Math. Z. 154, 265-273 (1977) Mathematische Zeitschrift

�9 by Springer-Verlag 1977

On Some Extensions of Bernstein's Theorem

Leon Simon*

Stanford University, California U.S.A.

A well-known theorem of Bernstein asserts that a C 2 function which satisfies the minimal surface equation on the whole of ~ 2 must be linear. This result was extended to a class of equations corresponding to parametric elliptic functionals by Jenkins [9], and to the entire class of C2(IR 2) functions whose graphs have quasiconformal Gauss map by Simon [11]. For higher dimensions the result was shown to be true for solutions of the minimal surface equation which are defined over the whole of ]R", n__<7, by Simons [13], using (in part) an argument of Fleming [7] and De Giorgi [3]. The result was shown to be false for n > 7 by Bombieri De Giorgi and Giusti [2].

We here wish to discuss the higher dimensional case (n > 3) for the same class of equations treated in the case n = 2 by Jenkins [9]; specifically, we consider the non-parametric Euler-Lagrange equation of a C 2'~ parametric elliptic function- al, with integrand not depending explicitly on the spatial variables. Our main result is that a Bernstein theorem always holds for such equations in case n = 3. The result is also shown to hold up to n = 7 provided the integrand of the associated parametric functional is close enough, in the C 3 topology, to the area integrand.

These results will be proved by first obtaining pointwise curvature estimates of a kind that were established for solutions of the minimal surface equation by Heinz [8] and extended to higher dimensions by Schoen, Simon, Yau [10] and Simon [12]. The main results appear in Theorem 1 and its corollaries.

w 1. Notation

n => 2 denotes a fixed integer;

~ R "+~ x~ =(Xo~, ... Xo~ X o ~ ( X 0 1 ~ . . . ~ X o ~ + l j ,

Bp(xo) = {xelR"+1: [x -Xo[ <p};

De(x;)= {x~":lx-x;I <p};

* Research partly supported by an NSF grant at Stanford University and a Sloan Fellowship

266 L. Simon

~ k denotes k-dimensional Hausdorff measure in IR "+ 2; ~o,,+ 1, ~ , denote Lebesgue measure in IR "+ 1, 1R" respectively;

will denote the set of functions F which are defined on IR "+~, which have locally H/flder continuous second partial derivatives o n / R " + I ~ {0} and which satisfy the conditions:

(i) f(#p)=#F(p), #>0 , pMR "+~,

(i) F(p)>=lp[, p~lR "+~,

(iii) ~ Fplpj(p)~i~j>lp]-~[~'[ 2, ~'=~-~Pl ~ p ' i , j = l

~IR "+1, p~lR"+l~{0}.

Associated with a given F e ~ we have the positive parametric elliptic functional F defined as follows:

If M is an oriented C 2 hypersurface in IR "+ 1, with continuous unit normal v and with ovf"(M)< 0% then

F(M) = y F(v(x)) dJf"(x). M

Notice that this definition depends on the choice of unit normal v; we therefore always take "oriented hypersurface M" to mean a pair M,v where v is a continuous unit normal for M.

Corresponding to the parametric functional F we have the non-parametric functional tp defined for C2(Dp(x'o)) functions u by

7J(u) = F(M,),

where M . = g r a p h u (with unit normal (-Du, 1)/1/1 +]Dul2). One easily checks that we can write

~(u)= S F(-Du(x), 1)a~"(x). D o (x6)

The Euler-Lagrange equation for extremals of this functional is

~G(-Du(x), 1)=0. (1) i = 1

Given a C 2 oriented hypersurface M with

J~F"(M~K)< ov for each compact K ~ I R "+1, (2)

we will let [[M]I denote the associated rectifiable current; that is, for any smooth n-form co with compact support in ]R "+ 1 we define

[[M-~ (co) = ~ co, M

On Some Extensions of Bernstein's Theorem 267

where the integral on the right is taken in the usual sense of differential geometry. (Actually, to be strictly precise we should write ~ i e co on the right,

M

where i is the inclusion map M ~ IR n + 1.) The boundary of [[M]I, denoted 3 I[M~, is the (n-1)-dimensional current defined by

0 [ M ~ (co) = [[M]] (do),

whenever co is a smooth (n -1 ) - fo rm with compact support in IR "+1. This current of course corresponds to the oriented set theoretic boundary 3M in case M is a compact manifold-with-boundary (by Stokes' theorem). We define the regular set, reg M, of M by

reg M = {x~M ~ s p t 0 [[M-~ : M is a C 2 hypersurface in some neighbourhood

of x}.

(Here spt c~ [[M~ = support of ~? I[M]] = IR "+j --~ U w,, where the union is taken over all open W c IR "+1 such that 3 [[M]I (co)= 0 whenever co has compact support in W.) The singular set, sing M, of M is defined by

sing M = ~r ~ reg M.

Notice that we always have M c r e g M (whenever M is an oriented C z hypersurface); if

Yt~"(sing M ~ spt 0 [[M]]) = 0, (3)

then after redefinition on a set of Yf"-measure zero, we can arrange that

M = reg M, sing M = .~r ~ M. (4)

o// will henceforth denote the collection of oriented C 2 hypersurfaces M satisfying (2), (3), and (4). #Z(Xo, p) will denote the set of M a q / with xo~M and spt 3 [[M]] c ]R" + 1 ~ Bp(xo).

We say M ~ # is F-minimizing in A (A any open subset of ]R "+1) if for each bounded open V with P c A we have

F (M ~ V) < F(N)

whenever N a ~ ' satisfies ~7 ~ A and 3 [[M c~ V~ = ~ [[N]]. ~ , where j > 0 , will denote the collection of F e Z with the property that

W;(s ingM,~ spt~ [[M]]) = 0

for every M~#Z which is F-minimizing in ]R "+1. One of the main results established in [1] is that ~ _ 2 = f f whenever n>2 .

For F ~ , JCZF(Xo, p) will denote the collection of M~#Z(xo, p) such that Mc~Bp(xo) is F-minimizing in Bo(xo) and such that Mc~Bp(xo)=DUMc~Bp(xo) for some open set UM~IR n+l. (In dealing with minimizing hypersurfaces in J/ZF(Xo, p) instead of rectifiable currents, we are losing no generality by virtue of the regularity theorem (see [1] Theorem (1.2)) and a well-known decomposition argument ([5, 4.5.17]).)

268 L. Simon

Jg~(x0, p) will denote the collection of MeqZ(x o, p) which can be represented in the non-parametr ic form X,+l=U(Xl . . . . . x,), (x I . . . . ,x,)eDp(x'o), where u is a C a function satisfying (1) on Do(x'o). Notice that each Meo/g'e(xo, p) is F- minimizing in D~(x'o) x IR ([1] L e m m a (2.1)), and hence J~'v(xo, p) c ~ e ( x o , p).

w 2. Main Results

Theorem 1. Suppose F e Y o. Then there is a constant c > 0 such that for any M e J g F ( X o, p) one has

~(~o) <-- c/p 2, (s) i = 1

where ~ci(Xo) , ...,~%(Xo) are the principal curvatures o f M at x o. I f F e ~ t then there is a constant c such that (5) holds for each M~J//'F(xo, P).

In particular, by letting p - , o o in (5), we deduce that any C 2 function u which satisfies (1) on the whole of lR ~ must be linear.

Before giving the proof of this theorem we wish to discuss some con- sequences. As we ment ioned in the introduction, we always have ~ - ~ "

n - - 2 - -

hence we can immediately deduce the following corollary.

Corollary 1. t f n = 2 then for any F e ~ there is a constant c such that (5) holds for any M E J g v ( x o , p).

I f n = 3 then for any F ~ f there is a constant c such that (5) holds for any M s J g ; ( x o, p). In particular, i f u is a C2(IR 3) solution of( l ) , then u is linear.

To describe some further consequences of Theorem 1, we introduce the area integrand A e ~ , defined by A(p)= [p[, p e n "+ 1. According to [1], Par t II, there is an t / > 0 such that if Fe~, ~ and if

n + l

iF(v) - A(v)l + IFp(v) - Ap(v)l + ~ IFp,pj(v) - Ap~v,(v)l i , j = l n + 1 (6)

+ ~ IFp,pjp~(V)-Ap,vjp~(v)l<~l i , j , k= l

for all vElR "+1 with ]v]=l , then F ~ , _ 6 � 8 9 where ( n - 6 � 8 9 1 8 9 (Actually, for each ~, > 0 there is i/ such that (6) implies FEJ~ ,_ 7+~>..) Thus we can deduce the following corollary from Theorem 1.

Corollary 2. In case n < 6 there is an t 1 > 0 with the following property: I f F e ~ - satisfies (6) then there is a constant c such that (5) holds for each Me.///dv(Xo, p).

In case n<=7 there is an 11 > 0 with the following property: I f F E Y satisfies (6) then ~here is a constant c such that (5) holds for each MEJ~'~,(xo, p). In particular, any C2(IR ") function satisfying (1) must be linear.

We now wish to prove Theorem 1. First we remark (see e.g. [1], Theorem (1.2)) that to prove an inequality of the form (5), it suffices to show that for each e > 0 there exists a constant 0e(0, 1) (depending on e and F) such that

On Some Extensions of Bernstein's Theorem 269

M c~ Bop (x 0) c { x: dist (x, HM) < ~ 0 p }

for some hyperplane H~ c lR "+ ~. Theorem 1 is thus an immediate consequence of the following lemma.

Lemma 1. Let ~>0 and F ~ o. Then there exists 0~(0,1) with the following property: I f M~.~e(xo, p), then for some hyperplane Hu we have xo6H~tclR "+~ and

M c~ Bop(Xo) c {x: dist (x, HM) < e Op}. (7)

In case F ~ , there is a 0~(0, 1) such that (7) holds (for suitable H M with xo~H M ~ IR,+ 1) whenever M ~Jd'F(x o, p).

Proof. If the first part of the lemma is false, then there is an e>0, F e ~ o and a sequence {M,} c dgv(Xo, p) such that

M~B(~/op(Xo)r } (8)

for every hyperplane H with x o e H c l R "+~. However, letting U~ be such that G ~ Bp(xo) = ~ c~ Bp(xo) , we know that from standard compactness, semicon-

tinuity, and regularity theorems (see e.g. [1], 1.1(33), Theorems (1.1), (1.2) and Remark (1) following Theorem (1.2)) we can deduce the following. There is an open UcBp(xo) and an MeJ/tF(xo, p) such that Mc~Bp(xo)=OU~Bp(xo) and such that

~e"+ I ( ( G ~ u ) u ( u ~ G))- - , o as k ~ (9)

for some subsequence {k} c {r} 1. But we are here assuming that F~go, hence (2~r ~M)c~Bp(xo)=~. Then xo~M and we have a tangent hyperplane H~t of M at x o. By (9) and [1, Remark (3) following Theorem (1.2)3 we know there is a 0o~(0,1) such that

M k c~ Bop(Xo) c {x: dist (x, H~t ) < e Op}

for all 0 < 00 and for all sufficiently large k. This contradicts (8). We now turn to the proof of the second part of the lemma. If this part is

false, then there is an e>0, F E ~ and a sequence {M~}cJ'p(xo, p) such that again (8) holds for every hyperplane H. Then let u~ be a C2(Dp(x'o)) solution of (1) such that graph G = M , and let

G = {x:x.+ 1 < u r ( x l , . . . , x . ) , (xl . . . . , x , ) s D , ( x ' o ) } ~ B,(xo).

We have (cf. the proof of the first part of the lemma) a subsequence {k} c {r} and an open U~Bp(xo) satisfying (9), and an MEJIF(Xo, p) satisfying MnBp(xo) =OUc~Bp(xo). Now we are given that F ~ , ~ , hence ~gg'~((M~M)~(Bp(xo))=O, We now recall (see [1] 1.2(18)) that if

i We know xo ~ r because ~"(M k c~ Bo(xo) ) ~ Jq'"(M c~ Br ) a.e. lim inf~(Mr c~ Br for each Ce(0, p) (see e.g. [1], Theorem (1.1) and 1.1 (28))

r ~ o ~

o-~(0 p), while

270 L. S imon

v~=(v] .... , v[,+ ~ ) = ( - D u , , 1)/]/1 +lDu~l 2,

then

n + l n + l

V ~ F ~ ~" ~ ~ c~ ~ v ~ i , j , l= l i , j=l

~(F;~;~(v ) ~ v ~ ~ - - r r J n + l / _ _ 0 ~

where 6~=(6'~ ... . . 6~) denotes the gradient operator on M~. Because of the convergence of M k n Bp(xo) to M described in [-1, Remark (3) following Theorem (1.2)], and because of the Harnack inequality given in Lemma (2.7) of [1], we know that if v denotes the unit normal of M pointing out of U then for each component M , of M we have either v~ + ~ ---0 or v.+ ~ > 0. Let us consider first the case when v~+~>0 at each point of a component M,. Let n denote the projection of N ~+1 onto IR ~ defined by zr(xl, . . . ,x~+~)=(x 1 . . . . ,x~). Then z~(M,) is an open subset of Do(x;). Also (cf. [13 Lemma (1.1)) 2, because ~ , - 1 ( ( ~ , ~M,)nBp(xo))=O, we can find an open set VcBp(xo) such that

M , n Bo(xo) = 0 Vn Bp(xo) = 0 l/~ Bo(xo) , (10)

and such that the outward unit normal of V has positive (n+ 1) st component at each point of M , r~Bo(xo). Next we want to show that if xo~M , ~ M , , then x~ is an interior point_ of the set ~ (M, nBp(x0) ). Indeed, if X o e i ~ , ~ M , and x;r (~(M, ~Bo(xo))), then it is not difficult to see (with the aid of (10)) that we would then have a whole vertical line segment contained i n / ~ , n Bp(xo). Since we are given 2/fl ((~r, ~ M , ) n Bp(xo) ) = 0, it would then follow that there is a vertical line segment contained in M,, thus contradicting the fact that v,+ 1 > 0 on M , . Thus if xo~/hr, ~ M , then we have x~Einterior (~(~r, r~Bp(xo)))" Then it follows that for suitably small cry(0, p), we have the representation

M , n (D ~(x'o) x IR) = graph u n (D ~(x') x IR),

" !

where u is a C 2 function defined on D,(xo)~ K, and where K is compact with H i ( K ) = 0 . (K in fact is simply zc ( ( /~ ,~M,) - ' n (D~(xo)x IR)).) Because K satisfies ~4~ we can assume (possibly by choosing a smaller a) that OD~(x'o)nK = ~. Fur thermore u must clearly satisfy the Euler-Lagrange equation on D~(x'o)

K. Then, by Theorem A of the appendix, we know that u can be extended to 2 / be a C (D~(xo)) function. Hence, we can deduce that M , nB~(xo) is a C 2

hypersurface. This is of course trivially true (for sufficiently small a) if xosM , to begin with.

We now consider the possibility that the component M, of M is such that !)n+ 1~-~0 on M , . In view of the fact that J ( f " - 1 ( ( ~ , ~M,)nBp(xo ) =0, it is quite easy to check that in this case we can write M , = A x/R for some closed subset A c IR". But the singular set of A x IR consists of a union of vertical lines, hence since ~((M,~M,)c~Bp(xo)=O, we deduce that M,c~Bo(xo) is a C 2 hyper- surface.

2 Not i ce tha t M, c M and J4f"(Mr~B~(xl))<ca" whenever x i s ~ ' ~ and a~(O,p-lxo-xll) (by [13, 1.1(33)); this fact is needed if one wishes to app ly L e m m a (1.1) of [11 to M ,

On Some Extensions of Bernstein's Theorem 27l

Thus we have shown that each component M , of M is such that M , c~ B~(xo) is a C 2 hypersurface for suitable a > 0 . From the fact that ~ " - z ( ( ~ r ~ M ) c~(Do(x'o)xlR))=O, no two of these hypersurfaces M , nB~(xo) can intersect non-tangentially. Furthermore no distinct two of these hypersurfaces can make contact which is tangential at each point, and no more than a finite number of components M , can intersect a given compact subset of Bo(xo). (See Lemma (2.4) of [1] and the latter part of the proof of Corollary (3.1) of [1].)

We thus finally deduce xoEM, and the proof is completed as for the first part of the lemma.

Appendix

Here, using an argument essentially due to Finn [6], we wish to consider removability of singularities of solutions of divergence-form equations

i = 1

on a C 2 uniformly convex domain f2clR". Here A~=Ai(p) are C ~,~ functions of p~lR" such that

IAi(p) [ <_ ca, pE1R", (A2)

Aipj(P)~i~j>O, PeN n, ~EIRn~{0}. A3) i , j = l

Notice that such structural conditions dearly hold in case Ai=Fp,, with F as in (1).

The theorem we want to prove is the following.

Theorem A. Suppose K is a compact subset of ~2 with ~ " - I ( K ) = 0 and suppose uECZ(~K) satisfies (A1) on f2~K. Then u extends to a C2(~) solution of(A1).

Remark. In the case of the minimal surface equation, a slightly more general theorem was proved by DeGiorgi and Stampacchia [4]. (In [4] an analogous theorem to that above, for the minimal surface equation, was proved in case ~2 was an arbitrary domain in 1R".)

Proof of Theorem A. Let e > 0 be given. Because gf" -1 (K)=0 we can use the definition of Hausdorff measure to find real numbers 61, . . . , 6 u > 0 and points

N N

x (1), ...,x<N>EK such that K c U Dal(x<~ and ~ c~7- 1 <e. Furthermore we know i = 1 i = 1

from standard theory of elliptic equations that we can find a C2(~2)c~ C1(~) . �9

function w such that ~ - -A~(Dw)= 0 and w--u on ~2. Writing this equation in i = 1 ~ X i

weak form we have

S Ai(Dw)Di(dx=O, (ECho(f2). (A4) ~2

272 L. Simon

Provided ~ = 0 in a neighbourhood of K, we also have

Ai(Du)D~ ~ dx = 0. (A 5)

We now let 71, i= 1, . . . ,N, be C 1 functions such that 7 i=0 on D~(x(i)), 7 i - 1 on JR"~D2~(x(i)), IDT~l<2/Oz on JRn and 716[0,1] on JR". Then we choose

= (I~1 7i ) g= arctan ( u - w) in (A4), (a5). Substracting (a4) and (A5) we then have

N

(1 +(u-w)2) -1 ~ (A.i(Du)-Ai(Dw))(Diu-Diw) [I ?fix f2 i=1 j = l

= - ~ Ai(Du)-A~(Dw 7~ Di?k arctan(u-w)dx. i= k = l j

j * k

Since 1

Ai(Du) _ Ai(Dw) = ~ d o -~Ai(Dw 4- t(Du -Dw))dt

=i ~ A:l,j(Dju-Djw)dt, 0 j = l

where

Atiej = Aipj(Dw + t(Du - Dw)),

this gives, by virtue of (A2),

S (1 + ( u - w)2) -1 A~pj(Diu-D,wl(Dju-Djwl ~k dtdt O.Q i , j = l =

N N

<n7~cl ~, S 1~ ?)lDykldX" k= l . Q j= 1

j ~ k

N

Since ~ (57-1 < e, we can let e ~ 0 (note that then max •i ~ 0). Thus we obtain, after i= l

using (A3) and Fatou's theorem on the left,

1 S ~ (l+(u-wl2) -~ ~ A~#D~u-D~wI(Dju-Djw) d~=~ O ~?~K i , j = l

Hence, by (A3), Du=Dw a.e. on f2~K. But u=w on ~ , and hence we deduce u =w on g2~K. Since w6C2(O) this completes the proof.

References

1. Almgren, Jr., F.J., Schoen, R., Simon, L.: Regularity and singularity estimates on hypersurfaces minimizing parametric elliptic integrands. Acta math. (To appear)

2. Bombieri, E., De Giorgi, E,, Giusti, E.: Minimal cones and the Bernstein problem. Inventiones math. 15, 24-46 (1972)

On Some Extensions of Bernstein's Theorem 273

3. DeGiorgi, E.: Una estensione del teorema di Bernstein. Ann. Scuola norm. sup. Pisa, Sci. fis. mat., III Ser. 19, 79-85 (1965)

4. De Giorgi, E., Stampacchia, G.: Sulle singolaritgt eliminabili delle ipersuperficie minimali. Atti Accad. naz. Lincei, Mem., C1. Sci. fis. mat. natur., VIII. Ser., Sez. I 38, 352-357 (1965)

5. Federer, H.: Geometric measure theory. Berlin-Heidelberg-New York: Springer 1969 6. Finn, R.: Isolated singularities of solutions of non-linear partial differential equations. Trans.

Amer. math. Soc. 75, 383-404 (1953) 7. Fleming, W.: On the oriented Plateau problem. Rend. Circ. mat. Palermo, II. Ser 2, 1 22 (1962) 8. Heinz, E.: l~ber die L6sungen der Minimalfliichengleichung. Nachr. Akad. Wiss. G/Sttingen IL

math.-phys. K1. 51-56 (1952) 9. Jenkins, H.: On 2-dimensional variational problems in parametric form. Arch. rat. Mech.

Analysis 8, 181-206 (1961) 10. Schoen, R., Simon, L., Yau, S.T. : Curvature estimates for minimal hypersurfaces. Acta. math.

134, 275~88 (1975) 11. Simon, L.: A H61der estimate for quasiconformal maps between surfaces in Euclidean space,

Acta math. (To appear) 12. Simon, L : Remarks on curvature estimates for minimal hypersurfaces. Duke math. J. 43, 545-553

(1976) 13. Simons, J.: Minimal varieties in Riemannian manifolds. Ann. of Math., II. Ser. 88, 62-105 (1968)

Received September 24, 1976