Plateau's Problem and the Calculus of Variations. (MN-35) [Course Book ed.] 9781400860210

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PLATEAU'S PROBLEM AND THE CALCULUS OF VARIATIONS

Copyright © 1989 by Princeton University Press All Rights Reserved

Printed in the United States of America by Princeton University Press, 41 William Street, Princeton, New Jersey 08540

The Princeton Mathematical Notes are edited by William Browder, Robert Langlands, John Milnor, and Elias M. Stein

Library o f Congress Cataloging in Publication Data Struwe, Michael, 1955Plateau's problem and the calculus of variations. (Mathematical notes ; 35) Bibliography: p. 1. Surfaces, Minimal. 2. Plateau's problem. 3. Global analysis (Mathematics) 4. Calculus of variations. I. Title. II. Series: Mathematical notes (Princeton University Press) ; 35. QA644.S77

1988

516.3'62

ISBN 0-691-08510-2 (pbk.)

88-17963

PLATEAU'S PROBLEM AND THE CALCULUS OF VARIATIONS

by

Michael Struwe

Mathematical Notes 35

PRINCETON UNIVERSITY PRESS

PRINCETON, NEW JERSEY

1988

To Anne

Contents A . The "classical" Plateau problem for disctype minimal surfaces.

I.

Existence of a solution 1. The parametric problem 2. A variational principle 3. The direct methods in the calculus of variations 4. The Courant-Lebesgue Lemma and its consequences 5. Regularity Appendix

IL

5 7 12 16 22 29

Unstable minimal surfaces 1. 2. 3. 4. 5. 6.

Ljusternik-Schnirelman theory on convex sets in Banach spaces The mountain-pass lemma for minimal surfaces Morse theory on convex sets Morse inequalities for minimal surfaces Regularity Historical remarks

33 41 52 60 66 78

B. Surfaces of prescribed constant mean curvature.

III.

The existence of surfaces of prescribed constant mean curvature spanning a Jordan curve in IR3 1. 2. 3. 4. 5.

IV.

The variational problem The volume functional "Small" solutions Heinz' non-existence result Regularity

Unstable

91 94 100 104 105

H — surfaces

1. H — extensions 2. Ljusternik-Schnirelman and Morse theory for "small" H — surfaces 3. Large solutions to the Dirichlet Problem 4. Large solutions to the Plateau problem: "Rellich's conjecture"

111 116 121 127

References

141

ix

Preface

Minimal surfaces and more generally surfaces of constant mean curvature - commonly known as soap films and soap bubbles - are among the oldest objects of mathematical analysis. The fascination that likewise attracts the mathematician and the child to these forms may lie in the apparent perfection and sheer beauty of these shapes. Or it may rest in the contrast between the utmost simplicity and endless variability of these remarkably stable and yet precariously fragile forms. The mathematician moreover may use soap films as a simple and beautiful model for his abstract ideas. In fact, long before the famous experiments of Plateau in the middle of the 19th century that initiated a first "golden age" in the mathematical study of minimal surfaces and through which Plateau's name became inseparably linked with these objects, Lagrange investigated surfaces of least area bounded by a given space curve as an illustration of the principle now known as "Euler-Lagrange equations". However, for a long time since Lagrange's derivation of the (non-parametric) minimal surface equation and Plateau's soap film experiments mathematicians had to acknowledge that their methods were completely inadequate for deeding with the Plateau problem in its generality. In spite of deep insights into the problem gained by applying the theory of analytic functions the solution to the classical Plateau problem evaded 19th century mathematicians- among them Riemann, Weierstrafi, H.A. Schwarz. To meet the challenge ideas from complex analysis and the calculus of variations had to merge in the celebrated papers by J.Douglas and T. Rad6 in 1930/31. But this was only the beginning of a new era of minimal surface theory in the course of which many significant contributions were made. Among other discoveries it was noted that the (parametric) Plateau problem may possess unstable solutions - which of course are not seen in the physical model - and in particular that the solutions to the Plateau problem in general are not unique. The question whether for "reasonable" boundary data the Plateau problem will always have only a finite number of solutions still puzzles mathematicians today. The most significant contributions are Tomi's result on the finiteness of the number of surfaces of absolutely minimal area spanning an analytic curve in ffi3 and the generic finiteness result of Bohme and Tromba. The existence of "unphysical" solutions in the parametric problem in the 60's led to a new approach to the Plateau problem by what is now known as "geometric measure theory". In the course of these developments the notions of surface, area, tangent space, etc. came to be reconsidered and the notion of "varifold" evolved which parallels the notion of "weak solutions" in partial differential equations, cp. Nitsche [1, § 2].

X

Plateau Problem - Preface

But also the theory of the parametric Plateau problem wets further pursued. Both to bring out the geometric content of the parametric solutions obtained (branch points, self-intersections) and - independently of the physical model - to explore the richness of a fascinating variational problem in its own right. In this monograph we will focus our attention on the interplay between the parametric Plateau problem and developments in the calculus of variations, in particular global analysis. For reasons of space we will at most casually touch upon the more geometric aspects of the problem. As far as classical results about the geometry of minimal surfaces or the geometric measure theory approach to minimal surfaces are concerned the reader will find ample material and references in J.C.C. Nitsche's encyclopedic book [1] or in the lecture notes by L. Simon [1], Our main emphasis will be on the power of the variational method. Notations * denotes duality; occasionally we also denote a certain normalization with an asterisque. These notes are divided into two parts with together four chapters, each divided into sections. Sections are numbered consecutively within each chapter. In crossreferences to other chapters the number of the section is preceded by the number of the chapter which otherwise will be omitted. These notes are based on lectures given at Louvain - La - Neuve and Bochum in 1985-86. I am in particular indebted to Reinhold Bohme, Stefan Hildebrandt, Jean Mawhin, Anthony Tromba, Michel Willem and Eduard Zehnder for their continuous interest in the subject which has been a major stimulus for my work. Special thanks I also owe to Herbert Graff for his diligence and enthusiasm at typesetting this manuscript with the ^^^-l^Ksystem. Finally, I wish to express my gratitude for the generous support of the SFB 72 at the University of Bonn.

Michael Struwe

Zurich, March 1988

PLATEAU'S PROBLEM A N D THE CALCULUS OF VARIATIONS

A. The "classical" Plateau problem for

disc-type minimal surfaces

I. Existence of a solution. 1, T h e parametric problem. Let T be a Jordan curve in JRn. The "classical" problem of Plateau asks for a disc-type surface X of least area spanning T. Necessarily, such a surface must have mean curvature 0. If we introduce isothermal coordinates on X (assuming that such a surface exists) we may parametrize X by a function X{w) = (X x (u)),..., X n (to)) over the disc

satisfying the following system of nonlinear differential equations

(1.1) (1.2) (1.3)

is an (oriented) parametrization of T.

Here and in the f o l l o w i n g e t c . , Euclidian SC.

and

• denotes the scalar product in

Conversely, a solution to (1.1) - (1-3) will parametrize a surface of vanishing mean curvature (away {torn branch points where V I ( u ) = 0) spanning the curve r , i.e. a surface satisfying the required boundary conditions and whose surface area is stationary in this class. Thus (1.1) - (1.3) may be considered as the Euler Lagrange equations associated with Plateau's minimization problem. However, (1.1) - (1.3) no longer require X to be absolutely area-minimizing. Correspondingly, in general solutions to (1.1) - (1.3) may have branch points, selfintersections, and be physically unstable - properties that we would not expect to observe in the soap film experiment. Thus as we specify the topological type of the solutions and relax our notion of "minimality" a new mathematical problem with its own characteristics evolves. In the following we simply refer to solutions of (1.1) -(1.3) as minimal surfaces spanning r. In this first chapter we present the classical solution to the parametric problem (1.1) - (1.3). Later we analyze the structure of the set of Jill solutions to (1.1) -(1.3). The key to this program is a variational principle for (1.1)-(1.3) which is "equivalent" to the least area principle but is not of a physical nature as it takes account of a feature present in the mathematical model but not in the physical solution itself: The parametrization of a solution surface. This variational principle is derived in the next section. Applying the "direct methods in the calculus of variations" we then

6

A. The classical Plateau Problem for disc - type minimal surfaces. •

obtain a (least area) solution to the problem of Plateau. At this stage the CourantLebesgue-Lemma will be needed. Finally, some results on the geometric nature of (least area) solutions will be recalled. It will often be convenient to use complex notation and to identify points with complex numbers Moreover, we introduce the complex conjugate and the complex differential operators

Note that hence anysolution holomorphic differential

X to (1.1) - (1-2) gives rise to a satisfying the conformality relation

cp. Lemma 2.3. Conversely, from any holomorphic curve satisfying the compatibility conditon

a solution

F dw to (1.1), (1.2) may be constructed. This relation between minimal surfaces and holomorphic curves is the basis for the classical Weierstrafi - Enneper representations of minimal surfaces in which constitute one of the major tools for constructing and investigating minimal surfaces, cp. Nitsche [1, 155 - 160] .

I. Existence of ft solution.

2. A variational principle. and let

Let be the Sobolev space of with square integrable distributional derivatives,

m

denote the

respectively the seminorm and norm in

For

let

denote the area of the "surface" Also introduce the class of

7

X, cp. Simon [1, p. 46]. is a weakly monotone parametrization of

-surfaces spanning

Note that the area of a surface X does not depend upon the parametric representation of X, i.e. (2.1) for all diffeomorphisms g of Hence by means of the area functional it is impossible to distinguish a particular parametrization of a surface X, and any attempt to approach the Plateau problem by minimizing A over the class is doomed to fail due to lack of compactness. In 1930/31 3esse Douglas and Tibor Rado however ingenuously proposed a different variational principle where the minimization-method meets success: They (essentially) considered Dirichlet's integral

instead of A. For this functioned the group of symmetries is considerably smaller; the relation (2.2)

8

A. The classical Plateau Problem for disc - type minimal surfaces. •

only holds for conformal diffeomorphisms g satisfying the condition

g

of

i.e. for diffeomorphisms

(2.3)

Now, A and D are related its follows: For (2.4) with equality iff X is conformal, i.e. satisfies (1.2). Conversely, given a surface parametrized by following result due to Morrey [2; Theorem 1.2]: Theorem 2.1: Let such that

we can assert the

There exists a diffeomorphism satisfies:

In particular, Theorem 2.1 implies that (2.5) We will not prove Morrey's e-conformality result. However, with the tools developed in Chapter 4 it will be easy to establish (2.5) for rectifiable cp. the appendix. By (2.5), for the purpose of minimizing the area among surfaces in it is sufficient to minimize Dirichlet's integral in this class. Moreover, we have the following Lemma 2.2: critical for D on

solves the Plateau problem (1.1) - (1.3) iff in the sense that

i) ii)

for any family of diffeomorphisms depending differentiably on a parameter

Proof:

and with

Compute

Hence the first stationarity condition i) is equivalent to the condition

X

is

I. Existence of ft solution.

9

which in turn is just the weak foim of the differential equation (1.1). By standard regularity results any weak solution of (1.1) will be smooth in B and (1.1) will be satisfied in the classical sense. It remains to show that for harmonic the stationarity condition ii) is equivalent to the conformality relations (1.2). This result requires some preparatory lemmata which we state in a slightly more general way than will actually be needed. Lemma 2.3: Let G be a domain in is harmonic. Then the function

and suppose

is a holomorphic function of

Proof:

Note that

may be written as a product

with component-wise complex multiplication and

the usual complex differential operators. Note that

Hence by harmonicity of

X

Then i.e. Moreover, Proof: Lemma with X isis 2.4: holomorphic. suppose Let conformal. Suppose there that for holds G anyisand differentiable a domain for in family with and of diffeomorhisms let

10

A. The classical Plateau Problem for disc - type minimal surfaces.

consider maps are injective and the rank of the differential

is maximal everywhere the

Since by choice of

the maps

in fact are diffeomorphisms

Compute by the chain rule:

Now

id implies that

while - labeling

I.e. where is defined It is now clear that as in Lemma 2.3. Thus If now again we consider rewrite the integrand as follows: (2.6)

i

is differentiable at

by letting

and we may

I. Existence of ft solution.

11

and the expression can only vanish for all r € C 1 (G; JR2) if $ vanishes identically in G, i.e. if Jl is conformal.

• To conclude the p r o o f o f Lemma 2.2 in view of Lemma 2.4 it suffices to remark that by (2.6) conformality of X also implies the stationarity condition ii) of Lemma 2.2. Hence the critical points of D in precisely correspond to the solutions of Plateau's problem.

Remarks 2.5. i) If X is harmonic on integrating by parts in (2.6) we obtain

B

, by Lemma 2.3 and upon

Thus, the conformality relations (1.2) may be interpreted as a natural boundary condition for the holomorphic function associated with X. Cp. Courant [1, p. 72 ft]. ii) Variations of the type i) in Lemma 2.1 may be interpreted as "variations of the dependent variables" i.e. of the surface X. Variations of the type ii) ("variations of the independent variables") correspond to variations of the parametrization of X. iii) By conformal invariance of D and the Riemann mapping theorem any minimizer of D in will be a critical point of D in the sense of Lemma 2.1. Indeed, by (2.6) it suffices to show that satisfies the stationarity condition ii) of Lemma 2.2 for all Suppose by contradiction that for some

with have But

Then for some

and

we

is conformed to B. Hence we may compose to obtain a comparison surface

with a conformal map with

The contradiction proves that

is critical for

D.

12

A. The classical Plateau Problem for disc - type minimal surfaces. •

3. The direct methods in the calculus of variations. We now proceed to derive the existence of a minimizer of D on -and hence of a solution to Plateau's problem (1.1) - (1.3), cp. Remark 2.4. iii) - from the following general principle: Theorem 3.1:

Let

Suppose that for any

M be a topological Hausdorff space, and let

the set

(3.1) is compact. Then there exists

' such that

In particular, E is bounded from below and lower semi-continuous on Proof:

M.

Let

and consider a sequence

of numbers

tending to

as

By compactness of for the nested sequence has non-empty intersection and there exists Clearly, that

for any

m and therefore letting

Since E does not assume the value bounded from below on M. lower Finally, semi-continuous. by (3.1) for any the set

in particular

we infer

and E is is open, i.e. E is

I. Existence of a solution.

13

Remark 3.2: In the work of M. Morse compactness of the sets M a in Theorem 3.1 defines the property of "bounded compactness" of E on M. This condition implies lower semi-continuity of E. However E cannot be continuous on M and simultaneously satisfy (3.1) unless M is locally compact: By (3.1) any set

must be relatively compact in M while by continuity

Ma is also open.

In applications a simple variant of Theorem 3.1 will often be encountered: Theorem 3.3: Suppose M is a sub-set of a separable Hilbert space H which is closed with respect to the weak topology on H. Let E \ M —> IR be a funtional which is sequentially weakly lower semi-continuous on M, i.e. which satisfies the condition

(3.2)

Also assume that there holds:

E is coercive, i.e. suppose that for any sequence

{ i m } in M

(3.3)

Then there exists a minimize! xo G M with

Theorem 3.3 is reduced to Theorem 3.1 by letting M be endowed with the weak topology on H. However, there also is a very natural direct and constructive proof of Theorem 3.3 which uses the concept of a minimizing sequence. P r o o f of Theorem 3.3:

and let coerciveness of

E

Let

be a sequence such that as By { x m } is bounded and hence weakly relatively compact.

Extracting a weakly convergent subsequence M also the weak limit

by weak closedness of

®o £ M. Finally, by (3.2)

and the proof is complete.

•

14

A. The classical Plateau Problem for disc - type minimal surfaces. •

Examples 3.4:

i) the norm in a Hilbert space

H with scalar product

is weakly lower semi-continuous. ii) on

More generally, let H such that

Then

be a continuous symmetric bilinear form

is weakly lower semi-continuous on

In particular, D is weakly lowerThen semi-continuous on Proof: Suppose By the Riesz representation theorem there exists

iii) Suppose there holds

H.

such that

is continuous and convex, i.e. for all

(3.4) Then

E is weakly lower semi-continuous on

Proof:

If

H.

weakly, by the Banach-Saks theorem

Hence, by continuity of E and (3.4)

I. Existence of ft solution.

Remark:

The inequality

for a convex functional

Example 3.5: Proof:

Hence for

15

is a special case of Jensen's inequality.

The functional D is coercive on

By the Sobolev inequality for

with

16

A. The classical Plateau Problem for disc - type minimal surfaces. •

4. T h e Courant Lebesgue Lemma and its consequences. ing chapter we have seen the importance of weak closedness of presence of the conformal group of the disc (4.1) acting on

. However, the

and conformal invariance of D cause problems.

Lemma 4.1: For let orbit of X. Then for any X the weak closure of Proof:

In the preced-

i) First consider

be the conformal contains a constant map.

Let

where

Clearly, as

for all Hence point wise in

uniformly away from B.

By conformal invariance of D moreover

while

and admits also a weakly convergent subsequence proves our claim forletregularbe functions. ii) For as above and define

By (2.2) and Example 3.5 is bounded in a subsequence To show that const it suffices to show that But by i) of this proof and conformed invariance of have :

This

and we may extract

D, with

we

I. Existence of ft solution.

17

which completes the proof.

In view of Lemma 4.1 the set cannot be weakly closed in However, equivariance of D with respect to G allows us to factor out the symmetry group. The most convenient way to do this is by imposing a three - point - Lemma condition4.2: on admissible functions. Note exists that (4.1) immediately~ implies: Given any triplesadmissible there a unique suchLet that Lemma 4.2 suggests to normalize as follows: and let be anfunctions oriented triple of distinct points on Define Then we obtain the following crucial result:

Lemma 4.3 The injection bounded subsets of are equicontinuous on

is compact, i.e.

For the proof we need the following fundamental lemma due to Courant [1, p. 101 ff.] and Lebesgue [1, p. 388]: Lemma 4.4: exists

we have:

Proof:

For any such that if s denotes

any arc length on

any

and

By Fubini's theorem

Since for all

for a.e.

and

there

18

we can

A. The classical Plateau Problem for disc - type minimalsurfaces.•

find

as claimed.

P r o o f of Lemma 4.3: there exists a number the points (4.2)

Let depending only on such that for all

We contend that the curve and there holds

, if

This statement is equivalent to the contended equicontinuity of D-bcvnded subsets of By a theorem of Arzlla - Ascoli the latter in turn is equivalent to the compactness of the injection Choose small enough such that any ball of radius contains at most one of the points Choose such that a ball of radius to in contains at most one of the three points Clearly, we may assume that Choose such that for any two points X, Y on at a distance there is a subarc with end-points X and Y contained in some ball of radius in (This is possiblefor any Jordan curve Otherwise, for sequences of points in with I any subarc joining with would intersect in a point By compactness of w» may assume In particular, the limits Y and X correspond to different parameter values of a given parametrization of But this contradicts our assumption that is a Jordan curve, i.e. a homeomorphic image of .) By choice of for with the subarc connecting X and Y and lying in a ball of radius in is unique and is characterized by the condition that contains at most one the points Now choose a maximal

Let

Denote

such that

be selected according to Lemma 4.4 satisfying

the points of intersection of with that subarc of with end-points which contains at most one of the points Also let and let be that subarc of connecting containing at most one of the points By monotonicity Moreover, by Holder's inequality:

I. Existence of ft solution.

By choice of any

Since

19

is contained in a ball of radius there holds

depends only on and the proof is complete.

In particular, for

and

while the latter only depends on

Lemma 4.3 immediately implies the following results: Proposition 4.5:

in Proof: in some

The set

is closed with respect to the weak topology

Consider a sequence By weak convergence,

such that

' weakly

is bounded and in particular for

uniformly in m. Lemma 4.3 now implies that (a subsequence) on Hence and is weakly closed.

uniformly

Together with coercivenes of Don (cp. Example 3.5) and weak lower semi-continuity of D on (cp. Example 3.4. ii)) Proposition 4.5 implies: Proposition 4.6: Suppose is a Jordan curve in such that Then there exists a solution to Plateau's problem (1.1) - (1.3) parametrizing a minimal surface of disc-type spanning Proof: Indeed, let be defined as above with reference to a conveniently chosen triple of points on Theorem 3.3 guarantees the existence such that of a surface

Moreover, for any by Lemma 4.2 there is a unique conformal diffeomorphism g of B such that By conformal in variance of D also and it follows that

Consequently, minimizes D over . Hence by Remark 2.5.iii) es a solution to the parametric form (1.1) - (1-3) of Plateau's problem.

furnish-

•

20

A. The classical Plateau Problem for disc - type minimal surfaces.

For later reference we also note the following compactness result: Proposition 4.7: IR the set

Suppose

T is a Jordan curve in HC1. Then for any

is compact with respect to the weak topology in H1'2(B;IR^) -topology of uniform convergence on dB.

and the C°(dB;

IR")

Proposition 4.7 is easily deduced from Proposition 4.5 using the coerciveness and weak lower semi-continuity of D on C*(T). Combining Proposition 4.7 and Theorem 3.1 would give an alternative proof of Proposition 4.6.

It remains to give a general condition for in Lemma 4.8:

C(T) to be non-void. This is contained

For any rectifiable Jordan curve

the class

Our proof rests on the following a-priori bound for the area of solutions to (1.1) (1.3): Theorem 4.9 (Isoperimetric inequality): Suppose T is a rectifiable Jordan curve in iZf* with length L(r) < oo. Then for any solution of (1.1) - (1.3) there holds the estimate The constant

4tt is best possible.

Cp. Nitsche [1, §323]. For our purposes it will be sufficient to establish the qualitative bound (4.3) for any C X (B; 2ZZ")-solution to (1.1) - (1.3). P r o o f of (4.3):

Multiply (1.1) by X and integrate by parts to obtain

where do denotes the one-dimensioned measure on dB, and normal and tangent vector fields to dB. Of course, by (1.3)

n and

r are unit

I. Existence of ft solution.

while by suitable choice of coordinates in

21

such that

This proves (4.3) with P r o o f of Lemma 4.8: Approximate by smooth Jordan curves in of class on This can be done its follows: Let be a homeomorphism First convolute with a sequence of non-negative vanishing for and satisfying Then letto obtain a sequence of smooth maps . In this way we generate a sequence of

-diifeomorphisms such that

Extending the parametrizations to harmonic surfaces we immediately see that for all m. By Proposition 4.6 there exist solutions to (1.1) - ( 1.3) for moreover, by Theorem 5.1 below But then Theorem 4.9 assures the bound (4.4) for large

m.

Now let generality we may assume that ence to the triples

With no loss of where we normalize with referSince for this normalization the constants

appearing in the proof of Lemma 4.3 clearly have a uniform lower bound (4.4) and the proof of Lemma 4.3 show that the surfaces are equicontinuous on

Hence a subsequence

uniformly on Note that Xis the maximum principle

weakly in

and

harmonic with and

Thus by

Lemma 4.8 and Proposition 4.6 finally yield the following existence result of Douglas [1] and Rad6 [1]: Theorem 4.10: Let be a rectifiable Jordan curve in a solution to (1.1) - ( 1.3) characterized by the condition

and

Then there exists

parametrizes a disc-type minimal surface of least area spanning

20

A. The classical Plateau Problem for disc - type minimal surfaces.

5. Regularity. The preceding considerations establish the existence of a solution to the parametric Plateau problem (1.1) - (1-3). In order to interpret this solution geometrically we now derive further regularity properties. Note that the regularity question is two-fold: First we analyze the regularity of the parametrization; then we turn to the question whether the parametrized surface is regular enough to be admitted as a solution to Plateau's problem, i.e. whether it is embedded (or at least locally immersed). While the first question is completely solved by Hildebrandt's regularity result [1] , for the second question a satisfactory answer can only be given in case n = 3 which corresponds to the physical case. In this case the results of Osserman [1] , Alt [1], Gulliver [1], Gulliver, Osserman, and Royden [1], Gulliver and Lesley [1], Sasaki [1] and Nitsche [1, p. 346] show that the solutions of Douglas and Rad6 will be free of interior branch points and hence will be immersed o v e r R — a n d even be immersed over B if T is analytic or has total curvature For extreme curves, i.e. curves on the boundary of a region which is convex or more generally whose boundary has non-negative mean curvature with respect to the interior normal, Meeks and Yau [1] even have proved that a least area solution to (1.1) - (1.3) parametrizes an embedded minimal disc. Related results were obtained independently by Tomi and Tromba [1] , resp. Almgren and Simon [1] . This extends an old result of Rad6 [2] for curves having a single valued parallel projection onto a convex planar curve. Simple examples show that without such additional geometric conditions on T in general least-area solutions to (1.1) - (1.3) need not be embedded. Below we briefly survey some of the most significant contributions to the regularity problem for parametric minimal surfaces and sketch some of the underlying ideas involved. Let us begin by recalling the fundamental regularity result of Hildebrandt [1] :

Theorem 5.1: Suppose T is a Jordan curve in IE", parametrized by a map which is a diffeomophism of 8B onto T. Then any solution to (1.1) - (1.3) belongs to the class Moreover, if solutionsare normalized by a three-point-condition, the norms of s o l u t i o n s t o (1.1) - (1.3) are uniformly a-priori bounded.

Hildebrandt originally required ; the improvement to is due to J.C.C. Nitsche [2] . An overview of the different proofs of the result is given in Nitsche [l.p. 283 ff.j Hildebrandt's approach is rather interesting in as much as it reveals the complexity hidden in the seemingly harmless equations (1.1) - (1.3). His basic idea is to reduce the boundary regularity problem for (1.1) - (1.3) to an interior regularity problem for an elliptic system by means of the following transformation: Suppose Let There is a diffeomorphism $ of class of such that 9 maps a normal neighborhood V of Qo on r to a normal neighborhood of 0 on the (new) X1- axis. Let (5.1)

I. Existence of ft solution.

23

By harmonicity of X, Y solves an elliptic system (5.2) with a bounded bilinear form whose coefficients of class depend continuously on Y. corresponds to the ChristoiFel symbols of the metric

By continuity, contains a neighborhood U of transformed surface Y thus satisfies the boundary conditions

while the conformality relations (1.2) and our choice of Neumann condition

in

The

give a weak form of the

By reflection across the function Y hence may be extended as a solution to an elliptic system like (5.2) with quadratic growth in the gradient (5.3) in a full neighborhood of The standard interior regularity theory (cf. in particular Ladyshenskaya - Ural'ceva [1, p. 417 f.]) now enables us to bound the second derivatives of Y—and hence of X - in in a neighborhood of in terms of the Dirichlet integral of X and its modulus of continuity. By Theorem 4.9 and Proposition 4.7 both these quantities are uniformly bounded for any solution X of (1.1) -(1.3) which is normalized by a three-point-condition. In view of Sobolev's embedding theorem an -bound for X implies a bound for in Returning to (5.3) the Calderon-Zygmund inequality yields that In particular, The complete regularity now is a consequence of Schauder's estimates for elliptic equations (5.2), cf. e.g. Gilbarg - Trudinger [1, Theorem 6.30].

In later chapters we will return to this aspect and actually see some of the techniques of elliptic regularity theory in performance, cp. Section II.5.

Now we direct our attention to the regularity of the parametrized surface. Note that by the conformality relations (1.2) any solution X of (1.1)-(1.3) will be immersed in a neighborhood of points where Definition 5.2: A point

is called a branch point of X iff

The behavior of X near a branch point can be analyzed by means of the following representation.

24

A. The classical Plateau Problem for disc - type minimal surfaces. •

Recall that if X given by

is harmonic, the components of the function

F

of

(5.4') are holomorphic over complex integration

B.

Conversely,

X

may be reconstructed from

F

by

(5-4") Moreover, conformality is equivalent to the relation (5.5)

( componentwise complex multiplication).

An interior branch point now may be characterized as a zero of the holomorphic vector function F. Since zeros of holomorphic functions are isolated this is also true for interior branch points of minimal surfaces X. Moreover, if X can be analytically extended across a segment C of X can have at most finitely many branch points on any compact subset of This observation leads to the following result of Douglas [1] and Rad6 [1]: Theorem 5.3: then

If

is a minimal surface bounded by a Jordan arc is a homeomorphism.

Proof: It suffices to show that is injective. Assume by contradiction that for Since X maps monotonically 1 is an open onto it follows that for where segment with end-points We may assume . Extending X by odd reflection across C we obtain a surface

which is harmonic in a neighborhood function

of

C

giving rise to a holomorphic

Moreover, on B so that also is conformal on But on Cso that on C and must vanish identically in Hence also const. = 0. In particular, X = 0 and . The contradiction proves the claim.

Definition 5.4: as a zero of F. Let

The order of a branch point

w of a surface

X is its order

be a zero of F of m—th order. Then aftefa rotation of coordinates

I. Existence of ft solution.

where

25

satisfies:

as a consequence of (5.5). Hence if

has the expansion

(5.6) in power series of An analoguous formula of course holds for if X is analytic in a neighborhood of in Using results of Hartman-Winter in [1] it is possible togive similar expansionsfor X near branch points on general provided is of class or cp. Nitsche [1, 381] for references. As a particular consequence of (5.6) we immediately deduce the following Theorem 5.5: Suppose is a Jordan curve of class and let be a solution to Plateau's problem (1.1) - (1.3). Then X has at most finitely many branch points. Moreover, the tangent plane to the surface X behaves continuously near any branch point. Let us now specialize formula (5.6) to the case n = 3. such that in powers of

I.e. locally, X through expansions for

There exist numbers

looks like an —sheeted surface over its tangent plane These sheets need not all be distinct, e.g if the power series only contain powers of for some

Definition 5.6: A branch point of a surface X is called a false branch point if there exists a neighborhood U of and a conformal mapping such that near Otherwise

wa is called a true branch point of

X.

The following result of Gulliver, Osserman and Royden [1] - cp. also Steffen-Wente [ 1, Theorem t.2 ] - excludes false (interior) branch points for minimal surfaces satisfying the Plateau boundary condition: Theorem 5.7: Suppose a minimal surface

is a rectificiable Jordan curve in cannot have false interior branch points.

Then

This result makes crucial use of Theorem 5.3. For solutions to (1.1) - (1-3) of least area in also true branch points can be excluded by means of the following argument due to Osserman [l].His results were completed and extended by Alt [1], Gulliver [1], Gulliver-Lesley [1] .

20

A. The classical Plateau Problem for disc - type minimal surfaces.

Theorem 5.8: Suppose minimizes D in C(T). Then not have true interior branch points. If in addition T is analytic then not have true boundary branch points, either.

X X

does does

The proof •uses the fact that near interior branch points w0 by (5.6) different sheets of X must meet transversally along a branch line through w0\ cp. Chen [1] . This allows to construct a comparison surface of less area by a cutting-and-pastingand-smoothing argument. Suppose for simplicity that x has a branch point at WOI and let

be a branch line of X along which X(7i(i)) = X(72(t)) while the u—derivatives of X along 71,72 are transverse. The figure illustrates the two - stage process of transforming X into a surface Y with the same area as X by a discontinuous 7 1 , 7 2 are transformation of the parameter space. (The successive images of indicated for clarity.)

After this change of parametrization has been carried out the reparametrized surface Y will still belong to the class C(r). Instead of a branch line, however, where two sheets intersect the new surface Y will have a contact line where two sheets touch. Moreover each sheet contains an edge along the contact line. Area can hence be reduced by smoothing off the edges. In this way we obtain a comparison surface Z e C ( T ) such that On the other hand, by (2.5) and since X by assumption minimizes D in C(T)

A contradiction. For branch points on analytic boundaries the same reasoning applies, cf. Gulliver Lesley [1]. The essential ingredient again is the existence of a branch line through the branch point. For smooth wires R. Gulliver [2] has recently presented an example of a smooth wire spanning a minimal surface with a boundary branch point which is not connected to a branch line of the surface.

I. Existence of a solution.

27

Finally, let us mention a result of Sasaki [1] and Nitsche [1, §380 , formula (156)] which relates the number and order of the branch points of a minimal surface X € IR? to the total curvature of its boundary T by the Gauss-Bonnet formula and hence permits to estimate the former independently of X. Theorem 5.9: Let T be a Jordan curve in 2R3 of class C2 with total curvature «(r). Suppose X £ C(r) solves (1.1) -(1.3) and has interior branch p o i n t s W j , of orders and boundary branch points e*** of orders Let K denote the Gaussian curvature of X. Then there holds the relation

In particular, if

any minimal surface spanning T is immersed.

Remark: Note that in case K(r) = 4 i a branched minimal surface X would have to satisfy K = 0, i.e. be a planar surface. Hence X could not have a branch point by the Riemann mapping theorem. The following example from Nitsche [1, §288] illustrates that in general even areaminimizing parametric solutions to the Plateau problem (1.1) - (1-3) may fail to be embedded (and hence will be physically unstable):

20

A . The classical Plateau Problem for disc - type minimal surfaces.

Foi curves like the depicted one the true physical solutions apparently can be described best by the methods of geometric measure theory, cf. Almgren [1]. However, there is a class of curves in iR3 where the least-area solution to the parametric Plateau problem can be shown to be a minimal embedding: these are the so-called extreme curves, i.e. Jordan curves on the boundaries of convex regions Q € iR 3 . More generally, one can also allow curves on boundaries of regions (1 with the property that the mean curvature of dd with respect to the interior normal is non-negative ("M—convex" regions). Convex or M-convex surfaces provide natural "barriers" for minimal surfaces (by the maximum principle for the non-parametric minimal surface eqution, cf. Nitsche [1, §579 fF.]). The following result is due to Meeks and Yau[l]; the existence of an embedded minimal disc was established independently by Tomi and Tromba [1] , resp. Almgren and Simon [1] : Theorem 5.10: Let fi be an M-convex region in 2R3 of class C2 and let T C dSi be a rectifiable Jordan curve, which is contractible in 0 . Then there exists an embedded minimal disc with boundary I\ Moreover, any solution X of (1.1)-(1.3) with X(B) C fi which minimizes D in this class is embedded. As a special case Theorem 5.10 contains the "existence" part of the following classical result of Rad6 [2]. The uniqueness is a consequence of the maximum principle. Theorem 5.11: Suppose T is a Jordan curve in 2R3 having a single-valued parallel projection onto a convex curve T in some plane P in IB?. Then there exist a unique minimal surface spanning T (up to conformed reparametrization). This surface is a graph over the region bounded by T in P. In higher dimensions a result like Theorem 5.10 is not known.

I. Existence of ft solution.

29

Appendix We establish (2.5). For simplicity we assume in addition that Proposition A«lt

Proof:

Let T be a

Jordan curve. Then

Let

, and

. Approximate X by embedded surfaces . We claim that for any there onto itself mapping dB monotonically

exists a map onto and such that (A.1) Since for any

while as

and any such

there holds

clearly

, we infer from (A.l) that

By density of implies the Proposition.

in

and continuity of

In order to prove (A.l) introduce the set the set

A the latter inequality

as the weak closure in

of

g is a diffeomorphism onto of normalized diffeomorphisms of Note Lemma A.2: Proof: where and

is weakly closed in we have

If

For any

any

weakly in since

is separable, a diagonal sequence

is weakly closed.

Next we show that any is in fact a uniform limit of diffeomorphisms. Indeed, a standard argument based on the Courant - Lebesgue Lemma 4.4 shows that bounded subsets of are equi-continuous: First, remark that whence bounded subsets of continuous on dB, cp. Lemma 4.3.

are equi-

30

A. The classical Plateau Problem for disc - type minimal surfaces. •

Next,given

any

there exists there is a radius

such that for any such that on

Since g is a diffeomorphism and since the set continuous on dB, for small . any such g maps the disc the "small" disc bounded by . Hence This proves equi-continuity of In particular, any monotonically onto

is equionto

bounded subsets of is a continuous map of onto and satisfying the three-point-condition

Thus, for weakly in

if

with

also we have

mapping Moreover,

(A.2) in the sense of distributions. We use this fact to compute uniformly there holds

with

First, since

. By (A.2) this equals

denoting the tangent derivative in counter clock-wise direction. Since monotonically onto we have with we may pass to the limit

maps Hence

in both integrals and finally arrive at the identity

I. Existence of ft solution.

31

as claimed.

I.e. proving A.2 and By Let P r ochoice Proposition oby f satisfies of whence Example (A.l) Proposition of be and minimizes by the 4.5 3.5 a we weak the minimizing hypothesis wehave EProposition. may lower A is.D coercive lassume (completed): semi-continuity among of sequence Lemma that onallsuch surfaces 2.4 Now with and that ofconsider respect Dirichlet's is conformal. the to and the integral functional But then Intopology. Eby particular, on Lemma

II. Unstable minimal surfaces 1. Ljusternik-Schnirelman theory on convex sets in Banach spaces. The method of gradient line deformations and the minimax-principle are the most general avaible tools for obtaining unstable critical points in the calculus of variations. Historically, the use of these methods can be traced back to the beginning of this century, cf. Birkhoff's [1] theorem on the existence of closed geodesies on surfaces of genus 0 . Through their famous improvement of BirkhofF's result the names of Ljusternik and Schnirelman [1] became intimately attached to these methods. In 1964 a major extension of these techniques was proposed by Palais [1], [2], Smale [l] and Palais - Smale [lj. Their fundamental work has found many applications and has inspired a lot of further research. One of the most significant contributions may be the well-known paper of A. Ambrosetti and P. H. Rabinowitz [1]. For the Plateau Problem, however, these methods still seemed inadequate, and for a long time analysts believed that the Palais-Smale condition would "never" be satisfied in any "hard" variational problem as the Plateau problem for minimal surfaces was considered to be. (Cf. the remarks by Hildebrandt [4, p. 323 f.]) As we shall see, by a simple variation of the classical concepts of Ljusternik-Schnirelman, resp. Palais and Smale the Plateau problem can be naturally incorporated in the frame of these methods. This extension of Ljusternik-Schnirelmann theory and its application to the Plateau problem was presented in Struwe [1]. In abstract terms we may regard this method as an extension of Ljusternik-Schnirelman theory to functionals defined on closed convex sets of (affine) Banach spaces and satisfying a variant of the Palais - Smale condition.

We now recall the pertinent ideas. Throughout this section we make the following Assumptions Let T be a Banach space with norm closed and convex. Suppose E : M —• IR admits a Frechet differentiate extension Definition 1.1:

At a point

measure the slope of E in

M.

let

34

A. The classical Plateau Problem for disc - type minimal surfaces. •

Remark 1.2.

i) If

then

ii) Since

and M is convex , g is continuous.

Definition 1.8: A point is called critical iff , Otherwise x is called regular. If x is critical, is called a critical value of E. If consists only of regular points, is called regular. Remark 1.4: If M = T by Remark 1.2 Definition 1.3 coincides with the usual definition of critical points.

Definition 1.5: Let is a pseudo gradient vector field for

. A Lipschitz continuous mapping E on M if the following holds:

i) ii)

There exists a constant c > 0 such that a) b)

Exactly as in the case M = T, cf. Palais [3, Chapter 3], we now establish

Lemma 1.6: Proof:

Let

There exists a pseudo-gradient vector field and choose

for E on

M.

such that

(1.1) By continuity, there exists a neighborhood we still have Hence E on

of

in M such that for while is a pseudo-gradient vector field for

The sets are an open cover of . Since is metric there exists a locally finite refinement of this cover, i.e. having the property that for any there is a neighborhood of and a finite collection such that for we have cf. Kelly [1, Thm. 8, p. 156; Cor. 35, p. 160]. Let be a Lipschitz continuous partition of unity subordinate to i.e a collection of Lipschitz continuous functions with support in such that for each and

II. Unstable minimal surfaces

35

E. g. we may let

Finally, define

where is associated to satisfies i) and ii) of Definition 1.7

Definition 1.7: holds: (P.S.)

by (1.1).

is Lipschitz continuous and

E satisfies the Palais-Smale condition on M if the following

Any sequence

in M such that is relatively compact.

uniformly, while

Remark 1.8: Again (P.S.) reduces to a variant of the well-known Palais-Smale condition (C), cf. Palais - Smale [1], in case The Palais-Smale condition crucially enters in the following fundamental "deformation lemma." For let

Lemma 1.9: Suppose E satisfies (P.S.) on M. Let and suppose N is a neighborhood of in M. Then there exists a number and a continuous one-parameter family of continuous maps I of M having the properties i) ii)

iff = 0,

or if

is non-increasing in t for any

or if x,

Hi)

For the proof we need the following auxiliary Lemma 1.10: families

Suppose

E satisfies (P.S.) on

M. Then for any

the

36

A. The classical Plateau Problem for disc - type minimalsurfaces.•

resp. for some constitute fundamental systems of neighborhoods of

Proof: By continuity of g, clearly each and each is a neighborhood of . Hence it remains to show that any neighborhood N of contains at least one of the sets Suppose by contradiction that for some neighborhood N of i we have N. Then for a sequence elements By (P.S.) the sequence accumulates point Hence for large m, a contradiction. for there exist sequences that By (P.S.) accumulates at does. A contradiction.

P r o o f o f Lemma 1.9: Coose numbers

and let

7/' be a Lipschitz continuous function on M such that in outside Also let a numberto be specified later, and choose a function if if

and any there exist at a critical Similarly, if such hence also

such that

and min be | such that

Define

where is the pseudo-gradient vector field constructed in Lemma 1.8. Since any critical point of E on M has a neighborhood where either or vanishes, e is Lipschitz continuous. Moreover, by convexity of Af, e satisfies (1.2) at any point

Now let

be the solution to the initial value problem

(1.3)

By convexity of M, may be constructed as a limit of approximate trajectories of (1.3) by Euler's method, cp. Struwe [1, Lemma 3.8].

II. Unstable minimal surfaces

If T is locally strictly convex and if the projection M obtained by letting

37

of

T onto

is locally Lipschitz continuous (e.g. if T is a Hilbert space), then a more instructive existence proof goes as follows: Extend e to T by letting

Now let be the solution to (1.3) on T which exists globally by Lipschitz continuity and boundedness of e. By (1.2) M is an invariant region for and the deformation may be obtained by restricting to Af. As

solves (1.3), each now is a continuous map from M into M and trivially satisfies i), ii) by our choice of Finally, if , either or for all . In the latter case, moreover, by choice of :

(1.4)

Now suppose all

or or the flow By boundedness

Then either for through as must traverse the annulus this will require

A This Thus and variant hence completes iii) will of inthe any bethe deformation satisfied event construction. weif obtain welemma let that 1.9 yields the following result:

38

A. The classical Plateau Problem for disc - type minimal surfaces. •

Lemma 1.11: Suppose E satisfies (P-S.) on M, and let relative minimum of E in M. Then there exists a number any we have

Proof:

Choose

By assumption there exists

and let

be a strict such that for

such that

be a minimizing sequence for

E in

If

the proof is complete. Otherwise and either or there exists such that for all m. In the first case, by (P-S.) accumulates at an element where . Since this contradicts the strict minimality of we are left with the second case. Choose and let be defined as in Lemma 1.9. Consider the sequence . Since it follows that

Moreover, since like (1.4) we obtain But since i this implies that m. The contradiction proves the lemma.

for large

m and

for large

Lemmata 1.9, 1.11 immediately yield the following variant of the classical" mountainpass-lemma": Theorem 1.12: Suppose E satisfies (P.S.) on M, and let be distinct strict relative minima of E. Then E possesses a third critical point distinct from is characterized by the minimax-principle (1.5') where

(1-5")

II. Unstable minimal surfaces

39

Moreover

and

is unstable in the sense that

is not a relative minimum of

Proof: i) By Lemma 1.11 that is a regular value of E, i.e. be as constructed in Lemma 1.9. By definition of

there is

Applying the map to P. while by iii) The contradiction shows that

E.

Suppose by contradiction Choose and let

such that

p by property i) of

the path

is critical.

ii) Now suppose that E possesses only critical points of energy which are relative minimizers of E in M. The set will then be both open and closed in hence there exists a neighborhood N of in M such that N and are disjoint. A fortiori, then and N will be disconnected for any Choosing corresponding to this N and and letting however by property iii) of we obtain a path

Since and since and N are disconnected, contradiction shows that E has an unstable critical point of energy

The

A slight variant of the preceding result is given in Theorem 1.13: Suppose E satisfies (P.S.), and let be two (not neccessarily strict) relative minima of E. Then either and are connected in any neighborhood of or there exists an unstable critical point of E characterized by the minimax-principle (1.5). Proof: Let be given by (1.5). If consists only of relative minimizers of E as in part ii) of the proof of Theorem 1.12 we deduce that for any sufficiently small neighborhood N of there holds for My Letting be as constructed in Lemma 1.9 corresponding to and choosing such that we obtain a path connecting

40

A. The classical Plateau Problem for disc - type minimal surfaces. •

with in Hence and the same connected component of in particular

and

both belong to

Along the same lines numerous other existence results for unstable critical points can be given. For our purpose, however, Theorems 1.12, 1.13 will suffice and we refer the interested reader to Palais [3] or Ambrosetti - Rabinowitz [1]. Theorem 1.13 is related to a result by Pucci and Serrin [1] .

II. Unstable minimal surfaces

41

2. The mountain-pass-lemma for minimal surfaces.

In order to convey

the preceding results to the Plateau problem we reformulate the variational problem in a more convenient way. At first we closely follow Douglas' original approach to the Plateau problem. Let assume that

be a reference parametrization of the Jordan curve is a homeomorphism.

Note that by (1.1) it suffices to consider surfaces are harmonic:

X

whose coordinate functions

By composition with and harmonic extension the space of monotone reparametrizations of More precisely, let defined by

We

may be represented by

be the harmonic extension operator in B,

h is linear and continuous by the maximum principle. Then the map coordinates on

Also let

denote

(2.1')

where y ox denotes the map (2.1")

(y o »)(«