One Dimmi Zel

8/13/2019 One Dimmi Zel

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On e d im en siona l V ar i a tiona l Pr oblem swh ose M in im izer s do n o t Sa t i sf y

the E u l er L agran ge E qua tion

J . M . B A L L V. J . M I Z E L

D e d i c a t e d t o Wa l t e r N o l l

w 1 I n t r o d u c t io n

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

I ( u ) = f f ( x , u ( x ) , u ( x ) ) d x (1.1)

in t hc s e t d o f abso lu t e ly con t inuo us func t ions u : [ a, b ] ~ R sa t is fy ing the endc o n d i t i o n s

u(a ) = or u(b ) = f l , (1.2)

dwh ere 0~ and /3 a re g iven cons tan ts . In (1 .1) , [a , b ] is a f in ite in te rva l , ' d eno tes ~ x '

a n d t h e i n t e g r a n d f =f ( x , u , p ) i s a s s u m e d t o b e s m o o t h , n o n n e g a t i v e a n d t osa t i s fy the r egu la r i t y cond i t ion

fpp > O. (1.3)

Th e s ign if icance o f the r egu la r i t y con d i t i on (1 .3 ) is t ha t , a s i s we l l kno wn , i tensu re s t he ex i s tence o f a t l e a s t one abso lu t e m in imize r fo r I i n d , p rov ided fa l so s at is fi es an ap p ro p r i a t e g row th co nd i t i on w i th re spec t t o p . F u r the r, i t imp l i e st h a t a n y L i p s c h i tz s o l u t io n u o f t h e i n t e g r a te d f o r m

x

f p = f f ~ d y +con s t , a .e . x E [a , b ] ( IE L)

o f th e E u l e r - L a g r a n g e e q u a t i o n i s in f a c t s m o o t h i n [ a, b ]. N o t w i t h s t a n d i n g t h e s efac t s and th e s t a tu s o f ( IEL ) a s a c la s s ica l neces sa ry cond i t i o n fo r a m in imize r,w e p r e s e n t a n u m b e r o f e x a m p l e s i n w h i c h I a tt a i n s a m i n i m u m a t s o m e u E ~ rbut u i s n o t s m o o t h a n d d o e sn o t sa t i s fy ( IEL) .


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326 J. M. BALL V. J . MIZEL

To s ee w h e r e t h e c l a ss i ca l a rg u m e n t l e a d i n g t o ( I E L ) m a y b r e a k d o w n , r e c a l lt h a t t h e a rg u m e n t r e l i e s o n c a l c u l a t i n g t h e d e r i v a t i v e

dZ ( u + t ~ 0 ) I t = o

b

= l i m f ( x , u (x ) + tg fx ) , u (x ) + tq J(x )) - - f ( x , u (x ), u (x ) ) dxt ~O t (1 .4)

fo r q~ a s m oo th func t i on s a t i s fy ing ~0 (a ) = ~0(b ) = O , an d conc lud ing t ha t s i nc eI (u + tg) i s m in im ized a t t = 0 t he de r i va t i ve i s z e ro ;viz.

b

f [L ~ L~ ] ax - o . 1 . 5 )

I f u E W l ~ ( a , b )t h is a rg u m e n t is cl e a r ly v a l i d , s i nc e b y th e m e a n v a l u e t h e o r e mt h e i n t e g r a n d o n t h e r i g h t - h a n d s id e o f ( 1. 4) i s u n i f o r m l y b o u n d e d i n d e p e n d e n t l yo f s m a l l t a n d c o n s e q u e n t l y o n e m a y p a s s t o t h e l i m i t t ~ 0 u s i n g t h e b o u n d e dc o n v e r g e n c e th e o r e m . H o w e v e r , i f i t is k n o w n o n l y t h a t t h e m i n i m i z er u b e lo n g st o d , t h e o n l y r e a d i l y a v a i la b l e p i e c e o f i n f o r m a t i o n w h i c h m a y a i d p a s s i n g t othe l im i t in (1 .4 ) i s th a tI (u) < o o . C o n s e q u e n t l y o n e i s t y p i c a ll y f o r c e d i n t o m a k i n ga s s u m p t i o n s o n t h e d e r i v a ti v e s o f f , t h e s e a s s u m p t i o n s b e i n g u n n e c e s s a r y f o r t h eex i s t ence o f a m in im ize r, so a s to pa s s t o t he l im i t . Mo re a l a rm ing ly, a d i f f i cu l tym a y a r i s e a t a n e a r li e r st a g e i n t h e a rg u m e n t t o d u e t h e p o s s i b i li t y t h a t n e a r s o m eu E d w i th I ( u ) < o o t h e r e m a y b e f u n c t i o n s v E d w i t hI ( v ) = cx~; in fac t ,i n t w o o f o u r e x a m p l e s w e a re a b l e t o s h o w t h a t f o r a l a rg e c la s s o f ~ EC~(a , b)t h e m i n i m i z e r s u a r e s u c h t h a tI (u + tq0 = cx~ fo rall t ~= O.

T h e p o s s i b i li t y t h a t a m i n i m i z e r u o f I in ~ r m i g h t b e s i n g u l a r w a s e n v i s a g e db y TO N EL LI, w h o p r o v e d a s t r ik i n g a n d l it tl e k n o w n p a r t i a l r e g u l a r i t y t h e o r e mt o t h e e f fe c t t h a t u i s a s m o o t h s o l u t i o n o f t h e E u l e r - L a g r a n g e e q u a t i o n o n t h ec o m p l e m e n t o f a c l o s ed s u b s e t E o f [ a , b ] o f m e a s u r e z e r o , a n d t h a t [ u ' (x ) ] = o of o r a ll x E E . H e t h e n g a v e a n u m b e r o f c r i te r i a e n s u r i n g t h a t t h e s e t E d o e s n o te x i s t a n d t h u s t h a t u EC~ b ] ). R em ar ks i n TONELLI [32] sugges t t ha th e d i d n o t k n o w o f a n y e x a m p l e s in w h i c h E i s n o n e m p t y, a n d w e b e li ev e t h a to u r e x a m p l e s a r e t h e f ir s t o f th i s t y p e . A p r e c is e s t a t e m e n t a n d p r o o f o f a v e r s i o no f t h e p a r t i a l r e g u l a r i t y t h e o r e m i s g i v en i n w 2 , w h e r e w e a l s o g a t h e r t o g e t h e ra n u m b e r o f r e s u l t s c o n c e r n i n g t h e e x i s t e n ce o f m i n i m i z e r s a n d f i rs t o r d e rn e c e s s a r y c o n d i ti o n s . I n t h i s c o n n e c t i o n w e m e n t i o n t h a t w e a r e u n a w a r e o f a n yin t eg ra l f o rm o f a f i r s t o rde r nece s sa ry cond i t i on t ha t i s s a t i s f i ed by eve ry min i -m i z e r u i n t h e a b s e n c e o f a d d i t i o n a l h y p o t h e s e s o n f .

O u r f i rs t e x a m p l e , g i v e n i n w 3 , is t h a t o f m i n i m i z i n g

s u b j e c t t o

I u ) f [ ( x 2 - u 3 ) ~ ( u ) 1 4 + e ( u ) 2 1dx ( 1 . 6 )0

u(0) = 0, u(1) ---- k , (1.7)


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Minimizers Nee d N ot Sa t is fy Euler-Lagrange Equ at ion 327

wh ere e > 0 , k > 0 . (As we po in t ou t a t t he end o f w 5 , t he po w er 14 i s t he l owes tfo r wh ich s ingu la r min im ize r s o f (1 .6 ) ex i st .) N o te t ha t i f 0 < k ~ 1 a nd e = 0

t h e n t h e m i n i m u m o f I is a tt a i n e d b yu ( x ) = min (x~ , k ) ; t he r e su l t s sum-m a r i z e d b e l o w s h o w th a t t h e s in g u l a r it y o f u a t x = 0 is n o t d e s t r o y e d p r o v i d e de > 0 i s su ff ic i en tly sma l l. Th e in t eg ra nd in (1 .6 ) has a s ca l e - inva r i ance p rop e r tyw h i c h a l l o w s o n e t o t r a n s f o r m t h e E u l e r - L a g r a n g e e q u a t i o n t o a n a u t o n o m o u sord ina ry d i f f e ren t i a l equ a t ion in t he p l ane , a nd th i s makes i t pos s ib le t o g ive ave ry de t a i l ed and com ple t e desc r ip t i on o f t he ab so lu t e m in imize r s u o f (1 .6 ) ,(1 .7 ) fo r a l l e and k . S om e o f t he m a in conc lus ions a r e t he fo l l ow ing (s ee e spec i a ll yT he or em 3 .12) . The re ex is t num be rs e0 = .002474 . . . . e* ---- .00173 . . . such th a t

2

(a ) for 0 < e < eo ther e ex is t tw o e lem en tary so lu t ion s k l (e ) x :~, k -2(e) x ~o f t he Eu le r-Lag rang e equa t ion o n (0, 1 ] ; ( b ) i f 0 < e < e* and k is su ff ic i en t ly

l a rge I a t t a in s an abso lu t e min imum a t a un ique func t ion u wh ich sa t i s f i e s2

u(x) , - ,~k2(e)x7as x - + 0 + , u E C ~ 1 7 6 a ndf , ( ., u ( . ) ,u ( . ) ) E L ~ ( 0 , 1 ) ,sotha t ( IEL) does no t ho ld : i f k = k -2 (e ) t henu(x) = k2(e) x~ ; (c) i f 0 < e < e*an d k is su ff ic i en t ly l a rge ( fo r exam ple , k ~ 1 ) t he re is no sm oo th so lu t ion o fthe Eu le r-Lag range eq ua t io n on [0, 1 ] s a t is fy ing the end con d i t i ons (1 .7) , and henceI d o e s n o t a t t a i n a m i n i m u m a m o n g L i p s c h it z f u n c t i o n s ; (d ) i f e > e * t h e n t h e r eis exac t ly one u t ha t m in imizes I and i t i s t he un iq ue sm oo th so lu t ion o f t he E u le r-Lag rang e e qu a t ion on [0 , 1 ] s a t is fy ing (1 .7 ) . Th e de t a i l ed s t ruc tu re o f the p hasepo r t r a i t t h a t l e ads t o t he se conc lus ions w ou ld have bee n ex t r em e ly d i f fi cu lt t od e t e r m i n e w i t h o u t t h e a id o f c o m p u t e r p l o t s, t h o u g h t h e se d o n o t f o r m p a r t o fthe p roo f s . S ince t he s ingu la r min im ize r s a r e sm oo th fo r x > 0 t he i r To ne l l is e t E cons is t s i n t he s ing l e end po in t (0 ) and theydo s a ti sf y t h e E u l e r - L a g r a n g eequ a t ion in t he s ense o f d i s t r i bu t ions ,i . e . i n i ts w e a k f o r m .

In w 4 we con s ide r t he ca se w hen f =f (u , p) d o e s n o t d e p e n d o n x . We f i r s tcon s t ruc t an f E Co~ 2 ) s a t i s fy ing (i n add i t i on to (1 .3 ))

]p]


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f = f e sa t i s fy ing (1 .8) such tha t fo r su i tab le k~ , k2 , I a t ta ins an abso lu te mini -m um sub jec t t o (1 .11) a t a un ique func t ion Uo wh ose Tone l l i s e t i s p r ec i se ly E .

A g a i n (1 .1 2 ) h o ld s . T h e s e t w o e x a m p l e s d e m o n s t r a t e t h e o p t i m a l i t y o f C o r o l -l a ry 2 .12 and the To ne l l i pa r t i a l r egu la r i t y t heo rem (T he ore m 2 .7) , re spec t ive ly.Aw a r e n e s s o f c o n d i t i o n s n e c e s s a r y f o r t h e v a l id i t y o f c h a i n r u l e c a l c u l at i o n s( [34] , [30], [27], [28]) in f lu enc ed ou r in i t ia l co ns t ruc t ion o f those exam ples ,t h o u g t h e p r o o f s p r e s e n t e d h e r e a v o i d t h i s i s s u e .

I n w 5 w e c o n s i d e r t h e p r o b l e m o f m i n i m i z i n g1

I u ) f [ (x* - l u ' l 9 + 21 dx ( 1 . 1 3 )--1

i n t he s e t ~ r o f abso lu t e ly con t inuo us fun c t ions on [ - -1 , 1 ](i.e. func t ions i n

W I ' 1 = W l a ( - - 1 , 1 )) s a ti s fy i n g t h e e n d c o n d i t io n s

u( --1 ) ---= k~, u(1) = k z , (1.14)

whe re s > 3 and e 2> 0 . (W e a l low s t o t ake non in t eg ra l va lues , even tho ug ht h e i n t e g r a n d i s s m o o t h o n l y i f s is a n e v e n in t eg e r.) W e s h o w ( T h e o r e m 5 .1 ) th a ti f s _>__27 then , p rov ide d - -1 ___ k l < 0 < kz ~ 1 an d s i s suff ic ien t ly smal l ,

eve ry min im ize r uo o f I in ~ r i s such tha tUo(X), -: Ix l :}sign x as x---~0, uo EC~ 1 ]) an d uo ~ W ~'p fo r 1 ~ p < 3 . I t f o l l ows tha t E = (0}a n d t h a t Uo d o e s n o t s a ti sf y t h e E u l e r - L a g r a n g e e q u a t i o n e i t h e r i n i ts w e a k o r

i ts i n t e g r a te d f o r m . F u r t h e r m o r e , i f 3 < q ~ 0 %

i n f I (v ) > i n f I (v ) = I (uo) . (1.15)vE wl q[~ /

T h i s r e m a r k a b l e f a c t i s k n o w n a s t h eLavren t i ev phenom enon (c f.LAVRENTmV[22], M A N ~ [25], CESARI [11]) , an d i ts occ urr en ce in aregular p r o b l e m h a s n o tp rev ious ly been n o ted ; i n the c i t ed r e f e r ences on ly t he ca se q - ---~ i s cons idered .I f s > 2 7 t h e n a n e q u a l l y s u r p ri s in g p r o p e r t y h o l d s ( T h e o r e m 5. 5) , n a m e l y th a tf o r a n y s e q u e n c e (Vm} ~ W l q f~ ~s u c h t h a t Vm(X --~ Uo(X)f o r e a c h x i n s o m ese t con ta in ing a rb i t r a r i l y sma l l pos i t i ve and nega t ive number s one hasI(v,,) --~

as m- -~ ~ . In pa r t i cu l a r, no min imiz ing sequence fo r I i n w ~ 'q f~ d can conve rgeto Uo. S ince conve n t iona l f i n i te - e l emen t m e tho ds fo r m in imiz ing I y i e ld suchsequences , i t f o l l ows tha t t hey ca nn o t i n gene ra l de t ec t s ingu la r min imize r s . S i -mi l a r ly, i f v, is a m in imize r o f , f o r examp le , an app a ren t ly i nno cuou s pen a l i zedfunc t iona l such a s

1

l u ) f [ x * - U6) 2 lU 'V + 2 + lU ' I3+ ]dx ( 1 . 1 6 )--1

i n ~ r wh e re y > 0 , t hen vn can no t conv e rge to Uo a s ~ -+ 0 -k . M ot iv a t ed bynum er i ca l exp e r imen t s o f BA LL & KNOWLES [6 ] we show a l so t h a t i f s > 27 ,3 ~ q 0 , k l , k2 a r e a rb i t r a ry t hen (T heo rem 5.8 ) I a tt a in s a min i -m um in W l'q f~ ~ r and any such min imize r u l is a smo o th so lu t ion o f t he Eu le r-L a g r a n g e e q u a t i o n o n [ - - 1 , 1]. ( N o t e t h a t s u c h p s e u d o m i n i m i z e r s d o n o t i ngene ra l ex i s t f o r (1 .6 ) , ( 1 .7 ) . ) The pseudomin imize r s can be r ega rded a s be ing


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a d m i s s i b l e m i n i m i z e rs o f I w i th r e sp e c t t o v a r io u s p e n a l t y m e t h o d s s u c h a s(1 .16 ). F ina l ly, we show (T he ore m 5 .9 ) t ha t fo r s < 26 a l l m in imize r s o f I i n

~ r a r e s m o o t h , a n d t h a t , a t le a s t f o r t h e c o r r e s p o n d i n g p r o b l e m p o s e d o n ( 0 , 1),s i n g u la r m i n i m i z e r s n o t s a ti s fy i n g t h e L a v r e n t i e v p h e n o m e n o n m a y e x i s t f o r26 ~ s < 27.

In a l l t he exam ples cons ide red we ana lyze w he the r o r no t t he m in imize r s s a ti s fyt h e w e a k o r i n te g r a te d f o r m s o f t he D u B o i s - R e y m o n d e q u a t i o n

d-~x(f - - u ' fv)= A . ( D B R )

T h e e x a m p l e s i n t h i s p a p e r w e r e m o t i v a t e d b y a t t e m p t s t o p r o v e t h a t m i n i -m i z e r s o f t h e t o t a l e n e rg y

I(u) = f W (x, Du(x)) ax (1.17)

o f a n e l a st ic b o d y s u b j e ct to a p p r o p r i a t e b o u n d a r y c o n d i t io n s a r e w e a k s o l u t i o n so f t h e c o r r e s p o n d i n g E u l e r - L a g r a n g e e q u a t i o n s

8x ~ OAk - - 0 , i = 1 . . . . n . (1 .18)

H e r e w e h a v e a s s u m e d t h a t t h e b o d y o c c u p i e s t h e b o u n d e d o p e n s u b s e t f 2 Q Ri n a r e f e r e n c e c o n f i g u r a t i o n a n d t h a t t h e r e a r e n o e x t e r n a l fo r c e s. T h e p a r t ic l ea t x E f2 i n t he r e f e r ence con f igu ra t ion is d i sp l aced tou(x)E 1% , an d Du(x)d e n o t e s t h e g r a d i e n t o f u a t x . O n e o f t h e c o m p l i c a t i o n s o f t h e p r o b l e m , w h i c hi s st il l open , i s t ha t t he s to red -ene rgy func t ionW (x, A)of t he m a te r i a l i s de f inedon ly fo r de t A > 0 and i s t yp i ca l ly a s sum ed to s a t is fyW (x, A)---~ooasd e t A --~ 0 + . T h e e x i s te n c e o f m i n i m i z e rs i n a p p r o p r i a t e s u b se ts o f t h e S o b o l e vsp ac e W 1'1 = WW (.Q; R )i s es tab l i shed in BA LL [2] for a c lass o f rea l is t icfunc t ion s IV, and cond i t i ons gua ran t ee ing tha t t he se min im ize r s s a t is fy o th e rf ir st o r d e r n e c e s sa r y c o n d i t i o n s a r e a n n o u n c e d i n B A L L[5]. I t i s k n o w n BALL[3] , BALL & MURAT [8 ]) t ha t even wh en W sa ti sf ie s f avo rab le con s t i t u t ive hyp o-theses such a s s t rong e l l ip t ic i ty, I m ay n o t a t t a in i t s min im um wi th in t he c l a s s o fsm oo th func t ion s , and in f ac t t ha t i f n i n f I(v) (1.19)v smooth v E W l q vEW I I

c a n o c c u r f o r a p p r o p r i a t e b o u n d a r y d i s p la c e m e n t s ~ . O f c o u r s e (1 .1 9 ) is a h i g h e r-d i m e n s i o n a l v e r s io n o f t h e L a v r e n t i e v p h e n o m e n o n . T h e d e f o r m a t i o n s r e s p o n s i b lehere f or LAVRENTIEV'S gap are those for w hich cav i ta t ion occu rs , tha t i s , ho lesf o r m i n t h e b o d y. C a v i t a t i o n c a n n o t o c c u r i f W s at is fie s th e g r o w t h c o n d i t i o n

W(x, A) >con st . [A 1~ fo r de t A > 0, (1.20)

fo r some p > n , by the Sobo lev em bed d ing theo re m (no r, i n f ac t , i f p = n ) .A n in t r i gu ing poss ib i l t y r a i s ed by ou r one -d im ens iona l exam ples is t ha t s i ngu la rm i n i m i z e rs a n d t h e L a v r e n t i e v p h e n o m e n o n m a y o c c u r f o r (1 .1 7 ) e v e n w h e n ( 1 .2 0 )ho lds , and tha t t he s ingu la r i t i e s o fD u m i g h t b e c o n n e c t e d w i t h t h e i n i t i a t i o n o ff r a c tu r e . M o r e w o r k n e e d s t o b e d o n e t o d e c i d e w h e t h e r th i s c a n h a p p e n u n d e r


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332 J . M . BALL V. J . MIZEL

O f cou r se , i f u s a ti sf ie s ( IEL) ( r e spec tive ly ( ID BR )) t hen u s a ti sf ie s (WE L) ( r e -spec t ive ly (W D BR )) . W e wil l see l a t e r t ha t t he conve r se i s f a ls e i n gene ra l ; w ha tis t r u e is t h a t , b y t h e f u n d a m e n t a l l e m m a o f t h e c a lc u lu s o f v a r ia t i o n s , ( W E L )i s equ iva l en t t o

Jr

fp (x , u (x ) , u (x ) ) =f f u d y + con s t , a .e . x E [a , b ] ,c

f o r a n y c E ( a, b ), a s i m i la r s t a t e m e n t h o l d i n g f o r ( W D B R ) .

T h e o r em 2 3

(i) Le t uE ~1 be a we ak re la t ive minim izer o f l an d suppose tha t f~( . , F t( .) , u ( . ) ) EL~(a , b ) whenever uE L~176 b ) wi thess sup l u ( x ) - - ~(x)] suff ic ient ly smal l .

xE[a b]Then u sa t i s f ies( IEL) .(ii) L et u E s4 be a s trong re la t ive m inim izer o f I a nd suppose th a t fx(Yc( .) , u( ) ,

u ( .) ) E L l (a , b ) whenever x E L~176 b ) wi thess sup l~(x) - - x [ suff ic ient lyx E [ a b ]

smal l . Then u sa t i s fies( I D B R ) .

P r o o f

( i) F o r 6 > 0 suff ic ien t ly sma l l an d G ( R c losed def ine

) 'G(x) = su p I f~(x , u(x) + t ,u ' ( x ) ) ] ,t E [ - O ~ ]F ~G

E (x) = { t E [ - -6 , 61 : I fu(x , u(x) t ,u ( x ) ) I - -

We c o n s i d e r t h e s e t - v a l u e d m a p p i n gE : x ~-~ E( x) .Clea r ly E ( x ) i s d o s e d f o ra . e . x E [ a, h i . Fu r the rm ore , fo r any c lo sed G ( R the s e t

{x E [a, b] : E (x ) Gno nem pty } = {x E [a , b ] : 7~ (x) - - 7 (x) = 0}

is me asu rab le ( s ince 7 a - 7R i s a me asu rab le func t ion ) . By a s t an da rdm easu rable se lec t ion theo rem (cf. CESARI [11 , p . 283ff ] ) there ex is t s a m eas ur-a b l e f u n c t i o n (x ~ t (x ) w i t h t ( x ) E E ( x ) a .e . x E [a , b ] . H en ce 7R(x) =[ fu(x , u(x) q- t (x) , u (x))]a .e . x E [a , b ] , so tha t ou r hyp oth es is is equ i -va l en t t o t he ex i s tence o f 7 EL l ( a , b ) s u c h t h a t

[f~(x, u(x),u'( x) ) I ~ ~,(x) a.e. x E [a, b]

for a l l ~ EL~176 b) wi th e s s sup [u (x ) - - f i(x)] suff ic ien t ly smal l . Th e resu l txE [a b]

no w fo l low s f ro m TONZLLI [31] ( see a l so CESARI [11, p . 61 ff ] , HESTENm [20 ,p. 196ff]) .

( ii ) T his fo l low s in a s imi la r wa y f rom TON~LLI[31 ] (se e also CESA_~ [11, p. 61 if]).A l t e rna t ive ly, one can deduce ( i i ) f rom ( i ) by a r educ t ion based on the i deatha t ~0 = 0 i s a w eak re l a t i ve m in im um o f

b

J ~) = f f ( x , u~o(x), u ~(x)) dxa

sub ject to ~0(a) = 90(b) = 0, w he reu , (x ) ~ f u (z ), z + ~ (z ) = x . [ ]


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Minimizers Need N ot Satisfy Euler-Lagrange Equ ation 333

Corollary 2.4 . L e t f = f l ( x , u ) + f 2 ( x , p ) . I f u E ~ i s a w e a k r e la t iv e m i n i m i z e ro f I then u sa t i s f ies( IEL) .

P roo f . I f f i 6L~~ b) t h e n fu (x , ? ,(x), u (x)) = (f l)u (x, ? ,(x))i s u n i f o r m l y b o u n d -e d . [ ]

Corollary 2.5 . L e t f = f l ( x , u ) + f 2 ( u , p ) . I f u E ~ is a s tr o ng re la ti ve m i n i-m izer o f I then u sa ti s f ies( I D B R ) .

P r o o f I f ~ EL ~ ( a , b ) t h e n fx (~ (x ) , u (x ) , u (x ) )= (jq)x (~(x),u(x) ) i s u n i f o r m l yb o u n d e d . [ ]

T h e a b o v e r e s u lt s a r e n o t a b l e f o r t h e l a c k o f a n y c o n v e x i t y a s s u m p t i o n s o n f .T h e g r o w t h a s s u m p t i o n s a r e a ls o c o n s i d e ra b l y w e a k e r t h a n t h o s e o f c o r re s p o n d i n gtheorem s kn ow n fo r mu l t ip l e in t eg ra l s . Fo r example , i n Th eore m 2 .30) the re isn o h y p o t h e s i s o n f / t h a t t h e re s u l t is t r u e w i t h o u t s u c h a h y p o t h e s is is su g g e s te db y t h e f a c t t h a t f p is b o u n d e d f o r a n y s o l u t i o n o f ( I E L ) . W e a r e n o t a w a r e o f a n yc o u n t e r e x am p l e s t o T h e o r e m 2 .3 i f t h e i n t e g r a b il it y h y p o t h e s e s a r e w e a k e n e dto re ad in p ar t ( i ) f , ( . , u( .) , u (-)) E L 1 a, b) , an d in par t ( i i) f x( , u( .) , u ( . )) EL l(a ,b).

W e no w desc r ibe r e su l ts i n wh ich f i s a s sumed convex wi th r e spec t t o p .

Theorem 2.6 . L e t u E W l ~ ( a , b )( = Lipsch i t z con t inuous func t ion s on[a, b])

be a we ak re l a t ive min im ize r o f I , and suppose tha t fp j, (x , u (x ) , p ) > 0 . fo r a l lx E [a, b] , p E I~. The n u E Ca([a,b]) and sa t i s f ies(EL) .

P r o o f Th i s i s s t an da rd and can be fo un d in CESARI [11, p . 57 ff] . [ ]

Le t R = l~ k ) ( - - oo ) L /{ + co} deno te the ex tended r ea l l ine wi th it s u sua l

topo logy. We de f ine Cl( [a , b] ; R) to be the se t o f con t inuou s func t ions u : [ a , b ]R such th a t fo r a l l x E [a , b]

u ( x ) de=_rimb_,0U(X + h)h -- u( x) (2.1)

ex is t s a s an e l em en t o f R (wi th the ap propr i a t e one - s ided limi t be ing t ake n i fx = a o r x = b ) , an d such tha tu : [a , b ] - + R i s con t inuous .

Theorem 2.7 (TONELLI Spa r t ia l regular i ty theorem). Le t fpp > O. I f u E ~ i s as t rong re l a tive min imize r o f I t hen u EC1([a, b]; R).

Befo re p rov ing T heo rem 2 .7 we no te som e consequences . C lea r lyu (x ) asdef ined in (2.1) coincides a lm os t everyw here wi th the der iva t ive of u in the senseo f d i s tr i b u ti o n s . T h e r e f o r e u n d e r t h e h y p o t h e s e s o f t h e t h e o r e m t h eTonel l i se t Edef ined by

g = {x E [a, b] : [ u (x)] = co}

i s a c losed se t o f me asure ze ro . The co mp lem en t [ a, b ] \ E i s a un io n o f d i s jo in tr e l a t ive ly ope r t i n t e rva l s Dj . By the op t im a l i ty p r inc ip le an d Th eore m 2 .6 , u


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334 J . M . BALL & V. J . MIZEL

i s a C a s o l u t i o n o f ( E L ) o n e a c h D j . B y T h e o r e m 2 .7 ,u (x ) t e n d s t o + o o o r - - c ~a s x t e n d s t o t h e e n d - p o i n t s o f e v e r y s u c h i n t e r v a l ( u n l e ss a E D j o r b E D j ) .

T h e s e c o n s e q u e n c e s o f T h e o r e m 2 .7 c o n s t i t u t e T O NE LL I'S s t a t e m e n t o f h i s t h e o r e m(TO N EL L I [3 1 I I , p . 3 5 9 ] ); o u r f o r m u l a t i o n i n c l u d e s t h e e x t r a r e m a r k t h a t u ' i sc o n t i n u o u s . T h e p r o o f w e g i v e, l i k e TO N E LL I'S , u s e s t h e l o c a l s o l v a b i l i t y o f ( E L ) ,b u t w e a v o i d h i s c o n s t r u c t i o n o f a u x il ia r y i n t e g r a n d s b y a p p l y i n g t h e f i e ld t h e o r yo f t h e c a l c u l u s o f v a r i a t i o n s . R e c e n t l y, C L A R KE & V I N T ER [ 13 , 1 4 ] h a v e p r e s e n t e dc e r t a i n e x t e n s i o n s o f TO N E LL I'S t h e o r e m t o t h e c a s e s w h e n f i s n o t s m o o t h a n du : [a , b ] - + R . T h e y h a v e a l s o s h o w n [1 5] t h a t i f f is a p o l y n o m i a l t h e n t h eTo n e l l i se t E i s a t m o s t c o u n t a b l e w i t h f in i te ly m a n y p o i n t s o f a c c u m u l a t i o n .

emma 2 .8 . Le t A C R2 be bounded , and le tM > 0 ,

e > O such that i f (Xo, Uo) E A , 1o~1 0 c a n b e c h o s e ns u f f i c i e n t l y s m a l l f o r ( b ) t o h o l d f o l l o w s b y a s i m p l e c o m p a c t n e s s a r g u m e n t ,u s i n g t h e r e l a t i o n s

8 u a u

u ( X o ; ~ , t ) = t , ~ ( X o ; ~ , t ) = 1 , ~ ( X o ; ~ , t ) = O ,

~ ~fl] (x o; o~, 8) = l. []

Proposition 2 .9 . TONELLI31 I I , p . 344ff ] ) .L e t m > 0 , ~ > 0 , M x > 0 .Thenthere exists e > O such th at i f Xo, Xl E [a, b] , O < xl -- Xo


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Minimizers Need No t Satisfy Euler-Lagrange Equation 335

h X o ) = U o , h x O = u ~m i n i m i z e r o f

o v e r t h e s e t

a n d m ax [ u(x) -- Uo] < ~,a n d ~ i s t h e u n i q u e a b s o l u t exE[xo,xd -~-

x l

i (u ) = f f ( x , u (x ), u (x ) ) dxXO

d ~ = ( U E w l l ( x o , X l ) :U Xo) = Uo, u ( x 1 ) = U l , m ax I~(x) -- Uo I< ~ .x~[xo,xd

P r o o f. L e t o r : m + 0 , A = [a, b] [ - -a , a ], M > m a x ( M ~ , 2 0 ) a n d l et0 < ~ 6 < M - - M ~ . L e t e > O b e c h os en a s i n L e m m a 2 . 8 , a n d s up p os e i na d d i t i o n t h a t 3 M e < 0 . L e t Xo, X l E [ a , b ] , O < x ~ - - x o ~ e , ]uol ~ m a n d

ul - - Uo ~ M~. N ote t ha t by in tegra t ing (2 .3) we have tha tX 1 X o

[u (x ; o~, f l ) - - Uo - - o~ - - f l ( x - - Xo) l = U o +M l X l - - X o)~ - ( M - - M x - - 6 ) ( x ~ - - X o ) > U a ,

u ( x l ; O, - - M ) 0 for f l E [ - - M , M ] there i s a u niq ue f lo E [ - - M , M ] such

t h a t u ( x l ; O , f l o ) = u l . Def ine h ( x ) = u ( x ; O, f lo ).Se t t ing x = x l i n (2 .5 )w eob ta in

[flo ] ----< 6 + M ~ . (2.6 )

Th erefor e , a gain by (2 .5) , for x E [Xo, x l ]

I ~ x ) - U o I < 6 + I ~ o I ) x - ~ o )

(26 + M 1) e < O-

N ow suppo se tha t v E C2([Xo, x l ] ) i s a lso a so lu t io n o f (EL) sa t i s fy ing

U 1 UoV ( X o) = u o , v ( x O = u la n d m a x I v ( x ) - - uo[ < p . Th en v (x- )xE[xo,Xd X~ -- X o

for som e ~ E (Xo, x l ) an d (x , v(~)) E A, and so app lying (2.3) wi th ( :~ , v(~))replac in g (Xo, Uo) we dedu ce th a t

] u ~ U ~ o ~ 6( x ) x ~ -In par t i cu la r,

[ v ( xo ) [ ~ M I + 6 < M .

f o r x ~ [X o , x l ] .

By the un iqueness o f flo we the re fo re have tha tv ( X o ) = f l o , an d thus v = ~ .To s h o w t h a t ~ m i n i m i z es I i n ~ , w e c o n s id e r th e o n e - p a r a m e t e r f a m i l y o f

so lu tio ns {u(.; or flo), Io~1 ~ m ). B y (2.5), (2.6) w e hav e

u ( x ; M , ri o) - - u o ~ M + ( ri o - - a ) ( x - - x o ) 2> M - - ( 2 a +M ~ ) e >


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3 3 6 J . M . B A L L V. J . M I ZE L

a n d

u ( x ; - - M , / 3 o ) - - U o < : - - m + (flo + 6 ) ( x - - X o) < : - - m +(26 + m l) e < - -Q,

0ufo r x E [xo, x d . Since ~-~ (x; o~, t3o) > 0 i t fol low s tha t f i is em be dd ed in a f ield

o f ex t r ema l s tha t s imply cove r s the r eg ion [Xo , x l ] [ u o - ~ , Uo + Q]. S incefvp > 0 i t fo l lows f rom W eie r s tr a s s 's fo rm ula(e.g. BOLZA [9, p. 91], CESAm [11,p. 72]) that

fo r a l l u E ~ , w i th equa l i ty i f and on ly i f u = 2 , w h ich conc ludes the p roof . [ ]

r o o f o f T h e o r e m2 .7 . Le t u E .d be a s t rong r e l a tive min im ize r o f I ; t hus the reex is t s 61 > 0 su ch th a tI (u) ~ : I (v)for a l l v E d wi th m ax ]u(x) - - v(x)[ < 61.

x E [ a b ]

L et ~ E [a , b], an d sup pose th a t

x 4= ~ x E [a b ]

S u p p o s e t h a t x 4 : b a n d t a k e Y l > Y w i t h Y l - Y s u ff ic ie n tl y s m a l l t h a t

m ax [u(x) -- u(x) l < 61ts : -~- Ch oos e M1 > M (x) . By (2.7) we can app ly Pro -

61pos i t ion 2 .9 wi th Xo = x , Uo = u(x) , Q = - f , u l =u(x l ) , where x l E (x , x l )satisfies

x l - ~ < e , u ( x l ) : ~ ~ ) - l ~ MX 1 - - X I

Le t f i be the corresp ond ing solu t ion of (EL) . Let ~ E d be def ined by f i(x) = f i(x)

if x E [x, xl ], fi(x) =u(x) o th erw is e . T h e n ~Eto,blmax ~(x ) - - u(x ) I =< -2-~I+ -2-62 -_ 62

a n d s o I ( f i ) - I ( u ) = [ ( u ) - [ ( u ) ~ 0 . S in ce u is t h e u n i q u e m i n i m i ze r o f

in ~7 i t fo l low s tha t ~ = u in [7, xa] and hence th a t u E C2([x , xa] ) . S imi-larly, i f ~ =4= a the n u E C2([Xo, x]) fo r som e Xo < x. In par t icu lar u is Li psc hitzin the ne ig hbo rhoo d o f any x E [a , b ] w i th M(Y) < oo , an d thus by Th eor em 2 .6i s C a in a ne ig hbo rhoo d o f any such x . S ince u i s d i f f e ren t i ab le a lm os t eve ryw herein [a, b] i t f ollo w s th at D d~f {x E [a, b] :M ( x ) < oo} i s a re la t ive ly ope n subse tof [a , b] of fu l l mea sure , an d t ha t u E Ca(D) .

L e t E ---- [ a , b ] \ D , a n d l et x o E E , s o t h a tM ( x o ) = O o . S u p p o s e t h a tXo E (a , b ). By an approp r i a t e r e f lec t ion o f the va r i ab le s x and /o r u we can sup-pose w i thou t lo s s o f gene ra li ty tha t t he re ex i s t po in t s y j ~ Xo- - w i th

U(Xo) -- u(yj)l im _ + oo .j -- > o o X O - - y j

Le t M 3> 0 , 6 > 0 be a rb i t r a ry and app ly Le m m a 2 .8 wi th Uo =u(xo). T h eso lut ion s {u(.; 0r M ) : Io~1 --< M } of (E L) fo rm a f ie ld o f ext remals s imp ly cover-


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Minimizers Need N ot Sa t is fy Euler-Lagrange Equ at ion 337

i n g s o m e n e i g h b o r h o o d o f(Xo, Uo)in R 2 . Th us , for Ix - - Xol suff ic ien t ly sma l lthere ex is ts a u n iq ue o~(x) w i th [0ffx)[ =< M such tha tu ( x ) = u ( x ;~ x ( x) , M ) ,

a n d b y t h e i m p l i c it f u n c t i o n t h e o r e m a n d ( 2. 4) o~ d e p e n d s c o n t i n u o u s l y o n x .Clea r ly o~(Xo) = 0 . W e c la im th a t o~(x) i s no nd ecre as in g ne ar Xo. In fa c t sup pos eth er e ex ist se qu en ce s a i---> Xo, bj---> Xo, e~---> Xo w ith a < bj < cj an d~(a~) = o~(ci) =l= o~(b,,). T h e n fo r larg e eno ug h j the s ol ut io nvj( x) ,~ef= (x ; o ffaj) , M ) ,aj =< x ~ cj, satisfie s vj (a j ) = u(a j ) , v j (b j ) q= u(b j ) , v j (c j ) = u(e j )a n dm ax l u (x ) - -vf lx) [ f f ( x , v~(x), vj(x ))d x ,a j a j

con t r ad i c t i ng o u r hy po thes i s t ha t u i s a s t rong r e l a t i ve min imize r. Th us ~ i s e i t he rnon dec re as ing o r non inc rea s ing nea r Xo , t he l a t t e r pos s ib i li t y i s exc luded byno t ing tha t by in t eg ra t i ng (2 .3 )(c f . (2 .5) ) we obta in

0 ~ y j ) 6 + M U(Xo) - - u (y i )X o - - y j X o - - y ]

so t ha t o r < 0 fo r j su ff ic i en tly l a rge . Th i s p rov es ou r c l a im . N ow le t x j - ~ xo ,z - -> Xo w i th x~ > z j. T he n fo r la rge en ou gh j ,

u xj) - u zj)x j - - ~

u (x f i ~x (x j) , M ) - - u ( z f i ~x (z j) , M )

u (Xj ; ~x (Z l) , M ) - - u ( z i ; ~x (z j) , M )

~ - - z j.= u (w j , o~ (z j) , M )

> M - - d ,

w h e r e xy ~ w j ~ z~and we hav e used (2 .3 ). Thus , s i nce M , ~ a r e a rb i t r a ry ,

l im u(x j ) - - u ( z j ) _ _ + c ~ . (2.8)- ~ x j - - z j

I n p a r t i c u l a ru ( x o )ex i st s i n the s ense o f (2 .1) and equa l s + c~ . A s imi l a r a rgu m en tapp l i e s i f xo = a o r xo = b . W e hav e thus show n tha tu ( x ) exis t s in the senseo f (2 .1) fo r a l l xC [ a, b ]. Th e con t inu i ty o f u a t Xo i s obv io us i f Xo E D , an dfo l lows s imp ly f ro m (2 .8 ) o the rwi se . [ ]

A s a n a p p l i c a t io n o f T h e o r e m 2 .7 w e p r o v e t h e f o l lo w i n g v e r s io n o f re s u lt s o fTONELLI [31, Vo l . I I , pp . 361 , 366 ], wh ich sh ou ld be c om pa red wi th Th eo rem 2 .3 .

T h e o r e m 2 .10 . L e t f pp > 0 a n d su p p o s e t h a t

f ( x , u , p )l i m o o

Ipr ~o~ iP lf o r e a c h x E [a , b ] , u E R .


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338 J .M . BALL & V. J . MIZEL

Le t u ( ') E s l be a s t rong re la t ive m in imizer o f I and suppose e i the r tha tfu( ' , u( ' ), u ' ( . )) E L ' (a , b) or that f~( . , u( . ) , u ' ( . ) ) E L l(a , b). The n u E Ca([a ,bl)

and sa t is f ies(EL) a n d ( D B R ) on [a, b].

Pr oo f . Le t DI be a m axim al re la t ive ly open in terval in D ---- [a , b] \ E . By Theo -rem 2 .7 , u E C a ( D r )an d sa t i s f ies (EL ) and thu s (D BR ) on D 1. I f fu( ' , u (. ), u ' ( . ))E L l ( a , b ) t h e n b y ( E L )

Ifp(X,u(x) , u'(x))] ~ con st . , x E D1 . (2.9)

If f~( . , u( .) , u '( . )) ELl(a , b ) t h e n b y ( D B R )

l u ' ( x ) f p ( X , u ( x ) , u ' ( x )) - - f ( x , u ( x ) ,u ' (x) ) [ ~ cons t . , x 6 D r. (2 .10)

By the fo l low ing lem m a, e i ther (2.9) o r (2 .10) impl ies tha t u ' is bou nd ed in D 1,a nd th us t ha t D l = D = [ a , b ] . [ ]

L e m m a 2 . 11. L e t f s a t i sf y t h e h y p o t h e s es o f T h e o r e m 2 .1 0 . T h e n

I f~ x , u , p ) [ ~ oo ,p L x , u , p ) - ] x , u , p ) - + o 0

as ]p ] - -~ oc , un i fo rm ly fo r x E [a, b ] and fo r u in com pac t se t s o f .

P r o o f . B y t h e c o n v e x it y o f f ( x , u , .) w e h a v e t h a t

f ( x , u , O) >= (x , u , p ) - - p fp (x , u , p ) ,

an d he nc e, fo r p =4= 0,

P ~ X ,~ Jp ( u , p) >= ( x i ~ 1' p) f (X,[plU,O)

Therefore , for f ixedx u

l i m fp (x , u, p) = 0 pfimoofp(x u, p) = -- oo . (2.11)

Bu t fp(X, u , p) i s increa s ing in p . Th us i fxj -+ x , u j -+ u , P i -- ->o o we havef o r p j > M ,

fp(xj , uj , p j ) ~ fp(x j , u j , M ),a n d s o

l i m in f fp (Xj , u , p j ) ~ fp (x , u , M ) .j- eo

Le t t ing M --> oo we deduce th a t t he f i r s t l imi t i n (2.11) i s un i fo rm fo r x , u incom pac t se t s; o the rwise the re w ou ld ex i st a convergen t sequence (x j, u j) and a

sequ ence Pl---> c~ suc h th at l im inffp(Xj, uj, pj) < c~. Th e case p ---> -- oo ist r ea t ed s imi l a r ly.

To p r o v e t h e s e c o n d a s s e rt io n o f t h e l e m m a w e n o t e t h a t

f ( x , u ,1) > = f ( x , u , p ) - - ( p - -1)fp(X, u , p ) ,


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Minimizers Ne ed N ot Sa t is fy Euler-Lagrange Equ at ion 339

an d hence , p ro v ide d p > 1 ,

f ( x , u , p ) p pp fp ( x , u, p ) - - f ( x , u , p ) > - -= p p - - 1 f ( x , u ,1 ) . ~ . p _ l

T h e r e f o r e , f o r f i x e dx , u ,

l im [ p fp ( X , u , p ) - - f ( x , u ,p)] = oo . (2 .12)p--- , ~

Tha t t he l im i t i n (2 .12 ) i s un i fo rm fo r x , u i n compac t s e t s fo l l ows a s above us ingt h e f a c t t h a t p ~ ( x , u , p ) - - f ( x , u , p )i s i nc reas ing in p fo r p > 0 . Th e ca sep - + - - c ~ is h a n d l e d s im i la rl y. [ ]

C o r o l l a r y 2 . 1 2 . L e t f = f ( u , p ) s a t i s f y f p p > 0 a n d

f ( u , p )l im - - - - o o f o r e a c h u E R . (2 .1 3)

I p L

I f u ( . ) E ~ i s a s tr o n g r e l a ti v e m i n i m i z e r o f I t h e n u ( .) EC3([a , b ] ) a n d s a t is f ie s( E L ) a n d ( D B R ) on [a , b] .

F i n a l ly, w e r e m a r k t h a t i f 1 < q < o o t h e n T h e o r e m 2 . 7 sti ll h o l d s ( w i t ht h e s a m e p r o o f ) i f w e r e p l a c e ~ r b y ~ r ~W ~ ' q ( a , b ) b o t h i n t h e s t a t e m e n t o fthe t heo rem and in t he de f in i t i on o f a s t rong r e l a t ive min imize r. Th i s is pe rh apso f i n t e r e s t s ince i n w 5 we show tha t m in imize r s i n ~r and ~ rw1 q(a , b ) m a ybe d i f f e r en t .

w 3 An integral with a scale invadanee property

I n t h i s s e c t io n w e c o n s i d e r t h e p r o b l e m o f m i n i m i z in g

1

I u ) = f [ ( x 2 - u 3 ) 2 ( u ) + e ( u ) 2 1d x ( 3 . 1 )0

sub jec t t o

u(O) = 0, u(1) = k, (3.2)

wh ere e > 0 an d k > 0 a re g iven .N o t e t h a t t h e i n t e g r a n d

f ( x , u , p ) = ( x 2 - -u3) 2 p14 q_ep2 (3.3)

in (3.1) sat isf ies

fpp ~ 2e > 0. (3.4)

T h e E u l e r - L a g r a n g e e q u a t i o n c o r r e s p o n d i n g t o ( 3 . 1 ) i s

d~xx (7(x2 -- u 3 ) z (u )13 + eu ) = - - 3 u Z ( x 2 - -u s) (u ) 14. (3.5)


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340 J. M. BALL V. J. MIZEL

I t i s eas i ly ver i f ied th at (3 .5) has an ex act solut ion u = ~:x] on (0 , 1] pro vide d

e = (1 -- ~3) (13~ca _ 7) . (3.6)

Def ine0(z) = (~)12 3 ( i - - r) (133 -- 7) .

Di ffe ren t i a t ing 0 we see tha t 0 a t t a ins i ts m ax im um in the in te rva l (~ , 1 ) a t the2 5

p o in t 3 - - - -. 86 8 9 28 a n d t h at 0 ' ( 3 ) > 0 f o r ~ < x < x * ,39 . . . .

0 '(~) < 0 fo r 3 < ~ < 1. D efine

e o = 0 ( z * ) = . 0 0 2 4 7 4 . . .

We h a v e t h u s p r o v e d

P r o p o s i t i o n 3.1 . I f 0 < e < eo t h e E u l e r - L a g r a n g e e q u a t i o n(3.5) h a s e x a c t l yt w o s o l u t i o n s i n(0, 1] o f th e f o r m u = k x ~ , k - > 0 ; t h e c o r r e s p o n d i n g v a l u e s o f - ks a t i s f y ~ < kt( e) 3 < 3 < k2(e) 3 < 1.I f e = e o t h e r e is j u s t o n e s u c h s o lu t i o n ,n a m e l y u = ( 3 *) 8 9x ~ ; i f e > e o t h e r e a r e n o s u c h s o l u t io n s .

Th e in te gra nd f in (3 .3) sa ti s f ies the scale invar iance pr op er ty

f ( 2 x , 2eu, 2 e- I p) = 2~( x, u , p)

f o r a ll 2 > 0 a n d a ll ( x , u , p ) , w h er e X = $ a n d ~ = - - ] .b y m a k i n g t h e c h a n g e o f v a r ia b l es

vV = U l i t , Z = ~ , q = V , X - - ~ e t .

x

Set t ing t = 1 / x in (3 .7) we obta in

f ( x , u , p ) = x e F ( z , q ) ,

w h e r e

(3.7)

We e x p l o i t t h i s

(3.8)

(3.9)

F z, q) f 1,zL ? ,z q ) . 3 . 1 0 )I t i s eas i ly ve r if ied tha t , fo r an y sm oo th in teg rand sa t i s fy ing (3 .7 ) , (EL ) i s t r ans fo rm -e d i n t o t h e a u t o n o m o u ss y s t e m

d zd t q z ,

(3.11)a F q

d t = F , - - a .

M ore p rec i se ly, i f 0 < a < b < oo and u i s a sm oo th so lu t ion o f (EL) on (a , b )sa t i s fy ing u ( x ) > 0 for a l l x G (a , b) , the n

q( t ) = 7 -1[ u(e t ) ] O-v) /v u , (e t ) , z ( t ) = e - t [u (e t ) ] I1~" (3.12)


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Minimizers Ne ed N ot Sa t is fy Euler-Lagrange Equ at ion 341

i s a sm oo th so lu t ion o f (3 .11 ) fo r l og a < t < l og b . Con ve r se ly, i f ( q, z ) is asm oo th so lu t ion o f (3 .11 ) de f ined fo r e~ < t < f l a nd sa t is fy ingz ( t ) < 0 fo r a l l

x E (0~,f l) the nu(x) ---- [x . z( logAx)] ~ (3.13)

is a s m o o t h s o l u t i o n o f ( E L ) f o r e ~ 0 i s a rb i t ra ry . Th ea rb i t r a ry cons t an t i n (3 .13 ) a r i s e s f rom the f ac t t ha t , s i nce (3 .11 ) i s au tonomous ,i f z ( t ) i s a so lu t ion so i sz ( t -k l og A) ; equ iva l en t ly, i fu(x) i s a pos i t ive so lu t iono f ( E L ) s o i sA - e u ( A x ) .Note t ha t (3 .11 b ) i s t he Eu le r-Lagrange equa t ion fo r t hein t eg ra l

j , o ( x ) . x ,

o b t a i n e d b y m a k i n g t h e c h a n g e o f v a ri a b le s ( 3 .8 ) in ( 3 .1 ). A s h a s b e e n p o i n t e dou t t o u s b y P. J . OLVER, the f ac t t ha t t he s ca le i nva r i ance p ro pe r ty (3 .7) imp l ie st h e e x i st e n c e o f a c h a n g e o f v a r ia b l e s m a k i n g ( E L ) a u t o n o m o u s is a c o n s e q u e n c eo f t h e t h e o r y o f L i e g r o u p s(cf. INCE [21, Ch ap . 4 ]) . W e r em ark tha t t he abov er e d u c t i o n t o a n a u t o n o m o u s s y s t em i s u s e d i n B A L L [3 ] a s a t o o l f o r s t u d y i n gt h e r a d i a l e q u a t i o n o f n o n l i n e a r e l a s ti c it y in n s p a c e d i m e n s i o n s , t h e a p p r o p r i a t ev a lu e s o f y, ~ b e in g 7 , = 1, ~ = n - - I .

F r o m n o w o n w e a s su m e t h a t f i s g i ve n b y (3 .3 ), a lt h o u g h i t w ill b e a p p a r e n tt o t h e r e a d e r t h a t m u c h o f w h a t w e h a v e t o s a y a p p l i es t o a g e n e r a l c l as s o f in t e -

g rand s s a t i sfy ing (3 .7) fo r su i t ab l e y, ~ . F o r l a t e r u se we n o te t h a t s i nceF ( z , q) : ~ ) 1 4 1 - - z 2 ) 2 z - 1 4 1 3 q 1 4 _ .[_ ~ ) 2 e z - 2 / 3 q 2 ,(3.14)

( 3 . 11 ) t a k e s t h e f o r m

d zd t - - q - - z ,dq (3 .15)

d t G(z , q ) ,

G (z, q) aef Fz - t- ~ Fq -- (q - - z) Fqzqq

q .._ 2.2 . ~ _ ) , l 1 - z z ) [ 1 3 q 7 - z 2 ) - 8 4 z ]q i , e z 4 _ ]3 . 1 6 )

3 z / 9 1 ( t ) 1 2 ( 1 - z 2 ) 2 q ' 2 + e z 4 J

W e s tu dy (3 .15 ) i n t he f ir s t qu ad ran t o f t he ( z , q ) p l ane . N o te t ha t so lu t ions o f(3 .15 ) i n t he f ir s t qu ad ran t co r r e sp on d to p os i t i ve so lu t ions u o f (3 .5) w i th u (x ) => 0 .I t i s c lear th a t an y m inim izer o f (3 .1) , (3 .2) sa t is f iesu (x) ~= 0 a.e. x C [0, 1],s in c e o th e r w i s e th e v a l u e o f I c o u l d b e r e d u c e d b y m a k i n g u c o n s t a n t o n s o m ein terva l .

B e f o r e p r o c e e d i n g w i t h t h e d e t a i ls o f o u r p h a s e - p l a n e a n a ly s is , t h e r e a d e r m a ywish to l o ok a t F igu re 3 .1 so a s t o s ee wh e re we a re head ing .

W e beg in by exam in ing the r e s t po in t s o f (3 .15 ) i n z > 0 , q > 0 . F ro m (3 .15 ),- - 3

(3 .16 ) t he se a r e ea s i ly s een to be g iven by q = z = k~ , wh e re k > 0 s a ti sf ie s


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342 J . M. BALL V. J . MIZEL

- - 2( 3 .6 ) , a n d c o r r e s p o n d t o t h e s o l u t i o n s u =k x f d i s c u s s e d i n P r o p o s i t i o n 3 .1 .

T h u s , f o r 0 < e < e o, t h e r e a r e p r e c i s e ly t w o r e s t p o i n t s , n a m e l y q = z = k l ( e ) ~

a n d q = z= k 2 ( e ) 3 , w i t h 7 < ~ 1 ( e) 3 Ts < 1. W e d e n o t e t h e s e p o i n t sb y P 1 a n d P 2 r e s p e c ti v e ly. W e s t u d y t h e n a t u r e o f th e r e s t p o i n ts b y l i n e a r iz a t io n .

- -3 - -3 - -3

T h u s l e t P d e n o t e a r e s t p o i n t q = z = k -~ . S e t t i n g z = k ~ - q - a , q = k ~ - + bg i v e s ( 3 . 1 5 ) t h e f o r m

w h e r e

a n d

( 3 1 k a - - 2 8 )

( 1 - - ~ a ) ( 1 4 - - 1 3 k 3 )

T h e e i g e n v a lu e s o f A a r e g i v e n b y

2 = ~l ( - -1 -4- 1/25 q- 36a(7~))

T h u s ,25

( i ) i f (~(k) < 36

( i i ) i f a ( k ) 25- - 36~

(iii) if --~-~25 o ' (k - ) < - -g -, 2 _ < 2 + < 0 ,

( i v ) i f a ( k ) = - - 3 , 2 _ = - - 8 9 2 + = 0 ,

(v) i f 2 < (~(~) , 2_ < 0 < 2+.

( 3 . 1 7 )

2 + , 2 _ a r e c o m p l e x ,

1 a n d A h a s a d o u b l e e l e m e n t a r y d i v is o r ,+ , 2 _ - - 6

A s i s w e l l k n o w n ( c f . H A RTM A N [ 1 9 , p . 2 1 2 , f t .] ) , c a se s ( i ) - (i i i) c o r r e s p o n d t o Pb e i n g a s in k , a n d c a s e ( v ) t o a s a d d l e - p o i n t . C a s e ( i v ) i s a c r i t i c a l c a s e w h e r e t h es t a b i li t y is d e t e r m i n e d b y t h e n o n l i n e a r t e r m s i n (3 . 1 7) , a n d w e d i s c u s s th i s p r e s e n t -l y. I n c a s e ( i ), P i s a f o c u s . I n c a s e ( ii ) P i s a n i m p r o p e r n o d e , a l l s o l u t i o n s o f ( 3 .1 5 )n e a r P a p p r o a c h i n g P w i t h s l o p e ~- a s t - + c ~ . I n c a se ( ii i) P is a n i m p r o p e r n o d ew i t h a s i n g le p a i r o f s o l u t io n s a p p r o a c h i n g P w i t h s l o p e 2 _ + 1 E ( 2 , ~_) a st --~ c x~ , a n d a l l o t h e r n e a r b y s o l u t i o n s a p p r o a c h i n g P w i t h s l o p e 2 + -]- 1 E ( ~ , 1 )a s t ~ c ~ . I n c a s e ( v ) t h e s lo p e o f t h e s ta b l e m a n i f o l d o f P a t P i s 2 _ 1 < 2 ,

t h a t o f t h e u n s t a b l e m a n i f o l d 2 + 1 > 1 . W e n o w n o t e t h a t t r ( k ) > - - 2

F i g . 1. T h e p h a s e - p l an e d i a g r a m f o r ( 3.1 5 ). S h o w n i n p a r ti c u l a r a r e th e s m o o t h s o l u t io no r b i t , w h i c h l e a v e s t h e o r i g i n w i t h s l o p e 3 /2 , a n d t h e s t a b l e a n d u n s t a b l e m a n i f o l d s o f P 2 .T h e a b s o l u t e m i n i m i z e r s o f I c o r r e s p o n d t o a p p r o p r i a t e p o r t i o n s o f th e d a s h e d c u r v e s( s e e T h e o r e m 3 . 12 ) .


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344 J . M . B A L L V. J . M I Z E L

( respec t ive ly ( r(k) < - -3 ) i f an d on ly i f

3932 - - 503 -~ 14 > 0 ( respec t ive ly G 0 )

2 5

w h e r e ~ - -- -~ 3 , a n d s in c e 3 9 G ~ t h is h o l d s i f a n d o n l y i f 3 > 3 =

25 + 1 /~-39 ( respec t ive ly 1: G 3*) . T he case a (k- ) = - -a cor resp on ds to 3 = 3* .

S imi l a rly, a (k ) > - - ~ ( r e spec tive ly a (k ) G --~6) i f an donly i f

32532 - 4273 q- 126 > 0 ( respec t ive ly G 0 ) ,

wh ich ho lds i f and on ly i f 3 > 3~ = . 86 63 4 . . . ( r e spec tive ly 3 G 3~) . W e l e t

e l = 0 (3~) = . 002473 . . . . W e have thus p rov ed

Proposi t ion3 .2 . L e t 0 G e G e o. T h e n P ~ is a s & k a n dP i s a s a d d l e p o i n t .

d zS i n ce ~ - = q - - z t h e f lo w i n t h e r e g i o n 0 ~ q < z is t o t h e l ef t, t h a t i n

the r eg ion 0 < z < q t o the r i gh t . W e a lso make f r eq uen t u se o f t he d i r ec t i ond z

of f l ow on the d i ago na l q = z , wh e re ~-~ = 0 , g iven in t he fo l l owing l em ma .

L e m m a 3 3

(i) L e t O < e < e o . T h e n G ( z , z ) > O f o r O < z < k l ( e ) ~ a n d f o r3

k2(e )2 < z < o0 , w h i l e G ( z , z ) < 0 f o r k ~ ( e ) 3 < z k'2(e)k .1

(ii) L et e = eo . Th en G(z , z ) > O fo r a l l z > O, z =t= (3*)~ .

(iii) L e t e > e o . T h e n G ( z , z ) > O f o r a l l z > O .

F o r t h e p u r p o s e o f s t u d y i n g th e e x i s te n c e o f p e r io d i c o r b i t s i t is c o n v e n i e n tt o i n t r o d u c e t h e n e w v a r i a b l er = Fq(z , q) . I t i s eas i ly ver i f ied , us ing the fac tt h a t F q q > O , t h a t (z, q ) - + (z , r ) m a p s z > 0 , q > 0 o n t o z > 0 , r > 0 a n dhas a smoo th inve r se . Thus (3 .11 ) i s equ iva l en t t o

d z d e f ~d t ---- q( z,r ) - - z = z ( z , r ) ,

(3.18)

d r F ~ z ,q z, r)) + ~ r ~ R z, r) .d t

A n e a s y c o m p u t a t i o n s h o w s t h a t

8 Z 8 R. . . .z q- 8r 89 (3.19)

I n t e g r a t i o n o f (3 .1 9 ) o v e r th e r e g i o n e n c l o s e d b y a n o n t r i v i a l p e r i o d i c o r h o m o -c l in i c o rb i t g ives a con t r ad i c t i on . We have thus p roved


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M i n im i z e r s N e e d N o t S a ti sf y E u l e r - L a g r a n g e E q u a t i o n 3 4 5

roposition3 . 4 . T h e s y s t e m( 3 . 1 5 ) h a s n o n o n t r i v ia l p e r i o d i c o r b i t a n d n o h o m o e l i n i cor bi t in z > O, q > O .

W e n e x t s tu d y t h e c o n t i n u a t i o n a n d a s y m p t o t i c p r o p e r t i e s o f s o l u ti o n s .

P r o p o s i t i o n 3 . 5 .L e t Zo > 0 , qo > 0 ,a n d l e t ( z ( t ) , q t ) ) d e n o t e t h e u n i q u e s o l u t i o no f ( 3 . 1 5 ) w i t h z O ) = Z o , q O ) ~ q o . T h e n z t ) , q t ) ) e x i s t s a n d r e m a i n s in z > O ,q > 0 o n a m a x i m a l i n te r v a l tm in , ~ ) , w h e r e - - o o ~tmin < 0 . A s t ~ c o ,

3 3

e i t h e r z t ) - + cx~ a n d q t ) - + cx~ o r z t ) , q t ) ) - + k - f , k 2 - ), a r e s t p o i n t . A st --~ train + e i ther z t ) , q t ) ) ~ 0 , O) o r z t ) --+ oo an d q t ) --> c ~- C Zo , qo) E

3 3

[ 0, ~ ) * or z t ) , q t ) ) - -~- k~ , k -~) , a res t po in t .

P r o o f . L e t t h e m a x i m a l i n t e r v a l i n w h i c h t h e s o l u t i o n ( z (t ),q t ) ) e x is ts a n d r e m a i n sin z > 0 , q > 0 be ( t ra in , t m ~ x ), w h e r e - -cx~ ~ tmin < 0 0 , q >= 0 f o r a l lt E [ 0 , tm a x) ( r e s p e c t i v e l y t E (tra in , 0 ] ) t h e n t m a x = o o ( r e s p e c t i v e l y t m i n = - - ~ ) ,a n d w e c a n a p p l y t h e P o i n c a r 6 -B e n d i x s o n t h e o r ycf. H A RT ~ A N [ 1 9 , p . 1 5 1 f t . ] ) .B y P r o p o s i t i o n 3 . 4 th e o n l y p o s s ib i l it i e s a r e t h a tz t ) , q t ) ) t e n d s t o a r e s t p o i n ta s t - + o o ( r e s p e c t i v e l y t - + - - ~ ) , o r t h a t th e ~ o -l im i t s e t ( r e s p e c t iv e l y c ~- li m its e t ) o f (z ( .) , q ( .) ) c o n t a i n s m o r e t h a n o n e r e s t p o i n t ( a n d t h u s 0 < e < e 0). T h el a t t e r c a s e c a n n o t o c c u r s i n c e P 1 is a s y m p t o t i c a l l y s t a b l e .

d zN e x t w e n o t e t h a t o n a n y o p e n t - i n te r v a l w h e r eq t ) ~ z t ) w e h a v e d -7 @ 0 ,a n d t h u s t h e o r b i t h a s t h e r e p r e s e n t a t i o n q =q z ) , w h e r e b y ( 3 . 1 5 )

d q q 2 [ (~_)12 1 - - z 2 ) [13q(7 - - z 2 ) - - 84z1q l l + e z 4 . ]d z - - 3 z q - z ) I_" 91(-~ ) 12 (1- - - 7 2 2q 1 2 _[_eZ4 I c le f H ( z , q , e ) .

3.20)

W e f i rs t e l i m i n a t e t h e p o s s i b i l i t y t h a tq z ) b e c o m e s u n b o u n d e d a s z -- ~ ~ E ( 0 , o o )e i th e r f r o m a b o v e o r b e lo w. B y g e n e r a l r e s u lt s o n o r d i n a r y d i ff e re n t ia l e q u a t i o n sw e w o u l d t h e n h a v eq z) --~ + cx~a s z ~ ~" -1- orq z ) ~ ~ a s z --~ ~ - - . I f

= ~ 1 , t h e n f o r q l a r g e a n d f o r z n e a r ~ w e h a v e

d q [ = q (32-)'2 ( 1 - z 2 ) [ 1 3 ( 7 - z 2 ) - 8 4 ~ ] e q'iS -< C q ,

q ) z,~ 3z 1 - - 91(-~) 12 (1 - - z2 ) 2 -t- q12

w h e r e h e r e a n d b e l o w C d e n o t e s a g e n e ri c c o n s t a n t . T h u s q i s b o u n d e d n e a r 5 ,a c o n t r a d i c t i o n . I f ~ = 1 , w e o b s e r v e t h a tq z ) s a t i s f i e s

d~ z ( ( q - - z ) f q - - F ) = - - 8 9F q , ( 3 . 2 1 )

t I t w i l l b e s h o w n i n P r o p o s i t i o n 3 .6 t h a t i n t h i sc a s e t m i n = - - oo an d C(Zo, qo) > 0 .


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3 4 6 J . M . B A L L & V. J . M I Z E L

w h e r e F is g i v e n b y ( 3 .1 4 ). ( T h i s is e s se n t ia l ly t h e D u B o i s - R e y m o n d e q u a t i o n f o r

I . ) N o w

/2 ~\13 14 4e 2q zFq=281~-z - 7 ( l - - z 2 ) 2 + - ~ -- - ~ - 2 " 3~ 9 ,a n d

~v(z, q) ae_f q __ z ) F 0 - - F

/ 2 q \ t a 14 Z2) 2( 1 3 l ~ z ) + 2 q 2 2

T h u s , f o r z n e a r 1 a n d q l a rg e ,

z 2 13[F ~ I < C ( q ' 3 ( l - - z ) 3 - + q )

14 13C ( q l 4 ( 1 - - 2 2 ) 2 - it-qi3)/Z

13< ~ C ( q l 4 ( 1 _ _ 2 .2 )2 _ [.. q 2 ) ~ ,

a n d s o b y ( 3 . 2 1 )

~ 13d~p z, q z)) CI ~o(z,q z ) ) I ~ .

T h u s ~ o ( z ,q z) ) is b o u n d e d n e a r z = 1 , w h i c h is a c o n t r a d i c t i o n .T h e c a s e w h e n (Z o, q o ) i s a r e s t p o i n t b e i n g t r i v i al , w e n o w c o n s i d e r t h e r e -

m a i n i n g ca s e s . F i r s t s u p p o s e t h a t q o < Z o. N o t e t h a t q = 0 , 0 < z < c ~ i sa n o r b i t o f ( 3 . 1 5 ), a n d t h a tG z ,q) > 0 i f z > 0 , q > 0 a n d z + q i s su ff i c i e n t ly

d zs m a l l . S i n c e - ~ - < 0 f o r q < z i t n o w f o l l o w s t h a t e i t h e rz t ) , q t ) )r e m a i n s

b e l o w t h e l in e q = z o n [ 0, tm ax ), a n d h e n c e b y t h e f ir s t p a r t o f t h e p r o o f t e n d st o a r es t p o i n t , o r t h a tZ to) = q to)f o r s o m e t o > 0 . I n t h e l a tt e r c a s e i t m a yh a p p e n t h a t z t O = q t l ) f o r s o m e t~ ~ t o , w i t hq t ) > z t ) f o r t o < t < t l .

/ 1

I f s o , t h e n b y t e m m a 3 . 3, 0 < e < e o a n dZ to) < -k~ e)~ < z t~)k2 8) ~-,s o t h a t , u n l e s s z t ) , q t ) ) ~ P~a s t - -~ o o w i t h o u t a f u r t h e r c ro s s i n g o f q = z ,z ( t2 ) = q ( t2 ) fo r so m e t2 > t~ . I f z ( t2 ) 0 , q > _ 0 f o rtmin ~ t ~0 a n d h e n c e t e n d t oP ~ a s t ~ - o ~ ; t h is i s i m p o s s i b l e a s P 1 i s a s i n k . T h u s b y P r o p o s i t i o n 3 . 4 ,z ( t2 ) > Z to), w h i c h i m p l ie s t h a t z t ) , q t ) )r e m a i n s i n a c o m p a c t s u b se t o f z > 0 ,q ~ 0 f o r 0 = < t < t m a x , a n d t h u s t en d s t o P ~ a s t - + o ~ .

T h e a b o v e c o n s i d e r a t i o n s s h o w t h a t , a s r e g a r d s t h e b e h a v i o r f o r t ~> 0 , i ts u f f i c e s t o e x a m i n e t h e c a s e w h e nq t ) > z t )f o r a l l t E [ 0, t m ~ ) a n d t h e c o r r e -s p o n d i n g s o l u t i o n c u r v eq z ) i s de f ined fo r a l l z => z0 . T o s ho w th a t tm~x ---- o o wee x a m i n e th e sl o p e o f t h e v e c t o r f ie ld o n th e l in e q = z , w h e r e /z > 1 . O n th i s


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M i n i m i z er s N e e d N o t S a ti sf y E u l e r - L a g r a n g e E q u a t i o n 3 4 7

l ine , as z - -> e%

1 7 13 ] 1rz12d z

= 2 1 ( - 1 ) 1 + o ,

( 1 ) 21 t h e r e e x i s t sh e r e t h e o t e r m i s i n d e p e n d e n t o f / z . H e n c e , p r o v i d e d / ~ 0 > ~ ,

> 0 s u c h t h a t i f z > k a n d /z ~ / ~ o t h e n ~ ( z ) < / ~ o n q = ~ z . C h o o s i n g

/ z > q (k )^ w e d e d u c e t h a tZ

~:(t) = ~r,a n d h e n c e t h a t t m a x = cx~ .We c o n s i d e r n o w t h e b e h a v i o r o fz t ) , q t ) )fo r t E ( tr ain , 0 ] . Su pp os e f i r s t

t h a t q o > Z o . I f q t ) > z t ) f o r a ll t E ( t ~ i n , 0 ] , t h e n e i t h e r i n f z t ) > O o rtE tm in,0]

z t ) -+ 0 a s t - - ~ tra in @ . I n t h e f o r m e r c a s e , s i n c eq t ) c a n n o t b e c o m e u n b o u n d e d

a s t ~ t r ai n + , t h e c u r v e li es i n a c o m p a c t s et o f z > 0 , q ~ 0 a n d w e m u s t h a v et h a t tmi~= - - o o a n d z t ) , q t ) )t e n d s t o a r e st p o i n t a s t - + - - o o . I fz t) --> 0

d qa s t---> t ra in + t h e n b y ( 3 .2 0 ) t h e c o r r e s p o n d i n g c u r v eq z ) satisfies ~z-z> 0

f o r s u f f i c i e n t l y s m a l l z > 0 , s o t h a tq t m i n ) d el i m q t ) e x i s t s . I f q t ~ ) > 0t ~ t r n i n +

t h e n b y ( 3 . 2 0 ) ~ > __C f o r s u f f i c i e n t l y s m a l l z > 0 , w h e r e C > 0 i s a c o n s t a n t ,d z = z

a n d i n t e g r a t i o n o f t h is i n e q u a l i t y g iv e s a c o n t r a d i c t i o n . T h u sz t) , q t ) ) -+ 0 , O)a s t - - > t r ai n + . O n th e o t h e r h a n d , i fq t ) = z t ) f o r s o m e t E ( tm i ., 0 ] t h e n

q t x ) . < z h ) f o r s o m e e a r l i e r t i m e .I t o n l y r e m a i n s , t h e r e f o r e , t o c o n s i d e r t h e c a s e w h e n q o < Z o. F i r s t , i f

q t) < z t ) fo r a l l t E t r a i n , 0 ]t h e n e i t h e r z t ) r e m a i n s b o u n d e d a s t ~ t ~ , + ,i n w h i c h c a s e tra in = - - o o a n dz t ) , q t ) )t e n d s t o a r e s t p o i n t a s t - -~ - - o o ,

qo r l i m z t ) = ~ . I n t h e l a t t e r c a s e , b y ( 3 . 2 0 ) , ~ zz < 0 , f o r z 2 > 7 , q < z ,

t- train

a n d s o a s t ~ tmi .q- q t )t e n d s t o a n o n n e g a t i v e l im i t , w h i c h w e d e n o t e b yC(Zo , qo) . N ex t , i f q to) = Z to)f o r s o m e t o E (tra in , 0 ] t h e n q ( i ) > z ( i ) f o r s o m et-E ( tm i ., t o) . W e h a v e a l r e a d y t r e a t e d t h e c a s e w h e nq t ) > z t ) fo r a l l t E

( t~ i n ,/ '] a n d t h u s i t r e m a i n s t o e l i m i n a t e t h e p o s s i b i l i ty t h a tq t j) = z t j) f o r a ninf in i te se q ue nc e t j - -> t ra in- l- , ~a n d o f c o u r s e th i s c a n o n l y o c c u r fo r 0 < e < e 0.T h e c o r r e s p o n d i n g o r b i t w o u l d s p i r a l e i th e r i n w a r d s o r o u t w a r d s a s t --~ t m i ~ + .I f i t s p i ra l le d i n w a r d s t h e n c l e a rl y w e w o u l d h a v e tm i, = - - e o a n dz t ) , q t ) ) - -~ P~a s t ~ - - c % w h i c h is i m p o s s i b l e s i n c e P ~ i s a s i n k . I t m u s t t h u s s p i r a l o u t w a r d s ,


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3 4 8 J . M . B A L L V . J . M I Z E L

a n d o f c o u r s e i t c a n n o t r e m a i n in a c o m p a c t s u b s e t o f z > 0 , q ~ 0 , s in c e o t h e r -w i s e i t w o u l d h a v e to t e n d t o P z a s t ~ t ~ i n + , w h i c h is c l e a r ly i m p o s s i b l e . F u r -

t h e r m o r e t h e o r b i t m u s t r e m a i n u n d e r t h a t p a r t o f t h e s t a bl e m a n i f c l d o f P zly ing in q > z , a nd soz ( t ) ~ 0a s t j ~ t ~ i ~ + . B u t t h e s o l u t i o n c u r v e(z , ( t ) , q , ( t ) )

o f ( 3. 15 ) s a t is f y in g z , ( 0 ) = 1, q ,( 0) : - - a p p r o a c h e s t h e z - a x is a s r - - -~ o o ,r

crossing q -----z a rb i t r a r i l y c lo se t o t he o r i g in , wh ich imp l i e s t ha tz(t j) is b o u n d e da w a y f r o m z e ro . [ ]

H e n c e ~ t ) - -~ 21 a s

o f (3 . 22 ) we ob t a in

N o t e t h a t P r o p o s i t i o n s 3 .2 , 3 .5 t o g e t h e r i m p l y t h a t w h e n e = e o t h e u n i q u e- - 3

f i xed po in t q = z = k ~ i s uns t ab l e .I t is p o s si b l e t o s p e c i f y m o r e p r e c is e l y t h e a s y m p t o t i c b e h a v i o r o f t h o s e

so lu t i ons o f ( 3 .15 ) s a t i s fy ing z ( t ) -+ o q(t ) -+ r a s t - + o o . F o r s u c h a s o l u -q(z)

t i o n w e h a v e s e e n i n t h e p r o o f o f P r o p o s i t i o n 3 .5 t h a t ~ i s b o u n d e d f o r l a rg e t .

S e t t i n g r z ( t ) w e s e e t h a t ( 3 . 1 5 ) b e c o m e s

= r - 9 9 ) ,

992 [ . (~ )1 2 ( ~ 2 1 ) [ 1399 (7 r 1 )__ 84r + e r ( 3 .22 )---= (1 - - 99) + T 91 (]) 12 (r __ 1)2 9~2 + eCX2 ,

defq( t )w h e r e 9 9 ( 0 - -z ( t ) a n d h e n c e as t ~ ~ ,

4 99(1 20= - - ~i 99) + o(1 ).2~t - + oo . L ine a r i z i ng ab ou t t he r e s t po in t r = O , 99 =

- ( ( t ) ~ C l e 20 ,

19( 0 - - ~ol


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Minimizers Nee d No t Sa t is fy Euler-Lagrange Equ at ion 349

P r o p o s i t io n 3 . 6 .E v e r y s m o o t h s o l u t io n u o f(3.5) w i th u (O) = O, u (O) > 0 co r-r e s p o n d s t o a s i n g l e o r b i t o f ( 3 . 1 5 ) i n z > O , q > 0 t h a t l e a v e s t h e o r i g i n z = q = 0

w i t h s l o p e -~ . T h e o n l y o t h e r o r b i t s o f ( 3 . 1 5 ) l e a v i n g t h e o r ig i n c o r r e s p o n d t o s o l u ti o n su o f (3.5) w i t h U (X o) = 0 f o r s o m e x o >0 ; t h e s e o r b i t s s a t i s f y

q t )l im z ( t ) = li m q ( t ) = 0 , l i m _ c x ~ . ( 3. 23 )

t ~ l O g x o + t ~ l o g x o + t * l o g x o +z ( t )

S o l u t i o n s (z(.), q(.)) o f (3.15) w h o s e o r b i t s h a v e a n u n b o u n d e d i n t e r s e c t io n w i t h0 < q < z c o r r e s p o n d p r e c i s e l y t o s o l u t i o n s uo f ( 3 . 5 ) w ith u(O) > O, u (O) > O,a n d t h u s s a t i s f y t l i m o z ( t )= oo , l im q( t ) = c ~ O, w he re c = c (z (O) ,q(0))

t ~ o o

i s a c o n s t a n t .

Pro of . Le t u be a sm oo th so lu t ion o f (3 .5 ) on som e in t e rva l [0 , a ] , a > 0 , s a ti s -fy ing u(0) = 0 , u ' (0) = ~ > 0 . T he nu ( x ) = ~ x q - o ( x ) , u ( x )= a~ + o(1) , as

3 1 1 3 1 1x --~ 0 + , a n d h e n c e z = o~-~ x -f + o(x2 -) , q = 3(o~-2- x -f + o(x-~ )) .T h u st h e c o r r e s p o n d i n g s o l u t i o n( z ( t ) , q ( t ) ) satisfies

l im z ( t ) = l im q ( t ) = 0, l im q ( t ) 3t ~ - o o t ~ - o o t ~ - o o z ( t ) 2 .

Th a t t h is so lu t ion i s t he s ame fo r any ~ > 0 (up to add ing a con s t an t t o t ) f o l -l o w s f r o m t h e s i m i l a ri ty t r a n s f o r m a t i o n ( 3. 13 ) a n d t h e u n i q u e n e s s o f so l u ti o n s t othe i n i t i a l va lue p rob lem fo r (3 .5 ) .

L e t u~(x) d e n o t e t h e u n i q u e s o l u t i o n t o (3 . 5) s a ti sf y in g u ~ ( 1 ) = 0 , u ~ ( 1 ) =f l > 0 ; t h i s co r r e spo nds t o a so lu t ion(za( . ) , qa( . ) )sa t i s fy ing

a n d

as t - * 0 + .

3[ f l( e t - 1) ,-+- o( e - 1)]~-

za ( t ) = e l = o(1) ,

q~(t ) = -~[f l(e t - -1) -F-o ( e t - - 1)]~- (fl + o(1 )) = o(1 ),

A l s o

l im qa( t ) l ira t 3t- ,o+ zr = t- ,o+2(e - - 1 ) = oo .

Le t 6 > 0 be su ff ic i en tly sma l l. I t fo l l ows f ro m Pro pos i t i on 3 .5 t ha tza(ta) = 6fo r som e min ima l t a > 0 . A l so , si nceqa(ta) > za(ta), t h e c o r r e s p o n d i n g i n t e r -

sec t ion a t x = eta of t he g rap h o f ua w i th 6 ~ x ] i s t ransve r sa l , and thus by

t h e i m p l ic i t f u n c t i o n t h e o r e m ta d e p e n d s c o n t i n u o u s l y o n ft. H e n c e a l soqa(ta)d e p e n d s c o n t i n u o u s l y o n ft. W e e x a m i n e th e b e h a v i o r o fqa(ta) as f l va r i e s f rom0 to oo . We f i r s t show tha t

l im qa(ta) -----oo . (3 .24)f l ~ e o


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3 5 0 J , M . B A L L & V. J . M I Z E L

S i n c e u ' ~ ( x )3 > 0 f o r a l l x ~ 0 , u s i s i n v e r t i b l e ; d e n o t e t h e i n v e r s e f u n c t i o n b yx a ( u ) .B y ( 3 . 5 ) x a ( . ) s a t i s f i e s t h e t r a n s f o r m e d e q u a t i o n

[ 1 9 ( x 2 - - u 3 ) 2 +e x ~ 2 ] x , . , = x . ( x z - -u 3 ) ( 2 8 x x . - - 3 9 u 2 ) , ( 3 . 2 5 )

w h e r e t h e s u b s c r i p ts d e n o t e d e r i v at iv e s w i t h r e s p e c t to u . T h i s e q u a t i o n h a s t h es o l u t i o n x ( u ) ~-~ 1 , u E [ 0 , 8 9 i n t h e n e i g h b o r h o o d o f w h i c h (3 . 2 5 ) c a n b e w r i t t e ni n t h e f o r m X . u - = h ( u , x , x . )w i t h h c o n t i n u o u s l y d i f f e re n t i a b l e . S in c e Y ( 0) =

1x o ( O ) - = 1 , ~ , ( 0 ) = O , ( X ~ ) u ( O ) = - ~ - ,i t f o l l o w s t h a t x ~ - + 1 i n C ~ ( [ 0 , 8 9 a s

f l ~ c o . I n p a r t i c u l a r, t o ~ 0 a s /3 - + o o . S i n c eq o ( t a )= 3 u # ( e t # ) 8 9u o ( e ' i s ) =t

~ T e ~-~3 2 t h i s g i v e s ( 3 . 2 4 ) .

2- (x a) , (~-~ eY ta )N e x t , l e t f ro ( x ) = f 12 u a( /3 -3 x ) , w h i c h a l s o s o l v e s ( 3 . 5 )n d sa t i s f i es h~( /33) = 0 ,

~ ( f f 3 ) = 1 . C l e a r l y u g ~ ~ i n C 1 ( [ 0 , 1 ] ) a s /3 ~ 0 q - , w h e r e ~ i s t h e u n i q u es o l u t i o n o f ( 3 .5 ) s a t i s f y i n g ~ ( 0 ) = 0 , ~ ' ( 0 ) = 1 . B u t~ 3 3 e t ai s t h e l e a s t v a l u e o f

x > / / 3 s u c h t h a t ~ a (x ) = ~ '} x :}, a n d t h u s t e n d s t o t h e le a s t p o s i ti v e r o o to f u ( x ) = ~ Z x { as / 3 -- > -0 + . T h u s

l i m q a ( t a )= 3 ( 6 ~ ) ~ ~ , ( ~ ) ,/3--*0+

w h i c h is t h e v a l u e o f q a t t h e in t e r s e c t i o n o f z = 3 w i t h t h e s m o o t h s o l u t i o n

or b i t l eav in g q -----z = 0 w i t h s l o p e 3 . W e h a v e t h u s s h o w n t h a t th e r e g i o n a b o v et h i s o r b i t in t h e s t r i p 0 < z 0 i s g i v e n t h e n( z ( t ) , q ( t ) ) = ( z r - -log Xo) , q o ( t - -l o g X o ) ) c o r r e -

s p o n d s b y ( 3 . 1 3 ) f f t o t h e s o l u t i o n u o f (3 .5 ) s a t is f y i n gU ( X o ) = O , u ' ( X o ) = / 3 X o - f ,a n d t h u s ( 3 . 2 3 ) h o l d s .

L e t u ~,, b e t h e u n i q u e s o l u t i o n o f ( 3 . 5 ) s a t i s f y i n g u ( 0 ) ---- y > 0 , # ( 0 ) = v > 0 .T h e n th e c o r r e s p o n d i n g s o l u t i o n (z~ .~ ('),q ~ , ~ ( . ) )o f ( 3 . 1 5 ) s a t i s f i e st l i _ m o z r , ~ ( t )

_--o~, t- .-~lim~ , ~ ( t) = 0 3 ~ _ y 8 9A s Y- - ~ 0 + , u ~, 1 ~ ~ i n C ~ ( [ 0 , 1 ]) a n d h e n c e ,3

f o r e a c h f i x e d t , z T. t ( t )~ k( t ) d~r ~- f (e ' ) - / . , de f 3- 1- u ~ ( e ) ~ ( d ) .l an d qv,~( t) -+ e t*~ = - f

C o n v e r s e l y, s u p p o s e t h a t ( z (. ), q ( .) ) i s a s o l u t i o n o f ( 3 .1 5 ) w h o s e o r b i t h a s a n u n -b o u n d e d i n t e rs e c t io n w i t h 0 < q % z . B y P r o p o s i t i o n 3 .5 , l imq ( t ) = c > = O .

t-+train+

L e t X o = e t r a i n . S u p p o s e t m i n ~ c ~ , S O t h a t X o > 0 . T h e n t h e c o r r e s p o n d i n gs o l u t i o n u o f ( 3 .5 ) w o u l d s a t is f y

l i r a v ( x ) ~ l i m z ( t ) = c ~ l i r a v ( x ) = c ,x-C Xo+ t-+to x-+Xo +

3

w h e r e v = u~ -, w h i c h i s i m p o s s i b l e . T h u s t m ~ = - o % XO = O , a n d s i n c e

l i m v ' ( x ) = cw e h a v e v ( x ) - + da s x - + X o + ,w h e r e d >= 0 i s a c o n s t a n t . B u tx - + O +i f d w e r e z e r o t h e n w e w o u l d h a v e

c ~ = l i m v ( x ) l i m v ' ( x )x - + O + X x - + 0 + - - '1 " - - r


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Minimizers Ne ed N ot Sa t is fy Eule r-Lagrang e Equa t ion 351

2 c

a con t r ad i c t i on . H enc e u (0 ) > 0 , u ' ( 0 ) = ~ ~ 0 . N ow i f c = 0 ,u ( x ) ~ u(0)

b y u n i q u e n e s s o f s o l u ti o n s t o ( 3 .5 ), a n d h e n c eq( t ) ~ O. He nce u ' ( 0 ) > 0 .I t f o l lo w s i m m e d i a t e l y f r o m t h e a b o v e t h a t f o r 6 > 0 s u f fi c ie n t ly s m a l l t h e

r e g i o n i n 0 < z < 6 , q > 0 b e l o w t h e s m o o t h s o l u t i o n o r b i t is c o m p l e t e l yf il led w i th o rb i t s co r r e sp on d in g t o so lu t i ons o f ( 3 .5 ) w i th u (0) > 0 , u ' ( 0 ) > 0 .I n p a r t i c u l a r th e r e a r e n o o t h e r o r b i t s l e a v i n g t h e o r ig i n . [ ]

W e n e x t a p p l y t h e r e s u lt s o f S e c t io n 2 .

T h e o r e m 3 . 7 . I a tt a i n s an a b so l u te m i n i m u m o n t h e s e t~1 ---- {u W u (0 , 1) :u(0) = 0, u(1) = k}. L e t u b e a n y m i n i m i z e r. I f e ~ e o t h e n u i s a C ~ s o l u t i o no f (3.5) o n [0, 1]. I f 0 < e ~ e o t h e n e i t h e r u is a C ~ s o l u t i o n o f(3 .5) o n [0, 1]

-- 2 1

or u i s a C ~ so lu t iono f (3 .5 ) o n ( 0, 1 ] w i t h u ( x ) ~ k x T , u ' ( x ) , ~ $ - k x - T a s x --> O + ,

w h e r e k s a t i s f i e s(3.6). I n a l l c a s e s u c o r r e s p o n d s t o a s i n g l e s e m i - o r b i t ( z ( t ), q ( t ) ) ,t E ( - - o 0 ] , o f (3.15), w ith z ( t ) > O , q ( t ) > O f o r a llt E ( - - c o , 0 ] .

P r o o f . T h a t I a t ta i n s a m i n i m u m o n d f o ll o w s i m m e d i a t e ly f r o m T h e o r e m 2 .1 .L e t u b e a n y m i n i m i z e r. B y T h e o r e m 2 .7 a n d t h e s u b s e q u e n t d i s c u s si o n t h e r e isa c l o s e d s e t E o f m e a s u r e z e r o o n t h e c o m p l e m e n t o f w h i c h u is a C 3, a n d h e n c esm oo th , so lu t i on o f ( 3 .5 ) . Le t D1 be a m ax im a l r e l a t i ve ly op en i n t e rva l in [ 0, l ] \ E ,a n d d e n o t e b y X o, x l t h e l e f t a n d r i g h t h a n d e n d p o i n t s o f D 1 r e s p e c ti v e ly. Weh a v e a l r e a d y n o t e d t h a t u ' ( x ) ~ 0 a .e ., a n d i t t h u s f o l l o w s f r o m T h e o r e m 2. 7tha t i f Xo ~ 0 ( r e spec t i ve ly x l =~ 1 ) t h en l imu '( x) = --}- ~ ( r e spec t i ve ly

x-e.-xo +

l i r a u ' ( x ) = + oo ) . I f u ' ( x ) w e r e z e r o f o r s o m e x E ( x0 , x l ) w e w o u l d h a v e ,x - x i - -

by un iqu enes s o f so lu t i ons t o ( 3 .5 ) , t ha t u = cons t , i n (Xo , x l ) an d t hus i nD1 ---- [0, 1 ], co n t ra d ic t ing k > 0 . T hu su ' ( x ) > 0 f o r a l l x E (Xo , x~ ) and ug e n e r a t e s a s o l u t i o n ( z ( t ) , q ( t ) ) , t E( l og xo , l og x 0 , t o ( 3 .15 ) w i thz ( t ) > O ,q ( t ) > 0 f o r a l l t E ( l og Xo, l og x i ) . Bu t by P rop os i t i on 3 .5 t he so lu t i on( z ( t ) , q ( t ) )ex i s ts f o r a l l t > l og Xo, an d t he r e fo r e

l i m u ' ( x ) = l i m ] q ( t ) z ( t ) - 8 9x~- 89< o 0 .x - ~ x l - -

t - l o g x t - -

H enc e x l ---- 1 . Su pp ose t ha t

l i m u'(x)-----x-* x o 2

- - o o ~ tm i, < l og x0 . Th en

l i m 2 q ( t ) z ( t ) - 8 9X o 8 9< oo ,t - - - ~ l o g x o

s ince Xo > 0 , y i e ld ing a con t r ad i c t i o n . Th e re f o r et m i n l o g X o. B y P r o p o s i t i o n3 .5 t he r e a r e t h r ee ca se s t o cons ide r. F i r s t , we m ay hav e( z ( t ) , q ( t ) ) - +(0 , 0) ast ~ l og X o+ . I f Xo > 0 t h i s i s imp oss ib l e s ince we wou ld t he n haveU(Xo) = 0a n d h e n c e u ( x ) - - - - 0 fo r a l l x E [0, 1 ]. I f Xo = 0 the n by P ro po s i t ion 3 .6 u

is C ~ o n [0, 1]. S e c o n d , w e m a y h a v ez ( t ) ~ o o a n d q ( t ) ~ c ~ 0as t ~ log Xo + .In t h i s c a se , by P ro po s i t i o n 3 .6 Xo = 0 an d u (0) > 0 , wh ich i s imp oss ib l e . Th i rd ,- - 3 - - 3

w e m a y h a v e X o = 0 a n d l i m( z ( t ) , q ( t ) )= (k~- , k~-) , a res t po int . In thist - - -> - - CO

- - 2

k x ~-, ,ase u is C ~ on (0, 1] w ith u ( x ) , - ~ u ( x ) ~ . ~ $ k x - ~ as x - ~ O + . [ ]


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3 5 2 J . M . B A L L & V. J . M IZ EL

A s a p r e l i m i n a r y r e s u l t s h o w i n g t h a t e v e r y m i n i m i z e r m u s t i n c e r ta i n c a s e s b es i n g u l ar w e p r o v e

e m m a 3 . 8 . Le t u m inimize I on d , and suppose tha t0 < ~x < f l < m in (1 , k )and

~ E 8 9 \13/ / 3 3 ) 2 0(,)14t 9 ~ ' a (1 - - ( f l - - . (3 .26 )

The n u x) > o~x fo r all x E 0, 11.

P r o o f . W e m o d i f y a n a r g u m e n t o f M A N L~ [2 5 ] ( se e a l s o C ES AR I[ ] a n d S e c t i o n 4 ) .I f t h e c o n c l u s i o n o f t h e l e m m a w e r e f a ls e t h e n t h e r e w o u l d e x i st a s u b i n t e r v a l( x l , x 2 ) o f [ 0, 1 ] s u c h t h a t

a n d

2o~x3 (1 - - fla)2f x , ( u , y + d x .x t

13L e t x = y. -r T h e n9

xT3x 2 2 / d u ~ 1 4

f X 4 ( U t ) 1 4 X = ( 9 ) 1 3 f \dy] d .x t

~I

a n d b y J e n s e n ' s i n e q u a l i t y t h e m i n i m i z e r o f t h i s i n t e g r a l s u b j e c t t o u I 9_ = o ~x ~,l y=x13

2_ 5 ~

u I 9 = ~ x ~ i s g p v e n b y t h e l i n e a r f u n c t i ~ 1 7 6= x p \ x p x l

T h e r e f o r e2 2 ~ 1 4

X 2f x , u , u ' ) d x > r ' s (1 - - fl 3 )2 f l x ~ - - o~ x ?

\1 3 / 9 9 Xll3o xp_

9 Xl 2 \ 1 4

X2 ( 1 - - - i X 1 / ~ /' ' ' ( 3 . 2 7 )

> [ 9 v l - - f l3 ) 2 X ~ f l - - 0 ~) ~+~ - ~ j


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Minimizers Nee d N ot Satisfy Euler-Lagrange Equ ation 353

Def ine v E ~r by

T h e n

v(x) =

I 2 3

x -~ 0 -


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354 J .M . BALL 8~; V. J. M ZEL

By Prop os i t ion 3 .6 ( see the fo rm ula fo r ~0 in the p r oo f o f The orem 3 .7 )

exis ts and i s f in i te . Therefore

1 2

Z u) = f x -WF d x = --3~o(z(0), q(0))0

3

= --3~p(k~-, q(0 )). [ ]

Since ~q(z , q ) = (q - - z ) Fqqand Fqq > 0 , i t fo l lows f rom L em m a 3 .9 tha t ,o f al l t rajec tories (z(-) , q( .)) o f (3.15) sat is fyin g z(0) = k an d l im( z ( t ) , q ( t ) ) =

t > oo3 3

(0, 0 ) o r (k~ , U2-) ( a r e s t po in t ) , t ha t co r re spond ing to an abso lu te m in im um3 3

of I h as e i ther the grea tes t va lu e of q(0) ~ k~- or the leas t va lue of q(0) =< k~- .So a s to d ec ide be tween these two poss ib i li ti e s it is conven ien t t o r e s t a t e Lem m a 3 .9in the fo l low ing way. D ef ine

I ' ( z , q ) = v /( z, q ) + 8 9 / F q ( ~ , ~ ) d ~ . (3.28)0

T h e n i f u s, u2 s a t is f y t h e h y p o t h e s e s o f L e m m a 3 .9 w i t h c o r r e s p o n d i n g s o l u t io n s(z,(.), qt(.)), (z2('), q2(')) o f (3.15),

3 3

I(Ul) -- I (u2 ) ---- - -3 [F (k T , q l (0 ) ) - - / ' ( k ~ , q2 (0 )) ]. (3.29 )

Note tha t by (3 .21 ) we have tha t a long so lu t ions o f (3 .20 )

d-~z i f ( z , q ) = - - 89 q ) - - Fa (z , z ) )

= - - ( q - - z ) M ( z , q , e ) , (3.30)

w h e r e M ( z , q , e ) > 0 fo r z , q > 0 . As an app l i ca t ion o f th i s i dea we p rove thefo l lowing p ropos i t ion .

W e deno te by (Zsm("), q sm(') ) t he sm ooth so lu t ion o rb i t , wh ich by P rop os i t ion

3 .6 leaves the o r ig in wi th s lope 3 ; t h i s o rb i t is un ique m od u lo add in g an a rb i t r a ryc o n s t a n t t o t , a n d w e c h o o s e f o r c o n v e n i en c e th e n o r m a l i z a t i o n c o r r e s p o n d i n g t othe sm oo th so lu t ion u of (3.5) sa t i s fy ing u(0) = 0 , u ' (0) = 1 .

roposition3.10. I f Zsm(t -+ o~, q~m(t) -- -> o0 a s t - + oo t hen fo r a ny k > 0t h e r e e x i s t s p r e c i s e l y o n e s o l u ti o n u o f(3.5) be long ing t oC~176 1]) a n d s a t i sf y i n gt h e b o u n d a r y c o n d i t io n s(3.2), an d u is the un ique m in im ize r o f I i n J f f.

Pro of . I f u is a sm oo th s o lu t io n of (3.5) o n [0, 1] sa t i s fy ing (3.2) th enu ( x ) > 0,u ' (x ) > 0 for a l l xC (0 , 1]. Otherw ise there wo uld exis t some Xo E (0, 1) wi thu ' ( x o ) ---- 0, an d he nc e u(x ) ~ U(Xo)b y u n i q u e n e s s , a c o n t r a d i c t i o n . T h u s a n ys u c h s o l u t io n i s r e p r e s en t e d b y a n a p p r o p r i a t e p o r t i o n o f t h e s m o o t h s o l u t i o n

3

orb it (Zsm(') , q~m(')), an d since this o rbi t cuts the l ine z = k-2- exa ctly, onc e theex i s t ence and un iqueness o f u i s a s su red .


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Minimizers Need N ot Sa t is fy Eule r-Lag range Equ a t ion 355

I t r e m a i n s t o p r o v e t h a tI ( u ) ~ I ( v )f o r e v e r y o t h e r v E ~ . I f t h is w e r e f a ls et h e r e w o u l d e x i st b y T h e o r e m 3 .7 a n a b s o l u t e m i n i m i z e r u~ o f / i n d w i t h Ux @ u ,

I ( u O ~ I ( u ) . Le t ( z~ (. ), q~ (. )) be t he co r r e s po nd ing so lu t i on o f ( 3 . 15 ); t hus3z ~ ( 0 ) ~ k ~ - . We k n o w b y T h e o r e m 3 .7 t h at w e m u s t h a v e 0 < e ~ e o a n d

- - 3 - - 3l im (Zl ( t ) , q ~ ( t ) ) = ( kT, k~-), and s i nce P, i s a s i nk w e a l so have k - = k2 i f

t -~ - - oo

0 ~ e ~ e0. S ince t he sm oo th so lu t i on o rb i t l ie s en t i r e l y ab ov e any such so lu t i on ,3

by ou r p r ece d ing d i s cus s ion we k no w tha t q~(0 ) ha s t he l e a s t va lue o f , 7(0) 0f o r q > z ,3 - - 3 - - 3 - - 3 - - 3 3 3

/ (k T , q t (0) ) ~- F (k ~-, k ~-) < l~(kT, qsm(k2-)) 0 w e h a v eny A > 0 , and t ha t ~A

2 ~ 3 ] 2 u A x) J A x ) T

l im U A ( X ) = l i m A 3 x - - 0 a n d l i mu x ( x ) = l im x / - - /A -~ o A -~ o A x ~ - ~ ~ -~ oo [ A x J

2 2

= x ~- l imz sm (t)~- = cx~. D efin e uo(x) ~ 0. T he n (Un}0_~A 0 , u ~ 0 . L e t v E ~ r v @ u , w i th

v(x ) ~ 0 f o r a ll xC (0 , 1 ] an d v(x ) 0 + ( w e h a v e a l r e ad y s e ent h a t a n y m i n i m i z e r o f I h a s t h es e p r o p e r t i e s ). I n o r d e r t o h a n d l e t h e s i n g u l a r it y

of the f i e ld a t the o r ig in d ef ine for 6 > 0 .

. x ) ,

x - - 0v~ x)= u ~ ) + - - - 7 - v 2 ~ ) -u 6)),

v x),T h e n

O _ < _ x < _ _ 6 ,

6__


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3 5 6 J . M . B A L L d~ V. J . M l Z E L

2~

l im f f ( x , v a, v ~ ) d x= O , i t f o l l o w s t h a t I (v~) - -~ I (v )a s 6 --~ 0 + , a n d w e o b -#--~0+

t a i n b y F a t o u ' s L e m m a t h a t

I ( v ) - - I ( u ) ~ f [ f ( x , V ( X ) , v ( x ) ) - - f ( x , v ( x ) , p ( x , v ( x ) ) )

- - ( v ( x ) - - p ( x , v ( x ) ) , f p ( x , v ( x ) , p ( x ,v(x) ) ) ] d x > O ,a s r e q u i r e d .

T h e o r e m 3 . 1 1 . T h e r e e x i s t s a n u m b e r e * s a t i s f y in g 0 < e * < e l < e o s u c h t h a t3 3

(i) i f 0 ~ e < e * t h e n (Zsm(t),qsm( t ) ) -+( k l 2 , k l 2 ) a s t ~ 0 %

3 3(i i) i f e = e* then (Zsm( t) , qsm( t ) ) -+ (/c~2 , k 2 ) as t - -~ oo , a nd

(iii) i f e > e * t h e n Zsm(t - + 0% qsm( t) -+ O0a s t ~ 0 0 .

Proo f . W e fi r st s h o w t h a t t h e r e e x is ts a m i n i m a l n u m b e r e * w i t h 0 < e * < e zs u c h t h a t ( ii i) h o l d s . I f e > e o t h e n Z ~ m ( t )- + 0 % q s m (t )- -~ O o a s t - + o o b y

P r o p o s i t i o n 3 . 5. T h u s s u p p o s e 0 < e ~ e 0 , a n d l e t k - = k 2 ( e ) i f 0 < e < e o ,

39 d e f i n eT = ( z * ) ~ i f e = e o ( f o r 3 a s i n ( 3 . 6 ) ) , a n d s e t z = / ~ 3 . F o r y > ~0_2_Z

v v ( x ) = k x 3 .T h e n b y d i r e c t c a l c u l a t i o n

clef la__(3~l4 ~ - 1 4 . . . .J ( ~ ) - s ~ , ~ t ~ t v ~ ) - I ( v , ) )

[ 1 1 3 ~ 2 ] [ 2 ~ 1---- 3 2 [~ i0 y ~ 3 9 4 y - - ~ 3 + 1 2 + ~ - 3 4 y _ 3 3

y l a 7 F 2+ + - 6 .

2 8 y - - 2 7 4 ), - - 3

T h e r e f o r e

w h e r e a = 1 . 5 2 3 7 8 . . . .

2 0 ) ' 2 1 8 ]

J 1.1) -----a32 + bv + c ,

b = - - 2 . 4 4 0 4 2 . . . . c = . 9 49 3 4 . . . . I t n o w f o l l o w s t h a tJ ( 1 . 1 ) i s n e g a t i v e i f 3 _ < 3 < ~ r+ , w h e r e 1 :_ = . 6 6 5 7 6 . . . . 3 + = . 9 3 5 7 8 . . . . S i n c e

3 > 3 _ i t f o l l o w s t h a t k -x~ d o e s n o t m i n i m i z e I i f e > 0 (3 + ) = . 0 0 1 9 6 0 3 . . . .3

T h e r e f o r e i f e > 0 (~ :+ ), t h e r e is s o m e s o l u t i o n ( z( .) , q ( .) ) o f ( 3 . 1 5 ) w i t h z ( 0 ) = ~ ,__3

q ( 0 ) : ~ k 2 a n d l i m ( z ( t ) , q ( t ) )= ( 0 , 0 ) o r a r e s t p o i n t , a n d t h i s c l e a r l y i m p l i e sl---~ - CO

t h a t Z s r n t ) ~o o , q sm (t) ~ o o a s t ~ ~ . D e f i n e e * t o b e t h e l e a st n o n n e g a t i v en u m b e r s u c h t h a t ( iii) h o l d s . S i n c e e l = 0 ( 3 1 ) = . 0 0 2 4 73 5 . . . i t f o l lo w s t h a t

0 < e * ~ e l , a s c l a i m e d .W e n e x t p r o v e th a t e * > 0 . I f n o t w e w o u l d h a v eZ s m t ) - -+ o o ,qsm(t) ---->o oa s t ~ o o f o r e v e r y e > 0 . B y P r o p o s i t i o n 3 .1 0 a ll m i n i m i z e r s o f I i n ~ r w o u l dt h e n b e s m o o t h f o r a n y k > 0 . B u t b y L e m m a 3. 8 t h i s i s f a ls e f o r e > 0 s u ff i-c i e n t l y s m a l l .


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M i n im i z e r s N e e d N o t S a t is f y E u l e r - L a g r a n g e E q u a t i o n 3 5 7

F o r t h e r e m a i n d e r o f t h e p r o o f i t is c o n v e n i e n t to m a k e t h e d e p e n d e n c e o n ee x p l i c i t b y w r i t i n g Z sm ( t) =Z~m(t,e) , q~ m (t) -----qsm(t , e ) ,a n d w h e r e a p p r o p r i a t e

qsm(z) = q,m(Z, e).U s i n g t h e i m p l ic i t f u n c t i o n t h e o r e m i t is e a si ly s h o w n t h a ti f Z~m(t ,~ ) = ~ > 0 , q ~m (t', ~ ) :4 = ~ t h e n t h e r e e x i s ts a s m o o t h f u n c t i o nt(e)d e f i n e d f o r e n e a r ~ s u c h t h a t Z ~m (t(e ), e ) = z . T h u s i fz ,~( t ,e * ) - + o o a s t - + o ow e a l s o h a v e zsm (t, e ) --~ o o a s t - * o o f o r e n e a r e * , c o n t r a d i c t i n g t h e m i n i m a l i t y

o f e * . L i k e w i s e , i f ( Z~ m (t, e * ) ,q~m(t,e*)) - -~ (k '~(e*)~ , k - l( e * ) ~ -) a s t - + o o t h e ns i n c e e * < e 1 w e h a v e q~m (t ,e*) < Z~m (t,e*)f o r s o m e t ; t h u s q~m (t , e ) e * a n d s o m e t ', a c o n t r a d i c t i o n . T h e r e r e m a i n s o n l y o n e

p o s s i b i l i t y, t h a t ( Z ~ m ( t,e * ) , q~m(t,e* )) ---~ (k-2(e*) -2-, k-2(e*) ~-) as t --* cx~, w h ic hp r o v e s ( i i ) .

W e n e x t r e m a r k t h a t f o r a n y e > 0 t h e s lo p e o f t h e v e c t o r fi el d o n t h e c u r v e1 4

q = 1 - '~ z e q u a l s , b y ( 3 . 2 0 ) ,

42

H z , , e = 3 9 z 2 ( 1 4 _ 1 3z 2 ) ,

14w h i c h i s p o s i t i v e i f

One Dimmi Zel

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