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    Phys ica 13D (1984) 181-194N o r t h - H o U a n d , A m s t e r d a m

    O N A N O N L I N E A R B O U N D A R Y V A L UE P R O BL E M ; T H E E Q U I L I B R IU M O FA C O N F I N E D P L A S M AR. C I P O L A T T IIns t i tu to de Matem at ica , Universidade Federal do Rio de Janeiro, 21944 R io de Janeiro , Bras i lR e c e iv e d 1 0 D e c e mb e r 1 9 83

    W e es tab l ish the ex is tence and un iqueness o f so lu t ions o f a non -l inea r boun dary va lue p rob lem wl f ich genera lizes ama th e ma t i c a l m o d e l fo r p l a s ma c o n f in e me n t .

    1. IntroductionI n t h is p a p e r w e d i s c us s t h e f o l lo w i n g b o u n d a r y v a l u e p r o b l e m :

    a u = x e K u ) i nula~ = 0 % 0 ~ R u n k n o w n ,f Px u) ~ /= I /h , I > 0 i s prescr ibed, ( I )

    wh ere 12 i s a bou nd ed ope n se t o f R N wi th sm oo th b ou nd ary 012 , h =# 0 i s a r ea l pa r amete r , K i s a c losedconv ex se t o f L2(12) and P r i s t he co r r espon d ing o r thogo na l p ro j ec t ion , f f i s t he ha rmo nic e x t ens ion to 12of a g iven fu nct io n qo ~ 0 in H1/2(012) .

    W he n K = ( v ~ L2 (f / ) ; v > 0 a .e . in 12} and ~p --- 1 on 012, we f ind the p las m a m od el pro ble m in t rod uce dby R. T em am [12] and s tud ied ex tens ive ly by severa l au thor s ( see [1, 5 , 9, 13 ]) .

    By an equ iva l en t decoup led fo rmula t ion o f ( I ) , we s tudy in sec t ion 2 the ques t ions o f ex i s t ence andun iqueness .

    I n s e c t i o n 3 w e d i sc u s s, a s a l i m i t c a s e o f a p r o b l e m o f t y p e ( I) , a f r e e - b o u n d a r y v a l u e p r o b l e m w h i c hw a s i n t r o d u c e d b y H e r o n a n d S e r m a n g e [7 ].

    Exce p t f o r sec t ion 3 , t h is a r t i c le i s pa r t o f a t hes is p r epare d a t t he U n ive r s i t6 Pa r i s -Sud a t O r say [3 ].

    2. An equivalent problemA necessa ry con d i t i on fo r ex i s t ence o f so lu t ions o f ( I) i s

    inf f o~, I

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    82 R Cipolatti The equilibrium of a confinedplasmaI f w e i n t r o d u c e t h e c o n v e x s e t

    K x = K n { v ~ L 2 f d) ; f v f = I / X ) ,t h e n 2 . 1 ) i s n e c e s s a r i l y v e r i f i e d i f

    K x q>. 2 . 2 )F o r a l l X ~ : 0 s u c h t h a t 2 . 2 ) is v a l i d , w e c o n s i d e r t h e f o l l o w i n g p r o b l e m : w e s e e k w , 0 ) ~ I - I~ f g ) R

    s a t i s f y i n g

    - a w =XPK~ w), i n ~ , 2 . 3 a )wlo~= 0 ;

    f PK w + Of f ) f = I IX . 2 . 3 b )T h e f o l lo w i n g l e m m a a s su r e s th e e q u i v a l e n c e o f p r o b l e m s I ) a n d 2 .3 ):

    Lemma 2 .1 . P r o b l e m I ) h a s a s o l u t i o n u ~ H i I T ) i f a n d o n l y i fi) K x @ q~;i i) 2 . 3 ) h a s a s o l u t i o n ~ 0 , 0 ) ~ H ~ ~ 2 ) X R .T h e s e s o l u t i o n s a r e r e l a t e d b y

    u = w + O f ,P, , u ) = P, ,~ w ) .

    Proof. L e t u b e a s o l u t i o n o f I ). F r o m faPK u) f = I /X i t f o l l o w s t h a t P~: u)~ Kx. L e t w = u - O f . T h e nw ~ H ~ ~ 2 ) a n d w e h a v e 2 . 3 b ).

    T o p r o v e t h a t w s a t is f ie s 2 . 3 a) , w e r e c a l l t h a t

    f u - e , , u ) ) o - e , , u ) ) < _ o , v o ~ : ,and

    S ~ w - p , ,~ w ) ) o - P , ,~ ~ ) ) _ < o , w ~ K ~ .T a k i n g v = Px~ w) n t h e f i r st in e q u a l i t y a n d v = Px u) i n t h e s e c o n d o n e , w e o b t a i n

    w h i c h i m p l i e s t h a t Pr u ) = P K x w ) .

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    83R . C i p o l a t t i / T h e e q u i l i b r i u m o f a c o n f i n e d p l a s m a

    Sinc e ~k is harm on ic , w sat is f ies (2 .3a) .T h e s a m e r e a s o n i n g s a r e v a l i d f o r t h e c o n v e r s e .

    2.2. An e x i s t e nc e r e su l tH e n c e f o r t h w e a s s u m e t h a t

    11 = in f Jafvtk < sup f vq, = l 2.v ~ K v ~ K 2W e w i l l s a y t h a t K v e r if i es t h e c o n d i t i o n ( H E ) i f : f o r a l l 1~ ]11 , 12[, { v ~ K: f awp = 1} i s bo un de d in L1(I2) .

    L e t A = { ~ 4 = 0 ; l 1 < I / ~ < 1 2 }T h e o r e m 2 .1 . U n d e r t h e c o n d i t i o n ( H E ) , p r o b l e m ( 2 .3 ) a d m i t s a t le a s t o n e s o l u t io n ( w , 0 ) ~ H ~ ( I ~ ) R ,fo r a l l A ~ A .

    T o p r o v e t h e o r e m 2 .1 w e n e e d t h e f o l l o w i n g g e n e r a l r e su l t :L e m m a 2 .2 . L e t H b e a re a l H i l b e r t s p a c e a n d l e t K c H b e a c l o s e d c o n v e x s u b se t . T h e n t h e f u n c t i o n a lJ : H ~ R d e f i n e d b y

    s v ) = < v - P K o ) ;

    i s c o n v e x a n d F r e c h 6 t - d i f f e i e n t i a b l e i n H , w i t h d i f f e r e n t i a l J ' = P K . M o r e o v e r , f o r a l l u a n d ~k i n H , t h er e a l f u n c t i o n .

    o ) = P , , u +

    i s L i p s c h i t z a n d i n c r e a s i n g i n R , s a t i s f y i n gl im ~-u(O) = in f (v ; ~k) ,O ~ - o o v ~ Kl i m ( 0 ) = s u p ( v ; i f ) .0 -- * + ~ v E K

    P r o o f . C o n s i d e r t h e f o l l o w i n g c o n v e x s . c .i f u n c t i o n a l :j : H ~ R ,j v ) = l v l l + I K O ),

    w h e r e I x d e n o t e s t h e s o - c a ll e d i n d i c a t o r f u n c t i o n a l o f K :i K v ) = { O , i f v ~ K ,+ ~ , o t h e rw i s e .

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    1 8 4 R . C i p o l a t t i / T h e e q u i l i b r i u m o f a c o n f i n e d p l a s m a

    T h e c o n v e x c o n j u g a t e fu n c t i o n a l o f j i s d e f i n e d b yj * ( o * ) = s u p ( ( o * ; o ) - j ( o ) } . ( 2 .4 )

    v E H

    I t is c le a r t h a t t h e s u p i n ( 2 .4 ) is a s s u m e d a t 0 e H w i t hv * e o + a l x ( V ) ,

    w h i c h i s e q u i v a l e n t t o 0 - - Px(o*) . H e n c ej * (o * ) = ( v * - P r ( o * ); e r ( o * )> = S ( o * ) .

    I t f o l low s ( s ee r e f . 6 ) tha t J i s con vex an d s . c .i . M oreo ver ,o * e O j (~ ) , i f a n d o n l y i f V e a j * ( o * )

    a n d w e o b t a i n0 s o * ) = e , , o * ) ) .

    T h u s J i s G a t e a u x - d i f f e r e n t i a b l e a t v * a n d s i n c e PK i s a c o n t i n u o u s o p e r a t o r , J i s F r e c h 6 t - d i f f e r e n t i a b l e .O n t h e o th e r h a n d , f r o m t h e C a u c h y - S c h w a r t z in e q u a li t y,

    t T . ( 0 I ) - ~ . ( 0 z ) I - < I IP r (u 0 1 4 ) - P K ( u + 024')11114'11-< 114'111101 - 0ela n d s o % i s L i p s h i t z .

    T o p r o v e t h a t % i s i n c re a s in g , w e n o t e t h a t(01 - 0 2 ) ( ( 0 1 ) - ( 0 2 ) ) = ( P r ( u + 0 1 4 ' ) - P r ( u + 024 ' ) ; (01 - 02 )4 ' )

    = ( P x ( u + 01 4') - P K ( u + 0 24 ') ; u + 01 4 - - U - - 024') >_ O .F i n a l l y , i t is n o t h a r d t o c h e c k t h a t

    l i ra T , ( 0 ) = i n f ( o ; 4 ' ) a n d l i m ~ ' , (0 ) = s u p ( o ; 4 ' ) . t0 - -* - oo o ~ K 0 - - , + oo v ~ KNote . T h e f i rs t p a r t o f t h e a b o v e l e m m a is a r e s u lt o f E . Z a r a n t o n e l l o [ 1 4 ]. T h e p r e s e n t p r o o f f o l l o w s a nidea o f B . M erc ie r [ 8 ] .Proo f o f theorem 2 .1 . S i n c e A ~ A , i t f o l l o w s f r o m l e m m a 2. 2 t h a t K x q ,. L e t e : H i ~ R b e t h e f o l l o w i n gf u n c t i o n a l :

    e ( w ) = l v w l 2 xt i n f a c t , o n e h a s f o r e x a m p l e : u + O C / - P x u + O / ) ; u - P K u + O / ) ) < 0 , V o e K . I t f ol lo w s t h a t 0 [ o ;~ k ) - l i m 0 ~ + ~ r , 0 ) ]

    r e m a i n s b o u n d e d a s 0 --* + o o, h e n c e s u p o ~ r o ; ~ k) < l i m 0 ~ + ~ r . 0 ) . T h e c o n v e r s e i n e q u a l i t y i s o b v i o u s .

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    R. Cipolatti The equilibriumof a confined lasmaW e c l a i m t h a t t h e r e e x is t s c ~ R s u c h t h a t

    e w ) > _ f l v w l 2 - c , V w ~ H ~ I ~ ) .T o t h is e n d , w e m u s t c o n s i d e r t w o c a s e s:i ) h > O: In this case

    e w ) >_ , v w l 2 - X l W I L , I P K xW ) I L + ~ I P K x W )I2 2f o r p , p * > 1 s u c h t h a t 1 / p + 1 / p * = 1.

    W h e n p = 2 N / N - 2 ) i f N > 2 a n d p > 2 i f N --- 2 w e h a v e b y t h e S o b o l e v i m b e d d i n g t h e o r e mIwlL P--< q lV W lL 2 , VW ~ H ~ f ~ ) .

    H e n c e

    e w ) > 1vw122 2 I VWlL=I PK x W) I Lr + ~ l e r x w ) 1 2 2>-- Vw l 2 , - - c 2 1 e x ~ w ) 1 2 r + ~ IP x~ w) 122.

    L e t s - - 2 / p ; t he n 0 < s < 1 a nd 1 - s + s / 2 ) -- l / p * a n d w e g e t1-s s] P r x W ) l g r < I P K x W ) I L ~ I P r ~ W) I L2.

    T h i s p ro v e s t h a t , u n d e r c o n d i t i o n H E )

    e w ) > 1Vwl22 + ~ le x ~ w ) 122 - c31er~ w)12~f o r s o m e c 3 ~ R , a n d 2 . 5 ) fo l l o w s b e c a u s e 0 < s < 1 .

    i i ) ~ < 0 : L e t g ~ K x f i xe d . S inc e

    f n w - P r ~ w ) ) g - P r x w ) ) < O , V w 6 H~ ~ )w e h a v e

    f ~ w - e , , ~ w ) ) e , , ~ w ) > _ _ 1 e ,~ w ) l b + , ~ w - e , , ~ w ) ) g .B u t i n t h i s c a s e

    e ( . , ) = ~V w ~ , + c l f , , ( w - ~ ' K ~ ( w ) ) , , , ~ ( w )

    185

    2 . 5 )

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    86 R Cipolatti The equilibrium of a confinedplasmaw i t h c 1 = - ~ > 0 , s o t h a t

    e w ) > 1Vwl~2 + c l f w - P K w ) )g + l v w l ~ 2 - c zl V w l o l g l L , . - c l lg lL2lP~x(w)IL~ + -~IPK~ (w ) 1~2_

    a n d ( 2 . 5 ) f o l l o w s w i t h c = c 2 1 g l ~ + ~qlglL~-IL e t { w . } b e a m i n i m i z i n g s e q u e n c e :

    w . ~ H ~ ( ~ 2 ) a n d e w , , ) ~ i n f { e z ) , z ~ H X o ~ 2 ) } .F r o m ( 2 .5 ) , { w . } i s b o u n d e d i n H ~ ( ~ 2 ) a n d w e c a n e x t r a c t a s u b s e q u e n c e ( a l s o d e n o t e d { w . } ) s u c h t h a t

    w . ~ w w e a k l y i n H ~ ( ~ ) ,w . ~ w i n L2(~ 2) .

    a n d w e e a s i l y s e e t h a te w ) < l i m i n f e ( w . ) = i n f e .

    I t f o l l o w s f r o m l e m m a 2 .2 t h a t w s a ti s fi e s ( 2 . 3 a ) a n d t h a t t h e r e e x i st s 0 ~ R s u c h t h a t

    T h i s c o m p l e t e s t h e p r o o f o f t h e t h e o r e m .R e m a r k 2 . 1. W h e n K i s a c o n e w i t h v e r t e x a t t h e o r ig i n , 0 c a n b e c a l c u l a t e d e x p l i c i t ly a s a f u n c t i o n o f w .I n f a c t , s i n c e a l l u ~ L 2 ( 2 ) c a n b e d e c o m p o s e d b y

    w he re K J- = { v ~ L2(~2); f v g < 0 , V g ~ K } is t h e o r t h o g o n a l c o n e o f K , w e h a v e f o r u = w + 0 ~ p, w h e r e(w , 0 ) ~ H~ (~ 2) R i s a so l u t i on o f (2 .3 ) , t ha t

    o=f u- + f s ~ w - P K x w ) ) P K x w )a n d s o

    R e m a r k 2 .2 . In t he l i m i t c a se , t ha t i s , I / ~ = 11 or I / ~ = 1 2, w e c a n n o t g u a r a n t e e i n g e n e r a l t h e e x i s t e n c e o fs o l u t i o n . F o r m o r e d e t a i l s , s e e r e f . 3 .

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    R. Cipolatt i / The equifibrium of a confined plasm a 1872.3. An uniqueness result

    I n t h i s s e c t i o n w e a s s u m e t h a t p h a s c o n s t a n t s i g n . H e n c e f o r t h ? h a n d ~ 2 w i l l d e n o t e t h e f i r s t t w oe i g e nv a l u e s f o r - A w i t h D i r i c h le t b o u n d a r y c o n d i t i o n s o n 0~2 a n d ~ 1, ~ 2 t h e c o r r e s p o n d i n g n o r m a l i z e de i g e n f u n c t i o n s .

    B e f o r e s t a t in g t h e u n i q u e n e s s t h e o r e m , w e c o n s i d e r t h e f o l lo w i n g c o m p l e m e n t a r y r e s u l t:L e m m a 2 . 5 . T h e e i g e n v a l u e s ~ * f o r t h e p r o b l e m

    - A u = h u , in I~,ulo~ = 0 % 0 u n k n o w n , ( 2 .6 )f o o u~ --~ ~ d F = 0

    a r e a l l p o s i t iv e s . M o r e p r e c i s e l y0 _ < x - - -

    M o r e o v e r 0 < ~ < ~ x and h~ = 0 i f an d on ly i f p i s con s tan t . I n th i s cas eX1 < X~ < h 2. (2.7 )

    Proof . L e t W = ( v ~ H x ( ~ ) ; v l0 a = 0 % 0 ~ R f r ee } a n d le t L : L 2 ( ~ ) ~ L 2 ( $ ) b e t h e c o m p a c t , s e l f- a d j o in to p e r a t o r d e f i n e d b y L f = u , w h e r e u i s th e u n i q u e s o l u t i o n o f

    f f f u . v v f , / v = f f l v , w .I t i s c l e a r t h a t u s a t i sf i e s t h e f o l l o w i n g b o u n d a r y v a l u e p r o b l e m :

    - Z ~ u + u = f , i n ~ ,u l a ~ = 0 %OuI S : = o

    Le t th < i t2 < g3 < the cha ra c te r i s t i c va lues o f L . t Th en the re ex i s t s v~ 4= 0 s uch tha tV = p i L v i

    t h a t i sI - A v ~ = t ~ - 1 ) v i ,u l o~ = 0

    ~1 f Ovi = .

    t S i n c e p h a s c o n s t a n t s i g n , w e h a v e t h < t t 2 , b y K r r i n - R u t m a n s t h e o r e m .

    2 . 8 )

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    88 R Cipolat t i / The equil ibrium o f a confined plas maH e n c e , A * is a n e i g e n v a l u e o f ( 2 . 6 ) i f a n d o n l y i f A * = tL i - 1 .

    F r o m ( 2 .8 ) w e n o t e t h a t

    f l v v i l 2 = ( P i - 1 ) s ,w h i c h g i v e s 2~* = / L i - 1 > 0 , V i .

    N o w , s i n c e H ~ ( $ 2 ) c W , i t e a s i l y f o l l o w s t h a t h ~ _< ?~ 1. M o r e o v e r , i t is a l s o e v i d e n t t h a t h '~ = 0 i fcp = c o n s t .

    C o n v e r s e l y , i f ~ = 0 , t h e n # 1 = 1 a n d t h e r e e x i s t s v :~ 0 s u c h t h a t- A v = O , i n ~ 2 ,

    v l a~ = 0 ~ ,~Tn ~=o

    T h e r e f o r e

    o= f lvo[2 -o f , lvo, 2,f o r c i n g v = c o n s t . , a n d s o q 0 = c o n s t .

    S u p p o s e n o w t h a t ~ = c o n s t . ~ 0 a n d t h a t h i > -- ~ '~ - T h e n 1 < / ~ 2 -< 2~1 + 1 a n d i t f o l l o w s f r o m ( 2 . 8 ) t h a t ,f o r s o m e v ~ O ,

    f o-a ,p)o .I f w e d e n o t e vo = v - 0~k , t h en vo ~ H ~ ( ~ 2 ) a n d

    f lV V o [ = ( 2 - 1 ) f f f o ( v o + O ~ ) = ( 2 - 1 ) f f f 2 - (1 ~ : -1 ) 0 2 ~ 2 1 ~ .H e n c e ,

    Xl v~

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    R Cipolatti / The equilibrium of a confined plasm a 89O n t h e o t h e r h a n d , f r o m t h e v a r i a ti o n a l c h a r a c t e r iz a t i o n o f h'~ w e h a v e

    f lv u l2 > _ _ h * 2 fu 2 , V u e W s u c h t h a t f Q u - - 0 . ( 2 . 9 )Le t w = ~ 2 + t ~ x w i t h t = - f~ep E/ fa ,~ 1 . T h e n l a w = 0 a n d ~*2f~w2

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    190 R. Cipolatti The equilibriumof a confined lasmaS i n c e

    f PK x W 1 ) - P K x W 2 ) ) W 1 - W 2 )~ -- - IP Ka W 1 ) -pKx W2)122,w e d e d u c e t h a t

    ) ~ [ P K x w a ) - P x x w 2 ) 1 2 < [ ~ T w , - ~7w212 < - ~ 2 [ P K x w , ) - eKx(w2)[~ 2a n d c o n s e q u e n t ly

    (1 - ) , / ) ,~ ) IP K~ (w,) - PK ~( w 2) 12 _< O.y t h e h y p o t h e s i s o n ) ,,PKx Wl )=PKx W2)

    s h o w i n g t h a t A w1 = Aw2 and s o w l = w2.R e m a r k 2 .3 . W h e n K i s a c o n e , i t f o l l o w s f r o m r e m a r k 2 . 1 t h a t u n d e r t h e h y p o t h e s i s o f t h e o r e m 2 .2 ,p r o b l e m ( I ) a d m i t s a n u n i q u e s o l u t i o n . T h i s i s t h e c a s e f o r t h e p l a s m a m o d e l - p r o b l e m . I n f a c t , f o r t h i sm ode l a nd us ing d i f f e r e n t me t hod s , J .P . P ue l s ho w e d th a t the re i s un iq uen es s fo r 0 < 7~ _< 7~2 [9] .

    I n a m o r e g e n e r a l s e t t in g , it is n e c e s s a r y t o i m p o s e c o n d i t i o n s o n K t o a s s u r e t h e u n i q u e n e s s o f 0 . O n es u c h c o n d i t i o n i s t h e f o l l o w i n g :A c l o s e d c o n v e x s e t K i n a H i l b e r t s p a c e H s a t is f ie s t h e c o n d i t i o n ( H U ) r e l a t i v e l y t o + i f, fo r a ll u + H ,t h e f u n c t i o n

    ~ , O ) = e ~ u + O + ) ; ~ )i s s tr i c t l y i n c r e a s i n g o n t h e s e t

    ] L l ( u ) ; C 2 ( u ) [ , V ~ (0 ),w h e r e

    L ~ u ) = i n f { O ; r u O ) > l l }i s s m a l l e r t h a n

    L 2 ( u ) = s u p { 0 ; % ( 0 ) < t 2 } .F o r e x a m p l e , w e h a v e

    L e m m a 2 .6 . L e t K b e a c lo s e d c o n e w i t h v e r t e x a t o r i g i n i n a H i l b e r t s p a c e H . T h e n K s a ti s fi e s t h eco nd i t io n (H U ) r e la t ive ly to ~k, fo r a l l ~k ~ H .

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    R Cipolatti The equilibrium of a confinedplasma 9P r o o f . Le t ru (01 ) = (02 ) = 14 : 0 . Th en

    e K u + o~ /, ~ ) - e K u + 0 2 ~ ) ; ~ ) = 0a n d s i n c e f o r a l l v ~ K

    u + o ,q , - P , , u + o , ~ ) ; v - e , , u + o ~ q , )) < _ o ,i t f o l l o w s t h a t

    P K U + O I ~ ) = P K U + 0 2 II /) .N o w , s i n c e ( P K ( v ) ; v ) = I Ie x v ) l l z fo r a l l v ~ H w e have

    e K U + O i g / ) ;u + O , q ~ ) = l l e K u + O i ~ ) l l =, i = 1 , 2w h i c h g i v e s

    0 1 - 0 ~ ) t = 0 .I n t h e g e n e r a l c a s e w e c a n s t a t e t h e f o l l o w i n g u n i q u e n e s s r e s u l t:

    Corollary. I f K s a t is f ie s t h e c o n d i t i o n ( H U ) r e l a t i v e l y t o ~ , t h e n , u n d e r t h e h y p o t h e s i s o f t h e o r e m 2 .2 ,p r o b l e m ( I ) h a s a n u n i q u e s o l u t i o n u ~ H 1 (~ 2) .R e m a r k 2 . 4. A s i t w a s p r o v e d b y G . S c h a e f f e r [ 1 0] , t h e r e is n o n u n i q u e n e s s f o r ~ > ~k2 i n t h e c a s e o f p l a s m am o d e l p r o b l e m . O n t h e o t h e r h a n d , t h e u n i q u e n e s s f o r h I _~o} ~ = I , I p r es c r ib ed ,

    ( I I )

    w h e r e h d e n o t e s t h e H e a v i s i d e f u n c t i o n0 i f s < 0 ,h ( s ) = 1 , i f s > 0 .

    T h i s i s a p a r t i c u l a r m o d e l o f m o r e g e n e r a l p r o b l e m s ( s e e r e fs . 4 , 7 ).A l t h o u g h i t i s n o t e s s e n ti a l, w e sh a l l a s s u m e f r o m n o w o n t h a t ~ ~ C 2 ( I ~ ) n C ( ~ ) a n d m i n r ~ k =

    m i n a u q o > 0 ; c o n s e q u e n t l y , 0 < I < f ~ i s a n e c e s s a r y c o n d i t i o n f o r e x i s te n c e o f so l u t i o n s o f ( II ) .

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    192 R. C ipolatti The equilibrium o f a confinedplasmaI n o r d e r t o o b t a i n o u r e x i s te n c e r e su l t, w e c o n s i d e r t h e f o l lo w i n g p e r t u r b e d p r o b l e m

    - ~ a u e = h ~ u e ) ,uE]o~ = OEcg h A u E ~ = ~ I ,

    in ~2,

    w h e r e e > 0 , h E i s d e f i n e d b y h ~ s ) = s + - s - e ) + .I f w e i n t r o d u c e t h e c l o s e d c o n v e x s e t

    K = v ~ L 2 ~ 2 ) ; O < v < I a . e . i n ~ } ,s o t h a t h e v ) = P E r V ) , t h e n ( 3 . 1 ) t a k e s t h e f o r m

    u E I o a = O E ~ ,f P e K U ~ ) q = e I .

    a .e . in I2 ,

    ( 3 . 1 )

    N o w , s i n c e p r o b l e m ( 3 . 2 ) i s o f t y p e ( I) , i t f o l l o w s f r o m l e m m a 2 .1 a n d t h e o r e m 2 .1 t h a t i t h a s a s o l u t i o nu, if 0 < I < fs~.T h e o r e m 3 . 1. S u p p o s e 0 < I < f s~ q' a n d l e t u e : (e > 0 ) b e a s o l u t i o n t o p r o b l e m 3 . 2. T h e n w e c a n e x t r a c t as u b s e q u e n c e o f { uE } w h i c h c o n v e r g e s in C 1 ' ~ ( ~ ) , 0 < a < 1 , t o a s o l u t i o n u o f (I I ).P r o o f . F r o m l e m m a 2 .1 , u e i s a so lu t ion o f ( 3 .2 ) i f a nd on ly i f u , = w~ + 0~q, w i th ( w~ , 0~) ~ H~( ~? ) Rs a t i s f y i n g

    { - A w E = 1 / e P x , w E ) , in 12, ( 3 . 3 a )w e l0 a = 0 ; P ~ x w e + 0 ~ ) ~ = e l , ( 3 . 3 b )

    w h e r e

    a n d

    Sin ce fo r a l l v ~ L2(~2)I 1 / e ) P ~ , v ) I = I P K ~ v / e ) l < - 1 a . e . i n ~2, w e h a v e , b y s t a n d a r d e l l ip t i c t h e o r y t h a t

    3 . 2 )

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    194 R Cipolatti The equilibrium of a confinedplasmaI f x ~ I a+ r e s p . : x ~ I 2 _ ) , w e c a n c h o o s e 8 x) > 0 s u c h t h a t

    - > 1 r e s p . : < 0 V O < e < 8 x )

    a n d s o1 / e) P ~ K U , ) X ) = 1 r e s p . : 1 / e) P ~K u ~ ) x ) = 0 ) .

    S i n c e 1 / e) P ~ K U , )~ X w e a k l y i n L 2 f a ) , i t f o l l o w s t ha tX = 1 a . e . i n 1 2+ r e s p . : X = 0 a . e . i n ~ 2 _ ) .

    O n t h e o t h e r h a n d , s i n c e u ~ H 2 I ~ ) , a w e l l - k n o w n r e s u lt o f G . S t a m p a c c h i a [ 11 ] s h o w s t h a t A u = 0 a . e .i n f g o -

    T h e r e f o r e X = h u ) a . e . i n ~2 a n d , b y 3 . 4 ) , u i s a s o l u t i o n o f I I ) .

    R e f e r e n c e s[1] H. Berestycki nd H. Brrzis , O n a f ree boun dary prob lem ar is ing in p lasma physics , No nl ine ar Ana l . th . Meth . A ppl . 4 , (1980)

    415-436 .[2] R. C ipo lat t i , Sur un pro b l rm e de valeur p ropre n on hneai re , Com ptes R endu s Ac. Sci . Par is 239 (1981) 455-458 .[3] R. C ipo lat t i , Con siderat ions sur un prob l~me no n l in rai re: l rqu i l ib re d un p la sm a conf ine, these 3e cycle , Unive rs i te de Par is X I ,1982.[4 ] R . C ip o l a t t i an d A. D amlam ian , No n co n v ex d u a l i t y fo r a f ree b o u n d ary p ro b l em: t h e p l asm a eq u i l i b r iu m, Co n t r i b u t i o n s t onon l inea r P .D.E., P i tm an Research Notes in M ath . No. 89 (Proc. In ter . Congress M adrid 1981) .

    [5] A . Dam lamian , Ap p l i ca t i o n d e l a d u a l i t 6 n o n co n vex e a u n p ro b l eme n o n l i n r a i re a f ro n t ie r e l i b re (6 q u i li b r e d u n p l asmaconfine) , C omp tes R end us Ac. Sci . Par is 286 (1978) 153-155 .[6] I . Ekelan d and R. T em am , Analyse Convexe et Prob l rm es Var iat ionnels (Du nod , Gran th ier -Vi l lars , 1974).[7] B. Hero n and M . Sermange, Non -conve x method s for compu t ing f ree bou nda ry equi l ib r ia o f ax ial ly sym metr ic p lasmas, Appl iedMathemat ics Opt imizat ion 8 (1982) .[8] B. Mercier , I n rq uat io ns Var iat ionnel les en M rcan ique (Publ ica t ions Ma thema t iques d Orsay , Par is , 1980) .[9] J . Puel , A f ree bo und ary nonl ine ar eigenvalue prob lem, Proc. o f the In t . Sym posium on Cont . M ech . and Par t ia l Dif feren t ialEqu at ions , Rio de Janei ro , 1977 , G.M . de La Pe nha and L.A. M edeiros, eds ., (Nor th -Hol lan d , Am sterdam , 1978) .[10] G. Schaeffer, No n un iqu eness in the equ i l ib r ium shape of a conf ined p lasma, Com m. in P .D.E. 2 (1977) 587-600 .[11] G . S tampac chia, Equat ion s el l ip t iques du second ordre h coeff icients d iscont inus (Presses de l Ur t iversi t6 de M ontreal , M ontreal ,1966).[12 ] R . Tem am, A n o n l i n ea r e ig en v a lu e p ro b l e m - t he sh ap e a t eq u i l i b r i u m o f a co n f i n ed p lasma , Arch . Ra t . M ech . An a l . 6 0 (1 9 76 )51.[13] R. Tem am , Rema rks on a f ree boun dar y value prob le m in p lasma physics, com m. in P .D.E. 2 (1977) 563 .[14] E. H. Z aran to nel lo , Pro ject ions on convex sets in a Hi lbe r t space and spec t ral theory , C ontr ibu t ions to Nonl in ear Func t iona lAnalysis , E .H. Zara n tone l lo , ed . (Academ ic Press, Ne w Y ork , 1971) .