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T H E U N I VE R S I T Y 0 F M I C H I GA. N COLLEGE OF LITERATURE, SCIENCE, AND THE ARTS Department of Mathematics Technical Report FREE BOUNDARY PROBLEMS IN THE CALCULUS OF VARIATIONS Leonard J. Lipkin ORA Project 07100 under contract with: NATIONAL SCIENCE FOUNDATION GRANT NO. GP-3920 WASHINGTON, D.C. administered through: OFFICE OF RESEARCH ADMINISTRATION ANN ARBOR April.i965

This report was also a dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in The University of Michigan, 1965o

ACKNOWLEDGMENTS I wish to thank Professor Lamberto Cesari for his teaching and guidance throughout the preparation of this dissertation. I also wish to thank Professor Maxwell 0. Reade for his reading of the manuscript. There are others, far too numerous to mention, who have made profound contributions to my education, both at Oberlin College and at The University of Michigan. To these people I express my deepest appreciation. This research was supported in part by National Science Foundation Grant GP-57 at The University of Michigan. ii

TABLE OF CONTENTS Page ABSTRACT iv INTRODUCTION 1 CHAPTER I, THE NON-PARAMETRIC PROBLEM 11 1.1. Basic Definitions and Theorems 11 1.2. The First Free Boundary Problem 17 1.3. The Free Boundary Problem for a Capstan Surface 30 CHAPTER II. THE PARAMETRIC PROBLEM 47 2.1. Basic Definitions and Theorems 47 2.2. The Main Lemma Concerning Boundary Values 56 2.3. Some Lemmas on Quasi-conformal Representations of Surfaces 77 2.4. Admissible Vectors and Minimizing Sequences 90 2.5. The Existence Proof 105 BIBLI OGRAPHY 126 11a1

ABSTRACT The purpose of this thesis is to study the problem of existence of an absolute minimum for two-dimensional integrals of the calculus of variations when the admissible surfaces satisfy certain variable boundary conditions, R, Courant has studied the existence theory for the special case of the Dirichlet integral, but for general integrals only some classical necessary conditions are known. For the non-parametric problem, we study integrals of the form I(z) = f(xlypz,zxzy)dxdy, where the integrand is continuous, convex B in (p,q), and satisfies the usual growth conditions. For admissible functions we take those which are continuous on a closed disk B, absolutely continuous in the sense of Tonelli on the interior of B, which map a fixed subarc y of the boundary B* of B onto a fixed arc r in E3, and such that the values on the complement of y in B* are bounded by fixed constants. Thus, these are surfaces whose boundaries have a portion spanning r, and the complementary portion is free on a fixed finite cylinder. By a convenient generalization of a process of L. Tonelli for smoothing surfaces, we prove that there is a function in this class which yields an absolute minimum for the integral. We are then able to extend this result to the case in which the boundaries of admissible surfaces are completely free in a fixed cylinder. iv'

Next we consider the same problem when the cylinder is replaced by a "capstan" surface, that is, a surface of revolution generated by a portion of a curve of the shape of, say, a branch of a hyperbola. The admissible surfaces are now defined on different Jordan domains contained in the disk B and containing a fixed disk D. We apply the Caratheodory theory of conformal mappings on variable domains to convert a minimizing sequence into a new minimizing sequence of functions all defined on the unit disk. We show that one of these sequences admits a uniformly convergent subsequence, and then by a conformal mapping we show that the image of the limiting function is again admissible, it is defined on the kernel domain, and it yields an absolute minimum. In the parametric problem we study integrals of the form Io(zG) /F(z,J)dG, where z(u,v) is a vector (surface) in E, J is the correspending Jacobian vector, and the function F(z,J) is a usual parametric integrand. We let T be a fixed torus, or similar manifold, in E3, and for admissible vectors we take continuous ones in the Sobolev space W2(G) which span T and "cover the hole" of the torus. Since Courant has shown that a minimizing vector need not have a continuous trace on the fixed manifold, we say that a surface spans m if for every sequence of points in the parameter domain approaching a point of the boundary, the corresponding sequence of points on the surface approaches To In order to say precisely what it means to "cover the hole," we prescribe a topological linking condition.

For the existence proof, we introduce a class of sequences (znJ of surfaces which are admissible, except that their boundaries approach T in the limit rather than lie on T. The number b = inf (lim IO(znG)), where the infimum is taken over all these sequences,.n- oo is (possibly) smaller than the infimum of IQ(z,G) over all admissible surfaces. We introduce an integral I(z,G), following C. B. Morrey, which dominates IO(z,G) and agrees in case the surface is quasih conformal. Minimizing I(z,G) in suitable classes;j(K), we obtain a sequence (zKJ of vectors such that I(zK,G) + b and IO(ZK,G) + 6. We are then able to show that a subsequence (zp] converges to an admissible vector zo. Appealing to a lower semicontinuity theorem of L. Cesari and L. Turner, zo yields an absolute minimum for I0(z,G). vi

INTRODUCTION The problem which will be discussed in this thesis is that of proving the existence of. iinimnizing functions for integrals of the calculus of variations when the boundary values for admissible functions are partly fixed and partly free, or totally free in a fixed manifold, More specifically, in the non-parametric case, consider a finite (closed) cylinder over a circle B and a set of functions z - z(x,y) defined on the disk B bounded by B, with the requirement that the graph of z traces out a continuous curve lying in the cylinder. Then we seek a class of such functions with the property that a given integral of the form I'(z) = F(x,y, z,z z )dxdy x y B assumes an absolute minimum in. this class. Alternatively, on a fixed subarc 7 of the boundary of B we may prescribe continuous boundary values which each function z = z(x,y) is to assume, and on the complementary arc we allow the values of z(x,y) to be continuous, but otherwise free in the cylinder. Instead of a cylinder, one may ask the same questions for, say, a capstan- shaped surface (Chapter I, Definition 3. 1), or for portions of such surfaces. 1

2 In the parametric case we admit more general manifolds: for example, a torus, Q.'a deformation of a cylinder or torus. Here, of course, we seek minimizing vectors in appropriate classes, and the integrals studied have the form I(z, G) F(z, J) du dv G 1 2 3 where z(u,v) (z (u,v),z (u,v), z3(u,v)), (u,v) in G, and J = (J, J, J ) is the vector of Jacobians relative to the mapping z z(u, V). These problems have been studied from the classical point of view for a long time, that is, studied under the assumption that a minimum exists. The first work goes back to Gauss, and the expression for the first variation was originally obtained by Poisson (1833) for problems of this type. In 0. Bolza's book, Vorlesungen uiber Variationsrechnung, there appears a brief study of necessary conditions of two-dimensional problems with variable boundaries, and there is also a guide to the early literature (page 668). H. A. Simmons [28] obtained expressions for both the first and second variations in the case of non-parametric surfaces whose boundaries are free in a capstan- shaped manifold, and he extended his results to higher dimensional problems [29].

3 E. A. Nordhaus [24] generalized the problem studied by Simmons to a Bolza problem. He obtained expressions for the first and second variations, and also the transversality conditions. For the existence theory in the case of parametric surfaces, Riemann and Schwarz asked for simply connected surfaces of relative minimum area, whose boundaries consist of one or more segments and one or more arcs on given planes. The existence theorem for minimal surfaces with partially free and partially fixed boundaries was given by R. Courant [10], and for totally free boundaries by R. Courant and N. Davids [12], and N. Davids [14]. Most of this material is summarized in the book by R. Courant [11]. One notices that although some necessary conditions are known for free boundary problems, the existence theory for general non-par:a metric integrals has not been studied, and little is known about the parametric problem except for the case of minimal surfaces where the integral in question is the Dirichlet integral D(z, G) (z+ zY) dx dy G We now describe the reutnis obtained on existence theory for general integrals in this thesis. If G is a domain in the plane, we shall denote by G its closure, by G its boundary, and by G~ its interior.

4 In Chapter I we treat the non-parametric problem in two forms. In the first we consider the class of functions z = z(x,y) which are continuous and ACT (Chapter I, Definition 1. 1) on a closed Jordan domain G, which assume given continuous boundary values on an arc y of the boundary G, and whose values on the complementary arc are bounded, but otherwise free; and therefore the corresponding portion of the boundary of the surface S: z = z(x,y) lies on the cylinder over G. We show that there is a function in this class which minimizes the integral (0. 1) I(z,G) = F(x,y,z, Z y)dxdy G where F(x,y,z,p,q) is continuous in (x,y, z,p,q), convex in (p,q), and satisfies the conditions F(x,y,z,p,q) > v +.(p +q), / >0 F(xy,z, 0, 0) f(x,y) In order to prove the existence theorem, we need to show that there is a minimizing sequence vWhich converges uniformly on the closed domain G to an admissible function. To this end, we select a minimizing sequence {z(x,y)}, n = 19 2, 3,..., and then define a convenient generalization of a leveling process due to L. Tonelli. By this process we obtain a new minimizing sequence in which the functions do not oscillate too badly. We are then able to show that the sequence is equicontinuous on the boundary, the

5 Dirichlet integrals are uniformly bounded, consequently the functions are equicontinuous on G, and there is a convergent subsequence. Using a lower semicontinuity theorem of L. Tonelli [311 and L. Turner [32], we prove that the limiting function yields an absolute minimum for the integral. In the second form of the non-parametric problem, we consider those functions z(x,y) defined on a closed Jordan domain G (depending upon z) whose boundary contains a fixed arc y, z such that z(x,y) is continuous and ACT on G, assumes g'ive-n boundary values on y, and such that the points (x, y, z(x, y)), (x,y) in G - y, lie on a fixed capstan surface. In such a class of functions we wish to minimize the integral (0. 1). This means that the problem is twofold: we must find an admissible domain G and an admissible function z(x,y) defined on G which yields an absolute minimum for I(z,G). The procedure is to begin with a minimizing sequence { zn(x,y), (x,y) E Gn }, n - 1, 2, apply a leveling process as before, and to map each domain G:. n conformally onto the unit disk B by a mapping (u (x,y), v (x,y)), arranging things so that three distinct fixed points of y are mapped on three distinct fixed points of B (the same points for each n). We then prove that the vector functions R (uv) = (x (u,v), y (u,v), z (x (u,v), y (u,v))) are equicontinuous on B, where x (u,v) and y (u,v) are the inverses of u (x,y)

6 and v (x,y), respectively. Next we are able to apply the Carath6odory theory of conformal mappings onto variable domains and we produce a limiting conformal map (x(u,v), y(u,v)) of B which we prove to be a homeomorphism of B onto an admissible domain G. After showing that the functions z (x (u,v), n (u,v)) actually converge uniformly on B to a function z(u,v), we prove that the function g(x,y) = z(u(x,y), v(x,y)) defined on G yields an absolute minimum for the integral. For both forms of the non-parametric problem we extend these results to prove the existence of minimizing functions when the boundaries are completely free on the fixed cylindrical and capstan surfaces. In Chapter II we treat the parametric problem for the integral I0 (z., Q) F(z, J) du dv Q 3 where z(u,v) is a vector in Euclidean 3-dimensional space, E J = (J, J, J3) is the vector of Jacobians, and F(z, J) is continuous in (z,J), convex in J, positively homogeneous of degree 1 in J, a Lipschitz function in z and in J, and satisfies the relation mlJi < F(z,J)< MIJI, 0< m< M

7 Let 2. be a fixed torus in E. (This replaces the capstan surface considered before. 7 could actually be a rather general manifold of which the torus and capstan surface are special cases.) For admissible vectors we turn to the space W2 of, S:.. L. Sobolev [30] (or the space PZ of J. W. Calkin [2] and C. B. Morrey [19]). Specifically, a vector z(u,v) defined on the unit square Q is admissible if it is of class W2(Q), continuous of the interior of Q, and if the surface S: z = z(u,v) "spans the fixed torus?T and covers the hole of 7." The last part of the admissibility condition needs a precise formulation. In order to define what is meant by "covering the hole," we prescribe a topological linking condition. Let H be a simple closed curve linking the solid torus. Then we may say that the surface S: z - z(u,v) "covers the hole" if there is a boundary strip Qh C Q such that every simple closed curve lying in Qh' which is homotopic to the boundary of Q, is mapped by z = z(u,v) onto a continuous curve in E linking H. Deciding what it shall mean for a surface S: z = z(u,v) to "'span 7-'' is a more delicate problem. Courant [11, p. 220] has given an example which shows that minimizing surfaces for some free boundary problems (specifically, for minimal surfaces) do not always have a continuous trace on the fixed manifold. Therefore, in order to formulate a reasonable problem, we shall say that the

8 boundary of S.z z(u,v) lies on, or S spans', if the shortest distance of z(u,v) from points of "' approaches 0 whenever (u,v) approaches (uO, vO), (u,v) E Q, (uOvO) E Q The lower semicontinuity theorem which we want to use (L. Cesari [6], L. Turner [32])requires uniform convergence of continuous vectors of class WI(Q). However, the example of Courant shows that we cannot in general obtain uniform convergence on Q. Therefore, our reasoning will have to rely on interior properties only, and we shall have to settle for uniform convergence on closed subdomains of Q (the compact-open topology). Even when such convergence is obtained, the difficult problems remain of proving that the limiting surface "covers the hole" and spans ~. The result which guarantees that the limiting surface spans 7 is proved in Section 2 of Chapter II. It essentially says that if a sequence of continuous vectors of class W2, whose boundary values lie on a manifold M, converges unifbrmly on every closed subdomain of Q, and if the norms in the space WZ are uniformly bounded, then the boundary values of the limit vector also lie on M. The linking is proved by a variant of an argument of Courant [11]. For the initial step in the existence proof we introduce, as does C. B. Morrey [21], the integral I(z,G) -=! F2(zJ) + (M+m) [(?)+F2] dudv G

9 where E, F, G in the integrand represent the usual fundamental quantities of the surface. This integral dominates!0(z,G) and agrees with I0(z,G) in case the vector z(u,v) is quasi-conformal (Chapter II, Definition 3.4). The integral I(z,G) is shown to satisfy the inequality <?,M mD(z, G) <'I(Z0 < D'(zM, G) 2 2 where D(z, G) (1I2 + lz 12)dudv G Therefore, a sequence {z } for which I(z,Q) is uniformly bounded will have a subsequence which converges weakly in the W norm (Section 4). In Sections 3 and 4 we enlarge the class of minimizing sequences to a class of generalized minimizing sequences. Thus 6 = inf (lim inf I0(z,Q)) n —oo where the infimum is taken over this larger class, is less than or equal to the original infimum. We show that there is a generalized { 00 minimizing sequence {; } of vectors of class C such that lim I (r,Q) = 6 and lim I(r,Q) 6 On n n -oo n -- oo In Section 5 we introduce the class ({(K) of admissible vectors of class WZ(Q) with

10 2 2 2 J (Z,Q) z + lz Iuvl + Iz )du dv < K Q and which map a fixed segment a into a closed set bounded away from the torus. We show that I(z, Q) attains a minimum in AZ(K), and applying the result of Section 4, we show that lim I(zk, Q) = 6, k — oo where zk is the minimizing function in 9(K). Using a'Dirichlet growth theorem" of C. B. Morrey, we are able to prove that a subsequence of { Zk}, say, { z }, converges uniformly on closed subdomains of kI on closed subdomains of Q - a and weakly in W2(Q). We then prove that the limiting vector z is actually continuous (and of class W2(Q)) on all of Qo, and apply the lemma stated earlier that the boundary of S: z = z(u,v) lies on the torus'. Finally we prove that the boundary of S links the fixed curve H, and z is the required minimizing vector, since z is admissible and IO(z,Q)< 6 < inf Io(, Q) where the infimum is taken over the class of all admissible vectors.

CHAPTER I THE NON-PARAMETRIC PROBLEM 1. Basic Definitions and Theorems In this chapter we shall be dealing with non-parametric surfaces, that is, with functions z = z(x,y) defined on an open set G of the Euclidean 2-dimensional plane. Definition 1.1. A function z = z(x,y) defined on G is said to be absolutely continuous in the sense of Tonelli in G, or ACT, provided z(x,y) is continuous in G and i) for almost all x the function z(x,y) of y alone is absolutely continuous in each closed interval contained in the set G(x) { (x,y) E G' X.= x} ii) for almost all y the function z(x, y) of x alone is absolutely continuous in each closed interval contained in the set G(y ~) { (x,y) E G y = y}; and iii) the partial derivatives P = a z, q - z, which p P ax qay exist almost everywhere in G are summable in G. As is usual in variational problems employing the direct methods, we shall need a theorem of closure for certain classes of functions. The theorem which will be used in this chapter is the 11

12 following one. Theorem 1.1. (Tonelli [31, ~2, No. 11]) Let { z (w) }, w G, n = 1, 2, 3,..., be any sequence of continuous ACT functions in the bounded open set G, let { z } converge to z(w) uniformly in each closed bounded set H C G, where z(w) is a given function defined in G, and let the partial derivatives p =-zn q =-z be L -integrable in G, a > 1, n ax n n ay n a and i pn + q ) dxdy < M < + oo, M a given constant. G Then z is continuous and ACT in G, the partial derivatives a a P = z, q = -Z are L -integrable in G, and I Pl dxdy< lim n l p nl dxdy G G 9 ql dxdy < lim |q I dxdy n —-0 oo n G G In the problems to be considered in this chapter, the admissible functions will be continuous, and therefore we shall eventually be faced with the task of proving that certain minimizing sequences are equicontinuous. We shall now describe a procedure, due to Tonelli [31, ~ 1, No. 6], which will be applicable in the existence theorems which we shall prove. This procedure replaces certain

13 sequences of functions by other sequences, and in our cases the new sequence turns out to be equicontinuous. Let f(x,y) be a function continuous on G, the closure of G, and ACT in G. Let n be a fixed integer, and let z., ZO < ZI <... < ZN, be all the numbers of the form -,k 0, +1, +2,..., such that the planes P. z = z. intersect n j J the surface z = f(x,y), (x,y) E G. The set S0 of the points (x,y) in G where f(x,y)= z0 is open, and we consider only those components g01' g02'.. on whose boundary g0 the function f has the constant value z0. Then we denote by f0 the function which is equal to z0 in each set gOs' s = 1, 2,..., and is equal to f otherwise. Thus fo f on G, and f is continuous on G and ACT on G. We now repeat the same process on f using the plane f0_ P1. We obtain a new function f with fl = fo = f on G, which is continuous on G and ACT on G. Repeating this process N times we obtain a function f with f = f on G, which is continuous on G and ACT on G, and we shall say that f has been obtained from f byja 1/n leveling. If a function f has been obtained by this process, then we say that f is 1/n leveled. For the equicontinuity theorems which we shall use, we need a restriction on the type of domain G.

14 Definition 1.2. A domain G satisfies condition (a) provided there is a number do > 0 such that for every point PO E G and every square Q~ of center P0, sides parallel to the x- and y-axes, and side length 2g with t < d0, we have Q G Thus, every Jordan domain satisfies condition (a), but 2 2 the set G: 0 < x +y < 1, for example, does not. We are now able to state our main equicontinuity theorem. Theorem 1.2. (Tonelli [31]) If G is a bounded open set satisfying condition (a), if {z (x,y) }, (x,y) E G, is a sequence - * of functions which are continuous on G, equicontinuous on G, ACT in G, with.(lPnl2 + Iqn 2) dxdy < A < + o, n = 1, 2,.. G for some constant A, and if each function z is 1/n leveled, n then the functions z are equicontinuous on G. n The final bit of preparation needed before stating and attacking our problem is a lower semicontinuity theorem for our integrals. There have recently been some very general theorems of this type given [22, 26], but the convergence of the functions which is used is not enough to guarantee that the limiting function is continuous.

15 Therefore, in this chapter we may just as well use a theorem of lower semicontinuity with respect to uniform convergence. The theorem is essentially that of Tonelli [31], but we use the more general form due to Turner [32]. Theorem 1.3. Let G be a bounded open set and let f(x,y,z, p, q) be a continuous function of (x,y, z,p, q) for (x,y) E G and all z, p, q. Assume that i) f(x, y, z, p, q) > N for some real constant N and all (x,y, z, p, q) E G X E3 ii) for every M > 0 there are positive numbers a, A, and L such that f(x,yz,p,q) > p(p l+ + Iq l+a)J for all (x,y, zp, q),e GX E, with 1z1 < M, IpI + jql > L; and iii) f is convex in (p, q) Let C be the class of all functions z(x,y) which are continuous and ACT on G and for which - ~ < I(z) - f(x,y,z,z,z ) dxdy < +oo. G Then I(z) is lower semicontinuous on G with respect to uniform convergence.

16 We complete this section with a lemma which will be useful in proving the equicontinuity of certain sequences of functions. Lemma 1. 1. If u(t), a < t < b, is an absolutely continuous function in [a,b] whose derivative u'(t) is L -integrable in [a,b] for some / > 1, and if u(t) has an oscillation > o in [a,bl, then b u'(t) I dt> ag/(b-a)31 a Proof. By the Schwarz inequality we have b b a < iu'(t)j dt < ( u(t) dt)1/ (b - a) a a and the lemma now follows immediately.

17 2. The First Free Boundary Problem In this section we shall state and prove an existence theorem in the case of a partially free and partially fixed boundary. We begin with the admissible functions defined on the closed unit disk B, then generalize to any Jordan domain, and finally reach a result concerning totally free boundaries. We shall consider the problem of minimizing the integral I(z) F(x,y,z Zy) dxdy B where we take as the elements of the class o of admissible functions z = z(x,y) those functions which are continuous on B, ACT on B, and which take on prescribed continuous boundary values =( ), < - 01 < 27r, on the arc': 1 < Q < Q r = 1]. Furthermore, there are constants a, b, a < b, such that for all (x, y) on the boundary arc K complementary to, z -= z(x, y) satisfies the relation a < z(x,y) < b. Thus, the portion of the boundary of the surface S z = z(x,y), (x,y) E B, which corresponds to the arc K, traces out a continuous curve on a finite (closed) cylinder over B. We assume that there is at least one admissible function for which the integral in question is finite. We now make an extension of Tonelli's 1/n leveling process. Let z(x,y) be an admissible function, and assume that it is already 1/n leveled. We let P. z - z, j - 0, 1,..., N, be the same planes used in the first leveling operation. Just as before, the

18 set SO of the points of B where z(x,y) / z0 is open, but now we consider those components g01, g02,.''' on whose boundary g0s the function z(x,y) has constant value z0 or whose boundary gO intersects only the arc K (and not y) and such that z(x,y) has constant value z0 on gO B. Then we denote by zO(x,y) the function which is equal to z0 on each set g and equal to z(x,y) otherwise. Thus z0(x,y) is still continuous and ACT, z0 = z on the arc Y, and on K we stillhave a < z0(x,y) < b. Now we repeat the process using the planes P1..., PN in succession finally obtaining an admissible function z(x,y). Theorem 2.1. Let F(x,y,p,q) be continuous for all (x,y) E B and all (p, q), let F(x,yp, q) be convex in (p, q), and assume that F(x,y,0,0) = 0 and F(x,y,p,q) > (p + q ). If there is an admissible function z for which I(z) = F(x,y,z,z ) dx dy B is finite, then the integral I(z) assumes an absolute minimum in the class Proof. Let L = inf F(x,y,z 9z ) dxdy, where the --—.-. x y B infimum is taken over the class of admissible functions. By hypothesis, 0 < L < +oo. Let { zn} be a minimizing sequence, and we may well assume that L < I(z ) < L + l/n < L + 1. Now we apply n to each z our generalized 1/n leveling process and obtain a new sequence, which we again call { z }. This new sequence is again

19 a minimizing sequence since F(x,y,z,z ) is non-negative, F(x,y, 0, 0) = 0, and the leveling process reduces the absolute value of z and z to 0 over the sets on which the leveling takes x y place. We shall first prove that the functions z are equicontinn uous on the boundary B. Since B is a compact set, it suffices to prove that given any e > 0 and any point w0 E B there is a number r0 > 0 such that for any circle C of center wO and radius r < r0, every function z has an oscillation < E on B r- C. Actually, given e > 0 and w0, it is enough to prove this for: some r0 > 0 and all n > N, N a fixed number. If wO is an interior point of T, there is nothing to prove since all z take on the same values along Y. Thus we consider the n two cases where w0 is an interior point of K, and where wO is an endpoint of K. The disk B satisfies condition (a), and we let hI be the number relative to B and this condition (Definition 1. 2). Case I. Assume that wO is an interior point of K. Let ho be the smallest of the numbers hI and the two distances from wO to the endpoints of K. We shall prove that the equicontinuity 0h. 1 property mentioned above holds for ro min[ h0o hO0exp(83 &L N a 8/e. Suppose this is not true. Then there is a function z, n n > N9 whose oscillation in the circle CO of center wO and radius r0 is > t. Then there are two pointsa wl, w2 E CO r B such

20 that z (w1) - znw2 ) > E. Then, say, z (w) > z (w) + 1/2, n 1 n 2 n1 n(W0 and we may assume that z (w2) < z (w) < Z (w) for there will surely be some point w in C0 with this property, and without loss of generality. it may be w0. Let QL denote the closed square of center w0, sides parallel to the x- and y-axes, and side length 2, r0 = ro< < 1 h0/Z. Let a, be the two points of intersection of Qa with K. Now since [z (wl)- z (w2)[ > e, we must have either j zn( a) - z (w) > 1/2 or Zn(a) - Zn(w)1 > /2 (a). Suppose that z (a) - z (w2) > c/2 1, Assume -z (w2) > Zn (a) + c/2. Then we claim that n nw0 there is a point s on Q~ such that z (7) > z (wO) + E/4. For if not, then for all w E Q B we have z (w)< z (w + C/4, n - w 0 while z (w) > zn(w0) + E/2 Hence z (w) > z (w) + t/2 - [z (w) + /4] + V/4 > z (w) + E/4 for all w E Qf C B. Thus if Pk } are the z-coordinates of the 1/n leveling planes, we have for some j (w)<) < z (w ) + s/4. Now we already have n nO0

21 (a) + e/2 < z (w2) < z (wO), and hence z (a) < z (w )+ r/4. Therefore, z has an oscillation > t/8 on Q - B, n _0 <. <. I. 2. Assume that z (a) > z (w2) + /2Z. Then we claim n n 2 that there is a point T e Q rn B with z (~) < z (w2) + r/4. If not, for all w E Q r- B we have z (w) > z (w) + E/4. Then n -- n2 there is a j such that just as before z (w) < P. [z (w2) + F/4] + e/4 > Zn (T) + E/4, and so z has an oscillation > ~/8 on Q r- B. n (b). Suppose that Iz (a) - z (w1) > g/2. Then precisely as in case (a), parts 1 and 2 above, we see that z has an oscillation > E/8 on Q r-N B. Therefore by Lemma 1. 1, for almost all ~ we have F(x, y, z, z )s > (z + zn )ds > ( E/s) (86) iQ Qi Integrating with respect to ~ in [O we have

22 L + 1 > F(x, y z,y )dx dy > (z2 + z2 )dx dy B B > (E/8) i (8) ldt = 83 lor Therefore, 2 h0 -3 3 - L + 1 > 8 log( ) > 8 e log(exp(8 3 (L + 2))) = L + 2 a contradiction. This proves the equicontinuity at every point w0 interior. to. K. Case II. Assume that w0 is an endpoint of the arc K (and hence also an endpoint of the arc y). Since all the functions z n agree on y (and at w0), there is a number h > 0 such that every z has an oscillation < C/8 in C r y,, where C is any circle with center w0 and radius r < h. If h0 is the number relative to condition (a), we may assume that h < h. We shall again 0 - prove that the equicontinuity statement at the beginning of this proof is true for ro - min[' h, ihexp(83 2 (L + 2))] N = 8/N Indeed, suppose it is not true. Then there is a function z n n > N, which has an oscillation > g on C r (Y uJ K) for some circle C with center w0 and radius r < rg and we may assume

23 that r = r0. Then there are two points w1, w2 E C Cr (yT K) with I z (w) - z (w2z) > F, and hence for at least one of these points, say wl, we must have I z (w1)- zn(w)I > r/2. Let us assume that z (w ) > z (w0) + C/2. We again consider all squares Qf with center w0, sides parallel to the x- and y-axes, and side length 2, ro = 0< f < f1 = h0/2. All these squares are contained in a circle C1 with center w0 and radius h0. Also, the boundary Q of Q must meet y in exactly one point and K in exactly one point. If a is the point of intersection of Q with Y, we know that z (a) < z (w0) + c/8 (see page 22). n n O We now claim that there is a point r E Qd C B such that z (7) > z (w ) + e/4. If this were not the case, we would have n n O z (w) < z (w ) + g/4. for all w E Q rC B, while n n 0 z (w) > z (w) + z/2. We draw an arc from w0 to Q so close n 1 n 0 to the arc y/ that for all w- on this new arc, z (w) < z (w0) + r/8. This, of course, is possible by the continuity of z. Then we have z n(w )> [zn(WO) + e/4] + /4 > z (w) + t/4 and, therefore, there is a number j as before with z (w)< Pj< Pj+ < zn(w1) n J j+l n

24 contradicting the fact that z is 1/n leveled in our generalized sense. Thus there is a point T E Q B with z ( ) > z (w ) + t/4, n n O while z (a) < z (w0) + c/8. Hence z has an oscillation > t/8 *n on CQ B for every L satisf'ying % Now exactly as before we are led to the contradiction L + 1 > L + 2. Therefore, we conclude that the functions z are equiconn tinuous on the boundary of the disk B. Thus by Theorem 1.2, the functions z are equicontinuous on B. By Ascoli's Theorem, there is a subsequence z which converges uniformly on B nk to a function z which is continuous on B and takes on the prescribed values ~ = - ( ) on y. Now by Theorem 1. 1 we can conclude that z is admissible, since for every n, (2 + z )dxdy < L + 1 nx ny B By Theorem 1.3, the functional I(z) is lower semicontinuous, so I(z)< lim I(z ) L nk-o0 k But since z is in our admissible class I(z) > L. Finally, I(z) - L, and z is the desired minimizing function. This proves the theorem.

25 We now state a slightly more general theorem than the preceding one. We consider the class C of functions z(x,y) defined on a closed Jordan domain G, which are continuous on G and ACT on G. Furthermore, each z(x,y) takes on prescribed continuous boundary values C- =(x,y) on an arc Y of the boundary G, and on the complementary arc K we have a < z(x,y) O, v real, such that F(x,y,z,p,q) > v+p(p +q ). Suppose further that there is a continuous function f(x,y) such that F(x,y, z, O, 0O) - f(x,y), and that there is at least one function z(x,y) in the class C described above such that I(z) =, F(x,y,z,z,z )cdxdy G is finite. Then the functional I(z) has an absolute minimum in. Proof. First note that I(z) > v measure (G) + S (z2 + z )dxdy > v measure (G) G for all z in the class C for which I(z) is finite. Hence, L = inf I(z) > - 00 zEC

26 and since by hypothesis there is a function z in ( with I(z) finite, we have -oo < L < +oo. Let { z } be a minimizing sequence. To this sequence we apply our generalized 1/n leveling process. The new sequence, { z }, again consists of members of the class. Furthermore, it is a minimizing sequence. Indeed, if we assume that L < I(n )< L + l/n < L + 1 then since F(x,y,z,z,Zn ) < F(x y,z,nz z ) (using the fact x y x y that F(x,y,z,O,O) = f(x,y) ), we have L < I(z ) < I(zn )< L + l/n< L + 1 - n- n - Since G is a Jordan domain, it satisfies condition (a). Therefore, exactly as in the proof of Theorem 2. 1, except that we replace.1 1 3-2 r0 min[~ h0, o h0 exp (8 t (L + 2))] by _1 1 =r min[2h0 a ho exp(8 r 2 (L + 2 - vmeas(G)))] we conclude that the functions z are equicontinuous on G n Since 2 I(z )> v meas (G) + p (z. + z )dxdy G x y we have

27 |(z + z )dxdy < l (I(z) - vmeas (G)) x y <,J (L+1 - v meas (G))-= A where A is a constant. Therefore, by Theorem 1.2 the functions z are equicontinuous on G, and by Ascoli's Theorem there is a n subsequence {z } which converges uniformly to a continuous nk function z on G. Furthermore, by Theorem 1.1 z is admissible (that is, in the class C ), and by Theorem 1.3 the functional I(z) is lower semicontinuous. Hence, I(z) < lim I(z )- L, while at k — oo k the same time I(z) > L. Therefore, I(z) - L, and z yields an absolute minimum in the class C. This proves the theorem. The results of Theorems 2. 1 and 2. 2 and their proofs enable us to prove the existence of a minimizing function for the problem in which the boundary values are left completely free in the cylinder over the boundary G of the Jordan domain G, where we, of course, take only a finite (closed) cylinder. Let E be the class of all functions z(x,y) defined on G, continuous on G, and ACT on G, whose boundary values lie in the finite cylinder S given by { (x9 yz) (xy) E G, a < z < b}

28 Theorem 2.3. Under the same hypotheses as in Theorem 2. 2, if the class C contains a function z such that I(z) is finite, then the functional I(z) assumes an absolute minimum in the class o Proof. Let { z } be a minimizing sequence. We may as sume that L inf I(z) < I(z) < L + 1/n < L + zE6 We divide the boundary G into two Jordan arcs a,. such that a and 3 have only their endpoints in common and a v 13 - G We may do this so that neither arc is degenerate. For the moment, let n be fixed. We consider the class. of all functions z which are continuous on G, ACT on G, and take on the values z (x,y) for all (x,y) E a. Then by Theorem 2.2 there is a function z (x,y) in this class which minimizes I(z) among all functions in Co. Therefore, I(. )< I(z ), and z is still in the class C. n n n Therefore, L < I(z )< I(z) < L + l/n< L + 1 so the sequence { z } is again a minimizing sequence. However, each z2 is l/n leveled in the generalized sense with respect to the fixed values on the arc a. Indeed, if one of them were not, we could apply the leveling process and obtain a function zOn such that I(zz ) < I(z ). Now exactly the same proof as in Theorem 2. 1 shows 0n n that the functions z are equicontinuous on /. n

29 Next, for each n, we may replace z by the function z which minimizes I(z) among all those admissible functions which agree with z along the arc 3. As before, L< I(z )< I(z )< L + l/n < L+ 1 and we may assume the functions z to be 1/n leveled. The proof of Theorem 2. 1 shows that the functions z are equicontinuous on a. Therefore, these functions are equicontinuous on G, and using the fact that the cylinder is finite, we may reason exactly as before. Thus we see that there is an admissible function z such that I(z) = L that is, I(z) takes on an absolute minimum in the class (.. This proves the theorem.

30 3. The Free 3oqundary Problem for a Cap_.stan Surface Definition 3. 1. By a capstan surface i we shall mean a surface of revolution generated by revolving a curve C in the (x, z)plane about the z-axis, where C can be represented by a twice continuously differentiable function x = f(z), a < z < b, f(z) > 0, having (a +b) the properties that f'(z) has precisely one zero z = f(Z) is 2 f symmetric in the line z - (a+b) and f"(z) > 0 for all z in the in2 terval a < z < b. Such a surface may be represented parametrically by the equations X = X(u,v), = Y(u,v) z = Z(uv), where 0 < u < r7, -T < v < + T. The functions X(u,v), Y(u,v), Z(uv), are assumed to be at least of class C, and X(u,v), Y(uv), are assumed to have positive real period rl in the variable u for values of v in the interval -? < v < + 7. We take the (x,y)-plane as the base 4' plane for, and we denote by D the intersection of with the (x,y)-plane. Thus, D is a circle, and we denote by D the disk I it bounds. Let D1 be the disk contained in, D with the property that the cylinder over /'1 t- -4.. D is tangent to A We shall be interested in Jordan D1 domains G with the property that G contains a fixed Jordan arc T lying in the annulus bounded by the circles D1 and D and

31 satisfying the relation D1 G C C D. It will be necessary to impose a certain restriction on the boundary G of G. This restriction is embodied in the following condition. Condition S(r0,a0,b0). Let r0, aO0 bO be given real r0 > 0, r /2 < ao, 7r/2 < b0. The domain G is said to satisfy conditicnb S(r, a0, b0) if for every point w e G - Y there are circular sectors S, S each withvertex at w and radius ro, a b having vertex angles a0 and b0, respectively, such that one of the sectors lies completely interior to G and one lies completely exterior to G (except for the point w). The domains in figures 1 and 2 below satisfy condition S(rO, a, b) with aO = b0o = r2, while the domains in figures 3 and 4 do not satisfy the condition for this set of numbers. 1 2 3X drst~w,,w,3tt,L

32 In the existence proof which follows, we shall use as one of our main tools the theory of convergence of conformal mappings onto variable domains (the Carathe'odory theory). We first recall the definition of the kernel of a sequence of domains and then state the Caratheodory convergence theorem. Definition 3. 1. Let { B } be a sequence of domains in the complex plane, each B containing a fixed point, say, 0. Then the kernel B K{ B} is defined by..the propertie.s: (a) If r-B contains no neighborhood of 0, then n n B {0} (b) If r- B contains a neighborhood of 0, then B n n is a domain such that (i) 0 E B; (ii) If E is any compact subset of B, then for n sufficiently large, E C B n (iii) If D is a domain satisfying (i) and (ii), then D C B. Such a domain always exists. Furthermore, if {B } is a subsequence of { B }, if B K{ Bn}, B' = K{ Bn }, then B' D' B. If actually B' - B for all subsequences { B nk then we say that the sequence of domains { B } converges to the' ---—' n domain B. We shall now state a convergence theorem which will be of use to us.

33 Lemma 3. 1. Let { f (w) be a sequence of conformal mappings defined on the disk B, let f (B) - G, D C G C D n n 1 n for all n and two fixed concentric disks D and D. If the sequence { f } converges uniformly on B to a function f, then f is a conformal mapping of B onto the domain G = K{ G }, and the sequence of domains { Gn } converges to the kernel G. Furthermore, on every compact subset of G we have lim f (z) - f (z) n n --- oo uniformly. Remark. This is a slightly different statement of the Caratheodory theorem than is usually found, for example, in Goluzin [16, p. 46] or Caratheodory [3]. The essential reason is that the usual statement is a necessary and sufficient condition, and it includes a normalization of the form f (0) - 0, f' (0) > 0. But n n since we are assuming uniform convergence of the conformal mappings (and hence the limit function is conformal) along with the uniform boundedness of the domains G, we do not need to make assumptions of this type. The proof of Lemma 3. 1 may be obtained from the two references noted above or more conveniently from the very general theorems of F. W. Gehring [15].

34 We wish. to consider functions z = z(x,y) defined on a Jordan domain G (depending upon z) with D1 C G C; D, and having the property that the set of points C: (x, y, z(x,y)), (x,y) e G, lies on the capstan surface i That is, the surface S' z = z(x,y), (xy) E G has its boundary on the capstan surface. z We let Y be a fixed Jordan arc lying in the annular region bounded by the concentric circles. D1 and D, and let r - ~(x,y) be a given continuous function defined on Ty, whose graph lies on ~_j. Denote by C the class of all functions z - z(x,y), defined on some Jordan domain G whose boundary G lies in the domain Z Z D - D1 and contains the arc y, such that z(x,y) is continuous on G, ACT on G, agrees with {(x,y) on y, and the points z Z (xy, z(x,y)), (x,y) G, lie on. We further assume that each domain G satisfies condition S(ro) a, by) for some fixed numbers r02 a0, b0. Theorem 3. 1. Let F(x,y,p,q) satisfy the same conditions as in Theorem 2. 1. Assume that there is at least one function z in such that I(z) = F(x,y, z z) dxdy is finite. Then the funcG z tional I(z) has an absolute minimum in G Proof. Let { n } be a minimizing sequence, where the domain of z is denoted by G. We may assume that n n

35 We apply Tonelli's 1/n leveling process de.scribed in Section 1 (not our generalized process) to obtain a new minimizing sequence { z }, each z having the same domain G as did z. By the n n n n Riemann mapping theorem, for each n there exists an analytic univalent function f mapping the interior of G onto the interior n n of the unit disk B in the w = (u, v)-plane, with f also analytic and univalent, mapping B onto G. These functions f, as we n n know from the Carathe'odory extension theory, may be extended to continuous functions on the closure of the domains G n Let Q., i = 1, 2, 3, be three distinct fixed points on the fixed arc yQ1 and Q3 being the eindpoints, and let P = z (Q), i = 1, 2, 3. The triples (f (Q1)' f (Q2)' f (Q3)) ni n 1 n' n 2 n 3 define one or the other of the two orientations of B. One of the orientations occurs infinitely many times. Make such a choice of orientation, and extract a subsequence, again called { f } 9 such that the orientation is the same for all n. Let (w1, w2, w3) be a triple of 3 distinct fixed points on B which defines the same orientation as above. Then there is, for each n, a unique linear fractional transformation B (w) of B onto itself sending f (Q ) n n 1 onto w., i = 1, 2, 3. Let h (w) = f (s (w)). Then h (w) is, 1.n n n n for each n,, a one-to-one conformal mapping of B onto G sen-irng: wo onto Q., i = 1, 2, 3. Furthermore, for each n, z (h (w.)) P i, 1 2 3. If we write h (w) (x (w), y (w)) rn n i nl n n n

36 and z (w) z (x (w), y (w)), we have a sequence of vectors n n n n 3 R (w) = (xn(w), y(w), z(w)) mapping B into E, such that the points R (w), w B, lie on A, and R (w.) p P n n 1 n i - 1, 2, 3. By the definition of the points P we have inf [mutual distances of the Pni] = do > 0 n We now assert that the vectors R (w) are equicontinuous n on B. For suppose that this is not the case. Then there is a number d > 0 and a sequence of pairs of points w' w" nm nm m = 1, 2,..., of B with jw' - wI" j- 0 as m — oo, and nm nm jR (w' ) - R (w" )_ > d for all m =1, 2,. m nm m nm By a convenient extraction of a subsequence and a renaming, we may assume that the sequence [nm] is the sequence [mn], that the sequence jw'. converges to a point w0 e B, and that ]w' - w I < 1/n, 1w" -Iw < 1/n, n 1, 2,. Now the n 0 n 0 point w0 is interior to one of the arcs defined by w1, w2, w3, say, to that arc wlW3 which contains w2. We may assume that all of the points w', w' are also interior to the same arc, and that n n the points in question are oriented (W1, w' " w'i w) The vector n n 3 functions R (w) are one-to-one on B, and the images under R n n of the two disjoint arcs w3w1 and w wI' are two disjoint arcs 3P1 and dameters n respectvely. n3'P nl " A -P I -P I I e% -F A n n ~ -"> 0 -, respectively.

37 Now let E > 0 be given. Choose N so large that 8/N< <, and choose 61 < 1/N. For the "fixed arc, " that is, the arc which is common to all the curves C R R (w), w e B, there is'. a n n number 62 > 0 such that any two points on the fixed arc which have a distance less than 62 lie on a subarc of diameter less than E (property of a continuous arc). Choose 6 = min [6 1d 6 diameter of fixed arc, 6a ], where "6 arises as follows. Given any e > 0 there is a number 64 > 0 such that if P, Q are points of the capstan surface A) of distance less than 6s if a is any arc lying in, joining P and Q such that the projection of a on the (x,y)-plane is contained wholly in one of the two sectors (possibly degenerate) of D determined by the center of D and the projections of P and Q, then the diameter of the projection of a on the (x,y)-plane is less than t/2. Now we claim that there is a number 6, 0 < 6 < 6 such that for n > N, any two points of C at a distance less than 6 belong to a subarc of diameter < e. Suppose, on the contrary, that this is not the case. Then there is a sequence of points P nl PnZ' n > N, such that dist (P, Pn2 ) -- 0 as n ~ o, while each of two arcs X n X with endpoints P, P2' X Knl X - C nl nZ nl n l n1 n has diameter > g. Since dist (P, P ) -- 0, we may assume nl nZ that dist (P, Pz ) < 1/n, and thus if nl n2 are, respectively, the projections of Pn, P2 on the (x,y)-plane, then n2~~~~~~~~

38 I nl -n2 I dist[ nl' n] < 1/n. Now by the choice of v5 nl and n cannot both lie on the fixed arc ly nl nZ Case I. Suppose that n' 1n2 are both on the arc complementary to y. Let us make the convention that X 1 does not contain the fixed part of C. Since diameter X n >, there is a point PnO on Xnl such that if PnO is its projection on the (x,y)-plane r no - rnZ2 > rn > 0, rnO - rnl > 0 for some constant r7 depending only on the curvature of the capstan surface. However, if we let n O then I l - -I 0, and this contradicts the fact that the domain Gn satisfies condition S(r0, ao' b 0) Case II. Suppose that r lies on the arc complementary nl to y, and ln2 lies on C. Let g0 be the endpoint of Y on the smaller of the two arcs bounded by rnl and 2. Since | 1nl- rn2' ~0 as n - oo we must have 1l ol 0 as n - oo and nZ- -[- 0 as n — oo. We may choose n so large that the fixed arc on the capstan surface between P0 and PnZ has diameter < C/2 (by the property of the continuous curve). Then the diameter of the arc P P0 is > C/2 for all n. Now we may reason exactly as in case I. This proves the assertion. Now we take s min[d0, d] and 6 > 0 as above. Then we claim that the arcs P Pt and P"P of C have a mutual nl n n n3 n distance > 6. If not, there would be points P', PIS of the first two arcs; at a distance < 6 sandcl therefore a subarc of C of n

39 diameter < e containing P' and P". Since C is simple, this n arc must contain either P' P" or P P, both of diameter > c, n 31 a contradiction. Let s0 be the minimum distance of w0 from wl and w3. Let us consider the family of arcs B of circumferences with center w0 and radius r, s < r < s, s = l/n, contained in n - 0 n B with endpoints on B. The images of the endpoints a, T of these arcs on C under the mapping R lie on P1P' and P"P n n In n 3 and hence at a mutual distance > 6. Introduce polar coordinates (r, 0) with w0 as pole. The region E covered by the arcs is given by is < r < s0; 0 (r) < 6<(r)]. Recalling that R (w) (x (w), Y(W), zn(w)), where x and y are the real and n n n n n n imaginary parts of a conformal mapping, we see that 2 2 2 2 (x (u,v) + x (u, v))du dv (y (u, v) + y (u, v))du dv u v B U v B B = dxdy < area of D - A G n since each G is contained in the disk D, and since the above inten grands are expressions for the Jacobian of the mapping (x (w), y (w)). Therefore, we have

40 2 2 L + 2A > I(z) + 2A > (z + z )dxdy + ZA - n n n G n 2 2 2 2 2 2 > (x +y + z + x +y + z )dudv - n n n n n n B u u u V V V 50 2 2 2 -2 2 2 2 > (X + y + zn + r [xn +n + z )rdrd >r \r r 0 0 0 n 1 0 02 > r dr (x + y + z )dO n 1 But we also have 2 2 2 o< \ (x + Y + zI) d < N{ (x +Y n z )dQ or 2 2 62 2(2 +z )dO > r nx + n- -- 7 1, 1n n 01 Combining this with the inequality at the top of thi.s page, we obtain 62 L + 1 +2A > log(ns 0)

41 a contradiction, since the right hand side approaches + oo as n approaches +00oo. Therefore, the functions R (w) are equicontinuous on B The equicontinuity of the vector functions R (w) on B n implies that the mappings hn(w) = (x(w)), y (w)) are equicontinuous on B. However, each h (w) is an analytic function on B and n continuous on B. Hence by the maximum-modulus principle, the functions h (w) are equicontinuous on B. Therefore, a subsequence n { hn converges uniformly on B to a function h(w) which is k analytic on B and continuous on B. Since each h maps the nk three distinct points w*, i = 1, Z, 3, of B onto the three distinct points Q., i = 1, 2, 3, respectively, of y, and since h(w) is continuous and h(Wo) = Q, i - 1, 2, 3, h(w) cannot be constant on 1 1 B. Therefore, h(w) is a one-to-one analytic function on B [169 p. 9], continuous on B, and h(w) maps one of the two arcs a1 w lW2W3, T2 = W3W1w2 onto the arc 7 in a one-to-one manner. We may choose the subsequence {h } so that each member maps, say, the arc a = a onto the arc Y. Then h(w) also maps a onto 7. We denote by G the image of B under the mapping h. Thus G = h(B) is a simply connected domain, D D GD D1, and G contains the arc y. By Lemma 3. 1 we conclude that G K{ G } and the sequence {G } converges to G.

42 -1 -1 Moreover, lim h () h () uniformly on compact subsets k —0oo nk of G. We now replace the sequence [nk] by the sequence [n] for the sake of simplicity. Next, we see that h maps B onto G Indeed, if ~~~~~~~~~ 1 E'cG and lim Ek =' I ~ G, then the points {h (:k) k — oo have an accumulation point w E B. Then a subsequence, again called rh l()k), converges to w. Since h(w) is a continuous mapping on B $ ~ ~ ~~- -l * lrim k = h(h l(h )) _ h(lim h (rk)) h(w ) k —Poo k —oo k —oo But since h(w) is a homeomorphism on B, w must be a point of B. Therefore, h maps B onto G Now we shall show that h is actually a one-to-one mapping of B onto G (and therefore of B onto G). Then we can conclude that G is a Jordan domain, since h will be a homeomorphism on B. Suppose, on the contrary, that there are distinct points w, wif on B such that h(wt) = h(w"), and we may assume that h is not constant on the arc w' w" - which does not contain the arc -l. a - h- (y). (We know by the "three point lemma"' [11, p. 103] that h is a one-to-one mapping of a onto y. ) Then there is a point w7 interior to the arc 3 such that h(wT)/ h(w'). By the uniform

43 convergence of the functions h toward h, we may choose N > 0 n so large that for every n > N we have |h (w)- h(w)I < r0/4, w = W1, 9 W O where r0 is the number relative to condition S(r0, a0, b0) Consequently, Ihn(w') - h (w,)| < r0/2 while Ih (w') - h (w0)1 and jhn(w") - hn(w ) are bounded away from 0. But this contradicts the condition S(r0, aO0 b0). Therefore, h is a one-to-one mapping of B onto G, and G is a Jordan domain satisfying condition S(ro, a0, b0) Also from the equicontinuity of the vectors R (w) on B n follows the equicontinuity on B of the fun'ctions z (w) - z (h (w)). n n n Since h (w) is conformal, z (w) is ACT on B. If necessary, n. n we may apply Tonelli's 1/n leveling process obtaining a new sequence, again called { z }, which is equicontinuous on B, ACT on B, and furthermore L + 1 > F(x,y,z,zn )dxdy G y n > (z2 + z )du dv n n B u v Thus by Theorem 1.2 the functions z- z (w) are equicontinuous n

44 on B, and so we may choose a subsequence, again called { z, which converges uniformly on B to a continuous and ACT function z (w). Now we write gn(x, y) z (h (xy)) g(x,y) = z(h- (x,y)) (x,y) E G = K{ G }. Thus we have shown that the functions gn(xy) converge uniformly on G to a continuous and ACT function g(x,y), and furthermore, the sequence { gn } is a minimizing sequence. The values of g(x,y) on the arc y are such that the points (x,y,g(x,y)), (x,y) e y, lie on the capstan surface. Thus, g is in the class 6 of admissible functions. By the lower semicontinuity theorem I(g)< lim I(gn) L n foo but since g is in, I(g) > L. Therefore, I(g) - L and the theorem is proved. We may now make the extension to slightly more general integrands just as in Theorem 2.2. Theorem 3.2. Let F(x, y, z,p, q) satisfy the same conditions as in Theorem 2.2. Then I(z) = F(x,y,z,z,z )dxdy assumes G z an absolute minimum in the class G (as long as there is a function z c C such that I(z) is finite).

45 Proof. The proof of this extension is reduced to the case of Theorem 3. 1 just as in the earlier proof of Theorem 2. 2. We may now use the final extension in Section 2 to prove the analogous existence theorem for boundaries totally free on a (closed) finite capstan surface. For this case we let C denote the class of functions z(x,y) defined on a Jordan domain G, D1 C G C D, such that z(x,y) is continuous on G, ACT on G, and such that Z Z the points (x,y, z(x,y)), (x,y) E G, lie on. Let each G Z Z satisfy condition S(ro, ao' b0)' Theorem 3.3. Under the same hypotheses as Theorem 2.3, the functional I(z) assumes an absolute minimum in the class C. Proof. We let { z } be a minimizing sequence, where z is defined on the domain G, and let qy be a connected subn n n arc of G with endpoints P and P lying, respectively, on n nl n3 the positive y-axis and the negative y-axis, with 7 lying in the right half-plane. Then there is also a point Pn2 which intersects the positive x-axis. We now proceed as in the proof of Theorem 3. 1. by mapping the three distinct points Pn,' Pnz Pn3 onto three distinct points w1, wZ, w3 of the unit circle B. The proof of Theorem 3.1 may now be repeated, except that we no longer need to make a special argument about equicontinuity at endpoints of a'fixed arc. 1! This proves the theorem.

46 Remark. It is clear that the theorems in this section are not really restricted to the case of a capstan surface which is a surface of revolution. The only features which we actually used were the fact that the curvature was bounded away from 0 and the fact that the projection of each admissible surface S e z = z(xy) onto the (x, y)-plane is a Jordan domain G containing a fixed disk D1 and contained in another fixed disk D, and satisfying Condition S(rop ao, bo0) Without a condition like S(rO, aO, bo) it is conceivable that a limiting (minimizing) domain has the form of Figure 5. If we were to enlarge the class of admissible domains to, say, simply connected domains, we would not be able to extend the Riemann mapping functions, used in the proof, to continuous functions on the closed domains.

CHAPTER II THE PARAMETRIC PROBLEM 1. Basic Definitions and Theorems In this chapter we shall consider the problem of minimizing the integral (1.1) I0(z,G) = F(z,J)dudv G z(u,v) = (z (u,v), z 2(u, v), z(u, v)) (u, v) E G 2 3 z z v v 1 2 1 U 1 U z I~iz z 3u u 1u u 2 U U 3 1 1 2 z z z z v v v v where the integrand F(z,J) satisfies the conditions: (i) F is continuous in (z,J) for all (z,J); (ii) F is positively homogeneous of degree 1 in J (iii) F is convex in J for each fixed z; (iv) there are numbers m, M, 0 < m < M, such that 47

48 mrJJ < F(z,J)< MIJI for all (z,J); and (v) there are numbers L1, L2 > 0, such that for all (z, J) IF(z,J) - F(z2, J)I < L z1l - z21 jF(z,J1) - F(z,J2)j < L2 I,1 - J21 It is known that properties (i) and (ii) imply that the integral I0(z, G) is independent of the representation (z, G) of the Freochet surface (L. Cesari [5]). Thus we shall consider all surfaces (vectors) to be defined on the unit square Q. It is also known that this integral is lower semicontinuous with respect to uniform convergence of continuous vectors of class W (Q), Definition 1. 1 (L. Cesari [6], L. Turner [32]), The existence proof is mainly a generalization of the technique of C. B. Morrey [21] used for the proof of existence of a minimizing function of the parametric problem for surfaces spanning a fixed Jordan curve in space; that result was first obtained for general integrands independently by L. Cesari [8], J. M. Danskin [3], and A. G. Sigalov [27]. We also incorporate the notions of R. Courant with regard to the formulation of the free boundary problem. We shall now give a brief discussion of the basic definitions and properties of the function spaces W (Q), the so-called Sobolev s spaces. The study and systematic use of these spaces in potential

49 theory, partial differential equations, and calculus of variations has been developed by many authors. Their use goes back to the work of B. Levi, and important contributions have been made by L. Tonelli, G. C. Evans, J. W. Calkin, C. B. Morrey, S. L. Sobolev, and others. A sizeable list of references may be found in Morrey [22]. The basic properties of these spaces may be found in [2, 19, 20, 22, 23, 30]. Definition 1. 1. A function z is of class W (Q), p > 1,. p _~ if z is of class L (Q) and there exist functions h1, h2 also of class L (Q), such that g (u, v)hi(u,v)dudv = - gi(v(,v)zu,v)dudv, i = l, 2 Q Q for all functions g of class C with compact support in Q, where I' ~ g v The functions h. are uniquely determined up to null functions, and furthermore, if z is of class W (Q) and z - z almost p * 1 everywhere, then z is also of class W (Q) and the same functions ho will serve for z in the above definition. The functions ho are 1 1 called the generalized derivatives of the function z, and we shall write z h, z = h, and call z and z the derivatives of u 1 v 2 u v z. In the language of the theory of distributions, the distribution derivative of the function z (as a distribution) is a function of class L (Q). Since we shall be concerned with vector functions, we shall p

50 say that a vector is of class W (Q) if each component is in that p class. Definition 1.2. The space W (Q) consists of equivalence classes of functions of class W (Q) under the equivalence relation p of equality almost everywhere. Definition 1.3. A function z is of class W (Q) if z is of class W (Q) and if each of its generalized derivatives up to p order m - 1 is of class W (Q). Analogous to Definition 1.2 above, we define the space Wm(Q). p By introducing the norms 2 2 2 > 2 2 7i- 1 z dudv+ (Izul + [z I )dudv W2Q Q L2(z,Q) + D(z,Q) and llz 2 1 L(z, Q) + D(z, Q) + zz2+21Z U|(Zu + Jz ZV )dudv 2 Q - L2(z, Q) + D(z, Q) + J(z, Q) the spaces W2(Q) and WZ(Q) become, respectively, Banach spaces (in fact, Hilbert spaces).

51 Theorem 1. 1. If z E W (Q), p > 1, then z has a p representative z which is absolutely continuous along almost all lines parallel to the coordinate axes, and the partial derivatives of z are representatives of the generalized derivatives of z. If z is a function bf class L (Q), p > 1, which is absolutely continuous p on almost all lines parallel to the coordinate axes, and if each first partial derivative of z is of class L (Q), then. is of class W (Q). p Theorem 1.2. Let z e W (Q). Then p (a) there exists a sequence { z } of Lipschitz functions such that z -- z in W (Q); n p (b) there is a function E L (Q ) such that z np n in L (Q ) for every sequence as in (a); (c) if z e W (Q), then there exists a sequence p z } of functions of class C (Q) such that z z in W (Q) and (d) if z Wm(Q), th m- 1 if also P P -) m in Wre()m- 1wz ~z in W (Q~) then q in W (Q. n p n p

52 Theorem 1.3. (a) If p > 1, then bounded families in W (Q) are conditionally compact with respect to weak convergence in W (Q); p. (b) if z - z weakly in W (Q), then z - z n p n rin- 1 m- 1 * strongly in W (Q) and k --' in W (Q ); and p n.. p (c) if mp > 2, then every function z W (Q) is p continuous (i. e., is equivalent to a continuous function); moreover, any set { z } of functions in W (Q), mp> 2, with uniformly p bounded norms, is a compact set in the space of continuous functions.-:Definition 1. 4. A function q is a Friedrichs mollifier (or a mollifier) if ~ is of class C (E), ~(u,v) > 0, c has compact support in the unit disk B, and 8S(u, v)dudv = 1. B Definition 1.5. If z is locally summable on an open set G, we define its k-mollified function z by z (w)- z(x)~ (x- w)dx, w e G - w e G B(w, r)C G} r r II" r B(w, r) where B(w,r) is the disk with center w and radius r,w = (u,v), x- (xlx2), and' (y) = r -(r y), = (,y Y2)

53 Theorem 1.4. Suppose z E L, S is a mollifier, and z denotes its 0-mollified function. Then r (a) z e C (E ) and its derivatives are formed by r differentiating under the integral sign; (b) z -~ z almost every.where as r -- 0; if z is continuous, the convergence is uniform; (c) IlzrIIL < llzlL P P (d) z - z in L (E ); r p (e) if z e W and h is a generalized derivative of p ko k k k D z of order < m, then D z h so that all such D z D z r r r in L (E ) as r -O 0 p (f) if z c W (G), then (e) holds for w e G and p r the convergence in (e) holds on each compact subset of Go Most of these results will be stated in the text which follows as they are needed. We have stated all the definitions and theorems in this section in terms of functions of 2 independent variables, 2 that is, functions defined on a subset of E. However, all that has been said holds for functions of n independent variables, except that -2 Theorem 1.3 (c) holds for mp > n, and r must be replaced by - n: r in the definition of ~ (y). Also we remark that any function in r

54 class L (G) becomes a function in class L (E ) by simply definP P ing it to be identically 0 outside of G. 1 For notational convenience, we shall write P2 for W2, PI for the class of functions in PZ which are absolutely continuous along almost all lines parallel to the coordinate axes, P"' for those functions of class PZ which are continuous, and H for W22 We propose to consider surfaces (vectors) whose boundaries are free on fixed manifolds. R. Courant has shown [11, p. 220] that there are smooth vectors which minimize Dirichlet's integral but whose boundaries are not continuous curves. Although there are some conditions known under which minimizing surfaces are continuous along, and up to, the boundary (see, for example, [10, 17]), we shall use the torus as our manifold for the sake of convenience, but we shall produce a general existence proof which holds even in case the manifold is the one of Courant's example. Thus, we shall make a precise definition of the concept of a surface having its boundary on a manifold. Definition 1.6. [11, p. 202] Let, be a (topologically) closedmanifold (i.e., closed connected point set) in the 3-dimensional space E, and let p? [z(u,v)] denote the shortest distance from the point z(u,v) to the manifold'b2. If z is a vector defined on Q, we say that the image of Q under z, or z(Q ), lies on if ((u,v) (un, vn), (u,v) e Q (un, v) e Q, implies

55 that p [z(u,v)] - O, for all such (u,v), (UO, Vo). (We shall sometimes say that "z lies on, " or "the boundary of z lies on.") Let 6 be a solid torus in E, and let H be a fixed circle linking T" and situated so that for every point p on H dist (p,') = d= constant > 0 and we take d small relative to the diameter of Z'. Our manifold will really be the surface of I, and the only reason for mentioning the solid torus above was to prescribe the correct homology class for H. Definition 1.7. We say that z links H, or z(Q ) links H, if there is a number r > 0 such that for every closed curve C, homotopic to Q, lying in the strip S C Q adjacent to Q having width r, z(C) is a closed curve linking H. For all the notions of topological linking and intersections we refer to Alexandroff-Hopf [1, pp. 413-425].

56 2. The Main Lemma Concerning Boundary Value s We first state a lemma of Reshetnyak [13, p. 747]. Lemma 2. 1. Let z be a vector of class P (Q), defined on Q and continuous on Q. Then there exists a sequence ({ (u, v)} of continuous piecewise-linear vectors which are nondegenerate in any triangle A C Q, and having the properties that as n- o, { } converges strongly in P2 to the vector z, while on Q ({ converges uniformly to z. Moreover, if z(u,v) is continuous on Q, then ( } converges to z uniformly on every closed subset F C Q whose boundary does not contain a point of Q. Furthermore, there is actually a sequence of continuously differentiable vectors satisfying all the conditions above. The goal now is to prove a type of closure theorem for admissible vectors, which will be instrumental in our treatment of free boundaries. We remark that p [z(u, v)]-* 0 uniformly as (u,v) -* Q if the boundary of z is on ~77 [11, p. 20Z]. In the following lemma we shall take Q to be the entire upper half-plane for the sake of simplicity. By a conformal mapping, the conclusion will hold for the case of the square. Let Q be the upper half-plane v > 0, and denote by Q' the set {(u,v) v > r}, r > O. We shallcall call P(Q) the

57 class of all vectors which are in P2(Q' ) for every Ti > 0 (i. e. 2 a in P2(Q) and continuous on Q0). Lemma 2.2. Let {z } be a sequence of vectors of class H2(Q1) which converge uniformly on each Q' to a vector z of class P"'(Q'), and assume that the image of Q: [-oo 0. Then the image of Q' under z also lies on Proof. Since for each p and each Q' C Q we have D(z, Q') < K, letting nr -- O we obtain D(z, Q) < K. By the lower semicontinuity of D(z, G) with respect to weak convergence in P2(G) [20, Chapter III, ~4], we have D(z, Q ) < 1irm D~z %,Q'z)R lim D(_z -Q ) < Z < oo p -o 00 p-00 P so that letting r < 0 we find D(z,Q) < K Let e > 0 be given. Then we may choose a number h, 0 < h <', such that in the strip Qh:[ [~~ < +u <; 0 < v < 2h] 4 we have D(z, Qh) < z We recall that since z and z are in class P (Q'), P 2 each is equivalent, respectively, to a function z, z which is absolutely continuous on almost all horizontal and almost all vertical lines. W.e maryra-ssue (since all z, z are continuous) [2, p. 181]

58 that z z z, and that the line v - h is a line of absolute p continuity for all the functions concerned. Let Sh: [h/3 < v< 2h; -1:. u < +1] = M. In this strip 4 we still have D(z, Sh) < V. By Lemma 2. 1 there is a continuously differentiable function (u, v) defined on Sh such that P2 (M) 2 * (ii) jz(u,v) -'(u,v)i < for all (u,v) ~ M (iii) |z(u,v) - (u,v)l < E, for all (u,v) c F where F = [h/3 + T < v < 2h - r; -1 + h/3 , (u,v) = 0 for j(u,v)J > r, SS (u, v)du dv = 1. By writing (U(u, v) we really mean the vector 2 E (OU 77', ), and when we write zT we really mean the vector (z (uv)qO (u,v), z (u, v) (u,v), z (u,v) 0(uv)). We shall always take T7 < h/6. Define 3h/2 1-h/3 r (u,v) - i(x,y)q (u - x, v - y)dxdy h/2 -l+h/3

59 00 Then we know (Section 1) that r (u, v) is a C vector and rr (u, v) --' (u,v) uniformly as r7 0. Thus we may choose rn so small that (iv) I r (uv) - r(u,v) < 2 Let J (r~ G)= = Ir+ zjr I + Jr [ )dudv G UU UV VV G [-1 +h/3 < u < 1 - h/3;h/3 + T < v < 5h/3 ]. Let h - h/3 + T. Then J(r,G) < + oo. We may choose a number h', h < h' < 5h/3, such that (v) D 1 I(r,G C <, G1 [-1 + h/3 N (2A 1) lz (u,h") - z(u,h")l < -1 < u < 1 We now obtain the following chain of inequalities:

60 u h' u i 5, L (u,h")du| < i 1 - (u, h')du dv I ~~0 hh'- hO h' |uj -< - | / ~ (u,h") dudv h' - h u h O (2. 2) h' |u| < 1 (u h) ( u)(u,v)) dudv h - h' u h O h' Ju! *-' - h *1 1 1~ r(uv)l dudv h' -h u h 0 First consider the second member of the right-hand side of (2. 2): h' juJ h' ju u h' lul (u, v) du dv < du dv (u v du dv h - h < l I 1( G + (2. 3u) < I l 2 E as follows by (i) and (vi), where 4 = h' - h Next we obtain

b~ > ~I Z:rJ IA IA IA IA IA IA IA A.I5 o1 ~1 _ I- _. I~,' i~. - I~--- -- - 11 _.~. - o.._ r~~l- rI o ~1~""~~~~-' -''U~~' o ~ ~ _.M ~'~ -- ~ I> > I;, r < K ~ ~~~ < e _~ I ~ ~~~~ ~',_. _ o -.._. O _,. ^ > + 5 - _~~~~~~~~~~~~ I~,_ --'"lH U) Ut ~ - - -3 I.l l:~.- - ~~~~~- > r I^ -- - + 1 ~~~~~~~~~~~~~~~~~~~_. CD CD~~~~~~~~~ - - ~ ~~~ ~~~ ~~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~... 5 - )-JI ~~~~~~~- ~.-.. —- "~ ~1~~~~~~~~~~~~~~~~~~~~~~~~i C-,- ~ ~ ~ ~ ~ ~ ~~~~~~~~~~~~~~~~~~~~~~-,-~ ~ _~~~~~~~~~~

62 Now using (iv) we obtain h' |u| h' |uI (u, h,')- (u, v)I du dv < 5 |(u,h") - r (u,h") dudv h 0 h O h' lul + NS. Ijr (u,h")- r (u,v) dudv h O hi Juj. S Ir(u, vU)- (uv) dudv h 0 h,,,I~~1 "'' I ul.+. lul u + I 2 + I lul vu U 1' (u, h")duo < (2 1 u + (lul)2 2 L (2 (2)' ) < [21u12 + U|2 ()2 + u12( ()2 ]~, since u| < i1 4 4 Therefore, for all u, -1 < u < 1, we have (since (u,v) is of class C ) )

63 4 44 + 1 1 1 Hence for lulj < 61 2 or [2 +( ) ~ (_)2 ]g 4 [2 + ( ) + () ] 4 )I we obtain (2. 9) h(u,h") -'(O0,h")j < Now combining (2.9) with (iii), we get (2. 10) jz(u,h")- z(O,h")j < jz(u,h") - ~(u,h")j + J (u,h") - rj0,h")J + I C(0,h") - z(0,h")J < t + ~ + e = 3e Finally, from (2. 1) and (2. 10) we conclude that for Jul < 6 and p> N P P (2 11) |z (Oh") - z (u,h")| < |z(Oih") - z(uh')l + lz(uhl) - zp(U h")j < 3 + E = 4. Next, let p > N be fixed. Since z e H2(Q) we have p z E P (Q') and furthermore z PZ(Qh on the strip Qh of P 2 P h height 2h. Thus there is a sequence { (u, v) } of functions of class C (Qh) such that (2. 12) 1 - z j - 0 as m -- o lJm p_ Po(Q_ )

64 We may assume that this norm is < e for all m. h" Let G (u) - | (u,v)|2dv Then G (u) is of class 0 C on the interval [-1 u << 1]. We may apply the mean value u theorem to 3 Gm(t)dt on the interval [0 < u < 6]. Thus there is 0 a number u, < < < 6, such that m - m6 0 6 (2. 13) 5G (t)dt- G (t)dt = G (u ) (t)dt m m m m m 0 0 0 where u depends upon m. We have 6 h" 6 h" 6 h" f I 1 |m(t,v)|Zdvdt} < 5 I +m P + }. t -, rn 0 0 0 0 0 0 6 h" so I j (t,v)jdvdt < + K + K Thus 0 0 6 h" 6 2 + 2K + K > m |t,V) dvdt v G (t)dt 0 0 0 - 6G (u r m 6 \ m(u mv)! dv 0

65 and hence 2 14 ) K2 )2 (2.14) | | m(Um')l dv < 0 Now the sequence of numbers {u } is bounded, so there is a subsequence f{Um} which converges to a number uO, 1 O < u < 6. Rename this subsequence {u } and assume that u- < U for every m, u t u and u - u o < l/Mr. (There is no loss of generality here because of the nature of the proof which follows.) We have h"f h" (2.15) { [z (u,v) dv}P<{ z (u,v)-' (uo v)IZdv}2 p 0 1 + { m(uv) - m(umv)l Zdv} + (ur h"dv 0 + us examine the second (U) integral on the right-hand side Let us examine the second integral on the right-hand side.

66 h" h" UO {%S | m(U, v)- I(,U v) Zdv}2 <{ (u, v)duI dv}2 0 0 u m h12 u0 <Y | I 2 mdudv}2 0 u m h" h +{ I z | dU dv v} 0u uvu h." Uo 2 2 U VU U threi0a sc u h" U that m > M implief2s {t/6-z Thduufdv < mt[M 2/6, andv u0 vu by the absolute continuity of the integral (and the fact that u t u there is an M such that m. > M2 implies du d 2 < /6. Thus for m > max[M M we have p P 2

67 h" (2. 17) U{ | |'muv)~ -- rm(u,v) lZdv < <2/3 0 Now examine the first integral on the right-hand side of (2. 15). First,we know that { (u,v) -- z (u,v) almost everywhere m p on Qh' Let E C Qh be the exceptional set. Then measure of (E) -- e(E) - 0, and Qh = E - B, B r E = ~. Then for almost all Uf [-1, 1], the set I = { (, v): 0 < v < 2h} has intersection with u B the set B_ = I r B, and B > O0 where I..I denotes oneU u u dimensional Lebesgue measure. In fact, for almost all ui, |B 2 Zh. u Assume first that u = u0 is one of these points. Then j Bu = 2h > 0 -U0 and' (uO, v) - z (u0, v) for almost all v E [0, 2h]. Therefore, Pv F_ (v) z V) (uO ) v) 1 2 O m pUo0 m0 o almost everywhere on [0) 2h] as m -- oo. Then we know that F (v) -- 0 almost uniformly on [0,2h]. That is, for every r7 > 0 m there exists a measurable set H C I2h = [0,2h] such that IZI - HI < 7H and lim F (v) - 0 uniformly on H. Let us write - H= I2h H. Then lim | F (v)dv = 0. Thus there is a number M such that m oo m 3 H m > M implies (2018){ |Fm(V)dv} _ { = Zp (u0, - m(uv)l 0dv} 2 < 2 /6 o H H

O ( H- O0 H~ t ~IApnp O z[' 0 n -d - ~ j ~T (0Z z 0 on On H- On 0 HnAd nA {p AP[np i zl + npAp[np z ] I I On On n H- On n H- 0 {fpAnp np z npAp [np z On I On 0n flAd n H- Ad A H- 0 dn a d Ad 4{nPnp Ap } np(A('n) z {npAp (A n) z On I I VetM aas aM LtlM!I a914 uo e1xu! 4 s'ITf a9tJ ~uLuuasxX H- 0 4{npAp J(A'0n) - (A'n) - f i }+ d H - H- 0 - z{npAp ] (A n) (An) A z - | ( | I I ~~~~~~~~~~~~~t A H H 0 I{np~p~(Acn) n (AP (A dn ) Z} {p( o61 o d1} I7~ ~ ~I z' n) -(A'0n) z | }-={Ap |(.,,G~n) AP(, (A' ~n) - | 0 | } "89XaN 89

69 Now since z E H (Q') there is a number r1 > 0 such that if p 2 I-Hi < ~1' then the whole right-hand side of (2.20) is < 18' By the same steps as in (2. 20), 0 -H 06 1 --, m,,, rn2 < { u0 m I2dudv + (1 - u0) | 5 du dK -H 0 -H u But since { ~m} converges in the P (Qh) norm to z and since the P2(Qh) norm of z is finite, it follows that the PZ(Qh) norm of M is finite for each m and, infact, uniformly bounded for all m. Therefore, there is a number r72 > 0 such that if I-HI M implies v 12 1 2 (2 22a) {z | |Iz (SuV) ( (uv) Idvdu} < | pQ) 11 ( (2,22), H K - p m PZ2{h} 18 0 -H Now let 7 = min[r71, 72]' M - max[M1, M2, M3, M4]. Then there is a subset H C IZh' I-HI < a, such that m > M implies A r Ci.122 2 2 2 (23) { z (u v)- {m(U Xvu ) dv 18 18 18 6 -H

70 Combining this with (2.22) we obtain h"l (2.24) { i zp (u v) N- vm(UuV)[Zdv}2 0 V O v i< { \ Zl (U, nV)^ re (u,v) dv -H + IZ (UoV) - (U,V)lZ dv} HPv -H 4 4 36 36 2 3 We now combine (2. 14), (2. 15), (2. 17), (2.24) to get that for:..m > M 21 2 2 (2. 25) { z (u0 v) dv} < - + + +K f3 3+' Pv 62 0 Now suppose that i = u0 is not one of the points for which |B |- 2h. Then we may choose a point u1 so close to u0, u1 < uO, that h" U0 rtl) 2 ~1 2 (2 26) | z I dudv}2 <Y p 6 vu 0 U1 1 We must show that (2. 17) holds now. Since u t uOP for m sufficiently large, say, m > Mr, we have ul < u < uA, and so _ ~ ehv 1 m 0

71 h",m 2 5, z 1 dudv}2 < 6 The rest of the inequality (2. 16) 0 u holds now with u0 replaced by ul. Therefore, (2. 17) holds with u0 replaced by ul. We note that if u0 is replaced by ul in (2. 24), the inequality still holds except that we may have to choose H differently. Further, (2. 14) does not depend upon u0 at all. Therefore, replacing u0 by ul in (2.25) we obtain ht1 2 3 (2. 27) { \|z, u.,v)l2dv} < ~ + + 11. p 3 3 0 x for either j = 0 or j - i. We now replace uo by uO, where uO is whichever point u0 or ul yields (2. 27). We point out that u0 depends upon p. Fr:om (2.27) we have h" (2.28) izp(uo~ h")- zp(u0O)j = Iz (u0,v)dvj 0 Pv 0 h1) < 1{ 5 v)Idv - 0 h'V 2 2 - r 62

72 Before proceeding further, we make the following observations. Let K (u,v) = z (u,v + 6 ) where 6 is a positive constant. Then the vector' (u,v) is continuous for v = 0, and the values p (u, 0) lie on a curve M whose distance from the manifold P P can be made arbitrarily small if 6 is chosen sufficiently small. Now as 6 - 0, p (u, v) - z (u, v), so we may prove the lemma by writing z i mtnstewai of f and Ql instead of M. Note that with this convention, z (u,v) has continuous boundary values on the p Hence by the triangle inequality and (2.28) we have (2.29) pi [z (Ug0hII)] < zp(uo' No ) - z ( )T + p [p(U0)] - p{U0 -)2 < n2r{ +... + 62 From (2.11) we have for p > N since IUO0 < 6, I zp(u0,h")- z(0,h")l < 4e By the triangle inequality, (2.30) p [z(O,h")] < [z(O,h") - z.(uoh"')l p, [zp(uO hi)] < 4 uo + { h + 3 2 9 62 and the right-hand side does not depend upon p. Now we recall that

73 1 1 62 = h' - h h"= h + (h' - h), h<. 4 [2 + (j)4 +(~) ]+ The second term on the right-hand side of (2. 30) yields 2 i 4 (2.31) {+ 1 + + }K{2 } { ++ (g) + )a2 -<{ + 3 t + ( +K) + ( + ( } <I-: + (E+ K)E[2E2 + r/2 + NR]} Therefore, pk [z(Oh")] can be made uniformly small. Since (O,h") was merely a convenient point on the line v - h", we conclude that the same estimate holds for any point (u,h"), -o0 < u < +00o This proves the lemma. Corollary 2.2. 1. Lemma 2.2 is true if we assume the weaker condition that z E Pc (Q'). P 2 Proof. The only place in which z c H (Q') is used is to P obtain relation (2. 8); namely, there exists a number u0 such that 2 1 z (u, z (uO0)j <'-h{2"z + 1 R p 0 p 3(0 -- 62 This relation is used in (2.29) to obtain oP [p(u h"')] < R

74 Now since z lies on the manifold ~b2, there is a number d, 0 < d < 2 such that P [z (u0 d)] <, where d depends upon p. Let D - Qd/Z. Then letting' (u,v) = (z * p )(u,v), where d/2'p P o0 is a mollifier and 71 is sufficiently small,' E C (D), and p hence' E H (D), and we have p [z (u,v)- (u,)[< for all (u,v) E D Now we may repeat the reasoning from (2. 12) to (2. 28), except that all integrals involved with respect to v have the form h" h"' (.. )dv instead of (.. )dv. Then we conclude that there is d/2 0 a number u0 such that I Cp(Uoh")- p(u0od)l < R Therefore, (2. 28') lz (u0,h") zp(- z,d)j < zp(UO h"J) (u h")j + Ip(u0ph")- p(U0 Pd) + j p(uo, d) - zp(u d) < 2e +-R Finally, (2 291) Pp~ [zp(uOhol] < Ip(uo0, )- z p(ud) l + PR [zp(uO d)] < 2e + R + e - 3 + R

75 From here we may repeat the reasoning from (2.28) to (2. 31) which gives the conclusion that z(Q ) lies on fl. This proves the corollary. Corbllary 2.2.2. Lemma 2.2 remains true under the weakened conditions: (1) The boundaries of the z are on manifolds M P P where M tends to a continuous manifold 7 in the sense that the p greatest distance from points on M to points on 7/ tends to O p as p -p 00o; (2) z E P"(Q') p 2 Proof. We must show that, under the remaining assumptions of the lemma, the relation PM [z. (uv)]O as v o p implie s p [Z(U, )] 0 as v 0 Let g(M, 7 ) be the greatest distance from any point of M to a??. By the triangle inequality we have pk [z(uYv)] < PM [z(u, )l + g(Mpm) p and hence pA [.(UV)] < lim PM [z(UM V)] p-. a oo p Therefore it is sufficient to investigate PM [z(unv)]. M plays the role of 1L in the lemma. Relation (2.31) shows that the bound

76 on PM [z(u,v)] is independent of p. Therefore, the same bound p holds for p [z(u, V)]. But this bound can be made arbitrarily small, uniformly in h". This completes the proof.

77 3. Some Lemmas Concerning Quasiconformal R epre sentations of Surfaces In this section we shall define A-admissible vectors for our problem and then show that each one of these may be replaced by a vector with a quasi-conformal representation which is again Aadmissible, and such that the new vector does not increase the integral Io. Definition 3. 1. A vector z defined on. Q is called A- admissible if (i) z E P'(Q); (ii) z(Q ) lies on a manifold whose greatest distance from the torus T is < d/4, d dist [H,Z]; (iii) z(Q ) links H (iv) I (z,Q)< +oo. We now state some definitions and a representation theorem from surface area theory. These may be found in Cesari [9, pp. 472486; 8, pp. 266-271]. Let S z z- z(u,v), (u,v) e Q, be a continuous surface (vector, mapping). For each point z0 E E which is in the graph of S, we denote by S(zO) ={w = (u,v) Q: z (w) = z}

78 The set S (z), z e [S] = graph of S, is a closed subset of Q, and hence its components y are subcontinua of Q (possibly single points of Q). Let us denote by G the collection of all continua -1 ~ Q which are components of at least one set S (z). The collection G has the following properties: (i) each point w e Q belongs to one and only one continuum Ty of G; (ii) G is the collection of maximal continua of Q on which the vector z = z(w) is constant; (iii) the collection G is an upper semi-continuous decomposition of Q. Definition 3.2. A surface S: z = z(w), w c Q, is called a base surface if for any continuum y e G the open set E -' is c onne cte d. Definition 3.3. A surface S z = z(w), w e Q, is called non-degenerate if (i) for any continuum y E G the open set Q- y Q is connected (ii) Y D Q for some y E G implies y- Q. The property in the definition of base surface and the two properties in the definition of non-degenerate surface are invariant under Frechet equivalence.

79 Definition 3.4. A representation S: z = z(w), w c Q, is called quasi-conformal if z E P"(Q) and z E P' (Q), and if 2 2 E = G, F = 0 almost every where in Q, where 3 3 3 E- = z z(w) G = z z(w), F z.(w)z(w), i-1U i=l V il U V z(w) = (z (w), (w), z (w)). Lemma 3. 1. (C. B. Morrey [18]; L. Cesari [9, p. 484]) Every non-degenerate surface S with finite Lebesgue area has a quasi-conformal representation S z = z(w), w E Q. For any representation of this kind we have Area (S) dw = EG - F dw (E, + G)- D(z,Q). Q Q Q Let z E P"(Q), let S: z - z(w) be a base surface, and let G = {g} be the upper semicontinuous collection of maximal continua on which z(w) is constant. Let { g} be the collection of all those g E G such that g r- Q f/'. Finally, let F {w w E g, g g { g}" Lemma 3.2. [4, p. 907]. F is closed. Let H- Q - F. Since Q C F and Q Q + Q, then H = Q - QF, and therefore H is open in the plane. Let { a.i be its (open) components, and denote by a. the boundary of a.o 1 1 Then there are finitely many or countably many a., and each p E ca. belongs to a continuum. g E { g}

80 Lemma 3.3. [4, p. 907]. Each a. is simply connected. Lemma 3.4. [4, p. 919] Let n be any positive integer. Then there exist at most finitely many cri such that diam{z(q.) >-. 1 n1 Now let z be an A-admissible base surface. Then by Lemma 3.4 there are finitely many components a., i - 1, 2,..., N, such that diam{z((a )} >'. (Note that there must be at least one i 4 such a., or else z would not be A-admissible. ) Lemma 3. 5. If diamz(a)} < 3d then the intersection number of z(a.) with H is 0. 3 Proof. A necessary condition that z(a.) have a non-zero intersection number with H is that some point of z(aj) touch H. But diam{z(aj)} = diam{ z( j)}< 3d and since a. rnQ = ~* * d dist[z(co r Q ) ] < - 3 4 Thus no point of z(a.) can touch H, and the intersection number is 0.

81 Now we separate { c.}, i = 1, 2,..., N, into two mutually exclusive subcollections C {! }, s = 1,. K S and = {pi' }1, r = 1,..., L, L + K = N. The collection C6 is defined in the following way. Let P0 be the center of gravity of?_, let II (H) be the plane containing the circle H, and let II' be the (unique) plane passing through P0, through the center h0 of H, and perpendicular to H (H). Then a.i e if and only if for any plane II passing through P0 and perpendicular to II' we have Ho z(a)/ (a It is possible that 3 =', but Q i ~ since z is A-admis sible. Lemma 3. 6. The intersection number of z(a'.' ) with H 1 r is 0 for all a' e r Proof. Since all' E there is a plane I satisfying r the above conditions with II r z( a!' ) ='. We have r maximum dist[z(a' ),<] < max dist[z(Q ), Z] < - i 4 r and d was assumed small:relative to the diameter of 77. Therefore, z(a' ) is not linked with H and so the intersection number of 1 r z(a." ) with H is 0. r

82 By Lemmas 3.5 and 3. 6, and by the hypothesis on z, k linking number{ z(Q *)H} = intersection number(z(q! ) H) O 0 s-l s Thus for one of the c', call it c1 the intersection number of s1 z(al) with H is / 0. Furthermore, z(al) "winds around the hole of the torus" since for any plane II satisfying the conditions on page 81, I z(l) We mention also the well-known fact that since c is a 11 bounded, open, connected, simply connected set in the plane, a1 is closed and connected. rf3 z. (...:'~ ~ra ~:~:~,~~~~~~~~~~l.f

83 We shall now describe a process for cutting off the "pinches"' from S: z = z(w), w E Q, leaving an A-admissible surface S: z = z(w), w E Q, which is non-degenerate. Such a surface S always has a quasi-conformal representation (Lemma 3. 1). The complement of -a1 in Q, (D (a1), is open in Q Thus C (a) ='- 3., i3. open in Q, I a finite or countable index 1 EI 1 1 set. We notice that. C a, and moreover,B C gi {gi} 1 1 1 i 1 Therefore, z is constant on 1i C g; Now we define a vector z (w), w c Q, as follows: z(w), w a (3.1 W)-(w) z(i, W E i Let I_ - {(u,v): u =, 0 < v< 1}, u 1 [0, 1]. That is, I- is the intersection of the line L_: u u- with Q. Let u u = n' ((1), an open linear set, and let F =I- r1 l, a closed linear set. Then 9 n F_, and f, u 1 u u I_- 0= - F.. The set (f_ may be written as u u u u = k —' J., J. disjoint open (intervals) components of r_, I a u 1 1 u i~I finite or countable index set. F. may be written as F._ - DX, u u XcA DX disjoint (closed) components of F.. We note that z(w) X u constant = c. for all w E J.; for if J, contains points in 3i and 1 1 1 13j i f j, it must contain a point of 13 (since 13, 1 are comJl 1 j ponents), and since 1. C a1 it also contains a point of',, a

84 contradiction. We recall that z E P1'(Q), and therefore z is equivalent to a function zO E P (Q). Since z is continuous, we may take z = zO EP1(Q) and z F P (Q) [2, p. 181]. Lemma 3. 7. If z is absolutely continuous on the segment I_ [u- = u, 0 < v < 1], then z is absolutely continuous on I. U U Similarly, z is absolutely continuous on I.. Proof. Let t > 0 be given. Then there exists a number 6 > 0 such that for every finite collection { (vi.,v )}, i = 1, 2,..., Ng v < v < v <... < vN, of non-overlapping intervals with 0 1-2 3- N N v -I Vil < 6 it follows that Z |z(u,v) - z(u, vi l) < e. For i=1 i-1 brevity we shall suppress u in the argument of z, u being fixed throughout. Case 1. v., vO F_. Then z(v.) z (v.) z(v. ) = (v. ) -1 u 1 1 i- 1 and hence 1I(v7l) -v(v.)l Iz(v.l) z(v )l. Case 2. v., v. 1 5.. If both are in jk for some k, -1 uk then j (v.) - Z(v l)I = Ick- ckl = 0 < z(v) - z(vl )l If Yvi. c,v e then v. E J, v E J r J s. Let 1 k i-l I 1 r i- 1rL J - (a,b ), J - (a,b ). Then r r r s s s v. < b < a < v. 1 s r 1 and b e, a E 1 Therefore, s r k

85 z (v.)- ((vi ) = Iz (ar ) z(b ) b - a I < V. s r <- i v i-1 Case 3. v. (9_, v. F_. Then v. F J C, for i1 u i-1 u 1 r k some r,k, and z(vi-l) z(v ). We let J (a,b ), so v < a < v < b, and ar EA. Thus i-i1 r 1 r r k I(v.) - Z(v )1 = jz(a) - z(V. )I and r i-U1 - i i-1 For vi e F-, vil E u the analogous result holds. Now combining cases 1, 2, 3, we obtain N N1 N2' j(v.) - i(v 1 )1 < l(v. )- (v- ) + jz(a. ) - z(b. )I 1 i=l 2 2 N3 + C z(a )z(v ) i3 1 3 33 By rearranging and renaming on the right-hand side, we obtain a sum of the form N z |z(v') - z(vi)l, where { (v! v)},i 1, i 1..' i i=l N is a collection of non-overlapping intervals with vI - v' I < 6. Thus the right-hand side is < g, and this proves the absolute continuity of z on I_. The proof is the same for I_.

86 Lemma 3.8. z E P2(Q). Proof. Q = a — CW (1) Let w E a1; then z(w)= z(w), so Iz(w)2 = z(w)|. Next let w E ( (i1). By our cutting process, z(w) is a point of z(Q ), and I z(Q')[ is uniformly bounded. Therefore, z(w)12 is uniformly bounded on (, and hence c E LZ(Q). By Lemma 3.7, z, z exist a.e. in Q. We must show u v that each is in LZ(Q). Let w0 = (uO, v0) be a point at which u and z are both defined. u Case 1. Assume w0 E al1 Then z(w) -z(w) in a neighborhood of w0. Therefore, (w0) = z (w0). Case 2. Assume w0 /3i for some i. Then z(w) - c in a neighborhood of w0. Therefore, z (w0) = 0. Case 3. Assume w0 C i. Then z(w0) = z(wO) = ci. (i) If (u0 + h, v) e 3i, then jZ(u0 + h,vO) - f(u0 v)O) = Ic - c I 0. (ii) If (u0 + h, v0) aci, then z(u0 + h v0) = z(u0 +h, v0) and hence I[(u0 + h, v 0) - z(u0,vO) = I z(u0 + h, v0) - z(u0, 0) (iii) Let (u0 +h,v0) e 3j j fi, and assume h > 0. Now L h 3. = I, L -v = v, is an open linear set, VO J v v 0 and (u0 +h,v0) belongs to one of its components, say J1, an

87 open interval. Let (u0 + h1,v0) be the left-hand endpoint of J1l so O < h1 < h. Then (u0 +hl,v0) E, so i(uo + h, vO) - z(uo, Vo) - fz(uO + h, v) - z(u, Vo) < j z(UOvO) - z(u +h12vO)I + Iz(u0 h1,vO) - z(u0 +h, v ) < Iz(u,v ) - z(uO+h1,vO)[ + [z(u0 + h1,v) - z(u0 + h1 O) < [ z(UO vO) - z(uO + hl1 vO)l and h, - 0 as h - 0. Moreover, since 0 < h1 < h, |(uo + h, vo) - z(u, Vo)l < z(uo + h1, vO) - z(uo, o)l h - h The analogous argument holds for h < 0. Now combining cases 1, 2, 3, we see that the absolute values of the difference quotients for z are bounded, point by point, by the absolute values of the difference quotients for z. Since the derivatives are known to exist almost everywhere in Q, we have that a. e. in Q (w < zw1 I,W) -(w) J< l zv(w) I Since z,z E L (Q), so are z, z. Therefore, we conclude u v 2 u v that z E P2(Q), and moreover, c PE(Q).

88 Corollary 3.8. 1. Z E P" (Q) 2 Proof. z(w) was continuous on Q, and so we may write [(uo + h, vo + k) - F(uo, vo) < jZ(uo + h, v0 k) -'(u0 + h, v0)[ + I j(u0 + h, v0) - F(uO' v0)[ and repeat cases 1,2,3 in the proof of the Lemma 3.8. to conclude that the right-hand side approaches 0 as (h,k) - 0. Therefore, z is continuous, and therefore, z E P't(Q). Therefore, we have shown that for any A-admissible vector z which represents a base surface S, we may choose a vector which is also A-admissible and such that the surface S: z - (w) is non-degenerate. Moreover, since J I - for all w ), cl' J being the Jacobian vector for z, and since IO(r,Q) < M J Il we see that Q IO (,p Q) < j{(zQ), and [S] C [S], where [S] denotes the point set of the surface S. Since S is non-degenerate, and since it has finite Lebesgue area, it has a quasi-conformal representation (Lemma 3. 1) g which is again an A-admissible vector. By the invariance of I0 under change of representation, we have I0(P,Q) = I0(z,Q) < I0O(z,Q). Lemma 3.9. [8, p. 271] Let S z = z z(w), w E Q, be a surface of class P' (Q). Then there is a base surface So z - z0(W), w E Q, such that aS- aS, [So] C [S], zO(W) E P2), and I0(Z0, Q) < In(Z, Q).

89 We collect the main results of this section. Theorem 3. 1. Let z- z(w), w E Q, be an A-admissible vector. Then there exists an A-admissible vector z - C(w), w c Q, such that I0( Q) < I0(z,Q), [z] C [z], and Y(w) is quasi-conformal, i.e., E =G, F- 0, a.e. in Q.

90 4. Admissible Vectors and Minimizing Sequences Definition 4. 1. A vector z = z(w), w c Q, is admissible (i) z C P2(Q~) (ii) z(Q ) lies on the torus -; (iii) z(Q ) links H; (iv) IO(z,Q)< +oo Let (4.1) L - inf I0(z,Q) where the infimum is taken over the class of admissible vectors. Our variational problem is to show that there is an admissible vector z such that I0(z,Q) = L. Definition 4.2. A sequence of vectors { z is called an P admissible sequence if each z satisfies the conditions of Definition 4. 1, except that z (Q ) is not necessarily on, but on a manifold p which approaches I as p -- 0o in the sense that the greatest distance of points of 7 from tends to 0 as.- p oo. Let (4.2) 6 = inf(lim inf I(z, Q)) p te i0 where the infimum is over all admissible sequences.

91 Definition 4.3. An admissible sequence {z } for which P I (z, Q) - 6 is called a generalized minimizing sequence. op Remarks. (1) Obviously 6 < L (2) We assume, as usual, that there is at least one admissible vector z with I (z,Q) < +oo. Lemma 4. 1. Every generalized minimizing sequence (g. m. s.) { z} may be replaced by a g.m.s. s. } such that each z is continuous on Q. Proof. For each p we choose a concentric square Q [0 < T < u < i-; T < v < 1- T], where T is chosen P P- - p p- p P so small that the (continuous) curve z (Q) links H and lies at P P a distance from 97 (and hence from t) going to 0 as p oo. p This is possible since { z is a g.m. s. By a conformal mapping u = u (x,y), v = v (X,y), we may map Q onto Q and obtain a P P p vector zp(Xy)- Z (u (X, y),v (xy)) defined on Q and of class P' (Q). The integral IO is known to ~ 0 be independent of representation, and so I0(Z,Q) I Io(z,Q ) < Io(z,Q). But I0(z, Q) 6- 6, and hence lim I(Z, Q) < 6 On the other p p 00 P p —, oo hand, { z } is an admissible sequence, so lim Io(Z,Q) > 6, Pp -00 Thus lim2 IO(Z,Q) = 6. Therefore, we may choose the sequence p -00

92 {z } such that IO( Q)6 p oP Let us now introduce the integral [21, p. 571] (4.3) I(z, G) = f(z, p, q)du dv G 1 2 3 1 2 3 where p = (z, z, z ), q = (z, z, z ) and U U U V V V 24.4) r2~z~) 2 hiM+m 2 E G 2 2 (4.4) f (z,p, q) _F (z,J) + (2~ ) [ (2) + F ] f > 0 2 2 E = IPp G= qlj F=p q, andwe notethat J =pXq, where "t*" denotes scalar product and "X" denotes vector product. Then f(z,p,q) is as smooth as F(z,J) - F(z,p X q), and more over (4. 5) ( )< f(z, p,q) < (I + q ) To see that (4.5) holds, we recall that ip X q12 = P121 2 2 (p q)n2 and by as sumption, F(z,J) = F(z,p X q)< MIJJ = MIp X q[ Thenwe have 2 2 M+m 2 EG_2 2 f (z,p X q) =F (z, p X q)+ ( 2 [% +F < M2 pX q 2+M[( -p q1 )2 + (p q)] < M 21 4 2

93 This demonstrates half of (4.5), and the verification for the other half is similar, As a consequence we see that (4. 6) 2 D(z, G) < I(z,G) < M D(z, G) 2 - - D and if z is quasi-conformal, I(z, G) = I0(Z G) The basic idea behind the introduction of the integral I is that we must always have (4. 7) I0(zG)< I(z,G) in addition to (4. 6). Moreover, we see that 2 E-G2 2 2 E-G 2 I2 EG- F + {(') +F2} = IJJ I+ ) +F E+G 2 and so (E-G)2 }2 {(E+G) IJ 1 Hence we may loosely say that the integral I(z, G) gives I0(z, G) plus "the average over G of the amount z misses being quasi-conformal.' Furthermore, it has the feature that boundedness of I(z,G) is equivalent to boundedness of D(z,G). Lemma 4. 2. I(z, G) is invariant with respect to conformal change of variables.

94 Proof. Let'(w) = (u,v) = x(u,v) + iy(uv), w = u + iv, be a conformal mapping of the (Jordan) domain A in the w = u + iv plane onto a (Jordan) domain B in the x + iy plane. Write 2%(u,v) = z(x(uv), y(u,v)), z e PF'(B). Let E, F, G correspond 0 2 to z, and let E, F, G correspond to 2iO.. Then (I(ziB)/ nM+m2 E- G2 -2 l(z,) = F (zJ ) + (M+m2 ) [( ) + F ]dxdy B = fl'V(Z)(J) [( + — ) - IJ I ] dxdy B But the integrals F(ZJ J),i (E +G), l B B B are known to be invariant. Therefore, the right-hand side of the expression above may be written 2 2 S 2(FZ~(,J Jr) | j(w) -4 + F | (w4) [ dudv A IV dv- A. )

95 Combining Lemma 4. 1 and Theorem 3. 1 we see that there is a generalized minimizing sequence {z } such that z c P"(Q)9 n 2 z E P' (Q), and z is quasi-conformal. n 2 n 3 Let F be a closed solid cylinder in E with the properties that the distance from F to t is a positive number 7, and if S is any surface whose boundary has greatest distance from - less than 7/2 and whose boundary links H, then any line parallel to a generator of F has a non- zero intersection number with S. Since the boundary of z links H, there is a point w E Q n n which is mapped by z into the interior of F, and a neighborhood U of w whose closure is mapped into F. Among all such neighn n borhoods U we choose one of maximal size, in the sense that for n every line parallel to a generator of F there is a point w c U which maps onto this line. The continuity of z assures the existence of such an open set U. In such a U, there is a point w such n n n that z (w ) lies on the axis of F. Now by a conformal mapping of n nL Q onto itself we may send w onto the center P0 of Q. Then the neighborhood U is mapped onto a neighborhood U of PO. Let us assume this has already been done and that z is the resulting function. Then z is still quasi-conformal, and the values of n Izn 2 Q) and I(z Q) have remained unchanged. Let c be the segment in Q which is parallel to the v-axis, has P0 as midpoint, and has iength, say, 1/8. Then a portion of

96 a is contained in our distinguished neighborhood of P' If a lies completely in this neighborhood, then the image z ({a) lies completely in F. If a does not lie completely in this neighborhood, then we make the following change. A portion of CT say a n lies in U. n n We shall map Q onto itself by a diffeomorphism d so that outside a thin rectangular strip R about a the mapping is constant, the segment ay is stretched onto the segment a, and such that if (e ) = z (d- (w)), then I(,R n)< I/n., I(z, R )< i/n. Hence n n n n n n n maps a into F, and since d is a diffeomorphism, n n IO(,Q) I (zn Q). Therefore, we have 0< I(P,Q)-I(z,Q) =I(z -R ) + I(, 9R ) (z,a-R ) I(z,R ) n n n Rn < 1/n Since I(z,Q) 6 and I (z Q) - 6 we still have I(,Q)- 6 n 0 n n and I ( Q) n I (z Q) 6 0 n 0 n Therefore, replacing n by z, we have a minimizing sequence { z } of vectors which map Y into F, and which are quaki-.confoarmal except on a rectangle R which may be chosen so thal its width approaches 0 as n oo. The sequence of numbers {I(z Q) is bounded. By relation (4. 6) this implies that { D(z Q) } is bounded, and.by [23 2] sois {L(z, S z Let R be a bound for Q

97 these numbers. Recall that Q is the square with vertices (0, 0), (0, 1), (1, 1), (1,0).'We now extend the domain of definition for all the z to the square Q1 with vertices (=1,-1), (-1,2), (2,2), (2,-1) by reflection in the sides of the square Q, and then by reflection in the sides of the four resulting squares. The new vector, which we again call z, is clearly still in class P'(Q1) and quasi-conformal. n Let p be a Friedrichs mollifier (Section 1). Then 00oo (i) p E C and has compact support jwl < P; (ii) > 0; (iii) q p(u v)dudv =- 1 B0(p) = {w w I < P } 0 B0 P) We shall always take p < 1/8. For each n we form the -mollified function Zn (u, ) (z * )(u, V) z ( ) n(uv B () (u, v) where "z " means the vector (z p,z 2 ~z P ) n and "z np n p n p n p nI means the vector (z * ~,z *i, z a ) P The following facts about zP are known [10, p. 14]: n pa oo (a) z e C (b) z - z uniformly on Q as P -;

98 (c) (p) = ), (z) (z ) n n-' n n n U u v v (d) IIn zn n d - O and IIz Z L as P --- u u L2 n n L2 2 v v 2 For each n we choose p - p (n) so small that ()n n n n n 2 u u Lz v v L 2 2 (1) zP(u, v) - z (u, v) < (1/) (dn/4) where d /4 is the greatest distance from points of z (Q ) to the n n torus Z, d -0 as n — oo, and d < d = dist[H, Z], for all n. n n Now we set y (u,v) = z (u,v), and we consider the domain rn n 10,ts?~~~~~ ~00 of n to be only Q. Then n (Q) is a continuous (in fact, C ) curve linking H (since it may be continuously deformed into z (Q ) without touching H), and the greatest distance from rn(Q ) to approaches 0 as n — oo. Thus the new sequence {n } is an admissible sequence. Furthermore, if p = p (n) is chosen sufficiently small, the segment o is mapped by n into the closed set F (page 96' ). We shall show that this new sequence is actually a generalized minimizing sequence. Lemma 4.3. I0(,Q) -6 as n- oo O n

99 Proof. i,Q) I (z I I [F(~nJ, ) - F(z,J )]dudvj ~Q n n < |F( n,J ) - F('Ja )] +'IF(n, J )- F(zJ )8 Q n n n n (4.8) < 2 LI J - J | + L i Z I (relation 1. 1 (')) Q n n < I J J j | + L1 (1/n)(dn/4) meas(Q). Q n n We obtain 2'3 2 3 g' ~ z z n n n n U U U U z I 2 3 3 n n: z z n n n n v v v u 33 2 3 2 z3 -n n n n n n n n n U V V U U V V U < 32 z 3 2 3 2 3 n. n n n n n n n U V U V V U V U 3 2 3 3 3 2 (4. 9) < z + z )z 4 -n n n n n n n U v v v U n n n n n n U U V V V U 2 3 3 3 3 z2 < 1Zn -z lie 2 K l n n n n n n U U V V V U

100 Now I2 2 3 2 27 3 | |n -n I l rn I < II I 1 r 11 n n - nn n n Q u u rv u u L2 v L2 (4. 10) < 3 z| + z| u u L2 v L2 <e (e + R), - n n where e -max(e, C ), page 98, and R is the bound given on n n I2 page 96. The same inequality holds for the other three members of (4.9). Thus | IJ'- JA I< 4 (t + R) n n Q f n Since IjJI (J1 J2 J3)< J1 + IJ + J3 I it folows that (4. 11) S - z I < 12 r (e + R) Q in n From (4.8) and (4. 11) we obtain (4. 12) I0 % Q) - Io(Z 2Q). < LZ (12E )(C + R) + L ( l/n)(d / and the right-hand side approaches 0 as n oo. Since I0(z,Q) 6, we have I 0(,Q) -- 6. This completes the proof. n n~~Of

101 We let E, G, F correspond to z, and E, G, F to n. Since z was quasi-conformal on Q - R, E = G, F - 0 n n n almost everywhere on Q - R (page 96). While r need not be quasi-conformal, as n -- oo oo becomes "nearly quasi-conformal" as the next lemma shows. Lemma 4.4. I(t,Q) -- 6 as n -oo n Proof. I,Q) - z(5 J) + (M m) [(_ G) + F ] Q (4. 13) < 5 F(C J) + Mm){ E-G — n 2 2{ i I + IFI} Q Q Q Using the fact that z is quasi-conformal on Q - Q - R n n n j IE-GI I E- E+G- GI Q Q n n (4. 14) < IE- E l + S I G - GI Q Q n n We obtain successively,

102 3 _ Z 3'1 )2 (i )2 ] ~n n Q jQ i n n < z 3i 2 ) (z )2j nu n Qi= u n (4. 14a) 3 i i i i < || 11 C || || 7- + 2 || i=l'u u L u u L 2 2 3 < ( t(f Z3z 11J + 211 11 -- n n n n i=l u u L2 u L2 i=1 1 fiii2 n) R as before. Similarly (4..14b) I G - GI <3 E (e +ZR) n Next,

103 SIFI=SI~~I i<,i zi z i |Ij Fj = I| F - F < r- rn n n n Q =l Q u v u v Q Q Q n n n 3 i i i o i I < - I (r 1 + + - z -nO n n n n n i=l u u v v v u n n 3 z (1] r i....Z i ~l~ l i "I'II z! ~ ) n l n'n n'!l 1 n < ~ { E ( r + R) + R} - 3 e ( +Z R) Combining (4.14a), (4.14b), (4.15) and substituting in (4.4), -~ 3){Sn | zI + I |F|} <;( —~m 1 6r ( 2 +2R)+3e (g +2R)} 2 2 2 n n n n Q Q n n < ( {6~ ( +2R)}l From (4. 13) we obtain I( 1n Q ) < 9 F(n,J) + tn[6( 2 )(En +R) (4. 16) Qn < I0(n Q ) + en[6( M )({ + 2RI]. -- n n n 2 n

104 By Lemma 4. 3 and the fact that g -- 0 as n -- oo, the n right-hand side of (4. 16) approaches 6 as n — oo. Therefore, lim I(n nQ )< 6 But I(,Q )< I(,Q ), so n n-0 n n nn n- -oo lin I(,nQ )> lim I (,Q ) =limI (Q )Q) 6 Thus forany -.n n O- 0 n n o n n n-oo n —~oo n —~ e > 0 we may choose n so large that 6- /2 < I( Q )< I(_,Q ) < I (,Q ) + 1/2 < 6 + /, that is, I(,Q ) -6 as n -- oo, and since meas(Q ) - meas Q, n n n I(S,Q) 6 as n --- oo We collect the main results of this section. Theorem 4. 1. There exists a generalized minimizing oo sequence O} of C vectors for I0(rQ). Furthermore, I(,Q) -- 6 as n --- oo, and the segment c is mapped into the closed set F for every n.

105 5. The Existence Proof Let { 3} be a sequence given by Theorem 4. 1. Denote by p the greatest distance from ~ (Q) (i.e., the greatest disn n tance of the manifold n on which P (Q ) lies) to g-. Then p -0 as n — ooo For the moment we fix n. Let 5(K) be n 2 the class of all vectors z e H (Q) having the properties (i) z(Q ) links H (ii) the greatest distance from z(Q ) to is < p (iii) The segment u (as before) is mapped by z into the closed set F; (iv) J(zQ) - uu(z + 2 1Zuv + Iz I )du dv < K. Q Then for K sufficiently large, n e (K). Theorem 5. 1. There exists a vector zK E 3(K) (K sufficiently large) such that I(z, Q) is minimized by zK among all vectors in - (K). Proof. { z } be a minimizing sequence for I in (K). Then by the remark oi. pate 96 and the definition of the 2 2 norm in H, page'5'0, the H norms of the vectors z are unip formly bounded. Thus, by the Sobolev imbedding theorem, there is a subsequence { z } which converges uniformly on Q and weakly in 2 tom 2avectorF in H (Q) to a vector zK E H (Q). For simplicity, assume [pm] - [p]. Weak convergence in H2(Q) implies weak convergence

106 in L2(Q) of the second derivative and strong convergence in LZ(Q) of the first derivatives [23, pp. 99, 100]. Since J(zQ) is lower semi-continuous with respect to this convergence, we have J(z, Q)< lim J(z,Q)< K K - p -- p-.oo and since I(z,Q) is lower semi-continuous, I(zKQ) < lim I(z,Q) K - 0 p p-oo Therefore, if zK behaves properly, it will be the desired minimizing function. First, since each z links H and since z (Q ) is P P bounded away from H, the uniform convergence of z toward ZK on Q implies that zK(Q ) may be continuously deformed into z (Q ) without touching H. Thus z K is linked. Secondly, by the uniform convergence of z toward zK on Q, given E > 0 there is a number P such that p > P implies zp (w) - ZK (W) < E for all w e Q. Thus dist[zK(w), Z] < dist[z (w), ] + [ zp(w) - ZK(W) 0 was arbitrary, so dist[zK, ] < Pn Finally, the uniform convergence again gives the fact that c is mapped into the closed set F. Therefore, ZK is a minimizing function in j (K).

107 Theorem 5.2. For each n we may choose a vector ZK n as above, Kn < K <..., such that lim I(zK,Q) = 6. n n+l K n —~ n Proof. For each n, choose K so that b c (K ). nn n n (See page 1L05). Since zK is a minimizing function for I(z, Q) in. n 3 (K), we have n I(zK,Q) < I( Pn') n The sequence {zK } is an admissible sequence for IO, so n.. 6 < lim I0(zK,Q) < lim I (z,Q)< lim I(,Q) rn —~00 n n -~o n n-~00 but lim I(q,Q) - lim I( q,Q) - 6. n n n —oo n —oo Thus lim I(z,Q)= 6 K n-o~o n We shall now begin the process of showing that a subsequence of { K } converges to a minimizing vector for our problem. n Lemma 5.1. [23, p. 100] Let z E P (G) and let ~ (| uz |2+ Iz 2)dudv< A(r/a), 0< r< a, O< p < 1 B(wO,r) for every w0 e G, where a is the distance from w0 to G. Then

108 Iw -w21 z(w) - z(w2)l < N( ), I< [wl - wZ < a a for every pair of points w1, w2 E G such that every point of the segment joining wl and w2 is at a distance > a from G, and -1 1 N A 2 2 3 2 Lemma 5.2. [20, p. 39] Let z E P2 on a disk B(P,R) with center P and radius R. Then there exist functions a (r), 0 a (r), b (r), n = 1, 2..., of class P' (and therefore of class n n 2 P') on each interval (r0,R) with 0 < r < R, such that the series -2 00 a a (r) 00 2 + ~ [a (r)cos nO + b (r) sinn n] 2 nl n n-l converges in L2(0, 2r ) to a function z(r, ) for each r, the convergence being absolute and uniform in 0 for almost every fixed r. Moreover, the function z(r, 0) is equivalent to z, and R 2 22 2 I a) oo n (a +b D(z, B(P, R)) = r[ + {al'2+ b+ n a Idr j2 n n 2 0 1= r Lemma 5.3. [30, pp. 103-1231 Let the functions {00' 10' 01 be given on the boundary of the disk B(P, R), and suppose that there is a function P e H (B(P,R)) such that n = =00' u n00' ar 10' av 1 on the boundary. Then there exists a unique function z H2(B(PR))

109 satisfying these boundary conditions and minimizing the integral, J(,B(P, R)) I (+ 21 I +Ji I 2)du dv 2 z 4 z +2 az +: B(P, R) among all such functions. The function z has continuous derivatives of all orders on the domain B(P. ) { w I w - PI < RI and satisfies the biharmonic equation 4 2 2 2 au Ou av av on B (P,R). Furthermore, z is the only function biharmonic on B (P,R) and satisfying the given boundary conditions. Theorem 5.3. Denote by Q' the (open) region Q - ao For each K, z satisfies the oondition n K D[zK' B(wO R)] < D[zK BB(w,a)] (R/a) K0 -- K 0 n n (5. 1) m 9 0 < R < a 2M for every disk B(w0,a) C Q'. Thus the vectors zK are equicontinuous n on every closed subd:omain of Q. Proof. To prove equicontinuity assuming that relation (5. 1) is true, we note that since I(zK,Q) - 6 (Theorem 5.2), there is a n constant C such that I(zK, Q) < C m/2. Thus n

110 D[zK B(wg a)] < D[rK Q1< Q] < C D 0 - K:-i K - n n n for all K, n = 1, 2,.. Then by relation (5. 1) n D[zK,B(wO,R)]< D[zK,B(w,a)] (R/a) < C(R/a) n n and so by Lemma (5. 1) we have I w - w21 x /2 (5.2) IZK (w l ZK (w2)I < N( a ) n n independently of K. But this says that the vectors zK are n equicontinuous on every closed subdomain of Q'. To prove relation (5. 1) we use the minimizing property of ZK and the relation (4.5) to see that n (5.3) 2mb[z, B (wO R)] < I[zK B(wO P R)] n n < I[gC, B(wo R)] < 2 MD[r B(Wo R)] 0 n where is the bih armonic function on B (wo PR) having boundary ~a~Z~~z KK n a(R' ) - (R, 8) (Lemma 5.3). Let (r, 0) be polar coordinates with pole at w0. By Lemma 5.2 we have

111 a (r) 00 Z (r, 0) + [a (r) cos nO + b (r) sin nQ],0 < r < a K. n n n n=l AO(r) 00oo (r, 0 ) 2 + Z [An(r)cos nO +B (r) sin n] O< r < R n=l Since ~ is biharmonic and has the same Dirichlet data as.z on K -n B (w0oR), one may easily obtain A...(r) - c (r/R)n + d (r/R)n+2 B (r) - en(r n + fr/R)n+ 92 where c, d e, f are constants defined by n n n n 2c = (n + 2)a -,, 2d = - n n n n n n n a = a (R) = Ra' (R) n > 0 n n n n and similar formulas for e and f for n > 0. Thus if we set n n (R) = D[zK,B(wO,R)] we obtain (R) < CD[,B(wo, R)] R 2 0 2 2 -22 2 2 = Cr| r{ + Z [A' +B' +r n (A +B )]}dr 2 n n n n d n=l < 2C'Rq (R), The ast membem of the inequaity follows by terwse compason The last member of the inequality follows by termwis compI I VarisoIn

112 Thus we have 0 (R) < ARO'(R) A > 1 or 0 < ARW'(R) - (R), which yields 1 1 1 _ F. -, - (R A0(R))=R -'(R)-R A (R) dR A R A(O'(R) AR (R)) > 0 Since V (0) = O, we have 1 1 _ _ R A? (R) < a (a) or D[zK,B(w, R)] < (-) /AD[z,B(w,a)] K 0 -- K 0 n n - _) D[z,B(w,a)] K 0 n Note: This proof may be found in [21, p. 573]. From Theorem 5.3 and the fact that the numbers D[zK,Q] n are uniformly bounded, as well as LZ[ZK, Q], we obtain the follown ing result.

113 Theorem 5.4. There exists a vector z e P2(Q) such that a subsequence { } of { ZK } converges weakly in P2(Q) to z and uniformly on every closed subdomain of Q' to z. Moreover, z(Q ) lies on the torus, z(Q ) links H, and I (z, Q) < 6. Proof. The statements about convergence follow from the above remarks. Thus z is continuous on every closed subdomain of Q', and we may take z e P'(Q). By the lower semicontivnudtV of IO, if R C Q' is any closed subdomain, IO(z,R) < lim I0(z,R)< lim I(z,Q) 6 -- 00 P --- p p- oo p-~oo The fact that z(Q ) lies on Z? (in the sense previously defined) follows from Corollary 2.2. 2. Thus it remains to show that z(Q ) links H. Let Q be any subrectangle of Q whose boundary Q contains the segment a. By [23, p. 99, Theorem 2. 11(b)], z -z weakly on Q, and by.[.Z3, p. 1 00 Theoein 2. lZ]", z -,z strongly.:in.LZ onQ. Thus there _s a subsequence, again called {z.}- such bthat. z -z a. e, (1-dimensional p P measure) on. Q Since z (or) is contained in a closed set F Ti P (bounded away from - ), for almost all w E c we have z(w) e F. On the other hand, z E PI(Q). Therefore, there is a point w0 E a and a line parallel to the u-coordinate axis, passing through wO, such that z is continuous along this line. Let us say wo - (U.OVO),

114 and the line L:v = v0 is the one in question. Since z(w0) E F, the distance from z(w ) to 77 is > 0. Thus there is a point near wO, say, (u1,v0), such that z(ul,v0) is at a distance > 0 from ~-. But z is continuous at (u1,v 0) so there is a neighborhood, say a square R: [u2 < u > 0 for all w E R. But as shown above, there is a strip S near Q such that z(w) is within, say, T/4 of Z- for all w in this strip. Thus, 0 < 3r < jz(u',v) - z(u",v)[ uf u! u" < z (u,v)du l <. (u,v)Idu Ul Ul u" < u' (u, v) E R, (u', v) e S. Since this holds for all v with v7 < v < v, 2j- - V3 v3 ut 0 < ~ T dv < ~ Iz (u,v)ldudv < ~ z (u,v) dudv V2 v ul" Q < ( z z(uv)K dudv)2 < (D(z,Q))2 Q Therefore, 0 < - D(z, Q) < I(z,Q)

115 We now assume for the moment that all our conditions are satisfied on the unit disk B, and we shall prove that z is linked. Then by a conformal mapping the result will hold for Q. Thus, since z(B ) lies on, there is a strip S0: [rO < r < 1; 0 < 0 < 27r such that z maps every closed curve in SO onto a curve whose distance from Z- is less than d/4 (d = dist[H, 2- ]). Hence for w E SO it must be that z(w) does not lie on H; in fact, the distance from z(w) to H is > Let e > 0 be given, E < d/4. Then there is a number r, 0 < r < 1, such that the circle r = r encloses all points w which map onto H under z, and such that the curve M:z -z(r,0), 0 < 0 < 2I, has its greatest distance from 7J" less than C/2. By the uniform convergence of z toward z on each closed subdomain of B' (B' corresponds to Q', i.e. a circular arc is omitted), we may choose p so large that iz (r 0) - z(r l0)1 < r/2, 0 < 0 < 27r Thus the curve M z- z (r 0) is at a distance < e from Z. Now suppose that the curve M z = z(r, ), 0 < 0 < 27, does not link H. Then neither does the curve M z = z (r,0), 0< ~ < 2r a?" P C link H (since they can be continuously deformed into each other without touching H). However, the curve M: z = z (1,0), 0 < 0 < 27, p P does link H, so the intersection number corresponding to the image under z of the ring r < r < 1 and the circle H is different p e -

116 from 0. Since I(z, B) > 0, there is a subdomain, say, B0: [r < b; 0 < 0 < 2] 0 O. Therefore, we have for some 3 > 0 < 3< I(z,B0 ) < lim I(z,B) p0 D p 0o so after extracting a subsequence we rename it {z } and we may assume that 0 <, < I(z,B0) for all p. Next, set F () z (r, 2r dr. Then F (Q ) EL(0,2), 2w 2w? 1 Fp(f )dO izp (r,0)2r dr dO < D(z, B) < A, O O b A = constant Now if F (0 ) > a. e. on (0,2wr), then we would have 2w7 F ( ) dO > A, a contradiction. Therefore, there is a set of 0 positive measure for which F (0) <A we let be in that set. Then 5izp (r, )12 r dr < A Hence for almost all r, b < r < 1,

117 1 blz (r,O )- z (1,0 )I2< bi(,z (r, 0 )drI2 r < b(l z r) (, | dr) r 1 < (1- r) |z (r,0 )I rdr p- P b r < (1 r) A bir Hence for almost all r, b < r< 1, (r,0 ) z (1,0 )12< (1 - r) and since z is continuous, this relation holds for all r in the p interval. Therefore, the oscillation on the radial. segment N [ 0 = -0; r < r < 1] is less than e/2 if r is chosen close p P [ - - enough to 1. The point z (1, 0 ) is on a manifold ~, the greatest distance from p to 2 goes to 0 as p- oo. Therefore, the values of z on N are at a distance less than V from P P 7G if p is large enough. But by the inequality above, r is independent of p, so we may rotate and obtain N - N for all sufp ficiently large p.

118 We now cut the ring Re:[r < r < 1, 0<0 < 27 along N to obtain a simply connected domain RI, whose boundary is mapped by z onto a continuous curve at a (greatest) distance p less than e from Z, and this curve is linked with H since the intersection number of z (R ) with H is not zero. (There can be p no point w e N such that z (w) lies on H, since the distance of p z (w) from' is < E, while the distance from H to - is p > E.) We now have I(z,R') = I(z,B) - I(z, B - R) p p p and I(z, B- R')> I(z, B) > > 0, p - p 0 - so that (z ) < I( z,B) - <3 p p Letting p- oo we have lim I(z,R')< 6-< p-cro n By a conformal mapping we can transform R' onto B and z into p a vector p defined on B with I(z, R' ) I(,B). The new sequence { } is an admissible sequence. Hence 6 < lim I(,B)- lim I(z R' < 6 - p-. O P p — oo P; i -

119 contradiction. Therefore, we conclude that z must link H. Now by a conformal mapping, the result holds on the square Q. Note: The proof of linking is a variant of that in [11, pp. 216, 217J. Theorem 5.5. IO(z,Q) = 6 Proof. The vector z is not yet known to be admissible since it is not necessarily continuous on the segment a. Let r > 0 be given. We choose a rectangle A containing a in its interior such that A C Q, and such that 2 (zD(7 A)< I This, of course, may always be done, by the absolute continuity of the integral. The boundary values of z on A are continuous, and therefore there is a harmonic function h whose boundary values agree with those of z on A, and moreover, h is continuous on A. We define a new vector h(w), w e A z' (w) - z(w), w e Q- At which is now continuous on Q0, linked with H (preceding theorem), and whose boundary lies on 7. That is, z1 is admissible. Furthermore,

120 D(z', A) = D(h, A) < D(z, A) and all these remarks hold for any such -re-cta-ngle: cnita:ined:in A. Therefore, 6 < I0(z',Q) = IO(z,Q-A) + I0(z',A) < I0(z, Q-A) + I(z',A) 0 was arbitrary, and so it must be that 6 < I0(z,Q). By the preceding theorem, 6 > Io(zQ). Therefore, I (z,Q) = 6. Now it remains only to show that z is continuous on the whole of Q 0. We shall prove that z is continuous on a closed subdomain &f. Qo - Whih, coiftaihns a. This is all that is necessary. We may take for such a subdomain a large (concentric) inscribed disk B. Lemma 5.4. [19, p. 42]. Let B be a region bounded by a finite number of circles. Let z be a vector of class PZ(B); whose boundary values are continuous as functions on B. Suppose also that there exists a number K > 1 such that D(z, G) < K D(H(z, G), G) for every subregion G, of B, which is of class K (see [19, p. 5]),

121 H(z, G) being the function harmonic on G and coinciding with z on G. Then z is equivalent to a function z which is continuous on B and takes onthese boundary values continuously. Lemma 5.5. If G is any Jordan region such that G C B and measure (G ) = 0, then IO(z,G) = lim I(z,G) Proof. By Theorems 5.2 and 5.5 we have (5.4) lim I(z,Q)6 = I (z,Q) p —*oo p We shall prove the lemma for every G C Qo and then it will be true for every G C B~ of the prescribed type. Furthermore, if G is a region such that there is a sequence of closed subregions R C G~ invading G, andlon each R, z - z uniformly, each p z being continuous on R, we have (by the lower semicontinuity p of Io) I(z, R) < lim I (z,R) < lim I (z,G) p p 0o 0 p p-roo pe-Oo Letting R invade G, this yields (5. 5) I0(z,G) < lim I0(z,G) p-1oo0 p Let ~ > 0 be given. By relation (5.4) above, there is a number N1 such that p > N1 implies

122 (5.6) < I(z,Q) - I (z, Q) < /3 3 p 0 In view of relation (5.5), there is a number N2 such that p > N2 implie s I(z, G) < I (z,G) + S or since I0(z,G) < I(zp, G), (5.7) - < I(z) G) - I0(z,G) Let G' Q - G. Since meas G = 0, I(z,Q) = I(z,G) + I(z,G'), and similarly for Io(z,Q), I(z,Q), I0(z,Q). We now construct a region G defined by G =G' mQ G' =w E G;: dist w,G > 17 Q' w E Q: dist w,Q Q > n} where Ti > 0 is chosen so small that (5.8) I0(z,G - G) < ) /3 Now Gi is a compact se-t, _meas G -, =an G2 is at a positive distance from G and from Q. Therefore, we may cover G by a finite grid of closed squares { }M whose sides are parallel to the coordinate axes, each square having the same sidelength, and whose diagonals are so small that if weQ. r G. 1 Ti then Q. G - and Q. Q =, 2,..M. 1 1

123 Then (by relation (5.5), changinlg the name G to Qi) IO is lower seemicontinuous oneach Q., and thus also on the closed set M GQ = *1 Qi C G'. Therefore, there is a number N3 such that p > N3 implies that IO(z GQ ) N, using relations (5. 6)-(5. 10), we have - < I (z G) - I(z, G) = I(z,Q) - I(z,G') - I (zQ) i I(z G') p 0 p p 0 0 + - (I(z,Q) - Io(zQ)) + (IO(z,G') - I(z,G')) P P < E/3 + (I0(z, GQ)- I(z,GQ)) + I0(z,G'-GQ) - I(z,G' - GQ) < C/3< r/3 + /3 - I(z,G' - G ) <,/3 + e/3 + /3 =- Therefore, lim I(z,G) = I0(zG), where G is a subdomain of Q of the proper type. Therefore, this also holds for subdomains of B.

124 Now if G is any subregion of the type of Lemma 5.4, we have by Lemma 5. 5 and the lower semicontinuity of D(z, G) (with respect to weak convergence in class P2) (5. 11) zD(zG) < lim -D(z,G) 2 Z p -00oo < lim I(z,G) -= I0(z,G) p- 0oo However, from the minimizing property of z (Theorem 5. 5) we obtain (using relation (5. 11) ) 2 D(z,G) 1. However, by Lemma 5.4, relation (5.12) implies rmz is continuous on B. Therefore, z is continuous on Q 0, z links H, and the boundary of z lies on -. Thus z is an admissible vector and Io(zQ) - 6 < L. But then since z is admissible, I0(z Q) > L. Thus I0(z Q) = L, and the existence theorem is proved. Remark. It can be seen now that the existence theorem holds for much more general fixed manifolds. For example, we could take any manifold whose complement in E is homeomorphic to the complement of a simple closed curve (or a torus). Thus a

125 capstan surface and a cylinder fall into this category. More generally, we could take a piece of a plane (or deformation of a plane) and attach an arc to the plane at the endpoints of the arc. The resulting manifold would be our fixed manifold. Moreover, the arc need not even be simple; the position of an admissible surface is described only by the intersection number of the surface with a fixed simple closed curve. However, the existence theorem gives no information as to the structure of the trace of the minimizing surface on the fixed manifold.

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