Periodic Solutions of Semilinear Elliptic Equations in A Strip

U LJ J"IW7LL-LU

L OqLcrr I L . V I I J U l l U npp,cic-ucrvrio

W. Ecckhaus and E . M . de Jager (ed8.l

Worth-Holland Publishing Company (19781

P E R I O D I C SOLUTIONS OF SEMILINEAR E L L I P T I C EQUATIONS I N A STRIP

Klaus Kirchgassner Math.Institut A Universitat Stuttgart S t u t t g a r t , W.Germany

A parameter-dependent

s e m i l i n e a r e l l i p t i c boundary v a l u e p r o b l e m i s c o n s i d e r e d i n a s t r i p . I t i s shown f o r some p a r a m e t e r i n t e r v a l t h a t , i f t h e n o n l i n e a r i t y s a t i s f i e s c e r t a i n symmetry c o n d i t i o n s , a l l "small" s o l u t i o n s a r e p e r i o d i c i n t h e unbounded v a r i a b l e . The method d e s c r i b e d i s g e n e r a l i s a b l e t o h i g h e r o r d e r e l l i p t i c equations. INTRODUCTION

The 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 i s c o n s i d e r e d

Here, A d e n o t e s t h e j j w o - d i m e n s i o n a l L a p l a c e a n , X a r e a l parameter, and f a C - f u n c t i o n o f i t s arguments which i s h o r i z o n t a l a t 0. One m i g h t c o n s i d e r ( 1 ) as a m o d e l e q u a t i o n f o r t h e N a v i e r - S t o k e s s y s t e m i f f i s c h o s e n t o b e u a y u , or one c o u l d c o n s i d e r ( 1 ) as t h e s t a t i o n a r y p a r t o f a r e a c t i o n d i f f u s i o n e q u a t i o n . I t i s q u i t e e a s y t o p r o v e , n a m e l y by r e s t r i c t i n g t h e c o n s i d e r a t i o n t o y - p e r i o d i c s o l u t i o n s of a g i v e n p e r i o d , t h a t (1) h a s a c o n t i n u u m o f b i f u r c a t i o n p o i n t s , p r o v i d e d f s a t i s f i e s c e r t a i n symmetry c o n d i t i o n s . L e t u s assume f o r a moment, t h a t (1) i s t h e s t a t i o n a r y p a r t o f some e v o l u t i o n e q u a t i o n i n t i m e a n d t h a t s e l e c t i o n o f c e r t a i n s o l u t i o n s o f (1) i s u n d e r s t o o d t h r o u g h t h e i r s t a b i l i t y - a n d i n s t a b i l i t y p r o p e r t i e s . Then t h e q u e s t i o n w h i c h p a t t e r n i s s e l e c t e d r e q u i r e s two m a i n a n s w e r s , namely t h e d e t e r m i n a t i o n o f a l l s o l u t i o n s and t h e s t u d y of t h e i r s t a b i l i t y . The n o n s t a n d a r d a s p e c t o f t h i s b i f u r c a t i o n p r o b l e m i s d u e t o t h e f a c t t h a t t h e d i f f e r e n t i a l o p e r a t o r s i n (1) a n d t h e domain Q a r e i n v a r i a n t u n d e r t r a n s l a t i o n s i n y - d i r e c t i o n . 7

K. KIRCHGXSSNER

8

Hence, p e r i o d i c i t y i n y w i t h a n y p e r i o d i s a n a d d i t i o n a l condition consistent with (1). I n t h i s respect (1) is the s i m p l e s t n o n t r i v i a l model for p r o b l e m s i n h y d r o d y n a m i c a l s t a b i l i t y , s u c h as t h e Bbnard- p r o b l e m , o r p r o b l e m s i n phase t r a n s i t i o n s which a r e i n v a r i a n t under t h e Euclidean g r o u p E(2) o f t h e p l a n e [ 4 1 , [ 7 1 , [ 8 1 . R e c e n t l y a l l s o l u t i o n s w i t h c e r t a i n symmetry p r o p e r t i e s h a v e b e e n d e t e r m i n e d s u c c e s s f u l l y by g r o u p - t h e o r e t i c m e t h o d s [ 8 1 . Howe v e r , t h e b a s i c assumption of p e r i o d i c i t y , though q u e s t i o n e d , has never been j u s t i f i e d mathematically. I n t h i s c o n t r i b u t i o n w e g i v e a p a r t i a l answer t o t h e q u e s t i o n r a i s e d a b o v e f o r t h e e q u a t i o n ( 1 ) . We c l a s s i f y , f o r v a l u e s o f X l e s s t h a n 4n2, a l l s o l u t i o n s o f ( 1 ) i n a s u i t a b l e n e i g h b o r h o o d o f 0 . For X l e s s t h a n n2 t h e t r i v i a l s o l u t i o n u = 0 i s l o c a l l y u n i q u e , f o r X b e t w e e n n 2 a n d 4n2 all"smal1" s o l u t i o n s a r e p e r i o d i c i n y , i f f h a s c e r t a i n symmetry p r o p e r t i e s . E x i s t e n c e o f " s i n g u l a r " s o l u t i o n s c a n b e shown i f a c o n d i t i o n f o r t h e g e o m e t r o f t h e b i f u r c a t i o n p i c t u r e i s met. For X g r e a t e r t h a n 4n' t h e p r o b l e m i s s t i l l u n s o l v e d . S i n c e t h e s t a b i l i t y q u e s t i o n h a s b e e n a n s w e r e d e l s e w h e r e we o m i t i t h e r e [ 31. W h i l e $ h e p r o o f of Theorem 2 appears e l s e w h e r e , we p r e s e n t a new p r o o f of Theorem 1 w h i c h , i n c o n t r a s t t o t h a t i n [ 51, c a n b e g e n e r a l i s e d t o h i g h e r o r d e r e l l i p t i c e q u a t i o n s w i t h c o e f f i c i e n t s d e p e n d i n g s m o o t h l y on x. I am i n d e b t e d t o D r . J . S c h e u r l e f o r many h e l p f u l d i s c u s s i o n s . UNIQUENESS

rt

Let d e n o t e t h e s p a c e o f t e m p e r e d d i s t r i b u t i o n s on 0 , d e f i n e t h e w e i g h t f u n c t i o n g k ( y ) = ( 1 t y 2 ) k / 2 f o r a n y n a t u r a l number k r N 0 , and c o n s i d e r t h e r e a l H i l b e r t s p a c e s

with t h e inner products

Moreover we n e e d

w i t h t h e norm

PERIODIC SOLUTIONS

9

w h e r e t h e infimum i s computed o v e r a l l r e p r e s e n t a t i o n s o f u of t h e f o r m u = caBgB, gBE L2(Q). It i s w e l l known ( c . f . [ 9 1 ) t h a t t h e Fourier tfansform F with respect t o y defines an isomorphism from H onto HOk. LEMMA 1

For X < (2)

n2

a n d for e v e r y k E N o ,

A X = ( A t X )

t h e continuous operator

HE-,H;

:

has a continuous inverse. P r o o f : C o n s i d e r t h e o p e r a t o r 3t' a$x t ( 1 - n2) a n d t h e c o r r e s p o n d i n g G r e e n ' s f u n c t i o n G ( x , C ; q ) f o r o-boundary c o n d i t i o n s a t x=O a n d x = l . For g EL2(R) r e s p . r ( Q ) , t h e function

(3)

f

v(x,n) =

G(x,S;n)g(S,n)

0

dS

l i e s i n L2(Q) r e s p . y(Q)a n d v a n i s h e s for x = o a n d 1. ( v h a s enough r e g u l a r i t y i n x t o d e f i n e t h e t r a c e . ) H e n c e 3 i s o n t o 6 i m p l y i n g & i %' t o b e i n j e c t i v e . The r e s t r i c t i o n I Y W I H m k = -k i s o n t o ~ - a ks c a n b e s e e n by d i f f e r e n t i a t i o n ofxv g , g,vE L z , a n d by t h e d e f i n i t i o n o f H q k . S i n c e H - ~ C 3: -k i s i n v e r t i b l e a n d i t s i n v e r s e + t I & i s c o n t i n u o u s . Moreover -1 (&-, g)(xo, ) = 0 for x 0 a n d xo = 1

r',

.

(4)

0

Now c o n s i d e r

(5)

,

Ah u = f

S e t u = Fv, f F-1 f ->g

d

f E HE

Fg, v -1 >

V L

=xZ$g, >

t h e n t h e sequence

U

y i e l d s t h a t Ax1 e x i s t s a n d i s c o n t i n u o u s i n H E . The e q u a t i o n s (4) a n d ( 5 ) i m p l y u € H i a n d t h u s t h e a s s e r t i o n . I n o r d e r t o f o r m u l a t e t h e u n i q u e n e s s r e s u l t we c o v e r R w i t h a s e q u e n c e of c o m p a c t a

K,

[0,1] x

[(a-l),

I? 1

,

11 E

z

Hm(K,) d e n o t e s t h e u s u a l S o b o l e v - s p a c e o f o r d e r m .

K. KIRCHGXSSNER

10

THEOREM 1 L e t b e f € C 2 ( R 3 , R ) , assume f ( 0 ) = 0 , Vf(0) 0 . If X < n 2 t h E n t h e r e e x i s t s a n E > 0 s u c h t h a t , for a n y two s o l u t i o n s ) (I) s a t i s f y i n g u , u E H $ ~ ~ ( E of

‘YP ‘ I u 1 ’ J it follows u

H2(Kj)

‘

3

J

E.

II;IIH2(Kj)

<

E

Proof: Define f j ( x , y ) = f ( x , y + j ) and suppose SUP II f J IIHO(K1) < m J f o r some f E HPoc(’sZ). Then we h a v e

4 c3 sup I I f J j

and t h e r e f o r e

(6)

s u p IIA,’fj

llH2(K1)

J

Lc !

IIHO(K1)

l t ( k1- 1 1 2

J

h o l d s . D e f i n e T : u + f ( u , a x u , a y u ) which, i n view of t h e s m o o t h n e s s a s s u m p t i o n s on f, i s a c o n t i n u o u s map f r o m H 2 ( K ) i n t o H o ( K ) for e v e r y compact s e t K C Q. M o r e o v e r , for every p > 0, t h e r e e x i s t s a 6 > 0 such t h a t IIT(u)

-

T(C) l l H ~ ( K l )<

PIIU

- iiII 2

H (K1)

v

if I I U I I ~ ~ ~ I~ I ~ u ) I , I ~ ~ )( ~ a r e l e s s t h a n 6 . Now l e t u a n d be s o l u t i o n s , o f ( 1 ) ; c h o & s e y p < 1 a n d E t o b e a c o r r e s p o n d i n g 6. U s i n g ( 6 ) we o b t a i n J

s u p I I A ~ \ T ( -~ ~T )( z ~ ) ) I I ~ ~ ( ~ ~ )

J < ypsup J

IIUJ

-

YlJ

II

2

H (K1)

which i m p l i e s t h e a s s e r t i o n . The method o f p r o o f c a n b e g e n e r a l i z e d i m m e d i a t e l y t o h i g h e r order uniformly e l l i p t i c operators L t X = A i n t h e s t r i p Q w i t h c o e f f i c i e n t s d e p e n d i n g s m o o t h l y on t h e bounded v a r i a b l e x t o g e t h e r w i t h homogeneous D i r i c h l e t b o u n d a r y c o n d i t i o n s . If X i s s u c h t h a t k e r ( L t X ) {Ol t h e n u = 0 is a n i s o l a t e d s o l u t i o n among a l l s o l u t i o n s w i t h u n i f o r m l y small H2m(Kj) -

PERIODIC SOLUTIONS

11

-.

norm - 2m b e i n g t h e o r d e r of L A s an example c o n s i d e r t h e t w o - d i m e n s i o n a l b o u n d a r y v a l u e p r o b l e m d e s c r i b i n g a l l timei n d e p e n d e n t p e r t u r b a t i o n s of p l a n e P o i s e u i l l e f l o w 121

(7)

A 2 J, + h ( - 2 a X r l ,

-

u,~,(AJ,)) + h a x + a r ( ~ J , )

-

aY+ax(A$))

a $ = o onaa Y where u o ( x ) x ( 1 - x ) , 'J, t h e stream f u n c t i o n a n d X t h e R e y n o l d s number. S i n c e ( 7 ) f a l l s i n t o t h e f r a m e w o r k o f t h i s a n a l y s i s we c o n c l u d e t h a t p l a n e P o i s e u i l l e f l o w i s a n i s o l a t e d s o l u t i o n of t h e Navier-Stokes system i n t h e s e n s e d e s c r i b e d above, as l o n g as t h e k e r n e l o f t h e d e r i v a t i v e o f ( 7 ) a t 0 i s { D ) . D e t a i l e d p r o o f s o f t h i s g e n e r a l i z a t i o n cam b e f o u n d i n a f o r t h coming p a p e r . J , =

T h e r e i s a n e x t e n s i o n o f Theorem 1 f o r e q u a t i o n ( 1 ) b e y o n d X n2. I n a p r e v i o u s p a p e r [ 5 1 i t was shown t h a t u n i q u e n e s s modulo k e r ( A + X ) h o l d s l o c a l l y f o r a l l XER. To b e p r e c i s e , l e t be

m

HE

u , s = o o r 2 k=l t h e i n d u c t i v e l i m i t o f t h e s a e s HS T h e n A X : "2 X + x0 i s always s u r j e c t i v e . I f X E (n'n',(n+!fj2n2), t h e k e r n e l of A X i s s p a n n e d by t h e f u n c t i o n s xs

'p2j-1

, ,

= s i n j r x c o s w.y J

w

j

-

j2n2}

1'2

,...,

s i n j n x s i n w.y j = 1 n 2' j J We d e f i n e t h e F o u r i e r c o e f f i c i e n t s o f u a n d a p r o j e c t o r a s follows 1

u . ( y ) = q y j s i n j n x u ( x , y ) dx J n o 1 = F(Uv(0)cP2"-l + - u;(0)'p2.) Pnu w

THEOREM 2

Assume t h a t X E ( n 2 r 2 , ( n t 1 ) 2 n 2 ) , n E N , t h e n t h e r e e x i s t s a, p o s i t i v e number E s u c h t h a t , g i v e n a n y t w o s o l u t i o n s u a n d u o f (1) s a t i s f y i n g

t h e two s o l u t i o n s c o i n c i d e . F o r t h e p r o o f see [ 5 ] . For a c o m p l e t e d e s c r i p t i o n o f a l l small s o l u t i o r i s o f ( 1 ) i t s u f f i c e s t o show t h a t " a b o v e " e v e r y

K. KIRCHGXSSNER

12

cp E k e r ( A t A )

t h e r e e x i s t s a t l e a s t one s o l u t i o n .

EXISTENCE What w e h a v e s a i d a b o u t t h e k e r n e l o f 15 t X s u g g e s t s t h a t , f o r X E ( v 2 , 4 7 r 2 ) , and u n d e r s u i t a b l e a s s u m p t i o n s on f , a l l "small" s o l u t i o n s o f ( 1 ) s h o u l d be p e r i o d i c i n y . T h a t a d d i t i o n a l c o n d i t i o n s on f a r e n e c e s s a r y i s shown by t h e e x a m p l e f ( a u ) 3 , for w h i c h n o y - p e r i o d i c s o l u t i o n e x i s t s , e x c e p t u 0 . Lex u s t h e r e f o r e assume

(8)

(a)

f(u,p,-q)

f(u,p,q)

(b)

f(u,p,-q)

-f(u,p,q)

and f ( - u , - p , - q )

f(u,p,q)

Consider t h e case 8 a ) , set u ( x , y ) v(x,wy) and d e t e r m i n e , f o r f i x e d X E ( 1 r 2 , 4 ~ 2 ) ,n o n t r i v i a l s o l u t i o n s of

(3)

B(w)v

t

v(0,z) where

B

a 2xx

XV t f(v,a v,WZv)= 0 X

v(1,z) t w

0

,

v ( x , . ) 27r - p e r i o d i c

2 a z2 z

We con i d e r (9) as a b i f u r c a t i o n p r o b l e m n e a r w = w 1 = (X - IT ) 1 / 2 a n d v 0 . I f we i m p o s e t h e f u r t h e r r e q u i r e m e n t t h a t v s h o u l d b e e v e n i n z , t h e o p e r a t o r B(w,), b e i n g s e l f a d j o i n t i n L 2 ( ( 0 , l ) x ( 0 , 2 ~ ) ) ,h a s a 1 - d i m e n s i o n a l k e r n e l f o r 0 w w 1 : H e n c e , by a w e l l known t h e o r e m [ l ] , w = 01, v is a bifurcation point.

3

Since,for every s o l u t i o n u, uc(x,y) = u(x,ytc) i s a solution as w e l l , o n e o b t a i n s a t w o ; d i m e n s i o n a l m a n i f o l d o f y - p e r i o d i c s o l u t i o n s o f (1) w h i c h i s m o d e l l e d o v e r k e r ( A t A ) ( c . f . [ 5 1 ) . The c a s e 8b) c a n b e t r e a t e d s i m i l a r l y . Hence we h a v e THEOREM 3

L e t b e X E ( ~ 2 ~ 4 1 a~n2d )assume o n e o f t h e c o n d i t i o n s ( 8 ) t o h o l d . Then t h e r e e x i s t s a p o s i t i v e number such t h a t , i f lPlul < E ~ ,and i f u i s a s o l u t i o n o f ( l ) , t h e n u i s p e r i o d i c i n y. Conversely, f o r every s a t i s f y i n g lPlul = E .

E

E ( O , E ~ Ia s o l u t i o n u o f

(1) e x i s t s

The c a s e X > 4 n 2 i s much more d i f f i c u l t t o s o l v e . The k e r n e l o f ( A + X ) c o n s i s t s of q u a s i p e r i o d i c f u n c t i o n s , i . e . f u n c t i o n s of t h e form u ( x , y ) = v ( x , u l y , a n y ) , where i s 2n - p e r i o d i c i n e v e r y z The s t u d y o f v(x,zl,,..,zn) t h e f u l l n o n l i n e a r e q u a t i o n l e a d s t o p r o b l e i s of small d i v i s o r s ( s e e [ 3 1 ) . N o t h i n g i s known a b o u t e x i s t e n c e .

...,

.

PERIODIC SOLUTIONS

13

The p o i n t X n2 may b e a p o i n t o f b i f u r c a t i o n f o r s i n g u l a r s o l u t i o n s , i . e . f u n c t i o n s whose f i r s t F o u r i e r component i s e i t h e r c o n s t a n t or n o n p e r i o d i c . L e t u s c o n s i d e r t h e s e t Sx = { u E X 2 / u b e l o n g s t o t h e compon n t o f s o l u t i o n s b i f u r c a t i n g a t (o,,~)}. I f u s h y A ~ ( 5 -n6 , n 2 + 6 ) f o r some 6 > 0, i s c o n f i n e d t o some domain D c ( n 2 , - ) x X 2 , and i f D fl ( { w I x X2) D, s h r i n k s t o {(W~,O)] 5 s approaches n2 from above, t h e n t h e r e e x i s t , f o r X E ( n , n + 6 ) , n o n t r i v i a l s o l u t i o n s o f a r b i t r a r y l a r g e i r r e d u c i b l e p e r i o d s . T hey c o n v e r g e with i n c r e a s i n g period towards a s i n g u l a r s o l u t i o n i n Xc. If t h i s g e o m e t r i c c o n d i t i o n is v i o l a t e d , s i n g u l a r s o l u t i o n s may n o t e x i s t . The p r o o f i s a c o n s e q u e n c e o f Theorem 2 a n d of t h e g l o b a l b i f u r c a t i o n r e s u l t of Rabinowitz [ 6 1 . REFERENCES

[11 C r a n d a l l , M . G . and R a b i n o w i t z , P . H . , (19711, B i f u r c a t i o n from s i m p l e e i g e n v a l u e s , J . F u n c t i o n a l A n a l . , 8 , p p . 321-340. 121 J o s e p h , D.D., (1976), S t a b i l i t y o f f l u i d m o t i o n s , I , 11, Springer-Verlag, Berlin.

[31 K i r c h g a s s n e r , K . ,

(1977), P r e f e r e n c e i n p a t t e r n a n d c e l l u l a r b i f u r c a t i o n i n f l u i d dynamics, i n Applic. o f i f u r c a t i o n t h e o r y , P . R a b i n o w i t z e d . , Academic P r e s s , pp.149-173.

141 K i r c h g s s s n e r , K . and K i e l h b f e r , H . ,

(1972), S t a b i l i t y a n d b i f u r c a t i o n i n f l u i d m e c h a n i c s , Rocky M o u n t a i n J . M a t h . , 3, p p . 275-318.

[5] K i r c h g s s s n e r , K . a n d S c h e u r l e , J . , (1977), On t h e

bounded s o l u t i o n s o f a s e m i l i n e a r e l l i p t i c e q u a t i o n i n a s t r i p , manuscript, t o appear.

[ 6 ] R a b i n o w i t z , P. H . ,

(1971), Some g l o b a l results f o r n o n l i n e a r eigenvalue problems, J . Functional Anal., 7, P P . 487-513.

[7] R a v e ch g , H . J . and S t u a r t ,

C . A . , (1976), B i f u r c a t i o n o f s o l u t i o n s w i t h c r y s t a l l i n e s ymm et ry, J . Mat h. P h y s . , 17, p p . 1949-1953.

181 S a t t i n g e r , D.H., (1977), Group r e p r e s e n t a c t o n t h e o r y , b i f u r c a t i o n t h e o r y , a n d p a t t e r n f o r m a t i o n , (1977), J . Functional Anal.,

t o appear.

[g] T r s v e s , F . , (1967), T o p o l o g i c a l v e c t o r s p a c e s ,

d i s t r i b u t i o n s a n d k e r n e l s , Academic P r e s s , N e w Y o r k .