lehner
Abstract. A d e s c r i p t i o n o f t h e a s y m p t o t i c d e v e l o p m e n t o f a f a m i l y o f m i n i -m u m p r o b l e m s i s p r o p o s e d b y a s u i ta b l e i te r a t i o n o f F - l im i t p r o c e d u r e s . A ne x a m p l e o f a s y m p t o t i c d e v e l o p m e n t f o r a fa m i l y o f f u n c t i o n a l s r e la t e d t o p h a s et r ans fo rma t ions i s a l so g iven .
Key Words. F - c o n v e r g e n c e , A s y m p t o t ic d e v e lo p m e n t s ,V func t ions .
A M S Classification. P r i m a r y 4 9 J 4 5 , S e c o n d a r y 4 1 A 6 0 .
O. Introduction
I t is v e r y c o m m o n , b o t h i n p u r e a n d a p p l i e d m a t h e m a t i c s , t o h a v e t o d e a l w i tha f a m i ly o f p r o b l e m s d e p e n d i n g o n a p a r a m e t e r e > 0, b e in g i n t e r e s te d i n t h ea s y m p t o t i c b e h a v i o r o f t h e p r o b l e m s a s e - , 0 . Ty p i c a l l y, i n t h e c a l c u lu s o fv a r i a t io n s w e a r e g i v e n a f a m i l y o f m i n i m u m p r o b l e m s
m in{ ~( u ) : u ~ X} (0 .1 )
a n d w e a r e i n t e r e s te d i n f i n d in g a l i m i t p r o b l e m
min {~-(u ): u e X}. (0.2)
O f c o u rs e , f r o m t h e p o i n t o f v i ew o f t h e c a l c u l us o f v a r i a ti o n s , t h e l i m i t p r o b l e mmu s t be such tha t i ts min imize r s a r e c lo se ly r e l a t ed t o t he po ss ib le l im i t po in t s o f
m inim izers {u~}~ of pr ob lem (0.1).A no t ion o f con ve rg ence fo r func t iona l s , wh ich is ve ry we ll su i t ed t o t he
v a r i a t i o n a l s e tt in g , is t h e w e l l - k n o w n F - c o n v e rg e n c e , i n t r o d u c e d b y D e G i o rg i[D F] . In f ac t, i f a func t iona l ~ i s t he F - l imi t o f t he ~ and i f u s a r e min imize r sof ~ an d u s ~ u , the n u is a minim izer of ~ ( see Se c t ion 1 for a prec ise s ta temen t ) .
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{l imi ts of minim izers} ~ {m inimizers of the F- l imi t} , (0 .3)
where the inc lus ion m ay we l l be p rope r, a s we can see by ve ry s imple an d na tu r a lexamples . Hence the F - l imi t , t hough ve ry use fu l , f a i l s i n gene ra l t o cha rac te r i zec o m p l e t e l yt h e a s y m p t o t i c b e h a v i o r o f t h e f a m i l y ~ .
Our r emark i s t ha t i n f ac t t he F - l imi t i s on ly the f i r s t s t ep toward thed e s c r ip t i o n o f t h e a s y m p t o t i c b e h a v i o r o f ~ , a n d t h a t w e m a y t r y t o p u r s u e f u rt h e rt h e d e s c r ip t i o n l o o k i n g f o r a n a s y m p t o t i c d e v e l o p m e n t
= o ~ o ) + e , ~ 1) + e 2 y a ) + . . . + e ko ~ k ) + o s k ), (0.4)
where the f i r s t -o rde r t e rm ff (o ) is j u s t t he F - l imi t Y o f the f am i ly Y~ and each
one o f the h ighe r-o rde r t e rms y (0 is a func t ion a l de f ined by a na tu ra l r ecu rs ivep r o c e d u r e o n t h e s p a c e ~ ,~ - 1 ) o f t h e m i n i m i z e r s o f f f ( ~ - l ) ( s e e t h e f o l lo w i n gsect ion ) . W hen a dev e lopm ent a s in (0 .4 ) ho lds , t hen we have the fo l lowings i tua t ion :
{ l imi ts of minim izers} c {m inimizers of ~(k)}
c {m inimizers o f c~ (k-1 ) } C ' ' C {m ini m ize rs o f ~,~t0)}. (0.5)
T h i s m a y p r o v i d e a c o n s i d e r a b l e i m p r o v e m e n t o f (0 .3 ), a n d i n s o m e c a se s m a yg iv e a c o m p l e t e c h a r a c t e ri z a t i o n o f th e a s y m p t o t i c b e h a v i o r o f ~ .
In Sec t ion 1 we g ive the gene ra l de f in i tions and theorem s ab ou t the no t io nof a sy m pto t i c d eve lop me nt o f a f am i ly o f func t iona l s . In Sec t ion 2 we i l l u s tr a t ethe gene ra l t heo ry by a s imple bu t n o t com ple te ly obv ious example , r e la t ed to thew e l l - k n o w n e x a m p l e b y M o d i c a a n d M o r t o l a [ M M ] . I t w a s in fa c t i n t h e c o n t e x to f th i s example th a t w e f ir s t go t t he idea o f cha rac te r iz ing the a sym pto t i c beh av io rth ro ug h a sequence o f func t iona l s j~(k) de f ined on nes t ed spaces . The idea tha t t hefunc t iona l s o~(k ) cou ld be th ou gh t fo rm a l ly a s an a sym pto t i c dev e lopm ent (wr i t t enas in (0 .4 )) was sug ges ted by an inc iden ta l r em ark by D e G io rg i . On the o the rh a n d , t h e f ac t t h a t s o m e n o t i o n o f a s y m p t o t i c d e v e l o p m e n t b y F - c o n v e rg e n c ecou ld be use ful , m us t h ave been k no w n m ore o r l e ss exp li c it l y by ma ny peop le
work ing in the f i e ld . For in s t ance , someth ing which i s c lose to an a sympto t i cd e v e l o p m e n t c a n b e f o u n d i n a w o r k b y B u t t a z z o a n d P e r e iv a l e [ B P ] , a n d a f ir s ta t t e m p t a t a d e f i n i ti o n c a n b e f o u n d a t t h e e n d o f a p a p e r b y M o d i c a [ M ] . A f t era ll , even the sca ling o f the func t iona l s in the f i rs t pape r b y M odic a a nd M or to la[ M M ] m a y b e th o u g h t o f a s a n u n c o n s c io u s o r d er - o n e d e v e lo p m e n t .
1. The Asym ptotic Developmentof a Fa m ily of Functionals by F Convergence
Let X be a topo log ica l space fo r wh ich the f i r s t ax iom of cou n tab i l i t y ho lds , and l e t
~ : X ~ R
be a f ami ly o f func t iona l s , w i th 8 a pos i t ive pa ram ete r.Fo r such a f ami ly the fo l lowing de f in i tion o f the F - l imi t i s we l l kn ow n (see
[ D F ] a n d [D M ] ) .
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A s y m p t o t i c D e v e l o p m e n t b y F - C o n v e rg e n c e 1 07
D e f i n i t i o n1 .1 . A func t ion a l ~ (o ) : X ~ I ] i s s a id t o be t he F (X - ) - l im i t o f t he f ami ly.~ a t a p o in t f i e X i f f t he fo l l owing s t a t em en t s a r e fu l fi ll ed fo r each sequ ence e s ] O :
(i) F o r e a c h se q u e n c e { @ = X w i t h us ~ ~ we ha vel im inf ~ , (us) > ~(o) (~).]--, ce
( ii) T he re ex is t s a sequ enc e {us} ~ X w i th us --* fi a n d
l im sup ~ (u s ) <_ ~m)(f i ).j--, ~
In t h i s s i t ua t ion we wr i t e
F ( X - ) l im ~ ( ~ ) = y m ) (f i) .e--~O
D e f i n i t i o n1.2 . W e wr i te
F ( X - ) l im ~ = g ( o ) i n E ~ X~ 0
i f ~ (o ) is t he F (X - ) - l im i t o f ~ a t e ach po in t f i e E .
W h e n n o c o n f u s i o n m a y a r is e w e o f t e n o m i t t h e s p e c if ic a t io n o f t h e s p a c e X .T h e i n t r o d u c t i o n o f F - c o n v e rg e n c e i n t h e c a lc u l u s o f v a r i a ti o n s i s j u s ti f ie d b y t h e
f o l lo w i n g w e l l -k n o w n r e s u lt , w h o s e e a s y p r o o f c a n b e f o u n d i n th e p a p e r s q u o t e da b o v e .
Theorem 1 .1 . L e t es ~ 0 b e a f i x e d s e q u e n c e , l e t{us} c X b e s u c h t h a t ~ , u j ) =m i n { ~ ( u ) : u ~ X } . I f ~ ~~ i s t h e F - l i m i t o f ~ o n t h e w h o l e s p a c e X a n d u j ~ f~ i nX , t h e n f i i s a m i n i m i z e r o f ~ < o ) a n d w e h a v e
l im ~ , u s ) = f f o ) f i ) .j ~ c e
U n f o r t u n a t e l y, w e m a y w e ll h a v e m a n y m i n i m i z e rs o f t h e F - l im i t , w h i c h a r eno t l im i t po in t s o f min im ize r s o f t he func t iona t s ~ . A t r iv i a l exam ple o f such ap h e n o m e n o n i s t h e f o l l o w i n g o n e .
E x a m p l e 1.1 , C o n s i d e r th e c a s e X = R a n d
~ u ) = ~ l u l .
We c a n e a s i l y c h e c k t h a t t h e f u n c t i o n s . ~ ( u ) F - c o n v e rg e t o t h e c o n s t a n t f u n c t i o ny~o) _ 0 . C lea r ly, eve ry po in t i n R is a min im um po in t o f y{0~, wh i t e t he on ly
l imi t po in t o f t he min imize r s o f ~ is t he po in t u = 0 .
N o w , t h e id e a is t h a t o f i n t r o d u c i n g a n o t io n o f a s y m p t o t i c e x p a n s i o n
= ~ o ) + ~ y , ) + . . . + ~k~ k~+ O ek)
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o f a f a m i l y J~ ( u ) i n s u c h a w a y t h a t t h e k n o w l e d g e o f t h e f u n c t i o n a l s ~ ( k ) g i v e sa d d i t i o n a l i n f o r m a t i o n o n t h e l i m i t p o i n t s o f m i n i m i z e r s . P r e c i s e ly : a n y l i m i t p o i n to f a s e q u e n c e o f m i n i m i z e r s u s w i ll a l s o b e a m i n i m i z e r o f e a c h o n e o f t h ef u n c t i o n a l s ~ (k ) a p p e a r i n g i n t h e d e v e l o p m e n t a b o v e . J u s t b y c o n s t r u c t i o n , t h esequen ce o f se t s
~//k = {m in im izers o f ~(k)}
w i l l b e n o n i n c r e a s i n g , a n d w e m a y h o p e t h a t i n s o m e c a s e s t h e m i n i m i z e r s o fs o m e t e r m ~ -(k ) o f t h e d e v e l o p m e n t a r e e x a c t l yl l t h e p o s s i b l e l i m i t p o i n t s o fmin imize r s u~ .
N o w, l e t u s d is c u ss h o w t o d e f in e a s u i t a b l e n o t i o n o f a s y m p t o t i c d e v e l o p m e n t .F o r i n st a nc e , s u p p o s e w e w a n t t o g i v e m e a n i n g t o t h e e x p r e s si o n
= y( o) + ~ ( 1 ) + o(8). (1.1 )
N a i v e l y, w e s h o u l d l ik e t o s a y t h a t ( 0 ,I ) is e q u i v a l e n t t o
l im _ ~ ( i ) ,s ~ O 8
w h e r e t h e l i m i t s h o u l d b e t a k e n i n t h e s e n s e o f F - c o n v e r g e n c e . U n f o r t u n a t e l y s u c ha d e f i n i t i o n m a k e s l i t t l e s e n s e , a s w e c a n c o n v i n c e o u r s e l v e s b y t r y i n g t o a p p l y i tt o s im p l e s it u a t io n s . F o r e x a m p l e , i t m a y h a p p e n t h a t ~ a n d j~ (o ) a r e f in i te o nd i s jo i n t d o m a i n s ( se e t h e c a s e i n [ M M ] ) . H o w e v e r, it tu r n s o u t t h a t w e c a n g i v ea s i m p l e a n d v e r y g o o d s u b s t i t u t e f o r t h e n a i v e d e f in i t io n :
Se t
~-(o) = F - l im ~ in Xe ~ O
(we a s sume the t he F - l im i t ex i s t s ) and a l so s e t
m o = in f ~(o ) , ~ o = {u ~ X : ~(~ = too}.X
D e f i n i t i o n 1 3W e s a y t h a t t h e f i rs t -o r d e r a s y m p t o t i c d e v e l o p m e n t
= ~(o) + s~(1) + o(e )
ho ld s , i f we ha ve
F - l ira ~ - m~ - ~ (1 ) in ago.~ 0 8
W i t h t h e n o t a t i o n a b o v e , a s s u m e t h a t m o < + oo a n d 0g o ~ .
(1.2)
F r o m n o w o n w e u s e t h e n o t a t i o n
~ , ) = ~ m 0
8
The de f i n i t i on (1 .2 ) i s mo t iva t ed ma in ly by t he fo l l owing ve ry s imp le r e su l t s .
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A s y m p t o t i c D e v e l o p m e n t b y F - C o n v e r g e n c e 1 09
T h e o r e m 1 . 2 . Supp ose the f i r s t -order a sym ptot ic deve lopm ent(1.2) holds, and letej .[ 0 be a sequence for wh ich there exists a sequ ence{u } = X , uj ~ f ii n X , and~ u j ) = min{Y~,(v):v ~ X }. Th en ~t ~ ago and ft minimizes ~ 1) in ago. M oreo ver, i fm~ deno tes the inf imum o f ~ on X and m 1 deno tes the inf imu m o f ~ 1) on ago we have
m~ = m o + e jm l + o e j) . (1.3)
Proof. T h e f a c t t h a t ~ e a go i s a c o n s e q u e n c e o f T h e o r e m 1 .1 . L e t v e a go a n d l e t{ @ c X b e a s e q u e n c e c o n v e r g i n g t o v i n X s u c h t h a t~ 1) v.~-- ,f f (1 ) (v ) : f romt h e F - c o n v e r g e n c e o f @ ~ 1 ) i t f o l lo w s t h a t s u c h a s e q u e n c e e x i s ts . F o r e a c h f i x e din d e x j w e h a v e th a t ~j~(1)(Va)>_ ff{~)(u.~j,,, a n d s o t h e F - c o n v e r g e n c e y i e l d s
f i r e ( v ) l i m ~ ( : ) '~ j (~ ;~ >_ l i m i n f ~ ( 1 ) .(~;I >- ~-(1)(~).j ~ + ~ j ~ + m
I n p a r t i c u l a r w e h a v e
j ~ + c o
T h i s r e a d s l i m j ~ + o ~ [ (m ~ j -mo)/ej - m : J = 0 , w h ic h imp l i e s (1 .3 ). [ ]
R e m a r k . N o t e t h a t (1 .3 ) i n g e n e r a l is t r u e o n l y f o r s e q u e n c e s ea 0 f o r w h i c h t h e r eis c o m p a c t n e s s f o r m i n i m i z e r s : i n p a r t i c u l a r , i t isfa l se t h a t
m~ = m o + em 1q- o(e), (1.4 )
a s t h e f o l l o w i n g e x a m p l e s h o w s .
L e t X = R , a n d c o n s i d e r t h e f o l lo w i n g fa m i l y o f f u n c t i o n a l s
i f u = 0
.7~(u) = - e 1 /2 i f u = 1 /e an d ~ i s ra t io na l ,{ e 2 o t h e r w i s e i n R .
O f c o u r s e e a c h f u n c t i o n a l ~ h a s a u n i q u e m i n i m i z e r , w h i c h i s 0 i f e i s i r r a t i o n a la n d i s 1 /e i f e i s r a t i o n a l . T h e o n l y l im i t p o i n t o f m i n i m i z e r s i s O , a n d w e h a v e
c o m p a c t n e s s o n l y f o r s e q u e n c e s e j 0 s u c h t h a t e j is d e f i n it i v e ly ir r a t io n a l . I n~21/2pa r t i cu l a r, ( 1 .4 ) i s f a l s e fo r a r a t i o na l s eq ue nc e e j J. 0 , f o r in t h i s ca se m , , = - v j ,w hile y (o ) _ 0 a n d ~.~-(1) _= 0.
T h e p r o c e s s a b o v e c a n b e i t e r a te d in t h e fo l lo w i n g w a y. S u p p o s e t h e a s y m p t o -t ic d e v e l o p m e n t o f t h e f i r s t o r d e r ( 1. 1) h o l d s , r e c a l l t h a t m i = in f,4 /o~(~ ) , s e t
d ~ 1 = { U E 0 ~ r f f 1 ) U ) = m l } ,
a n d s u p p o s e m a < + oo a n d ag~ r ~ . T h e n c o n s i d e r t h e f a m i l y o f f u n c t i o n a l s. ~ - 1 )
g
I f w e h a v e
y ( 2 ) = F - li m f f ~ ) i n a g l ,e-+ 0
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t h e p a r a m e t e r e : i f w e c o n s i d e r e~ a s a n e w p a r a m e t e r w e r e a c h t h is i n f o r m a t i o nin one s t ep on ly. Th e fo l lowing exam ple show s th a t i t is no t a lw ays so .
Ex am ple 1 .3. Le t X = [ - 1 , 1-] c R , and c ons ide r t he fo l l owing fami ly o f func -t i ona l s :
< u ) = 0 if u = 0 11 1ek if j U[ ~ ' 2 k- 1
F o r t h e d e v e l o p m e n t o f o r d e r z e r o w e h a v e t h e s a m e s i t u a t i o n a s b e f o re :F -co nve rges t o y (0 ) _ 0 , an d so q /o = [ _ 1 , 1 ], mo = 0 and we have no in fo rma -
t ion a t a l l . We have
{ i f u ~ ll ) ( u ) = 1 1ek -1 i f l u l~ , 2 .
O n the s e t ~//o t he f am i ly y~ l ) F -co nve rges t o t he fo l l ow ing func t ion a l :
.T/1)(u) = {01 if u ~ [- 8 9 1 8 9o t h e r w i s e in ~ o .
Hence we have oR* = I-_21_,89 an d m 1 = 0 . By indu c t ion on the ab ove p roces s w eget ~k = [ _ 1 /2k, 1 /2a] ,m k = 0 , and
y { k ) ( u ) = { 0 1 i f I x l < _ l / 2 k,othe rw ise in ogk- 1.
As ~ ff=~ q /k = {0} , on ly t he com ple t e a sy m pto t i c dev e lop m en t o f Y , g ives u s t hed e s i re d i n f o r m a t i o n a b o u t t h e l i m it p o i n t s o f m i n i m iz e r s.
Ex am ple 1 .4. I n som e cases ou r p roces s does no t work : t h is , we be li eve , ise s sen t i a ll y due to t he n eces s i t y o f t he r i gh t cho ice o fsca l ing i n o u r f a m i l y. F o rin s t ance , i f we m od i fy t he f ami ly o f Exam ples 1 .1 and 1 .2 a s fo l l ows :
~ u ) = e - ll ~ l u l ,
w e h a v e t h a t a ll th e t e r m s o f t h e a s y m p t o t i c d e v e l o p m e n t a r e t h e f u n c ti o n a lc o n s t a n t l y e q u a l t o z e r o , a n d s o w i t h t h i s i l l - o m e n e d s c a l i n g w e c a n n o t h o p e t oge t any in fo rma t ion a t a l l .
A s imi la r s i tua t ion i s the fo l lowing:
g ~ u ) = l u l + - .g
In t h is ca se t he F - l imi t is con s t an t ly equa l t o + az , and so we can no t deve lopour f ami ly fu r the r.
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O f cou r se a s imp le r e sca ling ob ta ined by m u l t i p ly ing the fun c t iona i s by o;e l imina t e s t he p rob lem , bu t i n gene ra l de t e rm ina t io n o f t he r i gh t s ca l ing is f a rf rom easy.
Remark A s i tua t ion a s t ha t o f Ex am ple 1 .5 m ay a r i se i f we have a c ou n tab l esequen ce o f func t iona l s {YJ} in s t ead o f a co n t in uo us f ami ly.
In t h i s case , once we h ave de f ined the func t iona l ~ - (o ) a s t he F - l imi t o f t hefami ly, we m ay de f ine ~ (~ ) a s t he F - l imi t o f t he s eq uence
o j
wh ere coj is a su i t ab l e van i sh ing sequence . In t h is ca se we d o no t have anatural
s c a li n g g i v en b y t h e p a r a m e t e r e , a n d w e m u s t c a r e fu l ly c h o o s e a s o r t o f o r d e ro f z e r o f o r t h e s e q u en c e ~ .
2 A Concrete Exa mp le of Asym ptotic Developm ent in aProblem Related to Phase Transitions
In t h is s ec t ion we s tud y a fami ly ~ o f func t iona l s ve ry c lo se ly r e l a t ed t o t hosea l r e ad y c o n si d e r e d b y M o d i c a a n d M o r t o l a [ M M ] , a n d a f te r w a r d s st u d ie d inm a n y p a p e r s, a ls o i n v ie w o f th e i r p h y s i ca l m e a n i n g i n t h e t h e o r y o f p h a s e
t r a n si t io n s , se e, f o r e x a m p l e , [ M I ] , [ M 2 ] , I S ] , [ F T ] , [ B ] , [ A B V ] , a n d [ B B ] .I n t h e p a p e r s q u o t e d a b o v e , t h e i n f o r m a t i o n a b o u t t h e a s y m p t o t i c b e h a v i o ro f t h e m i n i m i z e rs o f th e f a m i l y o f f u n c t io n a l s w a s o b t a i n e d s i m p l y b y c o m p u t i n gthe F- l imi t o f a su i tab le resca l ing o f the fam i ly i tse lf : in the se t t ing w e es tab l i shedi n t h e p r e v i o u s s e c ti o n t hi s c o r r e s p o n d s t o t h e fi r s t- o r d e r a s y m p t o t i c d e v e l o p m e n to f t he f ami ly. In t h i s s ec tion we show tha t , i n d imens ion on e , t he s eco nd -o rd e rd e v e l o p m e n t o f ~ 7 c a n a ls o b e c o m p u t e d a n d t h a t t h i s gi ve s f u r t h e r in f o r m a t i o na b o u t t h e a s y m p t o t i c b e h a v i o r o f th e m i n i m i ze r s. W e s h o w, f u r t h e r m o r e , w i t h a ne x a m p l e , th a t i n h i g h e r d im e n s i o n s t h e s e e m i n g l y m o s t n a t u r a l c o n j e c t u r e f o r t h esecond F - l imi t 2 (27 i s i ndeed f al se . F o r t he m om en t w e a re no t ab l e t o co n jec tu re
t h e c o r r e c t fo r m o f g ( 2 ) i n d i m e n s i o n g r e a t e r t h a n o r e q u a l t o 2 .T h r o u g h o u t t h is s e c t i o n w e h a v e X = L ~(f~ ), w i t h . q c R n a b o u n d e d o p e n
se t w i th C2- regu la r bou nd a ry . In t he s eco nd pa r t o f t he s ec t ion we r e s tr i c t ou r se lvest o n = l .
Sup pose we h ave a fun c t ion (o : R -~ [0 , + oo) w i th t he fo l l owing p ro pe r t i e s :
i ) ~ o e C ~(ii) Z = { u : ( p ( u ) = 0 } = [ a , b ] u [ c , d ] w i t h a < b < c < d .
( iii) T he re is a K > 0 such th a t (p(u) decrea ses for u < - K , bu t increases foru > K
b c d
8/11/2019 Anzellotti - Baldo - Asymptotic Development by Gamma Convergence
Le t g : ~?fl - * [ a , d ] be a l i p sc h i t z - con t inuou s fun c t ion , a nd de f ine ou r f ami lyo f func t iona l s a s fo l lows :
~ ( u ) = I e 2 f n , D u l 2 d x + f n c p ( u ) d x if u ~ H ~ ( ~ ) , u , o n = g ,
+ Go oth erw ise in Ll(f~) .
U p t o r e s ca l in g o u r f a m i ly o f f u n c t i o n a l s is th e s a m e a s in t h e p a p e r b y M o d i c aand M or to l a , bu t i n ou r ca se t he func t ion (p van i shes i n two in t e rva l s i n s t ead o ftwo po in t s . W e sha l l s ee t ha t t h i s d i f f e rence m akes i t pos s ib l e and in t e r e s t i ng toc o m p u t e a s t e p m o r e i n t h e a s y m p t o t i c d e v e l o p m e n t . H e r e w e a l s o h a v e ab o u n d a r y c o n d i t i o n w h i c h is n o t p r e s e n t i n [ M M ] , b u t t h is m a k e s o n l y a t e c h n ic a ldifference.
W e r e m a r k t h a t e a c h f u n c t i o n a l ~ h a s a t le a s t o n e m i n i m i z e r i n L~ (f ~) a n dtha t , und e r su i t ab l e hy po the s i s on the g row th o f q) a t i n f in i t y ( a supe r l i nea r g row this eno ugh ) , each fam i ly of m inim izers {u~}~ s re la t ive ly co m pa ct in Ll (f~) ( see [M 1] ,[FT-1 , and [B] ) . H enc e we a re lawfu l ly a l l owe d to speak o f l im i t po in t s o fmin imize r s .
h e o r e m 2.1. We h a v eF - l im ~ o ~ = i f (o )in Ll( f~) , where
g ~ ~ = ~ ~ u x ))dx .
J ~
P r o o f W e f ix a sequ enc e e j i 0 an d a fu nc t io n u ~ L~(~) .Firs t s tep .W e m u s t s h o w f ir st th a t , f o r e a c h s e q u e n c e{uj} c LI(f~)w i t h uj ~ u
i n L l (f~ ), t he fo l l owing ineq ua l i t y h o lds :
l im inf ~j (u j) >_ J(~j~ o~
T h i s f o l lo w s e a si ly f r o m F a t o u ' s l e m m a .S e c o n d s te p .We m u s t c o n s t r u c t a s e q u e n c e{uj} c L~ (~)w ith uj ~ u in L~(fl)
a n d s u c h t h a tl im ~j (u j ) - - ~ (~
j ~ ~
To do th i s , we f i r s t choose a s equence{vj} ~ W 1~(~) such th a t v j con ve rge to ui n L ~ (f~ ) a n d a l m o s t e v e r y w h e r e in f~. W e m a y s u p p o s e t h a tvjl~, = g( th i s can beob ta ine d by mo d i fy ing the o r ig ina l s equence v j on sm a l l ne ig hb orh oo ds o f ~?fl).
I f u is bo und ed , {v } c a n a l so b e t a k e n u n i f o r m l y b o u n d e d i n L ~~ a n d w eo b v i o u s l y h a v e
l im fa ~o v j)d x = f n ~ o u )dx .j ~ ~
I f u is n o t b o u n d e d , w e c a n o b t a i n t h e s a m e r e s u lt b y a p p r o x i m a t i n g t h et r u n c a t i o n s
W~(X}= m ax {- -m , min{u(x ) , m}}
8/11/2019 Anzellotti - Baldo - Asymptotic Development by Gamma Convergence
We c o n s i d e r t h e f u n c t i o n a l~o def ine d o n 4 ̀ 0 as fo l lows:
( 2 c o P a ( A u ) + 2 f IcI)(u) (I)(g)[ d~V f -i if Pn(Au) < + ~ ,~ ( 1 ) ( U )
L + oe oth erw ise in 4`(0),
wh ere ( l) (u) den otes the t race of ( I) (u) on Of~, wh ich i s wel l def ined wh ene ver theper imeter of A, i s f in i te .
Theorem 2 .2 . We h a v e
F l i ra -~(~) = ~ o ) in 4 ` 0
~;- 0
P r o o f Fix ej ~ 0 an d u ~ 4`0.Fi r s t s t ep . W e s h o w t h a t , f or e a c h s e q u en c euj -* u in Ll(f~), w e h a v e
l ir a in f ~ ( t ) -j ( - ; > 5 ( @ .j ~ + o c
W itho u t lo s s o f gene ra l i ty, we can r e s t r i ct ou r se lves to the case {@ c Hl ( f~ ) wi thus t~ = g , and we can a l so a s sum e tha t t he l imi t
l im ~-o ) . ,j-~ oe
exis ts an d is f in ite . In par t ic ular we ma y assum e th a t ~ ( . j j _< C.F r o m t h e in e q u a li t y
f n f a 1 f n f ~~slDu~l ~ + - ~0(uj) ~ 2 ID uj l~ '~ (u s) = 2 IV(q~(uJx))) l
g j
a n d t h e l o w e r s e m i c o n t i n u i t y o f th e t o t a l v a r i a t io n w e g e t t h a t9 o u e BV (D ) .Inpar t icu lar, the t race o f (q~ o u) on 8~ i s def ined. App lying a gain the s am e ine qua l i tyw e o b t a i n
infY ~))(u j) >_ 2 lim in f t ]D (~ u i)[im d xS ~ o o j ~ o e d f~
_> 2 ]D (qb ou )[+ [O(u) - q)(g)ld ~ / f " - ~ = ~ ( 1 ) ( u ) .
The l a s t i nequa l i ty i s a conseq uence o f the lower sem icon t inu i ty in L~(f2 ) o f thefunc t iona l
fnlDvl fov-h ld~~def ined fo rv ~ BV (f~)an d w i th h s Ll (Of~) f ixedsee [G]) .
S e c o n d s t e p . W e m us t exh ib i t a s equence {u j} converg ing to u in L~(f~) sucht h a t
l i m ~ o ) .'qq (uj) = /~(1)(U).j ~ + o ~
Th e o the r case be ing obv ious , we can a s sum e tha t ~~ < + oc.
8/11/2019 Anzellotti - Baldo - Asymptotic Development by Gamma Convergence
W e c an r educe t he p ro o f to a s i t ua t i on i n wh ich t he s e t ~?*A i s r egu l a r a ndi n t er s e c ts O f f t r a n s v e r s a l l y : t o d o t h a t w e c a n u s e a p p r o x i m a t i o n r e s u l ts a s in [ B B ] ,[ B ] , a n d [ O S ] .
As i n t he p ro o f o f T he o r em 2 .1 w e can ge t a s e quen ce { fj } c W 1, ~ (f~ ) suchtha t fjlo~ = 9 , f j - - u i n L l ( f l ) and a .e ., an d such t h a t t he fo l l ow ing e s t im a t e ho ld s :
IIDf~llL~ --<C ~ / 4
C o n s i d e r t h e f o l lo w i n g p a r t i t i o n o f n :
A j = { x ~ A . : d x , A c ) > K e ja n d d x , 3 n ) > K @ ,
V~ = {x ~ A . : d x , A c) <_ K @ w {x ~ f i : d x , On ) <_ K ej} ,
B j = n \ A Vj),w h e r e K is a s u i t a b le p o s i t i v e c o n s t a n t . W e n o w d e f in e
u~ x)= m ax{ m in{f~ (x) , b}, a} i f x e B~,
uj x) = m ax{ m in{f j (x) , d}, c} i f x E Aj .
F i n a l l y, w e m a y e x t e n d t h e f u n c t i o n u j w i t h f u n c t i o n s c o n s t r u c t e d a s i n [ B ] ,[ B B ] , a n d [ O S ] : t h e s e f u n c t i o n s a r e r e s p o n s i b l e , a t t h e l i m i t , f o r t h e p e r i m e t e ra n d t h e b o u n d a r y i n t e g r a l w h i c h a p p e a r in ~-(~ ). O n t h e o t h e r h a n d , t h e i n te g r a l s
o n A : a n d B j v a n i s h .A s i m p l if i ed d i s c u s si o n o f t h e i d e a s u s e d i n t h e p a p e r s a b o v e c a n b e f o u n din a r em ark o n p . 186 , o f [AB V] , whe re t he p r oo f o f c a se n = 1 i s ou t l i ned . [ ]
Wi t h t h i s l a s t F - l i m i t w e h a v e g a i n e d f u r t h e r i n f o r m a t i o n o n t h e a s y m p t o t i cb e h a v i o r o f t h e m i n i m i z e r s o f o u r f a m i l y o f f u n c t io n a l s , i n p a r t i c u l a r , w e c a n n o ws a y t h a t , f o r a l im i t p o i n t u o f m i n i m i z e r s , t h e s e t A u h a s m i n i m a l b o u n d a r y. I no t h e r w o r d s , b y e m p l o y i n g th e u s u a l n o t a t i o n w e h a v e
q/1 = { u ~ q / o : A ~ m i n i m i z e s H A ) },
w h e r e
H ( A ) : = 2 c o P n A )+ 2 ( ICo;~a - qb(g)[d ~ , , - ~.J oZ
i s de f i ned on t he subse t s o f ~ w i th f i n i te pe r im e t e r an d
m~ - - in f ~ (*~ = m inH A ) .a
A s w e h a v e s a id b e f o r e , w e a r e a b l e t o c o m p u t e a f u r t h e r s t e p in t h e a s y m p t o t i c
dev e lo pm en t o f o~ wh en ~ i s an op en i n t e r va l o f R , fo r exam p le , n = (0, 1).P u t
)Cuo if x = 0,g x)
u 1 i f x = l ,
8/11/2019 Anzellotti - Baldo - Asymptotic Development by Gamma Convergence
As a con sequ enc e o f ou r pos i t i on , fo r each 6 > 0 t he re ex i st s a a > 0and a sub se t o f pos i ti ve me asu re o f (x2 , x2 + 8 ) such tha tu(x) > c + or. As ujc a n b e s u p p o s e d t o c o n v e rg e t o u a lm o s t e v e r y w h e r e i n ~ , w e c a n f in d a p o i n tx ~ ~ ( X z , x 2 + 6 )s u c h t h a t u j { x * ) > cfo r j l a rge eno ugh .
F o r the sam e reaso n , fo r every 6 > 0 there is a p o in t x* e (x l - (5, Xx) sucht h a t u j ( x * ) < bfo r j l a rge e nou gh .
By d en ot in g I* the in te rv a l Cx* x *~~ 2 1 w e h a v e
= , ~ j Ai- - - ~O (U j) - m I + (U))2 + 25 r- - e j ' , , I ; ~ ' J '~ J , ) ~ \ I , ~ C , j \ i ~
(2.2)
Th e exp res s ion be tw een the squa re b rack e t s i s pos i t i ve : i n fac t , t he sum o f t he twoin t eg ra l s i s g r ea t e r t han o r equa l t o
2 f ~ ID (~ o uj) l ,
and th i s l a s t exp re s s ion i s g r ea t e r t han rn l becauseu j ( x T ) < ba n d u j ( x * ) > c .
Bec ause the las t in tegra l of (2 .2) i s a l so po s i t ive , we ge t th e es t im ate
u))2 d x < C ,, J f l \ l ~
henc e a s c5 i s a rb i t r a ry a nd the con s t an t C do es no t dep end on 6 we ge t t ha tu ~ H a( f l \ ( { x l} w {2} w {x2})) and tha t t he L2 -no rm o f t he de r iva t ive on th i s opense t i s domina t ed by the cons t an t C ( r eca l l t ha t t he func t ion u i s i ndeed cons t an tin th e in terv als (Xl, 2) an d (2, x2) .
To c o n c l u d e t h e p r o o f in t h e c a s e 2 E ( 0 , 1) w e o n l y n e e d t o s h o w t h a tu+(x2) = c an d u - ( x 0 = b .In fac t , this im plies th at u s ~ , w hile (2.2) gives thedes i red es t im ate on the m ini m um l imi t o f the sequ enc e g~(2)~ , jj. .
S u p p o s e b y c o n t r a d i c t i o n t h a t u + ( x 2 ) = c + h w i th h > 0 . B y u s i n g t h eregu la r i t y o f u p ro ve d abo ve we have tha t t h e re ex i st s a cons t an t f l > 0 such tha tu ( x ) > c + ~ hin (x2, x 2 + f l) an du ( x ) < c in (x 2 - f l ,x2) . As the u j converge to ua lm os t ev e ryw here , fo r each f i xed k ~ N we can f i nd an index Jk and two po in t s
Z 'k , Z ;, ~ ( X 2 - - l / k , x 2 + 1 / k )w ith z~, < z{ suc h th atJk -~ + 0 % c < Uj~(Z 'k) < C + 89a n d ujk(z ' i ) > c + 3h .In t h i s s i t ua t ion we have the e s t ima te
~ j, j ~ > - - 2 [ D ( ~ o u j ~ ) ld x + (u~) 2d x .
8/11/2019 Anzellotti - Baldo - Asymptotic Development by Gamma Convergence
T h e p a r t b e t w e e n t h e s q u a r e b r a c k e t s is p o s i t iv e b e c a u s euj~(z k) > c,w h i l e t h e l a s ti n t e g ra l i s g r e a t e r t h a n o r e q u a l t o
, [ h/3 ,~2 hZ k1~
A s th e l a s t e x p r e s s i o n g o e s t o + o e a s k ~ + o e w e g e t a c o n t r a d i c t i o n w i t h t h ea s s u m p t i o n t h a t
l i r a i n f ~ - ~ ) ( u j ) < + ~ .j ~ ~
W i t h a s im i l a r p r o o f w e c a n s h o w t h a t i f 2 = 0 o r 2 = 1 w e n e c e s s a r il y h a v e
l i ra i n f ~ ( 2 )j---~ § cO
a n d t h e f i rs t s te p o f th e p r o o f i s c o m p l e t e .Secon d s tep.W e e x h i b i t a s e q u e n c e { us } s u c h t h a t u j ~ u i n L I ( ~ ) a n d
l m ~ E ( 2 ) . ~
j-~ oo
We c a n a s s u m eff(2)(u)< + o c , b e c a u s e o t h e r w i s e t h e c o n s t r u c t i o n i s t r iv i a l , a n ds o w e h a v eu s H l ( f ~ \ { 2 } )a n d u ( O ) = u o , u ( 1) = u l , u - ( 2 ) = b , u + ( 2 ) = c .
We d e f i n e
(x) i f 0 ~ x < 2 ,
u j (x )= t l j (X- 2 ) i f 2 _ < x _ < 2 + ~ j ,
l ( u 1 - 2 -~ j 1 -2 ( x - 2 - r i f 2 + ~ y < x < _ l .
T h e f u n c t i o n s r/j a n d t h e s e q u e n c e Ca a r e d e f i n e d i n t h e f o l lo w i n g l e m m a .
L e m m a 2 . 4 . I t is possible to f ind increasing solutions o f the followin g sequence odifferential problems:
with ~ j --, O. Th e sequence 6 ab ove is defined by
fi~ = m e a s ( { s z [ b , c ](p~12(s)<_z)/2}).
O bv iou sly, g)) -~ O.
P r o o f C o n s i d e r a s o lu t i o n o f t he C a u c h y p r o b l e m c o r r e s p o n d i n g t o (P j) w i t h o u tt h e c o n d i t i o n t/j(~ j) = c . T h e s o l u t i o n o f t h is p r o b l e m i s g l o b a l l y d e f i n e d a n d i ss t r ic t l y i n c r e a s in g . W e c a l l ~i t h e t i m e t h i s s o l u t i o n t a k e s t o r e a c h t h e v a l u e c .M o r e p r e c i s e l y w e p u t ~ j = ( i n f { t ~ R : t /j (t ) = c } ).
8/11/2019 Anzellotti - Baldo - Asymptotic Development by Gamma Convergence
We have to show tha t t he sequence { j goesEj = {s e [b, c] : (pl/2(s) > e)/2}. W e h av e
f / ~ j ds~ = e l / 2 S ) - ~gj j
r ]g j d s f [ g j d s- - q ) l / 2 S ) +g~ j tS j AV b ,c ]k E j~ 0 1 / 2 S ) - }-E j 6 j
meas ( [b - c] \E j )<_ e)/2(c - b) +aj
~ - F. 1 /2 C - -b) + @
As the las t express ion v anishes as j --+ + so th e le m m a is prove d.
t o z e ro a s j - ~ + o o , P u t
W e a re now ready to co nc lude the second s t ep o f the p ro of o f Th eorem 2 .3 ,by e s t im a t ing the m ax im um l imi t o f o~2)(uj) (o f cour se the sequence u j convergesto u in Ll)f~)).
On (0, 1)\[2, 2 + ~j] we get
l ira sup (@2 + ~ p (uf l dxj ~ + o o . ) 0 , I ) \ [ s + { A L ~ ;j _ 1
_< lim sup ~ (u))2 d x + l i ra sup @ 2dx = u ) z dx.j ~ + a o , j 0 j ~ + o o d 2 + { i d O , 1 )\{ s
Fina l ly, on (2 , s + {f l we hav e the es t im ate
g~ ~ r 1 1 2 f [ tim s u p | | / (u) ) 2 + 25 ~o(uj)dx - - @ /2(s) dsj + + m k ~ L g j g j
= lim su p ffJ[(tl j)z + l~ q~(tlj) - 2 q jq~l/2(~lj)] d xj --+ + ae g j
= l i m s u p ( 'e~ [ , qr 2 - d x
l i m s u p a~ dx = O.j - + + o o
By add ing these e s t ima tes we ge t t he c l a im, and the p ro of is comple te . [ ]
In the mo re gene ra l s i tua t ion f l c R wi th n > 1, t he s imples t gene ra l i za t ionof the l imi t func t iona l wou ld be
f f n Vu ] d x if u E rk + oo o therw ise in ~1 ,
8/11/2019 Anzellotti - Baldo - Asymptotic Development by Gamma Convergence
~U = {u ~ ~gl c~ BV (f~) :( D u - DUle,A, ) ~ L 2( ~) ; ( I) (u) = ( I) (g) ~ - 1-a .e .
w h e n e v e r 9 e [ a, b [ u ] c , d ]; u - ( x ) = b a n d
u + ( x ) = c~ - 1 -a .e . o n 8 A , } .
H e r e u - a n d u § d e n o t e t h e t ra c e s o f u o n t h e t w o s id e s o f 0 *u . W e r e m a r k t h a ti n th e c a s e w h e r e 0 A , i s c l o s e d w e s i m p l y h a v e
= { u ~ J #~ c~ H ~ ( f ~ \ 0 A = ) : O ( u ) = q ~(g ) ~ - ~ - a .e . w h e n e v e r g e E a, b [ w ] c , d ] ;
u - ( x ) = b a n du + ( x ) = c ~ - 1 - a .e . o n ~ A , } .
T h e f o l l o w i n g e x a m p l e s h o w s t h a t t h is f u n c t i o n a l Z (=) c a n n o t b e t h e F - l i m i to f t h e s e q u e n c e Z ~ ~).
E x a m p l e 2 .1 . L e t ~ = { x ~ R 2 : l x ] < 1} a n dg ( y ) = c t on 0f~, ~ ( b , c ) . O n t h ei n t e g r a n d cp w e m a k e t h e f u r t h e r a s s u m p t i o n t h a t ~ q ) - 1 /2 ( s ) d s < + oo ( th i s i se s s e n t i a l l y a n a s s u m p t i o n o f t h e o r d e r o f z e r o o f (o a t c ).
I f w e c h o o s e e ~ ( b, e ) i n s u c h a w a y t h a t O ( c ) - O ( ~ ) < ~ ( c0 , w e h a v e
~,(1) = {u e Ll (f~) :u(x ) e Ec , d ]a.e. in f~}.
m 1 = 4rc(O (c) - (I)(~)).
I n p a r t i c u l a r , t h e f u n c t i o n u ~ - c i s i n ~ /(1 ). W h a t w e d o n o w i s b u i l d a s e q u e n c e{ uj} c H I ( F ~ ) w i t h u s -= g o n 8 f ~ a n d u s ~ uo~ i n L I ( ~ ) s u c h t h a t Y ( 2 )( u .~ = - f l ,w i t h f i > 0 . T h i s s h o w s t h a t t h e p o s i t i v e f u n c t i o n a l~ 2) c a n n o t b e th e F - l im i t o ft h e s e q u e n c e Z ~ 2).
L e t u s c o n s i d e r t h e s o l u ti o n o f t h e f o l lo w i n g C a u c h y p r o b l e m :
~( t ) = e ( 1 / 2 ) ( ~ ( t ) ) ,
2 . 3 )v ( O ) = c ~ .
v( t ) i s a s tr i c t l y i n c r e a s i n g f u n c t i o n o n t h e i n t e r v a l [ 0 , K ] w i t h K = S~ q ) -t / z ( s ) d s ,a n d v ( K ) = c .I f h ( x ) d e n o t e s t h e d i s t a n c e o f x ~ f~ f r o m 0 s w e d e f i n e
o t h e r w i s e i n f~ .
B y u s in g t u b u l a r n e i g h b o r h o o d c o o r d i n at e s , o b s e r v i n g th a t
O c ) - , ~ ) =J o , , , , ~ ; / / ~ ~ c /r ,
8/11/2019 Anzellotti - Baldo - Asymptotic Development by Gamma Convergence
= 2 ~ z f f ~ J { [ d v/t\ 2 L ~ )~51~e j \ \ e j / / J
e j \ \ e j / / k e j / )
- - 4 ~ ~ [ 'jo t - e j l q o t / 2 ( v ( t - ~ ) d v ( ~ ) \e j / / d t d t
= 4 7 C zq) i/2(V(Z )) ~ V('C)dz --= --4re zq0 (v(z)) r = -/ 3 .
I t is heu r i s t i ca l ly conv in c ing tha t s equences o f t h is k ind can be con s t ruc t e dw h e n e v e r O f] h a s a c u r v e d p o r t i o n , o r w h e n e v e r u j u m p s o n a c u r v e d s u rf a ce , a n dt h e n e g a t i v e t e r m w e g e t s e e m s t o d e p e n d o n s o m e m a n n e r o n a n i n t e g r a l o fthe cu rva tu re s o f t he su r face its elf. I n pa r t i cu l a r, a s imi l a r ph en om en on m ay occu r
a r o u n d t h e i n te r s e c ti o n s b e t w e e n i n t e r i o r j u m p s a n d t h e b o u n d a r y o f .Q , w h e r ew e m a y t h i n k t h e r e is s o m e c o n c e n t r a t e d c u r v a t u r e o f c 3A ,. A n y h o w , w e h a v e n og e n e r a l ly r e li a b le c o n j e c t u r e a b o u t t h e c o r r e c t f o r m o f th e s e c o n d F - l im i t i n ad i m e n s i o n h i g h e r t h a n o n e .
R e f e r e n c e s
[ ]
[ MT]
[ BV]
[B]
[BB]
[BP][ D M ]
[ D F ]
IF][FT]
[G]
L. Ambrosio: Metric space valued functions of bounded variation. Preprint, Scuola NormateSuperiore, Pisa, 1989.
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A s y m p t o t i c D e v e l o p m e n t b y F - C o n v e rg e n c e 1 23
[ M i ]
[ M 2 3
[ M M ]
o s ]
is]
L. Modica : Gra d ien t theory fo r phase t rans i t ions and the min imal in te r face c ri t e rion . Arch .Rat. Mech . An al. , 98 t987), 123 142.L . Mo dica : G rad ien t theory o f phase t rans i t ion s wi th bou ndar y con tac t energy. Ann . Ins t .
H. Poinc ar6 Anal . N on Lin6aire , 4 1987) , 487-512.L. Modica, S. M orto la: U n esem pio di F-converg enza, Boll . Un . Mat . I ta l . B 5) , 14 1977),285-299.N. Owen, P. S te rnberg : Nonconvex var ia t iona l p rob lems wi th an i so t rop ic per tu rba t ions , toappear.P. S te rnberg : The e ffect o f a s ingu la r pe r tu rba t io n on nonc onve x var ia t iona l p rob lems . Ph .D.Thesis , New York Universi ty, 1986.
