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Comparison and oscillation theorems for elliptic equations Allegretto, Walter 1969

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COMPARISON AND OSCILLATION THEOREMS FOR ELLIPTIC EQUATIONS by WALTER ALLEGRETTO B . A . S c , U n i v e r s i t y o f B r i t i s h C o lumbia, 19^5. A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY i n the Department • • ' o f -Mathematics We a c c e p t t h i s t h e s i s as c o n f o r m i n g t o the r e q u i r e d s t a n d a r d . The U n i v e r s i t y o f B r i t i s h Columbia A p r i l 1969. In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f the r e q u i r e m e n t s f o r an advanced degree a t the U n i v e r s i t y o f B r i t i s h C olumbia, I ag r e e t h a t t h e L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and Study. I f u r t h e r a g r e e t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g of t h i s t h e s i s f o r s c h o l a r l y p u rposes may be g r a n t e d by the Head o f my Department or by h i s r e p r e s e n t a t i v e s . I t i s u n d e r s t o o d t h a t c o p y i n g or p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . Department o f M 3.4Le Vt ^ i n cS. The U n i v e r s i t y o f B r i t i s h Columbia Vancouver 8, Canada Date A ^ C i l ^2) i i . T h e s i s S u p e r v i s o r : C. A. Swanson. ABSTRACT New comparison and Sturm-type theorems a r e e s t a b l i s h e d w h i c h e n a b l e us t o extend known o s c i l l a t i o n and n o r i - o s c i l l a t i o n c r i t e r i a t o : ( l ) n o n - s e l f - a d j o i n t o p e r a t o r s , (2) q u a s i - l i n e a r o p e r a t o r s , (3) f o u r t h o r d e r o p e r a t o r s o f a ty p e n o t p r e v i o u s l y -c o n s i d e r e d . S i n c e t h e c l a s s i c a l p r i n c i p l e o f Courant does n o t h o l d f o r some o f t h e o p e r a t o r s c o n s i d e r e d , t h e comparison theorems i n v o l v e , i n p a r t , new e s t i m a t e s on t h e l o c a t i o n o f t h e s m a l l e s t e i g e n v a l u e o f the o p e r a t o r s i n q u e s t i o n . A d e s c r i p t i o n o f the b e h a v i o u r ' o f the e i g e n v a l u e as the domain i s p e r t u r b e d i s a l s o g i v e n f o r such o p e r a t o r s by t h e use o f Schauder's "a p r i o r i " e s t i m a t e s . The Sturm-type theorems a r e p r o v e d by t o p o l o g i c a l a r g u -ments and extended t o q u a s i - l i n e a r as w e l l as t o n o n - s e l f - a d j o i n t o p e r a t o r s . The f o u r t h o r d e r o p e r a t o r s c o n s i d e r e d a r e o f a ty p e w h i c h does n o t y i e l d forms i d e n t i c a l t o t h o s e a r i s i n g i n second o r d e r problems. Some examples i l l u s t r a t i n g the t h e o r y a r e g i v e n . I i i . TABLE OF CONTENTS page CHAPTER I Second Order N o n - s e l f - a d j o i n t E q u a t i o n s . 1. I n t r o d u c t i o n 1 2. R e l a t i o n Between t h e Two Types o f O s c i l l a t i o n 2 3. A Comparison Theorem f o r L . 10 4. O s c i l l a t i o n C r i t e r i a f o r L . 28 CHAPTER I I Q u a s i l i n e a r E l l i p t i c E q u a t i o n s . 1. I n t r o d u c t i o n 34 2. A Sturm Theorem and a Comparison Theorem f o r L 35 3- O s c i l l a t i o n and n o n - o s c i l l a t i o n Theorems 43 4. Bounds on t h e F i r s t E i g e n v a l u e o f L 48 CHAPTER I I I F o u r t h Order E q u a t i o n s . 1. I n t r o d u c t i o n - 55 2. Comparison Theorems f o r F o r m a l l y S e l f - a d j o i n t O p e r a t o r s 57 3- O s c i l l a t i o n Theorems f o r F o r m a l l y S e l f - a d j o i n t O p e r a t o r s 64 4. N o n - o s c i l l a t i o n C r i t e r i a . 67 BIBLIOGRAPHY 70 ACKNOWLEDGEMENTS The a u t h o r would l i k e t o thank Dr. C. A. Swanson f o r s u g g e s t i n g the t o p i c o f t h i s t h e s i s , and f o r g i v i n g h i s a d v i c e and encouragement t h r o u g h o u t i t s p r e p a r a t i o n . He would a l s o l i k e t o ex t e n d h i s a p p r e c i a t i o n t o Dr. C. W. C l a r k who r e a d and c o n s t r u c t i v e l y c r i t i c i z e d the d r a f t form o f t h i s work. The generous f i n a n c i a l s u p p o r t o f t h e N a t i o n a l R e s e a r c h C o u n c i l o f Canada and o f t h e U n i v e r s i t y o f B r i t i s h C olumbia i s g r a t e f u l l y acknowledged. CHAPTER I SECOND ORDER NON-SELF-ADJOINT EQUATIONS 1. I n t r o d u c t i o n . I n t h i s c h a p t e r t h e second o r d e r l i n e a r e l l i p t i c o p e r a t o r L d e f i n e d by • n n (1.1) Lu - S D ( a . ,D.u) + E b.D.u + cu i , j = l 1 1 J 0 j = l J J w i l l be c o n s i d e r e d i n unbounded domains R o f t h e n - d i m e n s i o n a l E u c l i d e a n space E 1 1 . Our main r e s u l t s a r e a q u i t e g e n e r a l Sturm theorem as w e l l as a com p a r i s o n theorem f o r o p e r a t o r ( l . l ) . These w i l l e n a b l e us t o e a s i l y g e n e r a l i z e o s c i l l a t i o n theorems w h i c h a r e known f o r s e l f - a d j o i n t second o r d e r o p e r a t o r s . P o i n t s o f E 1 1 w i l l be denoted by x = ( x ^ . . . , x n ) and D^ w i l l s i g n i f y d i f f e r e n t i a t i o n w i t h r e s p e c t t o x^ . The c o e f f i c i e n t s a. . w i l l always be assumed r e a l and o f c l a s s C"'"(R) ; f u r t h e r m o r e t h e m a t r i x ( a . - ( x ) ) w i l l be t a k e n p o s i t i v e d e f i n i t e symmetric i n R . By 5 ) ( L 3 Q ) we s h a l l mean t h e c o l l e c t i o n o f a l l r e a l f u n c t i o n s o f c l a s s C (fi) 0 C(0) , where Q denotes any subdomain o f R . The c o e f f i c i e n t s - b . and c J w i l l a l s o be t a k e n r e a l . D e f i n i t i o n A ( c l a s s i c a l ) s o l u t i o n u o f Lu = f i n n i s a f u n c t i o n u i n ?)(L,0.) f o r w h i c h L u ( x ) = f ( x ) f o r e v e r y x i n • Q . D e f i n i t i o n A bounded domain N c R i s a n o d a l domain o f L i f f t h e r e e x i s t s a n o n - t r i v i a l s o l u t i o n u o f L u = 0 i n N such t h a t u = 0 on dN . N o t a t i o n R r = R 0 {x : x € eF1 and |x| > r} 0 = c l o s u r e o f Q i n t h e t o p o l o g y . D e f i n i t i o n The o p e r a t o r ( l . l ) i s o s c i l l a t o r y o f t y p e 1 (Osc 1) i f f f o r e v e r y r > 0 , L has a n o d a l domain i n R r . D e f i n i t i o n ^The o p e r a t o r ( l . l ) i s o s c i l l a t o r y o f type 2 (Osc 2) i f f f o r e v e r y r > 0 , e v e r y s o l u t i o n o f Lu = 0: i n R has a z e r o i n R r . D e f i n i t i o n The o p e r a t o r ( l . l ) i s n o n - o s c i l l a t o r y i f f t h e r e e x i s t s r > 0 such t h a t L has no n o d a l domains i n R r . 2. R e l a t i o n Between t h e Two Types o f O s c i l l a t i o n . We s h a l l now show t h a t i f L i s Osc 1 , t h e n i t i s i n f a c t Osc 2 under m i l d c o n d i t i o n s on the c o e f f i c i e n t s o f LV. and none a t a l l on the n a t u r e o f the b o u n d a r i e s o f the r e g i o n s i n v o l v e d . S p e c i f i c a l l y , we s h a l l show t h a t i f N i s a n o d a l domain o f L , t h e n e v e r y s o l u t i o n o f Lu = 0 i n N must v a n i s h somewhere i n N . I f t h e c o e f f i c i e n t s a r e c o n t i n u o u s i n N and L i s u n i f o r m l y e l l i p t i c t h e r e , i . e . , t h e r e e x i s t c o n s t a n t s m > 0 and k > 0 such t h a t 3. n n _ S a (x) U , > m Z 57 , | a . , ( x ) | < k , |b ( x ) | < k i , j = l 1 J i = l 1 1 1 1 ~ f o r e v e r y 5 = (^,...,§n) and f o r e v e r y x i n N , our r e s u l t f o l l o v / s i m m e d i a t e l y by methods i d e n t i c a l to. t h o s e o f P r o t t e r and Weinberger [ l ] . P r o p o s i t i o n 1.1 Assume t h a t . L obeys the above c o n d i t i o n s and t h a t N i s a n o d a l domain o f the o p e r a t o r L . Then e v e r y f u n c t i o n v i n ^ } ( L , N ) such t h a t L v = 0 i n N , must v a n i s h somewhere i n N . \ P r o o f : I f t h i s were n o t t h e . c a s e , we c o u l d f i n d a f u n c t i o n v i n (L,N) such t h a t L v = 0 i n N and, w i t h o u t l o s s o f g e n e r a l i t y , v > 0 everywhere i n N . L e t u be a n o n - t r i v i a l f u n c t i o n such t h a t L u = 0 i n N and u = 0 on dN . We d e f i n e a new f u n c t i o n w by w = ^ . Then c l e a r l y w i s i n ^ ) ( L , N ) and a g a i n we may assume w > 0 somewhere i n N . Now L(vw) = L ( u ) = 0- i n N_ , b u t n n n L(vw) = L ( v ) w - s D ( a . -D.w)v + 2 ( v b . - 2E a. .D.v)D.w. i , j = l 1 1 J J j=l J 1=1 1 J 1 J Hence n n n D.v - S D. ( a . .D.w) + £ (b . - 2 S a. , ~ - )D,w = 0 i n N . i , j = l 1 1 J J j = l J i = l 1 J v J But t h i s v i o l a t e s t h e c l a s s i c a l Hopf maximum p r i n c i p l e [ 2, p. 150] as w = 0 on 3N . The c o n t r a d i c t i o n p r o v e s t h a t v must 4. v a n i s h somewhere i n N . • We s h a l l show more g e n e r a l l y : Theorem 1.2 Assume t h a t N i s any n o d a l domain o f L i n R . Fur t h e r m o r e assume t h a t the c o e f f i c i e n t s c and b. a r e bounded i n N and t h a t one o f t h e s a t i s f i e s a ^ > y > 0 i n N , f o r some c o n s t a n t y . Then e v e r y s o l u t i o n o f L v = 0 i n N must v a n i s h somewhere i n N . * N o t e \ t h a t i f L i s u n i f o r m l y e l l i p t i c i n N , th e n i n f a c t a ^ (x) >_ m > 0 f o r e v e r y i = l , . . . , n and f o r e v e r y x i n N , as may be seen by c h o o s i n g a s u i t a b l e § i n the d e f i n i t i o n o f u n i f o r m e l l i p t i c i t y . T h i s shows t h a t Theorem 1.2 i s i n f a c t an e x t e n s i o n o f P r o p o s i t i o n 1.1. To p r o v e Theorem 1.2, t h e f o l l o w i n g p r o p o s i t i o n s w i l l _ f i r s t be shown: o P r o p o s i t i o n 1 .3 Assume w i s a f u n c t i o n i n C (N) and L(w) .'•> 0 i n N . Then t h e p o i n t s o f W where w = 0 cannot be minima o f w . P r o o f : I f w(x ) = 0 i s a minimum o f w i n N . t h e n o , : ? n a i j ( x o ) \ > 0 X s J — J . • • S i n c e the m a t r i x ( a . .(x )) i s p o s i t i v e d e f i n i t e , we may assume t h a t a t x Q i t I s d i a g o n a l w i t h p o s i t i v e e l ements. Then JU..^) D ± 1U(X 0) < 0 w h i c h i s i m p o s s i b l e , s i n c e • D j _ i u ( x 0 ) 1. 0 a ^ t ^ e minimum X Q f o r e v e r y i = l , . . . , n . The n e x t Lemma, t h e key p a r t i n the p r o o f o f Theorem 1.2, w i l l be p r o v e d by the use o f t h e c l a s s i c a l Schauder " c o n t i n u i t y " method [ J ] , Lemma 1.4 Assume t h a t u,v,w a r e f u n c t i o n s i n ( L j N ) and t h a t : x (a) L u >_ 0., u somewhere n e g a t i v e i n N , u = 0 on BN (b) w > 0 i n N , Lw > 0 i n N ( c ) L v >_ 0 i n N , v "somewhere p o s i t i v e i n N . Then v must v a n i s h somewhere i n N . P r o o f : Assume n o t . Then, s i n c e w > 0 i n N and u i s somewhere n e g a t i v e , t h e r e e x i s t s X q i n N and a > 0 such t h a t aw(x ) + u ( x ) = 0 , and L(aw + u) > 0 i n N . \Q' ^O • We now d e f i n e a f a m i l y o f f u n c t i o n s as f o l l o w s : w f c(x) = aw(x) + t u ( x ) + ( l - t ) v ( x ) f o r x e N and t e E"3 L e t T denote t h e s e t : T = {t : w, v a n i s h e s somewhere i n N} fl [0,1] . We s h a l l now show t h a t T i s a non-empty s e t w h i c h i s b o t h open and c l o s e d i n t h e i n d u c e d t o p o l o g y on [0,1] . By t h e c o n n e c t e d -ness o f [0,1] t h i s w i l l mean t h a t i n f a c t T = [0,1] and hence 0 € T , w h i c h i s c l e a r l y i m p o s s i b l e as aw + v > 0 i n N . F i r s t , T i s non-empty s i n c e 1 e T . To p r o v e T i s . c l o s e d , l e t {t.}!° -. be a sequence i n T w i t h l i m t . = t . Then L 1 1=1 l o t h e r e e x i s t x: e N such t h a t w, (x.') = 0 . As N i s compact, 1 z± 1 we may assume, w i t h o u t l o s s , o f g e n e r a l i t y , .that t h e r e e x i s t s a p o i n t x^ e N such t h a t l i m x. = x . Now, we a l s o have t h e ^ o l o ' e s t i m a t e : |wt (x) - w ( x ) | < | t . - t I l | u ( x ) | + | v ( x ) | } ) i o and t h e r e f o r e {w^ }*_-]_ converges u n i f o r m l y i n N t o w, i ~ o I n the i n e q u a l i t y : l w t ( X J ' - ' w t ( x o ^ " w t ( x i ^ + ' w t ^ x i ^ " ~ w t ^ x i ^ ' + ' w t ( x i ^ o o o o i i the f i r s t t e r m on t h e r i g h t hand s i d e tends t o z e r o w i t h i by u n i f o r m c o n t i n u i t y , and t h e second tends t o z e r o b y u n i f o r m convergence. Hence w, ( x ) = 0 . F i n a l l y , t o p r o v e T i s o ° open, l e t t € T and l e t w, have a z e r o a t x i n N . * ' o t o o C l e a r l y w, must have a z e r o i n N , as i t i s p o s i t i v e on dN . 0 S i n c e L(w, ) > 0 and w^ > 0 on oN. , by P r e p o s i t i o n 1.3 o o w, must have b o t h p o s i t i v e and n e g a t i v e v a l u e s i n N . Now, 7.. o o (^| C|u(x)| + | v ( x ) | } ) . Hence w^ a l s o w i l l have b o t h p o s i t i v e and n e g a t i v e v a l u e s i n N i f | t - ,t | i s t a k e n s m a l l enough. By t h e connectedness of- N j w t w i l l have a z e r o i n N . We have thus a r r i v e d a t th e c o n t r a d i c t i o n aw + v = 0 somewhere i n N . Our o r i g i n a l a s s u m p t i o n must be f a l s e and t h e r e f o r e v = 0 somewhere i n N . C o r o l l a r y 1.5 Assume' that t h e r e e x i s t s a function f i n §)(L,N) suchNshat E f > 0 ( o r < 0) i n N . I f u and v s a t i s f y t h e c o n d i t i o n s o f Lemma 1.4 , t h e n v must v a n i s h some-where i n N . P r o o f : A g a i n assume n o t . Then t h e r e e x i s t s a c o n s t a n t a such t h a t v > a > 0 i n N . Choose a c o n s t a n t £ such t h a t 3 > 0 and P ( ^ § < a.. D e f i n e a new f u n c t i o n w by w(x) = v ( x ) + p f ( x ) . C l e a r l y Lw > 0 and w > 0, i n N . By Lemma 1.4, v must be z e r o somewhere " i n N , and t h i s c o n t r a -d i c t i o n p r o v e s t h e c o r o l l a r y . I f L f < 0 i n N , l o o k a t - f . We a r e now i n a p o s i t i o n t o p r o v e Theorem 1.2: P r o o f o f Theorem 1.2: I t w i l l now be s u f f i c i e n t t o c o n s t r u c t a f u n c t i o n f such t h a t L f > 0 i n N, f €^)(L,N) . - F o r s i m p l i c i t y and w i t h o u t l o s s o f g e n e r a l i t y we may assume t h a t N l i e s i n t h e p a r t o f IT0, where x i > 0 » L e t f (x) = - x ^ , m a p o s i t i v e i n t e g e r t o be chosen l a t e r . We have, f o r x e N , 8. L f ( x ) = (m a,^ + m (bounded t e r m s ) ) and c l e a r l y f b e l o n g s t o (L,N) f o r eve r y v a l u e o f m . S i n c e a ^ i s assumed bounded away from z e r o i n N , L f > 0 f o r s u f f i c i e n t l y l a r g e m . Remarks The above r e s u l t s a l s o h o l d i f t h e m a t r i x ( a . . ( x ) ) p x j v i s m e r e l y assumed n o n - n e g a t i v e i n N , as l o n g as one o f t h e a.. i s p o s i t i v e i n N . A l s o i t w i l l be s u f f i c i e n t t o assume L e l l i p t i c i n any domain R such t h a t N c R , as t h e n > 0 i n N f o r e v e r y i = l , . . . , n as a t r i v i a l consequence o f t h e d e f i n i t i o n o f e l l i p t i c i t y . Some o f t h e s e r e s u l t s c o u l d p o s s i b l y be o b t a i n e d by u s i n g a more g e n e r a l f orm o f the maximum p r i n c i p l e . C o r o l l a r y 1.6 Assume t h a t u,v a r e f u n c t i o n s i n " ^ ( L , N ) and t h a t i n N • we have L u >_ f , L v >_ g w i t h f and g bounded f u n c t i o n s . F u r t h e r m o r e i f : ( i ) u = 0 on dN , u somewhere n e g a t i v e d . i n N ( i i ) v somewhere p o s i t i v e i n N >_ 0 i n N ( i v ) The c o e f f i c i e n t s o f L s a t i s f y t h e c o n d i t i o n s o f Theorem 1.2. Then v must v a n i s h somewhere i n N . P r o o f : I f v were always p o s i t i v e i n N , we c o u l d d e f i n e a new e l l i p t i c o p e r a t o r L-, , by ( i i i ) A ( u 3 v ) = f u V 9. L-jW- = (L Then L-jU = (L v )w w e "§)'(L,N) g ) u > f - g H = I f u > 0 and L^v >_ 0 i n N But as a consequence o f Lemma 1.4 and Theorem 1.2, v must v a n i s h somewhere i n N . C o n t r a d i c t i o n . C o r o l l a r y 1.7 Assume t h a t / u 3 v a r e f u n c t i o n s i n ' (L,N) and Lu = 0 i n N , u = 0 on BN , u n o n - t r i v i a l . L e t LgV = Lv + cv x = 0 i n N and c £ 0 bounded i n N . Then v must v a n i s h somewhere i n N i f t h e c o e f f i c i e n t s o f L s a t i s f y t h e c o n d i t i o n s o f Theorem 1.2. P r o o f : We may assume t h a t u i s always n e g a t i v e i n N , f o r i f i t were n o t , we c o u l d c o n s i d e r a subdomain o f N where i t was. I f v i s n e v e r z e r o we can assume v > 0 i n N . Then Lu = 0 , L v = -cv i n ' N and A(u,v) = 0 - c v u v = cvu > 0 i n N Hence v must v a n i s h somewhere i n N b y C o r o l l a r y 1.6, C o n t r a d i c t i o n . I n v i e w o f the above r e s u l t s , i t seems n a t u r a l t o concen-t r a t e on d e t e r m i n i n g some c o n d i t i o n s f o r t h e Osc 1 b e h a v i o u r o f L. 10. 3. A Comparison Theorem f o r L. I t i s w e l l known t h a t o s c i l l a t i o n r e s u l t s may be o b t a i n e d f o r an o p e r a t o r L i f we can r e l a t e L t o a n o t h e r o p e r a t o r L-^  . w i t h s u i t a b l e o s c i l l a t o r y b e h a v i o u r . We s h a l l do t h i s under t h e c o m p u t a t i o n a l l y s i m p l i f y i n g a s s u m p t i o n t h a t t h e c o e f f i c i e n t s o f L and t h e b o u n d a r i e s o f t h e domains i n v o l v e d a r e s u f f i c i e n t l y smooth. S p e c i f i c a l l y we s h a l l assume t h a t e v e r y bounded domain i n q u e s t i o n has a C M boundary [4, p. 128]' and t h a t t h e c o e f f i c i e n t s o f L a r e o f c l a s s C* i n a convex s e t c o n t a i n i n g t h e domain. \ l f G denotes any such domain, t h e n we know [4, p. 131] t h a t g e n e r a l i z e d s o l u t i o n s o f Lu = f i n G , u = 0 on 3G , a r e c l a s s i c a l ( i n f a c t CC°(G)) f o r f o f c l a s s 03 — — 1 C (G) and t h a t t h e g e n e r a l i z e d o p e r a t o r (L + \ ) ~ i s c o m p l e t e l y 2 2 1 c o n t i n u o u s i n t h e L norm as a map from L (G) t o H Q ( G ) , f o r X s u f f i c i e n t l y l a r g e [2, p. 199]. F u r t h e r m o r e , we s h a l l assume L t o be u n i f o r m l y e l l i p t i c . D e f i n i t i o n An o p e r a t o r A w i l l be c a l l e d p o s i t i v e i f f f e"2)(A) and f >_ 0 (a.e.) i m p l i e s t h a t A f >_ 0 ( a . e ) . P r o p o s i t i o n 1.8 F o r \ s u f f i c i e n t l y l a r g e , t h e g e n e r a l i z e d " o p e r a t o r (L + \ ) ~ ^ ~ i s p o s i t i v e . P r o o f : F i r s t assume t h a t f e C°°(G) , f >_ 0 . Then u = (L + \ ) _ 1 f i f f (L + \ ) u = f , u = 0 on dG i n t h e c l a s s i c a l sense. I t f o l l o w s t h a t u >_ 0 f o r \ s u f f i c i e n t l y l a r g e [ l ] . I f f >_ 0 (a.e) and f i s an a r b i t r a r y member o f 11. L (G) , we choose a sequence ^ f n ^ n _ i o f c (^) f u n c t i o n s such t h a t l i m f = f i n t h e L (G) norm and f n >_ 0 . Hence by c o n t i n u i t y , l i m (L + X ) _ 1 f n = (L + X ) _ 1 f , i n t h e L 2 ( G ) norm and t h e r e f o r e (L + X)~"*"f >_ 0 ( a . e ) . From now on we s h a l l always assume t h a t when the o p e r a t o r ( L + X) i s c o n s i d e r e d , X has been chosen s u f f i c i e n t l y l a r g e so t h a t t h e above r e s u l t s h o l d . By t h e smoothness•'" a s s u m p t i o n on t h e c o e f f i c i e n t s o f L we a r e a s s u r e d o f t h e e x i s t e n c e o f a C°° f u n d a m e n t a l s o l u t i o n f o r (L + X) , [2, p. 214]. S i n c e , f u r t h e r m o r e , t h e pr o b l e m (L + x ) u = f , u = 0 on SG has c l a s s i c a l s o l u t i o n s f o r e v e r y f o f c l a s s C°°(G) , a Green's f u n c t i o n K ( x , y ) may be c o n s t r u c t e d f o r (L + x) . C l e a r l y _ -i K ( x , y ) must be n o n - n e g a t i v e , o t h e r w i s e (L + X ) ~ c o u l d n o t be a p o s i t i v e o p e r a t o r . T h i s i s a f a c t w h i c h a l s o f o l l o w s f rom t h e maximum p r i n c i p l e . C o n s i d e r now t h e o p e r a t o r Jty : L-^(G) -*'C(£) d e f i n e d b y • y f j ( f ) = J K ( x , y ) f ( y ) d y G where p > n . Then [5> p. 2593 Jo i s a c o m p l e t e l y c o n t i n u o u s p o s i t i v e o p e r a t o r w h i c h has a p o s i t i v e e i g e n v a l u e w h i c h i s s i m p l e , b i g g e r t h a n t h e a b s o l u t e v a l u e o f any o t h e r e i g e n v a l u e , and whose c o r r e s p o n d i n g e i g e n v e c t o r i s n o n - n e g a t i v e i n G . F u r t h e r m o r e , has no o t h e r l i n e a r l y i n d e p e n d e n t n o n - n e g a t i v e e i g e n v e c t o r . From t h i s we may c o n c l u d e t h a t t h e o p e r a t o r L has a 12. s i m p l e r e a l e i g e n v a l u e X q , w i t h a n o n - n e g a t i v e eigenvector., s uch t h a t a l l o t h e r r e a l e i g e n v a l u e s a r e b i g g e r t h a n \ Q . Fur t h e r m o r e L has no o t h e r l i n e a r l y i n d e p e n d e n t n o n - n e g a t i v e e i g e n v e c t o r . D e f i n i t i o n The e i g e n v a l u e X w i l l be c a l l e d t h e s m a l l e s t  e i g e n v a l u e o f L . P r o p o s i t i o n 1.9 The e i g e n v e c t o r u a s s o c i a t e d w i t h X i s 1 o o p o s i t i v e i n G . \ P r o o f : -We have (L - X )u = 0 i n G . I f u were i d e n t -v o o . o i c a l l y z e r o on a non-empty s u b s e t S o f G , th e n n o t a l l o f i t s p a r t i a l d e r i v a t i v e s c o u l d be z e r o a t a l l p o i n t s o f S , as f o l l o w s by a t r i v i a l m o d i f i c a t i o n o f a r e s u l t o f K r e i t h [ 6 ] . Our n e x t c o n c e r n w i l l be t o de t e r m i n e bounds f o r X and i t s -o b e h a v i o u r as t h e domain G i s p e r t u r b e d . P r o p o s i t i o n 1 . 1 0 . [ l ] L e t w be any smooth f u n c t i o n , w > 0 i n G and w = 0 on 3G . Then \ < sup • r Lw(x) -j \5 - xeG 1 ITTxT J P r o o f : I f n o t , we c o u l d f i n d a f u n c t i o n w s u c h t h a t Lw < X Qw i n G w = 0 on SG. o r -13. (L <-• X )w < 0 i n G w = 0 on 3G S i n c e the extended o p e r a t o r (L + X ) " 1 i s p o s i t i v e and c o m p l e t e l y c o n t i n u o u s , we can c o n c l u d e by a Theorem o f K r e i n and Rutman [7, p. 65] t h a t t h e r e e x i s t s a f u n c t i o n v > 0 i n G such t h a t ( L * - X Q ) v = 0 i n G , v = 0 on CG where L* i s the f o r m a l a d j o i n t o f L . But t h i s i m p l i e s 0 > ( ( L - X Q)w,v) = (w,(L* - X Q ) v ) = 0 w h i c h i s i m p o s s i b l e . P r o p o s i t i o n 1.11 XQ >_ n , where u o denotes t h e s m a l l e s t L L* e i g e n v a l u e o f t h e f o r m a l l y s e l f - a d j o i n t o p e r a t o r — ~ — w i t h z e r o boundary c o n d i t i o n s . P r o o f : Under t h e r e g u l a r i t y a s sumptions a t t h e b e g i n n i n g o f t h i s s e c t i o n , we a r e a s s u r e d o f t h e v a l i d i t y o f C o u r a n t ' s P r i n c i p l e [8, V o l . 1, p. 398]. T h e r e f o r e _ i n f r B(u,u) -> ~ ueD 1 (u,u) J where n n D.(b.) 2 L a. . J J . U J J . U + - " GJ B(u,u) = S  D.uD.u (c  E — — — )u , i , j = l J 3 i = l 2 D = {u : u e C(G) , p i e c e w i s e C (G) and u = 0 on SG} L e t v be t h e e i g e n v e c t o r c o r r e s p o n d i n g t o X Q . We may assume II v || = ( v , v ) = l , and by the known r e g u l a r i t y p r o p e r t i e s o f 14. s o l u t i o n s of e l l i p t i c equations w i t h smooth c o e f f i c i e n t s , [ 4,p . l 3 l ] B(v,v) = ( ^ p ^ v,v) = (Lv,v) = X Hence X >_ \i o — ^o P r o p o s i t i o n 1.11 may be sharpened as f o l l o w s : P r o p o s i t i o n 1.12 I f S D i ( b i ) i s never zero i n G , x o > ^o ' then Proof: Let ^  L + L Lv = X v o i n G u = v = 0 on dG , u > 0, v > 0 i n G . L 4- L* As s t a t e d above, — ~ and L have only one p o s i t i v e l i n e a r l y independent eigenvector each. Now u + v > 0 i n G , hence i f u + v ^ cu f o r some constant c , u .+ v - cannot be an eigenvector of g . Therefore i f u + v ^ cu we have i Q J(u+v) 2 < ( (u+v) L(u+v) = VJL r 2 u + vLu + X^ uv + X„ oj oj v G G G G G and U, G (u-v) <: (u-v) L(u-v) = (i G G oj r u 2 - vLu- - X uv + X o, v G G adding we o b t a i n 15. G G G or u < \ o o Hence i t w i l l be s u f f i c i e n t to show t h a t i f ED^(b;^) i s never zero i n G then v ^ cu i n G . Now, i f SD i ( b i ) ^ 0 i n G and v = cu then and L + L* Lu = X Q U , — ^ u = u Q u (u,Lu) = (u, L g L u) or X Q = ^ o Therefore or n , n £ b.D u + i E D,(b .)u = (\ - u )u = 0 0=1 1 J ^ 0=1 J 3 n n 2 S b .D.u = - £ D.(b-)u j = l J J o=l J J Since u takes on i t s maximum a t a p o i n t i n s i d e G , t h i s i s impo s s i b l e . P r o p o s i t i o n 1.11 and 1.12 enable us to conclude: C o r o l l a r y 1 . l p L e t £Gt^t-l a f a m i l y o f domains whose diameters tend to zero w i t h t . Then the smallest eigen-value X ^ ( t ) corresponding to G. tends t o +<=> w i t h t . O u 16, Proof: We know [9, p.Y] t h a t t h i s i s the case f o r the eigen-values U 0 ( t ) °'£ ^ e operator — £ . C o r o l l a r y 1.13 then f o l l o w s from the f a c t t h a t ^("k) >. M 0 ( t ) f o r every t . Our next o b j e c t i v e i s to e s t a b l i s h a s u i t a b l e upper bound f o r X . n b ( x ) b j ( x ) D e f i n i t i o n • h(x) = - S i = l 2 d e t ( a i j . ( x ) ) b i x^here b^ i s \ the c o f a c t o r of + ~2~ ^ n "^^e m a t r i x b l 2 • e « b n 2 • b. b .  1,. •.. 3 n h 2" 2 D e f i n i t i o n n F[u] = r a. .D.uD u + i , j = l 1 J 1 J n u S b.D.u + (h + c)u' i = l 1 1 M[u] = I F[u] G f o r every f u n c t i o n u e D . The f o l l o w i n g Lemma i s known [10, p. 192]. 17. Lemma 1.14 I f there e x i s t s a s u f f i c i e n t l y smooth f u n c t i o n u , u ^ 0 , u = 0 on 3G , such that M[u] < 0 , then every s o l u t i o n v of Lv >_ 0 , v somewhere p o s i t i v e , must v a n i s h at a p o i n t i n G . From t h i s we can prove: Theorem 1.15 Assume that i n G the sma l l e s t eigenvalue of — i + h i s n o n - p o s i t i v e . Then i f D.. i s any smooth domain such t h a t G c D-^  , the sma l l e s t eigenvalue f o r L i n D-^  must be negative. \ Proof: Choose a smooth domain such t h a t G c D 2 c D 2 c D][ . Then i n D the smallest eigenvalue of the f o r m a l l y s e l f - a d j o i n t operator — ^ 1- h must be negative by the monotonicity property of eigenvalues (see Lemma 1.18). Therefore there e x i s t s a f u n c t i o n w i n ( L j D p ) such that w > 0 • i n D p , T T * w = 0 on oD c and ( % + h)w < 0 I n D« or ? 2 M[w] . = (wLw + hw ) < 0 D 2 Now l e t X be. the sm a l l e s t eigenvalue of L i n and v i t s a s s o c i a t e d p o s i t i v e e i g e n f u n c t i o n . I f X Q >_ 0 , then Lv >_ 0 i n D 2 and v > 0 i n , c o n t r a d i c t i n g Lemma 1.14. 18. C o r o l l a r y 1.16 I f u denotes the smallest eigenvalue of T T * 2 + h i n a domain G , then L has an eigenvalue < \yQ i n any smooth domain D-^  such t h a t G c D-^  . A determination of the c o n t i n u i t y of the s m a l l e s t eigen-value \ Q of L , as the domain v a r i e s inv.a reasonable f a s h i o n , w i l l enable us to say more than what i s s t a t e d i n C o r o l l a r y 1.16. We f i r s t have: P r o p o s i t i o n 1.17 Let w be any f u n c t i o n "^(L,G) such that w > 0 i n G and w = 0 on dG . Then , _ sup rLw(x)-[ o " xeG L w(x) J i f f w = cu f o r some constant c , where u i s the eigenvector corresponding to X . Proof: By P r o p o s i t i o n 1.10 we know tha t Xo - xeG L w ( x ) J C l e a r l y i f w = cu then the e q u a l i t y w i l l hold. Conversely i f the e q u a l i t y holds, then (L - X )w <_ 0 • Again l e t t i n g v be the p o s i t i v e eigenvector corresponding to L and X Q , we have: ((L - X Q)w,v) = 0 i . e . (L - X q ) W = 0 (a.e.) Therefore, (L - X Q)w =0 i n G . Hence w i s an e i g e n f u n c t i o n of L , but s i n c e X i s simple, there e x i s t s a constant c- such 19. t h a t w = cu . Lemma 1.18 I f G c . , G,G^ smooth domains, t h e n the s m a l l e s t e i g e n v a l u e o f L f o r G s t r i c t l y exceeds t h a t f o r G^ . P r o o f : L e t Lu = l u i n G , u = 0 on oG , u > 0 i n G L v = y,v i n G^ , v = 0 on oG^ , v > 0 i n G^ . Then i n G we have, "by a t r i v i a l c a l c u l a t i o n , 2 2 n T /U \ uLu u L v T r „ • p.. /U\ /Uv L ( _ ) = — - — g - - v S a. •D i ( - J D.(-) 2v . .v 2v^ i , j = l J 3 o r L ( ^ ) < (£b ( 2 X - li) 2v' — v 2v By P r o p o s i t i o n 1.17> 2 T ( U ) . . SUp r V 2 V ; - i . 0 . X < xeG [ — T j H < 2 X ~ ^ 2v T h e r e f o r e X > | i We w i l l now c o n s i d e r t h e problem o f t h e c o n t i n u i t y o f X Q as t h e domain v a r i e s . I t i s c l e a r l y s u f f i c i e n t t o c o n s i d e r t h e c o n t i n u i t y o f t h e e i g e n v a l u e s o f t h e o p e r a t o r L + X , where X i s a s u f f i c i e n t l y l a r g e c o n s t a n t so t h a t c ( x ) + X >_ 0 f o r ev e r y x . I t w i l l a l s o be u s e f u l t o i n t r o d u c e t h e ' f o l l o w i n g spaces and norms [8, V o l . I I , p. 332]. D e f i n i t i o n cm^y(^) w i l l denote t h e c l a s s o f a l l f u n c t i o n s u 2 0 . which have p a r t i a l d e r i v a t i v e s up to order m which are continuous i n G and a l l the m**1 p a r t i a l d e r i v a t i v e s s a t i s f y a Hfllder c o n d i t i o n w i t h exponent a . D e f i n i t i o n For every u e ^ m ( . a ^ G ^ w e d e f i n e where l i s ' an n-tuple (<£^,... ,1 ) of non-negative i n t e g e r s , \l\ = s t and DV s i g n i f i e s D ^ I D J 2 ... D*11 u . i *~1 Denoting by H [D mu] the smallest constant K w i t h the *Jtt t h p r o p e r t y t h a t a l l the m order d e r i v a t i v e s of u s a t i s f y a Holder c o n d i t i o n i n G w i t h exponent a ( 0 '< a < l ) and c o e f f i c i e n t K , we define: D e f i n i t i o n || u I j ^ = || u ||m + H a[D mu] . . Under the above r e g u l a r i t y c o n d i t i o n s we may assume tha t the Schauder "estimate to the boundary" [8, p. 3353 (1.2) I U H 2 4 a < K 1 ( | | u | | o + !| f ||a) w i l l h old f o r every C 2+a s o l u t i o n of Lu = f i n G w i t h zero boundary values. Since the "c" c o e f f i c i e n t of L may be taken p o s i t i v e , we can invoke the maximum p r i n c i p l e , .to reduce (1.2) to (1-3) l l u | | 2 4 a < 1^ (11 f ||a) 21. The constant K 1 depends only on G, a , the e l l i p t i c i t y constant of L , and the bounds on the c o e f f i c i e n t s of L , but not on the c o e f f i c i e n t s themselves. We a l s o f i n d convenient to introduce the concept of a "strong b a r r i e r f u n c t i o n " [8, V o l . I I , p. 3^0]. D e f i n i t i o n A strong b a r r i e r f u n c t i o n w^ corresponding to a p o i n t Q. on 3G i s a f u n c t i o n which i s of c l a s s C (G) fl C(G), non-negative i n G , zero only at Q and s a t i s f i e s L[WQ] >_ 1 i n G . \ Theorem 1.19 Let G be any bounded domain such that G = L)G , G c G , each G convex w i t h smooth boundary ^. m 3 m m+1 ' m and G-^  non-empty. Assume th a t at each p o i n t Q, on 3G there e x i s t s a strong b a r r i e r f u n c t i o n w^ . L e t 11-^^3 be the s m a l l e s t eigenvalue and a s s o c i a t e d normalized corresponding p o s i t i v e eigenvector f o r G m . Furthermore assume that a constant KT*" i n Schauder's estimate (1.3) can be chosen inde-pendent of G m . Then the operator L has an eigenvalue \x i n G which i s the l i m i t of the n m . The l i m i t of a subsequence of the , u n i f o r m l y i n the compacta of G , w i l l be an eigenvector corresponding t o \x . Proof: The convexity of Gffi i m p l i e s that a l l u m are of c l a s s C2+a^Gm^ f o r e v e r y a • Since G-^  i s open and non-void I t must co n t a i n a sphere. A l s o G i s bounded so i t may be placed 2 2 . i n s i d e a sphere. In any case, by Lemma 1.18, ^W^'i ^ s a monotone sequence bounded below. Hence we can f i n d a number (i such t h a t l i m u m = \s . Now without l o s s of g e n e r a l i t y ; assume || u j| (G ) = 1 a where II ° II (G ) i n d i c a t e s t h a t the norm i s to be taken over " " a x m G m . Then, which o b v i o u s l y reduces to \ w i t h K independent of m . By e q u i c o n t i n u i t y we can f i n d a subsequence of > a l s o c a l l e d C^}^ , and a f u n c t i o n u such that l i m u m = u u n i f o r m l y on compacta G , together w i t h t h e i r r e s p e c t i v e f i r s t and second p a r t i a l d e r i v a t i v e s . Hence i f x e G , then l i m ( L u m ( x ) ) = Lu(x) and l i m ( P - m u m ( x ) ) = u u ( x ) \ Therefore Lu = pu i n G . Note t h a t u >_ 0 . i n - G as each u^ i s p o s i t i v e i n G m . Once again by [6], we can s t a t e t h a t e i t h e r u > 0 i n G or u s 0 i n G . We c l a i m that i n f a c t u = 0 i s impossible. . To see t h i s , note t h a t (1.4) i m p l i e s t h a t the f i r s t and second p a r t i a l s of each u^ are bounded above by K i n absolute value. Hence f o r x,y e G m we have - u m ( y ) | = | s (8) (x. - y . ) | < Kn |x - y| where 6 denotes a p o i n t on the l i n e between x and y . Hence 23. .(1.5) | u m ( x ) l < Kn |x - y| + I u m ( y ) l I d e n t i c a l l y , (1.6) l D i u m ^ x ^ 1 Kn |x - y| + l ^ u ^ y ) ! f o r every i . A l s o || u m l|: (G r ) = 1 f o r every m . Hence we can III vJ!< Jil f i n d a constant c-^  independent of m such t h a t or. Now l e t G N denote a subset of G such that given any p o i n t x of G we can f i n d a p o i n t of G^ i n s i d e the sphere w i t h centre x and radius J l c 1 K n . Then f o r m s u f f i c i e n t l y l a r g e , we have from (1 . 5 ) ' t h a t | u m ( x ) | < K „ ( 1 ) + ™ L l u m ( y ) l m 1 N and, from (1 .6 ) Therefore I K H l ( G m > ! K » 1 < V And as a consequence 2k, Since l i m ( II u - u |L(G. T)) =0 we must have N " m " 1 x N 2^-< l l u J ^ C G ^ ) . F i n a l l y , set u = 0 on 3G . We c l a i m that w i t h t h i s d e f i n i t i o n u i s continuous i n G . For, l e t Q be an a r b i t r a r y p o i n t of dG and l e t e > 0 be chosen a r b i t r a r i l y . I f w^ i s the b a r r i e r f u n c t i o n a s s o c i a t e d w i t h Q , define a new f u n c t i o n w by w = e + PWQ w i t h 3 a p o s i t i v e constant to be l a t e r deters mined. Then L[w] = ce + PL[WQ] >^  p Now on 3G , w — u > 0 and i n G m ' m — m L(w + u m ) > 3 + u m u m Since || u || (G ) = 1 and u i s a bounded sequence, 3 can H i vJu. i l l Xxi be chosen so tha t L(w + u m ) >_ 0 - i n G m f o r every m . By the Hopf maximum p r i n c i p l e [2, p. 150], we conclude, that w + u m >_ 0 i n G m , or |u(x)| <_ w(x) = e -[- PWQ(X), f o r every x e G-But, f o r x i n G and near Q, , by the c o n t i n u i t y of w^ we must have |u(x)| < 2e . Remark domain Theorem 1.19 shows the existence of eigenvalues f o r any G of the above type. 25. eo Theorem 1.20 Assume now th a t G = D G , G non-empty, G, • • 1 • III JL bounded, G n c G w i t h G , \i , u , K and the c o n d i t i o n s ' m+1 m rrr m^ m upon them as f o r Theorem 1.19. Then again we can f i n d u,u such that Lu = uu i n G , u = 0 on 3G , u >_ 0 i n G and \i = l i m u m , i f we a l s o assume 3G = 3G . Proof: Again without l o s s of g e n e r a l i t y , assume that j| -II a^ Gm^ = 1 ' C l e a r l y once again {u m} i s a monotone bounded sequence. Proceeding e x a c t l y as i n Theorem 1.19* we f i n d that there e x i s t s a u , the l i m i t of a subsequence of {u^} such t h a t ' Lu = \m i n G . Fo l l o w i n g the steps of Theorem 1.19* we can s t a t e t h a t u ^ 0 i n G , as e x a c t l y the same procedure as before may be used to show the existence of a compact subset G* of G such that || u m H a(G*) f o r a l l m s u f f i c i e n t l y l a r g e . There: remains to show th a t u = 0 on 3G . To see t h i s , note that || u l!0_i™(G ) < K and hence the f i r s t d e r i v a t i v e s of are bounded i n G^ independently of- m . Then i f Q e 3G , |u(Q)| < |u(Q) - u m ( Q ) | + |u m(Q)| . Now, by t a k i n g m s u f f i c i e n t l y l a r g e , we can f i n d a p o i n t of dG a r b i t r a r i l y c l o s e t o Q. . Hence by e q u i c o n t i n u i t y and uniform convergence, we can conclude t h a t u(Q) = 0 Remark C l e a r l y \x i s the smallest eigenvalue of L i n G as i t s eigenvector i s p o s i t i v e i n G . 26. P r o p o s i t i o n 1.21 L e t ^ G i ^ i = i denote a f a m i l y o f c o n c e n t r i c spheres w i t h r ^ d e n o t i n g t h e r a d i u s o f G^ . I f t h e r e e x i s t s c o n s t a n t s 6 , p such t h a t 0 < 6 < i J f r . < sf> r. < fi t h e n t h i s f a m i l y s a t i s f i e s t h e c o n d i t i o n on the c o n s t a n t i n Schauder's e s t i m a t e imposed on the f a m i l y {G } o f Theorems 1.19 and 1.20. P r o o f : L e t G^ denote an a r b i t r a r y member o f the f a m i l y and 9^  t h e t r a n s l a t i n g and c o n t r a c t i n g b i j e c t i v e map from F/1 t o w h i c h maps the u n i t sphere U onto G^ . „ L e t y = ( y ^ , . . . , y ) and x = ( x ^ , . . . , x n ) denote a g e n e r i c p o i n t o f the domain and range space o f 0^ r e s p e c t i v e l y . Then 0 i may be e x p l i c i t l y g i v e n as: x. = y . r . + Y . J = l , . - 3 n where Y = (Y^**••>Y N) denotes t h e c e n t r e o f t h e sp h e r e s . F o r s i m p l i c i t y assume t h a t t h e o p e r a t o r L i s w r i t t e n i n non-d i v e r g e n t f orm, n n L u ( x ) = S Alm{x) D t o u ( x ) + S B t ( x ) D^(x) + C ( x ) u ( x ) Now l e t u be a f u n c t i o n i n ^) (L,G^) such t h a t u-= 0 on 3G^ and L u ( x ) = f ( x ) i n G± . D e f i n e new f u n c t i o n s g,h by g( y ) = u ( 6 i ( y ) ) and h ( y ) = f ( S i ( y ) ) . Then c l e a r l y we have g ( y ) = 0 on 3U and 27. l,m 1 to 1 • • 3 y t 3 y m i 1 * 1 3 y £ + C ( e i ( y ) ) g ( y ) = h ( y ) f o r y e U . These changes i n c o - o r d i n a t e s t r a n s l a t e our "many domains, one o p e r a t o r " p r o blem i n t o a "many o p e r a t o r s , one domain" p r o b l e m w h i c h can be d e a l t w i t h . Now by t h e a s sumptions on t h e r a d i i o f the i t i s c l e a r t h a t H o l d e r c o n s t a n t s and exponents can be s p e c i f i e d f o r t h e c o e f f i c i e n t s A £ m ( 6 i ( ) * B^( 9j_(°)) > a n d c(8^(°)) w h i c h a r e independent o f i . F o r example, f o r y1>y2 e Y '• \ . hi - y 2 l a l e ^ ) - e ± ( y 2 ) r a or. < ^ o U) where iu denotes s^ p p (J^Mifl2 A t o ^ x 2 ^ K a n d Q i g a , I 5 2 * V | x 1 - x 2 | a 7 sphere w i t h c e n t r e y and r a d i u s 3 . S i n c e an e l l i p t i c i t y c o n s t a n t f o r each o p e r a t o r may a l s o be t a k e n i n d e p e n d e n t l y o f i , we may c o n c l u d e t h a t H g l ! 2 4 < x (u) < K I ! h | | a ( G ) by the o r d i n a r y " e s t i m a t e t o t h e boundary", w i t h K n o t a f u n c t i o n o f i o r o f g . However, 28. sup Bg 3 n (y) = r sup i xeG. au(x') 3x ( T h e r e f o r e , 6 S U P x e G ± 3u 3x (x) sup - yeU Sg sy. (y) < B S U P - p xeG, cm ax, (x) Hence, from t h e s e and s i m i l a r r e s u l t s we can show the e x i s t e n c e o f two p o s i t i v e c o n s t a n t s k ^ , k 2 such t h a t \ I U I I 2 + a ( U ) > ^ l l u H ^ G , ) and || h || a(U) < k 2|| f ||a (G.) Combining our r e s u l t s , we o b t a i n l U I I ^ G i ) < K j | | f | | a ( 0 l ) w i t h a c o m b i n a t i o n o f a l l t h e p r e v i o u s c o n s t a n t s , and Inde-pendent o f i and u . 4. O s c i l l a t i o n C r i t e r i a f o r L We a r e now i n a p o s i t i o n t o a p p l y some o f t h e above r e s u l t s so as t o o b t a i n o s c i l l a t i o n c r i t e r i a f o r t h e n o n - s e l f - a d j o i n t o p e r a t o r L . As has been done b e f o r e [9], our aim w i l l be a c c o m p l i s h e d by f i n d i n g c o n d i t i o n s on t h e c o e f f i c i e n t s o f L 29. w h i c h w i l l assume t h e " m a j o r i z a t i o n " o f L by an o p e r a t o r w h i c h i s known t o be Osc 1 , o r c o n v e r s e l y , c o n d i t i o n s w h i c h w i l l e nsure t h a t L " m a j o r i z e s " a known n o n - o s c i l l a t o r y o p e r a t o r . We " s h a l l assume f o r s i m p l i c i t y t h a t t h e o p e r a t o r L has c o e f f i c i e n t s d e f i n e d on a s e t R c o n t a i n i n g the h a l f - s p a c e f x : x > 0} i n E . We r e c a l l t h a t 1 n — R r = R n {x : |x| > r} and we f u r t h e r d e f i n e as i n [11, p. 3] t h e cone C by C_ = f x : x > |x| cos a} and S r = R 0 f x : |x| = r ] L e t J\, (x) denote t h e l a r g e s t e i g e n v a l u e o f ( a . . ( x ) ) . A maj o r a n t [11] o f (a. . ( x ) ) i s a p o s i t i v e v a l u e d f u n c t i o n f o f — « i j c l a s s C^(0,o>) s u c h t h a t f ( r ) > m a x A ( x ) V - X € S r V ' F u r t h e r m o r e , any smooth f u n c t i o n g ( r ) such t h a t : g ( r ) > ^ [ c ( x ) + h ( x ) - \ 1 ] • SD (b ) w i l l be c a l l e d a m a j o r a n t o f c + h - — = ^ — w h e r e h i s t h e f u n c t i o n p r e v i o u s l y d e f i n e d . We a l s o i n t r o d u c e t h e s p h e r i c a l c o - o r d i n a t e s [12, p. 58] 30. n-1 X-, = r IT s i n Q. x = r cos 6-, 1 1 1 n -1-n - i x. = r cos 9 . Tf s i n 9- i = 2,...}n-1 I n— l+j. j — ] _ J Theorem 1.22 L I s Osc 1 i f t h e r e e x i s t s a cone C , ; b. a (0 < a < -?) and ( a . .( x ) ) , c + h - ED (-^) have m a j o r a n t s f , g c x j i i r e s p e c t i v e l y such t h a t .03 f ~~n~J^ = > \ r ^ f g C r ) + X a r " 2 f ( r ) ] dr = -x r \ f ( r ) J x a where X a i s t h e s m a l l e s t number f o r w h i c h the p r o b l e m ^ [ s i n n - 2 e i f^] + X a cp s i n n - 2 6 l = 0 0 < B± < a cp(a) = 0 has a n o n - t r i v i a l s o l u t i o n ( s u c h a X_, i s known t o e x i s t [ 1 3 ] ) . P r o o f : C l e a r l y t h e o p e r a t o r n L.u = - £ D . ( f ( r ) D - u ) . + g ( r ) u x 1=1 1 1 m a j o r i z e s — ~ + h . L e t S = [x : |x| < q] be a g i v e n sphere. We w i l l show t h a t L has a n o d a l domain o u t s i d e S , i . e . i n . W r i t i n g L^ i n h y p e r s p h e r i c a l c o - o r d i n a t e s , we n o t e [11] t h a t i t I s Osc 1 , and i n p a r t i c u l a r i t has n o d a l domains i n t h e form o f t r u n c a t e d cones. Now, by c h o o s i n g a-31. n o d a l domain N o f s u f f i c i e n t l y f a r from t h e o r i g i n , i t i s p o s s i b l e t o f i n d a sphere S-^ such t h a t § ( 1 ^ = 0 and N cz §1 I t f o l l o w s i m m e d i a t e l y t h a t L.^  w i l l have a n e g a t i v e s m a l l e s t L + L* e i g e n v a l u e f o r , and hence so w i l l — ^ + h and t h e r e f o r e L . L e t p = {Vi> • ' • >Vn) denote t h e c e n t r e o f S.^  and I i t s r a d i u s . C o n s i d e r t h e f a m i l y £ st^te[l «) o f c o n c e n t r i c spheres g i v e n by: 2 S t = {x : ' |x - p | 2 < ^ } t e [1,») and l e t \i{t) denote the s m a l l e s t e i g e n v a l u e o f L i n S^ . We d e f i n e t = sup [ t : u ( t ) < 0} C l e a r l y t e x i s t s . L e t t = l i m t m o n o t o n i c a l l y from below. ° o o n n ° Then S, = 0 S. and from t h e above r e s u l t s , o n n U ( t 0 ) = l i m u ( t n ) <_ 0 C o n v e r s e l y , l e t . t = l i m t m o n o t o n i c a l l y f rom above. Then^ S, = U S., and t h e r e f o r e t t • • o m m u ( t Q ) = l i m u ( t m ) > 0 . Combining t h e s e r e s u l t s , we see t h a t S, i s a n o d a l domain f o r L ^o S e v e r a l o t h e r theorems may now be p r o v e d w h i c h a r e analogous t o those known f o r s e l f - a d j o i n t o p e r a t o r s [9, 1 1 ] . They d i f f e r o n l y i n t h e c o n d i t i o n s w h i c h a r e imposed t o ensure the o s c i l l a t i o n o f t h e m a j o r i z i n g o p e r a t o r . These reduce t o d i f f e r e n t c o n d i t i o n s p o s t u l a t e d t o ensure the o s c i l l a t i o n o f an o r d i n a r y d i f f e r e n t i a l e q u a t i o n . I n p a r t i c u l a r , we can o b t a i n 32. n o n - s e l f - a d j . o i n t e q u i v a l e n t s o f t h e f o l l o w i n g theorems w h i c h a r e known [11] t o h o l d f o r s e l f - a d j o i n t o p e r a t o r s : Theorem 1.23 L i s Osc 1 i f dr n - l w \ r f ( r ) r 1 1 - 1 hJJ(r) [ g ( r ) + X a r _ 2 f ( r ) ] d r = - • 1 x 1 dt f o r some m > 1 , where h ( r ) = r t n _ 1 f ( t ) and a l l t h e o t h e r f u n c t i o n s a r e as p r e v i o u s l y d e f i n e d . P r o o f : The c o n d i t i o n s o f t h e Theorem ensure the o s c i l l a t i o n o f t h e r a d i a l e q u a t i o n i n t h e comparison o p e r a t o r [ 1 4 ] . The r e s t o f the p r o o f i s i d e n t i c a l t o t h a t o f Theorem 1.22. Theorem 1.24 Assume J\. (x) i s hounded i n R . Then L i s Osc 1 f o r n = 2 i f r 0 0 -2 r [ g ( r ) + \ r f ( r ) ] d r = - «. 1 \. ' and f o r n >_ 3 i f t h e r e e x i s t s 6 > 0 such t h a t : J i r 1 _ 6 [ g ( r ) + X a r " 2 f ( r ) ] d r = - «. P r o o f : I d e n t i c a l t o what vras done i n [11] f o r t h e s e l f - a d j o i n t c a s e . Theorem 1.25 L e t A (x) l A i i n R • T h e n L i s 0 s c 1 i f 55-l i m M f [ _ R 2 G ( R ) ] > A 1 [ x a + I S ^ L ] Proof: Again as was done i n [ l l ] . Remark Theorems 1.24 and 1.25 represent f u r t h e r extensions of c l a s s i c a l r e s u l t s such as Glazman's c r i t e r i o n [15, p. 158] f o r the Schr8dinger operator. To o b t a i n n o n - o s c i l l a t i o n r e s u l t s , we use the f a c t that L + L* L majorizes — g • C l e a r l y then, i f c o n d i t i o n s are so \ T i T * chosen that x — g i s not Osc 1 , n e i t h e r w i l l be L . This f a c t has alr e a d y been i m p l i c i t l y used by Swanson [16], and some r e s u l t s are given i n h i s paper. Remarks A l l the r e s u l t s i n t h i s Chapter have d e a l t w i t h o s c i l l a t i o n s "at » 11. Osc 1 behaviour may a l s o of course a r i s e due to the misbehaviour of the operator at a f i n i t e p o i n t . on the boundary of R . I t should therefore-be p o s s i b l e to extend a l l our r e s u l t s to some such s i t u a t i o n s by a s u i t a b l e transformation of co-ordinates. The r e s u l t s of S e c t i o n 2 were obtained not only f o r but e l l i p t i c equations b^ a l s o f o r a c l a s s of p a r a b o l i c equations. This leads us to b e l i e v e that i t may be p o s s i b l e to extend some of the l a t e r r e s u l t s to such operators and t h e r e f o r e o b t a i n Osc 1 c r i t e r i a f o r them. CHAPTER I I QUASILINEAR ELLIPTIC EQUATIONS I n t r o d u c t i o n Under c o n s i d e r a t i o n i n t h i s C h a p t e r w i l l be o p e r a t o r s o f t y p e (2.1) L u = L 1 u + c ( x , u ) where L^ denotes t h e l i n e a r e l l i p t i c o p e r a t o r d e f i n e d by: n n L,u = - I D.(a. D u ) + S b.D u 1 i , j = l 1 1 J 3 j=l 3 3 The domain.of L^ and t h e t e r m i n o l o g y d e s c r i b i n g the o s c i l l a t o r y b e h a v i o u r o f L w i l l be i d e n t i c a l t o t h o s e i n t r o d u c e d i n Cha p t e r I . Our main r e s u l t s w i l l be: ( l ) Comparison theorems o f Sturm's t y p e ; (2) O s c i l l a t i o n theorems, i . ' e . , s u f f i c i e n t c o n d i t i o n s on t h e c o e f f i c i e n t s o f L f o r L t o be Osc 2 ; and (3) Theorems g i v i n g bounds f o r t h e e i g e n v a l u e s o f L . Whenever a domain G c i s under c o n s i d e r a t i o n , we^ s h a l l assume t h a t c(x,§) i s a c o n t i n u o u s r e a l v a l u e d f u n c t i o n w i t h domain G x I , f o r some r e a l i n t e r v a l I c o n t a i n i n g the o r i g i n , and t h a t t h e p a r t i a l d e r i v a t i v e c 2 ( x , ? ) [ i . e . -|| ] e x i s t s as a c o n t i n u o u s f u n c t i o n i n 5-x I . We s h a l l denote by 5 ) ( G , L ) t h e c o l l e c t i o n o f f u n c t i o n s o f c l a s s C 2(G) 0 ^ ( S ) 35. which map G i n t o I . F i n a l l y , the c o n d i t i o n s on the c o e f f i c i e n t s of L-j_ w i l l be assumed to be those imposed at the beginning of Chapter I . 2. A Sturm Theorem and a Comparison Theorem f o r L . We begin by g i v i n g an extension of a Sturm theorem x^hich i s known to hold f o r l i n e a r operators. Then, under the assumpt-i o n that L-, i s f o r m a l l y s e l f - a d j o i n t ( i . e . b_. s 0 ) , we s h a l l prove a comparison theorem r e l a t i n g L to a l i n e a r operator. \ Theorem 2.1 Let G be a bounded domain of F/1 In which one of the a. . i s p o s i t i v e i n G and b . i s bounded f o r j = l , . . . , n . Furthermore assume that there e x i s t s a n o n - t r i v i a l f u n c t i o n u e ^ (G,L) such that Lu >_ 0 i n G , u <_ 0 i n G , u = 0 on 3G . F i n a l l y , l e t c 2(x,§) be monoton i c a l l y non-i n c r e a s i n g as a f u n c t i o n of § f o r each x . i n - G , and l e t c(x,0) _< 0 . I f v i s any f u n c t i o n i n ^ ( G , L ) such t h a t Lv >_ 0 i n G , v > 0 somewhere i n G , then v must v a n i s h somewhere i n G . Proof: I f not, then by the above assumptions v must always be p o s i t i v e i n G . For each x e G we define the f u n c t i o n f by f ( t ) = L ( t u ) f o r 0 <_ t <_ 1 . This i s p o s s i b l e as under the assumptions on c(x,§) , L ( t u ) i s w e l l defined f o r 0 <_ t <_ 1 . Then *1 f ( l ) - f ( o ) = o d z 36. or: r 1 L, u + [ c J x j t u ) d t ] u = Lu - c(x,0) >_ -c(x,0) o I d e n t i c a l l y : ^v + [ c 2 ( x , t v ) d t ] v _> - c(x,0) S e t t i n g c(u) = ,1 ,1 c„(x,tu)dt , c(v) = c ( x , t v ) d t we have: o o L^u >_ - c(u)u - c(x,0) jjV >_ - c ( v ) v - c(x,0) and u = 0 on SG , u < 0 • i n G , v > 0 i n G. Hence, A(u,v) = f - c(u)u - c(x,0) c ( v ) v - c(x,0) u v 1 = uv[ . ( c 0 ( x , t v ) - c (x , t u ) } d t ] + c ( x , 0 ) ( u - v) o Since t v ( x ) >^  t u ( x ) f o r every t e [0,1] , f o r every x e G , the f i r s t term on the r i g h t hand side i s non-negative by the f a c t that c 2 i s monotonically non-increasing. Since c(x,0) _< 0 , the second term a l s o I s non-negative. Therefore A(u,v) >^  0 i n G . This i s a c o n t r a d i c t i o n of the r e s u l t known from Chapter 1 [ C o r o l l a r y 1.6], f o r the c l a s s of l i n e a r operators of which L-^  i s a member. 37. Consequence Under a l l t h e above c o n d i t i o n s , i f L has a non-p o s i t i v e e i g e n v a l u e w i t h a n o n - p o s i t i v e e i g e n v e c t o r i n G , t h e n e v e r y v , as above, must v a n i s h somewhere i n G . Remark S i m i l a r r e s u l t s may be p r o v e d i f t h e c o n d i t i o n s i n Theorem 2.1 a r e r e v e r s e d i n s i g n . A comparison theorem w i l l now be p r o v e d i n t h e case t h a t t h e o p e r a t o r L i s f o r m a l l y s e l f - a d j o i n t Theorem 2.2 N Assume t h a t i n a bounded domain G t h e r e e x i s t f u n c t i o n s u and v such t h a t : : lu. = l^a + c* ( x , u ) >_ 0 i n G Lv = L^v + c ( x , v ) >_ 0 i n G where L, I a r e o p e r a t o r s o f t y p e (2.1) and L.^  i s f o r m a l l y s e l f - a d j o i n t . F u r t h e r m o r e , l e t t h e f o l l o w i n g c o n d i t i o n s h o l d : (a) u _< 0 i n G , u = 0 on 3 G , v somewhere p o s i t i v e i n G . (b) c*(x,§) m o n o t o n i c a l l y n o n - i n c r e a s i n g as a f u n c t i o n o f _ % f o r f i x e d x i n G . (c) c*(x,5) >_ c 2(x,§) f o r e v e r y (x,5) i n 5 x 1 (d) c ( x , 0 ) < 0 , c * ( x , 0 ) < 0 (e) u(L- Lu - -L-ju) <. 0 G 38. ( f ) The boundary o f G i s such t h a t Green's f o r m u l a may be a p p l i e d . (g) The domain o f c* c o n t a i n s t h e domain o f c . Then v must v a n i s h somewhere i n G . P r o o f : Assume t h a t v i s n e v e r z e r o . Then t h e f u n c t i o n i s w e l l d e f i n e d and smooth i n G . F u r t h e r m o r e , i t v a n i s h e s t o g e t h e r w i t h i t s f i r s t p a r t i a l s on 3G . We have: u 2v 2 uL, u 2 L n ( H _ ) = _ 1 - - u J.V - v S a. .D.(H) D.(H) l v 2 v y v 2 v 2 1 i , i i v v ' j W where (an--,-) denotes t h e m a t r i x a s s o c i a t e d w i t h L-, . Then, u s i n g c o n d i t i o n ( f ) and the assumed symmetry o f , we o b t a i n : uL^u - E a..D.(|) D,(H) G o r , as u and v o b v i o u s l y cannot be l i n e a r l y dependent i n G , r. ,.2 U v L l v < J u L l u G Now, once a g a i n we w r i t e : 1 ^ ( u ) + [ c 0 ( x , t u ) d t ] u >_ - c (x,0) o and L-jJv) + [ J c 2 ( x , t v ) d t ] v > _ - c ( x , 0 ) Set c ( u ) = c* ( x , t u ) d t , c ( v ) = I* c 0 ( x . t v ) d t ' o J o d 39. Then. u v [-c(x,0) - c ( v ) v j < | u ( L 1 u - -t-^ u) + (-c(x,0) -c(u)u)u G G G or (2.2) u V G r* p p r* [-c(x,0)j + c(x,0)u + u (c(u) - c ( v ) ) < u ( L 1 u - ^ 1 u ) G G G Now the f i r s t two terms on the l e f t hand s i d e are- c l e a r l y non-n e g a t i v e , and the t h i r d term may be expressed as: u 2 ( c ( u ) - c ( v ) ) = 1 G u 2 [ J ( c * ( x , t u ) - c 2 ( x , t v ) } d t ] d G G ° u [ {c (x,tu) - c * ( x , t v ) } d t + .1 G o + o { c 2 ( x , t v ) - c 2 ( x , t v ) } d t ] d G By the use of c o n d i t i o n s (b) and ( c ) , we see that i t i s a l s o non-negative. Hence the l e f t s i d e i n (2.2) i s non-negative w h i l e the r i g h t s i d e i s n o n - p o s i t i v e . The c o n t r a d i c t i o n e s t a b l i s h e s Theorem 2.2. Remark The same r e s u l t f o l l o w s i f we assume that: (a) u >_ 0 i n G , u = 0 on 3G , v < 0 somewhere i n G , and lu £ 0 , Lv < 0 i n G . (b) c*(x,?) monotonically non-decreasing as a f u n c t i o n of § . (c) c 2 ( x , ? ) _> c 2(x,§) f o r every (x,?) e G x I ... 4o. (d) c ( x , 0 ) >_ 0 , c*(x.O) >_ 0 . (e) \^(\ u - \ u ) 1 0 G ( f ) The boundary o f G such t h a t Green's f o r m u l a may be a p p l i e d . (g) The domain o f c* c o n t a i n s t h a t o f c . I t i s a consequence o f t h e s e r e s u l t s t h a t i f I a d m i t t e d a n o n - p o s i t i v e e i g e n v a l u e w i t h a n o n - p o s i t i v e e i g e n v e c t o r i n a domain G , t h e n L c o u l d n o t have i n a domain G^ such t h a t G c G^ , a n o n - n e g a t i v e e i g e n v a l u e t o w h i c h t h e r e c o r r e s p o n d e d an e i g e n v e c t o r p o s i t i v e I n G^ . The same c o n c l u s i o n h o l d s i f the c o n d i t i o n s a r e r e v e r s e d i n s i g n . Consequence Of p a r t i c u l a r i n t e r e s t i s t h e case c * ( x , u ) = Y(X)U, t h a t i s : t h e o p e r a t o r t i s l i n e a r . Then c l e a r l y c * ( x , 0 ) =' 0 -X-and Cg s a t i s f i e s a l l t h e above m o n o t o n i c i t y c o n d i t i o n s . F u r t h e r -more, under r e a s o n a b l e smoothness c o n d i t i o n s . o n t h e c o e f f i c i e n t s o f I and on t h e boundary o f G , t h e e i g e n v e c t o r c o r r e s p o n d i n g to t h e s m a l l e s t e i g e n v a l u e has c o n s t a n t s i g n i n G . I t i s c l e a r t h e r e f o r e t h a t i f the o p e r a t o r I admits i n a domain G a non-p o s i t i v e s m a l l e s t e i g e n v a l u e t h e n e v e r y f u n c t i o n v , o f c l a s s <^ (G,L) such t h a t L v i s o f f i x e d s i g n o r - z e r o i n G 'and v and L v do n o t have o p p o s i t e s i g n at" e v e r y p o i n t o f G , must v a n i s h somewhere i n G i f c ( x , 0 ) = 0 , c 2 ( x , ? ) <_ Y(X) f o r e v e r y x e G , f o r e v e r y % e I and' 41. J v(L-jV - l^v) £ 0 . G m 2 i As an example, l e t c 0 ( x , ? ) = y(x) - S g . ( x ) 5 where t h e g. d • i = l 1 a r e c o n t i n u o u s n o n - n e g a t i v e f u n c t i o n s . Theorem 2.3 Assume t h a t t h e o p e r a t o r t d e f i n e d i n an unbounded domain R by: l(u) = - S ;D (A D u) + Y ( x ) u i , j x o J has a smooth n o d a l domain o u t s i d e o f any g i v e n sphere and i n R . I f : c ( x , 0 ) = 0 , C2(X,E;) £ Y(X) f o r e v e r y § € I , f o r e v e r y x e R , and (A. .(x) - a. . ( x ) ) i s n o n - n e g a t i v e d e f i n i t e i n R , t h e n e v e r y f u n c t i o n v i n ^ ( R , L ) xvhich i s such t h a t L v = 0 i n R, must v a n i s h somewhere I n t h e complement o f any g i v e n sphere. P r o o f : S i n c e I has a n o d a l domain o u t s i d e o f e v e r y s p h e r e , i t i s c l e a r t h a t we may c o n s t r u c t a domain ' N o u t s i d e o f e v e r y sphere f o r w h i c h t h e s m a l l e s t e i g e n v a l u e i s n o n - p o s i t i v e . Then s i n c e o b v i o u s l y L v = 0 i n N and t h e r e f o r e v and Lv cannot have o p p o s i t e s i g n a t any chosen p o i n t o f N , r e g a r d l e s s o f y , the theorem f o l l o w s . A s m a l l v a r i a t i o n i n t h e c a l c u l a t i o n o f Theorem 2.2 l e a d s t o n o n - o s c i l l a t i o n c r i t e r i a w h i c h a p p l y t o the more g e n e r a l non-s e l f -ad j o i n t o p e r a t o r (2.1). We s t a t e one such r e s u l t . 42. Theorem 2.4 L e t t h e f o r m a l l y s e l f - a d j o i n t l i n e a r o p e r a t o r I be such t h a t t h e e q u a t i o n lv ~ IjV + y(x)v = 0 i s s a t i s f i e d i n R by a n o n - t r i v i a l f u n c t i o n v a l l o f whose z e r o s a r e i n s i d e a f i x e d sphere S . Then i f : (a) c 2 ( x , ? ) _> Y(X) f o r e v e r y x e R , f o r e v e r y % e I (b.) c(x , 0 ) =0 ( c ) u(-t- Lu - L-^u) <_ 0 f o r e v e r y bounded subdomain G o f R G and e v e r y u e ^ ( G , L ) , t h e o p e r a t o r L cannot have smooth n o d a l domains o u t s i d e the sphere P r o o f : The i d e a o f t h e p r o o f i s t o show t h a t i n t h e c l o s u r e o f a smooth n o d a l domain o f L e v e r y s o l u t i o n o f lv = 0 must v a n i s h a t l e a s t once. To do t h i s , we r e p e a t e x a c t l y t h e proof-o f Theorem 2.2 except f o r t h e use o f t h e f o l l o w i n g e s t i m a t e : ( c 2 ( x , t u ) - c 0 ( x , t v ) } d t = { c 0 ( x , t u ) - c 0 ( x , t u ) } d t + 2' o o + { c 2 ( x , t u ) - c ( x , t v ) } d t o t o r e p l a c e t h e one i n t h e p r o o f o f Theorem 2.3. 43. 3. O s c i l l a t i o n amtd N o n - O s c i l l a t i o n Theorems. We s h a l l b e g i n by g i v i n g two s p e c i f i c examples and then we s h a l l s t a t e o t h e r r e s u l t s o f a more g e n e r a l n a t u r e . Example 1 P u f f i n g ' s E q u a t i o n (Hard S p r i n g Case) [17, p . l 6 ] . The o r d i n a r y d i f f e r e n t i a l e q u a t i o n under c o n s i d e r a t i o n i s : ^ - y - k V = 0 . (k 2 > 0) dx^ I f xtfe choose xthe comparison e q u a t i o n : ^ - y = 0 d x 2 w h i c h has s o l u t i o n s y = + s i n ( x - a ) f o r e v e r y c o n s t a n t a , we see t h a t e v e r y s o l u t i o n o f D u f f i n g 1 s ' E q u a t i o n must v a n i s h i n e v e r y c l o s e d i n t e r v a l o f l e n g t h it . Example 2 M a t h i e u ' s E q u a t i o n [18, p. 401]. 'We c o n s i d e r t h e o r d i n a r y e q u a t i o n : ^ - g - \M - ( l + a cos 2 x ) y - c y 3 = 0 dx^ a x where a,b,c a r e c o n s t a n t s . M u l t i p l i c a t i o n by the i n t e g r a t i n g bx f a c t o r e reduces t h e e q u a t i o n t o : - ^ [ e b x ||] - e b x [ ( l + a cos 2 x ) y + c y 5 J = 0 so t h a t i n t h i s c ase: and ( x , y ) = - e b x [ ( l + a cos 2 x ) y + c y 5 ] c 2 ( x , y ) = - e [1+a cos 2x + J c y ] I f c > 0 , t h e n c p ( x , y ) £ - e b x [ l + a cos 2x] _< - e b x [ l - | a j ] We a r e t h e r e f o r e l e d t o t h e comparison e q u a t i o n : - T | t e t a | | ] - e b x [ l - | a|]y = 0 o r £ | + b f f + [1 - |a|]y = 0 w h i c h w i l l have o s c i l l a t o r y s o l u t i o n s i f b < 4(1 - |a|) . have thus p r o v e d t h e f o l l o w i n g theorem: Theorem 2.5 I f c >_ 0 and b < 4(1 - | a j ) , t h e n e v e r y s o l u t i o n o f M a t h i e u ' s e q u a t i o n must v a n i s h o u t s i d e o f e v e r y bounded i n t e r v a l . C o n v e r s e l y , I f c _< 0 , then c 2 ( x , y ) 2l - e b X [ l + a cos 2 t ] >_ - e b x [ l + |a|] and hence the comparison e q u a t i o n may be chosen t o be: 45. w h i c h w i l l have n e v e r v a n i s h i n g s o l u t i o n s i f b >_ 4(1 + | a | ) . Theorem 2.6 I f c £ 0 and b 2 >_ 4(1 + | a j ) , no s o l u t i o n o f M a t h i e u ' s e q u a t i o n can v a n i s h more t h a n once. To o b t a i n more g e n e r a l r e s u l t s , we make t h e f o l l o w i n g d e f i n i t i o n s s i m i l a r t o t h o s e o f Ch a p t e r 1, D e f i n i t i o n A f u n c t i o n f o f c l a s s C1(0,«>) i s a m a j o r a n t o f ( a ^ ) i f f f ( r ) > " ^ ( A ( x ) ) r where J\ (x) denotes t h e l a r g e s t e i g e n v a l u e o f ( a . . ( x ) ) . D e f i n i t i o n A smooth f u n c t i o n g such t h a t i s a m a j o r a n t o f Cg . Theorem 2.7 I f R c o n t a i n s a cone C^ (a > 0) , c ( x , 0 ) = 0 -j a and ( a j i ) > c 2 admit m a j o r a n t s f , g , r e s p e c t i v e l y , such t h a t 0 8 " 0 5 n 1 2 r n ~ [ g ( r ) + X a r " f ( r ) j d r = -dr + » 46. t h e n e v e r y s o l u t i o n o f - v. D. ( a . .D.u) + c ( x , u ) = 0 i , o - - J 3 i n R must v a n i s h o u t s i d e e v e r y sphere. Here X denotes th e same number w h i c h was d e f i n e d i n C h a p t e r 1 and i n [11]. P r o o f : Under t h e above c o n d i t i o n s , t h e o p e r a t o r n Lu = -i S D ± ( f ( r ) D i u ) + g ( r ) u has a n o d a l domain o u t s i d e e v e r y sphere. Our r e s u l t t h e n f o l l o w s i m m e d i a t e l y from Theorem 2.3. Theorem 2.8 Assume t h a t R c o n t a i n s t h e cone C a ( a > 0) and t h a t ( a j _ j ) > C 2 a d m i t m a j o r a n t s f , g , r e s p e c t i v e l y , s uch t h a t : eo eo dr , < CO , i- n _ 1h™(r)[g(r) + X a r - 2 f ( r ) ] d r f o r some number m > 1 , where h ( r ) = =, . Then t h e n J r t n - l f ( t ) same c o n c l u s i o n , as i n Theorem 2.7 h o l d s . P r o o f : A g a i n t h e c o n d i t i o n s on f and g a s s u r e t h a t the e q u a t i o n n - E D ( f ( r ) D , u ) + : g ( r ) u = 0 i = l . • • i s Osc 1. The r e s t f o l l o w s as i n Theorem 2.7. 47. I t i s c l e a r t h a t t h e o t h e r known o s c i l l a t i o n r e s u l t s f o r l i n e a r o p e r a t o r s may s i m i l a r l y be extended, and we t h e r e f o r e omit t h e i r p r o o f . Of c o u r s e , we can a l s o e x t e n d some o f t h e non-o s c i l l a t i o n r e s u l t s . Once a g a i n we draw a t t e n t i o n t o a paper by Swanson [19] where some such theorems a r e mentioned f o r a c l a s s o f f o r m a l l y " s e l f - a d j o i n t " o p e r a t o r s . I t , i s c l e a r t h a t s i m i l a r theorems e x i s t f o r o p e r a t o r s whose l i n e a r p a r t i s n o t s e l f - a d j o i n t . We s t a t e one as an example: Theorem 2.9 The o p e r a t o r (2.1) I s n o n - o s c i l l a t o r y i n R i f c(x , 0 ) = 0 , i f t h e d i f f e r e n t i a l e q u a t i o n : a d m i t s a s o l u t i o n a l l o f whose z e r o s a r e i n s i d e a f i x e d sphere n - S D.(A(r)D.v) + C ( r ) v = 0 i = l 1 X b. and SD ±(- 2^) < 0 , where A ( r ) < i n f xeS { ( / ^ ( x ) ) } r C ( r ) < i n f X€S i n f ? e l ( c 2 ( x , 5 ) } r and 7^, (x) denotes t h e s m a l l e s t e i g e n v a l u e o f ( a . . ( x ) ) . P r o o f : Immediate from Theorem 2.4. Remark Theorem 2.9 l e a d s t o o b v i o u s c o n d i t i o n s on t h e c o e f f i c -i e n t s o f (2.1) w h i c h w i l l ensure n o n - o s c i l l a t i o n and w h i c h we s h a l l o mit. 48. Remark The problem o f f i n d i n g s u f f i c i e n t c o n d i t i o n s f o r L t o be Osc 1 s t i l l remains open. 4. Bounds on t h e F i r s t E i g e n v a l u e o f L . We s h a l l c o n c l u d e t h i s c h a p t e r b y ' e x t e n d i n g t o the o p e r a t o r (2.1) some known e s t i m a t e s f o r t h e f i r s t r e a l e i g e n v a l u e o f u n i f o r m l y e l l i p t i c l i n e a r o p e r a t o r s . Theorem 2.10 \ L e t X be a r e a l e i g e n v a l u e f o r L w i t h r e a l e i g e n v e c t o r s u,v such t h a t u <_- 0 and v >_ 0 i n G . The two e i g e n v e c t o r s may be l i n e a r l y dependent. F u r t h e r m o r e , assume t h a t a l l f u n c t i o n s i n v o l v e d i n t h e d e f i n i t i o n o f L a r e C° and a l s o t h a t t h e boundary o f G i s 0°° . F i n a l l y , l e t c ( x , 0 ) = 0 and c 2(x,§) be m o n o t o n i c a l l y n o n - i n c r e a s i n g as a f u n c t i o n o f % . Then I f w i s any f u n c t i o n i n 1^ (G,L) su c h t h a t -w i s a l s o i n ^  (G,L) and such t h a t w > O l n G , w = 0 on 3G , we have: (o ^) S U P {lAzEL] > \ > i n f f L ( w ) 1 xeG L -w J 2. k 1 xeG 1 w J P r o o f : A g a i n we have: L ( u ) = L ^ ( u ) + [ J C g ( x , t u ) d t ] u = Xu «1 L(w) = L, (w) + [ c„(x,tw)dt]w -1 . J o d r 1 f 1 .= L, (w) + [ c ( x , t u ) d t j w + [ {c„(x,tw)-c p(x,tu)}dt] w 49. S i n c e tw > t u P 1 I^tw] + [ J c 2 ( x , t u ) d t ] w >_ L[w] Now, from t h e t h e o r y o f l i n e a r o p e r a t o r s [ l ] , we have: , . i n f r L l [ w ] + [^c g(x,tu)dt3w. i n f K - xeG 1 w J - xeG 1 w J To p r o v e the o t h e r i n e q u a l i t y i n ( 2 . $ ) , we .proceed i d e n t i c a l l y . We have: r 1 L-^v) + [ c 0 ( x , t v ) d t ] v = Xv o 2 L-^-w) + [ J c 2 ( x , - t w ) d t ] (-w) = L'(-w) and a g a i n we o b t a i n : r 1 ^ ( - w ) + [ c 2 ( x , t v ) d t ] ( - w ) ~ > _ L(-w) The f a c t t h a t v >_ 0 i m p l i e s t h a t X must be t h e s m a l l e s t eigen-v a l u e o f the l i n e a r o p e r a t o r Z d e f i n e d by: ,1 o £(f)(x) =L1(t)(x) + [j c 2 ( x , t v ( x ) ) d t ] f ( x ) Once a g a i n by [ l ] , X < S U P [ L 1 ( W ) + [ J o c 2 ( x ' t v > d t ] w } — X w C 1 1„ L, (-w) + [ f 1 c„(x, t v ) d t ] (-w) T / x s u p r l x J o 2 V ' ' K ' i . sup rL(-w)-| x 1 J i x L ~ ^ r - J 50. where i n the l a s t i n e q u a l i t y we use the f a c t t h a t -w <_ 0 . Remark. As t h e i n e q u a l i t i e s i n (2 .3 ) were p r o v e d i n d e p e n d e n t l y o f each o t h e r , i f o n l y the e x i s t e n c e o f u ( o r v ) i s known, the c o r r e s p o n d i n g I n e q u a l i t y i n (2 .3 ) w i l l s t i l l h o l d . • We a l s o remark t h a t t h e l o w e r bound f o r X a c t u a l l y o n l y r e q u i r e s w >_ 0 i n G , as i t i s a d i r e c t consequence o f t h e s i m i l a r bound i n [ l ] . Under the a s s u m p t i o n o f f o r m a l s e l f - a d j o i n t n e s s o f the l i n e a r p a r t o f the o p e r a t o r (2.1), o t h e r bounds may be found Theorem 2.11 ( " S e l f - a d j o i n t " c a s e ) . Assume now t h a t b. H 0 f o r a l l j , t h a t i s : L u = L-^u + c ( x , u ) - t D.(a.,D u) + c ( x , u ) . i , j xo J L e t Cg(x,5) be m o n o t o n i c a l l y n o n - i n c r e a s i n g as a f u n c t i o n o f § c ( x , 0 ) 1 0 . I f Lv = Xv i n G- w i t h _X r e a l , v > 0 i n G and v = 0 on 3G , then X <_ w where u denotes the s m a l l e s t e i g e n v a l u e o f L-^  + Cg(x,0) . CO / * P r o o f : L e t u e C. (G) . Then: uL-^u G G I K G G' u c ( x , v ) 2v o r X | u 2 < uL^u + P P1 u f [( c Q ( x , t v ) d t ) v + c ( x , 0 ) ] G G v o 51. T h e r e f o r e . r 2 , u < h u ^ c 2 ( x , 0 ) G G G and hence, X < i n f r J G U ( L 1 U + C 2 ( X ' ° ) U J ueC (G),u^o r 2 Gv Now t h e e i g e n v e c t o r c o r r e s p o n d i n g t o n i s a member o f t h e H i l b e r t space H^(G) , a space where by d e f i n i t i o n t h e CQ(G) f u n c t i o n s a r e dense i n t h e t o p o l o g y i n d u c e d by t h e norm: || u ||f = J ( S |D.u| 2 + |u| 2) G 1 F u r t h e r m o r e , f o r smooth f u n c t i o n s u,v v a n i s h i n g on SG , we have: K L - ^ u ) - (L l V,v)|•< J_S J a . ^ K l D . u l - l D j U - D j v l + + i D j v H ^ u - D j v D d G + C||u-v'||o|| u + v j | o < M(||u I^H u-v ||1 + || v I I J I u - v H ^ + C i l u-v ||o|| u +v|| o 2 * 2 *~ where || u || = J |u| , and M and C a r e c o n s t a n t s . I t i s G e v i d e n t t h e r e f o r e , t h a t the infimum t a k e n o v e r a l l C R O(G) f u n c t i o n s do es i n f a c t g i v e the s m a l l e s t e i g e n v a l u e o f 4- c 2 ( x , 0 ) Remark T h i s r e s u l t sharpens Theorem 2.2 f o r t h e s p e c i a l c h o i c e o f o p e r a t o r s l,L g i v e n by: 52. -Lu = L-^u + c 2 ( x , 0 ) u ; Lu = L^u + c ( x , u ) Theorem 2.12 Assume t h a t Lu = Xu , u _< 0 i n G, u = 0 on 3G Fu r t h e r m o r e assume c 2 ( x , ? ) i s m o n o t o n i c a l l y n o n - i n c r e a s i n g as a f u n c t i o n o f ? and t h a t c(x , 0 ) <_ 0 . Then X >_ p / where p denotes t h e s m a l l e s t e i g e n v a l u e o f L^ + c 2 ( x , Q ) . P r o o f : r 2 i* r f* 2 uL.udx + I u { c Q ( x , t u ) d t ] d x + u c ( x , 0 ) d x = X u dx But 1 G u G G uc(x,0)\>_ 0 and as. t >_ 0 , u _< 0 we have c 2 ( x , t u ) >_ c 2 ( x , 0 ) T h e r e f o r e , u 2 >_ [ u ( L 1 u + c 2 ( x , 0 ) u ) G G Combining t h e r e s u l t s o f Theorems 2.11 and.2.12 we a r e l e d t o the f o l l o w i n g c o n c l u s i o n : I f f o r any o p e r a t o r (2.1) w i t h s e l f - a d j o i n t l i n e a r p a r t , s uch t h a t c 2 ( x , ? ) i s monotone non-i n c r e a s i n g as a f u n c t i o n o f 5 and c(x , 0 ) i s n o n - p o s i t i v e f o r a l l x , t h e r e e x i s t s a r e a l e i g e n v a l u e X t o w h i c h t h e r e c o r r e s p o n d s b o t h a p o s i t i v e and n e g a t i v e e i g e n v e c t o r , t h e n t h i s e i g e n v a l u e must e q u a l the s m a l l e s t e i g e n v a l u e o f t h e o p e r a t o r d e f i n e d by: L.xu = - E D i ( a i j D j u ) + c 2 ( x , 0 ) u Consequence No o p e r a t o r o f t y p e L u = - E D.(a.,D.u) + c ( x , u ) 53. can have a r e a l e i g e n v a l u e i n a domain G t o w h i c h t h e r e c o r r e s -ponds b o t h a p o s i t i v e and a n e g a t i v e e i g e n v e c t o r i f c 2 ( x , | ) i s s t r i c t l y d e c r e a s i n g as a f u n c t i o n o f § and c ( x , 0 ) 0 . P r o o f : I f n o t , l e t X be such an e i g e n v a l u e o f L and u i t s p o s i t i v e e i g e n v e c t o r . By the above c o n s i d e r a t i o n s , we can c o n c l u d e t h a t t h e r e e x i s t s a f u n c t i o n w w h i c h i s p o s i t i v e i n G , v a n i s h e s on 3G and s u c h t h a t : - S D. ( a . .D w) + c p ( x , 0 ) w = Xw i , j o r r 1 E D.(a. .D.w) + [ c o ( x , 0 ) d t ] w = Xw i , j 1 1 J J o 2 and r 1 - E D.(a. -D.u) + [ c 0 ( x , t u ) d t ] u = Xu - c ( x , 0 ) . i , 0 1 1 J 3 o 2 M u l t i p l y i n g the above e q u a t i o n s by u,w r e s p e c t i v e l y , s u b t r a c t i n g and i n t e g r a t i n g , we o b t a i n : [J* ( c 2 ( x , 0 ) - c 2 ( x , t u ) } d t ] u w d G = |c(x,0)wdG G ° G Wow under the a s s u m p t i o n s , t h e l e f t - h a n d s i d e i s p o s i t i v e , w h i l e the r i g h t i s n o n - p o s i t i v e . C o n t r a d i c t i o n . As an example o f such o p e r a t o r s we may t a k e L u = - Z 1 ( a i - D . u ) + u - u 2 . 54. Remark I d e n t i c a l r e s u l t s h o l d i f the c o n d i t i o n s i n the above c o n s i d e r a t i o n s a r e r e v e r s e d i n s i g n . Remark The methods d e v e l o p e d i n t h i s c h a p t e r t o g e t h e r w i t h t h o s e d e v e l o p e d i n Chapter I may^perhaps^also be used t o d e s c r i b e the o s c i l l a t o r y b e h a v i o u r o f o p e r a t o r s d e f i n e d by: Lu = L-^u -f c(x,u,I)^yi,.. . *D n u) where L, i s a l i n e a r second o r d e r o p e r a t o r . CHAPTER I I I FOURTH ORDER EQUATIONS 1. I n t r o d u c t i o n I t has a l r e a d y been o b s e r v e d [10, p. 115], t h a t s o l u t i o n s o f the e q u a t i o n L U = i A t D i O ( a i J W D M u ) " A V ' l j V ) -+ * e D u - du . 0 \ need n o t v a n i s h i n the c l o s u r e o f a bounded domain G- f o r w h i c h t h e r e e x i s t s a f u n c t i o n v such t h a t L v = 0 i n G , v = 0 on 3G . I f we d e f i n e a n o d a l domain f o r t h e o p e r a t o r L t o be a bounded domain N f o r w h i c h we can f i n d a n o n - t r i v i a l f u n c t i o n w such t h a t : Lw = 0 i n N , w = w,. =0 on 3N. . i = 1,. . . ,n i -we can show t h a t t h e same b e h a v i o u r may o c c u r even when G i s a n o d a l domain f o r L . F o r example, t h e f u n c t i o n v = % ( l - cos2x) s a t i s f i e s : but any f u n c t i o n u = c o n s t a n t i s a l s o a s o l u t i o n o f La - 0 . p F o r t h i s r e a s o n , the o s c i l l a t o r y b e h a v i o u r o f f o u r t h o r d e r e l l i p t i c o p e r a t o r s w i l l be c o n s i d e r e d f r om a n o d a l domain 56. v i e w p o i n t o n l y . D e f i n i t i o n L i s o s c i l l a t o r y i f f L has a n o d a l domain o u t s i d e o f e v e r y sphere c e n t e r e d a t t h e o r i g i n . D e f i n i t i o n L i s n o n - o s c i l l a t o r y i f f L i s n o t o s c i l l a t o r y . O s c i l l a t i o n r e s u l t s f o r a s p e c i a l c l a s s o f f o u r t h o r d e r o p e r a t o r s have a l r e a d y been p r o v e d [9, p. ko]. The o p e r a t o r s c o n s i d e r e d t h e r e were o f a t y p e w h i c h g i v e s r i s e t o forms t h a t a r e I d e n t i c a l 'to th o s e o f second o r d e r o p e r a t o r s . We s h a l l c o n s i d e r o p e r a t o r s f o r w h i c h t h i s i s n o t t h e c a s e . S p e c i f i c a l l y , we s h a l l examine t h e o p e r a t o r s L, d e f i n e d by: (3.1) Lu = S D .(a. .a. ,D u) - E D. (b. .D u) + S c.D.u - du i , j , k , - t 1 0 1 J ^ k ^ • i , j 1 1 J J j 3 3 and whose c o e f f i c i e n t s a r e d e f i n e d i n an unbounded domain R o f E n . We s h a l l assume t h a t t h e c o e f f i c i e n t s a r e so smooth t h a t a l l the d i f f e r e n t i a t i o n s i n v o l v e d i n the o p e r a t i o n s d e f i n i n g L and L-^  , may be p e r f o r m e d , and t h a t t h e r e s u l t i n g p a r t i a l d e r i v a t i v e s a r e c o n t i n u o u s i n R . G i v e n any sub domain P o f R we s h a l l denote by ^ ( F , L ) ( o r ^ ( F , L 1 ) ) the c o l l e c t i o n o f f u n c t i o n s o f c l a s s (F) n C 2 ( F ) . F i n a l l y , we s h a l l assume t h a t ( a . . ( x ) ) i s a p o s i t i v e d e f i n i t e 57. symmetric m a t r i x i n R . The matrix (m. .(x)) w i l l be assumed to be e i t h e r p o s i t i v e d e f i n i t e or a symmetric m a t r i x a l l of whose terms are p o s i t i v e i n R . By the above assumptions, i t i s c l e a r that a t each p o i n t . of R the matrices (a. .(x)) , (m. .(x)) w i l l each have a p o s i t i v e eigenvalue. I f we f u r t h e r assume that: E a. j (x) > C S ^ . E rn. § . 5 j > c ± p. 5 ± S f o r every x e R and f o r every § = .. ., 1 ^ 5 when ( a 1 - ( x ) ) , '(m- .(x)) are p o s i t i v e d e f i n i t e , then the operators defined by (3-1) and (3-2) xd.ll be u n i f o r m l y e l l i p t i c . This w i l l a l s o be the case when the m a t r i x (m. .(x)) has p o s i t i v e e n t r i e s , i f we. — J assume that m. .(x) i s bounded away from zero f o r every — J i , j = 1,...,n . Under the above c o n d i t i o n s , and the f u r t h e r assumption that p. = c. = 0 j ~ l , . . . , n , the theory of Courant may be used to p r e d i c t f o r operators (3.1) and (3-2) the value of the smallest eigenvalue whose eigenvector s a t i s f i e s D i r i c h l e t boundary c o n d i t i o n s . 2. Comparison Theorems f o r Formally S e l f - A d j o i n t Operators. We s h a l l begin by r e l a t i n g f o r m a l l y s e l f - a d j o i n t operators of type (3-1) to those of type ( 3 - 2 ) . Since the m a t r i x (a. -(x)) i s symmetric, i t i s p o s s i b l e to reduce i t , by a s u i t a b l e 58. t r a n s f o r m a t i o n , to diagonal form w i t h the eigenvalues ^ j _ ( x ) a s e n t r i e s . Without l o s s of g e n e r a l i t y , we may assume ^j_( x)£^ i + 1( 1 — i ^  " " • J ~~' 1 • D e f i n i t i o n The ma t r i x (x) =(X (x)x..(x)) i s the c h a r a c t e r i s t i c m a t r i x of (a. . (x)) . C l e a r l y the c h a r a c t e r i s t i c m a t r i x i s non-negative symmetric and the r e f o r e d i a g o n a l i z a b l e . Theorem 3-1 " Let G be a bounded domain and u a f u n c t i o n of c l a s s ^ ) (G,L) such t h a t u = u^ = 0 on BG f o r i = l , . . . , n . Let L^u, L|U be given by: and L i u = £ D. .(m D. .u) 2 1,3 1 J 1 J I f m(x) >. X(x) where X(x) denotes the l a r g e s t eigenvalue of A U ) , then: 1 i p i (uL~u)dx > uL-^u)dx 2 G Proof: IuL^u G G G X j J Let U(x) denote the ma t r i x (D. .u(x)) . Then f o r any x i n cJ 59. (3'. 3) m(x) E (D . u ) 2 - ( Z a, ,D u ) 2 = m(x) Trace [ U 2 ] -1,3 1 3 • 1,3 J J • {Trace [ ( a ± ^ ( x ) - U ) ] } 2 L e t S denote the r e a l o r t h o g o n a l m a t r i x w h i c h d i a g o n a l i z e s (a. - ( x ) ) . S i n c e t h e t r a c e i s i n v a r i a n t under such t r a n s f o r m a t i o n s [20, p . l 8 ] , the l e f t - h a n d s i d e o f (3-3) e q u a l s : »(«) Trace [ ( s W l - (Trace [ S T ( a i . (x) ) S - s > ] } 2 T T where S denotes th e t r a n s p o s e o f S . L e t S US = (r\ . ( x ) ) . T S i n c e U i s symmetric, so i s S US , and t h e r e f o r e : m(x) E ( U . u ) 2 - ( E a. .D. . u ) 2 = m(x) Z "H? . - (S X • n. . ) 2 i , j X J i , J 1 J 1 J i , j 1 J i 1 1 1 > m(x) E n 2 . - E X.X T ^ T , . . The r i g h t - h a n d s i d e o f t h e i n e q u a l i t y i s n o n - n e g a t i v e , as may be seen by p e r f o r m i n g one more change o f c o - o r d i n a t e s and a p p e a l i n g to the i n v a r i a n c e under such t r a n s f o r m a t i o n s o f m a t r i c e s w h i c h a r e m u l t i p l e s o f t h e i d e n t i t y m a t r i x . C o r o l l a r y 3.2 L e t t h e o p e r a t o r L be as d e f i n e d i n (3.1) w i t h c- s 0 , and l e t J . L,u = E D (m(x) D. .u) - S D, (B, .D.u) - Eu . 1,3 d J 1,3 d J I f : ( i ) m(x) > X ( x ) 60. ( i i ) ( B . , ( x ) - b. . ( x ) ) i s n o n - n e g a t i v e ( i i i ) E ( x ) _< d ( x ) f o r ever}/ x i n a domain G , t h e n t h e s m a l l e s t e i g e n v a l u e o f L i n G cannot exceed t h a t o f . P r o o f : Under the g i v e n assumptions and by t h e use o f Theorem -3«1<> we have t h a t [ wL^w _> | wLw G G f o r any f u n c t i o n w o f c l a s s ^) (G,L) s u c h t h a t w = w^ = 0 on BG f o r i = l , . . . , n . The C o r o l l a r y i s t h e n an immediate consequence o f Courant's P r i n c i p l e . We have thus been l e d t o the problem o f d e t e r m i n i n g t h e b a h a v i o u r o f o p e r a t o r s d e f i n e d by o p e r a t i o n (3-2). T h i s w i l l be done by f i r s t comparing t h e o p e r a t o r L Q g i v e n by w i t h the o p e r a t o r L q o g i v e n b y L u = E D . , (m. .D . .u) oo ± j i i v x j 33 whose b a h a v i o u r can be more r e a d i l y d e t ermined. We i n t r o d u c e the f o l l o w i n g n o t a t i o n : 61. N o t a t i o n m i j ( x ) = \t^m±3^x^ .(x) . Z m - (x) A s i m p l e c a l c u l a t i o n t h e n g i v e s 'Mrs , i _ and L u - L u = 2 (m. .D. .u - m. J.D. .u) oo o . . v 10 i j i j ; J- * j u ( L u - L u) = 2 f m^D.uD.u - [ tJi.(D.n)2  v oo o ' . . J i j x o J • 0 0 -G l a J G G J" f o r any f u n c t i o n u e!>)(G,L) such t h a t u = u^ = 0 on 3G f o r i = 1,...,n . Theorem 3-3 I f th e m a t r i x < ' • * > ( ^ ( x ) ) - ( A l ( l > - . . A n ( x ) i s n o n - n e g a t i v e . i n a hounded domain G ,'then i n t h a t domain t h e s m a l l e s t e i g e n v a l u e o f L o Q i s n o t s m a l l e r t h a n t h a t o f L Q . P r o o f : The c o n c l u s i o n o f t h e Theorem i s an immediate.con-sequence of Cour a n t ' s P r i n c i p l e and the p r e v i o u s c o n s i d e r a t i o n s . Because o f our r e s u l t s on o p e r a t o r s o f t y p e (3-1)* the case m- -(x) = m(x) i s o f s p e c i a l i n t e r e s t . I n t h i s c a s e , Theorem 3.3 i J becomes: 62 C o r o l l a r y 3.4 I f t h e m a t r i x (m J ( x ) ) - ( E m ( x ) ) l i s non-i n e g a t i v e i n G , then the s m a l l e s t e i g e n v a l u e o f L u = E D . , (m D. .u) oo . . n v 3d i s n o t s m a l l e r than t h a t o f L o u =• E A j ( m D i J U ) ' P r o p o s i t i o n 3.5 The c o n d i t i o n s o f C o r o l l a r y 3.4 a r e f u l f i l l e d I f the m a t r i x ^ ( m 1 ^ ( x ) ) i s n o n - p o s i t i v e i n G . P r o o f : L e t X q be an a r b i t r a r y p o i n t o f G . S i n c e ( m l j ( x 0 i s s ymmetric, we may reduce i t t o d i a g o n a l form a t x Q by a s u i t a b l e change of c o o r d i n a t e s . S i n c e t h i s i s a c c o m p l i s h e d by an o r t h o g o n a l t r a n s f o r m a t i o n , the t r a c e o f ( m 1 ^ ( x o ) ) w i l l be l e f t unchanged. L e t t h e m a t r i x ( m l j ( x )) be g i v e n by: u n Then i f 5 = (5-^ A...,? N) denotes any n - t u p l e , t h e m a t r i x (3.4) i n t h i s s p e c i a l c a s e g i v e s r i s e t o t h e form: S U ? 2 - (E u. ).(E § 2) i ± x . i x j S i n c e <_ 0 f o r e v e r y i , t h e P r o p o s i t i o n f o l l o w s . F i n a l l y , o f i n t e r e s t i s t h e case m(x) = m(r) . Under t h i s a d d i t i o n h y p o t h e s i s , the s u f f i c i e n c y c o n d i t i o n t h a t 63. ( m 1 J ( x ) ) be n o n - p o s i t i v e may be f u r t h e r i n v e s t i g a t e d and s i m p l i f i e d . We have: ( m i J ( x ) ) = [ ^ l _ J U l , ] ( x . x . ) + | 5 i l d r 2 r d r r 3 1 3 d r r t h where x^ denotes t h e i c o o r d i n a t e o f x and I denotes t h e i d e n t i t y m a t r i x . S i n c e t h e m a t r i x ( x . x . ) i s n o n - n e g a t i v e , t o ensure t h e n o n - p o s i t i v i t y o f ( m l j ( x ) ) i n t h i s c a s e , i t i s s u f f i c i e n t t o assume: r < $1 < o d r 2 - Q 1" -To summarize some o f t h e s e r e s u l t s , we s t a t e : Theorem 3.6 Assume t h a t t h e r e e x i s t s a smooth f u n c t i o n m o f r such t h a t : ( i ) m ( r ( x ) ) _> X ( x ) , where ' \ ( x ) denotes t h e b i g g e s t e i g e n -v a l u e o f J\ (x) . p ( i i ) r i | < S < 0 d r 2 ~ d r -f o r e v e r y x e G . F u r t h e r m o r e assume t h a t t h e m a t r i x -(B. .(x) - b ( x ) ) i s n o n - n e g a t i v e and E <: d f o r e v e r y x <= G~ . Then the s m a l l e s t e i g e n v a l u e o f the f o r m a l l y s e l f - a d j o i n t o p e r a t o r L does n o t exceed t h a t o f L. , where L i s d e f i n e d by: a a . :, L (u) = 2 D (m D u) - E D (B. D.u) - Eu . • i , j 1 1 3 3 • I ! J J , 64. Remark The c o n d i t i o n s on the d e r i v a t i v e s of m are t r i v i a l l y s a t i s f i e d when m i s a constant f u n c t i o n . 3. O s c i l l a t i o n Theorems f o r Formally S e l f - A d j o i n t Operators. In t h i s s e c t i o n we s h a l l use the r e s u l t s of S e c t i o n 2 to o b t a i n s e v e r a l - o s c i l l a t i o n theorems. Before s t a t i n g any such theorems, we s h a l l make some b r i e f remarks on the behaviour of the smal l e s t eigenvalue of the operators under c o n s i d e r a t i o n . As a consequence of the s t r u c t u r e of the H i l b e r t space H Q i n which' eigenvectors l i e and of Ga'rding's I n e q u a l i t y [2, p. 1983* and i n view of the estimates on the L a p l a c i a n [12, p. 7], i t i s c l e a r that the sma l l e s t eigenvalue of f o r m a l l y s e l f - a d j o i n t operators of type (3«l) and ( 3 - 2 ) , must tend to i n f i n i t y as the domain i s so perturbed that i t s diameter tends to zero. Furthermore, by a long and tedious c a l c u l a t i o n f o l l o w -i n g e x a c t l y what was done by Courant and H i l b e r t [8, V o l . I , p. 421], we can assume that the eigenvalue v a r i e s c o n t i n u o u s l y when the domain G i s deformed "continuously" i n a sense s i m i l a r to that s p e c i f i e d by Courant and H i l b e r t . We s h a l l begin by c o n s i d e r i n g the operator defined by (3.1) w i t h b.. = c. = 0 ; i , j = 1,...,n . We s h a l l a l s o f o r s i m p l i c i t y assume th a t R contains•the half-space (x : x > 0} . 65. Theorem 3.7 The o p e r a t o r L g i v e n by (3.5) Lu = S / i j l ^ i j W ) " i s o s c i l l a t o r y i n R i f t h e b i g g e s t e i g e n v a l u e o f l\ (x) i s bounded above i n R by some number such t h a t ^ [ n X ^ ~ (n-l)uldt = « o • 1 1 where u denotes the s m a l l e s t e i g e n v a l u e o f t h e problem: ,4, \ 4^ =-u*(t) dt t e I * ( t ) = §|(t) = 0 t e 31 f o r some bounded I n t e r v a l I , and n-1 g ( t ) = rnin{d(x) : x e ( IT I ) x {t}} 1 P r o o f : We compare .(3.5) w i t h the o p e r a t o r L c d e f i n e d by 4 L c u = n X x S D ±u - g ( x n ) u By a t r i v i a l s e p a r a t i o n o f v a r i a b l e s and an a p p l i c a t i o n o f a theorem o f Glazman [15* p. 1 0 4 ] , we f i n d t h a t LQ i s o s c i l l a t o r y , w i t h n o d a l domains i n t h e shape o f r e c t a n g l e s . G i v e n any sphere S , we choose a n o d a l domain N o f . L and a sphere S-, such t h a t N c: and § D = 0 . C l e a r l y t h e s m a l l e s t e i g e n v a l u e f o r S-^  o f t h e o p e r a t o r d e f i n e d by (3.5) i s - n o n - p o s i t i v e . . I f we d e f i n e a f a m i l y o f c o n c e n t r i c spheres as was done i n C h a p t e r I , 66. we f i n d t hat one of them i s a nodal domain f o r L Theorem 3-8 The operator given by (3°5) i s o s c i l l a t o r y i f f o r r s u f f i c i e n t l y l a r g e , g ( r ) (n-1) \A > 0 and where A l i m r -* P and U j g A j a r e a s defined i n Theorem 3.7. Proof: The proof i s i d e n t i c a l to th a t of Theorem 3-7, except that a d i f f e r e n t theorem of Glazman [15 p. 100] i s used to ensure the o s c i l l a t i o n of the o r d i n a r y d i f f e r e n t i a l equation a r i s i n g i n the comparison. I t i s c l e a r that s e v e r a l other theorems may be proved by us i n g d i f f e r e n t o s c i l l a t i o n c r i t e r i a f o r f o u r t h order o r d i n a r y d i f f e r e n t i a l equations. As these theorems are analogous i n -statement and method of proof to Theorems 3-7 and 3-8, they w i l l be omitted. Several of these may be found i n [93. s e l f - a d j o i n t operators of type ( 3 -2 ) , by usi n g the assumptions of Theorem 3-3'on (^Hi) • S i m i l a r l y , o s c i l l a t i o n c r i t e r i a may be found f o r f o r m a l l y 67. 4. N o n - O s c i l l a t i o n C r i t e r i a . For the e l l i p t i c operator defined by the operation (3-1), the standard methods used to prove n o n - o s c i l l a t i o n c r i t e r i a f o r second order operators no longer work, as the operators do not giv e r i s e to forms a l l of which are p o s i t i v e d e f i n i t e . We s h a l l , n e v e r t h e l e s s , prove some n o n - o s c i l l a t i o n r e s u l t s f o r such operators under the assumption that the m a t r i x (b. .(x)) i s p o s i t i v e d e f i n i t e . ' Theorem 3-9 \ I f the ma t r i x (b. . - B. .) Is non-negative.in G , and the f u n c t i o n - d - £ ^ C j + v i s a l s o non-negative i n G , J 2 then the s m a l l e s t eigenvalue u of the operator £ i n G defined by cannot exceed that of the' operator L defined by (3.1). Proof: Let X be the s m a l l e s t eigenvalue of L and u i t s as s o c i a t e d eigenvector. We then have: Zv = - I D.(p. .D.v) - YV X u 2 '{(Z a ± .D± . u ) 2 +_Eb. D.uDu - E J'g 3 u 2 - du 2} G 2 and t h e r e f o r e X >_ (i , by Courant's P r i n c i p l e . 6Q\ It i s th e r e f o r e s u f f i c i e n t to f i n d c o n d i t i o n s such that the second order operator £ i s n o n - o s c i l l a t o r y to ensure t h a t L i s a l s o n o n - o s c i l l a t o r y . Such c r i t e r i a are as given i n [93 > and w i l l not he repeated here. C o r o l l a r y 3-10 I f the m a t r i x (m„. .) has p o s i t i v e c o e f f i c i e n t s i n G and the ma t r i x (l. . - 8. .) i s non-negative as i s the D.(p.) i g 1 J -f u n c t i o n - q - E 3 3 + y , then the smallest eigenvalue of £ cannot exceed that of the operator L-^  defined by ( 3 - 2 ) . We s h a l l now consider-the operator under the assump-t i o n that the ma t r i x (m- .) i s p o s i t i v e d e f i n i t e . We begin by s t a t i n g the opposite of Theorem 3.5. P r o p o s i t i o n 3.11 I f the ma t r i x (3-4) i s n o n - p o s i t i v e i n a bounded domain G , then the smallest eigenvalue of L Q i s not smal l e r than that of L oo Proof: I d e n t i c a l t o th a t of Theorem 3.5. P r o p o s i t i o n 3.11 enables us to prove: Theorem 3.12 Let (m> •) be a p o s i t i v e d e f i n i t e m a t r i x In G and assume th a t m a t r i x (3- i0 i s non-^positive i n G. Furthermore assume that the matrix (£,-.,• - B. •) i s non-negative and th a t the (p.) 1 3 ± 3 f u n c t i o n - q - 2 D. p u + y i s non-negative. Then the smallest eigenvalue of the operator £ cannot exceed that of L, . 69. P r o o f : Under t h e assumptions o f t h e theorem, (3.7 ) f u( S D. .(m. .D. .u)) > f u( 2 D. . (m. .D . .u)) Q X * J • ' G f o r e v e r y smooth f u n c t i o n u such t h a t u = u^ = 0 on SG f o r i = l , . . . , n . However, t h e r i g h t - h a n d s i d e o f (3.7) i s c l e a r l y n o n - n e g a t i v e . We may t h e r e f o r e employ t h e p r o c e d u r e used, t o p r o v e Theorem 3»9-We have a g a i n r e d u c e d t h e p r o b l e m o f f i n d i n g non-o s c i l l a t i o n c r i t e r i a f o r t o t h e problem o f f i n d i n g such c r i t e r i a f o r t h e o p e r a t o r <£ . For a c o l l e c t i o n o f such c r i t e r i a , we r e f e r t h e r e a d e r t o [ 9 ] . Remarks The methods d e v e l o p e d i n t h i s c h a p t e r a r e c l e a r l y a p p l i c a b l e t o h i g h e r o r d e r o p e r a t o r s d e f i n e d by: L u = 2 D D ( a .a D D u) , i,j,k,£ u 1 J K t 1 J • where m denotes a p o s i t i v e i n t e g e r . The problem o f m a j o r i z i n g a r b i t r a r y f o u r t h o r d e r o p e r a t o r s , and t h e problem o f o b t a i n i n g Osc 2 c r i t e r i a f o r f o u r t h o r d e r o p e r a t o r s r e m a i n open, as does t h e problem o f o b t a i n i n g o s c i l l a t i o n c r i t e r i a f o r h i g h e r o r d e r n o n - l i n e a r o p e r a t o r s . 70. BIBLIOGRAPHY [ l ] M. P r o t t e r and H. Weinberger, On the spectrum of General  Second Order Operators, B u l l , of the Amer. Math. S o c , 72, (1966), pp. 251-255.. [2] L, Bers, P. John, and M. Schechter, P a r t i a l D i f f e r e n t i a l  Equations, Proceedings of the Summer Seminar, Boulder, Colorado, 1957. [3] J- Schauder, Uber Li n e a r e E l l i p t l s c h e n D i f f e r e n t i a l g l e i c h u n g e n . Zweiter Ordnung, Math. Z., 38, (±93^), pp. 257-282. [4] S. Agmon, Lectures on E l l i p t i c Boundary Value Problems, Nosbrand, P r i n c e t o n , 1965. [5] M. A. K r a s n a s o l e s k i i , P o s i t i v e S o l u t i o n s of Operator  Equations, Noordhoff, Groningen, 1964. [6] K. K r e l t h , A Remark on a Comparison Theorem of Swanson, Proc. Amer. Math. Sco., to appear. [7] M. G. Krexn,and M. A. Rutman, L i n e a r Operators Leaving I n v a r i a n t a Cone i n a Banach Space, Usp'ehi Mat. Nauk 3 (19^8), pp. 3-95* Amer. Math. Soc T r a n s l . (1) 10 (1962), pp. 3-95-[8] R. Courant and D. H i l b e r t , Methods of Mathematical P h y s i c s , V o l . I and I I , I n t e r s c i e n c e , New York, 1966. [9.] V.-B. Headley, O s c i l l a t i o n Theorems f o r E l l i p t i c D i f f e r e n t i a l  Equations, Ph.D. D i s s e r t a t i o n , U n i v e r s i t y of B r i t i s h Columbia, 1968. [10] C. A. Swanson, Comparison and O s c i l l a t i o n Theory of L i n e a r D i f f e r e n t i a l Equations, Academic Press, New York and London, 1968. [11] V. B. Headley and C. A. Swanson, O s c i l l a t i o n C r i t e r i a f o r  E l l i p t i c Equations, P a c i f i c J . Math., to appear. [12] S. G. M i k h l i n , The Problem of the Minimum of a Quadratic  F u n c t i o n a l , Holden-Day, San F r a n c i s c o , 1965. 71. [131 [14] [15] [16] [17] [18]. [19] [20] E. A. Coddington and N. Levinson, Theory of Ordinary D i f f e r - e n t i a l Equations, McGraw-Hill, New York, 1955. R. A. Moore, The Behaviour of S o l u t i o n s of a L i n e a r D i f f e r - e n t i a l Equation of Second Order, P a c i f i c J . Math., 5 (1955) pp. 125-145. I. M. Glazman, D i r e c t Methods of Q u a l i t a t i v e S p e c t r a l  A n a l y s i s of S i n g u l a r D i f f e r e n t i a l Operators, I s r a e l Program, f o r S c i e n t i f i c T r a n s l a t i o n s , Daniel Davey and Co., New York, C. A. Swanson, N o n - O s c i l l a t i o n C r i t e r i a f o r E l l i p t i c Equations, Can. Math. B u l l . , t o appear. R. A. S t r u b l e . Nonlinea r D i f f e r e n t i a l Equations, McGraw-Hill, New York, 1962. N. M i n o r s k i , Nonlinear O s c i l l a t i o n s , Van Nostrand, New York, C. A. Swanson, Comparison Theorems f o r Q u a s i l i n e a r E l l i p t i c  D i f f e r e n t i a l I n e q u a l i t i e s , under p r e p a r a t i o n . M. Marcus and H. Mine, A Survey of M a t r i x Theory and M a t r i x  I n e q u a l i t i e s , A l l y n and Bacon, Boston, 1964. 

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