Existence and non-existence of minimal graphs

0.32 MB·

by  Qi Ding

EXISTENCE AND NON-EXISTENCE OF MINIMAL GRAPHS
QI DING,J. JOST, AND Y.L.XIN
7
1 Abstract. We study the Dirichlet problem for minimal surface systems in arbitrary
0 dimension and codimension, and obtain the existence of minimal graphs over arbitrary
2 mean convex bounded C2 domains for a large class of prescribed boundary data. This

n resultcanbeseenasanaturalgeneralizationoftheclassicalsharpcriterionforsolvability
a of the minimal surface equation by Jenkins-Serrin. In particular, we can construct a
J continuousfamilyofsolutionsofcodimension2tosuchDirichletproblemsbythemethod

6 of continuity. In contrast, we also construct a class of prescribed boundary data on
justmeanconvexdomains,forwhichDirichletproblemincodimension 2isnotsolvable.

] Moreover,westudytheuniquenessoftheseminimalgraphswhentheyarestrictlystable.
G
D
.
h
t
a
1. Introduction
m
[
The Dirichlet problem for minimal graphs is one of the classical problems in the theory

1
of nonlinear elliptic PDEs. It has been investigated for over a century, starting with
v
4 the fundamental work of Bernstein [3], Haar [10] and Rado [24], and it has inspired the
7 development of methods for solving nonlinear elliptic PDEs and the regularity of their
6
solutions. Many deep and important results were achieved, and for instance the papers of
1
Jenkins-Serrin [12] and Lawson-Osserman [15] can be considered as classics in the field.
0
. Nevertheless, this problem still poses difficult challenges when we move from the classical
1
0 case of surfaces in R3 to minimal graphs of arbitrary dimension and codimension. In this
7 paper, we make a systematic new contribution to that general problem.
1
: In codimension 1, the graphic function that describes a minimal graph over an n-
v
i dimensional domain Ω Rn satisfies the minimal surface equation
X ⊂
r (1.1) 1+ Du2 ∆u uiujuij = 0 in Ω.
a | | −
For n = 2, when the dom(cid:0)ain Ω is c(cid:1)onvex, the Dirichlet problem for (1.1) is solvable for
arbitrary continuous boundary data. This was achieved by successive efforts in the papers
of Bernstein [3], Haar [10] and Rado [24] already mentioned. On the other hand, the
Dirichlet problem is not necessarily solvable when Ω is non-convex, as pointed out by
Bernstein, and Finn [7] constructed such counterexamples. For n > 2, Gilbarg [8] and
Stampacchia [27] established the existence of solutions to (1.1) for smooth boundary data
and strictly convex smoothly bounded domains. Jenkins-Serrin [12] relaxed convexity of
the domain to mean convexity, and gave a sharp criterion for the solvability. When ∂Ω
is not mean convex, they found smooth boundary data for which the Dirichlet problem
of (1.1) is not solvable (see also [9]). (Also, in general, solutions are not unique; for some
recent interestingexamples of non-uniquetwo-dimensional minimalgraphsof codimension
3, see[25].) Thus,thesituation for codimension 1can beconsidered as well understood.
≥
Inhighercodimensions,however, thesituationismuchmorecomplicated, aswasshown
in a seminal paper by Lawson-Osserman [15]. In dimension n = 2, they showed the solv-
ability of the Dirichlet problem for the minimal surface system with arbitrary continuous
1
2 QIDING,J.JOST,ANDY.L.XIN
boundary data on a bounded convex domain in R2. Again, in general, these solutions are
not unique. In dimension n 4, they gave non-existence examples for Dirichlet problems
≥
on unit balls with higher codimensions (Theorem 6.1, [15]). It is therefore natural to in-
vestigate under which conditions on the boundary data the Dirichlet problem for minimal
graphs with dimension n> 2 and codimension m 2 can be solved.
≥
The first significant result for this problem was obtained by M.T.Wang [30] using the
mean curvature flow. Wang obtained the following result.
Let Ω be a bounded domain in Rn with convex C2-boundary and diameter l. Let there
be given a C2-map ψ :Ω¯ Rm with
→
(1.2) 8nlsup D2ψ +√2sup Dψ < 1.
| | | |
Ω ∂Ω
Then the Dirichlet problem for the minimal surface system with boundary values ψ admits
a smooth solution.
In the present article, we take the next step. We study the Dirichlet problem for mini-
mal graphs with arbitrary dimensions and codimensions, and obtain existence of minimal
graphs over arbitrary mean convex bounded C2 domain for a large class of prescribed
boundary data, see Theorem 4.3. While on one hand, our results include the classical
results of Jenkins-Serrin in the codimension one case, on the other hand, they provide an
alternative condition for the solvability of the Dirichlet problem for minimal graphs over
mean convex domains in higher codimensions.
Theorem 1.1. For any mean convex bounded C2 domain Ω and any ϕ C2(Ω), there
∈
are constants ǫ ,ǫ depending on the geometry of Ω and ϕ such that if the functions
1 2 C2(Ω)
| |
ψ1, ,ψm 1 satisfy
−
···
m 1 2
− 1
(1.3) sup D2ψα +sup Dψα ǫ ,
2
ǫ | | | | ≤
α=1(cid:18) 1 Ω Ω (cid:19)
X
then there is a smooth solution u= (u1, ,um) of the minimal surface system
···
gijuα = 0 in Ω
ij
(1.4) for α = 1, ,m,
(uα = ψα on ∂Ω ···
with g = δ + uαuα and ψm = ϕ.
ij ij α i j
P
The constants ǫ ,ǫ can in principle be computed explicitly for codimension 2 at least,
1 2
but since our values are presumably far from optimal, we do not bother to do so. Impor-
tantly, by the Lawson-Osserman counterexample, some such restrictions is necessary.
The main difference between our condition (1.3) and Wang’s condition (1.2) is that we
only need to impose a smallness condition on the second derivatives of m 1 out of the m
−
boundaryfunctions. Thus,whileWang’sresultcanbeconsideredinsomeintuitivesenseas
a perturbation of the case of linear boundary data (for which always a solution, the linear
extension of the boundary data, exists), our result can be considered as a perturbation of
the codimension 1 result, where, according to Jenkins-Serrin also always a solution can be
found.
EXISTENCE AND NON-EXISTENCE OF MINIMAL GRAPHS 3
When comparingthose results with the Lawson-Osserman counterexample, we are lead
to the question to what extent also non-perturbative results are possible. But this is a
question for future research.
If f = (f1, ,fm) is the solution of the parabolic system corresponding to (1.3),
···
the difficulty to show the above Theorem 1.1 consists in obtaining effective gradient
estimates for Dfm , when ψm is arbitrary. Assuming sufficiently small ψα for
C2 C2
| | | | | |
α = 1, ,m 1, we can in fact show that Dfα is small in the interior of the domain
··· − | |
using curvature estimates. Then we are able to choose a suitable auxiliary function to
estimate Dfm on the boundary. With the maximum principle, graph is area-decreasing
f
| |
along the flow, and this will enable us to show the long-time existence of the flow, which
then converges to the solution of (1.3). In codimension 2, we can construct a continu-
ous family of solutions by the continuity method for elliptic equations. The openness,
as always, uses the implicit function theorem; it relies on the solvability of the Dirichlet
problem in small neighborhoods of strictly stable minimal graphs.
On convex domains, we can control the constants to obtain an existence result for the
Dirichlet problem comparable with Wang’s theorem [30] mentioned above (see Theorem
4.5 for the proof).
Theorem 1.2. For a convex bounded C2 domain Ω, let ψ1, ,ψm be functions on Ω
···
with
m
(1.5) enlβ sup D2ψα +sup Dψα < β 1
0 0
| | | | −
α=1(cid:18) Ω Ω (cid:19)
X p
for each constant 1 < β 9, where l is the diameter of Ω. Then there is a smooth
0
≤
solution u= (u1, ,um) with sup det δ +uαuα < β of the minimal surface system
··· Ω ij i j 0
(1.3) with u= ψ on ∂Ω. (cid:16) (cid:17)
For any C2 map φ : Sn+k Sn Rn+1,n 4,k > 0 which is not homotopic to zero
as a map to Sn, Lawson-Osser→man [1∈5] constru≥cted boundary data on the unit ball in Rn,
for which the corresponding Dirichlet problem has no solution. Their construction gives
manynon-existenceexamplesforhighercodimensions(Theorem6.1,[15]). Insection6,we
also construct many non-existence examples for the Dirichlet problem for minimal graphs
in dimension > 2 and codimension 2 over any mean convex (but not convex) domain,
see Theorem 6.1 for concrete results. This implies that ǫ ,ǫ in Theorem 1.1 cannot be
1 2
independent of ϕ (in contrast to the convex case, see (1.2) and (1.5)).
C2(Ω)
| |
In the last section we consider the uniqueness of the Dirichlet problem for minimal
graphs over a mean convex domain. In [15] Lawson-Osserman showed that there exists a
real analytic function φ: ∂D R2 with the property that there are at least three distinct
solutions of the correspondin→g problem, where D is the unit disk in R2 (Theorem 5.1,
[15]). Moreover, one of these solutions represents an unstable minimal surface. Lee-Wang
[17] also considered the uniqueness. They proved a uniqueness theorem for nonparametric
minimal submanifolds whose graphic functions are both distance-decreasing and equal on
the boundary. We prove a new uniqueness result, Theorem 7.2, which states that if the
solution in Theorem 5.3 is strictly stable, then it is the unique strictly stable solution.
4 QIDING,J.JOST,ANDY.L.XIN
2. Preliminaries
For an open set Ω Rn, we consider a C2 isometric immersion X : Ω Rn+m. Then
⊂ →
X is a minimal immersion if and only if
n
∂ ∂X
(2.1) detgklgij = 0,
∂x ∂x
i,j=1 i (cid:18) j(cid:19)
X p
whereg = ∂X/∂x ,∂X/∂x , and thematrix (gij)is theinverse of (g ). Theimmersion
ij i j ij
h i
X is called non-parametric if it has the form X(x) = (x,u(x)) for some vector-valued
function u= (u1, ,um) : Ω Rm. In this case the system (2.1) becomes
··· →
n
∂
detg gij = 0 for j =1, ,n
kl
 ∂xi ···
(2.2)  Xi=1 (cid:16)p (cid:17) ,
 n ∂ ∂uα
 detg gij = 0 for α= 1, ,m
kl
∂x ∂x ···
i,j=1 i (cid:18) j(cid:19)
 X p

where gij are defined as above, and g = δ + m ∂ uα∂ uα.
 ij ij α=1 xi xj
By [23] (or [15]), the system (2.2) is equivalePnt to
n ∂2uα
(2.3) gij = 0 for α = 1, ,m.
∂x ∂x ···
i j
i,j=1
X
If m = 1, the above minimal surface system reduces to the following single equation
Du
(2.4) div = 0.
1+ Du2!
| |
De Giorgi [5] showed that Lipschitz soplutions to the minimal surface equation (2.4) are
smooth(seealso[22][27]forinstance). However, suchregularitycannotextendtoLipschitz
solutionsoftheminimalsurfacesystem(2.3)sinceLawson-Osserman[15]haveconstructed
non-parametric minimal cones, that is, nonsmooth Lipschitz solutions.
For any vector-valued function f = (f1, ,fm) C1(Ω,Rm), we define
··· ∈
2 2
sup df = sup df(x) = sup µ (x)µ (x),
i j
Ω x Ω x Ω,1 i<j n
(cid:12)^ (cid:12) ∈ (cid:12)^ (cid:12) ∈ ≤ ≤
where µ (x) n are(cid:12)thesin(cid:12)gularva(cid:12)luesofdf(cid:12)(x). Wecallgraph (strictly)area-decreasing
{ k }k=1 f
if sup 2df (<)1. Consider the Gauss map of the submanifold graph in Rm+n.
Ω f
≤
Then, the area-decreasing property means that the image under its Gauss map lies in the
(cid:12)V (cid:12)
largest g(cid:12)eodesi(cid:12)c convex subset of the Grassmannian manifold Gn,m (see Final remarks in
[14] for details).
Concerninginterior regularity, Morrey[20,21]showedthatC1 solutionsuofthesystem
(2.3) are smooth. Wang [30] showed the smoothness of Lipschitz solutions u to (2.3)
provided 2du 1 ǫ for some ǫ > 0. Allard’s regularity theorem holds for more
≤ −
general rectifiable stationary varifolds (see also [19][26]) when there is an upper bound of
(cid:12)V (cid:12)
the volum(cid:12)e ratio(cid:12)(as well as of the excess).
Assume ∂Ω C2. Let u be a smooth solution of (2.3) in Ω which is Lipschitz up to
∈
the boundary ∂Ω. If one blows the graph of u up at any boundary point, then any limit
EXISTENCE AND NON-EXISTENCE OF MINIMAL GRAPHS 5
is a Lipschitz graph over the half plane. From the arguments in 5.2 of [2] (see also [15]),
any limit is a tangent cone consisting of a finite number of half-planes (of dimension n),
each of which projects non-singularly into Rn. Thus such a limit consists of a single half-
plane, and the density is equal to 1 at each boundary point. Therefore, one can employ
2
the regularity theorem of Allard in section 4 of [2], and conclude u C1,γ(Ω) for any
∈
γ (0,1).
∈
Let ∆ and D denote the Laplacian and Levi-Civita connection of Rn, respectively. We
always denote
d(x) = d(x,∂Ω) = inf x y ,
y ∂Ω| − |
∈
and
Ω , y Ω d(y,∂Ω) > s and Ω , y Ω d(y,∂Ω) < s for any s > 0.
s { ∈ | } ∗s { ∈ | }
Unless the contrary is explicitly stated, we assume that the considered minimal graphs
or mean curvature flows have dimension n 2. For a vector-valued function f =
≥
(f1, ,fm), fi denotes its i-th component and
···
m
v , det δ + fαfα .
f v ij i j
u α=1 !
u X
t
The Einstein summation convention over repeated indices will beused. Greek indices α,β
take their values in the set 1, ,m .
{ ··· }
3. Boundary gradient estimates for minimal surface system
Let Ω be a bounded domain in Rn with ∂Ω C2 and diameter l. Then there is a
∈
constant r > 0 such that at any point p ∂Ω there is an exterior ball with a uniform
radiusr oΩutsideofΩ. Namely, forany p ∈∂ΩthereisauniqueopenballB (q) Rn Ω
Ω ∈ rΩ ⊂ \
centered at q and with the radius r such that B (r ) Ω = p.
Ω q Ω ∩
We have the following boundary gradient estimate which is similar to Theorem 3.1 in
[30], although the proof is different.
Lemma 3.1. Let Ω be a bounded domain in Rn as above, and ψ = (ψ1, ,ψm) C2(Ω).
··· ∈
Suppose the flow
∂fα
=gijfα in Ω
dt ij
(3.1) fα(,0) =ψα in Ω for α = 1, ,m,
 · ···
 fα(,t) =ψα on ∂Ω
·

exists smoothly onΩ [0,T), where (gij) is the inverse matrix of g = δ + fαfα.
× ij ij α i j
Then the following boundary gradient estimate holds:
P
(3.2) sup Dfα nl[f]1,T e1+rnΩl[f]1,T sup D2ψα +sup Dψα ,
| | ≤ | | | |
∂Ω [0,T) Ω Ω
×
where
[f] , sup v2(x,t) = sup det δ + fαfα .
1,T f ij i j
x∈Ω,t∈[0,T) x∈Ω,t∈[0,T) Xα !
6 QIDING,J.JOST,ANDY.L.XIN
Proof. We consider a point p ∂Ω; without loss of generality (after a translation), we
can assume p = 0 and B ( r∈E ) Ω = 0, where E = (0,0, ,1) Rn and r E =
rΩ − Ω n ∩ n ··· ∈ Ω n
(0,0, ,0,r ). We define a function
··· Ω
Θ
Sα(x,t) = α 1 e θρ(x) (fα(x,t) ψα(x))
−
θ − − −
(cid:16) (cid:17)
on Ω, where ρ(x) = x+r E r , and θ,Θ are positive constants to be defined later.
| Ω n|− Ω α
Obviously, ρ > 0 on Ω. Put y = x for i = 1, ,n 1 and y = x +r E . Note that
i i n n n
every eigenvalue of g is less than v2. For any·α·· 1−, ,m , we have ∗
ij f ∈ { ··· }
∂Sα δ y y
gij∂ Sα = gij Θ e θρ ij i j
∂t − ij − α − x+r E − x+r E 3
(cid:18) (cid:18)| Ω n| | Ω n| (cid:19)
y y
θΘ e θρ i j gij∂ ψα
− α − x+r E 2 − ij
| Ω n| (cid:19) i,j
X
y y θ δ
Θ e θρgij i j ij gij∂ ψα
(3.3) ≥ α − x+r E 2 − x+r E − ij
(cid:18)| Ω n| | Ω n|(cid:19) i,j
X
θ n
>Θ e θρ nsup D2ψα
α − v2 − ρ+r − | |
f Ω! Ω
θ n
Θ e θl nsup D2ψα ,
≥ α − v2 − r − | |
f Ω! Ω
where we have used the fact that the smallest eigenvalue of gij is larger than v 2. Let l
f−
be the diameter of Ω and θ = 1 + n v2. If we set
l r f
Ω
Θα = nl[f]1,T e1+rnΩl[f]1,T sup D2ψα
| |
Ω
with [f] , sup v2, then
1,T Ω [0,T) f
×
∂Sα
(3.4) gij∂ Sα > 0.
ij
∂t −
By the maximum principle, it is clear that Sα > 0 on Ω [0,T). Analogously to the above
×
argument, Θα 1 e θρ(x) +(fα(x,t) ψα(x)) > 0 for any (x,t) Ω [0,T). Thus
θ − − − ∈ ×
(3.5) fα((cid:0)x,t) ψα(x(cid:1)) < Θα 1 e θρ(x) for any (x,t) Ω [0,T).
−
| − | θ − ∈ ×
(cid:16) (cid:17)
Hence at the point p = 0, it follows that
(3.6) Dfα(p,t) Dψα(p) Θ for any t [0,T).
α
| − | ≤ ∈
Since p is an arbitrary point in ∂Ω, the proof is complete. (cid:3)
It is worth to note that the above estimate holds for any C2 bounded domain. In
particular, if Ω is convex, we can allow r in (3.2). Analogously to the above proof,
Ω → ∞
we have the following estimate.
Lemma 3.2. Let Ω be a bounded domain in Rn as inLemma 3.2, and ψ = (ψ1, ,ψm)
··· ∈
C2(Ω). Suppose u =(u1, ,um) satisfies
···
gijuα = 0 in Ω
ij
(3.7) for α = 1, ,m,
(uα = ψα on ∂Ω ···
EXISTENCE AND NON-EXISTENCE OF MINIMAL GRAPHS 7
where (gij) is the inverse matrix of g = δ + uαuα. Then the following boundary
ij ij α i j
gradient estimate holds:
P
sup Duα nl[u]1, e1+rnΩl[u]1,∗sup D2ψα +sup Dψα ,
∂Ω | | ≤ ∗ Ω | | Ω | |
where
[u] = supv2 = supdet δ + uαuα .
1, u ij i j
∗ Ω Ω !
α
X
We assume that ∂Ω has nonnegative mean curvature pointing into Ω. There is a small
constant d > 0 such that
0
d(x) , d(x,∂Ω)
is smooth for x Ω , y Ω d(y,∂Ω) < d and ∂Ω is an embedded C2-hypersurface
∈ ∗d0 { ∈ | 0} t
for all 0 t < d , where Ω = x Ω d(x,∂Ω) > t . ∂Ω then also has nonnegative mean
0 t t
≤ { ∈ | }
curvature (see Lemma 14.17 in [9] for example). Let φ = φ(d) be a C2-function on [0,d ]
0
with
φ 0 and φ 0.
′ ′′
≥ ≤
Let λ (d) denote the smallest eigenvalue of (d ) and
min ij n n
×
λ (d) , max 0, inf λ (d) 0,
− ( −Ω∗d0 min ) ≥
then ( d ) λ (d)I . Here, I is the unit n n matrix.
− ij n×n ≤ − n n ×
Let ϕ C2(Ω) and φ˜ = φ d+ϕ. For each wα C1(Ω) with α = 1, ,m 1, we
∈ ◦ ∈ ··· −
define
m 1
a , δ + − wαwα+φ˜φ˜
ij ij i j i j
α=1
X
for 1 i,j n, and let (aij) be the inverse matrix of (a ).
ij
≤ ≤
Assume that (Dφ˜,Dw1, ,Dwm 1) has singular values µ , ,µ with
− 1 n
··· ···
(3.8) µ µ 1 for any i = j.
i j
| | ≤ 6
Then at any fixed point p Ω , (by the uniqueness of the eigenvalues of (a )) we choose
∈ ∗d0 ij
a coordinate system such that
a = δ (1+µ2),
ij ij i
and µ21 ≥ µ22 ≥ ··· ≥ µ2n. From aii = 1+ mα=−11wiαwiα+φ˜iφ˜i = 1+µ2i, it follows that
(3.9) µP2 φ˜ 2.
i ≥ | i|
Lemma 3.3. Let φ be the function on [0,d ] defined as above. Then
0
(3.10)
2 n 1 n 1 φ
aijφ˜ij ≤ φ′λ−(d)(cid:18)(φ′)2 (cid:18)|Dϕ|2+ µ−21 (cid:19)+ 1+−µ21(cid:19)+ det′a′ij +nλ+(ϕ) on Ω∗d0,
where λ+(ϕ) = max{0,supΩ∗d0 λmax(ϕ)}, and λmax(ϕ) is the largest eigenvalue of the ma-
trix (ϕ ).
ij
8 QIDING,J.JOST,ANDY.L.XIN
Proof. Set
m 1
−
w = Dwα 2.
| |∗ v | |
uα=1
uX
Since 1+µ2 is the largest eigenvalue of (ta ), then obviously
1 ij
Dφ˜2 µ2 Dφ˜2+ w 2.
| | ≤ 1 ≤ | | | |∗
Hence combining (3.8) and (3.9) one has
n n
n 1 n 1
(3.11) φ˜ 2 µ2 − − .
| i| ≤ i ≤ µ2 ≤ Dφ˜2
Xi=2 Xi=2 1 | |
So we have
n 1
(3.12) |φ˜1|2 ≥ |Dφ˜|2− µ−2 .
1
Namely,
n
n 1
(3.13) (φ′)2d21+2φ′ϕ1d1+ϕ21 ≥ (φ′)2+ 2φ′diϕi +ϕ2i − µ−2 ,
i=1 1
X(cid:0) (cid:1)
where d = ∂d. For any 0 < ǫ < 1, by Cauchy-Schwartz inequality it follows that
i ∂xi
n
n 1
(φ′)2 1−d21 ≤− 2φ′diϕi +ϕ2i + µ−2
i=2 1
(cid:0) (cid:1) X(cid:0) (cid:1)
n n
1 n 1
(3.14) ≤ǫ(φ′)2 d2i + ǫ −1 ϕ2i + µ−2
i=2 (cid:18) (cid:19)i=2 1
X X
1 n 1
≤ǫ(φ′)2 1−d21 + ǫ −1 |Dϕ|2+ µ−2 .
(cid:18) (cid:19) 1
(cid:0) (cid:1)
Thus
1 1 1 n 1
(3.15) 1 d2 Dϕ2+ − .
− 1 ≤(φ)2 ǫ| | 1 ǫ µ2
′ (cid:18) − 1 (cid:19)
Recall d(x) = d(x,∂Ω). At the point p Ω , we may choose an orthonormal basis
∈ ∗d0
∂ ∂ , such that ∂ d = 1. There is a vector (τ , ,τ ) Rn 1 with
{ ρin=}−11∪τ{i2 =θα1}α−=1d,·21··,snu−c1h that ρ 1 ··· n−1 ∈ −
P ∂ n−1
(3.16) = d ∂ + τ ∂ .
∂x 1 ρ i θi
1
i=1
X
Choose ǫ = 1. With (3.15) we obtain
2
n 1
∂ ∂ −
d =Hess , = τ τ Hess ∂ ,∂ λ (d) τ2
(3.17) 11 d(cid:18)∂x1 ∂x1(cid:19) i j d θi θj ≥ − − i=1 i
(cid:0) (cid:1) X
2λ (d) n 1
=−λ−(d) 1−d21 ≥ −(φ′−)2 (cid:18)|Dϕ|2 + µ−21 (cid:19).
(cid:0) (cid:1)
Let e n be an orthonormal vector field tangent to ∂Ω at the considered point
{ i}i=1 t
p ∂Ω for 0 < t < d , and denote by e the unit normal vector field to ∂Ω so that e
t 0 n t n
∈
EXISTENCE AND NON-EXISTENCE OF MINIMAL GRAPHS 9
points into Ω . Since the function d is a constant on ∂Ω , then at p we get
t t
n 1
−
(3.18) D D (D e )T d = 0,
ei ei − ei i
i=1
X(cid:0) (cid:1)
and(D D D e )d= 0, where( )T denotestheprojectionontothetangentbundle
en en − en n ···
of ∂Ω . Since the boundary ∂Ω is mean convex and D d = 1 at p, one has
t t en
n n 1
−
∆d= (D D D e )d = D e ,e D d+(D D D e )d
ei ei − ei i − h ei i ni en en en − en n
i=1 i=1
(3.19) X X
n 1
−
= D e ,e = H 0.
− h ei i ni − ∂Ωt\∂Ω ≤
i=1
X
We shall now compute aijφ˜ . First, combining (3.8), (3.17) and (3.19), one has
ij
n 1 n µ2 µ2 n µ2
aijd = d = ∆d i d 1 d + i λ (d)
ij 1+µ2 ii − 1+µ2 ii ≤ −1+µ2 11 1+µ2 −
i=1 i i=1 i 1 i=2 i
(3.20) X X X
µ2 n 1 2 n 1 n 1
1 d + λ (d) λ (d) Dϕ2+ − + − .
≤− 1+µ21 11 i=2 1+µ21 − ≤ − (cid:18)(φ′)2 (cid:18)| | µ21 (cid:19) 1+µ21(cid:19)
X
Secondly, noting φ 0 and the definition of λ (ϕ) in this lemma, we have
′′ +
≤
n d2 n d2
aij φ d d +ϕ φ i +nλ (ϕ) φ i +nλ (ϕ)
′′ i j ij ≤ ′′ 1+µ2 + ≤ ′′ 1+µ2 +
(3.21) i=1 i i=1 1
(cid:0) (cid:1) X X
φ φ
′′ ′′
= +nλ (ϕ) +nλ (ϕ).
1+µ2 + ≤ deta +
1 ij
Combining (3.20) and (3.21), we obtain
aijφ˜ =aij φd +φ d d +ϕ
ij ′ ij ′′ i j ij
(3.22) 2 n 1 n 1 φ
≤φ′λ(cid:0)−(d)(cid:18)(φ′)2 (cid:18)|Dϕ|2+(cid:1) µ−21 (cid:19)+ 1+−µ21(cid:19)+ det′a′ij +nλ+(ϕ).
We complete the proof. (cid:3)
Now we state a result, which is sharp for m = 1.
Lemma 3.4. Let Sα be an (n m)-matrix with singular values s s 0 and
i × 1 ≥ ··· ≥ n ≥
n 2. If s √n 1 and s s 1 for i = j, then
1 i j
≥ ≥ − ≤ 6
1 2
2
(3.23) det δ + SαSα 1+ (Sα)2 .
ij i j ≤  i 
! !
α α i
X X X
 
10 QIDING,J.JOST,ANDY.L.XIN
Proof. Since s 1 for i 2, a straightforward computation implies
i ≤ s1 ≥
n
(1+s2)=1+ s2+ s2s2+ s2s2s2 + + s2
i i i j i j k ··· i
i=2 2 i n 2 i<j n 2 i<j<k n 2 i n
Y ≤X≤ ≤X≤ ≤ X ≤ ≤Y≤
=1+ s2 1+ s2+ s2s2 + + s2
i  j j k ··· j
(3.24) 2 i n i<j n i<j<k n i<j n
≤X≤ X≤ X≤ Y≤
≤1+ s2i 1+Cn1−is−12+Cn2−is−14+···+Cnn−−iis1−2(n−i)
2≤Xi≤n (cid:16) (cid:17)
=1+ s2i 1+s−12 n−i.
2 i n
≤X≤ (cid:0) (cid:1)
Note that s = 0 for j > m provided m < n. Denote n = min m,n . Then
j
∗ { }
n∗
(1+s2i)≤ (1+s21)1+ s2i 1+s−12 n∗−i
Yi=1 2≤Xi≤n∗ (cid:0) (cid:1)
=1+s21+ s2i 1+s−12 n∗−i+ s21s2i 1+s−12 n∗−i
(3.25) 2≤Xi≤n∗ (cid:0) (cid:1) 2≤Xi≤n∗ (cid:0) (cid:1)
≤1+s21+ s−12 1+s−12 n∗−i+ 1+s−12 n∗−i
2≤Xi≤n∗ (cid:0) (cid:1) 2≤Xi≤n∗(cid:0) (cid:1)
=1+s21+ 1+s−12 n∗−i+1.
2≤Xi≤n∗(cid:0) (cid:1)
Since s √n 1 √n 1 1, then
1
≥ − ≥ ∗− ≥
1+s−12 n∗−i+1 ≤ 1+ n 1 1 n∗−i+1
(3.26) 2≤Xi≤n∗(cid:0) (cid:1) 2≤Xi≤n∗(cid:18) ∗− (cid:19)
n∗ 1
n 1− n∗n∗1 − n n∗−1
= ∗ − = n ∗ 1 2n ,
n∗−1 1(cid:16)− n∗n−∗(cid:17)1 ∗ (cid:18)n∗−1(cid:19) − ! ≤ ∗
where we have used (1+ 1)i < e for i 1. So
i ≥
n∗
(3.27) (1+s2) 1+s2+2n .
i ≤ 1 ∗
i=1
Y
Now let us prove this lemma by induction. Clearly, the inequality (3.23) holds for
m = 1. Denote Sα = (Sα)2. Assume that (3.23) holds for m 1 with m 2. For
| | i i − ≥
pP