Open Collections

UBC Theses and Dissertations

UBC Theses Logo

UBC Theses and Dissertations

Comparison and oscillation theorems for elliptic equations and systems Noussair, Ezzat Sami 1970

Your browser doesn't seem to have a PDF viewer, please download the PDF to view this item.

Item Metadata

Download

Media
831-UBC_1970_A1 N68.pdf [ 3.24MB ]
Metadata
JSON: 831-1.0080501.json
JSON-LD: 831-1.0080501-ld.json
RDF/XML (Pretty): 831-1.0080501-rdf.xml
RDF/JSON: 831-1.0080501-rdf.json
Turtle: 831-1.0080501-turtle.txt
N-Triples: 831-1.0080501-rdf-ntriples.txt
Original Record: 831-1.0080501-source.json
Full Text
831-1.0080501-fulltext.txt
Citation
831-1.0080501.ris

Full Text

5&30 COMPARISON AND OSCILLATION THEOREMS FOR.ELLIPTIC EQUATIONS AND SYSTEMS by EZZAT S. NOUSSAIR B.Eng., B.'Sc, C a i r o U n i v e r s i t y , 1961, 1965. A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY i n the Department of 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 t h e 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 March 1970 In p r e s e n t i n g t h i s t h e s i s in p a r t i a l f u l f i l m e n t o f the r equ i r emen t s f o r an advanced degree at the U n i v e r s i t y o f B r i t i s h C o l u m b i a , I ag ree tha t the 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 s t u d y . I f u r t h e r agree tha 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 o f t h i s t h e s i s f o r s c h o l a r l y pu rposes may be g r a n t e d by the Head o f my Department o r by h i s r e p r e s e n t a t i v e s . It i s u n d e r s t o o d tha t c o p y i n g o r 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 owed w i t hou t my w r i t t e n p e r m i s s i o n . Depa rtment The U n i v e r s i t y o f B r i t i s h Co lumbia Vancouve r 8, Canada S u p e r v i s o r : C. A. Swanson. i i . ABSTRACT In the f i r s t p a r t o f t h i s t h e s i s , s t r o n g c o mparison theorems o f Sturm's type a r e e s t a b l i s h e d f o r systems o f second o r d e r q u a s i l i n e a r e l l i p t i c p a r t i a l d i f f e r e n t i a l e q u a t i o n s . The t e c h n i q u e used l e a d s t o new 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 c r i t e r i a f o r such systems. Some c r i t e r i a a r e deduced from a comparison theorem, and o t h e r s a r e d e r i v e d by a d i r e c t v a r i a t i o n a l method. Some o f our r e s u l t s c o n s t i t u t e e x t e n s i o n s o f known theorems t o n o n - s e l f - a d j o i n t q u a s i l i n e a r systems. A p p l i c a t i o n o f t h e s e r e s u l t s t o f i r s t o r d e r systems l e a d s t o c r i t e r i a f o r the e x i s t e n c e o f c o n j u g a t e p o i n t s . I n the second p a r t , comparison theorems a r e o b t a i n e d f o r e l l i p t i c d i f f e r e n t i a l o p e r a t o r s o f a r b i t r a r y even o r d e r . A d e s c r i p t i o n o f t h e b e h a v i o u r o f the s m a l l e s t e i g e n v a l u e f o r such o p e r a t o r s i s g i v e n 'under domain p e r t u r b a t i o n s by means o f Gar d i n g ' s i n e q u a l i t y . New 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 c r i t e r i a a r e . o b t a i n e d by v a r i a t i o n a l methods. S p e c i a l i z a t i o n o f our theorems to e l l i p t i c e q u a t i o n s o f f o u r t h o r d e r , and t o 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 s . y i e l d s v a r i o u s g e n e r a l i z a t i o n s o f known r e s u l t s . i i i . TABLE OF CONTENTS PAGE CHAPTER I - E l l i p t i c D i f f e r e n t i a l Systems •1. I n t r o d u c t i o n 1 2. Comparison Theorems f o r E l l i p t i c D i f f e r e n t i a l Systems 4 3. The Existence of a Conjugate M a t r i x 10 4. O s c i l l a t i o n C r i t e r i a 15 5. N o n o s c i l l a t i o n C r i t e r i a 30 CHAPTER I I - E l l i p t i c Equations of A r b i t r a r y Even Order 1. I n t r o d u c t i o n 37 2. Comparison Theorems f o r Formally S e l f -a d j o i n t Operators 40 3. General Remarks 42 4. Fourth Order Equations 44 5. Operators of A r b i t r a r y Even Order 49 6. N o n o s c i l l a t i o n C r i t e r i a 63 BIBLIOGRAPHY 68 / ACKNOWLEDGEMENT I am i n d e b t e d t o P r o f e s s o r C. A. Swanson f o r h i s v a l u a b l e s u g g e s t i o n s . a n d encouragement d u r i n g the p r e p a r a t i o n o f t h i s t h e s i s . H i s p a t i e n c e and u n s e l f i s h n e s s i n o f f e r i n g h i s time t o ensure my s u c c e s s can o n l y be r e a l i z e d by b e i n g h i s s t u d e n t . I would a l s o l i k e t o e x p r e s s my thanks t o Dr. J . G. Heywood who read 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. Many thanks a r e a l s o due t o M i s s Sara B a t e f o r h e r . e x c e l l e n t t y p i n g o f t h i s t h e s i s . The f i n a n c i a l s u p p o r t o f the U n i v e r s i t y o f B r i t i s h Columbia i s g r a t e f u l l y acknowledged. CHAPTER I • ELLIPTIC. DIFFERENTIAL SYSTEMS 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 q u a s i l i n e a r e l l i p t i c p a r t i a l d i f f e r e n t i a l o p e r a t o r £ d e f i n e d by n n (1.1) lu = - 'S D.[a^ . (x,u)D .u] + 2 S b.(x,u)D.u + c ( x , u ) u i , j = l 1 1 J 3 i = l 1 • 1 a i j = a j i 3 ^ 3 ^ ~ 3 ^ 3 ' ' ' * n ' w i l l be c o n s i d e r e d f o r x e G , u ( x ) e H , where G i s a non-empty bounded domain of n - d i m e n s i o n a l E u c l i d e a n space E n , and H i s a domain i n E™ c o n t a i n i n g the o r i g i n , m,n = 1,2,... Some of the r e s u l t s a r e comparison theorems o f Sturm's ty p e . These w i l l e n a b le us t o 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 l i n e a r s e l f - a d j o i n t . s y s t e m s . P o i n t s o f E n w i l l be denoted by x = ( x ^ , x 2 , . . . , x n ) and 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^ b y . The c o e f f i c i e n t s a. • , b. , c a r e r e a l symmetric mxm m a t r i x f u n c t i o n s o f c l a s s C 1(GxH) ,.where G i s the c l o s u r e o f G i n the E u c l i d e a n t o p o l o g y o f E n , and t h e mnxmn m a t r i x (a. .(x,u) ( i , j = l , 2 , . . . , n ] i s p o s i t i v e d e f i n i t e i n GxH . The domain D o f I i s d e f i n e d as t h e s e t o f a l l v e c t o r f u n c t i o n s u e C 2(G) n C^G) • w i t h range i n H . . The n o t a t i o n EP w i l l be used f o r the s e t o f a l l mxm m a t r i x f u n c t i o n s whose column v e c t o r s "V\ e D ,• i = 1 , 2 , . . . ,m. The c o n c l u s i o n o f the comparison theorem m T below concerns m a t r i c e s V e D w i t h the p r o p e r t y t h a t V LV i s p o s i t i v e s e m i - d e f i n i t e , where L i s t h e p a r t i a l d i f f e r e n t i a l o p e r a t o r d e f i n e d by (1.2) n LV = . - / £ ' D i [ A ± .(-x,V)D.V] + 1, 3=1 n 2 S B,(x,V)D,V + C(x,V)V , A . = A . where A. ., B C a r e r e a l symmetric mxm m a t r i x f u n c t i o n s o f c l a s s C^GxH1*1) . I t i s assumed t h a t t h e mnxmn m a t r i x f u n c t i o n (A. • ( x , V ) ) . ( i , j = 1,2,...,n) i s p o s i t i v e d e f i n i t e x n GxH™ Lemma 1.1 I f the mnxmn m a t r i x (A. . (x,§)) (1,3 = 1,2,...,n) -i-j i s p o s i t i v e d e f i n i t e f o r a l l (x,§) e GxH111 , the n a d i a g o n a l m a t r i x h(x,§) = ( ^ ^ ( x , ^ ) ) can be c o n s t r u c t e d such t h a t f o r any (x,§) e GxH m , t h e m a t r i x M(x,§) = (A ( x , ? ) ) (B ±(x,?) T (B i(x,§)) h(.x,?) i s p o s i t i v e d e f i n i t e i n GxH111 , where (IL) denotes the tr a n s - ' pose of the mxnm ma t r i x (B^) . i Proof: L e t T f c(x,§) = (t ±j(x,§)) , i , . j = 1,2,...,mn+k , k = l,2,...,m , be the (mn+k)x(mn+k) m a t r i x f u n c t i o n c o n s i s t -i n g o f the f i r s t mn+k columns and rows o f the m a t r i x M , and l e t T Q(x,§) = ( A ^ x , ? ) ) . Choose ' such t h a t : 1 mn h l l ( x ; S > - [ S e t T ^ x , ? ) ] " 1 £ t 1 , m w i ( * , S ) t £ j m w l ( x , e ) , where t | m n + i ( x J ? ) denotes the c o f a c t o r of t ^ ^ ^ ( x ^ ) i n ^ ( x , ? ) . I t i s c l e a r that ^ ( x , ? ) i s p o s i t i v e - d e f i n i t e i n ; GxH111 , and h-^(x, 5) > 0 (a c r i t e r i o n of Gantmacher i s used [ 1 0 ] ) . By i n d u c t i o n , h ^ ( x , § ) (t = l,2,...,m) can be chosen so t h a t mn+<t-l h ^ ( x , § ) > - [ d e t T ^ x , ? ) ] - 1 ^ t i j m n + ^ ( x , 5 ) t ^ m n + ( t ( x , ? ) , h^^(x,§). > 0 and each i s p o s i t i v e - d e f i n i t e . Hence M. i s p o s i t i v e - d e f i n i t e [ 1 0 ] . L e t f [ u ] , F[u,V] be the f u n c t i o n a l ? d e f i n e d by f [ u ] = J [• S (D u T ) a . , ( x , u ( x ) ) D . u + G- i , j = l 1 10 J 2 S ( L \ u T ) b , ( x, u ( x ) ) u + u T c ( x , u ( x ) ) u ] d x 1=1 1 1 JG • i , j = l 1 . 1 J J 2 2 (D.u T)B. (x,V(x))u + u T ( C ( x , V ( x ) ) u + u T h ( x , V ( x ) ) u ] d x i = l ' w i t h domains , and D fxD m , r e s p e c t i v e l y , where Dp denotes the set of a l l v e c t o r f u n c t i o n s u e C (G) w i t h range i n H such that u(x) vanishes i d e n t i c a l l y on BG , where oG i s the boundary of G . D e f i n i t i o n A m a t r i x V • i s s a i d to be conjugate r e l a t i v e to L i f f Y ±(x,V) = 0 i d e n t i c a l l y i n G f o r i = 1,2,... .,11 , where n Y ±(x,V) = £ [V TA,,(x,V)D V - (D.V) TA, ,(x,V)V] . j= l . ^ " J - L J ¥e s h a l l discuss l a t e r the existence of a conjugate matrix. 2. Comparison Theorems f o r E l l i p t i c D i f f e r e n t i a l Systems. In t h i s s e c t i o n we s h a l l o b t a i n Sturm type comparison theorems f o r a r b i t r a r y r e g u l a r domains. The f i r s t comparison theorem i s "weak" i n the sense that the c o n c l u s i o n a p p l i e s to G r a t h e r than G . For the weak theorem 3G i s r e q u i r e d only to be piecewise C 1 ( [ l ] , p. 1 2 8 ) . Theorem 1.2 I f ( i ) t h e r e 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 D^ , such t h a t f [ u ] £ 0 ; ( i i ) V e D m i s a c o n j u g a t e m a t r i x such t h a t V^LV i s p o s i t i v e s e m i d e f i n i t e i n G ; and ( i i i ) f [u] >_ F[u,V] I then det V ( x ) must v a n i s h a t some p o i n t i n G . P r o o f : Suppose t o t h e c o n t r a r y t h a t V ( x ) i s n o n s i n g u l a r f o r a l l x e G . Then t h e r e e x i s t s a u n i q u e w e C^(G) s a t i s f y i n g u ( x ) = V ( x ) w ( x ) i d e n t i c a l l y i n G . An easy c a l c u l a t i o n y i e l d s the f o l l o w i n g i d e n t i t y : n m n m (1.3) S (VD,w)±A, ,(x,V)(YD.w) + 2S(Vw.) XB, (x,V)VD .w i , j = l x" 1 J J i = l 1 J m n „ n + (Vw) h(x,V)Vw + E D.[(Vw) ± 2 A. .(x,V)(D .V)w] i = l 1 j = l 1 J • J - — T — - . 2 -T. ,„ , r N„ w ^ = F + [ u , V ] - (Vw) (LV)w + S w xY.(x,V)D. i = l 1 X where F +[u,,V] i s t h e i n t e g r a n d i n F[u,V] • , and Y^(x,V) i s d e f i n e d i n S e c t i o n 1. S i n c e Y i ( x , V ) = 0 i d e n t i c a l l y f o r i = 1,2,...,n', V T L V > 0 i n G , w(x) = 0 on 3G , and the m a t r i x M(x,V) i s p o s i t i v e d e f i n i t e , i t f o l l o w s from i n t e g r a t i o n o f the above i d e n t i t y o v er G and use o f Green's i d e n t i t y t h a t ( 1 . 4 ) F[u,V] >_ 0 , with equality i f f Vw = 0 i d e n t i c a l l y i n G . This i s impossible since, by hypothesis, u = Vw i s a n o n t r i v i a l function i n . Then the assumption that V(x) i s nonsingular throughout G leads to the contradiction f [u] _> P[u,V] > 0 . Theorem 1.3 Under the hypotheses of Theorem 1.2, and the further assumption that 3 G e , det V(x) vanishes at some point i n G . Proof: Suppose to the contrary that V i s nonsingular through-out G . Then there exists a unique w e sa t i s f y i n g ' u(x) = V(x)w(x) i d e n t i c a l l y i n G . Since the boundary of G i s of class , i t follows that ( [ 1 ] , p.. 131) u belongs to the Sobolev space H^(G) • This means that u belongs to the closure i n the norm || • ||^  defined by II u ||2 = J [|u|2 + £ |D u| 2]dx G' i = l of the class C-"(G) of i n f i n i t e l y d i f f e r e n t i a b l e vector functions with compact support i n G . Let {un} be a sequence of C^(G) functions con-verging to u i n the norm || • ||, . The following i n e q u a l i t y 7. f o l l o w s from the i d e n t i t y (1.3) : (1.5) F[u n,V] > J [^S (VTJ.w n) TA i j.(x,V)VD jw n G i , j —1 + 2S^VD iw n) TB i(x,V)(Vw n) + (Vw n) Th(x,V) (Vw n)]dx >_ 0 where w n i s the unique s o l u t i o n of u (x) ~ v ( x ) w n ( x ) j x e G Since A. • (x,V(x)) , B. (x,V(x)) , and K(x,V(x)) are u n i f o r m l y X (J X hounded i n G , i t f o l l o w s from the d e f i n i t i o n of F[u,V] that there e x i s t s a constant K > 0 such that ( 1 . 6 ) |F[u n,V] - F[u,V]| < K J L > E |D.^D.(u n-u) + D.(u* - u T)D.u| + J ^ D . u ^ - u ) ' u n ( u n " u ) + ( u n ": u ) u ' ] d x A p p l i c a t i o n of Schwartz ' s i n e q u a l i t y then gives |F[u n,V] - P[u,V]| < K ^ l l u J ^ + HU I I-L J HU^U I ^ f o r some constant K-^  > 0 . Since l i m || "u^-u l]-^  = 0 a s n - oo , we conclude from .(1.5) that F[u,V] > 0 . I f 8 . F[u,V] > 0 } we obt a i n the c o n t r a d i c t i o n f [ u ] > 0 , and hence F[u,V] = 0 . Let S denote a b a l l w i t h S c G } where S i s the clo s u r e of S , and define n P S [ u n ' V ] = I [(VD.w n) TA i j(x,V)VD j.w n + (VD j Lw n) TB 1(x,V)Vw n S i = l + (Vw n) Th(x,V)Vw n]dx Then (1.5) i m p l i e s (1.7) 0 < P s t u n J v ] < F[u n,V] 9 and a s i m i l a r argument to t h a t used before y i e l d s the estimate < K 1(||w n|| l j S + i i w i i l a S ) i i w n - w i r l a S where i s a p o s i t i v e constant, and the s u b s c r i p t S i n d i c a t e s that the i n t e g r a l s i n v o l v e d i n the norm || • ||^  , are taken over. S only. Since V - 1 ( x ) e C 1 ^ ) w = V - 1 u , and Wr = V - 1 u n , ' i t f o l l o w s that nam. |P g[u n,V] - P j u , V ] | = 0 as [I u n-u 1 ^ - 0 . • Since F[u n,V] - F[u,V] = 0 as || u R-u || - 0 from ( 1 . 6 ) , we see from (1.7) that Pg[u,V] = 0 . Since the m a t r i x M(x,V) i s p o s i t i v e d e f i n i t e i n G by hypothesis.,, we conclude that w(x) = 0 i d e n t i c a l l y i n S . Since S i s a r b i t r a r y , u(x) = V(x)w(x) = 0 throughout G and hence throughout G by c o n t i n u i t y . This c o n t r a d i c t s the hypothesis that u i s a n o n t r i v i a l f u n c t i o n of c l a s s D f . The c o n c l u s i o n of the' theorem f o l l o w s . Remark. When L i s symmetric ( i . e . B. = 0 x, f o r a l l . . . ^ x mxm i = l,2,...,n) , we can take h = 0 „ , where 0 i s the * ' mxm ' mxm mxm zero m a t r i x . The operator L reduces to (1.2') LV = - M D [A (x,V)D V] + C(x,V)V , A = A . i , j = l 1 1 J J I J j i . The f o l l o w i n g theorem i s s i m i l a r to a recent r e s u l t of Swanson [ 2 3 ] . The l a t t e r - i s obtained i f we put b^ = 0 ( i = 1,2,...,n) . / Theorem 1.4 ("Symmetric case"). I f ( i ) There e x i s t s a n o n t r i v i a l v e c t o r f u n c t i o n u e such that f [u] <_ 0 ; ( i i ) V e D m i s a conjugate m a t r i x such that V TLV i s p o s i t i v e s e m i - d e f i n i t e throughout G j and 1 0 . n m ( i i i ) J [ E D.u (a. . (x,u) - A (x,V).)D,u G • i , j = l 1 1 J . i j J n + 2 E D.u Tb.(x,u)u + u T ( c ( x , u ) - C(x,V) )u]dx >_ 0 i 1=1 1 1 then e i t h e r det V(x) vanishes at some p o i n t i n G or there e x i s t s a constant v e c t o r e ^ 0 such that u(x) = V(x)e . Proof: The proof i s s i m i l a r to that of Theorem 1 . 3 -Remark. Theorem 1 . 3 g e n e r a l i z e s previous r e s u l t s of Kuks [ 1 8 ] , and Swanson [23] to n o n - s e l f - a d j o i n t d i f f e r e n t i a l i n e q u a l i t i e s . I f m = 1 , Theorem 1 . 3 extends r e s u l t s of K r e i t h [ 1 7 ] , and Diaz and McLaughin [9] to q u a s i l i n e a r n o n - s e l f - a d j o i n t d i f f e r e n t i a l i n e q u a l i t i e s . I f . n = 1 , Theorem 1 . 4 extends a r e s u l t of Morse [ 2 1 ] . 3. The Existence of a Conjugate M a t r i x . The question of the existence of a conjugate m a t r i x V r e l a t i v e to the operator L defined by ( 1 . 2 ) r e q u i r e s i n v e s t i g a t i o n . • We s h a l l show that a conjugate m a t r i x always exi.sts f o r s e v e r a l n o n t r i v i a l cases of the operator L . For n = 1 , the d e f i n i t i o n ( 1 . 2 ) of L reduces to 11. (1.7) LV = ~(A(x,V)V')' + 2B(x,V)V' + C(x,V)V (x e I) , where I i s some r e a l i n t e r v a l . I t i s assumed that (1) A, B, C, and V are mxm m a t r i x f u n c t i o n s ; (2) A(x,V)- i s symmetric, p o s i t i v e d e f i n i t e and continuous f o r x e I , and f o r a l l values of the e n t r i e s of V ; (3) B(x,V) , and C(x,V) are symmetric and continuous f o r x on I and f o r a l l values of the e n t r i e s of V . D e f i n i t i o n . A m a t r i x V i s s a i d to be conjugate r e l a t i v e to L i f f Y(x,V) = 0 i d e n t i c a l l y i n I , where Y(x,V) = V x A ( x , V ) V - V / XA(x,V)V (see the d e f i n i t i o n given i n S e c t i o n l ) . P r o p o s i t i o n 1.5. I f V o i s a s o l u t i o n of the system / LV = 0 i n I Y(x o,V(x o).) = 0 , then any one of the c o n d i t i o n s ( i ) , ( i i ) , ( i i i ) , ( i v ) given below i m p l i e s that V Q i s a conjugate m a t r i x f o r L : ( i ) B(x,V(x)) = 0 " i d e n t i c a l l y i n I 12. ( i i ) ' BA commutes w i t h V"o and ; . ( i i i ) B i s a diagonal m a t r i x , A'(x,V(x)) e x i s t s , and. A cl commutes w i t h V and V 7 , where A' = A- j o o dx ( i v ) A i s a diagonal m a t r i x , A'(x,V(x)) e x i s t s , and B commutes w i t h V and V' . o o Proof: I f c o n d i t i o n ( i ) holds, then i t f o l l o w s from the symmetry of the matrices i n v o l v e d that Y'(x,v" o(x)) = 0 i d e n t i c a l l y i n I . Hence Y ( x , v ) = K f o r some constant K Since Y(x , Y 0 ( X 0 ) ) = 0 by hypothesis, i t f o l l o w s that K i s the mxm zero m a t r i x , and Y(x,V (x)) = 0 i d e n t i c a l l y i n I . I f c o n d i t i o n ( i i ) h olds, then Y ' ( x , V j = [ V 1A(x,V^)V / - V a A ( x , V )V ] ' v 3 o o v o o o v 3 o o = 2[V TB(x,V )V' - V / TB(x,V )V ] o v 3 o/ o o v 3 o' o = 2 B ( x , V o ) A - 1 ( x , V o ) [ V ^ A ( x , V o ) V ^ - V^ TA(x,V Q)V =. 2 B ( x , V o ) A ~ 1 ( x , V o ) Y ( x , V o ) . x , Hence [6] Y ( x , V Q ( x ) ) = [exp J B ( t , V Q ( t ) ) A ~ ± ( t , V Q ( t ) ) d t ] N , a where N i s a constant mxm matrix. Using the d e f i n i t i o n of V"0 , we conclude that N i s the zero mxm m a t r i x , and Y(x,V ) = 0 i d e n t i c a l l y i n I . I f c o n d i t i o n ( i i i ) holds, then 13. Y'(x,V 0) . A ( x , V 0 ) [ V ^ - v f v o ] ' On the o t h e r hand i t f o l l o w s from L V o = 0 t h a t Y ' ( x , V o ) Henc e: [ v V - V ' T V J ' = A _ 1 ( x , V ) [ 2 B ( x , V ) - A ' ( x , V ) ] [ v V - V / T V ] o o o o v ' o • v i o1 A ' oJ O O o o Henc e [6] A / ( t , V o ( t ) ) ) ] d t k , where N i s an mxm c o n s t a n t m a t r i x . But [vo<xo>Vo(xo> - Vo T(. xo) Vo( xo>1 - ^ o ' V ^ ^ V W ' From the above c o n s i d e r a t i o n s , we c o n c l u d e t h a t Y(x,V ) = 0 i d e n t i c a l l y i n I . I f c o n d i t i o n ( i v ) h o l d s , we can use a s i m i l a r argument to show t h a t Y(x,V ) = 0 i d e n t i c a l l y i n I . T h i s completes the p r o o f o f P r o p o s i t i o n 1.5. = 2B(x,V ) [ V 1 V / - V / J T ] . v ' oJ o o o o [V*(x)V^(x) - ( x ) V Q ( x ) ] = [exp J > X [ A - 1 ( t , V o ( t ) ) ( 2 B ( t , V o ( t ) ) -1^-I f the operator L defined by (1.2) i s l i n e a r , and i f we f u r t h e r assume that C(x) = C(x^) , B(x) = B(x^)--, and A ± i ( x ) = A i i ( x i ) >-^r(Aj±}=° .C1 + J") f o r a t least one s u f f i x i , then a conjugate matrix" V(x) of L can be sought i n the form of a m a t r i x V(x) = V(x^) , where V(x^) i s a con-jugate m a t r i x r e l a t i v e to the system of o r d i n a r y d i f f e r e n t i a l equations 1 1 • 1 the existence of which has been discussed i n P r o p o s i t i o n 1.5. In p a r t i c u l a r , f o r s t r o n g l y e l l i p t i c systems w i t h constant c o e f f i c i e n t s without mixed d e r i v a t i v e s a conjugate m a t r i x e x i s t s . Remark. I f . t h e operator L defined by (1.7) i s l i n e a r , 'then a s o l u t i o n V Q of the system LV = 0 i n I Y(c,V(c))' = 0 , e e l can be sought i n the form c f an mxm mat r i x whose column vectors are m l i n e a r l y independent s o l u t i o n s {v^,v 2,...,v m] of the vec t o r equation Lv = 0 s a t i s f y i n g the i n i t i a l c o n d i t i o n s v i ( c ) = e i , v ± ( c ) = 0 , 15. where i s the u n i t column v e c t o r and 0 i s the zero column vect o r . k. O s c i l l a t i o n C r i t e r i a . In t h i s s e c t i o n , we s h a l l f i n 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 of the s e m i - l i n e a r , n o n - s e l f - a d j o i n t m a t r i x d i f f e r -e n t i a l operator L defined by ( 1 . 2 ) , namely or d i n a r y s e l f - a d j o i n t d i f f e r e n t i a l systems. The operator L i s defined i n an unbounded domain n R of E For s i m p l i c i t y , we assume that R c o i n c i d e s w i t h E'Q . However, i t should be pointed out that the proofs given below could be extended to cover more general types of domains. 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 are'as given i n S e c t i o n 1. Not a t i o n . L e t Mn denote the mnxmn matrix n n T which imply that the matrix d i f f e r e n t i a l i n e q u a l i t y V LV >_ 0 ( as a form) i s o s c i l l a t o r y , g e n e r a l i z i n g previous r e s u l t s f o r (B ±(x,?)) T M c(x,?) = B±(x,5) h(x,5)+C(x,0 16. Let R r = [x € E n : |xf > r} T D e f i n i t i o n . The matrix d i f f e r e n t i a l i n e q u a l i t y V LV >_ 0 (as a form) i s sa i d to be o s c i l l a t o r y i n E Q i f f every con-jugate s o l u t i o n V of the i n e q u a l i t y has the pro p e r t y that det V(x) vanishes at some p o i n t i n R r f o r a l l r > 0 . D e f i n i t i o n . Two fu n c t i o n s § , i|i of c l a s s .C"^ "(0,«») are c a l l e d majorants i f there e x i s t s a C^(0,») p o s i t i v e f u n c t i o n 6 such that $(r) _> max sup '[X(x,5)l + 26(r) |x|=r ?eH m K r ) > max max sup [ T, S m (By K ( x , 5 ) ) k |x|=r §eH m i = l j = l 6 ^ r 1 x + n(x,§)] + 6(r) , k = 1,2, . . . ,m , where B ±(x,§) = ( b j k ( x , 5 ) ) ( j , k = 1,2,...,m) ( i = 1,2,...,n), X(x,?) denotes the l a r g e s t eigenvalue of the mnxmn ma t r i x (A. •) ( i , j = l , 2 , . . . , n ) , and |i(x,?) denotes the l a r g e s t eigenvalue of the ma t r i x C+h as defined i n . S e c t i o n 1. Notation. Let $ ( r ) l . mn ¥(x) = ( r = |x| * ( r ) L m where 1 ^ denotes the mnxmn i d e n t i t y m a t r i x , I denotes the mxm i d e n t i t y m a t r i x and 6 i s the mxmn zero matr i x . Notation. I f A, B are any two ma t r i c e s , then we w r i t e A > B (resp. _>) i f A-B i s p o s i t i v e d e f i n i t e (resp. s e m i - d e f i n i t e ) . P r o p o s i t i o n 1.6. I f $ , i|r are any majorants of (A. .) , C , r e s p e c t i v e l y ^ then W(x) > Mr,(x^§) f o r every (x,?) e E n x H m . Proof: Let (x,§) e E n x H m . Then ¥(x) > (X(x,§) +6(r).)l. mn (i|f(r) - 6 ( r ) ) l . m XC-x,?)!^ (B.(x,?)) B i ( x , ? ) . U(x,§)l. m >. M c(x,§) In Theorems 1.7, 1.8 and 1.9 below, we assume that 18. (A^ •) and C admit m a j o r a n t s $ , \|r r e s p e c t i v e l y . L e t be the s c a l a r o p e r a t o r d e f i n e d by n (1.8) l,v = - E D.[$(r)D.V] + * ( r ) v J ( r = |x|) . x ' i = l 1 1 D e f i n i t i o n . A bounded domain N i s a n o d a l domain f o r £^ 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 f u n c t i o n ' v such t h a t l^y = 0 i n N and v = 0 on dN . P r o p o s i t i o n 1.7- I f h a s a n c " ^ a ± domain w i t h p i e c e w i s e boundary i n t h e complement of e v e r y b a l l S > then the T m a t r i x d i f f e r e n t i a l i n e q u a l i t y V LV _> 0 i s o s c i l l a t o r y i n P r o o f : G i v e n r > 0 , we choose a n o d a l domain N c } and a n o n t r i v i a l f u n c t i o n v such t h a t -t^v = 0 i n N , and v = 0 on dN . L e t I be the d i f f e r e n t i a l o p e r a t o r d e f i n e d by n o u = - ^ D ^ ^ I ^ u ] + t ( r ) l m u , where u e H and I i s t h e mxm i d e n t i t y m a t r i x , and l e t f Q [ u ] = J* [ ^ D . u ^ r j I ^ u + u T 1 ! r ( r ) l m u ] d x . I f we choose u t o be t h e m-vector w i t h each component 1 9 . equal to v , then l^a = 0 , and f 0 [ u ] = 0 • T Let V be any conjugate m a t r i x such that V LV >_ 0 . We have to show that det V vanishes somewhere i n N . Since W(x) - M c(x,§) > 0 f o r every (x,.§) e E^xH111 , i t f o l l o w s that F[u,V] < f [u] , where F[u,V] i s defined i n Se c t i o n 1 . From the above c o n s i d e r a t i o n s , i t i s c l e a r that c o n d i t i o n s ( i ) , ( i i ) and ( i i i ) of ^ .Theorem 1 . 2 are s a t i s f i e d . Hence det V = 0 i n N . D e f i n i t i o n . The operator i s s a i d to be s t r o n g l y o s c i l l a t o r y i n E n i f f i t has a nodal domain w i t h C"^  boundary i n the complement of every b a l l S . P r o p o s i t i o n 1.7 enables us to extend well-known o s c i l l a t i o n c r i t e r i a f o r the s c a l a r operator to the ma t r i x T d i f f e r e n t i a l i n e q u a l i t y V LV > 0 . Theorem 1.8. Let $ ( r ) be bounded i n 0 <. r < os w i t h T upper bound K . Then the d i f f e r e n t i a l i n e q u a l i t y V LV >_ CL n i s . . o s c i l l a t o r y i n E i f ( 1 . 9 ) l i m sup r 2 i|r(r) < _ K ^ N" 2^ r -* co - 4 Proof: The hypothesis ( 1 . 9 ) i m p l i e s that the operator defined by (1.8) has a piecewise C 1 nodal domain a r b i t r a r i l y f a r from the o r i g i n [ 1 5 ] . Theorem 1.8 i s now an immediate 20. consequence of P r o p o s i t i o n 1.7. Theorem 1.8 g e n e r a l i z e s the K h e s e r - H i l l e Theorem [16] to ma t r i x d i f f e r e n t i a l i n e q u a l i t i e s . The f o l l o w i n g theorem g e n e r a l i z e s the Leightoh-Wintner Theorem [12]. Theorem 1.9- I f d - 1 0 ) r n - i d " ; r tn-H(t)dt = i t n j-$(t) i T then the matrix d i f f e r e n t i a l i n e q u a l i t y V LV > 0 i s o s c i l l a t o r y E 1 1 Proof: The hypothesis (1.10) i m p l i e s [15] that the operator t-^_ defined by (1.8) i s o s c i l l a t o r y . i n E 1 1 . Theorem 1.9 now f o l l o w s from P r o p o s i t i o n 1.7. S i m i l a r o s c i l l a t i o n theorems may be obtained by u s i n g other o s c i l l a t i o n c r i t e r i a f o r the operator . For a c o l l e c t i o n of such c r i t e r i a we r e f e r the reader to [15]. • We s h a l l now derive other o s c i l l a t i o n c r i t e r i a by us i n g a d i f f e r e n t technique which does not r e q u i r e a comparison theorem. 21. D e f i n i t i o n s and Notations. Let A i j.(x,V) = ( A ^ ( x , V ) ) , i , j = 1,2,...,n , k,£ = l,2,...,m , B i(x,V) = ( B ^ ( x , V ) ) , C(x,.V) ;= ( C ^ ( x , V ) ) , and h(x,V) = ( h ^ ( x , V ) ) . Define . A \ ^ ( r , e . , , . . . , e _) = A^1 IJT I 3 3 n - l y i j ( r , e 1 , . . . , e n _ 1 ) = B f (x,v(x)) c k 4 ( r , e 1 , . . . , e n _ 1 ) = c k ^ ( x , v ( x ) > H ^ ( r , 0 1 , . . . , 6 n _ 1 ) = h ^ ( x , V ( x ) ) , where r , 0 1,. . •., 0 n_ 1 are the h y p e r s p h e r i c a l p o l a r coordinates defined by. n-1 x-, = r TT . s i n 0 . , x = r cos 0n , -1- .1=1 1 n 1 n - i x j L = r cos 0 n _ ± + 1 ff. s i n 6j , i = 2,3,...,n-1 Let n n ^ r\ i — 1 n n Y * * ( r ) ^ " ( r ^ , . . . ^ ) +-B**(r,e n.....B n.i)]«> > 22. where fin i s the surface of the u n i t b a l l i n E n T Theorem 1.10. The m a t r i x d i f f e r e n t i a l i n e q u a l i t y V LV > 0 i s o s c i l l a t o r y i n E n i f there e x i s t s an i n t e g e r I , 1 _< Z <_ m , and a number q > 2 such that the f o l l o w i n g c o n d i t i o n s hold f o r every m a t r i x V w i t h det V(x) ^ 0 f o r a l l s u f f i c i e n t l y l a r g e x 2b (1) — f r n _ 1 a ^ ' t d r i s bounded above' f o r a l l b > 0 j b q b 2b -(2) -X f r n _ 1 ( 2 b - r)$ll(r)clr i s bounded above f o r b q b a l l b > 0 3 (3) For every 0 _< a < » } there e x i s t numbers p > 0 , 6 > 0 such that rv 1 ^ n-1 / \ , . j . J r y (rj.dr < -6 b q - 2 + P - a f o r a l l b s u f f i c i e n t l y l a r g e . Proof: Suppose to the c o n t r a r y that the d i f f e r e n t i a l T i n e q u a l i t y V IV > 0 i s not o s c i l l a t o r y . Then there e x i s t s a p o s i t i v e number a , and a conjugate matrix V(x) such that T V LV > 0 , and det V(x) ^ 0 i n R . Then a unique s o l u t i o n cl w(x) of u(x) = V(x)w(x) e x i s t s i n R o f o r any m-vector f u n c t i o n u(x) > By i n t e g r a t i n g i d e n t i t y (1 . 3 ) by p a r t s , and a p p l y i n g Green's formula, we o b t a i n the i n e q u a l i t y 23. (1.11) P [ u 3 V ] = J [ E (D u ) T A , (x,V)D,u a< |x|<2b- 1, j - l . . . 1 1 J - • J n + 2 E ( D . u ) T B . ( x , V ) u + u T ( C ( x , V ) 1=1 1 1 + h ( x , V ) ) ]u dx >_ 0 f o r any p i e c e w i s e C"*" f u n c t i o n u ( x ) on t h e annulus N = {x : a <_ |x| <_ 2b} such t h a t u ( x ) = 0 on the boundary 3W . I n p a r t i c u l a r choose u to be the v e c t o r f u n c t i o n u ( r ) = 0 r <_ a , u ( r ) = ( r - a ) e ^ a < r < _ a+1 , u ( r ) = e^ a+1 < r <_ b , u ( r ) = e t b < r <_ 2b u ( r ) = 0 r > 2b t h where i s the u n i t m-vector w i t h 1 i n the I p o s i t i o n and z e r o e l s e w h e r e . Choose c o n s t a n t s K-^  , such t h a t 1 r»2b "' n-1 11, \ , . „ ~ , . — J r a ( r ) d r <_ f o r a l l b > b q 'b 0 •1 r2b r n - l ( 2 b _ r ) p ^ ( r ) < K O f o r a l l b > 0 b q b ' ~ d T h i s i s p o s s i b l e by hypotheses ( l ) and ( 2 ) . Then 24. F[u,V] < J a + 1 c x ^ ( r ) r n - 1 d r + K ^ - 2 a (1-12) + J ( r - a ) P ( r ) r dr + Kgb^ i»a+l / \2 U / \ n-1, +| (r - a) Y ( r ) r d r "a r \ n-1, p2b ,2b-r\2 -M,, \ n-1, + j y ( r ) r dr + J (—^—) y ( r ) r d r a+1 b By hypothesis (3) there exists b - > a+1 such that the sum of the f i r s t 6 terms of the right member of the above in e q u a l i t y i s negative f o r a l l b > b Q . r Define f ( r ) = J y11(r)rn_1dr . Then b o d JL^L n—1 f ( r ) = y ( r ) r ~ , and l i m f ( r ) = -«> by hypothesis (3). r-*oo Now we choose b > b i n the d e f i n i t i o n of the vector function — o u as the l a s t root of f ( r ) . Then 2b (1.13) J ( 2 ^ £ ) 2 Y ^ ( r ) ^ - l d r b . u = [ ( ^ ) 2 f ( r ) ] f + ^ i i ^ r l f ( r ) d r < Q  u b Tt follows from- (1.12) that F[u,V] < 0 , contradicting ( l . l l ) . This completes the proof of the theorem. 25. The f o l l o w i n g theorem can be proved u s i n g the same argument as i n Theorem 1.10. T Theorem 1.11. The m a t r i x d i f f e r e n t i a l i n e q u a l i t y V LV >_ 0 i s o s c i l l a t o r y i n E n i f there e x i s t s an.integer I , 1 <_ I < m , and a number q <_ 2 such that the f o l l o w i n g c o n d i t i o n s hold f o r every m a t r i x V w i t h det V(x) =)= 0 f o r a l l s u f f i c i e n t l y l a r g e x : 1 2 ^ n—1 -L-L (1) —- f r a ( r ) d r i s bounded above f o r a l l b > 0 \ b q 'b _ 2b (2) f r n _ 1 ( 2 b - r)p' t" t(r)dr i s bounded above f o r b q "b a l l b > 0 i ( 3 ) For a > 0 \ n-1, J y ( r ) r dr = -» a When the operator L i s symmetric, i . e . IL = 0 f o r i = l , 2 , . . . , n , we can choose h = 0 . In t h i s case the f o l l o w i n g more general v e r s i o n of Theorems 1.10 and 1.11 hoid s. P r o p o s i t i o n 1.12. ("Symmetric" case). Under the hypotheses of Theorem 1.10 (or Theorem l . l l ) w i t h the weaker assumption that the m a t r i x (A. -) ( i , j = 1,2,...,n) i s p o s i t i v e semi-T d e f i n i t e , the m a t r i x d i f f e r e n t i a l i n e q u a l i t y V LV >_ 0 i s o s c i l l a t o r y i n E n . 26. Proof: Suppose to the con t r a r y that the d i f f e r e n t i a l i n e q u a l i t y V ^ L V >_ 0 i s not o s c i l l a t o r y i n F/1 . Then there e x i s t s a p o s i t i v e number a , and a conjugate m a t r i x V (x) such that V L V > 0 , and det V ( x ) ^  0 i n R . Then a unique s o l u t i o n — a w(x) of u(x) = V(x)w(x) e x i s t s i n R & f o r any m-vector f u n c t i o n u(x) . From i n e q u a l i t y (1.3) we can e a s i l y show that (1.14) F[u,V] = J [ ( D , u ) T A • ,(x,V)D U + u TC(x,V)u]dx >_ 0 a<Jx|<2b f o r any piecewise C"*" f u n c t i o n u(x) on the annulus N = (x : a <_ [xj <_ 2b] such that u(x) = 0 on the boundary BN . - I n p a r t i c u l a r we choose u as i n Theorem 1.10, and we proceed i n the same way to e s t a b l i s h the i n e q u a l i t y F[u,V] < 0 , which c o n t r a d i c t s ( l . l 4 ) . In the case n = 1 , the d e f i n i t i o n (1.2) of L reduces to (1.15) L V = - A ( A ( X , V ) A V ) + 2B(x,V).Av + c(x,V)V , " where " x e [a,») . 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 are the same as i n Section 3-Theorem 1.13. ("Ordinary case"). The m a t r i x d i f f e r e n t i a l T i n e q u a l i t y V LV > 0 i s > o s c i l l a t o r y i n [a,») i f there e x i s t s an i n t e g e r t , 1 <_ I <_ m , and a r e a l number q <_ 2 , such 27 that the f o l l o w i n g c o n d i t i o n s hold f o r every m a t r i x V(x) which i s nonsingular f o r s u f f i c i e n t l y l a r g e x : i 2 b 11 (1) — J A (x,V(x))dx i s bounded above f o r a l l b >_ b q b , 2b (2) — f (2b - x)B / t" L(x,V(x))dx i s bounded below f o r a l l b q b b >_ a j (3) I [C* 4(x,V(x)) + h W ( x , V ( x ) ) ] d x = — . Proof: The proof i s v i r t u a l l y i d e n t i c a l to that of Theorem 1 Remarks. ( i ) For q > 2 , Theorem 1.13 w i l l s t i l l hold i f we repla c e c o n d i t i o n (3) by the c o n d i t i o n (3') For any € >_ 0 , there e x i s t numbers p > 0 and 6 > 0 such that q Z ^ j V W ( x , V ( x ) ) + hll(x3Y(x))]dx < -6 f o r a l l b s u f f i c i e n t l y l a r g e . ( i i ) C o n d i t i o n (2) i n Theorem 1.13 i s weaker than c o n d i t i o n (2) i n Theorem 1.10 where i t was re q u i r e d that 1 2 b U — f (2b - r ) | B ^ ( x , V ( x ) ) |dx i s bounded above f o r a l l b > £ b q b ~ ( i i i ) In the "Symmetric case", i . e . B = 0 , Theorem 1.13 28. holds under the weaker assumption that the m a t r i x A i s p o s i t i v e s e m i d e f i n i t e . This f o l l o w s from P r o p o s i t i o n 1.12. I f we a l s o assume that the m a t r i x A(x) i s bounded above f o r a l l x s u f f i c i e n t l y l a r g e , then Theorem 1.13 reduces to a recent r e s u l t of Swanson [24]. Tomastik [27] considered the m a t r i x d i f f e r e n t i a l equation (1.16) - ^ ( A ( x ) ^ V ) + C ( x , V ( x ) ) V - 0 on the i n t e r v a l [a , a> ) . He assumed that A(x) i s continuous, symmetric, and p o s i t i v e d e f i n i t e on [a,») , and that C(x,V(x)) i s negative d e f i n i t e on [a,») . The f o l l o w i n g , theorem of Tomastik ([27], Theorem 2, p. 1430) w i l l be compared w i t h the s p e c i a l case B = 0 of Theorem 1.13. Theorem (Tomastik). The mat r i x equation (1.16) I s o s c i l l a t o r y i n [a,») i f ( l ) A(x) = t ( x ) l , where I i s the i d e n t i t y mxm ma t r i x , t ( x ) i s a p o s i t i v e s c a l a r function- and (2) J \(x,U(x))dx = -co a f o r every d i f f e r e n t i a l m a t r i x U(x) such that 29. X(U (x)U(x)) _> 6 > 0 f o r l a r g e x , where X i s the m - • T smallest eigenvalue of the ma t r i x U U , where U denotes the transpose of U . We n o t i c e that 1. • Tomastik's theorem re q u i r e s that A i s p o s i t i v e d e f i n i t e and C . i s negative d e f i n i t e . In the s p e c i a l case P = 0 of Theorem 1.13* we only r e q u i r e that A i s p o s i t i v e s e m i d e f i n i t e . 2. Condition (2) i n the above Theorem i s more general ( i n the n o n l i n e a r case), than c o n d i t i o n (3) i n Theorem 1.13. 3. The f o l l o w i n g example shows that c o n d i t i o n ( l ) i n the above Theorem and c o n d i t i o n ( l ) i n Theorem 1.13 overlap: Example: Set m = 2 , and l e t A, C be the 2x2 matrices A(x) = x 2 I , C(x) = / Then c o n d i t i o n ( l ) above i s not s a t i s f i e d . C o n d i t i o n ( l ) i n Theorem 1.13 i s s a t i s f i e d i f we choose q = 3 . Con d i t i o n (3') (see the Remark f o l l o w i n g Theorem 1.13) i s a l s o s a t i s f i e d p p ' p p since we can take p = \ , C (x) = -x , and h = 0 . Hence the ma t r i x d i f f e r e n t i a l i n e q u a l i t y VLV _> 0 i s o s c i l l a t o by Theorem 1.13. -x 1 1 -x 30. The d e f i n i t i o n of " o s c i l l a t o r y " given i n Se c t i o n 4 i s the same as the d e f i n i t i o n of o s c i l l a t o r y given i n [.9] i n the case that (1.2) i s l i n e a r . In the s c a l a r case, m = 1 , t h i s d e f i n i t i o n guarantees that a l l s o l u t i o n s of (1.2) have a r b i t r a r i l y l a r g e zeros. Hence a l l the c r i t e r i a obtained i n t h i s s e c t i o n , when r e s t r i c t e d to the case m = 1 , give o s c i l l a t i o n c r i t e r i a f o r s c a l a r d i f f e r e n t i a l i n e q u a l i t i e s . 5. N o n - o s c i l l a t i o n C r i t e r i a . In t h i s s e c t i o n , we ob t a i n a K n e s e r - H i l l e [16] non-o s c i l l a t i o n c r i t e r i a f o r the ve c t o r equation n n (1.17) IVL = - 2- D. [a. . (x,u)D .u] + 2 E b.(x,u)u + c(x,u)u = 0 . l,j=a 1 1 J J i = l 1 • defined i n an unbounded domain R of E n . No r e s t r i c t i o n s are r e q u i r e d on the shape of the domain R . The co n d i t i o n s on the c o e f f i c i e n t matrices a. . , b. , c are as given i n S e c t i o n 1, and u e H , where H i s a subdomain of E m c o n t a i n i n g the o r i g i n . D e f i n i t i o n . The operator I i s , s a i d to be n o n - o s c i l l a t o r y i n the domain R . i f there e x i s t s r > 0 such that the system (1.17) has no n o n t r i v i a l s o l u t i o n v a n i s h i n g on the boundary of any bounded n-dimensional domain belonging to R D {x : |x| > r } . D e f i n i t i o n . Let a ± j ( x , ? ) -b ±(x,§) ( ^ ( x ^ ) ) c ( x , 5 ) Let X be the smallest eigenvalue of the mnxmn matrix (a. .) = l , 2 , . . . , n , and p. be the sma l l e s t eigenvalue of the mxm ma t r i x c . D e f i n i t i o n . A f u n c t i o n g of class. C 1(0 J oo) i s s a i d to be a minorant of the ma t r i x c i f there e x i s t s a C"*"(0,») p o s i t i v e f u n c t i o n z such that m n ;(r) < min min i n f E s - i ( b ^ k ( x , ? ) ) 2 + n(x,§) Kk<m |x|=r §eH J=l i = l z. 1 . where b ±(x,§) = ( b ^ k ( x , ? ) ) , j,k = l,2,.:.,m , 1 = l , 2 , . . . , n D e f i n i t i o n . A f u n c t i o n f of c l a s s C^O,*) i s c a l l e d a minorant of (A-?-?) i f there e x i s t s a C"^ (0,«>) p o s i t i v e f u n c t i o n z such that 0 < f ( r ) <_ min i n f [\(x,§) - 2 z ( r ) ] 32. Notation. nix) = f ( r ) l . ran g ( r ) l . m > ( r = |x|) where I i s the mnxmn i d e n t i t y m a t r i x , I i s the mxm T i d e n t i t y m a t r i x , and 0 i s the mnxm zero mat r i x . P r o p o s i t i o n 1.14. Let f,g be minorants of (A. .) , C r e s p e c t i v e l y . Then the m a t r i x /7?(x,§) - 7l(x) i s p o s i t i v e s e m i d e f i n i t e f o r a l l (x,§) e E nxH . Proof: Let be the submatrix of' 711-71 c o n s i s t i n g of the 1 s t k rows and k columns. I t i s easy to check that det >_ 0 f o r k = l,2,...,mn . Using a c r i t e r i o n of , Gantmacher [10], we conclude that 7f[-7\ i s p o s i t i v e semi-d e f i n i t e . Theorem 1.15. Suppose the matrices (a. . ) , c admit minorants f,g r e s p e c t i v e l y , such that f i s bounded below i n R by some p o s i t i v e number \( o s c i l l a t o r y i n R i f Then the equation (1.17) i s non-(1.18) l i m i n f r 2 g ( r ) > - ( n~ 2) X o r -» ea 4 Proof: Suppose to the con t r a r y that there e x i s t s a bounded 3 3 -nodal domain N and a s o l u t i o n u of (1.17) such that s s u = 0 oh 3N . In view of P r o p o s i t i o n 1.14, an a p p l i c a t i o n s s . of Green's formula gives . n m 0 = J u i u = J [ E ) a, .(x,u )D,u N N i , j = l . 1 S• 1 J . s. J s s s. n + 2 . s u s b i ( x ' u s ) D i u s + U g C ( x , u s ) u g ] d x n Let u . , i = 1,2,...,m , denote the i t h component of the v e c t o r f u n c t i o n u . Hence the above i n e q u a l i t y i m p l i e s that m n o o (1.19) J E [_E X o(D iu ) d + g , ( r ( x ) ) u f . ] d X < 0 . N g j=L i = l Let u s J ( x ) = u s J ( r , 9 1 , . . . , 6 ^ ) , where r , 9 1 , . . . , 0 n _ are the h y p e r s p h e r i c a l p o l a r coordinates defined i n Section 4. Extend u . to a l l of R such/that i t i s i d e n t i c a l l y zero out-s j side N_ • We s h a l l use u . to denote the extended f u n c t i o n , b s j From the r e p r e s e n t a t i o n J > - 5 ( u s . ( r , . ) ) 2 d r = -2 A 8 j ( t . , . ) . ^ u a J ( t , . ) d r j V ^ d r , and by means of Cauchy-Schwartz i n e q u a l i t y we o b t a i n the i n e q u a l i t y 34. f r n " 5 ( u . ( r , - ) ) 2 d r < — i - * f r 1 1 " 1 ^ u . ) 2 d r s S J ~ (n - 2 ) 2 J s ^ r S J f o r j = 1,2,...,m . I n t e g r a t i n g the two sides of the above i n e q u a l i t y w i t h respect to the angular coordinates 6-^6g,...,0 n we o b t a i n (1.20) J [ I (D.u ) 2 - i ^ f - u 2.]dx > 0 N s i = l 1 S J' 4r^ S J f o r a l l j = l,2,...,m . The hypothesis (1.18) i m p l i e s that there e x i s t s a constant s Q > 0 such that 2 i w ( n - 2 ) 2 X o r g ( r ) > j provided that r >_ S q . Hence, from (1.20) we have J [ S X (D.u , ) 2 + g ( r ( x ) ) u 2 ,.]dx > 0 . N i = l . o J / s o J o . ' f o r a l l j = l,2,...,m . This c o n t r a d i c t s (1.19), and Theorem l f 1 5 f o l l o w s . Remark. When the operator I , defined by (1.17) i s symmetric, i . e . ' b^ = 0 f o r i . = l , 2,...,n , we can choose minorants f,g such that 35. 0 < f ( r ) <_ min i n f X(x,5) |x|=r §eH g ( r ) <_ i n f i n f n(x,§) . |x|=r §eH In t h i s case Theorem 1.15 g e n e r a l i z e s the K n e s e r - H i l l e non-o s c i l l a t i o n c r i t e r i o n [ l 6 ] to n o n l i n e a r p a r t i a l d i f f e r e n t i a l systems.. Remarks. The r e s u l t s obtained i n t h i s chapter can be a p p l i e d to the f i r s t order system A(x)V' = U - 2B(x)V (1.21) U' = C(x)V - 2B'(x)V , x > [ a , » ) , where A, B, C, V, U are mxm m a t r i x f u n c t i o n s , A(x) i's symmetric, p o s i t i v e d e f i n i t e and continuous on [a,») , and B ( x ) , C(x) are symmetric and continuous f o r x on [a,») . I t i s c l e a r that the above system i s equivalent to the system / LV = 0 , where L i s the matrix operator defined by (1.15). Let {V;U} be a s o l u t i o n of the system (1.21) such that V i s a conjugate matrix r e l a t i v e to L and V(a) = 0 , U(a) = E , where E i s an mxm constant matrix . I f v i s a nonzero o m-vector, then i t i s easy to see that the v e c t o r p a i r {v;u} 36. V ( x ) v o and u(x) = U ( x ) v o , i s a s o l u t i o n of u - 2B(x)v , C(x)v - 2B'(x)v , u(a) = 0 , v(a) = E V Q . Two p o i n t s s,t , of [a,<») are. s a i d to be (mutually) conjugate'with respect to (1.21), i f t h e r e . e x i s t s a ve c t o r s o l u t i o n ( v ( x ) ; u ( x ) ) of t h i s system w i t h v ( s ) = 0 = v(t)' and v(x) £ 0 oni [ s , t ] . From the c o n s i d e r a t i o n s above, i t • i s c l e a r that the o s c i l l a t i o n c r i t e r i a obtained i n Se c t i o n 4 give c r i t e r i a f o r the existence of conjugate p o i n t s . I n f a c t , i f L i s o s c i l l a t o r y , then f o r any s e [a,»), there e x i s t s t > s such that s,t are conjugate. We n o t i c e a l s o that the n o n - o s c i l l a t i o n c r i t e r i o n (1.15) gives s u f f i c i e n t 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 the system (1.21) which guarantee the existence of some i n t e r v a l [s,co) such that no two d i s t i n c t p o i n t s of i t are conjugate. Some of the r e s u l t s i n t h i s chapter can'be extended without d i f f i c u l t y to a c l a s s of p a r a b o l i c equations. given by v(x) = the system. A(x)v = u' = CHAPTER I I ELLIPTIC EQUATIONS OP ARBITRARY EVEN ORDER 1. I n t r o d u c t i o n . Comparison theorems f o r e l l i p t i c equations of a r b i t r a r y even order are e s t a b l i s h e d . S p e c i a l i z a t i o n to f o u r t h order equations y i e l d s o s c i l l a t i o n c r i t e r i a which gener-a l i z e r e s u l t s of A l l e g r e t t o [3] and Headley [13]-We a l s o o b t a i n new 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 of a r b i t r a r y even order which g e n e r a l i z e previous r e s u l t s on or d i n a r y d i f f e r e n t i a l equations [11], and p a r t i a l d i f f e r e n t i a l equations of second order. D e f i n i t i o n s and Notations. We s h a l l consider the l i n e a r •' e l l i p t i c operator L defined by (2.1) Lu H (-)m s D a(a a fjAi) - c ( x ) u , I P I = I a I =m whose c o e f f i c i e n t s are defined i n an unbounded domain G of n-dimensional Euclidean space E n . The d i f f e r e n t i a l operator D a i s defined as usual by D°u = D^ 1^ ... D^ n^ , n a = ( a ( l ) , a ( 2 ) , . . . . ,a(n)) , |a[ = Y, a ( i ) , where each 1=1 a ( i ) , i = l , . . . , n , i s a non-negative i n t e g e r . The c o e f f i c i e n t s aa8 a r e symmetric, i . i . &aQss'&Qa > a n d smooth enough so that 38. a l l the p a r t i a l d e r i v a t i v e s i n v o l v e d i n L e x i s t and are continuous i n G (the c l o s u r e of G i n the Euclidean topology on E n) . By D(F,L) we s h a l l mean the c o l l e c t i o n of a l l r e a l f u n c t i o n s of c l a s s C 2 i n(F) 0 C m(F) , where F denotes a subdomain of G . D e f i n i t i o n . A bounded domain N c G i s s a i d to be a nodal domain f o r L i f f 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 CL w e D(N,L) such that Lw = 0 i n N ,. D w = 0 on SN f o r a l l a w i t h |af <_ m-1 . 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 nodal domain outside of every sphere centered at the o r i g i n . The operator L i s assumed to be u n i f o r m l y s t r o n g l y e l l i p t i c i n G , i . e . there e x i s t s a p o s i t i v e constant . d Q such that f o r a l l x e G and f o r every % = (5^,...,§ n) . Let the f i n i t e set of m u l t i - i n d i c e s a be ordered, i n an a r b i t r a r y manner,, i n a sequence S = { c L ^ a ^ , . .. ,a^} , where = {CL±(l) , 0 ^ ( 2 ) , . . , . , a ± ( n ) ) . Each ^ ( p ) ( i = l , 2 , . . . , k ) (p = 1,2.,...,n) i s a non-negative i n t e g e r , n Y a. (p) = m , and k i s the number of the m u l t i - i n d i c e s a . 39-Corresponding to the sequence S , we can arrange the c o e f f i c i e n t s a a i n the form of a kxk matrix M„ defined by aB S J i J To each sequence ,• obtained by r e o r d e r i n g the elements of the sequence S , there e x i s t s a permutation a on the set of in t e g e r s (1,2, ,k} such that S 1 = (l) >aa(2)3 " ' 3(Xo(k)^ D e f i n i t i o n . Let TT be the kxk permutation m a t r i x defined by where 6 i J L = 1 , 6. . = 0 i f i ={= j . Lemma 2.1. The matrices M c = (a _ ) and M p = (a„ _ ), 1 J 1 a ( i ) a ( J ) corresponding to the sequences S, S^ r e s p e c t i v e l y , have the same set of eigenvalues. Proof: I t . i s easy to see that TT-"1" = (5 /. \, ) , where TT~"^  0 v cr(1 )t 3 denotes the in v e r s e m a t r i x . Elementary c a l c u l a t i o n s show that Ms1 = Hence Mg , Mg• are s i m i l a r m a t r i c e s , and the lemma f o l l o w s from well-known theorems, i n l i n e a r algebra. 40. D e f i n i t i o n . L e t X ( x ) be the l a r g e s t e i g e n v a l u e o f the c o e f f i c i e n t m a t r i x Mg . Under the above c o n d i t i o n s on the o p e r a t o r L , the Courant p r i n c i p l e [ 7 ] , ( [ 2 0 ] , p. 89), may be used t o p r e d i c t f o r the o p e r a t o r L the v a l u e o f the s m a l l e s t e i g e n v a l u e whose e i g e n v e c t o r s s a t i s f y 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 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 . We s h a l l b e g i n 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 o p e r a t o r s o f type (2.1) t o the o p e r a t o r L-^  d e f i n e d by (2.2) L x u = ( - ) m £ D a ( m ( x ) D a u ) ' - E ( x ) u I a I =m where x e G , and the c o e f f i c i e n t s m ( x ) , E ( x ) a r e smooth, enough so t h a t a l l t h e d e r i v a t i v e s i n v o l v e d i n the o p e r a t i o n d e f i n i n g L-^  e x i s t and a r e c o n t i n u o u s on 5 . F u r t h e r m o r e , we assume t h a t the o p e r a t o r L^ ''is u n i f o r m l y s t r o n g l y e l l i p t i c i n G . Theorem 2.2. L e t F be a bounded domain o f G and u a f u n c t i o n o f c l a s s D(F,L) such t h a t D au = 0 on 3F f o r I ct f <_ m-1 . L e t L, L 1 be t h e o p e r a t o r s g i v e n by (2.1), (2.2) r e s p e c t i v e l y . I f . ( i ) m(x) >_ X ( x ) f o r e v e r y x e F 3 41. ( i i ) E ( x ) _< c ( x ) f o r e v e r y x e F ; ( i i i ) The boundary 3F o f F 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 ; t h e n f uL,u dx > f uLu dx J F 1 ~ J F P r o o f : Under hypotheses ( i i ) a n d ' ( i i i ) we have f ( u L n u - uLu)dx > J [ '.2' m(x)(D°u) 2 - a „ R D a u D P u ] d x F F |a| = |p|'=m a p L e t Mg be the c o e f f i c i e n t m a t r i x c o r r e s p o n d i n g t o the o r d e r i n g s = (a-j^ct^,.. . ) . L e t V ( x ) be t h e row v e c t o r a l a 2 a k (D u, D u,...,D u) . I n vi e w o f t h e h y p o t h e s i s ( i ) and by the u se o f Lemma 2.1, we have t h a t 1c a 2 a „D au D^u = V M C V T < X 2 (D i u ) 2 \a.\ = \$\=m a p S ~ 1=1 < m(x) 2 ( D a u ) 2 , | a | =m and t h e theorem f o l l o w s . C o r o l l a r y 2.J>. Under the hypotheses o f Theorem 2-2, the 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 L i n F cannot exceed t h a t o f L^ . 42. P r o o f : By Theorem 2.2 we have t h a t ' J uL.u >_ J uLu F P f o r any f u n c t i o n u o f c l a s s D(F,L) such t h a t D u = 0 6n oF , J ot J <_ m-1 . The c o r o l l a r y - i s t h e n an immediate con-sequence o f Courant's p r i n c i p l e . J>. G e n e r a l Remarks. I n the n e x t s e c t i o n we s h a l l o b t a i n some o s c i l l a t i o n theorems f o r L . B e f o r e s t a t i n g o ur r e s u l t s , we s h a l l f i r s t make some remarks on t h e b e h a v i o u r of the 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 under c o n s i d e r a t i o n . I t i s well-known t h a t the e i g e n v e c t o r s o f the o p e r a t o r L , as d e f i n e d by ( 2 . 1 ) , on a bounded domain Cl o f E 1 1 w h i c h has s u f f i c i e n t l y smooth boundary, l i e i n the Sob o l e v space ( t h e c l o s u r e i n the norm || • IIM- d e f i n e d by 'I u "m W F ? ( D V ) 2 ' D X Cl |a|=m of t h e c l a s s o f C™(Q) o f i n f i n i t e l y d i f f e r e n t i a b l e f u n c t i o n s w i t h compact s u p p o r t i n Cl).. T h i s e n a b l e s us t o make use o f the s t r u c t u r e o f the H i l b e r t space , and o f G a r d i n g ' s i n e q u a l i t y ( [ l ] , p. 78) t o p r o v e the f o l l o w i n g p r i n c i p l e : 4?. M o n o t o n i c i t y P r i n c i p l e . F o r 0 < ,t < » , l e t Q^ . be a domain c o n t a i n e d w i t h i n a domain ft p f bounded w i d t h <_ t . I f 0 < r < s < co i m p l i e s ft.„ <= n fi ^ n } then the s m a l l e s t x S i S e i g e n v a l u e U 0 ( t ) o f the pr o b l e m Lu = u ( t ) u i n 0 t , D au = 0 on 3Q t , J ot | <_ m-1 i s monotone d e c r e a s i n g i n " t. , and l i m \i ( t ) = +» . t-0+ ° P r o o f : F o r the f i r s t p a r t we may adapt t h e argument i n . ([j]9 pp. 400-401). To p r o v e t h e second p a r t , l e t cp be a non-t r i v i a l f u n c t i o n o f c l a s s C ° ° ( n , ) . Extend cp c o n t i n u o u s l y O 0 t o a l l c f Q by s e t t i n g cp = 0 o u t s i d e ft^_ . A p p l y G a r d i n g ' s i n e q u a l i t y t o o b t a i n (cp,Lcp) n = J* [E a ^ D cp DPcp - c(x)cp ]dx > K-JI cp ||m - K2||cp || , where K^ and Kg a r e c o n s t a n t s and K-^  > 0 . Hence > K l I^.K 2>^-K 2 II CP II " 1 II Cf, || 2 " t m • 2 where the l a s t i n e q u a l i t y f o l l o w s from Poincare"'s i n e q u a l i t y , and K i s a p o s i t i v e c o n s t a n t . From the above i n e q u a l i t y , w h ich i s t r u e f o r a l l nonzero cp e c"(ft,) , and. i n v i e w o f 44. Courant's minimum p r i n c i p l e , w h i c h i s v a l i d f o r the type o f o p e r a t o r s under c o n s i d e r a t i o n ([20], p. 89), we see t h a t K.K ^ ( t ) 1 —w ~ K o • Hence l i m u _ ( t ) = +« . ° • t m d t~0+. °. • We can a l s o assume t h a t the s m a l l e s t e i g e n v a l u e 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 " c o n t i n u o u s l y " i n a sense s i m i l a r to t h a t s p e c i f i e d by Courant and H i l b e r t [7], and by R u d o l f Vyborny [28]. Remark. A p r o o f s i m i l a r t o the above has been r e c e n t l y g i v e n by Headley [ 1 4 ] . 4. F o u r t h O r d e r . E q u a t i o n s . The e q u a t i o n t o be c o n s i d e r e d i s t h e s p e c i a l c a s e m = 2 o f (2 . 1 ) , namely (2.3) Lu = S ^ ( a D^u) - e ( x ) u = 0 |a|=0|=2 .. a P * The g e n e r a l c o n d i t i o n s on L a r e as i n S e c t i o n 1 . The m a t r i x M s = (aa a ^ w i H denote as b e f o r e the kxk m a t r i x o f the c o e f f i c i e n t s o f L c o r r e s p o n d i n g t o the o r d e r i n g s = ( a ^ , . . . , o i j o f the m u l t i - i n d i c e s a , and X i s the l a r g e . s t e i g e n v a l u e o f M s . 45. N o t a t i o n . S ( x o J a ) = S a ( x Q ) = [x e E 3 1 " 1 : |x - X q | < a} We s h a l l assume t h a t the domain G c o n t a i n s an i n f i n i t e c y l i n d e r o f the form W x U n : x n > 0 3 > where x = ( x n J x o 3 . . . J x n ) e E 1 1 - and 6 > 0 o N 1' 2' ' n-1 N o t a t i o n . R^ = [ x e E n : |x| > r} Theorem 2 .4 . The o p e r a t o r L .given by ( 2 . 3 ) i s o s c i l l a t o r y i n G i f ' t h e l a r g e s t e i g e n v a l u e X ( x ) o f M g i s bounded above i n G by a number X. and CO J [ f ( t ) - u ]dt. = +• , O where \iQ denotes the- s m a l l e s t e i g e n v a l u e o f the p r o b l e m ( 2 . 5 ) X X Q n D ± cp(x) = n cp(x) , x e S 6 / 5 ( x o ) , Dacp = 0 on 3 S 6 / 5 ( x Q ) , |a| = 0, 1 , and f ( t ) = i n f ( c ( x ) : x = ( x , t ) , x e S 6 / 3 ( X Q ) } 4 6 . P r o o f : We compare t h e o p e r a t o r (2.3) w i t h t h e s e p a r a b l e o p e r a t o r I d e f i n e d b y n ,4 (2.6) -Cu = n X o _Z BTVL - f ( x n ) u The h y p o t h e s i s (2.4) i m p l i e s t h a t t h e 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 ' (2.7) n X o A - ( f ( x n ) - u o ) v = 0 i s o s c i l l a t o r y b y an a p p l i c a t i o n o f a t h e o r e m o f G l a z m a n [ 1 1 ] . F o r a r b i t r a r y r > 0 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 v o f (2.7) w i t h z e r o s o f o r d e r 2 a t x n = ^ i - ' ^ 2 3 w b e r e 6 2 > 6^ > r + 6 . I f cp i s a n e i g e n f u n c t i o n o f ( 2 . 5 ) c o r r e s -p o n d i n g t o t h e e i g e n v a l u e \xQ , t h e n t h e f u n c t i o n u Q ( x ) = v ( x n ) c p ( x ) , w h e r e x = ( ^ ^ X g , . . . , ^ ) , i s , b y d i r e c t c a l c u l a t i o n , a s o l u t i o n . o f ( 2 . 6 ) , w i t h n o d a l d o m a i n N o = S 6 / 3 ^ X \ x n : 6 1 < x n < 5 2 } ' I n f a c t , i f a = ( a - ^ a , . . . , a n ) = ( a , a n ) , t h e n D a u Q = D°' nv(x n) if- c p ( x ) : . ' Hence D a u = 0 v o n B'N f o r l a ! = 0,1 . Thus u h a s a o o . • o n o d a l d o m a i n N Q c •. i n f a c t , x e N Q i m p l i e s x| > x n > 6^ > r 47. Let N-^  be the c y l i n d e r defined by N l = S 2 8 / ^ o ) x { x n : 6 1 " 6 ^ < x n < 6 2 + 6 / 5 } ' Then % ^ R r • In f a c t , x e i m p l i e s |x| > x n > 6 1 - 5/j> > r. co We proceed now to construct a domain N w i t h C boundary [1]' such that N Q c N c ^ . Construct a C*(G) f u n c t i o n ijf(x) which s a t i s f i e s f ( x ) = 1 x e N Q f ( x ) = 0 x | ^ . Hence i/(x) i s a C™ mapping of onto the i n t e r v a l [0,1] . Let T c be the set of s i n g u l a r p o i n t s of i|f , i . e . T = (x e N-^  : grad i|r(x) = 0} . By a well-known theorem i n d i f f e r e n t i a l geometry [22], the set ty(T) has Lebesgue measure zero. Hence we can choose a number r Q e (0,1) such that {x : i|f (x) = r Q ] D T = $ .' Let • N = (x e N 1 • : 1 _> * (x) >_ r Q } . Then N has a c"3 boundary, as can be e a s i l y checked, and N c N . I f we define u (x) = 0 f o r x i N , then i t i s o o o easy to show that f u Lu dx < f U-.-tu dx = 0 . N * ~~ N. • ° 48. Hence the s m a l l e s t e i g e n v a l u e V q o f the p r o b l e m Lu = vu i n N , D au = 0 on 3N , |a| = 0,1 i s n o n p o s i t i v e . L e t t denote the d i a m e t e r o f ' N . Put t ' N ° = N . F o r each 0 < t < t choose N such t h a t o t • . •• , N ° = U I T , d i a m e t e r o f N < t , and t < t ' i m p l i e s t e ( o , t 0 ) t h a t N f c c N t , N t-4=N t' . L e t v ( t ) denote the s m a l l e s t ' e i g e n v a l u e of t h e p r o b l e m L u .= v ( t ) u i n N f c , D°u = 0 on 3 1 ^ , |a| = 0,1 . By the m o n o t o n i c i t y p r i n c i p l e and the c o n t i n u o u s dependence o f the e i g e n v a l u e on the domain d i s c u s s e d i n S e c t i o n 3* v ( t ) i s monotone d e c r e a s i n g i n t , and l i m v ( t ) = » . S i n c e t-0+ v ( t Q ) = v Q <_ 0 , t h e r e e x i s t s t * e ( 0 , t Q ] such t h a t v ( t * ) = 0. S i n c e N <= N <= N C= R^ 3 e q u a t i o n (2.3) i s . o s c i l l a t o r y i n G. T h i s completes the p r o o f o f Theorem 2.4. T h i s theorem reduces t o Glazman's theorem ( [ 1 1 ] , p. 104) i f n = 1 and a ^ ( x ) = 1 i n ( 2 . 3 ) * s i n c e f o r G = E n we may take cp = 1 , u Q = 0 i n ( 2 . 5 ) . S e v e r a l o t h e r o s c i l l a t i o n c r i t e r i a can be o b t a i n e d f o r the o p e r a t o r (2.3) by u s i n g d i f f e r e n t o n e - d i m e n s i o n a l o s c i l l a t i o n c r i t e r i a f o r the o p e r a t o r ( 2 . 7 ) . The f o l l o w i n g 49. theorem i s a K n e s e r - t y p e o s c i l l a t i o n c r i t e r i o n . Theorem 2.5.. E q u a t i o n (2.3) i s o s c i l l a t o r y i n G i f f o r s u f f i c i e n t l y l a r g e x and some 6 > 0 ,. P r o o f : The h y p o t h e s i s i m p l i e s t h a t 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 (2.7) i s o s c i l l a t o r y by Glazman's g e n e r a l i z a t i o n o f K n e s e r ' s theorem ([11 ] , p.96). The r e m a inder o f the p r o o f i s s i m i l a r to t h a t o f Theorem 2.5. T h i s theorem c o n t a i n s Glazman's g e n e r a l i z a t i o n ( [ 1 1 ] , p. 96) o f the K n e s e r - H i l l e theorem. Glazman's r e s u l t i s the s p e c i a l case n = 1 , a-^ = 1 o f our theorem. 5- O p e r a t o r s o f A r b i t r a r y Even Order. f ( x n ) - u >_ n \Q{a22 .+ 6 ) x "n ( a 2 = 5 N/2) . The d i f f e r e n t i a l o p e r a t o r to be c o n s i d e r e d i s the o p e r a t o r ( 2 . 1 ) , .namely Lu = (-) m |a|=iB|=m u) - c ( x ) u . The g e n e r a l c o n d i t i o n s on L a r e as i n S e c t i o n 1, e x c e p t t h a t the domain G w i l l be a l l o f E n . I t s h o u l d be p o i n t e d out t h a t t h i s a s s u m p t i o n on G i s n o t e s s e n t i a l , and the t e c h n i q u e 50. used below can be adapted t o c o v e r more g e n e r a l types o f unbounded domains. D e f i n i t i o n s and N o t a t i o n s . L e t X(x) denote, as b e f o r e , the l a r g e s t e i g e n v a l u e o f the c o e f f i c i e n t m a t r i x M_ =' ( a ^ ) as s a±a. d e f i n e d i n S e c t i o n 1. L e t x ( r , e 1 , . . • * e n _ 1 ) = X(x) , c ( r , e 1 , . . . , e n _ 1 ) = c ( x ) , A ( r ) = J x ( r , e 1 , . . . , e n _ 1 ) d n n , ^n f ( r ) = J* c ( r , e 1 , . . . , e n _ 1 ) d n n , ^n where r , 6-^ . • • , 0 n _ ] _ a r e t n e h y p e r s p h e r i c a l p o l a r c o o r d i n a t e s d e f i n e d i n Chapter I , and 0-n i s the s u r f a c e o f the u n i t b a l l m E Lemma 2 . 6 . I f u = u ( r ) i s an m-times d i f f e r e n t i a b l e f u n c t i o n f o r a l l r i n (0,<») , then the f o l l o w i n g i n e q u a l i t y h o l d s : (2 .8 ) , E ( D % ) S < ? m i c r k - m ( u ( k ) ( r ) ) 2 |a|=m k=l k • f o r r > 1 , where u^ = ~~Y u '3 a n c^ m k a r e P o s i t i v e d r c o n s t a n t s , k = l ,2,...,m. 51. n Proof: Let. a = ( a ( l ) , . . ,,a(n)) be a m u l t i - i n d e x , S a ( i ) =m 1=1 Then a(i) (j) a ( l ) - l A ( J + l ) , i i v r '• k = o t i l r . l < i < m , l < j < m , where the standard n o t a t i o n ^ a ( i ) - l ) ( a ( i ) - l ) ( a ( i ) - 2 ) ... ( a ( i ) - k + l )  k 1 X 2 x ... x k i s used. S u c c e s s i v e ' a p p l i c a t i o n of L e i b n i z ' s r u l e y i e l d s the i d e n t i t y (2.9) - D j ^ u ^ = a E X ) pJ ( x ^ r J u ^ + J ) ' / 1 k=l D ? ( i ) u = 1 ' P° (x n.„r)u^ ) . 1 k=l K 1 where P^.(x^,r) are fu n c t i o n s i n x ± and r , and |p£(x.,r)| < A J r ^ 1 5 (J = 0,1,...,m) f o r s u f f i c i e n t l y l a r g e r , where the A^ are constants, k = l,2,...,m. Since D°u = D ^ 1 ^ ® ^ . . . D ^ n ) u . Hence 52. k = l By s u c c e s s i v e a p p l i c a t i o n o f L e i b n i z ' s r u l e and use o f (2.9), we o b t a i n •D°u = S € k ( x 1 , x 2 , . . . , x n , r ) u ^ k ^ where ]^^(x^,x^,. . . ,x^,r) | <_ A^ . r k m f o r s u f f i c i e n t l y l a r g e r , where the A^ . a r e p o s i t i v e c o n s t a n t s , k = l , 2,...,m . Hence k=l K k=l K l k ' J < ? € 2 ( u ( k ) ) 2 + ? [ ( u ( i ) ) 2 + ( u ^ ) 2 ] , ~ k = l K i + J J i , j = l,2,...,m . U s i n g the e s t i m a t e s on €, , we o b t a i n S ( D a u ) 2 < ? m , r k " * ( u k ( r ) ) 2 a(=m k = l K D e f i n i t i o n . F o r each p a i r o f r e a l numbers {a,b} such t h a t 0 < a < b < » , l e t M b be the q u a d r a t i c f u n c t i o n a l d e f i n e d by l£[u] = J b ? [ m v r k - m A ( r ) ( u ( k ) ) 2 - u 2 f ( r ) ] r n - 1 d r a . a^.k=l K w i t h domain c o n s i s t i n g o f a l l u e C m ( a , b ) , where m, and 53. f ( r ) a r e as d e f i n e d above. N o t a t i o n . N b = (x : a < |x| < b} . a 2m P r o p o s i t i o n 2.7. I f u. i s a f u n c t i o n o f c l a s s C (a,b) such t h a t : (1) u ^ (a) = u ^ ^ b ) = 0 , i = 1,2,. . . ,m-l ; (2) M^[u] < 0 ; then the 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 L on N^ i s n o n p o s i t i v e . P r o o f : By C o r o l l a r y 2.3, the 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 L on N^ cannot exceed t h a t of L, , where L, i s d e f i n e d by L x v = ( - ) m _ 2 X(x) D^ av - c ( x ) v ' a | =m L e t v ( x ) = u ( | x | ) . Then ( v ^ v ) b = J [ S X ( x ) ( D a v ) 2 - c ( x ) v 2 ] d x < N b |a|=m a < C n f i [ A ( r ) m k r k - m ( u ( k ) . ) 2 - f ( r ) u 2 ] r ^ d r n a k = l K on account o f Lemma 2.6, where C n i s the s u r f a c e a r e a o f the u n i t sphere i n E 1 1 . Hence (u,L u) <_ 0 by h y p o t h e s i s . The 54. c o n c l u s i o n then f o l l o w s from Courant's p r i n c i p l e . The f o l l o w i n g lemma i s a well-known r e s u l t , i n the c a l c u l u s o f v a r i a t i o n s . I t i s o f t e n c a l l e d the Second Lemma of the C a l c u l u s o f V a r i a t i o n s . Lemma. 2 . 8 . I f u e _C[a;b] , and J B u ( r ) v ( 2 n ) ( r ) dr = 0 a 2 n f o r e v e r y f u n c t i o n v e C (a,b) w h i c h s a t i s f i e s , t he c o n d i t i o n s (A) v ^ ^ ( a ) = v ( x ) ( b ) = 0 , 1 = 0 , 1 , 2 , . . . , 2 n - l , 2 n - l t h e n u ( r ) = T. I . r 1 f o r a l l r e [a,b] , where a r e i=o 1 c o n s t a n t s . P r o o f : L e t cp ±(x) = x 1 , i = 0 , 1 , 2 , . . . , 2 n - l . I f v e C 2 n ( a , b ) whi c h s a t i s f i e s the c o n d i t i o n s (A) above, then i t i s easy to see t h a t J cp i(x)v ( 2 n ) d x = 0 , 1 = 0 , l , . . . , n - l . . a -We can dete r m i n e the c o n s t a n t s l^ , 1 = 0 , l , . . . , 2 n - l , so t h a t the d i f f e r e n c e 2 n - l u ( x ) " s ' ^ i c p i ( x ) = v Q ( x ) w i l l be o r t h o g o n a l t o a l l - the f u n c t i o n s cp^(x) , i = 0,1,2, 2 n - l . To do t h i s we must s o l v e the f o l l o w i n g system f o r c j c p ^ c p . ) + .c 1(cp 1,cp.) + ... + c 2 n - l ( ( p 2 n - l ' c p i ) = (u - v o , c p i ) , ( i = 0 , 1 , • • • ,2n-l) the d e t e r m i n a n t o f w h i c h i s g i v e n by b-a (b-a)' (b-ar (b-a) 3 3 (b-a) 2n 2n (b-a) 2n+l . 2n+l /, \2n /, v2n+l (b-a) (b-a) n 2n+l, /. An-1 (b-a) 4 n - l :(b-a) (2n)' 1 1 1 2 1 2 1 3 1 1 2n , 2n+l _1_ 2n 2n+l 1 4 n - l ( b _ a ) ( 2 n ) 2 [1121 ( 2 n - l ) ! ] 3 • (2n) ! (2n+l) ! . . . ( 4 n - l ) - ! 56, x x L e t V ] L ( x ) = J v ( t ) d t , and v f c ( x ) = J v k _ 1 ( t ) d t , a a k = l , 2 , . . . , n . Then v ^ e C 2 n ( a , b ) , v ^ = V q , and v ^ (a) = v ^ (b) = 0 , i = 0,1,•••,2n-l . . ' T h i s f o l l o w s from the f a c t t h a t (v ,cp^) = 0 , i = 0 , 1 , . . . , 2 n - l . Then b J u ( r ) v ( r ) dr = 0 a by h y p o t h e s i s . T h i s i m p l i e s t h a t b 2 n - l p J ( u ( x ) - E £ cp± )d = 0 . a i=o I t f o l l o w s from the above equation,- and the h y p o t h e s i s t h a t u ( x ) € C[a,b] , t h a t 2 n - l u ( x ) = E • l ± x x 1=0 Lemma 2.9- I f v = v ( r ) i s a f u n c t i o n d e f i n e d on the i n t e r v a l [a,b] , h a v i n g the p r o p e r t i e s ( i ) v ( r ) e C m - 1 [ a , b ] s ( i i ) v ( m ) ( r ) € L 2 ( a , b ) ; ( i i i ) v ^ 1 ? ( a ) ' = v ^ ^ ( b ) = 0 , 1 = 0,1,2,...,m-l ; Orr\ then f o r any 6 > 0 t h e r e e x i s t s a f u n c t i o n u e C (a,b) 5 7 . w h i c h s a t i s f i e s the c o n d i t i o n s u ^ ( a ' ) = u ^ - ^ b ) .= 0 , i = 0 , 1 , 2 , . . . , 2 m - l , and |M*(u) - M * ( v ) | < 6 . " P r o o f : L e t H denote the space o f a l l f u n c t i o n s v s a t i s f y -i n g the c o n d i t i o n s ( i ) , ( i i ) , ( i i i ) , above. F o r each p a i r o f f u n c t i o n s v,w i n H , d e f i n e • ' (v,w) = J b v ( m ) ( r ) w ( m ) ( r ) dr , and a l | v | | 2 = J b ( v W ( r ) ) 2 d r . a • • I t i s c l e a r t h a t ( , ) i s an i n n e r p r o d u c t on H . We s h a l l show t h a t H w i t h the i n n e r p r o d u c t d e f i n e d above i s a H i l b e r t space. L e t ( v i ( r ) , i = 1,2,...} be a Cauchy sequence i n H. w i t h r e s p e c t t o the above norm, i . e . , l i m || v. - v . ||2 =• l i m f ( v ( m ) - v ( m ) ) 2 d r = 0 . fm) S i n c e the sequence (vj* ( r ) , i = 1,2,. . . } i s a Cauchy sequence i n the H i l b e r t space L 2 ( a , b ) , i t converges to a f u n c t i o n h e L 2 ( a , b ) . D e f i n e v < r ) = 7 ^ T T jV - t ) m - 1 h ( t ) d t . a.. 58. Then v e C m _ 1 ( a , b ) , v ^ m ) ( r ) = h(r). and v(k)c(a) = 0 (k = 0',1,. . . , m - l ) . A l s o Wfr.\ - J^f^W = | J Y v ( k + l ) ( t ) _ v ( k + l ) a 1 • = |jVt)(v{k+2)(t) - v(k+2)(t))dt! a r ^ j j l f ( r - t l ^ l v f V ) - v ( k + 2 > ( t ) ) d t | (*) ! v ^ J ( r ) - v ^ ( . r ) ! = IJ* v ^ ^ ( t ) - v ^ + ^ ( t ) d t |  1Y a by the Cauchy-Schwartz i n e q u a l i t y and i n t e g r a t i o n by p a r t s . Hence the sequence { v | k ^ ( t ) , i = 1 , 2 , . . . } i s u n i f o r m l y con-v e r g e n t t o v ^ k ^ ( r ) f o r each k = 1 , 2 , . . . ,m - 1 . I n p a r t i c u l a r v ^ k ^ ( b ) = l i m v | k ^ ( b ) = 0 . I t f o l l o w s from the above c o n s i d e r -i—»oo a t i o n s t h a t v e H . We have shown t h a t e v e r y Cauchy sequence i n H converges t o some f u n c t i o n i n H . L e t H denote the s e t o f a l l r e a l v a l u e d f u n c t i o n s o f c l a s s C 2 m ( a , b ) s a t i s f y i n g u ^ k ^ ( a ) = u ^ k ^ ( b ) = 0 , k = l , 2 , . . . , 2 m - l . We s h a l l show t h a t H i s dense i n the H i l b e r t space H . I t i s enough t o show t h a t i f v e H i s o r t h o g o n a l t o H then v i s i d e n t i c a l l y z e r o . Suppose (v,u) = 0 f o r a l l u e § . I n t e g r a t i o n by p a r t s y i e l d s 0 = v ( m ) ( r ) u ( m ) ( r ) d r = J ^ r ) u ( 2 m ) ( r ) d r a I t f o l l o w s from Lemma 2.8 t h a t v ( r ) = 0 L e t v b e a g i v e n f u n c t i o n s a t i s f y i n g the hypo-theses ( i ) , ( i i ) , ( i i i ) , T h e n i t f o l l o w s from the above con-s i d e r a t i o n t h a t t h e r e e x i s t s a sequence u n e H such t h a t l i m || v - u ||2 = 0 . U s i n g the i n e q u a l i t y (*) we can show n-*eo t h a t u ^ ; converges t o v^ u n i f o r m l y f o r k = 0 , 1 , . . . ,m - 1 Then i t i s easy to see t h a t l i m |M Q[u 1 - M [ v ] | = 0 . a n a n-»co Theorem 2.10. The o p e r a t o r L d e f i n e d by Lu = ( - ) m S D a ( a ftDPu) - c ( x ) u |a|=|p|=m a P i s o s c i l l a t o r y i n F/1 i f (1) — a f , A ( r ) r n " 1 d r i s bounded above f o r a l l s u f f i c i e n t l y l a r g e b , and (2) J c(x) d x = + » , | x | > o where A ( r ) i s the l a r g e s t e i g e n v a l u e o f the c o e f f i c i e n t m a t r i x M c , as d e f i n e d i n S e c t i o n 1. Pr o o f : I t i s enough t o show t h a t f o r a r b i t r a r y a > 0 , L has a n o d a l domain i n the complement o f the b a l l S = [x e E n j x | < a] . We c o n s t r u c t a f u n c t i o n u as f o l l o w s : 6o. t m _ L e t w(t) = K J s (,1-s) ~ ds where K i s chosen o so t h a t w ( l ) = 1 . L e t u be d e f i n e d by u ( r ) = 0 0 < r <_ a u ( r ) = w ( r ~ a ) a < r < 2a x v a — u ( r ) = 1 2a < r <_ ^ u ( r ) = w ( ^ 2 l ) | < r _< b u ( r ) = 0 r > b , where b w i l l be chosen l a t e r so t h a t b/2 > 2a , and M b [ u ] < 0 . Then ^ " - ^ F A ( P ) ( u W j V ^ d r a k = l r ~ _ P 2 a v m k A ( r ) _1_ / J k ) , r - a _ u 2 n-1 m b m. „2k / , N „ _ 0 _ k = l b/2 r K b ^ K D where K-^ K 2 a r e c o n s t a n t s . The l a s t i n e q u a l i t y f o l l o w s from the f a c t t h a t the f i r s t i n t e g r a l on the l e f t hand s i d e o f the above i n e q u a l i t y i s bounded. Moreover, 61. b 2 a J u 2 f ( t ) t n " 1 d t = J ( W ( ^ ) ) 2 f ( t ) t n - 1 d t a a + f ( t ) t n " 1 d t + ^ ( w ( 2 b ^ ) ) 2 f ( t ) t n - 1 d t . 2a b/2 2a ^ L e t J (w( :*~*) ) 2 f ( t ) t n 1 d t = , where i s a c o n s t a n t w hich depends o n l y on a . Choose b Q > 4a l a r g e enough so t h a t b , b/2 „ -, (2.10) K-, + K Q / b m + 1 f A ( r ) r n _ 1 d r - K , - f f ( r ) r n ' 1 d r < -1 d b/2 9 ' 2a f o r a l l b >_ b Q . T h i s i s - p o s s i b l e by h y p o t h e s i s ( 2 ) . L e t y ( r ) = f t n " 1 f ( t ) d t ; b /2 o then l i m y ( r ) = +«• f o l l o w s from h y p o t h e s i s ( 2 ) . Choose b/2 r-*os i n the d e f i n i t i o n o f u t o be the l a s t r o o t o f y ( r ) . Then b/2 >_ b / 2 . , y ( r ) > 0 , f o r a l l r > b/2 , and C ( w (^ ) , 2 f ( r ) r n" l d r - C ( w ( ^ ) ) S ^ y dr " ' a = 0 + 4/b 'J>B/ w w ' ( 2 b ~ 2 r ) y ( r ) d r > 0 62. 2"b — 2 x* s i n c e y ( r ) > 0 f o r r > b/2 , and w ' ( — ^ — ) i s p o s i t i v e by the d e f i n i t i o n o f w ( t ) . From the above c o n s i d e r a t i o n s we o b t a i n M b [ u ] < 0 cl The f u n c t i o n u s a t i s f i e s the c o n d i t i o n s i n Lemma 2.9. 2m Hence a f u n c t i o n v e x i s t s w h i c h b e l o n g s t o C (a,b) and s a t i s f i e s the c o n d i t i o n s v ^ ( a ) = v ^ \ b ) = 0, i = l , 2 , . . . , m - l , M b [ v ] < 0 . Then the 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 L on N b i s a n o n p o s i t i v e by P r o p o s i t i o n 2.7. L e t N g = {x : a + 6 < _ | x | < _ b ] , and l e t P- 0(0 be the s m a l l e s t e i g e n v a l u e o f the pr o b l e m Lu = u ( 6 ) u i n N ^ + 6 , D°u = 0 on S N ^ + 6 , |ot | <_ m-1 Then \x (0) < 0 and p. (6) > u (0) . There e x i s t s 6 such o — o — o o t h a t M- G( 6 0) = 0 by the m o n o t o n i c i t y p r i n c i p l e f o r e i g e n -v a l u e s d i s c u s s e d e a r l i e r i n t h i s c h a p t e r , and t h e domain N a + 6 i s a n o d a l domain f o r L . T h i s completes the p r o o f o of Theorem 2.10. 63. C o r o l l a r y 2.11. L e t X(x) be bounded i n E by some number . Then the o p e r a t o r (2.1) i s o s c i l l a t o r y i n E 1 1 i f n <_ m+1 and ' J c ( x ) d x = +eo . |x|>o P r o o f : The p r o o f i s an immediate consequence o f Theorem 2.10. For n = 1 , m = 1 and c ( x ) >_ 0 C o r o l l a r y 2.11 g i v e s the c r i t e r i o n e s t a b l i s h e d b y A. W i n t e r [29]. For n = 1, Theorem 2 . 1 0 ' g e n e r a l i z e s a theorem o f Glazman ( [ l l ] , p. 104) to e q u a t i o n s w i t h v a r i a b l e c o e f f i c i e n t s . O ther o s c i l l a t i o n c r i t e r i a may be o b t a i n e d by com-p a r i n g the' o p e r a t o r L w i t h the o p e r a t o r L q d e f i n e d by m -k , ,k T , i \m d . / \ k-m d r. L u = (- ) Em, — ^ A ( r ) r — ^ u - f u . k = l dr dr 6. N o n o s c i l l a t i o n C r i t e r i a . I n t h i s s e c t i o n we d e r i v e s u f f i c i e n t c o n d i t i o n s f o r the o p e r a t o r d e f i n e d by (2.1) t o be n o n o s c i l l a t o r y i n an unbounded domain G i n n - d i m e n s i o n a l E u c l i d e a n space The boundary oG o f G i s supposed t o have a p i e c e w i s e con-t i n u o u s u n i t normal v e c t o r a t each p o i n t . F u r t h e r m o r e , we assume t h a t G i s c o n t a i n e d i n a h a l f space. O t h e r w i s e no s p e c i a l assumptions are needed regarding the shape of G D e f i n i t i o n . The operator L i s s a i d to be n o n o s c i l l a t o r y i n G i f f i t has no nodal, domain i n the complement of S r = {x : |x| <_ r} fl G f o r a l l r > 0 in Lemma 2.12. For any r e a l f u n c t i o n y €'C (0,eo) ; 2 . i i ) I z J f dx < — ^ — p J ( ^ ) 2 d x , o x 2 m - [(2m-l) ! ! ] 2 o dx m' where ( 2 m - l ) l ! = .(2m-1) (2m-3) . ..1 . Proof: f ^p-^-dx = 2/" x " 2 m d x y ( t ) | f dt . X o o -a/V<t)$dt f x- 2 max 0 t fc . co 2m-l o . • a z Hence by means of the Cauchy-Schwartz i n e q u a l i t y we o b t a i n o o Hence f x " 2 m y 2 ( x ) d x < J t — f (|Z)2 x 2 m - 2 d x o N ~ ( 2 m - l ) 2 o d x Repeating the above proceedure f o r the i n t e g r a l on the l e f t - h a n d 65. s i d e of the l a s t i n e q u a l i t y and so on,, we o b t a i n I n e q u a l i t y ( 2 . 1 1 ) . N o t a t i o n . Y ( X ) denotes the s m a l l e s t e i g e n v a l u e o f the c o e f f i c i e n t m a t r i x M s L e t C + ( x ) = max { c ( x ) , 0 } , where x = ( x - ^ X g , . . . , x n ) . For s i m p l i c i t y , we a r e g o i n g t o assume t h a t G c {x : x^ > 6 > 0} Theorem 2.13- L e t Y ( X ) be bounded below i n G by some p o s i t i v e number y such t h a t i • ? i J" ^ 6 ° ° ^ > Y o J , ? ( D a u ) 2 d x , !-a| = |6|=m N a p- ° N |a|=m where N i s any bounded domain, and u ^ 0 . Then the o p e r a t o r .L i s n o n o s c i l l a t o r y i n G i f 2m-1 ~ a •2 l i m sup t ^ J g ( t ) d t < ^ , t ~* oo t where g ( t ) = sup ( C + ( x ) , x = ( t , x ) e G] and « m = l 2 m - l l l l . P r o o f : Suppose to the c o n t r a r y t h a t L i s o s c i l l a t o r y i n G Then 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 r o f (2.1) w i t h a n o d a l domain N r c G n [x : fxf > r ] f o r a l l r > 0 . Then 66. 2.12) 0 = J u r Lu dx> Y J [ E ( D V ) 2 - ° M u 2]dx N • N • ' a ' = m ° r r r Let u r be the extension of u / to a l l of E n which i s i d e n t i c a l l y zero outside N r . Then d u „ g(x1 ) (2.13) I K T - n f ) 2 - ^ = N 3 X 1 ° r However J . [ ( T ^ r } " ~ Y ~ U r ] d x l d x  E n - 1 6 dx^ To 00 03 J s g ( x 1 ) u 2 d x 1 = 2 J s g ( x 1 ) d x 1 ; f i u d y i 2 I 6 ^ d y i I v § ( X l ) d X l u ±2 V * ^ ' ps=T y f " 1 ^ By hypothesis and the above inequality, we get o 2 co 2C£ o 3 2 a 2 « 2 i . ( ^ - ) 2 i 1 y l 67. where the l a s t i n e q u a l i t y f o l l o w s from Cauchy-Schwartz's i n e q u a l i t y . Prom the l a s t i n e q u a l i t y and Lemma 1, we have g ( x 1 ) L - y ~ u d x i ^ J . h ' d y i ' o To 6 3y^ Hence -,m 0 g(x-, ) 0 N 1 0 r which c o n t r a d i c t s ( 2 . 1 2 ) . This completes the proof of the theorem. 68. BIBLIOGRAPHY [ l ] S. Agmon, Lectures on E l l i p t i c Boundary Value Problems, Van Nostrand, P r i n c e t o n , 19&5-[2] W. A l l e g r e t t o and C. A. Swanson, Sturm Comparison Theorems f o r E l l i p t i c I n e q u a l i t i e s , B u l l . Amer. Math.Soc. 75 (1969), PP. 1318-1321. [3] W. A l l e g r e t t o , Comparison and Oscillation-Theorems f o r E l l i p t i c 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, 19^9. [4] L. Bers, F. 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 Summer Seminar, U n i v e r s i t y of Colorado, Boulder, 1957. [5] G. D. B i r k h o f f and M. R. Hestenes, N a t u r a l I s o p e r i m e t r i c Conditions i n the Calculus of V a r i a t i o n , Duke Math. J. 1 (1935), pp. 198-286. [6] 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-[7] 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. [8] J. B. Diaz and J . R. McLaughlin,- Sturm Comparison Theorems f o r Ordinary and P a r t i a l D i f f e r e n t i a l Equations, B u l l . Amer. Math. Soc. 75 (1969), pp. 335-339-[9] J- B. Diaz and J . R. McLaughlin, Sturm Separation and Comparison Theorems f o r Ordinary and P a r t i a l . D i f f e r -e n t i a l Equations, A t t i Accad. Naz.. L i n c e i Mem. • C l . S c i . F i s . Mat. Natur. Sez. I , Ser. V I I , V o l . IX (1969), pp. 135-194. [10] F. R. Gantmatcher, The Theory of M a t r i c e s , V o l . I , Chelsea, New York, 1959. 6 9 . [11] 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 , 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 , Danel Davey and Co., New York, 1 9 6 5 . [12] P. Hartman and A u r e l Wintner, On a Comparison Theorem f o r S e l f - a d j o i n t P a r t i a l D i f f e r e n t i a l Equations of. E l l i p t i c Type, Proc. Amer. Math. Soc. 6. (.1955), pp. 8 6 2 - 8 6 5 . [13] V. B. Headley, E l l i p t i c Equations of J Order 2m, J . Math. Anal. Appl. 25 ( 1 9 6 9 ) , pp. 5 5 8 - 5 6 8 . [ l 4 ] V. B. Headley, A Monotonicity Theorem f o r Eigenvalues, P a c i f i c J . Math. 30 ( 1 9 6 9 ) , pp. 6 6 3 - 6 6 8 . [15] 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. V o l . 2 7 , No. 3 . ( 1 9 6 8 ) , pp. 5 0 1 - 5 0 6 . [ l 6 ] E. H i l l e , N o n - o s c i l l a t i o n Theorems, Trans. Amer. Math. Soc. 64 ( 1 9 4 8 ) , pp. 2 3 4 - 2 5 2 . [17] K. K r e i t h , A Strong Comparison.Theorem f o r S e l f - a d j o i n t E l l i p t i c Equations, Proc. Amer. Math. Soc. 19 ( 1 9 6 8 ) , pp. 9 8 9 - 9 9 0 . " [ l 8 ] L. M. Kuks, Sturm's Theorem and O s c i l l a t i o n o f . S o l u t i o n s of S t rongly E l l i p t i c Systems, Soviet Math. Dokl. 3 ( 1 9 6 2 ) , pp. 24 - 2 7 . [19] W. Leighton, On S e l f - a d j o i n t D i f f e r e n t i a l Equations of Second Order, J. London.Math. Soc. 27 ( 1 9 5 2 ) , pp. 37-47 [20] S. G. M i k h l i n , The Problem of the Minimum of a Quadratic Function, Holden-Day, San F r a n c i s c o , 1 9 6 5 . [21] M. Morse, A G e n e r a l i z a t i o n of the Sturm Separation and Comparison Theorems i n n-space, Math. Ann. 103 ( 1 9 3 0 ) , pp. 5 2 - 6 9 . [22] A. Sard, The Measure of the C r i t i c a l Values.of D i f f e r e n t i a l Maps, B u l l . Amer. Math. Soc. 48 ( 1 9 4 2 ) , pp. 8 8 3 - 8 8 5 . 7 0 . [23] C. A. Swanson, Comparison Theorems f o r E l l i p t i c D i f f e r -e n t i a l Systems, P a c i f i c . J . Math. ( 1 9 7 0 ) . [24] 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 Nonlinear M a t r i x D i f f e r e n t i a l I n e q u a l i t i e s , Proc. Amer. Math. S o c , to appear. [25] 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. [26] 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, Canad. Math. B u l l . 12 ( 1 9 6 9 ) , pp. 2 7 5 - 2 8 0 . [27] E. C. Tomastik, O s c i l l a t i o n of Nonlinear M a t r i x D i f f e r -e n t i a l Equations of Second Order, Proc. Amer. Math. Soc. 19 ( 1 9 6 8 ) , pp. 1 4 2 7 - 1 4 3 1 . [28] Rudolf Vyborny, Continuous Dependence of Eigenvalues on the Domain, Lecture S e r i e s No. 42, U n i v e r s i t y of Maryland, The I n s t i t u t e f o r F l u i d Dynamics and Appli e d Mathematics, 1 9 6 4 . [29] A. Winter, A Condition of O s c i l l a t o r y S t a b i l i t y , Quart. Appl. Math., 7 ( 1 9 4 9 ) , pp. 115-117. 

Cite

Citation Scheme:

        

Citations by CSL (citeproc-js)

Usage Statistics

Share

Embed

Customize your widget with the following options, then copy and paste the code below into the HTML of your page to embed this item in your website.
                        
                            <div id="ubcOpenCollectionsWidgetDisplay">
                            <script id="ubcOpenCollectionsWidget"
                            src="{[{embed.src}]}"
                            data-item="{[{embed.item}]}"
                            data-collection="{[{embed.collection}]}"
                            data-metadata="{[{embed.showMetadata}]}"
                            data-width="{[{embed.width}]}"
                            async >
                            </script>
                            </div>
                        
                    
IIIF logo Our image viewer uses the IIIF 2.0 standard. To load this item in other compatible viewers, use this url:
http://iiif.library.ubc.ca/presentation/dsp.831.1-0080501/manifest

Comment

Related Items