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Perturbation of nonlinear Dirichlet problems Fournier, David Anthony 1975-02-05

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PERTURBATION OF NONLINEAR DIRICHLET PROBLEMS by DAVID ANTHONY FOURNIER B.Sc, , University of British Columbia, 1968 M.Sc, University of British Columbia, 1972 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in the Department of MATHEMATICS We accept this 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. Riddell - ii -ABSTRACT The solutions of weakly-formulated non-linear Dirichlet problems are studied when the data of the problem are perturbed in various ways. The data which undergo perturbations include the Lagrangian, the boundary condition, the basic domain, and the constraints, if present. The main conclusion states that the solution of the Dirichlet problem which minimizes the Dirichlet integral varies continuously with the data so long as it is unique. Detailed hypotheses are formulated to insure the validity of this conclusion for several large classes of problem. The hypotheses are not much stronger than the standard sufficient conditions for existence, in the generalized Lusternik-Schnirelman theory of these problems. - iii -ACKNOWLEDGEMENTS I would like to thank Ron Riddell for suggesting the problem treated here and for his Herculean efforts to make me unscramble my confused presentation of its solution. I would also like to thank the University of British Columbia and the National Research Council of Canada for their financial 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 Verification of Conditions [1.1] - [1.6] 32 CHAPTER 3 NONLINEAR DIRICHLET PROBLEMS WITH VARIABLE H0L0N0MIC CONSTRAINTS 3.1 Formulation of Theorem III 3.2 Construction of the Associated Standard Problem 3.3 Verification 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 this thesis is to study the behaviour of the solutions to nonlinear differential boundary value problems of Dirichlet type, when the data defining the problem are subjected to various perturbations. The basic result we shall obtain states that, under suitable restrictions, the solution of such a problem changes continuously with the data as long as the solution is unique. The conditions under which this conclusion is. valid are essentially that the usual sufficient conditions for existence in the variational theory of Dirichlet problems should hold uniformly in some sense as the given problem is perturbed. The differential equations which appear in the boundary value problems considered here are the Euler-Lagrange equations of multiple-integral 'Dirichlet' functionals defined on suitable Sobolev spaces. We consider such problems in their weak formulation, in which a solution is taken to be a distribution which satisfies the given boundary condition in an appropriate generalized sense, and which is a critical point of the Dirichlet integral restricted to such distributions. The nonlinearity of the Euler-Lagrange equation arises from the fact that the integrand defining the Dirichlet functional need not be quadratic in its arguments, but need only have a certain convexity in its dependence on the functions on which the functional is defined. For such problems, there are well known 'regularity' theorems asserting when a weak solution is in fact a smooth function and hence is, by a standard integration by parts argument, - 2 -a classical solution of the Euler-Lagrange equation. We shall not be concerned with this question, and we work entirely with weak solutions. The data defining such a problem appear to be of four kinds : (i) The 'Lagrangian', or integrand, of the Dirichlet functional; (ii) the 'Dirichlet data', or boundary conditions; (iii) the 'domain', i.e. the set over which the independent variables in the differential equation are allowed to run; (iv) the constraints, i.e. the set in which the dependent variables are required to lie. In principle we could formulate a global problem on sections of a subbundle of a smooth fibre bundle where the target subfibre represents (iv) and the base manifold represents (iii); and we ought to vary (iii) by pulling back along various embeddings into the base, vary (iv) by allowing the subbundle to vary, and vary (i) and (ii) at will, all simultaneously. This amount of generality presents technical obstacles which obscure the main phenomenon, and, in addition, treating particular cases of the above allows us to dispense with some of the assumptions in each case which are needed in the general case. Thus we shall take the following less general approach. In Chapter 1, we formulate and prove a theorem in the setting of a fixed fibre bundle over a fixed base manifold, where data (i) and (ii) are allowed to vary. In Chapter 2 we suppress (iv) by considering sections - 3 -qf a trivial vector bundle, and we hold (i) fixed, but we allow (ii) and (iii) to vary. In Chapter 3 we fix (i) and (iii) and allow (ii) and (iv) to vary. A simple example of the situation in Chapter 1 is the problem of minimal surfaces in ordinary Euclidean space, under variation of the boundary curve. The situation of Chapter 2 is illustrated by the problem of domain-perturbations for nonlinear elliptic boundary problems on domains of Rn . The situation of Chapter 3 is illustrated in the study of geodesies, or more general harmonic mapsfin imbedded submanifolds of IR We conclude our discussion by spelling out these examples in a little more detail in Chapter 4 . It should be noted that the significance of the uniqueness assumption in our main result is illustrated in all these cases, by well-known phenomena of jumping^.of the minimizing solution when unique ness breaks down. - 4 -CHAPTER 1 PARAMETRIZED NONLINEAR DIRICHLET PROBLEMS IN STANDARD FORM 1.1 Notation Generally we shall follow the notation of Palais [4]. For the reader's convenience we supply the following brief list : M compact C manifold of dimension n, possibly with nonempty boundary 9M ; u strictly positive smooth measure on M ; 00 E (total space of) C fibre bundle over M ; CO £ (total space of) C vector bundle over M ; (Pv^ product bundle M x IR ; k integer _ 0 ; k J (E) bundle of k-jets of sections of E ; C°°(E) set of all C°° -sections of E ; S(E) set of all sections of E ; p real number _ 1 ; P 00 L£(£) Banach space completion of ?C.(^) in the Sobolev p'th power norm on derivatives of order <_ k ; - 5 -I?(E) (for pk > n) Banach manifold of.all sections of E, each of which belongs to L^(£) for some open vector bundle neighbour hood £ in E ; b,s elements of L£(E) ; P 00 P L, (E), closed C submanifold of Lf(E) consisting of the closure in k b K L^(E) of the set of all sections s e L£(E) which agree with b in some neighbourhood (depending on the particular s) of 8M; [| || Finsler structure on the tangent bundle T(L£(E)); also,by abuse,induced structure on T*(L^(E)) ; 6 Finsler metric on L£(E) induced by || || ; k-jet extension map L£(E) —y LQ(J^(E)) > FB(E,E') set of all fibre-preserving 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 all k order Lagrangians on E, i.e. maps CO L : C (E) —> S(iO of the form L = F*°Jk for some F e FB(Jk(E) ,0^); F represents L . It should be noted that the norm in ^(5) depends on the volume element dy on M , but, by the.meaning of 'strictly positive smooth measure', all choices of u are equivalent to Lebesgue measure in all charts of M, so all induce equivalent norms in L^(C), and equivalent Finsler structures on L^(E) . - 6 -The Finsler metric is defined, technically, only on the path components of l£(E). We set 6(s,s') = 2 whenever , s, s' belong to distinct path components. The reader should also note that no.a priori smoothness assump tions are made on the maps F, hence on the Lagrangians L . Such assumptions will 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 fixed throughout the chapter : [la] A choice of M, y, and E ; [13] Choices of p > 1 and k _ 1 (pk > n if E is not a vector bundle); and a choice of || || 6 on L.P(E); on T(L£(E)), with induced [1Y] A locally 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 ri—> L of B —> Lgti, (E), such that each L extends to a C1 map l£(E) —> LjflE^) . We think of B in [ly] as a space of parameters r . Then [16] specifies an r-dependent family of Dirichlet-type boundary values on sections of E, and a C° global trivialization of the map ir : IE —> B, (r,s) H—> r, with the extra property that $ ^ is smooth on each fibre {r} x (E^ . We shall sometimes deviate from strict consistency, and identify the subset IE of L?(E) with the fibre {r} x IE , which is a .-. r k r subset of E C B x l£(E). Item [le] specifies an r-dependent family of Lagrangians L^, such that the integral 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 partial-function = f(r,.) is C1 on L£(E) for each r . The restriction g = f|E then has a C1 partial-function g = fr|E for each r ; and the critical locus of gr, r namely Kr = { S e Er : d^s) = ° is a well defined closed subset of -. By the standard parametrized  Dirichlet problem with data [la] - [le], we mean the problem, to describe - 8 -the (disjoint) union K = | (r,s) : r e B, s £ Kr j of the critical loci of all the gr» as a subset of IE a B x L^(E) . Of course the problem as posed is too general, and in the next section we shall put conditions on the data, in particular on their behaviour with respect to the parameter r in B, which will enable us to prove conclusions of a similar kind about the sets K . However, an f r important restriction has already been built in, by the provision in [16] that the disjoint union IE of the spaces E of the r-dependent varia tional problems should come to us equipped with the structure of a fibre space over B, trivialized by $ ^ . We are thus able to form a 'partial-function' g^ on B, for each t in the model fibre IF , namely by gt(r) = (go$)(r,t) = gr(()>r(t)) . Certain key technical considerations in what follows will be organized around these functions g^ . 1.3 Formulation of Theorem I Given a standard parametrized Dirichlet problem as defined in §1.2, we shall study the subset of each critical locus Kr on which the function g takes its minimum value, r To be precise, we begin with a family F of subsets of IF = L?(E), which is deformation invariant. This means invariant under K b ambient homotopy of IF , i.e., for any continuous map H : [0,1] x (F —> IF such that H(0,O is the identity on IF, and for any T e F, the set H^T) = | H(l,t) : t e T is also a member of the family F (cf. Browder [2 , Def(l.l)] ). Given F, we.define a corresponding family F(r) of subsets of each Er» by F(r) = j. S C Er : S = <|>r(T) for some T e F j Then the F-minimax of g is defined to be the function m^ on B, where for any r e B, mF(r) = inf sup g (s) ; r SeF(r) seS and the m^-realizing subset of K over r is defined to be Kp(r) = |(r,s) : s e Kr and gr(s) = nip(r) j- . More generally, for any subset C Cl B, the ny-realizing subset of K over C is the disjoint union KF(C) = U K (r) . reC - 10 -Our aim is to give conditions on the data under which m^ will be continuous on B, Kp(r) will be nonempty for each r, and Kp(C) will be compact in IE for any compact subset C of B . The existence of a point in Kp(r) is the main assertion of the generalized Lusternik-Schnirelman theory developed by Palais [5., 6], and Browder [1, 2, 3], where incidentally several examples of families F are given. In the following list of conditions on the data [la] - [le], the first states the standard hypotheses for existence in the Lusternik-Schnirelman theory, and the others mainly require that the existence hypotheses hold uniformly, in various senses, with respect to the parameter r . Note that some conditions refer to the functions g^, and some to the unrestricted f or f r [1.1] For each r e B, the function gr : E^ —-> IR is bounded below and satisfies condition (C) of Palais-Smale (i.e., any sequence Er for which §r(s_) xS bounded and ||dgr'(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 gr(s) <_R is compact in IE . [1.3] For each r e B and each R e IR, there exists a neighbourhood V of r in B such that the subset - 11 -IFR v = jt E (F : gt(r') '<_ R for some r* e V j is bounded in IF . [1.4] For each r e B and each bounded subset T of (F, there exists a neighbourhood V of r in B such that 6(*r(t), *r,(t)) < e for all r' E V and all t e T . [1.5] For each r e B, each bounded subset S of L^(E), and each e > 0, there exists a neighbourhood N of r in B such that |f(r,s) - f(r',s)| < e for all r' e N and all s E S . [1.6] For each r E B and each bounded subset S of L^(E), there exists A E IR such that ||dfr(s)|| <_ A for all s E S . With these preparations done, we can state our result : Theorem I Suppose a standard parametrized Dirichlet problem is given by data [la] - [Is] satisfying conditions [1.1] - [1.6]. Let F be a - 12 -deformation-invariant family of subsets of a single path component of IE, such that at least one element of F is compact and nonvoid. Then : (a) The F-minimax function m^ . of g is finite and continuous on B ; (b) For each r e B, the ny-realizing subset Kp(r) of K over r is not empty; (c) For each compact subset C of B, the mp-realizing subset Kp(C) of K over C is compact. REMARK. In case Kp(r) is a singleton {sp(r)} for each r in some open set N in B, so that Sp is a section of ir : IE —> B over N, it follows easily from (c) that Sp is continuous. Thus the critical point of which realizes a particular minimax value mp(r) varies continuously with r so long as it is unique. In the examples which we have in mind, this is the conclusion of principal interest. 1.4 Proof of Theorem I We first prove conclusion (b), by assembling several results in Browder [2]. For fixed r e B, let X denote the connected component of IE which contains, all the sets in the family F(r), and let h denote the restriction of g to X . Then X is a connected submanifold of the r CO C Finsler manifold (E , and is complete in the induced metric. Hence [2, Proposition 5.2, p.32] there exists a quasigradient field for h on X . The function h is bounded below and satisfies condition (C) on - 13 -X , on account of condition [l.l],and so by [2, Proposition(5.1), p.27], and the remarks following [2, Theorem 1, p.8 and Definition(2.1), p.18], all the hypotheses of [2, Theorem 1] are satisfied, except for the finite-ness of mp(r). But this follows from the existence of a compact nonvoid element in F , and Browder's Theorem 1 applies to establish (b). To prove (a) and (c), we apply [7, Theorem 1.2] with our quantities (E, (F, TT, g, and K in place of the "quantities E, F, p, f, D of that theorem. It requires that F be invariant under homeomor-phisms of (F, but this is used in the proof only to insure that the families F(r) are independent of the choice of trivialization x of p. Since we have defined our F(r) by a fixed trivialization $ "*" of TT , we do not need the homeomorphism-invariance of F to apply the cited result. Its conclusions are precisely the conclusions (a) and (c) of Theorem I; and one of its two hypotheses is just condition [1.2]. The other hypothesis, in our notation, becomes the following : for each r E B and each R e IR, there exists a neighbourhood V of r in B such that the family of functions Thus the proof of Theorem I will be complete as soon as we have shown how to choose V so that this equicontinuity assertion holds. Here we shall is equlcontinuous at r, where* as in [1.3] above, - 14 -use conditions [1.3] - [1.6]. Fix r E B and R E (R. By [1.3], there is a neighbourhood V of r in B such that T = IF is bounded in (F . By assumption in K,v [1<$], 4>r(T) is bounded in Er . By [1.4] with E = 1, there is a neighbourhood N1 of r such that <5(<j>r(t), <j>r,(t)) < 1 for all r' el and all t E T. Hence the set S1 = | <j>r, (t) : r' E N1, t E T } is bounded in L^(E) along with <J> (T) ; and so also is s 2 = | s E L^(E) : 6(s,s') < 1 for some s' £ S1 j Given E > 0, we apply [1.5] and [1.6] with S = S2 to find a neighbourhood N2 d of r such that f(r,s) - f(r'.,s)| < | (r' E N2, s E S2), and a number A > 1 such that dfr(s) <_ A (s £ S2) The we apply [1.4] to find a neighbourhood O N2 of r such that '• . 6(+r(t), *r,(t)) < min | 1, |£ | for r' E N3, t E T . 15 -Now choose any t e T = (FR v arid any r' e . In view of the. convention adopted in §1.1, that points s, s' in different path compo nents of k£(E) have 6(s,s') = 2, the last inequality implies, in particular, that the points s = <)>r(t) and s' = 4>rt(t) can be joined by a C1 path y in l£(E) with length y < min \ 1, 2A Any point on such a path can be no further than 1 from either endpoint, so lies in • Accordingly we can apply the first, second and last of the preceeding inequalities, to get |gt(r') - gt(r)| = |gr,(*t.(t)) - gr(4>r(t))| - |fr.(s*) - fr(s')l + |fr(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 is complete . CHAPTER 2 DIRICHLET PROBLEMS WITH VARIABLE DOMAINS 2.1 Formulation of Theorem II In this chapter we shall consider nonlinear Dirichlet problems in 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 interest is in the behaviour of the solution when both ft and a are perturbed. The restriction to real valued functions is for convenience of notation only, and the reader will easily see how to modify our state ments to cover the case of vector-valued functions. We start with a bounded open set ft° O IRn, with typical point x = (x^,...,xn) and with Lebesgue measure dco(x) = dx^...dx^ • An n-multi-index is an n-tuple a = (a^,...,an) of integers ou _ 0, and we will write |a| for the sum \ ou , and Da for the partial derivative i a. operator JYO/Sx.) . For any integer k _ 0, s, will denote the total number of a's having let < k, and u = (u ) i i , will denote a typical ° 1 ' — ' a |a|<k sk sk point in IR . Note that IR is just the fibre of the k-jet bundle jk(IR ) of the product bundle IR = ft° x IR over ft° , whose sections we Ho identify with the real valued functions on ft° . More generally, for any a o i product bundle IR over ft with fibre IR we have a natural - ft° - 17 P a p o & identification of Lf(fR ) with LMfi , IR ) and we shall use this k n° k identification freely in the following. s. With n ^ 1, k >_ 1, and fixed, let F : Q° x IR —> IR 2 be a C function. (Thus F is the principal part of an element of k FB[J (IR ), IR ].) Suppose F satisfies the following conditions for a certain p >^ 2, certain constants C, > 0, and a certain continuous function <E : RU —> R with <C(0) = 0 : k [2.1] For all x e 0° and all u e IR , F(x,u) | < C 1 + • I |uJ1 a <k a, [2.2] For all x, x' e and all u e IR , F(x,u) -F(x',u)| <_ (C(x - x') i+ i iu r |a|£k a o k [2.3] For all a, all xe.fi, and all u E IR , 9F 8u (x,u) < C |e|<k p o k [2.4] For all a, all x, x' e fi , and all u e (R , 3u~(x'u) ~ 9u~(x 'u) a a < (E(x - x') 1 + , J l$l<k o k. [2.5] For all a, g, all x e-fi°, and all u e.'R , - 18 -82F 8u aii- (x,u) < C 1 + I |u |P"2 [2.6] o k For all x e fi and all u, v e IR , |a|,|B|<k a 3 P |a|=k |P-2|„ |2 Conditions [2.1], [2.3], and [2.5] in particular imply that the integral hu(v) - F(x, j.(v)(x)) dco(x) ,o k is well defined for all v in the Sobolev space L^(fi°, IR), and that h° 2 is in fact a C function (cf. Lemma 2.1 in §2.2 below)'. Here the symbol J^Cv) i-s being abused to denote the principal part of the k-jet of the section of L^((R ) whose principal part is v . We shall consider k fi° 'restrictions' h = h^ obtained by replacing fi° in the above integral by various subdomains fi C fi° . To be precise, for all r in a suitable parameter space B, we consider an open subdomain fi^ CT fi° whose closure is obtained as the diffeomorphic image A(r)(M) of a fixed smooth manifold M . As r ranges over B, A(r) is supposed to change continuously in the space Diffeo[M,fi°] of Z9 diffeomorphisms of M into fi°, equipped with the topology of uniform convergence of all derivatives. We also suppose that a suitable boundary-function a^_ is given for each r. Our result is the following. - 19 -Theorem II Fix a bounded open subset ft° d (Rn, an integer k _ 1, and a o Sk 2 real number p _ 2. Let F : ft x (R —> IR be C and satisfy conditions [2.1] - [2.6] given above. Fix a locally compact Hausdorff space B and a compact 00 n-dimensional C manifold M with boundary 3M . Let r I—> a^ be continuous from B into L^(ft°,(R), and let ri—> 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 let h be the C real valued function on D defined by r r h (v) = r F(x, jk(v)(x)) da)(x) ft.. Then the following, conclusions hold : (a) The function m : B —> IR defined by m(r) = inf hr(v) veD r is finite and continuous on B ; (b) For each r e B, the set M(r) = e D : dhr(v) = 0 and hr(v) = m(r) is 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) | is a compact subset of B x LJ^IR^) . We shall deduce Theorem II from Theorem I of §1.2, by construc ting a suitable standard parametrized problem. It should be noted that our . conditions [2.1] - [2.6] can be weakened considerably by exploiting the Sobolev inequalities (cf. Browder [1, pp 25-29]). The techniques for doing this are well established and we shall not dwell further on the point. Before carrying out the reduction to Theorem I, we shall establish some properties of the functions h^ . 2.2 Preliminary consequences of the assumptions In the following we use the assumption that F satisfies [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(FA) : LP(Jk(VJ-> L(LP(Jk(V, Lj(V) and d2(F,). : LP(Jk(V) -> L2(LP(Jk(Rfi)), LJ(^,)) take bounded sets into bounded sets. (The reader is 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 v2 e LQ(J > and d(FA)(v)(v1) = 6Fv(V;L) d2(FA)(v)(vlsv2) = 62Fv(v19v2) . Proof : See Browder [1] For ft C Q° let H° : L^(Jk(IRfi)) —> IR be defined by Hfl(v) - FAv(x) du>(x) Let hn : Lj(Rfl) —> R be defined by h^ = Hfi°Jk Lemma 2.l(i) For each fi <rT , Hfi : L*J(Jk(lRfi)) —> IR is a C2 function such that (a) dH : L^(Jk(IRn)) —> LP(Jk(R ))* takes bounded sets into - 22 -bounded sets ; P k (b) dH is uniformly continuous on bounded sets of LQ(J " Q 2 Proof : The fact that H is C follows immediately from lemma 2.0. 2 Part (b) follows from the boundedness of d (F^) in lemma 2.0 together with the mean value theorem. Lemma 2.1(ii) o h : 'I_(Rfl) —> R is a c 2 function such For each 9, C 0, that (a) dh" : LP(Rfi) -> L!X>* and d2h": Lk<V •> L2(L£(Rn),IR) takes bounded sets into bounded sets (b) dh is uniformly continuous on bounded subsets of L^(IR^). Proof Since h^ = H^oj and jfe : .^(IR^) —> LP(Jk(IRn)) is a bounded linear map the proof follows immediately from lemma 2.1(i). Lemma 2.2 [dh"(vx) - dh"(v2)](Vl -> C Y DV(x) - D°v„(x) P du(x) i i, 1 £ for some constant C - 23 -Proof : From lemma 2.0 it follows that dh (v3)(vlsv2) 32F n'l lXl«l v-^(X'Jk(v3)(x))Ua(VV«)UB^k(v2>W) ft 0<_\ a I , I 8 | <k a 6 So [dhV-^ - dh^Cv^]^ - v2) ft J0 2 3 F 0<J a | , I B | <k a 6 ua ( j k (vrv2 > (x)H( j k(vrvz) (x)) dt dco(x) and by [2.6] f I f1 |D\(X) + t(D°'(v2-v1)(x))|P 2|Da(v2-v1)(x)|2 dt du(x) ft I a I =k •'0 | a | =k D v1(x) - D v2(x) K dx for some constant c Lemma 2.3 For vr v2, ar *2 E LP((R ) with V]_ E LJOR ') and Cdh"^) - dh^v^Kv^) > C !Ivl-v2~(al-a2)Il^p Lk; ara2 for some constant C independent of ft - 24 -• \'.-\ 1 . . Proof : By lemma 2.2 we need only show that there exists a constant independent of ft such that I ' |Da(v -v )(x)|P du)(x) | a | =k Jft > c <vl _ V " (al — a2y'' pj lai " a2N p  Lk Now v, - vn - (a, - a„) e LP(IR ),. . Therefore there exists a constant 12 1 2 k 0 independent of ft such that I |Da(V;L-v2)(x) - Da(ai-a2)(x)|P do>(x) _ c ||v1 - v2 - (a1 - a2)|| Now there exists a constant independent of ft such that • C. f _ f |Da(Vl-v9)(x) - Da(a1-a9)(x)|P du>(x) 1 1 |a|=k Jft . Z J I |a|=k |Da(V;L-v2)(x)|P dw(x) + j |Da(ai-a2)(x)|P dco(x) ,1 | a | =k |Da(Vl-V2) (x) |P da)(x) + ||ai - a2||Pp Therefore there exists a constant C independent of ft such that _ JD^v^vpCx)!11 da)(x) + | |a1-a21 |P > Cl lvrv2_(ara2) |a|=k 1 ft and the result follows - 25 -Lemma 2.4 Let {v^} De a bounded sequence in L^(IR^) such that v. e LP(IR ) with {a.] converging to a . Then if dh^(v.) —> 0, it x tc Si a.. x x i follows that fv.jj is a convergent sequence in 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 Lk [dh"(V;L) - dh"(v2)](V;L - v2) < 6 for some <5^, 62 to be determined. By lemma 2.3 62 - C lai-a2H P  Lk Since IvvW p - Hvrv2ll P  Lk Lk we get 6. > C |vrv2| |ara2iip iara2n P  Lk Since {v^} i-s a bounded sequence there exists 6-^ such that al " a2'' p < 61 Lk implies that - 26 -ivrv2NTp 'ara2MLpj la,-a„I |P - ClIv,-v„I|P I 1 2" p 1 2'ln k k Ce It follows that cl|v_-v2||Pp < ^|+62  Lk Setting 62 —^ we get v. - v. < e 1 i 3 T P it Lemma 2.5 the set Fix S C LMIR ) bounded and R e IR. Let h denote h . Then K o 6 v e Uo LP((R ) : h(v) < R weS k QO W is bounded in L, (IR ) . k Proof Fix v e LP(tR ) . Suppose v = t + w with t e LP(IR ) . Then k o/w VV k fio 0 h(v) = h(t + w) = h(w) + dh(w + ut) (t) du rl = h(w) + dh(w)(t) + [dh(w+ut) - dh(w)](t) du = h(w) + dh(w)(t) + By lemma 2.2 f1 1 1 ^ [dh(w+ut) - dh(w)](ut) du - 27 -> h(w) + dh(w)(t) + C ^ 1,, I|P A — ut r du u1 1 1 1 P 0 < = h(w; + dh(w)(t) + - IItl|p P II i i p Lk Since S is bounded, by lemma 2.1(ii) sup I | dh(w) ||= A < «> , weS sup | h (w) | = B < °° weS Therefore h(w +t)>-B-At • + - t LP P LP k k for all w e S, where C > 0, p > 0 . It follows immediately that h(w + t) < R with w e S implies that IItlI is bounded/ •1 1 1 1 p k Lemma 2.6 Fix S C LP((R ) bounded and R e |R . For any 0, C let k Q° T = { w|fi : w e S i.e. T consists of the "restriction" of S to fi . Then there exists 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 • Lk - 28 -Proof : Pick v e LP(IR ) . Then v = w + t for t e L^(IR0)n , and w k^fi'O is the restriction of some w, e Lp (IR ) to fi . Extend v to fi° by setting it equal to on fi° - fi . Call the extension . Then h" (vx) = h"(v) + F°j, (w, ) do) Jfi°-fi k 1 < R + | Q FcJk(W;L) do) By lemma 2.1(H) there exists a constant c e IR such that (wn )du> o Jk 1 < c for all w^ e S . Therefore h (vx) < R + c By lemma 2.5 there exists a constant A e IR with v, < A . Then it J • 1 Tp follows that VM £. A + sup I |w1 Lf w.eS k 1 1" p Lk and the result follows for B = A + sup I|wi|I w.eS LP 1 k - 29 -Lemma 2.7 For each ft , and w e LPQRfi), the restriction of h^ to LF(IR^) is bounded below, k ft w Proof : In the course of verifying lemma 2.5 we derived the inequality h(t + w) > - B - AlItlI + - II11 IP - LP P LP k k for t e LP(IR )n , and lemma 2.7 follows immediately from this inequality. 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 strictly positive smooth measure on M, and let E be the product bundle IR^. . [IB] p, k are already determined. Let || || be the standard norm on LJ^IE^), so that 6(s1,s2) = ||s1 - s 2 [ | . Lk [ly] B has been determined. For each r e B the map X(r) : M —> fir C fi° induces a map X(r)* : LP(fir,IR) —> LP(M,IR) given by A(r)*(v) = v°X(r). Recall 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—> br E X(r)*(ar) [16] Let b = the zero section of L^O^) > and tne maP r 1—> br as given above. Let $ : B x I^Orm)Q —> E be defined by $(r,s) = (r,s+br) so that <)>r(s) = s + br . OO For each r e M, let ar e C (M,IR) be defined by ar(y)X(r)*u(y) = y(y) . [le] For s e L^O^) and r e B, let Lrs(y) = ar(y)F(A(r)(y),Jk(soX(r)"1)(X(r).(y))) where we repeat that the symbol J^C') iS being abused to denote the principal 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 satisfies the hypotheses of Theorem I, we indicate how Theorem II follows. From the definition of Lf it follows that for s e L^OR^) that L s(y) dy(y) = M L°(SoA(r) 1)(v) du(x) ft r 31 -so that gr(t) = h^(toX(r) . Now in Theorem I let F = { {8.} : s e L£(Vo } Then mF(r) = inf g(t) = inf h (v) .= m(r) teE veD r r So mp(r) = m(r) and hence m(r) is finite and continuous on B . Also, M(r) = \ v e Dr : dhr(v) = 0 and hr(v) = m(r) •" = | v e Dr : dgr(voX(r) = 0 and gr(t°A(r)> = mp(r)| Therefore v e M(r) iff voA(r) e K^(r) from which it follows that M(r) is not void. Finally A*M(C) = |(r,v°A(r)) : r e C and v e M(r)j and from the above discussion it follows that A*M(C) = Kp(C) and from Theorem I it follows that A*M(C) is compact. Therefore in order to prove Theorem II we need only verify conditions [1.1] - [1.6] for the standard problem just 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 gr(t) = h r(t°X(r) ) and r by lemma 2.6 gr is bounded below. Now assume that xS a bounded sequence with ||dgr(t^)|| ->- 0 fir -III -1 It follows that ||dh (t^ACr) )|| 0 . By lemma 2.4 {t^oACr) } is a Cauchy sequence. Hence {t^} 1S a Cauchy sequence. Verification of [1.4] : Since <j>r(s) = s + b , 6(<|>r(s), cf)r, (s)) = | |s + br - s + br,| | \ = I lb - b ,1 I II r . r I I p . Lk = I f |Da(a oA(r))(y) - Da(a ,<>A(r'))(y)|P dy < I f |Da(a PA(r))(y) - Da(a oA(r'))(y)|P dy + i a >a(a °A(r'))(y) - Da(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 arbitrarily small for r' in some neighbourhood V"r of r . Verification 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} is bounded in LP(R. ). Now g (r1) < R r K. QO t — "r' -1 implies that h ( (t+b^, ) <>X(r') ) <_ R . From lemma 2.6 it follows that there exists a B £ IR such that | |(t + br,)oX(r')"1|| < B Then ItoXCr')"1! But since the sections b ,oX(r') are just the restrictions of the a ,'s to the various ft r' s it follows that sup ||b ,oX(r') r'eV r -1 and that there exists an A e (R with UtoXCrT1!! p < A for all t e Ft R,W and all r' e W . From the continuity of the map - 34 -r* 1—> X(r') it is easily seen that there exists a neighbourhood V C W of r and a constant A^ e IR such that | ItoXCr')"1!| < A and r' e V \ implies that I Itl I < A.. . From this it follows that IF,, TI is bounded. r 11 11 p 1 R,V Hi Verification of [1.5] For s e L^flR^ let sr e L^CR^ ) denote s°X(r) 1 . For ye M, let xr = 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(xr,jk(sr) (xr)) du>(xr) - F(x ,,jk(sr,).(xr,)) du(xr,) •"ft,-' Let 3 : ft^ —> be given by = X(r')°X(r) -1 Then x^, = 3(xr) and we rewrite the above as F(xr,jk(sr) (xr)) dco(xr) (A) (B) (C) F(f3(xr) ,jk(sr,) (3(xr)))d(3*0)) (xr) du(xr) [F(xr,jk(sr)(xr)) - F(xr,jk(sr,)(3(xr)))j F(xr>jk(sr,)(3(xr))) - F(3(xr),jk(sr,)(3(xr))) 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) xr •—> (xr,jk(sr,)(3(xr))) is a section in LP(Jk(lR^ )) . In fact it is just 3*(jk(sr,)) . 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, (s ,)) - s < e Jk r' r11 p L0 for all s e S and r' e Vr . Now (A) is the same as H (jk(sr)) - H r(3*(jk(sr,))) ftr And from the boundedness of dH which follows from lemma 2.1 (i) we get a bound on (A) which can be made arbitrarily small. 1 - 36 -(B) By condition [2.2] (B) is bounded by <C(x-3(xr))| 1 + I|ua(jk(sr,))(3(xr)) dco Now for any E > 0 there exists a neighbourhood W of r such that for r' c W , sup |(C(xr - 3(xr)) | < e . x efi„ r r (Recall that 3 depends on r'). Therefore (B) is bounded by 1 + I ha(3k(sr,)(3(xr))|p da) Since sr, is derived from s e S a bounded set, there exists a constant A e IR such that 1 + I IVVV)^))!1 fir v a dco < A for all s e S. It follows that (B) can be made arbitrarily small. (C) From the continuity of the map r I—> X(r) it is easily seen that the difference of the measure to - 3*w can be made uniformly small over fir by restricting r' to lie in a suitable neighbourhood of r . It follows that (C) can be made arbitrarily small. Verification of [1.6] : Let S C L^OE^) be bounded. Then the set { soX(r)"1 : s e S } - 37 -is bounded in ^ * ^ow dfr(s)(t) = dHfi'.(Jk(soA(r) 1)oJk(toX(r) From lemma 2.1(ii) it follows that sup ||dHn'(jk(soX(r)_1))|| < » ,. seS and since 1^0^) —> LP(Jk((Rfi )) given by tl—> Jk(t°A(r)_1) is bounded it follows that sup ||df (s)|| < seS Verification of [1.2] : Fix R E IR and C C B compact. Let {(r^,s^)} be a sequence with {(ri,si)} C K fl |(r',s) : r' e C and gr,(s) <R J . We need to show that {(r^,s^)} has a convergent subsequence. Since C is compact we can assume that {r__} is a convergent sequence and we assume that ^r__} converges to r E B . From conditions [1.3] and [1.4] it follows that {s^} is bounded. In order to proceed we need the following proposition whose proof will be found below. Lemma 2.8 Fix S CT Lk(R^) bounded and E > 0. Then for each r e B there - 38 -exists a neighbourhood V of r such that ||dg (s) - dg , (s)| | < e , j , r r for all r' £ V and s E S1 . From this it follows that since converges to r, {s^} is bounded, and dg (s.) = 0, that dgr(s.) converges to zero. From i ar -1 this it follows that dh (s^°A(r) ) converges to zero. Finally from lemma 2.4 we get that {s^°A(r) ^} is a convergent sequence and therefore {s^} is a convergent sequence. Proof of Lemma 2.8 : We will employ the same notation as in the verification of [1.5]. That is for s, t e L^ClE^), we let sr, tr £ 1^0^ ) denote s°A(r)_1, t°A(r) \ and for r, r' E B we let 3 : fir —> & , be given by 3 = A(r')°A(r)_1, and for xr £ Q.^ we let xr, = 3(xr) . Now dgr,(s)(t) = while dgr(s)(t) = n 5\(s ,)(x ,) W)(V> du,(V> nrt Jk r' r' 6Fj, (s )(x ) jk(tr)(xr> da)(xr> ftr Jkv x' v r' In order to compare dgr,(s)(t) and dgr(s)(t) we need to employ the coordinate system on J (R ) • In coordinates - 39 -So dgr,(s)(t) = 7 I —(x ,,j,(s ,)(x ,))oU (j. (t ,)(x ,) dw(x ,) L L 3u r' kv r' r' a Jkv r' r' r' a •'ft.,.? a = I ^-(6(xr),Jk(sr,)(6(xr)))oua(Jk(tr,)(3(xr)) d(B*co)(xr) dg (s)(t) - dg ,(s)(t) (A) (B) + 9F (x .j,(s ) (x )) n du v r'Jkv r' v r' ftr a ua(jk(tr)(xr))-ua(jk(tr,)(3(xr))) dto(xr) |f (xr,jk(sr)(xr)) - lf(xr,Jk(sr,)(B(xr))) a a x "a(Jk(tr.)(e(xr)) dco(xr) (C) + (D) + r L a J ftr*-V • I a * ftr 9F ^-(xr,jk(sr,)(3(xr))) - gJr(3(xr)fJk(8r,)(B(xr))) x ua(Jk(tri)(B(xr)) dco(xr) |f-(B(xr) ,jk(sr,) (3(xr))) -ua(jk(tr,) (3(xr)) d[co-3*co] (xr) As before we deal with each of these terms separately. Now (A) is the same as ft. dH r(jk(sr))(jk(tr) - 3*jk(tr,)) where - 40 -S*jk(tr,)(xr) = jk(tr,)(3(xr)) It is easily seen that for any 6 > 0 there exists a neighbourhood W of r such that I I Ji" Ct ) - 3*3% (t ,) I I < 6 1 1 Jkv r Jkv r" 1 1 p 0 for t , t , derived from t with I Itl I =1. From the boundedness of r r TP He • ftr dH which follows from lemma 2.1 it follows that (A) can be made arbitrarily small. (B) This term is the same as dH%;Jk(sr)) - dH%B*Jk(sr,)) **jk(tr.) . ftr From the uniform continuity of dH , the boundedness of 6*j, (t ,), K. 10 Jk(sr), and 3*jk(s'r,) and the fact that I |ik(sr) ~ 3*Jk(sr,)|| can L0 be made arbitrarily small it follows that (B) can be made arbitrarily small. (C) It is easily seen that for 6 > 0 there exists a neighborhood W of r such that |B(xr) - xr| = lACOoAO:)"1^) - xr| < 6 for all x e ft and all r' E W . From condition [2.4] it follows that r r L ' (C) is dominated by - 41 -ni J a JftT (CC3(xr)-xr) 1 +:I |ugakCsr,)C3(xr))) lP-1 uaak(tr,)(3Cxr)) 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. This 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 III In this chapter we shall consider a parametrized version of the Dirichlet problem described by Palais [4, pp 104-105, p 109]. The solution candidates are vector valued functions on a manifold M, Whose values are constrained to lie in a given submanifold W C IR , as well as to agree with those of a given function a on 8M, in case 3M is not empty. We shall study the solution when both W and a are permitted to vary. OO To be precise, we begin with a compact C manifold M, with th positive smooth measure y and possibly with boundary, and a k order Lagrangian L e Lgn^CO » where £ is the product vector bundle IR^ for some £ _ 2. We suppose that L satisfies the following conditions : [3.1] For some p with pk > n, L extends to a C.^ map : f£(£) —> LJO^) so that the integral h(v) = Lv dy M defines a real-valued function on LP(£) . [3.2] For any a±, &2 e LP(^) and any v± e a» v2 e Ll_^0a^ » [dh(V;L)-dh(v2)] (V;L-v2) > c| | (v1-v2)-(a1-a2) | |Pp - | [ (a^) | |Pp Lk Lk - 43 -for some constant c . [3.3] The map dh : LP(£;) —> L^CO* takes bounded sets to bounded sets. Next, let W be a closed C submanifold of (R with 8W = 0 . For each r in a parameter space B, the varied constraint manifold W will be obtained by acting on W with a diffeo-Z morphism A(r) of the ambient Euclidean space IR . Also, fixing a boundary-value function b on M with values in W, we obtain indepen dently - varied boundary functions a^ by first composing b with another Z diffeomorphism V(r) of IR which carries W onto W, then composing the result with A(r) . Notations like f(r)^b will be used for such a composite function, and by abuse, b may denote a section of the trivial bundle W^ and ^(r)^ the induced section. We shall require the maps A and V to be continuous from B % 00 Z into the space Diffeo(lR ) of all C diffeomorphisms of IR onto itself, with the topology of uniform convergence of each derivative on each compact set. Our result is the following. Theorem III CO Let M be a compact C manifold of dimension n, possibly with Z boundary, and with a strictly positive smooth measure u . With £ = IR^. , Z >_ 2, let L e Lgn^C?) satisfy 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 if 3M = 0). Let E =.W and b e LP(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 is a closed C submanifold of IR def Set W(r) = A(r)(W). Then E(r) = W(r) is a C°° subbundle of IR^ . Let ar = A(r)^(r)Ab , and D = L£(E(r)) . Let hr 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 is finite and continuous on B (b) For each r e B, the set - 45 -'M(r) = • j(r, v) : v'e Dr, dhr(v) = 0, and hr(v) = m(r)| is 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)| is a compact subset of B x LP(E) Theorem III will be„deduced from Theorem I of §1.3 by constructing a suitable standard problem. 3.2 Construction of the Standard Problem [la] Let M, u as given in §3.1, and E = W^ . [lg] p, k as determined in §3.1. Since W is a submanifold of (R , the inclusion W —> IR induces an inclusion ^(^) —> L£(IR^), We give LJ^W^) the induced Finsler structure || ||, and the corresponding Finsler metric. [ly] B as given in §3.1 . [16] b as given in §3.1 with b^ = Y(r)*b and <j> equal to the restriction of Viv)* to L^CE^ . [le] Lr = 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 is 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 ff , and and F(r) = -JS C L£(E)fe : S = ¥(r)*(T), some T e ¥} , r gr(s) = hr(A(r)A(s)) it 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, dhr(v) = 0 and hr(v) = m(r) = j(r, A(r)As) : s e ijJCE^ , dgr(s) = 0 and gr(s) = mF(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) : reC.se l£(E)b , dgr(s) = 0 V r and gr(s) = rtip(r) which is comapct when C is compact by (c) of Theorem I . Before proceeding with the verifications we prove a few lemmas about the map r l—> A(r)A . A standard assumption will 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 exists a neigh-bourhood V of r such that ||A(r')As - A(r)*s|| < e Lk for each r' e V and s e S . Proof : The proof follows easily from Palais [4, Lemma 9.9, p.31]. Lemma 3.2 If S C LP(£) is bounded then A(r).(S) is bounded. K. Proof : . This also follows from the above cited lemma 9.9. - 48 -Lemma 3.3 Fix r e B and S C L^CO bounded. Then there exists a neighbourhood V of r such that U A(r').(S) is bounded in 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 inf ||d(A(r)^)(s)|| > 0 . seS Proof : By lemma 3.2, A(r)(S) is bounded. Now ||d(A(r)^)(s)[| = ||d(A(r);1)(A(r)^)(s)|I"1 . Therefore we need only prove that for S C ^(5) bounded, and r e B that sup | |d(A(r)A) (s) | | < °° . Now for s E S and t E , SES d(A(r)J(s)(t)(x) = [6A(r)°s(x)](t(x)) , by Palais [4, Theorem 11.3, p.41]. Again by [4, Lemma 9.9, p.31], it follows that for [|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)|| < SES - 49 -3.3 Verification of conditions [1.1] - [1.6] It follows by the same techniques as employed in Chapter 2 that condition [3.2] implies that h : L^CO —> IR is bounded below and satisfies condition (C), and that for S C Lk(5) bounded and R -e IR , the set | s e LP(5)a : a e S and h(s) <_R is bounded in 1^(5) • Verification of [1.1] : Since g = hoA(r). , g is bounded below for each r, and by r * r the above remarks combined with lemma 3.2 it follows that for b e LP(E), and R c (R , the set | s e LP(E)B : g_.Cs) < R is bounded in and hence bounded in the Finsler metric on LJ^E) (Uhlenbeck [8]). Therefore in order to show that g__ satisfies condition (C) we need only show that if {s^} is a bounded sequence in LP(E)B (and hence bounded in L£(£), ), such that dg (s.) —> 0, then {s.} is convergent. Now dgr(s.) = dh(A(r)s,(si))od(A(r)A)(s.) . If dg(s ) —> 0 it follows from lemma 3.4 that dh(A(r)+ (s.)) —> 0, that {ACr^s^} is convergent and therefore so is {s^} . - 50 -Verification of [1.2] : Fix C C B compact and1 R E IR . Let {(r^, s^} be a sequence in K O {(r, s) : r e C and gr(s) £ R } . Since B is compact we can assume that {r^} converges to r e B . From lemma 3.3 it follows that the set {b } is bounded in Lf(£) . r. k x Now g (s) <_ R implies that h(A(ri)^ (s)) <_ R , where i A(r.),(s) E L?(E,)^ • From condition [3.2] it follows that the set l * k b r. {A(ri)A(s)} is bounded in 1^(0 • In order to proceed with the verification of [1.2] we need the following extension of the construction in Palais [4, pp 112-114]. co £ Fix r E B . Then A(r)(W) is a closed C submanifold of IR . r £ £ For each w E W let q (w) denote the orthogonal projection of IR = HRw CO onto TW . Then q is a C map of W into the vector space W i a z °° a L(IR , (R ), and since W is a closed C submanifold of IR , it extends co % z a to a C map of IR into the vector space L((R , tR ). If we define r r r 00 £ Q (x, v) = (x, q (v)), Q is a C fibre bundle morphism of £ = IR^ into L(5, O • As in [4, theorem 19.14], we define a map LJJa) —> L(LP(5), LP(C)) denoted by s I > p£. and given by Pg(t)(x) = Qr(s(x))(t(x)) . - 51 -In the above construction the map Pg was constructed by appealing to a general extension theorem. We wish to show that these maps r' can be constructed for r' in some neighbourhood of r such that Pg is "close" to P__ if r' is close to r . More precisely we have the following. Lemma 3.5 Fix r e B and S C ^£(0 bounded. There exists 6 > 0 and a method of defining the extensions of the such that for each r r1 e > 0 there exists a neighbourhood V of r with ||P - P || < e for s s all r' e V, and all s with distance(s, LP(E(r)))< 6 . k Proof : Let N be an arbitrarily large compact subset of W to be r a a determined. For r e B define the projection q : A(r) (N) —> L((R , (R ), as described above. For each point z e A(r)(N) extend q a finite distance along the normal directions to A(r)(N) at z in (R by making it constant. Now for r* "close" to r define qr : A(r')(N) -> LQR1, IR^) as above and extend it (shrinking N slightly if necessary) by making it constant along the normal directions determined by A(r)(N). The following diagram should clarify this argument. A(r) CN) --Extend along the *J normals A(r')(N) It is then easily verified'that the,maps P have the required property. This completes the proof of lemma 3.5. We resume the verification of [1.2]. First, dg (s^) = 0 i • implies that r. l dh(A(r.)Js.))oP = 0 i * I Let t. = A(r).(s.) . Then dMt.Mt. - t.) = dh(t.) r(P^)(t. " t.) + dh(t.) (I - P^Ct. - t.)} Now {t±} is bounded in 1^(5), and distance^, LP(A(r)(Er))fe —> 0 Therefore, there exists a sequence {u.^} in LP (A(r) (E.J )^ such that I|u. - t.II —> 0 . Consider the difference li_ _i i p Lk ||(I-P^)(t.- t.) - (I-PuMu. -u.)|| X 1 Li. k < Md-P^^-u.) - (t. -u.))|| r. x r + IKpt; -<)^i-V"TP+ l|(pt. -C^i-V-Np He 11 He Now [4, Theorem 19.14, 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 for a subsequence of {u^} which we assume is {u^} , and it follows that (I - P^Mt. - t.) || —> 0 . t. i J 1 ' P 1 Lk It follows that the difference dh(t.)(t. - t.) - dh(t.) xx 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(PtJ)(bi- Vj Since | \b± - b.|| —> 0 we get finally that |dh(ti)(ti - t.)| —> 0 Lk By [3.2] we can conclude that {t.} is a Cauchy sequence. Therefore {s^} is a Cauchy sequence, and Condition [1.2] is verified. We complete the verifications in the order [1.4], [1.3], [1.5], and [1.6] - 54 -Verification of [1.4] : Let T C L^CE) be bounded. Then T is intrinsically bounded and by Uhlenbeck [8], T is contained in a finite number of vector bundle neighbourhoods ^\^±) • Suppose that t E • Then £ Lf(n.) where n. is the vector bundle neighbourhood in E obtained by composing the map ^ —> E with the map ^(r) : E —> E . This induces a map LP(S.) —> LP(n.) and by lemma 3.1 there exists a K. X . KL X neighbourhood V of r such that ||v(r)^t - nr')*t|| < e Lk(V for all r' E V and t e T n l£(5.) • Since || || on E is an K. X admissable Finsler structure (Uhlenbeck [8]) the result follows. Verification of [1.3] : Fix -r E B and R e IR . Then gt(r') <_R implies that. h(A(r' )A°<f>r, (t)) <_R . By the remarks before the verification of [1.1] together with lemma 3.2 there exists a neighbourhood V., of r such that the set | (t) : hOKr'.^o^. (t)) < R is bounded in l^CO • By the argument used in the verification of [1.4], we can find a neighbourhood V C V1 such that [1.3] holds. - 55 -Verification of [1.5] : We have |f(r, s) - f(rT, s)| = |h(A(r)A(s)) - h(A(r«)*(s))| Now combining [3.3] with lemma 3.1 and the mean value theorem the result follows. Verification of [1.6] : This follows from [3.3] combined with the proof of lemma 3.4. This completes the verifications, and hence Theorem III is proved. - 56 -•'! CHAPTER 4 EXAMPLES 4.1 Perturbation of Minimal Surfaces In this example we have a fixed 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 locally compact topological space. Let F e FB[J1(IR^), Rl be given by i»3 2 3 Let the map B —> L^(R^) given by r I—> b^ be continuous, and let <j>r : I^(K^)0 —> Ll^\ be given by (J»r(s) = s + br . Then it is easily r seen that the standard problem determined by L^, <J>r, b^ satisfies [1.1] - [1.6]. Now if the set F of Chapter 1 is the set of singletons 2 3 in L]_("*jpo * ^t f°l-iOWS tnat for each r we are considering minimum values of the function g &r If br is a smooth section, so that its principal part carries 3 2 3 3M to a smooth curve in IR then a section belonging to L^((R^)br 3 defines a (generalized) surface in IR whose boundary is T . It is - 57 -well known, in this case, that a section which minimizes our Dirichlet integral g__ corresponds to a surface, of minimum area spanning . Moreover, the value of g__ agrees with the surface area just in this case; . Hence Theorem I applies to give conclusions : (a) There is at least one (generalized) minimal surface for each r , (b) the minimum surface area varies continuously with r, and (c) if for each r in a neighbourhood V C B the minimizing 2 3 section s__ is unique, then the map V —> L^(IR^) given by r I—> s^ is 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 in its second and third arguments and be denoted by (x1, u, r) I > a(xX, u, r) Let e FBtJ1^)., IR^ be given by Fr(x\ u, ii ) = Y l 2 I (u ) + a(x, u, r) i i 2 Let Lr e Lgn^OR^) be represented by F__ . Let B —> f^O^ given by 2 2 r I—> br be continuous. Let <|>r : L1(|RM)o —> Ll^IRM^br be Siven bv - 58 -s + b^ . Assume that a satisfies the following conditions (i) |a(x, u, r)I < C(r) 1 + u (ii) |a(x, u, r) - a(x, u, r')| <_ (^(r - r') 1 + u (iii) 3a 3u (x, u, r) 3 a < C(r) (l+ |u|' (iv) 0 < ••2-f (x, u, r) < C(r) 3uZ where C and are continuous functions of r, with C^(0) = 0 It is then easily verified that the standard problem determined by ' L , br, <f>r satisfies conditions [1.1] - [1.6]. The Euler-Lagrange operator associated with L is r 1 3a , N ~ L u +rr r, x., u . x.x. 2 3u l l ii Of course if a(r, x^^, u) = y(r)u for y : [0, °°) —> IR continuous we have the parametrized linear Euler Lagrange equation " I \ x + Y(r)u = 0 . i i In any case Theorem I applies again as in 4.1. - 59 -4.3 Domain Perturbations We shall employ the notation of Chapter 2 . Fix ft° C IRn and let F e FB[ JX((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 is C2 and satisfies the conditions (i) |a(x±_ u)| < C 1 + |u| (ii) \a(x±, u) - a(x[, u)| < C1(x± - x^) (iii) (Xi, u)| < C[ 1 + |u| 1 + |u (iv) 3u~ (xi' f -(x|, u) <_ C 1 + |u| > a2 (v) 0 < (x., u) < C 3u where C is a constant and C^ a continuous function on IR with C^O) = 0 . Let L e Lgn_ (IR ) be represented by F . Then it is clear that 1 OP F satisfies [2.1] - [2.6] of Chapter 2- with p = 2 . Let L(IRn) be the set of linear isomorphisms of (Rn over IR . Then for each r e L((Rn) the restriction of r to the closure of a smooth subdomain Q, C f2° is a diffeomorphism of Q into IRn . If ft is strictly contained in ft° , there exists a neighbourhood V of the identity in L((R ) such that - 60 -r(ft) C fi° for r e V . Let B = V, M = fi and A(r) = r|- . It is clear that X is continuous from B into Dif f eo(fi, IRn) . Let rl—> t>r be a continuous map of B —> L^CR^) . It is easily shown that the above problem satisfies the conditions of Chapter 2 and that Theorem II applies to it. 4.4 Perturbation of Geodesies This is an example of the type of problem treated in Chapter 3. £ Let M = [0, 1]. Let W C IR be a closed q dimensional subma-o 1 £ nifold of (R . Let F e FB[J (IR^) , (R^ be given by F(x, uj, uh = I (uh2 . 3 It is easily verified that F satisfies the conditions of Chapter 4.. The critical points of the map constructed with F correspond to geodesies on W in the Riemannian structure induced on W by the inclusion of W into IRq . Via the map A : B —> DiffeoflR^) defined in Chapter 4 we induce a continuous change in the Riemannian structure of W . The map $ : B -—> DiffeoOR*') varies the endpoints of the geodesies. 2 Fix a path component of ^(W^b . By Palais [4, Thm. 13.14, p.54], this is the same as picking a homotopy class of continuous maps M -—> W, - 61 -which we denote by H . Fix r e B and assume that for each r in some neighbourhood V C B, the minimizing geodesic assured by Theorem 111(b) is unique, say . Then we have shown that vr varies continuously with r , where variations of r in V correspond to variations of the Riemannian structure on W and of the end points of the geodesies, corresponding to br(0) and b^(l) . It is possible to change the above example to the case where M = S"^ . In this case we must assume that W is compact. We can also increase the dimension of M and let F represent "powers" of the Laplace-Betrami operator on W . For details see Palais [4, p.127]. - 62 -REFERENCES ' 1. F.E. Browder, "Existence theorems for nonlinear partial differential equations," Proc : Symp Pure Math 16, A.M.S., Providence, R.I., 1970, 1-60. 2. __, "Functional Analysis and Related Fields", F.E. Browder ed., Springer, New York, Heidelberg, Berlin, 1970, 1-58. 3. ,- "Infinite dimensional manifolds and non-linear elliptic eigenvalue problems", Annals of Math. 82 (1965), 459-477. 4. R.S. Palais, "Foundations of Global Nonlinear Analysis", Benjamin, New York, 1968. 5. , "Lusternik-Schnirelman theory on Banach manifolds",. Topology 5 (1966) 115-132. 6. , "Critical point theory and the minimax principle", Proc : Symp Pure Math 15, A.M.S. Providence, R.I., 1970, 185-212. 7. R.C. Riddell, "Nonlinear eigenvalue problems and spherical fibrations of Banach spaces", J. Functional Analysis, to appear. 8. K. Uhlenbeck, "Bounded sets and Finsler structures for manifolds of maps", J. Diff Geom. 7 (1972), 588-595. 

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