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On Green's function for the Laplace operator in an unbounded domain. Hewgill, Denton Elwood 1966

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ON GREEN'S FUNCTION FOR THE LAPLACE OPERATOR I N AN UNBOUNDED DOMAIN  by  DENTON ELWOOD HEWGILL B.Sc., U n i v e r s i t y o f B r i t i s h C o l u m b i a , 1963.  A THESIS SUBMITTED I N PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY  i n 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 t o t h e required standard  THE UNIVERSITY OP BRITISH COLUMBIA J u l y , 1966.  In presenting  t h i s t h e s i s i n p a r t i a l f u l f i l m e n t of the requirements  f o r an advanced degree at the U n i v e r s i t y of B r i t i s h Columbia, I agree that the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r reference and study.  I f u r t h e r agree that permission-for  extensive  copying of t h i s  t h e s i s f o r s c h o l a r l y purposes may be granted by the Head of my Department or by h i s representatives.  I t i s understood that copying  or p u b l i c a t i o n of t h i s t h e s i s f o r f i n a n c i a l gain shall, not be allowed without my w r i t t e n permission.  Department of The U n i v e r s i t y of B r i t i s h Columbia Vancouver 8, Canada  ii  Abstract Supervisor  ? r ; G o l i n W.  1  Clark.  T h i s t h e s i s I n v e s t i g a t e s the Green's f u n c t i o n s f o r T  the operator  defined  *  by  D {f e L ( E ) 2  Tt M -Af Here and  H^(E) E  i s a standard  Sobolev space,  i s a domain i n  I n p a r t i c u l a r we  2  f € o§(T).  for  :  | Af € L (E) }  assume t h a t  - a x i s , where  f o r some  cp  i s the,Laplaeian>  which i s taken to.be "quasi-bounded". E  l i e s i n the h a l f - s p a c e  arid i s bounded by the s u r f a c e obtained the  A  i s continuous,  by r o t a t i n g  M xj)  > 0  x  1  ep(x^) and  > 0 about  cp e L (0,+< ) k  B  1  k > 0. The  i s obtained  Green's f u n c t i o n  G(x,y,\)  f o r the operator  T + \  as the l i m i t o f the Green's f u n c t i o n s f o r the w e l l  lo$oiin\p'roiS.leim;on' tne.' truncated' 'domain  E^=Efl  [X£ < X],  Most  o f the expected p r o p e r t i e s o f the f u n c t i o n are developed i n c l u d i n g the i i i e q u a l i t y 0<G(i, where  K  y i  x ) <:.K(.p/3)  p =  |x-y|  i s the fundamental s i n g u l a r i t y f o r the domain. The; eigenvalues  i t i s shown t h a t  and  eigenfunctions  are c o n s t r u c t e d ,  and  i n *"X n ~* \ i where and  1^ n E  8 X 1 , 3  as  \ j  a  r  U  n  »  f o r each  n,  the e i g e n v a l u e s f o r the problem on  e  Furthermore, i t i s shown t h a t the  respectively.  values  X  E  x  eigen-  ) are: p o s i t i v e w i t h no f i n i t e l i m i t p o i n t , and the  c o r r e s p o n d i n g e i g e n f u r i c t i o n s are complete. . A d e t a i l e d c a l c u l a t i o n I n v o l v i n g the i n e q u a l i t y above showsIthat sbme'itisrate Hllbert-Schmidt kernel. t h a t the s e r i e s  S X^ ^' 2  (G^^^)  of  G(x,y,X)  Prom t h i s p r o p e r t y o f !  i s convergent.  o f t h i s s e r i e s t h r e e r e s u l t s are d e r i v e d .  G^ °^ k  displayed  is a I t follows  Prom the convergence The f i r s t  one i s an  expansion formula i n terms o f the complete s e t o f e i g e n f u n e t i o h s , and the second i s t h a t some i t e r a t e o f t h e G r e e n ' s f u n c t i o n tends ;  t o z e r o on the boundary. solution  H(x>X,f),  The l a s t one I s the c o n s t r u c t i o n o f the  f o r the boundary v a l u e problem  AH + XH = f H(x,X,f) ..- 0 f o r a '.Buffleiehtly regular  f  as on  x - 5E  E.  The f i n a l p r o p e r t y o f the Green's f u n c t i o n , namely, that  G(x,yyX)  the f a c t t h a t  tends t o z e r o on the boundary, i s proved u s i n g G^^ ^ 0  I s zero on t h e boundary, and  i n e q u a i i t i t e s e s t i m a t i n g the i t e r a t e s shown t o be unique.  certain  G(x,y, X)  i s also  The a s y m p t o t i c  formula  a g e n e r a l i z a t i o n o f t h e u s u a l a s y m p t o t i c f o r m u l a o f Weyl f o r t h e e i g e n v a l u e s , f i r s t , g i v e n by C. C l a r k , i s d e r i v e d f o r t h e s e q u a s i bounded domains.  F i n a l l y , t h e u s u a l a s y m p t o t i c f o r m u l a due  Carleman f o r t h e e i g e n f u n c t i o n s I s shown t o remain v a l i d .  to  Table of Contents  Introduction C o n s t r u c t i o n o f t h e Green's F u n c t i o n and some o f i t s elementary  properties  The E i g e n v a l u e s , a n d E i g e n f u n c t i o n s o f t h e Problem The i t e r a t e s o f t h e 1 Green 'js F u n c t i o n and their properties Boundary b e h a v i o u r and u n i q u e n e s s o f t h e Green's F u h c t i O h A p p l i c a t i o n s o f t h e Green's F u n c t i o n Bibliography  vl  Acknowledgment .  The a u t h o r w i s h e s t o thank h i s a d v i s o r Dr. G o l l n W. C l a r k 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 - h i s h e l p f u l advice  during; I t s p r e p a r a t l o h .  He a l s o w i s h e s t o thank  Dr. C.A. Swahsoh and E. G e r l a c h f 6 r t h e i r c a r e f u l , r e a d i n g o f t h e : draft f 0 ^ of this thesis. :  Furthermore, the f i n a n c i a l  assistance  o f the N a t i o n a l R e s e a r c h C o u n c i l t h r o u g h a s c h o l a r s h i p I s g r a t e fully  acknowledged.  1. 1.  Introduction  The e i g e n v a l u e problem f o r t h e L a p l a e i a n Au(x) + Xu(x)  =  0  x € E  u(x)  =  0  x e 3E,  ( -A),  namely  has "been t h e s u b j e c t o f much d i s c u s s i o n i n t h e l i t e r a t u r e . and H i i b e r t [5] domain  E  Courant  e x t e n s i v e l y i n v e s t i g a t e d t h e problem when t h e  i s bounded.  T l t c h m a r s h [10]> when he d i s c u s s e d t h e  S c h r K d i n g e r e q u a t i o n gave an a l t e r n a t e t r e a t m e n t o f t h e problem and extended many o f t h e r e s u l t s t o t h e whole p l a n e . R e i l i c h [ 9 ] was t h e f i r s t t o o b t a i n r e s u l t s " w h e n the': domain E  was unbo  d i f f e r e h t from t h e whole p l a n e .  t h a t when t h e doM^  along a given r a y ,  • thetp^blpbem^ had arSi:B^ete.;::set  ;  v  He ."showed  T  o f e i g e n v a l u e s ; Mol&anov [8] gener-  a l i z e d R e l l i c h ' s i r e M l t t o i n c l u d e 'domains w h i c h were "narrow a t i n f i n i t y " i n a w i d e r sense, S p e c i f i c a l l y , t h o s e domains w h i c h do n o t c o n t a i n ^ i n f i n i t e l y many d i s j o i n t s p h e r i c a l b a l l s jDf eq\aal "positive radius.  <31£zman: i n h i s r e c e n t book [7]•" gave a d i s c u s s i o n  of problems c l o s e l y domie>ted ed domains.  t o t h e e i g e n v a l u e problem f o r unbound-  S t i l l more r e c e n t l y , C l a r k [ 4 ] p u b l i s h e d a 'paper "An  embedding theorem i n f u n c t i o n spaces" w h i c h a l l o w s : t h e t r e a t m e n t '"of*an: a r b i t r a r y e l l i p t i c , o p e r a t o r i n s t e a d o f t h e L a p l a e i a n . The p r e s e n t d i s c u s s I 6 n w i l l c e n t e r around t h e e x i s t e n c e ;  and / p r o p e r t i e s 'of the, Green's f u n c t i o n  ( G P ) f o r the, problem.  The, g e n e r a l method^?Of a t t a c k and t h e arguments I n many cases  2.  f o l l o w t h o s e used by T i t c h m a r s h [ 1 0 ] . The f i r s t r e s u l t , whose.proof  i s an a p p l i c a t i o n o f t h e  A s e o l i - A r z e r a theorem t o t h e w e l l known case when t h e doiakiri I s bounded, i s t h e e x i s t e n c e o f t h e  GEV  I t i s important to note that  a l t h o u g h t h i s t c ^ h s t r u c t i p n g i v e s many o f t h e p r o p e r t i e s :1«^ch a GP  must p'oisseisSj i t does n o t show the. GP  boundary.  Prom t h i s ; : ' c o n s t r u c t i o n a b a s i c r e s u l t e a s i l y  namely t h a t t h e  GF  f o r t h e domain.  The!fundamehtal  I s the  tends t o zero a t the  GP  I s bounded by t h e fundamental  follows,  singularity  singularity ( a Bessel functloh)  f o r t h e rprdblem; on the. whole space;  This relation,  w h i c h i s h o t needed i n T i t c h m a r s h ' s development, forms a c r u c i a l l i n k i n the present a r ^ I n P a r t 3 we d i s c u s s t h e e i g e n v a l u e s and  eigenfunctions  o f th^;^prbblem,;br more p r e c i s e l y , t h o s e :of t h e o p e r a t o r  T  d e f i n e d by o8(T)H^(E)  n {f e L ( E ) ;  T f « -Af  2  | Af e  when  L^E)}  f e$ ( T ) ,  where , E ' 'igat.M^-e.s.' -tne.. "narrowness a t I n f i n i t y " c o n d i t i o n g i v e n ;  by C l a r k [ 4 ] , '„• The main r e s u l t t o b e p r o v e d I n t h i s section I s :  t h a t t h e e i g e h f u n c t l d h s ; of  T  are'} c o n t a i n e d i n  H^(E) . . Prom . t h i s  r e s u l t we c o n c l u d e t h a t 1 where t h e  X  b  X.„ „  -»  as-  X -» «  f o r each  n,  a r e t h e 'eigenvalues :for. tjbe'.truncat<ed .domain  E  ( s e e P a r t I I ) and t h e A . ".-  X  v  on  E.  are the eigenvalues  for  T  defined  n  T h i s r e s u l t appears t o be new  b u t not u n e x p e c t e d .  s e c t i o n c o h c l u d e s w i t h some d i s c u s s i o n about t h e  This  eigenfuhctions  n e a r t h e boundary. A t v a r i o u s ' p o i n t s i n t h e argument r e s t r i c t i o n s a r e a p p l i e d t o t h e domain„ for  Most o f t h e s e r e s t r i c t i o n s are Smoothness condition's  t h e boundary;  i s Introduced  however the;most i m p o r t a n t  r e s t r l c t i o n y whlteh  i n P i r t : 4 , i s an assumption about t h e r a t e a t w h i c h  t h e domain n a r r o w s ; a  T h i s c o n d i t i o n i s a's' f o l l o w s :  f o r ; example, when d i m e n s i o n o f the p o s i t i v e ; x ^ - a ^ i s ;  E=2,  l e t the boundary Tof  {x^,  and t h e s e t  ep(x^)}  x > 1  0,  i s a p o s i t i v e :cT&htiniibu.'si'-furi^tlotf;'.•' t h e n we assume t h a t e x i s t s an i n t e g e r  k; such t h a t  qp^Cx^)  the  GP,  i n s t e a d i n P a r t 4 we  w h i c h a r e much smoother.  An i m p o r t a n t  r  there  GP  directly  GP  which i s a  Prom the H l l b e r t - S c h m i d t p r o p e r t y  o b t a i n t h a t the s e r i e s  £  converges f o r some  series c o h ^  E.  as  theorem i n t h i s  Hilbert-Schmidt kernel.  dependent on t h e narrowness o f  cp  where  c o n s i d e r the i t e r a t e s ; o f  re^sfect: I S t h a t there;\e^B %B"' J^: ^%e^jB^^:\ot t h e r  be  i s Integrable to i n f i n i t y .  I t does riot Seem p o s s i b l e toi; t r e a t t h e .Tltchmarsh d o e S i  E  k  we  which IS  Once i t i s known t h a t t h i s  that the i t e r a t e which i s H i l b e r t -  S,chml,dt a c t u a l l y ^ t e n d s t o Z B r d ; o n t h e boundary.  This result^  we" show;iri P a r t 5 > i n t u r n i m p l i e s ; l i h a t t h e  i t s e l f tends t o  zero" a t t h e boundary  and  GF  as  4.  The. l a s t s e c t i o n , P a r t 6, d i s c u s s e s a p p l l c a t i o n o f t h e GF  t o a s y m p t o t i c p r o h l e m s f o r t h e e i g e n v a l u e s and e i g e n f u n e t i d h s .  The f i r s t a p p l i c a t i o n Is.',a p r o o f o f t h e a s y m p t o t i c •Jtytt*) (See Theorem 6.1  ~  J T(X) " E  ^  formula  dx  for notation),  w h i c h reduces t o t h e w e l l known  f o r m u l a o f Weyl, I . e . N( X) when t h e domain was  ~ E  TRF  1  a  r  e  a  E  has f i n i t e a r e a .  T h i s asymptotic  formula  f i r s t g i v e n hy C l a r k [3] f o r a s m a l l e r c l a s s o f domains.  second a p p l l c a t i o n p r o v e s  The  f o r m u l a due  C'arleman f o r .the e i g e n v a l u e s extends t o unbounded  E.  Throughout t h e work a l l lemmas and theorems a r e numbered s u c c e s s i v e l y , f o r example Theorem 2 . 4  means Theorem 4 P a r t 2 .  W i t h i n ; a section-/ e q ^ t l d h s and d e f i n i t i o n s a r e r e f e r r e d t o by number b u t i f w e w i s h t o r e f e r t o e d u a t i o n 6 o f P a r t 4 , would w r i t e e q u a t i o n ; ( 4 . 6 ) . :  R e f e r e n c e s a r e r e f e r r e d t o by  a u t h o r ' s name anci, a n u i b e r c d r r e s ^ b M i n g b i b l i o g r a p h y at the  end.  then  a we  the  t o h i s paper i n the  to  P a r t 2.  C o n s t r u c t i o n o f t h e Green's f u n c t i o n and some o f I t s e l e m e n t a r y p r o p e r t i e s .  In thissectionWe  s h a l l give a c o n s t r u c t i o n of the  Green's f u n c t i o n f o r t h e L a p l a c l a n on domain w h i c h w i l l he d e f i n e d below. Notations; Let  R  denote E u c l i d e a n n-space, l e t x=(x^, x ,  n  2  denote a t y p i c a l p o i n t I n norm.  Let  E  P^  and  let  |x-y|  3E.  For a given  E  x^)  be t h e E u c l i d e a n  denote a s i m p l y connected unbounded domain  w i t h boundary  ...,  R, n  define  E' = {p e E | d ( p , 9E) > b j , where and  b d  I s a f i x e d p o s i t i v e constant which determines denotes d i s t a n c e .  L = {X e where  K  X.| < K,  €  and  v  sometimes w r i t e  on  F i n a l l y we s e t E  x  = {x e E  Consider A  Furthermore, l e t and  |lm X| > v  or  Re \ < -2v},  are f i x e d i p o s i t i v e constants which determine  We w i l l v.  E',  L=*L( v)  t o I n d i c a t e t h e dependence o f  | x | < X, ±  i = l , 2, ...n}  X>0  .  t h e boundary v a l u e p r o b l e m  u(x) + X u(x) = 0 u(x) = 0  x e E x e SE  (1)  L. L  6.  where/ A L (E) 2  denotes t h e L a p l a c l a n .  We d e f i n e t h e o p e r a t o r  T  by: <£>(T) = H ( E )  n [f e L ( E )  0  Af  |  2  e L g E ) }  f  T f = -Af where  in  H^(E)  i f  f €  denotes t h e standard  (2)  o9(T)  Sobolev space (see e.g. Dunford  and Schwartz [6> P • 1652]) w i t h t h e norm  ilfll - [ J m  •  I  |D f(x)| dx] a  2  (3)  1 / 2  |a|<m  E  i n w h i c h we use t h e standard  notations  =1x7 i f K-^ I ct J -  1  Z  •  B  a = (a-L, a ,  a n n  2  space  T  L; ( E ) ,  j>  (4)  states that i f E 'is  f o r problem (1)  i s a s e l f - a d j o i n t operator  X , n = l , 2, 3, n  ...  ponding t o a n e i g e n f u n c t i o n L e t u s assume t h a t  i n the H i l b e r t  e t c . , each e i g e n v a l u e  corres^  u-(x). E  x :  I s s u f f i c i e n t l y regular to allow  us t o c o n s t r u c t a Green's f u n c t i o n i n t h e sense o f T i t c h m a r s h [10, We  n  and has a d i s c r e t e spectrum boh s i s t i n g o f i s o l a t e d ,  2  e^ig^hyalues  a)  .  The c l a s s i c a l t h e o r y bounded, t h e n  ..  G (x,y,X) x  Ch 1 4 ] .  f o r problem (1)  I n the present s e c t i o n  s h a l l need t h e f o l l o w i n g ; properties'/."of  G (x,y,X) s x  7 .  G (x,y,X)  h a s a s t a n d a r d s i n g u l a r i t y f o r x=y.  x  (See Lemma 2 . 7 ) . If  xie E > X ? some L , t h e n  G (x,y,X)  x  zero as y  approaches t h e houndary  define the operator G  x  x  f (x) = J E  G  tends t o  x  o f E . I f we x  ^ by  x  (5)  G ( x , y , X ) f ( y ) dy f ve L ( E ) x  2  X  X  t h e n we need two theorems: G  X  X• "n-W "'' (^n ^ ) " ^ SB  -  u  ( ) x  n  f  o  r  X non r e a l  (6)  and liO ,. f I! x  v " ||f I!  X € L ( v) .  where  1  x  (?)  A l l t h e s e r e s u l t s a r e proved i n T I t c h m a r s h [ 1 0 , Ch 1 4 ] e x c e p t f o r (7),  w h i c h i s o u r f i r s t lemma.  Lemma 2.1.  |lm x | >_ v > 0  Suppose t h a t e i t h e r  or that  Re X < - 2 v t h e n llGx^fll"•'.<  v"  1  llfll  .  (8)  P r o o f . . We'^knowT-itrpra' t h e c l a s s i c a l t h e o r y t h a t  G  x  ^  i s the  resolvent operator o f a non-negative s e l f - a d j o i n t operator L (E ) i . e . 2  X  G ^ x  Now i f  X=a+i0,  = (T-XI)" . 1  x  al<~-2v C O  and |p( < v,  then  T i n  8.  Hi*-xx I  >  ||T+2vi||  - lleill  > ||T+2vl|| But  since  IIT-XIH  >  T+2vI.\> 2 v l ,  TX), v,  and  v .  ||T+2vl|l>  so t h a t  2v.  Hence  therefore  ||Q II = IKT-XI)" !! 1  <  X;X  which i s equivalent to ( 8 ) .  One  v"  ,  1  can e a s i l y show, U s i n g t h e  r e s u l t s \ o f T i t c h m a r s h [ l O , Ch 1 2 ] , t h a t (8) h o l d s f o r |lm X| >  v > 0, The  c o m p l e t i n g t h e lemma.  f o l l o w i n g c o n s t r u c t i o n o f t h e Green's f u n c t i o n f o r  t h e problem ( 1 ) w i l l he g i v e n o n l y f o r t h e p l a n e .  A  c o n s t r u c t i o n i s a v a i l a b l e f o r higher dimensions.  The  similar argument,  up t o a p o i n t , w i l l be s i m i l a r t o ' T i t c h m a r s h ' s argument f o r the whole space  R| 2  requires;Us•to  however, t h e p r e s e n c e o f t h e boundary o f  be much more p r e c i s e about the n a t u r e o f  E  the  convergence. Let g(x) « g(x,u) =  ( h u<* I - \ (i -15>i «•<•»  r=|x-u|.  We  s u b t r a c t i n g the f u n c t i o n  x  Note:  P  Y  0  i f  remove t h e s i n g u l a r i t y from  T (x,y,X)  if  <  / where  2  s—  g(x);  we  G (x,y,X) x  i s dependent on  R,  r >  G (x,y,X) x  R, by  set -  since  g(x,y) g  is.  (9)  I  9.  Theorem 2.2.  The s e t  [X|X>0]  such t h a t t h e s e q u e n c e [ Q function  G(x,y,X)  (x,y,l)]  v x  has a subsequence  for  and  X>  x, y e E  0,  X  converges p o l n t w i s e t o a  f o r a l l x, y e E  and a l l X  n o t on t h e  Furthermore, given  t h e sequence; [ p ^ X y y , X) ]  and  x  X  k  non-negative p a r t of the r e a l a x i s ; L( v)  ^ k ^ k~*  E' c E,  converges u n i f o r m l y  X € L( v ) .  Proof. FIGURE 1  Let x, y, u e E Let  f ( x ) = G (x,y,X)  and  x  x  write  the c i r c l e  r = |x-u|  |x-u| ==r<R  formula f o r the region  r^R  and  'g-'('x') he as above; r ' = |y-u|,  be c o n t a i n e d i n  for  as i s f i g u r e 1  E . x  By  Green's  ( c f . T i t c h m a r s h [ 1 0 , p. 3 + ] ) , we  have (u,y,X) - g(u,y) = = - i :  f  G (x,y,X)dx Y  r<R +  X  J r<R  g(x,u) G ^ X j y y X ^ d x  + (10)  .  10. P ( x , y , X ) = G (x,y,X) - g(x,y)  Let  Y  forthis  Y  R,  and  F(x,u) = S u b s t i t u t i n g t h e s e r e l a t i o n s i n t o (10) ,>we o b t a i n P x  (u,y,X)  =  J E  G (x,y,X) F(x/u)dx x  X  = (G x (.",y,x), P ( - , U ) )  .  The f i r s t p r o b l e m i n t h e p r o o f i s t o show t h a t  P ( u , y , X) x  i s  u n i f o r m l y bounded f o r u , y e EX and X e L ( v ) (where X_ o- , ° ° ahd v a r e a r b i t r a r y f i x e d c o n s t a n t s ) . We f i r s t a p p l y t h e Q  P  Schwarz i n e q u a l i t y t o  x  t o obtain  < l|G (.,y,X)H  ir (u,y,X)l x  llF(-,u)||.  x  (11)  F o r t h e Second norm i n ( 1 1 ) , we have lF('.;yuj||  2  <:-4TT  ~ ir R  |gix,u; f <3X + 2|X| f | g ( x , u 1) 2| dx , r<R r<R 2  J  from t h e d e f i n i t i o n o f F ( x , u ) .  I!F(SU)|! K(u,R,|x|)  where  2  J  2  Therefore  < K(u,R,|x|),  i s bounded i f u  and  X  a r e bounded, t h e  d i s t a n c e between t h e boundary o f E  and u  and  The f i r s t term i n ( 1 1 ) ,  R  namely  i s bounded away from z e r o ; ||G ( • ,y,X) ||, X  as f o l l o w s :  i s greater than  i s more d i f f i c u l t t o e s t i m a t e .  O b s e r v i n g t h a t t h e form o f  P(u,y,X)  R,  One p r o c e e d s satisfies  11.  P (u,y,X) = G  Lemma 2.1 (note  x  I! r (u,.,x)l! < 2  v-  x  2  ^F(u,y)),  x  ||F(-,U)!|  we have  2  (12)  < v " K(u,R,|x|), 2  where e i t h e r  |lm X| > v > 0 P  the d e f i n i t i o n ; o f  !JG (^,-,X>(i  2  X  or  Re X <_ -2v  x  i n (9) one has  <  2|1 r ( u , , X ) ||'' +  and  2||g(u  2  x  v > 0.  From  Oil , 2  i  from which i t f o l l o w s t h a t  |l.G (u,-,X) ||  2  x  <  (l+v~ ) 2  K(u,R,|x|)  (13)  I n these formulas i t I s important t o note t h a t of  X.  K  i s independent  Combining these w i t h t h e Schwarz i n e q u a l i t y above, one  o b t a i n s the f i n a l  estimate  I r ( u , y , X ) | < f(l±v" ) K ( y , R , | X|) K ( U , R , | X|) ] 2  l  /  2  x  where  y  and  the d i s t a n c e  u  cannot be c l o s e r t o the boundary o f  R.  x  and  K  E  than  We can shorten t h i s i n t o  I r (u,yA)| where  ,  i • • , K ( i i y y  I s bounded i f f o r some g i v e n  >  . R , ] x ' | ) ( 1 4 )  E', X^ o  and  L . o  u, y e E  X e L . Q  We now a p p l y the same argument as T i t c h m a r s h [10, p . 3 5 ] * to arrive at  where  |u'-u| < 6 = 6(e)  independent  of  ( X ) I n o r d e r to:do  t h i s , o n e J u s t needs t o c o n s i d e r the r e p r e s e n t a t i o n o f  P , x  e q u a t i o n (10.), and make a p p r o p r i a t e e s t i m a t e s u s i n g t h e  X-uniform  hound ( 1 4 ) .  y  By 'symmetry a s i m i l a r r e s u l t h o l d s f o r the  We now  have t h e d e s i r e d e q u i c o n t i n u i t y I n  h u t we must a l s o : have i t i n the  u  and  D  x  r  x  x  8 3  fi  G (x,y,X) x  ®x  =  ( x( *'->'^>"• %(• G  u  y,  X v a r i a b l e , i n order to apply  the A s c o l i - A r z e l a theorem., . T h i s I s a c h i e v e d as f o l l o w s " resplvent equation  variable  by t h e  we have *y A ) )  •  The r i g h t hand :sid£ I s hounded, as" X tends t o I n f i n i t y , f o r u , y € E^. and X e L .,• by the* e s t i m a t e (13) and t h e Schwarz o ° i n e q u a l i t y . S i n c e t h e p a r t i a l d e r i v a t i v e ; o f /"^ w i t h r e s p e c t t o X  X  ;  i s u n i f o r m l y bounded,  P  x  w i l l be u n i f o r m l y c o n t i n u o u s I n X.  The above c a l c u l a t i o h s and remarks show t h a t t h e s e t o f [P (x,y,X)3  functions (E , 1  X  x  and  Q  L  Q  a s X — *, f o r x, y e E£ and X e L " o b e i n g f i x e d ) , i s e q u i c o n t i n u o u s I n each o f the  three variables separately.  The A s c o l i - A r z e l a theorem says t h a t  such a s e t i s compact, i . e .  there e x i s t s a f u n c t i o n P(x,y,X)  such t h a t subsequence  P  A  (x,y,\) tends t o P ( x > y , X ) , u n i f o r m l y f o r some k {X^} t e n d i n g t o I n f i n i t y , when x, y e Ejr and v  (  X €L. ;• O Now  by a s i m p l e d i a g o n a l I z a t i o n p r o c e s s , we can o b t a i n a  Q  13. sequence  [ p •• f*',y,"X) ]  x, y e E , axis.  w h i c h converges ( p o l n t w i s e ) f o r a l l  Y  .....  k  A  and V e"C  with  not O H t h e n o n - n e g a t i v e r e a l  X  Namely, l e t <  X .  o El o  <  X ,  . . .  -»  »  1  c E| c  ...  E  1  L .cL Q  1  ... - {X | Im x4=© o r X < 6 ] .  c  d ( p , a E ) > n -1.}  We c o u l d choose f o r example; E j t o he E' = {peE etc.  L e t [X(G,n)]  [ Px'('6'-n)'^ to  converges u n i f o r m l y when  x, y  I n f i n i t y such t h a t  and X  are  restricted  E^, X - and L.^. L e t [ X ( l , n ) ] be a subsequence o f Q  such t h a t to  be a sequence a p p r o a c h i n g  f nx(i  E£, X^  n  ) 3 converges u n i f o r m l y f o r . x , y > X  and L ^ , and so on.  X* = [X(n,n) ]  restricted  Then t h e d i a g o n a l sequence  [ P *;]  I s such t h a t  [X(6,n)i  x  converges;polntwise  for  d e s i r e d v a l u e s o f x , y , X. We n o t e t h a t t h i s sequence a l s o h a s the p r o p e r t y t h a t , g i v e n any u n i f o r m l y f o r • x, y e ,E  X  E», X  and L ( v ) ^ [ P l x  and X e L ( v ) . :  converges  T h i s completes the  proof  o f Theorem 2.2. A cohseqiiehee o f t h e above theorem I s t h a t we have a Green's f u n c t i o n f o r  E  w i t h the r e p r e s e n t a t i o n  ©(*,y>X)  =  P(x,y,X) - g(x,y) .  i^rthermbre,  r(v,y,\)  p  =  s a t i s f i e s the I n t e g r a l equation  J •  lfR  r<R  G(x,y,X)dx + X  J  r<R  g(x,u) G(x,y,X)dx  ( l 6 )  14. where  r=|x-u|  and  c o n t a i n e d i n E.  R  i s so s m a l l t h a t t h e c i r c l e  |x~u|<R i s  We s h a l l now examine some o f t h e elementary  p r o p e r t i e s o f t h e Green's f u n c t i o n . Theorem 2.3.  G(x y,X)  i s c o n t i n u o u s f o r x=f=y  s  G(x,y,X) where  l o g ~ + 0(1)  p = |'x-y|,  ..a. G ( x , y A )  +  dp Theorem 2.4.  p - 0  and a l s o  _ JL. (i)  =  as  and i t i s such t h a t  a s p - 0.  0  27rp G(x,y,X)  has c o n t i n u o u s p a r t i a l  derivatives  up t o t h e second o r d e r e x c e p t a t x=y, and {A + X] G ( x , y , X ) = 0  i f x + y •  The above two theorems a r e proved i n t h e same manner as i n t h e case o f bounded  E>  Theorem; 2.5. |lG(u;.,X)|! where Proof.  K  :<  2  v"  K(u,R |X|)  2  y  i s bounded i f u e E'  By i n e q u a l i t y (11) we have  °  and  X e L (v) . rt  lir (u, •, X) (I .< v " K(u,R, | X | ) . 2  2  x  Then f o r EV  j  IT  and  X  fixed  Q  (u>y,X)| dy 2  x  <  (E*=E' ) o X  v"  2  K(u,R,|x|)  (X>X ). 0  E* F i r s t l e t X -• »  t h r o u g h t h e sequence d e f i n e d i n Theorem 2.2.  We t h e n have J I r(u,y,X)| dy E*  <  2  v~  K(u,R, | x|)  2  where the r i g h t hand s i d e i s independent of E*  tend t o  E,  E*.  Thus,  one has hy F a t o u ' s lemma  || P(u,-,X)||  <  2  v"  K(u,R,|x|) .  2  I f we combine t h i s I n e q u a l i t y w i t h t h e d e f i n i t i o n o f result  ||g(x,-)ll = AR , 2  2  t h e n Theorem 2.5 w i l l  We d e f i n e t h e o p e r a t o r G  x  f(x)  =  G^  X  J - G(x,y,X) f ( y ) dy E  c o n t a i n e d i n some  Theorem 2.6.  :  If  P  and t h e  follow;  as f o l l o w s f e L (E). 2  The I n t e g r a l e x i s t s i n view o f e q u a t i o n (13) all  letting  tor a l l  xeE  and  L.  H(x,X,f) = - G f ( x ) , x  where  f € L (E) 2  and  X  I s n o t on t h e n o n - n e g a t i v e r e a l a x i s , t h e n ||H(.,X,f)|| < U+X]  v'  llfll ,  H(x,X,f) = f ( x )  H(x,X) = 0( | X | |X| > 6 > 0 We  1  1 / / 2  and  v ) _ 1  uniformly f o r  x e E  ,  Y  Im X = v =(= 0.  s h a l l prove t h e f i r s t r e s u l t and remark t h a t t h e o t h e r  two a r e proved by t h e u s u a l methods. Proof.  and  The f i r s t i t e m t o show i s t h a t  (See T i t c h m a r s h [10,  Ch 12]) .  H" (x,X,f) - H ( x , X , f ) x  16.  uniformly H -H  x € E£ , where o  x  x  f  J G(x,y,X) f ( y ) dy - J G ( x , y X ) f ( y ) dy .  =  x  H" (x,X,f) = - G ^ f ( x ) . l o w  x  E  E  x  I f we s e t G =0  outside  x  %-H  y  - "J  E  we have  x  [G(x,y,X) - G ( x y , X ) ] f ( y ) dy + x  5  E-E* J  [G(x,y,X) - G ( x , y , X ) I f ( y ) d y ; x  E* f o r any E* e q u a l t o some  E i . By t h e Schwarz i n e q u a l i t y t h e  second term t e n d s t o z e r o a s u'tilfbrmly o v e r  X -» »  since  G  x  tends t o G  E*. A p p l y i n g t h e Schwarz I n e q u a i l t y t o t h e f l r s t  term, one h a s J  | f i r s t term)  |G(x,y,X) .- G (x,y,X) 1 dy • J 2  x  E-E*  2  E-E*  {2||G(x,.:,x)li :+ 2||G (X^,X)!I } 2  *  |f| dx  2  X  J  |fi dx. 2  E-E* T h i s term t h e n c a n he made s m a l l '''uniformly i n X f e L (E),  x e E'  2  2.5".  and t h e r e s u l t s o f e q u a t i o n  (13) and Theorem  We how have t h e d e s i r e d convergence f o r . H . x  If J*  X e L(v)  |H (x,X,f)T dx 2  x  and E * c E , 1  v ~ J | f | dx 2  2  EJ' <: v -  I f we l e t X '-» »  we have by Lemma 2.1  x  E*  ;  hy n o t i n g  2  J  | f | d x = v " ||f|| . 2  2  2  t h r o u g h t h e sequence d e f i n e d I n Theorem 2.2,  IT* we have by t h e u n i f o r m convergence f o r H^v J : | H ( x , X , f ) | d x .<: v " | | f j j E* 2  I f we l e t E*  tend t o  E,  i|H(-;x,f)i| Remark;  <  2  2  proved above,  .  we have by PatoU' s lemma v"  1  ||f!|.  Q.E.B„  A t t h i s ' p b i h t i r t t h e argument we do n o t know t h e boundary  behaviour o f G(xiy,X). G^(x,y,X)  A l t h o u g h we know that- f o r each f i x e d  goes t o z e r o on t h e boundary a s y  X,  tends t o the  boundary, o u r convergence theorem (2.2) i s n o t s t r o n g enough t o i m p l y t h e r e s u l t f o r G(x,y, X ) . We now t u r n o u r a t t e n t i o n t o t h e f u n d a m e h t a l s i n g u l a r i t y f o r t h e domain and i t s r e l a t i o n t o t h e Green's f u n c t i o n . c o n s i d e r t h e case o f a gteneral d i m e n s i o n Lemma 2.T. ; ( B r o w n e l l  n.  [ 2 , p . 555, Lemma 2.1]) ;  There e x i s t s a r e a l p o s i t i v e f u n c t i o n for; a l l r e a l , p o s i t i v e is" r e a l - a n a l y t i c v l h  Here we  r  H  n  m  (r)  and a), and a l l i n t e g e r s  defined  n > l , which  ty and stibh t h a t t h e f o l l o w i n g holds?;  0  <  H  (r) <  0  <  HI:. ( r ) <  n -> 3  M„ ~ ^ " ^ exp(-wr/4) n  2  r  M  1 + log(l+(u)r)- ) 1  p  e  x  p  (  —  18.  where t h e , M 's  a r e c o n s t a n t s independent o f  n  u> and  r , and  o*  n  i s t h e volume• o f - t h e U n i t h a l l i n R.. The H ( r ) so d e f i n e d n n, ar ' i s c a l l e d t h e fundamental s i n g u l a r i t y f o r A-iu . /For t h e / d e t a i l s o f t h e p r o o f see B r o w n e l l ' s p a p e r [2, p. 555]. •' ,ThWoiejd:^B . r  Suppose t h a t  :  alml.t"df f u n c t i o n s  G(x,y,X)  G (x,y,X),  p=|x^-y|,  K(p<») 2  Lemma 2.7, and X=-tt) Proof.  Since  E  as a p o l n t w i s e  as " i n Theorem 2.2 :f o r  x  0 < G(x,y,X). :<; H ^ / p ) where  i s obtained  <  n=2.  Then  K(p «) :  I s t h e bound f o r  . fp)  given i n  cu > 0.  where  i s bounded t h e makimum p r i n c i p l e c a n be a p p l i e d  x  to prove 0 but f o r f i x e d  <  G (x,y,\) x  x , y e E,  <  n  G (x,y,X) x  goes t h r o u g h t h e sequence 0  <H ^(p) ,  x  G(x,y,X)  Theorem 2.9.  and  X,  G(x,y, X)  G(x>y,X)  d e f i n e d i n Theorem 2.2. <  H  .(p)  Note t h a t t h i s r e l a t i o n i m p l i e s t h a t for fixed  tends t o  as =  y  X  Thus  a l l x, y e E. G(xyy,X)  tends t o i n f i n i t y . G(y,x,X)  as  x, y e E.  tends t o zero,  1 9 .  Proof.  We know t h a t G (x,y,X)  =  x  for  x> y e E  and  x  X s u f f i c i e n t l y large.  Since the l e f t hand  ;  s i d e tends t o . ®(x,yyX) as/  G (y,x,X)  and the r i g h t hand / s i d e tenCs t o  X f tends t o i n f i n i t y V 1  G(y,x,X),  the: r e s u l t f o l l o w s .  The; f i n a l r e s u l t i n t h i s S e c t i o n i s G(Xyy>X)  Theorem 2 . 1 0 .  = G^ ^ (x,y, X) 2  and  G(x>yy X) == h i G^ ^") (x>y/ X) n+  for  X a n e g a t i v e number and  Proof.  x, y e E .  F i r s t we must e s t a b l i s h  the " r e s o l v e n t equation"? .; ' ( 1 6 )  (X-X») (G(-,x,X), G( • y y A ' ) |= G(y,x,X) - G(x,y,X'-) for  X  and  X'  negative.  The u s u a l p r o o f o f ( 1 6 ) r e q u i r e s t h a t ;  oh  the boundary;  boundary r e s u l t  G(xyyyX)  be zero  however we want this;lemma independeht o f the  so we proceed  differently.  We have (X-X») J  by E*:  G (s,x,X) G ( S , y , X ' ) d s = G (y,xyX) - G (x;y,X') ( 1 7 ) x  x  Green's theorem, s i n c e IS some f i x e d ':  E{ c E . •• o • x  x  E  x  i s bounded.  x  Let  x, y  € E*; where  The r i g h t hand s i d e o f ( 1 7 ) converges  2 0 .  to  t h e r i g h t hand s i d e o f ( 1 6 ) by Theorem 2 . 2 .  Thus we need t o  show t h a t t h e d i f f e r e n c e (18) t e n d s t o z e r o as extend  G (s,y,X) x  (G('xVi),.  by z e r o t o  G(- ,y, X 0 ) -  s£E .)  X  «.  (Here we  ;  x  ( G ( ' >*i X.)' * G ( • >Y> X')'.) x  x  ..-i ( G p , x , X ) - G ( . , x , X ) , G(.,y,X«)) +  (l ) 8  x  • + ( G ( - , x , X ) , G(-,y,X') - G ( . , y , X ' ) ) • x  x  By t h e Schwarz i n e q u a l i t y t h e f i r s t term o f t h i s e x p r e s s i o n i s l e s s than l!G( -,x,x) - G (>,x/X) j] x  2  • ||G(*,yX') II2  The second f a c t o r o f ( 1 9 ) i s bounded s i n c e fixed.  ( 1 9 )  y e E*  and  X'  is  ITbw cohsideir t h e f i r s t f a c t o r o f ( 1 9 ) / w h i c h e q u a l s  J G(s,x,X)  J  .  - G (s,x,X) x  I ds 2  E-E*  +  J  |G(s,x,X) - G ( s , x , X ) |  2  x  ds.(20)  E*  The l a t t e r t e r m o f ( 2 0 ) goes t o z e r o f o r each f i x e d  E*  goes'; t o I n f i n i t y , "since" G  E*.  x  converges u n i f o r m l y bh  as  X  Time  r e s u l t w i l l now be complete I f we can show t h a t t h e f i r s t term o f (20)  can be made s m a l l , i n d e p e n d e n t l y o f  choice of  X, by an a p p r o p r i a t e  E*.  | l s t term o f ( 2 0 ) | < 2  J  | G( s>x, X) | d s +. 2 J  E-E* <k  J  | G ( s,x, X) | d s  2  2  x  E-E* |K(pU))| ds 2  E-E* by Theorem 2 . 8 , where  p = | s-x|  and  X = -to ,  to > 0 .  Thus t h e  21. f i r s t term o f (20) can "be made s m a l l by p i c k i n g  E*, arid t h e  c h o i c e o f E* w i l l be independent o f X. A s i m i l a r argument can be s t p p i i ^ d t o t h e second term i n (l8) ^ so t h a t (18) approaches and e q u a t i o n (16) i s p r o v e d .  z e r o a s X -»  From e q u a t i o n (16) we hsLve ( G ( - , x , X ) , G(-,y,X')) iTtiw$:,le%/  =  ix-te)"^^;-?^) - G ( x , y , X ' ) ) 1  ^-)if t h e theorem i s p r o v e d .  symmetric i n x  and : y  b y Theorem 2.9.  show t h e r e s u l t s f o r h i g h e r  iterates.  Note t h a t  Gi s  Similar proofs  will  22; P a r t 3.  The  EigenvaiUe.s  o f t h e Problem. | In this section weShall  i n t r o d u c e c o n d i t i o n s on  that w i l l alldW c p h s t r u c t i o n of eigenvalues f o r j p t d b i e m : (2.1)  and  E  eigenfuhetions  I t t u r n s out t h a t a c e r t a i n c o n d i t i o n on  E  c a l l e d "narrowness a t i n f i n i t y " w i l l be s u f f i c i e n t , p r o v i d e d  E  R e l l i c h [9]  satisfieSx'ceHa^ Mpieanov ;[8]  gave " n a r r o w n e s s f a t  I n f i n i t y " conditions s u f f i c i e n t n  ^.o r':p:rbhl'em;'';'('2)l) t o have a d i s c r e t e ; s p e c t r a i  a c o n d i t i o n which; w e • S h a l l eigenvalues. I  The  6(X)  gave  use t o c o n s t r u c t the e i g e n f u n c t i o n  X>0  t h e r e e x i s t p o s i t i v e numbers  6(x)  —  d(x);  b)  d(X)'_••'/6(X) <M  c)  f o r each  +  b  d(X)  x—>  as  < »  x e E-E  for a l l  X  there e x i s t s a point  x  |x-y| < d(X)  and  E n [z  Condition I iniplies that  E  y  such  |z-y| < 6(X)'•} = 0.  I s narrow a t i n f i n i t y i n t h e  f o l l o w i n g sense; The  set  E  i s s a i d t o be "narrow a t i n f i n i t y " i f lim X-*»  where  p(A),  and  satisfying  a)  that  C l a r k [4]  c o n d i t i o n ( c a l l e d c o n d i t i o n I ) i s as f o l l o w s ;  C o r r e s p o n d i n g t o each and  and  for  p(E-E ) Y  =0,  A: an a r b i t r a r y s e t i n p(A) = sup d(x>SA). xeA  R, n  Is defined  by  2J.  I t i s clear that t h e spheres in^  p(A) i s t h e ;supremum o f t h e r a d i i o f A.  When T h ^ T  the operator  ( e q u a t i o n .2.2), w i t h c o n d i t i o n  Lemma 3 . 1 . L (E); 2  points;  T  t h e spectrum  d(T) I s d i s c r e t e arid has no f i n i t e  limit  R^(T) = ( X I - T ) "  1  continuous.  T h i s r e s u l t g e n e r a l i z e s t h e r e s u l t o f R e l l l e h [ 9 , p . 335] e l a s s ;pf dbmairis. a(T) => [ X ^ ] , S}ffieTeSX£;<..\fe.,£  Let n  E,'• we c a n concludes  f o r X £ CT(T) t h e r e s o l v e n t b p e r a t o r  t o &j.larger  \yr  on  I s a s e l f - a d j o i n t b p e r a t o r I n t h e H i l b e r t space  I s cpmpTetely /Remark;  I  and'  n  (  x  ) he t h e e i g e n v a l u e s  X^ ...  etc.  Let  and e i g e n f u n c t i o n s f o r t h e  ' problem; •,  j A u(x) + X u(x) = 0  x € E  x e a E  u(x) = 0 w h i c h a r e known t o ' e x i s t s i n c e  E  x  x  x  , i s bounded.  Since  we have, by elementary v a r i a t i o n a l p r i n c i p l e s . ( c f  E  x  c E  e.g. Glazman [ 7 ] ) ,  X •> X• . I n view o f t h e f a c t t h a t X „ i s a h o n - i n c r e a s i n g X,n — n . • >• x,n • : f U n b t l b h - o f X /: f o r each f i x e d " n, we have r  v  v  •\x h "* *n••— n x  Lemma 3-2. For  as  X - «  f o r each  n.  (2)  ( T i t c h ^ a r s n [10, p . 334] theorem 2 2 . 1 4 ) . p = l , 2y 3 ,  . ...... l e t H ( x ) have c o n t i n u o u s . p a r t i a l -  24. d e r i v a t i v e s up t o t h e second o r d e r and s a t i s f y t h e d i f f e r e n t i a l equation (A + X where  f  order.  and  As  q  p  - qjHp  -  t,  have c o n t i n u o u s p a r t i a l d e r i v a t i v e s o f t h e f i r s t  p -» »,  let H  tend t o a l i m i t  p  some g i v e n r e g i o n , and l e t  Xp  H(x) u n i f o r m l y over  tend t o a l i m i t  X.  Then  H(x)  has c o n t i n u o u s p a r t i a l d e r i v a t i v e s up t o t h e second o r d e r , and t h e equation (A + X - q) H(x) is  satisfied. The  i n mean  x,n v  same r e s u l t h o l d s i f we a r e merely g i v e n t h a t  square.  Lemma 5 - 3 . u  = f(x)  There e x i s t s a s e t o f f u n c t i o n s  ( x ) tends to' u„(x) ' n ' {X}  n  i n L ( E ) f o r each  x  sequence o f  (u (x)},  0  d  y  n  A u (x) + X n  n  the  and some sub-  t e n d i n g t o i n f i n i t y ( t h i s subsequence can be  p i c k e d from t h e subsequence g i v e n i n Theorem 2.2);  u  such t h a t  n  u (x)  = 0  n  moreover  x e E;  has c o n t i n u o u s p a r t i a l d e r i v a t i v e s up t o t h e second o r d e r ; u  Proof.  n  a r e o r t h o n o r m a l , and Extend  u  x  n  (x)  X  n  are p o s i t i v e .  by z e r o t h e E - E  x >  Since  H  25we have  r2  ,  K,Jl  and i f we t a k e  <  X > X .  1  >  h,n  +  we have  o 3 o f C l a r k [ 4 ] , t h e embedding map  Np#|fby;t^  i s completely continuous. bounded I n  H^(E),  In  Let  We  L (E). 2  H^(E)  S i n c e by (3) t h e sequence  u^^  i t must t h e r e f o r e have a subsequehce  u (x)  be t h e ^ i i m i t  n  is  convergent  o f t h i s c o n v e r g e n t su^seqiienc^ ; 1  s h a l l remove, by a d i a g b h ^ i i z a t i o n p r o c e d u r e , t h e r e s t r i c t i o n  t h a t t h e sequence chosen may subsequence sequence  X^  Xg  depend on  such t h a t  bf  X^  [u  ^\  x  such t h a t  [u  ••• /•  n.  F o r example, f i n d  ||u (x)' -'.''u' '^('x)f ••-»•;6". n  as  x  S i n c e we a l s o have  X „^X,.  x  2  ^  2. ..  c o n v e r g e s , and so on.  X • -» »  f o r each  V  h,  f o r each  result n.  we can a p p l y Lemma  n  A , H  obtain A u (x) + X  c  n  Ljemiaa 3 ; 2 , a l S 6 s ^ s t h a t derivatives^  u  u (x) =  n  n  ; R  x e E.  0 ,  \ d ! i l have c o n t i n u o u s p a r t i a l  t o the^second o r d e r .  Furthermore  orthonormal since <V  m>  u  = \  ( X,n> X,m> = m,n  ± m  A * -  u  0 0  u  a  converges',''-.then f i n d a sub-  We t h e n t a k e t h e d i a g o n a l o f t h i s p r o c e s s . t o show t h e  to  c Lg(E)  6  •  u  n  are  (4)  3-2  26.  Since J I X,nl X D. u ... -» D . u Inside 1 x,n i n v u  2 d x  = X,n < h - n X  i  f  X  ^ d> X  E  and  v  I f we now r e c a l l t h a t to  ||7 u|| £ X 2  n  X  x n  - X^  F a t b u ' s Lemma shows  we c a n sharpen t h e r e s u l t (5)  by a s i m p l e c o n t r a d i c t i o n argument.  i t follows that a l l the X Theorem 3.4;  E,  n  From t h i s  . "are'non-negative;.  L e t E / be such t h a t through every p o i n t o f t h e  boundary- p a s s e s a c i r c l e ; w h i c h l i e s o t h e r w i s e  entirely inside  E  , ( r o u g h l y ; t h i s Means t h a t t h e " p o i n t i s n o t t h e v e r t e x o f an outward-pointing  angle).  the " e i g e n f u n c t i o n "  Also, l e t E  be " s t a r - s h a p e d " .  Then  u : i s c o n t a i n e d " i n ' H^(E); . n o'  Remarks; 1.  The p r o o f w i l l be g i v e n f o r dim E=2, b u t h o l d s f o r a l l dimensions;  2.  The hypbthesis^  E, i s " s t a r shaped" can be d i s p e n s e d  w i t h e n t i r e l y by u s i n g a 3.  partition o f u n i t y .  The h y p o t h e s i s ; t h a t t h e boundary o f E  has I n t e r i o r  c i r c l e s , ; e t c . , c a n be weakened. The p r o o f s o f t h e a b o v e ; t h r e e remarks w i l l n o t be g i v e n  since  Theorem 3.4 w i l l b e ; S u f f i c i e n t f o r b u r purpose as i t i s p r o v e d ; P r o o f o f Theorem 3.4.  Write  u=u  n  f o r the present.  We  noteSthat  2f.  "by i n e q u a l i t y ( 5 ) •  u e H (E)  u ( x ) -• 0  g i v e s t o show t h a t  n  The p r o o f t h a t T i t c h m a r s h [ 4 , p. 9 9 ] as  hounded vjorfcs f o r o u r ease a l s o . we need'to show t h a t i n t h e ||||^ norm. Let  u  x -* 3E  Thus, t o complete t h e p r o o f ,  c a n be approximated by  We s h a l l approximate  f o r 6••'> 1;  then define  g ( x ) = g( |x| - R ) .  g ( x ) as R  C*(E) f u n c t i o n s  i n several steps. g(B) € G*(R^);  g(B) = 1  and  for B < 0,  We want t o show t h a t  R  can be made s m a l l f o r s u f f i c i e n t l y l a r g e  |u-g u|j R  u  g ( S ) be a f u n c t i o n such t h a t  0 1 g(B) < l j g(B) - 0  llu-gj^ujl^  f o r t h e case o f E  2  R.  Now  = J*|u-g u| dx + J l v C u - g g U ) | d x 2  R  2  E  E  (6)  v(u-g u)I dx.  u - g u | dx +  R  R  >R  >R The f i r s t i n t e g r a l i n ( 6 ) i s l e s s t h a n  4  u dx>  since  >R 0 < g ( x ) <. 1. R  Thus t h i s i n t e g r a l c a n be made s m a l l f o r l a r g e  s i n c e - u '€; L ( E ) .  C o n s i d e r now t h e second i n t e g r a l i n (.6)  2  v(u-g u) | d x < 2  | VU| d x + 2 • J  2  2  R  >R  >R  |v(g u)|^dx. R  |x">R  The f i r s t i n t e g r a l h e r e can be made s m a l l s i n c e  |vu| e L ( E ) . 2  Consider the remaining i n t e g r a l  J I v(g u) "dx  | v ( g u ) | dx R  R  R+l> x >R  x >R  ( 7 )  g  < 2  >R  2  |vu| dx + 2 2  R< x <R+1  u | v g | dx. R  R  28.  ' ' T h y - f i r s t - ' i n t e g r a l i n (7) can be made s m a l l , f o r s u f f i c i e n t l y large of  R> "since  g  < 1  R  and  |vuj € L ( E ) . 2  g (x) weMye' R  max  |vgt,(x)| -  R < | X | < R + I  ^ Hence  max  : max |g'(|x| - R ) | R<jxj<R+l (  max |g'(p) | . 0<B^1  v  | vg | R  is; l e ^  R.  which i t f o l l o w s t h a t large  R ,  |vg( |x| -R) |  :R<;|X-|-<R+I-  TT  Thus the^ l a s t integrai;LihvC7)  u  Prom t h e d e f i n i t i o n  can b e made a r b i t r a r i l y s m a l l , from  llu-gpull^  can be made s m a l l f o r s u f f i c i e n t l y  Thus we^may; ass:umey w i t h o u t Idss ;of g e n e r a l i t y , t h a t  h a s bounded IsuppdrtV T h e n e x t step i s t o  snow t h a t  u  c a n be a p p r b x i m a t e d ,  i n t h e H 113_ ho;M, w i t h f u n c ^ in u  E. W e S k r i b ^  or  t o be z e r o f o r a l l o t h e r v a l u e s o f x  Is " s t a r - s h a p e d ^ we have t h a t i f x e E, for E  0 <£< 1.  outside then  E.  L e t u ^ ( x ) t= u ( ( 1 - C )x) .  u ( ( l - £ )x) - u ( x )  as  u e  S-Q.  Since  (0, 0)  By u n i f o r m  T h i s means t h a t  u  E  into  continuity  i f |x| < R + 1.  we have, by a f u r t h e r a p p l i c a t i o n o f t h e  p r i n c i p l e ;bf u n i f orm' c o n t i n u i t y y %  Define;  (1- £ ) x e E  T h i s can be a c h i e v e d by t r a n s l a t i n g  i f necessary.  Hbweyery s i n c e  |xj>R+l^  l  "* i D  u  a  s  ^ "*  0  i = l , 2, ..., n.  29. Iji^-ul^ = J E  J  |u-u | dx+ 2  1  R+1  E  Ivtu-u-^^dx  R+1  can be made s m a l l , f o r s u f f i c i e n t l y s m a l l can assume, w i t h o u t  €  loss of g e n e r a l i t y , that  ,  and hence we  u  Is i n  To complete t h e p r o o f we need t o show t h a t can be approximated i n t h e norm o f H^(E). J^-  u ( e .C*(E))  To do t h i s , l e t  be t h e m o l l i f i e r f u n c t i o n ( c f . Agmon [ l , p.5])..  r e s u l t ( c f . Agmon [ l ] Theorem 1.5) for s u f f i c i e n t l y small Theorem 1.10)  since  J<£ u u  completes t h e p r o o f t h a t Lemma 3.5. If  u  llfll  2  tends t o u ,  as € - 0,  i n the This  e H^(E).  formula).  and  2  standard  u ( x ) e C^(E)  has compact s u p p o r t i n E.  (The P a r s e v a l f e L (E)  shows t h a t  A  £ . A f u r t h e r r e s u l t ( c f . Agmon [ l ]  shows t h a t  norm o f ' H^(E);  (^(E).  G  = I  n  = (f,'-u )., t h e n  lc l • n  2  n=o Lemma 3.5 c a n be p r o v e d by s l i g h t l y m o d i f y i n g  the proof  i n T i t c h m a r s h [ 4 , p.104]. Remark:  If f  and  g  a r e i n L g ( E ) we c a n show, b y a p p l y i n g  the P a r s e v a l f o r m u l a t o f + g,  that  oo  (f,g)= where  a  R  = (f, u ) R  I n n > n=o a  and b  b  R  = (g, u ) . R  30.  A s - a . summary o f the' pre d e e d i n g . r e s u l t s  we'may S t a t e t h e  f o l l o w i n g theorem: Theorem 3 . 6 .  L e t t h e domain  E, he "narrow a t i n f i n i t y " and  s a t i s f y c e r t a i n r e g u l a r i t y c o h d i t i o i i s , then t h e eigenfunctioris i n Lemma 3*3 c o n s t i t u t e a complete s e t o f o r t h o n o r m a l  constructed  f u n c t i o n s i n H^( E) "''''satisryl"hg;;the'~%#atibn' for of  x e E E.  and t h e boundary c o n d i t i o  u  A  = 0  n  u n  (  x  ) + X u ( x ) =. 0 n  n  on t h e boundary  . '] In; futui*fe;,~when  we r e f e r ; t o t ^  eigen-  f u n c t i o n s o f (2.1) wemean t h e e i g e n f u n c t i % r t s a n d e i g e r t y a i u b s t o f x  T  constructed  i n Lemma 3.1 and 3 . 3 -  f u n c t i o n s a r e complete we have shown (2)  Note:  since the eigen-  \^ = X'  n  (cf.  equation  ).  I n t h e next;, t h e b  i m p o r t a n t i n v e r s i o n ;'  ;prbperty o f Theorem 3 . 7 .  If X  i s riot ;bh\ t h e h ^  then  = Proof.  We s h a l l p r o v e (8) by l e t t i n g  (  X  8  )  go t o i n f i n i t y i n  F i r s t t h e r i g h t hand s i d e of; (9) converges t o t h e r i g h t hand s i d e of (8) b y t h e r e s u l t s g i v e h e a r l i e r i n t h i s p a r t . ; Nbw c o n s i d e r t h e d i f f e r e n c e  31.  (G(x, - ,X), u )  - ( G ( x ; - , X ) , u ^ ).  n  We can c o n s i d e r  U  x  G (x,y,X)  X n  t h e second I n n e r ; p r o d u c t s u )  (G (x,-,Xh u -u ^ ) n  x  !lG (x,-,X)|| l l u x  -  n  x  .  n  X).  u ^ J .  Since  goes t o i n f i n i t y , t h e second gbe>Vto i n f i n i t y .  (11)  term o f (11) i s bounded by  The f i r s t term i n t h e above i s bounded by Theorem n  x  as t h e domain o f I n t e g r a t i o n i n  By t h e Schwarz i n e q u a l i t y t h e second  2.13 ( i d e p e n d e n t o f  E  E x p r e s s i o n (10) i s now e q u a l t o +  n  x  E  (10)  n  t o be z e r o o u t s i d e  x  and t h u s we a r e a b l e t o u s e  ((G-G )(x,.,X),  x  H  u  n  " x n"  2.2 e q u a t i o n  goes t o z e r o as  u  term o f (11) goes t o z e r o as  I f we l e t E* = E£  where  E'  and  X  X  X Q  are  o a r b i t r a r y , t h e h t h e f i r s t term o f (11) can be w r i t t e n as [Gx-  J ;  u  G]  n  dy  E-E*  The second  J  [Gx-  • E*  '-"  .+  G]  u  R  dy .  term i n (12) goes t o z e r o f o r f i x e d  E*  ( 1 2 )  by Theorem  2.2.  By t h e Schwarz i n e q u a l i t y t h e f i r s t term i n (12) i s /less:'; t h a n ( J I V E-E*  G  '  J n E-E* p  r  <  ||(Gx-G)(x,.,X)!l  **}  u  0  • [ J  u*  1 / 2  -.1/2  dyj  E-E* Now  since  ||G (x,-,X)-G(x, •,*)!! x  (Theorem 2.2 and 2.5)  i s bounded independent  and t h e r e m a i n i n g p i e c e  J u E-E*  2  of  X  (y)dy  can  32.  be made as s m a l l as we p l e a s e ( s i n c e i  l m  <%n  u  X,n( )  =  x  G  u e  L ( E ) ) , we have  n  2  X n( >>  c o m p l e t i n g t h e p r o o f o f Theorem  u  x  3*7.  Wel'snall f i n i s h t h i s ; p a r t w i t h two lemmas on t h e e i g e n functions . Lemma 3 . 8 .  u ( x ) -» 0  as  k ./-» •  c o r r e s p o n d i n g t o an e i g e n v a l u e Proof.  u  where  u  i s any e i g e n f u n c t i o n  X.  has t h e r e p r e s e n t a t i o n ( c f . T i t c h m a r s h  equation  22.9.3)  J  u(y) W - i ^  u(x)dx  X J g(x,y) u ( x ) d x  +  r<R  i r R  ;5  r  £  (13)  R  2  where  [ l o g § - | (1 - ^ ) ]  g(x,y) = ^ =  arid  I f we a p p l y t h e Schwarz i n e q u a l i t y t o ( 1 3 )  l«(y)| < ^[ [TT"  since  J R„  r > R,  0  r = |x-y|.  <  r<R J* dx •r<R J* u d x ] 2  1  /  2  R"  1  + X[ r<R J u dx  1 / 2  2  -  2  ~  2  2  where  A  s  then  • r<R J g (x,y)dx]  + XAR] [ J u d x ] r<R  g (x,y)d;x = ( A R ) 2  r < R  i s a constant.  1  (14)  35.  C o n s i d e r t h e f o l l o w i n g d i a g r a m , where  r <_ R c E ;  FIGURE 2  Since  uCy-^y-j) = 0 ,  we have  u(y , y ) * - J y ix  D  2  u( -  t  2  y i  , t)dt.  Therefore 3 J  y  u(y , y ) | x  2  2  .<  (y  3  y <  - y )  R J  2  [D  t  u(  , t) ] d t 2  y i  3  [D  u] dt. 2  t  0 H Q W , i n t e g r a t i n g t h i s i n e q u a l i t y with respect t o b,  we g e t b J |u( , y g ) ! ^ ! y i  b <  I f we now use t h e f a c t t h a t  y  x  from  y,  R J  J  a  0  [D u(y , t) ] dt dy 2  t  |V u| € L«(E),  1  we have  x  a  to  34.  b  |  N y  y2) I d y  <  2  r  x  R o(l)  as  a - • .  'a I f t h i s expression has  r  ''idth:'re'^e"'b't' to , y , 2  one  b  j  J |u( , y g ) ! ^ : dy . c a y i  <  2  R o(l)(d-c) < R o ( l ) . 2  Thus, making the sides of :.'''khe'^¥qttare' touch the c i r c l e  r <_ R  (see figure 2), we have J  |u(x) 1 dx ..< 2  R  2  o(l)  as  y -  r<R Combining' t h i s r e s u l t With (14), we get |u(y)|  <  [ir~  Hence i t follows that  1 / 2  R  _ 1  + X A R] [R  u(x) * 0  as  2  o(l)]  1 / 2  as  y -  x -» «.  For the purpose of the next proof we assume (x,y) i s a point i n  Rg.  Lemma 3.9.  Suppose that the,boundary of  y = F(x).  Then f o r  the boundary  (x,y)  [x;F(x)], and  n,  i s givenflobalLly by  i n a region which lEtclttdes a ;piece of |u (x>y) | <K ' ,X^,, n  dent of  (x,y)  Remarks  The prbof of t h i s lenMa i s / a  arid  dim E = 2.  remark [ 1 0 , p. 1 0 8 ] Proof;  E  Cbhsider the foiibwlhg diagram  where  K  i s Indepen-  35FIGURE 3.  F(x)+36 - - - F(xV+26  F(x)+6 Ff  where  6  i s s u f f i c i e n t l y s m a l l , i . e . we a r e c o n s i d e r i n g a  s m a l l p i e c e o f t h e l o w e r boundary o f E.  Since  u  i s zero  x n  on t h e boundary, we have y u  X,n< >y>  u  X,n( ^^  x  - f"J  2  D  t  u  X,n( ^ x  d  t  Thus <  x  5  by t h e Schwarz i n e q u a l i t y . respect t o y F(x)+26 J ^ P(x)+6  from  6  I  [  D  tX,n( -^ u  X  2  d  t  I f we i n t e g r a t e t h i s e x p r e s s i o n w i t h  F(x)+6 t o F(x)+28 , F(x)+25 y  ( x , y ) d y < 35 ,1 dy J [ D F(x)+6 y 2  X j n  ]  then u ^ (x,t) ] dt. 2  t  x  n  (15)  1  I f we now i n t e r c h a n g e t h e o r d e r o f i n t e g r a t i o n i n t h e l a s t i n t e g r a l , we see t h a t  y  v a r i e s over a range n o t e x c e e d i n g  thus t h e l a s t i n t e g r a l i s l e s s t h a n y 9  6  2  J'  t t X,n( >*> D  u  x  J  2 d t  -  36>  36.  Next I n t e g r a t e  (15) with respect t o  x  over  [x^jXjj  this  gives x  P(x)+26  2  J dx x  x  J'  [u  ] d y '..< 2  x > n  9 6  P(x)+6  x  2  2  J  y J [ D  x  r  y  t  u  X j n  (x,t)  E  x  J I ^ J * * *  =  7  E  Thus i f  2  x  L e t u s f i r s t c o n s i d e r t h e l a s t term o f (16). from t h e case o f h b ^ d e d  ] d t dx. (16)  We know  that  ^X,n  v  *  X  X > X-, : —.  then  O  Hence  l i m sup , f I v u . I d x •••"< .'. \ : _ , 2  v  _. - «»  Y X  J  A  X  i»  n  v  —  A  « »  n  E  hut  since  X  Q  i s arbitrary  l i m sup j |v u Y - eo  since  X „ Y  - X  I s bounded by U  rt  ±  E  A  X  2  >  ~  n  n  X -».•>.. Thus t h e r i g h t hand s i d e o f (16)  as  9 6 X  _ ] d x •"..<'. X„,  Y  n  as  X  tends t o I n f i n i t y , but s i n c e  .-» u '•'• i h t h e mean, we have by F a t o u ' s theorem X,n n v  37. x  F(x)+26  2  J-; dx^  x  J  [>ii ] dy 2  n  9 6  :<  ^  2  .  - (17)  P(x)+6  1  We a r e now i n a p o s i t i o n t o make the r e q u i r e d on  u .  By/ T i t c h i r i a r ^  n  estimate  22.$ .3] we have  J.2. r<R  7 r R  where E.  r<R  r </R :.; i s a c i r c l e  Npw  vC'MIh| .  suppose  [ s ; | s-u|  that f o r x  Thus i f  K  in  |x-x | < £, o  which Is; Irai t u r n l e s s than  R]  P(x^) + 56/4  ( x , P( X ) Q  q  i f Q £^6/k<  + -|6)  •^6 min (M"* , 1 ) , l i e s 'between t h e curves 1  y « P(x) + 26. \ Now, I n e x p r e s s i o n R = ^6 min ( M  -  1  , 1) X ^  /Likewise  + ^6. and r a d i u s y = P(x) + 6  and  ( l 8 ) s e t u = ( x , F ( x ) + ^6) &  X^ .  2  |P(x+h)-P(x) |  then  Q  and  which i s contained i n  we have  [x^Xg]  F(x) + 26 >; P ( x ) - MC + 26 > F ( x ^ ) Hence the c i r c l e , center  1  Q  Note t h a t t h e c i r c l e  2  w i l l be I n the r e g i o n o f I n t e r e s t s i n c e  x j ^ X^ ^ 2  1  2  <.1  r < R  for a l l  By the Schwarz i n e q u a l i t y the f i r s t term i n ( l 8 ) i s . l e s s than -r^ i r R  [ J rj<R  dx • I  u (s)ds J 2  r<R: ' ;  1  /  2  n.  3 8 .  which i n turn l e s s than  hy i n e q u a l i t y (17)• Thus the f i r s t term of (l8) i s l e s s than Mg, X^,  Mere  .Mg : i B l n d e p e n d e h t of  5, u  and  n.  Again hy "the Schwarz inequality the second term of ( l 8 ) ;l;s?lefs: "thah. :  r<R  where  Cy  r<R  R  are Constants independent of  u, n,  Gomhining a l l these r e s u l t s , we have C, X_ 6 . I f we now assume that i n result  l ( >y)I < u  x  n  K  X  n  where  6 < 1, — ' K  and  6.  J'u (x) | <M:  2  X^ +  we have the desired  i s independent of  (x,y) and  39.  Part 4 .  The i t e r a t e s o f t h e Green•s f u n c t i o n and t h e i r p r o p e r t i e s .  F o r t h e sake o f s i m p l i c i t y we assume i n t h i s p a r t t h a t t h e domain ,  E  has a boundary formed by t h e x ^ - a x i s and a f u n c t i o n cp(0) = 0,  where  cp(x^) > 0  i f x^ > 0 ,  and  s a t i s f i e s t h e smoothness c o n d i t i o n s i n Theorem 4 . 2 .  ep(x ) 1  We a l s o assume  a l l c b n d i t l b h s i n p a r t 2 and 3 , so t h a t we s h a l l have a Green's function  G(x:,y>X)for  functions  E,  eigenvalues  X  and e i g e n -  u (x). n  n "_> 3  I f we a r e w b r k i n g I n a d i m e n s i o n v  E  f o r T>  n  t o be t h e domain I n R  n  formed;*by r e v o l v i n g  we s h a l l  ,cp(x^)  cohsider  about t h e  x^-axis. Beflnitilbh.  L e t t h e i t e r a t e s i o f t h e Green's f u n c t i o n be d e f i n e d  as f o l l o w s ; rf ^  (x,y,X) = G(x,y,X)  1  and  ^•ci^^iili'A) ^#(*i S' x),;&(-,^ )> :  ;  :  (iii).  These i t e r a t e s a r e w e l l d e f i n e d by T h e o r e m s " 2 . 5 arid 2 . 6 . Theorem 4 . 1 . on  E x E  The i t e r a t e  G^ ^(x,y,X) 2  (1)  2  x  and  I s continuous  and s a t i s f i e s  0<^ 'C^ where  f o r dim E = 3  X - -m , y.  w > 0,  and M  I s a c o n s t a n t Independent o f  .  4o. Lemma.  Let E  be an open s u b s e t o f a c y l i n d e r ( o f f i n i t e  s e c t i o n ) i n R„. Then, a s • n l  i  where  , ;  J  m  u -»y  |x-u|  E  a + £ > 1  Proof.,  , a f  ,^  -  |x-yr  and  6 •-» 0, • ° (  X  cross  !  )  i  «  f  +  P  <  n  •>  0(logl/6)  i f a + B = ri  =  0(6 ~ "" )  i f a + B > n,  n  a  B  6 = |u-y|.  I f u, y e E ,  t h e n t h e i n t e g r a l we a r e i n t e r e s t e d i n can  x  be b r o k e n up a s f o l l o w s : P dx J |x-u| |x-y| X+1 a  + P  a  E  Since  E  x  +  1  |* <3x J |x-u| |x-y| ~ X+1  E  / ) 2  P  E  i s bounded, we c a n a p p l y T i t c h m a r s h [ 1 0 , p . 3 2 3 ]  to the ' f i r s t i n t e g r a l i n ( 2 )  t o obtain  lim f / ^ U V y J |x-u| |x-y|P ^X+l  =  A  a  0(1)  a +.p < n  =  0(logl/6)  a + 0 = n  =  0(6  a + B > n .  n _ a  ~ ) P  To complete t h e p r o o f we need o n l y show t h a t t h e l a s t i n t e g r a l i n (2) i s bounded. |x-u| > 1  To see t h i s we n o t e t h a t , f o r  'arid - f x - y j ,>.1>.  E  ~ x+l E  x e E-E  x + 1  ,  i s c o n t a i n e d i n a i tube o f  f i n i t e c r o s s s e c t i o n , and a + 8 > 1 . P r o o f o f Theorem 4 . 1 . for  E  By t h e p r o p e r t y o f t h e f u n d a m e n t a l s i n g u l a r i t y  (Theorem 2.8), we have  41.  <. -J K ( ( B | X - Z | ) K ( u ) | y - 2 | ) d z ,  G^(x,y,X)  E where  K(tup) = M p  - 1  exp(-t«p/4) .  I f we how a p p l y t h e t r i a n g l e  (2) i n e q u a l i t y t o t h e I n t e g r a n d o f ,G\. *,  a (xyy,X) M' W(IIWI.)<• J j x - z l f y - z l E  then  (2)  2  <  M  • exp(-1(1x^1  exp(- « | W l / 8 )  2  +  |z-y|) )  J Ix-zlfy-zl E  <  M» e x p ( - uu|x-y|/8)  by t h e Lemma, s i n c e  a + 0 = 2 < n = 3.  F o r dim E = 2  Remarks;  G^(x,y,X)  <  However I f dim E = 4, as In  ( p r o o f b e l o w ) , we have  M exp(- uu|x-y|/2)  i f  m.y-1  .  (3)  and we t r y t o u s e t h e same method o f p r o o f  dim E = 3, we g e t G  ( 2 )  (x,y X) v  <  M e x p ( - «j|x-y|/8) l o g j-±^ry  •  CM  I n s t e a d o f (4) we c a n show G^'^Xyy/X) if  k > n,  <  M exp(- t B | x - y | / 2  and i f dim E = n > 3-  K  )  (5)  One p r o v e s t h i s fey a r e p e a t e d  a p p l i c a t i o n Of t h e method used t o p r o v e ( 1 ) . P r o o f o f (3) f o r dim E = 2 .  S i n c e , t h e Lerima does n o t a p p l y t o  t h i s case we proceed a s f o l l o w s ;  42.  We have 0 < G ^ ^ ( x , y , X ) < J K(t»|x-z| ) K(t»|y-z|) dz 2  E' where  K(cor) ='f(tur) e  - t u r  ,  and  = M ( I + i o ( i + r ))  f(?)  / ( 1+§  1  g  1  /  2  )  hy Lemma 2 . 7 . C o n s i d e r t h e I n t e g r a t i o n above o v e r t h r e e s e t s , namely N]X,U)" 6>, 1  and E*^= E ^ { H ( x , t B ~ ^ 6 ) U N ( y a r 6 ) } ,  N(y,u)" 6) 1  N(x,-"a) = {z ': | x - z | < a } , and  6 < 1  We need t h e f o l l o w i n g e s t i m a t e s  f o r f (?):  f(?)  £ M(l + log(l + 6 " ) ) « M , 1  1  Mg l o g  :|x-y| > 2 6 /  i s such t h a t  If ? > 6,  and i f § < 1 , t h e n  How c ^  ^(xyOB ^), -  i s bounded by  where  1  v  then f(§) < which  '  exp ( - t e j x - ^ I )  J  f(«j|x-z|) f(t«|y-z|) dz .  N(x,aj 6) _ 1  If that  z c N(x,to~" 6),  then  L  f (u)| x-z| ),•<_ M  |y-z| >v26 - a>v*6  ti)|x^z| .;<•. 6 < 1 ,  log|x-z| ~ , 1  2  or  uu|  i f we assume  y-z | .o 5( 2 u u - l ) .  f ( u u | y - z | ) <. M «  follows that  1  1  i s bounded by 2  f  .£  6,  «i > 1 .  follows  Also  from w h i c h i t  I f we s u b s t i t u t e t h e s e I n e q u a l i t i e s  i n t o t h e i n t e g r a l o y e r t h e s e t W(x,cu"" 6),  M,rM  from w h i c h i t  log  dz ,  then that i n t e g r a l  43. w h i c h is.Ybounded/. independent ;o f • oo  and  :  x,  as l o n g as  uo >. 1,  By symmetry a s i m i l a r r e s u l t h o l d s f o r N(y,uu 5) . -1  C o n s i d e r now'''the''.integral'over uo| x-z I > J less  and  <»|y-z| > 6.  If  z e E*  Therefore the I n t e g r a l over  then E*  is  than  exp(-uu|x^y|/2)M  2  | exp( ^<u( | x-z|• +.,J-zVy|')/2) dz E*  hy an argument s i m i l a r t o t h a t above. i s bounded independent | x-yj j> 26. 0 < and  E*.  of  uu, x  and  Hence t h e i n t e g r a l over y  i f OJ J>. 1,  fi'  E*  <..l, and  Combining t h e s e i n t e g r a l s , we have  G.( l(x,y,\)  <  2  |x-y| >. 26, Since  where  M exp(-tt)j;x-y|/2)  i f o > 1V  c  M  I s independent  6  6^ ^(x>y,X)  bf  x, y  i s continuous a t  2  x<=y,  (6)  arid  ou.  the r e s u l t  (3) i s p r o v e d . Theorem 4.2. (a)  (b)  i f E  I s a p l a n e r e g i o n such t h a t  there e x i s t s  a  t  o  > 0  such t h a t  < CD and sup t > t cp(t) — o f o r every B > 0, t h e r e e x i s t s a -pt cp(t) > kp e " t >t ,  k^ > 0  such t h a t  p u  Q  t h e n t h e r e e x i sst t s a constant integers  k .> 2,  or o  such t h a t f o r m> ou, — o  and  44.  I v | 0 >(x,y,X)| 2k  E where  2  dx dy  | l«p(s) |2k, "\Ss:y  <  0  E X = -to  Remark; Proof.  and  v  I s - a c o n s t a n t d ^ e h d i n g o n l y on  The p r o o f o f t h i s  theorem  f o r . dim E  F i r s t l e t US; show-^e; theorem f o r  2  co.  Is^due t o C l a r k .  k = 25  tfiai  Is,  ' c . b n M d ^ r G ^ ^ ( x , y - j i X ) . By a p p l y i n g t h e d e f i n i t i o n o f t h e I t e r a t e s Of t h e ^  2.9 (symmetry o f  G(x,y,X) ) ,  '•:dn'et can showV •  E By Theorem 4 . 1 ( E q u a t i o n  ( 3 ) ) We  have  Gi^(x>y,X).._.< M J exp(-co( |x-z| + |z-y| )/2)dz, ' E y=(y^y )  MOURE; 4 .  2  : x=(x^:y5c ^ 2  z^(zpz ) 2  By c o n s i d e r i n g f i g u r e 4 a n d / a p p l y i n g  t h e t r i a n g l e i n e q u a l i t y , we  can see t h a t (x]y, X)' .••<; M j e " ^  4  J J . exp(-(B| 'x^-z^ I ' A )  dz dz 2  1  0 0 If  we c a r r y b u t t h e I n t e g r a t i o n w i t h r e s p e c t t o  z  2  we g e t  » 5 .  co-  G  (2f)  (x,y,X) < H e  J tp( ) exp(-u)|x -z !/4)dz  _ a ) r / 4  r  Z l  1  1  1  .  (?)  0  The r e m a i n i n g i n t e g r a l must be a n a l y z e d i n t w o p a r t s , the f i r s t p a r t b e i n g over [ t , ,») Q  3  ( a )'.  where  t  t  :  and c o n s i d e r s  ep(z )  J  and t h e s e c o n d p a r t o v e r  i s t h e p o s i t i v e constant given i n hypothesis  Q  L e t X. l —> to t  v  [b, t ^ ]  t  exp(-(fl|x -z |/4)dz 1  1  1  Q  exp(-to|x -z |/4)dz  < M J*  1  1  1  < VLX  e  _  a  ,  x  l^  0  X Mg OPCX^) M,  ffcjV  and M  u s i n g h y p o t h e s i s (b) .  a r e : coristarits depending o n l y on to.  2  We now examine t h e second p a r t o f t h e I n t e g r a l , namely 'the--'one over  [ t , +«>) . Q  F i r s t we note t h a t h y p o t h e s i s sufficiently large -'(*)•;'  (a) i m p l i e s t h a t f o r  K,  max tp(t) exp(-K| t - ^ j ) t > t„. —. o  = ^(t^  for any  tj_ > t  Q  .  To see t h i s c o n s i d e r  D  t  [ cp(t) ^ ( - E j t - t ^ ) if  t > t  if  t  Q  and K  x  i s s u f f i c i e n t l y l a r g e , and  < t < t , where 1  ] := expC-Klt-^l)[V.q>"»--(t)--Kcp('t) ] < 0  K  i ssufficiently  large.  !  46.  This proves (*). A p p l y i n g (*) t o t h e I n t e g r a l Over  [t , >), Q  we have:  00  J  cp(z )  eixp(:-cfij^  1  ^O  (»  = J\'Mz±) e x p ( - 0 ) ^ - ^ 1 / 8 )  exp(-«)|z -x 1 |/8) d z ^ 1  < ipC^j) J \"e^C-»r*i-»iT/?)* :  i 1  tirhleh h o l d s f o r S u f f i c i e n t l y l a r g e  i  M ot^)  oo.  Thus we have, c o m b i n i n g t h e two e s t i m a t e s f o r t h e integrals, G^(x,y,X) < for  x^ 2 t  Q  (8)  cp(x ) exp(-uor/4) x  and s u f f i e i e n t l y l a r g e "oo;.  F o r r e a s o n s o f s i m p i i c i t y we I n t r o d u c e t h e f o l l o w i n g functions rcp(t )  i f t <; t  o  •81)  o  < q>(t)  I f t;> t  Q  .  Now by i n e q u a l i t y (7) we have .J  G % x , y , \ ) £ M ° exp(-»jr/4) (  i f ^ • .<; t  Q  .  (9)  Thus we have, by c o m b i n i n g ( 8 ) and ( 9 ) , G ' ( x , y , X) X (4  ^ ( x j exp( -tor/4j, 1  (10)  '  4 7 :  where  C o n s i d e r how t h e ! to  of (10)"over  E  with respect  x. J |  4  ^(x,y,X) | dx  M J | ^ (x ) |  <  2  1  E  2  exp(-«)r/2)dx  E •  0  6  M J"' 1 ^ 3 ^ ) |  expC-iBlx^j^ldx^  3  0 I f we a p p l y t h e same a n a l y s i s t o ' t h i s as we d i d t o e q u a t i o n  (7)>  we have ^ [ ^ ( x y y A ) ]  2  ^ /  I  ^ l ^ ) !  5  E w.  for sufficiently large  t h i s time w i t h respect t o  I f we i n t e g r a t e " t h i s e g r e s s i o n y,  J J |G^>(x,y,X)| dx-dy 2  we have ^  J  E E  \Wy^\ & 3  E 00  <  M  x  J I^  00  y  i  0  ) I^  <  Mg  J- | cp(s) | \ s  '  .  0  Thus we have shown t h a t t h e r e e x i s t s a p d s i t i V e number that  J* J | G C ) c ^ y , X ) | d x d y 4  E E fbr  again,  co > to...  2  :  1, M. } | c p ( s ) | d s 4  0  such  48.  I t i s now a s i m p l e m a t t e r t o extend t h e r e s u l t t o higher values of G  ( 6 )  k.  F o r example;  (x,y,\) = J G^)(x;z,X) G  ( 2 )  (z,yV\)dz .  E By e q u a t i o n s (3) and (10)  we have expi-wr^k)  0^ |(x;yil) < M 6  exp^-iijr^/ajdz;  E $ (x ) J  <  M  <  M ; <p ( x ^  x  exp(-«)(r +r )/4)dz 2  i  ,  ;exp(^oor/8)  and t h e r e f o r e J J |G  ( 6 )  ( x ; y , X ) I d x dy < M J J j $ ( ;  2  E E  ) | exp(-i«r/4)dx ^dy 4  X ] L  EE OD  < M J;!^(s)| ds y 6  0  hy t h e u s u a l arguments. values of  ky  The same c a l c u l a t i o n a p p l i e d t o l a r g e r  arid t h e p r o o f b f Theorem 4 . 2 i s complete, f o r }  dim E = 2. Remarks" b n e x t e n s i o n .Of Theorem 4 . 2 t o dlmerislOris l a r g e r t h a n two. The  case  dim E = 3;-I f we a p p l y t h e same procedure  we d i d t o (3)t  G ^ (dim E := 2)  G ^ ^ ( d l m E = 3)  to  u s i n g e q u a t i o n (1)  i n s t e a d of  we can show G( ?(?yy,X) 4  ^ M e~  mr/Q  J exp(-UJ| x - - z 1 / 8 ) d z . L  E  1  as  equation  49.  I f we now r e c a l l t h a t i n t h r e e d i m e n s i o n s r e v o l u t l b h formed hy r o t a t i n g  cpfx^)  E  i s the solid-of  about t h e x ^ - a x l s , t h e n by  a change'•'of v a r i a b l e s i n t h e above e q u a t i o n s and t h e u s u a l . c a l c u l a t i o n s , we have <*> 2ir op(z^) G^ ^(x,y,x) < M  e'  4  u , r  ^  exp(-a)|x -z |/8)rdr d > d z  J* J\J  8  1  1  0 0 0 U)_/o  < M e"  r  /  "!  ° J  .  exp(-u)|x -z |/8)  J  0  1  1  d £ dz  0  I f we I n t e g r a t e w i t h r e s p e c t t o x , J  i cp( z_)  1  we have  "  j W)((xx,,yy ,,xx))||2 dd x < M JJ' $K( ^ - )V ^ exp(-u)r/4)dx |G ( 4  2  4  X ] L  E  EE eo '  < M  J '.'$(x ) exp('.-co| x - y 1 / 4 ) dx^ 6  1  1  1  1  0 < M I f we i n t e g r a t e a g a i n ,  6  ^y^)  2  then , 00  IGV (  J  J  E  E •  ( x , y , X) I d x dy < 2  J  | q>( s) 8, | ds . 8  l«PlSj i  0  By s i m i l a r argument we can show t h a t J J E  where for  |G( l(x,y,X)dx  dy <  2 k  |cp(s)| ds, 4 k  0  E  t» > w\ f o r some — .o dim E > 3.  J  tu  o  and  k > 2. —  S i m i l a r r e s u l t s hold  H e n c e f o r t h we c o n s i d e r o n l y domains  E  which s a t i s f y  x  50. t h e c o n c l u s i o n o f Theorem 4.2. k  o  Thus t h e r e must e x i s t a  constant  "such t h a t I  J  I ^ V C x ^ - X ) ! '  2  ^  dy  <  «  3  EE t h a t i.s G o ) ( k  must be a H i l b e r t - S c h m i d t K e r n e l f o r some  k_. o  Hote t h a t , a l t h o u g h Theorem 4.2 i s s t a t e d o n l y f o r domains w h i c h l i e above t h e x£-axis, l i t a l s o h o l d s f o r domains'which have a s i m i l a r p i e c e below t h e x - a x i s , f o r example, one d e s c r i b e d 1  -<p^(Xj-')  where The  by  cp^ s a t i s f i e s t h e same c o n d i t i o n s as cp. n e x t s t e p i n o u r i n v e s t i g a t i o n " o f t h e boundary  behaviour of  G  i s t o Show; t h a t  t h e boundary f o r some  k.  heed t h e p r o p e r t i e s o f  X^  G^ l(x,y, X) k  tends t o zero a t  I n Order t o p r o v e t h i s r e s u l t we s h a l l and  UJ^(X)  d e v e i b p e d i n P a r t 3«  Unft»r%uhately I t does h o t seem p o s s i b l e t o Use t h e compactness o f eigeriftihLCtibnS; and  G^ ^ k  a l o n e t o c o n s t r u c t the^ e i g e h v a l u e s a j i d  Even though brie c a n show t h e e x i s t e n c e o f  u (x) such t h a t n  Au^(x) + X„uY(x) = 0, n* ' n n"...-. '  v  /  possiblei t o sho^ that  X  n  I t does n o t Seem  u ( x ) I s - z e r o on t h e boundary, and t h e r e b y n  avoid the c a l c u l a t i o n s i n P a r t 3. . U s i n g Theorem 3.7. (Green's f u n c t i o n i n v e r s i o n ) definition of  G ^ (x,y, X),  ...ca<^i;v,x).ii :  where  {u (x)} n  arid  u ) n  one c a n e a s i l y see t h a t -  (v;>r\f?');  >  ( X ) a r e : a s c o n s t r u c t e d i n P a r t 3• n  and t h e  Combining  51.  t h i s w i t h Theorem 4.2 we can prove t h e f a l l o w i n g re s u i t on t h e eigenvalues. Theorem 4 . 3 . k  Q  The s e r i e s  iS;idefinedby  Proof.  E  X  -2k  converges; i f  n  k:> k^,  where  t h e remark fblloWirig Theorem 4 . 2 .  F o r any set; o f orthonormal f u n c t i o n s / f o r example  {u (x) }, n  the Be'ssel i n e q u a l i t y h o l d s ; i . e i cb  I  I  G  n  l  <  2  II  Mi)  :where ; c  2  = (f, u ) .  n  R  n=0 By t h e i n v e r s i o n r e s u l t f o r d Thus If or  (X -X)*  u^y)  k  n  n  we have when  ;  f(x) =  <sl )(x,y,X) k  .  k = k^ . o N  | where  N  I X - X|- u (y) 2 k  n  jl0^> (> ,y,X)|| ,  <  2  n  2  n=0 I s as l a r g e as we p l e a s e .  Integrating t h i s expression with respect to N  I  IXn-Xl~2k  < J* J  n=0  we get  | G ( > ( x , y , X ) I d x dy, k  2  E E  which I s f i n i t e by Theorem 4 . 2 , i f X = : - « J enough, and: I S independent o f X.  y  oo, t h i s I m p l i e s t h a t  N.  £  X_  n=0  n  Hence  2  to i s l a r g e  and  E [X^XI  converges, which  < «.  Since  completes  Theorem 4 . 3 . Theorem 4 . 4 .  (The expansion f o r m u l a ) .  Let k  k^,  and l e t f (x)  52.  have s u f f i c i e n t d e r i v a t i v e s sb t h a t t h e ; f u h c t i b h s  are a i l cbntihhbus, contained i n Lg(E).;•'>  A ^ f (x)  A(Af(x)),  f , Af(x),  and tend t o zero a t t h e b o u n d a r y o f  E.  Then 00  f ( x ) =;;£ <?  where  h  c  rt  fe;^,.^') .  n=o Proof.  Let  c'^}-''m'y(t^\9 i u )  where  n  f^^x)  » -Af(x)  fM(x)  = -Af  ( 1  and  "%x)  I > 1.  By Giazman [7, Thebrem; 34, p . .90] we have; (Aui v) = (Av, u) where we s e t  iu,eL (Ej g  v = u  and b o t h  g(x) =  arid  v  a r e zero b n  aE;; T h u s ; i f  l±); .  and: u = f  n  v  00  Let  u  ^ c n=0  '••, we have  00  n  u (x) = n  J n n=0 c  ( k ) x  n"  k u  n( ) x  •  The method o f p r b b f w i l l be t o show t h a t (11) c o n v e r g e s a n d d e f i n e s a f u n c t i o n which i s i d e n t i c a l t o  f(x).  We f i r s t must show t h a t f o r some constant  I (^  2k  + D"  1  % (x) < C 2  n=0  unifbrmly f o r x € E' oo  n=0  As i n Thebrem 4.3 we have O  C  we have (*)  53.  We have by Theorem 2.6  if  X e L ( v ) > and l i k e w i s e  Since  ||G(x, -,X)|| < K .  oo  .  "  f p t x e EJ  .• . '  and  X e L , Q  we have  0  I |X - X | n  2 k  u  2 n  (x)  <K.  n=0 for  all  X € E  and  X e L .. S i n c e X„- », (*•) f o l l o w s . o n • •'"•'!.• We^ridw'^Qiisider t h e t a i l end,of t h e s e r i e s (11) . I f v  A_  0  X  N + 1  > 1,  t h e n by t h e Sdhwafz i n e q u a l i t y  n=N+l where 2 [c  n=N+l a„ = (X» n n  2k  v  ]  2  +1) X„ ' n  ok  n=N+l  < constant. —  Note t h a t t h e s e r i e s  converges, being the "Pourier"  i s i n L ( E ) by h y p o t h e s i s e  I n view o f  2  series of (*.)  and  which  f ( ) e L (E) k  2  t h e t a i l end o f t h e s e r i e s can be made s m a l l by t h e c h o i c e o f N , u n i f o r m l y f o r x e E i . Hence s e r i e s (11) d e f i n e s a c o n t i n u o u s function inside  E.  Also  n=o from w h i c h i t f o l l o w s  n=o  CO  1st*)!  2  <. I "°n ) ' I <K)  n=o  2  n=0  '  54. w h i c h i s bounded by Theorem 4.3  (note  k = k ). 0  A s i m i l a r argument shows t h a t m n=o epiiyef ges,-'in; 'mean. square * t o  g(x)  :  Schwarz I n e q u a l i t y , ' f o r any n  However we know t h a t _ , that  ( u , g) = c . n  n  E.  Hence, by t h e  n  = 0.  g-gj  lim ( u , m -» eo  over  (u , g_) v n m' = 'en . (m —> n') ,  from w h i c h i t f o l l o w s  x  Thus t h e f u n c t i o n  f ( x ) - g(x)  i s an  f u n c t i o n , a l l o f whose "Fourier'yp.e.fficI.entsV.Ivajiish;;--  Jjf  t h e P a r s e v a l Theorem uous i n s i d e  - g|| =  0.  2  Hence by  S i n c e t h e .integrand i s / c o n t i n -  E, i t must v a n i s h everywhere; i h s i d e E ;  cbmpiletes; t h e .proof o f Theorem  L (E)-  :  This  4:4^  We; aref how. i n a p o s i t i o n : t p p r o v e t h e main r e s u l t i n t h i s : p a r t , namely t h a t  Gl ^x>y,X) k  t e n d s t o z e r o as  The, p r o o f w i l l depend d i r e c t l y bh the^ k n o ^ e d g e / t h a t H l l b e r t - S c h m i d t and t h u s t h e s e r i e s method i s ^ d e r i v e d from T i t c h m a r s h [10> For  Theorem 4.5.  fixed  noh-hegative r e a l a x i s , :  t h e boundary Of Proof.  Let  E.  x € E  r = |x-u|  Q  p.  and f i x e d  G^(x,y,X)  (k > k  E X^  + 2  and d e f i n e  and  x -• dE. G^ ) k  cohvergent.  is \ The  106]. X e C  n o t oh t h e  t e n d s "to z e r o as dim E i =  2*)  y  approaches  55.  F(x,u) = - ^ where  R  fixed  Uj  i s such t h a t t h e c i r c l e  |x-u| < R  and g ( r ) has t h e f o l l o w i n g  g e G*(E)  ,  g(r) = 0 Since formula.for J.  log ~ g(r) ,  (G  ( k )  g(r) = 1  isinside  E for  properties:  r < R/2  and  T..> R .  G ^ " i s n o t s i n g u l a r , we have i n t h e G^ ^(x,y,X.)  and P ( x , u )  k  (x,y,x)  as f u n c t i o n s  Green's of x :  AP(x,u) - P ( x , u ) A G ^ (x,y,X) )dx  r<R  / \ k  r=R' the s i n g u l a r i t y o f P(x,u) G^ ^(u,y,X). k  at x = u  g i v i n g r i s e t o t h e term  C o n s i d e r t h e boundary term:  P(r) = 0  f o r r = R,  and  1 1 - |Z = o f o r r = R, b y t h e d e f i n i t i o n o f P. an ar boundary term o f (12) v a n i s h e s . Upon s u b s t i t u t i o n o f A G ( ( x , y , \ ) = - \ G ( ( x , y , \ ) - G ^ " * (x,y,X) k )  k )  Thus t h e  (k>l),  1  (k)  w h i c h f o l l o w s d i r e c t l y f r o m the d e f i n i t i o n o f G  '  v  and Theorem  2.6, i n t o ( 1 2 ) , we have G  ( k )  J  (x,y,X) =  [AP(x,u) + X P ( x , u ) ] o ( ( x , y , X ) dx k )  f<R + J p(x,u) . o ^ " ^ ( x y A } ' r<R J  d x  -  " C ?). 1  09  R e c a l l the r e s u l t  n  ?  a 0  b n  n  =  where  a  = (».) f  n  u  n  a  n  d  56.  t> = (&) ^ n  n  ) , which f o l l o w e d  d i r e c t l y from the  Parseval  Theorem.  Let d (u) n  and  = (AF(;,u) + X F ( - , u ) , u ) n  (G^('  r e c a l l that  Applying  and  d (u)  )  » (* -  >7,\),  these r e s u l t s t o (13) we  u n  - (F(-,u), u ) ,  n  n  X)"  n  k  u (y)  .  R  get  CO  G  ( k )  (u,y,X) =  £  d (u)  (X -  X)"  d (u)  (X -  X)"  n  n  u (y)  k  n  n=o cs  ,+  I  n  n  k + 1  u (y).  (14)  n  n=o We  want t o show t h a t  Consider the t o the  G^ ^(u,y,X)  goes t o zero as  k  second s e r i e s i n (14);  first.  By Lemma 3-9  near t o the boundary;  we  First  y -  aE.  a similar result will  have  |u (y)| < K X n  consider  the  n  t a i l end  for of the  apply y series  on the r i g h t hand s i d e o f (14), namely  n=N  n=N  <  -  n=N  n*o  ni=N  .  iiP(-,u)ii-(|x;  K  2 k +  *)  1 / 2  ,  n=N since  X  Now  by h y p o t h e s i s  k >  k  Q  + 2  and  so the  series  . _ Pk+4  S.X N  converges. so l a r g e  £  Thus g i v e n any  that  |Id (u) n  n=N  (X n  X)"  k + 1  u (y) n  > € > 0  we  can f i n d  an  57.  uniformly i n If  y, y  y  i n a neighbourhood o f t h e boundary.  i s s u f f i e i e n t l y c l o s e t o t h e boundary o f  E  we  have |d (u) ( \ - V X ) * u ( y ) | < 1  n  for  n  n  n=0, 1, . . N - 1 ,  since  t e n d s t o z e r o f o r each  |  n.  u  and  672N X  .  are f i x e d ^nd  u (y) Q  Combining a l l t h e s e e s t i m a t e s we g e t  *  II N-1  Xa (u) ( X - V x r * * u (y)| < n  n  n  n=o  d„(u)  (X -X)" n  k + 1  u (y) n  n=o  n=N < Thus E,  G^ )(x,y,X) x  and  +  goes t o z e r o as  K  when  N( £/2N)  X  €/2  y  =  £  .  t e n d s t o t h e boundary o f  are f i x e d .  We can p r o v e t h e f o l l o w i n g theorems i n a . s i m i l a r manner.  -  Theorem 4 . 6 . the  equation  as  x - SE\,  The f u n c t i o n AH + XH = f as l o n g as L (E)  E,  X  real  where  k = k . o,  and t h e boundary c o n d i t i o n f, Af',A(Af),;,., A ^ f  uous, c o n t a i n e d i n  2  H ( x , X , f ) = T ( G ( X , •, X) , f )  satisfies H(x,X,f)- 0  are a l l c o n t i n -  and tend t o z e r o a t t h e boundary o f  i s a complex number n o t on t h e n o n - n e g a t i v e . x  axis.  Proof.  H(x,X,f)  S a t i s f i e s the equation  AH -f XH = f  by Theorem  58.  2.6.  By t h e d e f i n i t i o n o f H ( x , X , f )  we have  00  H(x,X,f) =  £  c (x - X ) n  (15)  u (x) ,  _ 1  n  n  ri=o where  c  53 n  (f>  u n  )  b  y  the Parseval  Theorem.  As ;we showed i n t h e p r b o f . o f t h e e x p a n s i o n theorem (Theorem 4.4)  c n  ^  » X  c  k n  n >  . where  E-['.c ^) ] :  n  2  < * .  Thus,  s u b s t i t u t i n g t h i s r e s u l t i n t o ( 1 5 ) , we g e t 00  H(x,X.,f) =  I  c <*> X " n  k  n  (X - X ^ "  1  u (x). n  n=o •. We now ap^ply t h e same a n a l y s i s t o t h i s e q u a t i o n a s we d i d t o e q u a t i o n (14) i n t h e p r e v i o u s theorem, t o conclude goes t o z e r o on t h e boundary.  Note:  k - k  that  H(x,X,f)  w i l l be s u f f i c i e n t  to carry out the c a l c u l a t i o n s . -2k  Remark: E  Note t h a t Theorem 4 . 3 (2 7^  which l i e s i n the'half-space  f a c e o b t a i n e d by r o t a t i n g a conditions of  < *) G  tp*(x ), 1  cp, about t h e x - a x i s . 1  t h a t the. e i g e n v a l u e s o f  E  h o l d s f o r a domain  and I s bounded by t h e s u r where  cp* s a t i s f i e s t h e  T h i s f o l l o w s from t h e f a c t  dominate t h e e i g e n v a l u e s o f t h e s u r -  f a c e o f r e v o l u t i o n ( s e e Glazman [7, p.229]).  F u r t h e r m o r e Theorem  4.3 now I m p l i e s t h a t t h e o t h e r theorems o f : t h e s e c t i o n , namely, Theorem 4 . 4 , 4.5 and 4.6 h o l d f o r such a domain. are necessary  Some m b a i f l c a t i b n s  f o r example t h e s i n g u l a r i t y o f t h e function..',P(x,u)  i n Theorem 4 . 5 .  59.  P a r t 5.  Boundary b e h a v i o u r arid u n i q u e n e s s o f t h e Green's  Function.  We a r e now i n a p o s i t i o n t o p r o v e t h a t t h e Green's f u n c t i o n t e n d s t o z e r o at: t h e boundary b f  E.  We c o n t i n u e t o  make t h e assumptions o f P a r t 4, so t h a t some i t e r a t e  G^ )(x,y,X) k  s a t i s f i e s Theorem 4 . 5 . Theorem 5 . 1 . F o r to zero as Proof. follows:  y  X = -uu , co > 0  and  x e E,  t e n d s t o t h e boundary "of  G(x,y, x)  tends  E.  The p r o o f i s performed s t e p b y s t e p .  One p r o c e e d s as  Suppose f o r example t h a t G^)( jy,X) - 0 X  L e t us assume  (X  Q  y - 3E  i s negative) that  G^ ^(x,y,X )/ 0 2  Q  Thus t h e r e e x i s t s a  as  6  and a sequence  ^ ^(x,y ,X )> 2  n  as  0  y - BE . y  6 > 0  n  -» z e 3E  for a l l  and l*-y l n  > a > 0 ,  R e c a l l the estimate ( s i m i l a r r e s u l t s hold i t e r a t e s and d i m e n s i o n s ) f o r dim E = 2: 0 < G^ ^ (!x,y,-u) ) < M exp(-u>| x-y |/2) . 2  2  such t h a t ri  (l)  60.  Thus we have 0 <  2 )  (x,y >-u> ) ...CM exp( -t»| x - y | /2) 2  ;  n  n  < M exp(-ij) a/2) — a where  M  oo  i s a c o n s t a n t independent o f  Thebrem 4.1)".  '  From t h i s i t f o l l o w s t h a t t h e r e e x i s t s a '  o  X - ^ s -oo^)  (see equation 4.6 of  '. •  ...... i  such t h a t 0 <  G ^ ^ X ^ A - L )  < 6/4  for a l l  n.  (2)  By thebrem 2.10 we have D if  X  G ,(x,y,X) = G (2)  x  ( 5 )  (x,y,X)  i s n e g a t i v e , from w h i c h I t f o l l o w s t h a t '  -G  G^UyT^)  ( 2 )  (x,y ,X ) = n  0  X  l  J  G ^ . ( x , y , s ) d s . (3) n  S i n c e t h e p r o o f o f Theorem 4.5 c l e a r l y shows t h a t 0.^^-(3c,y ,X) n  uniformly f o r  X-^ "•  lim  -0  s <_ X  Q  as < 0,  y  R  - z e aE  we have  \  . [ G ^ ^ ( x , y .s)ds = 0 .  Thus from (j>) i t f o l l o w s t h a t as i G ^ ^ x ^ X ^  n -»*:»  - G ( > ( x , y , X ) | -..0,  (4)  2  n  0  w h i c h c o n t r a d i c t s e q u a t i o n s (1) and ( 2 ) . Thus  /^(x,y,\)  tends  6l. t o z e r o , as  y  3E,  f o r each  x and n e g a t i v e  X.  To p r o v e t h e f i n a l r e s u l t we u s e t h e f o l l o w i n g ' a r g u m e n t : By Theorem 2.10 and t h e same argument used t o p r o v e e q u a t i o n (4) i t follows; that | G ( x , y , X ) - G ( x , y , X ) - (X _- X ) G ) ( k , y , X ) 1 - 0 ( 2  n  1  as" . n. •-» ».  n  Q  ]  6  n  Q  The argument p r o c e e d s i n e x a c t l y t h e Same manner,  except f o r t h e i n t r o d u c t i o n o f t h e t e r m '.••"(• X^"-''X) Q  Which we a l r e a d y know goes t o z e r o as of  2  n  For higher  Q  values  k t h e p r o c e d u r e i i s c l e a r , t h a t i s y reduce t h e r e s u l t one  s t e p a t a t i m e u n t i l one has Theorem 5.2. if  n .-''»'.  G^ ^(x,y ,X )  G(x,y,X) -* 0  The Green's f u n c t i o n  G(xyy,X):  X i s negative.  Proof. negative  L e t Gj- and  a s y -» BE. is-unique f o r E ";•-.  G  2  he two Green's f a c t i o n s f o r t h e same  X and s e t f ( x , X) = G ( x , y , X ) 1  -^^i7^})^y^:i(<^^,)  •  By t h e r e s u l t s o f P a r t 2 we have ( A + X) f ( x , X ) - 0 if  x e E-  X i s n o t on t h e n o n - n e g a t i v e r e a l a x i s .  n e g a t i v e we can show  f ( x , X ) s 0,  I f X i s r e a l and  hy t h e maximum p r i n c i p l e . The  maximum-principle s t a t e s t h a t I f AU(X) + a(x) - u(x) = 0 f o r x i n a bounded opein s e t K ,  i f a ( x ) <_ 0 o n X, and i f u ( x )  62.  a t t a i n s i t s maximum a t an';interior" pbiiit''bf '"-K,..^ then' u ( x ) = 1  constant  on  f ( x , X ) j£ 0 point  x  Q  K . To a p p l y t h i s t o on  e E  E,  f(x,x)  suppose t h a t  so, w i t h o u t T o s s o f g e n e r a l i t y , t h e r e i s a  with  f ( x , X ) = a > 0.  Let  Q  K = E^  chosen so l a r g e t h a t ( c f . Theorem 2.6)  x  a/2  f o r any  for  |x| > X.  f(x ,X) = o Q  y  Then  |f(x,X)| < a  and t h e r e f o r e  f(x,X)  X.  Consequently  arid t h u s  on  K = E ,  he chosen a r h i t r a r l l y l a r g e ,  x  € E^  and  X  Is  |f(x,X)| <  x e 9K,  m u s t : a c h i e v e i t s maxlmttm a t  an I n t e r i o r p o i n t o f f ( x , X ) '.•= 0  Q  where  f ( x , X ) s c o n s t a n t oh  hy Theorem 5.1.  f(x,X) = 0  on  E.  Since  X  K, may  63.  P a r t 6.  Applications of the Green s:Function. 1  I n t h i s p a r t we S h a l l p r o v e two a s y m p t o t i c one f o r t h e e i g e n v a l u e s In;;part: 3.  1  properties,  and one f o r t h e e i g e n f u n c t i d r i s a s d e f i n e d  Throughout t h $ s p a r t we s h a l l assume'that  a l l t h e c o n d i t i o n s Of t h e p r e v l b u s i t h e o r e m s  E  satisfies  i n o r d e r t h a t we s h a l l  have a •": Green;' 'sifu n c t i o n w h i c h , i s 'zero i o n t h e l h o M d a r y o f ?  F u r t h e r m b r e y we:;assume t h a t t h e d i m e h s i b h o f  E vis  E.  2.  Let F ( x ) ^Mx where  - G(x,y,X)  p = |x-y[.  (1)  The f i r s t term on t h e r i g h t hand s i d e I s t h e  Green's f u n c t i o n f o r  A  I n the i w h o l e ^ p l  at  x=y: S i n c e a l l Green's f u n c t i o n s have t h e same'type o f s i n g u l a r i t y  as  p -• 0. ., OUr f i r s t t a s k i s t o o b t a i n hounds and  estimates"for the function respect t o  F(x,y,-u)  asymptotic  and I t s : d e r l y a t l v e s w i t h  u.  I f we s e t X = - j i where AF(x)  u  I s r e a l and p o s i t i v e , t h e n  = u F(x) .  Now we w i s h t o a p p l y t h e maximum p r i n c i p l e t o F ( x ) : however, since  E  I s n o t bounded we need t o know t h a t  By Theorem 2.8 we have t h a t  G(x) .-» 0,  and  goes t o z e r o a s : p  y.  Furthermore  H ^  as  F.(x) • •-•0  x  »  as  for fixed  x'•"-». •. X  so we have  64.  the r e q u i r e d r e s u l t t h a t  F(x)  -* 0  as  x -• «.  Now  i f we  the maximum p r i n c i p l e i n the u s u a l manner (see Titchmarsh p.169] we have t h a t of  F(x)  apply [10,  must assume i t s maximum on the boundary  E. If  x '.is i n t e r i o r to  G(x,y,-u) = 0 ,  and  E  and  y  i s on the boundary,  so  F(x,y,- ) = | i  (pj=fl - ( 2 r r )  .H^ ) 1  U  _ 1  K ( pjfl Q  i n the u s u a l n o t a t i o n s o f B e s s e l f u n c t i o n s ;  K v (t) o '  Now  is a  x  p o s i t i v e , s t r i c t l y decreasing function of hand s i d e l i e s between d i s t a n c e from regard  x  x  0  as f i x e d and  y U  all  for  x  and  K (a^), Q  where  y.  Since  v a r y i n g , we <:(2TT)  P  _ 1  right a  i s the I f we  now  obtain  K (aS)  (2)  Q  Is, continuous  at  x=*y  t h i s i s true  x=y. I f we  we  (2ir)~^~  Hence the  to=the n e a r e s t p o i n t oh the boundary.  0 < F(x,y,- ) for  and  t.  a g a i n a p p l y the r e s u l t s i n Titchmarsh  [ 1 0 , p.  can show - max  {(2TTU)"  1  K (aViI), Q  (^u / )" 1  2  1  a K^aJv)  }  < F^(x,y) < 0 where  P,, =  F.  The  higher d e r i v a t i v e s .  next  step i s t o extend  these r e s u l t s t o  170]  65.  Let  FJ  = D^F. p  From Titchmarsh [IG, ,-p\l-70] we know  l. ( >:y>-u) = - %  u~  x  u  K^p^u)  l / 2  y e BE .  We wish t o show' t h a t  F i r s t / b f •• afl:/..we; can  u p<*/d K  =  *JT-  show  . k/2 "  hy applying^.the';'e;'4uati:phs ;f i  tigt);^-v  ; ;  K ft) v  v K (t)  »  v  - t K _^(t) v  - t K^Ct)  .  Equationsv (4) can be, found i n WatsOn [ i i , / p . 7 9 Se'c. 3.71].  The  next step I s / t o d i f f e r e n t i a t e the l a s t term i n ( 1 ) , but by Theorem  2.10 D  k  G(x,y,X) = k!  and thus f o r  y e SE  be zero s i n c e  ^  G  ( k +  ^(x>y,X) ,  the d e r i V a t l v e Of the term  I s r e a l and  G^ ^(x,y,-a)= 0 k  G(x,y,-u) for  will  y e dE.  T h i s completes the 6ai;cuiati6n f o r e q u a t i o n (3) . ^\F^6m^"e^atid h^••':(4) I t i s easy t o see t h a t :  D  (t K k  t  However, /since t  k  K (t) fc  Is  K  k  (t)) = - t  k  K^Ct)  .  i s ; a! p o s i t i v e : f u n c t i o n , i t - f o l l o w s t h a t  a/positive" s t r i c t l y decreasing  fMcMbh  of  t.  Hence,  66.  if  y € 3E,  then a < ^  0 < F . (x,y,- ) 2  where  a  M  M  2  2^ »/^ - ^ 7 —  K  a  i s t h e d i s t a n c e between  x  (5)  and t h e n e a r e s t boundary  point. I n order, t o a p p l y t h e maximum p r i n c i p l e t o  F~  we  must f i r s t show t h a t  A  k  p  .  p  : +  k  (k-i).  p  M  •  <> 6  [(1.)/ H^ and G are.Green's f u n c t i o n s ( s e e e q u a t i o n 6 '• . s a t i s f i e s the d i s t r i b u t i o n equation  Since both F  1  "A F = u F +  6 '  .  We o b t a i n e q u a t i o n (6) by d i f f e r e n t i a t i r i g t h e d i s t r i b u t i o n k-times w i t h r e s p e c t t o  u,  (l)),  and o b s e r v i n g t h a t  D  8  equation  0.  As a s p e c i a l case o f (6) we.have * 2-M P  = P  P  2^u  +  2  P  n '  <> 7  We s h a l l work w i t h (7) i n o r d e r t o o b t a i n a bound f o r F . p  u s i n g t h e bounds a l r e a d y known f o r F  and  FY  H i g h e r r e s u l t s can  be shown by i n d u c t i o n . Suppose F  p <  > 0  Ort dE  T?2'\x  n  a  s  a  n  e  g  a  t i v e minimum i n s i d e  and i t tends t o z e r o f o r l a r g e  E  (note  x ) . Therefore  67.  but t h i s c o n t r a d i c t s e q u a t i o n (7) ,• so on  E.  Suppose n e x t t h a t  Pg  F g . ^ i s non^negative  . t a k e s on a v a l u e , i n s i d e  (5), then F g . ^  g r e a t e r t h a n t h e r i g h t hand s i d e o f e q u a t i o n must have a p o s i t i v e maximum i n s i d e  E,  E.  i.e.  and  hence o o „ K (a^ii) a K, (a,/u) F < - - F < max - { — — , ^, } . 2,uU U ~ M ^ % u 0  0  1  /  2  Thus we have K (ayu) 0  < F  (x,y,-u) < max{—  p  u  TT  M  a K,(a,/u)  A  a  —  ,  2  Kg(ay )  * •—r" 8 TT  2TT  u  —' 3» U  By an i n d u c t i v e argument we have , a K. (a^u) < max {-—-. i + 1 k - i / 2 ' 1=0,1,...k 1! 2 u 1  k  F  (x,y,- )  k  *  u  M  5  1 + i  K  1  /  d  We a r e now i n a p o s i t i o n t o extend t h e a s y m p t o t i c  formula  f o r t h e e i g e n v a l u e s ^ : g i v e n by C l a r k I n [3], t o Our domain .E. Let  T(X)  where between  A x  be any f u n c t i o n such t h a t 0  <  T(X)  <  A a(x) '  and  £  are p o s i t i v e constants;  k  + < £ :  , a(x) i s the distance  and t h e n e a r e s t p o i n t on t h e boundary, and  - i n t e g e r such t h a t t h e - I n t e g r a l o f  a(x) '  J\ J* [ G ^ (x,y,X) | d y dx < « E E 2  k  over where  E  k  i sa  i s f i n i t e and  m = [^]  68.  Is a l s o f i n i t e . . Furthermore.let  {  0  < o  x  X  p T  where  Q  = ( T,  |u | ),  u  n  "being the u s u a l e i g e n fu n c t i o n s d e f i n e d  n  In Part 3, Theorem 6 . 1 .  dim E = 2,  If  and  N (X) T  T(X)  and  a r e as  d e f i n e d above, then  -^JTW  f (X) T  dx .  E Proof.  If  X  and  X'  a r e not e i g e n v a l u e s , £ )  G(x,y,X) - G(x,y,X') =  n=o  u / x ) u (y) (X - X») f -  ( V ) (V ') X  by the r e s u l t s of the p r e v i o u s s e c t i o n s . X' = -u', _(„.  -  |JL and  where  1% u_(x)  )  I  M  n  n=o  |i  u (y) n  f  then  X  I f we l e t X = -u  are p o s i t i v e ,  , = | [K (pyn) -  and  then K (p^.)] Q  (^(V^) - F(x,y,-u) + F(x,y,-u') .  I f we d i v i d e b o t h s i d e s o f t h i s e x p r e s s i o n by fi'  tend t o  u,  j  u (x) u (y)  n=o ^ *n  +  and l e t  we have ( I f we can show the s e r i e s converges  uniformly with respect to  Z  u' - u  W  ,  u')  69.  To show t h e se^eB^'d6^verge's''un±tp^lj'^  c o n s i d e r t h e t a i l end,  of t h e s e r i e s :  C*riH* >' "  n=N  1  Now since^ (see  tends t o  u,  -u  can be c o n t a i n e d i n a s e t L  1  Q  P a r t 2 ) , and t h u s »  2/ \  u„(y)  by Lemma 3.5 and Theorem 2.5 ( n o t e independent o f y  and  y  I s f i x e d ) , where  [x\ . Thus t h e s e r i e s i s . u n i f o r m l y convergent,  s i n c e t h e t a i l end can be made s m a l l (by c h o i c e o f N) dent o f  K i s  indepen-  u' . Since  p K (p^u) 1  H• °- • • • and  K  ^ ( t )-^st » •rv  by l e t t i n g the  as  - 1  t -* 0,  2 ir u  we have  / \2  u fx)  y -• x  ,  i n expression ( 8 ) , i . e .  s e r i e s i s u n i f o r m l y convergent i n y  p  0.  We p r o v e t h a t  by a methbd s i m i l a r t o  t h a t w h i c h was used b e f o r e . T h e ^ n e x t I s t e p i s t o d i f f e r e n t i a t e t h e e x p r e s s i o n (9) t o  70. b u i l d up t h e f o l l o w i n g s e t o f e q u a t i o n s :  n=o  n -  I n o r d e r t o p a s s t h e d e r i v a t i v e t h r o u g h t h e summation s i g n we must show t h a t t h e r e s u l t i n g s e r i e s i s u n i f o r m l y convergent w i t h to the v a r i a b l e i n question.  respect  A g a i n c o n s i d e r t h e t a i l end o f a  Serle s 1Ike those i n (10); i . e . « V  2/ n  u  (  3  "2" where  k  i s a f r e e Index g r e a t e r than zero.  I , ^ r + 2  <  U+n) n=N ri™' w  v  where , K  1  Thus a s u s u a l  llo.(*;s-u)H <  k  V  (x +u) v A  1 K  >  n  n  I s bounded, s i n c e  x  i s f i x e d and  t h e s e r i e s I n ( l O ) Is'• u n i f o r m l y convergent i n  -y  e L .  u  f o r any  Q  "and hence;we ban d i f f e r e h t l a t e (9) w i t h r e s p e c t t o ;  p  Thus 'k>  2,  a s many  t i m e s as we p l e a s e . Let theorem.  T ( X ) be d e f i n e d a s i n t h e statement o f t h i s T ( X ) and I n t e g r a t e j  M u l t i p l y e x p r e s s i o n (10) by  this  g i v e s ( i f we can p a s s t h e I n t e g r a l s i g n t h r o u g h t h e summation) £  1*  T(X)  U  2  I  ( X )  +  P  ,  ^k¥-  ,  J  t(x  >  F  k-(i(x'x)dx-  (i:L)  E ;• We s h a l l show d i r e c t l y t h a t t h e i n t e g r a l and summation s i g n i n ( l l ) can be i n t e r c h a n g e d .  Let  71 •  .. 2  n=o< Consider  ,  T  0  u (x) 2  n  eo  dx  _  U_2,  E n=N  2/  0 9  _T  U„  <•  E oo  since  T(X)  n=o n u  2^ ^  ^  A  (x) + | i  '  dx  x  i s e v e n t u a l l y l e s s t h a n one.  I f we l e t m = [ j r * ] * '  t h e n t h e t a i l end! i s l e s s t h a n K X"  J J |G^)(x,,yvX)| dy E E  1  2  dx  Which I s bounded hy t h e h y p o t h e s i s on  k'.  Thus we c a n make t h e  o r i g i n a l d i f f e r e n c e as s m a l l a s we p l e a s e by a s u f f i c i e n t l y l a r g e choice of  N,  and hence we can pass t h e i n t e g r a t i o n s i g h t h r o u g h  t h e summation s i g n i n (11) f o r k >. k'. I f we use t h e d e f i n i t i o n o f  T^.,  namely  %  = ( > )> T  u  n  t h e e x p r e s s i o n on t h e l e f t hand s i d e o f (11) can be e x p r e s s e d as CO  I  ( X  n  +  u ) -  k  -  1  T  n  .  n=o We w i s h t o e x p r e s s t h i s t s e r i e s a s an I n t e g r a l i n o r d e r t o a p p l y  72.  7  a T a u b e r i a n theorem.  Consider  7 *  N (X)  f -HXP T  0  (  ^  *X  -  k+l  \  D  i.  jto  2  -/  |  .  (T- +  A  +  ... +  0  TJ)  6  3  Z. , ,•: . \ fc+i d=o J ^ (  X  +  Thus by combining t h i s r e s u l t w i t h (11) we have P N (X) J , -Tk+2 •  k "  dX  Q(X+U)  +  ( 4 7 r k (k  f  (k+l)!  T  +  1  1  J  >tf>  ( x ) F.  E  ^  r t ( x )  d  x  E  (x,x)dx.  (12)  L e t u s assume f o r t h e moment t h a t J T(X)  P^.^x^xjdx = 0(|i- "^)  ( u - •)  k  (13)  E  where  C  i s someL f i x e d ; p o s i t i v e number.  proved a f t e r t h e a p p l i c a t i o n - o f  T h i s r e s u l t w i l l be  t h e f b l i b W i n g T a u b e r i a n theorem t o  e x p r e s s i o n ( 1 2 ) , ( s e e T i t c h m a r s h [10, p.364]) . -;;lf.  0  where  a > 1  (x+y)"  x^  and 0 < $ < a ,  f(x)  then  ( ) x"- " P(a-p)rO) c r  g  6  1  .  Thus-:we have from e q u a t i o n (12) and t h e h y p o t h e s i s , e q u a t i o n (13),  73.  that  (\) ~ - T ^  N  + 2  ) _ J L _ f (x)dx . T ( k ) 4Tr(k+l)k J . T  r(2)  Hence  N j X ) ~.  T.f(x)d±--, E  :  w  w h i c h i s what we wished t o p r o v e . The theorem w i l l be complete i f we can show t h e h y p o t h e s i s (13).  R e c a l l t h e c o n d i t i o n on  T(X),  0 < T(X) ..<: A a ( x )  namely  k ,+ < f  k' where  A  and  £  a r e c o n s t a n t s and  i s i n t e g r a b l e over  E,  where  k' i s such t h a t a ( x )  a ( x ) i s t h e d i s t a n c e between  and t h e n e a r e s t p o i n t o n t h e boundary.  ~ (-gf)  v  e" '  which i s v a l i d f o r a l l  as  C  u~  5 / 4  t -  co,  ( s e e Watson [ 1 1 , s e c . 7 . 2 3 ) , t o t h e  v  bound we found f o r P ^ ^ ^ y V - i J i ) ,  <  I f we a p p l y t h e e s t i m a t e  .  •1/2  K (t)  a?  we see t h a t f o r  a^/jl >. 1 / 2 ,  e x p ( - V l i )  / 2  S i n c e t h e l a s t term i n t h e p a r e n t h e s e s dominates t h e o t h e r s . The g e n e r a l case f o r |  Plr  . (x,x)| <  F ^(x,x)  gives  k <  CVO ** )/ 2  x  1  4  a " k  1 / 2  e- ^ a  74. where  a^u 2 , 1 / 2 .  i s a c o n s t a n t and  C  Next we w i s h t o examine  |P. ( x , x ) |  f o r a^/ji <_ 1/2.  To; do t h i s we use t h e e s t i m a t e s  0( [ l o g t | )  K (t) Q  as  t -0  as  t  and  K (t)  = 0(t'^)  ±  T h e f i r s t t e r m I n the i n e q u a l i t y f o r :•••.•"..<; C u "  :  K  | log  u The I * ! term  ( i > 0)  1  a i  for  i ^> k-i/2 K  (  k i ) i  a^ul  .  gives rise t o  .  for a^jK  1 ^ _ k-i/2 "  1  C  c  C  <  C u"  fc  | l o g aVu|  -k  ^  Thus t h e f i r s t term dominates  (x,x) |  1/2.  gives  , . a . 4 I i/2  a  a^/u < 1/2. |F  *U.  v  K J a,/u) k  ( i > 1).  P^; •(x,x) K  °  0  and we have  f o r a^u < 1/2 .  We s h a l l how attempt t o e s t i m a t e t h e I n t e g r a l i n equation  (13). Let  E„m =» {x € E  a(x),/u <_ 2 ~ } m  I f we use t h e h y p o t h e s i s on applies t o E I J T(X) o  Q  < C  m = o, 1, 2, .....  T(x) and t h e above e s t i m a t e w h i c h  (a(x),/u <_ 1/2),  F ^ x ^ d x  1  then |a(x) o  k + £  u"  k  | l o g a(x)Vul dx  75which i s l e s s than C u^j] a(x) C-E o  k + C  | l o a ( x ) | d + | J|log>|a(*) E o g  k + £  X  - d x j. J  rt  Consider the f i r s t i n t e g r a l : J  a(x)  <  hut f o r x e E  | l o g a ( x ) | d x < C; I  k + £  f  C max a ( x ) ^ x e E. -1/2  a(x)  a(x)  a(x)  k + < f / 2  k  dx  dx,  E.  and Hence t h e maximum o v e r  E  C/k  of  a(x)  i s ; l e s s than  &  Also |  [logu|a(x)  k + £ :  dx  E '' IS: l e s s t h a n  G». p.~ ^.  P u t t i n g t h e s e t o g e t h e r we have  €  I | T(X);  <  ,(x,x)dx  G u "  k  ~ ^  Consider next t h e i n t e g r a l oyer m=0,  as  E  m + 1  |i >••«•;  - E  m  for  1, 2 y , . . . I . e .  E C  , -E m+1 m  G „-(2k+l)/4 E  |  a ( x )  k+£  a ( x )  f c - l / 2 -a(x)^u ^ e  ;  m+1- m E  We'' know f r o m t h e d e f i n i t i o n o f E , m  t h a t i f ^/€ E ^ - E ^  then  76  and so our i n t e g r a l over  G  M  -( + )A 2k  1  x  €  m  E m  ax  +i~  m  {a(x)  m+1"  E  E  I s l e s s than  E  k + £  "  e^*^}  m  J  a(x) dx, k  E  which i n t u r n I s l e s s than -(2k*l)/4  C  since  a(x) < u " / 1  I  J  T  m+1"  E  but  E  (x)  ^ H + £ - l / 2 exp(-2 - ) nl  { (  2  2  and  m  :L  a(x)^ u > 2 ~ . m  J a(x) dx k  )  1  Therefore ^ ^ - l / % p ( - 2  F ^ x ^ l C j C * ;  m  since 09  we have J  T  dx  P . k  < .C-  "k  f/2  w  E-E . o  i ^ ^ ^ ^ ' e x p t - ^ m=o  where the I n f i n i t e s e r i e s c l e a r l y converges. J E-E  T  P  k-u u  Cjf  Thus  -k-£/2  o from which I t f o l l o w s t h a t  J E  T  P  k . u dx  C' (u  <  <  C  u""*"^  + u  )  as  u  1  ) ,  a  77. The p r o o f o f t h e a s y m p t o t i c f o r m u l a i s now complete. Theorem 6.1 i s a generalisation o f t h e w e l l known a s y m p t o t i c  Remark:  [10> p . 1 7 2 ] ) .  formula f o r the eigenvalues (see Titchmarsh i s a hounded s e t we c a n s e t T(X) => 1 then  T  fora l l  n,  E.  from t h i s i t f o l l o w s t h a t  N(x) ; i s t h e number o f e i g e n v a l u e s l e s s t h a n  where for  = 1  N  throughout  T(X) -= 1  If E  I f T(X) => 1, N (X) « N(X), T  X.  Thus  Theorem 6.1 reduces t o N(X)  Theorem 6.2.  — 4ir  ~  a r e a E.  If  then M X) =- —  +  X  4ir  1 / / 2  )  f o r each  x.  TJ s i n g t h e re S u i t s and n o t a t i o n • o f Theorem 6.1, we have  Proof. for  0( X  dim E = 2 f n  -  to  (X +u) n  k + 1  ,  1  (-i) p  ,  k  %k7  ki  k  »  ^ -  Furthermore  2  oo  ioo  k+i k  4irk(k+l)|i *  n  /  t(^u)  +  V  k + 1  I l i J - F - .„(x,x) . v ,  ki  v  k-u  78.  Thus, s i n c e a(x)  F. ( x , x ) < C e x p ( - a ( x ) y u )  depends on J  i f  n  x  ilAL  0 (X+u)'  a(x)^/u > 1  o n l y , we have dx  ~  1  -•• - • h-  47rfc(k+l)  u  as  u - »  .  I f we a p p l y t h e T a u b e r i a n theorem used i n Theorem 6.1, $(X)  where  ~  — 4ir  as  X  T h i s r e s u l t Can be imprbved  » . to  •(x)- = f- + o(x ) 1/2  by the: methods o f T i t c h m a r s h [ 1 0 , p . 1 9 8 ] ,  then  79.  BIBLIOGRAPHY 1.  SHMUEL AGMON, L e c t u r e s on e l l i p t i c boundary v a l u e p r o b l e m s , P r i n c e t o n , New J e r s e y , Van N o s t r a n d (1965).  2.  F .H. BROWNELL, Spectrum o f the S t a t i c P o t e n t i a l S c h r o d i n g e r equation over  E . n  A n n a l s o f M a t h e m a t i c s , Volume 54, No. 3.  (Nov. 1951), P- 554-594. 3.  COLIN W. CLARK, Ah a s y m p t o t i c f o r m u l a f o r t h e e i g e n v a l u e s o f t h e L a p l a e i a n O p e r a t o r i n an unbounded domain, t o appear i n t h e B u l l , Amer. Math. Soc.  4.  COLIN W. CLARK,  An embedding theorem f o r f u n c t i o n s p a c e s ,  t o appear i n t h e P a c i f i c J . Math. 5.  R, COURANT AND D. HILBERT,  Methods o f M a t h e m a t i c a l P h y s i c s ,  I and I I , I n t e r s c i e n c e : New Y o r k ( l 9 5 3 i 6.  1962).  DUNFORD AND SCHWARTZ, L i n e a r O p e r a t o r s , I and I I ,  Interscience:  New Y o r k (19 BT'y 1963) . 7-  I.M. GLAZMAN,  D i r e c t methods o f q u a l i t a t i v e s p e c t r a l a n a l y s i s  of singular d i f f e r e n t i a l operators.  (Russian), Fizmatgiz:  Moscow (1963). 8.  A.M. MOLCANOV,  On, t h e c o n d i t i o n s f o r d i s c r e t e n e s s o f t h e  spectrum o f second o r d e r s e l f - a d j o i n t d i f f e r e n t i a l o p e r a t o r s ( R u s s i a n ) , Trudy Mosk. Mat. Obshchestva 2 (1953), 169-200.  8o. 9.  F. RELLICH,  Das Eigenwertproblem o f  i n , ''Essays;presented  Au+Xu = 0  i n Halbro^reri^  t o R. Courant" ..." The Courant A n n i v e r s a r y  Volume (1948), p. 329-344. 10. E.G. TITCHMARSH, E i g e n f u n c t i o n Expansions A s s o c i a t e d  with  Second Order D i f f e r e n t i a l E q u a t i o n s . P a r t I I , Oxford U n i v e r s i t y P r e s s : Oxford 11. G.N. WATSON,  (1958). Theory o f B e s s e l F u n c t i o n s Second E d i t i o n ,  Cambridge U n i v e r s i t y Press:(1944)»  

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