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UBC Theses and Dissertations

Perturbation of nonlinear Dirichlet problems Fournier, David Anthony 1975

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PERTURBATION OF NONLINEAR DIRICHLET PROBLEMS  by  DAVID ANTHONY FOURNIER  B.Sc, ,  U n i v e r s i t y o f B r i t i s h Columbia, 1968  M.Sc,  University  o f B r i t i s h Columbia, 1972  A THESIS SUBMITTED IN 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 the  required  t h e s i s as conforming t o  standard  ^ " U N I V E R S I T Y OF BRITISH COLUMBIA  In presenting this thesis  in partial fulfilment of the requirements for  an advanced degree at the University of B r i t i s h Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this  thesis  for scholarly purposes may be granted by the Head of my Department or by his representatives.  It  is understood that copying or publication  of this thesis for financial gain shall not be allowed without my written permission.  Department of The University of B r i t i s h Columbia Vancouver 8. Canada  Date  /J•  O '7.  Supervisor  :  - i i-  Dr. R.C. R i d d e l l  ABSTRACT  The  s o l u t i o n s o f weakly-formulated n o n - l i n e a r  are s t u d i e d when t h e d a t a o f t h e problem a r e p e r t u r b e d The  d a t a which undergo p e r t u r b a t i o n s  Dirichlet  problems  i n v a r i o u s ways.  i n c l u d e the L a g r a n g i a n , t h e boundary  c o n d i t i o n , the b a s i c domain, and t h e c o n s t r a i n t s , i f p r e s e n t .  The  main c o n c l u s i o n  s t a t e s t h a t the s o l u t i o n o f t h e D i r i c h l e t  problem which m i n i m i z e s the D i r i c h l e t the d a t a so l o n g as i t i s unique.  integral varies continuously  D e t a i l e d hypotheses a r e f o r m u l a t e d  i n s u r e the v a l i d i t y o f t h i s c o n c l u s i o n The  hypotheses a r e n o t much s t r o n g e r  for existence, problems.  i n the g e n e r a l i z e d  with to  f o r s e v e r a l l a r g e c l a s s e s o f problem.  than the standard  Lusternik-Schnirelman  sufficient theory  conditions  o f these  - iii  -  ACKNOWLEDGEMENTS  I would l i k e t o thank Ron R i d d e l l f o r s u g g e s t i n g t h e problem t r e a t e d h e r e and f o r h i s H e r c u l e a n e f f o r t s t o make me unscramble my confused p r e s e n t a t i o n o f i t s s o l u t i o n .  I would a l s o l i k e t o thank the U n i v e r s i t y of B r i t i s h  Columbia  and t h e N a t i o n a l Research C o u n c i l o f Canada f o r t h e i r f i n a n c i a l s u p p o r t .  - iv -  TABLE OF CONTENTS Page  INTRODUCTION CHAPTER 1  CHAPTER 2  PARAMETRIZED NONLINEAR DIRICHLET PROBLEMS IN STANDARD FORM Notation  1.2  Formulation  o f t h e Problem  6  1.3  Formulation  o f Theorem I  8  1.4  Proof o f Theorem I  12  NONLINEAR DIRICHLET PROBLEMS WITH VARIABLE DOMAINS 2.1  Formulation  2.2  P r e l i m i n a r y Consequences o f the Assumptions  20  2.3  Construction of the Associated Problem  29  2.4 CHAPTER 3  4  1.1  o f Theorem I I  V e r i f i c a t i o n of Conditions  16  Standard  [1.1] - [1.6]  32  NONLINEAR DIRICHLET PROBLEMS WITH VARIABLE H0L0N0MIC CONSTRAINTS 42  3.1  Formulation  3.2  Construction of the Associated Problem  3.3  o f Theorem I I I  V e r i f i c a t i o n of Conditions  Standard  [1.1] - [1.6]  45 49  - v -  TABLE OF CONTENTS  (Contd.)  Page  CHAPTER 4  BIBLIOGRAPHY  EXAMPLES 4.1  P e r t u r b a t i o n of Minimal Surfaces  56  4.2  P e r t u r b a t i o n o f t h e Operator  57  4.3  Domain P e r t u r b a t i o n s  59  4.4  P e r t u r b a t i o n o f Geodesies  60  62  - 1-  INTRODUCTION  The  purpose o f t h i s t h e s i s i s t o study  t h e behaviour  o f the  s o l u t i o n s t o n o n l i n e a r d i f f e r e n t i a l boundary v a l u e problems o f D i r i c h l e t t y p e , when the data d e f i n i n g t h e problem a r e s u b j e c t e d to v a r i o u s perturbations. restrictions,  The b a s i c r e s u l t we s h a l l o b t a i n s t a t e s t h a t , under s u i t a b l e the s o l u t i o n o f such a problem changes c o n t i n u o u s l y w i t h the  data as l o n g as the s o l u t i o n i s unique.  The c o n d i t i o n s under which  c o n c l u s i o n i s . v a l i d a r e e s s e n t i a l l y t h a t the u s u a l s u f f i c i e n t  this  conditions  f o r e x i s t e n c e i n the v a r i a t i o n a l t h e o r y o f D i r i c h l e t problems s h o u l d  hold  u n i f o r m l y i n some sense as the g i v e n problem i s p e r t u r b e d .  The  d i f f e r e n t i a l e q u a t i o n s which appear i n the boundary v a l u e  problems c o n s i d e r e d here a r e the E u l e r - L a g r a n g e integral  equations  of m u l t i p l e -  ' D i r i c h l e t ' f u n c t i o n a l s d e f i n e d on s u i t a b l e Sobolev spaces.  We  c o n s i d e r such problems i n t h e i r weak f o r m u l a t i o n , i n which a s o l u t i o n i s taken  t o be a d i s t r i b u t i o n which s a t i s f i e s  the g i v e n boundary c o n d i t i o n i n  an a p p r o p r i a t e g e n e r a l i z e d sense, and which i s a c r i t i c a l p o i n t o f t h e Dirichlet integral restricted of the E u l e r - L a g r a n g e d e f i n i n g the D i r i c h l e t  equation  t o such d i s t r i b u t i o n s . a r i s e s from the f a c t  The n o n l i n e a r i t y t h a t the i n t e g r a n d  f u n c t i o n a l need n o t be q u a d r a t i c i n i t s arguments,  but need o n l y have a c e r t a i n c o n v e x i t y i n i t s dependence on t h e f u n c t i o n s on which the f u n c t i o n a l i s d e f i n e d .  F o r such problems, t h e r e a r e w e l l  known ' r e g u l a r i t y ' theorems a s s e r t i n g when a weak s o l u t i o n i s i n f a c t a smooth f u n c t i o n and hence i s , by a s t a n d a r d  i n t e g r a t i o n by p a r t s argument,  - 2 -  a c l a s s i c a l s o l u t i o n of the E u l e r - L a g r a n g e concerned  equation.  We  s h a l l not  be  w i t h t h i s q u e s t i o n , and we work e n t i r e l y w i t h weak s o l u t i o n s .  The  d a t a d e f i n i n g such a problem appear to be of f o u r k i n d s :  (i)  The  'Lagrangian',  or i n t e g r a n d , of the D i r i c h l e t  (ii)  the ' D i r i c h l e t d a t a ' , o r boundary c o n d i t i o n s ;  (iii)  the 'domain', i . e .  the s e t over which the  functional;  independent  v a r i a b l e s i n the d i f f e r e n t i a l e q u a t i o n a r e a l l o w e d to run; (iv)  the c o n s t r a i n t s , i . e .  the s e t i n which the dependent  v a r i a b l e s are r e q u i r e d to l i e .  In p r i n c i p l e  we  c o u l d f o r m u l a t e a g l o b a l problem on s e c t i o n s o f  a subbundle of a smooth f i b r e bundle where the t a r g e t s u b f i b r e r e p r e s e n t s ( i v ) and  the base m a n i f o l d r e p r e s e n t s ( i i i ) ;  and we  ought to v a r y  p u l l i n g back a l o n g v a r i o u s embeddings i n t o the base, v a r y the subbundle to v a r y , and v a r y  ( i ) and  (iii)  ( i v ) by a l l o w i n g  ( i i ) at w i l l , a l l simultaneously.  T h i s amount of g e n e r a l i t y p r e s e n t s t e c h n i c a l o b s t a c l e s which obscure main phenomenon,  and,  by  the  i n a d d i t i o n , t r e a t i n g p a r t i c u l a r cases of the above  a l l o w s us to d i s p e n s e w i t h some of the assumptions i n each case which a r e needed i n the g e n e r a l case.  Thus Chapter  1,  f i b r e bundle  we  we  s h a l l take the f o l l o w i n g l e s s g e n e r a l approach.  In  f o r m u l a t e and prove a theorem i n the s e t t i n g of a f i x e d  over a f i x e d base m a n i f o l d , where d a t a  allowed to v a r y .  In Chapter  2  we  suppress  ( i ) and  ( i i ) are  ( i v ) by c o n s i d e r i n g s e c t i o n s  - 3 -  qf a t r i v i a l v e c t o r bundle, and we h o l d (iii)  to vary.  to v a r y .  I n Chapter 3  ( i ) f i x e d , but we a l l o w ( i i ) and  we f i x ( i ) and ( i i i ) and a l l o w ( i i ) and ( i v )  A simple example o f the s i t u a t i o n i n Chapter 1  i s the problem  of minimal s u r f a c e s i n o r d i n a r y E u c l i d e a n space, under v a r i a t i o n boundary  curve.  The s i t u a t i o n o f Chapter 2  i s i l l u s t r a t e d by t h e problem  of d o m a i n - p e r t u r b a t i o n s f o r n o n l i n e a r e l l i p t i c of  R  n  .  The s i t u a t i o n o f Chapter 3  of t h e  boundary  problems  on domains  i s i l l u s t r a t e d i n the study o f  g e o d e s i e s , o r more g e n e r a l harmonic m a p s i n imbedded s u b m a n i f o l d s o f f  IR  We c o n c l u d e our d i s c u s s i o n by s p e l l i n g out these examples i n a l i t t l e more detail  i n Chapter 4 .  uniqueness assumption  I t s h o u l d be noted t h a t the s i g n i f i c a n c e  i n our main r e s u l t i s i l l u s t r a t e d i n a l l these c a s e s ,  by well-known phenomena o f jumping^.of ness breaks down.  of the  t h e m i n i m i z i n g s o l u t i o n when u n i q u e -  - 4 -  CHAPTER 1  PARAMETRIZED NONLINEAR DIRICHLET PROBLEMS IN STANDARD FORM  1.1  Notation  G e n e r a l l y we s h a l l f o l l o w t h e n o t a t i o n o f P a l a i s reader's convenience we s u p p l y t h e f o l l o w i n g b r i e f  M  compact boundary  C  m a n i f o l d o f dimension  list  n,  [4]. F o r t h e  :  p o s s i b l y w i t h nonempty  9M ;  u  s t r i c t l y p o s i t i v e smooth measure on  E  ( t o t a l space o f )  M ;  00  C  f i b r e bundle over  M ;  CO  £ (Pv^ k  ( t o t a l space o f ) product bundle  C  v e c t o r bundle over  M x IR  M ;  ;  integer _ 0 ;  k J  (E)  C°°(E) S(E) p  P L£(£)  bundle o f k - j e t s o f s e c t i o n s o f s e t o f a l l C°° - s e c t i o n s o f s e t of a l l sections of  E ;  E ;  E ;  r e a l number _ 1 ; 00  Banach space c o m p l e t i o n o f ?C.(^) norm on d e r i v a t i v e s o f o r d e r <_ k ;  i n t h e Sobolev  p ' t h power  - 5-  I?(E)  ( f o r pk > n)  Banach m a n i f o l d o f . a l l  which belongs hood b,s P  £  in  elements  of  to  L^(£)  closed L^(E) b  [|  ||  C  L  E  s u b m a n i f o l d of  i n some neighbourhood  P  L f ( E ) c o n s i s t i n g o f the c l o s u r e i n K s e £( ) L  (depending on t h e p a r t i c u l a r  s t r u c t u r e on L  £( )  L  £( ) E  T(L£(E));  s) o f  8M;  also,by  T*(L^(E)) ;  induced by  E  k - j e t e x t e n s i o n map  which agree w i t h  E  s t r u c t u r e on t h e tangent bundle  F i n s l e r m e t r i c on  FB(E,E')  neighbour-  £( ) ;  abuse,induced 6  each o f  E ;  of the s e t of a l l sections  Finsler  E,  f o r some open v e c t o r bundle  0 0  L, ( E ) , k b  sections of  —  y L  Q (  || || ;  ^ (  J  s e t o f a l l f i b r e - p r e s e r v i n g maps o f  E  E  ) )  >  to  E' ;  CO  F^  map F  of  C ( E ) to  S(E )  induced by c o m p o s i t i o n w i t h  e FB(E,E') ;  til Lgn^(E)  s e t of a l l k  order  L a g r a n g i a n s on  E,  i . e . maps  CO  L  : C (E) — >  F  e F B ( J ( E ) ,0^);  It element  dy  S(iO  k  s h o u l d be noted on  M ,  F  o f t h e form represents  t h a t the norm i n  L = F*°J  a l l choices of  charts of  M,  u  ^(5)  L^( ). E  r  s  o  m  e  depends on t h e volume p o s i t i v e smooth  a r e e q u i v a l e n t t o Lebesgue measure i n a l l  so a l l i n d u c e e q u i v a l e n t norms i n  F i n s l e r s t r u c t u r e s on  o  L .  b u t , by the.meaning o f ' s t r i c t l y  measure',  f k  L^(C),  and e q u i v a l e n t  - 6-  The F i n s l e r m e t r i c i s d e f i n e d , t e c h n i c a l l y , o n l y on the path components o f distinct  path  The  l£(E).  We s e t 6 ( s , s ' ) = 2  whenever , s, s'  belong to  components.  r e a d e r s h o u l d a l s o note t h a t no.a p r i o r i  t i o n s a r e made on the maps  F,  smoothness assump-  hence on the L a g r a n g i a n s  L .  Such  assumptions w i l l be made below as they a r e needed.  1.2  F o r m u l a t i o n o f t h e Problem  We b e g i n by assuming s e v e r a l p i e c e s o f d a t a t o be g i v e n and h e l d f i x e d throughout  the chapter :  [la]  A choice of  [13]  Choices o f bundle); 6  on  M,  y,  p > 1  and  and  E ;  k _ 1  and a c h o i c e o f  (pk > n  i f E  on  T(L£(E)),  || ||  i s not a vector with  induced  L. (E); P  [1Y]  A locally  [16]  An element  compact H a u s d o r f f  b e l£(E)>  homeomorphism  space  a map  r  B ;  -> b  r  of  B —>  L, (E) , k  and a  - 7 -  6-bouncled s e t s to 6-bounded s e t s ;  [le]  A map  ri—>  to a  We [16]  C  s e c t i o n s of  {r} x (E^ .  B —>  l£(E)  —>  in  [ly]  B  and  We  a  C°  IE of r l£(E).  L?(E) k  L  extends  r .  Then  $ ^  ir : IE — >  i s smooth on each  from s t r i c t  w i t h the f i b r e  consistency, {r} x IE , r  on B,  fibre  and which i s a  L^,  integral  defines a function on  L  an r-dependent f a m i l y o f L a g r a n g i a n s  L s dy  = M  1  that  s h a l l sometimes d e v i a t e  f(r,s)  C  as a space of parameters  g l o b a l t r i v i a l i z a t i o n of the map  Item [ l e ] s p e c i f i e s such t h a t the  such t h a t each  LjflE^) .  w i t h the e x t r a p r o p e r t y  i d e n t i f y the subset .-. subset of E C B x  is  Lgti, ( E ) ,  an r-dependent f a m i l y o f D i r i c h l e t - t y p e boundary v a l u e s  E,  r,  of  map  1  t h i n k of  specifies  ( r , s ) H—>  L  f : B x L£(E) —>  £(E)  f o r each  partial-function  g  = f | r  (r e B,  r  r .  IR  The  whose p a r t i a l - f u n c t i o n  restriction  f o r each  E  s e LJJ(E)) .  r ;  and  g = f|  E  then has  = f(r,.) a  the c r i t i c a l l o c u s of  C  1  g , r  r namely  K  i s a w e l l defined Dirichlet  r  =  {  S  e  E  r  :  d  ^  s  c l o s e d subset of  problem w i t h d a t a  [la] -  )  =  -.  °  By  [ l e ] , we  the s t a n d a r d  parametrized  mean the problem, to  describe  -  the ( d i s j o i n t )  =  | ( r , s ) : r e B,  the c r i t i c a l l o c i o f a l l the  Of c o u r s e the problem s e c t i o n we  -  union  K  of  8  g» r  s £ K  j  r  IE a  as a subset o f  as posed  B x L^(E)  i s too g e n e r a l , and  s h a l l put c o n d i t i o n s on the d a t a , i n p a r t i c u l a r  b e h a v i o u r w i t h r e s p e c t to the parameter to prove c o n c l u s i o n s o f a s i m i l a r  f  r  B,  in  k i n d about  on  .  i n the next their  which w i l l enable  the s e t s  K  .  r  However, an  important r e s t r i c t i o n has a l r e a d y been b u i l t i n , by the p r o v i s i o n i n t h a t the d i s j o i n t  union  IE  o f the spaces  E  of the r-dependent  B,  t r i v i a l i z e d by  'partial-function'  g^  B,  on  $ ^ .  We  f o r each  [16]  varia-  t i o n a l problems should come t o us equipped w i t h the s t r u c t u r e o f a space over  us  fibre  a r e thus a b l e t o form a t  i n the model f i b r e  IF  ,  namely by  g (r) t  =  (go$)(r,t)  =  g (()> (t)) . r  r  C e r t a i n key t e c h n i c a l c o n s i d e r a t i o n s i n what f o l l o w s w i l l be o r g a n i z e d around  1.3  these f u n c t i o n s  g^ .  F o r m u l a t i o n o f Theorem I  Given a s t a n d a r d p a r a m e t r i z e d D i r i c h l e t §1.2,  we  function  problem  s h a l l study the subset of each c r i t i c a l l o c u s g  r  takes i t s minimum v a l u e ,  as d e f i n e d i n K  r  on which the  To be p r e c i s e , we b e g i n w i t h a f a m i l y IF = L ? ( E ) , K b  which i s d e f o r m a t i o n i n v a r i a n t .  ambient homotopy o f  IF ,  H(0,O  i s the i d e n t i t y on H^T)  =  i s a l s o a member o f the f a m i l y Given  F,  of subsets of  T h i s means i n v a r i a n t under  i . e . , f o r any continuous map  H : [0,1] x (F — >  such t h a t  F  IF  IF, and f o r any  | H(l,t)  T e F,  the s e t  : t e T  ( c f . Browder [ 2 , D e f ( l . l ) ] ) .  F  F ( r ) o f s u b s e t s o f each  we.define a c o r r e s p o n d i n g f a m i l y  E » r  by F(r)  Then the F-minimax o f f o r any  =  j. S C E  g  i s defined  =  F  inf SeF(r)  r  m^-realizing  subset o f  Kp(r)  =  C  f o r some  to be the f u n c t i o n  i s the d i s j o i n t  T e F j  m^  on  B,  where  K  g (s) ;  over  C Cl B,  r  r  =  U reC  i s defined  and  the  union  K (C) F  sup seS  |(r,s) : s e K  More g e n e r a l l y , f o r any subset over  r  r e B,  m (r)  and the  : S = <|>(T)  r  K (r) .  t o be  g ( s ) = nip(r) j- . r  n y - r e a l i z i n g subset o f  K  - 10 -  Our aim i s t o g i v e c o n d i t i o n s on the d a t a under which be continuous  on  B,  Kp(r) w i l l be nonempty f o r each  w i l l be compact i n IE  The  f o r any compact subset  existence of a point i n  generalized Lusternik-Schnirelman Browder  .  Kp(C)  B .  by P a l a i s  o f c o n d i t i o n s on the d a t a  s t a t e s the s t a n d a r d hypotheses  hypotheses  will  K p ( r ) i s the main a s s e r t i o n o f the  t h e o r y developed  I n the f o l l o w i n g l i s t  Schnirelman  r  of  and  [5., 6 ] , and  [1, 2, 3 ] , where i n c i d e n t a l l y s e v e r a l examples o f f a m i l i e s  are g i v e n . the f i r s t  C  r,  m^  F  [la] - [ l e ] ,  f o r e x i s t e n c e i n the L u s t e r n i k -  t h e o r y , and the o t h e r s m a i n l y r e q u i r e t h a t the e x i s t e n c e h o l d u n i f o r m l y , i n v a r i o u s senses, w i t h r e s p e c t t o the parameter  Note t h a t some c o n d i t i o n s r e f e r t o the f u n c t i o n s  the u n r e s t r i c t e d  f  or  g^,  and some t o  f r  [1.1]  F o r each and  the f u n c t i o n  g  : E^ —-> IR  r  s a t i s f i e s c o n d i t i o n (C) o f P a l a i s - S m a l e E  [1.2]  r e B,  f o r which  r  § ( _) s  r  x  S  bounded and  converges  to zero c o n t a i n s a convergent  F o r each  R e IR  K  H  : r e C  and  ( i . e . , any sequence ||dg '(s^)|| r  subsequence ) .  and each compact subset  |(r,s)  i s bounded below  C  of  B,  the subset  g ( s ) <_R r  i s compact i n IE .  [1.3]  F o r each  r e B  and each  V  in  such t h a t the subset  of  r  B  R e IR,  t h e r e e x i s t s a neighbourhood  - 11 -  IF  R  i s bounded i n [1.4]  For each  j t E (F : g ( r ' ) '<_ R  =  v  r e B  and each bounded subset V  of  r  in  6(* (t), * ,(t)) r  [1.5]  r'  For each  e > 0,  and a l l  r e B,  B  r' e N  each bounded subset  F o r each  r E B  exists  A E IR  and a l l  (F,  there  exists  E  N  of  of  L^(E),  r  in  and each  B  such t h a t  e  S . S  of  L^(E),  there  such t h a t  r  E  s  <  S  and each bounded subset  ||df (s)||  for a l l s  of  e  t h e r e e x i s t s a neighbourhood  for a l l  T  t e T .  |f(r,s) - f ( r ' , s ) |  [1.6]  j  such t h a t  <  r  V  E  r* e V  IF .  a neighbourhood  for a l l  f o r some  t  <_ A  S .  With these p r e p a r a t i o n s done, we can s t a t e our r e s u l t  :  Theorem I Suppose a s t a n d a r d p a r a m e t r i z e d data  [la] - [Is]  D i r i c h l e t problem i s g i v e n by  s a t i s f y i n g c o n d i t i o n s [1.1] - [ 1 . 6 ] .  Let  F  be a  - 12  deformation-invariant  f a m i l y o f subsets of a s i n g l e path component of  such t h a t a t l e a s t one  on  F  element o f  i s compact and  F-minimax f u n c t i o n  (a)  The  (b)  For each  i s not  empty;  m^  . of  g  nonvoid.  IE,  Then :  i s f i n i t e and  continuous  B ;  over  r  (c) Kp(C)  of  K  some open s e t  r e B,  the  n y - r e a l i z i n g subset  For each compact subset over  REMARK.  N,  -  N  C  of  B,  the  with  K  mp-realizing  In case in  B,  Kp(r)  i s a singleton  so t h a t  Sp  {sp(r)}  i s a s e c t i o n of  subset  Sp  r  i s continuous.  so l o n g as i t i s unique.  f o r each  ir : IE — >  which r e a l i z e s a p a r t i c u l a r minimax v a l u e  continuously  of  i s compact.  i t f o l l o w s e a s i l y from (c) t h a t  p o i n t of  C  Kp(r)  B  Thus the mp(r)  r  in  over  critical  varies  In the examples which  we  have i n mind, t h i s i s the c o n c l u s i o n of p r i n c i p a l i n t e r e s t .  1.4  P r o o f of Theorem I  We Browder IE  [2].  f i r s t prove c o n c l u s i o n For  fixed  r e B,  ( b ) , by  let  X  assembling s e v e r a l r e s u l t s i n  denote the connected component of  which contains, a l l the s e t s i n the f a m i l y  the r e s t r i c t i o n of  g  to  X .  Then  X  F(r),  and  let  h  denote  i s a connected submanifold of  the  r CO  C  F i n s l e r manifold  (E ,  [2, P r o p o s i t i o n 5.2,  p.32]  X .  The  function  h  and  i s complete i n the induced m e t r i c .  there  exists a quasigradient  i s bounded below and  field  for  s a t i s f i e s condition  Hence h (C)  on on  - 13 -  X ,  on account o f c o n d i t i o n  and the remarks all  following  mp(r).  p.27],  [2, Theorem 1, p.8 and D e f i n i t i o n ( 2 . 1 ) , p . 1 8 ] ,  But t h i s f o l l o w s from the e x i s t e n c e  F ,  element i n  and Browder's  To prove (a) and quantities D  (E,  (F,  ( c ) , we K  apply  [7, Theorem 1.2]  i n p l a c e of the " q u a n t i t i e s  I t requires  that  F  F(r)  our  I t s conclusions  and each  R e IR,  of  F  i s equlcontinuous at  Thus the p r o o f V  r,  e x i s t s a neighbourhood  p,  of  : of  p.  TT ,  the c i t e d  (a) and  V  x  $ "*" o f  to apply  a r e p r e c i s e l y the c o n c l u s i o n s  there  F,  to i n s u r e t h a t the  i n our n o t a t i o n , becomes the f o l l o w i n g  such t h a t the f a m i l y o f  to choose  only  and one o f i t s two hypotheses i s j u s t c o n d i t i o n  other hypothesis, E B  E,  be i n v a r i a n t under homeomor-  by a f i x e d t r i v i a l i z a t i o n  we do not need the homeomorphism-invariance  Theorem I ;  w i t h our  a r e independent o f the c h o i c e o f t r i v i a l i z a t i o n  S i n c e we have d e f i n e d  result.  of a compact nonvoid  a p p l i e s to e s t a b l i s h ( b ) .  but t h i s i s used i n the p r o o f  F(r)  families  Theorem 1  TT, g, and  (F,  of t h a t theorem.  phisms of  r  [2, P r o p o s i t i o n ( 5 . 1 ) ,  the hypotheses o f [2, Theorem 1] a r e s a t i s f i e d , except f o r the f i n i t e -  ness of  f,  [ l . l ] , a n d so by  (c) o f  [1.2].  The  f o r each r  in  B  functions  where* as i n [1.3]  of Theorem I  w i l l be complete  above,  as soon as we have shown how  so t h a t t h i s e q u i c o n t i n u i t y a s s e r t i o n h o l d s .  Here we  shall  - 14 -  use c o n d i t i o n s  [1.3] - [ 1 . 6 ] .  r E B  Fix of  r  in  [1<$],  B  4> (T) r  such t h a t  N  of  1  and a l l t E T.  By [ 1 . 3 ] ,  T = IF  r  r  .  such t h a t  =  1  i s bounded i n L ^ ( E )  s  =  2  N  d  2  | <j>, (t) r  along with  E > 0,  Given  r  1  r  r  t  E  S = S  2  1  j  to f i n d a  <  |  (r'E N , s ES ), 2  2  such t h a t  '• . 6 ( + ( t ) , * , ( t ) )  3  s' £ S  such t h a t  <_ A  The we a p p l y [1.4] to f i n d a neighbourhood  N ,  f o r some  we a p p l y [1.5] and [1.6] w i t h of  A > 1  and so a l s o i s  | s E L^(E) : 6 ( s , s ' ) < 1  r  E  f o r a l l r ' el  r  r' E N , t E T }  :  <J> (T) ;  df (s)  r'  there i s a  <5(<j> (t), <j> ,(t)) < 1  f(r,s) - f(r'.,s)|  for  By assumption i n  E = 1,  By [1.4] w i t h  V  Hence the s e t S  and a number  t h e r e i s a neighbourhood  i s bounded i n (F .  K,v  i s bounded i n E  neighbourhood  neighbourhood  R E (R.  and  T .  r  <  (s £ S ) 2  O  N  2  min | 1, | £ |  of  r  such t h a t  15 -  Now  choose any  c o n v e n t i o n adopted nents o f  k£(E)  in  t e T = (F  §1.1, t h a t p o i n t s  have  6 ( s , s ' ) = 2,  particular,  t h a t the p o i n t s  a C  y  1  path  in  l£( )  s, s'  the l a s t  s = <)> (t)  and  r  r' e  in  •  y  <  min  A c c o r d i n g l y we  the p r e c e e d i n g i n e q u a l i t i e s ,  |g (r') -  i n different  p a t h compo-  s' = 4> t(t) r  can be j o i n e d  by  \  1,  2A  1  from e i t h e r  can a p p l y the f i r s t ,  second  endpoint,  and l a s t  of  to get  g (r)|  t  t  =  | g , ( * . ( t ) ) - g (4> (t))|  -  |f .(s*) - f ( s ' ) l + |f (a') - £ (s)|  r  I n view o f the.  inequality implies, i n  p o i n t on such a path can be no f u r t h e r than  so l i e s  .  with  E  length  Any  arid any  R v  t  r  r  r  r  r  r  1  Thus of  {g  t  (Y(u))||  l  <_  -| + A- ( l e n g t h  : t e F  Theorem I  ||df  <  +  R,V  ||Y'(U)||  du  0  Y)  <  e.  } are equicontinuous at  i s complete  .  r  as c l a i m e d , and  the p r o o f  CHAPTER  2  DIRICHLET PROBLEMS WITH VARIABLE DOMAINS  2.1  F o r m u l a t i o n of Theorem I I  In  t h i s chapter we  s h a l l c o n s i d e r n o n l i n e a r D i r i c h l e t problems i n  which the c a n d i d a t e s f o r s o l u t i o n s are r e a l - v a l u e d f u n c t i o n s on a bounded domain ft d a .  Our  IR ,  a g r e e i n g on the boundary  n  interest  are perturbed.  i s i n the behaviour  8ft  with a prescribed f u n c t i o n  of the s o l u t i o n when both  ft  The r e s t r i c t i o n to r e a l v a l u e d f u n c t i o n s i s f o r  of n o t a t i o n o n l y , and  the r e a d e r w i l l e a s i l y see how  t o modify  and  a  convenience our  state-  ments to cover the case of v e c t o r - v a l u e d f u n c t i o n s .  We  s t a r t w i t h a bounded open s e t ft° O  x = (x^,...,x ) n  and w i t h Lebesgue measure  n - m u l t i - i n d e x i s an n - t u p l e w i l l write  |a|  ,  and  a. operator number of  JYO/Sx.) a's s  point i n jk(IR  )  IR  k  .  having °  For any i n t e g e r let < k, ' — ' s  .  and  1  Note t h a t  IR  k  of the product bundle  IR  a -  ft°  over  o  ft  and  we  f o r the p a r t i a l d e r i v a t i v e  a  k _ 0,  s,  u = (u ) i i , a |a|<k  w i l l denote the w i l l denote a  = ft° x IR  over  ft°  ,  total  typical  bundle  whose s e c t i o n s we  o  i d e n t i f y w i t h the r e a l v a l u e d f u n c t i o n s on  IR  D  An  ou _ 0,  i s j u s t the f i b r e of the k - j e t  H  product bundle  with t y p i c a l point  of i n t e g e r s  n  \ ou i  n  dco(x) = dx^...dx^ •  a = (a^,...,a )  f o r the sum  IR ,  ft°  .  More g e n e r a l l y , f o r any  i with f i b r e  IR  we have a n a t u r a l  -  a  P  identification  of  Lf(fR k  identification  freely  n°  )  17  p o & L M f i , IR )  with  and  we  s h a l l use  this  k  i n the  following. s.  With  n ^ 1,  k >_ 1,  and  fixed,  let  F  : Q°  x IR  —>  IR  2 be  a  C  function.  (Thus  F  i s the  principal  p a r t of an  element  of  k FB[J  (IR  certain function [2.1]  ), IR  ].)  p >^ 2,  Suppose  certain  s a t i s f i e s the  constants  For  all  x,  For  all  a,  9F 8u  [2.4]  For  all  3u~ a  [2.5]  following conditions for  > 0,  and  a certain  U  x'  <  C  e  1 +  and  F(x,u) - F ( x ' , u ) |  [2.3]  C,  <E : R — > R with <C(0) = 0 : k For a l l x e 0° and a l l u e IR F(x,u) |  [2.2]  F  For  ( x  all  a,  '  u )  a,  <  (x,u)  |uJ  u e IR  (C(x -  o xe.fi,  all  • I a <k  all  <_  ,  and  a,  x,  all  x'  o e fi ,  <  '  g,  o x e-fi°,  all  i  k  u E IR  r  iu  |a|£k  a  ,  C  ~ 9u~ a  ( x  ,  i+  x')  |e|<k  all  1  u )  and  (E(x -  and  p  all  x')  all  k  u e (R  1  +  u e.'R  ,  ,l$l<k J k.  ,  a  continuous  - 18 -  8 F (x,u) 8u a i i -  I  2  o fi  x e  For a l l  [2.6]  <  C  1 +  |u | " P  k  and a l l u, v e IR  2  ,  |P-2|„ |2  |a|,|B|<k  Conditions  [2.1],  a  3  |a|=k  P  [ 2 . 3 ] , and [2.5] i n p a r t i c u l a r imply t h a t t h e i n t e g r a l  h (v)  -  u  i s well defined f o r a l l v  ,o  F ( x , j . ( v ) ( x ) ) dco(x) k  i n t h e Sobolev space  L^(fi°, IR),  and t h a t  h°  2 is i n fact a symbol  C  J^Cv)  function i-  the s e c t i o n o f  ( c f . Lemma 2.1  i n §2.2  below)'.  Here the  b e i n g abused t o denote the p r i n c i p a l p a r t o f the k - j e t of  s  L^((R ) fi°  whose p r i n c i p a l p a r t i s  v .  We s h a l l  consider  k  'restrictions'  h = h^  o b t a i n e d by r e p l a c i n g  fi°  i n the above i n t e g r a l by  v a r i o u s subdomains fi C fi° .  To be p r e c i s e , f o r a l l r  i n a s u i t a b l e parameter space  B,  we  c o n s i d e r an open subdomain fi^ CT fi° whose c l o s u r e i s o b t a i n e d as the d i f f e o m o r p h i c image over  B,  A(r)  Diffeo[M,fi°]  A(r)(M)  o f a f i x e d smooth m a n i f o l d  M .  As  r  i s supposed t o change c o n t i n u o u s l y i n the space  of  Z  9  diffeomorphisms o f  M  into  fi°,  equipped w i t h the  t o p o l o g y o f u n i f o r m convergence o f a l l d e r i v a t i v e s .  We a l s o suppose  a s u i t a b l e boundary-function  r.  following.  ranges  a^_  i s g i v e n f o r each  that  Our r e s u l t i s the  - 19  -  Theorem I I F i x a bounded open subset ft° d o k F : ft x (R  (R ,  an i n t e g e r  n  2 C  S  r e a l number  p _ 2.  [2.1] - [2.6]  Let  —>  IR  be  k _ 1,  and  a  and s a t i s f y c o n d i t i o n s  g i v e n above.  F i x a l o c a l l y compact H a u s d o r f f space  B  and a compact  00  n-dimensional  C  continuous from from  B  into  manifold B  into  h  r  w i t h boundary  L^(ft°,(R),  and l e t  3M  .  Let  ri—>  r I—>  A.(r)  be  a^  be  continuous  Diffeo[M,ft°] . r e B,  F o r each and l e t  M  be the  h  r  C  2  set  ft = A(r)(M) r  and s e t  r e a l v a l u e d f u n c t i o n on  (v)  D  r  D  r  = L?(ft ,R) K r a  d e f i n e d by  F ( x , j ( v ) ( x ) ) da)(x)  =  k  ft..  Then the f o l l o w i n g , c o n c l u s i o n s h o l d : (a)  The  function  m(r)  i s f i n i t e and continuous on (b)  =  : B —>  inf veD r  =  IR  d e f i n e d by  h (v) r  B ;  r e B,  For each  M(r)  m  e D  the s e t  : dh (v) = 0 r  and  h (v) = r  i s not v o i d ; (c)  For each compact subset  C  of  B,  the s e t  m(r)  , r  - 20 -  A*M(C) > , | ( > v°X(r)) : r e C r  i s a compact subset  of  B x  LJ^IR^)  a s u i t a b l e standard  conditions  parametrized  from Theorem I problem.  Browder [1, pp 25-29]).  t h i s a r e w e l l e s t a b l i s h e d and we s h a l l n o t d w e l l  Before  Preliminary  The techniques  we  shall  h^ .  consequences o f the assumptions  [2.6].  Lemma 2.0 For each tt <Z tt° t h e map  ^2 C  f o r doing  f u r t h e r on the p o i n t .  In t h e f o l l o w i n g we use t h e assumption t h a t  extends to a  be noted t h a t our .  by e x p l o i t i n g t h e  c a r r y i n g out the r e d u c t i o n t o Theorem I ,  e s t a b l i s h some p r o p e r t i e s o f the f u n c t i o n s  2.2  o f §1.2, by c o n s t r u c -  I t should  [2.1] - [2.6] can be weakened c o n s i d e r a b l y  Sobolev i n e q u a l i t i e s ( c f .  v e M(r) |  .  We s h a l l deduce Theorem I I ting  and  map  Furthermore, the maps  F  satisfies  [2.1]-  21 -  d(F ) A  :  L P ( J  (  k  V  L(LP(J ( , L j ( k  J - >  V  V  )  and d (F,). : L P ( J ( ) 2  ->  k  V  take bounded sets into bounded sets. 2  Lg  L (LP(J ( 2  k  R f i  )), L J ( ^ , ) )  (The reader i s warned that the symbol  above means symmetric b i l i n e a r maps and should not be confused with  F i n a l l y f o r v, v^, and d(F )(v)(v ) A  1  =  v  p k e LQ(J  2  6F ( v  V ; L  >  )  and d (F )(v)(v v ) 2  A  Proof  :  H (v) fl  h  n  2  =  6 F (v 2  v  1 9  v ) . 2  See Browder [1]  For ft C Q°  Let  l s  : Lj(R ) — > R fl  l e t H° : L^(J (IR )) — > IR be defined by k  fi  F v(x) du>(x)  -  A  be defined by  h^ = H °J fi  k  Lemma 2 . l ( i ) For each fi <rT  ,  H  fi  : L*J(J (lR )) — > IR i s a k  fi  C  2  function  such that (a)  dH : L^(J (IR )) — > k  n  LP(J (R ) ) * k  takes bounded sets into  p L^)  - 22 -  bounded s e t s ; P (b)  dH  i s u n i f o r m l y continuous on bounded s e t s o f  Part  :  The f a c t t h a t  H  Q ( J  "  2  Q Proof  L  k  is C  f o l l o w s immediately 2  (b) f o l l o w s from t h e boundedness o f  from lemma 2.0.  d (F^) i n lemma 2.0  together  w i t h t h e mean v a l u e theorem. Lemma 2 . 1 ( i i ) o 9, C 0,  For each  h  : 'I_(R ) — >  R  i  s  a  c  2  fl  function  such  that dh" : LP(R )  (a)  fi  -> !X>* L  a n d  d 2 h  " k<V  •> L (L£(R ),IR)  : L  2  n  takes bounded s e t s i n t o bounded s e t s (b)  Proof  dh  Since  i s u n i f o r m l y continuous on bounded subsets o f  h ^ = H^oj  and  j  f  e  : .^(IR^) — >  l i n e a r map t h e p r o o f f o l l o w s immediately  L (J (IR )) P  n  from lemma 2 . 1 ( i ) .  Lemma 2.2  [dh"(v ) - d h " ( v ) ] ( x  >  f o r some c o n s t a n t  C  i  C  Y  i ,  2  V l  -  D V ( x ) - D°v„(x) 1  k  £  P  du(x)  L^(IR^).  i s a bounded  - 23 -  Proof  :  From lemma 2.0  dh  (v )(v 3  l s  i t follows that  v ) 2  3 F - ^ ' 2  n'l l l«l ft 0<_\ a I , I 8 | X  v  (  X  <k  k a  J  (  v  3 6  )  (  x  )  )  U  a  (  V V « )  U  B ^ k  (  v  2 > W )  So  [dhV-^ - dh^Cv^]^  - v ) 2  2 3 F  0<J a | , I B |  ft 0 J  <k u  6  a  a k r 2> H k r z (j  ( v  (x)  v  ( j  (v  v  ) (x))  dt dco(x)  and by [2.6] f I f1 | D \ ( X ) ft I a I =k •'0  + t(D°'(v -v )(x))| 2  D v (x) - D v (x) 1  K  2  P  2  1  | D ( v - v ) ( x ) | d t du(x) a  2  2  1  dx  | a | =k f o r some c o n s t a n t  c  Lemma 2.3 For  Cdh"^)  v  r  v , a 2  r  *  - d h ^ v ^ K v ^ )  f o r some c o n s t a n t  C  L ((R ) P  2 E  > C  independent  with  !I l- 2~ v  o f ft  v  V]  ( a  _  E  L J O R ')  l- 2 Il^p a  )  k  L  ;  and  a  r 2 a  - 24 -  • \'.-\  Proof  :  By lemma 2.2  independent o f  ft  1  .  .  we need o n l y show t h a t t h e r e e x i s t s a c o n s t a n t  such t h a t  I ' |D (v -v ) ( x ) | | a | =k f t a  P  du)(x)  J  >  c  <l v  V  _  "  (  l  a  —  a  l i  "  a  2 ' ' pj y  a 2  N  p L  Now  v, - v 1 2  - (a, - a„) e L (IR ),. . 1 2 k 0  independent o f  ft  such t h a t  I  _  Now  Therefore there e x i s t s a constant  P  n  |D ( a  c ||v  1  - v  -v )(x) - D ( a  V ; L  2  1  f 1  _ f |a|=k f t  I  1  independent of  ft  such t h a t  a  V l  9  P  1  9  Z  |D ( a  |D (  do>(x)  P  2  2  a  a  ,1  -a )(x)|  | D ( - v ) ( x ) - D ( a - a ) ( x ) | du>(x) .  J  |a|=k  a i  - (a - a )||  2  there e x i s t s a constant  • C.  k  V l  V ; L  -  -v )(x)| 2  V 2  ) (x) |  P  P  J  dw(x) + j  da)(x) +  ||  |D ( a  -  a i  a i  -a )(x)|  a || 2  dco(x)  P  2  P p  | a | =k  Therefore there e x i s t s a constant  _  |a|=k  1  ft  and the r e s u l t  JD^v^vpCx)!  follows  1 1  C  independent of  da)(x) + | |a 1 -a 2 1 |  P  ft  > Cl  such t h a t  l r 2 v  v  _ ( a  r 2 a  )  25  -  -  Lemma 2 . 4 Let  { ^} v  v. e L (IR )  with  P  x  tc  :  e  i  bounded sequence i n  a  {a.] c o n v e r g i n g t o  a..  Si  follows that Proof  D  L^(IR^)  a .  such t h a t  Then i f dh^(v.) — >  x fv.jj  Fix  x  i s a convergent sequence i n L^ftR^) .  e > 0.  Pick  1  A  so l a r g e t h a t  a. - a. W* < i j p k  1  ) - dh"(v )](  - v )  implies  ,  S,  11  i ,j > A  L  and [dh"(  for  some  <5^,  6  V;L  2  V ; L  to be determined.  2  6  2  -  <  2  By lemma  6  2.3  l ia  C  H  a 2  P L  k  Since  I v v W  p L  -  H r 2ll v  v  k  P  L  k  we get  > C  6.  Since  { ^} v  i-  sa  |v v | r  | a  2  bounded sequence  a  l  " 2'' a  L  implies  that  0, i t  <  a  a  there e x i s t s  p k  i r  r 2iip  6  1  a 2  n  P L  6-^  such t h a t  k  that  - 26 -  It follows  a  v  2  I 1  a  2"  P  - ClIv,-v„I| 1  p  L  T  Ce  P  2 ' l n  k  k  that  cl|v_-v ||P 2  L  Setting  la,-a„I |  ' r 2M pj  ir Np v  6  —^  2  < ^ |  p  6  +  2  k  we get  v. - v . 1  3  i  T  P  <  e  i t  Lemma 2.5 Fix  S C  LMIR ) K  bounded and  R e IR.  Let  h  denote  h  .  Then  ) . 0  Then  o6  the s e t  U  v e  L ((R ) : k QO W  h(v) < R  P  o  weS  i s bounded i n L, (IR ) . k  Fix  Proof  v e L (tR  h(v)  P  k  =  ) o/  .  Suppose V V  with  w  =  rl h(w) + dh(w)(t) +  =  h(w) + dh(w)(t) +  t e L (IR k o P  fi  h ( t + w)  =  By lemma 2.2  v = t + w  f 1  1  h(w) +  dh(w + u t ) ( t ) du  [dh(w+ut) - d h ( w ) ] ( t ) du  1 ^ [dh(w+ut) - d h ( w ) ] ( u t ) du  - 27 -  >  ^  h(w) + d h ( w ) ( t ) + C  1,,  — ut r du u P I  11  |  0  =  h(w; + dh(w)(t) + -  i i  p  k  L  S  A  IItl|p II  P  Since  P  <  1 1  i s bounded, by lemma 2 . 1 ( i i )  I | dh(w)  sup weS sup weS  ||=  | h (w) |  =  A  B  <  «> ,  < °°  Therefore h(w + t ) > - B - A t  • + L  for a l l  w e S,  h(w + t ) < R  where  with  C > 0,  w e S  P  L  k  p > 0 .  implies  t  P  that  P  k  I t f o l l o w s immediately  IItlI  that  i s bounded/  •1 1 1 1 p  k Lemma 2.6 Fix  S C  L ((R ) bounded and R e |R . T Q° = { w| : w e S P  k  i.e.  T  constant  implies that  of  S  to fi .  Then t h e r e  independent o f t h e c h o i c e o f fi such t h a t  V  £  {  ||v|| • k L  0, C  let  fi  c o n s i s t s of the " r e s t r i c t i o n " B e IR,  F o r any  V  E  < B  w^T  L  l%K  :  <  R  exists a  - 28 -  Proof  :  Pick  v e L (IR )  .  P  i s the r e s t r i c t i o n  v = w + t  w, e L p (IR  o f some  s e t t i n g i t equal to  Then  )  on fi° - fi .  h"  (v )  =  x  |  R +  By lemma 2 . 1 ( H )  J  J  W ; L  )  k  k  (w )du> 1  <  n  (v ) x  <  w  to fi° by .  Then  1  c e IR  such  that  c  R + c  there e x i s t s a constant •  V  M  £. A + sup w.eS 1  Lf  k  the r e s u l t  and  do)  A e IR w i t h  follows that  and  ,  Therefore  h  By lemma 2.5  (  v  n  F°j, (w, ) do)  there e x i s t s a constant  o  f o r a l l w^ e S .  J k  Extend  C a l l the e x t e n s i o n  h"(v) +  Fc  Q  0  t o fi .  Jfi°-fi  <  f o r t e L^(IR ) k^fi'O  I  |w 1"  p  1  L  k  follows f o r  B  =  A + sup  w.eS 1  I| i|I w  L  P  k  v, 1  T  p  < A .  Then i t  - 29 -  Lemma 2.7 For each ft , LF(IR^) k ft w  Proof  w e L QR ), P  fi  the r e s t r i c t i o n o f  h^  to  i s bounded below,  :  I n the c o u r s e o f v e r i f y i n g lemma 2.5  h ( t + w)  for  and  t e L (IR ) P  n  ,  > -  - B - AlItlI  L k  P  and lemma 2.7  we d e r i v e d  + - II11 P  I L  the i n e q u a l i t y  P  P k  f o l l o w s immediately from t h i s  inequality.  We a r e now ready to proceed w i t h the r e d u c t i o n from Theorem I I to Theorem I .  2.3  C o n s t r u c t i o n o f the A s s o c i a t e d  We c o n s t r u c t  [la]  M  has a l r e a d y been determined.  p,  k  on  LJ^IE^),  Problem  the s t a n d a r d problem by d e f i n i n g the d a t a [ l a ] - [ l e ]  smooth measure on  [IB]  Standard  M,  and l e t E  a r e a l r e a d y determined. so t h a t  Let  2  be any s t r i c t l y  be the p r o d u c t b u n d l e  Let  6(s ,s ) = ||s 1  u  || || 1  B  - s [ | 2  .  k  has been determined.  For each X(r)*  IR^. .  be t h e s t a n d a r d norm  L  [ly]  positive  : L (fi ,IR) — > P  r  r e B  t h e map  L (M,IR) P  X ( r ) : M — > fi  g i v e n by  r  C  fi°  A(r)*(v) = v°X(r).  i n d u c e s a map R e c a l l t h a t we  -  have a map  B —>  B —> L (M,IR) P  L (ft°,IR)  -  g i v e n by  P  r  > a  .  We d e f i n e a map  by  I—>  r  [16]  30  b  r  def E X(r)*(a ) r  L e t b = the zero s e c t i o n o f g i v e n above.  Let  $(r,s) = (r,s+b ) r  L^O^)>  $ : B x I^O so t h a t  r m  a  )Q —>  n  E  d  b  <)> (s) = s + b r  t  n  e  m a  P  r  1  —  >  b  a  s  r  d e f i n e d by  e  .  r  OO  For each  r e M,  let a  e C (M,IR)  r  a (y)X(r)*u(y) r  [le]  For  s e L^O^)  L s(y) r  =  a  nd  r e  B,  =  y(y)  be d e f i n e d by  .  let  a (y)F(A(r)(y),J (soX(r)" )(X(r).(y))) 1  k  r  where we r e p e a t t h a t t h e symbol  J^C')  i  S  b e i n g abused  the p r i n c i p a l p a r t o f t h e k - j e t o f t h e s e c t i o n  Assuming  s°X(r)  f o r t h e moment t h a t the s t a n d a r d p a r a m e t r i z e d  w i t h the above d a t a  s a t i s f i e s t h e hypotheses o f Theorem I ,  t o denote e L^Ot^ ).  problem  we i n d i c a t e  how Theorem I I f o l l o w s .  From the d e f i n i t i o n o f  L  L s ( y ) dy(y)  =  M  f  i t f o l l o w s t h a t f o r s e L^OR^)  L ° ( o A ( r ) ) ( v ) du(x) 1  S  ft r  that  31 -  so t h a t  g (t) = h^(toX(r)  .  r  F  =  Now i n Theorem I l e t  { .}  :  inf teE r  g(t) =  {8  s e L £ (  V  o  }  Then m (r)  =  F  So  mp(r) = m(r)  and hence  M(r)  Therefore  v e M(r)  m(r)  inf veD r  h (v) .=  m(r)  i s f i n i t e and continuous on  =  \ v e D  r  : dh (v) = 0  •" =  | v e D  r  : dg (voX(r) = 0  r  r  i f f v o A ( r ) e K^(r)  and  B .  Also,  h ( v ) = m(r) r  and  g (t°A(r)> r  from which i t f o l l o w s t h a t  = mp(r)|  M(r)  i s not v o i d .  Finally  A*M(C) = |(r,v°A(r))  above d i s c u s s i o n i t f o l l o w s t h a t follows that  A*M(C)  Therefore conditions  : r e C and v e M(r)j  A*M(C) = Kp(C)  and from the  and from Theorem I i t  i s compact.  i n order  [1.1] - [1.6]  t o prove Theorem I I  f o r t h e standard  we need o n l y  problem j u s t  verify  constructed.  2.4  [1.1] - [1.6]  Verification of Conditions  We shall carry out the verifications i n the order 1 , 4 , 3 , 5 , 6 , 2 Verification of [ 1 . 1 ] : _1  N  Fix  r e B  by lemma 2 . 6  g  t e L^OR^' . Then r i s bounded below.  r  and  Now assume that It follows that  fi  ||dh  r  x S  (t^ACr)  Cauchy sequence. Hence  {t^}  )||  a  r  r  ||dg (t^)|| ->- 0  bounded sequence with  a  -III  1 S  g ( t ) = h (t°X(r) ) and  0  . By lemma  r  -1  {t^oACr)  2.4  } is a  Cauchy sequence.  Verification of [ 1 . 4 ] : Since  <j>(s) = s + b  ,  r  6(<|> (s), cf) , (s)) r  r  =  | |s + b  r  - s + b ,| | r  \ =  I lb II  - b ,1 I r . r I I p. L  k  =  I f  |D (a oA(r))(y) - D (a ,<>A(r'))(y)| dy  <  I f  |D (a P ( r ) ) ( y ) - D (a o ( ' ) ) ( y ) | dy  a  a  a  P  a  A  P  A  > ( a °A(r'))(y) - D (a , °A (r')) (y) | dy a  +  i  a  M  r  It follows from the continuity of the functions  a  r I—>  P  and  r I—> A(r)  - 33 -  t h a t t h e above terms can be made a r b i t r a r i l y neighbourhood  V"  of  r  small f o r r '  i n some  r .  V e r i f i c a t i o n o f [1.3] :  Fix  r e B  that the s e t  implies that  and  R e IR .  {a , : r ' e W} r h  there e x i s t s a  "r'  Let  W  be a neighbourhood  i s bounded i n L (R. ) . P  K.  -1 ( (t+b^, ) <>X(r') ) <_ R .  B £ IR  Now  QO  From lemma 2.6  of  r  such  g (r ) < R t — 1  i t follows  that  such t h a t  | |(t + b ,)oX(r')" || 1  r  <  B  Then ItoXCr')" ! 1  But s i n c e t h e s e c t i o n s  b ,oX(r')  to t h e v a r i o u s ft  it  r'  s  sup r'eV  and t h a t t h e r e e x i s t s an  follows  a r e j u s t t h e r e s t r i c t i o n s o f the that  ||b , o X ( r ' ) r  A e (R w i t h  UtoXCrT !! 1  for a l l t e F  R,W t  -1  p  <  and a l l r ' e W .  From t h e c o n t i n u i t y o f t h e map  A  a ,'s  - 34 -  r*  it  1—>  X(r')  i s e a s i l y seen t h a t t h e r e e x i s t s a neighbourhood  constant  A^ e IR  V C  W  of  r  and a  such t h a t  | ItoXCr')" !|  < A  1  and  r' e V  \ implies that r  I Itl I  1 1  1 1  < A.. . 1  p  From t h i s i t f o l l o w s  that  IF,, R,V  i s bounded.  TI  Hi  V e r i f i c a t i o n of [1.5]  For ye  M,  let  s e L^flR^ x  let  = X(r)(y).  r  s  e L^CR^  r  Then f o r  f ( r , s ) - f(r»,s)  Ls  r  k  r  r  3 : ft^ —>  k  be g i v e n by =  Then  x^, = 3 ( x ) r  .  For  F(x ,,j (s ,).(x ,)) d u ( x , )  r  •"ft,-'  Let  1  , du  F ( x , j ( s ) ( x ) ) du>(x ) r  s°X(r)  [L s - L ,s] dp r r  =  dco -  r  denote  r, r' e B  M  Ls  )  X(r')°X(r)  -1  and we r e w r i t e the above as  r  r  r  F ( x , j ( s ) (x )) dco(x ) r  k  r  r  r  F(f3(x ) , j ( s , ) (3(x )))d(3*0)) (x ) r  k  r  r  r  [ F ( x , j ( s ) ( x ) ) - F ( x , j ( s , ) ( 3 ( x ) ) ) j du(x )  (A)  r  k  r  r  r  k  r  r  r  (B)  F ( x j ( s , ) ( 3 ( x ) ) ) - F ( 3 ( x ) , j ( s , ) ( 3 ( x ) ) ) dco(x )  (C)  F(3(x ),j,(s ,)(6(x ))) d[w-e*a>] (x )  r>  k  r  r  r  k  Jo..  r  r  r  r  We s h a l l deal with each of the above terms separately. (A) x •—> ( x , j ( s , ) ( 3 ( x ) ) ) r  r  k  r  r  i s a section i n L (J (lR^ )) . P  k  In fact i t i s just 3*(j (s ,)) . k  r  Furthermore, for s e S C L^OR^) bounded and s^, s^, as defined above there exists a neighbourhood  of r such that  3*(j,k (sr' ,)) - sr J  L  for a l l s e S and r' e V H  r  . Now  p < e  1 1  0  (A) i s the same as  (j (s )) - H (3*(j (s ,))) r  k  r  k  r  ft  r  And from the boundedness of dH  which follows from lemma 2.1 (i) we get  a bound on (A) which can be made a r b i t r a r i l y small.  - 36 -  1  (B)  By c o n d i t i o n  [2.2]  <C(x-3(x ))| r  Now f o r any  E > 0  (B)  1 + I|u (j (s ,))(3(x )) a  k  r  dco  r  t h e r e e x i s t s a neighbourhood  W  of  r  such t h a t f o r  cW ,  r'  sup |(C(x x efi„ r r (Recall that  3  Since  s ,  A e IR  such t h a t  r  i s derived  +  s e S.  (C)  k  a  s e S  I  .  r  by r e s t r i c t i n g  Let  (C)  o f [1.6]  S C  i s bounded by  (B)  r'  1  da)  p  a bounded s e t ,  I V V V ) ^ ) ) !  I t follows that  I t follows that  Verification  r  r  there e x i s t s a  dco  <  r I—> X ( r )  to - 3*w  constant  A  c a n be made a r b i t r a r i l y  From the c o n t i n u i t y o f t h e map  fi  (B)  h (3 (s ,)(3(x ))|  seen t h a t t h e d i f f e r e n c e o f the measure s m a l l over  e .  r ' ) . Therefore  a  v  r  <  r  I  +  from  1  fi  for a l l  - 3(x )) |  r  depends on  1  r  i s bounded by  small.  i ti s easily  can be made  uniformly  t o l i e i n a s u i t a b l e neighbourhood o f  can be made a r b i t r a r i l y  small.  :  L^OE^)  be bounded.  Then the s e t  { soX(r)"  1  : s e S }  - 37 -  i s bounded i n  ^ *  ^  df (s)(t)  it  since follows  dH '.(J (soA(r) fi  n  _ 1  k  L (J ((R P  ))  k  fi  )oJ (toX(r) k  that  ||dH '(j (soX(r)  1^0^) — >  1  k  i t follows  sup seS  and  w  =  r  From lemma 2 . 1 ( i i )  o  ))||  g i v e n by  <  » ,.  t l — >  J (t°A(r)  _ 1  k  )  i s bounded  that sup seS  Verification  ||df ( s ) | |  <  o f [1.2] :  Fix  R E IR  and  C C B  compact.  Let  {(r^,s^)}  be a sequence  with {(r ,s )} i  C  i  We need to show t h a t  Since  C  K  {(r^,s^)}  and [1.4]  : r' e C  and  i t follows  ^__} r  that  g ,(s) <R r  J .  has a convergent subsequence.  i s compact we can assume t h a t  sequence and we assume t h a t [1.3]  fl | ( r ' , s )  converges to {s^}  {__} r  r E B .  i s bounded.  i s a convergent From  In o r d e r  conditions to proceed  we  need the f o l l o w i n g p r o p o s i t i o n whose p r o o f w i l l be found below.  Lemma 2.8 Fix  S CT L ( R ^ ) k  bounded and  E > 0.  Then f o r each  r e B  there  - 38 -  e x i s t s a neighbourhood for a l l r ' £ V  and  From t h i s i s bounded, and  V  of ,j  E  S .  s  r  ||dg (s) - dg , ( s ) | | < e r r  1  i t follows  dg  such t h a t ,  that since  ( s . ) = 0,  that  converges t o  dg (s.)  r,  converges t o z e r o .  r  {s^} From  i a  this  i t follows that  lemma 2.4 {s^}  dh  we get t h a t  i s a convergent  r  -1 (s^°A(r)  {s^°A(r)  )  ^}  converges to z e r o .  Finally  from  i s a convergent sequence and t h e r e f o r e  sequence.  P r o o f o f Lemma 2.8 :  We w i l l That i s t°A(r) \  employ t h e same n o t a t i o n as i n the v e r i f i c a t i o n o f [ 1 . 5 ] .  f o r s, t e L^ClE^),  we l e t  and f o r r , r ' E B  3 = A(r')°A(r) , _ 1  £ 1^0^  r  3 : fi — >  5 n  n t  J  k  r'  denote  r  r  )  r'  (  s°A(r) , _ 1  be g i v e n by  x , = 3(x ) .  \ ( s , ) ( x ,) W  r  )  & ,  r  £ Q.^ we l e t  r  =  r  r  we l e t  and f o r x  dg ,(s)(t)  s , t  V >  Now  d u , (  V>  while  dg (s)(t) r  =  6  j , (s ) ( x ) k k x' r'  F  j  ft  J r  In order  to compare  employ t h e c o o r d i n a t e  v  v  dg ,(s)(t)  system on  r  J (R  ) •  ( t  r  and  ) ( x  r>  d a ) ( x  r>  dg (s)(t) r  In coordinates  we need t o  - 39  -  dg ,(s)(t) r  =  7 I  — ( x  , , j , ( s , ) ( x , ) ) o ( j . ( t ,)(x ,) dw(x ,) r' k ' r' a k r' r' r'  L 3u a •'ft.,.? a L  = I  U  v  J  v  r  ^-(6(x ),J (s ,)(6(x )))ou (J (t ,)(3(x )) r  k  r  r  a  k  r  d(B*co)(x )  r  r  So dg  ( s ) ( t ) - dg  9F  (A)  du  n  ft  r  (B)  v  a  ,(s)(t)  (x . j , ( s ) (x )) u ( j ( t ) ( x ) ) - u ( r' k r ' r' J  v  a  v  k  r  r  a  (t ,)(3(x )))  j k  r  r  dto(x ) r  | f (x ,j (s )(x )) - lf(x ,J (s ,)(B(x )))  +  r  k  r  r  r  a  k  r  r  a x " ( J ( t . ) ( e ( x ) ) dco(x ) a  (C)  +  r L a  ^-(x ,j (s ,)(3(x ))) r  J  k  r  r  k  r  r  r  9F  gJ (3(x ) J (8 ,)(B(x ))) r  r  f  k  r  r  ft *r  x u (J (t i)(B(x )) a  k  r  r  dco(x ) r  •  (D)  +  V | f - ( B ( x ) , j ( s , ) ( 3 ( x ) ) ) - u ( j ( t , ) ( 3 ( x ) ) d[co-3*co] ( x ) I ft a* r  k  r  r  a  k  r  r  r  As b e f o r e we  Now  (A)  d e a l w i t h each of these terms  i s the same as  ft.  dH where  r  (j (s ))(j (t ) k  r  k  r  3*j (t ,)) k  r  separately.  r  - 40 -  S*j (t ,)(x ) k  r  I t i s e a s i l y seen t h a t f o r any of  r  j (t ,)(3(x ))  =  r  k  r  r  6 > 0  t h e r e e x i s t s a neighbourhood  W  such t h a t  < 6  I Ik J i " rCt ) - k 3*3%r ("t ,)p I I 11 J  v  J  v  11  0 for •  t , t , r r ft  derived  from  t  I Itl I  with  T  P  =1.  From the boundedness o f  He  r  dH  which f o l l o w s from lemma 2.1  arbitrarily  i t follows that  (A)  can be made  small.  (B)  T h i s term i s t h e same as  dH%;J (s )) - dH%B*J (s ,)) **j (t .) k  r  k  r  k  r  .  ft  r  From t h e u n i f o r m c o n t i n u i t y o f  dH  ,  t h e boundedness o f  6*j, ( t , ) , K. 10  J (s ), k  and  r  3*j (s' ,) k  r  and the f a c t t h a t  I| i ( k  s r  ) ~ 3*J (s ,)|| k  can  r  L  be made a r b i t r a r i l y  small i t follows that  (B)  can be made  0  arbitrarily  small.  (C) W  of  r  I t i s e a s i l y seen t h a t f o r 6 > 0  such t h a t  |B(x ) - x | r  for (C)  t h e r e e x i s t s a neighborhood  all x r  r  =  lACOoAO:)" ^) - x |  e ft and a l l r ' E W . r  i s dominated by  1  r  From c o n d i t i o n  L  [2.4] '  <  6  i t follows  that  - 41 -  ni f t a J  lP-1 u a ( t , ) 3 C x ) ) (CC3(x )-x ) 1 +:I | u a C s , ) C 3 ( x ) ) ) r  r  g  k  r  a  r  k  r  (  r  dio  J  T  It follows from Holder's inequality and the above remark that  (C) can be  made a r b i t r a r i l y small. (D) I t follows from the fact that small that  co - 3*to can be made uniformly  (D) can be made a r b i t r a r i l y small.  T h i s completes the v e r i f i c a t i o n of conditions [1.1] - [1.6], and hence Theorem I I i s proved.  - 42 -  CHAPTER  3  DIRICHLET PROBLEMS WITH VARIABLE HOLONOMIC CONSTRAINTS  3.1  F o r m u l a t i o n o f Theorem I I I  In t h i s c h a p t e r we s h a l l c o n s i d e r a p a r a m e t r i z e d v e r s i o n o f t h e D i r i c h l e t problem d e s c r i b e d by P a l a i s  [4, pp 104-105, p 109].  The s o l u t i o n  c a n d i d a t e s a r e v e c t o r v a l u e d f u n c t i o n s on a m a n i f o l d  M,  constrained  as w e l l as t o agree  t o l i e i n a g i v e n submanifold  w i t h those o f a g i v e n f u n c t i o n  a  W C IR ,  on  8M,  i n case  W  and  a  s h a l l study the s o l u t i o n when both  Whose v a l u e s a r e  3M  i s not empty.  We  a r e permitted to vary.  To be p r e c i s e , we b e g i n w i t h a compact  OO  C  manifold  M,  with  th p o s i t i v e smooth measure Lagrangian some [3.1]  y  L e Lgn^CO »  £ _ 2.  and p o s s i b l y w i t h boundary, and a  where  We suppose t h a t  F o r some  p  with  f£(£) —>  LJO^)  £ L  k  order  i s the product v e c t o r bundle  IR^ f o r  s a t i s f i e s the f o l l o w i n g conditions :  pk > n, s  o  t  h  a  h(v)  t  t  h  L  extends t o a  C.^  map :  integral  e  Lv dy  = M  defines a  [3.2]  r e a l - v a l u e d f u n c t i o n on  For any a , & ±  [dh(  V ; L  )-dh(v )] ( 2  L (^) P  2 e  V ; L  L (£) . P  and any v e ±  a  »  L  1  2  1  2  L  a  2  - v ) > c| | ( v - v ) - ( a - a ) | | 2  e l_^0 ^ »  v  k  P p  - | [ ( a ^ ) | | L  k  P p  - 43 -  f o r some c o n s t a n t  [3.3]  The map  c .  dh : L (£;) — >  L^CO*  P  takes bounded s e t s t o bounded  sets.  Next, l e t W 8W = 0 .  with  be a c l o s e d  F o r each  constraint manifold  W  r  C  s u b m a n i f o l d o f (R  i n a parameter  space  B,  w i l l be o b t a i n e d by a c t i n g on  the v a r i e d W  with a d i f f e o -  Z morphism  A ( r ) o f t h e ambient  boundary-value  function  b  d e n t l y - v a r i e d boundary  E u c l i d e a n space  on  M  IR  .  Also, f i x i n g a  w i t h v a l u e s i n W,  functions  a^  by f i r s t  we o b t a i n indepen-  composing  b  w i t h another  Z diffeomorphism  V(r) of  the r e s u l t w i t h  A(r) .  IR  which c a r r i e s  Notations l i k e  composite f u n c t i o n , and by abuse, bundle  W^  b  W  f(r)^b  i n t o the space  Diffeo(lR )  W,  then  composing  w i l l be used f o r such a  may denote a s e c t i o n o f the t r i v i a l  and ^ ( r ) ^ t h e induced s e c t i o n . We s h a l l r e q u i r e t h e maps A and V  %  onto  t o be c o n t i n u o u s from  00  of a l l C  diffeomorphisms o f  Z  IR  onto  B itself,  w i t h the t o p o l o g y o f u n i f o r m convergence o f each d e r i v a t i v e on each compact set.  Our r e s u l t  i s the f o l l o w i n g .  Theorem I I I CO  Let  M  be a compact  C  m a n i f o l d o f dimension  n,  possibly with  Z boundary, Z >_ 2, with  and w i t h a s t r i c t l y p o s i t i v e smooth measure  l e t L e Lgn^C?)  pk > n ,  and s e t  s a t i s f y conditions  u .  [3.1] - [3.3]  With  £ = IR^. ,  f o r some  p .  - 44 -  h(v) =  M  Lv du  for  v e L?(C) fc  00 Let  I  W be a closed C  submanifold of IR  without boundary,  (W compact i f 3M = 0). Let E =.W and b e L ( E ) . Let B be a l o c a l l y compact Hausdorff space and l e t P  . o  T : B —> Diffeo((R ) be continuous maps such that, for each  r e B, ¥(r)(W) = W and  A(r)(W)  a  oo  i s a closed C Set  o  A : B —> Diffeo(lR )  and  submanifold of IR W(r) = A(r)(W).  def Then E(r) = W(r)  i s a C°° subbundle  of IR^ . Let a = A ( r ) ^ ( r ) b , and D = L£(E(r)) r  . Let h  A  r  denote  r the r e s t r i c t i o n of the Let  function  h to D r  F be a deformation-invariant family of subsets of a single  path component of L^(E)^ , such that F contains at least one compact non-void element. V(x)  For each  r e B, set  = | V C D_ : V = A ( r ) ^ ( r ) ^ ( T ) ,  Then the following conclusions hold : (a) The function  m : B —> R defined by  m(r) = i n f sup h (v) Vefl(r) veV r  i s f i n i t e and continuous on B (b)  For each  r e B, the set  some T e F j - .  - 45 -  'M(r)  = • j ( r , v)  : v'e D ,  d h ( v ) = 0, and  r  r  h (v) r  = m(r)|  i s not empty.  (c)  For each compact subset  M(C)  j ( r , v)  =  i s a compact subset of  B x  Theorem I I I  C C  : r e C  B,  the s e t  and  ( r , v) e M ( r ) |  L (E) P  w i l l be„deduced from Theorem I  of §1.3  by  c o n s t r u c t i n g a s u i t a b l e s t a n d a r d problem.  3.2  C o n s t r u c t i o n of the Standard  [la]  Let  M,  [lg]  p, k  u  as g i v e n i n §3.1,  as determined  the i n c l u s i o n We  give  Problem  W —>  LJ^W^)  i n §3.1. IR  and  E = W^  Since  W  .  i s a submanifold o f  induces an i n c l u s i o n  ^ ( ^ )  the induced F i n s l e r s t r u c t u r e  || ||,  —>  (R ,  L£(IR^), and  the  corresponding F i n s l e r metric.  [ly]  B  as g i v e n i n §3.1  [16]  b  as g i v e n i n §3.1  with  Viv)*  to  r e s t r i c t i o n of  [le]  L = LoA(r)* . . r  .  b^ = Y ( r ) * b L^CE^  .  and  <j>  e q u a l t o the  - 46 -  We deduce Theorem I I I from the application of Theorem I to the standard parametrized problem determined by the data  [la] - [le] . The  passage from Theorem I I I to Theorem I i s more direct than the passage from Theorem I I to Theorem I because we deal here with a fixed domain which enables us to construct a parametrized standard problem more closely related to the o r i g i n a l problem.  In fact since  fl(r) = iV C ti : V = A(r)*¥(r)*(T), some T e f f , and F(r)  = -JS C L£(E)  : S = ¥(r)*(T),  fe  some T e ¥} ,  r and g (s)  =  r  h (A(r) (s)) r  A  i t follows that m(r) =  inf sup h (v) VeP(r) veV r  =  inf sup g^(s) = m (r) SeF(r) seS F  and (a) of Theorem I I I follows from (a) of Theorem I Also we have M(r) =  = | ( r , v) : v e D , dh (v) = 0 and h (v) = m(r) r  j(r, A(r) s) A  r  r  : s e ijJCE^ , dg (s) = 0 and g (s) = m ( r ) | r  which i s non-empty by (b) of Theorem I. F i n a l l y ,  r  F  - 47 -  M(C)  =  =  | ( r , v) : r e C  { ( r , A(r)„s)  :  and  r e C . s e  ( r , v) e.M(r) j  l£(E)  , dg (s)  b  r  = 0  r  V  and  which i s comapct when  C  g ( s ) = rtip(r) r  i s compact by (c) o f Theorem I .  B e f o r e p r o c e e d i n g w i t h t h e v e r i f i c a t i o n s we prove a few lemmas r l—> A ( r )  about the map of t h e map  r  I—>  A  .  A s t a n d a r d assumption  w i l l be t h e c o n t i n u i t y  A(r) .  Lemma 3.1 Fix bourhood  V  S C of  r  L . ( 0 bounded and P  e > 0 .  such t h a t  ||A(r') s - A(r)*s||  <  A  L  and  e  k  f o r each  r' e V  Proof  The p r o o f f o l l o w s e a s i l y from P a l a i s  :  Then t h e r e e x i s t s a neigh-  s e S .  [4, Lemma 9.9, p.31].  Lemma 3.2 If  S C  L ( £ ) i s bounded then P  A(r).(S)  i s bounded.  K.  Proof  : . T h i s a l s o f o l l o w s from t h e above c i t e d lemma 9.9.  - 48 -  Lemma 3.3 Fix neighbourhood  Proof  :  r e B  and  V  r  of  S C  L^CO  such t h a t  bounded. U  Then t h e r e e x i s t s a  A(r').(S)  i s bounded i n L?(£)  T h i s f o l l o w s from lemmas 3.1 and 3.2.  Lemma 3.4 Fix  r e B  inf seS Proof  :  and  S C  1^(5) bounded.  ||d(A(r)^)(s)||  By lemma 3.2,  A(r)(S)  ||d(A(r)^)(s)[|  >  0 .  i s bounded.  Now  =  1  ||d(A(r); )(A(r)^)(s)|I"  T h e r e f o r e we need o n l y prove t h a t f o r S C that  sup | | d ( A ( r ) ) (s) | | < °° . A  Then  ^ ( 5 ) bounded,  Now f o r s E S  and  t E  1  and  .  r e B ,  SES  d(A(r)J(s)(t)(x)  =  by P a l a i s [4, Theorem 11.3, p.41]. follows that  f o r [|t||  A g a i n by [4, Lemma 9.9, p.31], i t  = 1,  | | (6A(r)°s) t| |  <  independent o f  s E S .  A  f o r some  A e (R,  [6A(r)°s(x)](t(x)) ,  sup j | d ( A ( r ) * ) ( s ) | | SES  <  A  < co  This implies that  - 49 -  3.3  Verification  It condition  of conditions  [1.1] - [1.6]  f o l l o w s by the same t e c h n i q u e s as employed i n Chapter 2  [3.2]  implies  s a t i s f i e s condition  h : L ^ C O —>  that  (C),  IR  and t h a t f o r S C  L k  that  i s bounded below and  (5)  bounded and  R -e IR ,  the s e t  | s e L (5) P  i s bounded i n  Verification  a  : a e S and h ( s ) <_R  1^(5) •  o f [1.1] :  Since  g = hoA(r). , r *  g r  i s bounded below f o r each  the above remarks combined w i t h lemma 3.2 and  R c (R ,  i t follows  r,  and by  that f o r b e L ( E ) , P  the set  |  s e L (E) P  i s bounded i n  a  n  d  B  :  g_.Cs) < R  hence bounded i n t h e F i n s l e r  metric  on LJ^E)  (Uhlenbeck [ 8 ] ) .  need o n l y  Therefore  i n order  show t h a t  i f {s^}  bounded i n  L£(£),  ),  t o show t h a t  g__  s a t i s f i e s condition  i s a bounded sequence i n  such t h a t  dg ( s . ) — >  0,  then  L (E) P  {s.}  B  (C) we  (and hence i s convergent.  Now dg (s.) r  If that  dg(s ) —> {ACr^s^}  0  =  i t follows  dh(A(r) ,(s ))od(A(r) )(s.) s  i  from lemma 3.4  i s convergent and t h e r e f o r e  A  that so i s  .  d h ( A ( r ) (s.)) +  {s^} .  —>  0,  - 50 -  Verification  o f [1.2] :  Fix in  K O  C C  B  compact and  { ( r , s) : r e C  assume t h a t the s e t  { ^}  g (s)£ R } . r  converges t o  r  {b } r. x  and  R E IR .  1  r e B .  Let  {(r^, s^}  Since  B  be a sequence  i s compact we can  From lemma 3.3  i t follows  that  i s bounded i n Lf(£) . k  Now  g  (s) <_ R  implies that  h ( A ( r ) ^ ( s ) ) <_ R , i  where  i A ( r . ) , ( s ) E L?(E,)^ l * k b {A(r ) (s)} i  •  From c o n d i t i o n  i s bounded i n 1 ^ ( 0  A  [3.2] i t f o l l o w s t h a t t h e s e t  r.  In order  •  t o proceed w i t h t h e v e r i f i c a t i o n o f [1.2] we need the  following extension  of the c o n s t r u c t i o n i n P a l a i s  [4, pp 112-114]. co  Fix For  each  r E B .  w E W  Then  r l e t q (w)  A(r)(W)  i s a closed  C  £  submanifold o f  denote t h e o r t h o g o n a l p r o j e c t i o n o f  IR .  £ £ IR = HR w  CO  onto  TW  .  Then  q  isa  C  W  a  z  L(IR , (R ) ,  and s i n c e  co  to a  C  map o f  W  %  IR  LJJa)  L(5, O  As —>  map o f  i s a closed  i n t o the v e c t o r  r r Q (x, v) = (x, q ( v ) ) , into  i  Q  r  W C  °°  space  i n t o the v e c t o r  space  submanifold of  IR ,  z  a  L((R , tR ) .  a  I f we  00  isa  C  f i b r e bundle morphism o f  •  i n [4, theorem 19.14],  L(L (5), L (C)) P  P  we d e f i n e a map  denoted by  Pg(t)(x)  s I  > p£. and g i v e n by  = Q (s(x))(t(x)) . r  i t extends define £ £ = IR^  - 51  -  In the above c o n s t r u c t i o n the map appealing  to a g e n e r a l  extension  theorem.  P We  was  g  constructed  by  w i s h to show t h a t these maps r'  can be  constructed  i s " c l o s e " to  for  P__  r'  if  r'  i n some neighbourhood o f i s c l o s e to  r .  r  such t h a t  More p r e c i s e l y we  P  have  g  the  following. Lemma  3.5 Fix  r e B  and  method of d e f i n i n g the  S C  ^ £ ( 0  extensions  bounded.  of  There e x i s t s  the  6 > 0  and  a  such t h a t f o r each r r - P || < e s s 1  e > 0 all  t h e r e e x i s t s a neighbourhood r ' e V,  Proof  :  and  Let  N  all  be  s  with  V  of  r  with  ||P  distance(s, L (E(r)))< k  6 .  P  an a r b i t r a r i l y l a r g e compact subset of  W  to  For  as d e s c r i b e d  above.  distance along it  constant.  as above and constant  r e B  For each p o i n t  for  extend i t  diagram s h o u l d  r*  " c l o s e " to  (shrinking  q  z e A(r)(N)  the normal d i r e c t i o n s to Now  along  d e f i n e the p r o j e c t i o n  N  r  define  slightly  : A(r) (N) — > extend  A(r)(N)  be  a  r determined.  for  q  L((R , (R ) ,  a finite  at  z  q  : A ( r ' ) ( N ) -> LQR ,  r  in  a  (R  by making 1  IR^)  i f n e c e s s a r y ) by making i t  the normal d i r e c t i o n s determined by  A(r)(N).  The  following  c l a r i f y t h i s argument. *J A(r)  CN)  --Extend a l o n g the normals A(r')(N)  I t i s then e a s i l y v e r i f i e d ' t h a t the,maps  P  have the r e q u i r e d  o f lemma 3 . 5 .  T h i s completes the proof  We resume t h e v e r i f i c a t i o n o f [ 1 . 2 ] . implies  First,  dg  (s^) = 0 i •  that  dh(A(r.)Js.))oP  r.  =  l  i * Let  property.  t. = A(r).(s.) .  0  I  Then  d M t . M t . - t.)  =  Now  {t } ±  Therefore,  dh(t.) ( P ^ ) ( . r  t  i s bounded i n 1 ^ ( 5 ) , there  I|u. l i _ - t_.iI Ii L  p  " t.)  and  e x i s t s a sequence  —>  0 .  + d h ( t . ) (I - P ^ C t . - t.)  }  distance^,  {u.^}  L (A(r)(E )) P  r  i n L (A(r) (E.J )^ P  f e  0  —>  such t h a t  Consider the d i f f e r e n c e  k ||(I-P^)(t.-  t.) - ( I - P  u  X  M u .  -u.)||  1  Li.  k  <  Md-P^^-u.) - t . -u.))|| (  +  r. x  IK t; - <r) ^ i - V " P p  T  Now [ 4 ,  Theorem 1 9 . 1 4 ,  He  +  l|(p  t. - C ^ i - V - N p 1  1  He  p.112] combined w i t h lemma 3 . 5 above i m p l i e s t h a t t h e  - 53 i. •  above terms converge t o z e r o .  By  [4, Theorem 19.15, p.113].  V  1  f o r a subsequence of  {u^} which we assume i s  (I - P ^ M t . - t . ) || —> t. i J ' P k 1  1  {u^}  ,  and i t f o l l o w s  that  0 .  L  I t f o l l o w s t h a t the d i f f e r e n c e  d h ( t . ) ( t . - t.) - dh(t.) x x 2 i  tends to z e r o .  Now  t . - t . = t . . + (b. - b.)  for  t . . e Lf  Therefore  d h ( t . ) (P  i  ) ( t . - t.) ,  + d h ( t . ) , ( P * ) ( 1b . - b . ) ,  dh(t.)  t  dh(t.)  Since  | \b  ±  - b.||  [3.2]  {s^}  we  P  tJ 0  )  (  b  i-  [1.6]  3  J  Vj  we get f i n a l l y  that  |dh(t )(t i  i  - t.)| —>  0  k  can c o n c l u d e t h a t  {t.}  i s a Cauchy sequence.  i s a Cauchy sequence, and C o n d i t i o n  We and  (  —> L  By  l  v  [1.2]  Therefore  is verified.  complete the v e r i f i c a t i o n s i n the o r d e r  [1.4],  [1.3],  [1.5],  -  Verification  -  5 4  o f [1.4] :  Let  T C  and by Uhlenbeck neighbourhoods  L^CE)  Then  i s c o n t a i n e d i n a f i n i t e number of v e c t o r bundle  ^\^±)  •  Suppose t h a t  where  ^  of  r  —>  E  L (n.)  P  X  t E  Fix  there e x i s t s a  <  e  k V (  X  K.  Since  (Uhlenbeck [8])  || ||  on  the r e s u l t  E  i s an  follows.  of [1.3] :  -r E B  r  and .  R e IR .  Then  g ( r ' ) <_R t  implies  that.  By t h e remarks b e f o r e the v e r i f i c a t i o n o f [1.1] t h e r e e x i s t s a neighbourhood  V.,  of  r  such t h a t  set  ( t ) : hOKr'.^o^. ( t ) ) < R  |  i s bounded i n we  E .  X  KL  t e T n l£(5.) •  t o g e t h e r w i t h lemma 3.2 the  E  such t h a t  structure  h(A(r' ) °<f> , ( t ) ) <_R A  in  ^(r) : E —>  and by lemma 3.1  P  .  L  admissable F i n s l e r  Then  w i t h the map  ||v(r)^t - n r ' ) * t | |  and  •  i s the v e c t o r bundle neighbourhood  t h e map  K.  V  n.  L (S.) —>  T h i s induces a map  Verification  i s i n t r i n s i c a l l y bounded  T  o b t a i n e d by composing  for a l l r' E V  T  [8],  £ Lf(n.)  neighbourhood  be bounded.  l^CO  •  By the argument used i n the v e r i f i c a t i o n o f [ 1 . 4 ] ,  can f i n d a neighbourhood  V C  V  1  such t h a t  [1.3]  holds.  - 55 -  V e r i f i c a t i o n o f [1.5] :  We have  | f ( r , s) - f ( r , s ) | T  =  Now combining  | h ( A ( r ) ( s ) ) - h(A(r«)*(s))| A  [3.3] w i t h  lemma 3.1  and t h e mean v a l u e theorem the r e s u l t  follows.  Verification  o f [1.6] :  T h i s f o l l o w s from [3.3]  combined  T h i s completes t h e v e r i f i c a t i o n s , proved.  w i t h the p r o o f o f lemma 3.4.  and hence Theorem I I I i s  - 56 -  •'!  CHAPTER  4  EXAMPLES  4.1  P e r t u r b a t i o n o f Minimal  Surfaces  In t h i s example we have a f i x e d domain and v a r y i n g boundary conditions.  oo  2 Let 2 IR ,  and  3 (E )  The f u n c t i o n s a r e v e c t o r  B  M C (R  be a compact  C  valued.  two d i m e n s i o n a l  submanifold o f  be a l o c a l l y compact t o p o l o g i c a l space.  F e FB[J (IR^), 1  Let  R l  be g i v e n by  i»3  2 Let the map <j> : I ^ ( K ^ ) r  B —>  0  —>  L  L^(R^)  - [1.6]. 2  in  L  b  e  r I—> b ^  g i v e n by  (J» (s) = s + b r  r problem determined by  Now i f the s e t  F  be continuous,  L^,  o f Chapter 1  .  r  <J>, r  and l e t  Then i t i s e a s i l y b^  satisfies  i s the s e t o f s i n g l e t o n s  3  ]_("*jpo *  values  g i v e n by  l ^ \  seen t h a t t h e standard [1.1]  3  ^  t  f°l-  of the f u n c t i o n  i O W S  t n  at  f o r each  r  we a r e c o n s i d e r i n g minimum  g r &  If 3M  b  r  i s a smooth s e c t i o n , so t h a t i t s p r i n c i p a l p a r t  t o a smooth curve  3 i n IR  then a s e c t i o n belonging  3 d e f i n e s a ( g e n e r a l i z e d ) s u r f a c e i n IR  whose boundary i s  to  T  .  carries  2 3 L^((R^)b  It i s  r  - 57 -  w e l l known, i n t h i s case, t h a t a s e c t i o n w h i c h minimizes integral  g__  corresponds  Moreover, the v a l u e o f case; .  each  t o a s u r f a c e , o f minimum a r e a spanning  g__  Hence Theorem I  our D i r i c h l e t .  agrees w i t h the s u r f a c e a r e a j u s t i n t h i s  applies to give conclusions :  (a)  There i s a t l e a s t one ( g e n e r a l i z e d ) minimal  surface f o r  (b)  the minimum s u r f a c e area v a r i e s c o n t i n u o u s l y w i t h  (c)  i f f o r each  r ,  r  i n a neighbourhood  2 section  s__  i s unique,  then the map  V —>  V C  B  r , and  the minimizing  3  L^(IR^)  g i v e n by  r I—> s^ i s  continuous.  4.2  P e r t u r b a t i o n o f the Operator  Let  M  be a smooth submanifold  of  (R  n  w i t h boundary  8M  and 2  Lebesgue measure . u . i n i t s second  Let  Let  a : M x (R x B — >  IR be  C  and t h i r d arguments and be denoted by (x , 1  Let  B = [0, °°).  e F B t J ^ ) . , IR^ 1  F (x\ r  u, r )  I  > a ( x , u, r ) X  be g i v e n by  u, ii ) l  =  I  Y  2 ) + a(x, u, r )  (u  i  i 2  Let  L  e Lgn^OR^)  r  be r e p r e s e n t e d by  F__ . 2  r I—> b  r  be continuous.  Let  <|> r  : ( L  1  |R M  Let  )o —  >  L  B —> 2  l^ M^ r IR  b  f^O^ b  e  S  i v e n  g i v e n by b  v  - 58 -  s + b^  .  Assume t h a t  a  satisfies  (i)  | ( x , u, r ) I  <  C(r)  (ii)  |a(x, u, r ) - a ( x , u, r ' ) |  a  3a 3u  (iii)  (iv)  0  (x,  u, r )  <  1 +  u  <_  C(r) ( l +  3 a < • • - f (x, u, r ) 3u 2  the f o l l o w i n g c o n d i t i o n s  <  ( ^ ( r - r')  1 +  u  |u|'  C(r)  Z  where  C  and  a r e continuous  I t i s then e a s i l y <f> r  satisfies  verified  conditions  r . l  we  operator associated with  u  x.x. i i  linear  " I \  Theorem I  for  C^(0)  = 0 by ' L  y  =  : [0, °°) — >  0 .  i  a p p l i e s a g a i n as i n  L  is  N  E u l e r Lagrange e q u a t i o n  + Y(r)u  x  i In any case  1 3a , + r r r , x., u 2 3u l  a ( r , x^^, u) = y ( r ) u  have the p a r a m e t r i z e d  with  [1.1] - [ 1 . 6 ] .  ~ L  if  r,  t h a t the standard problem determined  The E u l e r - L a g r a n g e  Of c o u r s e  f u n c t i o n s of  4.1.  IR  continuous  ,  b , r  - 59 -  4.3  Domain P e r t u r b a t i o n s  We let  s h a l l employ the n o t a t i o n o f Chapter  F e FB[ J ((R  ) , IR  X  F(x  a  F i x ft° C  i  and  n  by g i v e n by  , u, u  : ft° x (R — >  (R  )  is  C  <  v _ (u  =  |a(x _  u)|  (ii)  \a(x ,  u) - a ( x [ , u ) |  ±  ±  (iii)  ( ,  C  u)|  X i  x.  )  2  + a(x  and s a t i s f i e s  2  (i)  1 +  <  , u)  the c o n d i t i o n s  |u|  <  C[ 1 +  C (x 1  ±  <_  C  -  1 +  x^)  3u~  (  x  i ' a  0  (x|,  |u  |u| f  (iv)  (v)  IR  ft°  Q°  where  ]  2 .  u)  -  1 +  |u| >  2  <  (x., u)  <  C  3u  where C^O)  C  i s a c o n s t a n t and  a continuous  function  on  IR  with  = 0 .  Let F  C^  satisfies  L e Lgn_ (IR 1  )  [2.1] - [2.6]  the s e t o f l i n e a r restriction  OP  of  r  diffeomorphism  of  be r e p r e s e n t e d by of  Chapter  isomorphisms o f  (R  n  2-  over  F .  with IR .  Then i t i s c l e a r t h a t  p = 2 .  Then f o r each  to the c l o s u r e of a smooth subdomain Q  into  t h e r e e x i s t s a neighbourhood  IR  .  n  V  If  ft  is strictly  o f the i d e n t i t y  Let  in  Q, C  f2°  L(IR ) n  r e L((R ) n  is a  c o n t a i n e d i n ft° , L((R )  be  such t h a t  the  - 60  r(ft) C that  fi° X  for  r e V .  i s continuous  continuous map  of  Let  from  B —>  B = V, B  into  -  M = fi and  A(r) = r | - .  D i f f eo(fi, IR ) . n  4.4  2  and t h a t Theorem I I  P e r t u r b a t i o n of  n i f o l d of  (R  o  Let  be a  r  s a t i s f i e s the c o n d i t i o n s  Geodesies  t r e a t e d i n Chapter  £ IR  be a c l o s e d  1 £ F e F B [ J (IR^) , (R^  be g i v e n by  M = [0, 1 ] . .  t>  a p p l i e s to i t .  T h i s i s an example o f the type o f problem  Let  rl—>  L^CR^) .  I t i s e a s i l y shown t h a t the above problem of Chapter  Let  It i s clear  Let  F(x, u , j  W C  uh  =  I (uh  q  3.  d i m e n s i o n a l subma-  .  2  3 It i s easily v e r i f i e d  that  c r i t i c a l p o i n t s of the map W  F  s a t i s f i e s the c o n d i t i o n s of Chapter  constructed with  i n the Riemannian s t r u c t u r e induced on  IR  q  F W  The  c o r r e s p o n d to geodesies by the i n c l u s i o n o f  on  W  into  we  induce  .  V i a the map  A : B —>  DiffeoflR^)  d e f i n e d i n Chapter  a continuous change i n the Riemannian s t r u c t u r e of $  4..  : B -—>  DiffeoOR*')  W .  The  4  map  v a r i e s the endpoints o f the g e o d e s i e s . 2  F i x a path component of  ^(W^b  .  By P a l a i s  [4, Thm.  t h i s i s the same as p i c k i n g a homotopy c l a s s o f continuous maps  13.14, p.54], M -—>  W,  - 61 -  which we  denote by  neighbourhood i s unique, r  ,  where  V  H .  C  say  B, .  Fix  r e B  and assume t h a t f o r each  r  i n some  the m i n i m i z i n g geodesic assured by Theorem 111(b) Then we have shown t h a t  v a r i a t i o n s of  r  Riemannian s t r u c t u r e on  W  c o r r e s p o n d i n g to  and  b (0) r  in  V  correspond  v  r  varies continuously with  to v a r i a t i o n s of the  and o f the end p o i n t s of the  geodesies,  b^(l) .  I t i s p o s s i b l e to change the above example t o the case where M = S"^ .  In t h i s case we must assume t h a t  i n c r e a s e the dimension Betrami o p e r a t o r on  of  W .  M  and  let  F  W  i s compact.  We  can  also  r e p r e s e n t "powers" of the L a p l a c e -  For d e t a i l s see P a l a i s  [4, p.127].  - 62 -  REFERENCES '  1.  F.E. Browder, equations,"  " E x i s t e n c e theorems f o r n o n l i n e a r p a r t i a l d i f f e r e n t i a l Proc  : Symp Pure Math 16, A.M.S., P r o v i d e n c e , R.I., 1970,  1-60. 2.  _ _ ,  " F u n c t i o n a l A n a l y s i s and R e l a t e d F i e l d s " ,  F.E. Browder  ed., S p r i n g e r , New York, H e i d e l b e r g , B e r l i n , 1970, 1-58. 3.  ,- " I n f i n i t e d i m e n s i o n a l m a n i f o l d s and n o n - l i n e a r e l l i p t i c e i g e n v a l u e problems",  4.  R.S. P a l a i s ,  Annals  "Foundations  o f Math. 82 (1965),  459-477.  of Global Nonlinear A n a l y s i s " ,  Benjamin,  New York, 1968. 5.  , Topology  6.  " L u s t e r n i k - S c h n i r e l m a n t h e o r y on Banach m a n i f o l d s " , .  5 (1966) 115-132. ,  " C r i t i c a l p o i n t t h e o r y and t h e minimax p r i n c i p l e " , Proc :  Symp Pure Math 15, A.M.S. P r o v i d e n c e , R.I., 7.  R.C. R i d d e l l ,  " N o n l i n e a r e i g e n v a l u e problems and s p h e r i c a l  o f Banach s p a c e s " , 8.  K. Uhlenbeck, maps",  1970, 185-212. fibrations  J . F u n c t i o n a l A n a l y s i s , t o appear.  "Bounded s e t s and F i n s l e r  J . D i f f Geom. 7 (1972), 588-595.  s t r u c t u r e s f o r manifolds of  

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