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Comparison and oscillation theorems for elliptic equations Allegretto, Walter 1969

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COMPARISON AND OSCILLATION FOR E L L I P T I C  THEOREMS  EQUATIONS  by  WALTER ALLEGRETTO B.A.Sc, University  o f B r i t i s h Columbia,  A THESIS SUBMITTED I N PARTIAL FULFILMENT THE REQUIREMENTS FOR THE DEGREE OF  DOCTOR OF PHILOSOPHY  •  •  in  t h e Department  '  of Mathematics  We a c c e p t t h i s t h e s i s a s c o n f o r m i n g to the required standard.  The U n i v e r s i t y o f B r i t i s h April  1969.  Columbia  19^5.  OF  In p r e s e n t i n g an  this  thesis  in partial  advanced degree a t the U n i v e r s i t y  the  Library  I further for  shall  make i t f r e e l y  agree that  permission  s c h o l a r l y p u r p o s e s may  by  his representatives.  of  this  written  thesis  Date  A^Cil  f o r extensive  It i s understood gain  M 3 . 4 L e Vt ^ i n cS.  ^2  )  Columbia,  I agree  Columbia  shall  that  that  Study.  copying of this  be g r a n t e d b y t h e Head o f my  for financial  The U n i v e r s i t y o f B r i t i s h V a n c o u v e r 8, Canada  of B r i t i s h  a v a i l a b l e f o r r e f e r e n c e and  permission.  Department o f  f u l f i l m e n t of the requirements f o r  thesis  Department o r  copying or p u b l i c a t i o n  n o t be a l l o w e d w i t h o u t  my  ii. Thesis Supervisor:  C. A. Swanson.  ABSTRACT  New c o m p a r i s o n a n d S t u r m - t y p e t h e o r e m s a r e e s t a b l i s h e d which enable us t o extend criteria  to:  ( l ) non-self-adjoint operators, (3)  operators,  known o s c i l l a t i o n a n d  nori-oscillation (2)  quasi-linear  f o u r t h order operators o f a type n o t previously-  considered. Since the c l a s s i c a l  principle  o f C o u r a n t 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 c o m p a r i s o n t h e o r e m s 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  of the operators i n question.  eigenvalue  A description of the behaviour'  o f t h e e i g e n v a l u e as t h e domain i s p e r t u r b e d  i s also given f o r  s u c h o p e r a t o r s b y t h e u s e o f S c h a u d e r ' s "a p r i o r i " e s t i m a t e s . The  Sturm-type theorems a r e proved  by t o p o l o g i c a l  argu-  ments and extended t o q u a s i - l i n e a r a s w e l l as t o n o n - s e l f - a d j o i n t operators. The  f o u r t h order operators considered a r e of a type  does n o t y i e l d  forms i d e n t i c a l  t o those a r i s i n g  i n second  problems. Some e x a m p l e s i l l u s t r a t i n g t h e t h e o r y a r e g i v e n .  which  order  Iii. TABLE OF CONTENTS page CHAPTER I Second Order N o n - s e l f - a d j o i n t  Equations.  1.  Introduction  1  2.  R e l a t i o n B e t w e e n t h e Two T y p e s o f O s c i l l a t i o n  2  3.  A C o m p a r i s o n Theorem f o r L .  10  4.  Oscillation  28  Criteria for L .  CHAPTER I I Quasilinear E l l i p t i c  Equations.  1.  Introduction  34  2.  A S t u r m Theorem a n d a C o m p a r i s o n T h e o r e m f o r L  35  3-  Oscillation  43  4.  Bounds on t h e F i r s t E i g e n v a l u e o f  a n d n o n - o s c i l l a t i o n Theorems  48  L  CHAPTER I I I Fourth  Order  Equations.  1.  Introduction  2.  C o m p a r i s o n Theorems f o r F o r m a l l y Operators  3-  Oscillation Operators  4.  -  Theorems f o r F o r m a l l y  Non-oscillation Criteria.  BIBLIOGRAPHY  Self-adjoint  55 57  Self-adjoint 64 67 70  ACKNOWLEDGEMENTS  The a u t h o r w o u l d l i k e  t o t h a n k D r . C. A. Swanson f o r  s u g g e s t i n g t h e 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 e n c o u r a g e m e n t like  throughout i t s preparation.  t o extend h i s appreciation  constructively  criticized  He w o u l d  t o D r . C. W. C l a r k  the d r a f t form o f t h i s  who r e a d a n d  work.  The g e n e r o u s 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 Council  o f Canada and o f t h e U n i v e r s i t y  gratefully  acknowledged.  also  Research  o f B r i t i s h Columbia i s  CHAPTER  I  SECOND ORDER NON-SELF-ADJOINT EQUATIONS  1.  Introduction. In  L  t h i s c h a p t e r t h e second order l i n e a r  operator  defined by  (1.1)  Lu  -  •n n S D ( a . ,D.u) + E b.D.u + c u i,j=l j=l 1  will  1 J  0  J  E  .  11  Sturm theorem as w e l l These w i l l  Our main r e s u l t s  of the n-dimensional  are a quite  E  11  will  second order  be denoted b y  (l.l).  coefficients  a. . w i l l  furthermore the matrix  definite  symmetric i n R .  (a. -(x))  By 5 ) ( L Q ) 3  collection of a l l real functions of class denotes any subdomain o f  Definition  R .  in  x^ .  f o r which  u  positive  we s h a l l mean t h e C ( f i ) 0 C(0)  real.  ?)(L,0.)  D^  The  w i l l be t a k e n  The c o e f f i c i e n t s  A (classical) solution u  and  n  a l w a y s b e assumed r e a l a n d o f c l a s s  C"'"(R) ;  a l s o be t a k e n  operators.  x = (x^. ..,x )  s i g n i f y d i f f e r e n t i a t i o n with respect to  function  general  enable 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  Points of  will  R  as a comparison theorem f o r o p e r a t o r  w h i c h a r e known f o r s e l f - a d j o i n t  will  J  be c o n s i d e r e d i n u n b o u n d e d d o m a i n s  E u c l i d e a n space  Q  elliptic  of  Lu = f  Lu(x)= f(x)  -b . J  in  ,  where and  n  f o r every  c  i s a x  in • Q . Definition iff  A bounded domain  N c R  i s a n o d a l domain o f  there exists a n o n - t r i v i a l solution u = 0  such t h a t Notation  R  on  = R 0 {x : x € eF  0 = closure of  f o r every  Definition iff  R  r  2.  Q  and  |x| > r }  i n the  topology.  L  has a n o d a l domain i n  .  r  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 2 (Osc  r > 0 , every s o l u t i o n of  L u = 0:  in  R  2)  has a  such t h a t  L  h a s no n o d a l d o m a i n s i n  R  r  .  R e l a t i o n B e t w e e n t h e Two T y p e s o f O s c i l l a t i o n .  fact  s h a l l now  Osc 2  show t h a t i f L  is  Osc 1 , t h e n i t i s i n  u n d e r m i l d 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  and none a t a l l on t h e n a t u r e o f t h e b o u n d a r i e s involved. domain o f  S p e c i f i c a l l y , we  N .  I f the c o e f f i c i e n t s  i s uniformly e l l i p t i c k > 0  such that  Lu = 0  of  LV.  of the regions  s h a l l show t h a t i f N  L , then every s o l u t i o n of  somewhere i n  and  R  1)  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  r > 0  We  L  N  .  Definition exists  r > 0 ,  ^The  f o r every  zero i n  in  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  Definition iff  Lu = 0  of  dN . 1  r  u  L  in  i s a nodal N  must v a n i s h  are continuous i n  there, i . e . , there e x i s t constants  N  and m > 0  3. n S a i,j=l for  every  n _ Z 57 , i=l  (x) U , > m J 1  | a . , ( x ) | < k , |b ( x ) | < k ~  1  5 = (^,...,§ ) n  1  1  1  and f o r every  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  x  in  N , our r e s u l t  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 that  N  Assume  that . L  obeys t h e above c o n d i t i o n s  i s a n o d a l domain o f t h e o p e r a t o r  function  v  i n ^}(L,N)  somewhere i n  such that  L .  Lv = 0  in  Then  and  every  N , must  vanish  N . \  Proof: in  I f t h i s w e r e n o t t h e . c a s e , we c o u l d (L,N)  generality, function define  such that v > 0  such that  Lv = 0  N N .  Let  Lu = 0  N  and  u = 0  w = ^  .  w  in by  a n d a g a i n we may assume L(vw)  v  and, w i t h o u t l o s s o f  everywhere i n  a new f u n c t i o n  ^)(L,N)  in  find a function  u  be a on  dN .  Then c l e a r l y  w > 0  = L ( u ) = 0- i n  non-trivial We  w  somewhere i n  i sin N .  Now  N_ ,  but L(vw)  = L(v)w -  n n n s D ( a . -D.w)v + 2 ( v b . - 2E a. .D.v)D.w. i,j=l j=l 1=1 1  1 J  J  J  1 J  1  J  Hence n - S D. ( a . .D.w) i,j=l 1  1 J  J  n n D.v + £ (b . - 2 S a. , ~ - )D,w = 0 j=l i=l J  1 J  v  in  N .  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 H o p f maximum p r i n c i p l e  [ 2 , p. 1 5 0 ]  as  w = 0  must  on  3N .  The c o n t r a d i c t i o n p r o v e s  that  v  4.  vanish  somewhere i n • We  shall  T h e o r e m 1.2  N .  show more  Assume t h a t  F u r t h e r m o r e assume t h a t bounded i n  N  in  N must v a n i s h  ( x ) >_ m > 0  c  and  N .  in  R .  are a ^ > y > 0 Lv = 0  *  elliptic  i = l,...,n  N , a s may b e s e e n b y c h o o s i n g a s u i t a b l e  extension  L  Then e v e r y s o l u t i o n o f  f o r every  This  b.  satisfies  i s uniformly  of uniform e l l i p t i c i t y . an  y .  somewhere i n  Note\that i f L  in  i s any n o d a l domain o f  a n d t h a t one o f t h e  N , f o r some c o n s t a n t  a ^  N  the c o e f f i c i e n t s  in  fact  generally:  i n  N , then i n  and f o r e v e r y §  x  i n the d e f i n i t i o n  shows t h a t Theorem 1.2 i s i n f a c t  o f P r o p o s i t i o n 1.1.  To p r o v e T h e o r e m 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 b e shown:  o  P r o p o s i t i o n 1.3 L(w)  .'•> 0  Assume  in  N .  be minima o f  w .  Proof:  If  w(x ) = 0 o : ? s  the matrix  that at  x  Q  i s a function i n  Then t h e p o i n t s  a n  X J —J.  Since  w  of  W  i s a minimum o f , ij( o) \ x  >  C (N)  where  w  in  and  w = 0  cannot  N . then  0  • •  ( a . . ( x ) ) i s p o s i t i v e d e f i n i t e , we may  i t I s diagonal  w i t h p o s i t i v e elements.  assume  Then  JU..^) which i s impossible, for  every  since  D  ± 1  • j_i ( D  u  U(X ) < 0 0  x  ) 1.  0  0  a  ^  t  ^  e  minimum  Q  i = l,...,n .  The n e x t Lemma, t h e k e y p a r t i n t h e p r o o f will  X  be p r o v e d b y t h e u s e o f t h e c l a s s i c a l  o f T h e o r e m 1.2,  Schauder " c o n t i n u i t y "  method [ J ] ,  Lemma 1.4  Assume t h a t  that:  u,v,w  are functions i n  x  (a)  L u >_ 0., u  (b)  w > 0  (c)  L v >_ 0  Then  v  Proof:  somewhere n e g a t i v e  i n N ,  Lw > 0  i n N ,  must v a n i s h  on  BN  i n N  Then, s i n c e  there  exists ,  w > 0 X  q  and  i n N  i n N and  and  u i s  a > 0 such  L(aw + u) > 0 •  that  i n N .  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 (x) = aw(x) + t u ( x ) + ( l - t ) v ( x ) fc  T  u = 0  v "somewhere p o s i t i v e i n N .  aw(x ) + u ( x ) = 0 \Q' ^O We now d e f i n e  i n N ,  somewhere i n N .  Assume n o t .  somewhere n e g a t i v e ,  Let  ( L j N ) and  for x e N  and  denote t h e s e t : T = { t : w,  vanishes  somewhere i n N} fl [ 0 , 1 ] .  t e E"  3  We  s h a l l now  show t h a t  T  i s a non-empty  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 ness o f  [0,1]  this will  [0,1] .  mean t h a t i n f a c t  0 € T , which i s c l e a r l y i m p o s s i b l e as First, let  T  i s non-empty  {t.}!° -. 1 1=1  since  there exist  x: e N 1  T  such t h a t  By t h e c o n n e c t e d -  T = [0,1]  aw + v > 0  1 e T .  be a sequence i n  L  s e t w h i c h i s b o t h open  in  To p r o v e  with  and hence  T  N .  is. closed,  l i m t. = t . l o  w, (x.') = 0 . z 1  As  Then  N  i s compact,  ±  we may  a s s u m e , 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  point ^  x^ e N o  such that  l i m x. = x . l o  Now,  '  a  we a l s o h a v e t h e  estimate: |w  (x)|< |t.- t I  (x) - w  t  i and t h e r e f o r e In  +  |v(x)|})  {w^ }*_-]_ i ~  converges u n i f o r m l y i n  N  to  w, o  the i n e q u a l i t y :  l t w  the  l|u(x)|  o  o  (  X  first  J ' - '  w  t ( o^ " o x  w  t ( o  x  i ^  '  +  w  t^ i^ " ~ t ^ i^' o i x  w  x  +  '  t e r m on t h e r i g h t hand s i d e t e n d s t o z e r o w i t h  w  t ( i  i  x  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. open, l e t * ' Clearly  t  w,  Hence o  € T  w,  (x ) = 0 . F i n a l l y , t o prove o ° a n d l e t w, have a z e r o a t x in t o o  must have a z e r o i n  T  is  N .  N , a s i t i s p o s i t i v e on  dN .  0  Since w,  L(w,  o  ) > 0  and  w^  o  > 0  on  oN. , b y P r e p o s i t i o n  must h a v e b o t h p o s i t i v e a n d n e g a t i v e v a l u e s i n  N .  1.3 Now,  i ^  7..  o  o (^|  Hence N  w^  also w i l l  i f | t - ,t |  of-  N j  w  have a zero i n N . aw + v = 0  the c o n t r a d i c t i o n assumption  By t h e connectedness We h a v e t h u s a r r i v e d a t  somewhere i n N .  must be f a l s e a n d t h e r e f o r e  C o r o l l a r y 1.5 §)(L,N)  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  i s t a k e n s m a l l enough.  will  t  C|u(x)| + | v ( x ) | } ) .  v = 0  Our o r i g i n a l  somewhere i n N .  Assume' that t h e r e e x i s t s a function Ef > 0  suchNshat  ( o r < 0)  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  i n N .  , then  v  f i n  If u  and  v  m u s t v a n i s h some-  where i n N . Proof: that 3 > 0 w(x)  A g a i n assume n o t . a > 0  v > and  i n N .  P(^§  Then t h e r e e x i s t s a c o n s t a n t Choose a c o n s t a n t  < a..  = v(x) + pf(x) . v  d i c t i o n proves  the corollary.  such  Lw > 0  and  m u s t b e z e r o somewhere " i n If Lf < 0  We a r e now i n a p o s i t i o n t o p r o v e  w > 0,  such  that  D e f i n e a new f u n c t i o n  Clearly  By Lemma 1.4,  £  a  w  by  i n N .  N , and t h i s c o n t r a i n N , look at  - f.  Theorem 1.2:  P r o o f o f T h e o r e m 1.2:  I tw i l l  now b e s u f f i c i e n t  function  Lf > 0  i n N, f € ^ ) ( L , N ) . - F o r s i m p l i c i t y  f  such t h a t  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 part of  IT  0,  where  x  i  > 0 »  i n t e g e r t o be chosen l a t e r .  to construct a  N  L e t f (x) = - x ^ , m We h a v e , f o r x e N ,  lies  i n the  a positive  8.  Lf(x) and  =  clearly  Since for  (m a,^ + m ( b o u n d e d  f  a ^  belongs t o  (L,N) f o r every  value  of  i s a s s u m e d b o u n d e d away f r o m z e r o i n N ,  sufficiently large  Remarks  terms)) m .  Lf > 0  m .  The a b o v e 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)) x j  p  v  i s m e r e l y a s s u m e d n o n - n e g a t i v e i n N , a s l o n g a s one o f t h e a.. L  i spositive i n N . elliptic  in  N  A l s o i t w i l l b e s u f f i c i e n t t o assume  i n any domain  f o revery  R  i = l,...,n  definition of e l l i p t i c i t y . be o b t a i n e d  Assume  t h a t i n N • we h a v e functions.  v  (iii)  on  Proof:  that  dN ,  f u  3  u  results could possibly  are functions in"^(L,N) with  f  and  somewhere n e g a t i v e d . i n  >_ 0  V  The c o e f f i c i e n t s  v  u,v  somewhere p o s i t i v e i n  Theorem Then  Some o f t h e s e  L u >_ f , L v >_ g  A(u v) =  (iv)  as a t r i v i a l consequence o f t h e  g  and  bounded  Furthermore i f :  u = 0  (ii)  > 0  N c R , as then  b y u s i n g a more g e n e r a l f o r m o f t h e maximum p r i n c i p l e .  C o r o l l a r y 1.6  (i)  such t h a t  of  L  N  N i n N  s a t i s f y the conditions of  1.2.  m u s t v a n i s h somewhere i n N . If v  new e l l i p t i c  w e r e a l w a y s p o s i t i v e i n N , we c o u l d d e f i n e a  operator  L-, , b y  9. L-jW- = ( L  v  )w  w e "§)'(L,N)  Then g)u>f  L-jU = ( L and  L ^ v >_  0  in  Theorem 1.2,  v  Lu = 0  N  x  the  it If  in  somewhere  conditions  Proof:  ,  We may  N  on  BN , c £ 0  and  in  N  Contradiction.  in'  (L,N)  non-trivial.  bounded i n  Let  N .  i f the c o e f f i c i e n t s of  assume t h a t  u  consider  Then  L  i s always n e g a t i v e i n a subdomain o f  i s n e v e r z e r o we c a n assume Lu  u  and  v  satisfy  1.2.  o f Theorem  w e r e n o t , we c o u l d v  u = 0  N .  are functions  3  LgV = L v + c v = 0 must v a n i s h  somewhere i n  Assume t h a t / u v in  > 0  a s a c o n s e q u e n c e o f Lemma 1.4  But  N  f u  gH=I  must v a n i s h  C o r o l l a r y 1.7 and  -  =  0  ,  v > 0 in  Lv = -cv  N N  N , for i f  w h e r e i t was. .  Then  in' N  and A(u,v) =  Hence  v  must v a n i s h  0  -cv  u  v  somewhere i n  = cvu >  N  0  in  by C o r o l l a r y  N  1.6,  Contradiction.  I n v i e w o f t h e a b o v e r e s u l t s , i t seems n a t u r a l t r a t e o n d e t e r m i n i n g some c o n d i t i o n s  t o concen-  f o r t h e Osc 1 b e h a v i o u r o f  L.  10.  3.  A C o m p a r i s o n Theorem f o r It  for  i s w e l l known t h a t o s c i l l a t i o n  an o p e r a t o r  L  i f we  with suitable oscillatory computationally L  and  has  a  C  coefficients  of  the  domain.  \ l f G  [4,  p.  L  (G)  for  We  shall  we  shall  [4,  boundary  M  are of c l a s s  128]'  p. C*  d e n o t e s any  , are  An  e"2)(A)  and  P r o p o s i t i o n 1.8 operator Proof:  2  (in fact  norm a s a map [2,  uniformly  operator f >_ 0  \  classical  1  f  sense.  [l].  If  i f f  of  that  the containing  C °(G)) C  in  for  from  L  2 (G)  G , f  — 1 (L + \ ) ~  199].  p.  = f  know  of  is  to  H Q  1  class  completely  (G)  F u r t h e r m o r e , we  , shall  elliptic.  A  will  be  called positivei f f A f >_ 0  s u f f i c i e n t l y l a r g e , the  (a.e).  generalized"  i s positive.  F i r s t assume t h a t _  and  Lu  (a.e.) i m p l i e s t h a t  For  (L + \ ) ~ ^ ~  = (L + \ )  large  L  the  sufficiently  s u c h d o m a i n , t h e n we  s o l u t i o n s of  classical  sufficiently large  Definition  L-^ .  t h i s under  i n a convex set  t h a t the g e n e r a l i z e d operator  t o be  obtained  assume t h a t e v e r y b o u n d e d d o m a i n  and  L  do  domains i n v o l v e d a r e  3G  X  be  to another operator  on  assume  u  behaviour.  that generalized  continuous i n the  f  L  —  03  C  131]  relate  r e s u l t s may  s i m p l i f y i n g assumption t h a t the c o e f f i c i e n t s  Specifically  question  u = 0  can  the boundaries of the  smooth. in  L.  f e C°°(G) ,  (L + \ ) u = f  I t follows that f >_ 0  (a.e)  and  ,  f >_ 0 .  u = 0 u >_ 0 f  Then  on  dG  for  \  in  the  sufficiently  i s an a r b i t r a r y member o f  11.  L  (G) , we c h o o s e a s e q u e n c e  that  l i mf  continuity, and  = f  i n the  l i m (L + X )  therefore  _  ^ ^ _i f  n  L (G) 1  f  ( L + X)~"*"f >_ 0  f  (^)  c  norm a n d  = (L + X )  n  o  n  _ 1  f,  f  n  functions  >_ 0 .  i n the  such  Hence b y  L (G) 2  norm  (a.e).  From now o n we s h a l l a l w a y s assume t h a t when t h e o p e r a t o r ( L + X)  i s considered,  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 a b o v e r e s u l t s h o l d . on t h e c o e f f i c i e n t s C°°  of  L  B y t h e smoothness•'" a s s u m p t i o n  we a r e a s s u r e d  of the existence of a  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. 2 1 4 ] .  furthermore,  the problem  (L + x ) u = f , u = 0 on  classical  s o l u t i o n s f o r every  function  K(x,y)  f  of class  may b e c o n s t r u c t e d f o r  Since,  SG  has  C°°(G) , a G r e e n ' s (L + x) .  Clearly  _ -i  K(x,y)  must be n o n - n e g a t i v e ,  a p o s i t i v e operator.  otherwise  (L + X ) ~  c o u l d n o t be  This i s a f a c t which a l s o f o l l o w s from the  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(£)  defined by  •yfj ( f ) = J K(x,y) f ( y ) d y G  where  p > n .  Then  [5>  p.  2593 Jo  i s a completely  continuous  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 , bigger than  t h e a b s o l u t e v a l u e o f a n y o t h e r e i g e n v a l u e , a n d whose  corresponding has  eigenvector i s non-negative  in  G .  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 F r o m t h i s we may c o n c l u d e  that the operator  Furthermore, eigenvector. L  has a  12.  simple  r e a l eigenvalue  X  , w i t h a non-negative  q  such t h a t a l l other r e a l eigenvalues Furthermore  L  eigenvector.,  are b i g g e r than  h a s 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  \  Q  .  non-negative  eigenvector.  Definition  The e i g e n v a l u e  eigenvalue  of  w i l l be c a l l e d t h e s m a l l e s t  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  1  positive i n  X  u  associated with  o  X  i s  o  G .  \ Proof: ically  -We  (L - X )u = 0 v o o .  in  z e r o on a non-empty s u b s e t  partial by  have  S  G . of  If  u  were i d e n t -  o  G , then not a l l of i t s  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  a t r i v i a l modification of a result of Kreith  Our n e x t  concern  w i l l be t o d e t e r m i n e bounds f o r  S , as f o l l o w s  [6].  X  and i t s o  behaviour  as t h e domain  Proposition 1.10.[l] G  and  w = 0  on  <  \5 -  w  be any smooth f u n c t i o n ,  w > 0  Then sup • r L w ( x ) -j  xeG  1  ITTxT  J  I f n o t , we c o u l d f i n d a f u n c t i o n Lw < X w Q  or  i s perturbed.  Let  3G . \  Proof:  G  in  G  w = 0  w on  such that SG. -  in  13. ( L <-• X )w < 0  in  Since the extended operator c o n t i n u o u s , we c a n c o n c l u d e [ 7 , p. 65]  on  3G  i s p o s i t i v e and  1  b y a Theorem o f K r e i n a n d  that there exists a function in  Q  L*  w = 0  (L + X ) "  (L* - X ) v = 0 where  G  G ,  i s the formal a d j o i n t of  v > 0  in  v = 0  L .  on  Rutman  G  C  completely  such  that  G  But t h i s i m p l i e s  0 > ( ( L - X )w,v) = (w,(L* - X ) v ) = 0 Q  Q  which i s impossible.  P r o p o s i t i o n 1.11  X  Q  >_ n  , where  u  o  denotes the s m a l l e s t L L*  eigenvalue of the f o r m a l l y s e l f - a d j o i n t operator  — ~ —  with  zero boundary c o n d i t i o n s . Proof:  Under t h e r e g u l a r i t y assumptions 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 Principle  [8,  V o l . 1,  of the v a l i d i t y of  p . 398].  Courant's  Therefore  _ i n f r B ( u , u ) -> ~ ueD (u,u) 1  J  where B(u,u) =  n n D.(b.) L a. .D.uD.u .JJ.UJJ.U + + (c S a. E" — — — )u , i , j = l i = l 2 G J  2  3  J  D = {u : u e C(G) , p i e c e w i s e C (G) a n d u = 0 on Let  v  II 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 = ( v , v ) = l ,  to  X  Q  .  We may  SG} assume  a n d b y t h e 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 o f e l l i p t i c e q u a t i o n s 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 ^ Hence  v,v) = (Lv,v) = X  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 x  i s never zero i n G , then  I f SD (b ) i  i  o > ^o '  Proof:  Let ^ L + L  Lv = X v o  u = v = 0  on  dG  ,  in  G  u > 0, v > 0  in  G.  L 4- L* As s t a t e d above,  —~  and  L  have o n l y one p o s i t i v e  l i n e a r l y independent e i g e n v e c t o r each.  Now  hence i f u + v ^ c u  c , u .+ v - cannot be an  eigenvector of i J(u+v) Q  G  2  g  f o r some c o n s t a n t  u + v > 0 in G ,  .  T h e r e f o r e i f u + v ^ cu we have < (u+v) L(u+v) = VJL r 2 vLu + X^ oj uv + X„ oj v u + G G G G (  and U, (u-v)  G  <:  (u-v) L ( u - v ) = (i u oj G G  a d d i n g we o b t a i n  r  2  -  vLu- - X G  uv + X v o, G  15.  G or  u  o  < \  G  G  o  Hence i t w i l l be s u f f i c i e n t t o show t h a t i f never zero i n G in  G  and  then  v = cu  v ^ cu  ED^(b;^) i s  Now, i f S D ( b ) ^ 0  in G .  i  i  then L + L*  Lu =  Q  u = u u  — ^  ,  X U  Q  and (u,Lu) = ( u ,  L  g  L  u)  or  X  Q  = ^  o  Therefore n , n £ b.D u + i E D,(b .)u = (\ 0=1 ^ 0=1 1  J  J  - u )u = 0  3  or n n 2 S b .D.u = - £ D.(b-)u j=l o=l J  Since  u  J  J  J  t a k e s on i t s maximum a t a p o i n t i n s i d e  G , this i s  impossible.  P r o p o s i t i o n 1.11 and 1.12 enable us t o c o n c l u d e :  Corollary 1 .lp  Let  £ t^t-l G  diameters tend t o zero w i t h value  t .  X ^ ( t ) corresponding t o O  a  f  a  m  i  l  y  o  f  domains whose  Then t h e s m a l l e s t e i g e n -  G. u  tends t o  +<=> w i t h  t .  16,  Proof: values  We know [9, p.Y] t h a t t h i s i s t h e case f o r t h e e i g e n U ( t ) °'£ ^  operator  e  0  f o l l o w s from t h e f a c t t h a t  —£  C o r o l l a r y 1.13  .  ^ ( " k ) >. M ( t )  f  0  o  r  every  then  t .  Our n e x t o b j e c t i v e i s t o e s t a b l i s h a s u i t a b l e upper bound f o r X  .  Definition •  n b h(x) = - S i=l 2  (x)bj(x) det(a .(x)) ij  i + ~2~ b  x^here  b^  i s \ the c o f a c t o r o f  ^  n  "^^  e  matrix  l  b  2  • e «  b  n  2 • b.  b  .. 1,. •.. n 2" 2 3  h  Definition F[u] =  n n r a. .D.uD u + u S b.D.u + ( h + c ) u ' i,j=l i=l 1 J  1  J  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 t h e r e 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 ^ 0 ,u = 0 v  on  3G , such t h a t  o f L v >_ 0 , v  u ,  M[u] < 0 , then e v e r y s o l u t i o n  somewhere p o s i t i v e , must v a n i s h a t a p o i n t i n  G .  From t h i s we c a n p r o v e :  Theorem 1.15 —i  + h  such t h a t  Assume t h a t i n G  i s non-positive.  the smallest eigenvalue of  Then i f D..  G c D-^ , t h e s m a l l e s t e i g e n v a l u e  i s any smooth domain for L  i n D-^  must  \  be n e g a t i v e . Proof:  Choose a smooth domain G c D  Then i n  D  operator  — ^  1- h  exists a function  w  oD  c  and  D Now l e t  X  o f the formally s e l f - a d j o i n t  (see Lemma 1.18).  in (  %  (LjD ) p  T*  ? M[w] . =  .  ][  must be n e g a t i v e b y t h e m o n o t o n i c i t y  T  on  c D  2  the smallest eigenvalue  property o f eigenvalues  w = 0  c D  2  such t h a t  such t h a t  2  p  ,  or  2 (wLw + hw ) < 0  2  be. t h e s m a l l e s t e i g e n v a l u e  in D  there  w > 0 •in D  + h)w < 0 I n D«  i t s associated p o s i t i v e eigenfunction. Lv >_ 0  Therefore  and v > 0  in  of L If X  in Q  >_ 0 ,  and v then  , 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 T  If  u  denotes t h e s m a l l e s t e i g e n v a l u e  of  T *  2  +  i n a domain  h  i n any smooth domain  G , then  D-^  L  such t h a t  has an e i g e n v a l u e  < \y  Q  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 eigenvalue will We  \  of  Q  L , as t h e domain v a r i e s inv.a r e a s o n a b l e f a s h i o n ,  enable us t o say more t h a n what i s s t a t e d i n C o r o l l a r y  1.16.  f i r s t have:  P r o p o s i t i o n 1.17 w > 0  in  G  Let  w = cu  on  dG  L  w(x)  f o r some c o n s t a n t X  Then  J  c , where  u  i s the  eigenvector  .  By P r o p o s i t i o n 1.10 - xeG  o  X  Clearly i f  .  such t h a t  rLw(x)-[  _ sup  o " xeG  corresponding to Proof:  be any f u n c t i o n " ^ ( L , G )  w = 0  and ,  iff  w  w = cu  L  we know t h a t w(x)  J  t h e n the e q u a l i t y w i l l h o l d . (L - X )w <_ 0 •  i f the e q u a l i t y h o l d s , then  Conversely  Again l e t t i n g  be the p o s i t i v 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  L  and  X  Q  v  , we  have: ((L Therefore, of  - X )w,v) = 0  (L - X )w  L , but s i n c e  i.e.  Q  Q  X  =0  in  G .  (L - X ) W q  Hence  w  = 0  (a.e.)  i s an e i g e n f u n c t i o n  i s simple, there e x i s t s a constant  c-  such  19. that  w = cu .  Lemma 1.18  I f G  eigenvalue  of  L  . , G,G^  c  for G  smooth domains,  strictly  exceeds t h a t f o r  L e t Lu = l ui n G , u = 0  Proof:  L v = y,v i n G^ , v = Then i n G  we h a v e , "by a t r i v i a l  then the smallest  on  0  u > 0  oG ,  on  oG^ ,  v >  G^ . i n G  0  i n G^ .  calculation,  2 2 n /U \ uLu u Lv „ • p.. /U\ /Uv L(_) = — - — g -- v S a. • D ( - J D . ( - ) 2v . .v 2v^ i,j=l T  T r  i  J  3  or L(^)  <  2v'  —  v  (£b  1.17>  By P r o p o s i t i o n  (2X -  2v  li)  2  T ( ) r 2V -i U  . X  . SUp  < xeG  V  [  .  ;  — T j H  <  0  2  X  .  ~  ^  2v Therefore  X > |i  We w i l l  now c o n s i d e r  as t h e domain v a r i e s .  I ti sclearly  c o n t i n u i t y of the eigenvalues is  t h e problem o f t h e c o n t i n u i t y o f  every  x .  I tw i l l  s p a c e s a n d n o r m s [8,  Definition  c m  ^y(^)  s u f f i c i e n t t o consider the  o f the operator  a s u f f i c i e n t l y l a r g e constant  so t h a t  L + X ,  will  where  X  c ( x ) + X >_ 0 f o r  a l s o be u s e f u l t o i n t r o d u c e V o l . I I , p.  X  the'following  332].  denote t h e c l a s s o f a l l f u n c t i o n s  u  Q  20.  w h i c h have p a r t i a l d e r i v a t i v e s up t o o r d e r continuous i n G  1  a .  Definition  G  l  For every  u e ^ .a^ ^  \l\ = s t  (<£^,... ,1 )  DV  and  i *~1  Denoting by  w  e  d e f  m (  i s ' an n - t u p l e  integers,  which are  and a l l t h e m** 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  where  m  i  n e  of non-negative  signifies  D^IDJ  2  H [D u] t h e smallest constant m  *Jtt  u .  ... D*  11  K  with the  th property  that a l l the m  Holder c o n d i t i o n i n G coefficient  order d e r i v a t i v e s o f u  w i t h exponent  a  satisfy a  ( 0 '< a < l ) and  K , we d e f i n e :  Definition  || u I j ^  = || u || + H [ D u ] m  m  a  . .  Under t h e above r e g u l a r i t y c o n d i t i o n s we may assume t h a t t h e Schauder " e s t i m a t e  t o t h e boundary" [ 8 , p. 3353  (1.2)  < K (||u||  I U H  2  4  a  w i l l hold f o r every z e r o boundary v a l u e s .  1  C +a  o  + !| f || ) a  s o l u t i o n o f Lu = f  2  in G  Since the "c" c o e f f i c i e n t of L  with  may be  t a k e n p o s i t i v e , we c a n i n v o k e t h e maximum p r i n c i p l e , .to reduce  (1.2) t o (1-3)  llu||  2 4 a  < 1^(11 f || ) a  21.  The  constant  of  L , and  K  G, a , t h e e l l i p t i c i t y  a l s o f i n d convenient  Definition  to i n t r o d u c e the concept of a  A strong b a r r i e r function  Q.  on  3G  G .  G , zero only at  ^. m  and  G-^  w^  corresponding  Q  and  3  Let  ,  G m  G  C (G) fl C(G),  satisfies  L [ W Q ] >_ 1  c G  , each  G m  '  m+1  non-empty.  be any bounded domain such t h a t convex w i t h smooth boundary  Assume t h a t a t each p o i n t  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^.  Q,  Let  on  p o s i t i v e eigenvector f o r constant  KT*"  pendent o f  G  .  m  .  m  Then t h e o p e r a t o r  w h i c h i s the l i m i t o f the  o f the  Proof: class  n  m  .  (1.3) L  The C  to  G  \x i n  The l i m i t o f a subsequence G , w i l l be an  eigenvector  \x .  c o n v e x i t y of  2+a^ m^  can be chosen i n d e -  has an e i g e n v a l u e  , u n i f o r m l y i n t h e compacta o f  corresponding  corresponding  F u r t h e r m o r e assume t h a t a  i n Schauder's e s t i m a t e G  3G  11-^^3  be t h e s m a l l e s t e i g e n v a l u e and a s s o c i a t e d n o r m a l i z e d  G  to a  \  Theorem 1.19 G = L)G  3^0].  i s a f u n c t i o n which i s of c l a s s  non-negative i n in  the  themselves.  " s t r o n g b a r r i e r f u n c t i o n " [ 8 , V o l . I I , p.  point  constant  L , b u t n o t on  t h e bounds on t h e c o e f f i c i e n t s o f  coefficients We  depends o n l y on  1  f o r  e v e r  y  I t must c o n t a i n a sphere.  G  ffi  a  implies that a l l u  • Also  Since G  G-^  m  are of  i s open and  non-void  i s bounded so i t may  be  placed  22.  I n any c a s e , by Lemma 1.18,  i n s i d e a sphere.  monotone sequence bounded below. such t h a t  lim u  Now  without  II " ° II " a (G m )  G  Then,  m  x  .  ^  s  a  can f i n d a number (i  = \s .  m  where  Hence we  ^W^'i  j| (G ) = 1 a i n d i c a t e s t h a t the norm i s t o be t a k e n o v e r l o s s o f g e n e r a l i t y ; assume  || u  w h i c h o b v i o u s l y reduces t o \  with  K  independent o f  subsequence o f such t h a t  lim  m  .  > also called u m  =  Therefore u^  m  i s positive in u > 0  either u = 0  in  K  G G  G . .  m  or  i n absolute value.  m  Hence  lim (P-  Once a g a i n by u s 0  in  6  Hence f o r  u m  [6],  G .  we  We  with  ( )) =  uu(x)\  as each  can s t a t e t h a t  (1.4)  implies  a r e bounded above we  m  (8) (x. - y . ) | < Kn  denotes a p o i n t on t h e l i n e between  u  claim that i n f a c t  u^  x,y e G  a  Hence i f  x  m  u >_ 0 . i n - G  Note t h a t  second p a r t i a l s o f each  - u ( y ) | = |s where  and  i s i m p o s s i b l e . . To see t h i s , n o t e t h a t  t h a t the f i r s t and by  in  G , together  second p a r t i a l d e r i v a t i v e s .  l i m ( L u ( x ) ) = Lu(x)  Lu = pu  can f i n d  , and a f u n c t i o n  C^}^  u n i f o r m l y on compacta  u  t h e i r r e s p e c t i v e f i r s t and x e G , then  By e q u i c o n t i n u i t y we  x  have |x - y|  and  y .  23.  .(1.5)  |  ( )l  u  < Kn  x  m  |x - y| + I u ( y ) l m  Identically, (1.6)  l  Also  D  i  u m  ^  || u  ^  x  |x - y| + l ^ u ^ y ) !  l|: (G ) = 1  m  r  III  f i n d a constant  1 Kn  f o r every  m  f o r every .  i .  Hence we  can  Jil  vJ!<  c-^  independent o f  m  such t h a t  or.  Now  let  of  G  x  G  denote a s u b s e t o f  N  we can f i n d a p o i n t o f  and r a d i u s  .  1 J l c  K n  G G^  Then f o r  such t h a t g i v e n any p o i n t  i n s i d e t h e sphere w i t h c e n t r e 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 (x)| m  m  and, from  <K„  (  1  1  )  +  ™Llu (y)l m  N  (1.6)  Therefore  I K H l ( m > G  And as a  consequence  x  ! K » 1 < V  2k,  l i m ( II u "m  Since  - u |L(G. )) =0 "1 N x  N  2^-<  lluJ^CG^) .  Finally, set u = 0 u dG  on  3G .  i s continuous i n G . e > 0  and l e t  we must have  T  We c l a i m t h a t w i t h t h i s  For, l e t Q  by  w = e + PWQ  mined.  with  3  I f w^  Q , d e f i n e a new f u n c t i o n  a p o s i t i v e constant  3G  = ce  t o be l a t e r  , w — u > 0 m ' m —  Since  || u Hi  || (G ) = 1 vJu.  m  and u  ill  m  m  m  f o r every  Hopf maximum p r i n c i p l e [2, p. 150], we conclude, t h a t  m .  By t h e  w + u  m  >_ 0  , or | u ( x ) | <_ w(x) = e -[- PWQ(X),  But, f o r x must have  Remark domain  3 can  i s a bounded sequence,  Xxi  L(w + u ) >_ 0 - i n G  be chosen so t h a t  m  deters  and i n G m  m  G  w  + P L [ W Q ] >^ p  L(w + u ) > 3 + u u  in  i s the  Then L[w]  Now on  be an a r b i t r a r y p o i n t o f  be chosen a r b i t r a r i l y .  b a r r i e r function associated with  definition  in G  and n e a r  x e G-  Q, , b y t h e c o n t i n u i t y o f w^  we  | u ( x ) | < 2e .  Theorem 1.19 G  f o r every  shows t h e e x i s t e n c e o f e i g e n v a l u e s  o f t h e above t y p e .  f o r any  25. eo  Theorem 1.20  • •  1  Assume now t h a t  •  bounded, '  G  m+1 c G m w i t h  L u = uu  \i = l i m u Proof: j|  m  in  G ,  ,  III  G , \i , u , K rrr ^m m  n  upon them as f o r Theorem 1.19. such t h a t  G = D G  G  non-empty,  , i f we a l s o assume  on  JL  and t h e c o n d i t i o n s  Then a g a i n we can f i n d u = 0  G,  3G ,  u >_ 0  u,u  in  G  and  3G = 3G .  A g a i n w i t h o u t l o s s o f g e n e r a l i t y , assume t h a t  -II ^ m^ G  =  1  a  '  C l e a r l y once a g a i n  bounded sequence.  {u }  i s a monotone  m  e x a c t l y as i n Theorem 1.19*  Proceeding  we  f i n d that there e x i s t s a  u , t h e l i m i t o f a subsequence o f  {u^}  in  such t h a t ' L u = \m  Theorem 1.19*  G .  we c a n s t a t e t h a t  F o l l o w i n g the steps of  u ^ 0  in  G , as e x a c t l y t h e  same p r o c e d u r e as b e f o r e may be used t o show t h e e x i s t e n c e o f a compact s u b s e t all  m  G*  of  G  such t h a t  || u  m  H (G*)  sufficiently large. There: remains t o show t h a t  note that  || u  l! _i™(G )  <  0  a r e bounded i n  G^  K  u = 0  on  3G .  independently  of- m .  m  Now, by t a k i n g  m  Then i f Q e 3G ,  Remark  Q. .  .  Hence by e q u i c o n t i n u i t y and  convergence, we can c o n c l u d e t h a t  Clearly  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 o f  a r b i t r a r i l y close to  uniform  To see t h i s ,  and hence t h e f i r s t d e r i v a t i v e s o f  |u(Q)| < |u(Q) - u ( Q ) | + | u ( Q ) |  dG  for  a  u(Q)  =0  \x i s t h e s m a l l e s t e i g e n v a l u e  of  i t s eigenvector i s p o s i t i v e i n  G .  L  in  G as  26. P r o p o s i t i o n 1.21 spheres w i t h constants  Let  r^  denote a f a m i l y o f c o n c e n t r i c  G  =  denoting the radius  6 , p  such  that  < 6  < J  0 then t h i s  ^ i ^ i i  i  family satisfies  f  r . < f>  of  r.  s  G^ .  the condition  on t h e c o n s t a n t i n  9^  Let  G^  o f Theorems  d e n o t e a n a r b i t r a r y member o f t h e f a m i l y a n d  t h e t r a n s l a t i n g a n d c o n t r a c t i n g b i j e c t i v e map f r o m w h i c h maps t h e u n i t s p h e r e  y = (y^,...,y the be  {G }  1.20.  and  Proof:  exists  < fi  Schauder's e s t i m a t e imposed on t h e f a m i l y  1.19  I f there  )  and  U  onto  x = (x^,...,x )  domain and range space o f  0^  1  to  G^ . „ L e t denote a generic  n  F/  respectively.  Then  point of 0  may  i  e x p l i c i t l y given as: x.  where  =y. r . +  Y = (Y^**••>Y )  divergent  u  the operator  L  =  n S  A {x)  D  lm  t o  of t h e spheres.  For  i s w r i t t e n i n non-  n u(x) + S B (x) t  be a f u n c t i o n i n ^) (L,G^)  and  Lu(x)  g(y)  = u(6 (y))  = f(x)  g(y)  =  i  0  J = l , . - 3 n  form, Lu(x)  Now l e t  .  denotes t h e centre  N  s i m p l i c i t y assume t h a t  Y  on  3U  i n and and  G  ±  .  D^(x) + C ( x ) u ( x )  such that  D e f i n e new f u n c t i o n s  h(y) = f(S (y)) . i  u-= 0 g,h  on by  Then c l e a r l y we h a v e  3G^  27.  l,m  1  to  ••  1  3 y  t  3 y  m  i  + These changes i n c o - o r d i n a t e s operator" problem i n t o a w h i c h c a n be of the be  *  1  1  Now  t r a n s l a t e our  by  i t i s c l e a r that Holder  e  Y  2  '•  \  for y e U .  "many d o m a i n s ,  one  A  one  domain" problem  t h e a s s u m p t i o n s on t h e constants  and  radii  exponents  £ ( i ( ) * ^( j_(°)) > 6  B  9  can a n d  m  c(8^(°)) w h i c h a r e i n d e p e n d e n t o f 1  = h(y)  i  "many o p e r a t o r s ,  dealt with.  £  C(e (y))g(y)  s p e c i f i e d f o r the c o e f f i c i e n t s  y >y  3 y  i  .  For  example,  for  .  a or. hi  - y l  l e ^ )  a  2  -  e (y )r ±  2  < ^  where  iu  s  denotes  I sphere w i t h centre constant we  may  ^ 5  y  of  conclude  the o r d i n a r y i  p  2  * and  |x  V  radius  f o r e a c h o p e r a t o r may  or of  g  to^ 2^ K - x | A  x  a  1  a n d  Q  i g  a  ,  7  2  3  .  a l s o be  S i n c e an taken  ellipticity  independently  of  i ,  that Hgl!  by  (J^Mifl2  p  o U)  2 4 < x  "estimate .  (u)  < K  I ! h|| (G) a  to the boundary", w i t h  However,  K  not  a function  28.  Bg  sup  n  3  = ri  (y)  au(x') sup xeG. 3x (  Therefore, 6  P xeG S U  ±  3u 3x  (x)  -  sup Sg yeU s y .  < B  (y)  -  p  S  U  P  xeG,  cm ax,  (x)  Hence, f r o m t h e s e a n d s i m i l a r r e s u l t s we c a n show t h e e x i s t e n c e o f two p o s i t i v e  constants  k^,k  such  2  that  \ IUII  2  +  a  (U)  ^ l l u H ^ G , )  >  and || h || (U) < k || f || a  2  C o m b i n i n g o u r r e s u l t s , we  a combination  pendent o f  4.  i  and  < Kj||f || ( a  0 l  )  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-  u .  Oscillation Criteria for We  (G.)  obtain  l U I I ^ G i )  with  a  L  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 a b o v e  results  so a s 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 operator  L .  accomplished  As h a s b e e n done b e f o r e  [9],  our aim w i l l  b y 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  be L  29. which w i l l is  assume t h e " m a j o r i z a t i o n " o f Osc 1 , o r c o n v e r s e l y ,  known t o b e  ensure that  L  "majorizes"  L  by an o p e r a t o r  conditions which  fx 1  : x  defined  > 0} n —  in  on a s e t  E R  .  will  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 coefficients  which  R  L  has  containing the half-space  We r e c a l l  that  = R n {x : |x| > r }  r  a n d we f u r t h e r d e f i n e a s i n [11, C_ = f x : x  p. 3]  t h e cone  C  by  > |x| c o s a}  and S L e t J\, ( x ) m a«j o r a n t — class  [11]  = R 0 f x : |x| = r ]  r  denote t h e l a r g e s t eigenvalue of  C^(0,o>)  (i a . j. ( x ) )  such f(r) V  of  (a. .(x)) .  i s a p o s i t i v e valued  function  A f  of  that m a x  > A ( x ) - X€S ' V  r  F u r t h e r m o r e , any smooth f u n c t i o n  g(r)  will  > ^  be c a l l e d a m a j o r a n t o f  function previously defined. co-ordinates  [12,  p.  58]  g ( r ) such t h a t :  [c(x) + h(x) -  \  1  ] •  SD (b ) c + h - — = ^ — w h e r e We a l s o i n t r o d u c e  h  i s the  the spherical  30. n-1 X-, = r IT s i n Q. 1 1  x  1  x. = r c o s 9 I  = r c o s 6-, --  n  n-i . Tf n— l + j . j — ] _  1  s i n 9-  i =  J  2,... n-1 }  Theorem 1.22 L I s Osc 1 i f t h e r e e x i s t s a c o n e C , ; b. (0 < a < -?) a n d ( a . . ( x ) ) , c + h - ED (-^) have m a j o r a n t s c xj i i a  f,g  r e s p e c t i v e l y such t h a t  f x  where  X  a  ~~n~J^ = r \ f(r)  [sin n  cp(a)  e  i  + X  solution  cp s i n n  a  2 6  l  the problem  =0  0 < B  ±  < a  (such a  X_,  i s known t o e x i s t  [13]).  Clearly the operator  L. u = x  majorizes sphere.  note  =-  =0  has a n o n - t r i v i a l  i.e.  2  a  a x  f^]  2  + X r " f ( r ) ] dr  J  i s t h e s m a l l e s t number f o r w h i c h  ^  Proof:  .03 \ r^fgCr)  >  in  — ~ We w i l l .  + h .  -  n £ D1 . ( f ( r )1D - u ) . + g ( r ) u  1=1 Let  show t h a t Writing  [11] t h a t i t I s  L^  L  S = [ x : |x| < q]  be a g i v e n  has a n o d a l domain o u t s i d e  S ,  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  Osc 1 , a n d i n p a r t i c u l a r i t h a s 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, b y c h o o s i n g a -  31. nodal  domain  N  of  possible to find It  sufficiently  a sphere  f a r from the o r i g i n ,  S-^ s u c h t h a t  follows immediately that  L.^  will  § ( 1 ^ = 0  have a n e g a t i v e L  eigenvalue  for  L  p = {Vi> • ' • >V )  .  Let  , and hence so w i l l  radius.  Consider  the family  N cz §  1  smallest  + L*  — ^  + h  denote t h e c e n t r e  n  and  i t i s  £ t^te[l «) s  o  of  and t h e r e f o r e S.^  and  concentric  f  I i t s spheres  g i v e n by: S and We  let  t  = {x : ' |x - p |  \i{t)  t e [1,»)  denote the s m a l l e s t  eigenvalue  of  L  i n  S^ .  define t  Clearly ° Then  t  S,  o  exists.  = 0 S. n n  o  = sup [ t : u ( t ) < 0} Let  o  = U S., t m m  let. t  o  =  l  i  m  n  = lim u(t )  0  Conversely,  t  t  and from t h e above U(t )  S, t  2  < ^}  2  n  = l i mt  n  monotonically  Q  Combining these r e s u l t s ,  Several  other  from below.  results, <_ 0  monotonically  from above.  and t h e r e f o r e u(t )  °  Then^  • • = lim u(t ) m  we s e e t h a t  > 0 . S, ^o  i s a nodal  domain f o r L  t h e o r e m s may now b e p r o v e d w h i c h a r e  a n a l o g o u s t o t h o s e known f o r s e l f - a d j o i n t  operators  [9, 11].  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 e n s u r e the o s c i l l a t i o n different  of the majorizing operator.  conditions postulated  ordinary differential  equation.  These reduce t o  t o ensure the o s c i l l a t i o n o f an In particular,  we c a n 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 [ 1 1 ] t o h o l d f o r s e l f - a d j o i n t  Theorem 1.23  L  operators:  i s Osc 1 i f  dr n - l \ r f(r)  r  hJJ(r) [ g ( r ) + X r  1 1 - 1  _ 2  a  f(r)]dr = - •  w  1  x  f o r some  1  m > 1 , where  dt  h (r) =  r  t  n  _  1  and  f(t)  a l l the other  f u n c t i o n s a r e as p r e v i o u s l y defined. Proof:  The c o n d i t i o n s o f t h e Theorem e n s u r e t h e o s c i l l a t i o n o f  the r a d i a l equation i n the comparison operator  [14].  The r e s t o f  t h e 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 Osc 1  Assume J\. ( x )  i s hounded i n R .  0 0  r[g(r)  + \ r  i s  -2 f ( r ) ] d r = - «.  1  \.  f o r n >_ 3 J r i  Proof:  L  for n = 2 i f r  and  Then  1 _ 6  i fthere exists [g(r)  Identical  6 > 0  '  such t h a t :  + X r " f ( r ) ] d r = - «. 2  a  t o what vras done i n [ 1 1 ] f o r t h e s e l f - a d j o i n t  case.  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-  limMf  [ _  R  2  G  (  R  )  ]  >A  1  [ x  a  Proof:  A g a i n as was done i n [ l l ] .  Remark  Theorems 1.24 and 1.25  + I S ^ L ]  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] Schr8dinger  L  f o r the  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 t h e f a c t t h a t L + L* majorizes — g • C l e a r l y t h e n , i f c o n d i t i o n s a r e so \ T  chosen t h a t  x  i  T*  — 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 a l r 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 a r e g i v e n i n h i s paper. Remarks  A l l t h e r e s u l t s i n t h i s Chapter have d e a l t w i t h  oscillations  "at » . 11  Osc 1  behaviour  may a l s o o f c o u r s e  a r i s e due t o t h e m i s b e h a v i o u r o f t h e o p e r a t o r a t a f i n i t e p o i n t . on t h e boundary o f  R .  I t should therefore-be p o s s i b l e t o  extend a l l o u r r e s u l t s t o 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 o f S e c t i o n 2 were o b t a i n e d n o t o n l y f o r but elliptic  equations  b^ also f o r a class of parabolic  equations.  T h i s l e a d s us t o b e l i e v e t h a t i t may be p o s s i b l e t o extend some o f t h e l a t e r r e s u l t s t o such o p e r a t o r s 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  II  QUASILINEAR E L L I P T I C  EQUATIONS  Introduction Under c o n s i d e r a t i o n  i n t h i s Chapter w i l l  be o p e r a t o r s  of  type (2.1) where  Lu = L u + c(x,u) 1  L^  denotes t h e l i n e a r L,u  =  -  1  The  behaviour of Chapter I .  L^ L  and  (3)  1  J  3  operator  3  and t h e t e r m i n o l o g y  will  be i d e n t i c a l  (2)  Oscillation  on t h e c o e f f i c i e n t s  of  3  describing the o s c i l l a t o r y  t o those introduced i n be:  ( l ) Comparison  theorems, i.'e., L  for L  s h a l l assume t h a t w i t h domain  c(x,§)  G c  Osc 2 ; of  L .  i s under c o n s i d e r a t i o n ,  i s a continuous r e a l valued  G x I , f o r some r e a l i n t e r v a l  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 e x i s t s as a continuous f u n c t i o n i n the c o l l e c t i o n  theorems  sufficient  t o be  Theorems g i v i n g b o u n d s f o r t h e e i g e n v a l u e s Whenever a domain  5)(G,L)  defined by:  n + S b.D u j=l  Our main r e s u l t s w i l l  o f Sturm's t y p e ; conditions  n I D.(a. D u ) i,j=l 1  domain.of  elliptic  of functions  I  [i.e.  2  of class  function  containing the  c (x,?)  5-x I .  we^  -|| ]  We s h a l l d e n o t e b y C (G) 0 ^ ( S ) 2  35.  w h i c h map of  G  into  I .  F i n a l l y , t h e c o n d i t i o n s on the c o e f f i c i e n t s  L-j_ w i l l be assumed t o be t h o s e imposed a t the b e g i n n i n g  of  Chapter I .  2.  A Sturm Theorem and a Comparison Theorem f o r L . We b e g i n by g i v i n g an e x t e n s i o n o f a Sturm theorem x^hich  i s known t o h o l d f o r l i n e a r o p e r a t o r s . ion that  L-,  i s formally self-adjoint  p r o v e a comparison theorem r e l a t i n g  L  Then, under the  assumpt-  ( i . e . b_. s 0 ) , we  shall  to a l i n e a r  operator.  \  Theorem 2.1 o f the  Let  a. .  G  be a bounded domain o f  i s positive i n  G  and  F/  I n w h i c h one  1  b . i s bounded f o r  j = l,...,n .  Furthermore assume t h a t t h e r e e x i s t s a n o n - t r i v i a l  function  (G,L)  u = 0  u e ^  on  3G .  such t h a t  F i n a l l y , l e t c (x,§) 2  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 ) _< 0 . Lv >_ 0  in  If  G ,  somewhere i n Proof:  Lu >_ 0  v  §  in  f o r each  somewhere i n  G ,  in  G ,  non-  x . i n - G , and l e t  i s any f u n c t i o n i n ^ ( G , L )  v > 0  u <_ 0  be m o n o t o n i c a l l y  such t h a t  then  v  must v a n i s h  G .  I f n o t , t h e n by t h e above assumptions  positive i n  G ,  G .  f(t) = L(tu) assumptions on  F o r each  for  x e G  0 <_ t <_ 1 .  c(x,§) ,  must always be  we d e f i n e t h e f u n c t i o n  f  by  T h i s i s p o s s i b l e as under t h e  L(tu)  i s w e l l defined f o r  Then f ( l ) - f(o) =  v  *1 o  d z  0 <_ t <_ 1 .  36.  or: L, u + [  r  1  o  c J x j t u ) d t ] u = L u - c ( x , 0 ) >_ - c ( x , 0 )  Identically: ^v + [  Setting  c(u)=  c ( x , t v ) d t ] v _> - c ( x , 0 ) 2  ,1 o  ,1 c„(x,tu)dt ,  c(v) =  c (x,tv)dt o  we have:  L^u >_ - c ( u ) u - c ( x , 0 ) jjV and  u = 0 on  SG ,  >_ - c ( v ) v - c ( x , 0 ) u < 0 • in  G ,  v>0  A(u,v) = f - c ( u ) u - c ( x , 0 )  i n G.  Hence,  c(v)v - c(x,0)  v  u 1  = uv[ . ( c ( x , t v ) - c (x,tu)}dt] + c ( x , 0 ) ( u - v) o 0  Since  t v ( x ) >^ t u ( x )  f o r every  t e [0,1] , f o r e v e r y  x e G ,  t h e f i r s t term on t h e r i g h t hand s i d e i s n o n - n e g a t i v e by t h e f a c t that  c  2  i s monotonically non-increasing.  the second term a l s o I s n o n - n e g a t i v e . G .  Since  Therefore  c ( x , 0 ) _< 0 , A(u,v) >^ 0 i n  T h i s i s a c o n t r a d i c t i o n o f t h e 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 o f l i n e a r operators of which i s a member.  L-^  37.  Consequence  Under a l l t h e above c o n d i t i o n s , i f L  p o s i t i v e eigenvalue every  with a non-positive  eigenvector  v , a s a b o v e , m u s t v a n i s h somewhere i n  Remark  are reversed  G , then  G .  i n sign.  A comparison theorem w i l l the operator  Theorem 2.2 functions  L  N  u  i s formally  now b e p r o v e d i n t h e c a s e t h a t  self-adjoint  Assume t h a t i n a b o u n d e d d o m a i n and  v  are operators  self-adjoint.  there  o f t y p e (2.1)  in  G  i n  G  and  L.^  i s formally  Furthermore, l e t the f o l l o w i n g conditions  u _< 0  i n  G,  u = 0  on  exist  :  L v = L ^ v + c ( x , v ) >_ 0 L, I  G  such t h a t :  lu. = l^a + c * ( x , u ) >_ 0  (a)  i n  S i m i l a r r e s u l t s may b e 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  where  has a non-  3 G , v  hold:  somewhere p o s i t i v e i n  G . (b)  c*(x,§)  monotonically  for fixed  x  i n  (c)  c * ( x , 5 ) >_ c ( x , § )  (d)  c(x,0)  2  (e)  < 0  ,  L  G  as a f u n c t i o n o f _ %  G . f o r every  c*(x,0)  u ( L - u - -L-ju) <. 0  non-increasing  < 0  (x,5)  i n  5 x 1  38.  (f)  The b o u n d a r y o f  G  i s s u c h t h a t G r e e n ' s f o r m u l a may  be  applied. (g) Then  The d o m a i n o f v  must v a n i s h  Proof:  c*  contains  somewhere  Assume t h a t  v  in  t h e domain o f G .  i s never zero.  i s w e l l d e f i n e d and smooth i n  G .  where  u 2v  3G .  We  have:  2  -  n  v  Then t h e f u n c t i o n  Furthermore, i t vanishes  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 2 uL, u L (H_)=_1l 2v v  c .  u  y  2 v  J.V - v 1  2  S  a. .D.(H) D.(H) i,i i v ' j W v  ( --,-) d e n o t e s t h e m a t r i x a s s o c i a t e d w i t h a  n  L-, .  u s i n g c o n d i t i o n ( f ) a n d t h e assumed symmetry o f uL^u  E  -  a..D.(|)  , we  Then, obtain:  D,(H)  G or,  as  u  and  v  o b v i o u s l y c a n n o t be l i n e a r l y dependent i n r. ,.2 U v G  Now,  L  l  v  J  <  u  L  l  u  o n c e a g a i n we w r i t e : ^(u)  + [  1  c (x,tu)dt] u 0  >_ - c ( x , 0 )  o and L-jJv) + [ J  Set  c(u) =  ' o  c (x,tv)dt] v>_2  c* ( x , t u ) d t ,  c(x,0)  c ( v ) = I* c ( x . t v ) d t o 0  J  d  G ,  39.  Then. G  u v  [-c(x,0) - c ( v ) v j < | u ( L u  u  [-c(x,0)j  - -t-^u) +  1  (-c(x,0)  G  -c(u)u)u  G  or (2.2)  V  G Now  t h e f i r s t two  negative,  and  +  r* p p r* c(x,0)u + u (c(u) - c ( v ) ) < u ( L u - ^ u ) G G G 1  terms on the l e f t hand s i d e are- c l e a r l y  the t h i r d term may  u (c(u) - c(v)) =  2  1  G  G  .1  [  +  negative.  2  {c ( x , t u ) - c * ( x , t v ) } d t  {c (x,tv) 2  +  c (x,tv)}dt]dG 2  o  (b) and  ( c ) , we  Hence the l e f t s i d e i n (2.2)  right side i s non-positive. Theorem  c (x,tv)}dt]dG  o  G  of conditions  (c*(x,tu) -  ° u  By the use  non-  be e x p r e s s e d as: J  u [  2  1  The  see t h a t i t i s a l s o i s non-negative while  nonthe  contradiction establishes  2.2.  Remark  The  (a)  u >_ 0 and  same r e s u l t f o l l o w s i f we assume t h a t : in lu  G ,  £ 0 ,  u = 0  on  3G ,  Lv < 0  in  G .  (b)  c*(x,?)  (c)  c ( x , ? ) _> c (x,§) 2  monotonically 2  v < 0  somewhere i n  n o n - d e c r e a s i n g as a f u n c t i o n o f  f o r every  ( x , ? ) e G x I ...  G ,  § .  4o.  (d)  c(x,0)  (e)  \^(\  >_ 0 ,  c * ( x . O ) >_ 0 .  1  - \ )  u  u  0  G (f)  The b o u n d a r y o f  G  s u c h t h a t G r e e n ' s f o r m u l a may  be  applied. (g)  The d o m a i n o f  c*  contains that of  I t i s a consequence o f these a non-positive eigenvalue domain  G , then  G c G^  L  c o u l d n o t have i n a domain  eigenvector positive  In  conditions are reversed  that i s :  that i f  G^  .  G^  admitted in a  such  that  t o which there corresponded  The same c o n c l u s i o n h o l d s  an  i f the  i n sign.  Of p a r t i c u l a r i n t e r e s t i s t h e c a s e  the operator  I  with a non-positive eigenvector  , a non-negative eigenvalue  Consequence  results  c .  t  i s linear.  c * ( x , u ) = Y(X)U,  Then c l e a r l y  c * ( x , 0 ) =' 0  -X-  and  Cg  satisfies  a l l t h e above m o n o t o n i c i t y  more, u n d e r r e a s o n a b l e of  I  to the s m a l l e s t eigenvalue  G , the eigenvector  has c o n s t a n t  therefore that i f the operator  <^ ( G , L ) and  Lv  smallest eigenvalue such that  x e G ,  G  I  then  i f  f o r every  sign i n  corresponding  G .  every  function  sign or-zero i n  % e I  = 0 , and'  c (x,?) 2  G a non-  v , of class  s i g n at" e v e r y p o i n t o f  c(x,0)  I t i s clear  a d m i t s i n a domain  i s of f i x e d  do n o t h a v e o p p o s i t e  v a n i s h somewhere i n every  Lv  Further-  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  and on t h e b o u n d a r y o f  positive  conditions.  G  'and  v  G , must  <_ Y(X)  for  41.  J v(L-jV - l^v) G As  an  £  0 .  m c ( x , ? ) = y(x) - S g.(x)5 • i=l  example, l e t  d  are continuous  T h e o r e m 2.3 domain  x  c(x,0) = 0  e R  ,  and  then every in  g.  t  operator  d e f i n e d i n an  unbounded  by: = - S D i,j  (A  ;  a smooth n o d a l  If:  the  non-negative functions.  l(u)  has  where  1  Assume t h a t t h e  R  2 i  0  x  o  D u) J  +  Y  d o m a i n o u t s i d e o f any  ,  C (X,E;) £ 2  Y(X)  (A. .(x) - a. . ( x ) )  function  v  R, m u s t v a n i s h  (x)u  given  f o r every  s p h e r e and  in  § € I, for  R .  every  i s non-negative d e f i n i t e i n  in ^(R,L)  xvhich i s s u c h t h a t  somewhere I n t h e c o m p l e m e n t o f any  Lv  R =  ,  0  given  sphere. Proof: is  Since  I  c l e a r t h a t we  has may  have o p p o s i t e the theorem  A to  Lv  domain o u t s i d e o f every  c o n s t r u c t a domain ' N  sphere f o r which the since obviously  a nodal  smallest eigenvalue =0  in  s i g n a t any  N  and  o u t s i d e of  sphere, i t every  i s non-positive.  therefore  chosen p o i n t of  N  v  and  Then  Lv  , regardless  cannot of  ,  follows.  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 T h e o r e m 2.2  n o n - o s c i l l a t i o n c r i t e r i a which apply  s e l f -ad j o i n t  y  operator  (2.1).  We  t o t h e more g e n e r a l  s t a t e one  such  result.  leads non-  42.  Theorem 2.4  Let the formally s e l f - a d j o i n t l i n e a r operator  I  be s u c h t h a t t h e e q u a t i o n  is  satisfied i n  ~ IjV  R  by a n o n - t r i v i a l f u n c t i o n  + y(x)v  zeros are i n s i d e a f i x e d (a)  c ( x , ? ) _> Y(X)  (b.)  c(x,0)  sphere  S .  f o r every  2  Then  v  a l l o f whose  if:  x e R ,  f o r every  % e I  =0  u(-t- u - L-^u) <_ 0  (c)  = 0  lv  f o r e v e r y bounded subdomain  L  G  of  R  G and  every  u e ^(G,L) ,  t h e o p e r a t o r L c a n n o t have smooth n o d a l domains o u t s i d e t h e s p h e r e Proof:  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 v a n i s h a t l e a s t once. o f Theorem 2.2  L  every s o l u t i o n of  o  = 0  must  To do t h i s , we r e p e a t e x a c t l y t h e p r o o f -  except f o r the use of the f o l l o w i n g estimate:  (c (x,tu) - c (x,tv)}dt = 2  lv  {c (x,tu) - c (x,tu)}dt +  0  2' 0  o +  0  {c (x,tu) 2  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.  c(x,tv)}dt  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  shall  s h a l l b e g i n b y g i v i n g two  examples and  then  we  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 .  1  Example  Puffing's Equation  ordinary d i f f e r e n t i a l  ^  -y - kV  dx^  has  dx  -y =  2  solutions  p.l6].  The  =  0  .  (k > 2  0)  equation:  x  ^  [17,  (Hard S p r i n g Case)  equation under c o n s i d e r a t i o n i s :  I f xtfe c h o o s e t h e c o m p a r i s o n  which  specific  0  y = + sin(x-a)  f o r every constant  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 s ' E q u a t i o n must v a n i s h i n 1  every c l o s e d i n t e r v a l of l e n g t h  Example 2 ordinary  Mathieu's  Equation  [ 1 8 , p. 4 0 1 ] .  'We  c o n s i d e r the  equation:  - \M  ^-g dx^ where  it .  a,b,c  -  ( l + a cos 2 x ) y - c y  3  =  are constants.  M u l t i p l i c a t i o n by  the  integrating  bx factor  e  reduces - ^  so t h a t i n t h i s  [e  b x  the equation to: ||]  case:  0  a x  - e  b x  [ ( l a cos 2x)y + c y J 5  +  =0  (x,y) = - e  b x  [ ( l + a cos 2x)y +  cy ] 5  and c (x,y) = - e  [1+a c o s 2x + J c y ]  2  If  c > 0 , then c (x,y) £  - e  p  We  b x  [ l + a c o s 2 x ] _< - e  are t h e r e f o r e l e d t o the comparison  - T|te  t a  ||]  -  e  b x  [l  b  x  [l  - |aj]  equation:  - | a|]y  = 0  or £ |  which w i l l  +  b  f f  have o s c i l l a t o r y  [1 - | a | ] y = 0  +  solutions i f  have thus p r o v e d t h e f o l l o w i n g  Theorem 2.5  If  c >_ 0  s o l u t i o n of Mathieu's bounded  and  b  < 4(1 -  |a|) .  theorem:  b  < 4(1 - | a j ) , then  every  e q u a t i o n must v a n i s h o u t s i d e o f  every  interval. Conversely, I f  c _< 0 , t h e n  c ( x , y ) 2l - e 2  and hence t h e c o m p a r i s o n  b X  [ l + a c o s 2 t ] >_ - e  e q u a t i o n may  be c h o s e n  b  x  [l  +  t o be:  |a|]  45.  which w i l l  have n e v e r v a n i s h i n g  Theorem 2.6  If  c £ 0  and  Mathieu's equation can vanish  solutions i f b  b  >_ 4 ( 1 + | a | ) .  >_ 4 ( 1 + | a j ) , n o s o l u t i o n o f  2  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 definitions  s i m i l a r t o t h o s e o f C h a p t e r 1,  Definition  A function  f  of class  C (0,«>) 1  i s a majorant  of  (a^) i f f  f ( r ) > " ^  w h e r e J\  (x)  denotes t h e l a r g e s t eigenvalue o f  Definition  is  A smooth f u n c t i o n  a majorant o f  Theorem 2.7 and  contains  such  a cone  admit majorants  c  08  g  (a. . (x)) .  that  Cg .  If R  ( j i )> 2 -j a  (A(x))  r  dr + »  C^ a  (a > 0 ) , c ( x , 0 ) = 0  f,g , respectively, "  05  such  that  n 1 2 r ~ [g(r) + X r " f ( r ) j dr = n  a  46.  then every s o l u t i o n o f - v. D. ( a . .D.u) + c ( x , u ) = 0  i,o in  R  -  -  J  3  must v a n i s h o u t s i d e e v e r y s p h e r e .  Here  X  denotes  t h 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 a n d i n [ 1 1 ] . Proof:  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 = - S D ( f ( r ) D u ) + g ( r ) u i ±  i  has a n o d a l domain o u t s i d e e v e r y s p h e r e . f o l l o w s i m m e d i a t e l y f r o m Theorem  Theorem 2.8 and t h a t  Assume  ( j_j) > 2 a  C  that a  d  m  i  t  R  Our r e s u l t  then  2.3.  c o n t a i n s t h e cone  majorants  C  ( a > 0)  a  f,g , respectively,  such  that: eo  eo dr  f o r some number  ,  <  i- h™(r)[g(r) + X r - f ( r ) ] d r n_1  2  a  CO  ,  m > 1 , where  h (r) = n J  r  t  =, n - l  . f  (  t  Then t h e  )  same c o n c l u s i o n , a s i n Theorem 2.7 h o l d s . Proof:  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 that the  equation n - E D (f(r)D,u) +: (r)u = 0 i=l . • • g  i s Osc 1.  The r e s t f o l l o w s a s 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 l i n e a r o p e r a t o r s may their proof. oscillation  s i m i l a r l y b e e x t e n d e d , a n d we t h e r e f o r e  omit  Of c o u r s e , we c a n a l s o e x t e n d some o f t h e n o n results.  [19]  Swanson  results f o r  of formally  Once a g a i n we d r a w a t t e n t i o n t o a p a p e r b y  w h e r e some s u c h t h e o r e m s a r e m e n t i o n e d "self-adjoint" operators.  f o r a class  I t ,is c l e a r t h a t  similar  t h e o r e m s 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 a s a n  Theorem 2.9 c(x,0)  example:  The o p e r a t o r (2.1)  Is non-oscillatory i n  R  i f  = 0 , i f the d i f f e r e n t i a l equation: n - S D.(A(r)D.v) + C ( r ) v = 0 i=l 1  admits a s o l u t i o n a l l b.  X  o f whose z e r o s a r e i n s i d e a f i x e d  SD (- ^) < 0 , where  and  ±  2  inf A ( r ) < xeS C(r) <  and  sphere  7^, (x)  Proof:  Remark  inf X€S  r  {(/^(x))} inf ?el  (c (x,5)} 2  denotes t h e s m a l l e s t e i g e n v a l u e o f  Immediate f r o m Theorem  Theorem 2.9  i e n t s o f (2.1) omit.  r  (a..(x)) .  2.4.  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 -  which w i l l  e n s u r e n o n - o s c i l l a t i o n a n d w h i c h we  shall  48.  Remark  The p r o b l e m 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  b e Osc 1 s t i l l  4.  remains open.  Bounds on t h e F i r s t E i g e n v a l u e We s h a l l c o n c l u d e  operator  this  of  of uniformly e l l i p t i c  Theorem 2.10 \  linear  Let  u,v  X  L .  chapter by'extending  (2.1) some known e s t i m a t e s  eigenvectors  f o rthe f i r s t  to the  real  be a r e a l  such t h a t  eigenvalue  u <_- 0  and  for L  v >_ 0  i n  and a l s o t h a t t h e boundary o f = 0  and  function of that on  -w  c (x,§) 2  % .  i s  real  G .  The  0°° .  of L  are  Finally,l e t  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 i s a n y f u n c t i o n i n 1^  Then I f w  i s also i n ^  G  with  Furthermore,  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  c(x,0)  eigenvalue  operators.  two e i g e n v e c t o r s may b e l i n e a r l y d e p e n d e n t .  C°  to  (G,L)  and such t h a t  w>  (G,L)  O l n  G ,  such w = 0  3G , we h a v e :  (o ^)  S U  P {lAzEL] > \ >  xeG Proof:  f ( )1  i n f  L  -w  J  2.  k  1  xeG  L  w  w  1  J  A g a i n we h a v e : L(u)  = L^(u) + [ J  C g ( x , t u ) d t ] u = Xu  «1 L(w)  = L, (w) + [ . o 1  .= L, (w) + [  r  c„(x,tw)dt]w  J  1  c (x,tu)dtjw + [  d  f  1  {c„(x,tw)-c (x,tu)}dt] p  w  49.  Since  tw > t u P I ^ t w ] + [ J c ( x , t u ) d t ] w >_ L [ w ] 1  2  Now, f r o m 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 , . inf - xeG  L  r  K  l  [  w  ]  [l],  + [^c (x,tu)dt3w. w - xeG g  1  we i  n  have:  f  J  1  w  To p r o v e t h e o t h e r i n e q u a l i t y i n ( 2 . $ ) , we .proceed We  J  identically.  have:  r  L-^v) + [  1  c ( x , t v ) d t ] v = Xv  o  20  L-^-w) + [ J  c ( x , - t w ) d t ] (-w) = L'(-w) 2  and a g a i n we o b t a i n :  ^(-w) + [  The f a c t t h a t  v >_ 0  r  1  c ( x , t v ) d t ] ( - w ) ~ > _ L(-w) 2  implies that  value of the l i n e a r operator  Z  X  d e f i n e d by: ,1 + [j c ( x , t v ( x ) ) d t ] f ( x ) o  £(f)(x) =L (t)(x) 1  Once a g a i n b y X < —  S U  must be t h e s m a l l e s t eigen-  2  [l], P[ 1 L  (  W  )+  X  „ sup x  Jo  c  ( ' x  2  w  t  v  >  d  t  ]  w }  L, (-w) + [ f c„(x, t v ) d t ] (-w) x l Jo 2 ' ' ' i . sup L(-w)-| i x ~ ^ r 1  x  C 1 1  1  [  r  V  K  T /  r  J  L  J  50.  -w <_ 0 .  w h e r e i n t h e l a s t i n e q u a l i t y we u s e t h e f a c t t h a t  As t h e i n e q u a l i t i e s i n ( 2 . 3 )  Remark.  were p r o v e d  of each other, i f only the e x i s t e n c e of I n e q u a l i t y i n (2.3)  the c o r r e s p o n d i n g  remark t h a t t h e l o w e r bound f o r in  X  u  (or  will  still  i s known,  hold.  • We  also w >_ 0  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 ] .  l i n e a r p a r t of the operator  Theorem 2.11 for a l l  ("Self-adjoint" case).  Assume  now t h a t  b. H 0  j , that i s :  Cg(x,5)  be m o n o t o n i c a l l y  c(x,0) 1 0 . v = 0  If  on  eigenvalue Proof:  s e l f - a d j o i n t n e s s of the  ( 2 . 1 ) , o t h e r b o u n d s may be f o u n d  - t D.(a.,D u ) + c ( x , u ) . i,j o J  L u = L-^u + c ( x , u )  and  v)  a c t u a l l y only requires  Under the assumption o f f o r m a l  Let  independently  L v = Xv  3G , t h e n  of  x  non-increasing  in  G- w i t h _X  X <_ w  where  u  as a f u n c t i o n o f  real,  v > 0  CO /  *  u e C. (G) .  Then:  uL-^u G  u  I K  G  2v  c(x,v)  G'  G  or X |u G  2  <  uL^u G  +  P  uf v  [(  P  1  o  c (x,tv)dt)v + Q  G  denotes the s m a l l e s t  L-^ + C g ( x , 0 ) .  Let  in  §  c(x,0)]  51. Therefore.  r u 2 <, G and  2  G  G  inf  r JG ( 1  hence, X < ueC  Now  u^c (x,0)  h  H^(G)  r  G  corresponding  , a space where by  are dense i n t h e  L  (G),u^o  the eigenvector  space  U  topology  G  X  U  2 v  to  n  i s a member o f t h e  d e f i n i t i o n the  induced  || u ||f = J ( S  2( '°) J  U + C  by  |D.u|  C  Q(G)  Hilbert  functions  t h e norm: |u| )  2  2  +  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 , v ) | • < J_S J a . ^ K l D . u l - l D j U - D j v l lV  +  where  || u ||  2 =* J2 |u|  +  i D j v H ^ u - D j v D d G + C||u-v'|| || u + v j | o  < M(||u  I^H  , and  M  u-v and  ||  || v I I J I u - v H ^ + C i l u-v  C  are constants.  1 +  o  || || u v | |  It is  o  +  *~  G  evident  t h e r e f o r e , t h a t the infimum taken  do es i n f a c t g i v e t h e  Remark  This  of operators  smallest eigenvalue  r e s u l t s h a r p e n s Theorem 2.2 l,L  given  by:  over a l l of  C (G) RO  functions  4- c ( x , 0 )  f o r the  2  special  choice  52.  -Lu = L-^u + c ( x , 0 ) u ;  L u = L^u + c ( x , u )  2  Theorem 2.12  Assume t h a t  F u r t h e r m o r e assume a function of p  ?  c (x,?) 2  L u = Xu , u _< 0  in  i s monotonically  non-increasing  c(x,0)  and t h a t  <_ 0 .  denotes the s m a l l e s t eigenvalue  of  G, u = 0 on 3G  Then  as  X >_ p / w h e r e  L^ + c ( x , Q ) . 2  Proof:  r 2 i* r f* 2 1 u L . u d x + I u { c ( x , t u ) d t ] d x + u c ( x , 0 ) d x = X u dx Q  G  G  u  G  But  uc(x,0)\>_ 0  a n d as. t >_ 0 , u _< 0  we h a v e  c ( x , t u ) >_ c ( x , 0 ) 2  2  Therefore, u  2  G  >_ [ u ( L u 1  + c (x,0)u) 2  G  C o m b i n i n g 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 to the f o l l o w i n g conclusion:  I f f o r any o p e r a t o r  s e l f - a d j o i n t l i n e a r p a r t , such that 5  i n c r e a s i n g as a f u n c t i o n of all  x , there  c (x,?) c(x,0)  exists a r e a l eigenvalue  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 eigenvalue  must e q u a l  i s monotone non-  2  and  X  (2.1) w i t h  i s non-positive f o r to which  eigenvector,  the smallest eigenvalue  of the  there then  this  operator  d e f i n e d by: L. u x  Consequence  = - E D (a  No o p e r a t o r  i  i j  D u)  of type  j  + c (x,0)u 2  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  p o n d s b o t h a p o s i t i v e and strictly  decreasing  Proof:  I f not,  as  let  G  there  , v a n i s h e s on -  a negative  X  be By  s u c h an  and  to which there  §  and  i f  of  L  is  2  0 . and  t h e a b o v e c o n s i d e r a t i o n s , we w  corres-  c (x,|)  c(x,0)  eigenvalue  exists a function  3G  G  eigenvector  a f u n c t i o n of  p o s i t i v e eigenvector. conclude that  i n a domain  u  its  can  which i s p o s i t i v e i n  such that:  S D. ( a . .D w) i,j  + c (x,0)w = p  Xw  or  r E D.(a. i,j 1  1 J  .D.w)  1  + [ o  J  c (x,0)dt]w =  Xw  o  2  and  r -  E D.(a. i,0 1  M u l t i p l y i n g the and  1 J  G  (c (x,0) 2  +  [ o  3  G  right i s non-positive.  Z  respectively, subtracting  2  the  = -  an 1  - c(x,0) .  - c ( x , t u ) } d t ] u w d G = |c(x,0)wdG  under the assumptions, the  u  u,w  °  As  = Xu  0  obtain:  Wow  L  c (x,tu)dt]u 2  above e q u a t i o n s by  i n t e g r a t i n g , we [J*  -D.u)  1  left-hand  Contradiction.  example o f such o p e r a t o r s  (a -D.u) + u - u i  side i s p o s i t i v e , while  2  .  we  may  take  54.  Remark  Identical  results  h o l d i f t h e c o n d i t i o n s i n t h e above  considerations are reversed i n sign.  Remark those  The m e t h o d s d e v e l o p e d developed  i n Chapter  the o s c i l l a t o r y b e h a v i o u r  i n this chapter  L,  i s a linear  with  I may^perhaps^also be used t o d e s c r i b e o f o p e r a t o r s d e f i n e d by:  L u = L-^u -f c(x,u,I)^yi,.. where  together  second order  . *D ) u  n  operator.  CHAPTER I I I  FOURTH ORDER EQUATIONS  1.  Introduction It  of  has a l r e a d y  [10,  p. 115],  that  solutions  the equation  L  U  A  i  =  t  D  i O  there  exists  3G .  a  i J W M D  u  )  i n the closure  a function  I f we d e f i n e  bounded domain such  (  "  A V ' l j V ) -+  * e D u - du . 0  \  need n o t v a n i s h  w  been observed  N  v  o f a bounded domain  such that  Lv = 0  i n  G-  f o r which v = 0  G ,  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  f o r w h i c h we c a n f i n d a n o n - t r i v i a l  function  that: Lw =  0  i n N ,  w = w,. i  =0  on  3N. . i = 1,. . . ,n -  we c a n show t h a t t h e same b e h a v i o u r may o c c u r e v e n when 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  G  i s a  v = % ( l - cos2x)  satisfies:  but  on  any f u n c t i o n  u = constant i s also a solution of  L  a  - 0 .  p For  t h i s reason, the o s c i l l a t o r y behaviour of fourth  order e l l i p t i c  operators w i l l  be c o n s i d e r e d  from a nodal  domain  56.  viewpoint  only.  Definition of  L  i s oscillatory  i f f  L  has a n o d a l domain  every sphere centered a t the o r i g i n .  Definition  L  i s non-oscillatory i f f L  Oscillation  results  f o r a special  o p e r a t o r s have a l r e a d y been p r o v e d  i s not  are  Identical  [9, p. ko].  The  (3.1)  examine t h e o p e r a t o r s Lu =  S D i,j,k,-t  1 0  . ( a . .a. ,D ^ ^ 1 J  k  operators  We  i s not the case.  L,  order  to forms t h a t  'to t h o s e o f s e c o n d o r d e r o p e r a t o r s .  consider operators f o r which this shall  oscillatory.  c l a s s of f o u r t h  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  we  outside  shall Specifically,  d e f i n e d by:  u ) - E D. (b. .D u ) + S c.D.u • i , j j 1  1 J  J  3  - du  3  and  whose c o e f f i c i e n t s We  s h a l l assume  differentiations  a r e d e f i n e d i n an unbounded  that the c o e f f i c i e n t s  i n v o l v e d i n the operations d e f i n i n g  1  F i n a l l y , we  of  E  n  .  partial  L  and  L-^ ,  derivatives are  R .  G i v e n a n y sub d o m a i n (or^(F,L ))  R  a r e so smooth t h a t a l l t h e  may b e p e r f o r m e d , a n d t h a t t h e r e s u l t i n g continuous i n  domain  the c o l l e c t i o n  s h a l l assume  that  P  of  R  we  shall  denote by  of functions of class (a. . (x))  i s a positive  ^(F,L)  (F) n C ( F ) . 2  definite  57.  symmetric m a t r i x i n  R .  The m a t r i x  (m. .(x))  will  be  assumed t o be e i t h e r p o s i t i v e d e f i n i t e o r a symmetric m a t r i x a l l o f whose terms a r e p o s i t i v e i n  R .  By the above assumptions, of  R  the m a t r i c e s  eigenvalue.  (a. . ( x ) ) , (m. .(x))  w i l l each have a p o s i t i v e  I f we f u r t h e r assume t h a t :  E a. x)  > C S ^  j (  for  i t i s c l e a r t h a t a t each p o i n t .  every  x e R  and f o r e v e r y  .  E rn.  § =  §  .  5 j  >  c ±  .. ., 1 ^  5  p.  S 5 ±  when  (a -(x)) , 1  '(m- .(x)) a r e p o s i t i v e d e f i n i t e , then the o p e r a t o r s d e f i n e d by (3-1)  and  ( 3 - 2 ) x d . l l be u n i f o r m l y e l l i p t i c .  T h i s w i l l a l s o be  the case when t h e m a t r i x assume t h a t  m. .(x) —J  (m. .(x)) has p o s i t i v e e n t r i e s , i f we. —J i s bounded away from z e r o f o r every  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 a s s u m p t i o n that  p. = c. = 0  j ~ l , . . . , n , the t h e o r y o f Courant may  used t o p r e d i c t f o r o p e r a t o r s ( 3 . 1 ) and  be  ( 3 - 2 ) 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 a t i s f i e s D i r i c h l e t boundary conditions.  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 We  o f type  Operators.  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  (3-1)  t o those o f type  (3-2).  S i n c e the m a t r i x  i s symmetric, i t i s p o s s i b l e t o reduce i t , by a s u i t a b l e  operators ( a . -(x))  58.  t r a n s f o r m a t i o n , t o d i a g o n a l form w i t h t h e e i g e n v a l u e s entries.  Without  l o s s o f g e n e r a l i t y , we may assume  ^j_( ) x  a  s  ^j_( )£^ ( x  i+1  1 — i ^ " " • J ~~' 1 •  Definition  The m a t r i x  ( x ) =(X (x)x..(x))  c h a r a c t e r i s t i c matrix of  i s the  (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  therefore diagonalizable.  Theorem 3-1  "Let  c l a s s ^ ) (G,L) Let  L^u, L|U  G  be a bounded domain and  such t h a t  u = u^ = 0  on  BG  u  a function of  f o r i = l,...,n .  be g i v e n by:  and Liu =  If  £ D. .(m D. .u) 1,3  2  m(x) >. X ( x )  A U )  ,  1 J  where  G  i  (uL~u)dx > 2  p i uL-^u)dx  IuL^u G  Let  X ( x ) denotes t h e l a r g e s t e i g e n v a l u e o f  then: 1  Proof:  1 J  G  G  U ( x ) denote t h e m a t r i x  X j J  (D. .u(x)) . cJ  Then f o r any  x in  59.  (3'. 3)  m(x)  E (D 1,3  1 3  .u) •  2  - ( Z a, ,D 1,3 J  J  u) •  = m(x)  2  Trace  [U ]  -  2  {Trace [ ( a ^ ( x ) - U ) ] }  2  ±  Let  S  denote the r e a l orthogonal m a t r i x which d i a g o n a l i z e s  (a. -(x)) .  Since the t r a c e i s i n v a r i a n t under such  [20, p . l 8 ] , t h e l e f t - h a n d s i d e o f (3-3) »(«) where  S  T  Trace  [ ( s W l  equals:  - (Trace [ S ( T  denotes the transpose  of  transformations  S .  a i  . (x) ) S - s > ] } T S US  Let  = (r\  2  . (x)) .  T Since  U  i s s y m m e t r i c , so i s m(x)  E ( U .u) i,j X J  2  S US  , and t h e r e f o r e :  - ( E a. .D. . u ) i , J 1 J  1 J  > m(x)  The  2  = m(x) Z "H? . - ( S X • n. . ) i,j i 1 J  E n .  1  - E X.X  2  T^T,  r i g h t - h a n d s i d e of the i n e q u a l i t y i s non-negative,  seen by p e r f o r m i n g  one more c h a n g e o f c o - o r d i n a t e s a n d  to the i n v a r i a n c e under such m u l t i p l e s of the i d e n t i t y  C o r o l l a r y 3.2  . .  a s may  If:  (i)  m(x)  E D 1,3 >  (m(x) d  X(x)  are  matrix.  L  be  a s d e f i n e d i n (3.1)  . =  be  appealing  c- s 0 , and l e t L,u  1 1  transformations of matrices which  L e t the operator  J  2  D. .u) J  S 1,3  D, (B, .D.u) d  J  - Eu  .  with  60.  (ii)  ( B . , ( x ) - b. . ( x ) )  (iii) for L  E ( x ) _< d ( x )  ever}/ i n  i s non-negative  G  Proof:  x  i n a domain  G , then the s m a l l e s t eigenvalue o f  cannot exceed t h a t o f  .  U n d e r t h e g i v e n a s s u m p t i o n s a n d b y t h e u s e o f Theorem  -3«1<>  we h a v e t h a t [ wL^w _> | wLw G for  any f u n c t i o n  on  BG  G  w  of class  f o r i = l,...,n  consequence o f Courant's  .  ^ ) (G,L)  such that  w = w^  =0  The C o r o l l a r y i s t h e n a n i m m e d i a t e  Principle.  We h a v e t h u s b e e n l e d t o t h e p r o b l e m o f d e t e r m i n i n g t h e bahaviour be  of operators d e f i n e d by o p e r a t i o n  done b y f i r s t  comparing the operator  with the operator  L L  q  whose b a h a v i o u r  Q  u =  will  given by  E D . , (m. .D . .u) j i i x j 33 v  ±  c a n b e more r e a d i l y d e t e r m i n e d .  the f o l l o w i n g n o t a t i o n :  This  given by  o  oo  L  (3-2).  We  introduce  61.  Notation  m  ij( ) x  m  m  c a l c u l a t i o n then L  x  . Z - (x)  .(x)  A simple  \t^ ±3^ ^  =  gives  'Mrs , i _- m..D. .u) u - L u = 2 (m. .D. .u o . . 10 i ji j J- * j J  oo  v  ;  and u(L v  G  u - L u) = 2 f m^D.uD.u - [ tJi.(D.n) oo o ' . .J i j x o J • 0 0G G " l  f o r any f u n c t i o n for  a  J  u e!>)(G,L)  2  J  such t h a t  u = u^ = 0  on  3G  i = 1,...,n .  Theorem 3-3  I f the matrix  <'•*>  ( ^ ( x ) )  i s non-negative.in  -  (  A  l  (  l  >  a hounded domain of  L  - . . A  n  (  x  )  G ,'then  i n t h a t domain t h e  smallest  eigenvalue  Proof:  The c o n c l u s i o n o f t h e Theorem i s a n i m m e d i a t e . c o n -  o  Q  i s n o t s m a l l e r than  sequence o f Courant's P r i n c i p l e and t h e p r e v i o u s  Because o f o u r r e s u l t s m- - ( x ) = m ( x ) i J becomes:  i sof special  on o p e r a t o r s  interest.  that of  L  Q  .  considerations.  o f type  (3-1)*  I n t h i s case,  t h e case  Theorem 3.3  62 C o r o l l a r y 3.4 negative  in  I f the m a t r i x  (x)) -(Em i G , then the s m a l l e s t e i g e n v a l u e L  is  not  oo  o  P r o p o s i t i o n 3.5  is  Let  u  =•  The  the m a t r i x ^  Proof:  .  ED., . n  s m a l l e r than that  L  If  u =  E  1  s y m m e t r i c , we  orthogonal  A j  (  m  D  i J  U  '  )  i s non-positive i n  an a r b i t r a r y p o i n t o f  may  reduce i t to diagonal  transformation,  l e f t unchanged.  Since  Let the m a t r i x  (m  in  this  5 = (5-^ ...,? ) A  s p e c i a l case gives S i  Since  d e n o t e s any  N  <_ 0  ?  U ±  2  x  f o r every  rise  G .  - (E u. ).(E . i j  Since  form at  x  (m by  Q  (m ^(x ))  ) ) be  o  given  (x  a  will  1  l j  by be  by:  n  n-tuple,  to the  fulfilled  t h i s i s accomplished  (x  l j  are  G .  the t r a c e of  u  Then i f  of  of  s u i t a b l e change of c o o r d i n a t e s . an  non-  (m D. .u) 3d  v  be  q  (x))l is  J  c o n d i t i o n s o f C o r o l l a r y 3.4  (m ^(x)) X  (m  the m a t r i x  (3.4)  form:  § ) 2  x  i  , the 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 , of i n t e r e s t i s the case Under t h i s a d d i t i o n h y p o t h e s i s ,  the  m(x)  = m(r)  .  s u f f i c i e n c y condition that  0  63.  ( m ( x ) ) b e n o n - p o s i t i v e may b e f u r t h e r i n v e s t i g a t e d a n d 1 J  simplified.  We h a v e :  ( i J m  (  x  )  )  =  [  ^ dr  l r  2  _  JUl,] (x.x.)+|5il d  r  r  3  1  3  d  r  r  th where  x^  denotes the  the i d e n t i t y matrix.  i  coordinate of  Since the matrix  to ensure the n o n - p o s i t i v i t y o f sufficient  (m  < $1  dr  2  -  Q1  "  (x.x.)  (x))  I  denotes  i s non-negative,  i n t h i s case, i t i s  < o -  To s u m m a r i z e some o f t h e s e  Theorem 3.6  Assume  r e s u l t s , we  state:  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  of  such t h a t : (i)  m ( r ( x ) ) _> X ( x ) , w h e r e ' \ ( x ) v a l u e o f J\  (ii)  r  f o r every  eigen-  (x) .  d  r  x e G .  (B. . ( x ) - b  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 -  ( x ) ) i s non-negative  Then t h e s m a l l e s t e i g e n v a l u e does n o t e x c e e d t h a t o f (u) =  2 D i,j 1  (m D 1  3 3  and  E <: d  f o r every  of the f o r m a l l y s e l f - a d j o i n t L. a  L  denotes the b i g g e s t  p i | < S < 0 dr ~ 2  L  and  t o assume: r  r  l j  x  , where  L a  i s d e f i n e d by: . :,  u ) - E D (B. D.u) • I !J J ,  - Eu . •  x <= G~ . operator  64.  Remark  The c o n d i t i o n s on t h e d e r i v a t i v e s o f  s a t i s f i e d when  3.  m  i s a constant  m  are t r i v i a l l y  function.  O s c i l l a t i o n Theorems f o r F o r m a l l y  Self-Adjoint  Operators.  I n t h i s s e c t i o n we s h a l l u s e t h e r e s u l t s o f S e c t i o n 2 t o 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.  B e f o r e 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 t h e b e h a v i o u r o f t h e smallest eigenvalue  o f t h e o p e r a t o r s under c o n s i d e r a t i o n .  As a consequence o f t h e s t r u c t u r e o f t h e H i l b e r t space H  Q  i n which' eigenvectors  l i e and o f Ga'rding's I n e q u a l i t y  [2, p. 1983* and i n v i e w o f t h e e s t i m a t e s  on t h e L a p l a c i a n  [12, p. 7], i t i s c l e a r t h a t t h e s m a l l e s t e i g e n v a l u e s e l f - a d j o i n t operators  o f t y p e (3«l) and ( 3 - 2 ) , must tend t o  i n f i n i t y as t h e domain i s so p e r t u r b e d to zero.  of formally  t h a t i t s d i a m e t e r tends  F u r t h e r m o r e , by a l o n g and t e d i o u s 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. 4 2 1 ] , we can assume t h a t t h e e i g e n v a l u e domain  G  v a r i e s c o n t i n u o u s l y when t h e  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 t o t h a t  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 b e g i n by c o n s i d e r i n g t h e o p e r a t o r (3.1) w i t h  b.. = c. = 0 ;  s i m p l i c i t y assume t h a t  R  i , j = 1,...,n .  d e f i n e d by  We s h a l l a l s o f o r  contains•the half-space  (x : x  > 0} .  65.  3.7  Theorem  The o p e r a t o r  (3.5)  is  Lu =  oscillatory  in  S  R  bounded above i n  / i j l ^ i j W )  •  "  eigenvalue such  o f l\  (x)  is  that  ~ (n-l)uldt = «  [  u  by  b y some number  ^ nX^ where  given  i f the biggest  R  o  L  1  1  denotes the s m a l l e s t eigenvalue  of the problem:  ,4, \  ^4  dt *(t)  for  = §|(t)  some b o u n d e d I n t e r v a l  L u c  By a t r i v i a l  for  [15*  N c: S-^  n-1 ( IT I ) x { t } } 1  4  x  S D u ±  -  L  and  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  domain  of the operator  by  g(x )u  p. 1 0 4 ] , we f i n d  § D  defined  c  that  domains i n t h e shape o f r e c t a n g l e s .  , we c h o o s e a n o d a l  that  : x e  = nX  separation  theorem o f Glazman  S  t e 31  c o m p a r e .(3.5) w i t h t h e o p e r a t o r  We  with nodal  = 0  I , and  g ( t ) = rnin{d(x)  Proof:  t e I  =-u*(t)  N  = 0 .  of . L  Q  i s oscillatory, G i v e n any  and a s p h e r e  C l e a r l y the smallest  d e f i n e d b y (3.5)  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  L  S-,  such  eigenvalue  is- non-positive.  s p h e r e s a s was  sphere  .If  done i n C h a p t e r I ,  66.  we f i n d t h a t one o f them i s a n o d a l domain f o r L  The o p e r a t o r g i v e n b y (3°5) i s o s c i l l a t o r y  Theorem 3-8 r  sufficiently  i f for  large, g(r)  (n-1) \A > 0  and lim r -* A  where  and  P  UjgAj  Proof:  a  r  e  a  s  d e f i n e d i n Theorem 3.7.  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 3-7,  except  t h a t a d i f f e r e n t theorem o f Glazman [15 p. 100] i s used t o ensure the o s c i l l a t i o n o f 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 a r i s i n g i n the  comparison.  I t i s c l e a r t h a t s e v e r a l o t h e r theorems may be p r o v e d b y using 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 differential  equations.  As t h e s e theorems a r e analogous  in -  statement and method o f p r o o f t o Theorems 3-7 and 3-8, t h e y w i l l be o m i t t e d .  S e v e r a l o f t h e s e may be found i n [93.  Similarly,  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  s e l f - a d j o i n t o p e r a t o r s o f type ( 3 - 2 ) , by u s i n g t h e assumptions o f Theorem 3-3'on  (^Hi) •  67. 4.  Non-Oscillation Criteria. For t h e e l l i p t i c o p e r a t o r d e f i n e d by the o p e r a t i o n ( 3 - 1 ) ,  the s t a n d a r d methods used t o p r o v e 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 o r d e r o p e r a t o r s no l o n g e r work, as t h e o p e r a t o r s do n o t g i v e r i s e t o forms a l l o f w h i c h a r e p o s i t i v e d e f i n i t e .  We  n e v e r t h e l e s s , p r o v e 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 under t h e a s s u m p t i o n t h a t t h e m a t r i x  shall,  operators  (b. .(x)) i s p o s i t i v e  definite.'  Theorem 3-9  \ I f the matrix  and t h e f u n c t i o n  - d - £  (b. . - B. .) ^ j C  + v  Is non-negative.in  i s a l s o non-negative  G ,  in  2  J  then t h e s m a l l e s t e i g e n v a l u e  u  of the operator  £  in  G  d e f i n e d by Zv  = - I D.(p. .D.v) - YV  cannot exceed t h a t o f the' o p e r a t o r Proof:  Let  X  2  d e f i n e d by ( 3 . 1 ) .  be t h e s m a l l e s t e i g e n v a l u e o f  associated eigenvector. X u G  L  '{(Z  ±  2  +_Eb.  D.uDu - E  2  and t h e r e f o r e  and  u its  We then have:  a .D . u ) ±  L  X >_ (i , by Courant's  Principle.  J  '  3 g  u 2  du } 2  G ,  6Q\ It i s t h e r e f o r e s u f f i c i e n t t o f i n d c o n d i t i o n s such t h a t t h e second o r d e r 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 t o ensure t h a t  i s also non-oscillatory.  and w i l l n o t he r e p e a t e d  C o r o l l a r y 3-10 in  £  Such c r i t e r i a  here.  I f the m a t r i x  G  a r e as g i v e n i n [93 >  (m„. .)  has p o s i t i v e c o e f f i c i e n t s  and t h e m a t r i x (l. . - 8. .) i s n o n - n e g a t i v e as i s t h e D.(p.) function - q - E + y , then the s m a l l e s t eigenvalue of £ i g  3  1 J  3  cannot exceed t h a t o f t h e o p e r a t o r  We s h a l l now c o n s i d e r - t h e t i o n t h a t the m a t r i x  (m- .)  L-^  d e f i n e d by  operator  (3-2).  under t h e assump-  i s positive definite.  We b e g i n b y  s t a t i n g t h e o p p o s i t e o f Theorem 3.5.  P r o p o s i t i o n 3.11 bounded domain  I f the matrix  (3-4) i s non-positive i n a  G , then the s m a l l e s t eigenvalue  s m a l l e r than t h a t o f  of  L  Q  i s not  L oo  Proof:  I d e n t i c a l t o t h a t o f Theorem 3.5.  P r o p o s i t i o n 3.11 e n a b l e s us t o p r o v e :  Theorem 3.12  Let  (m> •)  and assume t h a t m a t r i x  ( 3 - 0 i s non-^positive i n  assume t h a t t h e m a t r i x  i  (£,-.,• - B. •)  (p.)  function eigenvalue  - q - 2 D.  u p  be a p o s i t i v e d e f i n i t e m a t r i x I n  1 3  Furthermore  i s n o n - n e g a t i v e and t h a t the  ± 3  + y  of the operator  G.  G  i s non-negative. £  Then t h e s m a l l e s t  cannot exceed t h a t o f  L, .  69.  Proof:  Under the assumptions of the  (3.7 )  f  u ( S D. .(m. .D. .u)) > Q * J •' X  f o r e v e r y smooth f u n c t i o n i = l,...,n  .  We  p r o v e Theorem  3»9-  G  such that  may  u = u^ = 0  on  SG  for  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  have a g a i n reduced the problem o f f i n d i n g  oscillation criteria for c r i t e r i a f o r the operator we  u ( 2 D. . (m. .D . .u))  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  non-negative.  We  u  f  theorem,  non-  t o the problem of f i n d i n g <£ .  such  For a c o l l e c t i o n of such  criteria,  r e f e r the reader to [9].  Remarks  The m e t h o d s 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  applicable to higher order operators defined Lu =  2 D D (a i,j,k,£ u  where  m  denotes a p o s i t i v e  1 J  .a K  t  1  clearly  by:  D D u) •  ,  J  integer.  The p r o b l e m 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 , a n d t h e p r o b l e m o f o b t a i n i n g Osc o p e r a t o r s remain open,  2 c r i t e r i a f o r fourth order  as d o e s t h e p r o b l e m o f o b t a i n i n g  c r i t e r i a f o r higher order non-linear  operators.  oscillation  70.  BIBLIOGRAPHY  [l]  M. P r o t t e r and H. Weinberger, On the spectrum o f G e n e r a l Second Order O p e r a t o r s , B u l l , o f the Amer. Math. S o c , 7 2 , ( 1 9 6 6 ) , pp.  [2]  251-255..  L, B e r s , P. John, and M. S c h e c h t e r , 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 , P r o c e e d i n g s o f t h e Summer Seminar, B o u l d e r , C o l o r a d o , 1957.  [3]  J - Schauder, Uber L i n e a r e E l l i p t l s c h e n  Differentialgleichungen  . Z w e i t e r Ordnung, Math. Z., 38, (±93^), pp. [4]  257-282.  S. Agmon, L e c t u r e s on E l l i p t i c Boundary V a l u e 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 o f O p e r a t o r E q u a t i o n s , N o o r d h o f f , Groningen, 1964.  [6]  K. K r e l t h , A Remark on a Comparison Theorem o f Swanson, P r o c . Amer. Math. Sco., t o appear.  [7]  M. G. Krexn,and M. A. Rutman, L i n e a r O p e r a t o r s L e a v i n g I n v a r i a n t a Cone i n a Banach Space, Usp'ehi Mat. Nauk pp.  3-95*  Amer. Math. Soc T r a n s l .  (1)  10  (1962),  pp.  3  (19^8), 3-95-  [8]  R. Courant and D. H i l b e r t , Methods o f M a t h e m a t i c a l 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 E q u a t i o n s , Ph.D. D i s s e r t a t i o n , U n i v e r s i t y o f 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 o f L i n e a r D i f f e r e n t i a l E q u a t i o n s , Academic P r e s s , 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 E q u a t i o n s , P a c i f i c J . Math., t o appear.  [12]  S. G. M i k h l i n , The Problem o f the Minimum o f a Q u a d r a t i c 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  E. A. Coddington and N. L e v i n s o n , e n t i a l Equations,  [14]  R. A. Moore, The  M c G r a w - H i l l , New  Theory o f O r d i n a r y York,  1955.  B e h a v i o u r of S o l u t i o n s o f a L i n e a r  e n t i a l E q u a t i o n o f Second Order, P a c i f i c J . Math., pp.  DifferDiffer-  5  (1955)  125-145.  [15]  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 o f S i n g u l a r D i f f e r e n t i a l O p e r a t o r 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 , D a n i e l Davey and Co., New Y o r k ,  [16]  C. A. Can.  [17]  Equations,  Math. B u l l . , t o appear.  R. A. New  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 Struble. Nonlinear  York,  D i f f e r e n t i a l Equations,  McGraw-Hill,  1962.  [18].  N. M i n o r s k i , N o n l i n e a r  Oscillations,  Van N o s t r a n d , New  [19]  C. A. Swanson, Comparison Theorems f o r Q u a s i l i n e 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 , under p r e p a r a t i o n .  [20]  M. Marcus and H. Mine, A Survey o f M a t r i x Theory and I n e q u a l i t i e s , A l l y n and Bacon, B o s t o n , 1964.  York,  Elliptic Matrix  

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