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Non-uniqueness in the equilibrium shape of aconfined plasmaDavid G. Schaeffer aa Messachusetts Institute of Technology, CambridgePublished online: 23 Dec 2010.

To cite this article: David G. Schaeffer (1977) Non-uniqueness in the equilibrium shape of a confined plasma ,Communications in Partial Differential Equations, 2:6, 587-600, DOI: 10.1080/03605307708820040

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COMM. IN PARTIAL DIFFERENTIAL EQUATIONS, 2(6) , 587-600 (1977)

NON-UNIQUENESS I N THE EQUILIBRIUM SHAPE OF A CONFINED

PLASMA ' David G . Schaeffer*

Messachusetts I n s t i t u t e of Technology Cambridge, Massachusetts 02139

R . Temam [3] has r e c e n t l y proved t h e e x i s t e n c e of

s o l u t i o n s f o r t h e equat ions of a model desc r ib ing t h e

equ i l ib r ium of a confined plasma. I n t h i s paper we

consider a s l i g h t l y modified vers ion [ 1 , 4 ] of Temam's

o r i g i n a l problem which, a f t e r some i n i t i a l r e d u c t i o n s ,

may be formulated a s t h e fol lowing non l inear D i r i c h l e t

problem:

(1) - A U = AU i n n + ( l a ) u 1 an i s a cons tan t ( t o be determined)

- ' Research supported i n p a r t under NSF gran t 22927 . * Alfred P . Sloan Fellow.

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Here R c R~ i s a bounded domain with a smooth

boundary r , A = P ( a / a x i ) i s t he Laplace opera tor ,

u+ = m a x ( o , u ) , and i s a given pos i t i ve constant

g rea t e r than the lowest eigenvalue A I of A on R

(with Di r i ch l e t boundary condi t ions) . The constant i n

( l a ) , denoted u f r ) , i s t o be determined by the i n t e g r a l

r e l a t i o n ( l b ) , i n which a/av denotes t he inward

d i rec ted normal. The plasma occupies t he region

it follows from (1) t h a t A i s t h e lowest eigenvalue

of A on R f p ) . The hypothesis A > A assures

u f r ) < 0 , so t h a t R f p ) i s properly contained i n

Temam [3 ] based h i s o r i g i n a l ex is tence proof on a

t h a t

R .

charac te r iza t ion of so lu t ions a s extrema of a c e r t a i n

v a r i a t i o n a l problem, and these ideas have been adapted

[1,41 t o y i e ld exis tence f o r (1) - ( lb) . In t h i s note we show t h a t these equations may

admit more than one so lu t ion , even i n t h e case of

so lu t ions coming from the absolute minimum of one of

the v a r i a t i o n a l problems. Consider the problem (1) - ( lb) on a domain with the shape of an hourglass , a

domain cons is t ing of two bubbles connected by a long

t h i n neck. More p rec i se ly , f o r o < E < I l e t aE

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NON-UNIQUENESS OF A CONFINED PLASMA 589

be a region i n t h e plane of t h e form SZE = SE U wE u @,

where S' i s t h e s t r i p

and 0; i s t h e r e f l e c t i o n of €7'' under t h e symmetry

opera t ion

Here B ~ ( x , ~ ) denotes t h e b a l l of r a d i u s r around

t h e p o i n t x g . We s h a l l show, f o r both of t h e

v a r i a t i o n a l formulat ions [ 1 , 4 ] which provide s o l u t i o n s

t o (1) - ( l b ) , t h a t a minimizing f u n c t i o n cannot be

symmetric under t h e operat ion ( 3 ) . Thus i f u (x, y ) i s

one s o l u t i o n of t h e v a r i a t i o n a l problem, then t h e r e -

f l e c t e d func t ion u (-x, g ) i s a second s o l u t i o n . I n

o t h e r words, non-uniqueness a r i s e s a s an example of

spontaneously broken symnetry.

We s h a l l assume t h a t A - X I i s bounded away

from zero by a c o n s t a n t n o t depending on E . The

r e l a t i o n (2) a l lows us t o o b t a i n upper and lower bounds

f o r A I t h a t do n o t depend on E , because t h e e igen-

va lues o f A v a r y monotonically wi th t h e domain.

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590 SCHAEFFER

1 LEPIMA : There i s a constant C , depen.ding only on A

and ;he diameter o f Q , such t h a t any s o l u t i o n of (1) -

( l b ) s a t i s f i e s

PROOF: The fundamental e s t imate f o r t h i s lemma i s

where c depends only on A and t h e diameter of R.

Assuming ( 5 ) , m u l t i p l y equat ion ( I ) , r e - w r i t t e n a s

by u - u(r) and i n t e g r a t e over Q. On i n t e g r a t i n g

by p a r t s on t h e l e f t and applying t h e Schwartz i n -

e q u a l i t y on t h e r i g h t we a r r i v e a t (4 ) .

To prove (5) we argue a s fo l lows . Sobolev 's

Lema i n two dimensions g ives u s t h e e s t i m a t e

f o r any p < m. But s i n c e ~ ( r ! < 0 , we have L -

J. owe t o X . Br6zis t h e observa t ion t h a t t h i s e s t i m a t e hoLas f o r any s o l u t i o n of (1) - ( l b ) , n o t j u s t f o r f m c ~ i m s which minimize t h e v a r i a t i o n a l problem.

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NON-UNIQUENESS OF A CONFINED PLASMA 591

u f; % ( R ) , so t h a t 1 1 u+ : XI 1Q)ll i s equivalent: t o + 1 , 0

. Moreover on mul t ip ly ing (1) by u+ and

i n t e g r a t i n g we see t h a t

On t h e o ther hand by Holder ' s i n e q u a l i t y

where 0 < 0 < S and p = ( 2 - 2 8 ) / ( 1 - 2 0 ) . Now it

1 follows from (1b) t h a t 1 + : L ( R 1 = A . Therefore

we may combine t h e above i n e q u a l i t i e s (wich a conven-

i e n t p > 2) t o conclude

I n t h i s way we may ob ta in an 2 p r i o r i bound on

11 u+ : L~ (RI 11, which i n t u r n provides t h e e s t i m a t e (5) 2 f o r [I u+ : L (Q) I / , a s R i s bounded. The proof i s

complete.

Let us apply (4) i n SE, t h e neck of ! IE. If

-1 5 x 5 1 , then

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the l a t t e r equa l i t y being a de f in i t i on . By the Schwartz

inequal i ty

Theref o re , given any pos i t i ve constant a , t he Lebesgue

measure

must be pos i t i ve , provided E i s s u f f i c i e n t l y small.

Indeed, f o r small E the measure of t h i s s e t w i l l be

nearly equal t o 2 .

Our s t r a t egy i n t he proof below w i l l be t o obtain

an upper bound f o r u ( r ) ,

f o r some pos i t i ve constant a . On combining t h i s with

the p o s i t i v i t y of (6) we w i l l conclude t h a t does

not c ross the neck of QE, provided E i s small enough.

( I t i s impossible f o r small E t h a t $ 2 ' ~ ) i s e n t i r e l y

contained i n s ~ , s ince A i s an eigenvalue of A

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NON-UNIQUENESS OF A CONFINED PLASMA

on S ~ ( P ) . The lowest eigenvalue of A on SE i s

B ~ E - ~ ) and the lowest eigenvalue on a proper sub-

domain would be even l a rge r . ) We then show t h a t the

assumption t h a t a minimizing funct ion i s symmetric

under ( 3 ) leads t o a contradict ion

( i ) . The formulation of Temam.

Temam [ 4 ] proves exis tence f o r (1) - ( lb) by

minimizing the func t iona l

over the s e t

Observe t h a t f o r a minimizing function u\ which

necessar i ly i s a so lu t ion of (1) - ( l b ) , we have

Let be the lowest eigenfunction of A , normalized

so t h a t / 4 d z = I-'. Since 4 E K , we have

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594

However, by t h e Schwartz i n e q u a l i t y

so t h a t

where m(REl is t h e Lebesgue measure of t h a t s e t . On

combining (8) and (9) we ob ta in an upper bound f o r

U * ( r ) of t h e form ( 7 ) . A s remarked above, i t fol lows

t h a t n i p ) cannot c r o s s t h e neck of Q E , i f E i s

small enough.

However Kinder lehrer and Spruck 121 have shown

t h a t f o r t h e abso lu te minimum of t h i s v a r i a t i o n a l

problem' ~ ( p ) i s connected. (The argument i s p re -

sented i n 51.4 of C41.) Therefore n f p ) must be

contained i n e i t h e r one bubble of o r t h e o t h e r ,

so i t i s impossible f o r t h e minimizing func t ion u f

t o be symmetric wi th r e s p e c t t o ( 3 ) . The argument i s

complete.

( i i ) . The formulation of Beres tycki - Brbzis .

Berestycki - Br6zis .[I] prove ex i s tence f o r (1) -

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NON-UNIQUENESS O F A CONFINED PLASMA

( lb) by minimizing the funct ional

over the convex s e t

I f p * i s a minimizing function f o r t h i s problem, then

-2 a solut ion of (1) - ( lb) i s given by u" A p * + uV(TI,

where ~ * ( r ) = Q ~ P * ) , and moreover by (1)

Arguing a s above, t h a t i s , considering the f i r s t

eigenfunction of A a s a t r i a l funct ion, we may

obtain the est imate

Again we conclude t h a t Q ( P ) cannot cross the neck of

E

It i s not known a t present whether R ( P ) i s

always connected f o r minima of the Berestycki-BrBzis

formulation of t h i s problem. However we present a

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596 S CHAEFFER

d i r e c t argument t o show t h a t f o r E small , Q ( p )

cannot be minimized by a function symmetric with respect

t o ( 3 ) . Suppose t o the contrary t h a t p * i s a

symmetric function t h a t minimizes ~ ( p ) . Let us wr i t e

p * = p l + p 2 , where p l ( x , y ) = 0 f o r x < 0 and p 2

vanishes i n the complementary half plane. Observe

t h a t 2 p 1 s a t i s f i e s the cons t r a in t s of the problem

and

Now Q I p * ) i s negat ive, an upper bound being provided

by (11) . We claim t h a t the second term i n (12) can be

made a r b i t n a r i l y small by choosing E small. Proving

t h i s claim w i l l show t h a t Q ( 2 p 1 ) < Q ( P V , cont rad ic t -

ing the hypothesis t h a t p * was a minimizing funct ion.

Let us choose E SO small t h a t the s e t (6) has

measure a t l e a s t 3 / 2 . Then there e x i s t po in ts

x c (h , 1 ) and x 2 t ( - 1 , -3) such t h a t s l i p ) does 1

not cross the l i n e s C ( x , y ) : x = x . 1 . Moreover i t i s 2

impossible f o r a component of ~ 2 ' ~ ' t o be contained

i n the region between these two l i n e s , so both

p1 (x, y ) and p 2 ( x , y l vanish i f -3; 2 3: ( $.

It follows from (10) and (5) t h a t

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NON-UNIQUENESS OF A CONFINED PLASMA

and t h i s i n e q u a l i t y provides an es t imate f o r

1 A : ( 1 . We may s u b s t i t u t e t h i s l a t t e r

e s t imate i n t o t h e argument of t h e paragraph con ta in ing

equat ion ( 6 ) t o j u s t i f y t h e fol lowing conclus ion:

Given any p o s i t i v e cons tan t 6 ,

has p o s i t i v e measure, provided E i s s u f f i c i e n t l y

smal l . Let x be any element of t h e s e t (14) , and 0

l e t

Since A - I P 1 i s harmonic on n - and vanishes a t a n ,

by t h e maximum p r i n c i p l e

But p 2 i s supported i n - , so

t h e e s t i m a t e f o r I l p 2 : L 1 ( n ) l l coming from (13) . The

proof i s complete.

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59 8 SCHAEFFER

I n [4] Temam proved t h a t t h e s o l u t i o n of (1) - ( l b ) i s unique provided X I < X < A2, where {Ai} a r e

t h e eigenvalues of A on R enumerated i n inc reas ing

o r d e r , and a s i m i l a r r e s u l t i s given i n [ I ] . I t seems

noteworthy t h a t i n our example X I - X 2 . To see t h i s

we argue as fol lows. Let p i be t h e e igenvalues of

A on

(Here we use t h e n o t a t i o n of t h e paragraph con ta in ing

equation ( 2 ) . ) The spectrum of A on 0 7 i s t h e 0

same po in t s e t p . but a l l m u l t i p l i c i t i e s a r e double , 2

s i n c e t h e union O0 Pt i s d i s j o i n t . I n p a r t i c u l a n 0

t h e lowest e igenvalue i s p I and has m u l t i p l i c i t y

two. F i n a l l y , nE i s only a smal l p e r t u r b a t i o n of

1 q r so we have A1 2 A 2 z p l . (Note t h a t ' 0 - 0 ' X I < A 2 < p 2 , by t h e minimax c h a r a c t e r i z a t i o n of

e igenvalues . ) Moreover i n our example we may take A

a r b i t r a r i l y c l o s e t o p1 . I n o ther words, our example

suggests t h a t t h e s e uniqueness r e s u l t s a r e probably

t h e b e s t p o s s i b l e without some a d d i t i o n a l r e s t r i c t i o n

on t h e domain Q .

We remark i n c l o s i n g t h a t by minimizing E l u l

o r Q ( p l i n t h e c l a s s of func t ions t h a t s a t i s f y t h e

c o n s t r a i n t s and a r e symmetric under t h e r e f l e c t i o n ( 3 ) '

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NON-UNIQUENESS OF A CONFINED PLASMA 599

one may ob ta in a s o l u t i o n of (1) - ( l b ) such t h a t n f p )

con ta ins two components. (Al l t h e es t imates of t h i s

paper apply without modif icat ion t o show t h a t R

cannot c r o s s t h e neck of R E f o r small E . ) Of course

t h i s s o l u t i o n does n o t r epresen t t h e absolute minimum

of t h e v a r i a t i o n a l problem.

ACKNOWLEDGEMENT --

I t i s a p leasure t o record my indebtedness t o

H . BrBzis, D . K inder lehre r , and R , Temam f o r con-

v e r s a t i o n s and correspondence on t h i s problem.

REFERENCES

(

1. H . Beres tycki and H . BrBzis, "Sur c e r t a i n s

p r o b l h e s de f r o n t i e r e l i b r e , " t o appear

i n Comptes Rendus.

2 . D . Kinder lehrer and J . Spruck, t o appear

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600 SCHAEFFER

3 . R . Temam, "A non-linear eigenvalue problem:

equilibrium shape of a confined plasma, "

Archive Rat. Mech. Anal. 60 (1975),

pp. 51-73.

4 . R . Temam, "Remarks on a f r e e boundary value problem

a r i s i n g i n plasma physics ," Corn. Pa r t .

Di f f . Eqn. , t h i s i s sue .

Received October 1976

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