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Perturbation of nonlinear Dirichlet problems 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 of B r i t i s h Columbia, 1968 M.Sc, U n i v e r s i t y of 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 the Department of MATHEMATICS We accept t h i s thesis as conforming to the required standard ^"UNIVERSITY OF BRITISH COLUMBIA In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British 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 British Columbia Vancouver 8. Canada Date /J• O '7. Supervisor : Dr. R.C. R i d d e l l - i i - ABSTRACT The solutions of weakly-formulated non-linear D i r i c h l e t problems are studied when the data of the problem are perturbed i n various ways. The data which undergo perturbations include the Lagrangian, the boundary condition, the basic domain, and the constraints, i f present. The main conclusion states that the s o l u t i o n of the D i r i c h l e t problem which minimizes the D i r i c h l e t i n t e g r a l v a r i e s continuously with the data so long as i t i s unique. Detailed hypotheses are formulated to insure the v a l i d i t y of t h i s conclusion for several large classes of problem. The hypotheses are not much stronger than the standard s u f f i c i e n t conditions for existence, i n the generalized Lusternik-Schnirelman theory of these problems. - i i i - ACKNOWLEDGEMENTS I would l i k e to thank Ron R i d d e l l for suggesting the problem treated here and f o r h i s Herculean e f f o r t s to make me unscramble my confused presentation of i t s s o l u t i o n . I would also l i k e to thank the U n i v e r s i t y of B r i t i s h Columbia and the National Research Council of Canada for t h e i r f i n a n c i a l support. - iv - TABLE OF CONTENTS INTRODUCTION CHAPTER 1 PARAMETRIZED NONLINEAR DIRICHLET PROBLEMS IN STANDARD FORM Page 1.1 Notation 1.2 Formulation of the Problem 1.3 Formulation of Theorem I 1.4 Proof of Theorem I 4 6 8 12 CHAPTER 2 NONLINEAR DIRICHLET PROBLEMS WITH VARIABLE DOMAINS 2.1 Formulation of Theorem II 16 2.2 Preliminary Consequences of the Assumptions 20 2.3 Construction of the Associated Standard Problem 29 2.4 V e r i f i c a t i o n of Conditions [1.1] - [1.6] 32 CHAPTER 3 NONLINEAR DIRICHLET PROBLEMS WITH VARIABLE H0L0N0MIC CONSTRAINTS 3.1 Formulation of Theorem I I I 3.2 Construction of the Associated Standard Problem 3.3 V e r i f i c a t i o n of Conditions [1.1] - [1.6] 42 45 49 - v - TABLE OF CONTENTS (Contd.) Page CHAPTER 4 EXAMPLES 4.1 Perturbation of Minimal Surfaces 56 4.2 Perturbation of the Operator 57 4.3 Domain Perturbations 59 4.4 Perturbation of Geodesies 60 BIBLIOGRAPHY 62 - 1 - INTRODUCTION The purpose of t h i s thesis i s to study the behaviour of the solutions to nonlinear d i f f e r e n t i a l boundary value problems of D i r i c h l e t type, when the data d e f i n i n g the problem are subjected to various perturbations. The basic r e s u l t we s h a l l obtain states that, under s u i t a b l e r e s t r i c t i o n s , the s o l u t i o n of such a problem changes continuously with the data as long as the s o l u t i o n i s unique. The conditions under which t h i s conclusion is. v a l i d are e s s e n t i a l l y that the usual s u f f i c i e n t conditions for existence i n the v a r i a t i o n a l theory of D i r i c h l e t problems should hold uniformly i n some sense as the given problem i s perturbed. The d i f f e r e n t i a l equations which appear i n the boundary value problems considered here are the Euler-Lagrange equations of m u l t i p l e - i n t e g r a l ' D i r i c h l e t ' functionals defined on s u i t a b l e Sobolev spaces. We consider such problems i n t h e i r weak formulation, i n which a s o l u t i o n i s taken to 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 given boundary condition i n an appropriate generalized sense, and which i s a c r i t i c a l point of the D i r i c h l e t i n t e g r a l r e s t r i c t e d to such d i s t r i b u t i o n s . The n o n l i n e a r i t y of the Euler-Lagrange equation a r i s e s from the fac t that the integrand de f i n i n g the D i r i c h l e t f u n c t i o n a l need not be quadratic i n i t s arguments, but need only have a c e r t a i n convexity i n i t s dependence on the functions on which the fun c t i o n a l i s defined. For such problems, there are w e l l known ' r e g u l a r i t y ' theorems as s e r t i n g when a weak s o l u t i o n i s i n fac t a smooth function and hence i s , by a standard i n t e g r a t i o n by parts argument, - 2 - a c l a s s i c a l s o l u t i o n of the Euler-Lagrange equation. We s h a l l not be concerned with t h i s question, and we work e n t i r e l y with weak so l u t i o n s . The data defining such a problem appear to be of four kinds : ( i ) The 'Lagrangian', or integrand, of the D i r i c h l e t f u n c t i o n a l ; ( i i ) the ' D i r i c h l e t data', or boundary conditions; ( i i i ) the 'domain', i . e . the set over which the 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 equation are allowed to run; (iv) the c o n s t r a i n t s , i . e . the set i n which the dependent va r i a b l e s are required to l i e . In p r i n c i p l e we could formulate a global problem on sections of a subbundle of a smooth f i b r e bundle where the target subfibre represents (iv) and the base manifold represents ( i i i ) ; and we ought to vary ( i i i ) by p u l l i n g back along various embeddings in t o the base, vary (iv) by allowing the subbundle to vary, and vary ( i ) and ( i i ) at w i l l , a l l simultaneously. This amount of g e n e r a l i t y presents t e c h n i c a l obstacles which obscure the main phenomenon, and, i n addition, t r e a t i n g p a r t i c u l a r cases of the above allows us to dispense with some of the assumptions i n each case which are needed i n the general case. Thus we s h a l l take the following l e s s general approach. In Chapter 1, we formulate and prove a theorem i n the s e t t i n g of a f i x e d f i b r e bundle over a f i x e d base manifold, where data ( i ) and ( i i ) are allowed to vary. In Chapter 2 we suppress (iv) by considering sections - 3 - qf a t r i v i a l vector bundle, and we hold ( i ) f i x e d , but we allow ( i i ) and ( i i i ) to vary. In Chapter 3 we f i x ( i ) and ( i i i ) and allow ( i i ) and (iv) to vary. A simple example of the s i t u a t i o n i n Chapter 1 i s the problem of minimal surfaces i n ordinary Euclidean space, under v a r i a t i o n of the boundary curve. The s i t u a t i o n of Chapter 2 i s i l l u s t r a t e d by the problem of domain-perturbations for nonlinear e l l i p t i c boundary problems on domains of R n . The s i t u a t i o n of Chapter 3 i s i l l u s t r a t e d i n the study of geodesies, or more general harmonic maps fin imbedded submanifolds of IR We conclude our discussion by s p e l l i n g out these examples i n a l i t t l e more d e t a i l i n Chapter 4 . I t should be noted that the s i g n i f i c a n c e of the uniqueness assumption 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 cases, by well-known phenomena of jumping^.of the minimizing s o l u t i o n when unique- ness breaks down. - 4 - CHAPTER 1 PARAMETRIZED NONLINEAR DIRICHLET PROBLEMS IN STANDARD FORM 1.1 Notation Generally we s h a l l follow the notation of Pa l a i s [4]. For the reader's convenience we supply the following b r i e f l i s t : M compact C manifold of dimension n, possibly with nonempty boundary 9M ; u s t r i c t l y p o s i t i v e smooth measure on M ; 00 E ( t o t a l space of) C f i b r e bundle over M ; CO £ ( t o t a l space of) C vector bundle over M ; (Pv̂  product bundle M x IR ; k integer _ 0 ; k J (E) bundle of k-jets of sections of E ; C°°(E) set of a l l C°° -sections of E ; S(E) set of a l l sections of E ; p r e a l number _ 1 ; P 00 L£(£) Banach space completion of ?C.(^) i n the Sobolev p'th power norm on der i v a t i v e s of order <_ k ; - 5 - I ? ( E ) (for pk > n) Banach manifold o f . a l l sections of E, each of which belongs to L^(£) for some open vector bundle neighbour- hood £ i n E ; b,s elements of L £ ( E ) ; P 0 0 P L, (E), closed C submanifold of L f ( E ) co n s i s t i n g of the closure i n k b K L^(E) of the set of a l l sections s e L £ ( E ) which agree with b i n some neighbourhood (depending on the p a r t i c u l a r s) of 8M; [| || F i n s l e r structure on the tangent bundle T ( L £ ( E ) ) ; also,by abuse,induced structure on T * ( L ^ ( E ) ) ; 6 F i n s l e r metric on L £ ( E ) induced by || || ; k-j e t extension map L £ ( E ) — y L Q ( J ^ ( E ) ) > FB(E,E') set of a l l fibr e - p r e s e r v i n g maps of E to E' ; CO F^ map of C (E) to S(E ) induced by composition with F e FB(E,E') ; til Lgn^(E) set of a l l k order Lagrangians on E, i . e . maps CO L : C (E) — > S ( i O of the form L = F*°J k f o r s o m e F e FB(J k(E) ,0^); F represents L . It should be noted that the norm i n ^ ( 5 ) depends on the volume element dy on M , but, by the.meaning of ' s t r i c t l y p o s i t i v e smooth measure', a l l choices of u are equivalent to Lebesgue measure i n a l l charts of M, so a l l induce equivalent norms i n L^(C), and equivalent F i n s l e r structures on L ^ ( E ) . - 6 - The F i n s l e r metric i s defined, t e c h n i c a l l y , only on the path components of l£(E). We set 6(s,s') = 2 whenever , s, s' belong to d i s t i n c t path components. The reader should also note that no.a p r i o r i smoothness assump- tions are made on the maps F, hence on the Lagrangians L . Such assumptions w i l l be made below as they are needed. 1.2 Formulation of the Problem We begin by assuming several pieces of data to be given and held f i x e d throughout the chapter : [la] A choice of M, y, and E ; [13] Choices of p > 1 and k _ 1 (pk > n i f E i s not a vector bundle); and a choice of || || 6 on L. P(E); on T(L£(E)), with induced [1Y] A l o c a l l y compact Hausdorff space B ; [16] An element b e l£(E)> a map r -> b of B — > L, (E) , and a r k homeomorphism - 7 - 6-bouncled sets to 6-bounded sets; [le] A map r i — > L of B — > Lgti, (E), such that each L extends to a C 1 map l£(E) — > LjflE^) . We think of B i n [ly] as a space of parameters r . Then [16] s p e c i f i e s an r-dependent family of D i r i c h l e t - t y p e boundary values on sections of E, and a C° global t r i v i a l i z a t i o n of the map ir : IE — > B, (r,s) H—> r, with the extra property that $ ^ i s smooth on each f i b r e {r} x (E^ . We s h a l l sometimes deviate from s t r i c t consistency, and i d e n t i f y the subset IE of L?(E) with the f i b r e {r} x IE , which i s a .-. r k r subset of E C B x l£(E). Item [le] s p e c i f i e s an r-dependent family of Lagrangians L^, such that the i n t e g r a l f ( r , s ) = L s dy (r e B, s e LJJ(E)) M r . defines a function f : B x L £ ( E ) — > IR whose p a r t i a l - f u n c t i o n = f ( r , . ) i s C 1 on L£(E) for each r . The r e s t r i c t i o n g = f | E then has a C 1 p a r t i a l - f u n c t i o n g = f r | E for each r ; and the c r i t i c a l locus of g r , r namely K r = { S e E r : d ^ s ) = ° i s a w e l l defined closed subset of -. By the standard parametrized D i r i c h l e t problem with data [la] - [ l e ] , we mean the problem, to describe - 8 - the ( d i s j o i n t ) union K = | (r,s) : r e B, s £ K r j of the c r i t i c a l l o c i of a l l the g r» as a subset of IE a B x L^(E) . Of course the problem as posed i s too general, and i n the next section we s h a l l put conditions on the data, i n p a r t i c u l a r on t h e i r behaviour with respect to the parameter r i n B, which w i l l enable us to prove conclusions of a s i m i l a r kind about the sets K . However, an f r important r e s t r i c t i o n has already been b u i l t i n , by the p r o v i s i o n i n [16] that the d i s j o i n t union IE of the spaces E of the r-dependent v a r i a - t i o n a l problems should come to us equipped with the structure of a f i b r e space over B, t r i v i a l i z e d by $ ^ . We are thus able to form a ' p a r t i a l - f u n c t i o n ' g^ on B, for each t i n the model f i b r e IF , namely by g t ( r ) = (go$)(r,t) = g r(()> r(t)) . Certain key t e c h n i c a l considerations i n what follows w i l l be organized around these functions g^ . 1.3 Formulation of Theorem I Given a standard parametrized D i r i c h l e t problem as defined i n §1.2, we s h a l l study the subset of each c r i t i c a l locus K r on which the function g takes i t s minimum value, r To be precise, we begin with a family F of subsets of IF = L?(E), which i s deformation i n v a r i a n t . This means invar i a n t under K b ambient homotopy of IF , i . e . , f o r any continuous map H : [0,1] x (F — > IF such that H(0 , O i s the i d e n t i t y on IF, and for any T e F, the set H^T) = | H ( l , t ) : t e T i s also a member of the family F (cf. Browder [ 2 , D e f ( l . l ) ] ). Given F, we.define a corresponding family F(r) of subsets of each E r » by F(r) = j . S C E r : S = <|>r(T) for some T e F j Then the F-minimax of g i s defined to be the function m̂  on B, where for any r e B, m F(r) = i n f sup g (s) ; r SeF(r) seS and the m^-realizing subset of K over r i s defined to be Kp(r) = | ( r , s ) : s e K r and g r ( s ) = nip(r) j- . More generally, for any subset C Cl B, the n y - r e a l i z i n g subset of K over C i s the d i s j o i n t union K F(C) = U K (r) . reC - 10 - Our aim i s to give conditions on the data under which m^ w i l l be continuous on B, Kp(r) w i l l be nonempty for each r, and Kp(C) w i l l be compact i n IE for any compact subset C of B . The existence of a point i n Kp(r) i s the main assertion of the generalized Lusternik-Schnirelman theory developed by P a l a i s [5., 6], and Browder [1, 2, 3], where i n c i d e n t a l l y several examples of f a m i l i e s F are given. In the following l i s t of conditions on the data [la] - [ l e ] , the f i r s t states the standard hypotheses f o r existence i n the Lusternik- Schnirelman theory, and the others mainly require that the existence hypotheses hold uniformly, i n various senses, with respect to the parameter r . Note that some conditions r e f e r to the functions g^, and some to the u n r e s t r i c t e d f or f r [1.1] For each r e B, the function g r : E^ —-> IR i s bounded below and s a t i s f i e s condition (C) of Palais-Smale ( i . e . , any sequence E r for which § r ( s _ ) x S bounded and ||dg r'(s^)|| converges to zero contains a convergent subsequence ). [1.2] For each R e IR and each compact subset C of B, the subset K H |(r , s ) : r e C and g r(s) <_R i s compact i n IE . [1.3] For each r e B and each R e IR, there e x i s t s a neighbourhood V of r i n B such that the subset - 11 - IF R v = j t E (F : g t ( r ' ) '<_ R for some r* e V j i s bounded i n IF . [1.4] For each r e B and each bounded subset T of (F, there e x i s t s a neighbourhood V of r i n B such that 6 ( * r ( t ) , * r , ( t ) ) < e for a l l r' E V and a l l t e T . [1.5] For each r e B, each bounded subset S of L^(E), and each e > 0, there e x i s t s a neighbourhood N of r i n B such that | f ( r , s ) - f ( r ' , s ) | < e for a l l r' e N and a l l s E S . [1.6] For each r E B and each bounded subset S of L^(E), there e x i s t s A E IR such that | | d f r ( s ) | | <_ A for a l l s E S . With these preparations done, we can state our r e s u l t : Theorem I Suppose a standard parametrized D i r i c h l e t problem i s given by data [la] - [Is] s a t i s f y i n g conditions [1.1] - [1.6]. Let F be a - 12 - deformation-invariant family of subsets of a s i n g l e path component of IE, such that at l e a s t one element of F i s compact and nonvoid. Then : (a) The F-minimax function m^ . of g i s f i n i t e and continuous on B ; (b) For each r e B, the n y - r e a l i z i n g subset Kp(r) of K over r i s not empty; (c) For each compact subset C of B, the mp-realizing subset Kp(C) of K over C i s compact. REMARK. In case Kp(r) i s a singleton {sp(r)} f o r each r i n some open set N i n B, so that Sp i s a section of ir : IE — > B over N, i t follows e a s i l y from (c) that Sp i s continuous. Thus the c r i t i c a l point of which r e a l i z e s a p a r t i c u l a r minimax value mp(r) v a r i e s continuously with r so long as i t i s unique. In the examples which we have i n mind, t h i s i s the conclusion of p r i n c i p a l i n t e r e s t . 1.4 Proof of Theorem I We f i r s t prove conclusion (b), by assembling several r e s u l t s i n Browder [2]. For f i x e d r e B, l e t X denote the connected component of IE which contains, a l l the sets i n the family F ( r ) , and l e t h denote the r e s t r i c t i o n of g to X . Then X i s a connected submanifold of the r CO C F i n s l e r manifold (E , and i s complete i n the induced metric. Hence [2, Proposition 5.2, p.32] there e x i s t s a quasigradient f i e l d for h on X . The function h i s bounded below and s a t i s f i e s condition (C) on - 13 - X , on account of condition [ l . l ] , a n d so by [2, Proposition(5.1), p.27], and the remarks following [2, Theorem 1, p.8 and D e f i n i t i o n ( 2 . 1 ) , p.18], a l l the hypotheses of [2, Theorem 1] are s a t i s f i e d , except for the f i n i t e - ness of mp(r). But t h i s follows from the existence of a compact nonvoid element i n F , and Browder's Theorem 1 applies to e s t a b l i s h (b). To prove (a) and ( c ) , we apply [7, Theorem 1.2] with our quantities (E, (F, TT, g, and K i n place of the "quantities E, F, p, f, D of that theorem. I t requires that F be invariant under homeomor- phisms of (F, but t h i s i s used i n the proof only to insure that the fa m i l i e s F(r) are independent of the choice of t r i v i a l i z a t i o n x of p. Since we have defined our F(r) by a f i x e d t r i v i a l i z a t i o n $ "*" of TT , we do not need the homeomorphism-invariance of F to apply the c i t e d r e s u l t . I t s conclusions are p r e c i s e l y the conclusions (a) and (c) of Theorem I; and one of i t s two hypotheses i s j u s t condition [1.2]. The other hypothesis, i n our notation, becomes the following : for each r E B and each R e IR, there e x i s t s a neighbourhood V of r i n B such that the family of functions Thus the proof of Theorem I w i l l be complete as soon as we have shown how to choose V so that t h i s equicontinuity a s s e r t i o n holds. Here we s h a l l i s equlcontinuous at r , where* as i n [1.3] above, - 14 - use conditions [1.3] - [1.6]. Fix r E B and R E (R. By [1.3], there i s a neighbourhood V of r i n B such that T = IF i s bounded i n (F . By assumption i n K,v [1<$], 4>r(T) i s bounded i n E r . By [1.4] with E = 1, there i s a neighbourhood N 1 of r such that <5(<j>r(t), <j>r,(t)) < 1 for a l l r' el and a l l t E T. Hence the set S 1 = | <j>r, (t) : r' E N 1, t E T } i s bounded i n L^(E) along with <J> (T) ; and so also i s s 2 = | s E L^(E) : 6(s,s') < 1 for some s' £ S 1 j Given E > 0, we apply [1.5] and [1.6] with S = S 2 to f i n d a neighbourhood N 2 d of r such that f ( r , s ) - f ( r ' . , s ) | < | (r' E N 2, s E S 2 ) , and a number A > 1 such that d f r ( s ) <_ A (s £ S 2) The we apply [1.4] to f i n d a neighbourhood O N 2 of r such that '• . 6 ( + r ( t ) , * r , ( t ) ) < min | 1, | £ | for r ' E N 3, t E T . 15 - Now choose any t e T = (F R v arid any r' e . In view of the. convention adopted i n §1.1, that points s, s' i n d i f f e r e n t path compo- nents of k£(E) have 6(s,s') = 2, the l a s t i n e q u a l i t y implies, i n p a r t i c u l a r , that the points s = <)>r(t) and s' = 4> rt(t) can be joined by a C 1 path y i n l£( E) with length y < min \ 1, 2A Any point on such a path can be no further than 1 from either endpoint, so l i e s i n • Accordingly we can apply the f i r s t , second and l a s t of the preceeding i n e q u a l i t i e s , to get |g t(r') - g t ( r ) | = | g r , ( * t . ( t ) ) - g r(4> r(t))| - |f r.(s*) - f r ( s ' ) l + |f r(a') - £ r(s)| < l + 1 ||df ( Y ( u ) ) | | | | Y ' ( U ) | | du 0 <_ -| + A- (length Y) < e. Thus {g : t e F } are equicontinuous at r as claimed, and the proof t R,V of Theorem I i s complete . CHAPTER 2 DIRICHLET PROBLEMS WITH VARIABLE DOMAINS 2.1 Formulation of Theorem II In t h i s chapter we s h a l l consider nonlinear D i r i c h l e t problems i n which the candidates for solutions are real-valued functions on a bounded domain ft d IRn, agreeing on the boundary 8ft with a prescribed function a . Our i n t e r e s t i s i n the behaviour of the s o l u t i o n when both ft and a are perturbed. The r e s t r i c t i o n to r e a l valued functions i s for convenience of notation only, and the reader w i l l e a s i l y see how to modify our s t a t e - ments to cover the case of vector-valued functions. We s t a r t with a bounded open set ft° O IRn, with t y p i c a l point x = (x^,...,x n) and with Lebesgue measure dco(x) = dx^...dx^ • An n-multi-index i s an n-tuple a = (a^,...,a n) of integers ou _ 0, and we w i l l write |a| for the sum \ ou , and D a for the p a r t i a l d e r i v a t i v e i a. operator J Y O/Sx.) . For any integer k _ 0, s, w i l l denote the t o t a l number of a's having let < k, and u = (u ) i i , w i l l denote a t y p i c a l ° 1 ' — ' a |a|<k s k s k point i n IR . Note that IR i s j u s t the f i b r e of the k - j e t bundle jk(IR ) of the product bundle IR = ft° x IR over ft° , whose sections we H o i d e n t i f y with the r e a l valued functions on ft° . More generally, for any a o i product bundle IR over ft with f i b r e IR we have a natural - ft° - 17 P a p o & i d e n t i f i c a t i o n of Lf(fR ) with LMfi , IR ) and we s h a l l use t h i s k n° k i d e n t i f i c a t i o n f r e e l y i n the following. s. With n ^ 1, k >_ 1, and f i x e d , l e t F : Q° x IR — > IR 2 be a C function. (Thus F i s the p r i n c i p a l part of an element of k FB[J (IR ), IR ].) Suppose F s a t i s f i e s the following conditions for a c e r t a i n p >̂  2, c e r t a i n constants C, > 0, and a c e r t a i n continuous function <E : R U — > R with <C(0) = 0 : k [2.1] For a l l x e 0° and a l l u e IR , F(x,u) | < C 1 + • I | u J 1 a <k a, [2.2] For a l l x, x' e and a l l u e IR , F(x,u) - F ( x ' , u ) | <_ (C(x - x') i+ i iu r |a|£k a o k [2.3] For a l l a, a l l x e . f i , and a l l u E IR , 9F 8u (x,u) < C |e|<k p o k [2.4] For a l l a, a l l x, x' e fi , and a l l u e (R , 3 u ~ ( x ' u ) ~ 9 u ~ ( x ' u ) a a < (E(x - x') 1 + , J l$l<k o k. [2.5] For a l l a, g, a l l x e-fi°, and a l l u e.'R , - 18 - 8 2F 8u a i i - (x,u) < C 1 + I |u | P " 2 [2.6] o k For a l l x e fi and a l l u, v e IR , |a|,|B|<k a 3 P |a|=k |P-2|„ |2 Conditions [2.1], [2.3], and [2.5] i n p a r t i c u l a r imply that the i n t e g r a l h u(v) - F(x, j . ( v ) ( x ) ) dco(x) ,o k i s w e l l defined for a l l v i n the Sobolev space L^(fi°, IR), and that h° 2 i s i n f a c t a C function ( c f . Lemma 2.1 i n §2.2 below)'. Here the symbol J^Cv) i - s being abused to denote the p r i n c i p a l part of the k-jet of the section of L^((R ) whose p r i n c i p a l part i s v . We s h a l l consider k fi° ' r e s t r i c t i o n s ' h = h^ obtained by replacing fi° i n the above i n t e g r a l by various subdomains fi C fi° . To be precise, for a l l r i n a s u i t a b l e parameter space B, we consider an open subdomain fi^ CT fi° whose closure i s obtained as the diffeomorphic image A(r)(M) of a f i x e d smooth manifold M . As r ranges over B, A(r) i s supposed to change continuously i n the space Diffeo[M,fi°] of Z9 diffeomorphisms of M into fi°, equipped with the topology of uniform convergence of a l l d e r i v a t i v e s . We also suppose that a s u i t a b l e boundary-function â _ i s given f o r each r . Our r e s u l t i s the following. - 19 - Theorem II F i x a bounded open subset ft° d (Rn, an integer k _ 1, and a o S k 2 r e a l number p _ 2. Let F : ft x (R — > IR be C and s a t i s f y conditions [2.1] - [2.6] given above. Fix a l o c a l l y compact Hausdorff space B and a compact 0 0 n-dimensional C manifold M with boundary 3M . Let r I—> a^ be continuous from B into L^(ft°,(R), and l e t r i — > A.(r) be continuous from B into Diffeo[M,ft°] . For each r e B, set ft = A(r)(M) and set D = L?(ft ,R) , r r K r a r 2 and l e t h be the C r e a l valued function on D defined by r r h (v) = r F(x, j k ( v ) ( x ) ) da)(x) ft.. Then the following, conclusions hold : (a) The function m : B — > IR defined by m(r) = i n f h r(v) veD r i s f i n i t e and continuous on B ; (b) For each r e B, the set M(r) = e D : dh r(v) = 0 and h r(v) = m(r) i s not void; (c) For each compact subset C of B, the set - 20 - A*M(C) > , | (r> v°X(r)) : r e C and v e M(r) | i s a compact subset of B x L J^IR^) . We s h a l l deduce Theorem II from Theorem I of §1.2, by construc- t i n g a s u i t a b l e standard parametrized problem. I t should be noted that our . conditions [2.1] - [2.6] can be weakened considerably by e x p l o i t i n g the Sobolev i n e q u a l i t i e s (cf. Browder [1, pp 25-29]). The techniques f o r doing t h i s are well established and we s h a l l not dwell further on the point. Before carrying out the reduction to Theorem I, we s h a l l e s t a b l i s h some properties of the functions h^ . 2.2 Preliminary consequences of the assumptions In the following we use the assumption that F s a t i s f i e s [2.1]- [2.6]. Lemma 2.0 For each tt <Z tt° the map ^2 extends to a C map Furthermore, the maps 21 - d(F A) : L P ( J k ( V J - > L ( L P ( J k ( V , L j ( V ) and d 2(F,). : L P ( J k ( V ) -> L 2 ( L P ( J k ( R f i ) ) , L J ( ^ , ) ) take bounded sets into bounded sets. (The reader i s warned that the symbol 2 p Lg above means symmetric bilinear maps and should not be confused with L^) p k Finally for v, v^, and v 2 e L Q ( J > and d(F A)(v)(v 1) = 6F v( V ; L) d 2 ( F A ) ( v ) ( v l s v 2 ) = 6 2 F v ( v 1 9 v 2 ) . Proof : See Browder [1] For ft C Q° let H° : L^(Jk(IRfi)) —> IR be defined by Hfl(v) - F Av(x) du>(x) Let hn : Lj(R f l) —> R be defined by h^ = H f i°J k Lemma 2.l(i) For each fi <rT , Hfi : L*J(Jk(lRfi)) —> IR is a C 2 function such that (a) dH : L^(J k(IR n)) —> LP(J k(R ) ) * takes bounded sets into - 22 - bounded sets ; P k (b) dH i s uniformly continuous on bounded sets of L Q ( J " Q 2 Proof : The fact that H i s C follows immediately from lemma 2.0. 2 Part (b) follows from the boundedness of d (F^) i n lemma 2.0 together with the mean value theorem. Lemma 2 . 1 ( i i ) o h : 'I_(Rfl) —> R i s a c 2 function such For each 9, C 0, that (a) dh" : LP(R f i) -> L!X>* a n d d 2 h " : L k < V •> L 2(L£(R n),IR) takes bounded sets into bounded sets (b) dh i s uniformly continuous on bounded subsets of L^(IR^). Proof Since h^ = H^oj and j f e : .^(IR^) — > L P ( J k ( I R n ) ) i s a bounded l i n e a r map the proof follows immediately from lemma 2 . 1 ( i ) . Lemma 2.2 [dh"(v x) - d h " ( v 2 ) ] ( V l - > C Y D V ( x ) - D°v„(x) P du(x) i i , 1 £ for some constant C - 23 - Proof : From lemma 2.0 i t follows that dh ( v 3 ) ( v l s v 2 ) 3 2F n'l lXl«l v - ^ ( X ' J k ( v 3 ) ( x ) ) U a ( V V « ) U B ^ k ( v 2 > W ) ft 0<_\ a I , I 8 | <k a 6 So [dhV-̂  - dh^Cv^]^ - v 2 ) ft J0 2 3 F 0<J a | , I B | <k a 6 u a ( j k ( v r v2 > ( x )H ( j k ( vr vz ) ( x ) ) dt dco(x) and by [2.6] f I f1 | D \ ( X ) + t(D°'(v 2-v 1)(x))| P 2 | D a ( v 2 - v 1 ) ( x ) | 2 dt du(x) ft I a I =k •'0 | a | =k D v 1 ( x ) - D v 2 ( x ) K dx for some constant c Lemma 2.3 For v r v 2 , a r * 2 E L P ( ( R ) with V ]_ E LJOR ') and C d h " ^ ) - d h ^ v ^ K v ^ ) > C ! I v l - v 2 ~ ( a l - a 2 ) I l ^ p Lk; a r a2 for some constant C independent of ft - 24 - • \'.-\ 1 . . Proof : By lemma 2.2 we need only show that there ex i s t s a constant independent of ft such that I ' |D a(v -v ) ( x ) | P du)(x) | a | =k Jft > c <vl _ V " ( a l — a 2 y ' ' pj l a i " a 2 N p L k Now v, - v n - (a, - a„) e LP(IR ),. . Therefore there e x i s t s a constant 1 2 1 2 k 0 independent of ft such that I | D a ( V ; L - v 2 ) ( x ) - D a ( a i - a 2 ) ( x ) | P do>(x) _ c ||v 1 - v 2 - ( a 1 - a 2 ) | | Now there ex i s t s a constant independent of ft such that • C. f _ f | D a ( V l - v 9 ) ( x ) - D a ( a 1 - a 9 ) ( x ) | P du>(x) 1 1 |a|=k Jft . Z J I |a|=k | D a ( V ; L - v 2 ) ( x ) | P dw(x) + j | D a ( a i - a 2 ) ( x ) | P dco(x) ,1 | a | =k | D a ( V l - V 2 ) (x) | P da)(x) + || a i - a 2 | | P p Therefore there ex i s t s a constant C independent of ft such that _ J D ^ v ^ v p C x ) ! 1 1 da)(x) + | |a1-a21 | P > Cl l v r v 2 _ ( a r a 2 ) |a|=k 1 ft and the r e s u l t follows - 25 - Lemma 2 .4 Let {v^} D e a bounded sequence i n L^(IR^) such that v. e LP(IR ) with {a.] converging to a . Then i f dh^(v.) — > 0, i t x tc Si a.. x x i follows that fv.jj i s a convergent sequence i n L^ftR^) . Proof : Fix e > 0. Pick A so large that i , j > A implies that and a. - a. W* < S , , 1 i j 1 1 p - 1 L k [dh"( V ; L) - d h " ( v 2 ) ] ( V ; L - v 2 ) < 6 for some <5̂ , 6 2 to be determined. By lemma 2 .3 6 2 - C l a i - a 2 H P L k Since I v v W p - H v r v 2 l l P L k L k we get 6. > C | v r v 2 | | a r a 2 i i p i a r a 2 n P L k Since {v^} i - s a bounded sequence there e x i s t s 6-̂  such that a l " a 2 ' ' p < 6 1 L k implies that - 26 - i vr v 2N Tp ' ara2MLpj l a , - a „ I | P - C l I v , - v „ I | P I 1 2 " p 1 2 ' l n k k Ce It follows that c l | v _ - v 2 | | P p < ^ | + 6 2 L k Setting 6 2 — ^ we get v. - v. < e 1 i 3 T P i t Lemma 2.5 the set Fix S C LMIR ) bounded and R e IR. Let h denote h . Then K o 6 v e U o LP((R ) : h(v) < R weS k QO W is bounded i n L, (IR ) . k Proof Fix v e LP(tR ) . Suppose v = t + w with t e LP(IR ) . Then k o / w V V k fio 0 h(v) = h(t + w) = h(w) + dh(w + ut) (t) du r l = h(w) + dh(w)(t) + [dh(w+ut) - dh(w)](t) du = h(w) + dh(w)(t) + By lemma 2.2 f 1 1 1 ^ [dh(w+ut) - dh(w)](ut) du - 27 - > h(w) + dh(w)(t) + C ^ 1,, I | P A — ut r du u 1 1 1  P 0 < = h(w; + dh(w)(t) + - I I t l | p P II i i p L k Since S i s bounded, by lemma 2 . 1 ( i i ) sup I | dh(w) | | = A < «> , weS sup | h (w) | = B < °° weS Therefore h(w + t ) > - B - A t • + - t L P P L P k k for a l l w e S, where C > 0, p > 0 . I t follows immediately that h(w + t) < R with w e S implies that I ItlI i s bounded/ •1 1 1 1 p k Lemma 2.6 Fix S C LP((R ) bounded and R e |R . For any 0, C l e t k Q° T = { w|fi : w e S i . e . T consists of the " r e s t r i c t i o n " of S to fi . Then there e x i s t s a constant B e IR, independent of the choice of fi such that V £ { V E w^T Ll%K : < R implies that ||v|| < B • L k - 28 - Proof : Pick v e LP(IR ) . Then v = w + t for t e L^(IR 0) n , and w k ^ f i ' O i s the r e s t r i c t i o n of some w, e Lp (IR ) to fi . Extend v to fi° by se t t i n g i t equal to on fi° - fi . C a l l the extension . Then h" (v x) = h"(v) + F°j, (w, ) do) Jfi°-fi k 1 < R + | Q F c J k ( W ; L ) do) By lemma 2.1(H) there e x i s t s a constant c e IR such that (wn )du> o J k 1 < c for a l l ŵ  e S . Therefore h (v x) < R + c By lemma 2.5 there e x i s t s a constant A e IR with v, < A . Then i t J • 1 T p follows that V M £. A + sup I |w1 L f w.eS k 1 1 " p L k and the r e s u l t follows f o r B = A + sup I|wi|I w.eS L P 1 k - 29 - Lemma 2.7 For each ft , and w e L PQR f i), the r e s t r i c t i o n of h^ to LF(IR^) i s bounded below, k ft w Proof : In the course of v e r i f y i n g lemma 2.5 we derived the i n e q u a l i t y h(t + w) > - B - A l I t l I + - II11 I P - L P P L P k k for t e LP(IR ) n , and lemma 2.7 follows immediately from t h i s i n e q u a l i t y . We are now ready to proceed with the reduction from Theorem II to Theorem I . 2.3 Construction of the Associated Standard Problem We construct the standard problem by defining the data [la] - [le] [la] M has already been determined. Let u be any s t r i c t l y p o s i t i v e smooth measure on M, and l e t E be the product bundle IR̂ . . [IB] p, k are already determined. Let || || be the standard norm on LJ^IE^), so that 6 ( s 1 , s 2 ) = | | s 1 - s 2 [ | . L k [ly] B has been determined. For each r e B the map X(r) : M —> fir C fi° induces a map X( r ) * : L P(fi r,IR) — > LP(M,IR) given by A(r)*(v) = v°X(r). R e c a l l that we - 30 - have a map B — > LP(ft°,IR) given by r > a . We define a map B —> LP(M,IR) by def r I—> b r E X ( r ) * ( a r ) [16] Let b = the zero section of L ^ O ^ ) > a n d t n e m a P r 1 — > b r a s given above. Let $ : B x I ^ O r m ) Q — > E b e defined by $(r,s) = (r,s+b r) so that <)>r(s) = s + b r . OO For each r e M, l e t a r e C (M,IR) be defined by a r(y)X(r)*u(y) = y(y) . [le] For s e L ^ O ^ ) and r e B , l e t L r s ( y ) = a r ( y ) F ( A ( r ) ( y ) , J k ( s o X ( r ) " 1 ) ( X ( r ) . ( y ) ) ) where we repeat that the symbol J^C') i S being abused to denote the p r i n c i p a l part of the k-jet of the section s ° X(r) e L^Ot^ ). Assuming for the moment that the standard parametrized problem with the above data s a t i s f i e s the hypotheses of Theorem I, we i n d i c a t e how Theorem II follows. From the d e f i n i t i o n of L f i t follows that for s e L^OR^) that L s(y) dy(y) = M L ° ( S o A ( r ) 1 ) ( v ) du(x) ft r 31 - so that g r ( t ) = h ^ ( t o X ( r ) . Now i n Theorem I l e t F = { {8.} : s e L £ ( V o } Then m F(r) = i n f g ( t ) = i n f h (v) .= m(r) teE veD r r So mp(r) = m(r) and hence m(r) i s f i n i t e and continuous on B . Also, M(r) = \ v e D r : dh r(v) = 0 and h r(v) = m(r) •" = | v e D r : d g r ( v o X ( r ) = 0 and g r(t°A(r)> = mp(r)| Therefore v e M(r) i f f v o A ( r ) e K^(r) from which i t follows that M(r) i s not void. F i n a l l y A*M(C) = |(r,v°A(r)) : r e C and v e M(r)j and from the above disc u s s i o n i t follows that A*M(C) = Kp(C) and from Theorem I i t follows that A*M(C) i s compact. Therefore i n order to prove Theorem II we need only v e r i f y conditions [1.1] - [1.6] for the standard problem j u s t constructed. 2 . 4 Verification of Conditions [ 1 . 1 ] - [ 1 . 6 ] We shall carry out the verifications in the order 1 , 4 , 3 , 5 , 6 , 2 Verification of [ 1 . 1 ] : N _1 Fix r e B and t e L^OR^' . Then g r(t) = h r(t°X(r) ) and r by lemma 2 . 6 g r i s bounded below. Now assume that x S a bounded sequence with ||dg r(t^)|| ->- 0 fir - I I I - 1 It follows that ||dh ( t ^ A C r ) ) | | 0 . By lemma 2 . 4 {t^oACr) } i s a Cauchy sequence. Hence {t^} 1 S a Cauchy sequence. Verification of [ 1 . 4 ] : Since <j>r(s) = s + b , 6(<|>r(s), cf)r, (s)) = | |s + b r - s + br,| | \ = I lb - b ,1 I II r . r I I p . Lk = I f |Da(a oA(r))(y) - D a(a ,<>A(r'))(y)| P dy < I f |Da(a P A(r))(y) - D a(a o A ( r ' ) ) ( y ) | P dy + i a > a ( a °A(r'))(y) - D a(a , °A (r')) (y) | P dy M It follows from the continuity of the functions r I—> and r I—> A(r) - 33 - that the above terms can be made a r b i t r a r i l y small f o r r' i n some neighbourhood V"r of r . V e r i f i c a t i o n of [1.3] : Fix r e B and R e IR . Let W be a neighbourhood of r such that the set {a , : r' e W} i s bounded i n LP(R. ). Now g ( r 1 ) < R r K. QO t — " r ' -1 implies that h ( (t+b^, ) <>X(r') ) <_ R . From lemma 2.6 i t follows that there e x i s t s a B £ IR such that | |(t + b r , ) o X ( r ' ) " 1 | | < B Then I t o X C r ' ) " 1 ! But since the sections b ,oX(r') are j u s t the r e s t r i c t i o n s of the a ,'s to the various ft r' s i t follows that sup ||b ,oX(r') r'eV r -1 and that there exists an A e (R with U t o X C r T 1 ! ! p < A for a l l t e F t R,W and a l l r' e W . From the continuity of the map - 34 - r* 1—> X(r') i t i s e a s i l y seen that there e x i s t s a neighbourhood V C W of r and a constant A^ e IR such that | I t o X C r ' ) " 1 ! | < A and r ' e V \ implies that I I t l I < A.. . From t h i s i t follows that IF,, T I i s bounded. r 1 1 1 1 p 1 R,V Hi V e r i f i c a t i o n of [1.5] For s e L^flR^ l e t s r e L^CR^ ) denote s°X(r) 1 . For y e M, l e t x r = X ( r ) ( y ) . Then for r , r' e B f ( r , s ) - f(r»,s) = M [L s - L ,s] dp r r Ls dco -r Ls , du r F ( x r , j k ( s r ) ( x r ) ) du>(xr) - F(x ,,j k(s r,).(x r,)) du(x r,) •"ft,-' Let 3 : ft^ —> be given by = X(r')°X(r) -1 Then x^, = 3(x r) and we rewrite the above as F ( x r , j k ( s r ) (x r)) dco(xr) (A) (B) (C) F(f3(xr) , j k ( s r , ) (3(x r)))d(3*0)) (x r) du(x r) [F ( x r , j k ( s r ) ( x r ) ) - F( x r , j k ( s r , ) ( 3 ( x r ) ) ) j F ( x r > j k ( s r , ) ( 3 ( x r ) ) ) - F(3(x r),j k(s r,)(3(x r))) F(3(x ),j,(s ,)(6(x ))) d[w-e*a>] (x ) Jo.. r dco(xr) We shall deal with each of the above terms separately. (A) x r •—> (x r,j k(s r,)(3(x r))) i s a section in L P(J k(lR^ )) . In fact i t is just 3*(j k(s r,)) . Furthermore, for s e S C L̂ OR̂ ) bounded and s^, s^, as defined above there exists a neighbourhood of r such that 3*(j, (s ,)) - s < e Jk r' r 1 1 p L0 for a l l s e S and r' e V r . Now (A) is the same as H ( j k ( s r ) ) - H r ( 3 * ( j k ( s r , ) ) ) ftr And from the boundedness of dH which follows from lemma 2.1 (i) we get a bound on (A) which can be made arbi t r a r i l y small. 1 - 36 - (B) By condition [2.2] (B) i s bounded by <C(x-3(xr))| 1 + I | u a ( j k ( s r , ) ) ( 3 ( x r ) ) dco Now for any E > 0 there e x i s t s a neighbourhood W of r such that f o r r' c W , sup |(C(xr - 3(x r)) | < e . x efi„ r r (Recall that 3 depends on r ' ) . Therefore (B) i s bounded by 1 + I h a ( 3 k ( s r , ) ( 3 ( x r ) ) | p da) Since s r , i s derived from s e S a bounded set, there e x i s t s a constant A e IR such that 1 + I I V V V ) ^ ) ) ! 1 fir v a dco < A for a l l s e S. I t follows that (B) can be made a r b i t r a r i l y small. (C) From the cont i n u i t y of the map r I—> X(r) i t i s e a s i l y seen that the di f f e r e n c e of the measure to - 3*w can be made uniformly small over fir by r e s t r i c t i n g r ' to l i e i n a su i t a b l e neighbourhood of r . I t follows that (C) can be made a r b i t r a r i l y small. V e r i f i c a t i o n of [1.6] : Let S C L^OE^) be bounded. Then the set { s o X ( r ) " 1 : s e S } - 37 - i s bounded i n ^ * ^ o w d f r ( s ) ( t ) = dH f i'.(J k(soA(r) 1 ) o J k ( t o X ( r ) From lemma 2.1(ii) i t follows t h a t sup | | d H n ' ( j k ( s o X ( r ) _ 1 ) ) | | < » ,. seS and since 1^0^) — > L P(J k((R f i )) given by t l — > J k ( t ° A ( r ) _ 1 ) i s bounded i t follows that sup ||df ( s ) | | < seS V e r i f i c a t i o n of [1.2] : Fix R E IR and C C B compact. Let {(r^,s^)} be a sequence with { ( r i , s i ) } C K fl | ( r ' , s ) : r ' e C and g r,(s) <R J . We need to show that {(r^,s^)} has a convergent subsequence. Since C i s compact we can assume that {r__} i s a convergent sequence and we assume that ^r__} converges to r E B . From conditions [1.3] and [1.4] i t follows that {s^} i s bounded. In order to proceed we need the following proposition whose proof w i l l be found below. Lemma 2.8 Fix S CT L k(R^) bounded and E > 0. Then for each r e B there - 38 - e x i s t s a neighbourhood V of r such that ||dg (s) - dg , (s)| | < e , j , r r for a l l r' £ V and s E S1 . From t h i s i t follows that since converges to r , {s^} i s bounded, and dg (s.) = 0, that dg r(s.) converges to zero. From i ar -1 th i s i t follows that dh (s^°A(r) ) converges to zero. F i n a l l y from lemma 2.4 we get that {s^°A(r) }̂ i s a convergent sequence and therefore {s^} i s a convergent sequence. Proof of Lemma 2.8 : We w i l l employ the same notation as i n the v e r i f i c a t i o n of [1.5]. That i s for s, t e L^ClE^), we l e t s r , t r £ 1^0^ ) denote s ° A ( r ) _ 1 , t°A(r) \ and f o r r , r' E B we l e t 3 : fir — > & , be given by 3 = A(r')°A(r) _ 1, and f o r x r £ Q.^ we l e t x r, = 3(x r) . Now d g r , ( s ) ( t ) = while d g r ( s ) ( t ) = n  5 \ ( s ,)(x ,) W ) ( V > d u , (V> n r t  J k r' r' 6 F j , (s )(x ) j k ( t r ) ( x r > d a ) ( x r > ftr J k v x' v r' In order to compare d g r , ( s ) ( t ) and d g r ( s ) ( t ) we need to employ the coordinate system on J (R ) • In coordinates - 39 - So d g r , ( s ) ( t ) = 7 I — ( x , , j , ( s ,)(x , ) ) o U ( j . (t ,)(x ,) dw(x ,) L L 3u r' k v r ' r ' a J k v r ' r' r' a •'ft.,.? a = I ^ - ( 6 ( x r ) , J k ( s r , ) ( 6 ( x r ) ) ) o u a ( J k ( t r , ) ( 3 ( x r ) ) d(B*co)(x r) dg ( s ) ( t ) - dg , ( s ) ( t ) (A) (B) + 9F (x . j , ( s ) (x )) n du v r ' J k v r ' v r' ftr a u a ( j k ( t r ) ( x r ) ) - u a ( j k ( t r , ) ( 3 ( x r ) ) ) dto(x r) | f ( x r , j k ( s r ) ( x r ) ) - l f ( x r , J k ( s r , ) ( B ( x r ) ) ) a a x " a ( J k ( t r . ) ( e ( x r ) ) dco(xr) (C) + (D) + r L a J ftr*- V • I a * ftr 9F ^ - ( x r , j k ( s r , ) ( 3 ( x r ) ) ) - g J r ( 3 ( x r ) f J k ( 8 r , ) ( B ( x r ) ) ) x u a ( J k ( t r i ) ( B ( x r ) ) dco(xr) |f-(B(x r) , j k ( s r , ) (3(x r))) - u a ( j k ( t r , ) (3(x r)) d[co-3*co] (x r) As before we deal with each of these terms separately. Now (A) i s the same as ft. dH r ( j k ( s r ) ) ( j k ( t r ) - 3 * j k ( t r , ) ) where - 40 - S * j k ( t r , ) ( x r ) = j k ( t r,)(3 ( x r ) ) I t i s e a s i l y seen that f o r any 6 > 0 there e x i s t s a neighbourhood W of r such that I I J i " Ct ) - 3*3% (t ,) I I < 6 1 1 J k v r J k v r " 1 1 p 0 for t , t , derived from t with I Itl I =1. From the boundedness of r r T P He • ftr dH which follows from lemma 2.1 i t follows that (A) can be made a r b i t r a r i l y small. (B) This term i s the same as d H % ; J k ( s r ) ) - d H % B * J k ( s r , ) ) * * j k ( t r . ) . ftr From the uniform continuity of dH , the boundedness of 6*j, (t , ) , K. 10 J k ( s r ) , and 3*j k(s' r,) and the fac t that I | i k ( s r ) ~ 3*J k(s r,)|| can L0 be made a r b i t r a r i l y small i t follows that (B) can be made a r b i t r a r i l y small. (C) I t i s e a s i l y seen that f o r 6 > 0 there e x i s t s a neighborhood W of r such that |B(x r) - x r| = lACOoAO:)" 1^) - x r| < 6 for a l l x e ft and a l l r ' E W . From condition [2.4] i t follows that r r L ' (C) i s dominated by - 41 - ni J a J f t T (CC3(xr)-xr) 1 +:I |u ga kCs r,)C3(x r))) lP-1 u aa k ( t r,) (3Cx r)) dio It follows from Holder's inequality and the above remark that (C) can be made arbitrarily small. (D) It follows from the fact that co - 3*to can be made uniformly small that (D) can be made arbitrarily small. T h i s completes the verification of conditions [1.1] - [1.6], and hence Theorem II is proved. - 42 - CHAPTER 3 DIRICHLET PROBLEMS WITH VARIABLE HOLONOMIC CONSTRAINTS 3.1 Formulation of Theorem I I I In t h i s chapter we s h a l l consider a parametrized version of the D i r i c h l e t problem described by Pa l a i s [4, pp 104-105, p 109]. The s o l u t i o n candidates are vector valued functions on a manifold M, Whose values are constrained to l i e i n a given submanifold W C IR , as well as to agree with those of a given function a on 8M, i n case 3M i s not empty. We s h a l l study the s o l u t i o n when both W and a are permitted to vary. OO To be precise, we begin with a compact C manifold M, with th p o s i t i v e smooth measure y and possibly with boundary, and a k order Lagrangian L e Lgn^CO » where £ i s the product vector bundle IR̂  f o r some £ _ 2. We suppose that L s a t i s f i e s the following conditions : [3.1] For some p with pk > n, L extends to a C.̂  map : f£(£) —> LJO )̂ s o t h a t t h e i n t e g r a l h(v) = Lv dy M defines a real-valued function on L P(£) . [3.2] For any a ± , &2 e L P ( ^ ) and any v ± e a » v 2 e Ll_^0a^ » [dh( V ; L)-dh(v 2)] ( V ; L-v 2) > c| | ( v 1 - v 2 ) - ( a 1 - a 2 ) | | P p - | [ ( a ^ ) | | P p L k L k - 43 - for some constant c . [3.3] The map dh : LP(£;) — > L^CO* takes bounded sets to bounded sets. Next, l e t W be a closed C submanifold of (R with 8W = 0 . For each r i n a parameter space B, the varied constraint manifold W w i l l be obtained by acting on W with a d i f f e o - Z morphism A(r) of the ambient Euclidean space IR . Also, f i x i n g a boundary-value function b on M with values i n W, we obtain indepen- dently - varied boundary functions a^ by f i r s t composing b with another Z diffeomorphism V(r) of IR which c a r r i e s W onto W, then composing the r e s u l t with A(r) . Notations l i k e f ( r ) ^ b w i l l be used for such a composite function, and by abuse, b may denote a section of the t r i v i a l bundle Ŵ  and ^ ( r ) ^ the induced section. We s h a l l require the maps A and V to be continuous from B % 00 Z into the space Diffeo(lR ) of a l l C diffeomorphisms of IR onto i t s e l f , with the topology of uniform convergence of 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 following. Theorem I I I CO Let M be a compact C manifold of dimension n, possibly with Z boundary, and with a s t r i c t l y p o s i t i v e smooth measure u . With £ = IR̂ . , Z >_ 2, l e t L e Lgn^C?) s a t i s f y conditions [3.1] - [3.3] for some p . with pk > n , and set - 44 - h(v) = Lv du for v e L?(C) M fc 00 I Let W be a closed C submanifold of IR without boundary, (W compact i f 3M = 0). Let E =.W and b e L P(E). Let B be a locally compact Hausdorff space and let . o o T : B —> Diffeo((R ) and A : B —> Diffeo(lR ) be continuous maps such that, for each r e B, ¥(r)(W) = W and A(r)(W) oo a i s a closed C submanifold of IR def Set W(r) = A(r)(W). Then E(r) = W(r) i s a C°° subbundle of IR̂  . Let a r = A ( r ) ^ ( r ) A b , and D = L£(E(r)) . Let h r denote r the restriction of the function h to D r Let 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. For each r e B, set V ( x ) = | V C D_ : V = A(r)^(r)^(T), some T e F j- . Then the following conclusions hold : (a) The function m : B —> R defined by m(r) = inf 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 - 45 - 'M(r) = • j ( r , v) : v'e D r, dh r(v) = 0, and h r(v) = m(r)| i s not empty. (c) For each compact subset C C B, the set M(C) = j ( r , v) : r e C and (r, v) e M ( r ) | i s a compact subset of B x L P ( E ) Theorem III w i l l be„deduced from Theorem I of §1.3 by constructing a s u i t a b l e standard problem. 3.2 Construction of the Standard Problem [la] Let M, u as given i n §3.1, and E = Ŵ  . [lg] p, k as determined i n §3.1. Since W i s a submanifold of (R , the i n c l u s i o n W — > IR induces an i n c l u s i o n ^ ( ^ ) — > L£(IR^), We give LJ^W^) the induced F i n s l e r structure || ||, and the corresponding F i n s l e r metric. [ly] B as given i n §3.1 . [16] b as given i n §3.1 with b^ = Y(r)*b and <j> equal to the r e s t r i c t i o n of Viv)* to L^CE^ . [le] L r = LoA(r)* . . - 46 - We deduce Theorem III from the application of Theorem I to the standard parametrized problem determined by the data [la] - [le] . The passage from Theorem III to Theorem I i s more direct than the passage from Theorem II 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 original problem. In fact since fl(r) = iV C ti : V = A(r)*¥(r)*(T), some T e f f , and and F(r) = -JS C L£(E)fe : S = ¥(r)*(T), some T e ¥} , r g r(s) = h r(A(r) A(s)) i t follows that m(r) = inf sup h (v) VeP(r) veV r = inf sup g^(s) = mF(r) SeF(r) seS and (a) of Theorem III follows from (a) of Theorem I Also we have M(r) = | ( r , v) : v e Dr, dh r(v) = 0 and h r(v) = m(r) = j(r, A(r) As) : s e ijJCE^ , dg r(s) = 0 and g r(s) = m F(r)| which is non-empty by (b) of Theorem I. Finally, - 47 - M(C) = | (r, v) : r e C and (r , v) e.M(r) j = { (r, A(r)„s) : r e C . s e l£(E) b , dg r(s) = 0 V r and g r(s) = rtip(r) which i s comapct when C i s compact by (c) of Theorem I . Before proceeding with the v e r i f i c a t i o n s we prove a few lemmas about the map r l—> A ( r ) A . A standard assumption w i l l be the continuity of the map r I—> A(r) . Lemma 3.1 Fix S C L. P(0 bounded and e > 0 . Then there e x i s t s a neigh- bourhood V of r such that | | A ( r ' ) A s - A(r)*s|| < e Lk for each r' e V and s e S . Proof : The proof follows e a s i l y from P a l a i s [4, Lemma 9.9, p.31]. Lemma 3.2 If S C L P(£) i s bounded then A(r).(S) i s bounded. K. Proof : . This also follows from the above c i t e d lemma 9.9. - 48 - Lemma 3.3 Fix r e B and S C L^CO bounded. Then there e x i s t s a neighbourhood V of r such that U A(r').(S) i s bounded i n L?(£) Proof : This follows from lemmas 3.1 and 3.2. Lemma 3.4 Fix r e B and S C 1^(5) bounded. Then i n f | | d ( A ( r ) ^ ) ( s ) | | > 0 . seS Proof : By lemma 3.2, A(r)(S) i s bounded. Now || d ( A ( r ) ^ ) ( s ) [ | = | | d ( A ( r ) ; 1 ) ( A ( r ) ^ ) ( s ) | I " 1 . Therefore we need only prove that f o r S C ^ ( 5 ) bounded, and r e B that sup | |d(A(r) A) (s) | | < °° . Now for s E S and t E , S E S d ( A ( r ) J ( s ) ( t ) ( x ) = [6A(r)°s(x)](t(x)) , by P a l a i s [4, Theorem 11.3, p.41]. Again by [4, Lemma 9.9, p.31], i t follows that f o r [|t|| = 1, | | (6A(r)°s) At| | < A < co for some A e (R, independent of s E S . This implies that sup j | d ( A ( r ) * ) ( s ) | | < S E S - 49 - 3.3 V e r i f i c a t i o n of conditions [1.1] - [1.6] It follows by the same techniques as employed i n Chapter 2 that condition [3.2] implies that h : L^CO —> IR i s bounded below and s a t i s f i e s condition (C), and that f o r S C L k ( 5 ) bounded and R -e IR , the set | s e L P ( 5 ) a : a e S and h(s) <_R i s bounded i n 1^(5) • V e r i f i c a t i o n of [1.1] : Since g = h o A ( r ) . , g i s bounded below for each r , and by r * r the above remarks combined with lemma 3.2 i t follows that for b e L P ( E ) , and R c (R , the set | s e L P ( E ) B : g_.Cs) < R i s bounded i n a n d hence bounded i n the F i n s l e r metric on LJ^E) (Uhlenbeck [8]). Therefore i n order to show that g__ s a t i s f i e s condition (C) we need only show that i f {s^} i s a bounded sequence i n L P ( E ) B (and hence bounded i n L £ ( £ ) , ) , such that dg (s.) — > 0, then {s.} i s convergent. Now dg r(s.) = d h ( A ( r ) s , ( s i ) ) o d ( A ( r ) A ) ( s . ) . If d g ( s ) — > 0 i t follows from lemma 3.4 that d h ( A ( r ) + (s.)) — > 0, that {ACr^s^} i s convergent and therefore so i s {s^} . - 50 - V e r i f i c a t i o n of [1.2] : Fix C C B compact and1 R E IR . Let {(r^, s^} be a sequence i n K O {(r, s) : r e C and g r ( s ) £ R } . Since B i s compact we can assume that {r^} converges to r e B . From lemma 3.3 i t follows that the set {b } i s bounded i n Lf(£) . r. k x Now g (s) <_ R implies that h ( A ( r i ) ^ (s)) <_ R , where i A (r.),(s) E L?(E,)^ • From condition [3.2] i t follows that the set l * k b r. { A ( r i ) A ( s ) } i s bounded i n 1 ^ ( 0 • In order to proceed with the v e r i f i c a t i o n of [1.2] we need the following extension of the construction i n Palai s [4, pp 112-114]. co £ F i x r E B . Then A(r)(W) i s a closed C submanifold of IR . r £ £ For each w E W l e t q (w) denote the orthogonal p r o j e c t i o n of IR = HRw CO onto TW . Then q i s a C map of W into the vector space W i a z °° a L(IR , (R ), and since W i s a closed C submanifold of IR , i t extends co % z a to a C map of IR into the vector space L((R , tR ). I f we define r r r 0 0 £ Q (x, v) = (x, q (v ) ) , Q i s a C f i b r e bundle morphism of £ = IR̂ into L (5 , O • As i n [4, theorem 19.14], we define a map LJJa) — > L ( L P ( 5 ) , L P (C) ) denoted by s I > p£. and given by Pg(t)(x) = Q r ( s ( x ) ) ( t ( x ) ) . - 51 - In the above construction the map P g was constructed by appealing to a general extension theorem. We wish to show that these maps r' can be constructed for r' i n some neighbourhood of r such that P g i s " close" to P__ i f r' i s close to r . More p r e c i s e l y we have the following. Lemma 3.5 Fix r e B and S C ^ £ ( 0 bounded. There exists 6 > 0 and a method of d e f i n i n g the extensions of the such that for each r r 1 e > 0 there e x i s t s a neighbourhood V of r with ||P - P || < e for s s a l l r ' e V, and a l l s with distance(s, L P ( E ( r ) ) ) < 6 . k Proof : Let N be an a r b i t r a r i l y large compact subset of W to be r a a determined. For r e B define the p r o j e c t i o n q : A(r) (N) — > L((R , (R ), as described above. For each point z e A(r)(N) extend q a f i n i t e distance along the normal d i r e c t i o n s to A(r)(N) at z i n (R by making i t constant. Now for r* "close" to r define q r : A(r')(N) -> LQR1, IR̂ ) as above and extend i t (shrinking N s l i g h t l y i f necessary) by making i t constant along the normal d i r e c t i o n s determined by A(r)(N). The following diagram should c l a r i f y t h i s argument. A(r) CN) --Extend along the *J normals A(r')(N) It 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 required property. This completes the proof of lemma 3 . 5 . We resume the v e r i f i c a t i o n of [ 1 . 2 ] . F i r s t , dg (s^) = 0 i • implies that r . l d h ( A ( r . ) J s . ) ) o P = 0 i * I Let t. = A(r).(s.) . Then d M t . M t . - t.) = dh(t.) r ( P ^ ) ( t . " t.) + dh(t.) (I - P ^ C t . - t.)} Now {t ±} i s bounded i n 1^(5), and d i s t a n c e ^ , L P ( A ( r ) ( E r ) ) f e — > 0 Therefore, there exists a sequence {u.̂ } i n L P (A(r) (E.J )^ such that I|u. - t . I I — > 0 . Consider the di f f e r e n c e l i _ _ i i p Lk | | ( I - P ^ ) ( t . - t.) - ( I - P u M u . - u . ) | | X 1 Li. k < Md-P^^-u.) - ( t . - u . ) ) | | r . x r + IKpt; - < ) ^ i - V " T P + l | ( p t . - C ^ i - V - N p He 1 1 He Now [4 , Theorem 1 9 . 1 4 , p.112] combined with lemma 3 . 5 above implies that the - 53 - i. • above terms converge to zero. By [4, Theorem 19.15, p.113]. 1 V f o r a subsequence of {u^} which we assume i s {u^} , and i t follows that (I - P ^ M t . - t.) || — > 0 . t. i J 1 ' P 1 L k It follows that the diffe r e n c e d h ( t . ) ( t . - t.) - dh(t.) x x 2 i tends to zero. Now t. - t. = t. . + (b. - b.) for t. . e Lf Therefore dh(t.) (P i ) ( t . - t.) , dh(t.) + dh(t.) , (P*)(b. - b.), t v 1 3 J dh(t.) l ( P t J ) ( b i - Vj Since | \b± - b.|| — > 0 we get f i n a l l y that | d h ( t i ) ( t i - t . ) | — > 0 L k By [3.2] we can conclude that {t.} i s a Cauchy sequence. Therefore {s^} i s a Cauchy sequence, and Condition [1.2] i s v e r i f i e d . We complete the v e r i f i c a t i o n s i n the order [1.4], [1.3], [1.5], and [1.6] - 5 4 - V e r i f i c a t i o n of [1.4] : Let T C L^CE) be bounded. Then T i s i n t r i n s i c a l l y bounded and by Uhlenbeck [8], T i s contained i n a f i n i t e number of vector bundle neighbourhoods ^ \ ^ ± ) • Suppose that t E • Then £ Lf(n.) where n. i s the vector bundle neighbourhood i n E obtained by composing the map ^ — > E with the map ^ ( r ) : E — > E . This induces a map L P ( S . ) —> L P(n.) and by lemma 3.1 there e x i s t s a K. X . KL X neighbourhood V of r such that ||v(r)^t - n r ' ) * t | | < e L k ( V for a l l r' E V and t e T n l £ ( 5 . ) • Since || || on E i s an K. X admissable F i n s l e r structure (Uhlenbeck [8]) the r e s u l t follows. V e r i f i c a t i o n of [1.3] : Fix -r E B and R e IR . Then g t ( r ' ) <_R implies that. h(A(r' )A°<f>r, (t)) <_R . By the remarks before the v e r i f i c a t i o n of [1.1] together with lemma 3.2 there e x i s t s a neighbourhood V., of r such that the set | (t) : hOKr'.^o^. (t)) < R i s bounded i n l^CO • By the argument used i n the v e r i f i c a t i o n of [1.4], we can f i n d a neighbourhood V C V 1 such that [1.3] holds. - 55 - V e r i f i c a t i o n of [1.5] : We have | f ( r , s) - f ( r T , s ) | = | h ( A ( r ) A ( s ) ) - h(A(r«)*(s))| Now combining [3.3] with lemma 3.1 and the mean value theorem the r e s u l t follows. V e r i f i c a t i o n of [1.6] : This follows from [3.3] combined with the proof of lemma 3.4. This completes the v e r i f i c a t i o n s , and hence Theorem III i s proved. - 56 - •'! CHAPTER 4 EXAMPLES 4.1 Perturbation of Minimal Surfaces In t h i s example we have a fi x e d domain and varying boundary 3 conditions. The functions are vector (E ) valued. 2 oo Let M C (R be a compact C two dimensional submanifold of 2 IR , and B be a l o c a l l y compact t o p o l o g i c a l space. Let F e FB[J1(IR^), R l be given by i » 3 2 3 Let the map B — > L^(R^) given by r I—> b^ be continuous, and l e t <j>r : I ^ ( K ^ ) 0 — > L l ^ \ b e given by (J»r(s) = s + b r . Then i t i s e a s i l y r seen that the standard problem determined by L^, <J>r, b^ s a t i s f i e s [1.1] - [1.6]. Now i f the set F of Chapter 1 i s the set of singletons 2 3 i n L]_("*jpo * ^ t f ° l - i O W S t n a t for each r we are considering minimum values of the function g & r I f b r i s a smooth section, so that i t s p r i n c i p a l part c a r r i e s 3 2 3 3M to a smooth curve i n IR then a section belonging to L^((R^)b r 3 defines a (generalized) surface i n IR whose boundary i s T . I t i s - 57 - w e l l known, i n t h i s case, that a section which minimizes our D i r i c h l e t i n t e g r a l g__ corresponds to a surface, of minimum area spanning . Moreover, the value of g__ agrees with the surface area j u s t i n t h i s case; . Hence Theorem I applies to give conclusions : (a) There i s at l e a s t one (generalized) minimal surface for each r , (b) the minimum surface area v a r i e s continuously with r , and (c) i f for each r i n a neighbourhood V C B the minimizing 2 3 section s__ i s unique, then the map V — > L^(IR^) given by r I—> s^ i s continuous. 4.2 Perturbation of the Operator Let M be a smooth submanifold of (Rn with boundary 8M and 2 Lebesgue measure . u . Let B = [0, °°). Let a : M x (R x B — > IR be C i n i t s second and t h i r d arguments and be denoted by (x 1, u, r) I > a ( x X , u, r) Let e F B t J 1 ^ ) . , IR^ be given by F r ( x \ u, ii ) = Y l 2 I (u ) + a(x, u, r) i i 2 Let L r e Lgn^OR^) be represented by F__ . Let B —> f ^ O ^ given by 2 2 r I—> b r be continuous. Let <|>r : L 1( | R M)o — > L l^ I RM^ br b e S i v e n b v - 58 - s + b^ . Assume that a s a t i s f i e s the following conditions (i ) | a ( x , u, r) I < C(r) 1 + u ( i i ) |a(x, u, r) - a(x, u, r ' ) | <_ (^(r - r') 1 + u ( i i i ) 3a 3u (x, u, r) 3 a < C(r) ( l + |u|' (iv) 0 < • • 2 - f (x, u, r) < C(r) 3u Z where C and are continuous functions of r , with C^(0) = 0 It i s then e a s i l y v e r i f i e d that the standard problem determined by ' L , b r , <f>r s a t i s f i e s conditions [1.1] - [1.6]. The Euler-Lagrange operator associated with L i s r 1 3a , N ~ L u + r r r , x., u . x.x. 2 3u l l i i Of course i f a(r, x̂ ,̂ u) = y ( r ) u for y : [0, °°) — > IR continuous we have the parametrized l i n e a r Euler Lagrange equation " I \ x + Y(r)u = 0 . i i In any case Theorem I applies again as i n 4.1. - 59 - 4.3 Domain Perturbations We s h a l l employ the notation of Chapter 2 . F i x ft° C IRn and l e t F e FB[ J X((R ) , IR ] by given by Q° ft° i v 2 F(x , u, u ) = _ (u ) + a(x , u) x. where a : ft° x (R — > (R i s C 2 and s a t i s f i e s the conditions ( i ) |a(x ±_ u)| < C 1 + |u| ( i i ) \a(x±, u) - a(x[, u)| < C 1 ( x ± - x^) ( i i i ) ( X i, u)| < C[ 1 + |u| 1 + |u (iv) 3u~ ( x i ' f - (x|, u) <_ C 1 + |u| > a 2 (v) 0 < (x., u) < C 3u where C i s a constant and C^ a continuous function on IR with C^O) = 0 . Let L e Lgn_ (IR ) be represented by F . Then i t i s c l e a r that 1 OP F s a t i s f i e s [2.1] - [2.6] of Chapter 2- with p = 2 . Let L(IRn) be the set of l i n e a r isomorphisms of (Rn over IR . Then for each r e L((Rn) the r e s t r i c t i o n of r to the closure of a smooth subdomain Q, C f2° i s a diffeomorphism of Q i n t o IRn . If ft i s s t r i c t l y contained i n ft° , there e x i s t s a neighbourhood V of the i d e n t i t y i n L((R ) such that - 60 - r(ft) C fi° for r e V . Let B = V, M = fi and A(r) = r | - . I t i s clear that X i s continuous from B into Dif f eo(fi, IRn) . Let r l — > t>r be a continuous map of B — > L^CR^) . I t i s e a s i l y shown that the above problem s a t i s f i e s the conditions of Chapter 2 and that Theorem II applies to i t . 4.4 Perturbation of Geodesies This i s an example of the type of problem treated i n Chapter 3. £ Let M = [0, 1]. Let W C IR be a closed q dimensional subma- o 1 £ n i f o l d of (R . Let F e FB[J (IR^) , (R^ be given by F(x, u j , uh = I (uh2 . 3 I t i s e a s i l y v e r i f i e d that F s a t i s f i e s the conditions of Chapter 4.. The c r i t i c a l points of the map constructed with F correspond to geodesies on W i n the Riemannian structure induced on W by the i n c l u s i o n of W into IRq . Via the map A : B — > DiffeoflR^) defined i n Chapter 4 we induce a continuous change i n the Riemannian structure of W . The map $ : B -—> DiffeoOR*') varies the endpoints of the geodesies. 2 F i x a path component of ^ ( W ^ b . By Pal a i s [4, Thm. 13.14, p.54], t h i s i s the same as picking a homotopy class of continuous maps M -—> W, - 61 - which we denote by H . F i x r e B and assume that for each r i n some neighbourhood V C B, the minimizing geodesic assured by Theorem 111(b) i s unique, say . Then we have shown that v r varies continuously with r , where v a r i a t i o n s of r i n V correspond to v a r i a t i o n s of the Riemannian structure on W and of the end points of the geodesies, corresponding to b r(0) and b ^ ( l ) . It i s possible to change the above example to the case where M = S"̂  . In t h i s case we must assume that W i s compact. We can also increase the dimension of M and l e t F represent "powers" of the Laplace- Betrami operator on W . For d e t a i l s see P a l a i s [4, p.127]. - 62 - REFERENCES ' 1. F.E. Browder, "Existence theorems for nonlinear p a r t i a l d i f f e r e n t i a l equations," Proc : Symp Pure Math 16, A.M.S., Providence, R.I., 1970, 1-60. 2. _ _ , "Functional Analysis and Related F i e l d s " , F.E. Browder ed., Springer, New York, Heidelberg, B e r l i n , 1970, 1-58. 3. ,- " I n f i n i t e dimensional manifolds and non-linear e l l i p t i c eigenvalue problems", Annals of Math. 82 (1965), 459-477. 4. R.S. P a l a i s , "Foundations of Global Nonlinear A n a l y s i s " , Benjamin, New York, 1968. 5. , "Lusternik-Schnirelman theory on Banach manifolds",. Topology 5 (1966) 115-132. 6. , " C r i t i c a l point theory and the minimax p r i n c i p l e " , Proc : Symp Pure Math 15, A.M.S. Providence, R.I., 1970, 185-212. 7. R.C. R i d d e l l , "Nonlinear eigenvalue problems and sp h e r i c a l f i b r a t i o n s of Banach spaces", J. Functional A n a l y s i s , to appear. 8. K. Uhlenbeck, "Bounded sets and F i n s l e r structures f o r manifolds of maps", J. D i f f Geom. 7 (1972), 588-595.

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