Non-existence, non-uniqueness and irregularity of ... - Project Euclid

NON-EXISTENCE, NON-UNIQUENESS AND IRREGULARITY OF SOLUTIONS TO THE MINIMAL SURFACE SYSTEM BY H. B. L A W S O N , Jr. a n d R. 0SSERMAN(1) University of California, Berkeley, California, USA

Stanford University, Stanford,California, USA

Table of contents w 1. w 2. w 3. w 4. w 5. w 6. w 7.

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The minimal surface system . . . . . . . . . . . . . . . . . . . . . . . . . A brief s u m m a r y of known results in eodimension one . . . . . . . . . . . . . . The existence of solutions of dimension two . . . . . . . . . . . . . . . . . . . The non-uniqueness and non-stability of solutions of dimension two . . . . . . . . The non-existence of solutions in dimensions >/4 . . . . . . . . . . . . . . . . . The existence of non-parametric minimal cones . . . . . . . . . . . . . . . . .

1 2 4 5 6 11 13

w 1. Introduction T h e s t u d y of r e a l - v a l u e d f u n c t i o n s whose g r a p h s are m i n i m a l surfaces has a long a n d rich history, a n d b y now we h a v e a f a i r l y p r o f o u n d u n d e r s t a n d i n g of t h e subject. I n c o n t r a s t , a l m o s t n o t h i n g is k n o w n a b o u t v e c t o r - v a l u e d f u n c t i o n s whose g r a p h s are m i n i m a l surfaces, a n d this p a p e r should e x p l a i n , a t least in p a r t , w h y t h i s is so. I t will be s h o w n t h a t m a n y of t h e deep a n d b e a u t i f u l results for n o n - p a r a m e t r i c m i n i m a l surfaces in codim e n s i o n one fail u t t e r l y in higher codimensions. W e shall be concerned p r i m a r i l y w i t h t h e Dirichlet p r o b l e m for t h e m i n i m a l surface s y s t e m on a b o u n d e d , c o n v e x d o m a i n ~ in R n. U s i n g a n old a r g u m e n t of R a d o , we shall show t h a t for n = 2 , t h e D i r i c h l e t p r o b l e m is solvable for a r b i t r a r y c o n t i n u o u s b o u n d a r y d a t a . H o w e v e r , we t h e n c o n s t r u c t e x a m p l e s to show t h a t these solutions are not unique in general. Moreover, we shall show t h a t such surfaces need not even be stable in c o n t r a s t w i t h t h e f a c t t h a t in codimension one, n o n - p a r a m e t r i c m i n i m a l surfaces are a b s o l u t e l y a r e a minimizing. W e shall t h e n show t h a t for n >~4, t h e Dirichlet p r o b l e m is not even solvable in general. (1) Work partially supported by NSF grants MPS-74.23180 and MPS 75.04763. 1 - 772904 Acta mathematica 139. Imprim6 le 14 Octobr9 1977.

2

H, B. LAWSON~ JR. AND R, OSSERMAN

I n fact, for each n ~ 4 it is possible to find a C ~ f u n c t i o n / : S ~ - l ~ I t k for some integer k, 2 ~~2. For the remainder of the paper we shall be concerned with the following. D I R I CH L E T P R 0 B L E M. Given a/unction r d ~ ~ It k o/class C s, 0 -~ s ~l, we/urther require that the area o/the graph o / / b e / i n i t e .

The study of the Dirichlet problem usually falls into two distinct parts, those of the existence and the regularity of solutions. I t is an unusual fact t h a t for the non-parametric minimal surface system more is known about regularity t h a n existence. T~EOREM 2.2. (Interior regularity; C. B. Morrey [22], [23]). A n y C 1/unction / which satis/ies the system (2.2) is real analytic.

THEOREM 2.3. (Boundary regularity; W. Allard [2]). Suppose that r is o/ class C s'~ /or 2 ~ s ~ ~ or s = w, and let / be any solution to the Dirichlet problem/or r in ~ . Then there is a neighborhood U o / d ~ such that/eCS'~(U fl ~). Note. Theorem 2.3 is deduced from the work in [2] as follows. Since ~ is strictly

convex and r is class C ~, the arguments in 5.2 of [2] show t h a t the graph F I of / in R n x R k, at each point of its boundary, has a tangent cone consisting of a finite number

H. B. LAWSON~ J R . AND R. 0 S S E R M A N

of half-planes (of dimension n), each of which projects non-singularly into R ~. Since each such cone is the limit of a sequence of dilations of FI, it is easy to see that the cone consists, in fact, of a single half-plane. I t follows that the density

O(llr111, x)= 89 at

each boundary

point x. Theorem 2.3 is then an immediate corollary of the Regularity theorem in [2, w4]. Remark. Perhaps it should be pointed out that if ~ is not convex, the Dirichlet

problem is not necessarily solvable in general even for n = 2 and r = s = w . In fact when n = 2, convexity is necessary and sufficient for a solution to exist corresponding to arbitrary C s boundary values [12]. When n > 2, convexity is still sufficient, but the precise necessary and sufficient condition is that the mean curvature of the boundary with respect to the interior normal be everywhere non-negative [17]. w 3. A brief s n m m a r y of kllown results in c o d l m e n s i o n o n e For comparison with our later results in higher codimension we list here some of the facts known about non-parametric minimal surfaces in codimension one. T H E o R E M 3.1. For k = 1, the Dirichlet problem is solvable /or arbitrary continuous boundary data. Furthermore:

(a) The solution is unique. (b) The solution is real analytic. (c) The solution is absolutely area minimizing, i.e., its graph is the unique integral current o/least mass in R "§ /or the given boundary (the graph o/4).

The existence for C~ boundary data follows from Jenkins and Serrin [17], and for general continuous boundary data from the a priori estimates in [5]. The regularity of Lipschitz solutions follows from de Giorgi [7]. (See [37] or [25].) The remainder of the theorem is classic. (See [21, pp. 156-7] for part (c).) For the purpose of completeness we mention the following strong removable singularities result. THEOREM 3.2. Let K be a compact subset o] ~ with Hausdor]/ (n-1).measure zero. Then any (smooth) /unction /: ( ~ - K ) - ~ R

which satis/ies the minimal sur/ace equation

(2.2)' in ~ - K extends to a (smooth) solution in all o / ~ . This was first proved by Bers [4] for n = 2 and K = {point}. The general 2-dimensional case was proved by Nitsche [27]. The result in arbitrary dimensions is the work of de Giorgi and Stampaeohia [8]. Note. I t has been known for some time that this last result is not true in general

THE MINIMAL SURFACE SYSTEM

codimensions. In fact, in [28] an example is given of a bounded function /:{(x, y)ER: 0 < x 2 + y~ ~~l, there exist solutions /EC~(~)N C~ Dirichlet problem/or any given continuous boundary/unction r d ~ R

to the

k.

Proo/. We begin by recalling the notion of a generalized parametric minimal sur/ace in R" having a given Jordan curve 7 as boundary. This is a map ~o: A ~ R ", where A = {z = x + iye C: [z I ~~R 2, it follows that A(Z~) ~>gR ~ for each i and (5.2) is established. We have shown that there must be two geometrically distinct parametric minimal surfaces with boundary 7, which satisfy condition (5.1). This establishes an example of the type claimed in the theorem with the exception that the curve is class C ~176 and the codimension is three.

10

H . B. LAWSO~N, J R . A N D R. 0 S S E R M A N

Y V

4o

JU

~ X

Figure 4

We now observe t h a t throughout the above argument it is possible to replace the curve y with a real analytic approximation. This is done b y replacing r with a sufficiently large piece of its Fourier e~pansion. Since the Fourier polynomials preserve such properties as

and

for (complex-valued) functions F on the circle, we have that the analytic approximation will also be o-invariant. We shall now indicate how to construct an example of codimension 2. The arguments are all essentially the same except that one begins with a map r 1 6 2

where R>~I and

40: dD-~It~ is a suitably parametrized double tracing of the curve pictured in Figure 4, where each loop is traversed once in each direction, and where the parameter is a multiple of arc length. The graph of r is invariant by the symmetries ok: IO-~R 4, k = 1, 2 where 01(u, v, x, y) = ( - u ,

- v , x, - y )

a,,(u, v, x, y) -- ( - u , v, - x , y). Let ~ be the graph of r If we again assume t h a t there are not two geometrically distinct minimal surfaces with boundary ~ satisfying condition (5.1), then the surface ]E given by the Douglas solution to the Plateau problem for ~ must be invariant under o 1 and 02. The linear function ~t= x has exactly one critical point p on ~]. This follows from an argument entirely similar to the proof of Proposition 5.2. Consequently, r

a~(p)=p,

and so p = 0 . The rest of the argument proceeds as above. The appropriate comparison


11

surface is given b y a map ~F = R~Fo where ~F0 is chosen to be linear on the lines u -~-v = c. For Ic ] ~~Re there is no solution to the Dirichlet Problem/or the boundary/unction ~ = R . r Remark 6.2. I t follows from the Implicit function theorem (cf. [26]) that the Dirichlet problem is always solvable for sufficiently small boundary data. Hence, given r as in Theorem 6.1, there is an re > 0 such t h a t for all r, Ir[ ~~crR~

(6.2)

where c~ i8 the volume o/ the unit ball in R ~. This is a direct consequence of the well known fact that r is monotone increasing and lim~_,o r

Iq B(xo, r))/%r p

(See [10], [13] or [20].)

Let us now suppose that for fixed R, there is a solution/R to the Dirichlet problem for Ca. Let Vn be the minimal variety given by the graph of ]n in R N where N ---2(n + 1) + k. B y Theorem 2.3 VR is regular at the boundary. We clearly have that dVR = {(x, Rq~(x)): [[x[[ = 1 } is contained in the sphere of radius l/l + R 2 about the origin in R N. Hence, it follows from (6.1) that l/i + R2 M(dVR). M(Vn) ~

V k-2 ~

2

for some fixed k-plane ~0 in R N, then M is a totally geodesic subsphere. The examples above show t h a t there is a positive lower bound for the "best" constant possible in this theorem. In fact, for the Hopf maps ~], ~', and 7/" above, there are planes v0 such that (v, v0) is constant and equal to 1/9, (1/8]/;/) and (7'.2 -1'.3-5V7i5) respectively. Remark. I t follows from the work of J. L. M. Barbosa ("An extrinsic rigidity theorem

for minimal immersions from S ~ into S~, '' to appear) t h a t there are no non-parametric (1) We should like to thank the MACSYMA program at M.I.T. and Bill Gosper of the Stanford University Artifieal Intelligence Laboratory.

16

H. B, LAWSOlff, JR. AND R. OSSERMAN

m i n i m a l cones of d i m e n s i o n t h r e e or less. Moreover, using this fact one can p r o v e t h a t e v e r y L i p s c h i t z solution of t h e m i n i m a l surface s y s t e m in t h r e e (or fewer) i n d e p e n d e n t v a r i a b l e s is real a n a l y t i c . Of course, b y t h e e x a m p l e s a b o v e such a s t a t e m e n t is false for t h e s y s t e m in four or m o r e variables. References

[1]. ALLARD, W. K., On the first variation of a varifold. Ann. o] Math., 95 (1972), 417-491. On the first variation of a varifold: B o u n d a r y behavior. Ann. o] Math., 101 (1975), 418-446. ALMGREN, F. J., JR., The theory o] vari]olds. Princeton mimeographed notes, 1965. BERS, L., Isolated singularities of minimal surfaces, Ann. o/Math. 53 (1951), 364-386. BOMBIERI, E., DE GIORGI, E. & MIRANDA, M., U n a maggiorazione a priori relativa alle ipersuperfici minimali non parametriche, Arch. Rational Mech. Anal., 32 (1969), 255-267. COURANT, R., Dirichlet's Principle, Con/ormal Mapping and Minimal Sur/aces. Interscience, N.Y., 1950. DE GIORGI, E., Sulla differentiabilitk e l'analiticitk delle estremali degli integrali multipli regolari. Mem. Accad. Sci. Torino, s. I I I , parte I, {1957), 25-43. DE GIORGI, E. & STAMPACCHIA, G., Sulle singolaritk eliminabili delle ipersuperfici minimali, Atti. Aecad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur., (ser. 8) 38 (1965), 352-357. DOUGLAS, J., Solution of the Problem of Plateau, Trans. Amer. Math. Soc., 33 (1931), 263-321. FEDERER, H., Geometric Measure Theory. Springer, N.Y., 1969. FEDERER, H. & FLEMING, W., Normal a n d integral currents. Ann. of Math., 72 (1960), 458-520. FINN, R., Remarks relevant to minimal surfaces and surfaces of prescribed mean curvature. J. Analyse Math., 14 (1965), 139-160. FLEMING, W., On the oriented Plateau problem. Rend. Cir. Mat. Palermo, 11 (1962), 69-90. HARVEY, R. & LAWSON, H. B., JR., Extending minimal varieties, Inventiones Math., 28 (1975), 209-226. HSIANG, W. Y., Remarks on closed minimal submanifolds in the s t a n d a r d Riemannian m-sphere. J. Di]]erential Geometry, 1 (1967), 257-267. HSIA~O, W. Y. & LAWSON, H. B. JR., Minimal submanifolds of low eohomogeneity, J. Di]]erential Geometry, 5 (1970), 1-37. JE~s, H. & SERRIN, J., The Dirichlet problem for the minimal surface equation in higher dimensions. J. Reine Angew. Math., 229 (1968), 170-187. LAWSON, H. B., JR. The equivariant Plateau Problem and interior regularity, Trans. Amer. Math. Soc. 173 (1972), 231-250. -Lectures on Minimal Submani]olds. I.M.P.A., R u a Luiz de Camoes 68, Rio de Janeiro, 1973. -Minimal varieties in real and complex geometry. Univ. of Montrdal, 1973. -Minimal varieties, pp. 143-175, Prec. o/Syrup, in Pure Mathematics, vol. X X V I I p a r t 1, Differential Geometry, A.M.S., Providence, 1975. MORREY, C. B., Second order elliptic systems of differential equations, pp. 101-160. Contri. butions to the Theory o] Partial Di]/erontial Equation. Annals of Math. Studies No. 33, Princeton U. Press, Princeton, 1954. -Multiple Integrals in the Calculus el Variations. Springer Verlag, N.Y., 1966.

[2]. - [3]. [4]. [5].

[6]. [7]. [8]. [9].

[10]. [11]. [12]. [13]. [14]. [15]. [16]. [17]. [18]. [19]. [20]. [21]. [22].

[23].


17

[24]. MORSE, M. & TOMPKINS, C., Existence of minimal surfaces of general critical type. Ann. o/Math., 40 (1939), 443-472. [25]. MOSER, J., A new proof of de Giorgi's theorem concernL~g the regularity problem for elliptic differential equations, Comm. Pure Appl. Math., 13 (1960), 457-468. [26]. NIRENBERG, L., Topics in Nonlinear Functional Analysis. Courant I n s t i t u t e Lecture Notes, 1973-4. [27]. NITSCHE, J . C . C . , l~ber ein verallgemeinertes Dirichletsches Problem fiir die Minimalfl/~chengleichtmg und hebbare Unstetigkeiten ihrer LSsungen, Math. Ann., 158 (1965), 203-214. [28]. OSSERMA~, R., Some properties of solutions to the minimal surface system for a r b i t r a r y codimension, pp. 283-291, Proc. o] Syrup. in Pure Math., vol. XV, Global Analysis A.M.S., Providence, 1970. [29]. - A Survey o / M i n i m a l Sur/accs. Van Nostrand, N.Y., 1969. [30]. - - - - Minimal varieties. Bull. Amer. Math. Soe., 75 (1969), 1092-1120. [31]. - On Bets' Theorem on isolated singularities. Indiana Univ. Math. J., 23 (1973), 337-342. [32]. RADO, T., On Plateau's problem. Ann. o] Math., 31 (1930), 457-469. [33]. - - - - On the Problem o] Plateau. Ergebnisse der Mathematik und iher Grenzgebiete, vol. 2, Springer, 1933. [34]. REILLY, R., Extrinsic rigidity theorems for compact submanifolds of the sphere. J. Di/]erential Geometry, 4 (1970), 487-497. [35]. SHIFFMAN,M. The Plateau Problem for non-relative minima. Ann. o/Math., (2) 40 (1939), 834-854. [36]. SIMONS, J., Minimal varieties in Riemarmian manifolds. Ann. o] Math., 88 (1968), 62-105. [37]. STAMPACCHIA,G. On some regular multiple integral problems in the calculus of variations. Comm. Pure Appl. Math., 16 (1963), 383-421. Received August 23, 1976.

2 - 772904 ,,lcta mathematica 139. Iraprim6 le 14 Octobre 1977.